Model based control of a flight simulator motion

Model based control of a flight simulator motionsystem

Model based control of a flight simulator motionsystem

PROEFSCHRIFT

ter verkrijging van de graad van doctoraan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir K.F. Wakkervoorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 10 december 2001 om 16.00 uur

door

Sjirk Holger KOEKEBAKKER

werktuigkundig ingenieurgeboren te Ermelo

Dit proefschrift is goedgekeurd door de promotor:Prof. ir O.H. Bosgra

Samenstelling promotiecommissie:

Rector Magnificus voorzitterProf. ir O.H. Bosgra Technische Universiteit Delft, promotorDr ir A.J.J. Van der Weiden Technische Universiteit Delft, toegevoegd promotorProf. dr ir M. Steinbuch Technische Universiteit EindhovenProf. dr ir J.A. Mulder Technische Universiteit DelftProf. dr ir P.M.J. van den Hof Technische Universiteit DelftProf. dr ir J.H. de Leeuw University of TorontoIr P.C. Teerhuis Technische Universiteit Delft

Ir P.C. Teerhuis heeft als begeleider in belangrijke mate aan detotstandkoming van het proefschrift bijgedragen.

Keywords: model based control, flight simulation, parallel motion systems

ISBN 90-370-0194-7

Copyright c 2001 by S.H. Koekebakker

All rights reserved. No part of the material protected by this copyright notice may be reproduced orutilized in any form or by any means, electronic or mechanical, including photocopying, recording orby any information storage and retrieval system, without written permission from the copyright owner.

Printed in the Netherlands by Ponsen & Looijen b.v.

To faith

We’re on a road to nowhereCome on inside

Takin’ that ride to nowhereWe’ll take that ride

Talking Heads, 1985

Running is no sportbut a way of travelling

displacing body and mind

Vrij naar Jan Knippenberg

Voorwoord

In de ruwweg zeven jaren, maanden, dagen en uren tussen de start van dit onderzoek en hetmoment waarop het verdedigd zal worden is veel gebeurd. Beginnend vanuit een kamertjein een studentenhuis in Delft kon ik me weinig voorstellen bij de grote, relatief eenzaamverrichte inspanningen waar kond van werd gedaan in het voorwoord van de verschillendeproefschriften. Onderzoek doen is leuk en zeker in het kader van een multidisciplinairproject werk je daarbij met veel mensen samen. Pas nadat het onderzoek na vier jaarafgerond maar nog niet helemaal opgeschreven was en naast een aantrekkelijke voltijdsbaan en een fantastisch gezin met inmiddels twee dochters vanuit Venlo het werk in grotergeheel vastgelegd moest worden, kon ik mij vinden in vele van die eerder gelezen frasen.Alhoewel een groot aantal mensen je op een of andere manier bijstaan tijdens de gehele pe-riode, moet het schrijfwerk toch grotendeels alleen werkend achter de computer gebeuren.

Hier een punt achter zettend is eindelijk de tijd ook gekomen om al deze mensen uit-drukkelijk te bedanken voor de hulp die ze me op verschillende wijze hebben geboden. Eenaantal mensen zal ik alleen als groep noemen gezien de vele veranderingen die zich overdeze lange periode hebben voorgedaan en het risico daarbij individuele namen over te slaan.

Ten eerste wil ik mijn promotor Okko Bosgra bedanken die me in staat heeft gesteldmijn promotieonderzoek te verrichten binnen zijn vakgroep. Vervolgens bedank ik mijndirecte begeleiders, Ton van der Weiden met zijn directheid en politiek gevoel voor tactwaar nodig in dit project en Piet Teerhuis met zijn aanstekelijke liefde voor de techniek enin het bijzonder de hydraulische systemen.

Zoals in de referenties terug te vinden is, heeft een grote groep studenten het laatste (an-derhalf of meer) jaar van de ingenieursopleiding een afstudeerproject in het kader van mijnonderzoek kunnen verrichten. Dit heb ik als zeer prettig ervaren. Zeker op het moment dater gelijktijdig vijf, zes enthousiaste mensen de ’Simona motion control room’ bij werktuig-bouwkunde bevolkten en we door slim tactisch spel een (audiovisio)hardloopwedstrijd kon-den winnen zonder als eerste aan te komen. Boris Rijnten, Vasken der Kevorkian, SanderBettendorf, Philippe Piatkiewitz, Jan van Hulzen, Maris Franken, Riad Al-Saidi, Etiennevan Zuijlen en Arne Scheffer bedankt voor jullie mede- en vooral samenwerking.

Alle groeps- en ex-groepsleden van de vakgroep Systeem- en regeltechniek wil ik be-danken voor de collegialiteit, ook in de periode dat ik al uit Delft weg was. Met name doorde combinatie van een grote groep promovendi, de ervaren staf en de rechtdoorzee supportgroep was er sprake van een levendige en productieve sectie. Marco Dettori bracht daarals langstzittende kamergenoot nog wat Italiaanse cultuur in die gecombineerd met CarstenScherers internationale en vooral ook wiskundige achtergrond tot prettig pittig smakendelunchdiscussies leidden. Thomas de Hoog wil ik ook speciaal noemen vanwege het feit dat

vii

viii Voorwoord

ik de template van zijn proefschrift en allerhande LateX-advies goed heb kunnen gebruiken.Door het interfacultaire karakter van het simulatorproject is er interactie met veel mensen

die ik als zeer leerzaam heb ervaren. Ik bedank alle mensen betrokken bij Simona, vooralook door de eigenlijk nooit ontbrekende en ook zeker noodzakelijke passie voor het makenvan een technisch complex apparaat waarbij altijd vele hobbels genomen moeten worden.

Ik wil een aantal mensen hiervan met name noemen waarmee ik door hun directe be-trokkenheid bij mijn onderzoek nauw samengewerkt heb. Sunjoo Advani, die met zijn en-thousiasme het Simonaproject als projectleider lange tijd over pieken en dalen getrokkenheeft en ook tot nu nog grote betrokkenheid bij dit onderzoek heeft getoond. Gert vanSchothorst, die, twee jaar eerder als werktuigkundig promovendus bij het simulatorprojectbegonnen, veel werk op het gebied van de modelvorming rond hydraulische systemen heeftverricht waarop ik met mechanica en regeling verder kon bouwen. Met Peter Valk kon ik,gegeven ook de gelijkgestemdheid op sportief gebied, eindeloze discussies voeren. Rensde Keijzer, Kees Slinkman, Rolf van Overbeek, Fred den Hoedt, John Dukker en Ad vander Geest brachten allen vanuit onze faculteit een praktisch technische insteek waar ik nogsteeds niet aan kan tippen. Ook de wizardous meet- en communicatiekastjes van de prettignuchtere Henk Huisman moet ik niet vergeten.

Buiten het werk brachten in de eerste jaren vooral de gezamenlijk gelopen ettelijkhonderden trainingskilometers, tientallen etentjes en (estafette)wedstrijden onder andere inLonden, Parijs, Barcelona en een tweede keer ook weer tactisch winnend rond het IJs-selmeer met het losse en toch ook weer vaste groepje jonge oude Delvers de nodige aflei-ding en zeker ook vriendschap. Daarnaast leefden ook veel andere vrienden, kennissen enfamilie mee. Allen bedankt!

Alhoewel de frequentie verlaagd is houdt nu een groep enthousiaste collega’s de sportivi-teit en de felle en gelukkig lekker oeverloze en daarmee gevarieerde discussies tijdens degezamenlijke etentjes erin. Ik wil graag hierbij ook mijn hoofd bij de researchafdeling vanOce, Jo Geraedts, bedanken voor zijn steun ten aanzien van mijn werk voor dit onderzoeken dan met name zijn motivatie om er vooral op een goede manier een punt achter te zetten.

Tenslotte kan ik van hieruit nooit mijn lieve Els, Hanneke en Ingrid voldoende bedankenvoor het geduld dat ze met me en het vertrouwen dat ze in me hebben gehad. Ik zal van’pappa z’n werk’ met vier PC’s en meer dan twintig meter stapels papieren archief op twaalfvierkante meter maar (g?)een museum maken.

Sjirk KoekebakkerVenlo, oktober 2001.

Summary

The extremely high safety requirements in air transportation require a proper understand-ing of human pilot behaviour under extreme weather conditions. Advanced research in thisarea must utilize flight simulator equipment that is able to reproduce critical flight condi-tions with great fidelity. The turbulence and wind shear phenomena in the lower part ofthe atmosphere contain a velocity spectrum that induces large forces on the aircraft over awide frequency range. Thus, motion reproduction and generation of desired accelerationsin a high quality flight simulator constitutes a complicated control task that must exploitthe capabilities of an available actuation system to its limits. Predominately, these systemsare comprised of six hydraulically actuated linear servo motors, similar to many roboticmanipulators. The control of modern robotic devices employs advanced model based con-trol strategies that allow higher performance to be achieved than less structured approaches,and can provide more insight into the system limitations. In flight simulator motion con-trol, application of such methods has been almost absent. One of the reasons might be thatthe motion systems have a degree of complexity, which does not allow exact modelling ordetailed models as part of the controller. Application has required an intermediate step ofextracting a reasonably detailed model, which described the most relevant dynamic charac-teristics.

The central problem of this research was to investigate which relevant system knowledgeshould be used in a model based control strategy, and to quantify the extent to which this im-proves the controlled dynamics of the simulator. First, the full design process was defined,structured and evaluated. This began with modelling and analysis, followed by control syn-thesis and, finally, testing the system. Many solutions and procedures of the steps takenwere given in literature, but an integral approach, having the specific properties and require-ments of flight simulation motion generation in mind, was still lacking. In this research,integratibility and applicability were the main arguments in choosing the most suitable al-ternative or, if necessary, newly proposed variant, of each step in the system/control designprocedure.

First, a relevant model structure for analysis and control through physical modelling wasderived. Next, the structure was evaluated by experiments. Model parameters were identi-fied and boundaries were provided to the extent for which the model validity holds. Then,the most appropriate model based robot control strategy was chosen and, given the particu-lar system properties and performance requirements, modified for this application. The nextstep was actual application within the simulator environment, after which the testing proce-dure was defined in order to quantify the performance with respect to the requirements, andto conventional systems.

ix

x Summary

Structural insight into the system kinematics and dynamics was required in order tomaintain at least the level of robustness of conventional control strategies, and simultane-ously increase motion performance and predictability. The physical modelling began withan analysis of the system kinematics and, as a contribution of this thesis, a method was pro-posed to exclude singular points from the operational workspace and, subsequently, safelyand efficiently determine platform pose from the actuator positions in real time. Calibrationof the physical system improved further its positioning accuracy by an order of magnitude.

The dynamics of the Stewart Platform of this simulator were expressed by a set of ex-plicit differential equations using platform coordinates. It was shown in theory and practicethat the rigid body modes resulting from interaction of hydraulics and mechanics form themost relevant motion system dynamics in simulator applications. The platform to actuatorcoordinate jacobian is a central operator here. The research showed also how the parallelactuation can be viewed from a new set of coordinates as six nearly independent hydraulicsystems driving a single mass.

The applied model based controller attained several goals, which could not be imple-mented conventionally. First, it directs the required forces for the desired accelerationsalong the appropriate vectors experiencing largely different masses. Then, the feedbackpaths of the positional errors also use these decoupled directions. Finally, the interactionbetween hydraulics and mechanics is minimised over the feed forward path primarily byapplying compensation of the required oil flow given the desired velocity. Implementationin practice showed an even higher bandwidth, less peaking, and a more equalised responseof the model based controller over each degree of freedom compared with a conventionalstrategy. Even more can be gained since model based control can more effectively usepredictive information from the vehicle simulation model.

Performance limitations to this technique can be found at frequencies where parasiticeffects such as structural flexibility, and hydraulic valve and transmission line dynamicsbecome significant, or when the system no longer behaves as being rigidly attached to thefloor. However, through the application of a light weight and rigid design of the movingplatform, and with the control strategy proposed in this thesis, the system bandwidth canbe raised to a sufficiently high level, thereby enabling a wider range of motion based flightsimulation research to be executed than heretofore.

Contents

Voorwoord vii

Summary ix

1 Introduction 11.1 Flight simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 The art of simulation . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Mechanism of motion perception . . . . . . . . . . . . . . . . . . 41.1.3 Historical perspective . . . . . . . . . . . . . . . . . . . . . . . . . 81.1.4 Motion simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 91.1.5 SIMONA project . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2 Motion control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2.1 Motion control structure . . . . . . . . . . . . . . . . . . . . . . . 171.2.2 Motion control specifications . . . . . . . . . . . . . . . . . . . . . 191.2.3 Conventional simulator motion control . . . . . . . . . . . . . . . 211.2.4 Robot control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.3 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.3.1 General problem statement . . . . . . . . . . . . . . . . . . . . . . 251.3.2 Structuring the problem statement;

approach in research . . . . . . . . . . . . . . . . . . . . . . . . . 251.4 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2 Mechanics of parallel driven motion systems 312.1 Parallel motion systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2.1 Notation and definitions . . . . . . . . . . . . . . . . . . . . . . . 332.2.2 Calculating velocity and acceleration by differentiation . . . . . . . 342.2.3 Parametrising orientation by euler angles . . . . . . . . . . . . . . 362.2.4 Parametrising orientation by euler parameters . . . . . . . . . . . . 382.2.5 Jacobian matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.2.6 Parallel, serial and singular configurations . . . . . . . . . . . . . . 412.2.7 Stewart platform definitions and assumptions . . . . . . . . . . . . 432.2.8 Stewart platform kinematics . . . . . . . . . . . . . . . . . . . . . 44

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xii Contents

2.2.9 Interpretation and use of the jacobian matrix, J l;x . . . . . . . . . . 452.2.10 Velocity and acceleration of the actuator joints . . . . . . . . . . . 472.2.11 The Simona flight simulator motion system kinematics . . . . . . . 49

2.3 Analysis of the Stewart platform kinematics . . . . . . . . . . . . . . . . . 502.3.1 Convergence NR-iteration . . . . . . . . . . . . . . . . . . . . . . 532.3.2 NR-convergence Stewart platform . . . . . . . . . . . . . . . . . . 542.3.3 Lipschitz condition on jacobian . . . . . . . . . . . . . . . . . . . 562.3.4 Exclusion of singular points . . . . . . . . . . . . . . . . . . . . . 582.3.5 Sufficient update frequency . . . . . . . . . . . . . . . . . . . . . 59

2.4 Modelling the mechanical part of the system dynamics . . . . . . . . . . . 602.4.1 General theory in modelling the dynamics of mechanical systems . 602.4.2 Example in modelling using Kane’s method . . . . . . . . . . . . . 642.4.3 Stewart platform dynamics . . . . . . . . . . . . . . . . . . . . . . 662.4.4 Basic Stewart platform mechanical model dynamics . . . . . . . . 662.4.5 Influence of the actuator inertial forces . . . . . . . . . . . . . . . 672.4.6 The Stewart platform model including actuator inertia . . . . . . . 692.4.7 Parasitic mechanical aspects: modelled as linear flexibility . . . . . 70

2.5 Chapter Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3 Hydraulically driven motion systems 753.1 The basic structure of hydraulic actuators . . . . . . . . . . . . . . . . . . 76

3.1.1 Leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.1.2 Basic hydraulically driven mechanical system model . . . . . . . . 79

3.2 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.2.1 Servo valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.2.2 Transmission lines . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.3 Integrated system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.3.1 Passive input/output pairs in flight simulator motion systems . . . . 903.3.2 The 1-d.o.f. hydraulically driven mechanical system . . . . . . . . 943.3.3 Additional mechanical modes . . . . . . . . . . . . . . . . . . . . 97

3.4 Hydraulically driven Stewart platform . . . . . . . . . . . . . . . . . . . . 1013.4.1 Basic model structure hydraulically driven systems . . . . . . . . . 1023.4.2 Dynamical and kinematical properties of the SRS-motion system;

Design aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043.4.3 Actuator inertial effects . . . . . . . . . . . . . . . . . . . . . . . . 1103.4.4 Analyzing the SRS hydraulically driven motion system model . . . 1123.4.5 Connection of a flexible foundation to the SRS system model . . . 116

3.5 Chapter Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4 Parameter identification and model validation 1194.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4.1.1 Calibrating the Stewart platform . . . . . . . . . . . . . . . . . . . 121

Contents xiii

4.1.2 Redundant measurement of the platform pose . . . . . . . . . . . . 1234.1.3 Stewart platform pose measurement in practice . . . . . . . . . . . 1264.1.4 Identification of the kinematical parameters . . . . . . . . . . . . . 1284.1.5 Results in calibrating the SRS motion system . . . . . . . . . . . . 131

4.2 Stewart platform model parameter identification . . . . . . . . . . . . . . . 1334.2.1 Gravitational force determination . . . . . . . . . . . . . . . . . . 1334.2.2 Identification of the inertial properties . . . . . . . . . . . . . . . . 136

4.3 Frequency response model validation . . . . . . . . . . . . . . . . . . . . . 1414.3.1 Frequency response dummy platform . . . . . . . . . . . . . . . . 1424.3.2 Additional dynamics into the higher frequency area . . . . . . . . . 147

4.4 Chapter Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

5 Model based control of the flight simulator motion system 1635.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1635.2 Another look at the control problem . . . . . . . . . . . . . . . . . . . . . 1645.3 Control Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1665.4 Inner loop pressure control . . . . . . . . . . . . . . . . . . . . . . . . . . 1685.5 Multivariable feedback linearisation . . . . . . . . . . . . . . . . . . . . . 173

5.5.1 Feedback linearising robotic manipulators . . . . . . . . . . . . . . 1735.5.2 Feedback linearisation of a Stewart platform . . . . . . . . . . . . 1745.5.3 Implicit state measurement requirements . . . . . . . . . . . . . . 1765.5.4 Outer loop control . . . . . . . . . . . . . . . . . . . . . . . . . . 177

5.6 Reference model based feed forward . . . . . . . . . . . . . . . . . . . . . 1785.6.1 Reference model based predictors . . . . . . . . . . . . . . . . . . 1795.6.2 Construction of the reference model based controller . . . . . . . . 181

5.7 Implementational issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1855.8 Performance quantification . . . . . . . . . . . . . . . . . . . . . . . . . . 185

5.8.1 Motion system evaluation methods and requirements . . . . . . . . 1875.8.2 New test procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 189

5.9 Experimentally evaluating performance . . . . . . . . . . . . . . . . . . . 1915.9.1 Experimental set up . . . . . . . . . . . . . . . . . . . . . . . . . . 1915.9.2 Characteristics of the Simona motion system . . . . . . . . . . . . 193

5.10 Chapter Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

6 Review and discussion on the results 2036.1 Flight simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2036.2 Motion system specifications . . . . . . . . . . . . . . . . . . . . . . . . . 2046.3 Theoretical modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2056.4 Experimental modelling and validation . . . . . . . . . . . . . . . . . . . . 2066.5 Control strategy and evaluation . . . . . . . . . . . . . . . . . . . . . . . . 208

7 Conclusion and recommendations 211

xiv Contents

7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2117.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

A Frequency domain measurements 215

B Derivation of actuator inertial properties 227

Bibliography 231

Glossary of symbols 241

Samenvatting en CV 247

Chapter 1

Introduction

”As man travels, he vibrates. For as he travels, he generally experiences accelerationsnot perfectly uniform in magnitude nor direction, and these he feels as varying forces orvibrations. Travelling faster, these vibrations become more severe, until he learns newways to reduce them. But the pioneers will vibrate considerably.” [23] These phrases byClark, originally used to motivate research into the tolerable limits of motion cues for humanbeings in the early development of the U.S. space program, exactly point at the aspect ofpiloted simulation, which is the subject of this research. The motion cues, which are aconsequence of travelling, can be stimulated without actually travelling, thereby enhancingperceived realism in a simulator. Further, since proper control of a vehicle requires sufficientreduction of vibration, the actual sensation of these resonances forms an essential part ofbasic skill training.

The tool to achieve this, is a motion system. By automatic control, the dynamics ofthe motion system will have to be transformed into the dynamics of the vehicle with itsenvironment. This requires knowledge of both systems. With the developments in computertechnology, more and more information can be directly incorporated in the controller. Givenfeasible trajectories generated by a reference generating system of e.g. an aircraft model inour case, this research considers the use of motion system knowledge in a model basedcontroller.

In this chapter, first an overview of the most important aspects of flight simulation will begiven in Section 1.1. Then, the specific requirements and approaches towards motion controlwill be discussed in Section 1.2. This will lead to the problem statement and approach ofthis research in Section 1.3. Finally, the outline of this thesis will be given in Section 1.4.

1.1 Flight simulation

First, the different aspects of the complex system of flight simulation will be discussed.With an emphasis on the motion aspect, the evolution of these systems over the years willbe sketched. Then, the Simona project centred around the development of a modern researchsimulator, in which this investigation was embedded, will be introduced. Finally, the specificfeatures of the six degrees-of-freedom Stewart-Gough Platform motion system, nowadaysgenerally used in flight simulation, will be stated.

1

2 1 Introduction

1.1.1 The art of simulation

The fact that simulation is often still considered an art instead of a science [25],[131], re-flects the fact that this complex mechanism of creating a virtual environment is not fullyunderstood yet. The flight simulator or more specifically piloted simulation is, however,widely used nowadays, especially in the area of aerospace and astronautics. Following Ad-vani [7], several areas of application can be recognised.

Area of application

Engineering simulators are used to evaluate the characteristics of a vehicle. In the devel-opment of a new aeroplane, the simulator is currently intensively used during the wholeprocess of design, due to their ability to predict problems and support flight clearance. Withsimulation, large reduction in flight testing could be achieved, which saved costs and al-lowed earlier certification [131]. Most engineering simulators are fixed base i.e. withoutinertial motion since motion cues are not always important and necessary.

However, the motion system is essential in predicting handling qualities, e.g. a pilotinduced oscillation (PIO) due the dynamics of the vehicle, in a proper way [121]. Lowfrequent modes of the aircraft can be damped more appropriately with motion, while vi-brations at higher frequencies amplified by the pilot in the loop, e.g. often modes due tostructural aeroelasticity, can usually only be detected if the controlled motion system leavesthese modes unaltered.

Training simulators are used to train pilots both in the procedural (e.g. flight management)as well as the basic skill tasks (e.g. manual control). In military applications the proceduraltraining is often done in fixed base simulators. The basic skill training is mostly performedin-flight since the discrepancy between the simulator and aircraft is considered too large inextreme manoeuvres and of course actual flight training is much more exciting. In com-mercial airlines, the economic attractiveness of motion base simulators is preferred overactual flight time and many hours are spent in training simulators. Proficiency checks andre-currency training i.e. skill examinations and transfer to another type of plane, are allowedby the regulating authorities without actually flying for pilots with over 500 hours experi-ence on an aircraft of the same group or 1500 (Europe) to 2500 hours (USA) grand total[25]. There is a tendency to further reduce this prescribed number of hours [143], whichallow this so-called Zero-Flight-Time (ZFT) training, but exact knowledge of the requiredsimulation fidelity is still lacking.

Research simulators are often applied for fundamental investigation into pilot/vehicle in-teraction and human perception research. These kind of simulators most often require thehighest level of performance w.r.t. the motion system as the resulting sensory input is es-sential in the manual control task [47].

Apart from this fundamental scientific work, the simulator has also proved valuablefor accident investigations [131]. Research simulators have been used to study the circum-

1.1 Flight simulation 3

stances of incidents due to clear air turbulence and other types of severe turbulence like gust,and come up with more beneficial strategies in control and better operational procedures.In these experiments, the motion system had to be capable of producing vibrations whichmade the pilot unable to read the instruments.

Advantages of simulatorsReviewing the area of application reveals the various specific qualities of the flight simula-tor. Over the years it has proved to have several advantages as compared to flying an actualaircraft.

-Cost effectiveness. Both with respect to initial investment as to operational costs, the flightsimulator, though still requiring a considerable amount of money (often over 10M$and about 1k$/h), is still an order of a magnitude less expensive than most publictransportation aircrafts. Further, the effectiveness of training and testing is in manycases even better.

-Safety requirements. The simulator is a safe tool to test and train in conditions, whichare dangerous in actual flight. Training simulators can safely be used to show pilotsthe best operating procedures in unusual circumstances. Inexperienced pilots canbe taught the bulk of basic skills and procedures avoiding the risk of putting theminto the air. Many potential flight design errors can be detected in advance. Further,the fact that accidents can be analysed in detail with a simulator, enhances flightsafety. An indirect safety aspect is the cost effectiveness enabling a thorough and stilleconomically feasible training program.

-Environmental considerations. Every hour spent in a simulator instead of an aircraft,directly saves a considerable amount of kerosine burnt in the air. Indirectly, environ-mentally less harmful flight operation can be tested and trained in a simulator.

Since the objective is not to duplicate, but to simulate an environment which creates thebest and most effective training and testing results [25],[117], the flight simulator shouldalso be considered as a training or testing tool in its own right, rather than as a substitute forthe aircraft [143]. In this respect the simulator is often more effective.

Training can be split up in parts and these modules can be taught separately, in a step bystep approach. The simulator provides an objective tool to evaluate different subjects underequivalent conditions as the repeatability of the system is much higher. A simulator is veryflexible in performing specific tests e.g. under desired weather conditions or at a certainairport.

Flight simulator componentsIn piloted simulation, a complex environment is created, which has to reflect the impressionof travelling in the actual vehicle with an emphasis on the perceived realism of the task tobe investigated or learned. This environment can be divided into several subsystems:

-Interior. First of all the static environment of the pilot has preferably the look of the inte-rior of the vehicle cockpit.

4 1 Introduction

-Visual system. The exterior out-of-the-window sight is simulated with an image generat-ing system in case of civil aircraft simulation often projected via a wide angle mirrorattached in front of the simulator cockpit.

-Instruments. The instruments in the cockpit give information about the state of the simu-lated vehicle and should actively be changed accordingly.

-Motion system. The required inertial motion, the actual accelerations of the system, canbe generated to some extent by actively moving the system.

-Control loading. The pilot controls to the vehicle often responds to the given input byfeedback of forces which reflect some of the external forces on the vehicle e.g. theaerodynamic force on the rudder, etc. These forces have to be generated artificiallyand in this case called control loading in a simulator and vehicles, where this has beendecoupled mechanically (fly-by-wire).

-Audio system. The generation of the sound a vehicle together with its environment pro-duces adds to the perceived realism in the simulator and, above this, is used in thecontrol task (gas throttle). In some cases, the audio system can also mask the para-sitic noise produced by other sub systems [10].

Apart from the static interior, all these subsystems add to the motion awareness and all cues(noticeable stimuli), which can be generated by these subsystems, have to be considered inevaluating a manual control task.

The main unknown remains to what extent the simulator environment reflects the in-the-air conditions and to what extent this is required to achieve a certain training or testobjective. First, the simulator is only as good as the data it contains i.e. the quality ofthe model representing the aircraft and the flight environment to be simulated. Secondly,inherent limitations of the system used to simulate can be the cause of mismatches. Themotion system of a simulator does posses one of the most obvious constraints with its verylimited amount of travel. Fortunately, the pilot can be made to perceive the motion desiredif the main cues are correctly simulated, allowing for some deviation in the others.

1.1.2 Mechanism of motion perception

Perception of motion has been studied for more than hundred years [88]. At that time,mainly experimental data was obtained. In the late 60’s of the past century models wereproposed, which described the human behaviour in a motion control loop [69],[94]. Withthe crossover model of McRuer et al. [94], the pilot acts according to a describing function,H(j!), over the frequency, !,

H(j!) = Kh�1j! + 1

�2j! + 1e��dj! : (1.1)

This model assumes only one main cueing input and considered additional dynamics of thesensory and response system as a simple time delay, e��dj!. The parameters, gain,Kh, leadtime constant, �1, and lag time constant, �2, are adapted such that a slope of -20 dB/decade


in the describing function frequency plot around the crossover frequency, ! c is attained ofthe open loop system of human and the system to be controlled, which usually results ina sufficient stability margin in the closed loop. This frequency is approximately 2 rad/s to5 rad/s in case of a flight control task (involving tracking and disturbance rejection) [53, 94,149]. With aircraft vehicle control, on which this model was experimentally validated, theopen loop transfer function with a double integrator, 1=(j!) 2, can be controlled with PD,proportional/differential action in which �2 < 1=(!c) < �1.

The optimal control model of Kleinman et al.[69] assumes that the pilot operates in anoptimal way, given all the perceived cues with their respective limitations. This motivatedthe research into the mechanism of motion perception ([159] and the references therein) witha model based approach in which the interaction between the different stimuli, e.g. inertialmotion and visual inputs, is taken into account. These models are still being refined [53]as the system is incompletely understood. Though the quantitative values are seen to bedependent on the subject, task and environment involved, the qualitative control behaviourof a well trained subject is reasonable consistent and reproducible.

Motion results in stimuli inputs to the sensory organs. Above some sensory threshold,the sensory organs start to fire pulses to the central nervous system. The subject perceivesmotion from the set of cues generated in this way. These cues have to be reasonably consis-tent relative to each other. Otherwise disorientation and motion sickness can occur. Somedeviations can be tolerated as long as the coherence functions of the subject from each cue tothe corresponding movement have sufficient overlap [148]. Advantage from these tolerabledifferences is taken in simulation, which allows perceived motion to be simulated withoutthe need for very large simulator excursions.

The main sensory mechanisms to detect motion as given in the AGARD Advisory report159, [2] are:

- Semicircular canals and Otoliths are called the vestibular organs. Both reside in theinner ear. The function of the semicircular canals is roughly that of rate gyros, fromwhich angular velocity can be measured. The otoliths sense the specific force whichis the combination of inertial acceleration and gravity.

- Tactile or somatosensory receptors permit sensing a change of force on the body, e.g. dueto a change in orientation. These sensors have a high-pass characteristic, which causessustained uniform pressure to be faded to a certain reference level. Especially the fin-gertips are very capable detecting high frequency vibrations.

- Proprioceptive and kinesthetic senses lead to information about the relative positions ofthe parts of the body as well as their movements i.e. the body acceleration. This isattained through the double sided muscular coupling of force and velocity/positionand an internal model of the body

- The eyes primarily detect motion through the peripheral retina. With uniform motion ofa wide visual field, it is possible to create a so-called self motion sensation, whichis called vection. The foveal area of the retina or central part is primarily associatedwith scanning and recognition i.e. the cognitive sense of self-motion.

6 1 Introduction

Advani [7] also mentions the auditory system in the ears, from which through sound motioncan be detected.

With the visual and auditory system of the flight simulator, the pilot can be isolated fromdetecting inertial motion through sound or sight. With the other organs, inertial motion cueswill be perceived as motion. In case of required high forces, the proprioceptive and tactilesenses can also be stimulated with a so-called g-suit, mainly used in military applications.Control forces and vibrations provided by the control loading system will also influence thetactile senses.

The vestibular system is considered to generate the main cues in the frequency areaaround the crossover frequency (:3 to 1Hz), essential to learn skill based control behaviour.This is the central issue in motion-based flight simulation. In order to recreate the skillbased behaviour, the pilot in the simulator must be provided with well coordinated visualand vestibular cues. Motion cues around the crossover frequency frequency area can onlybe stimulated with actual inertial motion and this is the main reason a motion system isessential in case of piloted control tasks in flight simulation.

In the low frequency area (� :1Hz), motion is much easier to detect through the visualinformation of position and orientation at the instrument displays and exterior screen. Somemismatch with inertial motion, with the already low accelerations related to these frequen-cies, is allowed, even slightly superthreshold, without the pilot noticing discrepancies.

Schroeder [126] notes that the review of different studies resulted in a wide range ofthreshold values. This is confirmed by the study reported in [54] in which it is shown that athreshold depends on several factors such as the frequency contents of the stimulus and thetask the subject has to fulfil i.e. mental load. Values range for the translational accelerationsfrom :08m=s2 in heave, no task involved to :7m=s2 in a very busy situation experiencingsurge (longitudinal direction). Rotational thresholds are found to be in a closer range of:8 deg =s2 to 1:5 deg =s2 for pitch and roll, though Schroeder [126] reports a range of:0035 deg =s2 to 2 deg =s2 for yaw motion (vertical axis rotation).

Hosman [53] provides some transfer functions for the vestibular system. Both the semi-circular channels as the otholits are modelled according to a function specified in the fre-quency domain:

�xout�xin

=K(�Lj! + 1)

(�1j! + 1)(�2j! + 1); (1.2)

where the ratio between (actual) input acceleration, �x in, and noticed acceleration (�xout inimpulses per second, ips), is given by a second order transfer function with gain, K, twolag time constants, �1 and �2, and a lead time constant, �L.

Although the models of the two vestibular organs, which give the sensitivity over the fre-quency, have the same structure, the characteristic is different. This is due to the parameters,which are given by Hosman [53] as in Table 1.1.

This means the sensitivity of the semicircular channels drops over the range from 0.025 Hzto 1.4 Hz after which it starts to drop again at 30 Hz, while the otholits become more sensi-tive from .16 Hz to .3 Hz, after which it starts to drop again at 10 Hz. This opposite effectin the relative important frequency range could maybe explain the fact that different con-clusions can be drawn towards the question whether translational accelerations or angular


Semicircular Channels OtholitsK 2 ips=(�=s2) 3:4 ips=(m=s2)�L 0:11 s 1 s�1 5:9 s 0:5 s�2 0:005 s 0:016 s

Table 1.1: The parameters of the models of the vestibular organs

accelerations are more important. In case of roll and lateral motion in large aircrafts and yawand lateral motion in helicopters, lateral motion cues are shown to be more important [126].In other cases reported in [54], [126] rotational cues are considered more relevant. Chunget al. [22] conclude that the translational and rotational motion relative to each other shouldnot differ in phase nor amplitude since these types of motion are coupled as a function ofplace.

The motion cues provided through the vestibular system with inertial motion, changethe describing function of the pilot [53], [149]. With the introduction of motion, the timedelay in the describing function is seen to drop from 0.23 s to 0.12 s in case of a disturbancerejection task. The crossover frequency a pilot can achieve is lower in the absence of iner-tial motion and, as a result, also the performance is. Hall [47] argues and confirms by anexample that, even with equivalent performance, the pilot can develop a different and falsestrategy if the cues in a simulator deviate from those in the aircraft. Especially in case ofrelatively inexperienced pilots, all cues are important in the learning process of picking themost relevant [143].

There is no doubt inertial motion is necessary in the appropriate perception of motion.Not much is known to what extent the inertial motion of the simulator is allowed to differfrom the aircraft. In the low frequency area it is concluded that high pass filtering at .08 Hz isacceptable [149]. The crossover region between .1 Hz and 1 Hz is likely to be most sensitiveto deviations. At higher frequencies, the situation is less clear. Pilots are seen to providelead well over 3 Hz and consequently still respond to motion with increasing gain in thisfrequency area. Further, pilots respond much earlier when inertial motion is present. In themore general requirements of the international standard ISO 2631 [138], which evaluateshuman exposure to whole-body vibration, it is noted that humans are most sensitive to heaveaccelerations in the frequency area between 4 Hz and 8 Hz. The bandwidth of many motionsystems in this frequency area makes these systems prone to errors exactly in the area wherethe level of comfort is most sensitive.

Over the years, many discrepancies in the motion systems of simulators have been con-sidered acceptable. There was no economical or technical feasible alternative. Other sub-systems were also unable to represent reality in a satisfactory manner. A pilot was requiredto fly an aircraft on the basis of instrument displays and external sight without motion cues.Being able to fly a simulator without appropriate inertial motion qualified since it is moredifficult than with an aircraft. In-flight training still formed a considerable part of the learn-ing process. This point of view is changing as can be seen by putting the current state of theart in a historical perspective.

8 1 Introduction

1.1.3 Historical perspective

In [4], Adorian et al. provide a nice overview of the development of the flight simulators overthe years. Almost directly with the introduction of the first aeroplanes, the first simulatorswere built. Already in those days, a motion system was part of the simulator. Around 1910,these motion systems had to be driven by hand by the instructor, or wind, and mainly hadto give the experience of a changing orientation around one to three axes. Soon the inertialmotion was provided automatically. The Ruggles Orientator, developed in 1917, had full ro-tation around three axes and additionally vertical movement, all driven by electrical motors.The most successful device, the Link trainer, developed in 1927-29, was used well into the50’s. It was driven pneumatically and operated by the stick and rudder in the cockpit. Thesimulated effects were adjusted by trial and error independently for ailerons, elevators andrudder. In this way, the aircraft’s coordinated behaviour could not be represented properly.

After 1930, for thirty years, the main developments concerned other subsystems than themotion system of the simulator. Early in the Second World Ward, hundreds of the so-calledCelestrial Navigation Trainers, massive bomber crew trainers, were built. A moving domeprovided for the ’correct’ out-of-the-window sight in case of a nightly atlantic ocean cross-ing. During the war, the visual system was developed further into actual image projectionsystems.

With the instruments in the cockpit, all basic flying behaviour, all engine, electrical andhydraulic systems could be simulated. The computation, at first pneumatically, could solvethe equations of motion of an aircraft with the introduction of the analog computer.

The first full aircraft simulator operated by an airline was installed in 1948 (a Boeing377 by PanAm). Most full aircraft simulators did not have a motion system at that time,which was justified by the statement that ”modern pilots should not fly by the seat of theirpants.” Partly, the control loading system was thought to compensate for the lack of motionand provide for a realistic feel.

With the improvement of flight test data and the increasing complexity of the aircrafts,the analog computer became the bottle-neck in the simulator. The demand for increasedfidelity and reliability motivated the introduction of the digital computer in the flight simu-lator system.

However, correlation was poor with flight tests concerning low damping, stability orcharacteristics of pilot induced oscillations. Further, strong visual cues supplied by theimproved wide field of view visual system were disorienting in the absence of motion cues.In 1959, the first motion research simulator was built at NASA, which also indicated theimportance of motion cues [10]. Analysis of loss of control in extreme operations motivatedthe research with centrifuge simulators [23]. The many motion deficiencies, due to thevery limited vertical stroke, the vibration noise and incorrect centrifugal forces, were sodistracting that most of the research was still performed with fixed base simulators.

Motion systems with alternative construction were designed. The introduction of thewide body transport aircraft, such as the B747, required lateral acceleration and led to thefour and six degrees-of-freedom (d.o.f.) motion systems. The first six-degrees-of-freedomresearch simulator at NASA Ames was put into operation in 1964. These systems were oftengiven too much stroke (up to 30 m), which was not necessary given the limited velocity andresulted in inferior dynamics of lengthy cables (low eigenfrequency). Also the turn-around-


bump due to coulomb friction prohibited unnoticeable fade to a central position.In the late sixties, the synergetic, fully parallel, hydraulically driven motion system came

into operation. The most commonly used is the Stewart-Gough platform [140]. Throughthe hydrostatic bearings introduced by Viersma [153] in 1969 in the first (3-d.o.f.) flightsimulator motion system at Delft University of Technology [12], these systems had virtuallyno coulomb friction. The first commercially available 6-d.o.f. motion system in 1977 withhydrostatic bearings was reported on by Baret in [13]. Nowadays, almost every simulatormotion system operates in this manner.

In the late 70’s, an attempt was made to classify the characteristics of motion systems inflight simulators [1], [67]. Travelling limits, response time (latency), roughness and noisewere considered most important. The regulating authorities such as the FAA (USA) andJAR (Europe), given confidence in the use of simulation, came up with required motionsystem specifications, which reflect the abilities of motion systems at that time [3], [25],[118], [139]. It is argued that the fact that these standards have been basically unchangedfor the past 15 years, has stopped virtually all fundamental development in flight simulators[143]. Further, since there are still no objective tests, no prescribed manoeuvres and nocriteria for accurate motion cueing, makes it difficult to establish new standards. There is,however, a market demand from the airlines to come up with additional knowledge to enablea decrease in required flight experience for zero flight time training.

In the 80’s and 90’s, the main developments concerned the rapid growing abilities inthe visual and computational systems. As the main selling argument in the 60’s came fromthe all but perfect though impressive motion systems, now most interest stems from the fastimproving, marketable, visual system. As in the past, higher standards for other subsys-tems, such as the quick response of the image generating systems now available [100], theincreased computational power enabling to include aeroflexible modes in the aircraft modeland more advance control strategies for motion control, will be a drive for higher fidelitywith respect to motion.

In [10] it is not expected that any major changes in motion travel will be made in flightsimulation motion systems. However, within the current structure improvements can be at-tained [7] and more often multi-stage, cascaded, motion systems can be observed, in whichthe need for motion cues (with the current 6-d.o.f. structure) and travel (additional for lateraland longitudinal travel over rails as in the National Advanced Driving Simulator [20], [92])have been decoupled but still have considerable interaction in the dynamics. Reduction oftime delays in motion system response and interaction is needed for solving more complexproblems with visual/motion/model mismatches [10]. At this moment, high frequency mo-tion cues (� 4 Hz) require special effects included in the motion system controller andphase lag in the crossover region is compensated for by additional lead (differential action)on the reference signals [10], [24], [143]. A more structural model based based approachwill be strived for in this research.

1.1.4 Motion simulation

As already mentioned, in flight simulation, the pilot relies on the perception of self-motionthrough several stimuli, and uses this perception to exercise control over the aircraft. From

10 1 Introduction

AircraftModel

Wash-OutFilter

MotionController

MotionSystem

Pilot

Experimentator

Continuous World

Situation

Simulation computer Motion Computer

Fig. 1.1: A block diagram representation of motion simulation.

the pilots point of view, the simulator should be able to replace the aircraft.In the control task, the pilot uses his visual perception to make a good estimation of

the aircraft’s long term attitude and velocity. The approximate frequency response of visualperception can be modelled as a first order low-pass filter with a break frequency of 0.1 Hz[92]. Fortunately, the inertial motion of the simulator has a minor role below this frequency.There is a potential payoff in tuning the motion cues to match the human physiologicalsensors’ responses, rather than attempting to match the aircraft motion themselves. If doneproperly, the effective provision of motion can be obtained in an envelope substantiallyreduced from that of the aircraft motion.

The stimulation through actual inertial motion with a simulator can usually be repre-sented by the block scheme given in Fig. 1.1. The pilot responds to simulator cues andthe task at hand provided through the experiment leader. With a model of the aircraft, it iscalculated what the actual aircraft motion would have done due to the action of the pilot,the situation (on the ground the aircraft has different dynamics in interacting with the trackthan in the air) and the situation dependent disturbances such as turbulence.

With washout, the calculated aircraft motion has to be translated into feasible motionsystem trajectories regarding the limited stroke, velocity and accelerations. Washout mustprovide some form of high pass filtering to limit the simulator excursions given the un-limited aircraft travel. Washout does usually separate the motion cues into high (”onset”)and low (”sustained”) frequency components. Further, by also introducing a scaling fac-tor of simulator motion, all onsets can be performed by the simulator without attaining theinherent limitations of the motion system.

Most sustained cues can not be represented through inertial motion of the simulator andhave to be ”washed out”. Sustained forward or lateral acceleration is sometimes representedthrough a trick already mentioned by Johnson in 1931 [4]. Through the help of the gravityforce, a backward tilt gives the same sensation as a forward linear acceleration. Problem


Fig. 1.2: A typical modern flight simulator equipped with a synergistic (hexapod) motionsystem.1

with this so-called tilt coordination is to attain the tilted pose without the pilot noticingi.e. pitching or rolling with subthreshold angular motion. In a critical survey, Advani [7]notices that the mostly used classical linear washout leads to considerable false cues in theimportant frequency area between 0.1 Hz and 1 Hz. In an overview of wash-out techniquesin [92] by Martin, also non-linear wash-out is mentioned, which however has not led tocompletely acceptable solutions so far.

The feasible washed-out motion trajectories are the reference input to the motion con-troller. The task of the motion controller here is to attain a one-to-one correspondencebetween the actual motion system trajectories and the desired feasible washed-out motiontrajectories. In the usual configuration, the references contain positional information, whichis also measured and can be used in feedback. Further, the desired acceleration profiles areprovided for, which can be applied in generating an appropriate feed forward.

As it is physically impossible to generate immediate acceleration with the motion sys-tem, imperfect cues are generated, which lag the desired cues. Enlarging the bandwidthdecreases the lag but has to be compromised with sufficient attenuation of false cues re-sulting from noise and excitation of parasitic dynamics, which mainly reside in the highfrequency area. Considering a more general setting than the one given in Fig. 1.1, can pos-sibly lead to a less severe compromise by e.g. allowing available predictive information inthe simulation model to be sent to the motion controller.

The flight simulator motion systems, nowadays, have almost invariably a so-called 6-d.o.f. synergetic construction. A motion system of this type is shown in Fig. 1.2. 1 Six

1Picture courtesy of Rexroth-Hydraudyne.

12 1 Introduction

actuators are connected in parallel to a base frame at the foundation with one gimbal, afreely rotating joint, and at the other end to the simulator with another gimbal. Kinemati-cally, the six lengths of the actuators can be driven independently. However, dynamically,this system is highly interactive i.e. an elongation force of one actuator generally induces re-action forces in all others. Further, prescribed platform motion requires coordinated motionof all actuators, which varies with the platform position.

The main advantages of this construction are the parallel connections resulting in a sys-tem with higher stiffness given a specific actuator and its simplicity. Six identical systemare easy to build and maintain. The stiffness is due to the fact that springs in parallel resultin higher stiffness while springs in series will result in a decrease. This is important in flightsimulator motion systems since the payload to be accelerated is usually relatively high. Withthe synergetic construction the static load per actuator will be lower and the eigenfrequen-cies will shift to higher values. The forces to be applied with the actuators are usually stillconsiderably high. Most flight simulator motion systems are therefore driven by hydraulicservo actuators, since these systems provide superior performance in generating high powerlong stroke linear motions [124]. Further, the use of hydrostatic bearings results in very lowcoulomb friction forces, which minimises the velocity reversal bump. As simulator washout motion reversal and aircraft motion do not coincide, an unnoticeable change of velocitysign is very important in flight simulation.

With the introduction of improved linear electrical drives, there is a trend to use elec-trically driven motion system if a simulator has relatively low payload up to � 2500 kg[24].

1.1.5 SIMONA project

Through a preliminary investigation for an upgrade programme of the older DUT 3-d.o.f.simulator, a new international centre for research into flight simulation techniques, Simona,was initiated in 1992. Simona stands for Simulation, Motion and Navigation, which reflectsthe areas of interest in the research programme of the three groups, who initiated this centre.Instead of an upgrade of the older simulator, which has been dismantled, the core of Simonawill be a full scale 6-d.o.f. flight simulator, the Simona Research Simulator (SRS) [6]. Thedevelopment of the SRS is also performed within Simona. The actual fabrication of thesimulator was started in 1994, after a large donation of the Dutch Government to enable amulti-discipline international centre with a high fidelity simulator.

The three groups involved, each have their specific research interest and contribution tothe project.

� The Simulation and Control Group of the Faculty of Aerospace Engineering performsstudies into flight control, the man-machine interaction of the pilot in the aircraftand modelling and real-time simulation of the aircraft and its systems. In the, nowongoing, realisation phase of the simulator the group takes part in the developmentof simulation software, the interior and the visual system. The Group of ProductionTechnology within Aerospace Engineering did the design of the integrated platform-cockpit shuttle and planned connections of projection platform and image projectionscreens, which will be connected in future.


Fig. 1.3: Artist’s impression of what will be the full operational Simona flight simulator

14 1 Introduction

Fig. 1.4: Simona motion system with empty shuttle on top, configuration B.


� The Mechanical Engineering Systems and Control Group of the Faculty of Design,Engineering and Production performs research into the design, development, mod-elling and control of hydraulic servo and motion systems. The 6-d.o.f. hydraulicallydriven motion system has been realised and tested and so have the first model basedcontrollers as part of this research. In future, alternative control concepts will have tobe developed. For one, to interconnect simulation models and motion control in anoptimal way. Further, to adapt to the changing characteristics of the motion systemwith e.g. a, to some extent flexibly attached, visual system.

� The Telecommunications and Traffic-Control Systems and Services Group of the Fac-ulty of Information Technology and Systems does research into navigation technologyand advanced cockpit display systems. The operational SRS can serve as an exper-imental set-up to test new navigation techniques and approaches to simulate thesetechniques appropriately. The group has been involved in the development of a real-time simulation platform.

An impression of what will be the full operational flight simulator is given in Fig. 1.3.The motion system consists of six hydraulic servo actuators which are connected in parallelat the base frame and simulator. The actuators together can drive the simulator in all threetranslational and rotational degrees of freedom. At the simulator side, the actuators are con-nected to the bottom of the shuttle, an integrated platform/cockpit made of the light weightmaterial TWARON/carbon. In front of the shuttle, a wide angle visual mirror projectionscreen will be attached. With projectors on top of the shuttle and a back projection screenin between, the pilot in the cockpit can be provided with a out-of-the-window view via themirror in front of the shuttle.

The new design of the integrated platform/cockpit and the use of light weight materialswill lead to a relative low centre of gravity and considerably less mass to be acceleratedas compared to conventional simulators. These properties strived for in the design of theconstruction, will have to result in favourable motion characteristics. In what sense theexpectedly 4 tons of the SRS instead of the 12 to 15 tons often reported for other systems[24], [92] can be taken advantage of, is one of the issues to be discussed in this research.

After separate testing of all six hydraulic actuators in a special experimental set-up in1994 [124], the integration of the motion system was complete early in 1997, and the firstfree-run tests of the integrated Simona motion platform were performed. The motion systemwas tested in three configurations in the Central Workshop at the faculty of MechanicalEngineering.

A. The first tests were performed with a steel platform, the dummy platform of 2200 kg.

B. In the autumn of 1997 the empty (only chairs) shuttle of 1700 kg was put on top of themotion system, as shown in Fig. 1.4.

C. Finally, tests were performed with the estimated final load of 4000 kg with the dummyplatform and additional masses (6 times 250 kg) on top, shown in Fig. 1.5, in themonths of May and June, 1998.

In the early spring of 1999, the motion system was to be operated at the building especiallymade to house the Simona centre, eventually with the full operational simulator. Main

16 1 Introduction

Fig. 1.5: Simona motion system with dummy platform and additional masses, configurationC.

1.2 Motion control 17

K

Psp

�y�u

- -

-

�

�w �z

Fig. 1.6: A standard control design structure

efforts are now in progress to complete the cockpit interior, the real-time simulation envi-ronment and the visual system.

1.2 Motion control

Having sketched the flight simulation process and the place motion generation has in thisenvironment, the motion control problem can be defined. First the process of motion gen-eration is put into a generalised framework. From this perspective the motion control spec-ifications are stated and the conventional approach towards the design of motion control inflight simulators is discussed. Then realising that the motion system of a flight simulator canbe considered a robotic system, of course with its specific demands and properties, some ofthe modern control strategies developed in robot control are put forward, since these could,with some modification, possibly be of benefit in flight simulation.

1.2.1 Motion control structure

A standard control framework, adopted in many textbooks on modern control [17], [89],[160], is given in Fig. 1.6. A controller, K, is provided with measurement signals, �y, andhas to stabilise a plant, Psp, with input signals, �u, such that the cost variables, �z, are minimalin some sense, despite the disturbance signals, �w.

The plant, Psp, is often called the generalised plant or standard plant since it usuallydoes not only consists of the plant to be controlled, the motion system in flight simulationmotion control, but can also contain weightings e.g. sensory thresholds weighting on thecost variable specific force or a reference model e.g. the aircraft to be simulated. Also theother entities can be viewed in a generalised way, e.g. reference signals can be incorporatedas disturbances �w and additional feedback paths can be taken in case of a robust controlproblem to describe a set of systems i.e. uncertainty.

In a general simulation motion control problem, as shown in Fig. 1.7, the performancecost variables, �zp, will be the difference of motion in the aircraft and in the simulator withthe pilot in-the-loop weighted with the ability of the pilot to perceive these differences,Wp. Further, motion system limitations should be penalised by additional cost variables,�zc. So the standard plant will consist of the motion system, the simulation model (Sim),the aircraft, the pilot and weighting functions. The generalised disturbances are both the

18 1 Introduction

Aircraft

Motion System

Pilot

Pilot

Wp

Motion Control

e? - -

-

6

-

�

�

-

�

�

-

-

�

�wda

�wr

�wdm

�yr �ym

�u

�zp

�zc

+

-

Sim Ww-

�

-

Fig. 1.7: The flight simulation motion control problem put in the generalised plant structure

tasks to be tracked, �wr and the external disturbances to be rejected. There will be bothexternal disturbances on the aircraft, �wda, such as forces due to turbulence, which shouldbe simulated, and external disturbances on the motion system, �w dm, such as friction forces,which are to be rejected.

The controller receives information from both the motion system, �ym, which has tobe stabilised and the other subsystems of the generalised plant to construct an appropriatereference to be tracked, �yr. In case of the motion controller in the conventional setting asgiven in Fig. 1.1 this will be the desired motion reconstructed by the wash-out filters, Ww .If the reconstruction of a feasible trajectory is taken as a part of the control problem, as isdone by Idan et al. in [58], the aircraft motion can be taken as the reference and the washout model becomes part of the motion controller to be designed. When the whole inertialmotion simulation is considered, the ’simulation controller’ is fed by actions of the pilot andthe experiment leader. The controller can act through the actuators of the motion system, �u.

To calculate an appropriate controller, a mathematical model of the subsystems in thestandard plant can be derived. If these models are sufficiently simple, e.g. linear time-


invariant, an optimal controller can be synthesised, which for instance minimises somemaximal amplification on some norm on the cost, �z, given a bound on the norm of thedisturbances, �w [160]. The flight simulation motion control problem is yet too complex tobe solved in this way, but some observations can be made from the structure of the standardplant given in Fig. 1.7.

The task of tracking a reference model is a particular element in flight simulation,which requires special consideration. Several techniques, like model following [145], modelmatching [156], the servo mechanism [33], etc., exist to achieve such tasks. Tracking re-sults through the servo-mechanism with linear time-invariant systems if the internal modelprinciple [40] is satisfied, which states that the (unstable) parts of the system to be trackedhave to be part of the controller. Also with the other techniques, knowledge of the systemto be matched or followed is incorporated in the controller (design). Although the referenceaircraft model is available in flight simulation, the specific properties of these models arenot yet taken into account in the motion controller design.

The reference tracking problem typically leads to a two degree of freedom controllerdesign problem [152], as given in Fig. 1.8, in which a feedforward,C 1, and a feedback part,C2, can be distinguished. With feedforward, ideally, the input, �u, required to achieve thedesired reference, �r, should be generated, i.e. an inverse model of the system, G. Usually,however, the system does not have a causal and/or stable inverse and therefore an inversemodel cannot be used. Further, the feedforward should not unnecessarily excite unmodelleddynamics in the system. Typically, this dynamics resides in the high frequency area, wherean inverse model would, moreover, usually result in high amplification.

The feedback part should take care of uncertainties such as unmodelled dynamics andother external disturbances and, if necessary, stabilisation of the plant using the error, �e, thedifference between the measured output, �y, and the reference, �r. Where feedforward canbe seen as specific shaping of the plant w.r.t. reference signals only, the feedback shapesthe sensitivity of the plant towards all disturbances. Feedback usually results in a frequencyarea (often the lower frequencies) where the controlled plant becomes less sensitive, an areawhere sensitivity is higher (around the bandwidth) and an area with unchanged characteris-tics (high frequency area) [89].

De Roover argues in [30] that a third degree of freedom in the design of the controllershould be considered. This should take care of the generation of an appropriate referencesignal. In case of the flight simulation motion control, the causality problem in the me-chanics is often overcome by generating an acceleration reference signal instead of positiononly. Calculating the forces required given a position reference would lead to a non-causaltransfer function. This is not the case with acceleration. It is questionable, whether an ac-celeration reference signal is sufficient to fully overcome the causality problem since thiswould imply that immediate force generation is possible.

1.2.2 Motion control specifications

The main task to be accomplished after wash-out, tracking of the reference accelerationsignals should not only be accomplished w.r.t. the asymptotic behaviour at infinitely longtime spans but especially the transient behaviour, the generation of appropriate onset mo-

20 1 Introduction

C2

C1

G- -?

6--

-

e e

-

+

+

+�r �e �y�u

Fig. 1.8: Two degree of freedom control structure

tion, is most relevant. By using (model) knowledge of the reference generating system, thedisturbed aircraft, the pilot and the plant, the motion system, use can be made of the twoadditional degrees of freedom in the controller to enhance performance [32].

As acceleration is the perceived unit, the performance of such motion control is muchmore sensitive as (disturbance) signals become faster, as compared to systems with po-sition as cost variable. Attaining higher bandwidths than required can make the systemunnecessarily sensitive. Though it is known that below approximately 0.1 Hz motion is notperceived through inertial movement, the required bandwidth is yet unknown. Of course,the crossover frequency area of the pilot/aircraft loop around 1 Hz should be simulated withclose correspondence but the question of whether the vestibular organs, which are sensi-tive up to 30 Hz, should be stimulated over this whole frequency area is not seen to beanswered in literature. It can be expected that striving for a bandwidth of 30 Hz will leadto compromising false cues due to parasitic dynamics of the motion system with requiredcues.

In summary, inertial motion control of the cockpit in flight simulators is directed atproviding appropriate cues in the following areas:

- The onset of the aircraft response to the actions of the pilot. Both shape (amplitude) andtiming (phase) of the response is important here. As the bandwidth of the pilot usuallydoes not exceed 1 Hz, in a first attempt, a bandwidth of an order of a magnitude highercan be strived for in this respect.

- A realistic generation of disturbances resulting from e.g. turbulence. At higher frequen-cies, above 2-3 Hz, the shape of these disturbances becomes most important while(linear) phase lags, small time delays, are not so relevant.

- The dynamics related to the aircraft resulting in vibrations. This can be both due to thepilot in the loop as the disturbances. This dynamics can change drastically with thesituation e.g. on the ground or airborne. This also means sufficient suppression or atleast minimal excitation of simulator related dynamics i.e. parasitic dynamics, whichdo not correspond with the pilot in-the-loop with the actual aircraft.

