Computer-aided Integrated Design of Mechatronic Systemschemori/Temp/Hussein/Thesis_Maira.pdf · the...

KATHOLIEKE UNIVERSITEIT LEUVENFACULTEIT INGENIEURSWETENSCHAPPENDEPARTEMENT WERKTUIGKUNDEAFDELING PRODUCTIETECHNIEKEN,MACHINEBOUW EN AUTOMATISERINGCelestijnenlaan 300B, 3001 Heverlee (Leuven), Belgium

COMPUTER-AIDED INTEGRATEDDESIGN OF MECHATRONIC SYSTEMS

Promotoren:Prof. dr. ir. H. Van BrusselProf. dr. ir. W. Desmet

Proefschrift voorgedragen tothet behalen van het doctoraatin de ingenieurswetenschappen

door

Maıra Martins DA SILVA

2009D05 April 2009

KATHOLIEKE UNIVERSITEIT LEUVENFACULTEIT INGENIEURSWETENSCHAPPENDEPARTEMENT WERKTUIGKUNDEAFDELING PRODUCTIETECHNIEKEN,MACHINEBOUW EN AUTOMATISERINGCelestijnenlaan 300B, 3001 Heverlee (Leuven), Belgium

COMPUTER-AIDED INTEGRATEDDESIGN OF MECHATRONIC SYSTEMS

Jury:Prof. dr. ir. D. Vandermeulen (chair)Prof. dr. ir. H. Van BrusselProf. dr. ir. W. DesmetProf. dr. ir. H. RamonProf. dr. ir. F. Al-BenderProf. dr. ir. J. SweversProf. dr. ir. O. Bruls (University ofLiege, Belgium)Prof. dr. ir. J. Van Amerongen (Uni-versity of Twente, The Netherlands)

Proefschrift voorgedragen tothet behalen van het doctoraatin de ingenieurswetenschappen

door

Maıra Martins DA SILVA

ISBN 978-94-6018-048-4D/2009/7515/31UDC 681.3∗J2 April 2009

c© Katholieke Universiteit LeuvenFaculteit Toegepaste WetenschappenArenbergkasteel, B-3001 Heverlee (Leuven), Belgium

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D/2009/7515/31ISBN 978-94-6018-048-4UDC 681.3∗J2

Acknowledgements

People talk about luck,but you produce that by working hard.

Prof. Arulkumaran

I can’t believe how incredibly lucky I am. To work in an internationallywell recognized research group, such as PMA, is an amazing opportu-nity. I am cheerful to declare that I could fulfill this dream in the nicestway.

I would like to express my sincere gratitude to Prof. Van Brusselfor accepting me into the PMA community, for trusting me such achallenging subject, for supporting me financially during the academicyear of 2004-2005, and for supporting my decision of using Oofelieduring the development of this work. It has been a great honor to bepart of his ambitious plan of building a Mechatronic Compiler.

I would also like to express my special gratitude to Prof. Desmet forsupporting me technically, financially and psychologically during thiswork. He always told me optimistic words. How can I forget when Ireceived a really tough review from one of my papers and Prof. Desmetspent his time trying to convince me that the paper was practicallyapproved with minor corrections? I am really thankful for workingwith such an extraordinary person.

Their kind advice, time, attention, support and trust have beenhighly appreciated. I also appreciated the fact that I have been involvedinto two European Projects: InMAR and NEXT, providing me not onlygood case studies but also the opportunity of exchanging knowledge

I

Acknowledgements

with very interesting people.I would like to extend my deepest gratitude to the jury members:

Professors Van Amerongen, Bruls, Ramon, Swevers and Al-Bender.Thank you for your valuable time, suggestions, comments, and crit-ical thoughts that have enabled me to improve the quality of thismanuscript. I would also like to express my thankfulness to Prof. Van-dermeulen for being the chairperson. I cannot forget to thank Prof.Vanherck, an incredible nice person, for translating my abstract toDutch.

I have to stress that I am deeply thankful for working with Prof.Bruls and Prof. Swevers. Surely, I have been learning from the best.Prof. Bruls introduced me to Oofelie and helped me throughout thisdoctoral programme answering my questions regarding multibody tech-niques and programming. Prof. Swevers helped me into the challengingcontrol field. As a structural engineer, modern control was a real chal-lenge for me. In this way, I am really grateful for receiving truthfulsupport from Prof. Swevers.

My doctoral programme has been supported by the K.U.LeuvenResearch Council (2008-2009) and by CAPES, the Brazilian Founda-tion Coordination for the Improvement of Higher Education Personnel,(2005-2008) to whom I gratefully acknowledge.

Back to Brazil, I will definitely miss my good Belgian, Cypriot,Byelorussian, Ukrainian, Italian, Indonesian and Spanish friends. Iwill miss the Happy Hour and the PMA weekends and, undeniably, Iwill miss Belgian beer. However, coming back to my home country isa joy. It has not been easy to stay away from those who I love.

Agradeco meu pai e minha mae, Osmar da Silva e Aparecida FatimaMartins da Silva, pelo apoio absoluto e pelo amor irrestrito. Pai emae, obrigada por tudo! Agradeco a minha irma, Anaı Martins daSilva, pela sua amizade e carinho. Nunca vou esquecer sua dedicacaodurante todos esses anos. A sua existencia me traz alegria todos osdias.

Finally, there are no words to express my gratitude and love to myhusband Leopoldo P. Rodrigues de Oliveira. Your love and kindnessmake me the luckiest person in the world.

Maıra Martins da SilvaApril, 2009

II

Abstract

This thesis concerns the optimal design of mechatronic systems, whichrequires that, among the structural and control parameters, an optimalchoice has to be made with respect to design specifications in an inte-grated way. To accomplish that, this work provides computer-aidedtools for the integrated design of mechatronic systems. Two mainchallenges have been identified when dealing with integrated design:modeling due to their multidisciplinary nature and optimization dueto their non-convex nature.

To address the first challenge, guidelines for modeling mechatronicsystems considering the system flexibilities, the system motion andthe control action have been proposed. These parametric models mayeasily contain several degrees-of-freedom and be unsuitable for controldesign purposes. To obtain a concise description of the system, modelreduction procedures have been employed. To address the second chal-lenge, two strategies have been investigated: the nested and the directdesign strategies. The nested design strategy combines nonlinear op-timization methods and model-based control design techniques. Thedirect design strategy considers, simultaneously, control and structuralparameters using non-linear or genetic optimization algorithms.

Three case studies have been exploited: a vibro-acoustic system andtwo pick-and-place machines, containing serial and parallel kinematicchains. The pick-and-place machines exhibit configuration-dependentdynamics, which inevitably affect the performance and the stability ofthe control system. Important issues have been identified when thedirect strategy is employed for designing a mechatronic system withconfiguration-dependent dynamics, suggesting that the nested designstrategy suits better this task. It can be concluded that the optimaldesign of mechatronic systems can only be accomplished when controland structural parameter are considered in an integrated way.

III

Beknopte samenvatting

Dit proefschrift heeft betrekking op het optimale ontwerp van mecha-tronische systemen dat vereist dat tussen de structurele en de con-troleparameters op een geıntegreerde wijze een optimale keuze moetgemaakt worden met betrekking tot de ontwerpspecificaties. Om dit tebereiken geeft dit werk computerondersteunde hulpmiddelen voor hetgeıntegreerde ontwerp van mechanische systemen. Bij de behandel-ing van een geıntegreerd ontwerp worden twee belangrijke uitdagingenvastgesteld: de modellering vanwege hun multidisciplinaire karakter ende optimalisatie vanwege hun niet-convexe aard.

Om de eerste uitdaging aan te pakken, zijn er richtlijnen voorhet moduleren van mechatronische systemen voorgesteld de rekeninghouder met de flexibiliteit, de beweging van het systeem en de con-troleactie. Voor het verkrijgen van een beknopte beschrijving van hetsysteem werden modelreductieprocedures gebruikt. Voor de tweedeuitdaging werden twee strategieen onderzocht: de genestelde en dedirecte ontwerpstrategie. De genestelde ontwerpstrategie combineertniet-lineaire optimalisatiemethoden met modelgebaseerde controleon-twerp technieken. De direkte ontwerpstrategie beschouwt gelijktijdigcontrole en structurele parameters gebruik makend van niet-lineaire ofgenetische optimalisatie algoritmes.

Drie case studies werden onderzocht: een vibro-acoustisch systeemen twee pick-and-place machines met serieel en parallel kinematischekringen. De pick-and-place machines hebben een configuratieafhanke-lijke dynamica die onvermijdelijk het gedrag en de stabiliteit van hetcontrolesysteem beınvloedt. Dit laat vermoeden dat een genesteld on-twerp beter past voor deze taak. Hieruit kan besloten worden dat eenoptimaal ontwerp slechts kan bekomen worden als zowel de controle alsde structuurparameters op een geıntegreerde wijze beschouwd worden.

V

List of symbols

Abbreviations

ANC : active noise controlASAC : active structural-acoustic controlBE : boundary elementCAD : computer aided designCAID : computer aided integrated designCAE : computer aided engineeringCMS : component mode synthesisCOE : control effortCPU : central processing unitdof : degree of freedomdofs : degrees of freedomEC : engine cavityFE : finite elementFEM : finite element methodFRF : frequency response functionGMP : global modal parametrizationLMI : linear matrix inequalitiesLPV : linear parameter-varyingLTI : linear time invariantLTV : linear time-varyingMBS : multibody systemsMIMO : multiple inputs multiple outputsPC : passenger cavityPID : proportional-integral-derivative

VII

List of symbols

PKM : parallel kinematic machinePKMs : parallel kinematic machinesSAP : sensor/actuator pairSIMO : single input multiple outputsSISO : single input single outputSPL : sound pressure level

Scalars

γ : Bound on ∞-norm of closed-loop transfer func-tions in H∞ control synthesis and analysis

η : Parameter indicating the relative position of anode between other two nodes

ζ : damping ratiofBW : bandwidth frequency [Hz]fI : Bellow this frequency the controller should have

integral action [Hz]fR : Beyond this frequency the controller should have

roll-off [Hz]mS : maximum of the sensitivity functionmSP

: maximum of the process sensitivity functionmT : maximum of the complementary sensitivity

functiont : timets : settling time

Vectors

δ : flexible modal coordinatesΨ : nodal rotationsλ : Lagrange multipliersη : modal coordinatesΦ : kinematic constraintsΓ : nodal displacementsρ0 : fluid densityθ : Varying parameters

VIII

List of symbols

θ : rigid modal coordinates representing the actua-tors related to the varying parameters

g : internal, external and complementary inertiaforces

p : Vector of polesp : nodal acoustic pressuresq : nodal coordinatesr : referencess : optimization variablesu : system inputsu : structural displacementsx : state space variablesy : system outputsw : disturbance, noise and reference signalsz : the signals to be minimizedz : Vector of zeros

Matrices

A : System matrix of a state-space modelB : Input matrix of a state-space modelB : Jacobian: the matrix of constraint gradientsC : Output matrix of a state-space modelC : Damping matrixD : Direct feed-through matrix of a state-space

modelDa : acoustic damping matrixDs : structural damping matrixI : Identity matrixK(s) : control systemK : stiffness matrixKa : acoustic stiffness matrixKs : structural stiffness matrixL : Boolean matrixL(s) : Loop gainM : mass matrix

IX

List of symbols

Ma : acoustic mass matrixMs : structural mass matrixP : Lyapunov matrixP : mechanical systemPa : augmented plantR : Rotation operatorS(s) : Sensitivity functionSK(s) : Control sensitivity functionSP (s) : Process sensitivity functionT(s) : Complementary sensitivity function

Shorthands

•T : Transpose of •• : Skew symmetric matrix• : Upper bound of •• : Lower bound of •min(•) : Minimum of •

X

Table of contents

Acknowledgements I

Abstract III

Beknopte samenvatting V

List of symbols VII

Table of contents XI

I Introduction and Modeling Approach 1

1 Introduction 31.1 Motivation . . . . . . . . . . . . . . . . . . . . . . 31.2 Challenges in modeling mechatronic systems . . . . 71.3 Challenges in optimizing/designing mechatronic

systems . . . . . . . . . . . . . . . . . . . . . . . . 81.3.1 Optimization problem statement . . . . . . 81.3.2 State-of-the-art in optimizing mechatronic

systems . . . . . . . . . . . . . . . . . . . . 131.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . 161.5 Methodology and case studies . . . . . . . . . . . . 171.6 Main contributions of this thesis . . . . . . . . . . 191.7 Outline of the dissertation . . . . . . . . . . . . . . 211.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . 24

2 Modeling of Mechatronic Motion Systems 272.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 27

XI

Table of contents

2.2 Simulation of an active flexible structure in a finite-element environment . . . . . . . . . . . . . . . . . 29

2.3 Simulation of reduced or simplified models andtheir controllers in a control design environment . 31

2.4 Co-simulation between dedicated virtual environ-ments . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.5 Integrated simulation in a unified environment . . 382.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . 40

3 Modeling of Mechatronic Motion Systems withVarying Dynamics 433.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 433.2 Short review on linear parameter-varying (LPV)

systems . . . . . . . . . . . . . . . . . . . . . . . . 473.2.1 LPV modeling . . . . . . . . . . . . . . . . 473.2.2 Interpolation of LTI models . . . . . . . . . 493.2.3 LPV stability analysis . . . . . . . . . . . . 52

3.3 Modeling of motion mechatronic systems withvarying dynamics . . . . . . . . . . . . . . . . . . . 533.3.1 Simulation of reduced models and their

controllers in a control design environment 563.3.1.1 Methodology . . . . . . . . . . . . 563.3.1.2 Case Study . . . . . . . . . . . . . 56

3.3.2 Co-simulation between dedicated virtualenvironments . . . . . . . . . . . . . . . . . 603.3.2.1 Methodology . . . . . . . . . . . . 603.3.2.2 Case study . . . . . . . . . . . . . 61

3.3.3 Integrated simulation in a unified environ-ment . . . . . . . . . . . . . . . . . . . . . . 623.3.3.1 Methodology . . . . . . . . . . . . 623.3.3.2 Case Study . . . . . . . . . . . . . 63

3.3.4 Comparison between the modeling strate-gies of Mechatronic Motion Systems withVarying Dynamics . . . . . . . . . . . . . . 64

3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . 65

XII

Table of contents

II Applications: Articles 67

4 Concurrent mechatronic design approach for ac-tive control of cavity noise 69Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . 711. Introduction . . . . . . . . . . . . . . . . . . . . . . . 722. Fully Coupled Vibro-Acoustic Modelling Approach . 74

2.1. From vibro-acoustic FE to state-space formu-lation . . . . . . . . . . . . . . . . . . . . . 75

2.2. Experimental Validation . . . . . . . . . . . . 842.3. Inclusion of sensor and actuators pairs (SAP)

models . . . . . . . . . . . . . . . . . . . . . 863. Concurrent Mechatronic Design of Active Systems . . 894. Conclusions and Future Work . . . . . . . . . . . . . 98Acknowledgements . . . . . . . . . . . . . . . . . . . . . 99Bibliography . . . . . . . . . . . . . . . . . . . . . . . . 100

5 Design of Mechatronic Systems With Configuration-Dependent Dynamics: Simulation and Optimiza-tion 105Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . 107I. Introduction . . . . . . . . . . . . . . . . . . . . . . . 108II. Modeling of mechatronic systems with configuration-

dependent dynamics . . . . . . . . . . . . . . . . . 110A. Parametric Finite-Element Model . . . . . . . . 111B. Model Reduction . . . . . . . . . . . . . . . . . 112C. Affine Models - Interpolation and Simulation . . 114

III. Test case: pick-and-place assembly robot . . . . . . 115A. Description of the Pick-and-Place Assembly

Robot Set-up . . . . . . . . . . . . . . . . . 116B. Modeling . . . . . . . . . . . . . . . . . . . . . . 116C. Experimental validation . . . . . . . . . . . . . 119

IV. Mechatronic Design Approach . . . . . . . . . . . . 123A. Control Design Approach . . . . . . . . . . . . . 123B. LTI PID Controllers . . . . . . . . . . . . . . . 125C. LPV PID Controllers . . . . . . . . . . . . . . . 127

V. Conclusions . . . . . . . . . . . . . . . . . . . . . . . 129Acknowledgements . . . . . . . . . . . . . . . . . . . . . 130Bibliography . . . . . . . . . . . . . . . . . . . . . . . . 130

XIII

Table of contents

6 Integrated structure and control design for mecha-tronic systems with configuration-dependent dy-namics 133Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . 1351 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1362 Modeling and optimization of mechatronic systems . . 139

2.1 Modeling . . . . . . . . . . . . . . . . . . . . . 1392.2 Stability Analysis . . . . . . . . . . . . . . . . . 1402.3 Multi-objective optimization . . . . . . . . . . 142

3 Pick-and-place robot: modeling details and controlalgorithms . . . . . . . . . . . . . . . . . . . . . . . 1443.1 Case Study . . . . . . . . . . . . . . . . . . . . 1443.2 Mechanical model . . . . . . . . . . . . . . . . 1453.3 Controller . . . . . . . . . . . . . . . . . . . . . 1483.4 Stability analysis . . . . . . . . . . . . . . . . . 149

4 Integrated structure and control design . . . . . . . . 1504.1 Case 1: Integrated design considering an LTI

PID controller . . . . . . . . . . . . . . . . 1544.2 Case 2: Integrated design considering an LPV

gain-scheduling PID controller . . . . . . . 1555. Conclusions . . . . . . . . . . . . . . . . . . . . . . . 158Acknowledgements . . . . . . . . . . . . . . . . . . . . . 159Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . 159Bibliography . . . . . . . . . . . . . . . . . . . . . . . . 161

7 Computer-Aided Integrated Design for MachinesWith Varying Dynamics 165Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . 1671. Introduction . . . . . . . . . . . . . . . . . . . . . . . 1682. Modeling and Control of Machines With Varying

Dynamics . . . . . . . . . . . . . . . . . . . . . . . 1702.1 Flexible Multibody Models . . . . . . . . . . . 1722.2 Model Reduction . . . . . . . . . . . . . . . . . 1742.3 Control Design for Linear Time-Invariant Mo-

tion Systems . . . . . . . . . . . . . . . . . 1772.4 Gain-scheduling Controller Derivation . . . . . 1792.5 Stability Analysis . . . . . . . . . . . . . . . . . 182

3. Pick-and-Place Machine: Modeling Details and Con-trol System . . . . . . . . . . . . . . . . . . . . . . 183

XIV

Table of contents

3.1 Mechanical Model . . . . . . . . . . . . . . . . 1833.2 Control System . . . . . . . . . . . . . . . . . . 185

4. Pick-and-Place Machine: Integrated Design . . . . . . 1885. Conclusions . . . . . . . . . . . . . . . . . . . . . . . 189Acknowledgements . . . . . . . . . . . . . . . . . . . . . 190Annex A . . . . . . . . . . . . . . . . . . . . . . . . . . . 191Bibliography . . . . . . . . . . . . . . . . . . . . . . . . 192

8 Computer-Aided Integrated Design of ParallelKinematic Machines 197Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . 1991. Introduction . . . . . . . . . . . . . . . . . . . . . . . 2002. Modeling and control of mechatronic systems with

varying dynamics . . . . . . . . . . . . . . . . . . . 2012.1 Flexible multibody systems . . . . . . . . . . . 2022.2 Model reduction methodology: a brief intro-

duction . . . . . . . . . . . . . . . . . . . . 2032.3 Spline-based feedforward for trajectory tracking 205

3. Pick-and-place parallel machine: modeling detailsand control design . . . . . . . . . . . . . . . . . . 2073.1 Mechanical model . . . . . . . . . . . . . . . . 2083.2 Validation of the control approach . . . . . . . 209

4. Pick-and-Place Machine: Integrated Design . . . . . . 2115. Conclusions . . . . . . . . . . . . . . . . . . . . . . . 212Acknowledgements . . . . . . . . . . . . . . . . . . . . . 213Bibliography . . . . . . . . . . . . . . . . . . . . . . . . 213

III Conclusions and Bibliography 217

9 Conclusions and future research 2199.1 Contributions and achievements . . . . . . . . . . . 2199.2 Main conclusions . . . . . . . . . . . . . . . . . . . 220

9.2.1 Modeling of mechatronic motion systems . 2209.2.2 Integrated design of mechatronic systems . 222

9.3 Future research . . . . . . . . . . . . . . . . . . . . 2259.3.1 Virtual prototyping for active vibro-acoustic

systems . . . . . . . . . . . . . . . . . . . . 2259.3.2 Model reduction of multibody systems . . . 225

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Table of contents

9.3.3 Fuzzy finite element method aiding controldesign . . . . . . . . . . . . . . . . . . . . . 225

9.3.4 Integrated design considering topology op-timization . . . . . . . . . . . . . . . . . . . 226

9.3.5 Integrated design of an industrial application 226

Bibliography 227

IV Appendices 245

A Review on Model Reduction 247A.1 Problem Statement . . . . . . . . . . . . . . . . . . 249A.2 Model Reduction via Modal Decomposition . . . . 251

A.2.1 Case study - Milling Machine . . . . . . . . 252A.2.2 Static Condensation - Guyan Reduction . . 252A.2.3 Modal reduction - Model Truncation with

Static Correction . . . . . . . . . . . . . . . 253A.2.3.1 Structure without rigid body modes 254A.2.3.2 Structure with rigid body modes . 256

A.2.4 Component Mode Synthesis (CMS) . . . . . 257A.2.4.1 Statically Complete Mode Sets . . 263A.2.4.2 Dynamic Component-Mode Su-

persets . . . . . . . . . . . . . . . 264A.2.4.3 The Well-Known Craig-Bampton

Method . . . . . . . . . . . . . . . 265A.2.4.4 Evaluation of the Dynamic Super-

sets . . . . . . . . . . . . . . . . . 265A.3 Model Reduction Techniques via

Approximation-data Based Methods . . . . . . . . 269A.3.1 Singular Value Decomposition Methods . . 270

A.3.1.1 Proper Orthogonal Decomposi-tion (POD) methods . . . . . . . . 270

A.3.1.2 Approximation by balanced trun-cation . . . . . . . . . . . . . . . . 271

A.3.1.3 Model Reduction for Control - Acase study . . . . . . . . . . . . . 274

A.4 Model Reduction via Global ModalParametrization . . . . . . . . . . . . . . . . . . . . 276

A.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . 279

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Table of contents

B Review on Flexible Multibody System Theory 281B.1 Choice of Coordinates . . . . . . . . . . . . . . . . 282B.2 Computer Implementation . . . . . . . . . . . . . . 284B.3 Mechanical Principle to Generate the Equations . . 284B.4 Modeling parallel and serial kinematic machines

using flexible multibody system . . . . . . . . . . . 286B.4.1 Parametrization of rigid body spherical

motion . . . . . . . . . . . . . . . . . . . . . 287B.4.2 Prismatic Joint: an example . . . . . . . . . 291B.4.3 The Elastic Beam: geometry and deformation 293B.4.4 Sliding joints . . . . . . . . . . . . . . . . . 294

B.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . 297

C Review on Time-Integration methods 299C.1 Integration methods for mechatronic simulation . . 299

C.1.1 Integration methods for multibody dynamics 300C.1.1.1 Integration methods for integrated

environment - A co-simulationscheme . . . . . . . . . . . . . . . 301

C.1.2 Integration methods for flexible multibodydynamics . . . . . . . . . . . . . . . . . . . 302

C.1.3 Integration methods for integrated environ-ment . . . . . . . . . . . . . . . . . . . . . . 305

C.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . 306

Curriculum Vitae 307

List of Publications 309

XVII

Part I

Introduction and ModelingApproach

1

Chapter 1

Introduction

1.1 Motivation

This thesis concerns the computer-aided integrated design (CAID) ofmechatronic systems. Among the various definitions of mechatronics,two of them summarize the main subjects of this thesis:

• Mechatronics can be defined as the science of motion control thatcomprises the control of desired motion (tracking control) and thecontrol of undesired motion (vibration cancelation) (Van Brussel,1996) and

• Mechatronic design is the integrated design of a mechanical sys-tem and its embedded control system (Van Amerongen, 2003).

The latter definitions reveal the aspects that should be includedduring the design of mechatronic systems: control actions, system mo-tions and flexibilities. Therefore, simulation tools for supporting thedesign of mechatronic systems should allow the dynamic modeling ofthe machine, the control system design and the closed-loop system eval-uation. The control system design can be more specifically describedby the definition of a control strategy, the selection and configurationof sensors and actuators, the control gains, etc.

The standard design approach, i.e. a sequential mono-disciplinaryapproach, has been used successfully in the past and is still used in caseswhere stiff structures are considered and performance requirements arenot so strict (Fonseca and Bainum, 2004). Nowadays, the designers

3

1 Introduction

need to cope with the ever increasing demand for faster and more ac-curate machines. Faster machines can be designed using smaller andlighter components, which may significantly modify the machine struc-tural resonances, but also by using high-bandwidth controllers, whichtypically yield to shorter response times. However, a high bandwidthcontroller also implies in a system which is sensitive to noise and toparameter variations (Skogestad and Postlethwaite, 1997). Therefore,systems with high-bandwidth controllers and under high accelerationsmay have their structural resonances excited, reducing the machine ac-curacy. From this reasoning, which is illustrated in Fig. 1.1, the mainmechatronic design tradeoff issue arises: how to design machines whichmight work at high speeds and accelerations and also have to fulfill thedemands on accuracy.

In line with this reasoning, the former definition of mechatronicsreflects how structural and control parameters should be consideredduring the design of mechatronic systems. In this way, optimal mecha-tronic design requires that, among the structural and control param-eters, an optimal choice has to be made with respect to design speci-fications in the different domains. In the control domain, the optimalcontrol design is a well understood problem and methods are widelyspread for linear systems. In spite of the advances in optimal con-trol design, optimal mechatronic design is still an open research area.There are, two main reasons for this fact: (i) the difficulty in modelingmechatronic systems due to their multidisciplinary nature and (ii)the difficulty in solving optimization problems involving struc-

Control

Bandwidth

Structural

Resonances

Figure 1.1: Mechatronic design tradeoff: higher bandwidthcontrollers and shifted structural resonances

4

1.1. MOTIVATION

0.51

1.52

2.5

0

10

20

30

40

60

70

80

90

100O

bjec

tive

Structural Parameter

Control Parameter

Figure 1.2: Standard design × Mechatronic design: () optimizedstructural system, () sub-optimal mechatronic system and (•)

optimal mechatronic system

tural and control parameters due to their non-convex nature.Nevertheless, in spite of the difficulties, optimal design of a mecha-

tronic system can only be achieved if control and structural parametersare considered concurrently. The standard design procedure typicallyleads to suboptimal designs. In the standard design approach, thestructure is first optimized based on the passive performance and then,subsequently, an active control system is derived. This can be exempli-fied by interpreting Fig. 1.2. This figure shows isolines representing theperformance evaluation, referred to as objective, in function of controland structural parameters for a case study. For example, the optimalstructural parameter, considering the passive performance, is 1.5 (rep-resented by an empty square in Fig. 1.2). The optimal control systemfor this structure yields a suboptimal design represented by a emptycircle in Fig. 1.2. The optimal design can only be achieved by con-sidering the structural and control parameters simultaneously, which isrepresented by a filled circle.

5

1 Introduction

Design Specifications

Layout Design

Preliminary Analysis and Optimization

Design of the Motion Units

Servomechanism Design

Detailed MechanicalDesign

Detailed ControlDesign

Detailed Design Phase

Conceptual Design Phase

Optimal Design

Figure 1.3: Various steps of control design (Mecomat, 2001)

The mechatronic design process can be, basically, divided into twophases (Mecomat, 2001): conceptual mechatronic design, wherethe space of possible design alternatives is explored, preliminarily an-alyzed and optimized; and detailed mechatronic design, where thedetailed coupled structural and control models are generated, and pre-cisely analyzed and optimized (Fig. 1.3).

During the conceptual mechatronic design phase, the layout, themotion units and the servomechanism are designed. The layout de-sign consists of generating and evaluating different morphologies and

6

1.2. CHALLENGES IN MODELING MECHATRONIC SYSTEMS

geometric layouts. The design of motion units consists of the specifica-tion of the auxiliary motion units, such as slideways, kinematic chains,motors and drivers. And finally, the servomechanism design evalu-ates and generates the possible control schemes for the driving system.These steps are accomplished by preliminary analysis and optimizations(Mecomat, 2001). The designer should analyse the possibilities and, ac-cording to the results of simple models, choose the best configurationsthat will be evaluated in the detailed design. Nemeth (2003) developeda new integrated computer-aided design system that supports the syn-thesis, analysis and optimization processes of the preliminary design of3-axis machine tool structures. Fleischer et al. (2008) discusses topol-ogy optimization of machine tools with parallel kinematics.

During the detailed design phase, the design of structural com-ponents and the control system should be carried out simultaneouslyaccording to the objective functions given in the design specifications.Therefore, a computer-aided integrated design approach requires simu-lation tools that enable the direct assessment of structural and controlparameters. Virtual prototyping can support the designer to evaluateand to optimize mechatronic systems during the detailed design phase.Obviously, this practice results in a decrease in test time and numberof physical prototypes, which results in a significant cost reduction.

1.2 Challenges in modeling mechatronic sys-tems

Due to its complexity, the design of a mechatronic system becomes amultidisciplinary task, integrating fields such as mechanics, electronics,control, thermodynamics, fluid dynamics, piezoelectricity, etc.

The demand for cost-efficient products, that can be introducedquickly in response to market needs, requires that these products aredesigned with minimal prototyping, relying on simulation to verify de-sign requirements. Therefore, concerning the optimal design of mecha-tronic systems, a multi-disciplinary simulation environment plays a keyrole (Diaz-Calderon, 2000; de Fonseca, 2000). However, it is still a bigchallenge to create simulation environments for complex mechatronicsystems, particularly with respect to the design process. Simulationmodels must allow the designer to combine models from different dis-ciplines into integrated system-level models, enabling models of sub-systems to evolve throughout the design process. An integrated envi-

7

1 Introduction

ronment should support the designer in different engineering disciplinesand provide simultaneous design and decision-making tools (Mecomat,2001). In spite of the demand, much work remains to be done on thedevelopment of a fully integrated design and optimization environmentfor mechatronic systems: the Mechatronic Compiler.

Still there is a gap between simulation software used for evaluationof mechanical structures and software used for controller design (VanAmerongen, 2003). Mechanical engineers are used to finite-element andmultibody packages to examine the dynamic properties of mechanicalstructures. It is only after a reduction to low-order models that theycan be used for control design. On the other hand, typical control-engineering software does not directly support the mechatronic designprocess because in the modeling process, the commonly used trans-fer functions and state-space descriptions often have lost their relationwith the physical parameters of the mechanical structure. In this way,simulation tools, which allow modeling of mechanical systems in a waythat the dominant physical parameters are preserved in the model andsimultaneously provide an interface to the controller design and eval-uation tools, are still required (Van Amerongen and Breedveld, 2003;Van Amerongen, 2003).

1.3 Challenges in optimizing/designing mecha-tronic systems

1.3.1 Optimization problem statement

A mechanical system, P, can be represented by a state-space modeldescribed by: [

xpyp

]=[

Ap Bp

Cp Dp

] [xpup

](1.1)

where Ap ∈ Rnp×np is the state matrix, Bp ∈ Rnp×ip is the inputmatrix, Cp ∈ Rop×np is the output matrix, Dp ∈ Rop×ip is the directtransmission matrix, xp ∈ Rnp is the state vector, yp ∈ Rop is theoutput vector and up ∈ Rip is the input vector.

Considering, for instance, that the input vector up is generated bya full-order output feedback controller K, for which the state-spacerealization is given by Eq. 1.2, a measure to evaluate the system andits controller can be expressed.

8

1.3. CHALLENGES IN OPTIMIZING/DESIGNING MECHATRONICSYSTEMS

PKuk

yk

up

u

yp

Figure 1.4: Scheme of the open-loop transfer function, L = PK

[xkyk

]=[

Ak Bk

Ck Dk

] [xkuk

](1.2)

where Ak ∈ Rnk×nk is the state matrix, Bk ∈ Rnk×ik is the inputmatrix, Ck ∈ Rok×nk is the output matrix, Dk ∈ Rok×ik is the directtransmission matrix, xk ∈ Rnk is the state vector, yk ∈ Rok is theoutput vector and uk ∈ Rik is the input vector of the controller K.

For this full-order output feedback controller, the input vector ofthe system, up, is provided by the output vector of the controller, yk,yielding yk = up and ok = ip (see Fig. 1.4).

The open-loop state-space representation of this mechanical systemand its controller can be derived by substituting up = yk = Ckxk +Dkuk into Eq. 1.1: xp

xkyp

=

Ap BpCk BpDk

0 Ak Bk

Cp DpCk DpDk

xpxkuk

(1.3)

In order to evaluate the dynamic response of a system and its con-troller, a suitable measure should be adopted f(P,K). This measurecan be associated with time-domain metrics, such as overshoot andsettling-time, and/or with frequency-domain metrics, such as stabilitymargins (Skogestad and Postlethwaite, 1997). In this way, this measureis a function of the open-loop transfer function, f(P,K) = f(PK) =f(L).

The integrated structure and control optimization problem canbe described by the following optimization problem (Lu and Skelton,2000):

minP,K

f(P,K)

s.t. h(P,K)(1.4)

9

1 Introduction

where the objective is to minimize a measure of the dynamic responseof the system, f(P,K), subject to some constraints h(P,K), which canbe described as functions of the system and its controller.

Among other issues, in order to evaluate f(P,K), structural andcontrol variables are multiplied yielding a non-convex optimizationproblem (Lu and Skelton, 2000). For instance, considering the me-chanical system and its controller described by Eqs. 1.1 and 1.2, thisissue can be clearly identified due to the presence of the terms BpCk,BpDk, DpCk and DpDk in the description of the open-loop state-spacerepresentation (Eq. 1.3).

Because of its non-convex nature, the integrated structure and con-trol optimization problem is a hard problem and methodologies arerequired to overcome the computational difficulty (Lu and Skelton,2000). According to Wujun and Changming (2005), there are mainlytwo numerical methods to perform the integrated structure/controllerdesign:

Nested design strategy: The nested design strategy combines non-linear optimization methods and model-based control design tech-niques, such as the ones based on linear matrices inequalities(LMI) and Ricatti approaches. In fact, this strategy consists oftwo optimization loops: the inner loop responsible for the con-trol derivation and the outer loop responsible for the closed-loopevaluation and for updating the structural parameters (see Fig.1.5). The inner loop can be performed using model-based controldesign techniques yielding a convex optimization problem. Theouter loop deals with a non-convex optimization problem whichrequires nonlinear optimization methods or the use of genetic al-gorithms. The nested design strategy converges when the outeroptimization loop converges.

In this way, the integrated design is performed in an iterativeway as depicted in Fig. 1.5. For a set of structural parameters,a model is derived using the suitable modeling approach. Thismodel is represented in Fig. 1.5 by a scheme containing rigid andflexible bodies, joints, external forces, etc. This model may con-tain several degrees-of-freedom, which are unsuitable for model-based control design. Therefore, the employment of a model re-duction technique is usually required. A controller is then derivedusing a model-based control design technique. This inner opti-mization loop is represented by the block-diagram scheme in Fig.

10


1.5.

It is important to highlight that, when using model-based controldesign, the order and the structure of the controller are mainlydependent on the order of the system model and the weightingfunctions used to shape the response. Thus, neither the ordernor the structure of the controller are known in advance. In thisway, the controller might be rather complex to be able to fulfilthe design requirements.

Direct design strategy: The direct design strategy considers, simul-taneously, the control and structural parameters using a numeri-cal method such as non-linear optimization algorithms or geneticalgorithms. These algorithms may require long calculation timespecially when several parameters are considered. To employ thisstrategy, the designer needs to access both structural and controloptimization variables. Concerning the control design, this con-dition is rather restrictive, because it requires that the controlstructure is known beforehand. A typical example of this ap-proach is the optimization of PID gains simultaneously with astructural parameter.

A scheme illustrating the direct strategy is shown in Fig. 1.6.The system to be optimized consists of both mechanical and con-trol subsystems and is illustrated by a scheme containing rigidand flexible bodies, joints, external forces, etc., representing themechanical subsystem, connected with a block-diagram scheme,representing the control subsystem. The dynamic response isevaluated and optimized using a optimization algorithm able tocope with non-convex problems.

It is important to highlight that, when using direct design strat-egy, the order and the structure of the controller should be knowin advance and may be not optimal. This strategy is relevantwhen the problem is well-understood and the designer is sure thatthe chosen control structure can cope with the requirements.

11

1 Introduction

aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

Force

Rigid body

Joint Flexible body

Flexible beam

Structure

Initial DesignStructural parameters

Str

uct

ura

l p

aram

eter

s

Convergence?

Yes

No

OptimisedMechatronic

System

ConstraintsOK?

Yes

NoConvergence?

Yes

No

OptimisedController

K-

Controller

P

Plant

Controller

Reference

+

Figure 1.5: Nested Design Strategy

12



Force

Rigid body

Joint Flexible body

Flexible beamK

Reference

+ -

Controller

ConstraintsOK?

Structure and its embedded controller

Yes

NoConvergence?

Yes

No

OptimisedMechatronic

System

Initial DesignStructural and control parameters

Str

uct

ura

l an

d c

ontr

ol

par

amet

ers

Figure 1.6: Direct Design Strategy

1.3.2 State-of-the-art in optimizing mechatronic systems

A classification of the optimization problems regarding the design vari-ables, constraints, objective functions, problem domain and environ-ment is provided by Roy et al. (2008). In summary, convex, single-objective, mono-disciplinary, linear optimization problems are inexpen-sive to solve whereas non-convex, multi-objective, multi-disciplinaryoptimization problems are expensive to solve. The aforementionedcharacteristics of an integrated design problem, which is a non-convex,multi-disciplinary, and often multi-objective optimization problem, arechallenging issues that have been partially treated in some references.

One of the methodologies to solve this expensive problem is theDesign for Control (Liu et al., 2001), which proposes designing themechanical structure of a machine by fully exploring the physical un-

13

1 Introduction

derstanding of the parameters, yielding simplified models for facilitat-ing the control design.

Reyer et al. (2001) have compared different control and structuraldesigns as well as optimization strategies. It has been shown that differ-ent strategies converge to different optimal solutions, often leading tosuboptimal solutions. The traditional method of sequentially optimiz-ing the structure and then designing the control has been called SinglePass Strategy. The method can be improved by repetitively designingand controlling until the coupling quantities match. This approach iscalled Iterative Strategy. A more sophisticated approach is the Decou-pled System Strategy, where the entire design and control system isoptimized while some set of coupling quantities is fixed. The Parti-tion Strategy treats the optimal mechanical design and optimal controldesign as separate subproblems while the Bilevel Strategy treats theoptimal control design as a subproblem. And finally, the ConcurrentStrategy optimizes the system treating all coupling quantities as vari-ables and solving it all at once. The conclusions of this work are thatthe Concurrent and Partition Strategies found the true optimum forthe general problem. However, the Concurrent Strategy could be verydifficult to solve and the Partition Strategy is solvable only when anappropriate coupling strategy could be determined. These strategieswere evaluated by Reyer and Papalambros (1999, 2000). The main con-clusion is that the typical separation of mechanical design and controldesign leads to a non-optimal system.

An experimental campaign regarding integrated design has beenperformed by Fathy et al. (2001). A simple experiment, consisting ofan electric motor receiving a constant voltage and driving a sheavethat pulls a mass, has been studied. The optimization problem is tomaximize the lifted mass and minimize the lift time. Experimentalresults showed that the optimum calculated using the standard designapproach has failed to render an optimal system because this approachneglected the coupling between the control and the structure.

Behabahani and de Silva (2007, 2008) have proposed a systematicmethodology for a detailed mechatronic design based on a MechatronicDesign Quotient. This method is a multi-criteria index, reflecting asystem-based evaluation of a mechatronic design. It aggregates sevenmain criteria: meeting the task requirements, component matching, ef-ficiency, intelligence, reliability, controller friendliness and cost. Guide-lines for the assessment of each criteria is given in Behabahani and de

14


Silva (2008).Most of the papers that can be found on the integrated struc-

tural/control optimization treat the problem by adopting the directstrategy whereby the entire controlled structure is analyzed in each it-eration of the optimization routine. The main problem of this approachis the excessive computation time, which grows exponentially with thenumber of structural design variables. Therefore, de Fonseca (2000)has proposed the Indirect Approach, in which approximation conceptsare used, to perform the integrated structural/control optimization ina three-axis milling machine design. A physical approximated modelwhich incorporates technological constraints, such as fatigue lives ormaximal motor torques, has been employed.

Ravichandran et al. (2006) solves the simultaneous plant-controllerdesign optimization of a two-link planar manipulator using a heuris-tic evolutionary algorithm. Structural parameters for the balancingof the planar rigid mechanism and the gains of a PD controller areoptimized simultaneously. Affi et al. (2007) optimizes the geometryand the control parameters of a motor-driven four-bar system usingmulti-objective genetic algorithm optimization. In this way, they haveobtained the Pareto Front and design tradeoffs can be evaluated. Wu-jun and Changming (2005) applied the mechatronic design approachto design an active damping guide roller to suppress elevator lateralvibrations. A ’sky-hook’ controller structure is chosen a priori. Mecha-tronic design has been rarely employed when considering active controlfor noise reduction. A simultaneous structural and control optimiza-tion of a flexible linkage mechanism for noise attenuation has beendescribed in (Xianmin et al., 2007). In that case, the aim is to reducethe structural-acoustic radiation of a flexible mechanism considering inthe objective function the weight of the structure, the vibration energyand the control system energy. The mechatronic design approach de-scribed by Ravichandran et al. (2006); Affi et al. (2007); Wujun andChangming (2005) and Xianmin et al. (2007) can be classified as DirectDesign Strategy according to the numerical methodology classificationdescribed in the previous subsection.

Rieber and Taylor (2004) have performed the integrated controlsystem and mechanical design of a compliant two-axes mechanism. Again-scheduling H∞ control design approach proposed by Apkarian andAdams (1998) has been applied to achieve compensation for the vary-ing mass distribution, to suppress structural bending, vibrations and

15

1 Introduction

friction disturbances, and to achieve small motion settling time. Thetradeoff between the mass and the settling times has been identified.To the best of our knowledge, the work reported by Rieber and Taylor(2004) is the only reference that treats integrated design of systemswith varying dynamics.

In order to overcome the difficulty of solving a non-convex problem(Eq. 1.4), Grigoriadis et al. (1996) and Skelton and Kim (1992) haveproposed the inclusion of a constraint that holds the closed-loop matrixconstant. This feature makes the structure/control problem convex,but it has an important drawback in that the mass matrix componentscannot be optimized. A more general structure for this problem isproposed by Lu and Skelton (2000) using mixed H2/H∞ performancecriteria. Camino et al. (2003) has proposed a convexifying algorithmeliminating the constraint. It is a very powerful approach based on lin-ear matrix inequalities, and the mass matrices can be optimized. A veryimportant assumption of this approach is that the mass, damping andstiffness matrices are affine in the parameters. None of the case studiestreated in this thesis can be modeled in this way. Therefore, this tech-nique is not considered in this work. The mechatronic design approachdescribed by Rieber and Taylor (2004); Grigoriadis et al. (1996); Skel-ton and Kim (1992); Lu and Skelton (2000), and Camino et al. (2003)can be classified as Nested Design Strategy according to the numericalmethodology classification described in the previous subsection.

In spite of its issues, the benefits of the integrated design approachhave been highlighted in several fields: motor-driven mechanisms de-sign (Affi et al., 2007; Ravichandran et al., 2006), machine tool design(de Fonseca, 2000; da Silva(b) et al., 2007; da Silva(c) et al., 2007;da Silva et al., 2008), active damping guide roller design (Wujun andChangming, 2005), smart structures (Behabahani and de Silva, 2008),active vibro-acoustic systems (de Oliveira et al., 2008), bioengineering(Zollo et al., 2007), among others.

1.4 Objectives

The scope of this doctoral research is to develop, combine and/or ex-tend simulation tools and methodologies for computer-aided integrateddesign of mechatronic systems. The developments described in thisthesis are related to the detailed design phase of a mechatronic sys-tem when detailed models of the mechanical and control systems are

16

1.5. METHODOLOGY AND CASE STUDIES

built, optimized and evaluated. Within this framework, three casestudies are considered to exemplify the benefits of integrated designfor mechatronic systems: a vehicle mock-up (study on cavity noise),a mixed (serial and parallel) kinematic pick-and-place machine and aparallel kinematic pick-and-place machine. In order to fulfil the mainobjective and illustrate the benefits of the methodology, the followinggoals are pursued:

• To reduce the gap between simulation software used for evalu-ation of mechanical structures and software used for controllerdesign;

• To contribute towards a mechatronic compiler;

• To extend the work of de Fonseca (2000), who described the useof a finite-element methodology to derive reduced models thatcould be employed for model-based control design;

• To provide modeling methodologies for mechatronic systems withconfiguration dependent-dynamics, such as parallel and serial ma-chine tools; and

• To illustrate the benefits of mechatronic design by means of casestudies.

1.5 Methodology and case studies

Mechatronic design has been performed for three case studies. Thesestudies comprise:

• the detailed modeling of the system using finite-element and/ormultibody methodologies;

• the use of a suitable model reduction technique to derive concisemodels used for model-based control design and/or closed-loopperformance evaluation; and

• the integrated design considering control and structural parame-ters simultaneously.

The first case study consists of a vibro-acoustic cabin mock-up (Fig.1.7). It consists of two closed cavities with concrete walls separated by

17

1 Introduction

a steel panel. The disturbance noise generated by an acoustic sourcein a cavity is transmitted to the other cavity through vibrations of thepanel. In order to attenuate the noise transmission, a collocated sen-sor/actuator pair is considered in a velocity feedback control strategy.The optimal mechatronic design is obtained by optimizing both vibro-acoustic plant features, such as the panel thickness, and control pa-rameters, such as the sensor and actuator placement and control gainsusing the Direct Design Strategy. The benefits of the mechatronic de-sign are also experimentally validated. This case study is treated inChapter 4 and in de Oliveira et al. (2008).

Figure 1.7: A vibro-acoustic cabin mock-up

The second and the third case studies are pick-and-place machines.In pick-and-place applications, a piece can only be picked and placedcorrectly if the position error is below a certain threshold. Vibrationsat the end of a set-point, which exceed this threshold, have to settlebefore the machine restarts (Rankers, 1997). The dynamic behaviorof these systems depends on their instantaneous spatial configuration.Modeling methodologies able to capture this behavior are proposed inChapter 3.

A pick-and-place machine, containing parallel and serial kinematicchains, is treated in Chapters 5, 6 and 7. It consists of an academicsetup of a three-axis motion system with three linear motors and arotary brushless DC motor (Fig. 1.8). Different modeling approachesare proposed in Chapter 3. These models are experimentally validatedin Part II. Direct and Nested Design Strategies, considering a feedbackcontrol, are discussed in Chapters 6 and 7, respectively.

A parallel kinematic pick-and-place machine, designed by FatronikFrance (Baradat et al., 2008), is treated in Chapter 8. It consists of

18

1.6. MAIN CONTRIBUTIONS OF THIS THESIS

a two degree-of-freedom system that should accomplish a prescribedtrajectory in a prescribed amount of time (Fig. 1.9). Due to this re-quirement, a feedforward control strategy is considered. The maximumacceleration can reach 300m/s2. A flexible multibody model is derivedand experimentally validated. The optimal mechatronic design is per-formed using the Nested Design Strategy. In this way, a feedforwardcontroller is derived for each set of possible structural parameters.

Figure 1.8: A mixed pick-and-place robot

Figure 1.9: A parallel pick-and-place robot (Baradat et al., 2008)

1.6 Main contributions of this thesis

This thesis inherits its motivation from the mechatronic design method-ology which is primarily market-driven due to the ever increasing de-

19

1 Introduction

mands for better products and reduced time-to-market. Simulation-based design plays an important role during the design phase, not onlyfor shortening development cycles and reducing design costs but also forenhancing product performances. In this way, the main contributionof this work is to provide tools for computer-aided integrated designof mechatronic systems. This thesis is written from the perspective ofa structural engineer within the integrated design of mechatronic sys-tems. A control engineer perspective can be found in Van Amerongenand Breedveld (2003).