- Given feasible reference trajectories, the position control should be such that errors in


the actuator position do not result in running the actuators out-of-stroke. The correc-tive action should not lead to false cues and should therefore typically be of a lowfrequency nature and thus have very limited bandwidth.

Not only the quality of the motion control is important, but also the predictability. In do-ing pilot/aircraft interaction experiments or in design of a training programm, it is relevantto know to what degree the simulated environment corresponds with the actual situation.Finally, the main requirement in doing piloted simulation experiments is safety, which de-mands stability of the motion control loop at all time.

1.2.3 Conventional simulator motion control

Most motion control schemes, reported on in flight simulation, focus on the kinematics ofthe mechanical system and the dynamics of independent hydraulically driven masses [12,24, 46, 51, 82]. Given a desired simulator trajectory, the required actuator trajectories can becalculated with the inverse kinematics (end-effector to actuator coordinate transformation)and with a fully parallel driven robot these relations can be given explicitly. By neglectingthe interaction between the actuators, each feedback loop can be designed given the transferfunction of a hydraulic actuator driving a mass. The linearised version from valve input,i, to position, q, as in [99, 153] consists of a lightly damped second order system in serieswith an integrator. This is described in the frequency domain as

�q

�i=

K

s(s2=!2o + 2�s=!o + 1); (1.3)

considering s = j!.So at low frequencies the system acts as a velocity generator and at higher values the

resonant characteristics of second order system plays a role with the eigenfrequency, ! o =pc=m, which depends on the driven mass, m, and the limited oil stiffness, c, often with

poor damping, � < :05.With low performance, the system can be controlled with proportional position feedback

of the integrator, neglecting the additional second order system. If a higher bandwidth hasto be attained, the resonance has to be damped sufficiently. :3 < � < :7 is preferred. Thiscan be done by pressure feedback. Consider the mechanical part of the system described by

�q =m

s2Apdp; (1.4)

with the driving force, fd, resulting from the product of the operational actuator area, A p,and the pressure difference, dp over the actuator compartments. Then, a pressure feedback,kdp, directly injects damping of �new = �old + kdpK(!o=2)=(mAp), without changingthe undamped eigenfrequency, !o. In this manner a closed loop bandwidth approximatelyequal to !o can be attained.

Of course the interaction in a fully parallel system can not be neglected without seriousdiscrepancies. Hoffman [51], however, already showed that the fully parallel, hydraulicallydriven, six degree-of-freedom motion system of a simulator with Stewart platform configu-ration has in every simulator position six independent directions, a set of eigenvectors, with

22 1 Introduction

the characteristics as described for the one degree-of-freedom system. Pressure feedback,therefore, still results in damping only. As the eigenfrequencies can vary considerably foreach direction and simulator position, the relative damping provided by pressure feedbackwill differ. The positional feedback will have to be tuned according to the lowest occurringsystem eigenfrequency. The resulting controlled system has a bandwidth corresponding tothis frequency and is often somewhat lower since sufficient damping of the faster modes canrequire overdamped characteristics of the slowest.

By using the acceleration reference signals as lead feedforward tuned according to aneutral simulator position2 [24, 46], the bandwidth up to which the system reacts properlyto these references can be extended. In principle, the second degree of freedom in the con-troller is used to create a local inverse of the controlled system up to a frequency somewhathigher than the bandwidth attained with feedback.

Although the parallel structure used in flight simulator motion systems requires some spe-cial consideration, the system can be viewed as a robotic manipulator for which a wide fieldof literature exists on control strategies, which make more extensive use of system/modelknowledge. Not until very recently, this is seen to be applied in simulator motion systems[20, 58, 82]. It seems worthwhile to review the robot control strategies, which can be of use.

1.2.4 Robot control

Although the dynamics of robot manipulators is highly nonlinear, a very specific structurecan be recognised. Most robot control strategies take advantage of this structure. In [136], aselection of articles provides a state of the art and an overview is given of the developmentsin robot control over the years. There are also numerous textbooks on the subject. In recentworks like [11, 31, 81, 102, 113, 115], modern issues such as passivity, robustness, tracking,elasticity and flexibility, are treated.

Computed torqueOne of the earliest applications of nonlinear control of robot manipulators was the method ofcomputed torque, which started in the early seventies. Using the Euler-Lagrange equationsof a rigid robot manipulator with n input torques, �� and output positions, �q given by

M(�q)��q + C(�q; _�q)_�q + �g(�q) = �� ; (1.5)

the required input torque can be calculated, compensating the coriolis/centripetal terms, C,and gravity, �g, measuring the positions, �q, and velocities, _�q, and filling in a new input vector,�a, instead of ��q, to end up with

M(�q)�q =M(�q)�a: (1.6)

2This position usually corresponds with all the actuators half stroke and is the position to which the wash-outfades


Since the inertia matrix, M , can be inverted, linear, parallel (decoupled) double integratorsystems result from �a to �q,

�q =1

s2In�n�a (1.7)

The double integrator systems can easily be controlled with any desired bandwidth by asimple proportional/derivative (PD) controller.

�a = Kp�e+Kd _�e+ ��qd (1.8)

with the positional error, �e = (�qd � �q), the desired positions, �qd, can be tracked with stepand ramp inputs.

In [77], several control strategies like the computed torque method, feedback linearisa-tion, resolved acceleration method and inverse dynamics are shown to be very similar andunifiable. In all cases, exact linearisation and decoupling is strived for. This can be doneboth in end effector or joint coordinates.

The main disadvantages of the computed torque method are its robustness properties.Both with inexact cancellation due to parameter uncertainties (e.g. an inexactly known in-ertia matrix) [9] and unmodelled dynamics, such as joint elasticity [85], this controller cango unstable.

Robust robot controlRobustness can be regained by adding an outer loop such as in the sliding mode controlof Slotine and Li [133] or the saturating controller applied by Spong [135]. Both methodsemploy local high gain, which is sufficiently high to track in the presence of uncertaintiesbut on the other hand is bounded. With the discontinuous sliding mode feedback method,high frequency unmodelled dynamics is easily excited and chattering is often reported inpractice. Transient response in saturating control is worse. Both properties do not seemmost suitable for flight simulation motion control.

Yet another robustifying approach is to compensate for the main nonlinearities and con-sider the system linear with some, possibly varying, uncertainty as was done by Van de Lin-den [146]. Robust control techniques developed for linear systems as described in textbooksas [132, 160] can then be used. Main problem is to arrive at a nonconservative descriptionof an uncertainty in the nonlinear plant at the level of the outer loop.

Passive robot controlAlternatively, use can be made of the passivity (’no energy generating’) property of a roboticsystem [11, 14]. Feedback with a passive controller results in a passive closed loop system,which under some assumptions has specific stability properties. The feedback controller cane.g. have a PD structure with possibly time varying gains (dependent on estimated mass ma-trix). W.r.t. feedforward, the computed torque structure can still be employed. This strategyeven generalises to a much larger class of systems [147] among which are the mechanicalsystems with possibly flexible structures [63, 66].

Most textbooks on robotic systems deal with rigid body structures only. The complex-ity of the system increases drastically in including deformations. The flexible structure has

24 1 Introduction

more degrees of freedom than control inputs. Trajectory tracking of a robot end-effector be-comes very difficult and requires accurate models [31]. The eigenfrequency of the flexiblemode is usually a hard constraint for the attainable performance such as bandwidth. Withpassive control, though stable, performance is often hard to specify.

Actuator dynamicsAlso passivity (control) of robot manipulators with dynamics in the actuators is addressedas by Arimoto [11]. The DC-actuator employed there is equivalent with a linearised versionof a hydraulic actuator [137]. Although the structure of the (voltage driven) DC-motor andthe hydraulic actuator may be equivalent, the parameters usually vary largely. Therefore,the DC-motor can often be considered relatively fast in comparison with the robot mechan-ics. On the contrary, the (finite oil stiffness of the) hydraulic actuator forms a relevant partof the dynamics in e.g. a flight simulator motion system around the achievable bandwidth.

The influence of the mechanics can nominally be decoupled in the hydraulics by localvelocity compensation [127]. By an inner feedback loop the hydraulic force generationbecomes relatively fast compared to the mechanics. This cascaded approach was success-fully implemented by Heintze and Van Schothorst [48, 124]. In another cascaded approachsuggested by Qu and Dawson [113], actuator-level robust control is achieved through arecursive or backstepping design [78]. Though this method is going through rapid develop-ment [128], not much practical applications have seen to be reported yet.

Parallel systemsParallel driven systems, such as the flight simulator Stewart platform motion system inFig. 1.2, are usually not discussed in considering robotic systems. Only recently, in the lateeighties, the advantages of parallel driven robotic manipulators were recognised and nowbecome more widely applied in industry [5, 50, 96]. Such systems are inherently stifferand due to the parallel actuation, less actuator weight has to be moved, which makes themsuitable for fast assembly lines.

The computed torque control laws for Stewart platforms presented by Liu et al. in [83],reveal some of the additional problems with application of model based controller for theparallel systems. The linearising control in end-effector coordinates requires matrix in-version (of the jacobian) and the forward kinematical problem has to be solved on-line.This means computing the platform position based on joint measurements, which is easyin serial robots. No closed form explicit solution, however, has seen to be found for thisproblem for Stewart platforms. Choice of linearising in joint coordinates does not solveany of these problems but actually adds the requirement of calculating derivative jacobianmatrices (solved in [38] by Dutre et al.).

Considering general forms of parallel or closed chain mechanisms, the situation be-comes even more difficult as discussed by Ghorbel in [44], where several parallel mecha-nisms and control strategies are classified. In some cases, it is not possible to describe sucha system with a minimal number of coordinates globally and due to this fact and the specificsingularities, which only appear in parallel systems, global stability results are generallyhard to derive, especially for tracking problems.

Many control strategies presented for parallel systems are designed for appropriate in-

1.3 Problem statement 25

teraction with an environment. Force or hybrid position/force control and creating a passivecompliance are often strived for [96, 103, 110, 116, 142]. Another field of application is thearea of active vibration control, mostly in space structure applications [42, 108, 93]. Thisrequires high bandwidth controllers, which, however, have very limited stroke and thus donot have to include typical position dependent nonlinearities.

Summarising, in principle, most of the robot control strategies take advantage of thespecific structure in the dynamics of mechanical systems. Though not often seen, the modelbased schemes are applicable to the parallel driven flight simulation motion systems, butall require some modifications in analysis and design. The main mechanical nonlinearitiescan be compensated for by a computed torque strategy, possible feedforwarded. Further,several alternatives exist to robustify the feedback loop. An important source of additionalrelevant dynamics can be expected from the hydraulic servo actuators, which are of mainconsideration in the conventional flight simulator motion control and usually not taken intoaccount in robot control.

1.3 Problem statement

Due to the rapid progress in all kinds of flight simulator subsystems such as simulationcomputing power, image generating systems, etc., also enhanced quality of flight simulatorinertial motion generation is desired. At least, high fidelity motion generation would enableresearch into the specific requirements of motion simulation, which are still not exactlyknown. In similar systems, such as robotic manipulators, advanced model based controlstrategies exist, which can attain higher performance than less structured methods, providemore insight into the systems limitations and possibly point at constructional properties,which can be taken advantage of. In flight simulator motion control, application of suchmethods has been almost absent. One of the reasons might be that the motion systems havea degree of complexity, which does not allow exact modelling or detailed models as part ofthe controller. Application requires an intermediate step of extracting a reasonably detailedmodel, which describes the most relevant dynamic characteristics.

1.3.1 General problem statement

The foregoing leads to the following problem statement for this research:

Investigate what relevant system knowledge should be used in a model based con-trol strategy and to what extent does this improve the controlled dynamics of a flightsimulator motion system.

1.3.2 Structuring the problem statement;approach in research

To solve this problem, the full design process has to be defined, structured and evaluated.This process starts with modelling and analysis, proceeds with control synthesis and finallytesting of the system. Many solutions and procedures of the steps to be taken are given

26 1 Introduction

in literature, but an integral approach, having the specific properties and requirements offlight simulation motion generation in mind, still lacks. In this research, integratibility andapplicability will be the main arguments in choosing the most suitable alternative or, ifnecessary, newly proposed variant, of each step in the system/control design procedure.

Since the overall investment is high and the fidelity of the motion system is consideredimportant, the full motion control design process of a flight simulator is allowed a fairlyelaborate procedure optimised towards the specific properties of the system at hand, ascompared to the industrial robot control design. Only, of course, if this leads to enhancedquality of the resulting system.

In the control design procedure employed in this research, the following fairly standardsubproblems in application directed systems and control, will be distinguished:

- Obtain a relevant model structure for analysis and control through physical modelling.

- Evaluate the structure by experiments, identify model parameters and provide boundariesfor the extent to which the model validity holds.

- Choose and, if necessary, modify the most appropriate model based robot control strategy,given the systems properties and requirements. Design and implement this controlleron the motion system of the Simona flight simulator.

- Define a test to quantify performance of the controlled system and use this test to comparewith the usual control design approach.

Related to the flight simulation application and the properties of the motion systems used,specific elements have to be emphasised in each of these steps.

Physical modellingModelling according to physical relations such as balance relations i.e. force balance equa-tions in mechanics and flow balance equations in hydraulics, has two important advantagesas compared to the so-called black box experimental identification. First, as the structureof the model points at the underlying physical properties of the system, analysis and (con-structional) design considerations are performed much easier. Secondly, modelling of anonlinear system such as the flight simulator motion platform, does not impose additionaldifficulties.

As already mentioned, the mechanics and hydraulics are considered the main subsys-tems to be taken into account. These subsystems specifically, and the integration of thetwo, will form the basis of the models to be derived. The mechanical system is, even ifthe parts are considered rigid, relatively complex. This complexity mainly stems from theparallel kinematical structure, which, however, has its influence on the systems dynamics.The kinematics of parallel systems, and of the Stewart platform specifically, will have to beanalysed.

Nowadays, the equations of motion of complex rigid body systems can often be gener-ated automatically in symbolic form. This way of working will be evaluated as opposed tothe derivation ’by hand’ for this system.

The models describing the dynamics of the mechanical system will have to be evaluatedfor use in a real-time model based control structure. Further, it will have to be analysed, how

1.3 Problem statement 27

choices in the design of the construction resulting in the specific kinematical and inertial(mass) properties, influence the dynamics.

The hydraulic servo actuators of the Simona motion system have been modelled andevaluated in detail in [124] by Van Schothorst. In this research, only models will be dis-cussed, which describe the most relevant dynamics for control. Finally, the properties of theintegrated hydraulically driven mechanical system will have to be analysed.

Experimental validationSince the kinematical structure of the system forms the basis of the dynamical properties ofthis motion system, some attention will have to be paid to the validity of the structure and theidentification of the parameters to be used. An, ’after construction’, calibration procedurewill be evaluated as opposed to setting tight (and often expensive) tolerance specificationsin fabrication.

Experimental identification procedures will have to be developed and applied to recon-struct the parameters such as dissipative terms, which are difficult to predict in advance,and evaluate others such as the inertial properties. The models will be evaluated throughcomparison with the experimental data.

The area of validity of the models is also an important aspect, which has to be checked.The cause and severity of possible parasitic effects will be investigated. Especially flexibil-ity is known to constrain the achievable control performance and difficult to predict.

Control strategyTo maintain structure in a model based controller for a complex system such as the motionsystem under consideration and to let a controller result, which can be implemented on areal-time application, modularization of the tasks is desired. The physical structure of thesystem provides a reasonable way to achieve this with a multi level controller in which foreach level specific tasks are defined in close relation with the other levels.

It should be possible to turn the hydraulic actuators into fast and smooth force generatorsby local inner loop pressure control as done by Heintze in [48]. The applicability of thisdesign step, which e.g. assumes accurate velocity measurement, should be evaluated.

Any standard robot control strategy, which assumes direct torque, can be put on top ofthis lower level. As first basic step, the computed torque schemes will have to be made fit foruse in parallel systems. This mainly concerns the choice of an appropriate set of coordinatesand evaluation of safe (convergent and stable) and sufficiently fast reconstruction of thesecoordinates.

In the outer level, the two degrees of freedom, smooth reference feedforward, possiblytaking reference model knowledge into account, and position feedback for stabilisation andpreventing the system from running out of stroke, should not interfere with each other inperforming their basic task.

Performance evaluationIn evaluation of the model based control strategy the feedback oriented conventional de-centralised controller will be taken as a reference. It is understood that, in practise, thisconventional controller can be improved by design of an additional feed forward path, but

28 1 Introduction

this usually leads to local improvement only and is either an ad hoc approach or takes asimilar strategy as in the more structured model based control design.

Evaluation test procedures exist [1], but are outdated and should be modified to currentstandards of control performance quantification. In this research, the following approachwill be taken. By assuming the high fidelity controller being capable of achieving a rea-sonably linear system response, the frequency response measurements are considered mostimportant. A set of other tests is mainly developed to evaluate the assumption of closedloop linear system response.

Further, it will be observed that the existing evaluation procedures have two main limita-tions. First, they have a local character and consequently do not show whether the controllerdoes adequately respond to the nonlinear characteristics of the system over the workspace.Secondly, the system is not evaluated performing its actual task, generating inertial motioncues in flight simulation manoeuvres. A modified test will, as well as possible, have toovercome these limitations.

1.4 Outline of the thesis

The thesis roughly follows the line of the reasoning for the approach sketched in the preced-ing section, to go through the full controller design process. Subsequently the subproblemsin modelling, identification, control and evaluation of the motion system/control design withthe flight simulation application in mind, will be treated.

Physical modellingBy considering the laws of physics, dynamical models of the flight simulator motion systemwill be obtained. In Chapter 2, the mechanical part will be discussed. Specifically, attentionwill be paid to the kinematics and dynamics of a parallel rigid body system. In Chapter 3,the basic structure of the hydraulic servo actuators and extensions to this model will first bediscussed. The integration of the hydraulic mechanical system, the full model of the motionsystem and its properties are the subject of Section 3.3 and its specifics in a Stewart platformconstruction are pointed at in Section 3.4.

’Open loop’ experimental verificationChapter 4 deals with the experimental validation of the physical models, the identifica-tion of model parameters and the parasitic ’additional’ dynamics. The identification of thestatic parameters in the kinematical structure, calibration through redundant measurementsis discussed in Section 4.1. Dynamic (inertial) parameter identification by modal analysistechniques is the subject of Section 4.2. Frequency response measurements are the mainingredients in the model validation considered in Section 4.3. This includes getting trackof the parasitic dynamics caused by flexibility in the system and its environment in Sec-tion 4.3.2, which concludes the discussion on the ’open loop’ experiments taken in thischapter.

Control strategyIn Chapter 5, the control strategy is built up by defining the general approach for multi level

1.4 Outline of the thesis 29

control in Section 5.3 after a short introduction to the main control aspects in Section 5.1and Section 5.2. Then, each part of this controller will be treated. In Section 5.4, the localpressure control of the hydraulic servo systems is outlined. Section 5.5 discusses the com-puted torque control control level to compensate for the multivariable, nonlinear mechanics.This requires an iterative coordinate reconstruction and is followed by the feedback path de-fined by the problem of position stabilisation. Interconnection with the trajectory generatingsystem with reference model based control, treated in Section 5.6, defines the forward pathat the outer level.

Closed loop experimental evaluationThe evaluation of the controller on the actual experimental set-up is the subject of the secondpart of Chapter 5 starting with discussing the implementational issues in Section 5.7. First,the procedure to test the performance of the controlled system is defined in Section 5.8.Then, this procedure is applied to the Simona motion system with the newly proposed con-troller, which is discussed in Section 5.9. As a reference, comparing with a conventionalcontroller strategy.

Final considerationsThe main aspects in the design of flight simulator motion system controllers evaluating theexperience of going through the full system/control design process defined are treated inChapter 6 giving a review and discussion on the results. Directed at the problem statement,defined in Section 1.3, the final Chapter 7 states conclusions towards the opportunities givenin this research by the application of a model based controller in flight simulation motioncontrol. Also recommendations are provided.

Chapter 2

Mechanics of parallel drivenmotion systems

The, fully parallel, hydraulically driven construction of a Stewart platform is often appliedfor flight simulator motion systems. In this research, the use of model based control for suchsystems is investigated. In this chapter, the first step in modelling and analysis is performedby considering the mechanics.

In general, models of parallel robots result in combined algebraic and differential equa-tions, which causes difficulties with simulation, analysis and model based control. Thisproblem will be stated in discussing some literature on parallel manipulators and modellingof mechanical systems in Section 2.1.

In Section 2.2, first a short introduction into basic kinematics will be given. Then, thestructure of kinematic parallel and serial singularities will be treated. The, less standard,parallel singularity causes local loss of controllability and observability of the mechanicalsystem and should be avoided. In Section 2.3, a new method for the Stewart platform isintroduced to ensure exclusion of these points from the working area of the manipulator.Moreover, this is also a necessary condition for guaranteed sufficiently fast convergence ofan iterative scheme in reconstructing the platform coordinates. For the motion system athand, sufficient accurate real time convergence is proven, which enables safe use of thisscheme in model based control later on in Chapter 5.

The platform coordinates are a proper choice in parametrization of the Stewart platform,thereby circumventing difficulties with combined algebraic and differential equations. InSection 2.4, which deals with the dynamics of the mechanical part of the motion system,it is shown that an explicit differential model results from this choice. Thereby, the use ofa suitable projection method, following Kane [64], is exploited to arrive at the equationsof motion. In this way, a proper starting point for simulation, analysis, model reductionand model based control is provided for. Some extensions, e.g. parasitic modes of thefoundation, are also included.

31

32 2 Mechanics of parallel driven motion systems

2.1 Parallel motion systems

The use of robot manipulators is widely spread in industry nowadays. Most of these ma-nipulators are constructed as a series connection of joints and links. The dual form of theserobots, the parallel manipulator, is less often seen to be applied. In (flight)simulation motionsystems however, the parallel construction is almost invariably in use. The Stewart platformis a 6 degrees-of-freedom (d.o.f.) parallel manipulator, which is applied in most of the cur-rent high fidelity flight simulators. It was named after Stewart [140] who illustrated the useof such a parallel structure for flight simulation in 1965. Nowadays, it is also referred toas Gough-platform, since it was Gough who presented the practical use of such a systemsomewhat earlier in 1949 rediscovering the structure already described around 1800 by themathematician Cauchy [95]. As a manufacturing and fabrication robot it is also named ahexapod.

There are several advantages in applying a parallel construction. This kind of manipu-lators have higher rigidity and accuracy due to the parallel force path and averaged link toend-effector error. The inverse kinematics (from end-effector to link coordinates) which isa problem in path generation of serial manipulators is easily solved in parallel robots. Thereare also disadvantages. The dual forward kinematics is a complex algebraic problem andhas in general more than one solution [114]. Modelling the dynamics is also less straightforward.

For several reasons, feedback control of these motion systems is still decentralizedi.e. per actuator without taking mechanical coupling into account. Setting higher standardsof motion realism in simulation will involve modern control strategies in order to fully bene-fit from recent constructional and computational improvements in flight simulators e.g. lightweight, low centre-of-gravity (c.o.g.) platforms, high frequency airplane dynamics simula-tion [6]. Also use of a Stewart platform as a more general robot, as presented by Nguyenet al. [104], will be enhanced with high performance control of motion, as these systemsusually require minimal time tasks.

By incorporating more structural system information i.e. a model into the controller,it is possible to achieve higher performance. Most modern control strategies are thereforemodel based in some way (directly, in design or evaluation). In this case the quality ofmotion depends on the fidelity of the model. Deriving a model of the mechanics of theStewart platform manipulator for analysis, design and control will be the subject of thischapter. To arrive at a model with structure from which insight can be gained, modellingwill be done based on physical laws.

Modelling the dynamics of a Stewart platform as a multibody system has been doneby Lee et al. [80] who claim to be the first to present a complete model and with moresimplifications by Do et al. [36] and Liu et al. [83]. Modelling the mechanics of this platformcan be done in several ways and with various objectives in mind. The equations of motioncan be derived by using the classical approach of Lagrange [80] or Newton-euler [36].

In general, deriving the equations of motion of a parallel manipulator results in com-bined differential and algebraic (constraint) equations (see e.g. Roberson and Schwertassek[120]). In simulation and control this formulation can cause difficulties, usually referred toas index problems [19]. Here it is shown that an explicit differential model for the Stewart

2.2 Kinematics 33

platform results if one makes the right choices in parametrization. Dependent variables areexplicit functions of the integrable differential equations. In this way index problems, etc.are circumvented.

By using a modern projection or dual basis method [79] like Kane’s [64], which havethe advantages of both the Newton-euler and Lagrange formulation but without the corre-sponding disadvantages [56], it will be shown that applying this approach can result in amodel from which more insight can be gained.

Together with the alternative parametrization, this is advantageous over the models ear-lier presented in literature, if one wants to apply a model for both analysis, simulation andcontrol. Model based feedback, however, is still more complex for parallel manipulators,since the dynamics are only described in end-effector coordinates and the measured signalsare link related. Next to the fairly low performance requirements of flight simulator motionsystems in the past, this will probably be the reason that in most of these motion systems,feedback controllers are decentralized (one SISO loop per parallel link i.e. actuator).

Through analysis of the derived model, some of the disadvantages of decentralized con-trol can be revealed. To apply a simple, but accurate model based controller in practise, onewould like to quantify the errors made by undermodelling, to be able to do robust analysisof the control scheme. The modelling approach taken here aims at a model from whichthe influences of different system parameters, like masses, inertias, velocity, gravity can beclearly separated.

2.2 Kinematics

After stating some notational issues, the fundamental formulas of mechanics to describe thekinematics will be introduced. Then the Stewart platform will be defined in order to derivea model of its specific kinematics. After some model analysis with the control objective inmind, this part will be concluded by a resume.

2.2.1 Notation and definitions

Capital symbols, X are used for matrices, �x for vectors, x for scalars. With some scalar(energy) functions X is used. �x � �y denotes the vector product which can also be writtenas ~X�y = (~Y )T �x where the vector product matrix, ~X, is a skew symmetric (X = �XT )matrix parametrized by the vector, �xT =

�x1 x2 x3

�, such that the result is the vector

product.

~X =

24 0 �x3 x2

x3 0 �x1�x2 x1 0

35 (2.1)

X � Y denotes vector wise product of the columns stacked in the matrices.The index �xn is used for the normalising operation �xn = �x= j �x j with j �x j=

p�xT �x.

Pxn denotes the projector to the (hyper)plane with normal vector �x n and can be constructed


Fig. 2.1: Projection of arbitrary vector, �s onto the plane with normalised normal vector, �x n.The projection matrix, Pxn , can be composed from the vector product (matrices,~Xn).

from the vector product matrix

Pxn = (I � �xn�xTn ) = ( ~Xn)

4 = ~Xn~XTn = �( ~Xn)

2: (2.2)

Projection matrices have some properties, which can be taken advantage of, like

Pxn = P Txn = P kxn ; (2.3)

with k > 0 and integer. In some cases it will be appropriate to use such a projection planefor construction of a frame as done in Fig. 2.1.

Motion can be described w.r.t. various frames. A matrix or vector described in someframe can have a superscript referring to this frame. For the inertial frame, G, or groundcoordinates, the index �xg will be used. As a function of the moving end-effector or platformframe, M , vectors will be denoted �xm. If a (rotation) matrix maps a vector into anotherframe it will be denoted as gRm if R maps from M to G.

The subscript index like �ai will be used to refer to the ith-actuator if also non actuatordependent variables appear in the equation.

2.2.2 Calculating velocity and acceleration by differentiation

An important part of kinematics is the ability to derive velocity and acceleration vectors ofany part of a (partly) moving construction. The basic formulas to do this will be derived inthis section.

The motion of a point (mass particle, joint, etc.) is usually most conveniently and in-variantly defined w.r.t. the body frame to whom it’s connected. The motion of a frame putin another frame generally consists of a translation, which can be described by a vector, �t,and a rotation for which a matrix, R, can be used. The orientation of a whole frame canbe described by a rotation matrix. A rotation matrix consists of perpendicular unit vectorswhich describe the basis of the frame into the other frame. As a result a rotation matrix, R

2.2 Kinematics 35

has the following property:

RTR = I (2.4)

Any real 3 � 3-matrix with this property and the property det(R) = 1 is a rotation ma-trix, which set is often referred as the SO(3)-group [77]. SO(3) stands for the Special(det(R) = 1) Orthonormal (RTR = I) group of matrices of size 3x3. With det(R) = �1also the mirror operation is included (transformation of right hand frames to left hand framesand vice versa). The position of a point �pm in frame M can now be described in frame Gby:

�pg = �tg + gRm�pm (2.5)

To describe the velocity of this point in the other frame one can simply differentiate thisequation. Some properties of the time derivative of the rotation matrix can be derived bydifferentiating (2.4). This results in skew symmetric matrices which can be parametrized bythe vector product matrix of the (thereby defined) angular velocity �! m.

RT _R = � _RTR = ~m (2.6)

with

~m =

24 0 �!3 !2

!3 0 �!1�!2 !1 0

35 (2.7)

Now, for the vector, which is rigidly attached to the frame M , _�pm= �0. Hence, by differen-

tiating (2.5) and using (2.6), one obtains

_�pg= _�t

g+ gRm ~m�pm = _�t

g+ ~g �pg = _�t

g+ �!g � �pg (2.8)

where the change of frame for the matrix ~, is given by ~g = gRm ~m mRg and usingthe inverse rotation given by mRg = (gRm)T with (2.4).

By differentiating (2.8) the acceleration of the point, still rigidly attached to the frameM ( _�pm = �0), can be calculated, using ~Pm�!m = ~m�pm:

��pg

= ��tg+ gRm( ~Pm)T _�!

m+ gRm(~m)2�pm

= ��tg+ ( ~P g)T _�!

g+ (~g)2�pg

= ��tg � �pg � _�!

g+ �!g � (�!g � �pg) (2.9)

In other cases, the point considered can already be moving in the frame M (�pm,�vmm =_�pm,�amm = ��p

m). Here, by differentiation of

_�pg= _�t

g+ gRm(~)m�pm + gRm�vmp (2.10)

the coriolis acceleration appears as the third term in

��pg

= �agp� +gRm�amm + 2 gRm~m�vmm

= �agp� + �agm + 2~g�vgm = �agp� + �agm + 2(�!g � �vgm): (2.11)

Where p� is a point connected toM momentarily at the same position as �p. Its acceleration,�agp� , is given by (2.9).


Fig. 2.2: Physical structure of the euler angle representation.

2.2.3 Parametrising orientation by euler angles

In stating the equations of motion, the state which describes the orientation, usually onlyappears in the rotation matrix. It is possible to parametrise the rotation matrix in severalways.

Although the rotation matrix consists of nine entries, its properties, given by (2.4), putconstraints on these entries. The orientation, i.e. the SO(3)-group, can locally be repre-sented by 3 parameters. It can, however, not be covered by a single coordinate chart of thissize [77]. Several parametrizations are used in literature. The euler angle description con-sists of three subsequent simple planar rotations around momentary axes. The euler anglesare not convenient for use in modelling the Stewart platform but are merely discussed forreference since it is a popular description in the aerospace community.

An often used example of euler angle rotation is physically represented in Fig. 2.2. Boththree unit vectors �nG;Mx;y;z of the ground frame,G, and moving frame,M , are drawn. A simpleplanar rotation around the x-axis has the following structure

Tx =

24 1 0 0

0 cos(�) � sin(�)0 sin(�) cos(�)

35 (2.12)

A vector, �xm, quantified in the axes of M can be represented in G by

�xg = R�xm = Tz;Ty;Tx; �xm (2.13)

Since the subsequent rotations are not commutative, it is important to define the order ofrotation. The order defined in (2.13) is the usual order applied in the aerospace community.

2.2 Kinematics 37

This means that the airplane, at first positioned so that the body axis is parallel to the fixedinertial frame, G, has to be rotated as follows:

- First rotate over an angle , the yaw angle, around the inertial �n gz-axis. In taxiing, thistypically will be the main angle of rotation.

- Then pitch over an angle � around the intermediate y ;-axis. This will usually be the mainangle to be changed in take-off. Note, e.g. from Fig. 2.2 that this axis is in general notan axis of the inertial or the moving frame.

- Finally roll around the airplanes’ longitudinal axis, �nmx , over an angle � to obtain its actualorientation.

The fully parametrized rotation matrix is then as follows:

R =

24 c c� c s�s�� s c� s s�+ c s�c�

s c� c c�+ s s�s� s s�c�� c s�

�s� c�s� c�c�

35 (2.14)

In this equation s = sin( ), c = cos( ) and so on. The three euler angles have thedisadvantage of a highly non-linear appearance in both the rotation matrix and the eulerangle velocity to angular velocity transformation. The latter can even become singular.Since, although any rotation matrix can be described by this parametrization, in some states(consider � = �=2 in Fig. 2.2) this chart does not allow three independent changes oforientation, i.e. the kinematical relationship between �� = [� � ]T and the angular velocityin the body frame, �!m, given by

�!m =

24 p

q

r

35 = �(��) _�� =

24 1 0 �s�

0 c� c�s�

0 �s� c�c�

3524 _�

_�_

35 =

24 1

00

35 _�+ T Tx0

24 0

10

35 _� + T Tx0T

Ty0

24 0

01

35 _ (2.15)

becomes singular. p, q and r are the components of the angular velocity, �!m, measured inmoving body frame. In an airplane this will usually be the output of the gyros.

In some experiments, a rotation matrix can be identified. With ��=2 < ; �; � < �=2,the euler angles can be calculated from such rotation matrix as follows:

sin(�) = �R(3; 1) (2.16)

sin( ) = R(2; 1)= cos(�)

sin(�) = R(3; 2)= cos(�)

As already said, the euler angle description of rotation is not convenient in describing themodel of the Stewart platform. This is mainly because evaluation of the high number of sineand cosine functions is both analytically and numerically relatively troublesome and this canfully be avoided in modelling the motion system mechanics by using euler parameters.


n

n

n

g

g

g

G

z

y

x

nµ

µsm

s

m

2sin(1/2 )µµ

smnµ x( )x nµ

smnµ x

sg

g

s

Fig. 2.3: Visualising the euler parameter representation by a rotation around the unit vector,�n�, over an angle, �. The lower right schematic drawing is the plane orthogonalto �n� through the points sg and sm.

2.2.4 Parametrising orientation by euler parameters

An attractive alternative of euler angles are the unit quaternions [130] called euler parame-ters. No singularities and, even more important, numerically convenient relationships withboth the rotation matrix and the angular velocity and as with the euler angles, an inter-pretation which can be visualised. This at the cost of an additional fourth parameter andconsequently an extra constraint equation.

The orientation of one frame w.r.t. another frame can always be described by a singlerotation over an angle �, about a unique axis with direction �n�. The four euler parameters,

��e =��0 ��T

�T, parametrise � and �n� as follows:

�0 = cos(1=2 �) (2.17)

�� =��1 �2 �3

�T= sin(1=2 �)�n� (2.18)

The constraint equation,

��Te ��e = 1; (2.19)

is a direct result from this definition. By not using the angle of rotation itself as a coordinatebut the sine and cosine instead, these geometric functions do not have to be evaluated any-more. And as a result we will see in the subsequent sections that in a full mechanical modelof the Stewart platform not one geometric function will appear.

2.2 Kinematics 39

With Fig. 2.3 the rotation of a vector, �sm, in the body frame, M , to its direction in theinertial frame, G, given as �sg , can be described using the unit vector, �n�, along the axis ofrotation and the angle of rotation, �.

The following derivation starts with constructing �sg from �sm adding the two componentvectors in the plane given in Fig. 2.3 and is directed towards the specification of the rotationmatrix.

�sg = �sm � (1� cos(�))(�n� � �sm)� �n� + sin(�)(�n� � �sm)

= �sm + 2 sin2(1=2�)�n� � (�n� � �sm) + 2 cos(1=2�)sin(1=2�)(�n� � �sm)

=�I + 2~�~�+ 2�0~�

��sm

= 2

24 �20 + �21 � 1=2 �1�2 � �0�3 �1�3 + �0�2

�1�2 + �0�3 �20 + �22 � 1=2 �2�3 � �0�1�1�3 � �0�2 �2�3 + �0�1 �20 + �23 � 1=2

35 �sm

=

24 ��1 �0 ��3 �2��2 �3 �0 ��1��3 ��2 �1 �0

3524 ��1 �0 �3 ��2��2 ��3 �0 �1��3 �2 ��1 �0

35T

�sm

= G(��e)LT (��e)�sm = R(��e)�sm (2.20)

So, rotation appears to be equal to two simple subsequent transformations, G and L,which are linear in the euler parameters. By Nikravesh et al. [107] it is shown that thisparametrization and using _RRT = ~!g leads to the following relation to calculate _��e or _��from �! and ��e,

_��e =1

2GT (��e)�!

g (2.21)

In only constructing the variable, _��, a reduced version is given by

_�� =1

2GTs (��e)�!

g; (2.22)

where the matrix, Gs, is given by the last three columns of G. This equation can be used toget from the angular velocity to the rotation matrix with help of an integration routine andinitial conditions on ��. Further, the fairly simple relations like

�!g = 2G(��e)_��e; (2.23)

�!m = 2L_��e and ��e = 1=2GT _�!g can be derived.

With angles �� < � < �, �� can be used as the (orientation) state from which �0 =p1� ��T �� is solved. In that case, a three parameter setting is used at the cost of possible

singularities. The nonsingular envelope is, however, twice as large as in case of using eulerangles.


Fig. 2.4: A 2-d.o.f. serial system in a singular configuration

It is possible to calculate euler parameters from any rotation matrix where j � j< �.

�0 =1

2

p(1 + tr(R)) (2.24)

�1 = (R(3; 2)�R(2; 3))=(4�0) (2.25)

�2 = (R(1; 3)�R(3; 1))=(4�0) (2.26)

�3 = (R(2; 1)�R(1; 2))=(4�0) (2.27)

where tr(R) =Pi=3

i=1 R(i; i) is the trace of the rotation matrix. Only in case of �0 = 0another solution strategy has to be applied. With the earlier given relationships, (2.14) and(2.16) betweenR and �� and vice versa, euler parameters can be calculated from euler anglesand vice versa.

2.2.5 Jacobian matrices

Connected parts of a mechanical system will result in less freedom of motion reflected byconstraint equations which can be explicit as in �y = �f(�x) or implicit as in �g(�x; �y) = �0.Such equations also constrain the time derivatives of these coordinates, i.e. the velocitiesand accelerations. In describing a mechanical system, e.g. stating its equations of motion,it is often convenient to state the motion of the system as a function of a limited number of(generalised) variables (coordinates and/or velocities).

If some variations or velocities can be described as a product of a (position-dependent)matrix and a vector of other variations this matrix will be called a (semi-)Jacobian matrix.The jacobian matrix between two sets of variables usually comes out naturally by a timedifferentiated version of an equation in which one of the sets is explicitly stated as in �y(t) =

2.2 Kinematics 41

�f(�x(t)),

_�y(t) =@ �f

@�x(�x(t)) _�x(t) = Jy;x(�x(t)) _�x(t) (2.28)

In mechanical systems such mappings often result in an indirect way from differentiationof the rotation matrix and using �! as velocity term. The angular velocity, �!, however, is notime derivative of any representation of attitude. E.g. given (2.8),

_�pg=�I gRm( ~Pm)T

� � _�tg

�!m

�= Jpg;x _�x (2.29)

Also matrices like Jpg;x will be termed jacobian, although this is not precise i.e. not a resultof (2.28), but just states one set of velocities as a function of another set.

2.2.6 Parallel, serial and singular configurations

In kinematics, jacobian matrices often play an important role in relating input/actuator andoutput/ end-effector coordinates. With parallel manipulators, it will be shown using someexamples, that these relations differ from those constructed from a serial manipulator. InFig. 2.4, a 2 degree-of-freedom serial joint robot is depicted. The end-effector coordinatesx and y can in the serial configuration explicitly be stated as functions of actuated rotatingjoints angles q1 and q2 as

x = l1 cos(q1) + l2 cos(q1 + q2) (2.30)

y = l1 sin(q1) + l2 sin(q1 + q2) (2.31)

And by differentiation�_x_y

�=

� �l1 sin(q1)� l2 sin(q1 + q2) �l2 sin(q1 + q2)l1 cos(q1) + l2 cos(q1 + q2) l2 cos(q1 + q2)

� �_q1_q2

�= J�x;�q�q (2.32)

At q2 = 0� �, the jacobian of this configuration becomes singular. At such configurations,physically, the actuators can only generate one d.o.f. motion/velocity of the end-effectorlocally. Further, it can be noted that, for each nonsingular configuration, two actuator posi-tions generate the same end-effector state.

The dual parallel configuration of a 2 d.o.f. robot is given in Fig. 2.5. In fully parallelsystems, the sliding actuator lengths can explicitly be stated as functions of the end-effectorcoordinates. With �xT = [x y] and �li(�x) = �x� �bi

klik =q�lTi�li (2.33)

and by differentiation

k _lik = �lTn;i _�x (2.34)

Thus JTl12;x = [�ln;1 �ln;2]. If the unit actuator direction vectors, �ln;i are in parallel, theconfiguration becomes singular. In this case, however, the end-effector can still move but


Fig. 2.5: A 2-d.o.f. parallel system in a singular configuration

φ

Fig. 2.6: A 3-d.o.f. system with serial and parallel singular configurations

not be forced in 2 d.o.f. and from the speed of the actuators, the end-effector velocity cannot be determined. In this system, in a nonsingular configuration, there are two end-effectorstates for each actuator position.

In general, a rigid body mechanical system, will have combinations of serial and parallelconnections as in Fig. 2.6. In this example, the actuator coordinates, �q, are no explicit func-tions of the end-effector coordinates, �x and v.v. Also both kind of singular configurationscan usually occur. An important observation in the following sections will be the fact thatthe Stewart platform flight simulator motion system is a purely parallel system.

The parallel manipulator construction of the Stewart platform is first defined. To de-rive the equations of motion, the velocity and accelerations should be described w.r.t. alimited number of generalized variations. This defines the kinematics after which the semi-equilibrium equations of the active and inertial forces can be stated.

2.2 Kinematics 43

2.2.7 Stewart platform definitions and assumptions

The Stewart platform (Fig. 2.7) consists of an end-effector body with mass, m �c, and 3x3inertia matrix Im�c w.r.t. the end-effector frame,M , connected to the centre of gravity (c.o.g.)of the body which has varying coordinates, �c, in the inertial frame,G. The end-effector bodyor platform is connected by six parallel actuators at the moving upper gimbal points withfixed coordinates, �ai, in the moving frame, M , to �bi in the inertial frame, G. The length ofthe six actuators can be varied. In describing a specific actuator the subscript i for the i th

actuator will be left away.The platform position is defined as

�sx =

��cs��

�(2.35)

With the last three euler parameters, ��, to define the rotation matrix, T (��), as R in (2.20),from the moving (platform) frame, M , to inertial (ground) frame, G and a scaling factors = 2kakmax equal to the maximum distance of the c.o.g. to any of the upper gimbalpoints. The platform speed is not defined as the derivative of platform position but as

_�x =

�_�c�!

�; (2.36)

where �! is the angular velocity of the platform. The platform (generalised) speed andposition are related through the jacobian, Jsx;x, using (2.22)

_�sx =

�_�cs_��

�=

�I 00 sGTs =2

� �_�c�!

�= Jsx;x _�x (2.37)

An actuator (Fig. 2.8) will be modelled as 2 bodies, the rotating cylinder and the movingpiston. The rotating body, with mass mb and a constant distance of rb of the c.o.g., bc, tothe connection of a 2-d.o.f.-rotational gimbal joint to the inertial frame at �b. The movingactuator body, i.e. the piston, with mass ma with a constant distance of ra of the c.o.g.,ac, is connected with a 3-d.o.f.-rotational gimbal joint to the platform at �a. With a 1-d.o.f. controlled sliding joint between these two bodies the length of the actuator can bevaried.

It is assumed that the inertia of the actuator bodies can be neglected around the actuatoraxis and to be uniform perpendicular to this axis. ia is the inertia of the moving actuatorbody at �ac and any axis perpendicular to the actuator. i b is the inertia of the rotating partof the actuator along any axis perpendicular to the actuator w.r.t. the connection point to theinertial frame (�b) .

With these assumptions also the case, often seen in practise, in which the moving part ofthe actuator not only slides at the connection with the rotating part but also rotates aroundthe �l-axis , and has only a 2-d.o.f. rotation gimbal in connection to the platform, results inthe same dynamics. Finally, the c.o.g.’s of the actuator parts are assumed to lie on the lineconnecting the upper and lower gimbal point.


Fig. 2.7: A schematic view of the Stewart platform

2.2.8 Stewart platform kinematics

The kinematics of the Stewart platform will be described by first defining the transformationof the platform to actuator coordinates. Then by differentiation also velocity and acceler-ation of all relevant points can be calculated as a function of the platform motion, whosevelocities will be taken as the generalized speeds.

Almost all vectors can be conveniently described in the inertial frame. Apart from �amiwhose time derivative in the moving frame is �0.

As shown in Fig. 2.7, the vector, �li, between the two attachment points of an actuatorcan be described by

�li = �c+ T �ami � �bi (2.38)

Now the squared length of the actuator, j �li j2= �lTi�li, and the unit vector in direction of

the actuator, �ln;i =�lijlij

can be calculated from the platform variables �cg and the orientationmatrix T = gRm which will be the only rotation matrix used.

The velocity of the length of the actuators can be calculated by projection of the velocityof the upper gimbal attachment point, �va, in the direction of the actuator, since

d

dtj li j=

d

dt

q�lTi�li =

�lTi �va

j �li j= �lTn �va: (2.39)

2.2 Kinematics 45

Using (2.8), the velocity of the upper gimbal points is given by

�vai = _�c+ �! � T �ami ; (2.40)

where �! is the angular velocity of the moving frame in the inertial frame. By projection ofthis velocity with (2.39) and filling in (2.40), the velocity of the actuator is described as afunction of platform velocities and poses.

_li = �lTn;i�vai =�lTn;i _�c+

�lTn;i(�! � T �ami ) (2.41)

With some reordering and written as matrix equation (�ln;i, �ami and �vai stacked in Ln, Am

and Va) for all the actuators the jacobian between the actuator and platform velocities isdefined.

_�l = LTn _�c+ (TAm � Ln)T �! = Jl;x _�x = LTnVa (2.42)

The semi-jacobian matrix, Jl;x, is one of the most important variables in the Stewart plat-form, relating the platform coordinates to be controlled and used as basic model coordinates,and the actuator lengths, which can be measured. Further, in transposed form, J l;x relatesthe actuator input forces with the platform forces as will be shown in deriving the platformdynamics. Note that with defining _�x

T= [_�c

T�!T ], �x does not exist.

2.2.9 Interpretation and use of the jacobian matrix, Jl;x

If the system has to be controlled by the actuators, the jacobian specifies how the controlinputs, the actuator forces, influence the platform (accelerations), which are, especially inflight simulation applications, often the variables to be controlled. Further, the measurableoutputs are often only the actuator lengths. The derivatives (actuator speed) of these outputsare given by the product of the jacobian and the platform speed.

There are two interpretations to the jacobian. In the force interpretation the rows of J l;xgive the (generalized) forces in the platform coordinates given a unit force in an actuator. Inthe velocity interpretation the columns of J l;x specify the velocity of the actuators requiredto have unit velocity of the platform.

In model based control, the inverse information is of interest. The measured variationsof the actuator have to be put in platform variations to calculate corrections in a modelspecified in platform coordinates. Each column of the inverse jacobian, J �1

l;x , specifies whatvelocity (angular velocity included) of the platform is necessary to have elongation of justone actuator while the others only rotate. Given measurable actuator velocities, the platformvelocities can be calculated.

The correction forces in a model based controller are also most conveniently calculatedas a function of platform coordinates. Each row of J �1

l;x specifies the forces necessary in theactuators to have unit force correction in platform coordinates.

The inverse jacobian appears in feedback linearising structures (like computed torque,etc.), which will be dealt with in the forthcoming chapters.

Another problem of a parallel manipulator, with only the link position measured, arethe forward kinematics. It is not known how to analytically calculate the platform position(without decision making about set of possible solutions) from link measurements.


l

a

b

ac

ra

rb

bc

Upper Gimbal

Lower Gimbal

Fig. 2.8: Stewart platform actuator link construction

The jacobian Jl;x( �sx) is an important element in the jacobian Jsx;l( �sx) from actuatorvariation to the platform position variation.

Jsx;l( �sx) = J�1l;sx( �sx) = Jsx;x( �sx)J

�1l;x ( �sx); (2.43)

where Jsx;x is given by (2.37).This jacobian provides a way to apply a Newton-Raphson iteration to calculate the solu-

tion provided one starts in a point sufficiently close to the solution and away from jacobiansingularities.

�sxj+1 = �sxj + J�1l;sxj

(�lmeasured � �lj) (2.44)

The condition number of J l;x also provides a measure for the controllability of the plat-form from the actuators which becomes uncontrollable at singularities of this matrix.

Further, most of the constraints of the platform are caused by the characteristics of theactuators like limited stroke, maximum speed and force. The jacobian plays an importantrole in translating these limitations into platform coordinates. E.g. given a maximum ve-locity, j v jmax, in extending an actuator either direction, the maximum velocity in the j’thplatform coordinate is given by the 1-norm of the respective row of J �1

l;x times the maximumactuator velocity.

_�xi;max = kJ�1l;x (i; �)k1 j v jmax=

6Xj=1

j J�1l;x (i; j) jj v jmax (2.45)

2.2 Kinematics 47

So, the maximal platform velocity along a specific coordinate can be calculated by addingall absolute values of the specific row of the inverse jacobian and multiplying with themaximum actuator velocity. In this case, every other platform coordinate is left unspecified.

In the more interesting case, with the other platform coordinates constraint to zero

_�xi;max =j v jmax =kJl;x(i; �)k1 =j v jmax =maxj

(j Jl;x(i; j) j); (2.46)

e.g. the maximum pure surge speed can be calculated by dividing the maximum actuatorvelocity by the largest number in the first column of the Jacobian. For position and acceler-ation, these formulas can be used as an approximation.

2.2.10 Velocity and acceleration of the actuator joints

If the inertia of the actuator joints is important or gimbal forces have to be calculated, thekinematical formulas have to be expanded with those to derive the velocities and accelera-tions of these joints.

The jacobian between the platform and the upper gimbal point velocity is defined by

�vai =�I T ( ~Ami )

TT T�_�x = Jai;x _�x (2.47)

To determine the inertial forces of the actuators, the jacobians, from gimbal point to thec.o.g.’s of the actuators, are also important. The angular velocity of the actuator perpendic-ular to the actuator, �!a, is defined by

�!a = �ln ��vaj l j (2.48)

Now the velocities of the c.o.g.’s of the actuator bodies �vac and �vbc can be stated as a functionof �va:

�vac = �va + �!a � (�ra�ln) = (I � ra

j l jPln)�va = Jac;a�va; (2.49)

and

�vbc = �!a � rb�ln =rb

j l jPln�va = Jbc;a�va; (2.50)

as becomes more clear by looking at Fig. 2.9. The acceleration of the actuators can becalculated by differentiating (2.42),

��l = Jl;x��x+ _Jl;x _�x = LTn_Va + _LTnVa: (2.51)

The derivative of the unit vectors, �ln;i, of each actuator, i, can be calculated with:

_�ln =d

dt

�l

j l j =_�l j l j ��l d

dtj l j

j l j2

=(I � �ln�l

Tn )

j l j �va =1

j l jPln�va (2.52)


ac

ra

bc

Upper Gimbal

Lower Gimbal

lrb n

va

vbc

vac ln

ln

ωa

ωa

ωa

ra ln ωax

l.ln

Fig. 2.9: Stewart platform actuator construction of link velocities. With the upper gimbalvelocity, �va, which can be derived from the platform velocity by (2.40), all otheractuator velocities can be constructed. The actuator speed, _l, is the projection of�va along the actuator �ln (2.41). The figure is best read by considering �va in theplane of the picture. Then, only the actuator angular velocity, �! a, given by (2.48),is out of the plane. In fact, it is orthogonal to the plane.

2.2 Kinematics 49

The acceleration of the actuator length consists of a term which is the projection of theacceleration of the upper gimbal in the direction of the actuator and a positive quadraticterm, which is the centripetal acceleration of the actuator.

��l = �lTn _�va + �vTa (1

j l jPln)�va (2.53)

So the acceleration of the actuator length is always positive if the platform is moving withconstant translational speed in any direction and constant orientation since _�va = 0 in thatcase. The acceleration of the upper gimbal can be derived directly with (2.9) and (2.40)

_�vai = ��c+ _�! � �ai + �! � (�! � �ai) = Jai;x��x� j �! j2 P!�ai; (2.54)

where the projection matrix, P! = (I+ �!n�!Tn ), projects the upper gimbal vector, �a i, on the

plane orthogonal to the normalised angular velocity direction of the platform, �! n.The acceleration of the c.o.g. of the moving actuator part also generates inertial forces

and can be written as a function of platform motion.

_�vac =d

dt�vac =

d

dt(Jac;a�va) = Jac;a _�va + _Jac;a�va (2.55)

So this jacobian needs to be differentiated. With (2.49) and (2.52),

_Jac;a =d

dt(I � ra

j l jPln) (2.56)

=ra

j l j2 �vTa�lnPln +

ra

j l j2 (Pln�va�lTn + �ln�v

Ta Pln)

Now with j Pln�va j2= �vTa Pln�va,

_Jac;a�va =ra

j �l j2 (j Pln�va j2 �ln + 2(�vTa

�ln)Pln�va); (2.57)

which shows a quadratic centripetal term in actuator direction and a coriolis term orthogonalto the actuator. Clearly, the piston acceleration, _�vac, only depends on the velocity, �va andacceleration, _�va, of the upper gimbal, given the positional coordinates.

The motion of the actuators has now been explicitly stated as a function of the platformcoordinates and its derivatives. The kinematic relations thus provide means to state themotion of the system as a function of a limited number of variables. Together with thedynamics i.e. semi-equilibria of active and inertial forces stated in Section 2.4, the equationsof motion result.

2.2.11 The Simona flight simulator motion system kinematics

In this thesis, any system connected to the environment through six parallel joints with five(passive) rotational and one (active) translational d.o.f. is considered a Stewart platform.The structure of the Simona flight simulator motion system design is much more specific. Itis schematically depicted from above in Fig. 2.10 and the measures are given in Table 2.1.


Fig. 2.10: The Simona flight simulator motion system kinematics represented schemati-cally in a top view in an ’all the actuators down’ position

Both the upper as the lower gimbal points virtually form a platform, which looks like twoblunt equilateral triangles rotated 180 degrees with respect to each other.

All the upper gimbal points as well as the lower lie on a circle. The circles lie in parallelplanes if the actuator lengths are equal. All the minimal and maximal actuator lengths areequal. The actuators and gimbal points, which are numbered 1 to 6, are pairwise distributedclockwise over the circle. (6,1), (2,3) and (4,5) for the moving upper and (1,2), (3,4) and(5,6) for the inertial lower platform. The distance between the two elements of the pairs arealso specified in the table. The upper pair (6,1) is connected on the positive x-side of thesimulator. The lower pair (3,4) at the negative x-side of the ground.

In most of the derivations, the motion system kinematics is assumed to belong to themost general class of Stewart platforms. This leaves room to adjust easily to a structurewhich differs from its design.

This concludes the kinematical modelling, which is used to derive the equations of mo-tion in modelling the mechanical system in Section 2.4. The nest section finishes the treat-ment of the kinematics with two issues, which will be specifically important for the modelbased control structure to be used in Chapter 5.

2.3 Analysis of the Stewart platform kinematics

A vast majority of the literature on parallel robotic systems is devoted to their kinematics.This is an important subject because in the design of the kinematics of a robotic system,

2.3 Analysis of the Stewart platform kinematics 51

Table 2.1: The Simona system parameters

variable value descriptionra 1:60m upper gimbal radiusrb 1:65m lower gimbal radiusdu 0:20m upper gimbal spacingdl 0:60m lower gimbal spacinglmin 2:08m minimal actuator lengthqmax 1:25m actuator strokeqop 1:15m operational actuator stroke

_lmax 1m=s maximum actuator speed

its manipulability, its workspace, etc. is to be fixed. This problem in combination with theapplication to flight simulation has been studied by Advani [7].

In this section, two new solutions towards kinematical problems will be presented,which have a direct relation with the ability to perform real time model based feedbacksafely on a specific kinematic design of a Stewart platform motion system.

- Given a specific design of the manipulator, one has to ensure the system can not be driveninto a singular configuration.

- Further, model coordinates have to be calculated from the measurements taken in realtime (e.g. a finite number of calculation steps) without the chance to have divergencebetween actual and calculated coordinates.

In parallel robots such as most flight simulation motion platforms, the position of thesystem is usually indirectly measured by the length of the actuators. The forward kinemati-cal problem of calculating the platform coordinates given the actuator lengths of the Stewartplatform is seen to be solved in roughly two ways in literature.

Using analytic techniques, the problem can be transformed to a set of combined poly-nomial equations whose roots have to be found to solve the forward kinematics e.g. [57].Although these equations can provide insight into the structure of the problem, closed formsolutions are only seen to be presented for special classes of platforms e.g. Merlet [97],which still is an active area of research [68]. Solving the roots of the equations, still leavesthe problem of choosing the actual pose. The complexity of this problem is reflected by thefact that it has been shown that the general version of the Stewart platform can have up to40 real solutions [34].

Secondly, for long, the forward kinematics has been tackled numerically by performingthe Newton-Raphson (NR) iteration scheme [35] already given in (2.44).

�sxk+1 = �sxk + J�1l;sx( �sxk)(

�lmeasured � �lk) (2.58)


As the actuator lengths, �l, are explicit functions of the platform coordinates, �sx, thejacobian, Jl;sx, is a function of platform coordinates.

Jl;sx(�x)(i; j) =@�li( �sx)

@ �sxj(2.59)

This latter method is preferred here as it is less involved to be implemented in a real-time model based controller where inversion of the jacobian is already part of the controlstructure. Convergence or convergence to the actual physical pose in this scheme is notguaranteed in general. As the forward kinematics of a Stewart platform have more than onesolution [57], and since singular points of the jacobian for unconstrained actuator lengthsexist [87], the iteration scheme does not converge globally to the right solution. This prob-lem has only recently been considered in literature [21], [70].

Chetelat [21] considers the problem of convergence in a general way by considering re-construction of coordinates of an implicit (matrix) function from which the image is knowne.g. a Stewart platform pose from length measurements. He points out that if one considersfunctions, of possibly interconnected vector equations, of polynomial degree <= 2, onehas a bijective (one-to-one) mapping. The length-function typically has a quadratic nature.If one however considers (2.38), the inproduct �lTi

�l will have a higher polynomial degreethan two due to the parametrization of the orientation. By using intermediate variables andadditional equations e.g. b = a2 and ab = 1 instead of a3 = 1, it is still possible to fulfilthe requirement. However, in this way more conservative results are obtained.

Chetelat further presents an algorithm to set out a path to calculate the platform positionwith initial guess far apart. In this case the larger difference in actuator length is split upin smaller parts with separate iterations which can easier proven to be convergent. In thissection, a direct approach will be presented, which is, however, specific for the iterationused for Stewart platforms.

A dynamic model of a parallel robot is described as a function of the platform positionand its derivatives [83]. To apply model based feedback, e.g. computed torque [105], basedon the actual platform state instead of the desired state one would like to guarantee bothconvergence of the NR-scheme and exclusion of the singular points (singular J-matrix) inthe work space. In this section an algorithm is presented with which this can be guaran-teed for general but known (inverse) kinematics of the Stewart platform at hand. Practicalrelevance is shown by application to the Simona flight simulator motion system.

First a general theorem on convergence of the Newton-Raphson iteration is considered.Then the jacobian, J , is shown to be Lipschitz i.e. its change w.r.t. two poses is bounded bya constant times the difference of the respective platform coordinates. With this condition itis possible to derive a radius in which the exclusion of singular points of a Stewart platformis guaranteed. Another singular point exclusion algorithm has also recently been presentedby Mertet [98] using the determinant of the jacobian. By gridding the work space withpoints from which a radius can be calculated, one can preclude a larger working volume upto the whole work space (for limited stroke actuators) from singularities.

Convergence of the NR-iteration scheme can also be guaranteed in a neighbourhoodof the solution if some conditions are satisfied. These conditions deal with the maximum


and minimum gain of J and again the Lipschitz condition. These can be calculated forStewart platforms. Exclusion of singular points of J is necessary to calculate a radius of theneighbourhood in which convergence is guaranteed.

From this radius, the maximum gain of J and the maximum speed of the actuator, asufficient update frequency of the iteration can be calculated above which (quadratic) con-vergence is guaranteed.

The parameters of the new Simona research simulator are used as an example whichshows that reasonable results can be obtained although the conditions derived are ratherconservative.

2.3.1 Convergence NR-iteration

A weak version of the Newton-Kantovorich theorem [109] given by Stoer [141] will beused. From this theorem convergence of the NR-iteration can be inferred. It is stated asfollows.

Theorem 2.1 Given: a set D � IRn, a convex set Do with exterior �Do � D and a function�f : D ! IRn which is continuous on D and differentiable with derivative D �f(�x) on Do.

If positive constants r, �, �, and h can be found for �xo 2 Do such that Sr(�xo) =f�x j k�x � �xok < rg � Do, h = (�� )=2 < 1, r = �=(1 � h) and if �f has the followingproperties:

a)

kD �f(�x)�D �f(�y)k � k�x� �yk 8 �x; �y 2 Do (2.60)

(This is called the Lipschitz condition)

b) (D �f(�x))�1 exists and k(D �f(�x))�1k � � 8 �x 2 Do

c) k(D �f(�xo))�1 �f(�xo)k � �

then

A) Starting at �xo the sequence �xk+1 = �xk � (D �f(�xk))�1 �f(�xk) k = 0; 1; : : : is well

defined and �xk 2 Sr(�xo) 8 k > 0

B) limk!1 �xk = �� exists, �� 2 Sr(�xo), and �f(��) = �0

C)

8 k � 0; k�xk � ��k � �h2

k�1

1� h2k

(2.61)

With 0 < h < 1 the iterates converge at least quadratically.


The proof of this theorem is given in [141].Roughly speaking, this theorem states that a solution �� can be found in the NR-iteration

(B) if the differential D �f(�x) does not vary too much (a)), is far enough from singularities(b)) and eventually does not jump too close to the boundary of the defined neighbourhood,at the first iteration (c)).

It also states that the iteration will not go out of a specified neighbourhood (A) andconverges at a certain speed (C). Conditions for a NR-iteration towards the Stewart platformcoordinates will be derived.

Chetelat [21] uses the fact that if one ensures that the function, f(�x), is not of polynomialdegree higher than two, then the jacobian is Lipschitz and in this case the Lipschitz constantcan be calculated using the infinity norm.

2.3.2 NR-convergence Stewart platform

Having stated the kinematical structure of the Stewart platform and given Theorem 2.1 onconvergence of a NR-iteration, it is now possible to investigate under what conditions thespecific NR-iteration of (2.58) will converge to the physical platform pose.

First an appropriate definition of the coordinates has to be given. The defined jaco-bian, Jl;x, is the description of platform translational and angular speed/variation to actuatorlength variations. To go from orientational parameter variations to angular speed dependson the parametrization used.

In this case the last three euler parameters, �� are used to parametrize orientation, �sx T =[�cT s��T ]T , where s is a scaling factor which can be used to get less conservative resultsin specifying a variation of �x since �c and �� = sin( 1

2�)�n� have different dimension (m and

rad). The scaling factor s = 2k�akmax will shown to be appropriate in the sequel.It will be assumed that after each iteration �� will be reset to zero. In that case the

relation between the angular velocity and the euler parameters is very simple, �! j ��=�0= 2I _��.Off course the rotation matrix in the iteration is now the multiplication of all the rotationmatrices calculated, Tk+1 = T (��k+1)Tk. The continuous function, �f , in Theorem 2.1, fromwhich the platform coordinates have to be found can now be given by k �l�i k�k�lik = �fi( �sx)where k�l�i k is the measured length of the ith actuator (fixed value for each iteration) and k�likis the length of the ith actuator given �sx (�li given by (2.38)). Since the measured length isfixed, the derivative function with scaled euler parameters, is only slightly different fromthe jacobian Jl;x given earlier in (2.42).

D �f( �sx)T = Jl;sx( �sx)T =

�Ln

2(TAm � Ln)=s

�(2.62)

Since it is a function of unit direction lengths, it is only defined for k�lk 6= 0.To derive the conditions (a,b,c)-st(ewart)pl(atform) for convergence of the NR-iteration

defined in Theorem 2.1 for the Stewart platform, the constants (�; �; ) can be specifiedusing the kinematics.

a-stpl) In this case the 2-norm of the matrix is taken which is equal to the largest singularvalue, �� and in the formulas d means an finitely small difference.


kJl;sx( �sx1)� Jl;sx( �sx2)k < stplk �sx1 � �sx2k (2.63)

kJl;sx( �sx1)� Jl;sx( �sx2)k =

dLn2d(TAm � Ln)=s

= ��(dJ) (2.64)

The Frobenius (semi)-norm is an upper bound on this norm and is advantageous incase of the Stewart platform since specific bounds can be derived on the matrix ele-ments as will be shown later on.

��(dJ) � kdJkF (2.65)

kdJkF =

vuut 6Xi=1

(kd�ln;ik2 + k2d(T �ai � �ln;i)=sk2 (2.66)

To derive a constant, stpl, the separate elements of the last equation will be describedas a function of �sx. This will be done in the Section 2.3.3.

b-stpl) To state the second condition from which a constant number � stpl has to be cal-culated also the 2-norm is used which can be upper bounded by one over the mini-mal singular value, �min, of the jacobian at some pose, �xo. By using the maximalvariation of the jacobian which has been calculated for the previous condition, theconstant, �, becomes an upper bound for the maximum gain of the inverse jacobianover a volume of poses, �sx.

kJ�1l;sx( �sx)k = (�min(Jl;sx( �sx)))

�1 � (2.67)

(max(�min(Jl;sx( �sxo))� kdJkF ; 0))�1 = �stpl (2.68)

c-stpl) To calculate �stpl, also use can be made of the singular values. Since any pointin the work space is a possible initial start of the iteration, the maximum conditionnumber of the scaled jacobian over the work space can be taken as the constant, � stpl.

kJ�1l;sx

�f( �sx)k � ��=�min (Jl;sx)kd �sxk = �stpl (2.69)

In the next part it will be shown that indeed the jacobian of a Stewart platform is Lips-chitz, as a-stpl) requires, as long as the actuators of the platform have minimal stroke strictlylarger than 0.


2.3.3 Lipschitz condition on jacobian

By constructively analyzing the kinematics of the Stewart platform the following theoremcan be derived.

Theorem 2.2 The Stewart platform jacobian, Jl;sx, is lipschitz i.e. a can be found suchthat

kJl;sx( �sx1)� Jl;sx( �sx2)k � k �sx1 � �sx2k 8 �sx1; �sx2 2 Do (2.70)

if k�li( �sx)k > � > 0 8 �sx 2 D; i 2 f1; : : : ; 6g: (2.71)

Note that the requirement of an actuator length larger than zero is an implicit constrainton the set of platform poses.

To prove Theorem 2.2, it is observed that jacobian, J l;sx, in (2.62) consists of twoelements: the unit actuator direction, �ln, and the vector product, (�ln�T �a), for each actuator.By bounding the difference between the unit actuator directions and vector products of twojacobians with a function linear in kd �sxk, Theorem 2.2 can be proven and a can actuallybe constructed.

In two steps d�ln will be bounded first.

1. Difference of an upper gimbal connection coordinate, d�x a, will be bounded given thefinite difference between two platform coordinates d �sx.

2. Then the difference between two actuator unit vectors, d�ln, given d�xa, will be con-sidered. See Fig. 2.11 in which the motion of actuator as a function of the movinggimbal point xa is schematically depicted.

With

�xa;i = �c+ T �ami (2.72)

the following bound directly follows

kd�xa;ik � kd�ck+ 2k�akkd��k �p2kd �sxk (2.73)

as the motion of a point due to rotation can be bounded easily with the reseted euler param-eter description as

kd(T �a)k = 2k�akkd��k (2.74)

As from Fig. 2.3, it can be observed that the absolute change of a vector due to rotation haslength 2 j sin(1=2�) j=j 2�� j.

The second bound takes into account that as the upper gimbal moves within a ball(Fig. 2.11), the maximum change of the actuator direction is achieved if the new �l, just


l

Φ

dxa

dln

ln

b

a

Fig. 2.11: Change of the unit vector in the direction of the actuator, �ln, as a function ofchange of the upper gimbal vector, d�xa.

touches the ball. In that case (�ln + d�ln) ? d�xa. With some geometry, sin(�) = kd�xak=k�lkand

cos(�) =

s1�

�kd�xakk�lk

�2

(2.75)

the following monotonous upper bounding function can be derived for d �ln.

kd�lnkmax = 2

r1� cos(�)

2�p2kd�xakk�lk (2.76)

Taking into account (2.73) gives kd�lnk � 2kd �sxk=k�lk. So, the actuator (minimum) lengthdirectly influences the bound, which can be put on kd�lnk. If the length of the actuator is notstrictly larger than zero, one can show that arbitrary small variations of the pose can resultin nondecreasing variations of d�ln i.e. not satisfying the Lipschitz condition.

Now, bounding the vector product is also possible. In general k�a � �bk � k�akk�bk and(�a + �c) � (�b + �d) = (�a � �b) + (�a � �d) + (�c � �b) + (�c � �d). Further, rotation does notchange the 2-norm, kT �ak = k�ak. Change of the moving gimbal due to rotation is boundedby kd(T �a)k = 2k�akkd��k = kds��k. Now,

kd(�ln � T �a)=k�akmaxk �

� kd�ln � T �ank+ k�ln � d(T �an)k+ kd�ln � d(T �an)k


�p2

k�lkkd�xak+ kds��k+ 2kds��kkd �sxkk�lk : (2.77)

So in this way every element of the jacobian matrix is explicitly bounded by the variationof the platform pose and the (explicitly platform pose dependent) actuator length. If aminimal actuator length is given, a specific stpl can be calculated.

Adding the bounds, for each of the actuators in the F-norm of the jacobian, results in anexplicit number for stpl. Assuming small platform pose variations i.e. second order effectsin the last equation are relatively small e.g. kd �sxk < :25, a minimal actuator length of 2mof the Simona platform gives

kdJkF �

vuut 6Xi=1

12 +

6Xi=1

(1:82 + :07)2kd �sxk

� 5:2kd �sxk (2.78)

Note that this stpl is valid for any Stewart platform having minimal actuator length of 2mindependent of gimbal point coordinates.

The limited difference of two jacobians given limited difference of the platform pose canbe used to derive guaranteed convergence of the NR-iteration but also to exclude singularpoints of the jacobian from part of the platform work space as will be shown in the nextpart.

2.3.4 Exclusion of singular points

Singular points of a Stewart platform are those platform poses at which the jacobian, J l;sxbecomes singular. At these points at least one platform pose variation will not result inactuator length variation and is therefore not supported by the actuators or uncontrollablefrom forces along the actuator directions. These points exist in the usual Stewart platformconfigurations if the actuators would not have length constraints [87].

Given a point �sxo it is possible to calculate the minimum gain of the jacobian, �min(Jl;sx).This value does not change more than the maximal variation of the jacobian. To be singular,�min should be zero. So with

�min(Jl;sx( �sx)) �

max(0; �min(Jl;sx( �sxo))� ��(dJ( �sx; �sxo))); (2.79)

and if kdJk < k �sx � �sxok < �min(Jl;sx), J�1l;sx( �sx) exists in the ball around �sxo in

platform coordinate space with radius rs � �min(J( �sxo))= . By calculating rs over 6-

dimensional ”boxes” d �sx = rs;min

q23

a grid is taken which precludes the whole work

space from singular points of the jacobian.


Algorithm

- Choose an expected minimal singular value of the jacobians, over the grid points.

- Take a grid such that the boxes with radius rs;min around the grid points fill the wholework space.

- Calculate the minimal singular value of the jacobian at every grid point.

- If the minimal singular value is larger than the expected value, there are no singular pointsin the work space, the algorithm finishes. If not, start another iteration choosing asmaller expected minimal singular value.

Lemma 2.3 If the work space of a robotic manipulator having a bounded Lipschitz con-stant, does not have any singular point this non singularity will be detected by Algorithm 7.1in a finite number of iterations.

Of course the boundary of the work space (in six dimensions!) should be known which is astand alone problem (treated by Haug et al. [86]). To calculate an upper bound for the gainof the inverse jacobian (�stpl) a finer grid should be taken. (This will increase calculationtime tremendously, e.g. n6 grid points extra.)

2.3.5 Sufficient update frequency

The bounds derived are rather conservative in most cases. However, by calculating a con-vergence radius for the NR-iteration in the practical example of the new Simona researchsimulator shows that it is possible to guarantee that this iteration can be used if a reasonablefrequency is used to update the platform pose. Further, the singular points can really beexcluded from the work space in this case.

To satisfy NR-iteration convergence the following constants were obtained using thekinematics of the Simona flight simulator motion system given in Table 2.1.

a-simona Using (2.78), assuming a minimal length of the actuator of lmin = 2:08m, resultsin simona = 5:2.

b-simona With gridding a minimal radius of rs;min = :09 is found. (About 200000 gridpoints had to be calculated to exclude singularity from the work space). The smallestsingular value found is 0:75 and with finer gridding a � simona = 2 can be guaranteed.

c-simona The first value of �f( �sxo) in any point can not be larger than �f( �sxo) � ��(J)kd �sxkTogether with the smallest gain this gives �simona = �kd �sxk � 3:5kd �sxk

Now with a bound kd �sxk � 0:02, hsimona = (�� )=2 = (3:5 �0:02 �2 �5:2)=2 = 0:23 andrsimona = �=(1� h) = (3:5 � 0:02)=(1� 0:23) = 0:11. Within the every ball with radiusrsimona, [a-simona] and [b-simona] should be guaranteed which is the case in the workspace given the operational stroke (not including actuator cushioning part). To guaranteeconvergence in the whole work space also non-singularity, etc. should be guaranteed further


outside the work space which needs lots of calculation (with an extra stroke of 0.15m,singularities can be obtained so rs;min becomes very small).

With a bound on the maximal speed of the actuator it is possible to calculate a minimalupdate frequency which guarantees kd �sxk < 0:02. Given k _lk < 1 m=s, now

�simona�t _lmax < kd �sxk (2.80)

and �t < 1=100 s follows. So with an update frequency fs > 100 Hz convergence of theNR-iteration is attained.

Lemma 2.4 The Stewart platform with the Simona motion system parameters has no sin-gular points in the work space and the NR-iteration with this platform will converge to theright platform pose if the update frequency is larger than 100Hz.

Summarising, to satisfy a general convergence theorem on Newton-Raphson iteration,one of the requirements is the Lipschitz condition on the derivative function, which wasshown to be satisfied for Stewart platforms. Variations of the jacobian can be bounded bythe variations of the platform coordinates.

Next to this requirement, the jacobian should not be singular in any point of the workspace. With the Lipschitz condition on the jacobian and gridding, volumes can be excludedfrom singularities.

Although only sufficient (and thereby conservative) conditions could be derived for con-vergence, it was shown to be possible to obtain an important convergence result in the prac-tical example of Simona flight simulator motion system. It will be one of the requirementsfor safe application of model based feedback.

2.4 Modelling the mechanical part of the system dynamics

After the thorough analysis of the kinematics of the parallel driven Stewart platforms in theprevious section, this section will focus on the dynamics of such systems. The aim of thissection is to show how to derive a limited number of differential, and possibly also somealgebraic, equations, which describe the motion of mechanical systems such as the motionsystem of a flight simulator as a function of the forces acting upon this system. It willbe assumed that these systems consist of rigid bodies. In the extensions these bodies arepossibly interconnected by springs and dampers.

2.4.1 General theory in modelling the dynamics of mechanical systems

Most theory described in this section can be found in many textbooks on mechanics androbotics, rigid multi body systems like [64, 81, 102, 106, 120, 129, 134]. Based on thisgeneral theory, a rigid body model of the Stewart platform, possibly with actuator jointinertia or linear ground flexibility, will be derived.

All mechanics treated here are based on the assumption of a semi-equilibrium given byNewton’s second law �f �m��p = 0 in an inertial frame for any mass particle. To describe the

2.4 Modelling the mechanical part of the system dynamics 61

acceleration, ��p, of all parts in a system as a function of a limited number of variables somekinematics has been introduced.

Having defined the kinematics of a mechanical system, its dynamics can be specified. Inthe equations of motion the semi-equilibria are described in a compact form as a function ofgeneralized velocities or coordinates. The integrals over the mass-particles of a body resultin inertia matrices and the active forces are projected along the velocities by a virtual workargument.

The equations of motion can be stated in several ways. First the Lagrange equations willbe given. Then it will be shown that from these equations the more simple Newton-eulerequations follow if one rigid body is considered. Finally the method based on Kane [64]will be introduced. This method in which all forces, active and inertial, are projected alonga limited set of generalized velocities, will be central in the derivation of the motion systemmechanical model.

Considering generalized velocities or coordinates ( _�z; �z), with the method of Lagrangethe difference between the kinetic energy,K(�z; _�z), and potential energy, P(�z), of a systemis called the Lagrangian, L(�z; _�z): L = K � P. For a system, which can be described by aminimal number of generalized coordinates �z, the Lagrange equations are given by:

d

dt

�@L

@ _�z

�� @L

@�z= �� (2.81)

The driving moments/forces, �� , are all the working forces (non inertial or conservative) pro-jected along the variations of the generalized coordinates. These are called the generalizedforces.

The kinetic energy can in general be described by

K =1

2_�zTM(�z) _�z; (2.82)

where the mass matrix, M(�z), is a symmetric positive semi definite matrix.It can be appropriate to apply a coordinate transformation by introducing the generalized

momenta (assuming a positive definite and thus invertible mass matrix)

�m =@L

@ _�z=M(�z) _�z: (2.83)

The implicit second order differential Lagrange equations (2.81) can now be transformedto a (nonlinear) state space description, i.e. the Hamiltonian equations of motion followingVan der Schaft [147]. A key role is played by the hamiltonian or total energy function, H,consisting of the addition of kinetic and potential energy,

H( �m; �z) =1

2�mTM�1(�z) �m+ P(�z) = �mT _�z � L(�z; _�z): (2.84)

Taking the partial derivative of (2.84) to �m for the first set of state equations and filling in(2.84) into (2.81) eliminating L for the second, results in

_�z =@H

@ �m( �m; �z) (2.85)

_�m = �@H@�z

( �m; �z) + �� : (2.86)


In this non energy dissipating system the increase in energy can be shown to be equal to theamount of work delivered,

d

dtH = _�z

T�� (2.87)

By observing that the work �W , done by forces �� , does not change if a change of coordinates�z, is applied, it is easy to show that projecting the forces along other coordinates is equal tomultiplying by the transpose jacobian.

�W = ��T ��z1 = ��T Jz1;z2��z2 = (JTz1;z2��)T ��z2 (2.88)

Considering such collocated pairs of forces and velocities as inputs (�� = J T (�z)�u) and out-puts �y = J(�z) _�z, a hamiltonian system results. The Stewart platform rigid body mechanicswill have this structure. The hamiltonian system structure does not require that the numberof input/output pairs and the number of generalized momenta/velocity pairs is equal. Thisallows to take into account parasitic resonant effects without losing the structural propertiesof a hamiltonian system.

Further, manipulation of the second order differential equation (2.81) can lead to

M(�z)��z +

�d

dt(M(�z) _�z � _�z

T

�@

@�ziM(�z)

�_�z

�+

@

@�zP(�z) = �� : (2.89)

Compactly written as

M(�z)��z + C(�z; _�z) _�z + �G(�z) = �� : (2.90)

with a mass matrix, M , a non linear coriolis/centripetal matrix, C, and a gravity vector, �G.If a rigid body (with mass, m, and inertia, I�c) is considered at its centre of gravity �c, the

Newton-euler equations result. The mass matrix is block diagonal in this case.

Kbody =1

2

�_�c �!

� � mI 00 I�c

� �_�c�!

�(2.91)

From the two blocks, two independent equations follow from (2.81). The impulse law:

� �f =d

dt(m

d

dt(�c)) = mI��c (2.92)

And the impulse moment law with the generalized forces �fg�c (moments in this case) isderived similarly from (2.81) taking the time derivative of the Lagrangians partial derivativeto the second set of (angular) velocities. The Lagrangian, in this case, only consists of thekinetic energy, which does not depend on the position.

� �fg�c =d

dt(gRmIm�c �!m) = gRmIm�c _�!

m+ gRm ~mIm�c �!m

= Ig�c_�!g+ ~gIg�c �!

g = Ig�c_�!g+ �!g � I

g�c �!

g (2.93)

In this way, the Newton-euler equations are easily stated for each rigid body in a system.However, extra equations with unknown internal forces result, in using this method to state


the equations of motion for a multi body system. Applying Lagrange, results in takingpartial derivatives of complex energy functions if the whole system is considered. These aredisadvantages which can be circumvented by using Kane’s method.

With Kane also generalized variations or velocities have to be specified. The generalformula

�f + �f� = �0 (2.94)

states the (semi-)equilibrium of the active forces, �f , and inertial forces, �f�, projected alongthe directions of the generalized velocities. Generalized velocities do not have to be deriva-tives of some positional coordinate e.g. in principle the angular velocity can be used assuch.

To calculate the over-all inertial forces, as in the Newton-euler approach, the specificinertial forces generated in the frame of each body can be stated. As in the Lagrangianapproach, a minimal number of equations results by writing the motion of the bodies as afunction of the generalized velocities and projecting each specific force from its local coor-dinates to the generalized ones. Also the active forces can first be stated in an appropriateframe after which projection follows. The projection in general consists of a change of co-ordinates i.e. a multiplication with a jacobian. This procedure can also be automated [65].Especially in fast modelling with the aim of simulation, this can be useful.

With this method it is possible to start with a strongly simplified system by calculatingpart of the (inertial) forces and separately adding other forces if a more accurate modelhas to be taken into account. Many of the specific merits of the method will become clearconsidering the modelling of the mechanical part of the Stewart platform including actuatorinertia.

The amount of generalized coordinates can exceed the number of d.o.f. of a system. Inthat case, next to the differential equations given by the semi-equilibria of the forces (2.94)constraint equations of velocities and position, generally stated as

A(�x; t) _�x+�b(�x; t) = �0; (2.95)

should be added. With mechanical systems having so called kinematic chains i.e. parallelmanipulators, such additional equations easily appear if the motion of some parts of thesystem can not explicitly be written as functions of the minimal number of coordinates(amounting to the number of d.o.f.). This will not be the case with the Stewart platform apartfrom the additional coordinate necessary to attain a global description of the orientation.

Systems where the constraint equations cannot be integrated to constraint position equa-tions are called non-holonomic. A parallel manipulator, like the Stewart platform, will occurto be a holonomic system. Since there are kinematic chains, it is often easier to state theequations of motion of a parallel manipulator by using constraint equations. In that case themanipulator is described as a serial system (with some of the joints disconnected). The par-allel connections are incorporated by adding constraints. A combined differential/algebraicdescription results. This kind of description causes difficulties (index problems etc., [19])in simulation and model based control. With these goals in mind during modelling, it ismore convenient to state the model in explicit differential equations if possible. Startingwith Section 2.4.3 in the second part of this section it is shown that this can be done with


φ

r

x

y

m

φ

r

x

y

m

Fig. 2.12: Example system consisting of one mass rotating in the plane on the left. On theright also a variable length, r, exists to the centre of rotation.

the Stewart platform. But first a simplified example will be given in Section 2.4.2 to clarifythe modelling method used.

2.4.2 Example in modelling using Kane’s method

To clarify the method of Kane, the dynamics of a simple example system will be modelled.The system is given in Fig. 2.12. A simple mass, m, can be rotated in the plane aroundsome point, which is taken as the origin in the inertial frame. It is assumed we do not knowhow to define the mass properties in rotation but only know the translational inertial forceequations:

�f�c =

�f�xf�y

�= �

�m 00 m

� ��x�y

�= �M��c (2.96)

We, however, want to choose the angular velocity, _�, defined in Fig. 2.12 as the generalisedvelocity for this 1-d.o.f.-system. The coordinates, x and y, and their derivatives can beformulated as a function of � and _� with

�c =

�x

y

�=

�r cos(�)r sin(�)

�(2.97)

and by differentiation

_�c =

�_x_y

�=

� �r sin(�)r cos(�)

�_� = Jc;� _�: (2.98)

The local inertial forces, �f�c can be projected along the generalised velocity by

f�� = JTc;��f�c (2.99)

Further differentiation of (2.98), provides us with the acceleration

��c = Jc;� ��+ _Jc;� _� (2.100)


with

_Jc;� =

� �r cos(�)�r sin(�)

�_�: (2.101)

This equation can be filled into (2.96), which can be filled into (2.99) to arrive at the simple

f�� = �mr2 ��; (2.102)

since JTc;� _Jc;� = 0 and JTc;�Jc;� = r2 in this example. The equilibrium equation of motion,

considering an active force already projected along _�, i.e. a moment fa, will be

f + f� = fa �mr2 �� = 0 (2.103)

In principle, also _�c could have been defined as the generalised velocity. The only con-straint in this choice is the ability to describe the motion of any body with this set of veloc-ities. In that case (2.96) would give the inertial forces and an additional constraint equation

_�cT�c = 0 (2.104)

would have to be satisfied. I.e. a combined algebraic and differential set of equations wouldresult. Simulating such a system is not trivial. E.g. in the example any finitely small stepsatisfying (2.104), a velocity orthogonal to �c, would enlarge the radius r.

In the case sketched, it is relatively easy to choose a minimal number of coordinatesand end up with differential equations only. But already with planar systems such as themanipulator given in Fig. 2.6, it is not clear how to choose three coordinates (minimal localdegrees of freedom) which globally describe this system.

The method presented seems somewhat overdone if applied on such an example but itcan easily be extended to the more complex cases. Consider for example the system ofFig. 2.12 with an variable radius (e.g. due to a non rigid spring/damper) connection.

If the force of the spring always acts along the radial direction it could be appropriate tochoose the polar generalised velocities, _�p, consisting of _r and _�.

Now

_�c = Jxp _�p =

�cos(�) �r sin(�)sin(�) r cos(�)

� �_r_�

�(2.105)

and the inertial forces follow from

�f�p = �JTxpm(Jxp��p+ _Jxp _�p) (2.106)

and using

_Jxp =

� � sin(�) _� �r cos(�) _� � _r sin(�)

cos(�) _� �r sin(�) _� + _r cos(�)

�(2.107)

gives the inertial forces more explicitly as

�f�xp = ��m 00 mr2

� ��r��

�� mr _�2

2mr _� _r

�: (2.108)


In this case nonlinear velocity forces like the quadratic centripetal force, mr _�2, and thevelocity product coriolis force, 2mr _� _r, naturally show up.

Including an additional body is relatively easy. E.g. the inertia, i tube of the tube drawnin Fig. 2.12, can be taken into account by adding the inertial force, f �

tube = itube ��.In the calculations, the projections using the jacobians often enable geometric simplifi-

cations. This will also be the case in using this method in modelling the Stewart platformincluding the actuator inertial properties.

With this example, the method of Kane has been made explicit and will be used in thenext sections to derive a model of the mechanical part of the Stewart platform motion systemdynamics.

2.4.3 Stewart platform dynamics

The general procedure of stating the equations of motion has now been given. In this part,this will be applied in modelling the Stewart platform. Kane’s method of projecting localsemi-equilibria (equations of motion) will be used to arrive at a compact description. Tostate the local equations, both Lagrange and Newton-Euler are applied wherever either oneis most appropriate.

With the choice of platform position/orientation as the generalized coordinates, all equi-libria can be written as explicit functions of these coordinates (and its derivatives). Themodelling approach, however, still enables a clear assignment of the contribution each partof the system has on the over all dynamics.

First a simplified model, with the platform as the only (rigid) body, will be derived.Then, the influence of the actuator inertial forces is quantified.

2.4.4 Basic Stewart platform mechanical model dynamics

The basic structure of the Stewart platform model results if one considers the platformalone, not taking into account the inertial forces of the actuators. Since the system in thiscase consists of only one body, the equations of motion are easily derived with Newton-Euler taking the velocity of the platform coordinates as the generalized speed. With theimpulse law, (2.92), the translational motion is described

Ln �fa +m�c�g = m�c��c; (2.109)

where �fa are the active forces generated by the actuators and �g is the gravity vector.And with the impulse moment law, (2.93), the rotational motion is described by�

TAm � Ln��fa = I�c _�! + ~I�c�!; (2.110)

with I�c = TIm�c TT . Combining these two results in the simplified model of the Stewart

platform.�Ln

TAm � Ln

��fa =

�m�cI 00 I�c

��c_�!

�+

�0 0

0 ~I�c

��_�c�!

��m�c�g�0

�(2.111)


In short

JTl;x( �sx)�fa =M�c( �sx)��x+ C�c( _�x; �sx) _�x+ �G�c; (2.112)

from which the mass matrix, M�c, the coriolis/centripetal effects, C�c, and gravity, �G, areclearly separated and the structure given in (2.90) is visible. Further, the jacobian, J l;x,already playing an important part in the sections on kinematics, also occurs in the equationsof motion directing the actuator driving forces.

In many cases, especially with large and heavy motion systems, these equations will suf-ficiently describe the rigid body dynamics of the mechanical part of the system. In practice,these dynamics will interact foremost with the hydraulic actuators, which will be modelledin the next chapter. This chapter concludes with some sections on extending the mechanicalmodel.

2.4.5 Influence of the actuator inertial forces

In some cases, the inertial forces (i.e. mass properties) of the actuators can not be neglected.With many heavy flight simulation systems they can, but with the light weight motion sys-tems requiring high performance motion this has to be reevaluated. Explicitly taking intoaccount the actuator inertial forces will complicate the model. However, this model can beused to attain an appropriate approximate version as will be shown in the next chapter.

Most of the assumptions considering the actuators have already been given consideringtheir kinematical properties in Section 2.2.11. See e.g. Fig. 2.8. The gimbals are assumed torotate frictionless. The inertia of both actuator parts rotating around the actuator directionis neglected and further assumed symmetric w.r.t. this axis. The mass properties of eachactuator part can therefore be described by two parameters, the mass and the inertia aroundan axis orthogonal to the actuator direction. Mass ma and inertia ia are taken for the uppermoving part of the actuator (mainly the piston), and (m b, ib) for the lower rotating part, thecylinder.

The inertial forces of the actuators can be split up in three parts: the gravitational forces,the inertial mass forces and the influence of its inertia. These forces will first be projectedon the upper gimbal points. Any force generated at this point is easily projected at thegeneralized platform velocities with (2.47) by

�fxa = JTa;x�faa : (2.113)

The gravitational forces are easily projected along the platform velocities. The lower partof the actuator using (2.50),

�fabg = JTbc;amb�g =mbrb

j l j Pln�g =�Gmb

: (2.114)

Clearly, in a position where the gravity vector is directed along the actuator (�ln) this forcewill not contribute. The contribution of the moving part is in that case maximal at �a as isshown by equivalently using (2.49),

�faag = JTac;ama�g = ma(I �ra

j l jPln)�g =�Gma

(2.115)


The inertial force generated by the mass at the c.o.g. of the moving part of the actuator iseasily described w.r.t. a frame at its c.o.g.

�fac = ma _�vac (2.116)

Projection of this force at the upper gimbal point using (2.49) and writing _�vac as afunction of _�va using (2.55) and (2.57), results in

�fama= ma(I � Pln +

(j l j �ra)2j l j2 Pln) _�va : : : (2.117)

+ mara

j l j2 j Pln�va j2 �ln + 2ma

(j l j �ra)raj l j3 (�lTn �va)Pln�va

= Ma _�va + Cma�va

The first term consists of a part in the direction of the actuator (I � P ln) where the massdirectly acts on the gimbal point. Perpendicular to this direction the influence gets smallerwith the squared ratio of distances to the lower gimbal point. The second and third term arethe quadratic velocity dependent coriolis and centripetal force.

The inertial forces generated by the inertias of the lower and upper part of the actuatorcan be taken together considering Lagrange and observing that their contribution to thekinetic energy is equivalent. With (2.48)

Kia;ib =1

2�!Ta �!a(ia + ib) (2.118)

=1

2�vTa

(ia + ib)

j l j2 Pln�va =1

2�vTaM

aia;ib

�va

The derivation of the following two equations can be found in Appendix B together witha proper choice for factoring the quadratic velocity terms. p a is the position of the uppergimbal point.

d

dt(Mia;ib) = � (ia + ib)

j l j3 (2�vTa�lnPln + Pln�va

�lTn + �ln�vTa Pln); (2.119)

�Kia;ib��pa

= � (ia + ib)

j l j3 ((�lTn �va)Pln�va +�ln�v

Ta Pln�va); (2.120)

With Lagrange (d=dt(M) _�v � @K=@�pga), (2.119) and (2.120) the inertial forces at pa result

�faia;ib =(ia + ib)

j l j2 Pln _�va �2(ia + ib)

j l j3 (�lTn �va)Pln�va =Mia;ib_�va + Cia;ib�va: (2.121)

The contribution to the mass matrix only exists at motion perpendicular to the direction ofthe actuator. Next to this, only coriolis and no centripetal terms appear. The coriolis forceis generated as a result of the inertia points in the opposite direction as the one generated by


the mass. This is due to the fact that the influence of the inertia decreases while that of themass increases as the actuator gets longer.

All the separate inertial force contributions of each actuator represented by f abg , faag ,fama

and faia;ib together with the active actuator force generated by the hydraulics, result intotal forces at the upper gimbals. Projection to the platform coordinates via the jacobian,Ja;x, puts these forces into place for the equations of motion.

2.4.6 The Stewart platform model including actuator inertia

The equation of motion of the Stewart platform including the inertia of the actuators canstill be put in form of

JTl;x( �sx)�fa =Mt( �sx)��x+ Ct( _�x; �sx) _�x+ �Gt( �sx) (2.122)

Where Mt, Ct and �Gt are given by

Mt =M�c +6Xi=1

JTai;x(Mma;i+Mia;i;ib;i)Jai;x (2.123)

Ct _�x = C�c _�x+6Xi=1

(JTai;x(Cma;i+ Cia;i ;ib;i)Jai;x _�x� j �! j2 (Mma;i

+Mia;i;ib;i)P!�ai)

(2.124)

�Gt = �G�c +6Xi=1

JTai;x(�Gma;i

+ �Gmb;i) (2.125)

In this model, the platform coordinates, velocities and acceleration are the only variablesand the effect of each term (mass, coriolis, centripetal, gravity, driving forces) and parameter(mass, inertia, centres of gravity, gimbal point) can clearly be distinguished. The model iswell defined in each state except for those in which any of the actuator lengths becomeszero. Note that this puts an implicit limitation to the validity of the model.

Adding the inertial influence of the actuators to the simplified model did not change thecompact form of six coupled second order differential equations. The equations, however,became much more complex which is not favourable in model based control in which themodel has to be calculated at high speed.

Ji claims in [61] that the actuator inertial effects can be seen as a change of the platformmass, inertia and c.o.g. This claim should be carefully interpreted as this change is not onlydependent on the position, but also on the direction of the motion i.e. not even valid at oneoperating point. E.g. in case of the Simona flight simulator, the mass of the actuators addmore to the mass matrix of simulator in heave than in the lateral directions of surge andsway.

With the equations given, it is possible to give bounds on the forces not taken into ac-count if the inertial forces of the actuators would be neglected. Although with conventional


motion systems this is often justified, the tendency towards light weight platforms makesthe actuator inertial forces more evidently come into play. Total neglection would resultin a too rough approximation in that case. Approximation by a constant additive term willshown to be more convenient.

2.4.7 Parasitic mechanical aspects: modelled as linear flexibility

The models presented only take into account the mechanical system as the rigid bodies.Together with the hydraulic actuators, this forms the most relevant part of the system dy-namics. However, it appeared that two kinds of parasitic mechanical effects due to flexibilitycould not be neglected.

First, the foundation on which the Simona motion system was tested in the CentralWorkshop was not entirely rigidly attached to the inertial world. Though relatively heavy(45 tons), the foundation formed by a block of concrete appeared to be in motion in theplane orthogonal to z if the simulator motion contained too much energy in the frequencyarea above > 5 Hz [41]. The foundation was not ideal for these tests, but in countrieslike the Netherlands, where buildings are often build on large piles in weak ground, similareffects can be expected if no special precautions are being taken.

Flexible effects like this, can be modelled by considering additional mass/ spring/ dampersystems in connection with the rigid body model. In Fig. 2.13, a representation of a foun-dation is given, which has two translational and one torsional spring/damper pot connectingthe mass/inertia of the foundation to the inertial world. Further, the motion system intro-duces forces through the lower gimbal points.

If the foundation is given, the three degrees of freedom: x f along the surge (x-)direction,yf along the sway (y-)direction and the angle f around the z-axis (yaw-motion), threeadditional equations of motion will result. At first instance, the interaction of the founda-tion with the inertial world will be modelled by decoupled spring/damper connections withspring stiffnesses cxf , cyf , c f , and damping coefficients bxf , byf , b f , which are used forthe respective directions. Now, the equations of motion of the foundation with massm f andinertia If can be modelled by using the additional matrix equation:

24 mf 0 0

0 mf 00 0 If

3524 �xf

�yf� f

35+

24 bxf 0 0

0 byf 00 0 b f

3524 _xf

_yf_ f

35 + (2.126)

24 cxf 0 0

0 cyf 00 0 c f

3524 xfyf f

35 = JTl;�xf

�fa

The jacobian of the general (six) foundation body parameters, �x fg to actuator lengthshas a similar structure as Jl;x.

Jl;�xfg =� �LTn �(B � Ln)

T�

(2.127)

In case only the aforementioned three degrees of freedom in the plane are possible a secondprojection is required through the selection matrix P f with unit vectors selecting x, y and


Fig. 2.13: Physical representation of the model used for the flexible attached foundation

.

P Tf =

24 1 0 0 0 0 0

0 1 0 0 0 00 0 0 0 0 1

35 (2.128)

Jl;�xf = Jl;�xfgPf (2.129)

Finally, the equation for the platform velocity has to be adapted since �b will start to vary.

_�l = LTn (Va � Vb) (2.130)

With the lower gimbal point velocities, _�vb stacked in Vb. If the model with the platformalone (or a constant mass matrix with respect to the moving simulator) is taken, the modelequation (2.111) does not have to be adapted. All together these equations form a model ofthe system with a platform attached to a non-inertial ground, which has the same validity asthe simple model (for all simulator poses except those for which (klk i = 0). This problemhas not seen to be considered in literature apart from cases in space technology [93, 108] inwhich the moving platform has very limited manoeuvrability and the actuators are used tosupress vibrations.

At first sight, the foundation, though also moving if the actuators force the simulatorto move, does not have too much direct interaction with simulator motion, since the basicmechanical model equation (2.112) looks the same. Indirectly, the following dependenciescan be observed.


- The motion of the foundation has influence on the actuator velocities. The hydraulicactuators form a port connection with the mechanical system through the active forcesand velocity. Generally, the active force will be influenced by the velocity due to acoupling in the dynamics of the (controlled) hydraulic system. This influence willtherefore affect the simulator motion. Note that this influence will be much larger ifthe actuators are position/velocity controlled than with force control in the relevantfrequency area.

- The platform pose can not be calculated from the actuator lengths anymore without cor-recting for the moving foundation.

- The jacobian, Jl;x, depends on the motion of the foundation since it depends on the unitdirection vectors of the actuators. These directions will change as the foundationmoves resulting in change of direction of the active forces acting on the simulator.This is a second order effect and is mostly compensated for by reconstructing theJacobian from the actuator length measurements. The motion in the plane of thefoundation will not influence the forces necessary to compensate for gravitationaleffects of the gross moving platform.

- The motion of the actuator (inertias) is directly influenced by the foundation. The equa-tions of motion for these parts have to be adapted and become somewhat more com-plex. If the actuator inertia can be neglected or taken as part of the constant massmatrix w.r.t. moving platform, this effect can also be neglected.

Given influences, both in the choice of the control strategy as in identifying the dynamicsof the system, the parasitic behaviour of the foundation should be taken into account.

The second important parasitic mechanical effect to be analyzed are the structural modesof the simulator itself. Especially in optimising the ability to realistically simulate the highfrequency vibrations in a vehicle, the unwanted resonances of the simulator should not behit up to noticeable level. In the design of the shuttle of the Simona simulator, the ratioof stiffness and mass has been optimized [8]. However, finite element models pointed outthat with this system one will not be able to extend the lowest structural vibrations above15-20 Hz

As a rule of thumb the flexible modes should only appear in a frequency area welldistinct from the area where the motion to be simulated has relevant energy and thus usuallyform an upper bound on the achievable bandwidths. Since there are only a limited numberof actuators, the moving platform loses functional controllability if structural deformationtakes place. In theory, one could well be able to apply a desired acceleration profile to thepilots head but one would in these cases structurally have to settle for vibration in other partsof the platform e.g. the large visual projection screen. Additional active damping devicescan probably be of help in making this restriction less tight.

The unneglectable deformations in the moving platform drastically change the structureof the system. Modelling using the projection methods like Kane’s is still possible if one canapproximate the modes by splitting up the previously rigid bodies in a limited number mass/spring/ damper systems i.e. smaller rigid parts connected by springs and dampers ([146]).Deriving these equations by hand, however, becomes very tedious and does in general not

2.5 Chapter Resume 73

result in a model from which analysis can take place more easily than from automaticallyderived models. In a master’s project as part of this research Rijnten [119] modelled thesystem, including a structural vibration, using the symbolic equation modelling software,Autolev [65]. Though simulation with such models can clarify systems behaviour, not muchstructural understanding of systems like a deformed Stewart platform can be gained sincethe number of variables in the model well goes above a thousand.

Other, more advanced, methods in modelling flexible structures [62, 63, 66, 129] oftenused in space applications with large flexible deformations, usually result in models, which,due to their complexity, can not directly be used in model based control. It was decided todo the actual analysis of the structural deformations in the Simona motion system using theactual data measured at the system. This will be discussed in Chapter 4. Further analysisand also simulation of the models derived will take place after introducing and integratingthe models of the hydraulic system to be discussed in the next chapter. This will be morerealistic since the integration of the hydraulics and the mechanics forms the most relevantpart of the dynamics considering the whole flight simulator motion system.

2.5 Chapter Resume

Since the kinematic model of the Stewart platform construction is fully parallel, the plat-form (end effector) coordinates and velocities can globally be used to describe the state ofthe mechanical system. It appeared most convenient to state orientation applying an eulerparameter (quaternion) like parametrization. Apart from the numerical advantages, an algo-rithm could be defined to evaluate the appearance of singular points in the work space of aparallel manipulator like the Stewart platform.

With this method, it finally became possible to proof that the SRS kinematical construc-tion has no singular points. This is not trivial, since it would have been possible to reachsuch a point with only 15 % more stroke of the actuators. Moreover, it could be proven thatthe iterative Newton-Raphson procedure to calculate the platform pose from the actuatorlengths converges anywhere in the SRS work space during any kind of possible (velocitylimited) motion. It converges sufficiently fast for safe use in a real time digital feedbackloop as will be done in the model based controller discussed in Chapter 5.

The dynamics of the Stewart platform was stated as a set of differential equations with-out algebraic constraints resulting from the kinematic chains in the system. This is possibleby writing the actuator motion explicitly as a function of platform motion. By using Kane’smethod of projecting forces onto the generalized velocities, the contribution of each inertialor active force e.g. stemming from the actuators piston mass, etc., can be quantified sepa-rately using local (simple) Newton-Euler like equations and (by projection) ending up witha limited number of equations of motion.

The models derived only describe the dynamics of the mechanical system. The observeddynamics of a flight simulator motion system will shown to be much better approximated bymodels which incorporate both the hydraulic and mechanical characteristics and their (twosided) interaction. This chapter provided for the mechanical modules for such models. Thenext chapter will treat the modelling of the hydraulic system itself and the interconnectedstructure of hydraulically driven mechanical systems.

Chapter 3

Hydraulically driven motionsystems

In the previous chapter, the mechanical system was modelled with input actuator forcesfrom which accelerations, velocities and positions of the motion system and possibly itsparasitic dynamics could be calculated. The energy necessary to move is supplied by theactuators. In robotics both electrically as hydraulically driven systems are seen to be appliedand the mechanical model can be merged easily to models of either kind of systems.

With the larger flight simulator motion systems almost invariably use is made of hy-draulic actuators as is also the case with the Simona motion system. Since both the energyand the cooling with fluid power technology can be supplied at an exterior hydraulic pump,a favourable high ratio of force delivered over the weight and size of the actuator can besustained well into the high power area. But even more importantly, the use of hydraulicsenables smoothly running (low friction, low wear) long linear actuators, which is necessaryin simulation requiring an unnoticeable change of sign in the direction of motion.

An important aspect of the motion systems under consideration is the fact that the dy-namics of the actuators heavily interact with the mechanics. So modelling either one ofthese phenomena does not provide much insight in the dynamics of such systems. In mosttextbooks treating the modelling and control of mechanical systems the dynamics of theactuators are at best considered as parasitic influences to be taken into account at high fre-quencies [11, 31, 113]. However, with (heavy) systems requiring a high amount of power,especially if driven hydraulically, the interaction between the actuators and mechanical sys-tem determines the ’rigid’ resonant modes. By using an integrated model, which will bepresented in Section 3.3, the most relevant dynamics is captured as will be shown in thesubsequent Chapter 4 in which evaluation with the experimental data takes place.

The basic integrated model can still have a relative simple structure although the re-sponse of such systems may already seem quite complex. The basic structure of the hy-draulically driven mechanical system allows the derivation of some important propertieslike passivity, discussed in Section 3.3.1, which can be of use in control and experimentalidentification. In this chapter, the dynamics of the hydraulic actuators will be modelled witha bilateral coupling to the mechanics through the energetic pairs of force and velocity at themechanical side and flow and pressure in the hydraulical part of the system.

75

76 3 Hydraulically driven motion systems

Fig. 3.1: Schematic drawing of a symmetrical hydraulic actuator.

The long stroke hydraulic actuators, which are typical for flight simulators, have beenmodelled in detail for motion control design by Van Schothorst in [124]. In Section 3.1, anoverview will be given of the parts of those models which proved to be essential as eitherbeing an explicit part in the model based controller considered in this research or in analysisexploring the limitations of such controllers. This last part usually requires an additionaldegree of detail in modelling discussed in Section 3.2. In this case, the dynamics of thehydraulic servo valve together with the transmission lines had to be taken into account.

For a more fundamental introduction into hydraulic systems, one is referred to textbookslike [99] and [153]. Most of the notation used here follows Van Schothorst [124]. Themodelling for control approach in hydraulic systems can also be found in references like[48] and [146]. In these references, however, the dynamics of the transmission lines wasassumed to be neglectable. In this research, this was not the case due to the long strokeand high performance requirements, and therefore a modelling for control approach withthe interplay of transmission lines dynamics and servo valve as the limiting factor, had to bechosen.

3.1 The basic structure of hydraulic actuators

The basic notion of the functionality of a hydraulic actuator can be observed by lookingat Fig. 3.1, although the actually used actuator looks far more complicated as shown inFig. 3.3. In Fig. 3.1, the piston connected to the load can be made to move by letting oilflow into the compartment of the cylinder on the left, � o1, and out of the compartment onthe right, �o2, or v.v. This flow will result in a pressure build up, Po1 � Po2, which throughthe reflected areas, Ap, will force the load to move. In a symmetrical actuator, this area,Ap, is equal on each side. In motion, the oil flows will also have to compensate for thechange in volume in the respective actuator chambers. This effect accounts for the two-sided coupling of flow/ pressure and/ or velocity/ force. Usually one also has to account forsome dissipation resulting from leakage flows.

The dynamics due to the hydraulic part of the system is caused by the compressibilityof the oil. The relative decrease in volume, � _V =V , causes a pressure rise, _P , linear with

3.1 The basic structure of hydraulic actuators 77

the effective bulk modulus of the oil, E.

_P = �E_V

V(3.1)

By taking into account the oil flow into and out of the compartments, the following equationscan be obtained, which describe the pressure dynamics of each chamber.

_Po1 =E

V1(�o1 ��l1 �Ap _q)

_Po2 =E

V2(�l2 ��02 +Ap _q) (3.2)

The sign of the oil flows due to leakage, � l1, �l2, are taken negative w.r.t. the main flows,�o1, �o2, usually controlled by a valve. The position, q, of the piston is set to zero at thepoint where the volumes of the actuator chambers are equal, Vm = V1 = V2. This willtypically be at or near half stroke position as drawn and will be referred to as mid position.As a result

V1 = Vm +Apq

V2 = Vm �Apq (3.3)

At this point it is most appropriate to apply a change of coordinates by taking mean anddifference pressures, flows and oil stiffnesses (Cm and dC), which are defined as follows�

Pm �m �lm CmdP d� d�l dC

�=

�12

12

1 �1

��Po1 �o1 �l1 E=V1Po2 �o2 �l2 E=V2

�(3.4)

The pressure dynamics of (3.2) can now be transformed to

_dP = 2Cm(�m ��lm �Ap _q) + dC=2(d�� d�l)

_Pm = Cm=2(d�� d�l) + dC=2(�m ��lm �Ap _q) (3.5)

The change of coordinates can be motivated by the fact that only the pressure difference,dP , directly affects the mechanics through the actuator force, f a = ApdP . Further analysisreveals that even more advantage can be taken of this transformation. First of all, dC = 0in the mid position, which fully decouples the two equations. Also in non mid position,the term d� � d�l is small or even zero which makes the second equation not only badlyobservable but also almost uncontrollable. So, in most cases a sufficiently accurate modelcan be obtained by only taking into account the structure given by

_dP = 2Cm(q)(�m � LlmdP �Ap _q); (3.6)

assuming a laminar leakage flow proportional to the pressure difference, � lm = LlmdP .Equation (3.6) provides one of the two main equations to describe the basic hydraulicallydriven mechanical system as given in the block scheme of Fig. 3.2.

Removing the dynamics of Pm not only simplifies the model but also reduces problems(e.g. numerical) in model based control of a hydraulic actuator due to the minor effect on


Fig. 3.2: Basic block scheme of a hydraulic actuator connected to a mechanical system.

the input/ output relation of this part of the dynamics. This model reduction step is oftenperformed without consistently showing what is to be neglected i.e. not considering thechange of coordinates.

The first order differential equation is nonlinear due to the position dependence of theoil stiffness, using (3.3) and (3.4),

Cm(q) =EVm

V 2m �A2

pq2=

E

Vm

�1

1�A2pq

2=V 2m

�; (3.7)

which attains the minimum at the mid position (q = 0).

3.1.1 Leakage

The leakage flows are considered laminar, which results in flows lineary dependent on thepressure drops. Assuming two reference pressures: Ps as the higher level supply pressureand Pt as the lower level tank pressure (which can chosen to be the 0-level), the leakageflows can depend on five different parameters, L i, given in the following equations.

�l1 = L1t(Po1 � Pt)� L1s(Ps � Po1) + Ln(Po1 � Po2) (3.8)

�l2 = �L2t(Po2 � Pt) + L2s(Ps � Po2) + Ln(Po1 � Po2) (3.9)

Not all terms will appear in every construction. E.g. in the double concentric actuator usedfor the SRS, schematically given in Fig. 3.3, the terms Ln and L1s do not exist.

Also in this case it is appropriate to consider a change in coordinates

�Ltm LsmdLt dLs

�=

�12

12

1 �1� �

L1t L1s

L2t L2s

�(3.10)

3.1 The basic structure of hydraulic actuators 79

The leakage can now be described as function of the new coordinates��lmd�l

�=

�Ln + (Lsm + Ltm)=2 dLt + dLs

dLt + dLs 4(Ltm + Lsm)

� �dP

Pm

�

+

� �dLs �dLt�4Lsm �4Ltm

� �PsPt

�(3.11)

It can be analyzed at what value the mean pressure, Pm, will stabilize, using these equationstogether with (3.5) with _dP = _Pm = 0.

In dropping Pm, only the main leakage parameter, L lm, influencing the resulting basichydraulic equation (3.6), is relevant. It can be constructed as

Llm = Ln + (Lsm + Ltm)=2 (3.12)

3.1.2 Basic hydraulically driven mechanical system model

In Fig. 3.2, a 2-port block scheme of (3.6) is given in connection with a mechanical system.This mechanical system with mass, M , viscous/ damping force constant, Bp, and possiblysome additional disturbance forces, Fext, can be described by the following equation

M �q = fa �Bp _q � Fext (3.13)

As all the parameters given are strictly positive, the hydraulic system can viewed as a ve-locity to force feedback with a first order system. The loop gain over the two integrators,

which is usually dominating, determines the undamped eigenfrequencyq2CmA2

pM�1.

Both leakage, Llm, and viscous friction/ damping,Bp will dissipate energy.It is not advisable to model the physical hydraulic actuator as a fully separate module.

E.g. some parts of the hydraulic system to be modelled like the mass of the piston/ cylin-der and the viscous friction should be made part of the mechanical system model to beconnected.

In case of the multivariable flight simulator, basically Fig. 3.2 holds, considering vectorpaths and matrices. To include platform coordinates, a mapping using the jacobian has tobe applied. The actuator forces fa = ApdP will effect the platform coordinates throughthe transpose jacobian JTl;x and the platform velocities will have to be projected along the

actuators by using _�q = _�l = Jl;x _�x. The undamped eigenfrequencies of the hydraulicallydriven mechanical system can be predicted by calculating the square roots of the eigenvaluesof the loop matrix gain,

p�(2CmA

2pJl;xM

�1JTl;x), assuming equal Ap and Cm for allactuators.

If this is not the case one should consider taking matrices with the specific actuatorvalues for Ap and Cm on the diagonal. All these variables, except for Ap, depend on thesimulator pose. These mass and oil stiffness dependent eigenvalues will be shown to reflectthe most prominent part of the dynamics of a hydraulically driven flight simulator motionsystem.

The structure given in Fig. 3.1 is a very rough simplification of an actual hydraulicactuator although the model derived will describe the main dynamical effects which can be


Fig. 3.3: Schematic drawing double concentric symmetric hydraulic actuator.

observed in practice. However, the way the controlled oil flow, �m (d�), behaves, has notyet been given and should also be studied to arrive at an appropriate model. This requiresthe modelling of the servo valve and transmission lines as will be discussed in the nextsection.

3.2 Extensions

In Fig. 3.3, a schematic drawing is given of the so-called double concentric symmetrichydraulic actuator, which is used in case of the SRS. This construction enables a symmet-ric actuator without the disadvantage of a piston at both sides. The arrows are drawn inthe direction where the flows and velocity take a positive sign. Note that e.g. the leakageflow, �l2�, is always negative. Next to the cylinder, the picture shows two other parts of ahydraulic servo actuator, which haven’t been discussed yet, the servo valve and the trans-mission lines.

3.2.1 Servo valve

The input flows, �i1 and �i2, given in Fig. 3.3 are determined by the servo valve. Modellingthe relevant dynamics between our control input signal (voltage), u, to the servo valve andthese oil flows is the subject of this section. The valve consists of an pilot valve and mainspool as given in Fig. 3.4. The main spool is controlled electronically using a control schemegiven in Fig. 3.5.