The main achievements of this work are:

• Guidelines for modeling mechatronic systems considering the sys-tem flexibilities, the system motion and the control action;

• The extension of a flexible multibody package to cope with themodeling of serial machines;

• The comparison of different techniques for deriving concise mod-els for mechatronic systems with configuration-dependent dynam-ics;

• The illustration of the benefits of the mechatronic design whendealing with active control of cavity noise;

• The derivation of parametric flexible multibody models for par-allel and serial pick-and-place robots including linear parameter-varying control system actions;

• The evaluation of system performance along the lines of themechatronic design using multi-objectives methodologies and al-gorithms;

• The employment of the nested and direct design strategies fordesigning a mixed kinematic pick-and-place robot concurrentlywith a feedback controller;

• The integrated design of a parallel pick-and-place robot and afeedforward controller.

20

1.7. OUTLINE OF THE DISSERTATION

1.7 Outline of the dissertation

This section gives an overview of the different parts/chapters and theirrelation in this dissertation. This thesis is mainly divided in four partscontaining nine chapters.

Part I introduces the main issues related to integrated design includ-ing guidelines for modeling mechatronic motion systems. It con-sists of three chapters.

Chapter 1 states the problem addressed in this thesis and framesit in the context of mechatronic design. It discusses the twomain challenges regarding the optimal mechatronic design: thedifficulty in modeling mechatronic systems due to their multidis-ciplinary nature and the difficulty in solving optimization prob-lems involving structural and control parameters due to theirnon-convex nature. Moreover, it clearly indicates the main ob-jectives and summarizes the main achievements of this work.

Chapter 2 presents an overview on modeling of mechatronic mo-tion systems. A mechatronic system model should contain theflexibility of some components, the system motion, and the con-trollers. Four methodologies for deriving mechatronic motionsystems models are addressed: the simulation of an active flex-ible structure in a finite-element environment; the simulation ofreduced or simplified models and their controllers in a controldesign environment; the co-simulation between structural designand control design environments; and the integrated simulation ofactive flexible systems in a unified environment. These modelingprocedures have been adapted in order to predict the behavior ofmechatronic systems with configuration-dependent dynamics inChapter 3.

Chapter 3 treats novel methodologies for modeling mechatronicsystems with configuration-dependent dynamics. Besides thechallenges treated in Chapter 2, such as the modeling of the flexi-bilities, the system motion, and the controllers; a large number ofmachines present varying dynamics that should be correctly pre-dicted during the design phase. Some important concepts andmethodologies used in this chapter are described in the Appen-dices A, B, and C. Based on these concepts and methodologies,three modeling approaches are proposed: the simulation of re-

21

1 Introduction

duced models, in several configurations or interpolated, and theircontroller in a control design environment; the co-simulation be-tween structural and control design environment; and the inte-grated simulation in a unified environment. These approachesare briefly introduced in this chapter. In Chapter 5, 6 and 7,these approaches are fully described and are used to model amixed kinematic pick-and-place machine (Fig. 1.8). In Chapter8, the latter methodology is applied to model a parallel kinematicpick-and-place machine (Fig. 1.9).

Part II is composed of published and submitted articles describing thebenefits of the integrated design for the case studies. It consistsof five chapters (articles).

Chapter 4 addresses, illustrates and experimentally validates thebenefits of the mechatronic design for active control of a vibro-acoustic system (Fig. 1.7). Active control is a potential solutionto many noise and vibration problems for improving the low-frequency performance. Cavity noise reduction as encounteredfor instance in aircraft cabins and vehicle interiors, is a typi-cal example. To cope with the mechatronic design approach, amethodology to derive a fully coupled mechatronic model thatdeals with both the vibro-acoustic plant dynamics as well as thecontrol parameters is described. This methodology provides areduced state-space model derived from a fully coupled vibro-acoustic finite-element model. Regarding noise reduction, op-timization results are presented considering both vibro-acousticplant features, such as thicknesses, and control parameters, suchas sensor and actuator placement and control gains.

Chapter 5 describes, in detail, a novel approach to model mecha-tronic motion systems with configuration-dependent dynamics:the co-simulation between structural and control design environ-ment. This approach is applied to a pick-and-place assemblyrobot (Fig. 1.8) and an experimental validation is carried out.Using the proposed approach, different control design techniquesand optimization methodologies can be applied considering notonly discrete configurations, but also continuous operation. Us-ing time-domain metrics, two control strategies are derived: alinear time-invariant proportional-integral-derivative (PID) con-troller and a linear parameter-varying PID controller. This is

22

1.7. OUTLINE OF THE DISSERTATION

further exploited in Chapter 6.

Chapter 6 considers the direct design strategy, which requiresa pre-defined control structure. Linear time-invariant and gain-scheduling PID controllers are addressed. A flexible multibodymodel is built containing a sliding joint, which allows the relativetranslational motion between flexible bodies. The derivation ofthis joint is described in Appendix B. This methodology is ex-ploited for the multi-objective optimization of a pick-and-placeassembly robot with a gripper carried by a variable-length flex-ible beam (Fig. 1.8). The resulting design tradeoffs betweensystem accuracy and control efforts demonstrate the advantageof an integrated design approach.

Chapter 7 employs the nested design strategy, which does notrequire a pre-defined control structure. A gain-scheduling con-troller derived from the interpolation of linear time-invariant con-trollers is considered. For each set of structural parameters, again-scheduling controller is derived for the pick-and-place as-sembly robot (Fig. 1.8). Eventually, the integrated design of thesystem, considering both structural and control parameters, isperformed.

Chapter 8 discusses the modeling and the control design of aparallel pick-and-place robot (Fig. 1.9). A flexible multibodymodel is built and a concise description of this model is obtainedthrough model reduction for each configuration. Based on a set oflinear time-invariant models, a feedforward controller is derived.The chosen feedforward controller design technique is based onthe optimization of splines and its constraints reflect the designrequirements. Considering a set of design requirements, the per-formance of the mechanical and control system is evaluated.

Part III presents the conclusion and the bibliography.

Chapter 9 states the most relevant conclusions of this work, sum-marizes the main achievements of this thesis, and presents somegeneral ideas for future work.

Part IV contains the appendices in which some theory and deriva-tions, employed throughout the thesis, are described. Finally,some biographical information about the author is given.

23

1 Introduction

Appendix A presents an overview, with dedicated examples, onmodel reduction. There are several model reduction approachesable to cope with different modeling methodology. This surveycomprises the model reduction of finite-element models, input-output representations and flexible multibody models. Thesetechniques are regularly applied throughout the thesis for thederivation of suitable models for mechatronic systems in Chap-ters 2, 3, 5, 7 and 8.

Appendix B presents an overview on flexible multibody tech-niques and the derivation of the sliding joint. This technique hasbeen employed for the modeling of the pick-and-place machinespresented in Chapters 3, 5, 6, 7 and 8.

Appendix C summarizes the time integration techniques ap-plied in the multibody field. Both co-simulation and integratedschemes are discussed. This is an important issue when model-ing mechanical systems with their embedded controllers which isconsidered in Chapters 3, 5, 6, 7 and 8.

The logical succession of the chapters is presented in Fig. 1.10. Sin-gle solid or dashed borders indicate chapters and appendices containingreviewed methodologies and theory. These sections can be skipped ifthe reader is familiar with the treated subject. Double borders indicatethe chapter containing contributions of the author. The outcomes ofthis research are described in these chapters.

1.8 Conclusions

The aims of this first introductory chapter are to clarify the objectiveand main achievements of this work as well as to describe the organi-zation of this dissertation.

The concept of mechatronic design is introduced as the integrateddesign of mechatronic systems considering structural and control pa-rameters. Two main challenges are identified when dealing with designof mechatronic systems: their modeling due to their multidisciplinarynature and their optimization due to their non-convex nature. Thelatter is treated in detail in Chapters 2, 3, 5 and 6 and the formeris exemplified via case studies in Chapters 4, 6, 7 and 8. Three casestudies are considered: a vehicle mock-up (study on cavity noise) andtwo pick-and-place machines.

24

1.8. CONCLUSIONS

Introduction

Chapter 1

Review onModel Reduction

Modeling of Mechatronic Systems

Chapter 2Modeling of Mechatronic Systems

with Varying Dynamics

Chapter 3

Chapter 6

Conclusions

Chapter 9

Time Integration Methods

Appendix A

Appendix C

Review on Flexible MultibodySystem Theory

Appendix B

Chapter 4

Chapter 7Chapter 5 Chapter 8

Co-simulationSerial Machine

Direct StrategySerial Machine

Nested StrategySerial Machine

Nested StrategyParallel Machine

Direct StrategyVibro-AcousticSystem

Figure 1.10: Thesis scheme: relations between the chapters

25

Chapter 2

Modeling of MechatronicMotion Systems

2.1 Introduction

Mechatronics is a multidisciplinary field. According to Van Amerongenand Breedveld (2003), the most important disciplines in mechatronicsare mechanical engineering, electrical engineering and software engi-neering. Therefore, due to its complexity, the design of a mechatronicsystem is a multidisciplinary task that should not only rely on the ex-perience of the designer but also on the support of dedicated tools.This chapter comprises a survey of the available simulation techniquesto aid the designer to predict the machine dynamics and optimize itsclosed-loop performance for systems of industrial complexity

Several references address the different approaches to model mecha-tronic systems. Giurgiutiu and Lyshevski (2003) indicate that, formodeling of electromechanical systems, the way that energy is stored,dissipated, transformed and transferred should be analyzed. In order toaccomplish this task, Giurgiutiu and Lyshevski (2003) use Maxwell’sequations, Lagrange equations of motion, Kirchhoff’s laws and New-ton’s laws to develop electromechanical models. Preumont (2006) de-rives the Lagrange equations for mechanical, electrical and electrome-chanical systems. Preumont (2006) also includes the modeling of smartsystems, using piezoelectric materials, and different control strategies.

In order to design and to evaluate mechatronic systems, multidis-ciplinary simulation tools are required. However, few tools can fulfil

27

2 Modeling of Mechatronic Motion Systems

this requirement, since most available commercial tools are highly ded-icated for a single field. In fact, there is still a gap between simulationsoftware used for evaluation of mechanical systems and software usedfor controller design (Van Amerongen, 2003). Mechanical engineersare used to finite-element and multibody packages to evaluate the dy-namic properties of mechanical systems. These models can contain alarge number of degrees-of-freedom (dofs). Therefore, it is just afterreducing the order of models that these models can be used for model-based control design. Some model reduction techniques are addressedin Appendix A. On the other hand, typical control engineering packagesdo not support directly the mechatronic design process since transferfunctions or state-space descriptions might have lost the relation withthe physical parameters of the mechanical system during the modelingprocess, or the models may no longer be parametric.

As mentioned in Chapter 1, a model that represents a mechatronicmotion system should contain the flexibility of some components, therigid-body motion, and the controllers. According to these require-ments, some approaches to model and evaluate mechatronic systemsare identified:

1. Simulation of an active flexible structure in a finite-element en-vironment, such as MSC/Nastran2004;

2. Simulation of reduced or simplified models and their controllersin a control design environment, such as Matlab/Simulink, or ina conceptual mechatronic design environment as, such as 20simand LMS Amesim;

3. Co-simulation between dedicated virtual environments, such asMatlab/Simulink and LMS/Virtual Lab. Motion;

4. Integrated simulation of the active flexible system in a unifiedenvironment, such as Oofelie.

Each one of these approaches is related to wide-spread modelingmethodologies such as finite-element and multibody system techniquesto evaluate the dynamical behavior of a mechanical system.

The basic principle of the finite-element (FE) method is the subdivi-sion of the whole problem domain into small subdomains, called finiteelements. Discretization techniques, such as the FE method, essen-tially transform systems described by partial differential equations into

28

2.2. SIMULATION OF AN ACTIVE FLEXIBLE STRUCTURE IN AFINITE-ELEMENT ENVIRONMENT

problems described by sets of simultaneous ordinary differential equa-tions. A review on the FE method is out of the scope of this thesis andcan be found in several references (Beer and Watson, 1994; Meirovitch,1980; Van Hal, 2004). Mainly, two strategies for modeling mechatronicsystems, which employ the FE method, can be identified: (1) the in-clusion of controllers in an FE environment and (2) the inclusion ofreduced/simplified models in a control design environment. The lattercan be accomplished using model-order reduction techniques, reviewedin Appendix A. Both strategies are reviewed hereafter in Sections 2.2and 2.3, respectively.

A mechatronic system is composed of rigid bodies, flexible bodies,joints, sensors, actuators and control units including the control algo-rithm. Due to these characteristics, formalisms developed in the field offlexible multibody dynamics appear to be especially suitable to modelthe mechanical part of mechatronic systems (Wasfy and Noor, 2003).A review on flexible multibody techniques can be found in AppendixB. An extension of those modeling methods is required to deal withcontroller dynamics. One option is (3) to use co-simulation betweentwo dedicated virtual environments, so that the time integration pro-cedure is based on a sequential analysis of the mechanical subsystemand of the control subsystem. Details about co-simulation approachesare described in Section 2.4. This scheme is often called weak or loosecoupling. An important drawback is that additional assumptions arerequired in the presence of algebraic loops (Samin et al., 2007). Asproposed in Bruls and Golinval (2006), a second option is (4) to use astrongly coupled modeling approach, so that a monolithic time integra-tor can be used, and no weak coupling assumption is required. Detailsabout this integrated simulation are described in Section 2.5.

2.2 Simulation of an active flexible structurein a finite-element environment

A large number of mechanical systems can be modeled by the FEmethod yielding the following second-order differential equations:

Mq(t) + Dq(t) + Kq(t) = Lif(t) (2.1)

where M ∈ Rn×n is the mass matrix, D ∈ Rn×n the damping matrix,K ∈ Rn×n the stiffness matrix, f ∈ Rm×1 the force vector, Li ∈ Rn×m

29


the Boolean matrix selecting the actuated dofs and q ∈ Rn×1 the dofs.Some FE packages, such as MSC/Nastran2004, allow the inclusion

of transfer functions in the FE model. According to MSC/Nastran2004,the relation between the dependent dof, qd, and the selected input dofs,qi, is given by the following transfer function:

(B0 +B1s+B2s2)qd +

∑i

(A0(i) +A1(i)s+A2(i)s2)qi = 0 (2.2)

where A’s and B’s are user-defined. Internally, this relation is simplyadded to a single row in Eq 2.1. More complex relations can be builtusing several transfer functions connected by fictitious dofs. This canbe done using the commands TF and EPOINT in MSC/Nastran2004.

For the sake of illustration, a clamped bar is considered as a casestudy. A scheme of the case study is depicted in Fig. 2.1. A disturbanceforce, f(t), is applied to node 19. xe is the vertical displacement of node19. A collocated velocity feedback controller is implemented using thetransfer function approach (Eq. 2.2), where the dependent and theselected input dofs are defined by the vertical motion of the node 4.The actuator applies a force, fc, proportional to the velocity, dxd/dt, atnode 4, fc = −Kdxd/dt. Figure 2.2 compares the frequency responsefunctions, (xe/f(s)), with and without the velocity feedback controller.The velocity feedback controller acts as a local damper since the forceis proportional to the velocity in vertical direction of the node 4. Thedamping effect can be verified in Fig. 2.2.

K, d/dt

f

f

x

x e

1 2 3 4 … … … 19

c

d

Figure 2.1: Collocated velocity feedback in a FE environment

In general, FE environments cannot handle system motion in thetime-domain simulation since the FE method is concerned with flex-ible bodies and small motions (deformations). This is an important

30

2.3. SIMULATION OF REDUCED OR SIMPLIFIED MODELS AND THEIRCONTROLLERS IN A CONTROL DESIGN ENVIRONMENT

0 50 100 15010

−5

10−4

10−3

10−2

Frequency [Hz]

Mag

nitu

de [m

/N]

without TFwith TF

Figure 2.2: Comparison between |xe/f(s)| with and without thevelocity feedback controller

drawback when modeling mechatronic systems by implementing thecontroller in the FE environment. Therefore, this methodology cannotbe directly applied for modeling active motion mechatronic systems,such as serial and parallel machines. Because of this reason, this tech-nique is not considered in this thesis.

2.3 Simulation of reduced or simplified mod-els and their controllers in a control designenvironment

The flexibility of some components of a mechatronic system can beconsidered by including reduced and/or simplified models described bystate-space models derived from FE models in a control design envi-ronment. In this thesis, the control design environments are classifiedin mainly three groups: block diagram tools, multiphysics tools basedon bond graph theory and modeling languages for general purposes.

Block diagram based tools, such as Matlab/Simulink and Easy5,are quite popular among control engineers since they allow the simu-lation and evaluation of dynamic systems via an output/input systemdescription. The block diagram represents the system modeling strate-gies, such as transfer functions and state-space models. These modelscan be derived using system identification and/or modeling techniques.

31


These tools do not support physical modeling in a direct way. And of-ten, the models have lost their parametrization and/or physical mean-ing.

Multiphysics tools based on bond graph theory, such as LMS Imag-ine.Lab AMESim and 20-sim, allow physical modeling in various phys-ical domains. The components of the model are described by analyticalmodels representing the hydraulic, pneumatic, electric or mechanicalbehavior of the system. The components can be connected to eachother via proper interfaces referred to as ports. Modeling in differentphysical domains requires that a core language be available to describea system in different domains. This is achieved by coupling models bymeans of the flow of energy, rather than by signals such as voltage,current, force and speed (Van Amerongen, 2003). A review on bondgraph theory and port-based simulation tools can be found in Breedveld(2003, 2004). Recently, Ligterink (2007) has combined a port-based ap-proach (20-sim) and an FE approach, which is highly desirable. Usingthis approach, the designer is able to evaluate mechanical subsystemsthat can be modeled using FE approach and electrical and control sub-systems that can be modeled using port-based approach.

A powerful modeling language Modelica (Otter and Elmqvist, 2001)is a freely available object-oriented language suitable for modelingmulti-domain problems. Models in Modelica are mathematically de-scribed by differential, algebraic and discrete equations. The packageis divided in several libraries: standard, magnetic, fluid, system dy-namics, bond graphs among others. The modeling methodology variesaccording to the library. More information about Modelica can befound on http://www.modelica.org/.

Mechanical systems can be modeled by the FE method yielding theEq. 2.1. These models can contain a large number of dofs. There-fore, it is just after reducing the order of models that these modelscan be used for active system evaluation and/or model-based controldesign in a control design environment. Using the Component ModeSynthesis (CMS) technique, described in detail in the Appendix A, thedisplacements of the physical coordinates, the dofs q ∈ Rn×1, are repre-sented in terms of component-generalized coordinates η ∈ Rk×1 usingthe classical modal transformation:

q = Ψη (2.3)

where Ψ ∈ Rn×k consists of a pre-selected dynamic component mode

32


superset: dynamic constraint-mode superset, dynamic attachment-mode superset or dynamic attachment-mode and k n (see detailsin the Appendix A).

Performing the transformation Eq. (2.3) and pre-multiplying byΨT , Eq. (2.1) can be modified to:

Mrη(t) + Crη(t) + Krη(t) = ΨTLif(t) (2.4)

where Mr ∈ Rk×k is the reduced mass matrix, Cr ∈ Rk×k the re-duced damping matrix, Kr ∈ Rk×k the reduced stiffness matrix, andLi ∈ Rn×m the Boolean matrix of which the matrix element is 1 whenrepresenting actuated dofs and 0 otherwise.

Equation (2.4) can be rewritten in state-space form as:

[ηη

]=[

0 I−M−1

r Kr −M−1r Cr

] [ηη

]+[

0M−1

r ΨTLi

]f(t) (2.5)

The output y ∈ Rm×1 can be calculated by

y = Lo

[Ψ0

] [ηη

](2.6)

where Lo ∈ Rm×n is a Boolean matrix which is 1 for the measureddofs and 0 for the rest. These state-space equations can be evaluatedin Matlab/Simulink, Easy5, LMS Imagine.Lab AMESim, 20-sim andother control design tools.

For the sake of illustration, the subsystem represented by a linearmotor and a flexible beam is considered as case study as shown in Fig.2.3. This subsystem represents the linear motor, the flexible beam andthe gripper of a pick-and-place assembly robot depicted in Fig. 1.8.The linear stiffness between the linear motor and the flexible beam isK=2.5·1010N/m, the motor slider mass is M=25.9kg, the beam lengthis l = 0.53m and the gripper mass is mg=1.25kg. The material prop-erties of the beam are: density ρ=7800kg/m3, Poisson’s ratio ν=0.3,damping ratio 0.01 and elasticity modulus E=2.12·1010N/m2. The xmand xg coordinates represent the motor and the gripper displacements,respectively.

The Craig-Bampton model reduction scheme (Craig, 1987) was ap-plied to the FE model when 1 constraint (static) mode, representingthe rigid-body motion, and 2 flexible modes, representing the beam

33


M

x

mg

K

l

Linear Motor

Fm

m

xg

Figure 2.3: Scheme of the FE model of the subsystem representedby a linear motor and a flexible beam

flexibility, are kept. Details about the Craig-Bampton model reductiontechnique can be found in Appendix A. The model was built using theStructural Dynamics Toolbox (SDT), but any FE tool can be used.This procedure yields a state-space model with 6 states, one input:linear motor force; and two outputs: the gripper and the motor accel-erations. The FRFs xg/Fm and xm/Fm, extracted from the state-spacemodel, are shown in Fig. 2.4. This model can be included in a Mat-lab/Simulink simulation where control schemes can be implementedand tested.

This reduced state-space model can be included in a control designenvironment where the controller can be designed and verified. Thissystem is treated again in Chapter 3 when mechatronic motion systemswith varying dynamics are considered. Controllers for this case studyare investigated in Chapters 5, 6 and 7.

In this section, model reduction based on modal transformation isconsidered to derive reasonably sized models. Other techniques, suchas approximation-data based methods (see Appendix A), can also besuccessfully employed for deriving reduced models from large FE mod-els. Since these methodologies rely on the construction of a subspacethat best approximates the collected data, the physical meaning of the

34


102

104

10−4

10−2

100

102

Mag

nitu

de [1

/Kg]

(a)

102

104

−400

−300

−200

−100

0

Pha

se [d

egre

e]

Frequency [rad/s]

102

104

10−3

10−2

10−1

100

Mag

nitu

de [1

/Kg]

(b)

102

104

0

50

100

Pha

se [d

egre

e]

Frequency [rad/s]

Figure 2.4: (a) xg/Fm and (b) xm/Fm

modeling may be lost during the model reduction which is an importantdrawback.

Modal reduction has been employed in several fields to derive con-cise models for control design and evaluation. Yan et al. (2008) presentsa quantitative analysis using the finite-element method for servo systemmodeling and reduction of hard disk drives. The modeling reductionprocedure consists of keeping the most important modes and has beenvalidated by experimental results. Yan et al. (2008) have shown thatthe modeling method can be used for the simulation and evaluation ofthe servo control to achieve integrated system design for mechatronicsystems. The dynamic modeling of serial and parallel machines canalso be performed using finite-element and modal reduction. Recently,a substructuring dynamic modeling procedure has been applied for themodeling of a flexible-link planar parallel platform by Wang and Mills(2006). This technique is based on the assembly of component modesets, extracted from finite-element models using CMS (Craig, 1987). Asimilar procedure was employed in the modeling of a 3-axis milling ma-chine by Van Brussel et al. (2001) for several discrete spatial configura-tions. This procedure has been described in Appendix A.2.4. Massonaet al. (2006) have presented a new method to improve the standard

35


modal reduction methods by taking into account a priori informationconcerning the potential structural modifications, thus allowing thesame transformation to be used throughout the optimization process.The approach consists firstly of extending the standard condensationbasis by a set of optimized static residuals and secondly, in eliminat-ing the associate coordinates from the reduced system. The proposedmethod can be used with any condensation procedure, including bothdirect reductions and component mode synthesis approaches with anykind of substructure natural modes; freefree, clamped or hybrid modes.For robustness analysis, internal and fuzzy finite-element method havebeen used. De Gersem et al. (2007) have suggested non-probabilisticmethods combined with the component mode synthesis technique toreduce the calculation time.

Modal reduction has been employed for deriving reasonably sizedfully coupled vibro-acoustic models in Chapter 4. This fully coupledmodel has been used for designing the active system considering struc-tural and control parameter along the lines of mechatronic design.

2.4 Co-simulation between dedicated virtualenvironments

A mechatronic system can be modeled as a flexible multibody sys-tem coupled with its control system (see a review in Appendix B).An extension of flexible multibody methods is required to deal withcontroller dynamics. One option is to use co-simulation schemes be-tween two dedicated virtual environments: a multibody package anda block diagram based tool. Commercial multibody packages, such asLMS/Virtual Lab. Motion and MSC/Adams, offer interface with blockdiagram based tools, such as Matlab/Simulink and Easy5. In this way,the commercial multibody packages are responsible for generating theequations of motion at each integration step. In other words, they areresponsible for deriving and solving the following equations of motion:

M(q)q + BTλ = g(q, q, t) (2.7)Φ(q, t) = 0 (2.8)

where q are the nodal coordinates, λ the Lagrange multipliers, M isthe mass matrix, which is not constant in general, g represents the

36

2.4. CO-SIMULATION BETWEEN DEDICATED VIRTUAL ENVIRONMENTS

internal, external and complementary inertia forces, Φ(q, t) are thekinematic constraints and B = ∂Φ/∂q is the matrix of constraint gra-dients. These equations are developed in Appendix B.

Typically, connecting elements, input and output signals, are de-fined in the multibody package and will be exchanged with a con-trol design software. Dedicated control design softwares, such as Mat-lab/Simulink and Easy5, are responsible for generating the actuatorforces or torques, y, according to the control input signals, u, in thefollowing way:

x = f(x,u, t) (2.9)y = h(x,u, t) (2.10)

where x are the state-space variables. The connection between Mat-lab/Simulink and a multibody package is typically performed by anS-function block (see Fig. 2.5).

Matlab/Simulink Multibody PackageMatlab Interface

(S-functions)

Figure 2.5: Co-simulation scheme: Matlab/Simulink and MultibodyPackages

Due to stability, accuracy, and performance issues, the com-munication rate should be chosen carefully. Generally, the solu-tion of these equations can be obtained in two ways (Adams, 2001;LMS.International, 2006):

1. Co-simulation mode : In this case, the integrators on both sides(control package and multibody environment) are running in par-allel. They exchange data (via connecting elements) at specifiedtime instants. In this way, the control package, using the valuesfrom the connecting elements (sensors), calculates the control ac-tuation and sends it back to the connecting elements (actuators).It is the responsibility of the multibody environment to integratethe mechanical system for a single time step with the specifiedinputs.

37


2. Block evaluation mode : In this mode, the multibody environmentacts as a block evaluator. In this way, the differential equationsdescribing the mechanical system are evaluated, and the outputsare sent to the control package. The equations are integratedsolely by the control package. From the control package point ofview, the multibody model is like a nonlinear block.

The latter approach provides very accurate results and avoids alias-ing. The drawback is that the time integration algorithms available inthe control packages must be used and they are not finely tuned forsolving mechanical systems. Hence, they have a tendency to fail forcomplex systems, particularly those with high frequency mechanicalsystem effects. Therefore, the second approach is less common in prac-tical applications (Zhou et al., 2007). Details about the integrationmethods for co-simulation schemes can be found in Appendix C.

Co-simulation schemes have been employed in several fields in or-der to predict the dynamic response of active systems. Levesley et al.(2007) describes the development and use of a multi-body co-simulationapproach for predicting the dynamic response of a vehicle containingmagneto-rheological semi-active dampers. In this work, road inputsand the tyre model are also implemented within Matlab/Simulink.Using the co-simulation capabilities, Prado et al. (2009) have built areal time remotely operated vehicle for inspection, maintenance, build-ing structures and surveillance in deep water offshore platforms. Co-simulation has been employed to model and evaluate active serial ma-chines by da Silva et al. (2007). This challenge is introduced in Chapter3 and treated in detail in Chapter 5.

2.5 Integrated simulation in a unified environ-ment

As proposed in Bruls and Golinval (2006), a second option is to usea strongly coupled modeling approach, so that a monolithic time inte-grator can be used, and no weak coupling assumption is required. Thisstrongly coupled formulation is available in the Oofelie finite-elementsoftware (Cardona et al., 1994). In theory, there is no difference be-tween this approach and the aforementioned Block evaluation mode.However, fine tuned implicit integration schemes can be implemented

38

2.5. INTEGRATED SIMULATION IN A UNIFIED ENVIRONMENT

in Oofelie, which are not available in Matlab/Simulink, avoiding failureduring the integration of the coupled equations.

The dynamic equations of a mechatronic system, consisting of amultibody model and a control system, have the general structure:

M(q)q + BTλ = g(q, q,w, t) + y (2.11)0 = Φ(q, t) (2.12)x = f(x,u, t) (2.13)y = h(x,u, t) (2.14)

Eq. (2.11) represents the dynamic equations of the mechanical system,Eq. (2.12), the kinematic constraints, Eq. (2.13), the state equationand Eq. (2.14), the output equation. The terms x, y and u are thestate-space, the output and the input variables, respectively. The termw, represents the disturbance, noise and reference signals vector. Thisterm has been added to the formulation in order to be in line with theterms used in the control field.

Figure 2.6 shows a scheme of the augmented plant Pa, which in-cludes the mechanical system P , and the control system K. Examplesof augmented plants can be found in Chapters 6 and 7. The notation isthe same one as adopted in Eqs. (2.11-2.14). The output system signalz and control signal inputs u can be described by combinations of thedisturbance, noise and reference signals w, the control signal outputsy, and the measurements from the mechanical system, which can bepositions q, velocities q, accelerations q or internal forces λ.

Pa

K

w z

y u

Figure 2.6: General scheme of the augmented plant and its controller

Equations (2.11-2.14) are coupled equations of motion and canbe solved numerically using an implicit time integration scheme, the

39


generalized-α method. A brief description of this time integrationscheme can be found in Appendix C. More details can be found in Brulsand Golinval (2006). Two examples of the integrated simulation in aunified environment can be found in Bruls and Golinval (2006): a ve-hicle equipped with a semi-active suspension and the motion and vi-bration control of a robotic arm. This technique has been employedalso to model and evaluate serial machines (da Silva(b) et al., 2009; daSilva(a) et al., 2009) and parallel machines (da Silva(c) et al., 2009).These references are placed on Part II in Chapters 6, 7 and 8.

2.6 Conclusions

Vibrations can cause position errors in highly dynamic mechatronicmotion systems. Therefore, the flexibility of some components shouldbe taken into account during the design phase. In summary, a modelthat represents a mechatronic system should contain the flexibility ofsome components, the system motion, and the controllers.

This task is not straightforward, since there is still a gap betweensimulation software used for evaluation of mechanical systems and soft-ware used for controller design. Alternatives to model and simulatemechatronic systems are described in this chapter. They can be mainlydivided into four methodologies: (1) simulation of an active flexiblestructure in a finite-element environment, (2) simulation of reduced orsimplified models and their controllers in a control design environment,(3) co-simulation between dedicated virtual environments, and (4) inte-grated simulation of an active flexible system in a unified environment.

The choice among these options relies on the system to be modeled,the designer expertise and the available tools. The most general ap-proach is the integrated simulation (4) where structure and controllerare simulated in a unified environment and integrated with the sametime-integration scheme. The approach (1) can not cope with motion,therefore, its applicability is rather limited. The approaches (2) and (3)are well established in the field of control design/evaluation. The maindrawback regarding (2) is that the simplified model should be able topredict correctly the mechanical dynamics. In this way, the designershould be able to choose the suitable model reduction strategy for eachcase study. The main issues regarding (3) are the coupling limitationsbetween the two environments which may result in lack of accuracyand in convergence problems.

40

2.6. CONCLUSIONS

Some of these methodologies are revisited and extended in Chap-ter 3, when mechatronic motion systems with configuration-dependentdynamics are considered.

41

Chapter 3

Modeling of MechatronicMotion Systems withVarying Dynamics

3.1 Introduction

Besides the challenges treated in Chapter 2, such as modeling of theflexibilities, the system motion, and the controllers; a large number ofmachines present varying dynamics that should be correctly predictedduring the design phase. The dynamics may vary, among others, dueto changes in the temperature, in the loads, and in the configuration.This chapter considers the modeling of mechatronic motion systemsof which the eigenfrequencies and mode shapes depend on the spa-tial configuration or operation positions, also referred to hereafter asparameters.

Considering their architecture, machines can have serial, parallelor mixed kinematic chains. The question which is the more suitablearchitecture for rapid machining is still open (Pashkevich et al., 2007).Theoretically, parallel kinematic machines have some advantages overthe serial ones since the principle no drive has to carry other drives,which can be achieved using parallel kinematics, yields good dynamics(Verl et al., 2006). In spite of these advantages, the required accu-racy is harder to be achieved because of the challenges in measuringthe tool center point and the non-homogeneous performance within theworkspace. Both architectures yield dynamics that may vary accord-

43

3 Modeling of Mechatronic Motion Systems with Varying Dynamics

ing to the configuration. The two following examples illustrate thisbehavior.

The first example is a mechatronic system that resembles a scissor-shape parallel kinematic machine. Figure 3.1(a) shows a picture ofthe actual setup and Fig. 3.1(b) depicts a scheme of the mechanismshowing the tool center point (tip) and how the mechanism is allowedto move. The configuration can be measured by the angle between thearms, θ.

Figure 3.2 shows two driving point frequency response functions(FRFs) for two different configurations. The tool center point (tip)was excited with white noise by a shaker. The FRFs are measuredwith an Hv-estimator, while the input and output signals are filteredwith Hanning windows. The first resonance, around 34Hz, is due tothe ground flexibility. The second resonance presents different valuesaccording to the configuration: 123Hz for θ=75o and 138Hz for θ=60o

illustrating that its dynamics depends on the configuration.The second example is a pick-and-place assembly robot, containing

serial and parallel kinematic chains. The setup is depicted in Fig. 1.8.The behavior of this machine exemplify the behavior of most indus-trial 3-axis machine-tools which have a serial kinematic architecturewith orthogonal linear joint axes along the x,y and z directions. Themodeling of this system is briefly treated in this chapter and their in-tegrated design is treated in Chapters 5, 6 and 7. The description ofthis setup can be found in Section III-A in Chapter 5, in Section 3 inChapter 7 and in Section 3.1 in Chapter 6.

This setup was identified for different lengths of the flexible beamand three FRFs are shown in Fig 3.3. The linear motor responsiblefor the x-direction motion is excited with a multi-sine excitation withrandom phase voltage signal (Paijmans et al., 2008). Two measure-ments are made: the encoder measurement (relative motor position)and the gripper acceleration in x-direction. An average over 5 experi-ments with different realizations of the random phases is taken in orderto obtain the FRFs. As it can be observed in Fig. 3.3, the dynamics ofthis mechatronic system vary and depend on the configuration. In thiscase, the configuration can be measured by the beam length, l. Thefirst resonance is 285rad/s for l = 0.41m, 355rad/s for l = 0.36m and430rad/s for l = 0.31m.

44

3.1. INTRODUCTION

q

(a)

Tip (tool centerpoint)

(b)

ControlledMotion

Figure 3.1: (a) Scissor-shape parallel kinematic machine and (b)Mechanism scheme

0 50 100 150 200 250 30010

−4

10−2

100

102

mag

nitu

de [1

/Kg]

0 50 100 150 200 250 300−200

−100

0

100

200

phas

e [o ]

freq [Hz]

60o

75o

Figure 3.2: Driving-point FRFs (acceleration of tip/ shaker force) indifferent configurations (60o and 75o)

45


102

103

10−8

10−6

10−4

Dis

plac

emen

t [m

/N]

(a)

102

103

−200

−150

−100

−50

0

Pha

se [d

egre

e]

Frequency [rad/s]

102

103

10−4

10−2

100

102

Acc

eler

atio

n [1

/Kg]

(b)

102

103

−600

−400

−200

0

Pha

se [d

egre

e]

Frequency [rad/s]

l = 0.53 ml = 0.43 ml = 0.33 m

Figure 3.3: (a) Gripper position and (b) Encoder measurement(relative motor position) for different beam lengths

The dynamic behavior of mechatronic motion systems withconfiguration-dependent dynamics can be considered linear for fixedconfiguration which can be described by scheduling parameters (e.g.the angle between arms of the scissor-shape pick-and-place machine andthe length of the flexible beam of the pick-and-place assemble robot).Moreover, the parameters, which represent the system configuration,do not depend on the states, inputs and outputs of the system. Thisclass of systems is referred to as linear parameter-varying (LPV) sys-tems (Paijmans et al., 2008)1. A review on LPV systems is given inSection 3.2.

Using concepts described in Chapter 2, Section 3.2, and AppendicesA, B and C; modeling methodologies for mechatronic motion systemswith configuration dependent dynamics are proposed and compared inSection 3.3.

1A Matlab toolbox has been developed by Paijmans et al. (2008). Details can befound at https://www.fmtc.be/.

46

3.2. SHORT REVIEW ON LINEAR PARAMETER-VARYING (LPV)SYSTEMS

3.2 Short review on linear parameter-varying(LPV) systems

LPV models depend linearly on parameters that are not dependenton the states, inputs and outputs of the system. LPV models are thecornerstone for dynamical performance evaluation of LPV systems inthe design phase and/or for model based control design, such as LPVcontrol synthesis (Apkarian et al., 1995; Apkarian and Adams, 1998;Gahinet et al., 1995).

3.2.1 LPV modeling

An LPV plant can be represented in state space form by the followingsystem equations:

x = A(l)x + B(l)uy = C(l)x + D(l)u (3.1)

where x ∈ Rn is the state of the system, u ∈ Rm and y ∈ Rp arethe input and the output, respectively, and l is a vector of varyingparameters. The state matrix A ∈ Rn×n, the input matrix B ∈ Rn×m,the output matrix C ∈ Rp×n and the direct transmission matrix D ∈Rp×m may vary according to parameters l.

According to Paijmans et al. (2008), an LPV model should satisfya set of requirements:

• an LPV model should represent the dynamical behavior withinthe parameter range;

• the order of an LPV model should be small;

• the parameter dependency should be as simple as possible; and

• the numerical conditioning of the LPV model should be good.

The latter requirements should be strictly fulfilled if the model isused for model based control design. The synthesis of LPV controllersstill faces a number of numerical issues, that can be overcome if a simpleand well-conditioned LPV model is used.

The aim in this chapter is to describe mechanical system withconfiguration-dependent dynamics, in other words, the parameters lare related to the system spatial configuration. In order to create these

47


LPV models, methodologies might be derived using the standard mod-eling techniques. FE models are not able to predict the varying dy-namics of mechanical system since they represent the system in a fixedspatial configuration. However, several reduced FE models can be de-rived for several discrete configurations and be used to derive an LPVmodel. Flexible multibody models can accurately represent the varyingdynamics of mechanical systems, but they might have a large numberof degrees-of-freedom and the parameter dependency is complex andmight not evident. Therefore, an LPV model can be derived if modelreduction techniques are employed to derived reduced order modelsfrom the FE models and/or flexible multibody models in several con-figurations. From this reasoning arises the necessity of applying LPVidentification techniques able to provide an LPV model from a set ofLTI models in different configurations.

According to Paijmans et al. (2008), experimental LPV identifica-tion techniques can be mainly classified in two kinds: (1) techniquesthat identify LPV models based on one set of measurement data on thesystem with time-varying parameters; and (2) techniques that identifyLPV models based on different fixed sets of measurements data on thesystem for different values of the varying parameter. The first kind ofLPV identification techniques can be classified into state-space basedtechniques (Lovera et al., 1998; Lee and Poolla, 1999) and input-outputform based techniques (Bamieh and Giarre, 2002). Verdult (2002) pro-posed an LPV state-space identification technique for MIMO systemsand multiple time-varying parameters. During the design phase, mea-surements are not always available, therefore, the required data can beextracted from models. In the case of the first kind of LPV identifi-cation technique, flexible multibody models can furnish the data withtime-varying parameters. In this way, using the technique proposedby Verdult (2002), LPV system with multiple inputs, multiple outputsand multiple time-varying parameters can be derived. In the secondkind of LPV identification techniques, an LPV model can be derivedfrom the interpolation between a set of LTI models, refereed hereafteras local models. These LTI models can be derived from measurementdata and/or simulation data of the system considering different valuesof the parameter. For instance, for a configuration-dependent dynamicssystem, LTI models can be derived from measurements or simulationperformed in different configurations. Generally, the first step of thetechniques based on the interpolation between LTI models is to adopt

48


the same invariant form for the models before the interpolation (Vander Voort, 2002; Yung, 2002; Paijmans et al., 2008). An importantdrawback of these techniques is that the LTI models should be interpo-late by a function of low order, which is not always possible. Moreover,no information about the system between the local models is consid-ered. In this way, the main hypotheses is that the system dynamicsbetween local models can be interpolated.

In line with the research presented by Paijmans (2007), this workadopts the second kind of LPV identification for deriving LPV models.This choice is mainly due the fact the LTI identification techniques,modeling procedures and model reduction techniques are fast, reliableand widespread in the industrial and academic groups. The interpo-lation technique proposed by Paijmans (2007), shortly described here-after, has been employed to derive LPV models in Chapter 3 and 5,and to derive gain-scheduling controllers in Chapter 7.

3.2.2 Interpolation of LTI models

An LTI model can represent not only a system plant, correspondingto the relation between the system output and the system input (Eq.1.1), or a controller, relating the actuation signal with the error signal(Eq. 1.2), but also the open-loop or the closed-loop plant of an activesystem (open-loop transfer function as in Eq. 1.3). Local models can bederived for each parameter value within its range. A set of these localmodels can be combined together yielding an LPV model described inan LPV state-space form (Eq. 3.1).

In order to obtain the SISO LPV model, an approach based onthe 4-step interpolation technique developed by Paijmans et al. (2008)is adopted. This interpolation procedure interpolates the pole andzero loci of the local LTI models. A single varying parameter, l, isconsidered. The steps of the interpolation technique are:

1. calculate the poles and zeros of the local LTI systems, and classifythem as complex or real and single or pair;

2. derive affine functions for the poles and the zeros in a way thatthe pole and zero loci of the affine functions match the poles andzeros of the local LTI systems;

3. calculate 1st order LPV state-space proper subsystem(containing1 pole and 0 or 1 zero) and 2nd order LPV state-space proper

49


subsystem (containing 2 poles and 0, 1 or 2 zeros) and;

4. concatenate the LPV subsystems yielding LPV gain-schedulingcontroller system.

For the applications considered in this thesis, an 1st order affinefunction is considered. For the poles p1 till pn, this affine relationequals:

p1(l)p2(l)

...pn(l)

=

p0,1

p0,2...

p0,n

+

p1,1

p1,2...

p1,n

f(l) (3.2)

where p0,1 till p0,n and p1,1 till p1,n are constants and f(l) is an an-alytical function of the scheduling parameter l. The first subindex isrelated with the order of the scheduling function. Since, only 1st-orderaffine functions are considered, the first sub-index can be 0 or 1. Thesecond sub-index is related with the pole that the interpolation needsto fit and it can vary from 1 to n, where n is the number of poles to befitted. Similar expressions are used to describe the varying zeros andgain.

Based on these affine functions, proper subsystems can be createdusing following rules (step 3):

• for all pairs of complex conjugated poles, a 2nd order subsystemis created;

• all pairs of complex conjugated zeros are added to the existing2nd order subsystem; if there are more complex zeros than poles,a 2nd order subsystem should be created containing 2 complexconjugated zeros and 2 real poles;

• for each remaining real pole, a 1st order subsystem is created;

• the remaining real zeros are added to the first and second ordersubsystems.

To illustrate this, a pair of complex poles (pi(l), pi+1(l)) and zeros(zi(l), zi+1(l)) combined into a 2nd order subsystem (A2

s, B2s, C2

s, D2s)

has the following form:

50


A2s(l) = Re

[pi(l) + pi+1(l) −pi(l)pi+1(l)

1 0

]B2s(l) =

[10

]C2s(l) = Re

[−zi(l)− zi+1(l) + pi(l) + pi+1(l)

zi(l)zi+1(l)− pi(l)pi+1(l)

]TD2s(l) = [1](3.3)

For a single pole (p(l)) and zero (z(l)), the following 1st order sub-system (A1

s, B1s, C1

s, D1s) is obtained:

A1s(l) = [p(l)] B1

s(l) = [1]

C1s(l) = [p(l)− z(l)]T D1

s(l) = [1] (3.4)

In the 4th step, all subsystems are combined in series (concate-nated). For two subsystems (As1, Bs1, Cs1, Ds1) and (As2, Bs2, Cs2,Ds2), these series yield the following state-space sentence:

Ac =[

As2 0Bs1Cs2 As1

]Bc =

[Bs2

Bs1Ds2

]Cc =

[Ds1Cs2 Cs1

]Dc = [Ds1Ds2] (3.5)

Since, the matrices Bs and Cs are independent on the schedulingparameter, the matrices products do not result on a system order in-crease. Eventually, if necessary, an affine expression of the gain factorcan be added to the system. The affine expression of the gain factormultiplies the matrix Cs and Ds increasing the system order.

Recently, De Caigny et al. (2008) have proposed a new technique toestimate LPV state-space models for MIMO systems whose dynamicsdepends on a single varying parameter. This technique is based on theinterpolation of LTI models for fixed operating points. The interpola-tion technique is formulated as a nonlinear least-squares optimizationproblem that can be solved by standard solvers. Aiming the sameissue, Lovera and Mercere (2007) have also proposed a technique forderiving MIMO LPV models for gain-scheduling control design fromdata generated by local identification experiments. In this case, thebalanced state-space matrices are direct interpolated. This can onlybe performed because of the unique properties of the balanced realiza-tions.

51


3.2.3 LPV stability analysis

The stability analysis of LPV systems can be performed usingLyapunov-based theory. For continuous systems, Lyapunov stabilitycan be formally defined by the following theorem (Lyapunov, 1966):

Let x be an equilibrium point of x = f(x,u, t). If there exists ascalar function V (x), with continuous first order derivatives such that:

• V (x) is positive definite

• V (x) is negative definite

• V (x)→∞ as ‖x‖2 →∞

then the equilibrium at the origin is globally asymptotically stable.For LTI systems, x = Ax + Bu, the Lyapunov function V (x) can

be defined as V (x) = xTPx. A necessary and sufficient condition toguarantee that the origin of an LTI system is an asymptotically stableequilibrium point is that V (x) < 0, which is equivalent to:

ATP + PA < 0, with P = PT > 0 (3.6)

which is an LMI problem (Gahinet et al., 1995).For LPV system, x = A(l)x + B(l)u, the Lyapunov function V (x)

can be defined as V (x) = xTP(l)x, which leads to the following suffi-cient condition:

A(l)TP(l) + P(l)A(l) +dP(l)dt

< 0, with P(l) = P(l)T > 0 (3.7)

For discrete-time LPV systems, x(k + 1) = A(l(k))x(k) +B(l(k))u(k), the Lyapunov function V (x(k)) can be defined as V (x) =x(k)TP(l(k))x(k). In this way the stability condition, 4V (x) =V (x(k + 1))− V (x(k)) < 0, is equivalent to:

A(l(k))TP(l(k + 1))A(l(k))−P(l(k)) < 0 (3.8)

The stability condition for LPV system needs to be checked withinthe parameter range, l ∈ [l, l], which leads to an infinite number of con-straints. Most of stability techniques described in the literature pro-pose alternatives to limit the set of LMIs. The most common stabilityanalysis technique for LPV system uses a constant quadratic Lyapunov

52

3.3. MODELING OF MOTION MECHATRONIC SYSTEMS WITH VARYINGDYNAMICS

function, which may lead to very conservative results. A survey aboutstability analysis techniques for LPV systems can be found in Paijmans(2007) and Symens (2004).