With the external connection to two reference oil compartments to the servo valve, theoil flow can be controlled by the valve opening. One external reference is the hydraulicpump supply pressure, Ps, which is 160Bar higher than the second reference in case of theSRS. This second reference, the tank pressure, Pt, can be considered zero. Both referencesare assumed to be constant in the sequel.

3.2 Extensions 81

Fig. 3.4: Schematic drawing of a three stage valve. The main spool position, Xm, is mea-sured electronically and controlled by feedback to the flapper positioning current,ia. The flapper/ nozzle mechanism internally stabilizes the positioning of the pilotvalve with position, Xs, with oil flows �ni. This is used to enable large oil flowsat the main spool, which is positioned with oil flows �mi. For a valve model,describing the most relevant high frequent dynamics as given in (3.17), only alimited number of variables given in the figure have to be taken into account.


Fig. 3.5: Valve main spool control scheme

Considering a turbulent oil flow through the servo valve, the main behaviour of thisflow can be described by a nonlinear algebraic relation depending on spool position and thesquare root of the pressure drop. In the higher frequency area (> 100Hz in case of the SRSservo valves), the dynamics of the spool positioning mechanism have to be considered too.

The spool is an important part of the servo valve. By positioning the spool, the valveopening can be set. The valve opening and pressure drop over the valve determine theoil flow through the valve to the actuator cylinder compartment itself. If large oil flows arerequired (typically larger than 100 l=min), the spool positioning itself requires an additionalhydraulic circuit, in itself an extra servo actuator controlled by a pilot spool. This multi-stage concept is applied in the SRS as flows up to 150 l=min are needed.

A schematic drawing of the valve is given in Fig. 3.4. The pilot spool (with positionxs), is positioned by an internal hydromechanical feedback mechanism called the flapper/nozzle. As already noted, the positioning of the main spool is performed by an electronicfeedback given in Fig. 3.5. This feedback uses the main spool position xm, or positioningerror �m to the current ia which forces the flapper position, xf , aside.

Over a broad frequency area, usually including the area of interest in positioning and/or forcing a mechanical system (servo-valve bandwidth in case of the SRS is 150 Hz), thetransfer function from desired to actual spool position, xm, can be considered a static unitygain.

So the main servo valve phenomena can be captured by considering the turbulent flowequations of the main four-way valve. With symmetric hydraulic actuators, one usuallyapplies symmetric 4-way valves. The main spool position, xm, together with the pressuredrops determine the flows, �i1 and �i2, to and from the actuator chambers from Ps and toPt, as is drawn in Fig. 3.4.

The equation describing a turbulent flow restriction is given by Merrit [99]

� = CdAm(x)

s2(Pin � Pout)

�(3.14)

In this case the discharge coefficient, Cd, and the oil density, �, can be considered constant.The geometric properties, Am(x) of the flow restriction depend (almost linearly) on thespool position. The flow gain evaluated w.r.t. the spool position varies with the square rootof the pressure drop and, moreover, this results in a hard non-linear change in gain if theflow changes sign and Pi1 or Pi2 are not both equal to 1=2Ps.

3.2 Extensions 83

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01−0.5

0

0.5

1

1.5

2Step response pilot valve

time (s)

x_s,

P_n

, f_p

, f_v

x_s

P_n

f_v

Fig. 3.6: A state step response of the model (3.17) of the pilot valve with the identifiedparameters given in Table 3.1.

As observed earlier, a more compact description can be arrived at at the actuator side inapplying a change of coordinates, using difference and mean flows and pressures. This isalso the case with the valve. With the important additional assumptions that

Pi1 + Pi2 = Ps; (3.15)

and unaltered geometric propertiesA(xm), and considering normalised pressures and flows,P = P=Ps, xm = xm=xx;max, � = �=�max, the maximum flow can be found by calculat-ing or measuring �max := f(xm; dPi;nom) = f(xm;max; 0). Now, one equation describesthe oil flow through the four-way valve

�im = f2(xm)

q1� sgn(xm)dPi; (3.16)

where the mean normalised valve flow is defined by �im = (�i1+�i2)=2. Again note that,due to the sign convention taken, this mean valve flow, � im, means the average amount ofoil going into transmission line 1 and out of transmission line 2.

With this equation, as proposed in [124] by Van Schorhorst, static geometric nonlinear-ities can be captured by the function f2(xm), which have been identified experimentally forall the six SRS servo valves. The sign-function (sgn) accounts for the switch in ports atthe four-way valve. Note that a reduction step by omitting one state in taking away meanpressure and difference flow at the actuator is also motivated at the input, the valve, sincethese variables are not involved in (3.16).

The main limitation in only using (3.16) in describing a valve lies in the high frequencyarea, as the spool position, xm, can not be positioned infinitely fast. The positioning mech-anism, as depicted in Fig. 3.4, can be modelled following Van Schothorst [124] by using the


variable value variable valuec1 4:29e3 c6 1e4c2 2:96e7 c7 9:93e-1c3 2:3e10 c8 4e3c4 1:13e11 c9 2:68e10c5 1e2 c10 1:27e2Kp 8 Kd 4:9e-4

Table 3.1: The Simona three stage valve parameters identified by Van Schothorst [124]

state space equation with normalized states given for the pilot valve:26664

�xf_xf_xs

d_Pn

37775 =

2664�c1 �c2 �c3 �c41 0 0 00 �1 �c5 c60 c7 1 �c8

37752664

_xfxfxsdPn

3775+

2664c9000

3775 ia (3.17)

All the coefficients, c1:::9 are positive and the numerical values, identified on the actualactuators of the SRS by Van Schothorst [124], can be found in Table 3.1. The model takesinto account the force balance and acceleration of the flapper, x f , the limited stiffness ofthe oil at Pn and a reduced force balance (velocity only) of the pilot spool, x s. The modelincorporates a right half plane zero as the flapper forces the pilot spool to move in negativedirection, which can be compensated only after pressure build up has taken place.

Further, the dynamics of the SRS pilot valve has four poles, among which is one lightlydamped resonance, at ca. 900 Hz and one pole at ca. 150 Hz. The step response of the pilotvalve is given in Fig. 3.6. Next to each other the flapper velocity (f v), position, xf , andpressure difference peak, Pn, in resonating at 880 Hz. The pilot spool moves slightly nonminimum phase and resonating along a first order response in approx. 6 ms to its final value.In this simulation the four states have been scaled to achieve a numerically more favourabledescription. xsc = diag(1e� 7; 1e� 3; 1; 10): The pilot spool lets oil flow (�m) to movethe main spool. Considering a dPm � 0, this can be modelled by a additional equation forthe third stage adding an integrator.

_xm = c10f1(xs) (3.18)

Also the quasi static functions f1(xs) and constants c10, have all been experimentally iden-tified [124].

As already noted, the position of the main spool is controlled with an electronic PD-controller (Fig. 3.5) with proportional (Kp) plus derivative (Kd) feedback given by

ia = Kp�xm �Kd_xm = Kp(u� xm)�Kdc10f1(xs) (3.19)

where u is the input signal sent to the valve.The additional state of this system in feedback results in a slightly resonating behaviour

(� = 0:3) at 150 Hz as opposed to the mainly first order like response of the pilot valve.

3.2 Extensions 85

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

x_m

x_s

P_n

f_v

Step response three stage valve

time (s)

x_m

, x_s

, P_n

, f_p

, f_v

Fig. 3.7: Model state step response of the three stage servo valve.

As can be seen from the state step response of the valve in Fig. 3.7, the high frequencybehaviour can hardly be noted in the input/ output relation. Up till 200 Hz, the valve canalso well be approximated by a second order system (e.g. via balanced reduction), whichwill lack physical interpretation, however. Taking a bode plot of the linearised version ofthe valve shows the close correspondence as shown in Fig. 3.8.

In hydraulic servo actuators, the valve characteristics are often limiting the achievableperformance of the system. In this case, the high bandwidth valves still introduce a phaselag of approx. 10� at 25 Hz and double this at 50 Hz. The combination with the longtransmission lines, which will be described next, form a risk to stability.

3.2.2 Transmission lines

A disadvantage of requiring long stroke actuators is the unavoidable use of relatively longtransmission lines. Transmission lines often show a lightly damped resonating behaviourand if the transmission lines are longer, the eigenfrequencies tend to be lower which is adisadvantage in striving towards a smooth high bandwidth control by the servo valve of theflow to and from the actuator. Moreover, if actuator pressure control (dP o) is intended tobe applied measuring the pressure, dPi, at the valve, the transmission line dynamics are inbetween. It will be shown that the pressure at the other side of a long transmission line, theoutput pressure, Po1;2, of the actuator, can deviate considerably from the measured inputpressure, Pi1;2, at frequencies ten times lower than the transmission line eigenfrequencies.

The transmission line dynamics can often be neglected if properly designed and withmoderate performance requirements. In [48, 146] practical examples can be found of hy-draulically driven mechanical systems (Brick laying robot, RRR-robot) in which the trans-mission lines can be omitted in modelling and control. In that case, the bandwidth of theservo-valve can be made reasonably lower than the lowest resonance resulting from the


101

102

103

10−2

10−1

100

101

Bodeplot of full and reduced (2nd order) model valve

Am

plitu

de

101

102

103

−400

−300

−200

−100

0

Pha

se (

deg)

Frequency (Hz)

Fig. 3.8: Bode plot of 5th-order linearised servo valve model and 2nd-order reduced ver-sion.

transmission lines. With [153], the lowest transmission line eigenfrequency,! t can roughlybe approximated by

Lt

co!t =

�

2(3.20)

With line length, Lt = 1:4m in case of the SRS and the wave propagation velocity, cogiven by

co =

sE

�=

r109

850� 1000; (3.21)

where E is the bulk modulus of the oil and � the density, this results in an estimated eigen-frequency of 180Hz.

In [124] it has been shown that these dynamics can not be neglected if one wants to applyhigh performance motion control on long stroke hydraulic actuators typically used in flightsimulation. The lightly damped resonances of the transmission lines together with the phaselag introduced by the limited bandwidth of the servo valve easily result in stability problems.By taking these phenomena into account in modelling, proper model based control can beapplied.

The main steps in modelling the transmission lines will now be described, following VanSchothorst [124] and Yang and Tobler [158]. Details can be found in these references. Themain conclusion will be that transmission lines can be modelled as a (theoretically infinite)set of parallel connected linear second order systems. Each second order system appoints a

3.2 Extensions 87

φo

oP

φi

Pi

Fig. 3.9: A causal two-port configuration for a transmission line.

specific vibrational mode with increasing eigenfrequency and, in approximating, all modesof interest are easily picked.

In choosing the input flow and output pressure as input and the input pressure and out-put flow as output, a proper causal solution will result as discussed by Van Schothorst in[124]. This two-port representation of the transmission line is given in Fig. 3.9. The exactcausal solution to a one dimensional distributed parameter model of a uniform rigid fluidtransmission line with laminar flow in the Laplace domain, is given by Yang and Tobler in[158] by�

Pi(s)�o(s)

�=

1

cosh(�(s))

�1 �Zc(s) sinh(�(s))

� sinh(�(s))=Zc(s) 1

��Po(s)�i(s)

�(3.22)

with the characteristic impedance, Zc(s), and wave propagation operator, �(s), of a singletransmission line, as in [158]. One can approximate this system by a finite dimensional statespace model by using the first n terms of an exact infinite modal sequence, which can bederived from (3.22).

This modal approximation technique leads to�Pi(s)�o(s)

�= �nk=1

�Pik(s)�ok(s)

�(3.23)

= �nk=1

1

s2 + cks+ dk

�acks+ bck �azks+ bzkask � bsk acks+ bck

��Po(s)�i(s)

�

Considering each second order system, an even more compact state space representationcan be derived [158], requiring only four parameters for the first mode and two additional(due to parameter dependencies) for each higher harmonic to be taken into account.

Each harmonic can then separately be described by�_Pik_�ok

�=

�0 (�1)k+1�1k

(�1)k�2k ��3k

��Pbk�ak

�+

�0 �4k

� �2k�1k�4k 0

��Pok�ik

�:

(3.24)

These parameters still have a clear physical interpretation, as given in [124]. A block schemerepresenting this model is given in Fig. 3.10 with for k = 1, t j = �j j = 1; : : : ; 4; t5 =��4�2=�1. Only one mode is given in this block scheme, as the structure of the others is


Fig. 3.10: Basic block scheme of the parallel modal transmission line model. Note that eachadditional mode, i, can be accounted for by an equivalent scheme connected inparallel through the sigma signs.

equivalent. The sigma-terms show the connection of these additional harmonics, which canbe taken into account by a parallel connection of similar block schemes.

To maintain approximate steady state gain for n = 1, which will be lost due to simpleneglection of all higher order terms, �4 can be set to ��1 if pressure and flow have beennormalised. The term

p�1�2 is equal to the undamped eigenfrequency of each mode and

�3 is the dissipating term, damping the resonance. In the hydraulic actuators of the SRSthe lowest eigenfrequencies of the transmission lines could be found in the range of 160�220 Hz and the first higher harmonic at a factor three higher. Given the bandwidth of thevalve of 150Hz, in this research each transmission line was modelled by taking only onemode into account

Apart from the resonating behaviour, the transmission lines result in a difference be-tween the pressure measured over the valve, dP i and the pressure dPo, over the actuatorchambers, which actually drive the mechanical system. This difference can become a prob-lem if one wants to apply a well defined force to accelerate with high precision. As will beshown, these differences occur at much lower frequencies than those where the transmissionline resonances occur.

In [124], it was argued that the state describing the mean pressure in the hydraulic ac-tuator, Pm, should be retained if transmission lines are to be taken into account and themodel is to be used over the whole stroke including the position dependent stiffness of thehydraulic actuator. But in applying model based control, it is unfavourable to include analmost uncontrollable and unobservable state. Therefore it is proposed to use an alternativemodel.

There are two separate transmission lines as shown in Fig. 3.3. The model of transmis-sion line 1 has inputs �i1 and Po1 and outputs Pi1 and �o1. The model of transmission line2 has inputs �i2 and Po2 and outputs Pi2 and �o2. The transmission line inputs Po1 andPo2 are constructed from the reduced hydraulic actuator model, (3.6), by

Po1 = 1=2(Ps + dPo)

Po2 = 1=2(Ps � dPo); (3.25)

3.3 Integrated system 89

which is a similar assumption as given in (3.15). Further, the other connection of this twosided coupling between the transmission lines and the hydraulic actuator is given by

�m = (�o1 +�o2)=2 (3.26)

At the other side of the transmission lines, the servo valve is connected. The models of thetwo can be merged by retaining (3.16) and using � i1 = �i2 = �im and dPi = Pi1 � Pi2.

By evaluating the differences between the reduced and full order model after connectingthe building blocks, this step can and will be validated. Integrating the set of submodelsintroduced for the mechanical and hydraulical part of a flight simulator motion system willbe the subject of the next section.

In this section two relevant extensions, the valve characteristics and transmission linedynamics, to the basic hydraulically driven mechanical system model were introduced. Ofcourse many more effects could be investigated. One, which should be mentioned here,but will not be discussed much further, are the hydraulic actuator cushioning areas at bothends of the stroke. By gradually disabling oil flow to and from the actuator, the cushioningprevents from high accelerations (max. 2g) occurring if an actuator for some reason runs outof stroke. This mechanism, however, also influences the system dynamics for at least thefirst and last 0:15m stroke of all the actuators. Design and evaluation of the SRS cushioningis discussed in [124].

3.3 Integrated system

Having introduced models of the mechanical part of a flight simulator motion system inChapter 2 and of the hydraulic servo actuators which drive the system in Section 3.1 andSection 3.2, the two can be connected. This results in a nonlinear model describing thehydraulically driven mechanical system.

An important system analytic tool, passivity analysis, will be used to derive some inter-esting properties of these non-linear models. E.g. it will be shown that the full multivariablenonlinear system model is passive from input oil flows to input valve pressure differencesirrespective of parasitic effects like transmission line resonances and/ or flexible mechanicalmodes. As a result proportional (or either other passive) input pressure feedback to valveoil flow can be applied without destabilising the system, as is of course already long knownheuristically.

In the sequel, the system characteristics due to the hydraulic system are best clarified byfirst considering a one degree-of-freedom actuator driving a single mass. Also the effect ofa parasitic mass or a non rigidly connected foundation is best understood by using such asimplified model first. The complexity of the fully non-linear six degree-of-freedom Stewartplatform model driven by its hydraulic actuators is considerable since it easily includes overfifty states. Most of the system properties seen in practise are, however, well understoodby using these kinds of models because the physical structure of the whole system has beenmaintained.


3.3.1 Passive input/output pairs in flight simulator motion systems

Passivity is a strong input/ output systems property, which can be derived for a large classof systems also inhibiting nonlinearities. It roughly tells that a system does not generateenergy by itself. From this property, stability is often easily derived, also in combinationwith feedback.

There exist a large amount of control strategies based on the passivity property e.g.found in [11, 66, 128, 147]. Mathematically it is defined here following Van der Schaft[147]. The inner product for the vector signals, �f and �g in the Ln2 -space, is taken as

< �f; �g >=

Z1

0

�ni=1fi(t)gi(t)dt: (3.27)

Definition 3.1 Let the causal input/ output system G map Ln2 ! Ln2 (thus stable). Then Gis passive if there exists some constant � such that

< G(�u); �u > � �; 8 �u 2 Ln2 : (3.28)

By introducing the extended L2e-space, the definition can be generalised to a larger sig-nal set and noncausal/ nonstable systems. A causal or nonanticipating system means thatG(�u)T = G(�uT ) 8 �u 2 Ln2e where the subscript T gives the truncation at time T . In theextended space, integration of (3.27) is required only up till time T.

Starting with small subsystems from which passivity can be derived, complex systempassivity can be evaluated by using the fact that feedback of a passive system with a passivesystem results in a over all passive system and two parallel passive systems together formone passive system. This does not hold for passive systems in a series connection.

Consider the feedback system of Fig. 3.11. With the assumption that (�e 1; �e2) 2 Ln2 !(�u1; �u2) 2 Ln2 and passive G1 and G2 i.e. < �yi; �ui > � �i for i = 1; 2, the feedbacksystem with inputs (�e1; �e2) and outputs (�y1; �y2) is also passive since

< �y1; �e1 > + < �y2; �e2 > = < �y1; �e1 + �y2 > + < �y2; �e2 � �y1 >=

< �y1; �u1 > + < �y2; �u2 > � �1 + �2 8�u1;2 2 Ln2 (3.29)

Notice that since this holds for a multivariable system, any input/ output combination, forwhich a system is passive, can be fed back by a passive system without influencing thepassivity of some passive input/ output combination somewhere else. Passivity of a parallelconnection of some passiveG1 andG2 with output �y = �y1+ �y2 and �u = �u1 = �u2 is provenby

< �y; �u > = < �y1; �u1 > + < �y2; �u2 > � �1 + �2 8 �u 2 Ln2 (3.30)

Passivity can be proven easily for some basic system structures. For the time varyingpositive gain ’system’ described by y(t) = k(t)u(t) with k(t) � 0, it can be proven byZ

1

0

y(t)u(t)dt =

Z1

0

u(t)k(t)u(t)dt � 0 8 u 2 L2 (3.31)


G1

G2

-- d

d� ?�

-

�

6

�e2

�y1�e1

�y2

�u1

�u2 +

+

+-

Fig. 3.11: A general feedback structure

For the stable linear first order system described by _y = (u�ky) with k � 0, or u = _y+ky,it follows fromZ

1

0

y(t)u(t)dt =

Z1

0

y(t)ky(t) +1

2

Z1

0

d

dty2(t)dt > �1

2y2(0) 8 u 2 L2 (3.32)

For linear systems passivity means that the real part of the transfer function in the frequencydomain will be positive. For passive linear single input single output systems the phase willthus vary between��=2.

The collocated input torque - output velocity pairs of a mechanical system, which can bedescribed in hamiltonian form, are passive for every initial condition. From the definition ofthe Hamiltonian (2.84) and assuming that the potential energy function is chosen as P(�z) �0 follows H � 0. With the mechanical system described by the generalised coordinates,�z, consider an output mapping �y = J(�z) _�z, and the input, �u, defined by the dual map,�� = JT (�z)�u. Now the system from input �u to output �y will shown to be passive. Consider(2.87), then

< �y; �u >T=< _�z; �� >T =

Z T

0

d

dtH( �m(t); �z(t))dt =

H( �m(T ); �z(T ))�H( �m(0); �z(0)) > �H( �m(0); �z(0)) 8 �u 2 L2e (3.33)

Take �y = _�q, �u = �fa, �z = �sx and J(�z) = Jl;sx( �sx) and this establishes passivity of therigid body Stewart platform (2.122) for the system considered from input actuator force,�fa to output actuator velocity, _�q. Moreover, it tells that passivity will be preserved in caseof connecting a large class of parasitic mass- spring- damper systems connected e.g. a nonrigidly attached foundation.

Having established passivity of the mechanical system, the system can be extended tothe hydraulics retaining the passivity property.

-Viscous friction/ damping Every Rayleigh function, R satisfying _�zT(@R)=(@ _�z)( _�z) � 0

is a passive system. Rayleigh functions often occur as dissipative feedback over pas-sive mechanical (flow) systems retaining passivity. A simple example of a Rayleighfunction is the viscous friction, bi _�qi in each of the actuators:

Rvact( _�x; �sx) = JTl;x( �sx)diag(bi _qi) = JTl;x( �sx)diag(bi)Jl;x( �sx) _�x: (3.34)


100

101

102

103

10−2

10−1

100

101

102

Bodeplot Phi_im−>dP_i,o

Am

plitu

de

dPi

dPodPo−dPi

100

101

102

103

−300

−200

−100

0

100

dPi

dPodPo−dPi

Frequency (Hz)

Pha

se (

deg)

Fig. 3.12: Bode plot of 7th-order linearised hydraulically driven mechanical system modelincluding transmission lines. Input valve flow, � im, outputs, dPi (full), dPo(dashed), dPo � dP i (dotted).

Equivalently damping induced through resonances can also be included without de-stroying passivity.

-The hydraulic actuator The model of the hydraulic part of an actuator, given in (3.6)can be considered as a stable first order system with a varying gain Cm(q). SinceCm(q) > 0 each linearisation of the hydraulics described by (3.6) results in a passivesystem for every position �q. Further, as shown as a part in the block scheme ofFig. 3.2, it is connected to the mechanical system as a negative feedback over a passivesystem. So each linearised hydraulic actuator connected to the nonlinear mechanicalsystem preserves passivity. The passive input/ output system of the actuator can bechosen from �m to dPo respectively. With the varying gain, the situation is somewhatdifferent:

< �m; dPo > =

Z1

0

d

dt

�dPo(t)

�1

4Cm(q(t)

�dPo(t)

�dt

+

Z1

0

dPo(t)

�Llm � d

dt

�1

4Cm(q(t))

��dPo(t)dt (3.35)

Due to the time derivative of the inverse actuator stiffness, this system is not alwayspassive. With an upper bound on the actuator velocity of e.g. 1 m=s and with apolynomial fit Cm(q) = b2q

2 + b1q + b0 = 99:2q2 � 9:8q + 104 [124], a leakageterm of 0.002 (0.01 is normal) will passify this system.

-The transmission lines The structure of the transmission lines models in the block schemegiven in Fig. 3.10 is clearly a parallel connection of passive systems which can be con-sidered as two stable linear first order systems in feedback. The actuator side of the


10−2

100

102

−100

0

100

10−2

100

102

−100

0

100

100

101

102

103

10−2

100

102

Frequency (Hz)10

010

110

210

3−100

0

100

Frequency (Hz)

Fig. 3.13: Bode plots from input valve flow (� im) to pressure difference (dPi) of linearisedhydraulically driven mechanical system models including transmission lines inupper/ mid/ lower position (q = :47; 0;�:47m) for full (full), reduced (n = 6,dashed) order models and difference (dotted).

port feeds back over the passive input/ output pair dPo and �m respectively and atthe valve side the input/ output pair is given by � im and dPi, considering the paralleltransmission lines over the pairs Pi1 and �i1 and Pi2 and �i2.

-The valve port dynamics The valve port dynamics modelled as turbulent flow port canbe considered as a positive time varying gain. If j dP i j� 1,

�im(t) = xm(t)p1� sgn(xm(t))dPi(t) = k(t)xm; (3.36)

implies k(t) � 0. This is a passive system which preserves passivity if used in a feed-back connection with e.g. constant gain from dP i to xm. As shown in Section 3.2.1,the valve spool positioning system cannot be considered a constant gain over an in-finitely large frequency area but inhibits dynamics which is clearly not passive frominput voltage (u) to output spool position (xm). At this point passivity based controlstrategies may cause possible (stability) problems.

Summarising, it has been established that the hydraulically driven nonlinear mechanicalsystem is passive with respect to a large number of input/ output pairs. Under moderateconditions, which are typically satisfied in practice, constant feedback of the pair (valvespool position, pressure difference measured at the valve) leaves the passivity properties ofthe nonlinear system unaltered. In fact, this kind of feedback will dissipate energy, whichdrives the system with badly damped resonances faster to its stable equilibrium.


Variable Description Value Unit�n Max. valve flow 25e�4 [m3=s]Ps Supply Pressure 160e5 [N=m2]Fmax = Ps �Ap Max. actuator force 40e3 [N ]_qmax = �n=Ap Max. velocity 1 [m=s]

Normalised Operators Description Value UnitAn = Ap _qmax=�n Piston area 1 [�]C1 = E�n=(V1Ps) Stiffness chamber 1 101 [1=s]C2 = E�n=(V2Ps) Stiffness chamber 2 104 [1=s]L1 = L1tPs=�n Leakage 1 4:32e�2 [�]L2 = (L2s + L2t)Ps=�n Leakage 2 5:06e�2 [�]Wpn = wp=Fmax Visc. friction 3:5e�4 [�]M�1n = Fmax=(Mp _qmax) Inv. Mass 10 [1/s]

Table 3.2: Parameters taken for a 1-d.o.f. hydraulic mechanical system. First the four nor-malising quantities are given. Then all the normalised values for of the operatorsare specified.

Variable Value Variable Valuet11 = 1737 t12 = 1254t21 = 650 t22 = 1184t31 = 30 t32 = 54

Table 3.3: Parameters taken for the two transmission lines.

3.3.2 The 1-d.o.f. hydraulically driven mechanical system

Further insight into the characteristics of the hydraulic mechanical models, derived in theprevious sections, can be obtained by filling in realistic values into the parameters, e.g. SRSdesign numbers, and performing an evaluation in the frequency (linearised models) andtime domain. First a one-degree-of-freedom hydraulically driven mechanical system withtransmission line dynamics will be considered. For these models, the difference in thedynamics of the actual driving pressure in the actuator and the measured pressure over thevalve will be evaluated. Further, the implications of the reduction step proposed by taking(3.6) instead of (3.2), assuming (3.25) and (3.26), can be quantified.

It is important to normalise the variables such as pressures (typically 107 in SI-units)and flows (typical values of 10�3 in SI) when building numerical models which containhydraulics. In Table 3.2, first the normalising values of these variables are given. Then thenormalised values of the operators taken for the models consisting of (3.13) together with(3.6) or (3.2) are specified at neutral position (at full stroke q = 0:625). With the mass takenof 4000kg and the typical values of the hydraulic actuators of the SRS, an undamped rigid


body mode eigenfrequency, frm of approx. 7Hz results.

frm =1

2�

r2CmMn

=1

2�

r2 � 4:1 � 106

4000=

1

2�

p(101 + 104)10 = 7Hz (3.37)

This corresponds with the lowest eigenfrequencies found for the SRS. Due to the normal-ising velocity of one, the normalised inverse mass can also be interpreted as the maximumacceleration to be attained, which in this case approx. amounts to gravity. The leakage termrelated to the damping in the actuator driving direction amounts to L lm = 2:35%, which isequal to the value experimentally found by Van Schothorst [124]. This is also the case forthe viscous friction term.

The model is completed by connecting the transmission lines using (3.24) and assumingan ideal valve input, �i1 = �i2 = �im. The parameters taken for the transmission linescorrespond to the values derived in [125] adapted with 10 % additional line length to arriveat the eigenfrequencies found in practice. The three resulting parameters, t 1k:::3k, for thetwo lines (k = 1; 2) are given in Table 3.3. Eigenfrequencies of the unconnected trans-mission lines therefore amount to 169 Hz and 194 Hz with damping of � t1 = 0:01 and�t2 = 0:02 respectively.

Influence of the transmission lines on input and output pressure dynamicsBode plots of this system from input �m to the outputs dPi and dPo are given in Fig. 3.12.Connection of the impedance of the hydraulic actuator to the transmission lines shiftedthe transmission line eigenfrequencies approx. 10 Hz higher. This shift of approx. 5%is also observed for the lower (rigid body) mode and in that case to lower values for theeigenfrequency resulting from the load together with the oil spring stiffness. This effectof interaction between the transmission lines and the actuator should not be neglected. Itmeans e.g. that the effective oil stiffness with the SRS-actuators will be 12% lower thanthe theoretically found value in case one identifies this hydraulically driven system with amodel structure not accounting for the transmission lines.

The phase of the passive input/ output pair (� im; dPi) clearly remains within �90�. Itis important to notice that although the pressures have a one-to-one correspondence in thelower frequency area, they start to deviate much earlier than the transmission line eigen-frequencies. This is due to the anti-resonance in the transfer function from � im to dPi atapprox. 60Hz. At 20Hz this amounts to 16% and at 30Hz already accounts for a differ-ence of 25%. Further, the resonating behaviour of the transmission lines is expected to bemuch (10 times) heavier observed in the measurable output, dP i, than it will be felt by themechanical system (through dPo).

Influence of the varying actuator stifness and model reduction stepSince the stiffness of the oil column depends on the actuator position, the eigenvalues, alsothose related to the transmission lines, will shift if the actuator moves. As argued in [124],this shift will be slightly different if the actuator model is reduced to (3.6). However, thereduced model describes the system reasonably well as can be observed in Fig. 3.13. Bodeplots of the system from input, �im, to the measurable output dPi in an upper/ mid andlower position are taken for the reduced and nonreduced models. Also the differences are


100

101

102

103

10−2

10−1

100

101

102

Bodeplot u−>dPi,dPo

Am

plitu

de

100

101

102

103

−400

−200

0

Frequency (Hz)

Pha

se (

deg)

Fig. 3.14: Bode plot of 8th-order linearised hydraulically driven mechanical system modelincluding transmission line and valve dynamics. Input valve input voltage, u,outputs dP i (full), dPo (dashed).

shown. The upper and lower positions are just within the operational area where the cush-ioning does not influence the actual system. Since the eigenvalues are relatively lightlydamped, very small differences in the eigenvalue shift locally result in larger deviations butthe general behaviour is described fairly well by the reduced order model. Both for theupper as the lower position the anti-resonance between the hydromechanical mode at ap-prox. 7Hz and the first transmission line related mode moves up from 60Hz to maximally90Hz.

Influence of the valve dynamicsThe model can be extended by adding the reduced linear second order model of the threestage valve dynamics derived from (3.19) and (3.17). In Fig. 3.14 the Bode plot of thismodel is given. The main effect of the valve is an additional lag in phase, which becomesprominent for frequencies > 100 Hz. Together with the high gains introduced with thetransmission lines resonances, the valve phase lag forms an important stability risc in feed-back design, which has to be taken into account.

In systems with moderate requirements, the valve bandwidth can be decreased in orderto filter most of the transmission line resonances. But also in that case, the phase lag ofthe valve will be dangerous in some frequency area e.g. in combination with resonancesdue to structural deformation. In the lower frequency region, the nonlinearities of the valve,e.g. due to (3.16), will form the most important factor.

In the time domain, the step response of this model is given in Fig. 3.15 on a 200 msrange. The steady state response to the input and output pressures is neglectable. Thismeans it is not very feasible to induce sustained accelerations with hydraulically driven


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2−4

−3

−2

−1

0

1

2

3

4

5

time (s)

Step response u−>dPi,dPo,dPo−dPi

Fig. 3.15: Step response of 8th-order linearised hydraulically driven mechanical systemmodel including transmission line and valve dynamics. Input valve input volt-age, u, outputs dPi (full), dPo (dashed). dPo � dPi (dash-dot). The high fre-quency resonances result from the transmission lines, the slower vibration withapprox. 0:16s period time is the rigid body mode.

mechanical systems. A quasi static gain of a moving actuator is attained from valve inputvoltage to actuator velocity. Of course in practice, the limited stroke is the main limitingfactor for the low frequency accelerations. In control design for acceleration (simulation),however, the virtually zero gain from valve input to force/ acceleration will be an importantfactor to be taken into account.

In both pressures the 160ms time period (7Hz) of the hydromechanical mode is mostprominent. The difference in response is mainly due to the transmission line resonances towhich the pressure measured over the valve is much more sensitive. Due to the interferenceof the two transmission line modes, the high vibrational amplitude comes up and falls withina 40ms time range.

3.3.3 Additional mechanical modes

The models presented in the previous paragraphs include the dynamics of the hydraulic ac-tuator(s) into the higher frequency areas of approx. 200 Hz. In deriving accurate models of ahydraulically driven mechanical systems over this broad frequency area, one can usually notassume the mechanics to be entirely rigid. In this section, two of such cases are examinedfor a 1-d.o.f. system. First, the possible nonrigidness of a foundation is taken into account.In the second case, the driven system consists of a main mass connected to a parasitic mass/spring/ damper.

A schematic drawing of hydraulically driven mass, ms, connected to a nonrigid foun-dation is given in Fig. 3.16. The foundation with mass, m f is fastened to the ground with a


Fig. 3.16: Schematic drawing of a hydraulic actuator moving a consisting of a single massms in 1-d.o.f. pushing from a non rigidly attached foundation, acting as a mass/spring/ damper system mf ; bf ; cf .

spring of stiffness cf and a damper, bf . The equations of motion become

ms�xs = AdPo (3.38)

mf �xf = �AdPo � bf _xf � cfxf

Although the actual force fa = ApdPo necessary to accelerate the mass, ms does notchange with the nonrigid foundation, the feedback path relating � _q = _x s � _xf changes. Incase of the structure with simple mass given in Fig. 3.2, the feedback path becomes � _q =ApdPo=(mss), assuming no viscous friction. Every pole of the feedback path becomes atransmission zero in the transfer function from input (valve flow) to output pressure, dP o.With the foundation the feedback path changes to

� _q =(ms +mf )s

2 + bfs+ cf

mss(mfs2 + bfs+ cf )ApdPo; (3.39)

considering the additional equation of motion,

mf �xf + bf _xf + cfxf + fa = 0: (3.40)

A complex pair of transmission zeros with the undamped eigenfrequency, ! f =pcf=mf ,

of the foundation alone results. The structure of the poles of this interconnected system issomewhat more complex. The foundation will add two complex poles and the transmissionzeros will always lie in between, since the passivity properties will be preserved. Usually themass of the foundation will be much higher than the mass to be moved. In this case, althoughthe respective eigenfrequencies can be close, there will not be a large shift in connecting themodels of the hydraulically driven mass,ms and the foundation. In Fig. 3.17 the Bode plotsof this model structure with three different parameter values are given. In all cases (1f,2f,3f)the model of the hydraulically driven mass is as specified in the previous paragraphs. Theparameters concerning the foundation are given in Table 3.4. � f is the relative dampinggiven by 2�f cf = !fbf .

In both cases 1f and 2f, in which the mass of the foundation is relatively large (therealistic cases), the nonrigid foundation results in a relatively small distortion of the open


Case mf=ms !f �f1f 10 2�10 0:052f 10 2�4 0:053f 1 2�10 0:05

Table 3.4: Parameters taken for the foundation.

100

101

102

103

10−2

10−1

100

101

102

Bodeplot Phi_im−>dP_i

Am

plitu

de

100

101

102

103

−300

−200

−100

0

100

Frequency (Hz)

Pha

se (

deg)

Fig. 3.17: Bode plots of input valve flow, � im, to pressure difference over the valve, dP i,with 1-d.o.f. hydraulically driven mass connected to a nonrigid foundation. Case1f (full), 2f (dashed), 3f (dash-dotted).

loop characteristics of the original system. The additional resonance and anti-resonance areclearly visible and also the, mostly oil spring stiffness-moveable mass related resonance isseen to be shifted slightly. In the third case, where the mass of the foundation and moveablemass are equal, the interaction between the modes is shown to be much higher. Althoughthis example is highly hypothetical in case of a foundation, these typical effects can occurif one would place two platforms on top of each other like McInroy et al. [93]. In allcases the transfer functions from �im to dPi clearly remain strict positive real i.e. passive.For the high and low frequencies, the response is not sensitive to the characteristics of thefoundation.

The second example of a possible parasitic resonating behaviour is schematically drawnin Fig. 3.18. The construction to be moved itself, e.g. the simulator, is nonrigid. In practicesuch a situation can occur if one connects a large projection screen in front of and heavyprojectors on top of the cockpit of a simulator. In this 1-d.o.f. case the parasitic behaviouris modelled as two masses connected through a spring/ damper. Although this schematicexample will not catch the nonlinear behaviour of a flexible distributed mass with grossspatial movement, some typical, linear parts of the characteristics can be investigated.


Fig. 3.18: Schematic drawing of a hydraulic actuator moving a mass m s in 1-d.o.f., whichis connected to a parasitic mass/ spring/ damper system mp; bp; cp.

Case mp=ms !p �p1p 1 2�10=

p2 0:05=

p2

2p 1 2�10p2 0:05=

p2

3p 3 2�10p2 0:05=

p2

Table 3.5: Parameters taken for the parasitic mass.

The equations of motion for this system become

ms�xs = cp(xp � xs) + bp( _xp � _xs) +ApdPo (3.41)

mp�xf = cp(xs � xp) + bp( _xs � _xp)

In this case the velocity measured at the actuator is equal to the inertial velocity of themass, ms. The transfer function from applied actuator force, f a = ApdPo, to the velocity,� _q = _xs changes to

� _q =mps

2 + bps+ cp

mss(mps2 + rmbps+ rmcp)ApdPo; (3.42)

with the mass ratio, rm = (ms + mp)=ms, since the actuator force is not the only forceon ms as also the spring and damper force are taken into account. Again, the zero at zerofrequency is retained and an additional complex pair is introduced. This time predicted atpcprm=ms. In principle, the structure of (3.39) and (3.42) is equal. As a result, one can not

discriminate between the case of a nonrigid foundation or a nonrigid structure to be moved,given the measurement from valve input to a pressure difference (dP i or dPo) or actuatorvelocity, � _q. To determine what causes the parasitic resonance, additional measurementshave to be taken e.g. from accelerometers attached to the foundation or simulator. Thisidentification problem will be considered in the next chapter.

In Table 3.5 the parameters of three cases are given for the flexible structure of Fig. 3.18.All the parameters of the hydraulic actuator are also taken equal to the ones presented inthe previous section (Table 3.2). The total moveable mass (m s + mp) is taken equal to

3.4 Hydraulically driven Stewart platform 101

100

101

102

103

10−2

10−1

100

101

102

Bodeplot Phi_im−>dP_i

Am

plitu

de

100

101

102

103

−300

−200

−100

0

100

Frequency (Hz)

Pha

se (

deg)

Fig. 3.19: Bode plots of input valve flow, � i, to pressure difference over the valve, dP i, with1-d.o.f. hydraulically driven structure consisting of two masses interconnected byspring and damper. Case 1p (full), 2p (dashed), 3p (dash-dotted).

4000 kg in all cases. The first case (1p) results in exactly the same model parametersas case 3f, considering (3.39) and (3.42). The transfer function from � i to dPi for theseparameters, plotted in Fig. 3.19, is almost the same (the viscous friction of the actuatorresults in very small differences). The other cases shown have somewhat more realisticbehaviour considering a resonant frequency twice as high (2f) and further taking 25% ofthe part of the moving object nonrigidly connected (3f). In reducing this part, the zerosand poles mostly related to the parasitic resonance shift back to lower values towards theoriginal eigenfrequency of the hydraulically driven load since there is less interaction.

The effect of interaction will be similar to the parasitic resonating spring/ damper/masses treated in this section if one measures the flow to pressure/ velocity transfer func-tion of one actuator in case of a hydraulically driven Stewart platform where more than oneactuator is connected to the system. This structure will be treated in the next section.

The parasitic resonance shows a moderate peak in the frequency domain in the moreor less realistic setting of case 3p. In open loop the characteristics of the standard hy-dromechanical mode will dominate. With feedback this will change and will change moredrastically if performance specifications require a bandwidth close to these parasitic effects.This problem will be more closely looked at in Chapter 5 where the closed loop system isvalidated.

3.4 Hydraulically driven Stewart platform

Having considered the hydraulic actuator and its most relevant properties to be taken intoaccount in modelling in a 1-d.o.f. setting and considering the description of the 6-d.o.f. me-


b

b

-L -B

Mt

-1

C

RR

JTl;x

-A

A

Jl;x��6

6

-6- -

?

?

�-?

?

- -

�

6

-� b

b

�fa

_�q _�x��om

�dP o

Fig. 3.20: Basic structure hydraulically driven motion system.

chanical system model of the Stewart platform introduced in Chapter 2, the two can becombined. As interaction will be the most prominent additional effect to be taken into ac-count, the structure of this phenomenon will be analyzed first, using a simplified model. Theinertial effect of the actuators will be treated in the hydraulically driven setting. Finally, thefull model will be considered performing analysis in the frequency domain (through locallinearisation) for several platform poses using numerical design values of the SRS for themodel parameters.

3.4.1 Basic model structure hydraulically driven systems

As a resume, the basic structure of a hydraulically driven parallel motion system is givenin Fig. 3.20. Through the servo valves and transmission lines, oil flows to the actuatorcompartments, ��om, can be regulated. Strong coupling with the mechanics results throughactuator velocities, _�q, requiring a in/decreasing amount of oil in the compartments. Togetherwith a small leakage term, �L, which provides for some damping, the nett flow will resultin in/decreasing pressure rises, _�dP o, through the oil stiffness, C. Actuator pressure timespiston area,A, will force the system to accelerate. With parallel systems the actuator forces,�fa, have to be transformed to platform forces though the transposed jacobian, J T

l;x. Someadditional damping force results from the viscous friction along the actuators, B. The ac-celeration of the actuators can be calculated as the nett sum of forces times the inverse massmatrix,M�1

t , considered along the platform coordinates. Coulomb, coriolis and centripetalforces are not shown in this system and neither are the gravity terms.

The mass matrix as seen from the actuators consists of the simulator pose, �sx, dependentjacobian, Jl;x( �sx) as defined by (2.42), and an almost constant simulator mass matrix, M t,

M�1act = Jl;xM

�1t JTl;x: (3.43)

This matrix determines the coupling between the actuators. Given similar hydraulic actu-ators in equal position, all other matrix operators can be considered as one operator timesthe identity matrix. So, the eigenvectors of the linearised version of this system can be cal-culated from the mass matrix. This still holds if decentralized feedback (diagonal feedbackstructure) is applied. As the mass matrix is symmetric the eigenvector matrix can be made


unitary. This results from the singular value decomposition

M�1act = U�UT (3.44)

With the basic model for the hydraulically driven Stewart platform, it will be shown that thematrix,U , can be used in a state transformation matrix, which decouples the 12 th order sys-tem into 6 second order systems each having the same structure as the 1-d.o.f. hydraulicallydriven mechanical system.

Considering for a moment �dP o and _�q as state1, the state space equations for Fig. 3.20become �

_�dP o��q

�=

� �CL �CAM�1actA �M�1

actB

� ��dP o_�q

�+

�C

0

��om (3.45)

The state transition matrix can be decomposed as� �CL �CAM�1actA �M�1

actB

�=

�U 00 U

��CL �CA�A ��B

��UT 00 UT

�: (3.46)

This can be used to decouple the system using the assumption that all matrices involved,apart from the mass matrix, have a scalar times identity structure,�I . From this observation,a decoupled system results if the system is described w.r.t. a new set of inputs �om =UT �om and outputs dP o = UT �dP o, _q = UT _�q.

If lower captions are used for the diagonal elements of the respective operators, e.g. bfrom B = bI , the six decoupled second order systems can be described by the state spaceequations "

_dP o;i�qi

#=

� �cl �ca�ia ��ib

� �dP o;i_qi

�+

�c

0

��om;i; (3.47)

with i = 1; : : : ; 6. So the system can be described with six second order systems in parallel.Moreover, exciting the system with an oil flow, ��om, directed along one of the columns, �u i,of U will only result in response of the pressure, �dP o, and velocity, _�q along this very samedirection, �ui. Therefore the following definition makes sense.

Definition 3.2 Each column of the matrix, U , is defined as the rigid body modal direc-tion of the hydraulically driven Stewart platform. The mode, eigenfrequency/ damping, ofeach subsystem, (3.47), is defined as a rigid body mode of the hydraulically driven Stewartplatform.

The eigenvalues of the subsystems can be calculated from the respective determinants,

�i = �p�ic(l

2

pc=�i +

b

2

p�i=c� ja); (3.48)

1The most appropriate choice for the state equations in modelling e.g. in numerical simulation, explicit modelbased control, are of course the platform pose, �sx and velocity, _�x as state instead of �q and its derivative and requirevariational actuator stiffness. The model which results from that choice will be given with equation (3.66) inSection 3.4.4


where the product of l and b is assumed to be neglectable. The eigenfrequencies can beconsidered the rigid modes and are mainly determined by the hydraulic stiffness, c, andthe singular values of the inverse mass matrix, � i, taken w.r.t the actuators. These singularvalues can be interpreted as the inverse of the generalised masses, which the actuators willhave to accelerate moving along the six orthogonal directions described by the columns ofthe unitary decoupling matrix, U .

The characterisation with U can already be found by Hoffman [51], who uses this de-coupling to analyse a flight simulator motion system, a Stewart platform, in combinationwith a decentralised feedback. It has not seen to be applied after that time. However, aswill be shown in the sequel, both in model analysis, identification and decoupling control itcan be very appropriate, since the decoupling is seen to hold in practice even in the face ofmoderate additional variations/ dynamics.

The parallel structure makes the calculation of the directions in U simpler. It can becalculated from a ’scaled’ jacobian, Jm without inversion by assuming M�1

t does not varyand can be calculated in advance

Jl;x( �sx)M�1t JTl;x( �sx) = Jm( �sx)J

Tm( �sx)

= U( �sx)�1=2( �sx)V T ( �sx)V ( �sx)�1=2( �sx)UT ( �sx) (3.49)

As shown in the previous sections, the stiffness of the actuators, C, also varies withthe position of the actuator. With different stiffness values, the decoupling with U willnot be exact anymore. However, in many practical cases, as will be analyzed by takingnumerical design values in the next sections and experimentally found parameters in the nextchapter, a strongly decoupled system results calculating U and � only from the platformpose dependent Jacobian and a constant inverted platform mass matrix M t.

In the next section, first, the specific properties of the SRS motion system will be dis-cussed. Then, possible neglection of the variation of M t and appropriate choice of its valuewill be treated.

3.4.2 Dynamical and kinematical properties of the SRS-motion sys-tem; Design aspects

Having introduced the generic formulas to describe the mechanics and hydraulics of a (flightsimulator) motion system, additional insight can be gained by considering a specific designexample of such a system and analyzing the Simona Research Simulator characteristics.The analysis presented in this section can be done before actually building a motion systemand can/ should therefore be taken into account in the process of design. The parameters forthe kinematics of the SRS were already given in Table 2.1.

The inertial properties of the SRS-motion system discussed here will be the values cal-culated in the design of a fully operational flight simulator. In that case the centre of gravity(cog) of the moving body excluding the actuators is predicted 1 m above the upper centregimbal point (cgb). The cog will be taken as the origin of the moving reference frame. Thevector from cog to cgb will thus be given by

�xmcgb =�0 0 1

�T(3.50)


It is not required to take the c.o.g. as the origin but usually specific design values, such asthe mass matrix are more easily interpreted. Further, an important point in simulation, thedesign eye point (DEP) of the pilot is predicted very close to this point, only :2m higher.

�xmDEP =�0 0 �0:207 �T (3.51)

As discussed, flight simulation usually requires a neutral position of the motion system. Thehigh pass wash-out filter characteristics should keep the motion system not to far away fromthis point to prevent the actuators from running out of stroke. The most straight forwardchoice of the neutral position is at half stroke of all the actuators. The moving referenceframe of the SRS is in that case 3:38m above the lower centre gimbal point, which is takenas the origin of the inertial grounded frame (or its nominal position in case of a flexiblyattached foundation).

�xgneutral =�0 0 �3:38 �T (3.52)

Since the motion system will normally be close to this point2 the (dynamic) characteristicsare often specified at this point. The jacobian in neutral position, �sxn, is given by:

Jl;x( �sxn) =

26666664

0:1943 �0:4269 �0:8832 0:3386 1:6047 �0:7011�0:4668 �0:0452 �0:8832 �1:2204 �1:0955 0:70110:2725 0:3817 �0:8832 �1:5589 �0:5091 �0:70110:2725 �0:3817 �0:8832 1:5589 �0:5091 0:7011

�0:4668 0:0452 �0:8832 1:2204 �1:0955 �0:70110:1943 0:4269 �0:8832 �0:3386 1:6047 0:7011

37777775

(3.53)

Each column of this matrix specifies at what velocity the actuator should run in case onlyone of the platform states is to be moved with unit velocity ( _�l = Jl;x _�x). The columns,in this specific pose, are orthogonal except for the pitch/ surge and roll/ sway direction.This strongly influences the hydromechanical modes of the system evaluated along the plat-form directions. The rows of the matrix specify the platform forces felt in case of a unitforce of one of the actuators. These rows are not orthogonal showing strong coupling ofneighbouring actuators and strongly influence the mass matrix evaluated along the actuatorcoordinates.

Recalling the discussion on the jacobian in Section 2.2.9, it is possible to calculate whatcan be the maximum velocity in a certain platform direction given a maximum actuatorvelocity, given the jacobian. If the infinity norm (max) of the actuator velocity vector isgiven, the 1-norm (added absolute values) of the rows of the inverse jacobian specify themaximum attainable platform velocity.

k _�xik1 = kJ�1i;� k1k_�l1 (3.54)

Given a maximal applicable actuator force, a similar formula can be used to calculate themaximal attainable platform force (take 1-norm of each row of the jacobian). In both cases

2Directing the gravitational vector properly often results in a certain off set from neutral as an airplane is usuallyoperated slightly pitched.


10−1

100

101

10−2

10−1

100

101

102

Frequency [Hz]

Acc

eler

atio

n lim

its [m

/s2,

100

deg/

s2]

x,y z m_z m_x,m_y

Fig. 3.21: Approximate accelerational limits for sinusoidal platform motion considered atthe neutral position, along the platfrom directions, surge (x), sway (y), heave (z),roll (mx), pitch (my) and yaw (mz).

it should be noted that the other platform variables are allowed to vary. A more appropriatebound is therefore given by the maximal attainable velocity/ force in a specific direction notmoving/ forcing the others.

For the velocity this bound can be found by

_�xi;max = _�lmax=kJ�;ik1 (3.55)

And for the force

�fxi;max = �fa;max=kJ�1i;� k1 (3.56)

The maximal actuator force and velocity are parameters, which are determined by the spe-cific design of the hydraulic system. The maximum valve flow, �max, together with thehydraulic actuator operational area, Ap, determine the maximum velocity

_qmax = �max=Ap =SRS 1m=s (3.57)

The maximal applicable actuator force can be found by multiplication of the supply pres-sure, Ps, with this area, Ap,

fa;max = PsAp =SRS 40 kN (3.58)

The operational area is clearly a trade-off parameter in design for velocity vs. force given aspecific available amount of energy. There are, however, also other constraints in choosingthis area e.g. radial stiffness, and of course also the stroke of the actuator limits motion.A first estimate of this last limit can be found by using (3.55) from which the jacobian to


Pos. displacement x y z ��x ��y ��z[m; deg] 1.34 1.10 0.74 22.3 20.9 45.2

Neg. displacement x y z ��x ��y ��z[m; deg] -1.05 -1.10 -0.69 -22.3 -30.0 -45.2

Est. displacement (1 it.) x y z ��x ��y ��z[m; deg] 1.34 1.46 0.71 23.1 22.5 52.9

Max. velocities _x _y _z !x !y !z[m=s; deg=s] 2.14 2.34 1.13 36.8 35.7 81.7Max. force fx fy fz mx my mz

[kN,kNm] 47.7 46.3 212.0 159.1 174.1 168.1Max. acceleration �x �y �z _!x _!y _!z[m=s2; kdeg=s2] 13.5 12.9 55.4 1.39 1.24 1.01

Freq. of pos. to vel. constr. x y z ��x ��y ��z[Hz] 0.32 0.39 0.29 0.31 0.32 0.35

Freq. of vel. to acc. constr. x y z ��x ��y ��z[Hz] 0.99 0.87 7.71 5.85 5.56 2.01

Table 3.6: SRS-properties considered at neutral position and c.o.g.

euler parameter velocities for the rotational variation can be calculated. Iteratively using thisformula (Newton-Raphson) and �lmin;max, the least available stroke, a precise limit canbe found very fast as opposed to a bilinear search. This Newton-Raphson iteration usuallyrequires three or four steps, ending up far below measurable error in a well designed system.With a given mass matrix, it is also possible to calculate the maximal attainable acceleration.The specific mass matrix (design values) of the SRS is discussed in the next section. Allthese values calculated for the SRS in the neutral position are specified in Table 3.6.

Given sinusoidal inputs, the velocity and positional constraints limit the maximal attain-able acceleration for lower frequencies. For the 1-d.o.f. hydraulic actuator, Viersma [153]specifies exact performance limit diagrams in the velocity domain. In Fig. 3.21 approximateperformance diagrams for the 6-d.o.f. motion system are given in the platform coordinatesusing the limits specified in Table 3.6. The precision of the approximate velocity and accel-eration constraints for the sinusoidal motion is favourably influenced since the maximumvelocity of the sinusoid is attained at neutral position where the velocity constraint is exactand the maximal acceleration is attained (with limited positional offset at these high fre-quencies) at zero velocity with no centripetal and coriolis accelerations. Gravity, however,results in a more serious offset to be taken into account. Given the force which needs tobe applied to compensate gravity, the least available maximal force required to move in aspecific direction determines the maximal attainable acceleration.

The frequencies where the positional constraint, pmax, and velocity constraint, vmax,meet, fpv, and velocity and accelerational constraint meet, fav, are given by fpv = vmax

2�pmax

and fpv = amax

2�vmax. A first estimate of fpv is equal for all platform directions and does only

depend on the actuator specifications: ~fpv =vact;max

2�pact;max=SRS

12�0:625

= 0:25Hz. From


−1.5 −1 −0.5 0 0.5 1 1.50

0.5

1

1.5

2

2.5

3

3.5

−y [m]

−z

[m]

Lower Gimbal 1 Lower Gimbal 2 (6)

Upper Gimbal

Fig. 3.22: Envelope and largest circle within this envelope of a 2-d.o.f. parallel planar ma-nipulator. Parameters taken as for actuator 1 and 6 of the SRS except for mergingof the upper gimbals.

Fig. 3.21 it can be observed that the actual constraint will be attained at a frequency whichis somewhat higher. It is interesting to observe that already from :3 Hz and higher it is notso much the finite stroke of the actuators which limits the (flight) simulation but the velocityconstraint. For these frequencies it is of no use to build a larger motion system. One shouldfirst choose a larger valve.

Only looking at heave, roll and pitch, the operational area of the actuators could havebeen chosen somewhat smaller as velocity is the main constraint for these directions. Forsurge and sway, however, the force constraint comes in at much lower frequencies. In thisrespect the jacobian (kinematical structure) of the Stewart platform is also a trade-off designparameter where e.g. with the SRS neutral position high surge velocity can be attained(twice as high as the maximal actuator velocity since the actuators are mainly rotating)compromised with limited applicable force in this direction.

The specific values given in Table 3.6 should be treated with extreme care since theycan vary considerably over the motion envelope. Especially the motion envelope itself hasa complex non convex structure. Consider a simplified (planar) system with two actuatorshaving limited stroke manipulating a point. In Fig. 3.22 this structure is drawn with thelower gimbals the same distance (2:5 m) apart as actuator 1 (2,4) and 6 (3,5) in the SRSand the same constraints on the minimal and maximal length (2:08 m,3:33m). The uppergimbals have been merged (are :2m apart in the SRS). Choosing the neutral point somewhatlower in this construction, would lead to a considerable growth of the maximal attainablesway moving straight away from this point. But the constraint towards the attainable heavewould become even more asymmetric in doing so. Thus changing the neutral point in thisway will probably not be a appropriate decision.

Appropriately describing the motion envelope of the general 6-d.o.f. Stewart platform


in task space is still an open problem [27]. The inverse problem of checking whether aspecific task space envelope fits in a specific kinematical structure can be interesting for(flight) simulation and can sometimes be solved. Consider e.g. a normed task space for the2-d.o.f manipulator of Fig. 3.22. With the 2-norm, (

p�xT �x), this space describes a ball and

the maximal radius (norm) of the circle can be found in the half stroke neutral position andis exactly equal to this half stroke (0:625m).

With the general Stewart platform, exactly the same bound can be found if the platformis not allowed to rotate since the upper gimbals in that case are also only allowed to move inthis same ball. So the maximal ball of translational motion of the SRS which is applicablecan be found in the all actuator half stroke neutral position and has radius r = 0:625m. Forany ball larger than this, translations can be found which run (some of) the actuators out ofstroke.

In case of orientational variations together with translation, a radius can be found whichsatisfies the constraints but is not necessarily maximal. Recall the motion of any uppergimbal

�xa;i( �sx) = �c+ T (��13)�ai (3.59)

which can be bounded by

k�xak2 �p2k �sxk (3.60)

In case of the SRS kak = 1:6m when using the centre upper gimbal point as origin (andgets larger/worse in using DEP or cog). So in case of the SRS r = :44m and this means amaximal rotation of � = 5:7�. It would be interesting to know whether it is possible to usemore structure concerned with the interaction between rotation and translation in order toarrive at possibly less conservative results.