Recently, a sufficient condition for the stability of an LPV systemhas been provided by Amato et al. (2005) taking into account a bound∆ on the rate of parameter variation. This technique is applied inthis thesis to analyze the stability of the LPV systems considered inChapter 7.

According to Amato et al. (2005), for a given maximal rate of vari-ation ∆, the parameter space is divided into ν intervals. The size ofthe interval is such that in one discrete time step, the parameter l(i)can only jump into the next interval:

|l(i+ 1)− l(i)|Ts

≤ ∆ (3.9)

where Ts is the sampling period and i = 1 . . . ν. A simplified notationfor the theorem presented in Amato et al. (2005), which states a suffi-cient condition for stability of an LPV system, is proposed in Paijmans(2007) and described hereafter. Considering a discrete-time LPV sys-tem described by x(i+1) = A(l(i))x(i), if there exist i = 1 . . . ν positivedefinite constant matrices P(j), such that the following linear matrixinequalities (LMIs) are satisfied for all i = 1 . . . ν and j = −1, 0, 1:

A(l(i))TP(i+ j)A(l(i))−P(i) < 0A(l(i+ 1))TP(i+ j)A(l(i+ 1))−P(i) < 0

(3.10)

then the system is uniformly asymptotically stable for all time-varying realization of the parameter l satisfying constraints on therange and rate of the parameter variation. Due to the notation simpli-fication, the first LMI of the first interval (i = 1 and j = −1) and thelast LMI of the last interval (i = ν and j = 1) are not valid and shouldbe removed.

3.3 Modeling of motion mechatronic systemswith varying dynamics

The aim of this chapter is to propose methodologies to predict thevarying dynamic behavior and the control system action of serial and

53


parallel machines during the design phase. According to the method-ologies for modeling mechatronic systems, described in Chapter 2, threeapproaches, are proposed:

1. Simulation of reduced models, in several configurations, and theircontroller in a control design environment (see Section 3.3.1).Eventually, these models can be interpolated yielding an LPVmodel;

2. Co-simulation between dedicated virtual environments (a briefdescription can be found in Section 3.3.2 and a detail descriptioncan be found in Chapter 5); and

3. Integrated simulation in a unified environment (a brief descriptioncan be found in Section 3.3.3 and a detail description can be foundin Chapters 6 and 7).

The most common issue when modeling parallel kinematic machines(PKMs) is to treat the closed kinematic chains (Merlet, 2000; Ganovski,2007; Davliakos and Papadopoulos, 2008). Modeling approaches forparallel robots are discussed in detail in Merlet (2000). Ganovski (2007)has employed multibody system theory to model and simulate redun-dantly actuated parallel manipulators. PKMs can be modeled usinganyone of the proposed methodologies (approaches 1, 2 and 3). Forinstance, dynamic substructuring techniques, which are based on theassembly of component mode sets extracted from finite-element modelsusing CMS for each configuration, can be applied in the same way asit can be applied for serial machines (see Section 3.3.1). Consideringthe approaches (2) and (3), the PKMs can be modeled using multi-body packages including flexible bodies and conventional joints, suchas prismatic and/or revolute joints. An industrial PKM, a pick-and-place robot, is modeled using flexible multibody technique in Chapter8.

On the other hand, the issue when modeling serial machines is topredict correctly the relative motion between flexible components. Forinstance, the flexible beam of the pick-and-place machine (Fig. 1.8)can move in the y-direction which leads to time-varying boundary con-ditions. The standard approach is the use of substructuring dynamictechniques as described in Section 3.3.1 and in de Fonseca (2000) andWang and Mills (2006). Using this approach, the machine can only beevaluated in discrete configurations. CMS cannot be directly employed

54


to evaluate a serial machine in time-domain, since it is not possible torepresent the flexibilities using a single mode set for each componentdue to their time-varying boundary conditions. In this way, commercialsoftwares, such as LMS/Virtual Lab. Motion and MSC/Adams, can-not be directly used to model serial machines, since the flexible bodiesare included using CMS technique. To overcome this issue, Zaeh andSiedl (2007) proposes an integrated finite-element and multibody simu-lation for modeling the varying boundary conditions. This technique isrelatively time-consuming and no control integration is foreseen whichare important drawbacks in the context of mechatronic system mod-eling. The approach (2) proposes a co-simulation scheme where affinereduced models are evaluated in a control design environment. Thisapproach is briefly described in Section 3.3.2 and in detail in Chapter5. Using this methodology, serial machines can be modeled and sim-ulated concurrently with their controllers in time-domain. In order toguarantee the convergence and accuracy, the integration between thecontrol design environment and the multibody package is performedeither as block evaluation mode, where the control design software isresponsible for the integration of the equations, or as co-simulationmode, where each program uses its own solver and only inputs andoutputs are exchanged at a pre-defined rate, which should be high dueto the varying dynamics. These alternatives are relatively accurate buthighly time-consuming.

A more general framework can be created using the strongly cou-pled modeling approach proposed by Bruls and Golinval (2006). Thisapproach (3) is available in the Oofelie finite-element software (Car-dona et al., 1994). This flexible multibody package is based on finite-element coordinates and the connection between flexible bodies canbe described by dedicated joints. The derivation of a sliding joint en-abling the translational motion between flexible bodies is describedin Appendix B. This joint has been implemented in Oofelie enablingthe time-domain evaluation of serial machines. Moreover, the stronglycoupled modeling approach proposed by Bruls and Golinval (2006) hasbeen extended to deal with linear parameter varying controllers in Sec-tion 3.3.3.

Section 3.3.4 presents a comparison between the proposed modelingstrategies.

55


3.3.1 Simulation of reduced models and their controllersin a control design environment

3.3.1.1 Methodology

de Fonseca (2000) has proposed the use of CMS to derive LTI reducedmodels from FE models to model serial machines. This methodologyis exploited, via dynamic supersets evaluation, for extracting reducedmodels from FE models of a milling machine in Appendix A. In this sec-tion, this approach has been employed for deriving LTI reduced modelsfrom FE model of a serial pick-and-place in several local configurations.

3.3.1.2 Case Study

FE models have been built for describing the x- and y-direction mo-tion of the pick-and-place robot (Fig. 1.8), considering the frame andthe carriage as rigid bodies, as described in Subsection 2.3. StructuralDynamics Toolbox (SDT) has been used to create the FE models. Ascheme of the FE models is shown in Fig. 3.4. The dynamics of thecarriage, the frames and the linear motor do not depend on the configu-ration and their components remain the same within the configurationspace. On the other hand, the dynamics of the flexible beam dependon the configuration since the length of the beam, li, can vary from0.53m to 0.33m. Because of the beam length changes, a varying mass,mi = (0.53− li)πr2ρ, is added to the fixed-end of the beam, in order tokeep the beam mass constant within the configuration space. The nom-inal beam radius, r , is 12mm. A very stiff spring, Kc = 2.5e11N/m,represents the connection between the linear motor and the flexiblebeam. The other parameter values are specified in Subsection 2.3. Asdepicted in Fig. 3.4, the actuator force, generated by the linear motor(x-direction), is applied to the linear motor mass (action) and to thecarriage (reaction).

Each FE model has 49 dofs which is not a large number in FE mod-eling terms but it still requires further reduction for model-based con-trol design. A reduced model has been derived from each FE model us-ing the Craig-Bampton model reduction technique, which is describedin Appendix A. Two constraint modes are kept representing the inputdofs: the motor force applied at the motor and at the carriage. Be-sides these modes, two normal modes are also kept, yielding a modalbase with 4 modes, Ψ ∈ R49×4. Applying the modal transformation,

56


q = Ψη, to the set of second order differential equations describing theFE model (Eq. 2.1), a reduced model can be derived:

Mrη(t) + Krη(t) = ΨTLif(t) (3.11)

where Mr ∈ R4×4 is the reduced mass matrix, Kr ∈ R4×4 the reducedstiffness matrix, η ∈ R4×1 the component-generalized coordinates andf(t) ∈ R1×1 the force vector, and Li ∈ R49×1 the Boolean vector whichis equal to 1 in the position representing the linear motor dof, -1 inthe position representing the carriage dof and null in the other vectornodal positions. The product between the Boolean vector and the forcevector represents the action (at the linear motor) and the reaction (atthe carriage) forces of the motor force.

Equation (3.11) can be rewritten in state-space form:

[ηη

]=[

0 I−M−1

r Kr −M−1r Cr

] [ηη

]+[

0M−1

r ΨTLi

]f(t)

(3.12)where Cr = 2ξΩ, where ξ = 0.05 is the damping and Ω are the eigen-frequencies of the reduced problem.

The output signals y ∈ R3×1 can be calculated by

y = Lo

[Ψ0

] [ηη

](3.13)

where Lo ∈ R3×49 is a boolean matrix which is 1 for the measureddofs and 0 for the rest. In this case, the measured dofs are the motorposition, the carriage position and the gripper position. This proce-dure results in a reduced 8-state space-state model with 1 input: thelinear motor force Fm, and 3 outputs: motor position xm, the carriageposition xc and the gripper position xg.

The encoder of the linear motor measures the difference betweenthe positions of the motor and the carriage, xm−xc. The FRFs (xm−xc)/Fm and xg/Fm, obtained using this approach, are shown in Fig.3.5 for three beam lengths.

This approach follows the methodology proposed by de Fonseca(2000) for deriving dynamical models for configuration-dependent dy-namics. Using this approach, the system can be evaluated at severaldiscrete configurations but it cannot be used to predict the systemperformance during the actual motion. Therefore, an extension of this

57


M m

MM

carriage

frame Mframe

li

Kc

i

m g

motor

K1 K

12K

22K

2

Fm

Fm

Figure 3.4: Scheme of the FE pick-and-place robot

102

103

10−8

10−6

10−4

Mag

nitu

de [m

/N]

102

103

−300

−200

−100

0

Pha

se [d

egre

e]

(a)

Frequency [rad/s]

102

103

10−8

10−6

10−4

Mag

nitu

de [m

/N]

102

103

−300

−200

−100

0

Pha

se [d

egre

e]

(b)

Frequency [rad/s]

l=.53ml=.43ml=.33m

Figure 3.5: (a) FRF - (xm − xc)/Fm and (b) FRF - (xg)/Fm for li =0.53, 0.43, 0.33m

work is proposed to cope with this drawback. Using the interpolationtechnique proposed by Paijmans et al. (2008) and described in Section3.2.2, an LPV model can be derived from this set of LTI reduced mod-els. Each LTI SIMO model is divided into 2 LTI SISO models since the

58


chosen interpolation technique can only deal with SISO models. Theinput of the LTI SISO models is the motor force, Fm, and the outputcan be either (1) the encoder value, xm−xc or (2) the gripper displace-ment, xg. These two sets of LTI SISO models are then interpolatedyielding two LPV models. The analytical function f(l) is chosen to beequal to l, the beam length, since the frequencies vary according to thisproperty. According to Eq. 3.5, the linear interpolation of poles andzeros yields a state-space system which is quadratically dependent onthe parameter l. If a 1st-order affine gain is considered, the interpo-lation yields a state-space system which is cubically dependent on theparameter l.

The set of LTI SIMO models consists of 20 equidistant configura-tions, li ∈ [0.33, 0.53], where i = 1, 2 . . . 20. The poles of these 20 LTISIMO models are illustrated by the black markers in Fig. 3.6. It canbe observed that the system has one rigid-body mode (the pole near tothe origin) and three flexible modes. The rigid-body mode representsthe linear motor motion. The first flexible mode varies according tothe beam length while the second and third are independent on theconfiguration. After the derivation and interpolation of the LTI SISOmodels, two LPV SISO models are created. Both have the same poles,but the zeros are different since they represent different measurementpositions. The poles of the configuration l=0.42m are illustrated by thegrey circles in Fig. 3.6. These poles are estimated through the LPVSISO model. As it can be concluded from Fig. 3.6, the poles of thissingle configuration can be predicted by the interpolated LPV model.

These LPV models can be evaluated in the time-domain using thepdsimul command in Matlab. An alternative for this approach is toimplement these LPV models in Matlab/Simulink, using S-functions.Each LPV SISO model should be implemented in an S-Function ofwhich the inputs are the motor force (the input to the state-spacemodel) and the beam length (the scheduling parameter). The ABCDmatrices are updated every time step according to the beam length.Both options can be used to evaluate the system and its embeddedcontroller in time-domain.

59


Figure 3.6: (black O ∗ ×) Poles of the set of LTI models and(Grey ©) Poles of the interpolated model for l=0.42m

3.3.2 Co-simulation between dedicated virtual environ-ments

3.3.2.1 Methodology

In order to avoid repetition, this approach is briefly introduced in thissection. A detailed description can be found in Chapter 5.

One way of modeling mechatronic systems is to use flexible multi-body models and co-simulation schemes as described in Section 2.4.In general, the controller is performed in a control design environ-ment, such as Matlab/Simulink, while the flexible multibody modelis implemented in a multibody environment, such as LMS/VirtualLab. Motion and MSC/Adams. In most commercial flexible multi-body packages, the flexibility is included via CMS (see Section A.2.4),which is based on a set of modes extracted from a FE model withpre-defined boundary conditions. These packages cannot describe therelative translational motion between flexible bodies. Due to this rel-ative motion, the boundary condition, i.e. how and where the bodiesare connected with each other, varies continuously. In this way, there isno single set of modes able to represent the flexible body correctly; andtherefore, CMS cannot be considered to model a flexible body undervarying boundaries conditions. The relative translational motion be-

60


tween two flexible bodies can be described by flexible multibody pack-ages based on finite-element coordinates as described in the followingSubsection 3.3.3.

To overcome this problem, a modeling methodology, inspired by theapproach described in the previous subsection, is proposed (da Silvaet al., 2006; da Silva(a) et al., 2007; da Silva(b) et al., 2007; da Silva(c)et al., 2007) and briefly described hereafter (a detailed description canbe found in Chapter 5).

The system is divided in two parts:

1. a subsystem of which the boundary conditions do not depend onthe configuration and

2. a subsystem with time-varying boundary conditions.

The former subsystem can be modeled using any commercial multi-body package which allow the modeling of rather complex systems.The latter can be modeled using an LPV model. Both can be cou-pled using co-simulation. A 3-step methodology to model the subsys-tem with configuration-dependent dynamics, yielding an LPV model, isproposed: (i) a parametric high-order finite-element model of the sub-system is elaborated, (ii) local reduced models are extracted at severalconfigurations using a linear model reduction technique in the sameway as performed in the previous subsection 3.3.1 and (iii) an LPVstate-space model is built by affine interpolation between poles, zerosand gains extracted from reduced models in several discrete configura-tions. This 3-step methodology is illustrated in Fig. 5.2. The choseninterpolation technique is described in Section 3.2.2.

3.3.2.2 Case study

A model has been built to simulate the pick-and-place assembly robotmotion in x- and z-direction (Fig. 1.8). A full description of this modeland its experimental validation can be found in Chapter 5. Accordingto the proposed modeling methodology, the system is divided in twosubsystems. The boundary conditions of the subsystem containing theframe, the two linear motors which drive the y-motion, the carriage,their bushings and joints do not depend on the configuration. Thissubsystem can be modeled in a commercial multibody environment,such as LMS Virtual.Lab Motion. The boundary conditions of thesubsystem containing the flexible beam and the linear motor which

61


drives the x-motion dynamics depend on the configuration and aremodeled using the aforementioned 3-step methodology.

Both subsystems can be integrated through co-simulation betweenLMS/Virtual Lab. Motion and Matlab/Simulink (see Fig. 5.5).Since the coupling between the two subsystems is performed in Mat-lab/Simulink, the proposed simulation scheme can also be used to eval-uate the system and its embedded controller in the time-domain.

3.3.3 Integrated simulation in a unified environment

3.3.3.1 Methodology

In order to avoid repetition, this approach is briefly introduced in thissection. A detailed description can be found in Chapters 6 and 7.

As introduced in Section 2.5, formalisms developed in the field offlexible multibody dynamics appear to be especially suitable to modelmechatronic systems (Wasfy and Noor, 2003). In particular, the nonlin-ear finite-element approach described in Geradin and Cardona (2001)is a general and systematic technique to model and simulate artic-ulated systems with rigid and flexible components. An extension isproposed for dealing with active systems, yielding Eqs. 2.11-2.14. Thissection is focused on providing all the necessary features to simulateserial and parallel kinematic machines embedded with LTI and/or LPVcontrollers in an integrated flexible multibody environment.

The modeling of the mechanical system of parallel machines can beperformed using standard elements available in Oofelie; such as revo-lute and spherical joints, and rigid and flexible elements. The modelingof the mechanical system of serial machines requires a correct modelingof the translational motion between flexible components. An element,referred to as sliding joint, allows the relative translation between tworigid bodies, while no rotation is allowed. A simplified derivation ofthis element is described in Chapters 6 and 7. A full derivation of thiselement is described in Appendix B. This derivation follows the nota-tion adopted by Geradin and Cardona (2001) when deriving prismaticand revolute joints. This element has been implemented in Oofelie andused for time-domain evaluation of serial machines in Chapters 6 and7.

Besides the modeling of the mechanical system of the machines,the modeling of an active machine requires the inclusion of controlalgorithms. Both LTI and LPV control strategies are considered in

62


this work. Bruls (2005) have proposed and implemented an integratedmodeling approach for dealing with structure and control simulationconcurrently. This work has been extended to include not only LTIcontrollers but also LPV controllers.

Equations (2.13)-(2.14) can be used to describe controllers in LTIstate-space form and controllers in LPV state space form:

x = A(l)x + B(l)uy = C(l)x + D(l)u (3.14)

where l is the scheduling parameter. In the cases considered in thisthesis, the LPV controllers are built using the interpolation techniquedescribed in Section 3.2.2. Considering poles, zeros and gains describedby a 1st-order affine function, the state-space matrices can be describedby a state-space which is cubically dependent on the scheduling func-tion f(l):

A(l) = A0 + A1f(l) + A2f(l)2 + A3f(l)3

B(l) = B0 + B1f(l) + B2f(l)2 + B3f(l)3

C(l) = C0 + C1f(l) + C2f(l)2 + C3f(l)3

D(l) = D0 + D1f(l) + D2f(l)2 + D3f(l)3

(3.15)

The f(l) represents the configuration of the machine that can be,for instance, interpreted as the distance between two nodes or the anglebetween two beam elements in the flexible multibody environment.

3.3.3.2 Case Study

A flexible multibody model has been built to simulate the pick-and-place robot motion in x- and y-directions in Oofelie (Cardona et al.,1994). The connection between the linear motor and the flexible bodyis done via a sliding joint which allows the translational motion of theflexible beam in y-direction while the linear motor is allowed to movein x-direction. All components are modeled as rigid bodies, exceptfor the flexible beam. The actuator force, generated by the linearmotor (x-direction), is applied to the linear motor mass (action) and tothe carriage (reaction). Details about its derivation and experimentalvalidation can be found in Chapters 6 and 7.

Flexible multibody models can contain a large number of dofs and,therefore, may be unsuitable for model-based controller design pur-poses, such as pole-placement and H∞ control design, since the order

63


of the controller is related to the order of the model. A model-orderreduction technique has to be applied in order to derive a concise de-scription of the flexible multibody model. Among the several model re-duction approaches (Antoulas and Sorensen, 2000), a technique basedon Global Modal Parameterization (GMP) has been chosen because itprovides direct access to the reduced stiffness and mass matrices widelyemployed for machine design evaluation. A short introduction on theGMP technique and its application on the case study are presented inChapter 7.

3.3.4 Comparison between the modeling strategies ofMechatronic Motion Systems with Varying Dy-namics

The approach (1) described in Section 3.3.1 is an extension of themethodology proposed by de Fonseca (2000). de Fonseca (2000) hasproposed to derive reduced LTI models from FE models via CMS indifferent configurations. Using this approach, the machine cannot beevaluated during actual motion, when its dynamics are varying. Thisis an important drawback that can be overcome by the derivation of anLPV model, which can be performed by interpolating a set of reducedLTI models. The selected interpolation technique fits a linear functionon the varying poles, zeros and gains of the LTI reduced models. ThisLPV model can be implemented in Matlab/Simulink and, therefore, itcan be evaluated with its embedded control. Both parallel and serialmachines can be modeled with this technique.

The approach (2) described in Section 3.3.2 divides the mechanicalsystem in two subsystems: (i) a subsystem with boundary conditionswhich do not vary and (ii) a subsystem in which the boundary con-ditions vary depending on the spatial configuration. The subsystem(i) model can be built using any commercial multibody model. Thesubsystem (ii) model can be described by an LPV model implementedin Matlab/Simulink. This LPV model can be derived by the interpo-lation of reduced models extracted from FE models. Both subsystemscan be interconnected via co-simulation. Since the co-simulation isimplemented in Matlab/Simulink, a control design environment, themechanical system can be evaluated concurrently with its embeddedcontrollers. This methodology can be employed to model both paral-lel and serial machines. Details about this approach can be found inChapter 5.

64

3.4. CONCLUSIONS

Both approaches rely on the derivation of affine reduced models.Therefore, non-linear effects can not be included. This is an impor-tant drawback is since non-linear effects, such as local dampers andgyroscopic effects, may have major impact on the machine dynamics.

Finally, the approach (3) described in Sections 2.5 and 3.3.3 con-siders the implementation of special elements in a unified environment,Oofelie. In order to model the mechanical system of serial machines; asliding joint, which allows the translational movement between flexiblebodies, has been implemented in this environment. Moreover, sinceLPV controllers are considered in this work, an extension of the workproposed and implemented by Bruls and Golinval (2006) has been im-plemented. In this way, not only LTI controllers but also LPV con-trollers can be considered during the modeling of mechatronic systems.Details about the modeling approach can be found in Chapters 6 and 7.This approach can include non-linear effects such as gyroscopic effectsand local dampers.

The choice among these options depends on the system to be mod-eled, the designer expertise and the available tools. The third optionis the most general approach since the structure and the controllerare simulated in a unified environment and integrated with the sametime-integration scheme. All methodologies allow the derivation of re-duced LPV models, which can be employed on the derivation of LPVcontrollers.

3.4 Conclusions

This chapter treats the modeling of mechatronic systems withconfiguration-dependent dynamics. For this kind of machines, be-sides the modeling of the flexibilities, the system motion, and the con-trollers; the varying dynamics should be correctly predicted. This kindof mechatronic systems can be described by LPV models. A reviewon the dynamical behavior, modeling and stability of LPV systems isaddressed.

The dynamics of parallel and serial machines are typically depen-dent on the configuration. During the machine design phase, the cor-rect prediction of this behavior may support the engineer to evaluateand improve the dynamical behavior of this kind of machines. Threemethodologies are proposed and briefly described in this chapter. Theyare presented in detail in Part II.

65


In Part II, besides the modeling of the mechanical systems, con-trol design considering feedforward and feedback strategies is treated.Eventually, integrated design considering structural and control param-eters is performed.

66

Part II

Applications: Articles

67

Chapter 4

Concurrent mechatronicdesign approach for activecontrol of cavity noise

Leopoldo P.R. de OliveiraMaıra M. da SilvaPaul SasHendrik Van BrusselWim Desmet

69

4 Journal of Sound and Vibration 314(3-5) (2008) 507-525

Paper published in the Journal of Sound and Vibration:

L.P.R. de Oliveira, M.M. da Silva, P. Sas, H. Van Brussel, W.Desmet, Concurrent mechatronic design approach for active control ofcavity noise, Journal of Sound and Vibration 314(3-5) (2008) 507-525.

Author’s contribution: This work is the result of a fruitful col-laboration between L.P.R. de Oliveira and M.M. da Silva. Theauthor’s contributions are mainly related to:

• the model reduction approach yielding a state-space representa-tion of the coupled vibro-acoustic model; and

• the optimization approach considering control and structural pa-rameters concurrently yielding the main results of this work.

The contributions of Leopoldo P.R. de Oliveira are mainly related to:

• the derivation of the fully coupled vibro-acoustic model, includingthe interpretation of Eq. 4.14, which furnishes the modal basefor the reduced model;

• the inclusion of the actuator dynamics into the optimization prob-lem; and

• the experimental validation of the passive and active vibro-acoustic system.

70

Abstract

Active control is a potential solution to many noise and vibration prob-lems for improving the low-frequency performance. Cavity noise reduc-tion as encountered for instance in aircraft cabins and vehicle interiors,is a typical example. However, the conventional design of these ac-tive solutions may lead to suboptimal products, since the interactionbetween the vibro-acoustic plant dynamics and control dynamics is usu-ally not considered. A proper way to design such active systems wouldbe considering control and plant parameters concurrently. To copewith this approach, a methodology to derive a fully coupled mecha-tronic model that deals with both the vibro-acoustic plant dynamicsas well as the control parameters is proposed. The inclusion of sensorand actuator models is investigated, since it contributes to the modelaccuracy as it can present frequency, phase or amplitude limitationsto the control performance. The proposed methodology provides a re-duced state-space model derived from a fully coupled vibro-acousticfinite element model. Experimental data on a vibro-acoustic vehiclecabin mock-up are used to validate the model reduction procedure.Regarding noise reduction, optimization results are presented consider-ing both vibro-acoustic plant features, such as thicknesses, and controlparameters, such as sensor and actuator placement and control gains.A collocated sensor/actuator pair is considered in a velocity feedbackcontrol strategy. The benefits of a concurrent mechatronic design whendealing with active structural-acoustic control solutions are addressed,illustrated and experimentally validated.

71


1. Introduction

The demands for improvement in sound quality and reduction of noisegenerated by vehicles are steadily increasing, as well as the penaltiesfor space and weight of passive solutions. Active solutions have thepotential to enhance the dynamic performance beyond the passive per-formance which may allow a lighter and improved product [1].

Demonstrations about the viability of active noise control (ANC)and active structural-acoustic control (ASAC) in cavity noise appli-cations, including automotive interior noise reduction, have been de-scribed by several authors [1-6]. A relatively new development in ASACis the use of decentralized controllers, i.e., systems with sensors and ac-tuators connected as independent pairs in feedback loops, rather thanthrough a centralized control unit. This technique has received con-siderable attention [7-11], mainly because of its advantages over thecentralized strategy in terms of practical realization (simpler connec-tions and savings on cabling) and the system transducer fault tolerance[11]. The importance of the proper placement of sensors and actuatorshas also been highlighted in [9, 11-13].

Nowadays, virtual prototyping techniques are being developed inorder to support the design process and to improve product perfor-mance, while reducing design costs and shortening development cycles[14]. In order to bring active solutions to the level of industrial ap-plications, the designer needs numerical tools that allow the inclusionof sensors/actuators and control strategy in the virtual product designand optimization. In this way, the design of active solutions for noisereduction should be performed along the lines of a mechatronic designapproach. For the purpose of this study, mechatronic design is definedas the approach that deals with the integrated design of a mechanicalsystem and its embedded control system [15]. This approach has beenillustrated by performing a concurrent optimization for a 3-axis ma-chine tool considering control and structural aspects, resulting in animproved system performance [16]. For active noise control, the con-current mechatronic approach has been rarely employed. Recently, asimultaneous structural and control optimization of a flexible linkagemechanism for noise attenuation has been described [17]. In that case,the aim is to reduce the structural-acoustic radiation of a flexible mech-

72

1. INTRODUCTION

anism considering in the objective function the weight of the structure,the vibration energy and the control system energy. It is claimed thatthe integrated approach improved significantly the acoustic radiation(performance) and the controller inputs (effort) for that case study.

In this paper, a concurrent mechatronic approach to active controldesign for interior cavity noise reduction, as encountered for instancein automotive interior applications, is proposed using simulation andoptimization. The benefits of this methodology are demonstrated ona vibro-acoustic cabin mock-up (Figs. 4.1 and 4.2). It consists of asimplified car cavity with concrete walls to provide well-defined acous-tic boundary conditions, thus reducing uncertainties during the vibro-acoustic modelling phase. The system is divided into two closed cav-ities: the passenger compartment (PC) and the engine compartment(EC). A rectangular clamped steel panel resembles the firewall, allow-ing the disturbance noise generated by the acoustic source in the ECto be transmitted to the PC. The PC main dimensions are 3400 x 1560x 1270mm; the EC is 800 x 1100 x 750mm and the firewall is 895 x 545x 1.5mm (Fig. 4.2). A structural sensor/actuator pair (SAP) placedon the firewall realizes the control signals for noise reduction in thePC. One of the challenges resides in deriving reasonably sized mod-els that integrate the structural, acoustic and electrical componentsalong with the control algorithm. Moreover, the presence of distinct

Figure 4.1: Photo of the experimental set-up: the vibro-acousticcabin mock-up

73


paths (fluid-structure-fluid) imposes the necessity of dealing with fullycoupled vibro-acoustic models. In order to fulfil this requirement, afully coupled finite element (FE) model of the vibro-acoustic systemis reduced and exported as a state-space model into Matlab/Simulink.The inclusion of sensor and actuator models, which can be realized inthis environment, contributes to the model accuracy, since their owndynamics may change the original system response significantly.

Firewall

Passenger Compartment (PC)

Engine (EC)Compartment

1560

800

750

1100

3400

1270

x

y

z

895

545

microphonessound source

Figure 4.2: Schematic view of the system under study (dimensionsin mm)

The modelling approach for the fully coupled vibro-acoustic systemand its experimental validation are presented in Section 2. The inte-grated design of the active system is treated in Section 3. Finally, someconclusions are addressed in Section 4.

2. Fully Coupled Vibro-Acoustic Modelling Ap-proach

Vibro-acoustic systems can be modelled using Computer Aided Engi-neering (CAE) tools such as finite element (FE) and/or boundary ele-

74

2. FULLY COUPLED VIBRO-ACOUSTIC MODELLING APPROACH

ment (BE) methods. In order to improve the prediction of the struc-tural behaviour in the presence of fluid loads, simulation procedureshave been proposed [18-20], where the influence of the fluid (modelledwith BE) is added to the original structural FE models. The presentcase study, however, requires not only the fluid load on the structure,but also the interaction between the structural vibrations and the pres-sure field. In other words, the vibro-acoustic model should be fullycoupled. To cope with this, a coupled vibro-acoustic FE/FE modellingapproach is adopted [21]. As a result, any combination of structuraland acoustic inputs/outputs can be used for the control design, e.g.,an acoustic source in the EC, structural sensors and actuators on thefirewall and a microphone in the PC.

Another advantage of using a fully coupled vibro-acoustic approachis the accuracy of the estimated closed-loop performance, as an uncou-pled analysis can overestimate the controller efficiency [22]. It is alsorequired that the modelling approach fits into an optimization loop,as the design of an active control system usually requires the settingof some controller parameters (e.g. sensor and actuator positions andcontrol gains).

One of the coupled FE/FE formulations is the Eulerian, in whichthe structural degrees of freedom (DoFs) are displacement vectors,while the acoustic DoFs are expressed as scalar functions. The latter isusually the acoustic pressure, but can also be the fluid velocity poten-tial [23-27]. If pressure is adopted, the system of equations yields non-symmetrical mass and stiffness matrices, posing a disadvantage to FEsolvers. The choice of velocity potential as acoustic DoF also presents adrawback, as the vibro-acoustic coupling terms populate the dampingmatrix, yielding a symmetric but complex model, which is computa-tionally more expensive than the non-symmetric one [28]. Moreover,the modal base resulting from the non-symmetric eigenproblem caneasily be handled by the modelling procedure, as will be described inmore detail in the next section. Therefore, a displacement/pressureEulerian formulation is adopted hereafter.

2.1. From vibro-acoustic FE to state-space formulation

Usually, control design and simulation is performed in a dedicated time-domain environment, raising the necessity of deriving a compatiblerepresentation of the system under study. An appropriate approachwould be a modal representation of the FE model in a state-space

75


(SS) formulation. This representation is a mathematical model of aphysical system as a set of input, output and state variables related byfirst-order differential equations, providing a convenient and compactway to model and analyze systems with multiple inputs and outputs.A number of control design tools are available for systems describedin this form such as Linear-Quadratic-Gaussian (LQG) design, linear-quadratic state-feedback regulator design (LQR) and H∞ controllersynthesis. The purpose of this paper is to deliver tools to allow thedesigner to perform a concurrent mechatronic design; therefore the SSrepresentation suits better this objective.

A first step in the FE modelling of vibro-acoustic systems is thedefinition of appropriate meshes for the acoustic and structural com-ponents. Coincident structural and acoustic meshes are adopted overthe coupling boundary resulting in a simplified procedure [29]. Thefrequency range of interest is limited to 0-200Hz to reduce the compu-tational effort during the modelling procedure. It may not be represen-tative for all interior acoustic problems, but is sufficient to demonstratethe proposed technique and to provide general insights. Moreover, thischoice is not a limiting factor, since this technique is valid as far as FEmodels can be used.

The size of the structural elements is chosen such that the highest-order mode (at 160 Hz: see Fig. 4.3b and Table 4.1), is representedby at least 6 linear elements. The structural mesh (Fig. 4.3a) has 2004-noded shell elements, yielding 1026 DoFs since the borders of thefirewall are clamped. The chosen 4-node shell element was an isopara-metric quadrilateral element with the evaluation of the forces at thecentroid of the element (QUAD4). This element may exhibit lockingeffects for trapezoidal shapes [30]. Due to the characteristic of the ge-ometry, a rectangular mesh was employed avoiding this phenomenon.Experimental validation, showed hereafter, confirms the model accu-racy. However, locking phenomenon should be addressed properly whenmore complicated geometries and meshes are involved. The 1.5mm-thick firewall presents 12 modes between 0 and 200Hz (Table 4.1). Theelement type chosen for the acoustic mesh is the 8-noded brick ele-ment. The resulting mesh, with 26050 elements, and the mode shapeat 192.5Hz are depicted in Fig. 4.4. The total number of acoustic DoFsis 23196. With respect to the element size, this acoustic model exhibitsa minimum of 6 linear elements per wavelength up to 500Hz. Table 4.1shows the resonance frequencies for the coupled vibro-acoustic model

76


(b)(a)

A

0

Figure 4.3: Firewall: (a) FE mesh and (b) uncoupled mode at 160Hz[5,1]

(a) (b)

-A

A

0

Figure 4.4: Acoustic cavities: (a) FE mesh and (b) uncoupled modeat 192.5Hz

and the uncoupled structural and acoustic components. It shows alsothe mode shapes in terms of the number of half wavelengths in the x-,y- and z- directions for the uncoupled modes. In a coupled FE/FEapproach, the effect of the fluid on the structure dynamics can be con-sidered as a pressure load on the wetted surface. For a system withns structural DoFs and na acoustic DoFs, the structural differentialequation takes the form of Eq. (4.1).

77


Table 4.1: Resonance frequencies for coupled and uncoupled systems

vibro-acoustic uncoupled uncoupledmodes structural modes acoustic modes

modes freq. freq. half-wavelength freq. wavelength] [Hz] [Hz] [y,z] [Hz] [x,y,z]

1 0 0 EC - [0,0,0]a

2 0 0 PC - [0,0,0]b

3 35.3 34.2 [1,1]4 49.1 48.3 [2,1]5 52.5 52.1 PC - [1,0,0]6 75.6 75.6 [3,1]7 85.9 86.0 [1,2]8 99.4 99.4 [2,2]9 101.8 101.6 PC - [2,0,0]10 110.5 110.6 PC - [0,1,0]11 112.5 112.6 [4,1]12 122.4 122.5 [3,2]13 122.9 122.8 PC - [1,1,0]14 137.2 137.2 PC - [0,0,1]15 145.9 145.9 PC - [3,0,0]16 151.3 151.3 PC - [2,1,0]17 155.7 155.7 [4,2]18 157.1 157.2 PC - [1,0,1]19 159.0 158.7 EC - [0,1,0]20 160.0 159.9 [5,1]c

21 164.5 164.5 [1,3]22 176.0 176.0 PC - [0,1,1]23 177.8 177.8 [2,3]24 182.9 182.8 PC - [2,0,1]25 184.1 184.1 PC - [1,1,1]26 192.5 192.5 PC - [2,1,1]d

27 198.9 199.0 [3,3]28 199.4 199.7 PC - [4,0,0]

aEC = Engine CompartmentbPC = Passenger Compartmentcmode depicted in Fig. 4.3dmode depicted in Fig. 4.4

78


(Ks + jωDs − ω2Ms)u(ω) + Kcp(ω) = Fs(ω) (4.1)

where Ks, Ds and Ms ∈ Rns×ns are, respectively, the stiffness, dampingand mass matrices of the structural component, Kc ∈ Rns×na is thecoupling matrix, u ∈ Rns×1 is the vector of structural displacementDoFs, p ∈ Rna×1 is the vector of nodal acoustic pressures and Fs

∈ Rns×1 is the structural load vector.In a similar way, the structural vibrations provide an acoustic ve-

locity input and therefore must be taken into account in the acousticmodel as:

(Ka + jωDa − ω2Ma)p(ω) + ω2Mcu(ω) = Fa(ω) (4.2)

where Ka, Da and Ma ∈ Rna×na are the acoustic stiffness, damping andmass matrices, Mc ∈ Rna×ns is the coupling matrix and Fa ∈ Rna×1

is the acoustic load vector. For the sake of brevity, any frequencydependent function ‘h(ω)’ is represented just as ‘h’ hereafter.

Using the relation Mc = −ρ0KTc [31-34], where ρ0 is the fluid den-

sity, the combined system of equations, known as the Eulerian FE/FEmodel, yields:

([Ks Kc0 Ka

]+ jω

[Ds 00 Da

]− ω2

[Ms 0−ρ0KT

c Ma

]) up

=

Fs

Fa

(4.3)

Based on Eq. (4.3) it is clear that the resulting vibro-acoustic sys-tem is coupled, though it is no longer symmetric. As a consequence ofsuch non-symmetric nature, the solution of the associated undampedeigenproblem is computationally more demanding and results in differ-ent left and right eigenvectors:

[Ks Kc0 Ka

]ΦRr = ω2

r

[Ms 0−ρ0KT

c Ma

]ΦRr , (4.4)

r = 1, . . . , na + ns

79


ΦLTr[

Ks Kc0 Ka

]= ω2

rΦLTr[

Ms 0−ρ0KT

c Ma

], (4.5)

r = 1, . . . , na + ns

where r is the index of the coupled natural frequency ωr and ΦL andΦR ∈ R(ns+na)×1 are, respectively, the left and right coupled modes.

Moreover, it has been indicated [35] that, for the Eulerian formu-lation, the left and right eigenvectors, can be related as:

ΦL r =ΦLsrΦLar

=ΦRsrω2

rΦRar

r = 1, 2, . . . ns + na

(4.6)where the indexes a and s represent, respectively, the acoustic andstructural DoFs.

A common practice in solving such vibro-acoustic problems is theuse of component mode synthesis (CMS). It consists of expanding thestructural DoFs in terms of a set of Ns uncoupled structural modesΦs ∈ Rns×1 (without any acoustic pressure load along the couplinginterface), as well as expanding the acoustic DoFs in terms of a set ofNa

uncoupled acoustic modes Φa ∈ Rna×1(acoustic boundaries consideredrigid at the wetted surface). The structural and acoustic expansionsbecome, respectively,

u =Ns∑r=1

qsrΦsr = Φsqs (4.7)

p =Na∑r=1

qarΦar = Φaqa (4.8)

where qs ∈ RNs×1 is the vector of modal amplitudes related to thestructural DoFs, qa ∈ RNa×1 is the vector of modal amplitudes relatedto the acoustic DoFs, Φs ∈ Rns×Ns is the structural modal matrix, Φa

∈ Rna×Na is the acoustic modal matrix and r is the index representingthe number of the mode.

This procedure yields non-symmetrical coupled modal stiffness andmass matrices [34]. Therefore, obtaining the modal SS representationof a reduced model derived from CMS can be a difficult task, since it

80


is necessary to invert the coupled modal mass matrix (which is non-diagonal) and the coupling matrix should be fully available. An alter-native to describe a modal SS for a fully coupled vibro-acoustic systemis to apply a variable substitution to the coupled eigenproblem relatedto Eq. (4.3) [36]. This procedure is detailed hereafter.

Substituting the component mode expansions in Eqs. (4.7) and(4.8) into Eq. (4.3) and pre-multiplying the structural and acousticparts of the resulting matrix equation, respectively, with the trans-pose of the structural and acoustic modal vectors yields the undampedmodal representation:

[ΦTs KsΦs ΦT

s KcΦa

0 ΦTaKaΦa

] qsqa

−ω2

[ΦTs MsΦs 0

−ρ0ΦTaKT

c Φs ΦTaMaΦa

]qsqa

=

ΦTs Fs

ΦTaFa

(4.9)

The homogeneous system of equations related to Eq.(4.9) can bewritten as:

ΦTs (Ks − ω2Ms)Φs ΦT

s KcΦa

ω2ΦTaKT

c Φs − 1ρ0

ΦTa (Ka − ω2Ma)Φa

qsqa

=

00

(4.10)

Since each uncoupled mode is normalized with respect to the un-coupled mass matrices, Eq. (4.10) yields:

Ω2s − ω2I ΦT

s KcΦa

ω2ΦTaKT

c Φs −1ρ0

(Ω2a − ω2I)

qsqa

=

00

(4.11)

where Ωs ∈ RNs×Ns and Ωa ∈ RNa×Na are, respectively, the structuraland acoustic diagonal matrices of uncoupled natural frequencies.

Equation (4.11) still results in a non-symmetric eigenproblem andis therefore expensive to solve. The first line of Eq. (4.11) leads to:

qs = ω2(Ω2s)−1qs − (Ω2

s)−1ΦT

s KcΦaqa (4.12)

81


Applying the substitution qs = ω2qs in Eq. (4.12) yields:

qsqa

=[

(Ω2s)−1 −(Ω2

s)−1ΦT

s KcΦa0 I

]qsqa

(4.13)

Using Eq. (4.13) it is possible to rewrite Eq. (4.11) as a symmetricsystem of equations in qs qaT :

[Ts TT

cTc Ta

]qsqa

=

00

, (4.14)

Ts = I− ω2(Ω2s)−1 ,

Tc = ω2(Ω2s)−1ΦT

aKTc Φs ,

Tc = − 1ρ0

(Ω2a − ω2I)− ω2ΦT

aKTc Φs(Ω2

s)−1ΦT

s KcΦa .

The coupled modal vector Φ ∈ R(ns+na)×(Ns+Na), resulting fromthe eigenproblem associated with Eq. (4.14) on qs qaT , can beinterpreted as the left eigenvector ΦL of the eigenproblem in Eq. (4.6)on qs qaT . The right eigenvector ΦR can be retrieved using Eq.(4.6).

Since the uncoupled bases Φa and Φs result from symmetric eigen-problems, solving Eq. (4.14) may seem less demanding when comparedto the solution of Eqs. (4.5) and (4.6). However, the reduction on thecomputational effort is rather small, as to accurately represent thecoupled modes, it is necessary to retain a higher number of uncoupledmodes. Nevertheless, the advantage of this method is the possibility ofusing dedicated software for each component uncoupled modal analysis.

Eventually, the structural and acoustic DoFs u pT can be pro-jected using the modal base (ΦL and ΦR) and the modal coordinate qusing the following expansion:

up

=Ns+Na∑r=1

qr ΦRr = ΦRq (4.15)

Moreover, the left and right eigenvectors are normalized such that:

ΦTL

[Ms 0−ρ0KT

c Ma

]ΦR = I (4.16)

82


ΦTL

[Ks Kc0 Ka

]ΦR = Ω2 (4.17)

ΦTL

[Ds 00 Da

]ΦR = Γ (4.18)

where I, Ω2 and Γ ∈ R(Ns+Na)×(Ns+Na) are, respectively, the iden-tity, the squared coupled natural frequencies and the modal dampingmatrices.

Applying the modal expansion described by Eq. (4.15) into Eq.(4.3) and pre-multiplying it by ΦT

L, Eq. (4.3) can be re-written as

ΦTL

[Ks Kc0 Ka

]ΦRq + ΦT

L

[Ds 00 Da

]ΦRq

+ΦTL

[Ms 0−ρ0KT

c Ma

]ΦRq = ΦT

L

Fs

Fa

(4.19)

Using the relations described by Eqs. (4.16), (4.17) and (4.18), Eq.(4.19) can be described in a modal state-space form:

qq

=[ 0 I−Ω2 −Γ

] qq

+[ 0

ΦTLB

]FsiFai

(4.20)

uopo

= [ CΦR 0 ] q

q

(4.21)

where B ∈ R(na+ns)×Ni is a matrix with ones on the Ni desired in-put DoFs and zeros everywhere else, Fsi ∈ RNsi×1 is the structuralinput load vector, Fai ∈ RNai×1 is the acoustic input load vector (withNsi + Nai = Ni), uo ∈ RNso×1 is the structural output vector, po∈ RNao×1 is the acoustic output vector (with Nso +Nao = No), and C∈ RNo×(na+ns) is a matrix with ones on the No desired output DoFsand zeros everywhere else. In this formulation, the role of B and C isto select, respectively, columns from ΦT

L and rows from ΦR accordingto the desired inputs and outputs DoFs.

Applying the aforementioned procedure, the original 24192 DoFs(23196 unconstrained acoustic and 1026 unconstrained structural) havebeen reduced to a SS model with 2 × (Ns + Na) DoFs, related to thekept modal amplitudes q and their derivatives q, with force and vol-ume velocity as inputs and displacement and pressure as outputs. The

83


Table 4.2: Kept structural modal amplitudes (Ns) for differentfirewall thickness

firewall thickness 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5[mm]Ns 94 47 29 20 15 13 10 7 7

number of kept acoustic modal amplitudes is the same for all config-urations, since the cavity compartments are not modified during theoptimization procedure. Thus, Na = 78, i.e. the number of uncou-pled acoustic modes with a natural frequency up to 400 Hz, which isadequate to represent the acoustic system in the frequency range ofinterest (0-200Hz). In order to represent the structure (firewall) in thefrequency range of interest, Ns may vary according to the number ofmodes occurring from 0 to 400Hz. Table 4.2 shows Ns for several fire-wall thicknesses. Considering the nominal 1.5mm firewall, the totalnumber of states is 214 (2 × (78 + 29)). Fewer states would lead toinaccuracies within the frequency range of interest.

The validity of the reduced model is illustrated by comparing FRFsfrom the original model with the reduced model (Fig. 4.5). The systeminputs are volume velocity applied in the EC (acoustic input) and forceapplied on the firewall (structural input); and the outputs are pressuremeasured at the PC (acoustic output) and displacement measured atthe firewall (structural output). The good correlation between thereduced SS and the direct FE models validates the model reductionprocedure.

2.2. Experimental Validation

The FRFs derived from the SS model are compared with the FRFsmeasured on the cabin mock-up. The considered FRFs include struc-tural and acoustic inputs and outputs. As depicted in Fig. 4.6(a), thestructural excitation is performed with a LDS shaker (model V201/3),the force transducer is a PCB 208C04 and the accelerometers arePCB 352C67. Figure 4.6(b) shows the LMS acoustic source (modelE-LMFVVS) placed at the EC. The microphones used are B&K 4188.The vibro-acoustic system has been excited with white noise. TheFRFs are measured with an Hv estimator, while input and output sig-

84


50 100 150 20010

−2

100

102

(a)

Freq.[Hz]

FR

F [P

a.s/

m3 ]

50 100 150 200

10−2

100

(b)

Freq.[Hz]

FR

F [P

a/N

]

50 100 150 200

100

102

(c)

Freq.[Hz]

FR

F [(

m/s2 )/

(m3 /s

)]

50 100 150 200

100

102

(d)

Freq.[Hz]

FR

F [(

m/s2 )/

N]

Figure 4.5: Comparison between (- -) FE and (-) SS FRFs: (a)Acoustic/Acoustic, (b) Acoustic/Structural, (c) Structural/Acoustic

and (d) Structural/Structural

nals are filtered with Hanning windows.Figure 4.7 shows a comparison between the experimental and the

simulated (derived from the SS model) FRFs. The material propertiesadopted for this model (nominal case) are: speed of sound in the airco = 344.7m/s, air density ρ0 = 1.185kg/m3, firewall density (steel) ρs= 7800kg/m3 and elasticity modulus E = 2.33·1011Pa. A single modaldamping ratio of 0.35% is applied in the state-space model.

As it can be seen, the resulting FRFs present a good agreement upto 150Hz. Discrepancies above this frequency arise among others fromthe lack of accuracy in determining the exact place of the disturbancesource, sensor/actuator pairs and microphones and from assuming thedisturbance source as an ideal point source. Such mismatches are ex-pected and reflect a limitation in the FE modelling rather than in the

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(a) (b)

Figure 4.6: Experimental setup: (a) view from the passengercompartment with shaker and sensors and (b) view from the engine

compartment with sound source

use of the reduced models in closed loop form, which are the focal pointof this work.

2.3. Inclusion of sensor and actuators pairs (SAP) models

For ASAC simulations, the models must integrate not only structuraland acoustic components, but also sensors, actuators and the controlleralgorithm. The importance of including detailed information about thecontroller and the secondary paths is critical for an accurate assess-ment of the actual performance, since sensor and actuator dynamicscan present frequency, phase and amplitude limitations. In order tocope with this strategy, the reduced model of the system, derived fromthe aforementioned methodology and described in a state-space rep-resentation, is included into the control system design environment(Matlab/Simulink), where the interaction between structure and sen-sor/actuator can be taken into account.

In this case study, sensors and actuators are, respectively, ac-celerometers and inertial-shakers. Appropriate accelerometers are se-

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50 100 150 20010

−2

100

102

(a)

Freq.[Hz]

FR

F. [

Pa.

s/m3 ]

50 100 150 200

10−2

100

(b)

Freq.[Hz]

FR

F. [

Pa/

N]

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100

102

(c)

Freq.[Hz]

FR

F. [

(m/s2 )/

(m3 /s

)]

50 100 150 200

100

102

(d)

Freq. [Hz]

FR

F. [

(m/s2 )/

N]

Figure 4.7: Comparison between (-) SS and (- -) experimental FRFs:(a) Acoustic/Acoustic, (b) Acoustic/Structural, (c)Structural/Acoustic and (d) Structural/Structural

lected, for the frequency range of interest, such that the voltage sig-nal generated by these devices can be considered proportional to themeasured quantity. However, for the inertial-shakers, a more detailedmodel for the electro-mechanical coupling within the actuator and itsinteraction with the structure must be taken into account. The inter-action between an electrodynamic shaker and the structure under testhas been an issue since the very beginning of modal test methods (seee.g. Ref. [37,38]) and is still a subject of research [39-41].

Figure 4.8 shows the electromechanical model of an inertial-shaker.The mechanical model (Fig. 4.8a) comprises the moving mass mi

= 0.03kg, the suspension stiffness ki = 29.6N/m and damping ci =0.1N/(m/s); the moving mass displacement ui, the structure connectingpoint displacement us and the electro-magnetic force Fe. The electro-

87


E

R L

EI bemfFe

m i

k c

u

usi i

i

(a) (b)

structure

Figure 4.8: Electromechanical model of an inertial shaker: (a)Mechanical and (b) Electrical model

magnetic force is proportional to the current I in the circuit, Fe = kfI,where kf = 4N/A is the force-current constant.

The electrical model (Fig. 4.8b) includes the current I, the voltageinput E, the circuit resistance R = 4Ω, the inductance L = 5µH andthe voltage generated by the moving coil Ebemf . The latter can bewritten in terms of the voltage constant kv = 4V/(m/s) and the relativevelocity between the structure connecting point and the moving mass,Ebemf = kv(us−ui). Equations (4.22) and (4.23) describe the dynamicsof this coupled electro-mechanical system operating in voltage mode(ideal power amplifier).

RI + LI + Ebemf = E (4.22)

miui + ci(us − ui) + ki(us − ui) = Fe (4.23)

As proposed in [42], the inertial-shaker/structure interaction canbe modelled as a moving mass and an active interface, that includesthe mechanical suspension and the electro-magnetic force (Fig. 4.9).Given the shaker model, the input voltage and the connecting pointdisplacement, it is possible to estimate the force acting on the structureat the interface point (Fint). Substituting the current value I in Eq.(4.22) into Eq. (4.23) and neglecting the inductance L, since it isusually small [43], yields

Fint =kfRE −

[kfRkv + ci

](us − ui)− ki(us − ui) (4.24)

88

3. CONCURRENT MECHATRONIC DESIGN OF ACTIVE SYSTEMS

Fint-Fint

u i

E

su

mi

u i su

seismic

mass

active

interfacestructure

Figure 4.9: Inertia-shaker active interface model

Figure 4.10(a) shows the structural FRF defined by us/Fint of thevibro-acoustic model and (b) a comparison between the idealized forceinput defined by Fid = kfE/R, constant over frequency, the simulatedload provided by an inertial-shaker, Fint, and the actual measured force,Fexp. It can be seen that, in the low frequency range and in the vicin-ity of structural resonances, the force level drops, as a result of theshaker/structure interaction. The inclusion of the actuator model inthe simulation, allows the assessment of a phenomenon inherent tothe use of such electrodynamic devices, i.e., the force drop-off aroundresonances frequencies. The drops in the excitation force can lead toerrors in the experimental FRFs [40,41] but mainly, when the activecontrol system is concerned, can result in overestimated authority andperformance of the active solution [44].

The shaker model can be externally connected to any DoF of thefirewall, with the advantage of the SS model of the passive plant re-maining unchanged (Fig. 4.11b). This is a useful structure for opti-mization as the SAP positions, i.e. the SS inputs, can be variables ofthe optimization procedure.

3. Concurrent Mechatronic Design of ActiveSystems

The main objective of the considered active control system is to mini-mize the noise transmitted from the EC to the PC. Among the param-

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20 40 60 80 100 120 140 160 180 200

10−6

10−4

FR

F [m

/N]

(a)

20 40 60 80 100 120 140 160 180 200

0.6

0.8

1

Freq. [Hz]

For

ce [N

]

(b)

Fid

Fint

Fexp

Figure 4.10: (a) System driving point FRF and (b) input forces

F p

collocated control loop

performance

-G s

a

Fint

(EC) (PC)

(firewall)

ECPC

(a) (b)

Eus

Figure 4.11: Control scheme for the closed loop vibro-acousticsystem (a) positions of sensors and actuators and (b) block diagram

eters addressed during the design of an active system is the controllerdesign; more specifically, the definition of a control strategy, the selec-tion and configuration of sensors and actuators, the parameters setting,etc. In a concurrent mechatronic approach, plant dynamic parameterscould also be taken into account, aiming at an improved active system

90


Am

pli

tud

e [m

/s]

Frequency [Hz]

3

Figure 4.12: Disturbance spectrum - acoustic input

design.Due to its relatively simple implementation, a time-invariant col-

located velocity feedback is selected. In this application, the feedbackgain on the structural SAP is optimized with respect to the pressure atthe driver’s ear, rather than the firewall vibration. This ASAC strat-egy is applicable when the acoustic source is transmitted into a cavitythrough a limited number of structural paths [1,45]. Figure 4.11(a)shows a scheme of sensor and actuator positions and Fig. 4.11(b) showsa scheme of the adopted ASAC with the structural sensors and actua-tors involved in the control loop and the acoustic sensors and actuatorsrelated to the performance evaluation.

As a disturbance signal, an acoustic source in the EC that resem-bles engine noise is used (Fa in Fig. 4.11b). At constant speed, thecharacteristic frequency content of engine noise is a combination of thefundamental frequency (rotating speed), its harmonics and the back-ground noise. During a run-up, these frequencies are swept, excitinga broad spectrum. After analyzing a series of time signals from realengine run-ups, the average amplitude of the disturbance signal wasdefined as depicted in Fig. 4.12.

Since the control strategy is selected, the control design consists ofdetermining the position of the SAP and the velocity feedback gain.The structural design parameter is the thickness of the firewall, as itdirectly affects the vibro-acoustic behaviour of the system.

The metrics adopted to evaluate the design are the system perfor-mance, the control effort and the structural firewall mass. The per-

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0.5 1 1.5 2 2.5 3 3.5 4 4.5

60

70

80

90

Thickness [mm]

SPL

[dB

]

Figure 4.13: Passive performance for several firewall thicknesses

formance of the system is defined as the sound pressure level (SPL),in dB, at the driver’s ear position (Eq. 4.25) and the control effort isdefined as the applied control effort (COE), in V, (Eq. 4.26).

SPL = 20 log(

prms2 · 10−5

)(4.25)

COE = Gurms (4.26)

where p, in Pa, is the acoustic pressure, G, in V/(m/s), is the velocityfeedback gain and u, in m/s, is the structural velocity.

In a conventional design procedure, the structure is first optimizedbased on the passive performance and then, afterwards, an active con-trol system is designed. Figure 4.13 shows the passive performancefor different firewall thicknesses (from 0.5mm to 4.5mm with 0.1mmstep). Different performances occur due to the coupling between thestructural and acoustic resonances. A good coupling between theseresonances allows the noise at the EC to be transmitted through thefirewall to the PC more efficiently, decreasing the performance, as oc-curs for instance for firewalls around 2.0mm. On the other hand, thenoise at EC will not be efficiently transmitted when the resonances andmodes are not strongly coupled, increasing the performance, e.g. forthe 1.5mm firewall (Table 4.1). The markers on Fig. 4.13 represent thetypical plate thicknesses readily available on the market, and thereforewill be considered as the only feasible choices hereafter.

The presented optimization problem adopting a concurrent mecha-tronic design approach assumes that the controller is performed by a

92


single collocated SAP. Therefore, the variables are the firewall thick-ness, the velocity feedback gain and the position of the SAP. This op-timization problem deals with continuous variables, i.e. the feedbackgain, and discrete variables, i.e. the firewall thickness (discrete valuesreadily available on the market) and the SAP position (node locationson the firewall FE model). In addition, this problem is non-convex andnonlinear. Since the model is of reduced size, an extensive search isperformed for all possible configurations, comprising: all free nodes ofthe firewall as possible positions for the SAP (171 positions as indi-cated in Fig. 4.14), different thicknesses of the firewall (1.0, 1.5, 2.0and 2.5mm) and several feedback gains (from 0 to 10kN/(m/s)).

Multi-objective optimization problems, usually, have conflicting ob-jective functions. Therefore, the derivation of a single cost function asa weighted summation of those objectives [46] is not a trivial task sinceit may cause a huge impact on the optimal design.

A more comprehensive strategy is to find the tradeoffs among sev-eral objectives. The Pareto plot represents the best obtainable com-promises between all the conflicting objective functions [47]. This plotshows the feasible and infeasible design regions in the objective space.Figure 4.15 shows the feasible region in the design space limited by thedesign constraints and its mapping to the objective space. The lowerborder between the feasible and infeasible regions in the objective spaceis the Pareto front. It contains the possible optimal combinations ofthe objectives. Objectives out of the border may lead to infeasible orsuboptimal designs, i.e., for a solution belonging to this border it is notpossible to improve one objective function without worsening anotherone [47]. Eventually, the solution derived by any single cost functionis captured by the Pareto front. In this way, the designer can chooseone single solution belonging to the Pareto front that suits better otherdesign criteria.

Figure 4.16 shows, for different firewall thicknesses, the Pareto plotconsidering performance and control effort. All feasible configurations,derived from an extensive search for various SAP locations and feedbackgains, are shown in this Pareto plot. The first conclusion that can bedrawn is that the lightest option (1.0mm) presents unsatisfactory pas-sive and active performances. Therefore, considering performance andeffort as design criteria, three configurations may be suitable: 1.5, 2.0,and 2.5mm firewall. Among them, the lighter configuration, 1.5mm,presents the best passive performance (72.4dB as already indicated in

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Figure 4.14: Possible SAP positions on the firewall: the nodes onthe firewall FE model

Fig. 4.13).According to a conventional design sequence, a natural choice would

be to select the thinner firewall (1.5mm), resulting in the lightest design

94


design space objective space

designvariable #1

Pareto Front

des

ign

var

iable

#2

objectivefunction #1

obje

ctiv

efu

nct

ion #2

mapping

feas

ible

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ible

Figure 4.15: Mapping from the design to the objective space

10−2

100

60

70

80

90

1.0mm

COE [V]

SP

L [d

B]

10−2

100

60

70

80

90

1.5mm

COE [V]10

−210

0

60

70

80

90

2.0mm

COE [V]10

−210

0

60

70

80

90

2.5mm

COE [V]

Figure 4.16: Performance and control effort for different firewallthicknesses - Pareto plot

and the best passive configuration. However, when the closed-loopperformance is analyzed, the performance improvement of the lightestconfiguration is rather limited compared to the other configurations.For instance, requiring that the performance of the active system shouldbe below 70.0dB, the lightest configuration barely achieves such target(Fig. 4.16). Considering this target, the thicker configurations wouldachieve higher noise reduction; despite their lower passive performance.For a 70.0dB target the lightest design is the 2.0mm firewall. Moreover,it is possible to reach the same performance with less control effort by

95


selecting the 2.5mm firewall.In summary, a conventional design sequence would lead to 1.5mm

firewall with the best possible closed-loop performance of 70.2dB, whilethe concurrent mechatronic approach would lead to 2.0mm firewall withclosed-loop performance up to 68.8dB or even 2.5mm achieving up to64.1dB. The concurrent mechatronic approach delivers better resultsthan the conventional design sequence, since the system passive per-formance is not considered independently from the control dynamics.

Additionally, Fig. 4.16 shows that some SAP positions with highvelocity feedback gain value, i.e. high effort values, can deterioratethe system performance compared with the passive performance. Thisphenomenon occurs because high velocity feedback gains may clampthis position, modifying the dynamic behaviour of the firewall, shift-ing natural frequencies and modes. Thus, the coupling between theacoustic and structural resonances can be amplified, deteriorating thesystem performance.

The system performance, according to the collocated SAP position,depends strongly on the firewall thickness. Figure 4.17 shows the bestachievable performance for each SAP position on the firewall (z andy-directions) for different thicknesses. As mentioned before, this be-haviour can be explained by the fact that different thicknesses lead todifferences in the structural resonance frequencies and, consequently,variations on the vibro-acoustic coupling. In this way, different vibro-acoustic modes may have a stronger contribution on the transmissibilityprocess, resulting in distinct topologies for the optimum surfaces. Asit can be observed in Figs. 4.16 and 4.17, the same performance can beachieved by different configurations. These results justify the mecha-tronic design approach and illustrate the limits in the conventionaldesign methodology for active vibro-acoustic applications.

Figure 4.17 results from a priori derivation of reduced models foreach firewall thickness. Each firewall thickness requires a reducedmodel. The required calculation time for deriving a reduced modeldepends a.o. on the modal density (see Table 4.2). Using a PentiumIV, with a processor of 1.4GHz, the CPU time varied from 580s forderiving the reduced model for the 2.5mm firewall to 750s for derivingthe reduced model for the 1.0mm firewall. The model reduction proce-dure is performed just once for each firewall thickness, since all possibleSAP positions are kept during the model reduction procedure. Oncethe reduced model is derived, the Pareto plot can be built finding the

96


Figure 4.17: Sound pressure level at each SAP position (withoptimal gain) for different firewall thicknesses

best gain for each possible SAP position. Using the same processor,this derivation took about 1500s for each reduced model.

The numerical results have been verified experimentally by com-paring the passive and active sound spectra at the driver’s ear positionfor the 2.0mm firewall. Figure 4.18 shows the sub-optimal case, wherethe SAP is placed closer to the border (node 130 in Fig. 4.14) andthe feedback gain is set to 466V/(m/s). Figure 4.19 shows the globaloptimal solution, with SAP at node 87 and feedback gain 466V/(m/s).Table 4.3 summarizes the SPL and the noise reduction for the passive,sub-optimal and optimal solutions depicted in Figs 4.18 and 4.19. Thegood agreement between experiment and simulation corroborates theresults presented and emphasizes the benefits of the proposed concur-rent mechatronic approach.

97


Table 4.3: Experimental and simulated SPL at the driver’s ear forpassive, sub-optimal and optimal solutions

Solution Experiments SimulationSPL [dB] Reduction [dB] SPL [dB] Reduction [dB]

Passive 76.7 - 78.0 -Sub-optimal 75.8 0.9 76.6 1.4

Optimal 67.9 8.8 68.8 9.2

50 100 150 20040

60

80

100

120

140(a)

Freq.[Hz]

Pres

sure

[dB

]

50 100 150 20040

60

80

100

120

140(b)

Freq.[Hz]

Pres

sure

[dB

]

Figure 4.18: Sound spectra at driver’s ear position for (-) passivesystem and (- -) suboptimal active system : (a) Experimental and (b)

Simulation

4. Conclusions and Future Work

A concurrent mechatronic approach for ASAC, considering a fully cou-pled vibro-acoustic system with SAP models, has been presented. Thisgeneral approach allows the inclusion of any kind of controller that usesstructural or acoustic sensors and actuators. The modelling procedurewas validated by correlating the direct FE and the reduced SS models.Eventually, the vibro-acoustic model reduction procedure was exper-imentally validated for a system that resembles a passenger vehicleinterior.

The benefits of this approach have been exploited in some investi-gations considering structural and control parameters as the firewall

98

ACKNOWLEDGEMENTS

50 100 150 20040

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100

120

140(a)

Freq.[Hz]

Pres

sure

[dB

]

50 100 150 20040

60

80

100

120

140(b)

Freq.[Hz]

Pres

sure

[dB

]Figure 4.19: Sound spectra at driver’s ear position for (-) passivesystem and (- -) optimal active system : (a) Experimental and (b)

Simulation

thickness, the velocity feedback gain and the position of the SAP.The first conclusion that can be outlined is that optimal passive per-formance systems may have inferior closed-loop performance. Con-sequently, an optimal design can only be achieved when consideringstructure and control concurrently.

Considering that an ASAC modelling procedure is a multi-disciplinary assignment, distinct objectives arise from these disciplines.Capturing the design tradeoffs, using for instance the Pareto front, canassist the designer to gain better insights into the problem.

Comparisons between experimental and simulation results for thepassive, sub-optimal and optimal solutions showed good agreement con-firming the benefits of the proposed concurrent mechatronic approachfor ASAC design.

Given the potential of piezoelectric materials for active control pur-poses, a next step in this study will be the inclusion of distributed sen-sors and actuators in the methodology. As optimization variables, notonly the placements and the control gains, but also the shape of thepiezo-patches may then be considered.

Acknowledgements

The research of Leopoldo P. R. de Oliveira is supported in the frame-work of a bilateral agreement between KU Leuven and University of Sao

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Paulo. The support for the research of Maıra M. da Silva is providedby CAPES, Brazilian Foundation Coordination for the Improvement ofHigher Education Personnel. The research presented in this paper wasperformed as part of the FP6-Integrated Project InMAR, ”IntelligentMaterials for Active Noise Reduction”. We are also grateful to LMSInternational for the technical support and encouragement.

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[19] K.A. Cunefare, S. De Rosa, S., An improved state-space method forcoupled fluid-structure interaction analysis, J. Acoust. Soc. Am.105 (1999) 206-210.

[20] S. Li, A state-space coupling method for fluid-structure interactionanalysis of plates, J. Acoust. Soc. Am. 118 (2005) 800-805.

[21] L.P.R. Oliveira, A. Deraemaeker, J. Mohring, H. Van der Auweraer,P. Sas, W. Desmet, A CAE modeling approach for the analysis ofvibroacoustic systems with distributed ASAC control, Proceedingsof ISMA2006, Leuven - Belgium, September 2006, pp. 321-336.

[22] J.I. Mohammed, S.J. Elliott, Active control of fully coupledstructural-acoustic systems, Proceeding of Inter-Noise 2005, Rio deJaneiro - Brazil, 2005, pp. 1-10.

[23] W. Desmet, B. Pluymers, P. Sas ,Vibro-acoustic analysis proce-dures for the evaluation of the sound insulation characteristics ofagricultural machinery cabins, Journal of Sound and Vibration 266(2003) 407441.

[24] P. Sas, C. Bao, F. Augusztinovicz, W. Desmet, Active control ofsound transmission through a double panel partition, Journal ofSound and Vibration (1995) 180(4) 609-625.

[25] G. Pan and D.A. Bies, The effect of fluid structure coupling on thesound waves in an enclosure: theoretical part, J. Acoust. Soc. Am.2(1987) 691-706.

[26] G.C. Everstine, A symmetric potential formulation for fluid-structure interactions, Journal of Sound and Vibration 79 (1981)157-160.

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[28] W. Desmet, D. Vandepitte, Finite element method in acoustics,Seminar on Advanced Techniques in Applied and Numerical Acous-tics - ISAAC17, Leuven - Belgium, September 2006, pp. 1-48.

[29] J.P. Coyette, Y. Dubois-Plerin, An efficient coupling procedure forhandling large size interior structural-acoustic problems, Proceed-ings of ISMA-19, Leuven - Belgium, September 1994, pp. 729-738.

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[33] S. De Rosa, G. Pezzullo, L. Lecce, F. Marulo, Structural acousticcalculations in the low frequency range, AIAA Journal of Aircraft,31(6)(1994), 1387-1394.

[34] W. Desmet, A wave based prediction technique for coupled vibro-acoustic analysis, PhD Thesis, Katholieke Universiteit Leuven, Me-chanical Engineering Department - PMA, 1998.

[35] J. Luo, H.C. Gea, Modal sensitivity analysis of coupled acoustic-structural systems, Journal of Vibration and Acoustics 119 (1997)545-550.

[36] Sysnoise rev. 5.5 User’s Manual, LMS International, Leuven, Bel-gium, 2000.

[37] K. Unholtz, Vibration testing machines - Shock and VibrationHandbook, McGraw-Hill Book Co., New York, v.2, pp. 25.1 - 25.74,ed.1, 1961.

[38] G.R. Tomlinson, Force Distortion in Resonance Testing of Struc-tures with Electrodynamic Vibration Exciters, Journal of Soundand Vibration 63 (1979) 337-350.

[39] K.G. McConnell, Vibration Testing: Theory and Practice, JohnWiley and Sons, NY, EUA, 1995.

[40] T. Olbrechts, P. Sas, D. Vandepitte, FRF measurement errorscaused by the use of inertia mass shakers, Proceedings of the 15International Modal Analysis Conference, IMAC , 1997, pp. 188-194.

[41] P.S. Varoto, L.P.R. de Oliveira, On the Force Drop-off Phenomenonin Shaker Testing in Experimental Modal Analysis, Journal ofShock and Vibration 9 (2002) 165-175.

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[42] S. Herold, H. Atzrodt, D. Mayer, M. Thomaier, Integration of differ-ent approaches to simulate active structures for automotive applica-tions, Proceedings of Forum Acusticum 2005, Budapest - Hungary,2005, pp. 909-914.

[43] N.M.M. Maia, J.M.M. Silva,Theoretical and Experimental ModalAnalysis, John Wiley and Sons Inc., 1997.

[44] L.P.R. de Oliveira, P.S. Varoto, P. Sas, W. Desmet, A state-spaceapproach for ASAC simulation, Proceedings of the XII InternationalSymposium on Dynamic Problems of Mechanics (DINAME 2007),2007, pp. 1-10.

[45] P.A. Nelson, S.J. Elliot,Active Control of Sound, Academic Press,1992.

[46] S. De Rosa, A. Sollo, F. Franco, K. A. Cunefare, Structural-Acoustic Optimisation of a Partial Fuselage with a Standard FiniteElement Code, 7th AIAA/CEAS Aeroacoustics Conference and Ex-hibit, Maastricht, Netherlands, May 28-30, 2001; Collection of Tech-nical Papers. Vol. 1 (A01-30800 07-71) AIAA-2001-2114.

[47] M. Gobbi, F. Levi, G. Mastinu, Multi-objective stochastic optimi-sation of the suspension system of road vehicles, Journal of Soundand Vibration 298 (2006) 1055-1072.

104

Chapter 5

Design of MechatronicSystems WithConfiguration-DependentDynamics: Simulation andOptimization

Maıra M. da SilvaWim DesmetHendrik Van Brussel

105

5 IEEE/ASME Trans. Mechatronics 13(6) (2008) 638-646

Preliminary results have been presented at 2007 IEEE/ASME Interna-tional Conference on Advanced Intelligent Mechatronics (AIM2007).This paper has been nominated as best student paper finalist. Paperpublished in the IEEE/ASME Transactions on Mechatronics:

M.M. da Silva, W. Desmet, H. Van Brussel, Design of Mecha-tronic Systems With Configuration-Dependent Dynamics: Simulationand Optimization, IEEE/ASME Transactions on Mechatronics 13(6)(2008) 638-646.

Comments to the reader : Section II-C briefly presents an inter-polation technique to derive linear parameter varying models. Thistechnique is explained in more detail in Section 2.4 in Chapter 7. Thecase study described in Section III-A is also presented in Section 3in Chapter 7 and in Section 3.1 in Chapter 6. The results describedin Section IV are further exploited in Chapter 6. Bibliographicalinformation about the authors has been omitted.

Further discussion: Issues arose during the preliminary defenceare described and commented:

• The term multiobjective problem in page 124 is not appropriatedand should not be considered.

• The definition of the Total Variation (TV) can be found in Skoges-tad and Postlethwaite (1997). This term is related to the qualityof the responses and can be easily correlated with frequency-domain metrics. Large TV values are associated with large over-shoots and longer settling times. Therefore, this metric has beenselected instead of other time-domain metrics.

• No gain-scheduling approach has been employed to derive an LPVPID controller because the aim of this paper was to support thedirect design strategy. Further results and issues related to thedirect design strategy are tackled in Chapter 6.

• It is believed that the results of LPV PID could be improved ifthe same interpolation technique described in Section II-C hadbeen used. This is also applicable for the results in Chapter 6.

106

Abstract

This paper considers the simulation and optimization of mechatronicsystems with configuration-dependent dynamics. A modeling method-ology, able to capture the varying dynamics and the embedded controlsystem actions, using affine reduced models and cosimulation, is pro-posed. In this way, mechatronic systems with configuration-dependentdynamics can be evaluated during the design phase. This methodologyis applied to a pick-and- place assembly robot and an experimental val-idation is carried out. The mechatronic design approach, which takesinto consideration structural and control parameters, is considered. Us-ing time-domain metrics, two control strategies are derived: a lineartime-invariant proportional-integral-derivative (PID) controller and alinear parameter-varying PID controller. Finally, design tradeoffs areevaluated in a truly mechatronic approach.

107


I. Introduction

A large number of mechatronic systems may have their eigenfrequen-cies and mode shapes dependent on the instantaneous spatial configu-ration, which inevitably affects the performance and the stability of thecontrol system [1], [2]. In particular, mechatronic systems, such as ma-chine tools, cartesian mechanisms, and pick-and-place machines, can beclassified as systems with configuration-dependent dynamics since therelative motion between their flexible components can lead to time-varying boundary conditions. A typical example is a pick-and-placerobot with a gripper carried by a flexible beam (see Fig. 5.1). Fastmovements may excite the eigenfrequencies of the flexible beam, whichare dependent on the beam length, causing vibrations and decreasingthe positioning accuracy.

Figure 5.1: Pick-and-place machine used as test case

The demands for faster and more accurate machine tools, carte-sian mechanisms, and pick-and-place machines, which can be classi-fied as mechatronic systems with configuration-dependent dynamics,are steadily increasing. These requirements are conflicting since fastmovements may excite the machine eigenfrequencies, deteriorating itsaccuracy. Active control schemes are generally implemented to enhancethe machine dynamic performance. Therefore, as the performance ofactive machines depends on both structural and control parameters,

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I. INTRODUCTION

their design should be performed along the lines of a mechatronic de-sign approach [3]. The benefits of this approach, which deals with theintegrated design of a mechanical system and its embedded control sys-tem, have been highlighted in several fields: motor-driven mechanismsdesign [4][6], machine tools design [7], active damping guide roller de-sign [8], smart structures [9], active vibro-acoustic systems [10], andbioengineering [11], among others.

Simulation-based design has been extensively used, not only forshortening development cycles and reducing costs, but also for enhanc-ing product performance. According to the mechatronic design ap-proach, simulation/optimization should be employed by the engineeras an effective design tool, enabling the direct access to structural andcontrol parameters. Therefore, regarding the design of mechatronicsystems with configuration-dependent dynamics, two steps should beaccomplished: 1) the simultaneous simulation of the mechanical systemand the controller and 2) their integrated design/optimization.

The modeling of mechatronic systems with configuration-dependentdynamics has been treated in several references [7], [12], [13]. Themain difficulty during the modeling phase arises from the time-varyingboundary conditions. The standard approach is the use of substructur-ing dynamic techniques, which are based on the assembly of componentmode sets extracted from finite-element models using component-modesynthesis (CMS) for each configuration [7], [12]. Therefore, the machinecan only be evaluated in discrete configurations. An integrated finite-element and multibody simulation has been proposed by [13]. Thistechnique is relatively time consuming and no control integration isforeseen, which are important drawbacks.

There are mainly two control strategies that can be employedfor systems with configuration-dependent dynamics: (1) robust lin-ear time-invariant (LTI) controllers that can be explicitly designed totake into account the dynamical variations as uncertainties or (2) linearparameter-varying (LPV) controllers that can adapt according to theparameter variations.Modal LTI controllers have been applied to con-trol configuration-dependent dynamics of a flexible-link manipulator[14] and a planar parallel platform [15]. A robust LTI controller anda gain-scheduling controller (LPV) have been designed for a pick-and-place assembly robot by [1] and [16].Adaptive controllers for a two-linkrobot with a time-varying payload have been derived by [2].

This paper presents a methodology to model and simulate a mecha-

109


tronic system with configuration-dependent dynamics, using cosimula-tion between a commercial multibody environment, LMS Virtual.LabMotion, and Matlab/Simulink. Using the hereafter proposed approach,different control design techniques and optimization methodologies canbe applied regarding the mechatronic system, and considering not onlydiscrete configurations, but also continuous operation.

In this way, structural and control parameters can be taken intoconsideration during the design phase. In order to demonstrate thetime-domain capabilities of the proposed modeling methodology, LTIand LPV control strategies are derived using time-domain metrics.However, the modeling methodology is not restricted to this controldesign approach, and can be employed to evaluate any control designtechnique, e.g., robust and gain-scheduling controllers.

The paper is organized as follows. Section II presents the proposedmodeling methodology, able to capture not only the configuration-dependent dynamics, but also the control actions. A pick-and-placeassembly robot was chosen as test case (see Fig. 5.1). Its setup andmodel are described and experimentally validated in Section III. Us-ing time-domain metrics, LTI and LPV proportional-integral-derivative(PID) controllers are derived. Design tradeoffs are evaluated in atruly mechatronic approach, exemplifying the benefits of the integratedmechatronic design in Section IV. Finally, some conclusions are drawn.

II. Modeling of mechatronic systems withconfiguration-dependent dynamics

One way of modeling mechatronic systems is to use flexible multibodymodels and cosimulation schemes. In general, the controller is im-plemented in Matlab/Simulink, while the flexible multibody model isimplemented in a multibody environment. However, a mechatronicsystem with configuration-dependent dynamics requires a model ableto capture this dependency. This can be performed, among others,by creating an updated finite-element model for each integration stepor implementing a sliding joint for the flexible body connection [17].Neither of these options are available in the standard commercial multi-body environments.

To overcome this issue, a novel modeling methodology is proposed.First, the system is divided in two parts: (1) a subsystem of whichthe dynamics does not depend on the configuration and (2) a sub-

110

II. MODELING OF MECHATRONIC SYSTEMS WITHCONFIGURATION-DEPENDENT DYNAMICS

system with configuration-dependent dynamics. The former can bemodeled using any commercial multibody package. The latter can bemodeled using an LPV model. Both can be coupled using cosimula-tion. A three-step methodology (see Fig. 5.2) to model the subsys-tem with configuration-dependent dynamics, yielding an LPV model,is proposed.

A three-step methodology (see Fig. 5.2) to model the subsystemwith configuration-dependent dynamics, yielding an LPV model, is pro-posed.

1) A parametric high-order finite-element model is elaborated.

2) Local linear models are extracted at several discrete configurationsusing a linear model reduction technique.

3) An LPV state-space model is built by affine interpolation betweenpoles, zeros, and gains extracted from reduced models in severaldiscrete configurations. This LPV model can be implemented inMatlab/Simulink, which, for each integration step, reevaluatesthe state-space model depending on the parameter.

Each one of these steps is described in detail hereafter.

A. Parametric Finite-Element Model

A high-order finite-element model of the subsystem with configuration-dependent dynamics is, initially, created. Any commercial finite-element software can be used for deriving this model. Since severaldiscrete models for different configurations are necessary, a good ap-proach to generate these models is to create a parametric high-orderfinite-element model. This model can be described by a parametervarying second-order differential equation

M(l)q(l) + C(l)q(l) + K(l)q(l) = L(l)f(l) (5.1)

where l is a vector of varying parameters, i.e., the configuration vector,M is the mass matrix, C is the damping matrix, K is the stiffnessmatrix, q are the physical degrees of freedom, f is the applied forcevector and L is the input force influence matrix, indicating how theinput forces act on the structure.

111


Figure 5.2: Modeling of systems with configuration-dependentdynamics: three-step methodology

The accuracy of the reduced models extracted from the parametricfinite-element model relies on the accuracy of the high-order finite-element model, which may depend on the available information duringthe design phase and the chosen model reduction technique. For adiscrete configuration i, a model can be described evaluating (5.1) forl = li. Several discrete models can be extracted varying the parametervector within its feasible range.

B. Model Reduction

The CMS provides an appropriate solution for the reduction of a finite-element model [18]. It is a form of substructure coupling analysis inwhich the dynamic behavior of each substructure is formulated as asuperposition of modal contributions. This modal contribution consistsof preselected component modes of the following types: normal modes,rigid body modes, constrained modes, attachment modes, inertia reliefmodes, and inertia relief attachment modes.

There are combinations of the rigid body modes, constrainedmodes, attachment modes, inertia-relief modes, and inertia-relief at-

112

II. MODELING OF MECHATRONIC SYSTEMS WITHCONFIGURATION-DEPENDENT DYNAMICS

tachment modes that generate a superposition of the modes, sufficientto determine exactly the static response of a component subjected toexternal forces applied at boundary nodes. Any of them may be sup-plemented by dynamic modes: fixed interface, free interface, or hybridinterface defined by the normal modes [7].

For a discrete configuration i, the model can be described by a setof second order differential equations:

M(li)q(li) + C(li)q(li) + K(li)q(li) = L(li)f(li) (5.2)

For the sake of brevity, any parameter-dependent function h(li) isrepresented just as h, in this subsection. In any CMS technique, thedisplacements of the physical coordinates q are represented in terms ofcomponent coordinates η using the classical modal transformation:

q = Ψη (5.3)

where the transformation Ψ consists of pre-selected component modes.The reduced model is built using the Craig-Bampton method [18].

According to this method, the modal transformation Ψ, defined by(5.4), is composed of the constrained modes, Ψc, and the normal modeswith fixed-interface, Φk

Ψ =[

Ψc Φk

]. (5.4)

Performing the modal transformation, the equation of motion canbe written in component coordinates η [19].

η + 2ξΩη + Ω2η = µ−1ΨTu f (5.5)

where Ω are the modal frequencies, ξ is the modal damping ratio, µis the modal mass, Ψu = LTΨ where L is the force influence matrix[19]. The component coordinates are then represented by the degree offreedom θ, representing the actuators, and the modal coordinates φ

η =[θφ

]. (5.6)

The output equations for a set of sensors, represented by the matrixLy, can be defined by

y = LTy q = LTy Ψη = Ψyη. (5.7)

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The state-space equations can be written by (5.8) and (5.9)

x =[

0 I−Ω2 −2ξΩ

]x +

[0

µ−1ΨTu

]f (5.8)

y =[

Ψy 0]x (5.9)

where x = [η η]T are the state variables.This model reduction technique is applied to extract several reduced

models for discrete configurations.

C. Affine Models - Interpolation and Simulation

A set of reduced single-input single-output (SISO) models, which canbe built from the procedure described in the previous section, is usedto build an LPV model in the configuration space:

x = A(l)x + B(l)uy = C(l)x + D(l)u (5.10)

where x is the state of the system, u and y are the input and the output,respectively, and l is a vector of varying parameters. In order to keepthe conventional state-space notation, the input u = f is adopted.

The technique used to create the LPV model relies on a linear in-terpolation of discrete poles, zeros and gains [16],[20]. This approachis suitable for models with few poles and zeros, justifying the model re-duction step. This fact is not a limiting factor, since few poles and zerosare usually sufficient to represent the motion of a mechatronic systemwithin the frequency range of interest. Equation (5.11) illustrates thetechnique applied for the vector of poles:

p1(l)p2(l)

...pn(l)

=

p0,1

p0,2...

p0,n

+

p1,1

p1,2...

p1,n

f(l) (5.11)

where p1 till pn are the poles of the system, p0,1 till p0,n and p1,1 tillp1,n are constants and f(l) is an analytical function of the schedulingparameter l. Similar affine functions have been made to describe thevarying zeros and gains. Based on these affine functions, proper 1st

and 2nd order state-space subsystems can be derived.

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III. TEST CASE: PICK-AND-PLACE ASSEMBLY ROBOT

From a pair of complex poles (pi(l), pi+1(l)) and a pair of zeros(zi(l), zi+1(l)), a 2nd order state-space subsystem can be derived by:

A2s(l) = Re

[pi(l) + pi+1(l) −pi(l)pi+1(l)

1 0

]B2s(l) =

[10

]C2s(l) = Re

[−zi(l)− zi+1(l) + pi(l) + pi+1(l)

zi(l)zi+1(l)− pi(l)pi+1(l)

]TD2s(l) = [1] . (5.12)

From one pole (p(l)) and one zero z(l), a 1st order state-space sub-system can be derived by:

A1s(l) = [p(l)] B1

s(l) = [1]

C1s(l) = [p(l)− z(l)]T D1

s(l) = [1] . (5.13)

These subsystems are then concatenated, yielding a varying state-space model. The concatenation can be performed between two sub-systems (As1, Bs1, Cs1, Ds1) and (As2, Bs2, Cs2, Ds2):

Ac =[

As2 0Bs1Cs2 As1

]Bc =

[Bs2

Bs1Ds2

]Cc =

[Ds1Cs2 Cs1

]Dc = [Ds1Ds2] . (5.14)

All subsystems can then be concatenated, yielding an LPV state-space system that is quadratically dependent on the affine function f(l)[16]. The gain is eventually added to the system.

This LPV model can be implemented in Simulink using S-functions.For each integration step, the S-functions compile the state-space modelfrom the LPV model depending on the parameter vector.

III. Test case: pick-and-place assembly robot

In this section, the pick-and place assembly robot and its model aredescribed. The model is then experimentally validated.

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A. Description of the Pick-and-Place Assembly RobotSet-up

The chosen test case is an industrial three-axis pick-and-place machine(see Fig. 5.1). The Y-motion is gantry driven by two linear motors andthe X-motion over the carriage is also driven by a linear motor. Thevertical Z-motion is actuated by a rotary brushless dc motor that drivesa vertical beam by a ball screw/nut combination. The position of thelinear motors and the beam length are measured with optical encodersand the acceleration at the gripper in the X-direction is measured withan accelerometer.

The objective of a pick-and-place machine is to move the gripper asaccurate and fast as possible in a point-to-point motion. However, fastmovements of the linear motor will excite the eigenfrequencies of theflexible beam, which may vary during the movement, since the beamlength is continuously changed.

B. Modeling

A model has been built to simulate the pick-and-place assembly robotmotion in X- and Z-direction. The Y-motion is not considered in thispaper. According to the proposed modeling methodology, the systemis divided in two subsystems:

1. a subsystem of which the dynamics does not depend on the con-figuration; and

2. a subsystem with configuration-dependent dynamics.

The subsystem containing the frame, the two linear motors whichdrive the Y-motion, the carriage, their bushings and joints does not de-pend on the configuration and is modeled in LMS Virtual.Lab Motion,a commercial multibody environment. Figure 5.3 shows the multibodymodel of this subsystem.

The total mass of the frame and of both linear motors (which drivesthe Y-motion) is 169kg. The carriage weighs 13.9kg. These values canbe found in the machine manual. The frame suspension is attachedto the ground by 8 connecting points. The frame suspension stiffnessand the damping constant are 2.6e7N/m and 2500Ns/m, respectively.A spring and a damper represent the connection between the carriage

116


and the frame and their values are 9.2e6N/m and 1000Ns/m, respec-tively. The stiffness and damping values are adjusted to match theexperimental data.

The dynamics of the beam and the linear motor which drives theX-motion depend on the configuration and are modeled using the afore-mentioned 3-step methodology. Firstly, a parametric high-order finite-element model of this subsystem is created. Figure 5.2(A) shows ascheme of the parametric dependence of the model for several dis-crete positions. There is one rigid mode, the lateral motion in theX-direction. In this figure, the connection between the linear motorand the beam is modeled as a linear stiffness K=2.5e11N/m, M=25kgrepresents the motor mass, x the rigid-mode degree of freedom andmi = (0.53 − li)πr2ρ the equivalent mass of the part of the beamthat is connected to the linear motor. The gripper mass is mg=1.2kg.The nominal beam radius is r=0.012m. The material properties of thebeam are: density ρ=7800kg/m3, Poisson’s ratio ν=0.3, damping ra-tio 0.01 and elasticity modulus E=2.12e11N/m2. The beam length, li,can vary from 0.33m to 0.53m. A parametric finite element model iscreated using the Structural Dynamics Toolbox (SDT) in Matlab. Foreach configuration, a finite element model was derived from the para-metric one modifying the beam length. The flexible beam is modeledwith 100 beam elements, which guarantee good accuracy for the finiteelement model and for the reduced model.

Figure 5.3: Multibody model of the subsystem that does not dependon the configuration

117


The suggested model reduction technique is applied for 20 finite-element models representing 20 configurations (between 0.33m and0.53m). These reduced models are then interpolated in order to gen-erate the LPV model. There is no rule to decide on the amount ofconfigurations that should be evaluated. However, it is important tohighlight that the proposed interpolation technique is based on the lin-ear relation between the scheduling parameter and the system polesand zeros; therefore, this fact should be verified in advance. In thiscase study, the poles and the zeros of these 20 configurations seem tovary in a linear way according to the configuration. Higher-order in-terpolations should be considered if the poles and the zeros seem notto vary in a linear way (see [20]).

Each model has one degree of freedom for the actuator (linear mo-tor) and 2 modal coordinates, representing the first and the secondnatural frequencies. This single-input-multiple-output (SIMO) modelhas the force applied by the motor as input, Fm, and the gripper andmotor accelerations as outputs, ag and am respectively. Fig. 5.4 illus-trates the frequency response functions (FRFs) of the system at threedifferent beam lengths (l = [0.53, 0.43, 0.33m]).

These SIMO models are divided into two SISO models since theinterpolation technique can only be applied for a set of SISO mod-els. The proposed interpolation technique has been applied yieldingtwo SISO LPV models. The chosen analytical function f(l) is l, themeasurement of the beam length since a linear relation between thesubsystem eigenfrequencies and the beam length can be verified in Fig.5.4. S-functions have been implemented in Matlab/Simulink in orderto simulate these LPV SISO models (see Fig. 5.5).

There are two inputs for these S-functions: the force of the linearmotor, Fm, and the beam length, l. For each integration step, theS-functions compile the ABCD-model from the LPV model dependingon the beam length.

The LMS Virtual.Lab Motion interface with Matlab/Simulink al-lows time-domain simulations of arbitrary mechanisms coupled withtransfer functions or state-space models described in Matlab/Simulink.Therefore, both subsystems are integrated through co-simulation be-tween LMS Virtual.Lab Motion and Matlab/Simulink (see Fig. 5.5).The force generated by the motor, Fm, is applied to the motor (S-function input in Matlab/Simulink for the LPV model) and to the car-riage (in Motion) with opposite directions (action and reaction forces).

118


The encoder of the motor measures the difference between the positionsof the motor, xm (S-function output in Matlab/Simulink for the LPVmodel), and the carriage, xc (in Motion). The motor and the gripperpositions, xm and xg, are calculated by integrating twice the motor andthe gripper acceleration, am and ag, respectively. The FRFs obtainedusing this approach for three beam lengths are depicted in Fig. 5.6.

The controllers of the Y-motion and the X-motion are independent;therefore, they can be designed separately. The description of the Y-motion modeling is omitted because the main concern, in this work,is to control the X-motion. The Y-motion modeling can be performedusing the same methodology since it also depends on the configuration.The system division would be performed in a similar way: the subsys-tem (1) would consist of the carriage, the linear motors and the flexiblebeam; and the subsystem (2) would be the frames.

C. Experimental validation

In order to validate the model, comparisons between simulated and ex-perimental results were performed. The setup is identified for four dif-ferent lengths of the beam, based on FRF measurements using multisineexcitation [20], [21]. The linear motor responsible for the X-direction

Figure 5.4: (a) ag/Fm and (b) am/Fm for l = 0.53, 0.43, 0.33m

119


Figure 5.5: Co-simulation scheme between LMS Virtual.Lab Motion(grey background) and Matlab/Simulink (white background)

motion is excited with a multisine voltage signal with random phases.Two measurements are acquired: the encoder position, xm − xc, andthe gripper acceleration, ag. An average over 5 experiments with dif-ferent realizations of the random phases is performed in order to obtainthe FRFs.

Fig. 5.7 shows the comparisons for four beam lengths (l = 0.53,0.41, 0.36 and 0.33m with the first resonance frequency at 32, 45, 56and 66Hz, respectively). The comparisons are similar for other beamlengths. Differences at low frequencies are mainly due to the lack offriction in the model and differences in high frequency are mainly dueto a mismatch in the sensor position; in the model, the sensor is placedon the end of the flexible beam and in the set-up, it is placed on thegripper. For controlling the motion of mechatronic systems, such as ma-chine tools and mechanisms, a control bandwidth around 30Hz is typ-ically expected (as discussed in Section IV-A). Consequently, a modelable to capture accurately the machine dynamics up to 100 Hz is usu-ally sufficient. The proposed model can be used for control design andevaluation.

Since the approach generates a new state-space model for each time-integration step, it is possible to perform a trajectory with continuously

120


Figure 5.6: (a) (xm − xc)/Fm and (b) xg/Fm for l = 0.53, 0.43,0.33m

varying beam length. A value proportional to the measured currentinput was used as input signal to the model, since the current and themotor force are typically proportional for a linear motor. The encodermeasurements obtained by the experiment and by the simulation arecompared in Fig. 5.8. In this experiment, the beam length does notchange significantly (about 20% of the total range), since, in the actualmachine, the velocity of the X-motion is much higher than the velocityof the Z-motion. However, even this limited range analysis can assurea confident correlation between the model and the real system, sincethe dynamics vary quite smoothly.

121


Figure 5.7: Comparison between the simulated (- - dashed line) andthe experimental FRFs (- solid full line): (a) (xm − xc)/Fm (b)

ag/Fm

Figure 5.8: Encoder (motor position): comparison between thesimulated and the experimental motor position for the same current

input (a) from 0 to 8s and (b) from 6 to 6.8s

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IV. MECHATRONIC DESIGN APPROACH

IV. Mechatronic Design Approach

A mechatronic design approach has been employed considering as vari-ables: the radius of the flexible beam (discrete, values: 0.010, 0.012,0.016, and 0.020 m) and the gains of the PID controller (continuous).An optimal PID controller is designed for each possible radius of theflexible beam. Therefore, structural and control parameters are be-ing considered and tradeoff designs can be evaluated for several activesystems in a truly mechatronic design approach.

By choosing the control strategy a priori, the size of the optimiza-tion problem is reduced. LTI and LPV control strategies are exploitedhereafter for designing a PID controller for the X-direction motion. Itis necessary to highlight that the Z-direction motion is added to themodel, but no control design is carried out for it, since its dynam-ics do not depend on the configuration. The Z-direction movement isconsidered ideal, i.e., the applied motion is perfectly tracked.

The following subsections describe the chosen control design ap-proach based on time-domain metrics, the LTI PID controllers andLPV PID controllers. Design tradeoffs considering structural and con-trol parameters are highlighted for both control strategies.