In some cases it is favourable to define a (hyper) ellipsoid instead of a ball in whichthe task space states are allowed to move. E.g. in case of Fig. 3.22 it would be appropriateto choose a larger allowable variation in y-direction. Also this problem is in some casestractable in optimising a norm given quadratic constraints, which can be transformed tolinear matrix inequalities. The problem has to made convex, however. This can be attainedby replacing the constraints w.r.t. the minimal actuator lengths by convex approximationsin the task space e.g. a planes in the tangent space of the lmin balls. The ellipsoidal form isattained by multiplying �xa by a (positive definite) scaling matrix S.

E.g. maximize � = �xT �x such that

(S�x� �b)T (S�x� �b) < l2max (3.61)

(and smaller l2min) which can be transformed to�S �x�xT �bT�b+ l2max � 2�bTS�x

�> 0 8 k�xk � � (3.62)

By use of the S-procedure [18], the ’uncertainty’ ball of �x’s can be removed from the equa-tions, leaving a tractable problem.


Parameter Description Value Unitma Mass upper part actuator (piston) 120 [kg]mb Mass lower part actuator (cylinder) 135 [kg]ra Distance upper gimbal to cog piston 0:7 [m]rb Distance lower gimbal to cog cylinder 0:5 [m]ia Inertia upper part actuator wrt cog 20 [kgm2]ib Inertia lower part actuator wrt lower gimbal 36 +mbr

2b [kgm2]

Table 3.7: Inertial parameters SRS-actuators.

Limitations of a hydraulically driven motion system with specific kinematical and dy-namical properties have been quantified in this section. This knowledge can be used indesign of the system itself but also in the design of experiments in which it is often impor-tant not to approach such limits to closely.

3.4.3 Actuator inertial effects

In model based control choice of an appropriate model is important. The model has todescribe the control relevant aspects of the system. On the other hand it should not be toocomplex since it has to be calculated on a real-time control computer. W.r.t. modelling, thequestion arises whether the mass properties of the actuators have to be taken into account.And if so, how this should be done in a convenient way.

With the currently used simulator motion platforms the weight of the actuators is rela-tively low (ca. 250 kg) w.r.t. the platform (up to 12:5 tons). In these cases the weights donot contribute significantly to the motion of the total system. As the designed weight of thesimulator was reduced drastically (to ca. 3:2 tons) with the Simona research simulator, onehas to reevaluate this assumption.

Let’s look into the mass contribution of the system. The one body simulator has ac-cording to the design (finite element modelling) roughly the following mass matrix, M �c,in platform coordinates w.r.t the c.o.g. which is calculated 1 meter above the centre uppergimbal point.

M�c =

26666664

3200 0 0 0 0 00 3200 0 0 0 00 0 3200 0 0 00 0 0 7000 0 5000 0 0 0 7000 00 0 0 500 0 8000

37777775

(3.63)

The inertia of the simulator is somewhat higher than the weight but this does not necessarilymean that the impact is higher since the actuators are connected on a circle with radiuslarger than 1m. The main axes of inertia do not exactly correspond to the platform bodyaxes. The y-axis does, as the system can be mirrored w.r.t the xz-plane. The projectors onthe upper back of the simulator are the main reason for the xz cross term.


Case/Actuator 1 2 3 4 5 6Neutral position lmid lmid lmid lmid lmid lmid

Extreme pitch/surge lmin lmin lmax lmax lmin lminExtreme twist (yaw) lmax lmin lmax lmin lmax lminExtreme roll/sway lmid lmin lmin lmax lmax lmid

Table 3.8: Actuator lengths for platform poses considered.

Case/Platform pose x y z �1 �2 �3Neutral position 0 0 -3.38 0 0 0

Extreme pitch/surge 2.00 0 -2.43 0 -0.32 0Extreme twist (yaw) 0 0 -3.02 0 0 -0.52Extreme roll/sway 0.14 0.41 -3.25 0.20 -0.01 0

Table 3.9: Platform poses considered.

Recalling the additive mass matrix term, Ma;n, in (2.123), given by

Ma;n =

6Xi=1

JTai;x(Mma;i+Mia;i;ib;i)Jai;x; (3.64)

which consists of the contribution of all six actuators. In neutral position it becomes the fol-lowing, using the relations determining (2.122), and the inertial parameters for the actuatorgiven in Table 3.7.

Ma;n =

26666664

500 0 0 0 �525 00 500 0 525 0 00 0 675 0 0 00 525 0 1400 0 0

�525 0 0 0 1400 00 0 0 0 0 1325

37777775

(3.65)

The actuators contribute considerably to the total mass matrix (about 15 %). The terms inthe upper right and lower left block contribute to a lower cog of approx. .1 m. Remarkableis the fact that the weight experienced in moving in z-direction will be higher than in xor y-direction, due to the actuators. The actuator mass matrix will vary as a function ofplatform position since the actuators move w.r.t. the platform. These variations, however,are rather small. By taking a constant mass matrix in platform coordinates into the modelbased controller, a relatively simple but accurate choice is made.

If a Stewart Platform with even less relative weight is used, the full model should betaken into account. Apart from extra calculational effort this does not result in additionalproblems if the modelling steps described, are taken.

With respect to the nonlinear quadratic velocity terms in the model describing the corio-lis and centripetal terms, the constraints on the velocities are important. The most restrictive


100

102

100

102

−1000

−500

0

100

102

−100

0

100

100

102

100

102

100

102

Fig. 3.23: Bode plots from input valve voltage (u) to pressure difference (dP i) of linearisedmultivariable hydraulically driven mechanical system models. Masses: Simonadesign values. Rows 1,2 gives response of input 1 to outputs 1 (full), 2-6 (dotted).Rows 3,4 give responses of the diagonal elements along the decoupled rigid bodymodal directions (1,1:full), (2,2-6,6:dotted). Column 1: mid position, Column2: extreme pitch/ surge, Column 3: extreme twist, Column 4: extreme roll/sway. X-axis: frequency (Hz).

constraint, which holds for both pitch and roll of 35�=s = 0:61rad=s in the neutral position,was taken as the rotational velocity bound in calculating the influence of these nonlinearterms in [119] and it was found that these only contribute up to ca. 500N for the SRS.

3.4.4 Analyzing the SRS hydraulically driven motion system model

Having introduced the specific parameters of the SRS, the SRS-model can be analyzed con-sidering the interaction between the hydraulics and mechanics in this 6-d.o.f. system. It willbe evaluated whether the hydromechanical rigid body modes describe the main character-istics of the system. Along the rigid body modal directions, it is expected that the systemlocally behaves as six independent 1-d.o.f. hydraulic actuators moving a mass. This has tobe checked also as is the predictability of these directions. The influence of the main me-chanical nonlinearity, the platform pose dependent variation of the jacobian, J lx, has to bequantified.

The state equations, constructed from Fig. 3.20, describing the nonlinear dynamical


frb 1 2 3 4 5 61: (y; �x) (x; -�y) (�z) (�y; -x) (�x; y) (z)

[Hz] 4.5 4.5 7.6 13.3 13.5 14.82: (x; z; -�y) (y; �x; -�z) (�z; -�x) (�y; -x; z) (�x; y) (z)

[Hz] 4.1 5.8 9.1 15.5 16.1 17.43: (�z; -z ) (y; �x; �y) (x; -�y; �x) (�y; x) (�x; �z ; -y) (z; �z; �y)

[Hz] 3.6 4.5 4.5 16.1 16.3 17.24: (x; -y; -�y) (y; x; �x; -�y) (�z) (�x; z; -y) (�y; x) (z; -�x; x)

[Hz] 4.9 5.4 7.7 12.0 14.3 18.0

Table 3.10: Model eigenfrequencies, frb, in Hz of the rigid body modes and roughly therigid body modal directions. 1: Neutral position, 2: Extreme pitch/ surge, 3:Extreme twist (yaw), 4: Extreme roll/ sway. �i is the rotation around the ith

axis.

model from the output oil flows, ��om, are given by��x_�dP o

�=

�M�1t JTlx( �sx) 0

0 Ch( �sx)

�: : : (3.66)

�� bI aI

�aI �lI��

Jlx( �sx) 00 I

� �_�x�dP o

�+

�0I

��om

�+ : : :

: : :+M�1t

�G( �sx) + C( �sx; _�x)

0

�

In this case it is assumed that there are similar actuators (operational area, a, viscous friction,b and leakage, l) easily replaced by the respective matrices, A;B;L otherwise. By takingthe platform pose, �sx; as state variable, the model is valid for singular poses as well. Onlyif actuators would reach zero length, the jacobian is not defined.

The two-sided coupling with all (�om,dPo) pairs to the transmission lines and valves canreadily be made. For a reduced but fairly good model, each transmission line has four statesand each valve has two. Further, the six platform pose coordinates, �sx, have to be calculatedthrough integration of the velocities. The model thus consists of 12 + 6(2 + 4) + 6 = 54states.

This model with the SRS-parameters has been analyzed for several poses. Togetherwith the neutral position, three extreme poses were taken into account. Actuator lengthsare given in Table 3.8. The extreme pitch/ surge and twist only consist of extreme actuatorpositions. As the oil stiffness varies parabolically with the position, the extreme roll/ swaypose is important since it contains actuators with maximal (extreme position) and minimal(mid position) stiffness.

The platform poses, which are given in Table 3.9, do result from these actuator lengths.The euler parameters can be translated to angles by taking 12� per :1 up to :6 with 1�


Case/mode 1 2 3 4 5 6Neutral position 8.8 8.8 3.0 1.0 1.0 0.8

Extreme pitch/surge 14.8 7.4 3.0 1.0 0.9 0.7Extreme twist (yaw) 19.0 12.2 12.2 0.9 0.9 0.8Extreme roll/sway 9.0 7.8 3.6 1.8 1.0 0.6

Table 3.11: Generalised masses (tons). The large differences in (generalised) mass easilyresult in interaction. E.g. consider a force along the 8.8 ton direction in neutralposition, which is only 1� misaligned into the 0.8 ton direction. This will resultin 20% interaction. In more extreme positions e.g. pitch/ surge this will evenamount to more than 40%!

accuracy. In the first two rows of Fig. 3.23 the Bode plots of the linearised models in thesepositions are given from the first valve input voltage to the various pressures, dP i, takenover the valves. The hydromechanical modes are visible and are highly interacting in thefrequency area between 1 Hz and 30 Hz. The phase remains within �90 � for the (1; 1)transfer function up till the point where the valve itself starts to be relevant (> 70 Hz).Above 30Hz the measurable interaction strongly fades as the nondiagonal terms lack zerosin the transfer functions, which can be interpreted as springs in between input/ cause andoutput/ point of measurement.

In the last two rows the transfer functions have been considered from the other inputs andoutputs (only diagonal terms) resulting from the approximate decoupling matrix U definedin (3.44). This matrix was calculated from the approximate mass matrix (not taking intoaccount the variation due to the specific inertial properties of the actuators) and does nottake into account the varying actuator stiffness. Still the modes in the system can clearlybe distinguished. In the hydromechanical frequency interaction area, only one mode pertransfer function results in a single shift in phase of 180�. All the nondiagonal transferfunctions in these coordinates are small over the full frequency area considered.

Considering stability from a control point of view, the connection of a hydraulic circuitstill enables a simple passivity preserving dissipating feedback from pressure/ force suppliedto the mechanical system to the oil flow through the valve instead of velocity feedback toforce in a mechanical system.

The eigenfrequencies related to the hydromechanical rigid body modes of the system aregiven in Table 3.10. There is considerable variation for the different rigid body modes bothfor a specific pose as for the different poses considered. As each extreme position consideredwas only given a specific variation of the platform pose w.r.t the neutral position, the rigidbody modal directions in platform coordinates, given by J �1

lx U , can still be interpreted.E.g. in extreme pitch/ surge, symmetric involving x; z; �y and asymmetric modes involvingy; �x; �z. If all platform coordinates are varied this structure will be lost.

Looking at the Bode plots in Fig. 3.23, it can be observed that building up a unit pressurein a specific direction will take very different input values (more than a factor 10 variation).Very slight misalignment (wrong input) will easily result in large coupling. This is roughlylinear with the generalised masses, which affect the eigenfrequencies through a square root.


Case/actuator 1 2 3 4 5 6Neutral position 18 18 18 18 18 18Retraction gain -1.08 -1.08 -1.08 -1.08 -1.08 -1.08Extraction gain +0.90 +0.90 +0.90 +0.90 +0.90 +0.90

Extreme pitch/surge 26 67 -42 -42 67 26Retraction gain -1.12 -0.57 -1.19 -1.19 -0.57 -1.12Extraction gain +0.86 +1.29 +0.76 +0.76 +1.29 +0.86

Extreme twist (yaw) -24 48 -24 -24 48 -24Retraction gain -0.87 -1.21 -0.87 -1.21 -0.87 -1.21Extraction gain +1.11 +0.72 +1.11 +0.72 +1.11 +0.72

Extreme roll/sway 21 32 25 6 11 18Retraction gain -1.10 -1.15 -1.12 -1.03 -1.05 -1.09Extraction gain +0.89 +0.83 +0.86 +0.97 +0.95 +0.90

Table 3.12: Static load as percentage of maximal load Fmax = PsA and the resulting nor-malized valve gain discontinuity.

The generalised masses, 1=�i from (3.44), are given in Table 3.11.Already in the neutral position, the SRS is expected to have very different characteristics

along the rigid body modal directions. In the sway/ roll and surge/ neg. pitch directions thesystem will experience approx. 9 tons where the sway/ neg. roll and surge/ pitch directionsonly actuator forces to accelerate 1 ton of load have to be applied. In the extreme positionsthe largest mass ratio (mass matrix conditioning) along the different modes grows from nineto more than twentyfour in extreme twist. Although the jacobian, J lx, has a main influenceon this variation, its conditioning is not a very well predictor for this factor. It grows from4:5 to 5:2 going from the neutral position to extreme twist.

With the SRS design parameters, gravity requires approx. 18 % of supply pressure inthe neutral position. The platform contributes 15 % and the moving parts of the actuators3 %. The contribution of the lower parts is almost neglectable. This percentages can riseextensively in other platform poses. In the extreme surge/ pitch position considered, actuator2 and 5 will have to carry two third of the theoretically maximal weight. In practice, twothird is relatively high to take care of the static forces only. This means that the SRS designweight and height of the c.o.g. are in fact maximal for this motion system to operate safely.

Another important aspect of the static loads on the actuators is the nonlinear characteris-tic of the valve as given in (3.16). A preload results in a discontinuous valve voltage to flowgain as �im = u

p(1� sgn(u)dPi) for normalized variables. In Table 3.12 the required

preload for the four platform poses considered is given with the resulting valve gains foractuator extension and retraction. Negative percentages for the load mean that retractionpreload has to be applied to statically balance the platform. The all actuator minimal lengthposition is not the global minimum for the potential gravitational energy. As a result, thesystem will not always go to this pose if for some reason the preload disappears e.g. pumpsupply pressure drop to tank pressure in shut down.

All together, the hydraulically driven Stewart platform model can be transformed to six


Direction Mfii [tons; tons m2] !fi [Hz] �fixf 45 2�7 0:05yf 45 2�7:5 0:05!f 80 2�6 0:10


independent 1-d.o.f. hydraulically driven dynamic and static masses for each pose usingthe variable unitary decoupling matrix, U( �sx), describing the rigid body modal directions.Analysis of the SRS motion system design showed considerable differences in the rigidbody eigenfrequencies among these 1 d.o.f.-systems for the SRS.

3.4.5 Connection of a flexible foundation to the SRS system model

The hydraulically driven Stewart platform model of (3.66) can be extended with a modelincluding parasitic dynamics such as a flexible foundation. This model is used to predictin what way the dynamics of the system is influenced by such distortion. A model of aflexible foundation was introduced in Section 2.4.7 described by (2.126). Combination ofthese models leads to the following state equation

2664

_�xf��xf��x_�dP o

3775 =

2664I 0 0 00 M�1

f 0 0

0 0 M�1t 0

0 0 0 Ch

3775 : : : (3.67)

26642664

0 I 0 0Cf �Bf � JTf BJf �JTf BJlx JTf A

0 �JTlxBJf �JTlxBJlx JTlxA

0 �AJf �AJlx �L

37752664

�xf_�xf_�x�dP o

3775+

2664

000I

3775 ��om

3775

The foundation is allowed to move in the horizontal plane and introduces three addi-tional modes, foundation surge, xf , sway, yf , and yaw, !f , described by six states (�xf ,_�xf ). As an example, the parameters given in Table 3.13, are filled into the equations. Thesedescribe roughly the characteristics of the foundation on which the system was tested in thecentral workshop. The mass matrix of the foundation has numbers on the diagonal, whichare ten times higher than those of the platform. The stiffness matrix and damping matrixare chosen such that the undamped eigenfrequencies and damping factors specified for thefoundation alone result. The high mass of the foundation in relation to the platform letsthese frequencies only slightly altered in connecting the motion system.

In Fig. 3.24 of Appendix A, the system is evaluated in the frequency domain from 1Hzto 50Hz. The upper left bode amplitude plots shows the model actuator pressure differenceresponses to the oil flow into the first actuator (not all 36 responses as the other 30 are


100

101

10−1

100

101

100

101

10−1

100

101

100

101

10−1

100

101

Frequency (Hz)10

010

1

10−1

100

101

Frequency (Hz)

Fig. 3.24: Bode amplitude plots of linearised multivariable hydraulically driven mechanicalsystem models without transmission lines but with flexible foundation. Upperleft: neutral position from first input actuator flow �om1 to all output pressures( �dP o). Upper right: neutral position along the (original) rigid body modal direc-

tions from all �om to �dP o, nondiagonal terms dashdotted. Lower left: neutralposition along the (original) rigid body modal directions from all �om to theaccelerations (m=s2) (�xf ; �yf ; _!f ) of the foundation, nondiagonal terms dashed.Lower right: extreme pitch/ surge along the (original) rigid body modal direc-tions from all �om to dP o, nondiagonal terms dashdotted.

similar) in the neutral position. The complex dynamics of the hydromechanical modesprevents one from recognising the parasitic dynamics. Using the decoupling transformationU calculated from the platform mass matrix the bode amplitude plots in the upper right partof the figure results. In the diagonal responses the fact that there are parasitic modes isdirectly visible. Three additional modes show up separately along the eigenvectors of thelowest three original platform modes. Further, the nondiagonal terms are not zero anymorethough still quite small.

As discussed in Section 3.3.3, both a flexible foundation as a mass/ spring/ damper-system connected to the platform result in similar characteristics evaluated at (combinationsof) actuator coordinates. To evaluate whether the foundation is causing parasitic dynamics,one can measure accelerations at the foundation itself. The model predicts the accelerationfrequency response at the neutral position given in the lower left part of Fig. 3.24 of Ap-pendix A. This was calculated by multiplying the velocity frequency response of the modelby j!. Remarkable is the fact that the peaks in the response, which are most prominent, are


not in all cases mainly the original modes of the foundation. Input along the higher platformeigenfrequencies are expected to result in relatively high (approx. 15 Hz) frequency vibra-tions and of the three main transfer functions the yaw platform direction to yaw foundationmodal response peaks are maximal at approx. the original platform yaw mode.

The fact that the parasitic modes influence only three platform modes is not generallytrue. In the lower right part of Fig. 3.24, the system is evaluated again along the originalplatform eigendirections. But now the platform has been put into the extreme surge/ pitchmode. The fact that the platforms c.o.g. is now approx. 2 m in front of the c.o.g. of thefoundation makes the sway mode of the platform interact with both the foundation yawand sway mode and through the foundation yaw mode with the platform yaw mode. Theresponse along this direction shows four resonances. Also the decoupling is almost com-pletely lost between 4Hz and 15Hz.

So, parasitic dynamics such as a flexible foundation can have an important influenceon the motion system characteristics. This is not always directly visible in the measurableresponses (inputs and outputs of the actuators).

3.5 Chapter Resume

The motion systems rigid bodies interacting with the hydraulic actuator stiffnesses, thehydromechanical rigid body modes, describe the main characteristics of the hydraulicallydriven motion system dynamics. The eigendirections of these modes can be quite accuratelypredicted by taking the eigenvectors of the approximate actuator mass matrix consisting ofthe variable jacobian, Jlx( �sx), and the platform mass matrix. Along the rigid body modaldirections the system locally behaves as six independent 1-d.o.f. hydraulic actuators movinga mass in every platform pose.

Under mild conditions, these systems are passive evaluating pairs of the oil flow andpressure difference at the valve or at the actuator of the hydraulic servos or (collocated)combinations of these pairs, no matter what parasitic flexible mechanical systems are to beincluded. Any passive feedback over these pairs will result in stability of the full nonlinearsystem.

The valve dynamics causes loss of passivity mostly relevant in combination with thehigh frequency (e.g. transmission line) modes. The transmission lines, though mainly re-sulting in high frequency dynamics, cause a considerable shift of the hydromechanical rigidbody modes and introduce large differences at relative low frequencies (10 times smallerthan the transmission line modes) between the measurable pressure at the valve and theequivalent force (pressure difference times operational area) the actuator induces into themotion system.

Parasatic dynamics such as a flexible foundation or an additional mass/ spring/ dampersystem connected to the moving platform is easily included in the models introduced. Thisdynamics can have considerable influence on the systems characteristics and is most easilyrecognized evaluating along the hydromechanical rigid body modes. Only measuring at theactuators, the cause of the parasitic dynamics is not identifiable.

Chapter 4

Parameter identification andmodel validation

The models presented in the two previous sections were derived from the laws of physics.Where numerical examples were given, the parameter values were taken from design val-ues or rough approximations of what could be expected in practice. These models can bevaluable for the structural analysis of (parallel, hydraulically driven) motion systems. Thestrengths of these models should, however, be proven by experimental validation. Properchoice of model parameter values is required in using the models to predict the characteris-tics of the actual system and enable direct comparison of model and experimental response.

Usually, the parameter choice is also to be based on experiments. Experiments fromwhich the parameters can be identified. Small differences between model and practice,i.e. a well validated model structure with proper parameter values is also important in theuse of a model based motion controller.

Identification can also be performed with more general model structures which are es-pecially suited for parameter identification. In nonlinear systems it is very hard to findconvenient model structures. The physically motivated structure of the nonlinear modelspresented in this thesis, however, enables one to investigate what influence changes in thedesign of the system will have on its dynamics and helps in pointing at the relevant partsof the statics and dynamics in a complex system as the flight simulation motion systems athand. This also helps in the design of proper experiments.

As already discussed, the motion system (model) consists of kinematics and dynamicsrelated to the mechanics i.e. the Stewart platform, and the hydraulic actuators which drivethe system. Further some additional parasitic dynamics can be expected from a nonrigidfoundation and a flexible platform/simulator. The identification of the dynamics specificallyrelated to the hydraulic actuators has already been discussed in detail by Van Schothorst[124] and will not form an explicit part of this thesis though the hydraulic model struc-ture presented does. Of course the dynamics resulting from the interaction between thehydraulics and mechanics will be, since the models presented in the previous chapter ap-pointed this effect as the most relevant.

This chapter will have to provide a good starting point for model based control on theactual system. The gap between the theoretically derived model structures in the previous

119

120 4 Parameter identification and model validation

chapter and the actual motion system to be controlled will be bridged in three steps.

� Parameter estimation. The calibration method for the Stewart platform kinematicalmodel will be presented, implemented and evaluated in Section 4.1. Next, in Sec-tion 4.2, the experiments to infer the dynamic model parameters for the mechanicsare set up, implemented and evaluated. Dynamic experiments to infer the positionand orientation of measurement equipment consisting of accelerometers and rate gy-ros will be treated in Section 4.3.2.

� Model validation. The model characteristics with the parameters estimated will bevalidated in the frequency domain in Section 4.3. Especially the conclusion in theprevious chapter that the hydromechanic rigid body modes are most relevant and thatthe system can be considered locally as six independent 1-d.o.f.-hydromechanic sys-tems along the rigid body modal directions, will have to be evaluated experimentally.If this conclusion is justified in practise, the models, which describe this dynamics,are expected to provide a good starting point for model based control.

� Quantification of parasitic dynamics. The cause of the main parasitic dynamicsobserved in Section 4.3 will be clarified in the final part of this chapter, Section 4.3.2,by additional measurements showing vibration of the foundation and non rigid shut-tle. Together with the hydraulically generated parasitic dynamics stemming from thevalve and transmission lines, these parasitic dynamic effects are to be considered thelimitations to the control scheme based on the basic model.

All experiments discussed in this chapter are open loop based i.e. no feedback is in-volved apart from a very low bandwidth controller, which is to stabilise the system withoutrelevant influence on the information inferred from measurements for calibration/identification(higher frequencies or static relations).

4.1 Calibration

In this section the parameters of the kinematical model of the simulator motion system willbe identified (calibrated) and the kinematical model itself will be validated. In robot calibra-tion [15], parameter identification of both the kinematics and dynamics of the mechanicalsystem is referred to if calibration is considered. In this thesis, calibration will assume to beconcerned with the identification of the kinematical model parameters or of specific mea-surement equipment. Parameter identification will generally refer to the model parameters.It is most convenient to perform calibration first as this step can be made independent of thesystem dynamics.

Calibration is a very important topic in robotics. Especially if positional accuracy isrelevant since this can often be improved by an order of a magnitude [15]. Further, onecan expect that most experimental effort will have to be put into calibration as related tothe time spend in testing the motion control strategies in practice [52]. In flight simulatormotion systems positional accuracy is clearly not the main issue but indirectly it affects theaccelerational accuracy and is therefore considered in this thesis.

4.1 Calibration 121

The basics of calibration can be found in textbooks such as [101] and books containingselections of articles published such as [15]. The framework adapted in this thesis is basedon an article of Hollerbach and Wampler [52]. In this article, a large set of robot calibrationmethods is unified considering joints and selecting closed kinematic loops. Further, a num-ber of numerical issues is discussed, which provide a theoretical basis on the identifiabilityproblems of kinematical parameters in practice. This method easily integrates the paralleldriven structures but does not explicitly do so by presenting equations which are explicitin the end effector coordinates. The approach of Zhuang in [161] generalizes the idea ofmaking use of residuals in redundant equations for parallel structures in identifying the un-known parameters i.e. calibrating such systems and these ideas will also be incorporated inthe calibration procedure.

4.1.1 Calibrating the Stewart platform

In this research no specific new theory will presented considering the calibration of a Stewartplatform. The contribution will be a fully worked out applicable method consisting of acombination of known only slightly modified steps, which, not often seen in literature, isbrought into practice and is put into a general framework. A large part of the procedurefollowed has been worked out as a part of two master projects [16] [151] in calibrating theSRS motion system with the dummy platform of Fig. 1.5. Here the results for the practicalpart of the calibration will involve the calibration of the motion system with the SRS-shuttleof Fig. 1.4.

The calibration procedure proposed consists of only two basic steps.

� A redundant measurement procedure for platform pose reconstruction.

� Identification of the kinematical parameters of each platform leg separately using theredundancy of the actuator length measurement.

Only a small part of the literature on calibration is devoted to the parallel manipula-tors. These systems differ from most serial systems considering the number of kinemat-ical parameters involved, which is high. For the Stewart platform, a kinematical model,schematically given by Fig. 2.7, was presented by (2.38) from which

klmi + loik2 = (T �ai + �c� �bi)T (T �ai + �c� �bi) i = 1; : : : ; 6 (4.1)

follows, which involves 42 unknown kinematical parameters. Seven for each equation con-sisting of an unknown offset length, loi, which together with the measured length, lmi rep-resents the total length of the i’th leg. The unknown upper and lower gimbal vectors, �a iand �bi, each consist of three parameters. These parameters will be identified in this sectionafter redundantly measuring the platform pose consisting of the rotation matrix, T and thetranslation, �c.

This kinematical model assumes that all the gimbals are perfect ball and u-joints asschematically given by Fig. 2.8 and the prismatic joint axis of the sliding actuator exactlyintersects the two gimbal points. In [154, 155] a more general kinematical structure isconsidered, allowing each axis of rotation in one gimbal to cross the other at a specific


distance not perfectly orthogonal. This model involves 132 unknown kinematic parameters,which would heavily complicate the calibration problem. Validation of the simple modelafter calibration can point out whether this is necessary.

Reducing the number of unknown parameters to be identified in one step usually helpsthe conditioning of the problem to be solved and therefore improves the accuracy of theresults. In [162] by Zhuang and Roth, the separate calibration of each leg of the Stewartplatform is made possible by assuming a redundant measurement of the platform pose. Thiswill leave one redundant actuator length variation measurement per actuator from whichthe seven unknown kinematical parameters in �a i, �bi and loi will have to be identified. In[162] also the calibration of the offset length, loi, is decoupled from the others, but as thisstrongly reduces the number of measurements, which can be taken into account, this part ofthe procedure has not been adopted.

The solution method to identify the kinematical parameters is iterative with only localconvergence, like the forward kinematical problem discussed in Section 2.3. In [59, 60] amethod is presented to calculate every solution to the calibration of �a i and �bi nonredundantlytaking only six measurements into account and using a seventh measurement in decidingwhich solution is the right one. As it involves rootfinding of a twentieths order polyno-mial equation, this method is usually numerically not suitable for full (noisy) calibration inpractice but it can be used for validation of the calibration procedure used here.

The mobility of a system, roughly its degrees of freedom, can be specified with themobility index, M as done by Hollerbach and Wampler [52]. It can be calculated given thenumber of links,L, including the base link and the number of constraints,D i, for each joint.Putting the method of Zhuang and Roth [162] for the Stewart platform actuator of Fig. 2.8,in the framework given in [52], the mobility index, M , of a separate leg of the platform is

M = 6(L� 1)��Ji=1Di = 3(4� 1)� (4 + 5 + 6) = 6 (4.2)

The four links consist of base, lower and upper actuator part and the platform. There isa u-, prismatic and a ball joint. The calibration index, C, a redundancy measure given byHollerbach and Wampler [52], can also be calculated as the total number of sensors, S, isseven, assuming an independent pose measurement, with

C = S �M = 1: (4.3)

This means there is only one redundant measurement involved from which all seven kine-matical parameters have to be constructed. As in system identification of a dynamic misosystem with several parameters, this is only possible if more than one, here at least seven,measurements are taken and if the inputs, the platform poses chosen, can be made suffi-ciently rich to make the parameters identifiable.

As opposed to the discussion of serial manipulator examples given in [52] for systemswith C = 1, the number of measurements in the joints in a parallel system is one insteadof the number of measurements for the end-effector. An advantage of C = 1 is that allequations involved and the residuals resulting, are of the same type. This is favourable inconditioning.

In applying a calibration method in practice, the desired precision of the result shouldbe known since as a rule of thumb the measurement equipment should be an order of a

4.1 Calibration 123

−20

2 −2

0

2

−1

0

1

2

View in perspective

x−y

−z

−2 −1 0 1 2−1

0

1

2

3Front view

−y

−z

−2 −1 0 1 2−1

0

1

2

3Side view

x

−z

−2 −1 0 1 2−2

−1

0

1

2Top view

x

−y

Fig. 4.1: Two calibration measurement frames for the case of the Stewart platform. Thefixed frame consists of three points. With the SRS they were chosen somewherebetween the lower gimbals. The moving frame consists of three points chosensomewhere on the upper gimbal blocks. Knowledge on the exact positioning ofthe measurement frame points is not necessary for this method.

magnitude more precise [15]. For the measurement of the relative position of the prismaticjoint a linear position transducer of the Temposonic type is used. The accuracy of thesesensors is 0:1 mm (resolution with [55] 0:009 mm). A calibrated motion system withan positional accuracy of 1 mm will be strived for. The independent measurement of theplatform pose discussed next will have to be done with equipment of � :0:1 mm accuracyalso.

4.1.2 Redundant measurement of the platform pose

The redundant measurement of the platform pose is based on the method presented by Gengand Haynes [43]. One defines two measurement frames e.g. as in Fig. 4.1. One fixed onthe ground and one on the moving platform/simulator using three points for each frame.The motion of a point with respect to the fixed frame can be determined by measuring thedistance of this point to the three points fixed to the ground if the distance between thefixed points is known and if the moving point remains in a known area w.r.t. plane spannedby the three points. With three additional length measurements (two to the second movingpoint and one to the third) the orientation can also be determined if the distance between themoving points is known.

In case of the calibration of the shuttle of the SRS the distance between the movingpoints fixed to the shuttle are not precisely known and can not easily be measured directlysince the shuttle shape does not allow so. This problem is solved by taking three distancemeasurements for each moving point (nine in total) from which the relative position of themoving points can be calculated. The relative distance of these points is redundant if more


Fig. 4.2: Determination of the position of a moving point in space using three length mea-surements. In case of the SRS, the three fixed points, G i, can be thought in be-tween the lower gimbal blocks. Each moving point, P i, was taken as part of theupper gimbal blocks.

than one pose measurement is taken. This redundancy can and will be used to decide ona confidence level for the measurements in the weighted least squares calibration of thekinematical parameters. As only one distance to a specific point can be measured at thetime this redundancy does not take significant additional effort to be measured since theplatform has to be moved to a specific pose three times at least (using three pairs of threemeasurements instead of three, two and one measurement respectively).

In Fig. 4.2 a top view of a three point distance measurement with three lengths, l i,i = 1; : : : ; 3 is given to determine the position of a moving point P i with respect to theframe defined by the three points Gi, i = 1; : : : ; 3. Point, G1, is taken as the origin and theline throughG1 and G2 as the y-axis. Using the cosine formula:

l22 = l21 + b22 � 2l1b2cos(�) = l21 + b22 � 2b2y1; (4.4)

y1 can be calculated. By letting the third fixed point, G3, define the fixed measurementxy-plane, similar application of the cosine rule will lead to a parameter, s. This parameter,s, is the distance and direction of the origin to the point S(sx; sy; 0) in Fig. 4.2. So also thefollowing holds

spa23 + b23

=sx

a3=sy

b3(4.5)

Further, the line in the xy-plane through S, orthogonal to the line throughG 1 and G3, runsthrough (x1; y1; 0), i.e. direction @y=@x = �a3=b3 can also be given by

y1 � sy

x1 � sx= �a3

b3: (4.6)

Combining these equations to get rid of sx and sy leads to

b3y1 + a3x1 = s

qa23 + b23 (4.7)

4.1 Calibration 125

So in applying the cosine rule, s can be replaced by x 1 and y1, from which x1 can be solved

l23 = l21 + a23 + b23 � 2l1 cos(�)qa23 + b23

= l21 + a23 + b23 � 2sqa23 + b23

= l21 + a23 + b23 � 2(b3y1 + a3x1) (4.8)

Finally j z1 j can be determined from

l21 = x21 + y21 + z21 (4.9)

The sign of z1, above or below the plane of measurement, should be known in advance. Inthis way the three vectors from the fixed frame to the moving points, �p i, can be calculated.The frame defined through Gi given in Fig. 4.2 is convenient in determining the vectors tothe moving reference points �pi. In identifying the kinematical parameters it will be moresuitable to define a slightly different fixed and moving frame from G i and Pi respectively,which are almost equal to the ground frame and moving frame defined for the SRS (Fig. 2.7).

The new fixed reference measurement frame origin, Gm, is given by a translation

�oGm= (�g1 + �g2 + �g3)=3 (4.10)

with respect to the frame given in Fig. 4.2. The orientation remains unchanged. The movingreference measurement frame origin, Pm, is specified w.r.t Gm accordingly with an origin(translation �cpg)

�oPm = (�p1 + �p2 + �p3)=3� �oGm= �cpg (4.11)

and the orientation, with the line through P1 and P2 chosen parallel to the y-axis, consistsof three unit vectors along the new axes. With �p ij = �pj � �pi, �npy = �p12=

p�pT12�p12,

�npz = (�npy � �p13)=p

�pT13�p13 and �npx = �npy � �npz . These together form the rotationmatrix from Pm to Gm

TGm;Pm = [�npx �npy �npz ]: (4.12)

And this is the pose measurement we were looking for. In short it is given by (�c pg , Tpg),which basically can be interpreted along the same lines as the pose defined for the Stewartplatform.

All together the used platform pose measurement requires nine absolute length mea-surements and the knowledge of the absolute distance between the fixed reference points.The quality of the calibration procedure heavily depends on these measurements. The po-sitioning of the reference points themself is not that important. A rough idea will help inselecting a proper initial estimate in the calibration and the conditioning of the optimizationproblem to be solved is improved if all the distances to be measured (e.g. in Fig. 4.2) areabout the same. Further the platform reference points have to reside at a predefined side ofthe plane defined by the fixed reference points.


Fig. 4.3: In practise, the measurement device could not be used to take direct measurementof the length between two points at an angle (� 6= 0). A funnel (a cone shapedutensil) had to be placed in between. The picture shows that the measurementcable will not run directly from Gi to Pi but will follow the rounded edges of thefunnel from Gi to D. This requires funnel length measurement compensation.

4.1.3 Stewart platform pose measurement in practice

The length measurements to reconstruct an independent, redundant platform pose in case ofthe SRS-motion system have been performed with relatively low cost string encoders madeby AMS Gmbh. A measurement cable winds onto an accurately machined cable drum. Acoil spring delivers a constant pull in force to maintain cable tension. An incremental rotaryencoder on the drum gives a digital output. The resolution of these encoders is 0.03 mm andthe calibrated relative precision 0.1 mm. The standard version of this equipment does notallow the cable to be used at different angles w.r.t. the output point. This would damage thedevice.

To allow calibration for an absolute length measurement, also for measurements takenat an angle, a funnel was designed. As shown in Fig. 4.3, this slightly affects the lengthmeasurement between Gi, which is chosen at the bottom of the funnel and P j . In practicethis difference can take values up till 4 mm (R=15 mm) and should be corrected.

Considering Fig. 4.3, lpg , the length between Pj and Gi is equal to

lpg =p(X 0)2 + Z2 (4.13)

the measured length will be equal to

d1 +R� =p(X 0)2 + Z2 � 2RX 0 +R� (4.14)

By using the measured lengths to reconstruct a first estimate of �p i the required correction

4.1 Calibration 127

can be approximated by taking

� � �0 = tan

�X 0

Z

�(4.15)

or more precise (in this case not necessary since R=lpg was small, only used for evaluation)reconstructing � from

Z tan(�) = X 0 +R

�1� cos(�)

cos(�)

�(4.16)

It is assumed that the direction (Z) of the funnels (in the frame Gm) is known.The choice for the approximate positions of the reference measurement points is given

in Fig. 4.1. Considering Fig. 2.10, the fixed reference points (the encoders) have beenplaced in between the lower gimbal pairs (4,5), (2,3), (6,1) respectively. This allows thefunnels to point along the z-axis of the measurement (and ground) frame. The upper, movingreference points have been taken in between (slightly lower: 80 mm) the same pairs ofupper gimbal points. In most platform poses, the lengths l pg will therefor not be the same(slightly less favourable conditioning) but in practice prevention from actuators running intothe measurement cables is more important. As a result, the fixed ground frame and platformframe almost coincide with the measurement reference frames. This helps in interpretationof numerical results but does not improve the calibration itself.

To allow measurement in between a favourable maximum and minimum length the lim-ited (2 m) range string encoders were enlarged with steel cables (as used for fishing!) whichwere calibrated with the string encoders themselves. Two sets of these additional offset ca-bles were used (forGi to Pj with i = j approx. 1.5 m and 2.0 m in case of i 6= j). The offsetof the string encoders themselves is also necessary to allow absolute measurement and wasmeasured at a small, approx. 60 mm, retracted up right position with a sliding calipers tothe funnels. The distance (� 1:8m) between the string encoders could be measured with alarge (2 m range) sliding calipers.

Careful attention in performing these (offset) measurements largely improved the cali-bration. By calibration before and after a measurement session also some drift compensation(1.4 mm for one string encoder) could be applied. Another important point of considerationwas the dry cable friction in the funnels. Before each measurement, the cables had to bevibrated a little bit to get rid of play.

To allow proper choice of the sensor range to be used, the set of platform poses, ap-proximate since the actual kinematics is not known in advance, used for calibration can besimulated. In Fig. 4.4 all the positions of upper gimbal 1 are given for all 64 combinationsof setting the six actuators in a minimum length position of 20 cm above minimal and belowmaximal stroke. These positions are sufficiently far away from the cushioning area of theactuators to be able to hold the system stable at these positions in taking a measurement.

It is assumed that this set of positions will give relatively high variation in order toenhance the identifiability of the kinematical parameters. From the front view it can be ob-served that the positions are roughly the extremes of the 2-d.o.f. structure given in Fig. 3.22which are rotated if considered from the side view. The fact that some points are relatively


−20

2 −2

0

2

0

1

2

3

View in perspective

−2 −1 0 1 20

0.5

1

1.5

2

2.5

3Front view

−y

−z

−2 0 2 40

0.5

1

1.5

2

2.5

3Side view

x

−z

−2 0 2 4−2

−1

0

1

2Top view

x−

y

Fig. 4.4: Positions of upper gimbal 1 for calibration.

close to each other in this picture does not imply that they will contain the same informationsince the orientation of the platform, which will be different, is not shown.

If compared to the unconstraint actuator, it can be observed that especially the range ofangles to be reached by the platform actuators is relatively limited. E.g. the angles of the linebetween the lower gimbal point 1 and the dots in Fig. 4.4. This will affect the conditioningof the calibration as will be discussed in the next sections.

4.1.4 Identification of the kinematical parameters

After the determination of the platform poses, this information plus the actuator displace-ment measurements can be used to identify the kinematical parameters. Taking an initialestimate of these parameters in (4.1), each measurement regarding the j’th actuator, willhave a residual. One can try to change the parameters in order to minimize these residuals.Using the quadratic form of (4.1), the jacobian of kinematical parameter variations, � �k, toresidual variation, ��r, has a favourable linear structure

Jrk =@�r

@�k= Xo +�7

i=1Xiki (4.17)

with the kinematical parameters, �k, of one actuator defined by

�kT = [�aT �bT loj ] (4.18)

and the i’th row (measurement) of the jacobian

Jrk(i; �) = 2[(lm(i) + lo(i)) (�cTi Ti +�bTTi + �aT ) (�cTi + �aTT Ti +�bT )]: (4.19)

4.1 Calibration 129

It is immediately clear from the last equation that every row can be split in a non kinematicalparameter dependent row

Xo(i; �) = 2[lm(i) �cTi Ti �cTi ] (4.20)

from which the matrix, Xo, can be constructed and parts which are linearly dependent ononly one kinematical parameter e.g.

X1(i; �)�k(1) = 2[0 1 0 0 T Ti (�; 1)]�a(1): (4.21)

A weighted least squared solution to the linearised problem of minimising the residualsis given by

��kwls = (JTrkWJrk)�1JrkW �r (4.22)

In robot calibration, this is usually effective since the equations are only mildly nonlinear[52]. The weighting matrix, W , can be used to discriminate between the measurements.This can be appropriate since the confidence level for the different platform pose measure-ments can differ. If one of the string encoders showed e.g. a little hysteresis due to frictiononce in a while, the redundancy in the platform pose measurement can be used to detectthis. With

Wii =1

0:5 + 1000 j lPP �mlpp j(4.23)

the confidence is upper bounded by 2 and goes to zero if high difference is detected betweenthe measured distance, lpp, and the mean distance, mlpp , between two points, Pi.

Iteration on (4.22) is similar to a Newton Raphson iteration. For convergence, it helps ifthe jacobian is Lipschitz, far from singular, well conditioned and starts with a well choseninitial estimate. Due to the linearity in the parameters, the jacobian is clearly Lipschitz.Further the structure of each parameter (X i) is almost equal. The choice of platform posescan be used for conditioning. E.g. not varying the orientation, T , will result in a singularjacobian, since all solutions with only a translational vector added to the actual solutions ofboth �a and �b will have equal residuals.

The singular values of the jacobian, Jrk, are also called the observability indices of thecalibration problem. In a comparative study cited by Hollerbach and Wampler in [52] it wasconcluded that optimal conditioning of these indices or maximising the minimal index gavegood quantative results.

The calibration problem of the actuator kinematics was idealised by an unconstrainedsimulation i.e. no constrains of platform poses to be chosen and all actual parameters zero.Choosing all (729) combinations of the tri state (-1,0,1) for translation and (-.5,0,.5) for theeuler parameters results in a jacobian with condition number two at the solution, which isalmost optimal. By Hollerback and Wampler [52] it is referred to a research concluding thecondition number should be better (lower) than 100 to have reasonable results. With theSRS Stewart Platform to be calibrated with constrains on the actuator lengths the conditionnumber increases to sixty using the 64-state set given in Fig. 4.4.


−0.01 0 0.01−0.01

00.01

0

0.01

0.02

0.03

view in perspective

−0.02 −0.01 0 0.01 0.02−0.01

0

0.01

0.02

0.03front view

−y

−z

−0.02 −0.01 0 0.01 0.02−0.01

0

0.01

0.02

0.03side view

x

−z

−0.02 −0.01 0 0.01 0.02−0.02

−0.01

0

0.01

0.02top view

x

−y

Fig. 4.5: SRS calibration result with the design structure on a 1:100 scale and the differ-ences 1:1.

0 20 40 60−5

0

5

0 20 40 60−5

0

5

e_le

ngth

[mm

]

0 20 40 60−5

0

5

0 20 40 60−5

0

5

e_tr

ans

[mm

]

0 20 40 60

−5

0

5

calibrated system0 20 40 60

−5

0

5

non calibrated system

e_ro

t [m

eps]

Fig. 4.6: Positional errors for all 64 platform poses used for calibration made by the non-calibrated and the calibrated system. The first row shows length measurementprediction errors of the actuators length measurement given the redundantly mea-sured pose. The next rows give the transformation of these errors to translational(2nd row) and rotational (3rd row) prediction errors of the platform pose.

4.1 Calibration 131

4.1.5 Results in calibrating the SRS motion system

The calibration procedure leads to kinematical parameter (vectors) defined in the referencemeasurement frames. By construction of the forward kinematics of the newly found param-eters for the actuators in half stroke, the transformation, �sx gp, between the two measurementframes can be calculated. All upper gimbal positions can now also be referred to the groundmeasurement frame, Gm.

�aGm = �cgp + Tgp�aPm (4.24)

As the parameters were not constrained to a plane, the simulator ground and movingframe are not trivially defined anymore. As for the reference measurement frames, theground frame origin is chosen as the mean of the lower gimbal points. The ground frame x-direction is chosen as the addition of the vectors between the origin and gimbal point 1 and6 and the xy-plane is formed by making the vector from the origin to the mean of gimbals 2and 3 part of it. This defines the transformation to a ’design’ ground frame.

Both the upper as the lower gimbals can now be referred to the design frames. In Fig. 4.5the gimbal points found through calibration are given in a design frame which is a 100times compressed making the differences, which are less than 0.1 %, visible. As opposedto the dummy platform [16] [151], no large deviations are found in the construction of theshuttle. It is now interesting to see what the differences are in predicting the actuator lengthmeasurements i.e. using the design values and the calibrated values. These deviations forall six actuators can also be transformed to the deviations in platform pose using

� �sx = J�1l;sx( �sx)�

�l (4.25)

In Fig. 4.6 it is shown that the positional accuracy of the calibrated platform has almostimproved an order of a magnitude.

The accuracy of the noncalibrated system is not better than 5 mm in actuator lengths,2 mm in translation and 10 mrad in orientation. With the calibrated system this improvesas shown in detail in Fig. 4.7 to �.75 mm in actuator length predictability, apart from oneoutlier also predicted with upper reference point distance measurement. In the more im-portant platform pose translational and orientational accuracy the errors are reduced to lessthan 0.5 mm and 2 mrad respectively.

The initial estimate in the calibration procedure was more than another magnitude worse(> 100 times) since the pose of the measurement reference frame was only roughly (within100 mm) taken into account. Still the iteration on the kinematical parameters convergedquickly within 4 steps for each actuator leg. Convergence is not guaranteed with suchinitial errors, but since the procedure can be done off line, as opposed to the platform posereconstruction treated in Section 2.2.11, this is not so important.

One important aspect, the direction of the gravitational field has not been identified yet.This can be done using an additional measurement device (e.g. an inclinometer) as has beendone in [16] [151]. But the pressure (force) measurements in the actuators can also be usedto detect this as shown in the next section.


0 20 40 60−2

−1

0

1

2

3Variation upper triangle measurement

erro

r [m

m]

0 20 40 60−2

−1.5

−1

−0.5

0

0.5

1Errors in actuator length prediction

resi

dual

s in

[mm

]0 20 40 60

−1

−0.5

0

0.5

1Translational pose prediction errors

pose no.

resi

dual

[mm

]

0 20 40 60−1.5

−1

−0.5

0

0.5

1

1.5Orientational pose prediction errors

pose no.re

sidu

al [m

eps]

Fig. 4.7: Positional errors for all 64 platform poses used for calibration made by the cali-brated system in detail. Also showing the variation in upper reference point dis-tance measurement, which is used to calculate the confidence levels.

ResumeThe following conclusions can be drawn w.r.t. the calibration procedure described.

� It has been shown that the kinematic mechanical parameters of a Stewart platformused as a flight simulator motion system can be calibrated resulting in a system with apositional accuracy better than 1 mm (for DEP). This is an order of a magnitude moreprecise than before calibration.

� The fact that the kinematical model itself is accurate within these error bounds for theSRS is another important conclusion.

� Further, it means that with calibration such motion systems can be made more ac-curate without putting more extreme tolerances on the fabrication and constructionalprocess.

� The two step calibration procedure has been shown to work in practice.

� For the first step, measuring relative platform pose variations w.r.t. unknown, butplatform and inertial world fixed, reference frames is sufficient redundant informationnecessary for calibration.

� The parallelism of the platform can be used in the second step to identify the 42kinematical parameters in six separate actuator groups. This enhances accuracy.

� It is important to realise that the result strongly depends on the accuracy of someof the measurements taken. These are the absolute length measurement of, and thedistances in between, the redundant sensors.

4.2 Stewart platform model parameter identification 133

As the calibration is concerned with sub mm precision on a sup m motion system, it isrecommended that the procedure is rerun after each reconstruction of a platform e.g. inmoving the SRS motion system to another site.

4.2 Stewart platform model parameter identification

Now the kinematic model has been validated and its parameters have been calibrated, theparameters concerning the dynamics of the motion system can be identified. The mainunknowns in the Stewart platform model are its mass properties.

First, the experiments to reconstruct the parameters related to the gravity forces, thelocation of the centre of mass and the weight itself will be discussed. Then, the identificationof the inertial parameters is treated. These can be determined experimentally without muchconcern about the precise dynamics of the hydraulics. In this way, a reasonably accurateapproximate model, i.e. (2.112) with a general mass matrix, of the mechanics presented inChapter 2 taking into account the actuator intertia as in (2.122) can be identified.

In the model based control structure to be presented in the next chapter, this part of themodel will be important since the forces required for the accelerations to be applied forsimulation will for most be generated by feed forward which requires an accurate model.Identification of the hydraulics, considering the separate actuators, has been discussed in[124]. The model parameters concerning the hydraulics are therefore assumed to be known.Validation of the dynamics resulting from the interaction in the hydraulically driven Stewartplatform will be discussed in the next section.

The set up of the hydraulically driven motion system supplied with the six valve inputsand measurement equipment to determine both pressure and positional information of theactuators, the legs, provides a complex but sufficiently rich environment by itself to performalmost all experiments required. The only additional measurement equipment which isused in this research are accelerometers and rate gyros which do more directly reflect thesimulation performance on the platform and with which one can more accurately appointthe cause of parasitic dynamics resulting from flexible modes in the shuttle or foundation.

4.2.1 Gravitational force determination

The parameters to be identified, the static mass experienced and the position of the centre ofgravity, can be used to incorporate a gravity compensation in a controller. This is importantsince gravity forces form a considerable part of the total force (up to 70 % of full pressurerequired as specified in Table 3.12). Further, a reasonable mass matrix results from takingthe centre of gravity as the origin of the moving body axes in the dynamic model.

Considering the model (2.122) in a static pose, leaves

�Gt =

��fg

�mg = ( �fg � �co)

�= JTl;x( �sx)

�fa (4.26)

with the gravity force, �fg = Tgg [0 0mg]T , transformed to the fixed lower gimbal frame by

the rotation matrix, Tgg . Usually the matrix Tgg is approximately the identity matrix.


z X

l1 2

3l l

f3f2f1

fg

oc

z X

l

1

2

3

l

l

f3

f2

f1

fg

oc

1c

1c

2c

Fig. 4.8: Simplified planar example of identifying mass and centre of gravity. Through thethree actuator forces of a parallel 3-d.o.f. system the gravity vector (length anddirection) plus line of action can be identified with two force equilibria and onemoment equation. By rotating the system and identifying another (rotated) lineof action, the intersection of both lines provides for the position of the centre ofgravity. In a spatial 6-d.o.f.-system, also two measurements are sufficient but thetwo lines of action can cross in practice. By identifying the point, which is closestto a set of lines of action, a more robust estimation can be attained.

Further, the moments, �mg, are induced by the vector product of this force with the vectorof the upper centre gimbal point to the centre of gravity, �c o. The vectors, �fg , the gravity forcei.e. platform mass times gravity and the centre of gravity, �c o, have to be identified.

From Fig. 4.8, it can be observed that one can only identify the part of �c o, which isorthogonal to �fg considering one pose as the moments result from a vector product i.e. aline of action can be identified on which the centre of gravity point resides. By measuringthe forces required in other static poses in which the platform has been rotated (around avector which is at least partly orthogonal to the gravitational direction), a set of lines canbe determined from which �co can be obtained. In theory all these lines will intersect at thispoint but in practice they will cross. The point which is closest to all lines in some sensecan be taken as best guess for the centre of gravity. Here the least squares optimal solutionof all (Euler) distances will be taken.

Consider a specific static pose measurement of the pressure differences at the valves,[ �fTg �mT

g ] = A �dPT

i J( �sx), assuming the input and output pressures are equal statically. Avalid point of the line containing the centre of gravity, ��o, can be found by

��o = � 1�fTg

�fg�fg � Png �mg ; (4.27)

with �ng = �fg=(mg) andq

�fTg�fg = mg. Now, the line of action can be described by

��o+��ng . Every line, (��o+��ng)i, found for a pose (�ci; Ti), can be transformed back to theplatform coordinates in one specific pose e.g. the neutral pose.

(�on + �n�ngn)i = T Ti (��o + ��ng)� �ci (4.28)


Configuration A B Cm [tons] 2.60 2.10 4.25�co(x) [m] 0.00 -0.12 0.00�co(y) [m] 0.00 0.00 0.00�co(z) [m] -0.23 -0.26 -0.45

Table 4.1: The Simona Research Simulator gravitational parameters

The vector, �dlp, with smallest distance of some point, �xp, to such a line is given by

�dlp = Pnng (�xp � �on) (4.29)

The vector, �co, to the point, which minimizes the sum of squared distances to n of suchlines, is given by

�co =

�1

n�ni=0Pnng;i

��1

1

n�ni=0Pnng;i�oni (4.30)

The conditioning of this optimization problem greatly depends on the ability to perform ameasurement in which the summation of the projection matrices, Pnng;i , equally span the3D-space i.e. whether the platform has been rotated sufficiently.

Due to the limited stroke of the actuators of the Stewart platform, rotation can not beperformed freely. In fact, performing valid pressure measurements can only be done ifthe actuators are out of the cushioning area (approx. 15 cm from max or min). Further,incorporating the mass of the actuators in the total mass is less tolerable with large rotations.With the SRS, the fast gravitational force identification procedure consisted of measuringpressures in five poses: neutral, +roll, -roll, +pitch and -pitch from neutral considering thelimits: (lmin + :15,lmax � :15).

The more elaborate procedure takes into account all 64 poses used in the kinematiccalibration. Over time, the mass and position of the centre of gravity of a simulator underconstruction, as the SRS, is likely to change. Therefore this procedure is to be repeatedmore often than full kinematic calibration.

The conditioning of the matrix to be inverted in (4.30) is 12 and as a result the x- andy-position of the centre of gravity can be found approx. 10 times as accurate as the z-position. Fortunately, as this point is mainly used for gravitational compensation, the x- andy-position are also much more important.

The gravitational forces of the three SRS system configurations evaluated in this re-search,

A) The dummy platform alone,

B) The shuttle and

C) The dummy platform with additional load


have been identified. The results are given in Table 4.1.In the poses considered, the observed gravity forces are varying related to a mass of

approx. 20 kg due to the relative motion of actuators w.r.t. the platform. In all cases thec.o.g. is found above the centre gimbal plane. As the dummy platform has symmetricity, itis natural to find the c.o.g. in the origin of the xy-plane. The c.o.g. of the shuttle, which issomewhat to the back, is expected to move to the front if the visual system will be added.

Measurements are best taken with a very slowly moving platform (e.g. sinusoidal motionin z-direction < :5Hz, amplitude < 2mm over a whole number of periods) to eliminatemost of the friction forces. Coulombic friction forces are small for the prismatic joint, theactuator with hydrostatically bearings, [124] but the rotation of the gimbal joint also hasfriction. These friction forces together are seen to be as large as 250 N.

Both the calibration procedure as the gravitational force determination require a simplecontroller to stabilize the system and moving the system to a specific platform pose and holdit there. The standard local (actuator per actuator) pressure feedback [153] in combinationwith positional PI-feedback with low bandwidth (1/10 of lowest eigenfrequency in neutralposition i.e. approx. 0.5 Hz) will usually be sufficient.

To determine the inertial properties of the system, discussed in the next section, exper-iments preferably require vibrations of higher frequency, e.g. approx. 4 Hz. This will stillbe attainable with a local control structure as no harsh requirements will be stated towardscross talk in the identifying procedure.