A. Control Design Approach

During the control specification, time- and/or frequency-domain char-acteristics are used to evaluate closed-loop performance. The frequencyresponse functions of the open-loop and closed-loop transfer functions,such as sensitivity and complementary sensitivity functions, are typi-cally used to characterize closed-loop performance [22]. The commonfrequency-domain metrics are gain and phase margins (GM and PM ,respectively), the maximum value of sensitivity and complementarysensitivity functions (MS and MT , respectively). It is easier to evalu-ate the feedback performance in the frequency-domain. However, fortime-variant systems, such as mechatronic systems with configuration-dependent dynamics, frequency-domain metrics are not ready availablesince there is no plant that represents the system for all simulation timeand/or configurations.

The idea in this paper is to design a controller for a time-variantsystem using the total variation (TV ), a time-domain metric, to eval-uate the performance of the system. The TV of a function g(t) can bedefined as the largest sum of variations for any subdivision of t [22].

123


Considering a step response (see Fig. 5.9), the total variation and theexcess variation (EV ) can be defined as:

TV =∑i

vi EV =TV

v0(5.15)

In the cases considered hereafter, v0=1, then TV = EV .

Figure 5.9: Total variation (TV ) and excess variation (EV ) of g(t)

The TV was chosen rather than the settling time and the over-shoot since it offers a tradeoff between these two time-domain metrics.Using frequency-domain performance criteria, an upper bound in MT ,has been a frequent design specification. Indeed, MT < 2 guaranteesGM ≥ 1.5 and PM ≥ 29 [22]. The TV of the step response in thetime-domain can be quite well correlated to MT and will be used as acriterion to evaluate the performance of the system [22].

MT ≤ TV ≤ (2n+ 1)MT (5.16)

where n is the order of the complementary sensitivity function. Most ofthe system dynamics can be represented by low-order models, suggest-ing that TV provides a good approximation to MT [22]. Consequently,TV should be smaller than 2 and as close as possible to 1, in order toguarantee good performance.

The optimization problem is described by a multiobjective problemwhere the difference between the reference input, r, and the position

124


of the gripper, xg, should be minimized for all m time steps during thesimulation time (see in Fig. 5.10).

minS∈<n

f(S) =12

m∑i=1

(ri(S)− xgi((S)))2 =12

m∑i=1

Fi(S)2 (5.17)

where the n×1 vector S is the variables vector which describes thePID gains. The beam radius are considered as a discrete parameter inSections IV-B and IV-C.

This optimization problem was solved using the Levenberg-Marquardt optimization method. This method combines the steepestdescent method with the Gauss-Newton method [23], aiming to findS = s1, s2, . . . , snT that minimizes a least-squares problem, f(S):

minS∈<n

f(S) =12

m∑i=1

Fi(S)2 (5.18)

where Fi(S) are the objective functions. The solution can be founditeratively:

Si+1 = Si − (J(Si)TJ(Si) + λiI)−1J(Si)T f(Si) (5.19)

where J is the Jacobian matrix of the function f(S), I is the identitymatrix and λi is a constant that ensures the positive definiteness of(J(Si)TJ(Si) + λiI). This method has been largely used for trackingproblems and parameter estimation [23].

B. LTI PID Controllers

Firstly, the optimization problem was solved for a fixed radius(0.012m), considering a step input as a reference signal. Table 5.1shows the gains of the optimal PID (KP , KI , KD) resulted from theoptimization for different beam lengths. The total variation for themotor position measured by the encoder (TV 1) and for the gripperposition (TV 2) are also shown.

As it can be observed in Table 5.1, the optimal PID controllers fordifferent lengths are not the same. In order to choose an LTI PIDcontroller for the system, considering the beam length variation, a

125


Figure 5.10: Simulink model used for the optimization loop

Beam KP KI KD TV 1 TV 2

length [m] [N

µm] [

N

µm · s] [

N · sµm

] [µm] [µm]

0.53 0.461 0.068 0.007 1.17 3.220.43 1.078 0.028 0.009 1.17 3.720.33 1.813 0.000 0.011 1.20 3.60

Table 5.1: Optimal LTI PID and TV for different beam lengths andnominal radius (0.012m)

Figure 5.11: Pulse input while the length of the beam iscontinuously changed

126


pulse train was chosen as input while the beam length was continu-ously changed, according to Fig. 5.11. The optimization problem iscarried out in the same way, using (5.17).

The flexible beam is a commercially available product and its radiuscan be 0.010, 0.012, 0.016 and 0.020m. The optimal LTI PID gains andthe total variations (TV 1 and TV 2) for the nominal radius and for thecommercial available ones is shown in Table 5.2. Clearly, there is atradeoff between the positioning of the motor and the vibration of thegripper. Thinner beams lead to better motor position performance andworse gripper position performance than thicker beams. Consideringthat the total variation of the gripper should be as close as possible to2, the best design option would be the thicker beam (0.020m). How-ever, considering the structural and control design space, no designguarantees GM ≥ 1.5 and PM ≥ 29o, i.e., TV 2 <2 for all possiblestructural modification.

Radius KP KI KD TV 1 TV 2

[m] [N

µm] [

N

µm · s] [

N · sµm

] [µm] [µm]

0.010 0.761 0.010 0.012 1.10 4.620.012 1.091 0.085 0.010 1.15 3.800.016 1.783 0.075 0.011 1.23 3.110.020 2.180 0.201 0.011 1.26 2.45

Table 5.2: Optimal LTI PID gains and TV for different radius

C. LPV PID Controllers

Another alternative to cope with tighter performance requirements forthis time-variant problem is using an LPV controller. In this way,the control can be modified according to the configuration. Firstly,a gain-scheduling controller is created linearly interpolating the opti-mized gains for discrete configurations (5.20). For the nominal beamradius (0.012m), the gains are linearly interpolated from the 3 config-urations shown in Table 5.1. The results obtained using this approachare shown in Table 5.3. The indexes 0 and 1 represent the constantcoefficients of the linear interpolation function, as described for theproportional gain:

127


KP = KP0 + l ·KP1. (5.20)

As it can be noticed, the LPV PID controller, using linearly interpo-lated gains, does not present any performance improvement comparedto the LTI PID controller using the optimized gains over the chosentrajectory, shown in Table 5.2.

These interpolated LPV PID gains may be a good “hot start” foran optimization problem considering the LPV PID gains (KP0, KP1,KI0, KI0, KD0 and KD1) as variables. The gains calculated from thisoptimization are referred to as the optimized LPV gains and the gainscalculated from the interpolation are referred to as the interpolatedLPV gains. Table 5.3 shows that the controller using the optimizedLPV gains achieved better performance than the interpolated LPVgains and LTI PID gains. Moreover, it can be observed that by choos-ing the optimized LPV PID, a thinner beam (0.012m) yields similarperformance than the best design option (0.020m) when the LTI PIDis considered (Table 5.2).

Methodology index KP KI KD TV 1 TV 2

[0,1] [N

µm] [

N

µm · s] [

N · sµm

] [µm] [µm]

time-invariant - 1.091 0.085 0.010 1.15 3.80

interpolation 0 4.044 -0.011 0.017 1.22 3.851 -6.760 0.340 -0.020

optimization 0 4.044 -0.011 0.015 1.16 2.721 -7.981 0.338 -0.024

Table 5.3: LTI, interpolated LPV and optimized LPV PID gains andTV for the nominal radius (0.012m)

Finally, Table 5.4 shows the total variations TV 1 and TV 2 foreach commercially available flexible beam when the LPV PID gains areoptimized. For the sake of brevity, the gains are omitted. The sametradeoff design between the positioning of the motor and the vibrationof the gripper can be verified: thinner flexible beams lead to bettermotor position performance and worse gripper position performancethan thicker flexible beams; however, using the LPV PID controller,tighter performance requirements can be fulfilled. These relations canonly be verified through simulation during the design phase.

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V. CONCLUSIONS

Radius TV 1 TV 2[m] [µm] [µm]

0.010 1.11 3.650.012 1.16 2.720.016 1.23 2.320.020 1.27 2.15

Table 5.4: TV for different radius considering the optimized LPVPID controller

V. Conclusions

A methodology to model and simulate mechatronic systems withconfiguration-dependent dynamics, using affine reduced models andcosimulation,was presented. This methodology was applied to a pick-and-place assembly robot. The system was divided into two subsys-tems: a subsystem of which the dynamics does not depend on theconfiguration and a subsystem with configuration-dependent dynam-ics. The former was modeled using LMS Virtual.Lab Motion andthe latter using an LPV model derived from the linear interpolationof reduced finite-element models. Both subsystems are coupled usingcosimulation. Since the cosimulation environment is Matlab/Simulink,the methodology allows also the inclusion of the control actions in thesimulation. The model was experimentally validated demonstratingthat this methodology is able to provide a good approximation of thereal behavior of a mechatronic system with configuration-dependentdynamics.

Using the simulation capabilities, integrated design was performedconsidering control and structural parameters. These results exemplifythe benefits of the mechatronic design approach since the active systemdesign tradeoffs were identified. For instance, in closed-loop simulation,thinner flexible beams lead to better motor position performance andworse gripper position performance than thicker flexible beams. Theadvantage of using the mechatronic design approach is to enable theevaluation, not only of the qualitative behavior, such as the previousanalysis, but also quantitative metrics, such as the TV values.

Finally, using time-domain metrics, LTI PID, and LPV PID con-trollers were designed for each possible structural modification. It hasbeen observed that better performance is reached using the LPV PID

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controller derived from an optimization problem, which minimizes theTV of the gripper position and considers the PID gains as variables.Therefore, the same performance can be reached by smaller diametersusing LPV controllers, as by thicker diameters using LTI controllers.

Acknowledgements

The work of M. M. da Silva was supported by the Brazilian Founda-tion Coordination for the Improvement of Higher Education Personnel,CAPES. The authors would like to thank the reviewers for their con-structive comments and suggestions.

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[2] P.R. Pagilla, B. Yu and K.L. Pau, Adaptive Control ofTime-Varying Mechanical Systems: Analysis and Experiments,IEEE/ASME Transactions on Mechatronics, vol. 5, no. 4, pp.410-418, December 2000.

[3] H. Van Brussel, P. Sas, I. Nemeth, P. De Fonseca and P. Vanden Braembussche, Towards a Mechatronic Compiler, IEEE/ASMETransactions on Mechatronics, vol. 6, no. 1, pp. 90–105, March2001.

[4] W.J. Zhang, Q. Li, and L.S. Guo, Integrated Design of Mechani-cal Structure and Control Algorithm for a Programmable Four-BarLinkage, IEEE/ASME Transactions on Mechatronics, vol. 4, no.4, pp. 354-362, December 1999.

[5] Z. Affi, B. EL-Kribi and L. Romdhane, Advanced mechatronic de-sign using a multi-objective genetic algorithm optimization of amotor-driven four-bar system, Mechatronics, vol. 17, pp. 489–500,2007.

[6] T. Ravichandran, D. Wang and G. Heppler, Simultaneous plant-controller design optimization of a two link planar manipulator,Mechatronics, vol. 16, pp. 233–242, 2006.

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[8] F. Wujun, Z. Changming, Active Damping Guide Roller DesignBased on Integrated Structure/Controller Optimization with aStructured Controllers. Proc. of the ASME 2005 InternationalDesign Engineering Technical Conferences and Computers, and In-formation in Engineering Conference, Long Beach, CA, Sep., pp.1–5, 2005.

[9] S. Behabahani and C.W. de Silva, System-Based and Concur-rent Design of a Smart Mechatronic System Using the Concept ofMechatronic Design Quotient (MDQ), IEEE/ASME Transactionson Mechatronics, vol. 13, no. 1, pp. 14-21, February 2008.

[10] L.P.R. de Oliveira, M.M. da Silva, P. Sas, H. Van Brussel, W.Desmet, Concurrent Mechatronic Design Approach for Active Con-trol of Cavity Noise. Journal of Sound and Vibration, vol. 314, no.3-5, pp. 507–525, 2008.

[11] L. Zollo, S. Roccella, E. Guglielmelli, M. Chiara Carrozza andP. Dario, Biomechatronic Design and Control of an Anthropo-morphic Artificial Hand for Prosthetic and Robotic Applications,IEEE/ASME Transactions on Mechatronics, vol. 12, no. 4, pp.418–429, August 2007.

[12] X. Wang, J.K. Mills, Dynamic Modeling of a Flexible-Link PlanarParallel Platform using Substructuring Approach. Mechanism andMachine Theory, vol. 41, pp. 671–687, 2006.

[13] M. Zaeh, D. Siedl, A New Method for Simulation of MachiningPerformance by Integrating Finite Element and Multi-Body Simu-lation for Machine Tools. Annals of the CIRP, vol. 56, no. 1, pp.383–386, 2007.

[14] A. Konno, M. Uchiyama, M. Murakami, Configuration-DependentVibration Controllability of Flexible-Link Manipulators. The Inter-national Journal of Robotics Research, vol. 56, no. 4, pp. 567–576,1997.

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[15] X. Wang, J.K. Mills, Active Control of Configuration-DependentLinkage Vibration with Application to a Planar Parallel Platform.Proc. of the 2005 IEEE International Conference on Robotics andAutomation, Barcelona, Spain, April, pp. 4327–4332, 2005.

[16] B. Paijmans, W. Symens, H. Van Brussel and J. Swevers, AGain-Scheduling-Control Technique for Mechatronic Systems WithPosition-Dependent Dynamics, Proc. of American Control Confer-ence, Minneapolis, USA, June 14-16, Paper ThB04.4, 2006.

[17] O.A. Bauchau and C.L. Bottasso, Contact Conditions for Cylin-drical, Prismatic, and Screw Joints in Flexible Multibody Systems,Multibody Systems Dynamics, vol. 5, no. 3, pp. 251–278, 2001.

[18] R.R. Craig, A Review of Time Domain and Frequency DomainComponent Mode Synthesis methods, Proc. of the Joint MechanicsConference, Albuquerque, USA, June 24-26, pp. 1–30, 1985.

[19] A. Preumount, Vibration Control of Active Structures: An Intro-duction, 2nd ed. Dordrecht, The Netherlands: Kluwer, 2002.

[20] B. Paijmans, W. Symens, H. Van Brussel, J. Swevers , Identificationof Interpolating Affine LPV Models for Mechatronic Systems withone Varying Parameter. European Journal of Control, vol. 14, no.1, pp. 16–29, 2008.

[21] J. Schoukens and R. Pintelon, System identification: a frequencydomain approach, IEEE Press, Piscataway, 2001.

[22] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control :Analysis and Design, Wiley Publishers, Chichester, U.K., 1997.

[23] J. Nocedal and S. Wright , Numerical optimization, New York:Spring Verlag, 2000.

132

Chapter 6

Integrated structure andcontrol design formechatronic systems withconfiguration-dependentdynamics

Maıra M. da SilvaOlivier BrulsWim DesmetHendrik Van Brussel

133

6 submitted to Mechatronics

Paper submitted to Mechatronics:

M.M. da Silva, O. Bruls, W. Desmet, H. Van Brussel, Inte-grated structure and control design for mechatronic systems withconfiguration-dependent dynamics, Mechatronics.

Comments to the reader : The case study described in Section3.1 is also presented in Section III-A in Chapter 5 and in Section3 in Chapter7. The full multibody model described in Section 3.2is also partially described in Section 3.1 in Chapter 7. The theoryrelated with stability analysis, which is described in Section 2.2, is alsopresented in Section 2.5 in Chapter 7. Table 6.1 has been divided intwo to fit the thesis borders.


• It is believed that the results of the gain-scheduling PID could beimproved if the same interpolation technique described in Section3.2.2 had been used. This is also applicable for the results inChapter 5.

• Case 2 is not much better than case 1 because of the linear choicemade for the gain-scheduling PID, which is related to the previousissue.

134

Abstract

This paper considers the optimal design of mechatronic systems withconfiguration-dependent dynamics. An optimal mechatronic design re-quires that, among the structural and control parameters, an optimalchoice has to be made with respect to design specifications in the dif-ferent domains. Two main challenges are treated in this paper: thenon-convex nature of the optimization problem and the difficulty inmodeling serial machines with flexible components and their embed-ded controllers. The optimization problem is treated using the directoptimization strategy which considers simultaneously structural andcontrol parameters as variables and adopts non-convex optimizationalgorithms. Linear time-invariant and gain-scheduling PID controllersare addressed. This methodology is exploited for the multi-objectiveoptimization of a pick-and-place assembly robot with a gripper car-ried by a variable-length flexible beam. The resulting design tradeoffsbetween system accuracy and control efforts demonstrate the advan-tage of an integrated design approach for mechatronic systems withconfiguration-dependent dynamics.

135


1 Introduction

An optimal mechatronic design requires that, among the structuraland control parameters, an optimal choice has to be made with respectto design specifications in the different domains [1]. In spite of theadvances in optimal control design, optimal mechatronic design is stillan open research area. There are, mainly, two reasons for that: (i)the difficulty in solving optimization problems involving structural andcontrol parameters due to their non-convex nature and (ii) the difficultyin modeling mechatronic systems due to their multidisciplinary nature.

Considering that P represents the structural system plant and Kthe control system, the integrated structure and control optimizationproblem can be described by the following optimization problem:

mins∈Ω

f(L(sp, sk)) (6.1)

where s is the vector of structural, sp, and control, sk, variables;Ω is the feasible solution set and f is a measure of the system dy-namic response which depends on the open-loop transfer functionL(s) = P(sp)K(sk). Among other issues, structural and control vari-ables are multiplied in order to evaluate the open-loop transfer func-tion, resulting into a non-convex optimization problem. There is nocomputationally tractable approach to solve Eq. 6.1 due to the com-plex and non-convex nature of the optimization [2]. According to [3],there are mainly two numerical strategies to perform the integratedstructure/controller design: the nested and the direct optimization.

The nested optimization strategy combines nonlinear optimizationmethods and model-based control design techniques, such as the onesbased on linear matrices inequalities (LMI) and Ricatti equations. Inother words, for each set of structure parameters, a controller, with ageneral structure, is designed using model-based control design tech-niques. The nested optimization strategy has been employed for de-signing a gain-scheduling control to achieve compensation for the vary-ing mass distribution, to suppress structural bending, vibrations andfriction disturbances, and to achieve shorter motion settling time [4].Recently, [5] has proposed a general platform for designing serial andparallel machines based on the nested optimization strategy. The main

136

1 INTRODUCTION

drawback of this strategy is that the variables are not optimized simul-taneously.

The direct optimization design considers simultaneously the con-trol and structural parameters, using numerical methods, such as non-convex optimization algorithms or genetic algorithms. It can be usedwhen the control structure is known beforehand. An important draw-back of this approach is the excessive computation time, which growsexponentially with the number of structural design variables. The di-rect optimization strategy has been used, among others, for optimizinga two-link planar manipulator and a PD controller [6], for optimizingthe geometry and the control parameters of a motor-driven four-barsystem [7], for optimizing structural and control parameters for cavitynoise reduction [8] and for designing structural and control parametersof a flexible linkage mechanism for noise attenuation [9]. In contrastwith the control strategies exploited in this paper, most references re-garding direct optimization strategies do not address the design of lin-ear parameter-varying (LPV) controllers. Because of the non-convexnature of the direct optimization strategy, a LPV system cannot bedescribed as a polytopic system [10]. This is an important drawbackwhen designing LPV controllers using frequency-domain metrics, sincethe infinite set of inequalities imposed by the parameter variation can-not be reduced to a finite one. For this reason, time-domain metricsare considered in this paper, which demands time-domain simulationand evaluation of the system under study.

Besides the complex and non-convex nature of the optimizationproblem, the integrated structure and control design is a challengingtask due to the difficulty in modeling some mechatronic systems. Thisis the case of serial machines with flexible components, such as Carte-sian mechanisms, milling machines and pick-and-place machines. Inthese machines, the relative motion between flexible components leadsto time-varying boundary conditions, so that the eigenfrequencies andmode shapes are not constant but depend on the spatial configuration.The dynamic modeling of serial machines has been treated in recentreferences [11-14]. A dynamic substructuring procedure has been ap-plied for the modeling of a flexible-link planar parallel platform in [12]using Component Mode Synthesis (CMS) [15]. A similar procedurewas employed in the modeling of a 3-axis milling machine in [11] forseveral discrete spatial configurations. However, CMS can not be di-rectly employed to evaluate a serial machine in time-domain, since it

137


is not possible to represent the flexibilities using a single mode set foreach component due to their time-varying boundary conditions. Sincemost commercial multibody packages use CMS to include flexibility,the modeling of serial machines is not a straightforward task. Alter-natives have been proposed in [13,14], integrating dedicated softwaresuch as commercial finite element and multibody packages. However,these techniques are rather time-consuming.

This paper concerns the integrated design of serial machines withconfiguration-dependent dynamics considering the direct optimizationstrategy. Two control strategies are compared: linear time-invariant(LTI) PID and gain-scheduling PID. To simulate and evaluate serialmachines in time-domain properly, an innovative feature has been im-plemented in Oofelie, an open source finite element software [16]. Thisfeature, referred to as sliding joint, is not available in most commer-cial multibody packages. It allows the relative translation motion be-tween flexible bodies. To simulate and evaluate gain-scheduling con-trollers, Oofelie capabilities have been extended to include controllersdescribed by LPV state-space equations. Based on this mechatronicdesign tool, able to simulate serial machines and different controllersin time-domain, a multi-objective optimization strategy is proposed forthe integrated design of the mechanical structure and the controller.This approach is very useful to evaluate tradeoffs among conflictingobjectives in mechatronic applications. The selected control design ap-proach does not directly guarantee stability, which should be assessedafter the control derivation. To cope with this requirement, stabilityanalysis is included as a set of constraints in the optimization problem.

The paper is organized as follows. The general methodology forthe modeling, stability analysis and optimization of mechatronic sys-tems with configuration-dependent dynamics is described in Section 2.The integrated design methodology is applied to an industrial 3-axispick-and-place assembly robot. Section 3 presents the case study de-scription, its mechanical modeling, as well as various control algorithmsand stability analysis. In Section 4, the integrated structure and con-trol optimization is developed. The results demonstrate the benefits ofthe mechatronic design approach. Finally, some conclusions are drawnin Section 5.

138

2 MODELING AND OPTIMIZATION OF MECHATRONIC SYSTEMS

2 Modeling and optimization of mechatronicsystems

2.1 Modeling

This paper presents a methodology to model and simulate a serial ma-chine system with configuration-dependent dynamics in time-domain,using nonlinear flexible multibody dynamics. The approach describedin [17], which is a general and systematic technique for the simulationof articulated systems with rigid and flexible components, is selected.For mechatronic systems, an extension of those modeling methods isrequired to deal with the controller dynamics. One option is to use acoupled modeling approach, so that a monolithic time integrator canbe used, and no weak coupling assumption is required [18,19]. Thisstrongly coupled formulation has been adopted for the present devel-opments.

According to [17], a flexible multibody system can be describedusing absolute nodal coordinates. Hence, each body is representedby a set of nodes and each node has its own translation and rotationcoordinates. The various bodies of the system are interconnected bykinematical joints, which impose restrictions on their relative motion.If the nodal coordinates are gathered in a vector q, the joints are thusrepresented by a set of m nonlinear kinematic constraints:

Φ(q, t) = 0 (6.2)

According to the Lagrange multiplier technique, the formulation of theconstrained equations of motion requires the introduction of a m × 1vector of Lagrange multipliers λ.

The dynamics of the controller can be represented by a nonlinearstate-space model with state variables xk and control signal outputvariables yk. In this way, the dynamic equations of a mechatronicsystem consisting of a multibody model and a control system (see Fig.6.1a), have the general structure:

M(q)q = g(q, q,w, t)−BTλ+ yk (6.3)0 = Φ(q, t) (6.4)

xk = f(xk,uk, t) (6.5)yk = h(xk,uk, t) (6.6)

139


Eq. (6.3) represents the dynamic equations of the mechanical system,Eq. (6.4) the kinematic constraints, Eq. (6.5) the state equation andEq. (6.6) the output equation. M is the mass matrix, which is not con-stant in general, g represents the internal, external and complementaryinertia forces, B = ∂Φ/∂q is the matrix of constraint gradients, yk de-notes the actuator forces or torques generated by the control action,uk represents the input signals to the controller and w represents thedisturbance, noise and reference signals vector. Equations (6.3-6.6) arecoupled equations of motion and can be solved numerically using animplicit time integration scheme. Typical applications are described in[18].

Figure 6.1a shows a scheme of the augmented plant Pa, which in-cludes the mechanical system P described by Eqs. (6.3)-(6.4), and thecontrol system K described by Eqs. (6.5)-(6.6). The notation is thesame as those adopted in Eqs. (6.3-6.6). The output system signal zand control signal inputs uk can be described by combinations of thedisturbance, noise and reference signals, w, the control signal outputs,yk, and the measurements from the mechanical system, which can bepositions q, velocities q or accelerations q. The objective is to design acontroller K that minimizes the signal z. Figure 6.1b shows a schemeof the selected control strategy which is described in Section 3.3. Inthis case, w includes only the reference signal, r, and z involves thetracking error, r− p, and the force generated by the controller, g. Thegripper position is referred to as p.

In the case of mechatronic systems with configuration-dependentdynamics, LPV controllers are widely employed [4,5,20]. Using thesimulation framework described by Eqs. (6.3-6.6), LPV controllers canbe included adapting the ABCD matrices according to time-varyingparameters l(t):

xk(t) = Ak(l(t))xk(t) + Bk(l(t))uk(t)yk(t) = Ck(l(t))xk(t) + Dk(l(t))uk(t)

(6.7)

2.2 Stability Analysis

When stability is not guaranteed directly by the control design ap-proach, it needs to be assessed eventually. The stability analysis ofLPV systems can be performed with Lyapunov-based theory. Recently,a sufficient condition for the stability of LPV systems has been provided

140


Pa

K

w z

y uk k P

K

r

y uk k

p

Pa

pmotor

vmotor

(a) (b)

r

++

-w zr-p

g

g

Figure 6.1: (a) Scheme of the augmented plant Pa, which includesthe mechanical system P, and the control system K and (b) Scheme

of the augmented plant and the control system of the case studydescribed in Section 3.3

in [21] taking into account a bound ∆ on the rate of parameter varia-tion. The system under verification should be described as a discrete-time LPV system x(i+ 1) = A(l(i))x(i), where the varying parameterdescribed in continuous-time, l(t), is represented in different time stepsi by l(i).

For a given maximal rate of variation ∆, the parameter space isdivided into ν intervals. The size of the intervals is such that in onediscrete time step, the parameter l(i) can only jump into the nextinterval:

|l(i+ 1)− l(i)|Ts

≤ ∆ (6.8)

where Ts is the sample period and i = 0 . . . ν. A simplified nota-tion for the theorem presented in [21], which states a sufficient con-dition for stability of an LPV system, is proposed in [22] and de-scribed hereafter. Considering a discrete-time LPV system describedby x(i + 1) = A(l(i))x(i), if there exist i = 1 . . . ν positive definiteconstant matrices P(i), such that the following LMIs are satisfied forall i = 1 . . . ν and j = −1, 0, 1:

141


A(l(i))TP(i+ j)A(l(i))−P(i) < 0A(l(i+ 1))TP(i+ j)A(l(i+ 1))−P(i) < 0

(6.9)

then the system is uniformly asymptotically stable for all time-varying realization of the parameter l, satisfying constraints on therange and rate of the parameter variation. Due to the notation simpli-fication, the first LMI of the first interval and the last LMI of the lastinterval are not valid and should be removed.

2.3 Multi-objective optimization

The objective of this work is to perform a direct optimization consider-ing structural and control parameters. As stated in the Introduction,this is a non-convex and non-linear optimization (Eq. 6.1). Moreover,this optimization problem is usually composed of distinct objectives,such as minimizing the tracking error and minimizing the motor ef-fort. The latter characteristic suggests that multi-objective optimiza-tion strategies, which aim to find tradeoffs among several conflictingobjectives, should be considered. Indeed, the derivation of a singlecost function as a weighted summation of multiple objectives is nota trivial task. This non-convex, non-linear and multi-objective opti-mization problem is computationally demanding; therefore, a suitableoptimization strategy should be selected.

In the context of mechatronic systems, the vector s of design vari-ables may include structural as well as control parameters. The opti-mization problem is stated as [23]:

mins∈Ω

fi(s) i = 1, ..., nf (6.10)

where fi (i = 1, .., nf ) denotes the set of objective functions, whereasΩ is the feasible solution set defined by the inequality constraints

hi(s) ≤ 0 i = 1, ..., nh (6.11)

Let us note that the functions fi and hi are typically computed fromsimulation results. The objective functions evaluate the dynamic sys-tem performance addressed by the non-convex optimization problem(Eq. 6.1).

A solution s∗ is a non-dominated solution or Pareto-optimal solutionif there is no other solution s such that fi(s) ≤ fi(s∗) for all i and

142


design space objective space

designvariable #1

Pareto Optimal Solutions

des

ign

var

iab

le #

2

objectivefunction #1

ob

ject

ive

fun

ctio

n

#2

mapping

W

Figure 6.2: Pareto plot: mapping from the design to the objectivespace

fj(s) < fj(s∗) for at least one j. Figure 6.2 illustrates the feasibleregion in the design space limited by the design constraints, and itsmapping to the objective space resulting into the Pareto plot. The lowerleft border between the feasible and infeasible regions in the objectivespace actually represents the Pareto-optimal solutions.

An attempt to solve this non-convex, non-linear, multi-objectiveand computationally demanding optimization problem can be per-formed using stochastic methods, such as evolutionary algorithms.Among evolutionary algorithms, the Non-Dominated Sorting GeneticAlgorithm proposed in [23] has been improved in [24] yielding an al-gorithm used for multi-objective problems: NSGA-II. In this refinedalgorithm, the population is ranked according to the individual’s non-domination criterion before the selection. In other words, an individualis compared with every other individual in the present population andalso with the non-dominated individuals from the previous populationto find if it is dominated. Eventually, a large fitness value is assigned tothe non-dominated solutions. This process is repeated to find the sub-sequent non-dominated solutions and it stops when all individuals inthe present population are dominated by the non-dominated individu-als from the previous population yielding the Pareto-optimal solutions.The diversity of the population is preserved by a crowded-comparisonapproach, which guarantees a good spread of solutions.

143


y

z

x

frame

linear motors

carriage

flexible beam

gripper

Figure 6.3: Pick-and-place machine used as test case

3 Pick-and-place robot: modeling details andcontrol algorithms

3.1 Case Study

This integrated design methodology is applied to an industrial 3-axispick-and-place assembly robot with a gripper carried by a flexible beam(Fig. 6.3). The fast movements of this machine may excite the vibra-tions of the variable-length flexible beam. The Z- and the X-motionare performed by a ACM H-drive system, which is a Cartesian robotbased on three linear motor motion systems, produced by Philips. TheZ-motion is gantry driven by two linear motors and the X-motion, overthe carriage, is also driven by a linear motor. The vertical Y-motion isactuated by a rotary brushless DC-motor which drives a vertical flexi-ble beam by a ball screw/nut combination. The position of the linearmotors and the beam length are measured by optical encoders, andthe acceleration at the gripper in the X-direction is measured by anaccelerometer. The objective is to move the gripper as accurately andfast as possible along a prescribed trajectory in the working area.

A model has been built to simulate the pick-and-place assemblyrobot motion in X- and Y-direction. The Z-motion is not consideredin this work.

144

3 PICK-AND-PLACE ROBOT: MODELING DETAILS AND CONTROLALGORITHMS

Table 6.1: Inertia values of the rigid bodiesrigid mass moment of inertia [kgm2]

bodies [kg]) Ixx Iyy Izzframes 169.0 1.0 · 103 2.0 · 103 1.0 · 103

carriage 13.9 1.0 · 102 1.0 · 101 1.0 · 102

linear motor 31.0 - - -gripper 1.25 - - -

Table 6.2: Center of gravity of the rigid bodiesrigid center of gravity [m]

bodies x y zframes ±0.57 0.53 0.00

carriage 0.00 0.53 0.00linear motor 0.00 0.53 0.00

gripper 0.00 0.00 0.00

3.2 Mechanical model

A flexible multibody model has been built to simulate the pick-and-place robot motion in X- and Y-directions (see Fig. 6.3) accordingto Eqs. 6.3 and 6.4. All components are modeled as rigid bodies,excepted the flexible beam. The inertia of the sliders of the linearmotors, responsible for the Z-direction motion, is added to the bodiesrepresenting the frames; the gripper is modeled as a concentrated mass.Tables 6.1 and 6.2 show the inertia values and center of gravity for allrigid bodies.

A specific feature, a sliding joint, has been implemented to enablethe translational relative motion between flexible bodies. This slidingjoint is responsible for the translational motion in Y-direction betweenthe flexible beam and the linear motor (see Fig. 6.4). According to theTimoshenko theory, displacements Γ and rotations Ψ are treated asindependent fields in the beam. In a single element with two nodes n0

and n1, an arbitrary point can be represented by the non-dimensionalcoordinate along the beam η ∈ [0, 1]. For linear shape functions andunder the assumption of small rotations, the positions and orientations

145


Node J

x

y

z

linear motor

flexible beam

Node 0

Node 1Node 0

Node 1

L

Figure 6.4: Scheme of the translational movement between thelinear motor and the flexible beam

of this point are expressed in terms of the nodal coordinates

Γ(η) = (1− η)Γ0 + ηΓ1 (6.12)Ψ(η) = (1− η)Ψ0 + ηΨ1 (6.13)

Using again the small rotations assumption, the axis of the beam andof the sliding joint are close to the y-axis. If the linear motor is rep-resented by a node J , the sliding joint is thus modeled by five kine-matic constraints between the nodal coordinates Γ0 = [x0 y0 z0]T ,Γ1 = [x1 y1 z1]T , ΓJ = [xJ yJ zJ ]T , Ψ0, Ψ1 and ΨJ

Φ1 = (1− η)x0 + ηx1 − xJ = 0 (6.14)Φ2 = (1− η)z0 + ηz1 − zJ = 0 (6.15)

Φ3,4,5 = (1− η)Ψ0 + ηΨ1 −ΨJ = 0 (6.16)

where η is computed as η = (y0−yJ)/L, with L, the total length of thebeam element. Figure 6.4 shows a scheme of the linear motor (rigidbody) and the flexible beam (flexible body) in two configurations illus-trating their behavior during the translational motion in Y-direction.

A general scheme of the pick-and-place robot model is shown inFig. 6.5. The actuator force generated by the linear motor, is appliedto the linear motor mass (action) and to the carriage (reaction). Thenominal machine specifications are described hereafter. The springstiffness and the damping value between the carriage and the frameare, respectively, K1 = 9.15 · 106N/m and D1 = 1042Ns/m. The frame

146


flexible beam

gripper

sliding joint

motor

M

M

frameM

carriageM

1K 1,D

1K 1,D

2K 2,D

2 2,D

2K 2,D

2K 2,D

3D

x

y

z

-FmotorFmotor

frameM

K

Figure 6.5: Scheme of the flexible multibody model of theX-direction motion of a pick-and-place machine

suspension is connected to the ground by four connecting points. Thestiffness and the damping of these connections are, respectively, K2 =5.3 · 107N/m and D2 = 5204Ns/m. The damping D3 = 100Ns/mrepresents the connection between the linear motor and the carriage.The flexible beam has a nominal diameter of 24mm. The materialproperties are: density ρs = 7800kg/m3, Poisson’s Ratio ν = 0.3,damping ratio 0.01 and elasticity modulus E = 2.1 · 1011N/m2. Themass and inertia values can be found in the machine manual. Thestiffness values have been adjusted according to experimental data (Fig.6.6).

This flexible multibody model has hundreds of degrees-of-freedom andits dynamics vary according to the configuration. Thus, a full descrip-tion of the dynamic equations cannot be directly included in this sec-tion. For the sake of completeness, a reduced model extracted from thefull flexible multibody for a given configuration (l = 0.53m) is presentedin Appendix A. Alternatively, a lumped model has been proposed in[25]. This lumped model can be used for the evaluation of the nominalsystem over the configuration space.

Figure 6.6 shows the comparison between simulated and measuredFRFs for four beam lengths (l = 0.53, 0.41, 0.36 and 0.33 m with the

147


Figure 6.6: Comparison between the simulated (- - dashed line) andthe experimental FRFs (– full line) (a) Motor position/Motor force

(b) Gripper acceleration/Motor force

first resonance frequency at 32, 46, 57 and 63 Hz). The curves are ingood agreement, which confirms the validity of the model.

3.3 Controller

For machines with configuration-dependent dynamics, two controlstrategies can be adopted: (1) LTI controllers, that can be explicitlydesigned to take into account the dynamic variations as uncertainties,like robust controllers designed using µ-synthesis or (2) linear time-varying (LTV) controllers, that can adapt according to the parametervariations, such as LPV controllers [4] and LPV gain-scheduling con-trollers [20].

In this work, a PID control scheme is implemented for controllingthe X-axis motion using measurements of the motor position. Both LTIand LPV gain-scheduling PID controllers are described by the followingstate-space representation

xk = [0] xk + [ −1 1 0 ] uk (6.17)

148


yk = [−KI(l)] xk + [ KP (l) −KP (l) −KD(l) ] uk (6.18)

where the input vector uk = [r, pmotor, vmotor] collects the referenceinput, the motor position and the motor velocity; and the output vectoryk = [g] represents the motor force. The gripper position is referredto as p. The performance of the system is measured by the gripperposition accuracy.

A linear dependence on l is selected for the gain-scheduling PID. Inthis way, the multiobjective optimization can still handle the numberof variables. In this way, the vector of scheduling parameters l simplyrepresents the beam length l = [l(t)].

Figure 6.1b shows a scheme of the augmented plant Pa, which in-cludes the mechanical system P, and the control system K. In thiscase, the signal w represents the reference signal, r; and the signal zrepresents the tracking error, r − p, and the actuation force generatedby the controller, g. The control signal inputs uk are [r, pmotor, vmotor].

The same strategy can be applied for modeling and controlling theZ-direction motions, but this design is not considered in this work.An imposed motion assures that the Y-direction motion follows theprescribed trajectory.

3.4 Stability analysis

A state-space description of the machine model, for a given configu-ration, is obtained by using a model-order reduction technique [26].For l=0.53m, the state-space model is described in Appendix A. Thisstate-space model is then combined, in a feedback fashion, with thestate-space description of the PID controller (Eqs. 6.22 and 6.18),yielding the closed-loop state-space model of the system with its em-bedded controllers for a given configuration. This procedure should berepeated for each configuration considered during the stability analysis.

These closed-loop state-space models are then discretized consider-ing a sampling frequency of 2000 Hz. The frequency range of interest isfrom 0 to 400 Hz, where the first three system resonances are located.In this way, the sampling rate can guarantee good measurements, ifthey are eventually performed, in the frequency range of interest.

In order to guarantee that the system is uniformly asymptoticallystable for a parameter variation between 0.33 and 0.53m (the beamlength) and bounded rate by 10.0m/s, the parameter space should bediscretized in ν = 40 intervals (l(i+ 1)− l(i) < 0.005). The feasibility

149


problem described by the LMIs (in Eq. 6.9) can be solved using theLMI toolbox available in Matlab [27]. Since 40 intervals should beevaluated, 41 closed-loop state-space models should be extracted foreach configuration considered (between 0.33 and 0.53m).

4 Integrated structure and control design

In pick-and-place applications, the position error should be kept belowa specified threshold. The diameter of the beam has a direct influ-ence on the vibration of the effector and is thus considered as a designvariable. The other parameters are associated with the particular PIDcontrol strategy, as described below. The set of variables is collectedin the vector of design variables, denoted as s.

In order to mimic point-to-point movements, a pulse train is chosenas a reference signal for the optimization problem. As illustrated inFig. 6.7, the beam length, l(t), evolves during the simulation, so thatthe different eigenfrequencies are excited in various configurations. Forthis reference input, the simulation takes about 110s CPU time usinga Pentium IV, with a processor of 1.4GHz.

Taking these aspects into account, the minimization of two objectivefunctions is considered.

The first objective function f1 represents the weighted squared errorbetween the gripper position, p(t, s), and the reference signal, r(t), andis computed according to

f1(s) =1ce

∫ 1

0ζ(t)(r(t)− p(t, s))2dt (6.19)

where t refers to time, ce = 4 ·10−4 is a constant and ζ(t) is a weightingfunction. The value of constant ce is chosen in such a way that thevalues of f1 are normalized between 0 and 1. Too large or too smallthe objective or constraint values can lead to numerical errors duringthe optimization procedure. Therefore, it is a common practice tonormalize these values.

The weighting function ζ(t), shown in Fig. 6.7, is adopted topenalize longer settling times. A basic weighting function, ζ∗(t) =tan(α) ∗ t + 1/T − tan(α) ∗ T/2, where t is the time within each step

150

4 INTEGRATED STRUCTURE AND CONTROL DESIGN

0 0.2 0.4 0.6 0.8 14.8

5ζ(t)

0 0.2 0.4 0.6 0.8 1

0

5

10

x 10−4

r(t)

[m]

0 0.2 0.4 0.6 0.8 10.3

0.4

0.5

l(t)

[m]

Time [s]

Figure 6.7: The weighting function ζ(t), the reference input r(t) andbeam length variation during the simulation l(t)

interval and T represents the total simulation time associated with thestep input (in the present case 0.2 s) and α is the curve steepness. Anillustration of this basic weighting function is depicted in Fig. 6.8a.An attempt to define such weighting functions is addressed in [6], butno clear guidelines are proposed. In the way it is presented here, theangle α can be adjusted such that the weighting function penalizesmore large overshoot or longer settling time. An example illustratesthe behavior of this basic weighting function for a second-order systemwith bandwidth of 10 Hz and a damping factor of 0.01. The referenceinput (a step) and the system response are shown in Figs. 6.8b and6.8c, respectively. Figures 6.8d and 6.8e show two different weightingfunctions and the respective weighted squared errors. The higher α,the lower is the penalty on the overshoot and the higher is the penaltyon the settling time. For the case study presented in this work, theangle α is chosen to be 60o, illustrated in Fig. 6.7, which penalizeslonger settling times.

151


0 0.5 10.8

0.9

1

1.1

Time

ζ* (t)

(a)

0 0.5 10

0.5

1

1.5

2x 10

−3

Ste

p in

put [

m]

Time [s]

(b)

0 0.5 10

0.5

1

1.5

2x 10

−3

Sys

tem

res

pons

e [m

]

Time [s]

(c)

0 0.5 1

0.8

1

1.2

1.4

ζ* (t)

Time [s]

(d)

0 0.2 0.4 0.6 0.8 10

1

2

3

x 10−3

Wei

ghte

d sq

uare

d er

ror

[m2 ]

Time [s]

(e)

α

Figure 6.8: (a) the basic weighting function ζ∗(t), (b) the step inputr(t), (c) system response, (d) two different weighting function and (e)

their respective weighted error length variation l(t)

The second objective function f2 represents the maximum force (g)required by the controller during motion, i.e.

f2(s) =max |g(t, s)|

cf(6.20)

where cf=500N is a constant for normalizing f2. The maximum forcedelivered by the present motor is 500N, which motivates the cf valuechoice. Normally, a linear motor is selected based on the maximumforce required by the controller to perform a desired motion. Sincethis is an expensive item, the present motor is kept, enforcing that themaximum required force is below 500N. In this way, the values of f2

can only vary between 0 and 1 (normalized objective), which will beenforced by the constraint h2, defined below.

The responses evaluated during the optimization, such as the grip-per position and the motor position, are obtained with the simulation ofthe flexible multibody with its embedded PID controller (Eqs. 6.3-6.6).

Two sets of constraints are imposed to avoid infeasible gains (≤0)

152


and large actuation forces (≥500N):

h1(s) : sl ≤ s ≤ su (6.21)h2(s) : f2 ≤ gu (6.22)

where sl and su assure that the control gains are always positive andthat the structural parameter varies according to the available commer-cial options (from 0.02 to 0.04m); and gu is guaranteeing a maximumrequired force below 500N.

A set of constraints is imposed to guarantee that the system isuniformly asymptotically stable for parameter variations between 0.33and 0.53m and rate bounded by 10.0m/s:

h3(s) :

P(l(i)) > 0A(l(i))TP(i+ j)A(l(i))−P(i) < 0A(l(i+ 1))TP(i+ j)A(l(i+ 1))−P(i) < 0

(6.23)

where i = 1, . . . , 40 and j = −1, 0, 1.This constraint is evaluated in discrete-time; where i represents the

time step, l is the beam length, P(l(i)) are positive definite matricesand A(l(i)) represented the discretized closed-loop state space system(the reduced plant embedded with the controller).

For multi-objective optimization problems with 3 and 4 variables,described below, the adopted initial population and the populationsize are equal to 30 individuals (solutions); whereas for problems with7 variables, the initial population and the population size are equalto 40 individuals. Around 25 individuals are expected to be foundon the Pareto optimal solutions in both cases. The inverse crossoverprobability is chosen to be 0.85, which guarantees the inclusion of newindividuals in the optimization process. Finally, the maximum num-ber of iterations is 30, which is large enough to observe the algorithmconvergence, which means that all individuals in the present popula-tion are dominated by the non-dominated individuals from the previouspopulation.

Two control strategies and optimization problems are analyzed asfollows:

Case 1 The gains of an LTI PID are optimized simultaneously withthe diameter of the flexible beam. These results are comparedwith the nominal case (d=24mm), where only the controller isoptimized.

153


Case 2 The gains of an LPV gain-scheduling PID are optimized simul-taneously with the diameter of the flexible beam. Comparisonsbetween the integrated design considering the LTI PID and theLPV gain-scheduling PID controllers are reported.

4.1 Case 1: Integrated design considering an LTI PIDcontroller

Considering the reference input in Fig. 6.7, the gains of an LTI PIDcontroller are optimized. Firstly, the nominal case (d=24mm) is con-sidered resulting in a 3-variable optimization problem

s = KP ,KI ,KD

Secondly, the gains are optimized simultaneously with the beam diam-eter resulting in a 4-variable optimization problem, i.e.

s = KP ,KI ,KD, d

The multi-objective optimization problem is stated as

mins

[f1(s)f2(s)

](6.24)

subject to

h1(s)h2(s)h3(s)

Figure 6.9 shows the Pareto optimal solutions and Fig. 6.10 showsthe values for the design variables associated with each individual onthe Pareto optimal solutions for both nominal case and integrated de-sign. It can be observed from Fig. 6.9, that the inclusion of the struc-tural variable improves the overall design. For instance, consideringthe same f1 value (see the squares in Figs. 6.9 and 6.10) and increas-ing slightly the beam diameter (12.5%), the value of f2 is considerablyreduced (from 0.58 to 0.35). This reduction actually means that themaximum required force by the controller is lower than the nominalcase, i.e., the same level of performance can be achieved with a smallermotor. On the other hand, considering the same f2 value (see thecircles in Figs. 6.9 and 6.10) and increasing the beam diameter, thevalue of f1 is considerably reduced. In general, the optimal integrated

154


0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.4

0.5

0.6

0.7

0.8

0.9

1

f2

f 1

Nominal case (LTI PID)Integrated design (LTI PID)

Figure 6.9: Pareto optimal solutions considering the LTI PID: (o)nominal case and (x) integrated design

solutions have resulted in thicker diameters (see Fig. 6.10d). How-ever, since the maximum force required by the system is related toboth beam diameter and PID gains, thicker diameters are not alwaysyielding higher values of the maximum force (see the squares in Figs.6.9 and 6.10). Moreover, the design constraint, h2, is respected in thewhole objective space, which means that the present motor can be keptand the maximum required force is below 500N.