4.2.2 Identification of the inertial properties

Basic concept

The main unknown parameters left to be determined are those contained in the mass ma-trix, the inertial properties of the system. In short, these parameters will be estimated byconsidering force and acceleration (through position) measurements and evaluating thesesignals in a domain where the basic equation force is equal to mass times acceleration inmatrix form, �f = M��x, holds. The mass matrix, M , can be deduced by performing sixapproximately sinusoidal vibrations which span the 6-d.o.f. coordinate space with sufficientacceleration (position) and force amplitude to be measured accurately and sufficiently smallpositional amplitude to have a system with only moderate nonlinear effects. It is assumedthat no relevant parasitic flexible modes are hit at this harmonic.

Basically from calculation of the platform forces �fp = JTl;x( �sx)A�dP i and position

( �sx = �f�1(�l)), the amplitudes of the ground harmonic signal parts can be determined. Theamplitudes of the accelerations, ��xa, resulting from the ground harmonic with frequency ! sare easily obtained from the positional as j ��xa j=j !2s �xa j. Now, taking all experimentstogether in a matrix equation,

Mt = Fap �X�1a ; (4.31)

Mt can be identified, with for the i’th experiment M t��xai � �fapi and the force basic har-monic amplitudes, �fap, stacked in the columns of Fap and stacking the accelerational am-plitudes, ��xa, into �Xa.


0 0.5 1

−6

−4

−2

0

2

4

6

x 10−3

pitc

h (3

eps,

m)

0 0.5 1

−1

−0.5

0

0.5

1

x 10−4

0 0.5 1−0.2

−0.1

0

0.1

0.2

time (s)

mpp

(40

kNm

,kN

)

0 0.5 1−0.03

−0.02

−0.01

0

0.01

0.02

0.03

time (s)

Fig. 4.9: Decomposition of platform forces and position into ground harmonic and residualsignals in a vibrational (pitch) test of the dummy platform without additional load(configuration A.) for mass matrix determination. Upper left plot gives positionalground harmonics for the second euler parameter (pitch in three times the eulerparameter, eps) and the ground harmonics for the other d.o.f.’s. Upper right arethe residual positional measurements during the test. Lower left are the platformforces/moments ground harmonics, normalised to the maximum force of one ac-tuator (times 1m for the moments). The lower right plot provides the measuredresidual forces and moments.


Estimation in the presence of nonlinear terms

To evaluate under what conditions a reasonable estimate of the SRS motion system massmatrix, Mt, results, the nonlinear model will be considered. As proposed in the previouschapter, the mass matrix is assumed constant in the platform frame i.e. the relative motion ofthe actuator parts is not taken into account. Referring to (2.122) and including a dissipatingvelocity term, Bt( �sx) _�x,

Mt( �sx)��x+ Ct( �sx; _�x) _�x+Bt( �sx) _�x+ �Gt( �sx) = JTl;x( �sx)�fa

it is immediately clear that even in order to identify the mass matrix, M t, locally at some�sx, dynamic experiments (��x 6= 0) have to be performed in which nonlinear terms are

unavoidable. Fortunately, as the term, Ct, is quadratic in the velocity it does for mosthave second and higher harmonic signal components since 2 sin(�) cos(�) = sin(2�) and2 cos2(�) = (1 + cos(2�))). So in convolution with the ground harmonic, C t almost dis-appears. Further, the ground harmonic function can be split up into two parts. One in phasewith the position and one shifted over �=2 rad, or the part which is in phase with velocity.By only considering the part of the signal (force, position) which is in phase with position,also the dissipating forces drop out.

In Fig. 4.9 the necessary measurements are provided for one typical experiment out ofsix in estimating the mass matrix. Only taking the ground harmonic into account allowsaccurate determination of the ground harmonic accelerational amplitude. As the groundharmonics along all degrees of freedom are taken into account, no severe requirements holdfor the controller during the experiment.

Another cause left of calculating a biased estimate of the mass matrix will be the dryfriction terms. Also these are velocity (sign) related and will also mainly influence theground harmonic which is �=2 out of phase.

In practice, the system will not move in one platform coordinate only. Often the parasiticmotion will also be out of phase. The system will therefore move along an ellipsoidaltrajectory instead of a line as can be observed from Fig. 4.9, which shows phase shiftedpositional ground harmonics. This will be a possible cause of still having friction like termsin phase with the ground harmonic used in the estimation procedure.

Another nonlinear term to be considered is the position dependent jacobian, J l;x( �sx).Linearising in a static pose, �sxo, with static gravitational actuator forces, �fao results in

Mt��x = JT ( �sxo)� �fa +�6i=0

@Jl;sx( �sxo)

@sxi( �sxo)�sxi �fao (4.32)

With �sxi sufficiently small, these terms can be neglected. To check what is sufficientlysmall, the partial derivative of the jacobian, J l;sx to the states �sx is to be evaluated. Asgiven in (2.42), this jacobian consists of two kind of vectors. All the unit direction vectors,�lni of the actuators and the vector products, T �a i� �lni. Using the time derivative of �ln givenin (2.52), and decomposition with (2.47) the partial derivative to �sx is given by

@�ln@ �sx

=1

klkPlnJai;x�I 00 J!�

�(4.33)


with the transform from euler parameter variations to angular velocity given by J !� =2G(�) (2.23). Now

@(T �a� �ln)

@ �sx= T �a� @�ln

@ �sx+@T

@ �sx�a� �ln (4.34)

where @T=@ �sx follows from _T = ~!T . Validation is easily done by calculation of thedifference between two jacobians for which the pose is slightly altered.

Experiment design

Calculation of each @Jl;sx=@sxi-matrix in the neutral position gives matrices with roughlythe same gains as Jl;sx itself. With the gravity forces at the same magnitude as the dynamicsforces, the relative effect of the variation (to compensate for gravity) in the linearisationis nearly the same as the ratio of positional over accelerational amplitudes. As the testfrequency becomes higher, this ratio decreases quadratically.

In practice the test frequency of the ground harmonic, ! t is chosen somewhat belowthe lowest open loop ’rigid’ eigenfrequency. In this manner, it possible to successfullyperform the test with a simple non model based controller. In case of configuration A.and B., !t = 2�4 and for the dummy with additional load it was taken somewhat lower!t = 2�=0:3. At these frequencies it is possible to perform the measurements with reason-able pressure (up till 25 % of full load) and velocity (up till 20 % of maximum velocity)amplitudes by requiring harmonic reference signals for each platform coordinate in a sep-arate test of positional amplitude of � 10mm in the translational and :003 for k� 13k inrotational directions. Also an equivalent ratio of measurement errors in position (0.1 mm)and pressure (�25 mV noise on�10V scale) in relation to the amplitudes is attained in thisway (both 1%). Typical measurement signals, for the forces, S f , and positions, Sp, for sucha test are given in Fig. 4.9. The actual test length was t l = 5 s and the measurements weresampled at 500Hz.

Determination of the ground harmonics

The harmonic coefficients of the positional and force (pressure) measurements contain theinformation of the mass properties. They are calculated by using a matrix, H , with a basisfunction in each column over the time span 0; : : : ; t l sampled at the same frequency

H = [1 t� :5tl sin(!tt) cos(!tt) sin(2!tt) cos(2!tt) : : : ] (4.35)

Now the coefficient matrices for the platform forcesCf and positionsCp are the least squareestimates found by

[Cf Cp] = (HTH)�1HT [Sf Sp] (4.36)

For the ground harmonic (third, �c3, and fourth, �c4 rows of the coefficient matrices), thephase, �p, of the platform coordinate, which is tested, is calculated. The columns �fap and��xap of the matrices Fap and �X of (4.31) are now found by

[ �fTap ��xTap] = real(e�j�p([�c3f + j�c4f �c3p + j�c4p])) (4.37)


Configuration Place in 6x6 A B CMass matrix Mt

mx [tons] (1,1) 2.6 2.0 4.3my [tons] (2,2) 2.6 2.0 4.3mz [tons] (3,3) 2.7 2.3 4.3

Ixx [tons m2] (4,4) 2.3 2.6 4.1Iyy [tons m2] (5,5) 2.3 2.8 4.0Izz [tons m2] (6,6) 3.8 3.5 6.7Ixz;zx [tons m2] (4,5),(5,4) 0.0 0.3 0.0M15;51 [tons m] (1,5),(5,1) 0.0 0.0 0.1

Table 4.2: The Simona Research Simulator identified nonzero inertial parameters. Theseare mainly appearing on the diagonal of the mass matrix, M t, apart fromthe Ixz;zx term in case of the shuttle (configuration B.). At least 1 % error(25 � 50 kg) can be expected since this is the maximum singular value of thenonsymmetric part of the estimated mass matrix.

In this way, a maximal part of the ground harmonic for each test direction is taken intoaccount and only the part of the force and positional signal which are in phase with thismotion.

In Fig. 4.9 it is shown that a significant part of the forcing signal does not belong to theground harmonic and will not be taken into account in the mass matrix estimation.

Experimental results of the mass matrix estimation

The estimated mass matrices for the three configurations are given in Table 4.2. An advan-tage of taking the identified centre of gravity as origin in the moving body frame is that themass matrix becomes almost diagonal. The dummy platform has the advantage of havingits main axes of inertia along the platform frame axes. For the shuttle, configuration B, thexz-plane is a plane of symmetry (making the y-axis a main axis of inertia, running I yz;zyand Ixy;yx zero) but also has a cross term Ixz;zx of approx. 250 kg in case of an emptyshuttle.

In principle, the mass matrix should be symmetric. The nonsymmetricity of the iden-tified mass matrix can give some impression of the errors made due to measurement inac-curacy, etc. In case of configuration A., the maximum singular value of the non symmetricpart is 26 kg(m2) and in B. and C. it amounts to 50 kg i.e. 1 % of the estimate itself for thedummy platform and somewhat higher for the shuttle. This means no more accurate resultswere obtained than specified in Table 4.2.

Resume

� A very compact test sequence was presented, which allows estimation of the relevantdynamic mechanical parameters of the Stewart platform in the presence of hydraulic

4.3 Frequency response model validation 141

actuators.

� In practice five static poses (at least two in theory) measuring pressure and position,allowed identification of the total mass experienced and the position of the centre ofgravity.

� With six persistently exciting periodic motion tests, the inertial properties of the mo-tion system could be identified through a (6 by 6) mass matrix.

� By only taking the ground harmonic in phase with positional motion, velocity re-lated (friction) and higher order terms (nonlinearities) have minimal influence on theprocedure.

� Together with the earlier identified hydraulic actuator parameters, these mechanicalparameters allow a reasonably tight fit to practice of the complex multivariable non-linear hydraulically driven mechanical system model.

In the next section, this will be validated in comparing the responses of the models de-rived with the parameters determined with ’open loop’ motion system frequency responsemeasurements.

4.3 Frequency response model validation

At this point, the model structures have been formulated and the model parameters havebeen estimated. With this information, the calibrated models can be evaluated by compar-ison of model and actual responses. In this section the model open loop system character-istics will be validated in the frequency domain. Especially with mechanical systems, alsohydraulically driven, many important properties are revealed considering responses to sinu-soidal inputs.

Characterising nonlinear systems by describing functionsWith frequency response measurements not only the input-output relation of a linear systemcan uniquely be described but also many nonlinear system structures can be characterizedby performing harmonic excitation. E.g. a sinusoidal input describing function identifica-tion procedure approximately characterises a nonlinear system by considering amplitudeand phase of the response as a function of frequency and input sinusoidal amplitude, asextensively discussed in Van Schothorst [124]. This only requires a filter hypothesis, whichsays that the non-linear system has low-pass characteristics.

Further, the linear dynamics of a non-linear system (for some operating point) can bedefined as the describing function of the system as the amplitude approaches zero i.e. issufficiently small. The describing function can be found measuring a frequency responsesuch that the amplitude of the response is flat. This requires the design of an input amplitudefilter found e.g. by iteratively measuring the frequency response.

Appropriate experiments characterising hydraulically driven mechanical systemsSystems experiencing structural vibrations such as mechanical systems are often charac-


terised experimentally using modal analysis techniques. A large part of the modal analysisalso heavily relies on measuring frequency responses [39, 90]. With modal analysis, the ve-locity response of a mechanical system to force excitation is called mobility. With a singledegree of freedom (one mode, vibration) e.g. a mass-spring-damper system, the mobilityfrequency response has a zero at zero and at infinite frequency and a complex pole pairassuming reasonably low damping with an undamped eigenfrequency at the square root ofthe mass vs. spring stiffness ratio.

Similar responses can be expected measuring the response of our motion system frominput valve voltage to the pressure difference over the valve. As discussed in the previouschapter, the hydraulically driven mechanical system has roughly a mobility like character-istic if the response of the pressure difference over the actuator compartments is consideredfrom input actuator valve flow. Measuring from input valve voltage sets a low pass filterin series with this system thereby fulfilling the filter hypothesis. Viscous damping not onlydamps the resonance but also shifts the zero at zero frequency to the left in the complexplane. As shown in Section 3.3.1 considering the models, the response of the multi degreeof freedom multi input valve/output pressure difference of the motion system at hand is ex-pected to have a mobility like characteristic with multiple modes. Also with parasitic effectsfrom transmission lines or mechanical flexibility, this does not change, apart from inducingadditional anti-resonance and resonance pairs.

4.3.1 Frequency response dummy platform

The models of the previous chapter appointed the rigid body modes as the main characteris-tics of the hydraulically driven mechanical system. In this section, this will be evaluated bythe actual frequency responses of the dummy platform (configuration A. with no additionalload). In this configuration relatively little influence of parasitic effects, e.g. stemming fromflexible deformation, is expected to enter the frequency area of interest i.e. the frequencyarea up till 50Hz where the rigid body modes reside.

With the inertial parameters found in the previous section and the parameters for the hy-draulic actuators given in Table 3.2, the theoretic basic model structure of the hydraulicallydriven motion system given in Section 3.4.1 is specified. Calculation of the eigenfrequenciesof the rigid body modes in the neutral position for the dummy platform, configuration A.,results in 6:8 Hz for a surge/pitch and sway/roll mode, 13:6 Hz for a yaw mode, 17:7Hzfor heave and 23:4Hz for the different sign surge/pitch and sway/roll modes.

With a HP frequency analyzer ([91]) the relation at the neutral pose from each valvesteering voltage to each actuator pressure difference was measured between 5 Hz and50Hz by a swept sine at 1:5% of full input voltage. With the analyzer 201 logarithmicallyspaced datapoints per input output relation were obtained. As this analyzer only resolvesone input output relation at the time, 36 separate experiments were performed. Most of thefrequency domain measurements are given in 6x6 form in Appendix A.

The systems pose was stabilized by a moderate position feedback of gain k p = 0:2resulting in a bandwidth of � 0:05 Hz. At 5 Hz the influence of this feedback can be� 1% and grows at the first resonance to maximally 2%. At the frequencies larger than10 Hz it will be as little as 0:1% and decrease quadratically. This influence is estimated


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Fig. 4.10: Bode plots of the highly interacting responses from the first valve input voltageto the six pressure input differences (dPi), measured at the dummy platform(configuration A.) at the left. On the right side the model response (18 th order)is given incorporating the rigid body modes and three additional modes for thefoundation. Six by six separated Bode plots can be found in Fig. A.1 and Fig. A.2of Appendix A

by taking the measured ’closed loop’ gain, km, and recalling the open loop response gain,kg , to be measured is found through kg = km(1� kckm)

-1 with kc, the position loop gainincluding the transfer function from (normalized) pressure to position. For small (complex)values, kckm is the relative gain variation.

j kckm j = kpAdpnormkm(m!2)�1 (4.38)

= 0:2 � 25 � 10-4 � 200 � 105 � 7=(8:8 � 103(2�5)2)= 0:008

The measured Bode plot of the measurement from the first valve input to the six pressuredifferences is given in the left column of Fig. 4.10. The predicted rigid body modes arepresent and additionally some parasitic resonances are present between 10� 13 Hz and at


Direction Mfii [tons; tons m2] !fi [Hz] �fixf 45 2�10:5 0:05yf 45 2�10:0 0:035 f 140 2�12:5 0:05


approx. 30Hz.Additional measurements at the floor, which will be discussed in the next section,

pointed out that the vibrations at 10� 13Hz could be assigned to the foundation. By mod-elling the foundation as a 3-d.o.f mass/spring/damper system as modelled in Section 3.4.5with the parameters of Table 4.3, an 18th-order model results.

In general the damping structure of mechanical systems is hard to predict theoretically.The damping of the first three rigid body modes is well described by the experimentallyfound values for leakage and viscous friction of the hydraulic actuators. For the three highestfrequency modes some additional damping had to be introduced.

This can be explained by the fact that the gimbal rotation is probably also dissipatingenergy and not explicitly taken into account. Further, it is observed in modal analysis thatdamping does not always act as the linear viscous effect, which is part of the model usedhere. Alternatively, hysteretic damping structures are proposed in [39, 90], in which thefrequency dependence of the dissipation is changed. A drawback of that structure is that itcan not be incorporated in a state space (time-domain) model. Therefore hysteretic dampinghas not been included in the models proposed in this thesis for motion systems.

The Bode plot from the first valve input to the six pressure differences of the (6 � 6)18th-order model is given in the right column of Fig. 4.10. The measured transfer functionsare fully shown in Fig. A.1 and Fig. A.2 of Appendix A. The model gives a reasonablyclose description of the characteristics observed from the measurements. Both the phase asamplitude frequency responses are very much alike. Note that only the phase of the firstinput to the first output remains within �90� necessary for passivity.

The largest discrepancy can be observed at 30 Hz where the measurements point outan additional resonance, which is not incorporated in the model. Limited stiffness of theactuators in the radial direction are a possible cause of this vibration but this could not beconfirmed by some additional measurements of an accelerometer waxed at the actuators.Further, the additional phase lag coming in at frequencies � 40 Hz can be assigned tothe valve characteristic. The model including this effect will be evaluated with the shuttle(configuration B.). The frequency responses of that system were measured up till 500Hz.

Decoupled response by coordinate transformation

In the frequency responses of Fig. 4.10 is it not immediately clear how many modes are to beused to the describe the 6�6 system. Now we can use the basic model structure we obtainedfor hydraulically driven mechanical systems in Section 3.4.1. An important system propertyof the model is the coordinate transformation along the rigid body modal directions, which


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Fig. 4.11: Bode plots along the six estimated decoupling rigid body modal directions (with-out parasitic vibrations) from redirecting the input valve voltages and valve pres-sure differences measured (left) and modelled (right). In the upper left, thelargest singular value over the frequency of the error system regarding all thenondiagonal elements is given in the dashed plot. Six by six separated Bodeplots can be found in Fig. A.3 and Fig. A.4 of Appendix A

decouples the system into six independent systems. Each of which incorporates one of therigid body modes.

With additional parasitic modes this decoupling is not exact anymore but in the previouschapter it was found that at least in the neutral position, this transformation still puts themain modes on the diagonal. This transformation with an unitary matrix, U , can be derivedfrom the mass matrix evaluated from the actuators as defined in (3.44). A quick impressionfor these modes can attained by evaluating the singular values for each separate frequencymeasured.

In Fig. 4.11, the frequency responses of the diagonal elements of the 6� 6 transformedtransfer function are plotted. (Full transfer functions are shown in Fig. A.3 and Fig. A.4of Appendix A. In the left column, the measured response is given and on the right, themodel response is drawn. Still, the model and the measured response have very much thesame characteristics. Having the rigid body modes decoupled on the diagonal of the transferfunction enables a much easier interpretation. E.g. the phases of the system are now clearly�90� from which the conclusion can be drawn that the linearised system is passive at thispoint of operation as was predicted from the non-linear system. As predicted, the dynamics


of the foundation and other additional mechanical resonances do not influence this property.Of course, the valve characteristics will make the system lose its property of being strictpositive real and cause the additional phase shift at the higher frequencies of the measuredresponse.

In this decoupled form, the two frequency responses, the measured response and theone derived from the model, can be compared in more detail. Further, it is easier to adjustthe model in this form in order to let the model more tightly fit reality. For example thedamping structure is most conveniently changed in these coordinates since each rigid modecan be varied independently. Also parameters like massmodes (eigenvalues massmatrix)can be adjusted. In the model, the mass viewed along the z-direction can be changed toa 13% higher value to have higher correspondence of the modelled and measured z-mode.Probably the actuator masses are somewhat higher than expected.

Important is the fact that the non diagonal part of the responses is almost neglectableas the largest singular value of this part of the transfer is much smaller than the rigid bodymodes, which appear separated on the diagonal transfer functions. In the measured re-sponses it is clear that there is some distortion at the resonance frequencies coming into thelow gain parts of the other modes.Resuming, the model which describes

� the rigid body mechanical system

� together with the basic hydraulic system,

� and in this environment some additional dynamics from the non rigid foundation,

gives a reasonable description of the system over the frequency range, which will be im-portant for control (5� 50 Hz). Both the rigid modal eigenfrequencies and directions canbe predicted from the model structure together with the kinematic and inertial model pa-rameters identified. The damping structure had to be adapted somewhat using the measuredfrequency responses. Something which is not unusual since the damping structure of amechanical system is hard to predict.

Frequency domain identification

A more tightly fitted local description of the system at the neutral pose can be obtained byfrequency domain identification. A flexible tool was used to identify a multivariable modelfrom frequency domain data with the method of De Callafon et al [28] as already reportedin [75]. No reasonable results could be obtained, however, from directly fitting the original6 by 6 by 201 datapoints. Results improved drastically by separately identifying the rows ofthe data transformed by the coordinate transformation, U . Rows instead of columns weretaken as the singular directions could be predicted more closely at the directly measuredoutput than at the input through the valve.

As these models will usually contain redundant modes from non exact decoupling, es-pecially at the dominant lowest rigid modes at 6:8 Hz, the full multivariable model can bereduced. By using the block balancing approach of Wortelboer [157], the reduction problemcan be transformed to finding a good multivariable representation of each observed mode.


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Input filter shuttle frequency response measurements

Fig. 4.12: Bode amplitude plot of the 6th-order experiment notch filter.

The resulting 20th-order model describes both the six rigid modes as four additionalparasitic modes, including the modes at 30 Hz which were not incorporated in the moreglobal model with physical interpretation. The largest singular value of the difference be-tween model and data at each frequency at which the data was measured is equivalent withthe dashed line in the upper left of Fig. 4.11. Thus the error remains well below the relevantdynamics.

With the inverse coordinate transformation, U T , this model can be transformed to theactuator coordinates from which the original measurements were taken. The error is ap-proximately 10%.

More research into identification of the flight simulator motion system has to point outin what way models or model parameters can be identified which provide a tight descriptionof the system on a global scale, e.g. over all platform positions.

4.3.2 Additional dynamics into the higher frequency area

With the shuttle on top of the motion system, Configuration B., it was decided to measurethe frequency responses over the wide frequency area between 5 Hz and 500 Hz againwith a sine (up) sweep of 201 logarithmically spaced frequencies. In this way the possibleflexible modes of the shuttle and the influence of the transmission lines on the pressuredynamics, could be observed and evaluated with the system characteristics predicted by themodels obtained in the previous chapter.

In performing the experiments, more care w.r.t. the dummy platform measurements hadto be taken not to hit resonances to badly. Especially because the pilot seats, experiencingsome play in the mounting, appeared to have their resonant behaviour close to the mainrigid modes of the shuttle. Moreover, the high frequency area, containing the transmissionline dynamics, did not allow, already for the audible noise level alone, input voltage levelsnecessary to measure the low frequecy rigid mode area with sufficient signal/noise ratio.

A 6th-order experiment filter was designed to obtain a sufficiently flat response level,which is also favourable in spreading the relative effect of possible nonlinearities. The mea-sured bode amplitude plot of this filter is drawn in Fig. 4.12. Having the poles of the filter


!p1 !p2 !p3 !n1 !n22�5 2�35 2�100 2�7:8 2�42�p1 �p2 �p3 �n1 �n20:7 0:7 0:7 0:25 0:25

Table 4.4: The parameters of the input frequency response measurement filter.

appear just before the lightly damped (notching) zeros on 7:8Hz and 42Hz, gradually re-duces the amplitude level and above 100Hz filters more rigorously. The filter was designedin the continuous time domain and converted into a digital version with a sample frequencyof 5kHz by a bilinear transformation (in Matlab) for implementation on the motion controlcomputer. By measuring the response of the filter implemented on the computer, the systemfrequency responses can be adequately compensated for. The input level of the experimentscould be set on 2 % but the filter already starts to reduce this level at 6Hz.

The continuous version of the experiment filter, Uf (s) is given by

Uf (s) =(s2=!2n1 + 2�n1s=!n1 + 1)

(s2=!2p1 + 2�p1s=!p1 + 1)�

(s2=!2n2 + 2�n2s=!n2 + 1)

(s2=!2p2 + 2�p2s=!p2 + 1)(s2=!2p3 + 2�p3s=!p3 + 1)(4.39)

with the parameters specified in Table 4.4.The direct measurements of the system with the shuttle from the actuator inputs to the

input pressure differences show an even higher number of resonances than those with thedummy platform. However, the structure of the system characteristics is revealed by de-composition along the eigen directions of the rigid modes predicted through the (rigid body)model of the shuttle using the mass properties identified by the experiments discussed in theprevious section. This requires pre and post multiplication of the measured 6 by 6 transferfunction by a constant unitary matrix Ush;n and its transpose.

Also with the shuttle and over the larger frequency area, this transformation puts thedominant responses on the diagonal, which show only one rigid mode per direction, whileall 36 directly measured responses contain every mode in principal. This result and themodel predicted frequency responses are given in Fig. 4.13 and in Fig. A.7 and Fig. A.8of appendix A all plots are given seperately, while Fig. A.5 and Fig. A.6 of Appendix Aprovide the Bode plots for the untransformed frequency data.

Due to the nondiagonal inertia matrix, there are more platform directions involved ineach mode considering the neutral pose than in case of the dummy platform. Still, in thispose, one can discriminate between symmetric (x; z; �-related) and non-symmetric (y; �; -related) modes. Premultiplying the Ush;n-matrix, whose columns describe actuator dis-placements, by the inverse jacobian results in a column wise description of the modal direc-tion in the platform coordinates. Rescaling the columns to a (2-)norm of one, by postmulti-


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Fig. 4.13: Bode plots of the shuttle response redirected along the six estimated decouplingrigid body modal directions considering the valve input voltages and the valvepressure difference responses measured (left) and modelled (right). Six by sixseparated Bode plots can be found in Fig. A.7 and Fig. A.8 of Appendix A

plying with Ncsc gives:

J�1lx Ush;nNcsc =

26666664

0:987 0:000 0:000 0:083 �0:078 0:0000:000 0:988 �0:032 0:000 0:000 �0:064�0:026 0:000 0:000 0:828 0:676 0:0000:000 0:153 0:089 0:000 0:000 0:988�0:159 0:000 0:000 0:554 �0:733 0:0000:000 �0:022 0:996 0:000 0:000 �0:139

37777775

(4.40)

The eigenfrequencies, which belong to these modal directions, are 7:3, 7:3, 14:7, 20:3, 22:0,24:2 Hz. The (60th-order) model predicts these frequencies (and its directions) appropri-ately as are the transmission lines anti-resonances at approximately 70Hz and the charac-teristic due to the (anti)-resonances at 200Hz. This does not include the exact behaviour atthe resonant transmission line frequencies since only the actuator cylinder pressure differ-ence and not the absolute pressures were taken into account as was discussed in the previouschapter.


The transfer function from valves flow to input pressure differences is passive despitethe transmission lines, flexible shuttle modes and moving foundation. In Fig. 4.13 thisclearly shows by the phase moving between ��=2 rad. The valve voltage to valve flowcharacteristic with two poles modelled at 150Hz results in the additional phase shift, which(together with the transmission line resonances and flexible shuttle modes) can result instability problems when using pressure feedback.

Another property, which clearly shows, is that the hydraulic actuators are velocity gen-erators at low frequencies below the rigid body resonance. The Bode plots show that thisrequires different pressure levels since the mass experienced along the modal directionsvaries. Above the resonant frequencies, the actuators become pressure difference derivativegenerators and in this respect the fact that all actuators are similar causes indifference fordirection considering the amplitudes. A dual effect will show in considering the velocity(accelerating) behaviour over the frequency using constant valve input amplitude.

The characteristics of the foundation having resonant frequencies at 10:2, 10:7 and12:7Hz do show the same characteristics in the model and measurements. The model pa-rameters and the evaluation of the foundation causing these resonances were obtained byperforming measurements at the foundation itself, which will be discussed in the next partof this section.

The main discrepancy between the model and the actual system is the omission of theflexible modes which show at 33Hz, again, as with the dummy platform probably due tothe radial stiffness of the actuators. Further, new resonances show up most prominently at43; 57 and 65Hz and are still visible around 100Hz. These resonances were most probablycaused by the flexibility of the shuttle. To confirm this, accelerometers measurements wereperformed at several points of the shuttle.

Although the characteristics of the pressure measurements in Fig. 4.13 are for mostwell contained in the model, the model does not predict the measurements very accuratelyconsidering the numerical values. This probably requires a more thorough identificationprocedure and/or more research into the nonlinear behaviour of the system. The Bode re-sponse is based on a linearised version of the model. E.g. the nonlinear valve characteristicscould be a cause of some of the discrepancies.

The vibrating foundation

As already shown in several Bode plots, the vibrating foundation is part of the model ac-cording to the equations presented in Section 2.4.7. In principle, the parasitic resonance,which shows up in e.g. Fig. 4.13, can be caused by any flexibly attached mass at eitherside of the platform. In experiments and demonstrations it was readily verifiable that alsothe floor was moving if the motion system was made to vibrate at frequencies approaching10Hz.

To confirm that it was actually the concrete foundation, which is approx. 20 times asheavy as the shuttle of Configuration B., which causes the specific behaviour, and to helpidentifying the model parameters and structure, experiments were set up measuring theacceleration at the floor itself. Creating the experimental environment to perform the mea-surements for closer analysis of parasitic mechanical behaviour of the system (so also those


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Fig. 4.14: Bode plot amplitude responses of measured (dashed) and modelled accelerationsalong the xf (upper left), yf (upper right), x2f at (0,-2) (lower right) and con-structed angular acceleration around the z-axis, ! (lower left) due to input valvevibration along the first three ’rigid modes’.

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Fig. 4.15: Bode plot phase responses of measured (dashed) and modelled accelerationsalong the x (upper left), y (upper right), x2 at (0,2) (lower right) and constructedangular acceleration around the z-axis, ! (lower left) due to input valve vibra-tion along the first three ’rigid modes’.


in the shuttle itself) have been part of a masters’ project described in [41].Use was made of the (at that time) new compact accelerometers of Analog Devices, the

ADXL051, which evolved from airbag system initiators and now measure accelerations at arange of �5g at a resolution of 5mg from DC into reasonable high frequencies (10 kHz).Three of these sensors were mounted orthogonally in a small package, such that accelerationmeasurement in all three directions, �xf , �yf and �zf was possible.

This package was mounted on the floor in between the motion system on three placesperforming 6(x6) experiments in total assuming that the floor itself was rigid but flexiblyattached to the inertial frame. Right in between the lower gimbal points as the frame givenin Fig. 2.13, all three accelerations were measured, again while performing a sine sweepat the valve input voltage for each actuator at the time (between 5Hz and 500Hz usuallystopping the experiment if no motion could be observed anymore above 50Hz). Again, themotion system provides a nice 6-d.o.f. force inductor to perform the required experiment 2.Recording both accelerations and the forces induced is required in identifying the massand spring stiffness, which can hardly be identified if only the resonant behaviour in thee.g. pressure dynamics is taken into account.

At the coordinates (0; -2)[m] and (2; 0)[m] taken in the frame of Fig. 2.13, measurementswere performed in x and z and z direction respectively. All measurements in z-directiondid not show any acceleration (apart from gravitation), allowing to take only three directions(xf ,yf , f ) into account.

In Fig. 4.14 and Fig. 4.15, the bode amplitude and phase plots of the model and mea-surements in the xf and yf direction are given as well as for the x2f and reconstructed fdirection due to the inputs along the rigid modal directions most concerned with the motionto be observed. Clearly, in Fig. 4.14 and Fig. 4.15 both the rigid shuttle modes can be ob-served along with the mode due to the fact that the foundation is not rigidly attached to thefloor. In the neutral state, there is no interaction between the rigid shuttle modes due to thefloor but in other platform poses it can.

Although the accelerometers were relatively noisy, the characteristics observed in themodels can be recognized in the measurements. The model fails to identify a secondarymode in x-direction (at approx 12 Hz) which is hardly affecting the yaw mode (as it doeswith the model). The most relevant behaviour is, however, well described by the model. Noadditional effort has been done to more accurately describe the dynamics of the foundationas the system had to be moved to another building in the end.

The parameters used in the model of the foundation are given in Table 4.5. The total sys-tem eigenvalues, which are due to this part of the mechanics, are shifted somewhat w.r.t. thevalues given here for the foundation alone. The direction of this shift naturally depends onthe eigenfrequencies of the rigid body modes of the platform (up in case of x and y, downin case of ).

The experiments point at the importance of taking into account the dynamic behaviourof the construction of the building, foundation, best already in design, in order to prevent

1This sensor is not available anymore and has been replaced by the ADXL105 having higher resolution andtemperature compensation

2As already put forward, the hydraulic actuators are no force generators by themselves. Probably, the experi-ments could be enhanced by using some sort of force control loop.


Direction Mfii [tons; tons m2] !fi [Hz] �fixf 45 2�10:5 0:035yf 45 2�10:0 0:03 f 182 2�12:5 0:07


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Fig. 4.16: A typical frequency response with a Bode plot of measured and reconstructed(from the other measurements) acceleration approx. along the y-axis in the backleft floor position due to a sinusoidal sweep of the input voltage at the 4 th actua-tor.

this part of the system influence the characteristics of a flight simulator motion system in arelevant frequency area as it did with the experiments performed at the Central Workshopof mechanical engineering.

Accelerating a flexible shuttle

At this point only measurements were presented, which indirectly provide insight in the be-haviour of acceleration in the shuttle. By mounting a (minimal) number (of six) accelerom-eters onto the shuttle, the rigid body motion, felt by a person sitting in the simulator, can beobserved. Additional sensors can provide insight into the parasitic flexibility resulting fromthe shuttle itself.


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Fig. 4.17: Bode amplitude plots of the platform acceleration response due to the valve in-puts along the rigid body modal input directions. Left column �x; �y; �z, and on theright _!x; _!y; _!z. For each plot one of the modal direction is not dashed. It isclear that strong interaction and hard to analyze responses result from evaluationalong the platform directions. Much stronger decoupling will result evaluatingalong the rigid body modal directions at the output also.

Reconstruction of platform acceleration

To reconstruct the platform acceleration, first the gain, direction and positional parametersof the accelerometers used, have to be identified. By using a method in which the procedurein identifying the platform mass matrix and centre of gravity are combined, the sensors canbe characterised. In general, an accelerometer with gain k, mounted at position, �r, w.r.t. areference frame and directed along the unit vector, �n, will output a signal, s, determined by

s = k�nT (��c� �g + _�! � �r + �! � (�! � �r)) (4.41)

where ��c and _�! provide the reference frame translational and rotational acceleration.By performing sinusoidal motion, the accelerational amplitudes, (��c; _�!)i, in six direc-

tions, which span the motion space, can robustly be derived from the position measure-ments. By correlating with the basic sinusoidal frequency, ! t, noise and the quadratic


velocity term will be very small as in (4.36) and will be neglected. (4.41) also holds forthe ground harmonic amplitudes. With some manipulation using (4.41), the accelerometerground harmonic output amplitudes of all the experiments, �s, can be written as

�s = A �m; (4.42)

the multiplication of a known matrix, A, of platform accelerational ground harmonic am-plitudes of which each i’th row is given by

Ai;� =h�aT��c;i �aT_�!;i

i; (4.43)

and the measurement device vector, �m, with the yet unknown accelerometer parameters,direction, �n, gain, k, and line of action, �r, given by

�m =hk�nT (k�n� ( �g

!2t� �r))T

iT(4.44)

Gravity gives an offset to the line of action of the sensor which is inversely proportional withthe squared frequency, !2

t , at which the test has been performed. This offset results fromthe fact that gravity is constant in the inertial frame and the direction of the accelerometers,�n, moves (and thus rotates) with the platform. If e.g. pitch rotation occurs, gravity willshow at the accelerometer measuring in x-direction proportionally to the sine of the anglerotated. For small angles the sine operation can be dropped and the angle decreases withthe squared frequency with constant rotational acceleration. By inversion of A, the gain, k,and direction, �n, of the accelerometers can be determined.

�m = A�1�s (4.45)

The position, �r, can not, just as in the determination of the centre of gravity. Using onlythe linear terms (not the quadratic velocity term), the sensor can be anywhere on a lineparametrized by the free variable �, described by

�r(�) = ��o + ��n (4.46)

with

��o = � 1

�mT11 �m11

( �m11 � P�n �m21) (4.47)

where [ �mT11 �mT

21]T = �mT .

With at least two accelerometers, not measuring in parallel directions, in a measurementdevice, the position of this device can be defined by the point closest to both lines identi-fied as �r(�). As in the identification of the centre of gravity (4.30), incorporation of moremeasurements (accelerometers) can easily be done. In case of the shuttle measurements, adevice with three accelerometers was attached in several positions, �po, as given in Fig. 4.19.For each position, the accelerometer lines of measurement were identified and from theselines �po.


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Fig. 4.18: Bode plot of the diagonal elements of the input valve voltage to output pressuretransfer function along the rigid body modal directions reconstructed from theaccelerometer measurements. As in Fig. 4.13 the system is strongly decoupledand each plot shows only one rigid body mode. The transfer function to output(actuator) pressure difference lacks the transmission zeros at approx. 70Hz andshows phase shift much earlier in comparison with input pressure difference asoutput plotted in Fig. 4.13.

Three positions were chosen at the shuttle floor, the part of the shuttle which was ex-pected to be the most rigid (highest flexible eigenfrequencies). Just behind the pilot’s chairsin the middle, in the back at port and starboard. At the front side, the ’chin’ of the shuttle,just below the place where the window is to be placed, in the middle and at the starboardside, the redundant measurements were taken. The identified positions, using 4Hz sinu-soidal motion, are given in Table 4.6. These positions give some idea of the placing of theaccelerometers. In further calculation also the relative positioning of the measurement lines,�r(�), and their direction should be taken into account.

Assuming a rigid body shuttle floor, the rigid body frequency response can be identifiedby measuring at least six accelerometer responses at the floor. To include some redundancyfor evaluation, seven sensors were used in measuring the response to the sine sweep of allthe actuators independently. x, y and z for the mid and y and z for the (port and star) back


Position 1:mid 2:port back 3:star back 4:front mid 5:front starx [m] -0.44 -1.33 -1.32 1.21 1.11y [m] -0.18 -0.40 0.43 0.07 0.89z [m] -0.11 -0.11 -0.12 -0.39 -0.39

Table 4.6: Positions identified in the shuttle for the accelerometer device relative to thecentre of gravity in platform coordinates.

xy

13

24

5

Fig. 4.19: Approximate positions of the accelerometer device in the shuttle xy plane as alsogiven by Table 4.6.

positions. Port y is used for evaluation. The other six are used to construct the platformresponse by

��x =M�1p �s (4.48)

where each row ofMp is given by �mT of (4.45). With the platform response, ��x, it is possibleto reconstruct all the (linear) parts of the acceleration resulting from the rigid body motionat all positions of the shuttle.

Reconstructing rigid body acceleration, pressure and velocity dynamics using accelerom-eters.

Sine sweeps from 5Hz up to 500Hz with 201 logarithmically spaced frequency points weremade with 150mV amplitude send through the experiment filter of Fig. 4.12. All measure-ments presented have been compensated for this filter.

Reconstruction of rigid body redundant accelerometer measurement.In Fig. 4.16, a typical accelerometer frequency response to a sinusoidal valve voltage inputis given. It is the response of the accelerometer measuring in y-direction at the port backposition. In dashed form the reconstructed response is plotted.

Clearly visible are the rigid body resonance at 8Hz (sway/roll), 15Hz (yaw) and 24Hz(roll/sway). At 10Hz the resonance due to the foundation can be observed. Modes, whichare not identified yet, are visible at 35; 57 and 80Hz. Irrespective of the high number ofresonances, the two responses correspond quite well. Apart from the valve phase drop, the


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Fig. 4.20: Bode amplitude plot along the rigid body modal directions of valve input to thevelocities along these directions reconstructed by the accelerometer measure-ments. Again these plots are decoupled showing only one rigid mode per plotand further the convergence to the typical hydraulic actuator static velocity gaingoing to frequency zero shows.

transfer function phase start to drop below �=2 around 57Hz and further at 80Hz possiblydue to flexibility between the input (actuator) and the sensor.

Reconstructed dynamics of input valve voltage to platform accelerations.Step by step the responses can be manipulated such that a structured response results whichcan be analyzed more easily. In a first step, Fig. 4.17 provides the responses of the plat-form pose accelerations to the inputs along the rigid body directions. Almost all platformdirections (apart from the yaw-direction) are influenced by more than one rigid body mode.In the neutral pose, where the measurements where taken, it is possible to discriminate be-tween symmetric and non-symmetric modes (x; z; �) and (y; �; ).

Output pressure dynamics.More fully decoupled responses (up till the flexible resonances) can be achieved by con-sidering the reconstructed output pressure differences or velocities along the rigid bodydirections. The output pressures are reconstructed using a constant premultiplication givenby

�dP o = UTA�1J�Tl;x M��x (4.49)

containing the unitary modal direction matrix, U , the actuator area matrix, A, the jacobian,Jl;x, and the identified mass matrix, M .


actuator 1 2 3 4 5 6yaw 1 -1 1 -1 1 -1chin up 1 0 0 0 0 1torsional x-axis 1 -1 -1 1 1 -1

Table 4.7: Input directions flexible mode measurements.

In Fig. 4.18 it is shown that all the rigid modes have been decoupled in this way. Itlooks like the six accelerometer measurements allow a reconstruction of the (linear partof the) output pressure difference characteristics. The rigid modes appear on the diagonalelements of the 6x6-transfer function. For the lower frequencies, there is a considerabledifference in gain between the different directions due to the variation in mass (and thus thepressure necessary to move with the same speed). Above the rigid mode eigenfrequenciesthe gains are expected to be equal apart from the effects due to the parasitic modes.

The three assumably flexible modes at 43, 57 and 65Hz mainly appear along the di-rection of the most stiff two rigid modes. Along the pitch/surge mode direction the 43and 65Hz resonances are experienced. The 57Hz vibration comes in along the roll/swaydirection.

The pressure transfer function constructed from the accelerometers assuming a rigidconstruction should be passive apart from the influence of the transmission lines (and ofcourse the valve characteristics). Passive behaviour is certainly lost at approx. 60Hz wherethe phase along the direction of the pitch/surge mode (6) drops another � rad.

Velocity dynamics.Another alternative to reconstruct a decoupled transfer function is to use the velocities alongthe rigid body modal directions. These frequency responses are plotted in Fig. 4.20. Theseplots show the fact that for low frequencies, the hydraulic actuator is a velocity generator.This enables easy evaluation of the differences in the rigid mode resonance gains. It alsoshows that for the most stiff modes, the acceleration at the frequencies above the rigid bodymodes is the highest. Probably the (high frequency) flexible modes appear mainly alongthese directions due to the fact that they are best measured along these directions.

Flexible phenomena.A closer inspection of these flexible modes was done by evaluation of the additional ac-celerometer measurements taken at the front of the shuttle. From visual inspection of theshuttle, it can be expected that the weakest part of the shuttle appears at this point wherea large hole has been provided for the front window. Possibly torsional or chin up likedeformation could be induced. By comparison of the actual accelerations measured at thefront and those reconstructed from the (back) floor assuming rigid motion, this has beeninvestigated.

In Fig. 4.21 three examples of actually measured and reconstructed frequency responsesfrom inputs along some specific directions are given, some of which are expected to induceenergy for the flexible modes. These directions are given in Table 4.7.


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Fig. 4.21: Bode plots of measured (dotted) and reconstructed rigid part of the acceleration atfront middle in approx. y (upper row), x (middle row) and y (lower row) directiondue to input vibration along the yaw, ’chin up’ and ’torsional x-ax’ directionsrespectively. Left column provides amplitude and right column provides phaseresponses. Differences in measured and reconstructed plots can point at flexiblemodes.


In the yaw direction it can be observed that the accelerometer output at front can be pre-dicted fairly well by the accelerometers at the back of shuttle on the floor. However, noneof the flexible modes to be investigated (43,57,65 Hz) is very pronounced in this direction.For the chin up input, the reconstructed and measured output for the accelerometer mountedfor the x-direction result in totally different frequency responses. In the reconstructed fre-quency response, it is not expected that the modes at 43 and 65Hz will be observed whilethey appeared to be very pronounced in the measurements at this point. Finally, with thetorsional x-axis input it is shown that around 57Hz the phase of the bode plots starts todiffer 180�. The measurements have a complex zero pair like characteristic at 50Hz whilethe reconstructed response has not.

In many ways, the reconstructed and measured responses differ. Too much to closelypredict the characteristics of the flexible modes very accurately at this point. In some di-rections, it seemed the accelerometers were not sufficiently excited by the experiments per-formed. If exact knowledge of the flexibilities is required, more thorough modelling to-gether with further experiments is necessary. In this research the parasitic resonances weremainly investigated in order to give a proper view on the extent to which the models notincluding these effects are valid.

Resume

� Through frequency response model validation it was shown that the model structureproposed fairly well describes the characteristics of the actual system.

� Especially the rigid body modes and directions resulting from the interaction of hy-draulics and mechanics, which were predicted rightly to dominate the response, arewell caught by the model.

� Low frequency (� 10Hz) parasitic behaviour could be pinpointed to the dynamics ofthe floor, also well described by the model including the floor.

� Accelerometers could be used to reconstruct rigid body platform accelerations, veloc-ity and pressure dynamics and point at possible flexibilities, after careful identificationof accelerometer line of action and gain.

� The shuttle flexible modes could be assigned to part of the high frequent parasiticresonances but could not yet be included in the model structure. This will also requiremore rigorous measurement to allow accurate mode shape identification.

4.4 Chapter Resume

In this chapter it has been shown that the parameters of the models derived on physical lawscould be identified by performing a very limited number of experiments. The responsesof the kinematical model of the motion system did have a very close match with the mea-surements taken. The characteristics of the dynamics modelled for the hydraulically driven


mechanical system does correspond fairly well to the actually observed dynamics up tillreasonable high frequencies (� 30Hz) if the dynamics of the foundation is taken into ac-count. The properties due to the valve and transmission lines in the hydraulic system arestill important for the over all motion system. But at the frequencies where these effect havetheir main influence (� 30Hz) additional parasitic resonances due to the mechanics haveto be taken into account, especially in case of the light weight shuttle.

Chapter 5

Model based control of the flightsimulator motion system

5.1 Introduction

In previous chapters it was shown through theoretical modelling and experimental verifica-tion that in systems like the Simona flight simulator motion system, the dynamics due to thehydraulic actuators and the parallel driven mechanical system with six degrees of freedomform an integrated structure with typical characteristics, which can be modelled with rea-sonable complex equations. This is in practice the most elaborate part of design in using amodel based control strategy as laid down in this chapter.

Referring to the discussion in Chapter 1, the eventual task of high performance simulatormotion control is improved motion realism. With flight simulator motion systems one wantsto provide the pilot in the simulator with appropriate generalized specific forces i.e. bothrotational and translational accelerations plus gravity [92]. Particularly in case of tight pilotcontrol, e.g. landing, phase lag of realized versus desired accelerations should be minimal.In this way smaller differences between simulator and in-the-air flight conditions will exist.

In flight simulation, the pilot relies on the perception of self-motion through severalstimuli, and uses this perception to exercise control over the aircraft. Stimuli cueing systemsinclude the visual system, the motion system, the audio system, control loading and theaircraft instruments stimulation. In the control task, the pilot uses his visual perception tomake a good estimation of the aircraft’s attitude and velocity. The approximate frequencyresponse of visual perception can be modelled as a first order low-pass filter with a breakfrequency of 0.1 Hz [92], which is rather slow.

Motion in the higher frequency area is, however, mostly perceived by the pilot’s vestibu-lar and tactile sensors, which are sensitive to specific forces and angular accelerations.These signals are processed rapidly by the central nervous system and, therefore, give the pi-lot lead information. With this lead information, the pilot can react more quickly to changesin the vehicle motion state. This information is most important at the higher frequencies(above the bandwidth of visual perception). These high-frequency motions are, in simula-tion, often called onset-cues.

163

164 5 Model based control of the flight simulator motion system

Because of the high-frequency nature of these onsets, it is important that the motionsystem has sufficient bandwidth. Moreover, it is important that the time-delay in the motionsimulation is kept as small as possible, since an onset is also time-critical. If the onset issimulated noticeably late, simulator training quality will be decreased considerably. Timedelays, furthermore, lower the pilot-vehicle crossover frequency and may require the pi-lot to adapt by applying lead compensation. The reduction of the adaptation required isprecisely one of the goals of future flight training [7]. Therefore, a primary objective ofa flight simulator motion system is to provide the pilot in the simulator with appropriaterepresentations of, what one could call, the ”generalized forces” i.e. both translational androtational accelerations and gravity. This has to be attained by driving the simulator withsix parallel hydraulic servo actuators. With the models of the SRS, an important basis forhigh performance motion has been attained further helped by construction design for con-trol including low-mass, low centre-of-gravity, and a high-stiffness structure. Its potentialcan only be fully exploited by advanced controller design techniques such as used in thearea of robotics.

Another necessary requirement is the application of appropriate hardware components,like fast multi-processor Digital Signal Processor (DSP) boards, and software which per-forms automatic DSP-code generation from higher level programs e.g. Matlab/Simulink.Thereby, it is possible to use highly structured complex control strategies which take modelknowledge into account. This includes fast iteration on controller design methods con-sisting of setting the specifications, analyzing the system, synthesising the controller andimplementing on the actual experimental set up.

5.2 Another look at the control problem

Since the motion controller to be designed has to be implemented on a complex system, theprevious chapters dealt with extensive modelling, which took place with the control objec-tive in mind, even before construction [74], [124]. In the earlier stages measurements couldbe done on an experimental set up in which each hydraulic actuator was tested separately.Further, tests were performed with a dummy platform and the empty shuttle replacing theeventual simulator on top of the motion system [75].

Model based control strategies such as the computed torque like methods are directlyapplicable for mechanical systems modelled in the standard structure of explicit differentialequations whose states can be measured and whose control inputs are the torques which arealigned to the model coordinates. So, in applying a computed torque like technique, at leastthe following issues have to be regarded.

- Model equations: A model-based method requires modelling. Modelling parallel sys-tems typically can result in combined constraining algebraic and differential equa-tions. For instance, the motion of an actuator mass in the Stewart Platform, will befully dependent on the motion of the platform itself. Preferably the model equationswill have to be parametrized in such a way that an explicit differential structure re-sults.

5.2 Another look at the control problem 165

- Model coordinates: Parametrization of the model also includes choice of the model co-ordinates. In the Stewart Platform one can choose end-effector/ platform coordinates,input coordinate/actuator length or any other coordinate system to describe the 6 de-grees of freedom. Parallel systems are usually and most conveniently modelled inend-effector coordinates. In general these are not the coordinates which can be di-rectly measured or steered.

- Extent of modelling: Every system can be modelled in several ways and in various de-grees of accuracy and complexity. In case of the mechanics of the Stewart Platformthe simplified model of (2.112) could be taken. But next to the simulator body alsothe actuators themselves have mass properties which can not always be neglected.

- Measurements: A full state measurement is assumed. In mechanical systems this meansthat a number of independent positions and velocities equal to the number of degrees-of-freedom of the system has to be measured. On top of this, not only position andvelocity measurement are assumed to be taken, but it is also assumed that these quan-tities directly represent the model coordinates.

With the Simona motion system only actuator lengths (next to actuator pressure) aremeasured. So not only the velocity but also the appropriate model position or platformpose have to be constructed somehow.

- Control inputs: The same goes for the control inputs. These are assumed to be the gen-eralised forces along the model coordinates. These would be the platform forces incase of the Stewart Platform. Since the actuator forces have several (position de-pendent) components in platform coordinates, these can not be considered to be therequired control inputs using a standard feedback linearisation scheme without anyfurther manipulation.

Also the forces will have to be generated by an actuator which in practice cannotapply required forces instantaneously. Actuator dynamics has to be looked into. Incase of the Simona motion system, the actuators are hydraulic servo systems. Thiskind of actuators can not be considered force generators as such.

Direct application of the feedback linearising control on the motion system requiresconsideration of the various problems observed. Other aspects also need to be looked into.In the sequel several issues will be discussed, and where necessary the control strategy willbe modified.

By analysis of both theoretic models derived from basic physical laws as experimentalmodels based on measurements taken, an inventory of the relevant control problems can beput together.

� Control objectivesFlight simulation or fooling a pilots motion awareness basically forms a control prob-lem with mixed objectives. The system should provide for the accelerations beingsimulated without running out of stroke. This problem is mainly left to a host whichhas to come up with feasible trajectories but the motion controller still has to bothtrack reference accelerations as to stabilize platform pose.


� Hydraulics.Control of the long-stroke hydraulic actuators used in flight simulator motion systemsis not easy since the phase lag introduced by the servo valve together with the non-negligible high frequent transmission line resonances form a stability problem [125].

� Hydraulics/MechanicsFurther, the bilateral coupling of the hydraulic mechanical system with strong energyexchange via pressure/flow and force/velocity introduces mechanical pose dependentresonances with interaction over the actuators. As was shown, the dynamics resultingfrom this interaction forms the most relevant part of the system.

� MechanicsWith the actuators mounted in parallel to the simulator, the construction forms a Stew-art platform. Due to the resulting kinematic loops, care should be taken to model thissystem with only explicit differential equations [74]. This required modelling in ap-propriate coordinates i.e. the platform pose. As only the actuators lengths are beingmeasured, a transformation to platform pose was required to be able to apply modelbased control. This transformation is, dual to the actuator trajectory generation of se-rial robots, explicitly known from platform pose to actuator length but not injective.

� Trajectory generationThe reference acceleration and pose will be calculated at a low sample rate host com-puter which incorporates a complex airplane model. References have to be introducedsmoothly to the motion controller. But smoothing the signal should not result in re-sponses with too much phase lag since the timing of on-set motion is an importantpart of simulation quality.

In the next section, a control strategy will be described, which takes into account theafore mentioned problems and resulted in a controller which could be implemented on areal-time control computer connected to the motion system.

5.3 Control Strategy

The control strategy will be targeted at generating the appropriate accelerations for the sim-ulation and at the same time stabilising the platform pose. Taking into account the refer-ence generating system, e.g. the airplane, enables sufficiently smooth accelerating trajecto-ries. Now, acceleration of the simulator mechanical system results from the forces applied.Through a model of the mechanical system the necessary forces can be calculated. Minimalinvasive but stabilising position feedback is attained through use of a decoupled coordi-nate system. The hydraulic actuators can generate the required forces but in the previouschapters we observed that they face strong interaction from the mechanics. By supply ofappropriate oil flow the hydro-mechanical coupling can be minimized and local pressurefeedback enables the desired force generation.

By design of a control structure which has four levels, each of which have their ownspecifications in close relation with the other levels, one can circumvent having to solve

5.3 Control Strategy 167

Fig. 5.1: Multi level control of the Simona Motion System.1. Inner loop feedback: Each hydraulic actuator is feedback transformed into aforce generator by the inner loop feedback using the input pressure, dp i, and esti-mated velocity, _qi, of an actuator. Input to these subsystems are the desired inputpressures, dpi;d.2. Feedback linearisation: The desired pressures are calculated using a model (2a)of the multivariable mechanics, which calculates gravity, centripetal and coriolisforce compensation and uses the mass matrix to calculate the required forces foracceleration. This block uses the measured platform state (pose, �sx, and estimatedvelocity, _�x) to do this feedback linearisation step. This state is calculated on linefrom the measured actuator position and velocity in an iterative manner (2b).3. Outer loop feedback: This is outer loop state feedback of the platform coor-dinate and velocity errors, the difference between the desired �xs dr and _�xdr andreconstructed coordinates. Corrective accelerations, ��xc, result from this block andare fed to to feedback linearisation block (2.).4. Reference model-based feedforward: Smooth and possibly predictive acceler-ation, velocity and position, ��xdr; _�xdr; �xs, are reconstructed from the desired ac-celerations supplied by the host, ��xd. They are output to the feedback linearisationblock to be used for feed forward.

a too complex set of problems at once. Further, it is shown that this structure leads to animplementable controller.

General idea

Looking at Fig. 5.1, in which the control structure is schematically depicted and described,we consider the following control levels with reference to the applied control theory [76].

Level 1. Local hydraulic pressure control loops [49], [124], [127].

Level 2. Multivariable feedback linearisation [71], [133].

Level 3. Outer loop position stabilisation [113].


- V - - - - - -?

6

�

?

�

d d

-Llm -B

M 1

-Ap

CR R

�u �� _�x�f

Ap- -�dp

�J

JT-

�_�q

�fa

Fig. 5.2: Basic structure hydraulically driven motion system.

Level 4. Reference model based feed-forward [72], [111], [144].

In short, the actuators are turned into pressure generators by local controllers. Thesecontrollers receive their reference pressure from a feedback linearisation loop in which pres-sures can be calculated necessary to track desired accelerations. Desired accelerations arepartly corrections which are required to stabilise the pose of the simulator and for mostthe cues generated to provide the pilot with reasonable motion awareness. As these cueshave to be smoothed but not delayed, a reference model based controller has to calculateappropriate cues for the feedback linearisation controller.

The next sections will describe the different control levels more closely.

5.4 Inner loop pressure control

To turn the hydraulic actuators into nice force generators two of the afore mentioned controlproblems have to be solved at this level. Feedback of the pressure can result in stability prob-lems since the relatively long transmission lines cause badly damped resonances togetherwith phase lag of the valve. Further, the coupling between the mechanics and hydraulicsresults in the pose and load dependent rigid modes of the system.

As shown in Fig. 4.13, for the shuttle the rigid modes can be observed in the frequencyarea between 7 Hz and 25 Hz. At 200 Hz the transmission lines cause peaking and at 75 Hzalso a notch results. The valve has a bandwidth of 150 Hz which can clearly be observed bylooking at the phase. Flexibility in the mechanics caused some additional parasitic modesbetween 40 Hz and 80 Hz.

The coupling from which the rigid modes result, can be dealt with using the controlmethod introduced by Sepehri [127] and successfully applied by Heintze [49]. A hydrauli-cally driven motion system basically has the structure given in Fig. 5.2. Through the valves,V , the oil flows, ��, can be steered by the inputs, �u. The required oil flows are mainly de-termined by the speed at which the volumes in the actuators have to be filled. These areequal to the velocities, _�q, of the actuators times the area of the piston, Ap. Together withthe oil loss due to the leakage Llm, the net oil flow difference cause the pressures �dp to risethrough the hydraulic oil stiffness, C.

_�dp = C(V �u� Llm �dp�Ap _�q) (5.1)

5.4 Inner loop pressure control 169

V - - -?

6

�

d

-Llm

-Ap

CR

�u ��

Ap-�dp

-

� _�q

�faV �1dd- Kdp

A�1p

-1

?

?- - -6

6

-�dpd

Inner loopcontroller

Fig. 5.3: Basic structure inner loop controllers. On the right, the hydraulic part of the sys-tem structure of Fig. 5.2 is given. On the left, the inner loop controller, the cas-cade dp structure, compensates for the influence of the mechanics and controls thepressure, �dp, which results in a theoretically arbitrarily fast first order response de-pendent on the controller feedback gains in Kdp. In case of the SRS, _�q can not bemeasured and has to be reconstructed.

The acceleration of the actuators is determined by the inverse mass matrix, M �1, whichcauses the interaction, times the forces supplied by the actuators minus the viscous frictione.g. b due to each hydraulic bearing, B = J Tl;xbIJl;x, with Jl;x as the pose dependentjacobian.

M��x = JTl;xAp�dp�B _�x (5.2)

The actuator and platform velocities are related through _�q = Jl;x _�x. Gravity forces and theless relevant coriolis and centripetal forces are assumed to be dealt with at the higher levels.

In Fig. 5.3, the basic structure of an inner loop pressure controller, which decouples themechanics from the hydraulics, is given together with the hydraulic part of the system ofFig. 5.2. By compensation of the oil flow due to actuator velocity, the hydraulics can bedecoupled from the mechanics.

A smooth 50 Hz bandwidth pressure generator was obtained by filtering the pressurefeedback signal properly for a one degree of freedom set up in which the hydraulic actuatorwere all tested separately. As the inner loop controller should not interact with the mechan-ics its characteristics could be designed and tested in this setting at first instance [124]. Inthis way the hydraulic servo actuators, which usually are considered velocity engines, areapproximately turned into force generators. However, as already put forward, the measuredpressure difference (at the valve) does not reflect the applied force over the full relevantfrequency area due to the transmission lines. Already at 30 Hz, this difference amounts to25%.

The structure of the implemented inner loop controller as given in Fig. 5.4 is somewhatdifferent from the basic structure given in Fig. 5.3. The structure implemented on the singlehydraulic actuator as was used by Van Schothorst [124] is taken. As the velocity of thehydraulic actuator is not measured directly it has to be reconstructed and three filters, C 1,C2 and C3 have been added to deal with the transmission lines and the valve dynamics.


Roughly the filters remove signal content from the frequency area above 100Hz wherethe valve and transmission line dynamics are most prominent and only mildly change thecharacteristics up to 30Hz where the rigid body dynamics resides.

The linear dynamic filters, C1; C2; C3, used for the platform are given in the frequencydomain by

C1(s) =1

1(2�95)2

s2 + 2�0:32�95

s+ 1(5.3)

C2(s) =1

12�120

s+ 1(5.4)

C3(s) =

1(2�200)2

s2 + 2�0:22�200

s+ 1

1(2�400)2

s2 + 2�0:22�400

s+ 1

11

(2�150)2s2 + 2�0:6

2�150s+ 1

(5.5)

and for the shuttle

C2s(s) =1

12�120

s+ 1

1(2�65)2

s2 + 2�0:0152�65

s+ 1

1(2�65)2

s2 + 2�0:132�65

s+ 1(5.6)

a notch had to be added due to the parasitic resonance of the mechanical system at thisfrequency, which should not be hit. Implementation was done through the standard bilineartransformation to digital filters running at 5kHz.

The flexibility of the shuttle, which enters well above 30 Hz has still to be taken intoaccount. An illustrative picture is given in Fig. 5.5. Nyquist plots of the characteristicloci are given reconstructed from the 6x6 frequency domain measurements taken for theshuttle by taking the eigenvalues of the measured transfer function, T , ��(T (j!)), for eachfrequency, !. Recall that since the multivariable system can approximately be decoupledby a unitary matrix at each operating point, Nyquist plots can be constructed, which can beanalyzed as in the SISO case as given in (3.47).

In the upper left part of Fig. 5.5, no feedback filter is applied and a pressure feed-back gain of kdp = 0:35 is chosen. All rigid body modes and flexible resonances remain’circling’ in the right half plane as our passivity analysis of Section 3.3.1 predicted. Theadditional phase lag of the valve rotates the transmission line dynamics into the left halfplane encircling the stability point,�1.

The gain of kdp = 0:35, is well below the achievable gain of kdp = 0:75 reported onin the SISO case [124]. For the SISO case this gain could safely be used in combinationwith the dynamic filters. The parasitic dynamics due to the flexibility of the shuttle, how-ever, complicates the separation of low (rigid body modes, foundation) and high frequentphenomena (valve and transmission lines). The upper right plot of Fig. 5.5 still shows aresonance (at 65Hz) encircling the stability point. An additional notch solves the stabilityproblem as shown in the lower left and right plots of Fig. 5.5 although the resonances at 43and 57Hz become more prominent.

In theory, measured velocity compensation should remove all influences, also the flex-ibilities, from the pressure dynamics. As velocity could only be estimated with limitedaccuracy, a considerable amount of this compensation was done with the required velocityinstead. This leaves the pressure feedback dynamics for a large part unchanged.

5.4 Inner loop pressure control 171

C3 Kdp C2

K _q

_qobs

C1

e e- - - - -6

6

�

-

�

�

idpi;d

dpi

_qref

q

�dpi

+++-

�

?

Fig. 5.4: Implemented structure inner loop controllers with velocity observer and additionalfilters, Ci. The output, i, of this controller part is still an intermediate controlvariable, which is input to the valve flow compensation module of (5.7).

−10 0 10 20−10

−5

0

5

REAL

IMA

G

−5 0 5 10−10

−5

0

5

REAL

IMA

G

−5 0 5 10−10

−5

0

5

REAL

IMA

G

−1 −0.5 0 0.5−1.5

−1

−0.5

0

0.5

REAL

IMA

G

Fig. 5.5: Largest characteristic locus for pressure feedback with just feedback gain (k dp =:35) (upper left), feedback filter applied (upper right), feedback filter with notchfor the flexible platform (lower left and right).


101

102

10−1

100

101

102

−300

−200

−100

0

Fig. 5.6: Bode plot of pressure feedback filter (C2KdpC1) with flexible platform.

Given the normalized stiffness, C = 2Cm � 220, similar as in Table 3.2, this results ina bandwidth of 2Cmkdp

2�= 12:2Hz in case only the first order pressure system is considered.

A Bode plot of the resulting feedback controller is shown in Fig. 5.6.As discussed in Section 3.2.1, the valve has a nonlinear characteristic given by (3.16),

especially in the face of load variations. Van Schothorst [124] proposes a control structure

u =f�12 (i)q

1� sgn(i)dpi

(5.7)

to compensate for these effects. As this was shown to be successful experimentally, it wasmade part of the inner loop controller for the SMS.

Velocity estimationThe positive velocity compensation loop requires knowledge of the actuator velocity. Po-sition and (indirectly through pressure difference) applied force are measured. Further, itis known at what velocity the actuators should run. Again, in the thesis of Van Schothorst[124] a range of velocity observers is presented. The most simple version consists of dif-ferentiating the position signal (with some low pass filtering). With the application of theincreased resolution position measurement [55], this version seemed feasible. However, theposition measurement appeared to have a (complex deterministic) error of 100�m in ampli-tude (and� 4:3mm period length) which could result in severe vibrations running at highervelocities.

Therefore it was decided to use the other sources of information as well, i.e. the mea-sured pressures and the required actuator velocities, which can be calculated by �v des =Jl;x _�xdr. A simple second order filter was used to attain cut off at both low (no bias) and

5.5 Multivariable feedback linearisation 173

high (less noise) frequencies for all three inputs of the velocity reconstructor.

_qobs = (1� �)

1(2�6)2

s�q �dpi+ 2�0:7

2�6svdes + sq

1(2�6)2

s2 + 2�0:72�6

s+ 1+ �vdes (5.8)

The acceleration of the actuator, �q, had to be reconstructed at the higher level using theinformation on the mass properties of the system and the measured pressure differencesat the valves. The velocity compensation gain was set at k _q = 0:9, just safely below thetheoretic value of 1 as the velocity valve gain is still slightly nonlinear. Further, in practicethe feedforward part, �vdes, had to be set to 50% , i.e. � = 0:5.

The parametrization of the inner loop control, which was implemented, has more or lessbeen attained through loop shaping. A more structural approach has been taken in [150] bydoing inner loop control design using robust control techniques. This unfortunately did notlead to controllers which could be implemented safely yet. Additional effort in this directionis required.

5.5 Multivariable feedback linearisation

Considering the actuators as smooth force generators, a task of the upper level is to comeup with proper reference forces. These can be calculated using the model of the mechanicalpart of the system. A feedback linearisation structure results.

5.5.1 Feedback linearising robotic manipulators

General feedback linearisation as described in [105] can for rigid body robotic manipulatormodels be given in a computed torque control structure [133] or an inverse dynamics controlstructure [31]. In [77] it is shown that many of such differently named control strategies infact lead to the same control structure.

For a rigid body serial robotic manipulator, the controller is constructed as follows. SeeFig. 5.7. Given the general equations of a mechanical model of such a system

M(�q)��q + C(�q; _�q) +G(�q) = �� (5.9)

in which similar terms appear as in (2.112). The torques/forces, �� are assumed to drive thesystem. In the computed torque controller all terms in (5.9) are compensated for accordingto

�� =M(�q)(��qd + ��c) + C(�q; _�q) +G(�q) (5.10)

which leaves decoupled double integrators.This linear system is achieved by compensation with terms, which depend on the mea-

sured state and is therefore called feedback linearisation.The desired accelerations, ��qd, can be steered directly, and the double integrators can be

stabilised independently by the outer loop, e.g. a PD position feedback controller for eachinput-output channel


Fig. 5.7: Feedback linearising control of a robotic manipulator

��c = K(s)(�qd � �q): (5.11)

As the feedback linearising control strategy provides a suitable feedforward path fordesired accelerations and turns the non-linear multivariable structure into an easy to controldecoupled double integrator system, the method should be able to deal with most of thespecifications set for the flight simulator motion system [83].

However, in the theory presented quite some assumptions are implicitly made. Thesehave to be checked in applying the method. In the next section it will be put forward thatwith parallel robotic systems the assumed model structure does not fit in general. Thecontrol method has to be modified, which requires additional analysis.

5.5.2 Feedback linearisation of a Stewart platform

Also with parallel motion systems, the force generators should be used to control the non-linear and multivariable mechanics. With model based calculation of the required forces toaccelerate along the desired path, ��xd, given a measured pose and velocity, the system is tobe provided with both feed forward and decoupled feedback linearised correction paths tobe used by the higher level controllers.

The proposed control structure is given in Fig. 5.8. This structure differs from the stan-dard computed torque controller of a mechanical system. In modelling for control the par-allel Stewart platform configuration, one has to take care of generating an explicit set ofdifferential equations [74]. Since this is only possible taking the platform pose as the gen-eralised coordinates, the controller has to incorporate an algorithm which calculates these


Fig. 5.8: Modified feedback linearising control structure

coordinates from the measured actuator lengths, �l, and translates desired platform forcesinto required actuator pressures.

The actuator lengths, �l can be calculated from the platform pose �sx.

�l = �f( �sx) (5.12)

As discussed in Chapter 2, in measuring the actuator lengths, the platform pose has to bereconstructed iteratively e.g. by (2.58).

�sxk+1 = �sxk + J�1l;sx( �sxk)(

�lmeasured � �lk) (5.13)

With the jacobian, Jl;sx, which was defined by

Jl;sx(�x) =@�l

@ �sx(5.14)

In Chapter 2 it was shown that this iteration converges sufficiently fast if some requirementsare fulfilled.

The desired actuator pressures can be constructed by calculation of a platform massmatrixM , coriolis and centripetal forcesC _�x and gravity �G and filling in the desired platformaccelerations, ��xd for the simulation plus the corrective accelerations, ��xc, from the outer loopposition control. These are functions of the reconstructed platform pose and velocity. Thesimplified model of the Stewart platform taking into account the actuator inertia in additionto the platform mass matrix given in (2.112) will be considered and further the actuatorforces are approximated by �fa = Ap �dP o � Ap �dP i, whereAp is the normalised operationalarea of the hydraulic actuator. Multiplying (2.112) by J �T

l;x , the desired pressure is given by

�dP i;d = (ApJl;x)�T (Mt(��xd + ��xc) + C�c( _�x; �sx) _�x+ �G�c): (5.15)


In the first approach, filling in design values for these parameters like masses, inertias andcentres-of-gravity already results in reasonable performance by a considerable reduction ininteraction. This was further improved by taking identified model parameters. With theparameters of the kinematical model of (4.1) calibrated in Section 4.1.4 the jacobian, J l;x,can be calculated, as follows from (2.42). The identified platform mass matrixM t of (4.31),including actuator intertia constant w.r.t. the platform, is taken instead of the more limitedone body mass matrix M�c of (2.112). By using the identified center of gravity in (4.30) asorigin of the platform coordinates, the model parameters become relatively simple. E.g. asseen in (4.26), the gravity vector force, G �c only consists of the identified mass, mg , timesgravitational acceleration g in z-direction.

The inverse jacobian calculation was performed at 1kHz twice as required for the im-plicit state measurement. Inversion was implemented through a LU-factorisation with fullpivoting as given in [112].

Gravity compensation is easily made explicit for the actuators by use of (the third col-umn of) the inverse jacobian information as

�G = �J�Tl;x (�; 3)mg; (5.16)

Easily derived from (2.111).For coriolis and centripetal forces, only the nonlinear term given in (2.111) at platform

force level by

�fc = �! � (I�c�!) = ~I�c�! (5.17)

was taken into account. Thereby neglecting the (small) actuator related nonlinear inertialterms.

5.5.3 Implicit state measurement requirements

As the iteration (5.13) reconstructing the state is part of the feedback loop, it has to con-verge at all times and moreover sufficiently fast in order to prevent the system from goingunstable. This problem has been considered in detail in Section 2.3. Summarising, the con-vergence properties of the implicit state measurement by the NR-scheme can be looked intoby considering the weak Newton-Kantovorich theorem given by Stoer [141], stating that aNR-scheme results in a well-defined sequence in a limited area with a solution (limit point)to which it converges with a guaranteed speed if the following three properties hold.

1. The Lipschitz condition on the jacobian This condition implies that a limited differ-ence between two platform poses should result in a limited difference between thetwo jacobians at those points.

kJl;sx( �sx1)� Jl;sx( �sx2)k � k �sx1 � �sx2k (5.18)

2. ’Away’ from singular positions This means that the smallest gain of the jacobian shouldbe sufficiently far away from zero:


kJ�1l;sx( �sx)k � � (5.19)

With Stewart Platforms this is not trivially true. Singular points exist in the construc-tion if the actuator lengths are not constrained by limited stroke [97]. For example, ifa platform is moving down until all legs are in the XY-plane, no forces in Z directionor moments around X and Y axis are possible.

Singularity can be excluded by considering a nonsingular point and the Lipschitzcondition. This excludes a (possibly small) volume from singularity, and the wholeworkspace can be proved to be free of singularities by gridding. In this way theworkspace of the Simona motion system was proved to be free of, and even far enoughfrom singular points.

3. ’Near’ to solution The third and last condition requires an initial guess which is not toofar away from the real solution:

kJ�1l;sx(

�lmeasured � �lk=0)k � �: (5.20)

With the described control structure implemented on a digital computer with givensample rate (> 100 Hz) and limited speed of the system, the previous solution canbe used as an initial guess which can be proved to be close enough. The hydraulicactuators are physically limited in speed.

Since the three conditions can be proved to be true for the Simona motion system, as wasshown in detail in Chapter 2, the proposed control structure can be used without stabilityproblem concerning the iterative part of the controller. Given the speed of convergence, thesolution is close enough (within sensor accuracy) after two steps. The update frequency iseasily attained by calculating two iterations at 1 kHz.

5.5.4 Outer loop control

Considering Fig. 5.8, the transfer function from our input, the desired accelerations, �x d, tothe platform pose without the feedback pathK(s), does contain unstabilized double integra-tor paths. The outer loop controller,K(s), will have to stabilise the simulator pose to preventthe actuators from running out-of-stroke. As the feedback linearising controller decouplesthe mechanics into separate double integrators, the outer loop can generate correction ac-celerations resulting from a PD-structure (kpos + kvels) to stabilise these integrators.

The output of the outer loop controller, K(s), can be considered as an extra desiredacceleration, additional to ��xd, to correct the errors in platform pose. These correction accel-erations, ��xc, should not exceed human sensory thresholds [54] i.e. generate no noticeablefalse cues. Therefore the correction should ideally be sufficiently smooth (filtered) andonly requires limited bandwidth (well below 1 Hz). In practice, a bandwidth of 2Hz waschosen necessary to achieve sufficient suppression of the disturbances and for most the un-modelled dynamics. With the normalised signals, the feedback gains are easily chosen at


kpos = 158 = (2�2)2 and kvel = 17:5 for a damping of � = 0:7 for each platform direc-tion. The measured platform velocity is constructed from the actuator velocity estimationmultiplied by the inverse jacobian. The angular velocity, which follows from this multipli-cation can then be used to calculate euler parameter velocities.

The outer loop controller are split up into translational and rotational platform coordi-nate controllers. The outer loop for the translational directions is given by

��cc = kpos(�cd � �c) + kvel(_�cd � _�c) (5.21)

For the rotational directions, the nonlinear structure for the parameters used for the de-scription of the rotation should be used. In this case the euler parameters are taken and as�� = GTs _�!=2, equivalent to (2.22), a double integrator structure of the euler parameters canalso be considered in the platform structure of Fig. 5.8. The appropriate corrective angularacceleration to generate a corrective euler parameter acceleration can be found through

_�!c = 2G(��e)��e;c; (5.22)

where the corrective euler parameter acceleration, ��Te;c = [��0;c ��Tc ], is taken as the P(I)D

control structure on the reduced euler parameters

��c = kpos(��d � ��) + kvel(_�� _��) (5.23)

and the reconstruction of ��0;c is found by

��0;c = �(��T��c + _��Te_��e)=�0; (5.24)

twice differentiating (2.19). Of course, using this strategy, rotations, �, should satisfy� < �. A disadvantage of this structure is the nonlinear structure of the euler parame-ters. This results in different response on similar rotational errors in different poses. In [45]an alternative method is proposed, which controls difference rotation matrices but it is notclear yet how to incorporate integral control in this structure.

As the pressure generators could not be made an order of a magnitude faster than theouter loop, the references for the outer loop (the desired velocity and position) were madeto lag by using a second order filter with the estimated bandwidth of the acceleration gen-eration (12Hz). Otherwise the outer loop would start to compensate errors, which werealready taken into account.

Although a P(I)D-structure of the outer loop controller robustifies the system [113],explicit robust control will have to be used to more accurately deal with the varying systemconditions one encounters working within a real-time environment. A first step towardssuch a controller was done as part of a master thesis research [122], but this did not lead tocontrollers, which could be implemented safely. So also at this point many research aspectsare still open.

5.6 Reference model based feed forward

The motion control computer is provided with desired simulator accelerations ( ��xd) andposes ( �sxd) by a host, which controls the over all simulation (including visual, instrumental

5.6 Reference model based feed forward 179

and acoustic stimuli). These signals are generated by a model of the vehicle to be simulatedand the subsystem with the wash-out filters which translate vehicle motion into feasiblesimulator motion. Although the host comes up with new set points at a relative low updatefrequency (ca. 60 Hz), the fact that the systems being simulated are known, will enable areasonable prediction of the next set point.

The reference model based control has the task to deal with the set points and futurepredictions in a proper way. Using knowledge of the set points supplied (mainly the factthat the signals do not contain information at frequencies higher than 30 Hz) a smoothinterpolation filter provides a suitable reference acceleration to the feedback linearisationlevel together with a smooth jerk (derivative acceleration) signal which can be used as leadsignal in the same feed forward channel.

With the reference model based control considerable improvement can be achievedw.r.t. phase lag or delay in simulating on-set of abrupt (e.g. landing bump) and fast varyingmotion (e.g. turbulence) [111].

The objective of predictive reference model based feed-forward control is to reducelatencies to effectively zero. This is achieved by using knowledge of the vehicle to be sim-ulated, known as the reference model, to guide the simulator by feed-forward and feedbackcontrol, which also accounts for the motion system dynamics.

Providing the motion controller solely with the desired system’s output results in la-tencies, since pure feedback has limited bandwidth. This observation led to the key ideathat predictive knowledge from the fully available simulation model should be obtained andused in an appropriate way.

To obtain a solution that can be implemented on and properly integrated with the avail-able subsystems, namely the host and motion computer, the problem was split into thefollowing:

� reference model prediction

� reference model based feed forward control design

� systems interconnection

These problems will be treated subsequently now.

5.6.1 Reference model based predictors

The simulation model can be used as a reference signal generator. The only uncertain factoris the pilot. However, due to the relatively slow response by the human operator, it is quiteeasy to predict with reasonable accuracy the future accelerations over a short period of time(30-50 ms). The major problem is due to the complexity of the vehicle model, making itdifficult to calculate future values within the real-time environment. Approximate modelscan however be used, and several methods have been studied and evaluated [29].


Approaches

The model approximation methods ranged from the very general method of series expansion(with little knowledge of the simulation model), to the fast-time modelling approach, inwhich the structure of the simulation model is taken explicitly into account.

� Taylor series expansion.With the most recent data points of the simulated signals, a polynomial is fitted. Ex-trapolation of this polynomial results in a prediction of the future values of the signal.

� Identified linear model predictor.A linear state-space model is identified from the non-linear time-variant vehicle model.With the identified model and its state, a future prediction can be made.

� Self tuning predictor.The previous model parameters can be updated by an adaptation using a stochasticmodel and correlation methods.

� Fast time modelling (FTM).The simplified model of the equations of motion of the aircraft can be updated by thecoefficients calculated by the original (and more complex) simulation model. In theconstruction of the predicted motion, these coefficients are assumed to be constant.

The advantage of the first three methods is that they can be used more independent ofthe specific vehicle to be simulated. However, the expansion method and self tuning predic-tor appeared to suffer heavily from high frequent distortion like a turbulence model. Oneconstant linear approximation model was not able to come up with high quality predictionsat all operation points. The FTM method proved to be best suited for prediction over limitedtime horizons. It revealed constant performance under different test conditions.

The parameters transferred to a linear model of the equations of motion of the aircraftare:

� Aerodynamic coefficients,

� Thrust,

� Mass parameters,

� Aircraft state vector.

With these parameters together with the wash out filter parameters (which do not have mucheffect on the short horizon as they merely act as high pass filters) the required accelerationof the simulator can be obtained

The total of forces and moments considered acting on the aircraft is a summation ofthose generated by the engine model, aerodynamic model, landing gear model and turbu-lence model. As forces immediately result in accelerations considering the equations ofmotion, the prediction horizon should be obtained by the ability to predict the future forces


smoothly over some time frame. As thrust, aileron, elevation and rudder steering and land-ing gear model have reasonable time constants, this can be done with respect to the engine,aerodynamic and landing gear forces. The turbulence related forces vary considerably fasterbut as the dependence on the pilots actions can be assumed constant over short horizons, thestochastic nature can be simulated in advance to deliver future predictions.

5.6.2 Construction of the reference model based controller

An important aspect with respect to the reference model based control is that there is nodirect response of the acceleration (or force generation) resulting from the control inputs,�u. This is due to the presence of the valve dynamics and the finite oil stiffness in thecolumn. With the application of model-based control, the desired motion outputs (namely,the specific forces and angular accelerations) can be driven more directly. Therefore thedesign of a feed forward signal becomes more straight forward.

First, the requirements for the reference model based control task will be given. Then,the different approaches will be outlined.

Requirements

In the ideal case, the multiple-level controller would result in a transfer function from the de-sired platform accelerations, ��xd to actual accelerations ��x equal to a fast first-order response,determined by the time constant of the pressure feedback.

In practice, however, several other aspects lead to deficiencies in the system. First of all,the velocity compensation is not perfect and, as a result, the pressure feedback gains cannotbe increased to arbitrarily high values. Nonetheless, the hydraulic actuators can be turnedinto ’force generators’ with a bandwidth of about 2-3 times the lowest natural frequency ofthe rigid-mass system with finite oil spring stiffness. In the case of the SRS, this naturalfrequency, with a design load of 4000 kg, is � 4-7 Hz. Finally, the limited bandwidth of theservo valves accounts for an additional latency of about 5 to 10 ms as was shown in Fig. 3.7.

Furthermore, the reference accelerations should not have a frequency content higherthan 20-30 Hz: Undesirable deformations of the simulator result in parasitic resonancesslightly above this frequency range. The pilot’s visual system is highly sensitive to vibra-tions of the visual display system optics.

Approach

A study of literature resulted in three options for a trajectory tracking control approach.

� Servo Control. If little use can be made of knowledge of a reference model or of thesystem to be controlled, the servo control of Desoer [33] still results in the robustasymptotic tracking of a reference signal. Transient response quality, which is con-siderably relevant in motion simulation where onsets play an important role, cannotbe guaranteed however. The hydraulically driven motion system has tracking qualityw.r.t. step signals of the position if position feedback is applied due to the physicalstructure given in Fig. 5.2.


� Model Matching. Vehicle simulation can also be seen as trying to match the dynamicsof the ’washed out’ vehicle model by the dynamics of the simulator motion system.Model matching [156] can be applied with feed-forward, feed back, or both.

� Preview Control. With preview control [144], predicted references can be used in acontroller in order to yield limited phase lags for systems not having stable inverses.

The scheme, given in Fig. 5.1, results in tracking for the long term of the platform posewhich is stabilized by the outer loop position feedback. Further, as already pointed out, theinner loop feedback and feedback linearising control result in first order responses of thesystem, from desired to actual accelerations nominally. This system can be described by

Gnom(s) =1

�s+ 1I; (5.25)

where the time constant � is determined by one over the product of pressure feedback andthe oil stiffness times valve gain, CV . The oil stiffness, C, varies slightly with the positionof the actuator[124]. The valve gain also shows some non-linearity.

The immediate unit (step) response of a system can be obtained by precompensationwith an inverse system. Because the system Gnom(s) is strictly proper, the inverse is not aproper stable system. Two approaches can be taken to overcome this problem.

� First, knowledge of the reference system can be used to determine the derivative ac-celeration, the ’jerk’ signal d=dt(��xd). If

��xr = �d

dt(��xd) + ��xd (5.26)

is used as feed forward, both the phase lag and high frequent amplitude decrease arecompensated for. As high frequency components tend to be amplified in this way, oneneeds to be sure that the reference signal does not contain high frequent componentswhich could excite the parasitic resonances of the motion system. This could bereduced somewhat by filtering the reference acceleration with first order filter witha shorter time constant than � , which results in a faster but still lagging response.However, as � is not even exactly known and is not constant, the compensation canbe wrong in the mid-frequency area i.e. the bandwidth area of the motion system of10-20 Hz.

� Alternatively, using a prediction of the future acceleration over a horizon equal tothe time constant of the controlled motion system can be used to compensate for thephase lag over the relevant frequency area without unnecessarily amplifying the highfrequency area. The reference to the feedback linearising controller would then be

�xr(t) = �xd(t+ �): (5.27)

Also with this approach high frequency reference signal will not be compensated forproperly, but at least will not be amplified.


In either case, the connection of the host to the motion computer has to be such thatthe reference signal does not have a significant high frequency content. This is especiallyrelevant since the host is operating at a relative low sample rate w.r.t. the motion computer.Caution has to be taken to prevent aliasing effects from mapping into this area. This will bediscussed next.

Host/Motion computer connection

The host computer generates the simulator feasible trajectories on a sample frequency whichis usually set at 60 Hz. In earlier days even 30 Hz was seen to be used. In near future, oneis likely to be able to set this frequency at 120 Hz.

According to Shannon’s theorem, reference signal frequencies higher than half the sam-ple frequency, > 0:5fn, can not be discriminated from the area 0 : : : 0:5fn. In practice,aircraft model time constants should be well below this frequency. This alone motivates ahigher sampling frequency as also flexible effects of large aircraft or rotorcraft need to betaken into account in simulation models in the future.

Extensive literature [26] towards signal processing and filtering in the area of multi-rate signal processing exists. Since the motion system control computer requires (smooth)signals at 1-5 kHz, the lower-frequency sampled signals of the host computer will have tobe interpolated. Ideally, if the signal x(k) to be interpolated would be known from time, t,in the interval �1 to 1, filtering with the well known anti-causal filter h(t),

xc(t) =

1Xk=�1

h(t� k=fn)x(k)

=

1Xk=�1

sin(�(fnt� k))

�(fnt� k)x(k); (5.28)

would exactly reconstruct a continuous signal xc(t) with only frequency content in relevantarea without any deformation in this area. In practice none (or at best only part) of the fu-ture is known; different techniques exist to approximate h(t). Every filtering technique thentypically not only reduces high frequency signal contents, but also introduces phase lags.There are always shortcomings, which the designer must be aware of.

Choice of polynomial interpolation techniqueIn this research, several filtering techniques like Finite Impulse Response (FIR) filters, Infi-nite Impulse Response (IIR) filters and Cubic Polynomial Reconstruction (CPR), have beenconsidered. Unlike FIR and IIR filters which are designed in the frequency domain, polyno-mial interpolation techniques can be designed in time domain. This has some advantages:

� time domain criteria can be used,

� non-linear design is possible,

� interpolation functions are continuous time functions, so they are independent of theinterpolation factor.


The last point is considered important because in case of inter-connecting systems withlarge (> 50) ratios between their sampling rate (as in the case of host-motion computerintegration), CPR interpolation techniques can still be implemented as low order filters.This is unlike the FIR-filters. Further, no redesign has to done as the interpolation factorchanges, unlike the IIR-filter design.

In polynomial interpolation on the interval tk to tk+1 of order p, a function

f(�) =

pXi=0

ai�i (5.29)

is defined on the local time interval � from 0 (tk) to 1 (tk+1). The coefficients ai of thepolynomial are chosen such that the value of the signal at the sample time points, andpossibly the time derivatives thereof, satisfy certain constraints. With a third-order CPR,which was used to filter the acceleration signal, the four coefficients can for example bespecified by defining the value f(0) = x(k � 1) and f(1) = x(k) and their first timederivative _f(1) = x(k)� x(k � 1) and _f(0) = x(k � 1)� x(k � 2). If a prediction of thesignal x(k + 1) is known. the interpolation can be shifted.

With the position signal interpolation of a fifth-order function, (C)PR can be used toallow the accelerations (second order time derivatives) to correspond to the desired valuesalso. In case signals from the host would arrive late, or not at all, precaution should be takennot to extrapolate the polynomials. This could run the signal off-line very quickly. Here,by taking the next sample equal to the previous one for the position the system graduallycomes to a stop.

With CPR, the host and motion computer communicate the reference signals with a sim-ple low-order method, which has relatively little high-frequency contents. The method caneasily be combined with the use of the time-derivative of the acceleration for feed forward.This ’jerk’-signal can be extracted explicitly at every time point. Predicted time points canbe incorporated directly to reduce the phase lag.

Resume concerning the reference model based controlIn this section a number of approaches to reduce time delays or lags in the response ofsimulation motion systems were presented. A synergistic procedure was proposed in whichsystem knowledge, of both the vehicle model to be simulated and the motion system tobe accelerated, is used to have virtually immediate response to the desired motion. By adivision of sub-tasks, the modular simulation structure can be maintained.

Further research with a full operational simulator will be needed in order to point outexactly what delays are allowed while having no influence on the perceived realism of sim-ulation. Having a simulator motion system as the SRS, which is able to operate nominallywith low latencies, will help enabling this research.

Resume concerning the multi level control structureMotion control of a complex non-linear multivariable system like the Simona ResearchSimulator can be performed by using a control strategy in which the controller is structuredin multiple levels. Each level has its own specifications in close relation with the level it

5.7 Implementational issues 185

communicates with. In this way a set of various control problems can be solved more orless separately. Further, the control structure enables implementation on a multi-processordsp-system.

5.7 Implementational issues

The multiple level controller has been implemented on a real-time multi-processor DSPmotion computer [37] connected to the motion system with configuration C., a temporarydummy platform (ca. 4 tons). In this set-up, as shown in Fig. 5.9, one C40-processor has toperform all communication with the outside world (bottle-neck w.r.t. sampling frequency)and could be run at 5 kHz. Also the coprocessor, which calculates the inner loop control,runs at 5 kHz, necessary to deal with the relevant fast actuator dynamics.

In this respect the multiple level structure pays-off since the other levels, especially thefeedback linearising control ran on yet another coprocessor at 1 kHz which is just sufficientto go through all the algorithms involved.

Design of the control structure was performed in the user friendly environment of Mat-lab/Simulink1 from which c-code can be generated automatically and connected with user-written code in so called S-functions. The four processor c-code for the full model basedcontroller consisted of about 9500 lines automatically generated code from the simulinkmodel and 1850 lines of user written code (of which 600 lines came from the standard tem-plates). Though probably not optimal in efficiency, a very user friendly environment hasbeen enabled, which allows reasonable complex control structures to be implemented. Inthis way, rapid prototyping of complex controllers, as presented in this research, becomesfeasible. Going from a Simulink model to a controller running in a real time environmenttakes about 10 minutes.

5.8 Performance quantification

The evaluation procedure of the multiple level controller implemented on the flight simu-lator motion system SRS is presented in this section. The results, as partly also describedin [73], were obtained through experimental measurements with the dummy platform withadditional weights added, which amounted to the predicted final weight of 4 tons as inFig. 1.5.

The motion-base must be able to guarantee a high level of performance throughout itsworkspace. To do this, first a test procedure and a framework was defined whereby thedynamic characteristics could be quantified. A basis for the procedure is the long existingAGARD Advisory Report AR-144 [1], which quantifies several independent tests includingthe measurement of describing functions, dynamic thresholds, noise levels and the hystere-sis. This set of tests gives insight into several linear and non-linear properties of a controlledmotion system in both the time and frequency domains. The AGARD tests had to be mod-ified to include the high-frequency dynamics, and extended to enable characterisation of

1Matlab and Simulink are registered trademarks of the MathWorks, Inc.


Fig. 5.9: Motion controller hardware setup: The model based controller is implemented ona multi-C40-processor board. A master processor has to deal with all the I/O andforms the bottle-neck w.r.t. the necessary 5 kHz sampling time for the hydraulicloops. Apart from I/O the master only performs some safety checks on the out-going signals. The two slave processors calculate through the multi level controlstructure given in Fig. 5.1 On the first coprocessor (slave 1), the six inner loopcontrollers (module 1 of Fig. 5.1) are calculated at 5 kHz. They receive their ref-erence (pressure) from the second coprocessor (slave 2), whose task is to processthe mechanical model based control (modules 2 to 4 of Fig. 5.1). Finally, a thirdco-processor (slave 3), was used to generate the host signal (at 100Hz).

the system throughout its operating area. In order to arrive at a standardised approach, it isproposed that the latter be achieved by performing a set of path tracking tests. Furthermore,it was noticed that in order to enable an accurate characterisation of a high-performancemotion system, both the experimental set up and test method have to be critically observed.

Since the goal of the SRS simulation facility is to develop and validate new simulationtechnologies and to investigate human-machine interface design concepts, the performancedemands on the motion cueing capability are higher than current devices. The high level ofperformance has to be guaranteed throughout the workspace of the motion system.

Qualifying or even quantifying the performance of a non-linear, multivariable, physicalsystem is complex, especially if one wishes to compare the results with other configurationsor simulators. Ideally, first a model structure is chosen which describes the characteristicsof such systems. Then, the parameters of this model structure are identified, which requiresexperiments or testing. Finally, the models are classified by assignment of a measure or costfunction. The performance classification will at best produce results as good as the methodand devices used to measure it. The identification of rigorous models, which describe all

5.8 Performance quantification 187

systems characteristics, generally does not lead to accurate results [84]. Therefore, mo-tion systems are often classified by performing several tests to identify specific parameters,sometimes assigned within independent simplified model structures.

In case of motion simulation, the objective cueing performance should preferably bemeasured by the ability to properly stimulate the human vestibular system [54]. This con-sists of two organs; the otoliths and the semicircular canals. Consider the perception modelsgiven in Section 1.2 and the parameters provided in Table 1.1. The otholiths are primarilyand almost proportional sensitive to linear specific forces, the so-called ”regular” unit. Thetime derivatives of the specific forces, called the ”irregular” unit or ”jerk”, are also detected.The semicircular canals are proportionally sensitive to rotational acceleration in some fre-quency area (1:44� 30Hz) but are also proportional to rotational speed at lower frequency(0:026�1:44Hz). These sensitivities form the basis of the use of acceleration as the primarymetric to measure the performance of a motion system.

5.8.1 Motion system evaluation methods and requirements

At present, only one standard method to characterise the performance of a motion systemis known to exist. This is described in the AGARD Advisory Report 144 [1]. Although thisreport dates back to the end of the seventies, the technique is still up to date. Additionalaviation requirements, like JAR-STD-1A8 [118] and FAA 120-40C5 [3] generally, overlapAR-144 [1] and are basically less stringent. In robotics, even though rapid developments areunderway in design and testing parallel robots, these have not led to a generally acceptedcharacterisation standard. The AGARD report defines tests to measure

- the describing function as a frequency domain evaluation,

- dynamic threshold as the time domain response,

- noise levels to characterise parasitic motion,

- and hysteresis to identify hard non-linearities.

It uses these parameters to describe the performance in one prescribed point within theworkspace. As this is an international standard, which has been applied to several motionsystems [46], [123], it will be used as a basis in further expansion to evaluate the perfor-mance of our mechanism.

The AGARD method does have some limitations.

- First of all, it operates in only one point of the workspace, the neutral point. The dexterityin this point is often minimal [7], implying that the excursion forces are well withinthe operational limits of the motion system, and the limited non-linearity in this pointimplies that simple controllers can still perform reasonably well.

- Secondly, the simulator upper-gimbal centroid is taken as the point of reference, but thispoint is almost never the pilot’s head reference location.

- Furthermore, only a limited bandwidth of up to 10 Hz is tested in the AR-144 procedures.


- No evaluation of the limitations of the experimental set up is included. Especially inmeasuring the noise levels and hysteresis in high-performance motion systems, thediscrimination between measurement noise and system-generated parasitic motionshould be possible.

- Finally, the separate tests do not guarantee a level of performance of the system beingoperated during actual flight motion simulation. This evaluation is lacking in AR-144.

To improve the performance characterisation this section will therefore propose somemodifications to the original AGARD method in order to enable the calibration of high-performance motion systems. The newly proposed test should not be unnecessary extensiveto remain economically feasible.

The objective of this section on performance quantification is to characterise the per-formance of a high-fidelity flight simulation motion system throughout its workspace in anefficient manner. This can not be achieved with existing methods. Therefore, a first attemptis made to modify the test procedure in order to overcome the current limitations. For thesake of practicality and standardisation purposes, the experimental set up is considered to bepart of the procedure. Application to the Simona motion system will lead to a preliminarycharacterisation and an evaluation of the procedure.

Although tests exist which are independent of the system at hand, it is preferred totake the specific characteristics of a simulator motion system into account in designing aprocedure.

- A motion system is a hydraulically or electrically-driven mechanism in which the uncon-trolled dynamics usually have resonance frequencies with a low level of damping.A test procedure should point out whether the degree of damping of the controlledsystem is sufficient.

- A purely mechanical system is considered to have immediate force generation and, asa result, direct acceleration. If the dynamics of the actuators is relevant, and this iscertainly the case with hydraulics (no direct feed-through), then there will be a limitedbandwidth to be quantified.

- The use of a synergistic Stewart Platform implies that all actuators have to move to per-form pure motion of one platform degree-of-freedom. Interaction in moving alongthe different degrees of freedom easily results if not compensated for. This should betaken into account in testing. Further, the interaction and reflected masses depend onthe platform pose.

- The hydraulic actuators are symmetric, enabling similar force generation in both direc-tions. Asymmetricity can still occur if there is a pre-load, as is usually the case ifgravitational forces exist. Hydrostatic bearings are applied to reduce friction, therebyreducing hysteresis, turn-around bump and parasitic motion. As a result, there areseveral sources of possible non-linear behaviour of the system, which should be con-sidered.

5.8 Performance quantification 189

In the design of a modernised test procedure an attempt will be made to take thesecharacteristics into account.

5.8.2 New test procedure

In setting up the test procedure to describe the dynamic properties of a controlled flightsimulator motion system, the AGARD Advisory Report 144 [1] is taken as a basis. Inthe following it is discussed, which part of AR-144 is considered important, what relevantmodifications were made, and what extensions were taken into account.

Describing functionWith the describing function method, the response of a system is given along two axes, infrequency and amplitude. As extensively discussed in Van Schothorst [124], the describ-ing function method can be used to characterise a large class of non-linear systems in thefrequency domain. It is stated there that hydraulically-driven mechanical systems belong tothis class even if hard non-linearities such as Coulomb friction are relevant. If no effectslike Coulomb friction are present, as should be the case with high performance motion sys-tems with hydrostatic bearings, then the sinusoidal input describing function converges tothe frequency response of the linear dynamics for small input signals. This assumption hasto be checked, e.g. by the hysteresis and threshold tests described below.

Considering the frequency response of a linear system, several properties can be de-duced. Bandwidth can be calculated, stated as the -3 dB and/or the �45 � point. The ”peak-ing” or degree of damping of the controlled system can be found, and cross talk can beanalyzed. By taking a sinusoidal input amplitude of about 10 percent of the system’s posi-tional, speed or acceleration limits, the non linear effects are observed to have a relativelyminor influence.

By using an analog frequency analyzer, a broad frequency spectrum can be taken intoaccount and highly accurate frequency responses can be measured. This is considered to bepreferable to the method described in AGARD in which an extensive number of discrete-frequency measurements have to be taken. The limitations of this technique are howeverthat one should measure within the (amplitude) range where the system can be consideredlinear, and that no higher harmonic response is taken into account.

Threshold responseWith step response measurements, the response of the system can be analyzed in the timedomain. It is considered an easily applicable test to infer the non-linearity of the systemwith respect to different amplitudes. Originally, these kinds of tests were used to measurethe lowest amplitude or threshold acceleration to which the system still responded. Withmodern motion systems having virtually no friction, the motion-base will respond to anyamplitude and the problem at very low amplitude responses mainly results from the accuracyof the measurement apparatus.

By considering a very simple first-order linear system response model with time delay,


0 0.5 1 1.5 2 2.5 3

−1

−0.5

0

0.5

1

acce

lera

tion

(m/s

2)

0 0.5 1 1.5 2 2.5 3

0

0.05

0.1

time (s)

posi

tion

(m)

Fig. 5.10: Representative input signal, acceleration and position, for the dynamic thresholdtest.

the following structure results:

G(s) =K

�s+ 1e-�ds (5.30)

This is often a convenient structure to approximate the motion system accelerational re-sponse. If e.g. a feedback linearisation and inner loop cascade-dp is applied, the motionsystem acceleration roughly follows the first order presssure controlled dynamics. The dy-namics at higher frequencies and computational implementation mostly influences phaseand can therefore often appropriately be described by one time delay.

The parameters of this model are identified from the different step responses. The timedelay, �d, is the time that the system takes to respond, and the time constant, � , is given bythe time taken from this point to reach 63% of the final value.

With flight simulator motion systems, the acceleration step response is considered im-portant for good cueing. To measure this response, a trajectory as given in Fig. 5.10 canbe applied. One should notice that this cycled trajectory actually consists of eight differ-ent acceleration step responses. In this research, a response starting with zero velocity isconsidered.

Path trackingTo evaluate the system in its normal operating mode, and to analyze whether the propertiesmeasured at one operating point can be extended to a relevant part of the workspace, a new

5.9 Experimentally evaluating performance 191

test is introduced. Advani [7] already introduced a set of benchmark manoeuvres to checkthe design of the kinematics of a motion system. These include standard manoeuvres dur-ing flight simulation training which are considered critical in the utilisation of the simulator.Some 30 manoeuvres, responses to heavy turbulence, rejected take-off, landing, etc., areincluded. The ability of the controlled system to track these manoeuvres is considered. Atthis moment, the mean and standard deviation of the acceleration error, after being compen-sated by the time delay, and the time delay itself are taken as the parameters to be measured.With this test, two aforementioned limitations of the AGARD procedure are accounted for.

Further testsThe AGARD procedure also includes noise levels and hysteresis tests. With high perfor-mance systems, the acceleration noise levels are very close to perceivable thresholds of 1%of gravity, and become hard to measure. This is because the measurement apparatus alsointroduces measurement noise, which should be discriminated from system-generated par-asitic motion. With the gyro measurement taken with respect to rotation, the situation evenbecomes more severe as this signal has to be differentiated. With the hysteresis test, inwhich the system is made to track very low-frequency sinusoids, the situation is the same.It was concluded that at this moment these tests can (and should) only be used to checkwhether the system’s noise and hysteresis are below measurable thresholds e.g. to investi-gate if turn-around-bumps can be noticed.

5.9 Experimentally evaluating performance

In this section the results of the model based controller will be evaluated using the pro-posed test method and with the reference of a conventional controller with a decentralisedpressure/position feedback for each actuator.

5.9.1 Experimental set up

As explained in the previous sections, the test procedure consists of several trials in whichthe controlled motion system is made to perform a series of manoeuvres. During thesetests, the response of the system has to be measured. First, the design and calibration of theexperimental set up will be considered.

Performance test set upThe experimental configuration is given in Fig. 5.11. It consists of the Simona motionsystem, which is controlled by the motion computer. With the actuator extensions beingmeasured, together with the kinematics of the system, the platform pose (position and ori-entation) can be calculated on-line. In fact the system’s response with respect to the ac-celerations can also be calculated from these measurements. However, as the signals thenneed to be differentiated twice, the high-frequency component of the reconstructed acceler-ation becomes corrupted by measurement noise. Therefore, an independent data acquisitionsystem is used to measure the systems accelerating response.


Fig. 5.11: Motion control evaluation test setup: Independent of the motion control systemgiven in Fig. 5.9 implemented on the motion computer, the accelerations andangular velocities of the system are measured at the motion platform by anotherdata acquisition system (also a dSpace computer). The motion controller anddata acquisition system are synchronised by a frequency generator which is usedto trigger the start of each run and detect possible discrepancies between thesystem clocks.

With a reference measurement package consisting of three accelerometers and threerate gyros attached to the dummy platform, the local specific forces and angular velocitiesare measured. This still requires one differentiation step in order to calculate the angularacceleration.

As measurements of the motion computer and the data acquisition computer will beused in the evaluation of the test procedures, the time scale of both systems has to be syn-chronised. To achieve this, an external reference signal from a frequency generator was fedto both systems. This sinusoid was used to trigger the start of each run. The test setup isdepicted in Fig. 5.11.

Calibration measurement systemReconstruction of the actual acceleration of the platform in all six degrees-of-freedom can-not be done with the measurements taken by the reference package alone. Some additionalinformation has to be known in order to process the data properly. The gain and orientationof the measurement devices, and also their relative position, are required. Some (but not all)of these parameters can be calibrated by off-line measurements. As part of this research,


another approach was taken.In the low-frequency region, the accelerations of the platform can be reconstructed with

the actuator extensions and the kinematics of the platform as also done in Section 4.2.2.By performing a manoeuvre at a specific frequency (e.g. at f=4 Hz), a high signal-to-noiseratio can be obtained. Especially if only this component of the signal is taken into account,the non-linear components can be neglected. This frequency should be chosen such thatboth the accelerometers and the position transducers have an approximately equal signal-to-noise ratio, taking into account the accuracy and resolution of the measurement devices.Manoeuvring in six independent platform directions assures spanning the whole motionspace. Here, we assume only rigid-body motion of the platform, and a rigidly-attachedbase. This assumption is not valid in the case of the current placement of the Simonamotion system (on the floating concrete floor plate) at frequencies higher than 8 Hz.

The position and orientation of the reference package in platform coordinates can be re-constructed taking the procedure presented in the previous chapter (4.46) for the accelerom-eters used in identifying the flexible dynamics. The identification was improved further byincorporating the dynamics of the anti-aliasing filters.

This procedure was applied to the reference package attached to the dummy platformmoved by the Simona motion system. The accelerometers and gyros appeared almost per-fectly aligned with the axes of the moving reference frame. The gains also correspondedto the earlier off-line calibrated values (approx. 3.6V/g). The position of the lines of mea-surement for the accelerometers were also very much the same with respect to their rel-ative design positions. The absolute position of the reference package was identified at683mm� 1mm below the centre of gravity (which is 233mm below centre upper gimbalpoint). The x-y coordinates (forward and to the side) depend on the accelerometer at handwhich should be compensated for in the following test procedure. As this easy to applyprocedure was successful, it could be used in the future to reconstruct the relative positionin the interior of the simulator.

5.9.2 Characteristics of the Simona motion system

Given the description of the test procedure and the experimental set up, the resulting perfor-mance characteristics of the Simona motion system will be outlined.

Frequency responseFirst, the application of the describing function test will be discussed. In Fig. 5.12, boththe amplitude and phase frequency characteristics of the Simona motion system with themultiple level controller from 0.5 to 50 Hz are shown. All plots are separately shown inFig. A.9 and Fig. A.10 of Appendix A with direct comparison of the model based controllerto a conventional controller and in Fig. A.11 of Appendix A also the error response of bothcontrollers to reference signals is given.

It can be seen that the -3 dB point can be found at 13 to 15 Hz for the ’non horizontal’directions of pitch (omy), roll (omx) and heave (z). In surge (x), sway (y) and yaw (om z),the bandwidth is lower due to the motion of the current floor foundation. Except for yaw,


Direction Response time (ms) Direction Response time (ms)Surge 32 Roll 35Sway 32 Pitch 35Heave 33 Yaw 34

Table 5.1: The Simona Research Simulator latency with a model based controller.

the�45� bandwidth can be found at approx. 5 Hz. The peak amplitudes remain well below2 dB. The most relevant parasitic motion is measured between pitch due to surge and viceversa, and roll due to sway and vice versa.

For reference purposes, a conventional approach was used to control the same motionsystem. The frequency response is given in Fig. 5.13. The same pressure feedback gainis applied, however, since there is no compensation for the different natural frequencies,some responses, like surge and sway, demonstrate peaks much more than others (like pitchand roll, which are over damped). The bandwidth is twice as low, and the peaking andinteraction are twice as high with this control approach. The describing function enablesthe characterisation of some of the parameters, which are considered most important incontrol, like bandwidth, damping and interaction in a multivariable system.

Threshold responseThreshold tests have been performed for all platform directions for four different amplitudesin acceleration (1, 0.4, 0.1, 0.05 m/s2) for both negative as positive directions. The responsein the surge direction is shown in Fig. 5.14. Note that the signal amplitudes have been scaledto a desired final value of 1. As can be seen the response becomes less regular for the loweramplitudes, but the time delay (approx. 10 ms) and time constant (approx. 22 ms) do notvary too much and correspond with the�45� bandwidth (� = 1=(2�fbandwidth)) measuredwith the frequency response. Also, the smooth response without too much peaking waspredicted by the frequency response.

The characteristic parameters for the other platform directions are given in Table 5.1. Inall cases, a time delay of about 10 ms can be observed. Given the sample frequency of thehost processor of 100 Hz, such values can be expected. It should be noted that the responsesin the rotational directions were much more corrupted by measurement noise, as the signalhad to be differentiated. Especially for the low amplitude responses of 0.05 and 0.1 rad/s2,no exact time constant or delay could be extracted.

Path trackingTo check whether the characteristics as measured in one operating point would still be pre-served while performing relevant flight manoeuvres, the path tracking test was introduced.

Testing Characteristic ManoeuvresAt the KLM flight crew training centre, 31 simulator training-critical manoeuvres wereflown in a Boeing 747-400 simulator and the aircraft model responses registered. These


100

101

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>m

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Fig. 5.12: Actual model based control motion system frequency response. The Bodeplot of the closed loop system from required to actual (translational and rota-tional) accelerations applying the model based controller in the neutral position.The most important interactions are also given (dashed) and rotations are mul-tiplied by the upper gimbal radius. Highest bandwidths are attained with pitch(omy), roll (omx) and somewhat lower with heave (z). For the directions surge(x), sway (y) and yaw (omz), where the foundation is flexible and not includedin the model used to design the controller, some energy does not accelerate theplatform above 4Hz. The phase of the yaw direction does lag the others. Maybethe gyro filter was not compensated for properly. All responses are reasonablyflat without peaking more than a few percent above 100. Interaction, especiallybetween pitch to surge and roll to sway and vice versa is unexpectedly high uptill 30 % above 4Hz. The six by six Bode response plots directly comparing themodel based and conventional controller can be found in Fig. A.9 and Fig. A.10of Appendix A


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Fig. 5.13: Actual conventionally controlled motion system frequency response. TheBode plot of the closed loop system from required to actual (translational and ro-tational) accelerations applying the basically conventional control strategy in theneutral position. The most severe interactions, between pitch (omy) and surge (x)and roll (omx) and sway (y) are given (dashed) and the rotations are multipliedby the upper gimbal radius. As expected, with the conventional controller it is notpossible to tune the bandwidths of the platform accelerational loops w.r.t. eachother. Further, a compromise has to be found between peaking (up till 6 dB)of the lowest bandwidth loops of surge (x) and sway (y) and the overdampedresponse of pitch (omy) and roll (omx). The 4 Hz bandwidth of surge (x) andsway (y) is also a maximum attainable bandwidth using this control structure, theamount of peaking given and the lowest eigenfrequency of the platform. The sixby six Bode response plots directly comparing the model based and conventionalcontroller can be found in Fig. A.9 and Fig. A.10 of Appendix A


+/−1.0 m/s2+/−0.4 m/s2+/−0.1 m/s20.05 m/s2

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14−1.5

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Fig. 5.14: Short response times of the model based controller of a large range of accel-erational amplitudes. The normalised threshold or acceleration step responsemeasurements are given over a range of amplitudes for the most difficult surgedirection.

manoeuvres included a wide range of dynamic conditions, including take-offs, normal andhard landings, engine seizures, and response to heavy turbulence. These manoeuvres wereused for research into kinematical design of motion platforms by Advani [7] and referencemodel based feedforward control within a Masters project as part of this research by Pi-atkiewitz [111].

It was decided to use a set of these manoeuvres, which were dynamically most challeng-ing, as a benchmark for the SRS motion platform. Five of the thirty-one manoeuvres wereselected, since these represented the most motion-demanding conditions.

These were:

1. Response to maximum clear air turbulenceThe motion system is made to move in a stochastic manner experiencing vibrationsup to 10 Hz. Also an airpocket is included which requires almost full speed of theactuators (:9 m/s with vmax=1 m/s).

2. TaxiingFast vibrations introduced through the landing gear have to be experienced in simu-lator during which it has to move gradually through surge, sway and yaw motion.

3. Landing with cross windIn the air, the simulator has to experience the stochastic nature of cross wind. At


landing three instances of landing gear touch down bumps require fast accelerationpeak generation after which the landing slide introduces even faster vibrations up to15 Hz.

4. Rejected take offImmediate negative surge together with tilt coordination (in which gravity is madeto introduce long term breaking force) move the simulator through a wide area ofoperating conditions w.r.t the platform pose.

5. Rotation during take off rollLimited amount of motion also has to be introduced smoothly in the asymmetric(roll) motion. Faster resonances (before take off) during the first seconds have to bereplaced by the more smooth vibrations in flying.

To test the system with the load conditions of a full operational SRS the control strategyproposed was evaluated with a dummy platform of 4000 kg on top of the SRS motion sys-tem depicted in Fig. 1.4. The host computer was replaced by an additional processor on themotion computer running at a lower sampling rate of 100 Hz. In the ’host’-processor thetracks for platform pose and acceleration are made available and send to the motion com-puter control processors at demand. The original tracks were aircraft model output. By aplain wash out these trajectories were made fit for the SRS. Actuator stroke had to be within1.25 m and velocity within 1 m/s. To test the motion system with the control strategy thesesignals were considered the feasible trajectories which had to be tracked as close as possible.

Evaluation of the path trackingEach manoeuvre has a duration of 16 seconds, however, due to the limited hydraulic powersupply (temporarily) available at the Central Workshop of Mechanical Engineering this du-ration is just manageable. In Fig. 5.15, the measured heave response together with thereference acceleration, are depicted.

The measured heave signals correspond very closely to the desired acceleration. If itis considered on a short time frame an almost constant lag of approximately 30 m can beobserved. The main contribution of the lag in response is due to the smoothing filter at 30 Hz(63 % rise time of 20 ms). This motivates the use of a prediction horizon of 20-30 ms, inwhich case the lag in response could very well be reduced to a situation of ’virtual zero timedelay’. In this chapter a number of approaches to achieve this were presented.