4.2 Case 2: Integrated design considering an LPV gain-scheduling PID controller

Considering the reference input in Fig. 6.7, the gains of an LPV gain-scheduling PID and the beam diameter are optimized simultaneously.This optimization problem leads to 7 design variables

s = KP0,KP1,KI0,KI1,KD0,KD1, d

As performed previously (Eq. (6.24)), both objective functions f1 andf2 are considered. Figure 6.11 shows the Pareto optimal solutions forthe aforementioned LTI PID controller (case 1) and for the LPV gain-scheduling PID controller (case 2). It can be observed that the LPVgain-scheduling PID can perform slightly better than the LTI PID,since the solutions of the latter are located below the solutions of the

155


0 10 20 300

2

4

6x 10

5 (a)

KP [N

/m]

Optimal Solutions0 10 20 30

0

5

10x 10

5 (b)

KI [N

/ms]

Optimal Solutions

0 10 20 300

2000

4000

6000

8000(c)

KD [N

s/m

]

Optimal Solutions0 10 20 30

0.02

0.03

0.04(d)

d [m

]Optimal Solutions

Figure 6.10: Optimal solutions considering the LTI PID: (o)nominal case and (x) integrated design

former. The variations of the optimal beam diameter along the Paretofront are illustrated in Fig. 6.12. There is no significant differencebetween the solutions set.

It can be observed from Figs. 6.11 and 6.12 that the LPV gain-scheduling performance and solution set are rather similar to the per-formance and solution set presented by the LTI PID. The main reasonfor this is that the objective function related to the tracking error met-rics (f1) does not take into account how homogeneous the results areover the configuration space. It is expected that machines, such aspick-and-place robots and milling machines, present the same dynamicproperties over the whole configuration space.

In order to illustrate this behavior, two particular optimal solu-tions, highlighted in Fig. 6.11b, are considered. The system responsesof these particular optimal solutions are depicted in Fig. 6.13. It canbe observed that both systems behave similarly and their f1 value (Ta-ble 6.3) are almost the same. However, a more detailed analysis indifferent periods reveals that the weighted squared error values are not

156


0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.4

0.5

0.6

0.7

0.8

0.9

1

f2

f 1

(a)

0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58

0.5

0.55

0.6

0.65

f2

f 1

(b)

LTI PIDLPV Gain−scheduling PID

Figure 6.11: Pareto optimal solutions of the integrated designconsidering the LTI PID and the LPV gain-scheduling PID

controllers: (a) the objective space and (b) zoomed area on theobjective space

0 5 10 15 20 25 300.015

0.02

0.025

0.03

0.035

0.04

d [m

]

Optimal Solutions

Nominal Case (LTI PID)Integrated Design (LTI PID)Integrated Design (LPV PID)

Figure 6.12: Optimal beam diameters: nominal case and integrateddesign considering the LTI PID and the LPV gain-scheduling PID

controllers

157


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−6

−4

−2

0

2

4

6

8

10

12

x 10−4

Time [s]

Dis

plac

emen

t [m

]

Reference InputIntegrated design (LPV PID)Integrated design (LTI PID)

Figure 6.13: System responses considering the two particular cases

Table 6.3: Weighted squared error value in different periods of timePeriod of time [s] f1

PID [0-0.2] ]0.2-0.4] ]0.4-0.6] ]0.6-0.8] ]0.8-1.0]1. LTI 0.1214 0.1199 0.1137 0.1104 0.1069 0.57232. LPV 0.1160 0.1157 0.1123 0.1126 0.1124 0.5690

homogeneous. Table 6.3 describes the weighted squares error for each0.2 s of the simulation. For case 1, the LTI PID, the mean value is0.1145 and its standard deviation 0.061; while for the case 2, the LPVPID, the mean is 0.1138 and the standard deviation 0.019. In this way,the LPV gain-scheduling control strategy assures a more homogeneousdynamic behavior over the configuration space.

5. Conclusions

This paper addresses the integrated design of structural and control pa-rameters of serial machines with flexible components. A multi-objectiveoptimization framework has been developed based on a general simu-

158

ACKNOWLEDGEMENTS

lation tool for flexible multibody systems embedded with nonlinearcontrollers.

The methodology is exploited for the optimization of a pick-and-place assembly robot with a gripper carried by a variable-length flexiblebeam. The model involves a sliding joint that connects the flexiblebeam to the rigid frame. The beam diameter and the gains of LTIand LPV gain-scheduling PID controllers are optimized according to adirect optimization strategy.

These results reveal the benefits of the mechatronic design approachsince the active system design tradeoffs are identified. The qualitativestatement that the optimal integrated solutions result in thicker diam-eters seems to be predictable. Actually, any thicker diameter wouldimply vibration reduction. However, the quantitative results achievedin this framework are not that simple to foresee. Using the proposedmethodology, one can decide which beam diameter should be selectedand predict the closed-loop response of such a complex mechatronicsystem in time-domain. In this way, not only qualitative behavior, butalso quantitative metrics, such as overshoot and settling time, can beevaluated over the configuration space during the design phase.

For future work, tighter performance requirements can be achievedusing more advanced control strategies, such as robust and optimalcontrollers, with the nested optimization strategy. It is expected thatthese control strategies benefit more from the gain-scheduling designthan the PID used in this work.

Acknowledgements

The research of Maıra M. da Silva is supported by CAPES, BrazilianFoundation Coordination for the Improvement of Higher EducationPersonnel. The research presented in this paper was performed aspart of the Marie Curie RTN project: A Computer Aided EngineeringApproach for Smart Structures Design (MC-RTN-2006-035559) andthe EU project: NEXT (IP 011815). The scientific responsibility isassumed by its authors.

Appendix A

The model-order reduction technique described in [26] has been ap-plied to the full flexible multibody model, described in Section 4.1,

159


for each configuration considered in the stability analysis (41 configu-rations from l = 0.33 to 0.53m). This appendix reports the reducedmodel for l = 0.53m. The reduced mass and stiffness matrices are builtaccording to a selected set of modes: 1 rigid-body mode and 3 keptflexible modes, yielding 4 by 4 matrices. A state-space model can bederived from the reduced mass and stiffness matrices, the modal ma-trices and the chosen modal damping factor. Eventually, the flexiblemultibody has been reduced from hundreds of degrees-of-freedom toa state-space model with 8 states, 1 input (the motor force) and 2outputs (the motor position and the gripper position). Details on themodel-reduction technique and its application on the case study canbe found in [5].

The minimal realization state-space model of this reduced model isgiven by the following equations:

xp = Ap(0.53)xp + Bp(0.53)upyp = Cp(0.53)xp + Dp(0.53)up

(6.25)

where

Ap(0.53) =

−3.7 2.0 · 103 −9.6 · 10−18 −1.8 · 10−15 . . .4.4 · 10−4 −3.0 −1 −1.3 · 10−17 . . .−5.6 3.9 · 104 −6.0 · 10−5 −4.8 · 10−1 . . .−2.2 · 10−4 1.5 1.0 · 10−2 −2.6 · 103 . . .6.0 · 10−6 −8.4 · 10−2 −6.8 1.7 · 105 . . .−2.0 · 10−6 −3.8 · 10−1 −6.4 · 101 1.6 · 106 . . .3.4 · 10−6 4.5 · 10−1 6.9 · 101 −1.7 · 106 . . .−1 2.8 · 10−7 5.1 · 10−6 −1.3 · 10−1 . . .

. . . 1.2 · 10−14 −9.5 · 10−16 −2.7 · 10−14 3.7 · 10−7

. . . 5.8 · 10−17 −6.9 · 10−18 4.2 · 10−17 −5.8 · 10−10

. . . 2.4 · 10−13 −1.7 · 10−14 −5.2 · 10−13 7.2 · 10−6

. . . −9.7 1.8 · 10−15 7.1 · 10−14 −9.8 · 10−7

. . . 8562 −9.0 · 102 −4.1 · 10−11 5.7 · 10−4

. . . 8.2 · 104 −8.5 · 103 9.6 5.3 · 10−3

. . . 4.2 · 105 −4.5 · 104 7.4 · 101 −2.3 · 10−2

. . . 3.1 · 10−2 −3.3 · 10−3 5.4 · 10−6 −1.7 · 10−9

160

BIBLIOGRAPHY

Bp(0.53) =

−3.7 · 10−2

4.4 · 10−6

5.6 · 10−2

−1.5 · 10−5

8.6 · 10−3

8.1 · 10−2

−8.3 · 10−2

−6.1 · 10−9

Cp(0.53) =[−2.5 · 10−13 2.4 · 10−7 3.9 · 10−5 −9.9 · 10−1 . . .−2.5 · 10−13 −7.7 · 10−1 −1.0 · 10−4 9.9 · 10−1 . . .

. . . −1.5 · 10−1 1.6 · 10−2 −3.2 · 10−5 1.0

. . . 1.5 · 10−1 −1.6 · 10−2 3.2 · 10−5 1.0

]

Dp(0.53) =[

00]

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[5] da Silva MM, Bruls O, Swevers J, Desmet W, Van Brussel H.Computer-Aided Integrated Design for Machines With Varying Dy-namics. Machine and Mechanism Theory (submitted).

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[7] Affi Z, EL-Kribi B, Romdhane L. Advanced mechatronic designusing a multi-objective genetic algorithm optimization of a motor-driven four-bar system. Mechatronics 2007; 17:489–500.

[8] De Oliveira LPR, Da Silva MM, Sas P, Van Brussel H, Desmet W.Concurrent mechatronic design approach for active control of cavitynoise. J Sound and Vibration 2008; 314(3-5): 517:525.

[9] Jianwei L. Study on the integrated structural design and noiseattenuation of flexible linkage mechanism . Mechatronics 2008;18(3):153–158.

[10] Apkarian P, Gahinet P, Becker G. Self-Scheduled H∞ Control ofLinear Parameter-Varying Systems: A Design Example . Auto-matica 1995; 31:1251–1261.

[11] Van Brussel H, Sas P, Nemeth I, De Fonseca P, Van den Braembuss-che P. Towards a mechatronic compiler. IEEE/ASME Transactionson Mechatronics 2001; 6(1):90–105.

[12] X. Wang, J.K. Mills, Dynamic modeling of a flexible-link planarparallel platform using substructuring approach. Mechanism andMachine Theory 2006; 41:671–687.

[13] da Silva MM, Desmet W, Van Brussel H. Design of mechatronic sys-tems with configuration-dependent dynamics: simulation and opti-mization. IEEE/ASME Trans. on Mechatronics 2008; 13(6):638–646.

[14] Zaeh M, Siedl D. A New Method for Simulation of Machining Per-formance by Integrating Finite Element and Multi-Body Simulationfor Machine Tools. Annals of the CIRP 2007; 6(1):383–386.

[15] Craig RR. A Review of Time Domain and Frequency Domain Com-ponent Mode Synthesis methods. Proc. of the Joint MechanicsConference, Albuquerque, USA, June 24-26 1985.

[16] Cardona A, Klapka I, Geradin M. Design of a new finite elementprogramming environment. Eng Comput 1994; 11:365-81.

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[17] Geradin M, Cardona A. Flexible multibody dynamics: a finite ele-ment approach. Chichester: John Wiley & Sons; 2001.

[18] Bruls O, Golinval JC. The generalized-α method in mechatronicapplications. J of Applied Mathematics and Mechanics 2006;86(10):748–58.

[19] Samin JC, Bruls O, Collard JF, Sass L, Fisette P. Multiphysicsmodeling and optimization of mechatronic multibody systems.Multibody Systems Dynamcis 2007; 18(3):345–73.

[20] Paijmans B, Symens W, Van Brussel H, Swevers J. A gain-scheduling control technique for mechatronic systems with position-dependent dynamics. Proc of American Control Conference, Min-neapolis, USA, June 14–16 2006.

[21] Amato F, Mattei M., Pironti A. Gain scheduled control for discrete-time systems depending on bounded rate parameters. Int J of Ro-bust and Nonlinear Control 2005; 15:473-94.

[22] Paijmans B. Interpolating gain-scheduling control for mecha-tronic systems with parameter-dependent dynamics. PHD Thesis:Katholieke Universiteit Leuven; 2007.

[23] Srinivas N, Deb K. Multi-objective function optimization using non-dominated sorting genetic algorithms. J of Evolutionary Computa-tion 1994; 2(3):221–248.

[24] Deb K, Agrawal S, Pratap A, Meyarivan T. A fast elitist non-dominated sorting genetic algorithm for multi-objective optimiza-tion: NSGA-II. IEEE Trans on the Evolutionary Computation2002; 6(2):182–197.

[25] De Caigny J, Demeulenaere B., De Schutter J., Swevers J. Polyno-mial spline input design for LPV motion systems. Proceedings of10th IEEE International Workshop on Advanced Motion Control,Trento, March 26–28 2008.

[26] Bruls O, Duysinx P, Golinval JC. The global modal parameteri-zation for non-linear model-order reduction in flexible multibodydynamics. Int. J. Numer. Meth. Engng 2007; 69:948-977.

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[27] Gahinet P, Nemirowski A, Laub AJ, Chilali M. LMI Control Tool-box. The mathworks: Natick, MA, 1995.

164

Chapter 7

Computer-AidedIntegrated Design forMachines With VaryingDynamics

Maıra M. da SilvaOlivier BrulsJan SweversWim DesmetHendrik Van Brussel

165

7 submitted to Mechanism and Machine Theory

Preliminary results, presented at the 9th International Conference onMotion and Vibration Control (MOVIC), have been selected to bepublished in a Springer book:

M.M. da Silva, O. Bruls, B. Paijmans, W. Desmet, H. Van Brussel,Computer-Aided Integrated Design for Mechatronic Systems WithVarying Dynamics, In H. Ulbrich & L. Ginzinger (eds.) Motionand Vibration Control, Selected Papers from MOVIC 2008, ISBM978-1-4020-9437-8, Springer, The Netherlands, 2008, 53–62.

Paper accepted to be published in the Mechanism and MachineTheory:

M.M. da Silva, O. Bruls, J. Swevers, W. Desmet, H. Van Brus-sel, Computer-Aided Integrated Design for Machines With VaryingDynamics, Mechanism and Machine Theory.

Comments to the reader : The case study described in Section 3is also presented in Section III-A in Chapter 5 and in Section 3 inChapter 6. The full multibody model described in Section 3.1 isalso described in Section 3.2 in Chapter 6. The theory related withstability analysis, which is described in Section 2.5, is also presentedin Section 2.2 in Chapter 6.


• The sampling frequency in Section 3.2 should be 2000 Hz insteadof 2048 Hz.

• Frequency domain requirements are kept the same for all possiblemechanical systems. These requirements reflect the system designrequirements which are usually defined in early design phases.

• The model parameters have been manually tuned to fit the ex-perimental results.

166

Abstract

This paper discusses the integrated design of mechatronic systems withvarying dynamics, such as serial and parallel machine tools. This char-acteristic affects the machine stability and performance. A computer-aided integrated design methodology is proposed and validated on apick-and-place robot. It consists of two main steps: (i) the deriva-tion of reduced models from a flexible multibody model and (ii) thesystematic robust control design. Eventually, the integrated design ofthe system, considering both structural and control parameters, can beperformed.

167


1. Introduction

The ever increasing demands for faster and more accurate machinetools give rise to conflicting design requirements since fast movementsmay cause vibrations, deteriorating the machine accuracy. Moreover, alarge number of machines presents varying eigenfrequencies and modeshapes. For instance, in machine tools, Cartesian mechanisms and pick-and-place machines, the relative motion between flexible componentsleads to time-varying boundary conditions, so that the eigenfrequen-cies and mode shapes depend on the configuration. This nonlinearphenomenon inevitably affects the stability and the performance of thesystem [1-4].

Simulation-based design has been extensively used, not only forshortening development cycles and reducing design costs, but also forenhancing product performance. Since the performance of machinetools depends on both structural and control parameters, their designshould be performed concurrently along the lines of a mechatronic de-sign approach, which deals with the integrated design of mechanicalsystems and their embedded controls [5-8]. A computer-aided inte-grated design approach requires simulation tools enabling the directaccess to structural and control parameters. These tools should allowthe dynamic modeling of the machine, the control system design andthe active system evaluation.

Dynamic modeling of serial and parallel machines has been treatedin several references [1,9]. Recently, a dynamic substructuring model-ing procedure has been applied for the modeling of a flexible-link planarparallel platform in [10]. This technique is based on the assembly ofcomponent mode sets, extracted from finite element models using Com-ponent Mode Synthesis (CMS) [11]. A similar procedure was employedin the modeling of a 3-axis milling machine in [1] for several discretespatial configurations. CMS cannot be directly used to evaluate a se-rial machine in time-domain, since it is not possible to represent theflexibilities using a single mode set for each component due to theirtime-varying boundary conditions. An alternative for modeling serialmachines using affine reduced models has been investigated in [7,8].These affine reduced models and the control system, whose dynam-ics depend on the configuration, are implemented in a co-simulation

168

1. INTRODUCTION

scheme with a commercial multibody package. These alternatives arerelatively time-consuming, which is an important drawback when con-sidering optimization-based design. The present work exploits a fullycoupled simulation approach, overcoming this important issue.

An integrated control system and mechanical design dedicated toa compliant two-axes mechanism has been presented in [2]. For eachvalid set of structural parameters, a linear parameter varying (LPV)controller was derived and evaluated. A more general platform has beenproposed for parallel kinematic machines using a CAD interface, finiteelement and multibody analysis, and optimization in [12,13]; however,no LPV controllers are treated. A simultaneous engineering frameworkfor designing mechatronic systems has been addressed in [1], propos-ing methodologies from the conceptual design to the detailed design ofmachine tools. A revision of the latter work is necessary for the in-clusion of recent and relevant developments on computer-aided designand control strategies. For the case study considered in this work (Fig.7.1), the gains of linear time-invariant and linear parameter varyingPID controllers are optimized simultaneously with structural param-eters in [7,8]. In order to achieve tighter performance requirements,model-based control design has been selected and derived. Prelimi-nary results, reported in [14], are further investigated in the presentwork. Important issues as stability analysis, graphical interpretationsof the methodologies, the interpolation of linear time-invariant systemsand the description of the control design parameters are fully exploitedhereafter.

The present work concerns the computer-aided integrated design ofmachines with varying dynamics. A general framework based on a sim-ulation platform, where active machines can be evaluated, is proposed.Reduced-order models, suitable for model-based control design, can beextracted from the flexible multibody models. A systematic model-based control design approach, based on frequency-domain metrics, isalso derived. The proposed methodology is applicable to serial and par-allel machines and uses state-of-the-art robust control design methodsthat can cope with varying dynamics. Eventually, the system dynamicperformance can be evaluated, considering not only discrete configu-rations but also continuous operation, using active flexible multibodymodels.

The remainder of the paper is organized as follows. The generalmethodology for modeling and control design of mechatronic systems

169


with varying dynamics is described in Section 2. This methodologyis applied to an industrial 3-axis pick-and-place assembly robot witha flexible gripper arm (Fig. 7.1). Section 3 presents its dynamicalmodel and control design. In Section 4, design tradeoffs are evaluatedconsidering the closed-loop system performance for different structuralconfigurations. Finally, some conclusions are drawn in Section 5.

frame

linear motors

carriage

flexible beam

gripper

y

z

x

Figure 7.1: Pick-and-place assembly robot

2. Modeling and Control of Machines WithVarying Dynamics

A mechatronic system is composed of rigid bodies, flexible bodies,joints, sensors, actuators and control units. Sensors and actuators canbe either considered as ideal forces or torques sources, or included ex-plicitly by their models. For this paper, ideal actuators are considered.Due to these characteristics, a mechatronic system can be modeled asa flexible multibody system coupled with its control system (shown inFig. 7.2). The multibody formalism chosen in this work is introducedin Section 2.1.

Flexible multibody models can contain a large number of degrees of

170

2. MODELING AND CONTROL OF MACHINES WITH VARYING DYNAMICS


External Moment

Revolute Joint

Flexible body

Flexible beam

ABCD

Reference

+-

Controller

Rigid body

Sliding Joint

Motor

Spring

Damper

Sensors

Figure 7.2: General scheme of a flexible multibody and its controller

freedom (dofs) and, therefore, may be unsuitable for model-based con-troller design purposes, such as pole-placement and H∞ control design,since the order of the controller is related to the order of the model.A model-order reduction technique has been applied in order to de-rive a concise description of the flexible multibody model. Among theseveral model reduction approaches [15], a technique based on globalmodal parameterization (GMP) has been chosen because it provides di-rect access to the reduced stiffness and mass matrices widely employedfor machine design evaluation. A short introduction on the GMP tech-nique is presented in Section 2.2 and details can be found in [15].

For time-variant systems, two kinds of control strategies can beused: (1) linear time-invariant (LTI) controllers, that can be explicitlydesigned to take into account the dynamical variations as uncertainties,like robust controllers designed using µ-synthesis or (2) linear time-varying (LTV) controllers, that can adapt according to the parametervariations, such as LPV controllers [2,17], LPV gain-scheduling con-trollers [3,4] and adaptive controllers [18]. In this paper, a LPV gain-scheduling controller, obtained by interpolating LTI robust controllers,is considered. These LTI controllers are referred hereafter as local con-trollers. The local controllers are derived for several specific values ofthe scheduling parameter via an extension of the four-block H∞-control

171


design approach [19] described in Section 2.3.A compact way of describing a gain-scheduling controller is by the

following linear parameter-varying (LPV) state-space form:

x = A(θ)x + B(θ)uy = C(θ)x + D(θ)u (7.1)

where x is the state of the controller, u and y are, respectively, theinput (error signal) and the output (actuation) and θ is a vector ofvarying parameters which represents the configuration space. Thereare several interpolation techniques to derive an LPV system. A surveyon these techniques can be found in [20]. The interpolation techniquechosen in this work, based on the linear interpolation of poles andzeros of each local controller is described in Section 2.4. The stabilityof the resulting closed-loop system is not directly guaranteed by thegain-scheduling control design procedure, but can be verified using, forinstance, Lyapunov-based theory, which will be addressed in Section2.5.

2.1 Flexible Multibody Models

Formalisms developed in the field of flexible multibody dynamics ap-pear to be especially suitable to model mechatronic systems [21]. Inparticular, the nonlinear finite element approach described in [22] isa general and systematic technique to model and simulate articulatedsystems with both rigid and flexible components. An extension of thosemodeling methods is required to deal with controller dynamics. Oneoption is to use co-simulation, so that the time integration procedureis based on a sequential analysis of the mechanical subsystem and ofthe control subsystem, using two different programs, see e.g. [7,8]. Inthis case, each program uses its own solver and only inputs and out-puts are exchanged at a pre-defined rate. This scheme is often calledweak or loose coupling. Due to stability, accuracy, and performanceissues, the communication rate should be chosen carefully. An impor-tant drawback is that additional assumptions are required in presenceof algebraic loops [23]. As proposed in [24], a second option is to usea strongly coupled modeling approach, so that a monolithic time inte-grator can be used, and no weak coupling assumption is required. Thestrongly coupled formulation, which is available in the Oofelie finiteelement software [25], has been chosen for the present developments.

172


According to [22], a flexible multibody system can be described us-ing absolute nodal coordinates. Hence, each body is represented by aset of nodes and each node has its own translation and rotation coor-dinates w.r.t a fixed reference frame. The various bodies of the systemare interconnected by kinematical joints, which impose restrictions ontheir relative motion. If the nodal coordinates are gathered in a vectorq, the joints are thus represented by a set of m nonlinear kinematicconstraints:

Φ(q, t) = 0 (7.2)According to the Lagrange multiplier technique, the formulation of

the constrained equations of motion requires the introduction an m×1vector of Lagrange multipliers λ.

The dynamics of the controller can be represented by a nonlinearstate-space model with state variables x and control signal output vari-ables y. In this way, the dynamic equations of a mechatronic systemconsisting of a multibody model and a control system, have the generalstructure:

M(q)q = g(q, q,w, t)−BTλ+ y (7.3)0 = Φ(q, t) (7.4)x = f(x,u, t) (7.5)y = h(x,u, t) (7.6)

Eq. (7.3) represents the dynamic equations of the mechanical system,Eq. (7.4), the kinematic constraints, Eq. (7.5), the state equation andEq. (7.6), the output equation. M is the mass matrix, which is notconstant in general, g represents the internal, external and comple-mentary inertia forces, B = ∂Φ/∂q is the matrix of constraint gradi-ents, y denotes the actuator forces or torques generated by the controlaction, u represents the control input signals and w represents the dis-turbance, noise and reference signals vector. Equations (7.3-7.6) arecoupled equations of motion and can be solved numerically using animplicit time integration scheme [24].

Figure 7.3 shows a scheme of the augmented plant Pa, which in-cludes the mechanical system P , and the control system K. The no-tation is the same one adopted in Eqs. (7.3-7.6). The output systemsignal z and control signal inputs u can be described by combinationsof the disturbance, noise and reference signals w, the control signaloutputs y, and the measurements from the mechanical system, whichcan be positions q, velocities q, accelerations q or internal forces λ.

173


Pa

K

w z

y u

Figure 7.3: General scheme of the augmented plant and its controller

2.2 Model Reduction

In linear structural dynamics, CMS provides an appropriate solutionfor the reduction of a finite element model. In CMS, the dynamicbehavior of each component (e.g. frame, slide, ram, carriage, etc.) isformulated as a superposition of modal contributions. A classical modaltransformation is used [1,11], where the modal basis is constructedby the combination of various sets of modes (e.g. static modes, rigidmodes, eigenmodes, etc.). During the procedure, the connecting dofsat the interface of each component should be kept, therefore, yieldinga reduced model that may require further reduction (see an examplein [1]).

Approximation-data based methods, such as singular value decom-position (SVD) based approximation methods (e.g. balanced trunca-tion method for linear system and proper orthogonal decomposition(POD) method for non-linear system) and Krylov-based approxima-tion methods (e.g. balanced realization) have also been applied for thereduction of large dynamical systems [15]. Considering systems withvarying dynamics, the POD method has been used for reducing theorder of large LTV models derived from partial differential equations[26]. The application of balanced truncation for LTV systems has beendiscussed and exemplified in [27,28]. Since these methodologies relyon the construction of a subspace that best approximates the collecteddata, the physical meaning of the modeling may be lost during themodel reduction which is an important drawback. The design of bothserial and parallel machines often relies on the evaluation of dynamicalcharacteristics in the configuration space, such as the stiffness matrix[29,30]. Therefore, a model reduction technique able to keep the rele-

174


vant information during the procedure is preferred.Bruls et al. [16] have proposed a model-order reduction technique

based on Global Modal Parametrization (GMP) which considers theglobal modes extracted from the whole mechanism and parameterizedaccording to the mechanism’s configuration. This approach yields amore concise model than the one provided by CMS and provides directaccess to the reduced stiffness matrix and other dynamical character-istics. Therefore, this technique is chosen in this work and is brieflydescribed hereafter (see details in [16]).

Considering the passive flexible multibody system (Eqs. 7.3-7.4,y = 0) and the augmented coordinates v = [q λ]T , the GMP isdefined as the mapping between the augmented coordinates v, andthe modal coordinates η. A mechatronic system generally undergoeslarge-amplitude rigid motions, which can be represented by rigid modalcoordinates θ, and small-amplitude superimposed deformation, whichcan be represented by flexible modal coordinates δ. From the practicalpoint of view, the rigid coordinates can be conveniently defined as theactuators dofs, with the advantage that those coordinates will explicitlyappear in the reduced model.

The reduction procedure is based on the transformation betweenaugmented coordinates, v, and modal coordinates, η. In general, due tothe nonlinear kinematics of the machine, the relation v(η) is nonlinear.However, in the neighborhood of a given configuration θ0 which is notdeformed (δ0), we have the incremental relation

∆v = Ψ∆η = Ψr(θ0)∆θ + Ψf (θ0)∆δ (7.7)

with ∆(•) = (•)−(•)0. In this expression, the configuration-dependentrigid and flexible mode shape matrices Ψr and Ψf are defined for smallmotions around the selected configuration θ0.

In a given configuration, the mode shapes are constructed by acomponent-mode synthesis using the linearized form of the equation ofmotion. The mass and stiffness matrices can be rewritten as: Mrr Mrg Mri

Mgr Mgg Mgi

Mir Mig Mii

, Krr Krg Kri

Kgr Kgg Kgi

Kir Kig Kii

(7.8)

where the indexes r, g and i represent rigid, constraint and internaldofs, respectively.

175


During the reduction procedure, θ and qg should be kept sincethey represent the rigid body dofs and the constraint dofs respectively.Constraint dofs are required when additional external loads, besides theones related to the rigid body motion, are used; for instance, a distur-bance force at the end-effector. The internal dofs, vi, can be condensedduring the reduction by selecting fewer lower-order flexible modes torepresent their dynamics. According to the pre-selected modes: rigidbody modes Ψr, constraint modes Ψg and lower-order flexible internalmodes Ψl; the modal transformation is defined as Ψ = [Ψr Ψg Ψl].

The rigid modes Ψr should satisfy: Krr Krg Kri

Kgr Kgg Kgi

Kir Kig Kii

Ψr =

000

(7.9)

The constraint modes Ψg are static deformations obtained when therigid dofs are fixed and a unit displacement is imposed to the constraintdofs: [

Kgg Kgi

Kig Kii

]Ψg =

[gg

0

](7.10)

where gg is the force vector required to impose an unit displacementto the constraint dofs.

And the internal modes Ψi = [Ψl Ψh], divided into lower-order(represented by the index l) and higher-order (represented by the indexh) modes, are the normalized eigenmodes when rigid and constraintdofs are fixed:

(Kii − ω2kM

ii)Ψuik = 0 (7.11)

where ωk is the kth eigenfrequency associated with the kth mode shape,Ψuik . The model-order reduction technique relies on a truncation of the

higher-order internal modes.After transformation in modal coordinates, the equations of motion

(Eqs. 7.3-7.4, y = 0) become [16]:

Mηη(θ)η + Cηη(θ)η + Kηη(θ)η = gη(θ) (7.12)

where gη(θ) denotes the actuator forces, Mηη is the reduced massmatrix, Cηη is the reduced damping matrix and Kηη is the reducedstiffness matrix. Equation (7.12) can be rewritten in state-space form:

176


[ηη

]=[

0 I−(Mηη(θ))−1Kηη(θ) −(Mηη(θ))−1Cηη(θ)

] [ηη

]+

+[

0−(Mηη(θ))−1

]gη(θ)

(7.13)

For a given configuration θ0, this equation defines a low-order lin-earized model which can be used for control design, as described inSections 2.3 and 2.4.

2.3 Control Design for Linear Time-Invariant Motion Sys-tems

The demands for increasing the throughput of machine tools give riseto performance requirements such as limited settling-time and over-shoot when an aggressive trajectory reference is applied. These per-formance specifications should be achieved via a systematic controllerdesign framework able to support the designer to derive controllers thatguarantee system stability and performance.

In an integrated control design environment, the translation of thethese specifications to the control design requires a systematic designframework which can be provided by H∞-control design techniques.A stabilizing control Kc is derived by minimizing the H∞-norm of thesystem closed-loop transfer function matrix, γ = ‖Mc‖∞ [31]. The per-formance specifications are translated to weighting functions that aug-ment the system model yielding the so called augmented plant model(Pa in Fig. 7.3).

Van de Wal et al. [19] propose for high-dynamic mechatronic mo-tion systems a 4-block H∞-control design approach to select appro-priate weighting functions to derive the augmented plant, Pa, for atypical set of performance specifications. This scheme inherently ex-hibits some robustness against uncertainty; e.g. mismatches on theresonances, which is especially beneficial for positioning devices withresonant behavior.

An extension of the 4-block H∞-control design approach has beenproposed in [32] in order to specify explicitly the maximum of the sensi-tivity, mS , the maximum of the process sensitivity, mSP

, the maximumof the complementary sensitivity, mT and the bandwidth frequency,

177


fBW , through weighting functions. The sensitivity, process sensitivity,complementary sensitivity definitions are formally given below. In thiswork, the bandwidth frequency is defined as the frequency where theopen-loop gain, L = KcP, crosses unity from above for the first time.The aim of the weighting functions is to adequately shape the loop-gain L providing not only sufficient stability margins (around fBW ),but also demanding the control to have integral action at low frequencyand to roll-off at high frequency. The former requirement is necessaryto suppress low-frequency and constant disturbances, and the latterto minimize the effect of measurement noise and to be robust againstmodel uncertainty.

The chosen performance frequency-domain metrics (mS , mT , mSP

and fBW ) are correlated with time-domain metrics. The peaks of theclosed-loop transfer functions, mS and mT , are related to the stepresponse overshoot and settling-time. The value mT correlates quitewell with the total variation of the output which can be defined asthe largest sum of variations for any interval [8,31]. Therefore, limitedvalues of mT and of mS impose limits on overshoot and settling time.The speed of the response, measured by the step response rise time, isrelated to fBW . In general, a larger bandwidth yields a smaller rise timebut also an increased sensitivity and reduced robustness. Consideringinput disturbance rejection, the smaller mSP

, the lower is the influenceof the disturbance. Often, a tradeoff has to be made between thesevalues since it is rather difficult to obtain perfect tracking (mS = 1)and perfect input disturbance rejection (small mSP

).A scheme of the control structure with the weighting functions,

proposed by [32], is shown in Fig. 7.4, where w1 is the reference trackingsignal, w2 is the disturbance signal, z1 is the weighted tracking error, z2

is the weighted control signal and rSPis an additional weighting factor,

that can be adjusted to improve the response quality. The weightedclosed-loop transfer function Mc can be described as

[z1

z2

]= −

W1S

mS

W1SPrSP

W2SKmTmSrSP

W2T

mSPmT rSP

[ w1

w2

]= Mc

[w1

w2

](7.14)

where S = (I+PKc)−1 is the sensitivity function, SP = SP is the pro-cess sensitivity function, SK = KcS is the control sensitivity function

178


and T = PKcS is the complementary sensitivity function.

K P+

-

W1

W2

S

1m rT SP

1mSP

w1

w2z2z1

1m

yu +C

Figure 7.4: Scheme of the control structure with the weightingfunctions

The shaping filters W1 and W2, illustrated in Fig. 7.5, are typicallychosen in order to guarantee that the controller contains integral actionup to fI = fBW /4 and roll-off at the higher frequencies (fR = 4fBW ).The shaping filters can be described by the following expressions:

W1(s) =s+ 2πfI

s(7.15)

W2(s) = α2|GfBW |s2 + 2ζ(2πfR)s+ (2πfR)2

s2 + 2ζα(2πfR)s+ (α2πfR)2(7.16)

where |GfBW | is the gain of the system at fBW , ζ indicates the dampingcoefficient and is considered to be 0.7 and α = 10 guarantees a cut-offat a fairly high frequency fI = 10fR, making W2 a proper function. If‖Mc‖∞ ≤ 1, the performance specifications are met.

2.4 Gain-scheduling Controller Derivation

From the procedure described in the previous section, a set of LTI con-trollers can be derived for local configurations. These LTI controllers

179


f I f BW f BW f R f Raf[Hz] f[Hz]

1

-20dB/dec +40dB/dec

|W(s

)|1

|W(s

)|2

Gf BW

Figure 7.5: Magnitude of the shaping filters W1 and W2

are then combined together yielding a gain-scheduling controller rep-resented in an LPV state-space form (Eq. 7.1). In order to obtain thisLPV gain scheduling controller, a technique developed by Paijmans etal. [20], which interpolates the pole and zero loci of the LTI controllers,is used. The 4-step interpolation procedure is:

1. calculate the poles and zeros of the local LTI systems, and classifythem as complex or real and single or pair;

2. derive affine functions for the poles and the zeros in a way thatthe pole and zero loci of the affine functions match the poles andzeros of the local LTI systems;

3. calculate 1st order LPV state-space proper subsystem(containing1 pole and no or 1 zero) and 2nd order LPV state-space propersubsystem (containing 2 poles and no, 1 or 2 zeros) and;

4. concatenate the LPV subsystems yielding LPV gain-schedulingcontroller system.

For the considered application a 1st order affine function is consid-ered. For the poles p1 till pn, this affine relation equals:

p1(θ)p2(θ)

...pn(θ)

=

p0,1

p0,2...

p0,n

+

p1,1

p1,2...

p1,n

f(θ) (7.17)

where p0,1 till p0,n and p1,1 till p1,n are constants and f(θ) is an ana-lytical function of the scheduling parameter θ. Similar expressions areused to describe the varying zeros and gain.

180


Based on these affine functions, proper subsystems can be createdusing following rules (step 3):

• for all pairs of complex conjugated poles, a 2nd order subsystemis created;

• all pairs of complex conjugated zeros are added to the existing2nd order subsystem; if there are more complex zeros than poles,a 2nd order subsystem should be created containing 2 complexconjugated zeros and 2 real poles;

• for each remaining real pole, a 1st order subsystem is created;

• the remaining real zeros are added to the first and second ordersubsystems.

To illustrate this, a pair of complex poles (pi(θ), pi+1(θ)) and zeros(zi(θ), zi+1(θ)) combined into a 2nd order subsystem (A2

s, B2s, C2

s, D2s)

has the following form:

A2s(θ) = Re

[pi(θ) + pi+1(θ) −pi(θ)pi+1(θ)

1 0

]B2s(θ) =

[10

]C2s(θ) = Re

[−zi(θ)− zi+1(θ) + pi(θ) + pi+1(θ)

zi(θ)zi+1(θ)− pi(θ)pi+1(θ)

]TD2s(θ) = [1] (7.18)

For a single pole (p(θ)) and zero (z(θ)), the following 1st ordersubsystem (A1

s, B1s, C1

s, D1s) is obtained:

A1s(θ) = [p(θ)] B1

s(θ) = [1]

C1s(θ) = [p(θ)− z(θ)]T D1

s(θ) = [1] (7.19)

In the 4th step, all subsystems are combined in series. For the twosubsystems (As1, Bs1, Cs1, Ds1) and (As2, Bs2, Cs2, Ds2), these seriesyield the following state-space sentence:

Ac =[

As2 0Bs1Cs2 As1

]Bc =

[Bs2

Bs1Ds2

]Cc =

[Ds1Cs2 Cs1

]Dc = [Ds1Ds2] (7.20)

181


Since the matrices Bs and Cs are independent on the schedulingparameter, the matric products do not result in a system order increase.Eventually, if necessary, an affine expression of the gain factor canbe included into the system. The affine expression of the gain factormultiplies the matrix C and D increasing the system order.

2.5 Stability Analysis

The stability analysis of LPV systems can be performed usingLyapunov-based theory. Recently, a sufficient condition for the sta-bility of an LPV system has been provided in [33] taking into accounta bound ∆ on the rate of parameter variation. For a given maximalrate of variation ∆, the parameter space is divided into ν intervals. Thesize of the interval is such that in one discrete time step, the parameterθ(i) can only jump into the next interval:

|θ(i+ 1)− θ(i)|Ts

≤ ∆ (7.21)

where Ts is the sampling period and i = 1 . . . ν. A simplified nota-tion for the theorem presented in [33], which states a sufficient con-dition for stability of an LPV system, is proposed in [32] and de-scribed hereafter. Considering a discrete-time LPV system describedby x(i+1) = A(θ(i))x(i), if there exist i = 1 . . . ν positive definite con-stant matrices P(j), such that the following linear matrix inequalities(LMIs) are satisfied for all i = 1 . . . ν and j = −1, 0, 1:

A(θ(i))TP(i+ j)A(θ(i))−P(i) < 0A(θ(i+ 1))TP(i+ j)A(θ(i+ 1))−P(i) < 0

(7.22)

then the system is uniformly asymptotically stable for all time-varying realization of the parameter θ satisfying constraints on therange and rate of the parameter variation. Due to the notation simpli-fication, the first LMI of the first interval (i = 1 and j = −1) and thelast LMI of the last interval (i = ν and j = 1) are not valid and shouldbe removed.

182

3. PICK-AND-PLACE MACHINE: MODELING DETAILS AND CONTROLSYSTEM

3. Pick-and-Place Machine: Modeling Detailsand Control System

The proposed design methodology is applied to an industrial 3-axispick-and-place assembly robot with a gripper carried by a flexible beam(Fig. 7.1). The fast movements of this machine may excite the vi-brations of the variable-length flexible beam of which the first reso-nance frequency varies between 30 and 70Hz. The Z-motion is gantrydriven by two linear motors and the X-motion over the carriage is alsodriven by a linear motor. The vertical Y-motion is actuated by a ro-tary brushless DC-motor which drives the vertical flexible beam by aball screw/nut combination. The position of the linear motors and thebeam length are measured using optical encoders, and the accelerationat the gripper in the X-direction is measured using an accelerometer.The objective is to move the gripper as accurately and fast as possiblealong a prescribed trajectory in the working area.

3.1 Mechanical Model

A flexible multibody model has been built to simulate the pick-and-place robot motion in X and Y-directions (see Fig. 7.6). All componentsare modeled as rigid bodies, except for the flexible beam. The actuatorforce, generated by the linear motor (X-direction), is applied to thelinear motor mass (action) and to the carriage (reaction). The framesand the carriage masses are, respectively, 169.0 and 13.9Kg. The lin-ear motor weights 25.9Kg and the gripper 1.25Kg. The spring stiffnessand the damping value between the carriage and the frame are, respec-tively, K1 = 9.15e6N/m and D1 = 1042Ns/m. The frame suspension isconnected to the ground by four connecting points. The stiffness andthe damping of these connections are, respectively, K2 = 5.3e7N/mand D2 = 5204Ns/m. The damping D3 = 100Ns/m represents theconnection between the linear motor and the carriage. The flexiblebeam has a nominal diameter of 24mm. The material properties are:density ρs = 7800kg/m3, Poisson’s ratio ν=0.3, damping ratio 0.01 andelasticity modulus E = 2.1 · 1011N/m2.

The modeling of serial machines requires the description of transla-tional movements between flexible components which can be modeledusing sliding joints. Most of the commercial multibody software candeal with translational movements between rigid bodies but not be-

183


flexible beam

gripper

sliding joint

motor

M

M

frameM

carriageM

1K 1,D

1K 1,D

2K 2,D

2 2,D

2K 2,D

2K 2,D

3D

x

y

z

-FmotorFmotor

frameM

K

Figure 7.6: Scheme of the flexible multibody model of theX-direction motion of pick-and-place machine

tween flexible ones. Therefore, a sliding joint has been derived andimplemented in the Oofelie finite element software. Its formulation isdescribed in Annex A. Between the motor and the flexible beam, thereis a sliding joint which allows the Y-direction movement of the flexiblebeam.

The model-order reduction technique described in Section 2.2 hasbeen applied to this flexible multibody model, reducing it from about50 dofs, to a state-space model with 8 states (1 rigid-body mode and3 flexible modes kept during the model reduction), 1 input (the mo-tor force) and 2 outputs (the motor position and the gripper posi-tion). During the machine design, physical prototypes may not beavailable; therefore, the accuracy of the reduced model can be checkedcomparing the FRFs provided by the reduced model and the full ini-tial model. In this work, experimental measurements were available;therefore, comparisons between the experimental and the simulated(reduced model) FRFs are performed in order to check the accuracy ofthe reduced model. Figure 7.7 shows these comparisons for two beamlengths (l = 0.41m with the first resonance frequency at 45Hz andl = 0.36m with the first resonance frequency at 56Hz). The curvesare in good agreement. Differences in low frequency are mainly dueto the lack of friction in the modeling. Differences in high frequency

184


are mainly due to a mismatch in the sensor position; in the model, thesensor is placed on the end of the flexible beam and in the set-up, it isplaced on the gripper.

102

103

10−8

10−6

10−4

Mag

nitu

de [m

/N]

102

103

−200

−150

−100

−50

0

Pha

se [d

egre

e]

Frequency [rad/s]

102

103

10−4

10−2

100

102

Mag

nitu

de [1

/Kg]

102

103

−600

−400

−200

0P

hase

[deg

ree]

Frequency [rad/s]

Figure 7.7: Comparison between the simulated (- -) and theexperimental FRFs (–): (a) Motor position/Motor force (b) Gripper

acceleration/Motor force

3.2 Control System

A gain-scheduling controller is derived for the pick-and-place modelconsidering the length of the flexible beam as the varying parameter,θ = l. The augmented plant (Pa in Fig. 7.3), is shown in Fig. 7.8. Thedesign of the weighting functions reflects the desired control specifica-tions according to the guidelines presented in Section 2.3.

The frequency-domain metrics, used in the definition of the weight-ing functions Eqs. 7.14, 7.15 and 7.16, are derived based on 2nd ordersystem with the closed-loop bandwidth fBW=20Hz and the settling-time ts=0.15s. The complementary sensitivity function for a conven-tional 2nd order system can be described by [31]:

T (s) =(2πfBW )2

s2 + 2ζ1(2πfBW )s+ (2πfBW )2(7.23)

185


Plant P

Encoder

GripperPosition

W1

-

+

-

+

mS

-1

+

+

W2

w

y

z

u

(m r )SPT

-1

mS

-1

mSP

-1

Pa

Figure 7.8: Augmented plant: w are the reference signals anddisturbances, z are the signals to be minimized, y are the control

signal output and u are the control signal input

where the system damping ratio ζ1 can be estimated by:

ζ1 =− ln(0.02)ts(2πfBW )

(7.24)

Considering a second-order system with settling-time ts=0.15s anddamping ratio ζ1 = 0.21, the peak of sensitivity and complemen-tary sensitivity are mT=2.46 and mS=2.64, respectively. Therefore,reasonable desired values of mT and mS should be 1< mT <2.46and 1< mS <2.64. For this work, the desired values were chosenmT=mS=1.5. The maximum of the process sensitivity is chosen tobe mSP

=0.0001, a rather small value. The weighting factor rSP=0.5

guarantees satisfactory response quality.The H∞-control derivation is performed for four local configura-

tions yielding four linear-time invariant (LTI) controllers. The result-ing controllers (see Fig. 7.9) are then interpolated via the methodologydescribed in Section 2.4 yielding a gain-scheduling controller.

Figure 7.10 shows the response of the full flexible multibody model(about 50 dofs), described in Section 3.1, and its gain-scheduling con-troller. The desired motion in X-direction is a pulse train (Displace-ment in X shown in Fig. 7.10) which is applied when the length of

186


101

102

103

103

104

105

106

Mag

nitu

de [N

/m]

(a)

101

102

103

−200

−100

0

100

200

Pha

se [d

egre

e]

Frequency [rad/s]

101

102

103

102

104

106

108

Mag

nitu

de [N

/m]

(b)

101

102

103

−200

−100

0

100

200

Pha

se [d

egre

e]

Frequency [rad/s]

Figure 7.9: Local LTI controllers for four equidistant values ofl ∈ [0.33, 0.53]m

the beam is continuously varying from 0.33 to 0.53m yielding a totalmovement of 0.2m (D. in Y shown in Fig. 7.10). The gain-schedulingcontroller adapts its gains according to the beam length. It can be ob-served that the vibrations are quite well damped throughout the wholeconfiguration space.

For a given configuration and controller, a closed-loop state-spacemodel can be derived. This state-space model is then discretized con-sidering a sampling time of 1/2048s. In order to guarantee that thesystem is uniformly asymptotically stable for a parameter variationbetween 0.33 and 0.53m and rate bounded by 10.0m/s, the parameterspace should be discretized in ν = 40 intervals (θ(i+1)−θ(i) < 0.005).The feasibility problem described by the LMIs (Eq. 7.22) is verified us-ing the LMI toolbox available in [17].

187


0 0.2 0.4 0.6 0.8 1 1.2

−5

0

5

10

15

x 10−3

Dis

plac

emen

t in

X [m

]

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

time [s]

D. i

n Y

[m]

Figure 7.10: Active system response to a pulse train in X-directionwhile the length of the beam is continuously varying

4. Pick-and-Place Machine: Integrated Design

The final aim of this work is to demonstrate the potential of the de-scribed methodology to design along the lines of a mechatronic designapproach.

Due to the parametric nature of flexible multibody models, struc-tural modifications, e.g. modifications on dimensions, stiffness and ma-terial characteristics, can be easily implemented. On the other hand,the frequency-domain requirements (fBW , mS , mT and mSP

), usuallyremain the same since they reflect the desired system performance in-dependent of its mechanical components.

For the considered case study, structural modifications are evalu-ated in frequency and time-domain. The diameter of the beam has adirect influence on the vibration of the end-effector and it is thus consid-ered as a design variable. Four case studies with different diameters areconsidered (0.020, 0.024, 0.028, 0.032m), in an attempt to eventuallyreflect commercially available beam diameters. For each case study, again-scheduling controller is derived, using the same frequency-domainrequirements described in Section 3.2. Table 7.1 shows the bounds ofthe first resonance frequency for each case study; and the mean, µ(γ),and the standard deviation, σ(γ), of the H∞ closed-loop norm, γ, for

188

5. CONCLUSIONS

Table 7.1: 1st resonance frequency bounds and H∞ closed-loop normfor each case study

diameter 1st resonance frequency H∞ closed-loop norm[m] minimum [Hz] maximum [Hz] µ(γ) σ(γ)

0.020 22 50 1.16 0.200.024 30 70 1.02 0.070.028 41 92 0.98 0.040.032 51 116 0.95 0.02

ten equidistant closed-loop systems. The second and the third reso-nance frequencies remain the same for all case studies since they arerelated to the carriage, the frame and their suspension.

The lower the first resonance frequency, the higher the H∞ closed-loop norm. The frequency-domain requirements are only fully fulfilledin the whole configuration space by the larger diameter. These re-sults are quite intuitive: larger diameters present less vibrations. Theadvantage of using the aforementioned methodology is to enable thedirect access not only to the qualitative behavior but also to quantita-tive metrics (e.g. H∞ closed-loop norm, overshoot, settling-time, etc.).Therefore, the designer can carry out a tradeoff study considering thedesign requirements and the implementation costs.

Time-domain metrics such as overshoot and settling time, canbe evaluated in the whole configuration space using the initial flexi-ble multibody model and its embedded gain-scheduling controller (ford=0.024m see Fig. 7.10). The maximum overshoot and maximumsettling-time happen when the length of the beam is maximum and,therefore, the resonance frequency is lower. Table 7.2 shows the max-imum overshoot and maximum settling time for each case study. Thevalues quantify the intuitive conclusion that smaller diameters presentworse performance than larger diameters.

5. Conclusions

A simulation platform and control design guidelines for the systematicdesign and evaluation of mechatronic systems with varying dynamicshas been proposed and implemented for a pick-and-place robot.

The proposed methodology relies on the derivation of a concise de-

189


Table 7.2: Time-domain metrics for each case studydiameter time domain

[m] maximum overshoot [%] maximum settling-time [s]0.020 62 0.250.024 60 0.200.028 56 0.180.032 55 0.15

scription of the passive system that is used during the model-basedcontrol design. To accomplish this, a model-order reduction technique,based on Global Modal Parametrization, is employed on a flexiblemultibody model. The good agreement between the simulated reducedmodel and experimental FRFs show that the model-order reductiontechnique is able to provide concise and fairly accurate models.

A systematic controller design framework is proposed. A gain-scheduling controller, based on the interpolation of local H∞ con-trollers, is derived and the closed-loop stability is eventually checked.The chosen control design scheme for deriving the local H∞ controllers,the four-block H∞-control problem, allows the specification of somefrequency-domain requirements explicitly.

Keeping the same frequency-domain requirements, the closed-loopperformance of different machine designs can be evaluated in frequencyand time-domain. Therefore, using the proposed methodology, thedesigner is able to predict the machine dynamics and the control actionsin order to evaluate the performance of active machines with varyingdynamics.

Acknowledgements

The research of Maıra M. da Silva is supported by CAPES, BrazilianFoundation Coordination for the Improvement of Higher EducationPersonnel, which she is gratefully acknowledged. Olivier Bruls is sup-ported by the Belgian National Fund for Scientific Research (FNRS)which is gratefully acknowledged.

190

ANNEX A

Annex A

For modeling of the pick-and-place robot, a specific element has beendeveloped for the sliding joint connecting the flexible beam to the linearmotor. Figure 7.11 shows a scheme of the linear motor (rigid body)and the flexible beam (flexible bode) in two configurations illustratingtheir behavior during the translational motion in Y-direction.

Node J

x

y

z

linear motor

flexible beam

Node 0

Node 1Node 0

Node 1

L

Figure 7.11: Scheme of the translational movement between thelinear motor and the flexible beam

According to the Timoshenko theory, displacements r and rotationsψ are treated as independent fields in the beam. In a single elementwith two nodes n0 and n1, an arbitrary point can be represented bythe non-dimensional coordinate along the beam η ∈ [0, 1]. For linearshape functions and under the assumption of small rotations, the posi-tions and orientations of this point are expressed in terms of the nodalcoordinates

r(η) = (1− η)r0 + ηr1

ψ(η) = (1− η)ψ0 + ηψ1

Using again the small rotations assumption, the axis of the beam andof the sliding joint are close to the y-axis. If the linear motor is rep-resented by a node J , the sliding joint is thus modeled by five kine-matic constraints between the nodal coordinates r0 = [x0 y0 z0]T ,r1 = [x1 y1 z1]T , rJ = [xJ yJ zJ ]T , ψ1, ψ2 and ψJ

Φ1 = (1− η)x0 + ηx1 − xJ = 0Φ2 = (1− η)z0 + ηz1 − zJ = 0

Φ3,4,5 = (1− η)ψ0 + ηψ1 −ψJ = 0

191


where η is computed as η = (y0 − yJ)/L, with L, the total length ofthe beam element.

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[31] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control: Analysis and Design, Wiley Publishers, England, 1997.

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[32] B. Paijmans, Interpolating gain-scheduling control for mecha-tronic systems with parameter-dependent dynamics. PhD thesis:Katholieke Universiteit Leuven; 2007.

[33] F. Amato, M. Mattei, A. Pironti, Gain scheduled control fordiscrete-time systems depending on bounded rate parameters. In-ternational Journal of Robust and Nonlinear Control 15 (2005) 473–494.

195

Chapter 8

Computer-AidedIntegrated Design ofParallel KinematicMachines

Maıra M. da SilvaJan SweversJ. De CaignyOlivier BrulsM. MichelinCedric BaradatOlivier TempierW. DesmetH. Van Brussel

197


Paper submitted to Mechanism and Machine Theory as a shortcommunication:

M.M. da Silva, Jan Swevers, J. De Caigny, Olivier Bruls, M.Michelin, Cedric Baradat, Olivier Tempier, W. Desmet, H. VanBrussel, Computer-Aided Integrated Design of Parallel KinematicMachines, Mechanism and Machine Theory.

Comments to the reader : This short communication extends thework presented in Chapter 7. The modeling procedure and modelreduction technique theory, which are described in detail in Sections2.1 and 2.2 in Chapter 7, are summarized in Section 2.1 and 2.2.


• The value of the experimental results has been questioned. In-deed, the improvement of the feedforward signal is rather limited(50%), because of some model mismatches.

198

Abstract

This paper discusses the integrated design of parallel machine tools,which exhibit varying dynamics. This characteristic affects the ma-chine stability and performance. A computer-aided integrated designmethodology is proposed and validated on a pick-and-place robot. Thedesign methodology consists of two main steps: (i) the synthesis ofreduced-order models suitable for control design from flexible multi-body models and (ii) the systematic guaranteed optimal tracking con-trol design based on a feedforward signal. Eventually, the integrated de-sign of the system, considering structural modifications, is performed.

199


1. Introduction

Mechatronic design deals with the integrated design of mechanical sys-tems and their embedded controllers [1-4]. A computer-aided inte-grated design platform, which enables the direct assessment of struc-tural and control parameters via simulation, has been proposed in [4].This tool allows dynamic machine modeling, control system design andclosed-loop system evaluation through simulation using specified vali-dated tests. This platform has been used for the computer-aided inte-grated design of an industrial 3-axis serial pick-and-place machine witha flexible gripper arm and its robust feedback controller in [4]. Thispaper presents the computer-aided integrated design of a two-degree-of-freedom (2-dof) parallel kinematic pick-and-place machine and itstracking controller. In this way, the design of both serial and parallelkinematic machines can be performed using the proposed simulationtool. The case study in this paper is the prototype Par2, a 2-dof pick-and-place machine, which has been designed and built by FatronikFrance [5]. The prototype is depicted in Fig. 8.1a and a scheme show-ing the actuators and elements is illustrated in Fig. 8.1b.

Both serial and parallel kinematic machines present configuration-dependent dynamics which inevitably affect the stability and the per-formance of the system [1,6-8]. The correct modeling of the varying dy-namical behavior and the control design are essential steps in computer-aided integrated design of such machines. The dynamic modeling andcontrol of PKMs have been treated in several recent references [9-11].For model-based feedback control design, dynamic models of low com-plexity, including the rigid body and hydraulics dynamics, have beenderived in [9]. Using the Lagrangian formalism, [10] has derived explicitequations of motion for parallel robots. A simplified dynamic model ofa Stewart platform-based parallel kinematic machine has been used toderive a model-based feedforward controller in [11]. According to [11],model-based control is an essential step to maximize the performanceof PKMs, but the research on this topic is insufficient. The main reasonfor that is the difficulty to derive reasonably sized sufficiently accuratemodels of PKMs.

This paper describes systematic procedures: (i) for deriving re-duced models, suitable for model-based control design, extracted from

200

2. MODELING AND CONTROL OF MECHATRONIC SYSTEMS WITHVARYING DYNAMICS

(a)

yz

x

Motor 1

Motor 2

Active arms

Passive arms

Platform

Couplingsystem

Spherical joints

q2

q1

(b)

Figure 8.1: (a) Prototype of the Par2: the case study and (b) ascheme of the Par2: dofs and elements

flexible multibody models and (ii) for deriving a feedforward signal,based on a spline optimization, which can cope with varying dynam-ics. The remainder of the paper is organized as follows. The generalmethodology for modeling the active system and designing the feed-forward controller for machines with varying dynamics is described inSection 2. This methodology is applied to a 2-dof pick-and-place ma-chine (Fig. 8.1a). Section 3 presents its dynamical model, the controldesign and experimental validation. In Section 4, design tradeoffs areevaluated considering the system performance for different structuralconfigurations. Finally, some conclusions are drawn in Section 5.

2. Modeling and control of mechatronic systemswith varying dynamics

A mechatronic system is composed of rigid bodies, flexible bodies, jointsand control units [4]. It can be modeled as a flexible multibody systemcoupled with a control system (see Section 2.1). Flexible multibodymodels usually contain a large number of dofs and, therefore, may beunsuitable for controller design purposes [1], since the complexity of thecontroller is related to the size of the model. In order to derive a concise

201


description of the flexible multibody model, a model-order reduction,based on global modal parameterization (GMP) [12], is applied (seeSection 2.2).

A typical control structure for motion systems is shown in Fig. 8.2,where P represents the plant, Kfb the feedback controller and uffthe feedforward signal. The feedback controller guarantees stabilityand disturbance attenuation, while the feedforward ensures accuratetracking of the reference r. In this work, only the feedforward signaluff design is derived.

y+

_K

++u

u

Pru

fbfb

ff

Figure 8.2: Feedback and feedforward structure

A general framework to synthesize optimal polynomial splines forrigid motion systems has been proposed in [13,14]. This frameworkhas been extended in [15], by the inclusion of discrete-time linear time-invariant state-space system models in the optimization framework. Inthis way, the input of the motion system can be optimized regarding thetrajectory tracking error, the boundary and the bound constraints onthe system input and output. This approach has been extended to treatlinear parameter-varying models (LPV) in [16] and further exploitedto treat multiple-input-multiple-output (MIMO) systems hereafter (seeSection 2.3).

2.1 Flexible multibody systems

Formalisms developed in the field of flexible multibody dynamics ap-pear to be especially suitable for deriving models of mechatronic sys-tems. In particular, the nonlinear finite element approach described in[17] is a general and systematic technique for the medeling and evalu-ation of articulated systems with rigid and flexible components. Thestrongly coupled formulation [4], which is available in the Oofelie finiteelement software [18], has been chosen for the present case study.

202


A flexible multibody system can be described using absolute nodalcoordinates [17]. Hence, each body is represented by a set of nodesand each node has its own translation and rotation coordinates. Thevarious bodies of the system are interconnected by kinematic joints,which impose restrictions on their relative motion. These restrictionscan be formulated as a set of m nonlinear kinematic constraints:

Φ(q, t) = 0 (8.1)

where q represents the nodal coordinates of the model.According to the Lagrange multiplier technique, the formulation of

the constrained equations of motion requires the introduction of a m×1vector of Lagrange multipliers λ. The dynamics of a feedback controllercan be represented by a nonlinear state-space model with state variablesxfb and output variables ufb. Hence, the strongly coupled equationsof the mechatronic system have the general structure [4]:

M(q)q = u(q, q,w, t) + ufb + uff −BTλ (8.2)0 = Φ(q, t) (8.3)

xfb = f(xfb,q, q,w, t) (8.4)ufb = h(xfb,q, q,w, t) (8.5)

Eq. (8.2) represents the dynamic equations of the mechanical system,Eq. (8.3), the kinematic constraints, Eq. (8.4), the controller state equa-tion and Eq. (8.5), the controller output equation. M is the massmatrix, which is not constant in general, u represents the internal, ex-ternal and complementary inertial forces, B = ∂Φ/∂q is the matrix ofconstraint gradients, ufb denotes the actuator forces or torques gener-ated by the feedback control action, uff denotes the actuator forcesor torques generated by the feedforward control action, and w repre-sents the disturbance, noise and reference signals vector. The vector wincludes the reference vector r. Equations (8.2-8.5) are coupled equa-tions of motion and can be solved numerically using an implicit timeintegration scheme [19].

2.2 Model reduction methodology: a brief introduction

In linear structural dynamics, component-mode synthesis (CMS) pro-vides an appropriate solution for the reduction of a finite element

203


model. In CMS, the dynamic behavior of each substructure is for-mulated as a superposition of modal contributions. As described in [4],a more drastic reduction, based on the GMP and including all bodiesand joints, has been proposed for flexible multibody systems [12].

The total motion q in a flexible multibody system can be decom-posed into rigid motion qr and elastic deformation qf in the followingway

q = qr + qf (8.6)

Considering the passive flexible multibody system (Eqs. 8.2-8.3)and the augmented coordinates qa = [q λ]T , the GMP is defined asthe following mapping

(θ, δ) 7−→[

qλ

]=[ρ(θ)

0

]+[

Ψqδ(θ)Ψλδ(θ)

]δ (8.7)

where θ are the independent parameters related to the actuation, ρ isthe mapping between the rigid motion and the independent parameters,qr = ρ(θ), δ are the modal coordinates, and the Ψqδ and Ψλδ are theflexible mode shape matrices which depend on the configuration.

The dofs can be rearranged in qa = [θ qg qia]T , where θ are the

independent parameters (they should be kept since they represent theactuators), qg are the constraint dofs (they should be kept in case ad-ditional external loads are required) and qia are the remaining internaldofs including the Lagrange multipliers (they can be condensed dur-ing the reduction procedure). Accordingly, rigid modes Ψθ, constraintmodes Ψγ and internal modes [Ψι Ψε], divided into lower and higher-order modes, can be calculated (details in [4,12]). The model reductionrelies on a truncation of the higher-order internal modes.

Performing the modal transformation, q = Ψη, where η = [θ δ]T

and Ψ = [Ψθ Ψγ Ψι], the equations of motion (Eqs. 8.2-8.3, uff =0) yield the reduced model:

Mηη(θ)η + Cηη(θ)η + Kηη(θ)η = uη(θ) (8.8)

where uη(θ) denotes the actuator forces, which includes uff and ufb.Mηη, Cηη and Kηη are the reduced mass, damping and stiffness matri-ces, respectively. Equation (8.8) can be rewritten in state-space form:

204


x =[

0 I−(Mηη(θ))−1Kηη(θ) −(Mηη(θ))−1Cηη(θ)

]x+

+[

0−(Mηη(θ))−1

]u (8.9)

y = Lq = L[

Ψ 0]x (8.10)

where x = [η η]T and the matrix L determines which states (or acombination of the states) are going to populate the vector y.

For a given configuration θk, this equation defines a low-order lin-earized continuous-time state-space model. With a zero-order hold onthe input, this model can be converted to a discrete-time one, yieldingxk+1 = Akxk + Bkuk and yk = Ckxk + Dkuk, and used for controldesign, as described in Section 2.3.

2.3 Spline-based feedforward for trajectory tracking

The objective is to design a smooth feedforward signal uff over a totalsimulation time tK , such that the error between plant output yk and thereference trajectory rk is kept below a certain threshold. The subindexk is related to the time step, which can vary from 0 to K. The sampleperiod of the discrete system is Ts = tK/K. Since only the feedforwardsignal is derived, uff is referred to as u hereafter.

In order to ensure smoothness and to describe the optimizationproblem as a linear program, [15] has proposed a piecewise-linear con-tinuous parametrization of the M-th derivative of the feedforward signalu(M)(t), using B-splines of order 2:

u(M)(t) =K∑k=0

u(M)k βk(t) (8.11)

where βk(t) denotes a B-spline of order 2 (see [15]). In this way, u(t) isa polynomial spline of order M+2. In order to define this polynomialspline, Eq. 8.11 needs to be integrated M times, yielding K+M+1independent variables.

In order to guarantee the feedforward signal smoothness, two cri-teria are evaluated: the infinity norm of u(M+1) and the one norm ofu(M+2). The former keeps the maximum absolute value of u(M+1) small

205


and the latter penalizes erratic and sharp behavior. Both criteria canbe described as linear goal functions augmented with constraints [20].

The linear optimization program yields [15]:

minu,w,v,x,y

w + γ

K−1∑k=1

vk (8.12)

−w ≤ u(M+1)k ≤ w , k = 1 . . .K (8.13)

−vk ≤ u(M+2)k ≤ vk , k = 1 . . .K − 1 (8.14)

u(m)0 = 0; u(m)

K = 0 , 0 ≤ m ≤M (8.15)−u ≤ uk ≤ u , k = 0 . . .K (8.16)

xk+1 = Akxk + Bkuk , k = 0 . . .K − 1 (8.17)yk = Ckxk + Dkuk , k = 0 . . .K (8.18)

x0 = x0 (8.19)−ε ≤ rk − yk ≤ ε , k = 0 . . .K (8.20)

where u is the feedforward to be designed, w and v are the variablesto be minimized in the goal function, which are related to the lineargoal functions that guarantee the signal smoothness, x and y are thesystem state and the output variables at every time step. Values ofγ ∈ [0.001, 0.1] avoid erratic and sharp signals. Inequalities 8.13 and8.14 describe the linear goals related to the infinity norm of u(M+1)

and the one norm of u(M+2), respectively. Equation 8.15 and Ineq.8.16 specify the boundary conditions and the maximum value of thefeedforward signal. Equations 8.17, 8.18 and 8.19 are related to thesystem model, which may present varying dynamics, and its initialconditions. The state-space matrices can be adapted at every timestep; therefore, coping with the simulation of systems with varyingdynamics. It is important to highlight that in order to obtain a linearoptimization problem, the variation of the state-space matrices needsto be known in advance, so that Eqs. 8.17 and 8.18 are linear in theoptimization variables. This is usually the case in trajectory planningapplications, such as described in this paper. And finally, Eq. 8.20guarantees that the tracking error is below a threshold.

206

3. PICK-AND-PLACE PARALLEL MACHINE: MODELING DETAILS ANDCONTROL DESIGN

3. Pick-and-place parallel machine: modelingdetails and control design

The proposed design methodology is applied to a 2-dof parallel kine-matic pick-and-place machine and its feedforward controller (Fig. 8.1a).Two servo motors drive the active inner arms (see Figs. 8.1b and 8.4a)and the platform moves in the x-z plane. In order to guarantee thisplanar motion, the rotations of the passive inner arms are coupled bybelts. The motor positions are measured by encoders (see θ1 and θ2

in Fig. 8.4a). The outer arms are connected by spherical joints. Thearms are made of carbon fiber and the platform of aluminium. Moredetails on this prototype and its design can be found in [5].

The objective is to move the platform according to a pre-definedtrajectory, shown in Fig. 8.3, reaching accelerations of 300m/s2. Thetrajectory starts from the point (0.35,-0.70) and goes to point (-.035,-0.70). The required motor measurements for the nominal case study(Case 1 in Table 7.1) are also shown in Fig. 8.3. This trajectory isconsidered when evaluating the integrated design in Section 4.

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−0.7

−0.69

−0.68

−0.67

−0.66

x−direction displacement [m]

z−di

r. d

isp.

[m]

0 0.05 0.1 0.15−0.4

−0.2

0

0.2

0.4

time [s]

x−di

r. d

isp.

[m]

0 0.05 0.1 0.15−0.71

−0.7

−0.69

−0.68

−0.67

time [s]

z−di

r. d

isp.

[m]

Figure 8.3: Pre-defined trajectory

207


3.1 Mechanical model

A flexible multibody model has been built to simulate the pick-and-place robot motion. The arms are modeled using beam elements andthe platform as a rigid body. The present prototype has the followingdimensions: the length of the active and passive inner arms is 0.39m;the length of the active and passive outer arms is 0.87m; the distancebetween the motors is 0.35m and the platform measures 0.10×0.085m(xy-directions). The platform, the inner arms and the outer armsweigh 0.50, 0.82 and 0.14kg, respectively. The inertia of each mo-tor is 0.19kgm2. The beam stiffness has been adjusted such that thefirst two resonance frequencies and mode shapes match experimentalresults (errors < 1%). The first two mode shapes of the system, in aspecific configuration, are illustrated in Figs. 8.4b and 8.4c. The valuesof resonance frequency are omitted for confidentiality reasons.

(a) (b)

yz

x

motor 1

motor 2

inner arms

outer arms

(c)

platform

q2

q1

Figure 8.4: (a) Scheme of the Par2: the inner and outer arms, (b)mode shape related to the 1st resonance frequency and (c) mode

shape related to the 2nd resonance frequency

The full flexible multibody model, which contains more than thou-sand dofs, cannot be used for model-based control design. For this rea-son, reduced models have been derived using the technique describedin Section 2.2. The reduced models contain 8 states, 2 inputs (the mo-tors torques) and 4 outputs (2 encoder measurements and the platformposition in x- and z-directions).

In order to evaluate the accuracy of the reduced model, a com-

208

3. PICK-AND-PLACE PARALLEL MACHINE: MODELING DETAILS ANDCONTROL DESIGN

parison between time-domain responses (encoder measurements) is de-picted in Fig. 8.5. The torques, also shown in Fig. 8.5, have beenapplied in both full and reduced models. For every time step (every0.001s), a reduced model is extracted from the full flexible multibodymodel at the desired configuration and discretized (sampling frequencyis 1000Hz). Considering a simulation time of 0.155s, 155 reduced mod-els are extracted, discretized and evaluated during the simulation. Thegood agreement between the responses demonstrates that the modelreduction procedure is adequate for deriving accurate and reasonablysized models for this case study.

0 0.05 0.1 0.150

20

40

60

80

time [s]

Enc

oder

[o ]

full model − θ

1

full model − θ2

red. model − θ1

red. model − θ2

0 0.05 0.1 0.15−200

0

200

time [s]

Tor

que

[Nm

]

Motor 1Motor 2

Figure 8.5: Comparison between the full flexible multibody modeland the reduced model

3.2 Validation of the control approach

In order to experimentally validate the control design methodology, arectilinear trajectory in x-direction is selected. The x-direction accel-eration can reach up to 200m/s2. Based on this trajectory, the desiredencoder measurements can be calculated. The desired motor measure-ments (the references r in Eq. 8.20), are shown in Fig. 8.6. Firstly,reduced models have been extracted from the full flexible multibody

209


model for each configuration involved during the trajectory evaluation.Secondly, based on these reduced models, a feedforward controller, theinput signals u, has been derived using the model-based control designdefined by Eqs. 8.12-8.20.

It is desired that the order of the polynomial describing the in-put signals u(t) is at least 4, yielding M=2. The total simulationtime is tK=0.4s, the sample period of the discrete system is Ts=0.001and K=400. A reduced model has been extracted for every time step,yielding 400 reduced models. These models are used for evaluating thesystem dynamics (Eqs. 8.17-8.18). The parameter γ=0.1 is selectedavoiding erratic behavior. The initial states are null (x0 = 0 in Eq.8.19) and the initial and final values of the inputs u and their deriva-tives are null (u(0,1,2)

0 = 0 and u(0,1,2)K = 0 in Eq. 8.15). The maximum

torque is u=300Nm (Eq. 8.15). The tracking error between the ref-erences rk and the encoder measurements yk should be smaller than0.025 (ε=0.025 in Eq. 8.20). The solution of this optimization problemconsists of the state variables xk, w, v, the feedforward signals uk, thesystem outputs yk and the tracking error rk − yk in all time steps (k= 0. . .K). This optimization problem is modeled in Matlab using theYalmip toolbox [21] and solved using Mosek, a commercial code forsolving large-scale convex problems.

The input signals u have been used as feedforward signal in an ex-perimental campaign. The references r, the encoder measurements (seeθ1 and θ2 in Fig. 8.6) and the error between these signals are depictedin Fig. 8.6. Two strategies have been experimentally compared: thefeedback strategy and the combined feedback and feedforward strat-egy. The same feedback algorithm (PID) and gains are applied forboth strategies. The feedback signals are the encoder measurements.The gains have been tuned empirically and are not optimal. However,this non-optimality is not an issue, since the main objective of this com-parison is to evaluate the feasibility and performance of the proposedcontrol strategy.

As it can be concluded from Fig. 8.6, the feedforward signal hasimproved the tracking behavior, reducing the error by 50%. In practice,the errors presented by the proposed control strategy are still large1.The main reason for this fact is the model mismatches2. Therefore,

1The present machine accuracy is higher than the ones shown in Fig. 8.6, due tothe different feedback algorithm approach (see [5]).

2The present machine had been modified before the experimental campaign.

210

4. PICK-AND-PLACE MACHINE: INTEGRATED DESIGN

in spite of the errors presented by both strategies, the achieved errorreduction promoted by the feedforward signal is promising and will befurther investigated.

0 0.1 0.2 0.3 0.40

20

40

60

80

θ 1 [o ]

0 0.1 0.2 0.3 0.40

20

40

60

80

θ 2 [o ]

0 0.1 0.2 0.3 0.4−5

0

5

Err

or [o ]

time(s)0 0.1 0.2 0.3 0.4

−5

0

5

Err

or [o ]

time [s]

reference feedback strategy feedforward + feedback strategy

Figure 8.6: Comparison between the two control strategies: feedbackand combined feedback and feedforward strategies

4. Pick-and-Place Machine: Integrated Design

In order to illustrate the potential of this design platform, the sys-tem design has been re-evaluated considering structural parametersand control design objectives and constraints. The design objective isto cope with a pre-defined trajectory depicted in Fig. 8.3. Some casestudies, described in Table 8.1, are evaluated considering as structuralvariables the length of the arms. The required encoder measurementsare derived for each case study depending on its geometry.

Nearly all optimization parameters described in the previous sectionare considered for this problem. The total simulation time is the onlyparameter that has changed. For this trajectory, the total simulationtime is tK=0.155s, the sample period of the discrete system is Ts=0.001and K=155.

The aim is to evaluate the tradeoff between the maximum required

211


torque and the tracking error, shown in Table 8.1. The maximum re-quired torque is defined by the maximum value of uff and the trackingerror is defined as the maximum error between the desired and the cal-culated platform position. Cases 3, 4 and 5 present worse performancesthan the Case 2 (the nominal case). Case 1 presents better trackingbehavior but requires more torque. It can be concluded from Table 8.1that modifications on the length of the arms cause major changes in thesystem performance. In this way, using the proposed simulation plat-form, design tradeoffs of PKMs can be evaluated quantitatively duringthe design phase.

Table 8.1: Tracking error and maximum required torque for thedifferent case studies

Set of structural parameters Tracking MaximumCase Active Arms [m] Passive Arms [m] Error Torque] Inner Outer Inner Outer [m] [Nm]1 0.23 1.03 0.23 1.03 0.013 2462 0.39 0.87 0.39 0.87 0.015 2223 0.55 0.71 0.55 0.71 0.019 2224 0.71 0.55 0.71 0.55 0.020 2295 0.87 0.39 0.87 0.39 0.025 334

5. Conclusions

Recently, a simulation platform and control design guidelines for thesystematic design and evaluation of mechatronic systems with varyingdynamics have been proposed in [4]. The aim of this work is to extendand illustrate this approach for designing parallel kinematic machinesand feedforward controllers.

The proposed approach relies on the derivation of a concise descrip-tion of the passive system that is used during the model-based controldesign. To accomplish this, a model-order reduction technique, basedon Global Modal Parametrization, is employed on a flexible multibodymodel. A model-based feedforward signal, based on a spline optimiza-tion, is derived. In order to illustrate the potential of the proposedmethodology towards the integrated design of mechatronic systems

212

ACKNOWLEDGEMENTS

with varying dynamics, some possible machine designs have been eval-uated.

Some improvements can be considered in the proposed approachand will be performed in the future. The feedforward has been de-rived considering the tracking error of the motor encoders because,during the experimental campaign, no measuring system was availableto acquire the platform position. Improving the model accuracy, byincluding the frame model, and deriving the feedforward consideringthe platform tracking error are promising alternatives to enhance thesystem accuracy. Moreover, using this design platform, optimizationscan be carried out enhancing the system performance.

Acknowledgements

This work is the result of a fruitful collaboration between some part-ners involved in the EU project NEXT (IP 011815). The research ofM.M. da Silva has been supported by K.U. Leuven (Research CouncilScholarships programme)and by CAPES, Brazilian Foundation Coor-dination for the Improvement of Higher Education Personnel, whichshe gratefully acknowledges.

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[16] J. De Caigny, B. Demeulenaere, J. De Schutter, J. Swevers, Poly-nomial spline input design for LPV motion systems. Proceedings ofthe 10th IEEE International Workshop on Advanced Motion Con-trol, Trento, Italy, March 26-28, 2008.

[17] M. Geradin, A. Cardona, Flexible multibody dynamics: a finiteelement approach. 1st ed. England: John Wiley & Sons; 2001.

[18] A. Cardona, I. Klapka, M. Geradin, Design of a new finite ele-ment programming environment. Engineering Computations 11(4)(1994) 365-381.

[19] O. Bruls, J.-C. Golinval, The generalized-α method in mecha-tronic applications. Journal of Applied Mathematics and Mechanics86(10) (2006) 748–758.

[20] S. Boyd, L. Vandenbergue, Convex optimization, Cambridge Uni-versity Press, 2004.

[21] J. Lofberg, YALMIP : A Toolbox for Modeling and Optimizationin MATLAB, Proceedings of the CACSD Conference 2004, Taipei,Taiwan (http://control.ee.ethz.ch/ joloef/yalmip.php).

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Part III

Conclusions andBibliography

217

Chapter 9

Conclusions and futureresearch

This chapter summarizes the main achievements and conclusions of thisthesis. It provides also some recommendations for future research.

9.1 Contributions and achievements

Two main challenges have been identified when dealing with the de-sign of mechatronic systems: modeling due to their multidisciplinarynature and optimization due to their non-convex nature. This thesisaims at tackling those challenges. As a result, the following generalcontributions can be identified:

• Guidelines for modeling mechatronic motion systems consideringthe system flexibilities, the system motion and the control action;

• Guidelines for optimizing structural and control parameter of ma-chines with varying dynamics using nested or direct optimizationapproaches;

• The extension of a flexible multibody package to cope with themodeling of serial machines;

• The comparison of different techniques for deriving concise mod-els for mechatronic systems with configuration-dependent dynam-ics;

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9 Conclusions and future research

• The extension of a flexible multibody package to cope not onlywith LTI controllers but also with LPV controllers.

These general guidelines and tools have been used in some casestudies and allowed the following achievements:

• The illustration of the benefits of the mechatronic design whendealing with active control of cavity noise;

• The derivation of parametric flexible multibody models for par-allel and serial pick-and-place robots including linear parameter-varying control system actions;

• The evaluation of system performance along the lines of themechatronic design using multi-objective methodologies and al-gorithms;

• The employment of the nested and direct optimization for de-signing a serial pick-and-place robot concurrently with a feedbackcontroller;

• The mechatronic design of a parallel pick-and-place robot and afeedforward controller.

9.2 Main conclusions

9.2.1 Modeling of mechatronic motion systems

A model that represents a mechatronic system should contain the flex-ibility of some components, the system motion, and the controllers.This task is not straightforward, since there is still a gap between sim-ulation software used for evaluation of mechanical systems and softwareused for control design. Mechanical engineers are used to finite-elementand multibody packages to evaluate the dynamic properties of mechan-ical systems. On the other hand, control engineers are used to blockdiagram tools that do not support directly the mechatronic design pro-cess since transfer functions or state-space descriptions might have lostthe relation with the physical parameters during the modeling process.Alternatives to model mechatronic systems can be mainly divided intofour methodologies: (1) simulation of an active flexible structure ina finite-element environment, (2) simulation of reduced or simplified

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9.2. MAIN CONCLUSIONS

models and their controllers in a control design environment, (3) co-simulation between dedicated virtual environments, and (4) integratedsimulation of the active flexible system in a unified environment.

The dynamics of parallel and serial kinematic machines are typicallydependent on configuration. For this kind of mechatronic systems,besides the modeling of the flexibilities, the system motion, and thecontrollers; the varying dynamics should be correctly predicted. Threemethodologies have been proposed:

1. The first methodology is based on deriving reduced LTI modelsfrom finite-element models via CMS in different configurations.Using this approach, the machine can be evaluated in any fixedconfiguration. However, this strategy fails to represent the ma-chine during the actual motion, when its dynamics are varying.This is an important drawback which can be overcome by thederivation of an LPV model. LPV models can be derived by in-terpolating a set of reduced LTI models. The interpolation tech-nique fits a linear function on the varying poles, zeros and gainsof the LTI reduced models. This LPV model can be implementedin Matlab/Simulink and, therefore, it can be evaluated with itsembedded controller.

2. The second methodology divides the mechanical system in twosubsystems: (i) a subsystem in which the boundary conditions donot vary and (ii) a subsystem in which the boundary conditionsvary according to the spatial configuration. The subsystem (i)can be modeled using any commercial multibody package. Thesubsystem (ii) can be described by an LPV model implementedin Matlab/Simulink. This LPV model can be derived by the in-terpolation of reduced models extracted from FE models. Bothsubsystems can be interconnected via co-simulation. Since theco-simulation is implemented in Matlab/Simulink, a control de-sign environment, the mechanical system can be evaluated con-currently with its embedded controllers.

3. The third methodology considers the implementation of specialelements in a unified environment, Oofelie. In order to modelthe mechanical system of serial machines; a sliding joint, whichallows the translational movement between flexible bodies, hasbeen implemented in this environment. Moreover, not only LTI

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controllers but also LPV controllers can be included during themodeling of mechatronic systems. Flexible multibody models cancontain a large number of dofs and, therefore, may be unsuitablefor model-based controller design purposes. For this reason, amodel reduction technique for flexible multibody models needsto be considered.

Both parallel and serial machines can be modeled using these ap-proaches. The choice among these options depends on the system tobe modeled, the designer expertise and the available tools. The thirdoption is the most general approach since the structure and the con-troller are simulated in a unified environment and integrated with thesame time-integration scheme. All methodologies allow the derivationof reduced LPV models, which can be employed on the derivation ofLPV controllers.

9.2.2 Integrated design of mechatronic systems

The proposed modeling approaches and model reduction techniqueshave been used in the integrated design of several case studies. Con-clusions can be drawn for each case study:

The vibro-acoustic system: Integrated design, considering struc-tural and control parameters as the firewall thickness, the velocityfeedback gain and the position of the SAP, has been employed forreducing cavity noise in Chapter 4. The first conclusion that canbe drawn is that optimal passive performance systems may haveinferior closed-loop performance. Consequently, an optimal de-sign can only be achieved when considering structure and controlconcurrently.

Considering that an ASAC modeling procedure is a multi-disciplinary assignment, distinct objectives arise from these dis-ciplines. Capturing the design tradeoffs, using for instance thePareto front, can assist the designer to gain better insights intothe problem.

Comparisons between experimental and simulation results for thepassive, sub-optimal and optimal solutions showed good agree-ment confirming the benefits of the proposed concurrent mecha-tronic approach for ASAC design.

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9.2. MAIN CONCLUSIONS

The mixed pick-and-place machine: Both direct and nested de-sign strategies have been exploited in this case study.

As described in Chapter 6, an important issue raises when con-sidering the direct design strategy of mechatronic system withconfiguration-dependent dynamics. Because of the non-convexnature of the direct optimization strategy, an LPV system can-not be described as a polytopic system. Therefore, the infiniteset of inequalities imposed by the parameter variation cannot bereduced to a finite one when designing the controller or evalu-ating the system stability. For this reason, time-domain metricsare selected in this work, which demands time-domain simula-tion and evaluation of the system under study. For the directdesign strategy, a multi-objective optimization framework hasbeen developed based on a general simulation tool for flexiblemultibody systems embedded with nonlinear controllers. The re-sults reveal the benefits of the mechatronic design approach sincethe active system design tradeoffs are identified. The qualitativestatement that the optimal integrated solutions result in thickerdiameters seems to be predictable. Actually, any thicker diame-ter would imply vibration reduction. However, the quantitativeresults achieved in this framework are not that simple to foresee.Using the proposed methodology, one can decide which beam di-ameter should be selected and predict the closed-loop response ofsuch a complex mechatronic system in time-domain. In this way,not only qualitative behavior, but also quantitative metrics, suchas overshoot and settling, can be evaluated over the configurationspace during the design phase.

The nested design strategy relies on the derivation of a concisedescription of the passive system that is used during the model-based control design in Chapter 7. To accomplish this, a model-order reduction technique, based on Global Modal Parametriza-tion, is employed on a flexible multibody model. A systematiccontroller design framework is proposed. A gain-scheduling con-troller, based on the interpolation of local H∞ controllers, is de-rived and the closed-loop stability is eventually checked. Thechosen control design scheme for deriving the local H∞ con-trollers, the four-block H∞-control problem, allows the specifica-tion of some frequency-domain requirements explicitly. Keepingthe same frequency-domain requirements, the closed-loop perfor-

223


mance of different machine designs can be evaluated in frequencyand time-domain. Therefore, using this strategy, the designer isable to predict the machine dynamics and the control actions inorder to evaluate the performance of active machine with varyingdynamics.

Comparisons between direct and nested design strategies are re-ported in the literature for LTI systems. However, no comparisoncan be found about LPV systems. As reported in Chapters 6and 7, different approaches have been employed for deriving thecontroller: time-domain (direct design strategy) and frequency-domain metrics (nested design strategy). In this way, compar-isons between the active system performance are not straightfor-ward.

Tighter performance requirements can be achieved using moreadvanced control strategies, such as the ones derived by the four-block H∞-control problem, which can only be considered whenusing the nested design strategy. Moreover, the direct designstrategy for this case study lacks robustness and may vary if thereference input changes, which are important drawbacks. Be-cause of these reasons, the nested design strategy presents morerobust and reliable results for this case study.

The parallel kinematic pick-and-place machine: Nested designstrategy has been considered to extend the work described inChapter 7 and illustrate the design of parallel kinematic machinesand feedforward controllers.

Model-based control is an essential step to maximize the perfor-mance of PKMs and it is considered in Chapter 8. In spite ofits potential, there is a lack of literature on model-based controldesign for parallel kinematic machines. The main reason for thisis the difficulty in deriving reasonably sized accurate models ofPKMs. This issue is overcome, in this work, by using the pro-posed simulation platform. Concise models have been derivedusing a model-order reduction technique and used for designingmodel-based control. A model-based feedforward signal, based ona spline optimization, is derived and experimentally validated. Inorder to illustrate the potential of the proposed methodology to-wards the integrated design of mechatronic systems with varyingdynamics, some possible machine designs have been evaluated.

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9.3. FUTURE RESEARCH

9.3 Future research

9.3.1 Virtual prototyping for active vibro-acoustic sys-tems

In order to bring active solutions to the level of industrial applica-tions, the designer needs numerical tools that allow the inclusion ofsensors/actuators and control strategy in the virtual product designand optimization. In this way, the design of active solutions for noisereduction should be performed along the lines of a mechatronic designapproach. Moreover, sound quality metrics should be included in thisplatform not only for product quality assessment but also for activesound quality control design.

Given the potential of piezoelectric materials for active control pur-poses, a natural future step, in the work presented in Chapter 4, is tostudy the inclusion of distributed sensors and actuators in the method-ology. As optimization variables, not only the placements and the con-trol gains, but also the shape of the piezo-patches may be considered.In order to accomplish this, the researcher needs to overcome issuessuch as model reduction in multiphysics, multivariable control design,etc.

9.3.2 Model reduction of multibody systems

The selected model reduction technique for multibody system, basedon the Global Modal Parametrization, has been developed to furnishreasonably sized models for model-based control design. The reducedmodels are described in terms of modal coordinates. The changes ofthe modal coordinates for a system following a certain path can onlybe calculated up to a given precision. Therefore, in a simulation loop,the initial and the last modal coordinates may be different yieldingirregular behavior. In this way, this technique should be further inves-tigated in order to provide reduced models not only for model-basedcontrol design but also for reducing the calculation time of multibodysimulations.

9.3.3 Fuzzy finite element method aiding control design

In the fuzzy finite element method, uncertain geometrical, material andloading parameters are treated as fuzzy values. The modeling of uncer-

225


tain parameters as fuzzy values is necessary when it is not possible touniquely and reliably specify these parameters either deterministicallyor stochastically.

Uncertainty is also the main issue in robust control design. Modeluncertainty raises when system gains or other parameters are not pre-cisely known, or can vary over a given range. Examples of real pa-rameter uncertainties include uncertain pole and zero locations anduncertain gains. In the control field, these uncertainties are roughlyestimated. The use of approaches/ideas employed in the fuzzy finiteelement method could be beneficial during robust control design. More-over, these approaches/ideas could be extended also to cope with multi-body systems and related techniques.

9.3.4 Integrated design considering topology optimiza-tion

Topology optimization considering structural-related objectives andconstraints are standard procedures in automotive and aircraft compa-nies. Results on topology optimization of multiphysic systems have alsobeen reported recently. Issues as the amount of optimization variables,optimization algorithms and modeling difficulties have been tackled bythe academic community. Nevertheless, very few results can be foundtreating topology optimization of mechatronic systems.

9.3.5 Integrated design of an industrial application

Milling and turning machines present stability problems generated byperiodic cutting forces acting in the machine structure and chatter vi-brations. The prediction of chatter vibrations is essential as a guidanceto the machine tool user for an optimal selection of depth of cut andspindle rotation, aiming at maximum material removal without vibra-tions.

It is highly desirable to identify these phenomena during the designphase. Moreover, active solutions have been widely employed to controland reduce these issues. The benefits of the integrated design for suchcomplex cases have not been fully investigated.

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244

Part IV

Appendices

245

Appendix A

Review on ModelReduction

Models of dynamic systems are useful primarily for two reasons: (1) forsimulation and (2) for control design purpose. The steps of a standardactive system design are shown in Fig. A.1, as proposed by Preumont(2002). Firstly, a mechanical system, some performance objectives anda specification of the disturbances applied to the system are identified.The choice of the proper type and location for a set of sensors and ac-tuators is the second step. The next step is to develop a model, usuallyusing finite element (FE) method and/or multibody (MBS) techniques.The resulting models can contain a large number of degrees of freedom(dofs) and, therefore, may be unsuitable for model-based controller de-sign purposes, such as pole-placement and H∞ control design, sincethe order of the controller is related to the order of the model. Im-plementing a high-order controller is still a challenging task since thecost of hardware and the time of computation grow proportionally withthe controller order (de Fonseca, 2000; Antoulas and Sorensen, 2000).The following steps regard the implementation and optimization of thecontroller.

Therefore, low-order models, obtained from model reduction proce-dures, are required for controller design in the development of a virtualengineering environment for mechatronic system design. Any impreciseformulation of the reduced or condensed model may affect the predicteddynamic behavior of the whole system.

In the literature, there are several model reduction approaches withrespect to modeling methodologies and/or model representations. For

247

A Review on Model Reduction

Figure A.1: Various steps of control design (Preumont, 2002)

instance, large-order FE models are traditionally reduced to lower-order models either by removing some of the insignificant physical coor-dinates such as in the Guyan techniques (Guyan, 1965) or by evaluatingthe modal contributions of a substructure such as Component-ModeSynthesis (CMS) (Hurty, 1965; Craig, 1987; Hintz, 1975).

In the control design field, the model is usually represented in astate-space form and, if required, its order is reduced via one of the re-duction methods currently employed by the control community such asapproximation by balanced truncation and Hankel norm approximation(Antoulas and Sorensen, 2000). These methods, in essence, approxi-mate a large dynamic system with a fewer number of state variableswhile making minimal change on the input-output characteristics.

A survey on model reduction techniques, with dedicated examplesfor mechatronic motion systems, is presented in this chapter. Thesections are divided according to the employed modeling technique:

248

A.1. PROBLEM STATEMENT

• Model reduction techniques based on modal decomposition aredescribed in the Section A.2. These techniques are generallyemployed by structural engineers when modeling using the FEmethod. In a more general view, they can be employed to re-duce the size of systems represented by a large set of differentialequations.

• Model reduction techniques via approximation-data based meth-ods are described in the Section A.3. These techniques are well-established in the control design field. They can be employed forboth linear and non-linear systems.

• Model reduction via global modal parametrization technique isbriefly described in the Section A.4. This technique is employedin order to reduce the order of flexible multibody systems.

These techniques are repetitively applied throughout the thesis, es-pecially, for the derivation of suitable models for mechatronic motionsystem in Chapters 2 and 3. Due their different nature and usability,the examples are not the same for all techniques. For model reductiontechniques based on modal decomposition and via approximation-databased, the case study is a FE model of a milling machine described inthe Section A.2.1. This case study can not be used for the later modelreduction technique, since the model reduction technique via globalmodal parametrization is a technique developed for flexible multibodymodels. This technique is not exemplified in this chapter, since thistechnique is employed in Chapters 3, 7 and 8. Reasonable sized modelsfor model-based control design from flexible multibody models are de-rived for parallel (Chapter 8) and serial (Chapter 7) kinematic machinesusing the model reduction technique via global modal parametrization.

Firstly, the model reduction problem is formally introduced in Sec-tion A.1.

A.1 Problem Statement

A large number of mechanical systems can be represented by second-order differential equations:

Mq(t) + Dq(t) + Kq(t) = f(t) (A.1)

249


where M is the mass matrix, D the damping matrix, K the stiffnessmatrix, f the external force vector and q the coordinates associatedwith the dofs. A state-space representation of this system can be de-rived considering the state-space variables x = [q q]T .

x =[ q(t)

q(t)

]=[ 0 I−M−1K −M−1D

][ q(t)q(t)

]+[ 0

M−1

]f(t) (A.2)

In this way, a state-space representation can be expressed in matrixform:

x(t) = Ax(t) + Bu(t)y(t) = Cx(t) + Du(t) (A.3)

where A ∈ Rn×n is the state matrix, B ∈ Rn×m the input matrix, C ∈Rp×n the output matrix, D ∈ Rp×m the direct transmission matrix,x ∈ Rn the state vector, y ∈ Rp the output vector and u ∈ Rm theinput vector.

A simplified notation can also be defined by

Ξ =(

A BC D

)(A.4)

where Ξ ∈ R(n+p)×(n+m) represents the system.The application of the Laplace transformation to this linear system,

considering null initial conditions (x(0) = 0), yields the following I/Orelation in the frequency-domain:

Y(s) = (C(sI−A)−1B + D)︸︷︷︸G(s)

U(s) = G(s)u(s) (A.5)

where G(s) is the transfer function matrix.The model reduction problem is to approximate the dynamical sys-

tem, described by Eq. A.3, by a reduced-order system

˙x(t) = Ax(t) + Bu(t)y(t) = Cx(t) + Du(t)

(A.6)

where A ∈ Rr×r, B ∈ Rr×m, C ∈ Rp×r, D ∈ Rp×m, x ∈ Rr, y ∈ Rp

and u ∈ Rm.