In Fig. 5.15 part of the response during a landing with cross wind is shown. Accelerationpeaks can be observed as the three landing bumps occur. These are well represented by thecontrolled motion system. The frequency content of the signal is clearly changing as thewheels start to roll over the landing track (more high frequency rumble required). Theerrors are well below 2% of gravity. As the high-frequency part of the error signal can besignificant (due to measurement noise and also anti-alias filtering, starting at 30 Hz), it ishard to draw conclusions with respect to this frequency region.

The figure shows that the motion system reacts without gain decrease on the stochasticnature of wheel landing track contact at least up to vibrations of ca. 8 Hz (which can be seenduring the time span 8.05 s: : : 8.45 s). The ’noise’ in the measured signal is mainly caused


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Fig. 5.15: Path tracking in landing with cross wind manoeuvre with the model based con-troller considering accelerations in the platform heave (z-direction). Upper plotgives the desired accelerations in heave direction and the lower one gives theactually measured accelerations (off-set 2m=s2 brought in on purpose for com-parison)

by the application of servo valve dither. As said, the top-top values remain however wellbelow 0.02 g, which almost unnoticeable.

The other manoeuvres, not depicted here, show comparable characteristics. This provesthat this motion system control and design strategy resulted in a high-quality system. Theuse of a predictive reference model based control will even enable a synergistic connectionbetween host and motion computer, which will considerably reduce latencies.

The two aforementioned controllers, (1) the multiple level controller and (2) the con-ventional controller, were compared using the manoeuvres. The characteristic parameters,given in Table 5.2, show again that the multiple level controller has higher performance withrespect to the conventional one.

Further testsFurther, as already pointed out, noise level and hysteresis tests are also part of this proce-dure. As both were near or below measurable or noticeable levels, no exact values can begiven here. The hysteresis tests showed values less than 0.3 mm. This value was measuredgoing through a sinusoid with 0.4 m of amplitude and period time of 5 minutes, and wasstill decreasing as the period time was increased. It was observed that higher values of hys-


Manoeuvre Control Delay error Mean error Std error(ms) (m=s2) (m=s2)

Landing Model based 32 0.032 0.037Conventional 62 0.042 0.041

Taxiing Model based 34 0.022 0.019Conventional 62 0.041 0.033

Table 5.2: Comparison path tracking results in heave (delay, mean error, standard deviation(std)).

teresis appear if the dither - a high-frequency vibration in the pilot valve of the actuator - isnot properly tuned. Therefore, this test should be part of the procedure to guarantee someupper bound.

The noise levels measured with respect to the translational acceleration are acceptable,though close to measurable levels of 0.01 to 0.02 g. It should be noted that with the ex-perimental set-up used, no predictions with respect to high-frequencies (� 50 Hz) can begiven. The noise levels in the rotational directions could not be measured as there was toomuch measurement noise caused by differentiating the angular velocity. Therefore, moreaccurate gyros should be used here. The frequency responses measured in Fig. 5.12 do not,however, predict higher sensitivity in the rotational directions compared to the translations.

5.10 Chapter Resume

Almost invariantly, in every test performed, the model based controller did a far better jobthan the reference conventional controller. Higher bandwidths (8 Hz for 90 � phase lag)were measured in the frequency domain along with less peaking (max: 2 dB). Further,higher suppression of interaction was also attained, though somewhat less than expected. Inthe time domain, fast responses (response time between 30 and 35ms) were observed overa large range of amplitudes including 5mg.

In the conventional controller, a compromise had to be settled for in which some loopshave high bandwidth (still not more than 4 to 5 Hz) and too severely damped responseand other loops show already relatively high peaking up till 6 dB. Only in heave direction,in the neutral position, performance is almost equal to the model based control approach.Fortunately, for the conventional control approach, most accelerations in the simulated ma-noeuvres were required along the heave direction.

In this chapter also a test procedure was proposed to characterise the performance of ahigh-fidelity flight simulation motion system throughout its workspace in an efficient man-ner. The procedure was evaluated with the Simona motion system, which, although in pre-liminary form, already shows favourable properties due to its design and the model basedcontrol approach. The main parameters of such a system appear to be given by its lineardynamics. Measuring the frequency response is an efficient way to characterise this. Othertests such as threshold step responses, path tracking and hysteresis, can be used to quantify


to what extent these properties are maintained throughout the full operating area. Furtherresearch has to be done to identify the noise levels with higher accuracy.

Some modifications to the current standard outlined in AGARD-144 can yield a betterquantification of high-performance motion systems.

Performance of the motion system was defined by the degree of motion realism attained.There are no measures known which exactly quantify this. More research into human per-ception has to point out how this has to be done. The Simona Research Simulator couldplay a role in attaining this goal.

Chapter 6

Review and discussion on theresults

The previous chapters went over the full system/control design process of flight simula-tor motion systems. Before drawing the final conclusions towards this research, the mainconsiderations in this process will be discussed.

It has been investigated to what extent the use of relevant system knowledge in the mo-tion control strategy improves the controlled dynamics of a flight simulator motion system.This required structuring the research in several subproblems. Defining quality i.e. settingspecifications, obtaining system knowledge through theoretical modelling, quantifying andverifying these models and the relevant dynamics by experiments. Definition of a model-based but implementable control strategy and a test to validate and compare the obtainedclosed loop dynamics with conventionally controlled motion systems. In literature vari-ous solutions to many of these subproblems are proposed. Integratibility in the full designscheme and actual applicability have been the main arguments in this research in the choicefor the most suitable available alternative or, in some cases, newly proposed variant.

6.1 Flight simulation

Flight simulation is about creating a complex environment, which can be used to train pi-lots, evaluate flight system characteristics, etc. in a system which is not actually flying.Using a flight simulator is cheaper, safer, environmentally less harmful than flying an air-craft and training or evaluation can more easily be defined, standardised, modularised andrepeated. Since these advantages become more and more important, there is a continuingmarket driven tendency towards the use of flight simulators. However, requiring higher stan-dards in simulation quality. Major breakthrough in technology, mainly due to the increasein computer power, concern the visual system and the ability to take complex flight char-acteristics into account in the aircraft (environment) modelling. Motion is one the stimuliwhich can never be perfect given the limited dexterity of motion systems kinematics. Thisforemost constrains the ability to duplicate the low frequency part of manoeuvres. This canpartly be compensated for by current visual systems. The attainable quality of simulating

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204 6 Review and discussion on the results

high frequent vibration or onset motion is determined by the controlled dynamics of themotion system together with the simulation model. By design of the Simona motion systemconstruction, expectedly favourable properties such as low mass and inertia, low centre ofgravity and high rigidity were strived for. Design of a suitable controller to exploit this, isthe next step. This has been investigated in this research.

6.2 Motion system specifications

Perceived motion or realism of motion simulation is a subjective measure which can notexactly be specified. Research into motion perception models is one of the arguments intrying to attain a relatively high performance motion system. Flight simulator users explic-itly ask for quantification of required motion cues and accuracy necessary to attain a certaintraining quality. Literature provides for quite some indications in what range specificationsshould be tighter. First of all, predictable characteristics of the controlled motion systemover its full working area should be strived for to be able to perform well defined simulatortraining experiments. This alone motivates the use of as much relevant system knowledgeas possible.

Control specifications should result from the requirements set, the internal structure ofthe system to be controlled and the properties of the external signals which are expectedto perturb this system. A fundamental difference in simulator motion systems w.r.t. usualrobotics, is the fact that acceleration instead of positional accuracy is most important. In thelow frequency range (preferably below :1Hz) position stabilisation to prevent the actuatorsrunning out of stroke should prevail, but as from this frequency on, acceleration shouldbe tracked. Since the simulator is running free, tracking of acceleration reference signalsis the major task, which has to be compromised with unmodelled dynamics, uncertaintiesand measurement noise. Disturbance rejection is less evidently necessary. If the system isknown, feed forward tracking can be used. As uncertainty grows, feedback will have to beapplied.

Although there is no difference in the ratio between desired and attained position oracceleration, errors or sensitivity to external signals are to be weighted with the squaredfrequency (!2) if one evaluates accelerational w.r.t. positional accuracy in the frequencydomain. This means sensitivity to noise is much more important in simulators than it is inrobotics. As a system is typically sensitive to noise around the bandwidth, it will be moredifficult to obtain a high bandwidth simulator motion system. Use of system knowledgethrough model-based control, can prevent from unnecessary amplification of noise.

In the usual setting, the controlled motion system is in serial closed loop connectionwith the aircraft model and pilot. Both the aircraft and pilot are not exactly specified, how-ever, an aircraft typically has its highest relevant frequency modes at 2-5Hz and pilots willclose the loop in order to obtain a bandwidth of .3 to 1Hz. This is attained by providinglead over a decade around this frequency (so up till 3 Hz). To have minimal influence onthe pilot-aircraft(-model) loop, phase lag of the controlled simulator motion system shouldbe virtually zero in this frequency area, which otherwise would influence relative stabilityof this loop. This requires a bandwidth of an order of a magnitude higher (� 10 Hz).The frequency response of an aircraft evaluated w.r.t. position can be misleading as this

6.3 Theoretical modelling 205

response quickly drops above the highest mode while acceleration response remains con-stant. Apart from these loop characteristics, realism might also be improved by simulationof vibrations well above these frequencies such as turbulence, rotorcraft resonance, taxiingwheel to ground rumble and impact on landing bump. Amplitude characteristics should beunchanged in simulation of these kind of manoeuvres.

Finally the multivariable system should not introduce parasitic coupling in the otherdegrees of freedom and as this depends on the relative position, the response in each degreeof freedom should be similar.

To summarise, a controlled motion system should preferably be able to simulate ref-erences with frequency content up to 1 Hz with virtually zero time delay and an order ofa magnitude higher without amplitude attenuation. Use of system knowledge can help inattaining a predictable response without unnecessary noise amplification.

6.3 Theoretical modelling

Modelling is the first step towards model based control but also helps in analysing the char-acteristics of the open loop system, the response of the conventionally controlled systemand picking the relevant dynamics. Theoretic physical modelling enables analysis beforeactually building a system and helps in attaining an experimentally fitted model of lim-ited complexity describing the relevant dynamical features and pointing at the physics fromwhich these features result. In describing the dynamics of a flight simulator motion system,the most relevant dynamics can be found in the mechanical system, hydraulic drives and theconnection between the two.

The mechanical system is, unlike the usual robotic manipulators, a fully parallel sys-tem. This leads to some dual properties if compared with the serial connected systems. Insingular points, the manipulator becomes unsupported in one or more degrees of freedom,while in serial connected systems these become fixed. Kinematically, the parallelism posesproblems in describing the position as an explicit function of the actuated coordinates, whilein series connections these problems can occur using end effector coordinates. In describingthe dynamics, the equations of motion of a mechanical system, not choosing appropriate co-ordinates then leads to combined differential algebraic equations which results in additionalproblems in simulation or model based control. In the most general mechanical systemsboth problems occur together with possible nonholonomy, in which case it is not clear ifor which coordinates can be chosen to have a global description in explicit form. In theStewart Platforms, i.e. the flight simulator motion systems, only parallelism has to be takencare off and its dynamics can be described explicitly as a function of platform coordinates.Of course still choices have to be made in describing the spatial rotation. The use of quater-nions or Euler-parameters leads to a computationally simple description also favourable insystem analysis although an additional but easy to handle, constraint equation results.

Symbolic multibody equation of motion solvers are suitable to generate a simulationmodel in short time and without to much difficulties. These solvers, however, as they handlethe most general form of mechanical systems, do not easily recognise special properties, likethe possible explicit form of certain systems like the Stewart Platforms. Large numbers ofscalar variables result with systems of limited complexity like the motion system. They are


not suited to represent automatically appropriate vector and matrix equations or to take intoaccount equivalence between the different actuator legs. This makes system analysis moredifficult.

Using a projection method, the equations of motion of hexapod flight simulator motionsystem can be written down preserving insight. Further, this shows that in this case bothEuler-Newton as Lagrange, whatever is appropriate, methods can be used in stating thelocal equations of motion.

System analysis of multibody mechanical system shows the following. The strong cou-pling in the system evaluated along the platform coordinates is mainly due to the coupledtransposed jacobian w.r.t. the centre of gravity mapping the actuator forces onto the platformwith specific mass and inertial properties. The jacobian varies with the platform positionand forms the main non-linearity in the mechanical system. Choice of appropriate kinemati-cal parameters, mass and inertial properties in the design of the construction should result inreasonable conditioning of the mass matrix evaluated at the actuators. Inertial properties ofactuators are not very relevant in conventional systems due to high mass payload. With theSimona motion system the one body mass matrix structure is not enough. Varying relativeactuator inertia is, however, still neglectable. The centripetal and coriolis forces will remainrelatively small due to the limited velocity of the actuators and reasonably low centre ofgravity.

Connection of hydraulics and mechanics results in the most relevant modes, the plat-form pose dependent rigid modes. Direction of eigenvectors of these modes only dependon the mass matrix of mechanical system in which the jacobian plays an important part.Eigenfrequencies typically vary three to four times in magnitude as the eigenvalues of themass matrix vary ten to twenty times.

The collocated input/output pair of valve position and valve pressure difference of abasic set of hydraulic actuator models connected to a general non-linear mechanical systemis passive. This is not the case with taking resulting actuator output pressure or platformacceleration as an output. The property is also lost if valve dynamics can not be neglectedas is the case with the range where transmission line dynamics plays a role.

Passive feedback of pressure difference will not cause stability problems with any flexi-ble mode (e.g. position dependent rigid modes, fundament or simulator parasitic dynamics,(transmission lines dynamics)). Direct feedback of pressure to flow e.g. leakage providesfor an additional dissipative term like viscous friction as feedback from velocity to forcedoes in purely mechanical systems. Total energy consists of both the mechanical kineticpart as the hydraulic pressure oil stiffness part.

6.4 Experimental modelling and validation

The kinematical model forms a basic part of the dynamics. Validation of the kinematicsand identification of the parameters is the first step in identifying a full model. Redundantmeasurements are required to identify the parameters. It is important to distinguish measure-ments in calibration, which should be accurate vs. those which do not matter. Combiningtwo existing methods with limited adaptation, first redundant measurement of platform pose

6.4 Experimental modelling and validation 207

and secondly identification of the kinematic parameters, results in closely fit (within evalu-ation accuracy) kinematics. This improves positional accuracy by an order of a magnitudeand can be attained by low cost length measuring devices. The location of the devices andtheir connection on moving and rigid base is basically not important though it can influencethe sensitivity of the method. Important are the absolute lengths between the devices andthe absolute measurement of length itself.

In evaluating simulator acceleration response, the location of the accelerometers is highlyrelevant. Gain, direction and location can be identified using the kinematical model of thesystem with reasonable accuracy. Conditioning filters should also be known and taken intoaccount in evaluating measurements.

In this research, the motion system has been tested with three different load condi-tions. The unloaded dummy platform (2250 kg), the empty shuttle (1680 kg) and finally thedummy platform with full operational flight simulator load (4000 kg).

Already with a basic hydraulic/mechanical system model, the most relevant open loopdynamics can quite closely be predicted. This was confirmed by measuring the result ofsine sweeps with a frequency analyser. With the Simona motion system this rigid bodydynamics typically showed around five to thirty Hertz. Dynamics, earlier obtained from asingle actuator and resulting from the valve and transmission lines is still valid, confirmed,in the multivariable system but appears for most in the high frequency area (larger than fiftyHertz) .

With the dummy platform additional resonances could be observed in the mid frequencyarea, eight to fifteen Hertz, which resulted from a non rigid foundation at the test site asconfirmed by additional measurements with accelerometers attached to the ground. Byassuming a planar mass/spring/damper system in the fundament this additional dynamicscould be modelled, which led to a reasonable fit to the measurements taken at both thepressure dynamics in the actuator and the acceleration measurements at the ground. Withthe loaded dummy platform the dynamics of the foundation changes since a relatively higherdissipation (possibly relatively more work done by friction) can be observed leading tooverdamped characteristics. Design of the foundation at a simulator motion system test siteshould be done in careful consideration and always be evaluated properly after the motionsystem has been set up.

Accelerometer measurements in the newly designed shuttle reveal deformation at thefrequencies where resonances occur in the pressure dynamics. Though stiff (> 40Hz), themodes of the system can be very lightly damped and occur in a frequency area, which waspreviously used to attenuate pressure feedback (introducing phase lag losing the passivityproperty). Further, one of the modes of the shuttle (at 65 Hz) appeared to be almost un-observable in the valve pressure dynamics as the transmission line dynamics block (pair ofcomplex zeros) at this point. The number of measurements and the ability of the data ac-quisition system should be enlarged to identify such mode shapes and other characteristicsof a flexible system more closely. Flexible deformations can be expected to occur at muchlower frequencies with a large projection screen attached. Further, a less exciting experi-ment should be designed in case of a full operational simulator to prevent such a systemfrom any harm done.

If flexibility of the fundament can be modelled precisely, it can be compensated for,


but this is doubtful. The flexibility of the simulator results in a principle compromise sinceonly six coordinates can be functionally controlled. Problems with these kinds of parasiticmodes should for most be dealt with in design of the system. Not afterwards in control.

6.5 Control strategy and evaluation

With a conventional control strategy one uses pressure feedback to introduce extra dampingto the rigid mode resonances. With additional position feedback, a bandwidth close to thisresonance frequency can be obtained. As in this case, decentralised feedback is used, noexplicit compensation for each direction, which typically has a range of the rigid modefrequencies, is taken into account and worst case compensation, usually overdamping thesystem with respect to the variation in platform pose, will have to be settled for. Feedforward can at best locally provide for adequate amplification of desired high frequencyreferences. If the system is operated and testing is only performed in or close to any neutralposition, this can be sufficient but still can require quite some tuning. Tuning can be reducedif a model is taken into account in design.

With a model-based control strategy, one tries to compensate for the directionally dif-ferent rigid modes varying as a function of the position by taking into account the varyingjacobian and platform mass matrix. Taking into account the integration of hydraulics andmechanics will of course form the basic part of this strategy. To attain a controller, which isstructured and can be implemented in different modules, it was split up in the following subtasks.

- Inner loop decentralised actuator pressure control

- Partial feedback linearisation

- Coordinate reconstruction

- Reference model based feed forward

- Outer loop position stabilisation

As acceleration in the mid-frequency area is important, the hydraulic actuators shouldbe able to deliver the required force to attain this acceleration. With the so-called cascade-dp control one compensates for the oil necessary to move at actual velocity and feeds backthe pressure necessary to attain required force. With this principle, a bandwidth of two tothree times the rigid mode frequency can be obtained. Velocity compensation principallydecouples the influence of the mechanics on the hydraulics.

The lowest rigid mode frequency can be found at� 5Hz with the fully loaded Simonamotion system in the neutral position. It will typically change to lower values moving toother positions.

As the velocity is not measured directly, it has to be reconstructed by an observer. Bothposition and pressure measurements and the desired velocity can be used to calculate therequired compensation. Differentiation of position is simple but very limited in frequencydue to measurement errors. Pressure can be used with higher frequencies but requires the

6.5 Control strategy and evaluation 209

full multivariable mechanical model which is not exact, adds multiple uncertainties and isfurther very sensitive to friction which is small but existent in the motion system. Even withexact velocity reconstruction, velocity compensation will not be exact due to uncertainty inthe valve characteristics.

Also platform state reconstruction has to be performed since the model is only explicitin platform coordinates. An iterative scheme is used to calculate the platform pose fromactuator position measurement. Use in feedback requires convergence at all time for stabil-ity and guaranteed accuracy in a very limited amount of iterations if applied in real time.Sufficient conditions on the convergence (speed) of a general Newton-Raphson scheme canbe translated to the general Stewart Platform. The jacobian used in the scheme is shown tobe Lipschitz under mild conditions. With the Simona motion system, convergence in twoiterations within measurement device accuracy can be guaranteed given the limited actuatorspeed and a sufficiently high update frequency (100Hz), which was easily attainable.

With partly feedback linearising control of the mechanical system, one strives for twoobjectives. First, from feed forward of desired accelerations, required forces are calculatedtaking the varying mass matrix, non-linear velocity terms and gravity into account. Further,feedback of positional and velocity error is performed in a theoretically decoupled set ofcoordinates. Translation of desired platform forces to actuator forces requires inversion ofthe jacobian, which, however, can also be used with the coordinate reconstruction.

In the conventional structure of motion simulation, the series connection of controllerand motion system ideally requires an infinitely fast controlled motion system. Since theaircraft model and wash-out filters are incorporated in the simulation program, referencemodel knowledge can be used to generate a predicted acceleration over a certain amount oftime. 30-50 ms would be sufficient to attain a situation in which the acceleration onset canbe felt exactly at the required instant of time with a finitely fast motion system. This will bepossible since the model based controlled motion system behaves in a predictable manner.

As positional accuracy is not the main objective of the controller, the position feedbackshould only be used in preventing the system from running out of stroke and attaining theright accelerations in the low frequency area. Especially at the bandwidth of the positionalfeedback, care should be taken not to distort the appropriately generated accelerations.

A model based controller is able to enlarge the bandwidth of the accelerations to be sim-ulated by a factor two w.r.t. the directions of the rigid mode with the lowest frequency andalso compared to the conventional controlled system. This was confirmed in this research.By limiting the bandwidth in the other directions to attain equal response in any direction,unnecessary excitation of unmodelled dynamics, amplification of noise and coupling be-tween rotation and translation is avoided.

The dynamics of the foundation at a test site, which is not explicitly taken into account,limits the performance of the system. Success of compensation is doubtful since the effectis seen to vary a unpredictable manner w.r.t. payload characteristics.

System feedback control enforces reasonable linear response. With this assumption,the characteristics of the system can be tested by measuring the frequency response. Theassumption can be verified by performing some additional tests. In case of a motion sys-tem, the linearity can be tested with respect to a set of different amplitude accelerationstep responses and a standardised number of realistic manoeuvres representative for motion


through the workspace of the system. Exact quantification of the noise would require moresophisticated methods and measurement devices. Moreover, the noise seems to vary overtime.

A modern standardised test method is required to evaluate the performance of a simula-tor motion system. In this research some building blocks to set up such a test were provided.Main aspect in such a test should be the predictability of a system i.e. guaranteed simulationquality. The model based approach taken in this research helps to attain this.

Chapter 7

Conclusion and recommendations

7.1 Conclusion

Central to this conclusion is the main problem statement to investigate what relevant systemknowledge should be used in a model based control strategy for a flight simulator motionsystem and to what extent this results in improved controlled dynamics.

In this thesis, the path of choice became physical modelling through balance equationsof the hydraulically driven mechanical system, identification of the model parameters, ex-perimental evaluation of the model structure and explicit use of the model in the controller.

An important first observation, considering the mechanical construction of Stewart plat-form like motion system, is their limited complexity if compared to general higher degreeof freedom manipulators. The dynamics can explicitly be described as a function of a fixedset of platform coordinates unlike many combined serial/parallel constructions. This alsolimits the complexity of a model based control strategy.

The mapping between the variations along the platform coordinates and actuator coordi-nates, the jacobian, is an important parameter in control of multi degree of freedom manip-ulators such as the Stewart platform. Most of the calculations in the model based controllerpresented for the flight simulator motion system are directed towards this parameter.

In parallel systems, controllability is lost if the jacobian loses rank in some position,unlike serial robotic systems. In this thesis, a method was presented to exclude these kindof singular positions from the reachable envelope of a parallel system considering limitedstroke actuators. For the Simona Research Simulator (SRS) it was proven that such points,although not far away, can just not be reached.

If the position of a parallel system is only measured along the actuator coordinates,an inverse problem reconstructing the platform coordinates has to be solved in real timeusing the model based control strategy. In this thesis a method was presented to proveconvergence of the specific Newton-Rapshon iteration chosen anywhere in the reachableenvelope of the platform. It was proven that it does and it does sufficiently fast for theSRS. The platform coordinate reconstructor could therefore be made a central module inthe model based controller, which was implemented.

The platform pose dependent jacobian together with the platform coordinate relatedmass matrix of the system recover the specific masses as seen through the actuators. In

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212 7 Conclusion and recommendations

most flight simulator motion systems, and also the SRS, large variations of the specificmasses, more than an order of a magnitude, can be expected. Therefore, only misdirectinga force 1% easily results in a ten times higher accelerational error of 10%. Feedforwardcontrol therefore requires highly accurate models.

The kinematical model structure of the system was shown to be accurate within tensof mm. Calibration of the model parameters improved the positional accuracy an order ofa magnitude. The structural model properties of the dynamics of the hydraulically drivenmechanical motion system were also confirmed. However, estimation of the system massproperties can probably be improved despite the scheme developed to robustly derive themass properties of such systems in the presence of dissipation or some nonlinear effects.

The specific mass directions are almost equal to the rigid body modal directions con-necting the hydraulic actuators. In analysis and identification of hydraulically driven motionsystems, knowledge of these directions proved to be very helpful to decouple the, at firstsight, highly interactive dynamics. Uncontrolled hydraulically driven systems are usuallylightly damped and this increases the already large difference along the rigid body modaldirections in the neighbourhood of the resonance frequencies. The rigid body resonancemodes form the main dynamics in the most relevant frequency area for flight simulation uptill 30Hz.The conventional control structure robustifies the decentralised positional feedback throughdissipating energy by pressure feedback thereby damping the resonance peaks. In this thesisthis strategy proved right by showing that feedback of the input valve pressure differenceto valve flow is feedback of a passive transfer function, also in the presence of parasiticdynamics e.g. due to transmission line resonances or mechanical flexibilities. However,no directionality can be accounted for in this structure. It was shown that this leads tolimited bandwidth systems, which still have to compromise overdamped directions withsome peaking of others.

With the model based strategy, the already known cascade dp structure was chosen,which maintained local decentralised dissipating pressure feedback by inner loops and de-coupled hydraulics from mechanics. This leaves the directionality to be dealt with by themultivariable feedback of the mechanics. Through implementation of this controller onthe SRS it was shown that this indeed leads to a system with higher bandwidth and moreequalised response in each degree of freedom.

The attainable improvement through the model based controller chosen is limited in sev-eral ways. Not all dynamics was or could be made part of the model used in control. In test-ing the SRS, of course, the dynamics of the foundation appeared in the relevant frequencyarea due to the non ideal experimental environment. But this should not be a problem in aspecific simulator building. However, it can be concluded that the absence of such dynamicsshould be confirmed, tested, in any place where motion systems are to be used or evaluated.

The flexible deformations of the system are a more severe problem. This causes loss offunctional controllability, strived for in simulation. Adding additional objects, e.g. projec-tion screens, will further limit the attainable performance.

The phase lag of the hydraulic valve usually limits the attainable feedback pressurebandwidth. With fast valves one can choose this bandwidth in between the rigid bodymodes and the transmission lines resonances but exactly in this frequency area the flexible

7.2 Recommendations 213

resonances show up if velocity is not precisely compensated for, as is impossible in practice.The physical modelling in this research enabled a limited complexity model based con-

trol structure for flight simulator motion systems by choosing only the most relevant dynam-ical effects to be incorporated. Also the parameter identification procedure could be madevery compact in this way. The model based controller proved better in almost all respects ifcompared to a conventional control structure implemented on the actual motion system.

Summarising, the full design process proposed was shown to be applicable and leads tomore predictable characteristics and higher performance of flight simulator motion systems.

7.2 Recommendations

This thesis systematically went through the full design process of motion control design offlight simulator motion systems but it still leaves many points of research open for morethorough investigation. Some at the high abstract level of the over all simulation, some atthe lower level of detail though in practise often the most troublesome.

First of all, specifications towards the design of motion controllers in simulation shouldbecome more precise in stating in what way the fundamentally non exact motion simulationcharacteristics should be compromised for. E.g. how much more accelerational noise canbe settled for in attaining one additional Hertz of bandwidth? What class of signals willhave to be tracked by the motion system? Usually there is much information in this area,structure in the trajectories, but not yet very well specified. Short time spectral propertiesof the reference signals is relevant in this sense since otherwise the important aspects, highfrequency content, of e.g. a landing bump, would not be revealed.

These specifications should form the basis for a standardised test to measure the per-formance and predictability i.e. robustness of a motion system. In this research a numberof tests were introduced to measure the characteristics of a controlled motion system in amore modernised manner as compared to the current flight simulation standards. But onlyproviding more specific guidelines, weighting factors w.r.t. desired properties of a motionsystem, will enable the design and analysis of the right test and can lead the way towardsthe most effective control strategy.

Information from the flight simulation model can also be used more conveniently in thecontrol scheme itself. Limited short response time of the required pressure results in laggedaccelerational response of the motion system as compared to what was required in simu-lation. As the spectrum of the desired accelerations has almost always limited bandwidth,predictive information theoretically is sufficient to exactly attain the required response. Inthis thesis it was shown how this information can be used. Flight simulation models shouldbe set up in such a way that this predictional information can be communicated to the mo-tion system controller. It is the most viable way to attain virtual zero time delay response insimulation.

Dynamics, not taken into account by the model based controller, occurred in practise.The most important part was shown to be caused by the flexible deformations of the system,which in future will become even more relevant. Procedures to identify the specific structureof these parasitic resonances will have to be set up and applied. With more specifically

214 7 Conclusion and recommendations

directed identification techniques in general, parameter estimation of the system model canprobably be improved further.

The control strategy should be extended in order to deal effectively with the uncertainor identified flexible dynamics. One could even consider applying additional actuators,to dissipate energy stemming from local resonant modes, to solve the current problem offunctional controllability in the presence of deformations. Also additional sensors, suchas the accelerometers applied in identification, could be considered in an extended controlstructure. After all, acceleration felt in the cockpit is what really matters in motion cues forflight simulation.

Dealing with uncertainty in general in control requires robust feedback/feed forwardschemes. Models can never be exact. Unfortunately many modern robust control techniquesare not mature enough at the moment to deal efficiently with nonlinear six d.o.f. hydrauli-cally driven motion systems. In the side line of this research some first steps towards robustcontrol of such systems were taken but this did not lead to satisfying results yet. Still, at-taining robustly guaranteed properties in simulation is important in flight training and makesmore effort in this direction worthwhile.

W.r.t. the specific model based control structure chosen, the interaction between the in-ner and outer loops appeared to be more severe than expected. More exact velocity compen-sation, central in the decoupling of mechanics and hydraulics, would directly enhance theproposed control structure. This requires more accurate velocity estimation first. E.g. dueto a structural deviation in the position measurement the increased resolution could not yetbe used in more precise velocity reconstruction though it can probably be compensated for.Next, enhanced control over the valve, the oil flow to and from the actuators, would be ofhelp. Especially the nonlinear characteristics of the valves often appeared to be troublesomein practise. E.g. each valve has to be activated at all time by a high frequent dither signal toprevent stick of one of the spools. In setting the dither amplitude one had to be very carefulnot to hit transmission line dynamics (too high) or otherwise let sticking effects occur (toolow). An adaptive setting can improve this.

Over all this thesis provides on one hand many leads towards more scientific investiga-tions and on the other hand laid down an improved design procedure, which can be applieddirectly by the practioneer. And that’s what’s research is all about.

Appendix A

Frequency domain measurements

215

216 A Frequency domain measurementshd

p_i1

hdp_

i2hd

p_i3

hdp_

i4hd

p_i5

hu_1

hdp_

i6

hu_2 hu_3 hu_4 hu_5 hu_6

Fig. A.1: Bode amplitude plots dummy platform from valve voltage to valve pressuremeasured and modelled (dashed). Frequency runs form 5Hz to 50Hz withticks at 10Hz, 20Hz and 30Hz. Amplitude from 0:5 to 50 with ticks at 1 and 10.

217hd

p_i1

hdp_

i2hd

p_i3

hdp_

i4hd

p_i5

hu_1

hdp_

i6


Fig. A.2: Bode phase plots dummy platform from valve voltage to valve pressure mea-sured and modelled (dashed). Frequency runs form 5Hz to 50Hz with ticks at10Hz, 20Hz and 30Hz. Phase from �700� to 100� with ticks from �135� to90� every 45�.


p_i1

hdp_

i2hd

p_i3

hdp_

i4hd

p_i5

hu_1

hdp_

i6


Fig. A.3: Bode amplitude plots dummy platform from valve voltages to valve pressuresalong the rigid body modal directions measured and modelled (dashed). Fre-quency runs form 5Hz to 50Hz with ticks at 10Hz, 20Hz and 30Hz. Amplitudefrom 0:5 to 50 with ticks at 1 and 10.

219hd

p_i1

hdp_

i2hd

p_i3

hdp_

i4hd

p_i5

hu_1

hdp_

i6


Fig. A.4: Bode phase plots dummy platform from valve voltages to valve pressuresalong the rigid body modal directions measured and modelled (dashed). Fre-quency runs form 5Hz to 50Hz with ticks at 10Hz, 20Hz and 30Hz. Phasefrom�700� to 100� with ticks from�135� to 90� every 45�.


p_i1

hdp_

i2hd

p_i3

hdp_

i4hd

p_i5

hu_1

hdp_

i6


Fig. A.5: Bode amplitude plots shuttle from valve voltage to valve pressure measuredand modelled (dashed). Frequency runs form 5Hz to 500Hz with ticks at 10Hzand 100Hz. Amplitude from 0:03 to 30 with ticks at :1, 1 and 10.

221hd

p_i1

hdp_

i2hd

p_i3

hdp_

i4hd

p_i5

hu_1

hdp_

i6


Fig. A.6: Bode amplitude plots shuttle from valve voltage to valve pressure measuredand modelled (dashed). Frequency runs form 5Hz to 500Hz with ticks at 10Hzand 100Hz. Phase from�700� to 100� with ticks from�180� to 90� every 90�.


p_i1

hdp_

i2hd

p_i3

hdp_

i4hd

p_i5

hu_1

hdp_

i6


Fig. A.7: Bode phase plots shuttle from valve voltages to valve pressures along therigid body modal directions measured and modelled (dashed). Frequencyruns form 5Hz to 500Hz with ticks at 10Hz and 100Hz. Amplitude from 0:03to 30 with ticks at :1, 1 and 10.

223hd

p_i1

hdp_

i2hd

p_i3

hdp_

i4hd

p_i5

hu_1

hdp_

i6


Fig. A.8: Bode phase plots shuttle from valve voltages to valve pressures along therigid body modal directions measured and modelled (dashed). Frequencyruns form 5Hz to 500Hz with ticks at 10Hz and 100Hz. Phase from �700� to100� with ticks from�180� to 90� every 90�.

224 A Frequency domain measurementsax

ayaz

aex

aey

ax_d

aez

ay_d az_d aex_d aey_d aez_d

Fig. A.9: Bode amplitude plots closed loop around heavy weight dummy platformfrom desired to actual ccelerations model based controller (-) and conven-tional controller (dashed). Frequency runs form 0:5Hz to 50Hz with ticks at1; 2; 3; 5; 10; 20; 30Hz. Amplitude from 0:025 (�32dB) to 2:5 (8dB) with ticksat �30;�20;�6;�3; 0; 3; 6dB.

225ax

ayaz

aex

aey

ax_d

aez


Fig. A.10: Bode phase plots closed loop from desired to actual accelerations modelbased controller (-) and conventional controller (dashed). Frequency runsform 0:5Hz to 50Hz with ticks at 1; 2; 3; 5; 10; 20; 30Hz. Phase from �180�to 180� with ticks from�180� to 180� every 45�.

226 A Frequency domain measurementse_

axe_

aye_

aze_

aex

e_ae

y

ax_d

e_ae

z


Fig. A.11: Bode amplitude plots closed loop from desired to error accelerations modelbased controller (-) and conventional controller (dashed). Frequency runsform 0:5Hz to 50Hz with ticks at 1; 2; 3; 5; 10; 20; 30Hz. Amplitude from�32dB to 14dB with ticks at �20;�6;�3; 0; 3; 6dB.

Appendix B

Derivation of actuator inertialproperties

In this appendix essentially the derivation of (2.119) and (2.120) is performed. Further, itis shown how to choose a factor N in the quadratic velocity term to make _M � 2N skewsymmetric. This last property is important in passivity based control.

First the derivation of the derivative mass matrix equation (2.119).

d

dt(Mia;ib) = � (ia + ib)


�lTn + �ln�vTa Pln) (B.1)

With

d

dtj l j= d

dt

p�lT �l =

�lT �vaj l j = �lTn �va (B.2)

and

_�ln =d

dt

�l

j l j =_�l j l j ��l d

dtj l j

j l j2

=(I � �ln�l

Tn )

j l j �va =1

j l jPln�va: (B.3)

the required steps can be taken

d

dt(Mia;ib) =

d

dt((ia + ib)

j l j2 Pln) = (ia + ib)

�d

dt

�1

j l j2�I +

d

dt

� �ln�lTn

j l j2��

= (ia + ib)�2�lTn �va j l j

j l j4 I �( 1jljPln�va

�lTn + 1jlj�ln�v

Ta Pln) j l j2 �2�lTn �va j l j �ln�lTnj l j4

= � (ia + ib)


�lTn + �ln�vTa Pln): (B.4)

N is defined as a factor in the factorisation of the nonlinear velocity term C(�x; _�x). Butthis factorisation is not uniquely defined by requiring that C(�x; _�x) = N(�x; _�x) _�x.

227

228 B Derivation of actuator inertial properties

Taking

�faia;ib =Mia;ib_�va �

2(ia + ib)

j l j3 (�lTn �va)Pln�va: (B.5)

The nonlinear velocity term, � 2(ia+ib)jlj3

(�lTn �va)Pln�va can be factored into N(�x; �va)�va in aninfinite number of ways. Two possible factors N are in this case:

N1 = �2(ia + ib)

j l j3 (�lTn �va)Pln (B.6)

and

N2 = �2(ia + ib)

j l j3 Pln�va�lTn (B.7)

And all factorisations �N1 + (1 � �)N2, with � 2 IR are valid. Only one factorisationresults in a skew _M � 2N .

Take

N =1

2(N1 +N2): (B.8)

Then

_M � 2N =2(ia + ib)

j l j3 (Pln�va�lTn � �ln�v

Ta Pln): (B.9)

Check through _M � 2N = �( _M � 2N)T that the skew symmetry is there (P Tln

= Pln).Now the derivation of the partial derivative of the kinetic energy actuator inertia given

by

@Kia;ib

@�pa= � (ia + ib)

j �l j3��lTn (�v

Ta Pln�va) + �vTa Pln(

�lTn �va)�: (B.10)

Define a partial derivative of a vector, �x to a vector, �yn�1 as:

@�x

@�y=h

@�x@y1

@�x@y2

: : : @�x@yi

: : : @�x@yn

i: (B.11)

Kinetic energy of the actuator intertia was derived as the scalar term:

Kia;ib =1

2�vTa

(ia + ib)

j �l j2 Pln�va (B.12)

=1

2�vTa

(ia + ib)

j �l j2 �va �1

2(ia + ib)

��vTa

�l

j �l j2� �

�vTa�l

j �l j2�; (B.13)

with

Pln = (I � �ln�lTn );

�ln = �l= j �l j : (B.14)

229

The upper actuator gimbal point, �pa, depends on the variables �c and �� in the same way as �l:

�pa = �c+ T (��)�a; (B.15)

�l = �c+ T (��)�a� �b: (B.16)

Therefore

@�l

@�pa= I: (B.17)

Now

@Kia;ib

@�pa=

1

2(ia + ib)�v

Ta �va

@h

1j�lj2

i@�pa

� (ia + ib)

��vTa

�l

j �l j2� @h �vTa �l

j�lj2

i@�pa

: (B.18)

So two partial derivatives to scalars have be examined further. The first

@h

1j�lj2

i@�pa

=@�

1�lT �l

�@�pa

=�2�lTj �l j4 =

�2�lTnj �l j3 ; (B.19)

and the second

@h�vTa

�l

j�lj2

i@�pa

=�vTa j �l j2 �2�vTa �l�lT

j �l j4 =�vTaj �l j2 � 2

�vTa�ln�l

Tn

j �l j2 = �vTa Pln1

j �l j2 ��vTa

�ln�lTn

j �l j2 : (B.20)

Filling in these equations in (B.18) results in, using (�xT �y) = (�yT �x),

@Kia;ib

@�pa= �1

2(ia + ib)�v

Ta �va

2�lTnj �l j3 � (ia + ib)

��vTa Pln

1

j �l j2 ��vTa

�ln�lTn

j �l j2� �

�vTa�ln

j �l j

�

= � (ia + ib)

j �l j3��vTa I�va

�lTn � �vTa�ln�l

Tn �va

�lTn + �vTa Pln(�vTa�ln)�

= � (ia + ib)

j �l j3��lTn (�v

Ta Pln�va) + �vTa Pln(

�lTn �va)�: (B.21)

And this was what was to be determined.

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Glossary of symbols

241

242 Glossary of symbols

Arabic symbols (large)Variable Unit ExplanationA [m2] Actuator operational area

[m] 3x6 matrix with upper gimbal (to c.o.g.) vectors stackedAp [m2] Actuator operational areaB [m] 3x6 matrix with lower gimbal vectors stackedB,Bp [Ns=m;Ns] Viscous friction (matrix)C [kg=s,kgm2=s] Nonlinear coriolis and centripetal termsCi [N=m5;�] Oil column stifness chamber idC [N=m5] Difference oil column stifness chambers 1 and 2Cm [N=m5] Mean oil column stifness chambers 1 and 2E [N=m2] Effective bulk modulus of the oilG [ ] Ground frame or intertial frame

[�] 3x4 transformation as part of rotation R = GLT

GTs [�] 3x3 �! to �� transformationI [ ] Identity matrixI�z [kgm2] 3x3 platform inertia matrix evaluated at c.o.g.Jy;x [ ] Jacobian matrix between two sets of variables y and xJl;x [�;m] Jacobian from platform velocities to actuator velocitiesK [ips=(circ=s2;m=s2)] Semicircular and vestibular organ model gain

[m=(sA)] Hydraulic actuator model gainKd As=m Valve main spool differential feedback gainKp A=m Valve main spool proportional feedback gainKh [nd] Crossover model pilot gainL [ ] 4x3 eul. par. transformation as part of rotation R = GLT

Ln [�] 3x6 matrix with all unit actuator directions vectors stacked

Llm [m5=N ] Oil leakage constant due to pressure difference actuator chambersLt [m] Transmission line lengthM [kg,kgm2] Mass matrix

[ ] Moving frame[-] Mobility index

Mact [kg; kgm2] Simulator mass matrix actuator coordinatesMt [kg; kgm2] Simulator mass matrix platform coordinates


Variable Unit ExplanationdP [N=m2] Net generated actuator pressurePi1 [N=m2] Oil pressure at valve side transmission line 1Pi2 [N=m2] Oil pressure at valve side transmission line 2Pm [N=m2] Mean actuator chamber pressurePm1;2 [N=m2] Main valve spool pressuresPn1;2 [N=m2] Pilot valve spool pressuresPo1 [N=m2] Oil pressure at actuator chamber 1Po2 [N=m2] Oil pressure at actuator chamber 2Ps [N=m2] Supply oil pressurePt [N=m2] Tank oil pressurePxn [�] 3x3 projection matrix onto plane with normal �xnR [�] 3x3 rotation matrixBRA [�] 3x3 rotation matrix from frame A to BT [�] 3x3 rotation matrix platformTx;y;z [�] 3x3 rotation matrix around one axis of the frameU [�] Unitary (rigid body modal direction) matrixV [m3] General volumeV1 [m3] Oil volume actuator chamber 1V2 [m3] Oil volume actuator chamber 2Vm [m3] Mean oil volume actuator chambers 1 and 2Va [m=s] 3x6 matrix with upper gimbal velocity vectors stackedX general matrix (capital)~X vector product matrix

Arabic symbols (small)Variable Unit Explanationac [�] Upper actuator body c.o.g.�ai [m] Vector from c.o.g. simulator to ith upper gimbal pointbi [Ns=m] Viscous friction coefficient actuator ibxf;yf; f [Ns=m;Nms=rad] Viscous friction coefficients foundationbc [�] Lower actuator body c.o.g.�bi [m] Vector from ground frame origin to i th lower gimbal pointc [N=m] Hydraulic actuator stifnessc1;::: ;10 [ ] Valve model parametersco [m=s] Wave propagation velocitycxf;yf; f [N=m;Nm=rad] Stifness coefficients foundation�c [m] Vector from ground frame origin to c.o.g. simulatord [ ] Finitely small variationdl [m] Lower gimbal spacingdu [m] Upper gimbal spacing�e [ ] Error vector

[ ] General external signaldp [N=m2] Pressure difference over hydraulic actuator compartments


Variable Unit Explanationf� N;Nm Inertial force�fa N 6x1 vector active actuator forces�f(:) [ ] Vector function�g(:) [ ] Vector function�g [N ,Nm] Gravitational termsfd [N ] Actuator driving forcefrm [1=s] Rigid body mode eigenfrequencyi [A] Valve input currentia [kgm2] Inertia upper actuator body aroundib [kgm2] Inertia lower actuator body around around gimbal pointj [�] Imaginairy unit,

p�1kdp [Am2=N , ] (Normalised) pressure feedback gainli [m] Length ith joint (serial), actuator (parallel) manipulator�l [m] Vector with all actuator platform lengths�li [m] Vector form the ith lower to upper gimbal�ln [�] Normalised unit direction vector actuatornG;Mx;y;z [m] Unit direction vectors

direction in ground,G, and moving frame, Mm [kg] Mass (platform)ma [kg] Mass upper actuator bodymb [kg] Mass lower actuator bodymf [kg] Mass of the foundation�m [kgm=s; kgrad=s] Vector of generalised momentan� [ ] Euler paramter unit vector along axis of rotation�pg [m] Vector to point p in frame G�pm [m] Vector to point p in frame Mq [m] Actuator extensionqi [m] Rotation angle ith joint serial manipulator�q [m,rad] Robotic manipulator positional coordinates�qd [m,rad] Desired positional coordinatesra [m] Length from c.o.g. upper actuator body to upper gimbalrb [m] Length from c.o.g. lower actuator body to lower gimbals [rad=s] Sloppy Laplace operator as j!

[m] Scaling factor, k�a jmax�sx [m] Vector with platform coordinatest [s] Timet1;::: ;5 [ ] Transmission line model parameters�tg [m] Translational vector in frame Gu [V ] Actuator input voltage

[ ] General system input�va [m=s] Upper gimbal point velocity vector�vmm m=s Velocity vector of a point in a moving frame


Variable Unit Explanationx [ ] General scalar

[m] Surge directionxf [m] Valve flapper position

[m] Foundation surge displacementxm [m] Valve main spool positionxs [m] Valve pilot spool position�xa [m] Vector from ground origin to upper gimbal point�xn [ ] General normalized vector (index n) with length one�x [ ] General vector (bar on top)_�x [m=s; rad=s] Vector with translational and angular velocity platform�xg [m] Vector in inertial (ground) frame�xm [m] Vector in platform (moving) frame�xin [(rad;m)=s2] Semicircular and vestibular input (rotational) acceleration�xout [(rad;m)=s2] Noticed semicircular and vestibular (rotational) accelerationy [m] Sway direction

[ ] General system outputz [m] Heave direction�z [m; rad] Vector with generalised (positional) coordinates

Greek symbols (small)Variable Unit Explanation� [ ] Bound on first Newton Raphson iteration� [ ] Relative damping factor

[ ] Bound on inverse lowest singular value NR-iteration�� [rad] Vector of euler angles� [ ] Infinitely small variation�0;1;2;3 [�] The specific four euler parameters�� [�] Vector with last three euler parameters 1, 2, 3��e [�] Vector with all four euler parameters�m [V ] Valve spool position error~� [�] Cross product matrix with euler parameters [ ] Lipschitz constant� [rad] Angle in general

[rad] Last euler angle around airplane roll axis� [rad] Euler parameter angle of rotation� [rad] Second euler angle around intermediate pitch-axis� [kg=m3] Oil density� [ ] Singular value


Variable Unit Explanation�� [N ,Nm] Generalized torques, forces�1 [s] Crossover model pilot lead time constant

[s] First semicircular and vestibular lag time constant�2 [s] Crossover model pilot lag time constant

[s] Second semicircular and vestibular lag time constant�L [s] Semicircular and vestibular lead time constant�d [s] Crossover model pilot pure time delay�� Nm;N Generalised torques! [rad=s] Frequency!o [rad=s] Hydraulic actuator eigenfrequency�! [rad=s] Angular velocity vector platform�!a [rad=s Angularo velocity vector actuator orth. to actuator length�1;::: ;5 [ ] Transmission line model parameters [rad] First euler angle around ground frame yaw axis

Greek symbols (large)Variable Unit Explanation�i1 [m3=s] Oil flow from valve into transmission line 1�i2 [m3=s] Oil flow from valve out of transmission line 2�l1 [m3=s] Leakage oil flow outof actuator chamber 1�l2 [m3=s] Leakage oil flow into actuator chamber 2�lm [m3=s] Leakage oil flow from actuator chamber 2 into 1�m [m3=s] Mean oil flow into chamber 1 and out of chamber 2d� [m3=s] Difference oil flow �o1 ��o2~ [rad=s] Cross product matrix with angular velocity

Caligraphic symbols (large)Variable Unit ExplanationG [N;Nm] Generalised gravitational forcesH [kgm2=s2] Hamiltonian function (addition kinetic and potential energy)K [kgm2=s2] Kinetic energy functionL [kgm2=s2] Langrangian function (difference kinetic and potential energy)P [kgm2=s2] Potential energy function

Samenvatting en CV

De hoge veiligheidseisen in de luchtvaart vereisen een goed begrip van het gedrag van depiloot in extreme weersomstandigheden. Vernieuwend onderzoek op dit gebied moet ge-bruik kunnen maken van vluchtsimulatoren die in staat zijn om met hoge nauwkeurigheidkritische condities tijdens een vlucht te simuleren. Verschijnselen zoals turbulentie en zij-wind in de lagere delen van de atmosfeer vertonen een snelheidsspectrum dat grote krachtenintroduceert over een breed scala aan frequenties. Het reproduceren van de benodigde be-weging, het voelbaar maken van de inertiele versnellingen en het juist uitrichten van dezwaartekracht in geavanceerde vluchtsimulatoren, is dus een uitdagende regeltaak die hetuiterste uit het systeem moet halen. Voor vergelijkbare systemen, zoals robots, bestaanmoderne modelgebaseerde regelstrategieen, die hogere prestaties kunnen behalen dan min-der gestructureerde methoden, meer inzicht geven in de begrenzingen van het systeemen wellicht op constructieve eigenschappen van het systeem kunnen wijzen waar gebruikvan kan worden gemaakt in het ontwerp. Deze regelstrategieen worden nog vrijwel niettoegepast bij bewegingssystemen van vluchtsimulatoren. Een van de redenen zou kunnenzijn dat deze systemen zo complex zijn dat exacte of zeer gedetailleerde modellen geenonderdeel van de regelaar konden en kunnen worden gemaakt. Toepassing ervan in ditonderzoek vereiste een tussenstap waarin een minder gedetailleerd model moest wordengeextraheerd dat toch de meest relevante dynamica moest beschrijven.

De vraag welke relevante systeemkennis gebruikt zou moeten worden in een modelge-baseerde regelstrategie en in welke mate deze strategie bijdraagt aan het geregelde gedragvan een bewegingssysteem van een vluchtsimulator, is het centrale onderzoeksthema vandit onderzoek geweest.

Om hierop een antwoord te geven moest het volledige regelaarontwerpproces wordengedefinieerd, gestructureerd en geevalueerd. Veel van de tussenstappen toegepast in dit pro-ces kunnen al in de literatuur worden gevonden maar een integrale aanpak, waarin ook despecifieke eisen ten aanzien van een vluchtsimulator werden meegenomen, ontbrak nog. Bijde keuze van bestaande dan wel nieuw voorgestelde bouwstenen in dit proces van systeem-en regelaarontwerp, vormden integreerbaarheid en praktische toepasbaarheid de voornaam-ste argumenten.

Door middel van modelvorming op basis van fysische balansvergelijkingen is eerst eenrelevante modelstructuur ten behoeve van systeemanalyse en regeling afgeleid. Daarna isdeze structuur geevalueerd met behulp van experimenten. Daarbij vond identificatie van demodelparameters en het in kaart brengen van het geldigheidsgebied van de gekozen struc-tuur plaats. Vervolgens is de meest geschikte modelgebaseerde regelstrategie gekozen enaangepast voor toepassing op dit bewegingssysteem met zijn specifieke parallelle struc-

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248 Samenvatting en CV

tuur. Ontwerp en implementatie van deze regeling op het bewegingssysteem van de Simonavluchtsimulator was de volgende stap. Tenslotte is een testprocedure gedefinieerd om deprestatie van dit geregelde systeem te kunnen kwantificeren en te kunnen afzetten tegen deprestatie van een conventioneel geregeld systeem. Gegeven de specifieke toepassing vanvluchtsimulatie moesten verschillende stappen in dit generieke ontwerpproces nader wor-den uitgewerkt.

Opbouwen van vergaand inzicht in de kinematica en dynamica van het systeem wasnodig om zowel de robuustheid van een conventionele regeling te behouden als ook deprestatie en de voorspelbaarheid te verhogen. Met fysische modelvorming kon kwalitatiefinzicht worden opgebouwd en gelijkertijd gaf deze structuur de mogelijkheid om de param-eters van het zich niet-lineair gedragende bewegingssysteem experimenteel te identificerenen de te implementeren regelaarstructuur te modulariseren. Met deze identificatie en ex-perimentele evaluatie kon de vereiste nauwkeurigheid voor het model worden bereikt dienodig was om de prestatieverhoging met de modelgebaseerde regelaar ook in de praktijk tekunnen behalen.

Fysische modelvorming van bewegingssystemen begint met de analyse van de sys-teemkinematica. In dit onderzoek wordt een nieuwe methode gepresenteerd waarmee hetvoor het eerst mogelijk werd om te garanderen dat singuliere punten buiten het werkgebiedvan het systeem vallen. Hierop verder bouwend kan ook nagegaan worden of een iteratiefschema om de platformcoordinaten te berekenen, gegeven de gemeten lengte van de actua-toren, convergeert en snel genoeg convergeert voor implementatie in de werkelijke regeling.Hiermee kon voor het Simona bewegingsysteem de garantie worden geboden dat dit deelvan de regelstructuur veilig toegepast kon worden. Inzicht in de kinematica gaf verder demogelijkheid om door middel van calibratie de positioneringsnauwkeurigheid van het alnauwkeurig gespecificeerde systeem met een factor tien te verhogen waarmee direct ookhet kinematisch model geevalueerd was.

Van het bewegingssysteem in de vorm van een Stewart platform is afgeleid dat deze totde klasse van volledig parallelle systemen behoort waardoor de dynamica als een stelsel ex-pliciete differentiaalvergelijkingen beschreven kan worden vanuit de platformcoordinaten.Zowel in theorie als in de praktijk kon aangetoond worden dat de modes van de stijvelichamen volgend uit de interactie tussen hydraulica en mechanica de meest relevante sys-teemdynamica vormen bij simulatortoepassingen. De relatie tussen platform- en actua-torcoordinaten beschreven door een jacobiaan vormt hierbij de centrale operator. Volgenduit deze analyse kan het systeem, met op het eerste gezicht sterke interactie, in elke standlokaal vanuit de juiste coordinaten gezien worden als zes hydraulische actuatoren die elkvrijwel onafhankelijk een massa aandrijven.

Met de voorgestelde modelgebaseerde regeling kunnen een aantal zaken bereikt wor-den die conventioneel niet mogelijk zijn. Ten eerste kunnen de krachten benodigd voor degewenste versnelling aangepast worden voor de standsafhankelijke richtingen waarin ont-koppeld de sterk verschillende massa’s gevoeld worden. Vervolgens kunnen de toch nogoptredende positioneringsfouten ook langs deze richtingen zonder interactie teruggekop-peld worden. Tenslotte wordt de interactie tussen de hydraulica en de mechanica gemini-maliseerd door te compenseren voor de oliestroom die nodig is om de gewenste snelheidaan te houden. Implementatie van deze structuur heeft aangetoond dat hiermee een hogere

Samenvatting en CV 249

bandbreedte, minder opslingering en een meer over de vrijheidsgraden geegaliseerde re-spons kan worden bereikt dan met de minder gestructureerde conventionele regeling. Verderis aangegeven hoe nog meer bereikt kan worden als ook voorspellende informatie vanuit hetvoertuigsimulatiemodel aan de regeling wordt aangeleverd.

De te behalen prestaties van de voorgestelde regelaarstructuur worden eerst begrensddoor het optreden van parasitaire effecten ten gevolge van mechanische vervormingen, flex-ibiliteiten. Ook met de combinatie van de altijd iets vertraagde olietoevoer en zingendeleidingen moet op nog wat hogere frequenties rekening gehouden worden. Door in hetontwerp van een simulator een laag gewicht en hoge stijfheid na te streven is echter aange-toond dat met de voorgestelde regelaarstuctuur over een breed spectrum kritische simulatie-experimenten zeer realistisch uitgevoerd kunnen worden.

250 Samenvatting en CV

Curriculum Vitae

Naam Sjirk Holger Koekebakker25 oktober 1967 Geboren te Vierhouten (Gemeente Ermelo).1980 - 1986 VWO aan het Agnes College te Leiden.1986 - 1993 Werktuigbouwkunde aan de Technische Universiteit Delft.

Afgestudeerd vanuit de vakgroep Systeem- en Regeltechniek.Titel afstudeerwerk: ”Decoupling by means of H1;F ;application to a wafer stepper”. Afstudeerwerk uitgevoerdop het Philips Natuurkundig Laboratorium te Eindhoven.

1993 - 1994 Onderzoeker bij het TNO Productcentrum i.h.k.v.de vervangende dienst.

1994 - 1998 Promovendus bij de vakgroep Systeem- en Regeltechniekvan de faculteit Werktuigbouwkundeaan de Technische Universiteit Delft.

1999 - Onderzoeker bij de Group Research Technologyte R & D Venlo van Oce Technologies B.V.

Model based control of a flight simulator motion

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