250

A.2. MODEL REDUCTION VIA MODAL DECOMPOSITION

The order r n of the system should be such that

‖Y(s)− Y(s)‖ = ‖G(s)U(s)− G(s)U(s)‖‖Y(s)− Y(s)‖ ≤ ‖G(s)− G(s)‖‖U(s)‖ < ε‖U(s)‖

(A.7)

where ε is the error tolerance. Eventually, Eq. A.7 expresses‖y(s) − y(s)‖ < ε‖u(s)‖. Antoulas and Sorensen (2000) have definedthe requirements for a model reduction technique as:

• the derivation of an automatic and computationally efficient pro-cedure to create reduced models;

• the preservation of the stability and passivity; and

• a satisfactory error tolerance.

Model reduction techniques such as modal reduction (Craig, 1987;Koutsovasilis and Beitelschmidt, 2008), balancing and truncation (An-toulas and Sorensen, 2000; Gugercin and Antoulas, 2004), Krylov sub-space methods (Antoulas and Sorensen, 2000; Gragg, 1972; Gragg andLindquist, 1983; Grimme, 1997; Koutsovasilis and Beitelschmidt, 2008),among others are based on the following transformation:

x = Tx (A.8)

where T ∈ Rn×r. Hence, the model order-reduction problem is tofind a suitable transformation to fulfil the requirements (Antoulas andSorensen, 2000). Recently, Koutsovasilis and Beitelschmidt (2008) hascompared different model reduction techniques for a large FE model,indicating that Krylov subspace method may also became a suitabletool for large dynamic systems.

A.2 Model Reduction via Modal Decomposi-tion

The principle of the FE methodology is the subdivision of the wholestructure into individual analytical elements such as beams, platesand shells, which could be assembled to provide the complete set ofequations. Discretization essentially transforms vibration problems de-scribed by partial differential equations into problems described by sets

251


y

z

x

Figure A.2: Case study - Milling Machine (de Fonseca, 2000)

of simultaneous ordinary differential equations. Finally, the structuralequations can be written as Eq. A.1, which can easily reach severalthousands dofs. The following subsections describe a case study andselected model reduction methodologies for FE models.

A.2.1 Case study - Milling Machine

The case study in this section is a three-axis high-speed milling ma-chine for super-finishing of dies and moulds (de Fonseca (2000)). TheFE model, shown in Fig. A.2, has over 30000 dofs. The motion alongthe x-axis, y-axis and z-axis of the machine are, respectively, the lat-eral motion, the frontal motion and the vertical motion of the tool.This machine model has been developed in the framework of the Brite-Euram project KERNEL II (Annex, 1995; de Fonseca, 2000). All thecalculations were performed using MSC/Nastran2004.

The machine can be divided into four components: the ram, theY-axis slide, the X-axis slide, and the base. These components arevisualized in Fig. A.6.

A.2.2 Static Condensation - Guyan Reduction

Guyan reduction technique is often firstly considered for reducing FEmodels. This model reduction methodology eliminates the dofs that

252


are not excited and have no mass. Usually, these dofs are called slavesq2 and the dofs kept are called masters q1 (Preumont, 2002). Neglect-ing the damping, a system derived from the FE methodology can bedescribed as

[ M11 00 0

][ q1

q2

]+[ K11 K12

K21 K22

][ q1

q2

]=[ f1

0

](A.9)

The second line of Eq. A.9 yields the relation q2 = −K−122 K21q1.

In this way, the total number of dofs can be written as

q =[ q1

q2

]=[ I−K−1

22 K21

]︸︷︷︸

T

q1 = Tq1 (A.10)

In order to eliminate the slaves nodes the transformation q = Tq1.The reduced mass and stiffness matrices can be described by M =TTMT and K = TTKT, respectively.

The Guyan reduction methodology was applied to the milling ma-chine model (de Fonseca, 2000). The model of each component: theram, the Y-axis slide, the X-axis slide, and the base; were reducedby using the procedure described hereafter. The reduced models havebeen assembled together in the nominal position (the nominal positioncan be visualized in Fig. A.2).

The differences in the eigenfrequencies between the complete modeland the reduced model are shown in Fig. A.3. In summary, the higherthe eigenfrequency the higher the error. Theses results are quite poorcompared with the dynamic reduction using Component Mode Synthe-sis that is described in Section A.2.4. Guyan reduction is the defaultmodel reduction method in MSC/Nastran2004.

A.2.3 Modal reduction - Model Truncation with StaticCorrection

This modal reduction technique is based on a projection of the dofsin the modal space. It can be used not only for model reduction ofFE models but also for any linear model representation for which themodal characteristics can be extracted.

A general modal transformation can be described by:

253


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

Modes

Eig

enfr

eque

ncie

s E

rror

(%

)

Figure A.3: Eigenvalues error (%) for the reduced model applyingGuyan reduction

q = Tqred (A.11)

where q are the initial dofs, qred are the kept dofs, and T is the re-duction matrix. The following subsections describe how the matrix Tcan be derived for structures with and without rigid-body modes. Thedescriptions are based on Preumont (2002) and Balmes (2004).

A.2.3.1 Structure without rigid body modes

Firstly, a modal transformation, q = Φη, is applied on the equationof motion (Eq. A.1), neglecting the damping. Φ is the modal matrix,which is the matrix of eigenvectors defined by [φ1 φ2 . . .φm]. Pre-multiplying the resulting equation by ΦT , the equation of motion canbe rewritten as

diag(µi)η + diag(µiω2i )η = ΦT f (A.12)

where diag(µi) is the modal mass matrix, ΦTMΦ; diag(µiω2i ) is the

modal stiffness matrix, ΦTKΦ; and η is the vector of modal ampli-tudes: the kept dofs.

254


Considering the system described by Eq. A.1, the steady stateresponse to a harmonic excitation f = Feiωt is q = Qeiωt. Substitutingthe excitation and the response into Eq. A.1, the transfer function,G(ω), can be defined:

Q(ω) = [−ω2M + iω(D) + K]−1F(ω) = G(ω)F(ω) (A.13)

Considering the undamped system, the transfer function matrix,G(ω), can be represented as:

G(ω) = [−ω2M + K]−1 =n∑i=1

φiφTi

µi(ω2i − ω2)

(A.14)

For ω = 0, G(0) is defined as the static flexibility matrix and canbe described as:

G(0) = K−1 =n∑i=1

φiφTi

µiω2i

= Φ(ΦTKΦ)−1ΦT (A.15)

For a limited frequency band ω < ωb, m modes can be selected insuch a way that ωb ωm. In this way, when low-frequency responseis considered, the transfer function can be split into the contributionof the low-frequency modes (i ≤ m), which respond dynamically, andhigh-frequency modes (i > m), which respond statically.

G(ω) ≈m∑i=1

φiφTi

µi(ω2i − ω2)

+n∑

i=m+1

φiφTi

µiω2i

(A.16)

Using Eq. A.15, Eq. A.16 can be rewritten in such a way that thehigh-frequency modes do not appear explicitly:

G(ω) ≈m∑i=1

φiφTi

µi(ω2i − ω2)

+ K−1 −m∑i=1

φiφTi

µiω2i

(A.17)

In the low-frequency band, the static contribution of the high-frequency modes is refereed as to residual mode. Its addition con-tributes to improve the low-frequency response prediction, especially,the lower responses values. This, eventually, helps the correct predic-tion of the system’s anti-resonances.

255


Considering the state-space model definition in Eq.A.3, the reduc-tion matrix T, Eq. A.11, can be written as

T = [φ1 . . . φm K−1B−m∑i=1

CφiφTi Bµiω2

i

] (A.18)

A.2.3.2 Structure with rigid body modes

If the structure has rigid body modes, the displacement can be splitinto rigid and flexible contributions as

η = ηr + ηe = Φrηr + Φeηe (A.19)

where Φr and Φe are the matrices whose columns are the rigid bodymodes and the flexible modes, respectively. In the same manner thatwas done for the structure without rigid body modes, for a limitedfrequency band ω < ωb, m modes can be chosen according to therelation ωb ωm. In this way, the transfer function can be split intothe contribution of the rigid-body modes, the low frequency modes,which respond dynamically, and high-frequency modes, which respondstatically.

G(ω) ≈r∑i=1

φiφTi

−µiω2i

+m∑

i=r+1

φiφTi

µi(ω2i − ω2)

+n∑

i=m+1

φiφTi

µiω2i

(A.20)

The last part of the right side of the Eq. A.20 is called residualmode, R, and can be rewritten as

R =n∑

i=m+1

φiφTi

µiω2i

=n∑

i=r+1

φiφTi

µiω2i

−m∑

i=r+1

φiφTi

µiω2i

(A.21)

And it can be rewritten independent from the high frequencymodes:

R = Φe(ΦTe KΦe)−1 −

m∑i=r+1

φiφTi

µiω2i

(A.22)

In this way, Eq. A.20 can be rewritten independent from the highfrequency modes

256


G(ω) ≈=r∑i=1

φiφTi

−µiω2i

+m∑

i=r+1

φiφTi

µi(ω2i − ω2)

+ R (A.23)

So, using the static truncation with static correction, the reductionmatrix T can be written as

T = [φ1 . . . φr φr+1 . . . φm Φe(ΦTe KΦe)−1B+

−m∑

i=r+1

CφiφTi Bµiω2

i

] (A.24)

Figure A.4 shows a frequency response function of the tool tip ac-celeration response in the x-direction to a unit torque at the x-axismotor of the 51-dofs model of a milling machine (full line) (de Fonseca,2000), of the reduced model with 9-dofs without static contribution(dotted line) and with static contribution (dotted point line). The re-duced model with static reduction represents better the response of the51-dofs models, especially near the anti-resonances.

A.2.4 Component Mode Synthesis (CMS)

CMS is a subtructuring technique where the system equations of motionare firstly obtained for each individual substructure (component), andthen, they are coupled subject to the given constraints (Meirovitch,1980).

Subtructuring is an essential methodology to predict the dynamicbehavior of complex systems, since the dynamic properties of differ-ent subsystems can be coupled yielding the behavior of the whole sys-tem. The dynamics properties of the subsystem can be extracted the-oretically or experimentally by different organizations and/or researchgroups. According to Maia and Montalvao e Silva (1997), the mainsteps involved in a substructuring technique may be described as:

1. partitioning of the whole physical system model into a numberof substructures with a proper choice of connections and interiorcoordinates;

2. derivation of the respective subsystem models, either by a theo-retical or an experimental approach;

257


40 60 80 100 120 140 160 180 200

100

105

Acc

eler

atio

n (m

/s2 )

40 60 80 100 120 140 160 180 2000

50

100

150

200

Frequency (Hz)

Deg

ree

(o )

complete modelred. model with static comp.red. model withot static comp.

Figure A.4: Comparison of frequency response function of acomplete model and reduced models with and without static

contribution

3. formulation of the subsystem equations of motion which are gen-erally made by using physical or modal coordinates and, if pos-sible, without requiring the knowledge of the dynamic propertiesof the remaining components forming the global structure;

4. construction of reduced-order equations for the global structureby invoking interface compatibility and force equilibrium condi-tions previously established for the different components models.

Previous studies have shown that CMS allows a simple combina-tion of substructures with known dynamics characteristics consideringtheir relative position (de Fonseca, 2000). Therefore, it provides anappropriate solution for the reduction of FE models (Craig, 1987) forsystems with configuration-dependent dynamics. Examples and appli-cations are described in Section A.2.4.4 and Chapter 3.

In any CMS technique, the displacements of the physical coordi-nates, the dofs q, are represented in terms of generalized modal coor-

258


dinates η of the components using the classical modal transformation:

q = Ψη (A.25)

where the transformation Ψ consists of pre-selected component modesof the following types: normal modes, rigid body modes, constrainedmodes, attachment modes, inertia relief modes, inertia relief attach-ment modes. Some of these sets are described and their availabilityin FE software MSC.Nastran is briefly discussed hereafter and also inde Fonseca (2000). A more comprehensive overview with strict mathe-matical derivations of the CMS methods is given in Craig (1987). Forthe description of the component modes, the following subscripts’ de-scriptions apply:

q total set of dofs of a component;

i interior coordinates set (see Fig. A.5a);

b boundary coordinates set that are fixed (no motion allowed) duringcomponent normal modes calculation (see Fig. A.5a);

c boundary coordinates set that are free (motion is allowed) duringcomponent normal modes calculation (see Fig. A.5a);

r reference dofs used to determine rigid-body motion (see Fig. A.5b);

e excess boundary coordinates (r is sufficient to restrain the componentfrom rigid body motion - see Fig. A.5b);

g set of dofs used to calculate the constrained modes (defined here-after) and

a set of dofs used to calculate the attachment modes (defined here-after).

The number of dofs related in the various sets of coordinates is designedas nq, ni, nb, etc.

(i) Normal ModesNormal modes, Φq, of an oscillating system are defined as defor-

mation patterns at the eigenfrequencies, ωq. They present a varietyof different shapes depending on which frequency is excited. Normal

259


e i i r

b=r+e

(b)(a)

Boundary coordinates

Interior coordinates

Figure A.5: (a) Definition of the boundary and interior coordinatesdofs and (b) Scheme of the boundary, rigid, excess and interior dofs

(Craig, 1987)

modes and eigenfrequencies are obtained from the eigenproblem of theform:

(K− ω2qM)Φq = 0 (A.26)

where ωq is the qth eigenfrequency and Φq the associated mode shape.The modes may be normalized and assembled as columns of the

modal matrix Φn = [φ1 φ2 . . .φp], where n is denotes normal modes.Component normal modes may be classified as fixed interface

modes, free interface modes, hybrid interface modes, and loaded in-terface modes depending on the boundary condition and loads. Forinstance, the boundary dofs of the fixed interface modes are clampedwhile the boundary dofs of the free interface are free. Hybrid inter-face modes present some free and fixed boundary conditions and loadinterface modes present loaded boundary conditions.

For free-interface modes, Eq. A.26 is employed directly, with andbeing the full component stiffness and mass matrices. Then, Φn hasthe form

Φn =[ Φin

Φbn

](A.27)

where i denotes interior coordinates and b boundary coordinates. Thedefinition of the boundary and interior dofs are shown in Fig. A.5.

260


For fixed-interface modes, Eq. A.26 is applied only to interior co-ordinates, yielding

Φn =[ Φin

0

](A.28)

In a CMS analysis a truncated subset of Φn, referred to as keptnormal modes Φk, is used. These normal modes are used to supplementstatic modes defined below.

Nastran note: MSC/Nastran2004 calculates these normal modeswhen a method and an eigr(l) card is present. The boundary condi-tions are set by specifying the appropriate b- and c-set dofs. The b-setdetermines which nodes will be fixed during the analysis and the c-set which nodes will be free during the analysis. The default methodused in MSC.Nastran when performing a dynamic transformation of asuperelement is the fixed-boundary approach, where all exterior dofsbelong to the b-set of the superelement.

(ii) Rigid-Body ModesThe rigid-body modes are defined by Eq. A.29 based on reference

dofs for r rigid-body modes, e excess boundary dofs and i interior dofs.[ Kii Kie Kir

Kei Kee Ker

Kri Kre Krr

][ Ψir

Ψer

Irr

]=

[ 000

](A.29)

So the rigid-body modal matrix Ψr (np× nr) can be rewritten as

Ψr =

[ Ψir

Ψer

Irr

]=

[−[ Kii Kie

Kei Kee

]−1[ Kir

Ker

]Irr

](A.30)

Nastran note: MSC/Nastran2004 calculates the rigid-body modesusing the dof specified by the support-entry or by normal modes cal-culated from 0 Hz.

(iii) Constraint ModesA constraint mode is defined as the static displacement resulting

from an imposed unit displacement on one dof of the g-set (constraintcoordinates), while the other coordinates of this set are restrained (zerodisplacement). The reaction force is represented by R. The dofs that donot belong to the set g are force-free. In the case that g = e, the modal

261


matrix is called matrix of redundant constraint modes. Constraintmodes have been used to supplement fixed-interface normal modes byCraig (1987). Ψg (np× ng) can be defined by Eqs. A.31 and A.32:[ Kii Kig Kir

Kgi Kgg Kgr

Kri Krg Krr

][ Ψig

Igg0

]=

[ 0Rgg

Rrg

](A.31)

Ψg =

[ Ψig

Igg0

]=

[ −K−1ii Kig

Igg0

](A.32)

Nastran note: It should be noted that the g-set is not related withthe b-set. The boundary conditions are specified by b- and c-set dofswhen MSC/Nastran2004 is calculating the normal modes.

(iv) Attachment ModesAn attachment mode is defined by the static deformation of the

system when applying an unit force at one dof of the a-set, while keep-ing the other nodes of this set force-free. Attachment modes have beenused to supplement free-interface normal modes by Rubin (1975). Inthe case that a = e, the modal matrix of attachment modes. Ψa

(np× na) can be defined by Eqs. A.33 and A.34:[ Kii Kia Kir

Kai Kaa Kar

Kri Kra Krr

][ Ψia

Ψaa

0

]=

[ 0IaaRra

](A.33)

Ψa =

[ Ψia

Ψaa

0

]=

[ [ Kii Kia

Kai Kaa

]−1[ 0Iaa

]0

](A.34)

Nastran note: The attachment modes can be calculated inMSC/Nastran2004 by setting the parameter resvec on and specifyinga user-set called U6.

(v) Inertia-Relief ModesInertia-relief modes should be included in a component that has

rigid-body degrees of freedom, i.e. when the r -set is not empty. In thisway, the static response is complete. They correspond to the rigid-bodyresponse of a free component. Craig (1987) defines inertia-relief modesas static displacements of a component, which is loaded by d’Alembert

262


forces due to rigid-body motion and which is supported such that thestiffness matrix is not singular. In this way, the inertia relief modesΨm (np× nr) are given by:

Ψm =

[ Ψim

00

]=

[ K−1ii (MiiΨir + MieΨer + Mir)

00

](A.35)

Nastran note: MSC/Nastran2004 adds the inertia-relief modes tonormal modes set when resviner or inrlm are switched on.

(vi) Inertia-Relief Attachment ModesInertia-relief attachment modes are calculated applying a unit force

on the entire boundary dofs, one by one, and equilibrating these forceswith d’Alembert inertial forces. The inertia-relief attachment modes,Ψn (np× nb), are orthogonal to rigid body modes and defined as

Ψn =

[Ψim

Ψbn

]= (PTGP)

[0Ibb

](A.36)

where G is the flexibility matrix, introduced in Eq. A.15 and P isgiven by

P = I−MΨrrΨTrr (A.37)

A.2.4.1 Statically Complete Mode Sets

There are combinations of the aforementioned mode sets (Ψr, Ψa, Ψg,Ψm and Ψn) that generate a superposition of the modes sufficient todetermine exactly the static response of a component submitted toexternal forces applied at boundary nodes. These combinations definea static superset.

Hintz’s method of constraint modes superset is defined as follows(g=e):

Ψg = [Ψr Ψg Ψm] =

[ Ψir Ψig Ψim

Ψer Ieg 0Irr 0 0

](A.38)

Hintz’s method of attachment modes superset is defined as follows(g=a):

263


Ψa = [Ψr Ψa Ψm] =

[ Ψir Ψia Ψim

Ψer Iea 0Irr 0 0

](A.39)

The inertia-relief mode superset defines the third statically com-plete set of modes.

Ψn = [Ψr Ψn] (A.40)

Ψg, Ψa and Ψn may be supplemented by dynamic modes: fixed-interface, free-interface, or hybrid-interface defined by the normalmodes. These three static component mode supersets span the samesubspace. The combination of a static superset and some dynamicmodes defines a dynamic superset.

A.2.4.2 Dynamic Component-Mode Supersets

Elastic normal modes are used to supplement one of the static compo-nent mode supersets.

A dynamic constraint-mode superset can be defined as

Ψg = [Ψg Φk] (A.41)

A dynamic attachment-mode superset can be defined as

Ψa = [Ψa Φk] (A.42)

And a dynamic attachment-mode superset as

Ψn = [Ψn Φk] (A.43)

These dynamic supersets are equivalent since the static supersetsspan the same subspace.

Nastran note: This equivalence is, probably, the reason whyMSC/Nastran2004 only supports the dynamic constraint mode super-set.

264


A.2.4.3 The Well-Known Craig-Bampton Method

The well-known Craig-Bampton method only employs a subset of thedynamic constraint-mode not using the inertia-relief modes and fixed-interface normal modes to supplement the static constraint mode su-perset. This is the explanation why some others combination can beobtain more accurate results. This choice yields the following dynamicsuperset:

ΨCB = [Ψr Ψg Φk?] (A.44)

where Φk? are normal modes with fixed-interface nodes.

A.2.4.4 Evaluation of the Dynamic Supersets

CMS was applied for evaluating the dynamics of a three-axis highspeed-milling machine, a representative example of a complex mecha-tronic system. This machine model was developed in the frame of theBrite-Euram Project KERNEL II Annex (1995). The construction ofthe three-axis milling machine allows a natural definition of the com-ponents (substructures) as the ram, Y-axis slide, the X-axis slide, andthe base (Fig. A.6). Each one of these components was submittedto a CMS method in order to obtain reduced component models thatare independent of the machine position. These reduced componentscould be assembled together in any desired position resulting in theassembled model. This assembled model was submitted once more toa CMS method. An overview of the model reduction method approachused by de Fonseca (2000) is shown in Fig. A.6.

As a general rule, boundary points that are rigidly connected toother components should be placed in the fixed set of dofs (b-set), whileexternal points, which are free in the assembled model should be placedin the free set of dofs (c-set). However, for systems with configuration-dependent dynamics (e.g. the case study - milling machine), two re-quirements are imposed on their reduced component modes (de Fon-seca, 2000):

• they should be independent from the relative positions in whichthey are assembled and

• they should be independent from each other, so that, a later mod-ification of a single component only affects the reduction processof this component.

265


Figure A.6: Overview of the applied reduction method by deFonseca (2000)

Considering the latter statement, if the rule based on the configurationfor the definition of the fixed and free interface dofs were followed, a newreduction should be performed for each different spatial configurationof the machine, as different nodes on the guideways are then connectedwith the adjacent components. Consequently, the component reductionwould not be independent from the model synthesis, and all advantagesassociated with the use of the CMS techniques, would be lost. So,mixed-boundary configurations will be not considered in this work. Allboundary nodes will be treated as fixed-set of dofs or as free-set of dofs.

During the reduction processes, the same number of dofs were keptfor each component and they are described in Table A.1.

Three different CMS approaches were re-evaluated:

1. The components were reduced using the Craig-Bampton method.

266


Degrees of Freedom Ram Y-axis X-axis BaseInterior dofs 1714 3496 3436 19738Exterior dofs 324 408 138 472

Total number of dofs 2038 3904 3574 20200Dofs of the reduced model 334 418 148 482

Table A.1: Interior and exterior dofs for the components used for thefirst step of the reduction

So, a subset of the static constraint mode set, [Ψr Ψg], was sup-plemented with 10 fixed-interface normal modes. This approachdid not consider the inertia-relief modes.

2. The same static constraint mode subset chosen in the Case 1 wasused but the dynamic complement was made by 10 free-interfacenormal modes. This approach also did not consider the inertia-relief modes.

3. A complete static constraint mode was considered [Ψr Ψg Ψm]. The ram, the Y-axis slide and the X-axis slide are componentsthat have rigid-body modes, so 6 inertia-relief modes were added.In order to keep the same dofs in all the cases only 4 free-interfacenormal modes were added to supplement the set dynamically.Differently, the base is support by spring and has no rigid-bodymodes consequently the inertia-relief modes did not add any im-provement on the quality of the reduction. In this case, a subsetof the static constraint mode set [Ψr Ψg] and 10 free-interfacenormal modes were employed.

After this first reduction step, the components were assembled anda new reduction procedure was realized. Only 9 external dofs were kept,which were the rotations around the three motor axes and the six dofsof the tool. The later are kept to allow the evaluation of the cuttingforces at the machine. The reduced model was built using 9 constraintmodes, 36 normal modes and 6 inertia-relief modes. The rigid bodymodes were obtained as a linear combination of the constraint modes(Craig, 1987).

The comparison between these different approaches is shown in Fig.A.7. It can be concluded that the inclusion of inertia-relief modesindeed contributes to the accuracy of the reduced model.

267


Figure A.7: Comparison between different dynamic supersets

Considering the milling machine study case, the reduced stiffnessand mass matrices and the modes are exported to Matlab, and a state-space model with 102 dofs is derived, which is still too large to be usedfor controller design purposes. Consequently, further reduction wasstill required for control design purposes. There are several methodsimplemented in Matlab, and this final reduction procedure is describedin detail in Section A.3.

Nastran note:The reduced stiffness and mass matrices andthe modes can be exported to a text file when the commandsEXTSEOUT(EXTID = number of the superelement,DMIGPCH) andPARAM, POSTEXT, YES are included in MSC/Nastran2004. Fromthese data, a state-space model can be derived. A formal descriptionof the state-space model derivation is given in Chapter 2.

268

A.3. MODEL REDUCTION TECHNIQUES VIAAPPROXIMATION-DATA BASED METHODS

A.3 Model Reduction Techniques viaApproximation-data Based Methods

In the design of robust controllers for complex systems, model reduc-tion plays a fundamental role. According to the specifications of thedesign of control systems, a small model that can reproduce the sys-tem dynamics is required not only to speed up the simulation process,but also to make the control design feasible. Modern control methods,such as LQG or H∞ , result in a high-order controller, generally of thesame order as the system, which means that a good model reductionalgorithm may reduce the complexity of the whole system.

Sometimes, the model reduction techniques suitable for large sys-tems, e.g. CMS described in Section A.2.4 cannot derive a sufficientlyreduced model for control design proposes. For instances, problemscontaining non-linearities, damping, varying dynamics, among otherscannot be reduced using CMS.

Approximation-data based methods rely on the construction of asubspace that best approximates the collected data, i.e., that best ap-proximates the output/input relation. These techniques have been usedfor reducing large models (thousands of dofs), small models (with hun-dreds of dofs), linear and non-linear models to a lower-order modelcontaining less than 20 states. Therefore, these techniques are well-known in the control design field. An overview about some techniquescan be found in Antoulas and Sorensen (2000); Gugercin and Antoulas(2004). In this reference, three main categories of model reduction areidentified: methods based on (1) singular value decomposition (SVD),(2) Krylov subspace and (3) the combination of aspects from both SVDand Krylov (see Table A.2).

Due its nature, the physical meaning of reduced models extractedusing approximation-data based methods may be lost. Since these tech-niques approximate an model that best fits the input/output relation,the terms of the model may have no physical meaning. For instancethe terms of the approximated state-space model will be not related tothe mass, stiffness nor modal matrices. During the design phase, thismay be an important drawback since physical characteristics, such asstiffness matrices and modal participation factor, may be relevant forevaluating the design of both parallel and serial machines.

Hereafter, a short description of the SVD methods, includingProper Orthogonal Decomposition (POD) methods and balanced trun-

269


SVD KrylovNonlinear Systems Linear

RealizationPOD methods Balanced truncation Interpolation

Empirical grammians Hankel approximation LanczosArnoldi

SVD-Krylov

Table A.2: Overview of approximation methods (Antoulas andSorensen, 2000)

cation methods, are stated (see Section A.3.1). A survey about Krylovsubspace methods and its combination with SVD can be found in An-toulas and Sorensen (2000).

A.3.1 Singular Value Decomposition Methods

The SVD-based approximation methods are based the Singular ValueDecomposition (SVD). These values are used to decide which is impor-tant information to be kept during the approximation procedure. Thesingular values, σi, of a matrix A are defined by σi =

√λi(M∗M),

where λi() represents the eigenvalue extraction procedure and ∗ repre-sents the adjoint matrix. The adjoint matrix can be obtained by takingthe transpose and then taking the complex conjugate, denoted by , ofeach entry M∗

i,j = Mj, i.Any matrix M ∈ Rn×m can be decomposed in 3 matrices (Antoulas

and Sorensen, 2000):

M = ZΣY∗ (A.45)

where Z and Y are unitary and orthogonal matrices and Σ =diag(σ1, . . . , σn) ∈ Rn×m is a diagonal matrix with singular values.

A.3.1.1 Proper Orthogonal Decomposition (POD) methods

Considering the system described by Eq. (A.4) and a fixed input u(t),the state trajectory at certain time steps, tk, is given by:

X = [x(t1) x(t2) . . .x(tN )] ∈ Rn×N (A.46)

270


where n is the number of states. One can evaluate which singularvalues are important to the response and which are not contributingto the response when extracting and organizing the singular valuesof the matrix X. Small singular values can be elimitated and a low-order approximation of this system can be computed, X = ZΣY∗ =ZkΣkY∗k, where k n.

The model-order reduction can be performed via the transformationx(t) = Zkξ(t), where ξ(t) ∈ Rk. Details about POD can be foundin (Kerschen et al., 2005; Antoulas and Sorensen, 2000; Azeek andVakakis, 2001; Feeny and Kappagantu, 1998).

A.3.1.2 Approximation by balanced truncation

There is a set of invariants called the Hankel Singular Values (HSV)which can be attached to every linear, constant, finite-dimensional sys-tem. These invariants play the same role for dynamic systems as thesingular values play for constant finite-dimensional matrices. It hasobserved that the HSVs of many systems decay rapidly. Hence, verylow-rank approximations are possible and accurate low-order reducedmodels can be derived.

A system Ξ, realized by (A,B,C,D), is balanced, if the solutionsP and Q, known as controllability and observability grammians respec-tively, of the Lyapunov equations (Eqs. A.47 and A.48) satisfy P = Q= diag(σ1, . . . , σn) with σ1 ≥ σ2 ≥ . . . ≥ σn > 0.

AP + PAT + BBT = 0 (A.47)

ATQ + QA + CTC = 0 (A.48)

The σ1, . . . , σn are the HSVs of the system Ξ and they can be foundby extracting the square roots of the eigenvalues of the product PQ,Eq. A.49.

σi(Σ) =√λi(PQ) (A.49)

The energy transfer index, E, from the input vector u to the out-put vector y can be described by Eq. A.50, considering the initialconditions x(0) = x0.

E = sup

∫∞0 y(t)Ty(t)dt∫ 0−∞ u(t)Tu(t)dt

=1

‖ x0 ‖

n∑i=1

σ2i x

20,i (A.50)

271


From Eq. A.50, it is possible to conclude that just the higher HSVscontribute for the energy transfer of the system. Then a model reduc-tion procedure can be proposed where higher HSVs are kept.

The system can be reorganized as described in Eq. A.51, the sub-system (A11,B1,C1,D) contains the higher HSVs and the subsystem(A22,B2,C2,D) contains the lower HSVs. This reorganized systemis called the balanced realization of the system. Figure A.8 shows ascheme of the subsystems (A11,B1,C1,D) and (A22,B2,C2,D).

τ : (A,B,C,D) 7−→ (TAT−1,TB,CT−1,D) =([A11 A12

A21 A22

],

[B1

B2

],[

C1 C2

],D)

(A.51)

Figure A.8: Model reduction scheme (Preumont, 2002)

A transformation, x = T−1x, can be performed to compute thebalanced realization of the system (see Eq. A.51)

The reduced order model, Ξ, is obtained by a simple truncation(A, B, C, D) = (A11,B1,C1,D). The transformation T can be derivedfrom the Cholesky factors of the solutions of the Lyapunov equations:

P = UU∗ Q = LL∗ (A.52)

Denoting the SVD of U∗L = ZΣY∗, a balanced realization can beperformed by the transformation T:

T = Σ−12 Y∗L∗

T−1 = UZΣ−12

(A.53)

This transformation is asymptotically stable and minimal (Gugercinand Antoulas, 2004). The drawback of this methodology is that theHSV computation involves the solution of two linear matrix equations(Lyapunov equations) which may be computationally demanding.

272


Balanced truncation can also be based on other techniques, butLyapunov equations. Applying the same basic principle of balancedtruncation, the gramians P and Q can be found as follow (Gugercinand Antoulas, 2004; Benner, 2007):

Classical Balanced Truncation The Gramians P and Q express re-spectively the controllability and the observability of the systemΞ. They are calculated solving the Lyapunov equations (Eqs.A.47 and A.48).

LQG Balanced truncation The Gramians P and Q express respec-tively the controllability and the observability of the close-loopsystem based on LQG compensator. They are calculated solvingthe algebraic Riccati equations (Eqs. A.54 and A.56).

AP + PAT −PCTCP + BBT = 0 (A.54)

ATQ + QA−QBTBQ + CTC = 0 (A.55)

Balanced Stochastic Truncation The Gramian P is the solution ofthe Lyapunov equation (Eq. A.47). The Gramian Q is the rightspectral factor of power spectrum of the system Ξ and can befound by the solution of the algebraic Riccatti equation A.57

ATQ + QA−QBTwBwQ + CTDDTC = 0 (A.56)

where A = A−Bw(DDT )−1 and Bw = BDT + PCT

Positive-Real Balanced Truncation This method is based onpositive-real equations. The Gramians P and Q are found solvingthe positive real Ricatti equation:

AP + PAT −PCT R−1CP + BR−1BT = 0 (A.57)

ATQ + QA−QBT R−1BQ + CT R−1C = 0 (A.58)

where R = D + DT and A = A−BR−1C.

Bounded-real Balanced Truncation Based on bounded real lemma,the Gramians are calculated solving the bounded real Riccatiequations (Eqs. A.59 and A.60).

273


AP+PAT +BBT +PCT R−1p CP+BDT R−1

p DBT = 0 (A.59)

ATQ+QA+CTC+QBR−q 1BTQ+CTDR−1q DTC = 0 (A.60)

where Rq = I−DTD, Rp = RTq and A = A + BR−1DTC.

H∞ Balanced Truncation is a closed-loop balancing based on H∞compensator.

Frequency-weighted versions These versions attempt to approxi-mate the model in the interesting frequency range requiring asmall weighted error (similar to Eq. A.7). Given some inputweight Wi and some output weight, the weighted error can bedescribed by

‖Wo(s)(G(s)−Gr(s))Wi(s)‖∞ ≤ ε (A.61)

All described approaches for balanced truncation methods preservestability. Moreover, Positive-Real Balanced Truncation also preservespassivity, i.e. the system can not generate energy, and BalancedStochastic Truncation preserves the minimum phase property. Detailsand comparison about these methodologies can be found at Gugercinand Antoulas (2004) and at the references herein.

A.3.1.3 Model Reduction for Control - A case study

After the final CMS reduction was applied to the FE model of themilling machine, this model was converted to state space and importedinto Matlab environment in Section A.2.4.4. The model has 102 states,which is not yet suitable for model-based control design. So, modelreduction used for robust control design was applied. Figure A.9 showsthe Hankel singular values for the states of this system. As can beobserved, there are states that influence more the system than others.So this kind of approach is reasonable for this problem. Figure A.10shows the FRF from the motor torque to the tool displacement (a) andto the tool acceleration (b) of the model with 102 states (in solid line)and the reduced model with 18 states (in dashed line). The 18 statesselected to compose the reduced model are able to predict the systemresponse in the desired frequency range. There were three rigid bodymodes and more six flexible modes that were kept. The methodologyapplied resulting a reduced model with good accuracy until 125 Hz.

274


0 20 40 60 80 1000

2

4

6

8

10

Order

Han

kel S

ingu

lar

Val

ue

Figure A.9: Hankel Singular Value to the state space system with102 states of the milling machine

0 100 200 30010

−10

10−5

100

Frequency (Hz)

FR

F (

1/N

)

(a)

0 100 200 300−800

−600

−400

−200

0

200

Frequency (Hz)

Deg

ree

(o )

0 100 200 30010

−2

100

102

Frequency (Hz)

FR

F (

1/kg

m)

(b)

0 100 200 300−1000

−500

0

500

Frequency (Hz)

Deg

ree

(o )

102 states model18 states model

Figure A.10: FRF from the motor torque to (a) the tooldisplacement and (b) to the tool acceleration of the model with 102

states and the reduced model with 18 states

275


A.4 Model Reduction via Global ModalParametrization

Bruls et al. (2007) have proposed a model order reduction techniquebased on Global Modal Parametrization (GMP) which considers theglobal modes extracted from the whole mechanism and parameterizedaccording to the mechanism’s configuration. This approach yields amore concise model than the one provided by CMS and provides directaccess to the reduced stiffness matrix and other dynamical characteris-tics. Therefore, this technique is considered in Chapter 3 and is brieflydescribed hereafter (see details in Bruls et al. (2007)).

Firstly, a brief introduction on flexible multibody system is givenin this section, a more formal and detailed introduction can be foundin the following chapters. According to Geradin and Cardona (2001),a flexible multibody system can be described using absolute nodal co-ordinates, q. Hence, each body is represented by a set of nodes andeach node has its own translation and rotation coordinates w.r.t a fixedreference frame. The general equations of motion have the followingstructure:

M(q)q + BTλ = g(q, q, t) (A.62)Φ(q, t) = 0 (A.63)

Eq. (A.62) represents the dynamic equations of the mechanical system;and Eq. (A.63), the kinematic constraints. M is the mass matrix,which is not constant in general, g represents the internal, external andcomplementary inertia forces, B = ∂Φ/∂q is the matrix of constraintgradients and λ are the Lagrange multipliers which may be interpretedas the internal forces generated by the constraints.

Prior to the description of the model reduction technique based onmodal global parametrization for flexible multibody systems, a intro-duction on the eigenvalue analysis is required. The generalized eigen-value problem associated with Eqs. A.62 and A.63 can be describedas [

K BT

B 0

] [uλ

]= ω2

[M 00 0

] [uλ

](A.64)

where K is the tangent stiffness matrix (formally defined in Chapter

276

A.4. MODEL REDUCTION VIA GLOBAL MODALPARAMETRIZATION

2), M is the mass matrix, B the constraint gradients matrix, u and λare the amplitudes and ω is the frequency of the harmonic solution.

The solutions of Eq. A.64 are the same for the following constrainedeigenvalue problem:

(K− ω2M)u = 0subject to Bu = 0 (A.65)

According to Rayleigh’s theorem (Geradin and Cardona, 2001),there are (n−m) solutions of the generalized eigenvalue problem givenby (

ω21,

[Φu

1

Φλ1

]), . . . ,

(ω2n−m,

[Φun−m

Φλn−m

])(A.66)

and 2m additional solutions given by

(−∞,

[0eλ1

]),

(∞,[

0eλ1

]), . . . ,

(−∞,

[0

eλm

]),

(∞,[

0eλm

])(A.67)

where eλk = [0 . . . 0 1 0 . . . 0], the value 1 is in the k -thposition.

Considering the flexible multibody system (Eqs. A.62-A.63) andthe augmented coordinates v = [q λ]T , the GMP is defined as themapping between the augmented coordinates v, and the modal coor-dinates η. A mechatronic system generally undergoes large-amplituderigid motions, which can be represented by rigid modal coordinatesθ, and small-amplitude superimposed deformation, which can be rep-resented by flexible modal coordinates δ. From the practical point ofview, the rigid coordinates can be conveniently defined as the actuatorsdofs, with the advantage that those coordinates will explicitly appearin the reduced model.

The reduction procedure is based on the transformation betweenaugmented coordinates, v, and modal coordinates, η. In general, due tothe nonlinear kinematics of the machine, the relation v(η) is nonlinear.However, in the neighborhood of a given configuration θ0 which is notdeformed (δ0), we have the incremental relation

∆v = Ψ∆η = Ψr(θ0)∆θ + Ψf (θ0)∆δ (A.68)

277


with ∆(•) = (•)−(•)0. In this expression, the configuration-dependentrigid and flexible mode shape matrices Ψr and Ψf are defined for smallmotions around the selected configuration θ0.

In a given configuration, the modes shapes are constructed by acomponent-mode synthesis using the linearized form of the equation ofmotion. The mass and stiffness matrices can be rewritten as:

Mrr Mrg Mri

Mgr Mgg Mgi

Mir Mig Mii

, Krr Krg Kri

Kgr Kgg Kgi

Kir Kig Kii

(A.69)

where the indexes r, g and i represent rigid, constraint and internaldofs, respectively.

During the reduction procedure, θ and qg should be kept sincethey represent the rigid body dofs and the constraint dofs respectively.Constraint dofs are required when additional external loads, besides theones related to the rigid body motion, are used; for instance, a distur-bance force at the end-effector. The internal dofs, vi, can be condensedduring the reduction by selecting fewer lower-order flexible modes torepresent their dynamics. According to the pre-selected modes: rigidbody modes Ψr, constraint modes Ψg and lower-order flexible internalmodes Ψl; the modal transformation is defined as Ψ = [Ψr Ψg Ψl].

The rigid modes Ψr should satisfy: Krr Krg Kri

Kgr Kgg Kgi

Kir Kig Kii

Ψr =

000

(A.70)

The constraint modes Ψg are static deformations obtained when therigid dofs are fixed and a unit displacement is imposed to the constraintdofs: [

Kgg Kgi

Kig Kii

]Ψg =

[gg

0

](A.71)

where gg is the force vector required to impose an unit displacementto the constraint dofs.

And the internal modes Ψi = [Ψl Ψh], divided into lower-order(represented by the index l) and higher-order (represented by the index

278

A.5. CONCLUSIONS

h) modes, are the normalized eigenmodes when rigid and constraintdofs are fixed:

(Kii − ω2kM

ii)Ψuik = 0 (A.72)

where ωk is the kth eigenfrequency associated with the kth mode shape,Ψuik . The model-order reduction technique relies on a truncation of the

higher-order internal modes.After transformation in modal coordinates, the equations of motion

(Eqs. 7.3-7.4, y = 0) become (Bruls et al., 2007):

Mηη(θ)η + Cηη(θ)η + Kηη(θ)η = gη(θ) (A.73)

where gη(θ) denotes the actuator forces, Mηη is the reduced massmatrix, Cηη is the reduced damping matrix and Kηη is the reducedstiffness matrix. For a given configuration θ0, this equation definesa low-order linearized model which can be used for control design, asdescribed and illustrated in Chapter 7.

A.5 Conclusions

A non-exhaustive overview about model reduction has been described.These concepts and techniques are going to be applied throughout thethesis.

Techniques based on modes evaluation, such as Component-ModeSynthesis and Global Modal Parametrization, has the advantage of pro-viding direct access to the reduced stiffness and mass matrices widelyemployed for machine design evaluation. The design of both serialand parallel machines often relies on the evaluation of these dynamicalcharacteristics in the configuration space.

Approximation-data based methods, such as Singular Value De-composition based approximation methods and Krylov-based approxi-mation method, have also been devised for the reduction of large linearand non-linear dynamical systems. Since these methodologies rely onthe construction of a subspace that best approximates the collecteddata, the physical meaning of the modeling may be lost during themodel reduction which may be an important drawback when designingmechatronic motion systems. Nevertheless, these methodologies pro-vide reliable and accurate reduced models for a large range of systemswhich techniques based on modes evaluation may fail.

279


The choice of the reduction technique always rely on the modelingmethodology, the required accuracy, the designer expertise and softwareavailabilities.

280

Appendix B

Review on FlexibleMultibody System Theory

While the FE method is concerned with flexible bodies and smallmotions (structural and modal analysis), multibody systems mostlyconcerns rigid/flexible bodies and large motions. The dynamics ofmultibody systems is based on classical mechanics and is character-ized by different formalisms (Wasfy and Noor, 2003). Independentfrom the adopted formalism, MBS comprises a set of elements such asrigid and/or flexible bodies, bearings, joints and supports, springs anddampers, active force and/or position actuators (Schiehlen, 1997).

The motion of a multibody system is governed by a set of equations,comprising differential and algebraic equations, derived from dynamicalequations of motion. The differential equations are an expression of thephysical laws and the algebraic equations consider the constraints ofthe system (Costa, 1992).

Considering these characteristics, several formalisms were devel-oped for different approaches considering mainly the choice of coor-dinates, the mechanical principle to generate the equations and thecomputer implementation. The choice among these options relies onthe problem to be modeled, the designer experience and the softwareavailabilities. A short survey about these approaches is described here-after. A more detailed survey can be found in Wasfy and Noor (2003).

281

B Review on Flexible Multibody System Theory

B.1 Choice of Coordinates

The generalized coordinates are a set of coordinates that allow a fullgeometric description of the system with respect to a reference frame(Preumont, 2006). However, this representation is not unique and doesnot always have a simple physical meaning. The choice of coordinatescan influence the number of equations, the computer efficiency and theusability of the software. Among the choices available in the literature,there are: the minimal coordinates, the relative coordinates (or jointcoordinates), the Cartesian coordinates, the finite-element coordinates(or natural coordinates) and the mixed coordinates. Figure B.2 showsthe different choices of coordinates for a 4-bar mechanism (Bruls, 2005).

q q1

q2

q3

(b)(a)

(d)(c)

(x , y , )1 1 q1

(x , y , )3 3 q3

(x , y , )2 2 q2

(x , y )2 2

(x , y )3 3

(x , y )1 1

(x , y )5 5

(x , y )6 6

(x , y )4 4

Figure B.1: Coordinate Systems: (a) minimal, (b) relative, (c)Cartesian, (d) finite-element (Bruls, 2005)

The minimal coordinates approach relies on the description of thesystem using the minimal set of degrees of freedom. This approachgenerates a system of equations free of algebraic constraints. Relativecoordinates can be employed using recursive descriptions. Both mini-mal and relative coordinates are used to formulate a minimum numberof equations. The eventual system of equations using these approaches

282

B.1. CHOICE OF COORDINATES

may be smaller but the kinematical and dynamic description of thesystem is more complicated. However, since the minimal number ofequations is found, it is a desirable method for several applications.Theory and examples about minimal coordinates are described in Ver-linden(a) et al. (2005); Verlinden(b) et al. (2005).

The third approach relies on the description of location and rota-tion using a set of Cartesian coordinates. It has the advantage thatthe equations of motion can be obtained in a systematic way. For eachrigid body in the system, six coordinates are sufficient to describe itsposition and orientation. When using minimal, relative and Cartesiancoordinates, the motion of flexible bodies can be described by the rigidbody motion and additional deformation introduced by the flexibility.This is usually performed by the introduction of the floating frame ofthe reference which defines an intermediate frame to represent the rigidbody motion. Assuming linear elastic behavior, additional flexible co-ordinates can represent the body deformation which is considered tobe small. This procedure is usually combined with low order models,extracted from finite-element models, to represent the deformations.Using CMS, the flexible body is represented by few interface nodal coor-dinates and internal modal coordinates. Most available software, suchas ADAMS (Adams, 2001), SIMPACK and LMS Virtual.Lab Motion(solver DADs) (LMS.International, 2006), uses Cartesian coordinatesand the inclusion of flexible bodies is performed via floating frame ofthe reference and low-order models extract from finite-element models.

Using the Cartesian coordinates, the coupling between the elasticcoordinates and the Cartesian coordinates is limit. Assumptions onsmall deformations are the main drawback of this approach. Moreover,the description of motion between flexible and rigid bodies may becomemore cumbersome due to the condensation of nodes. In this way, thefinite element coordinates are the most suitable set of coordinates forsystems under large deformation or containing joints connecting flexibleand rigid bodies. Finite element coordinates are redundant and con-straints can be applied using boolean identification. Samcef/Mecanoand Oofelie are software based on finite elements coordinates. Whenusing finite element coordinates mainly two reference frames can beused: the inertial frame and the corotational frame. In both strategies,the absolute nodal translation and rotation are defined with respect tothe frame. The corotational frame creates an additional intermediateframe to describe small amplitude elastic deformation.

283


Finally, the mixed coordinates are extremely useful for problemswith different natures since it can mixture several coordinates systems(see an example in Garcia-Vallejo et al. (2003)).

B.2 Computer Implementation

The implementation of a MBS code can be made numerically or sym-bolically. In the numerical approach, a set of numerical data includingthe topology, constants and initial values are used to generate the sys-tem of equations. These equations are numerically integrated accordingto the initial conditions. In the symbolic approach, a symbolic set ofdata is used to generate symbolic system of equations. A symbolicsystem of equations is the output of this approach and can be used tocalculated the numerical solution (Samin and Fisette, 2003).

In spite the believe that the symbolic approach is more suitablefor small application, several industrial application has been describedin the literature (Samin and Fisette, 2003; Verlinden(a) et al., 2005;Verlinden(b) et al., 2005; Hiller and Kecskemethy, 2001).

B.3 Mechanical Principle to Generate theEquations

Among the mathematical formalism used in MBS, it is possible to iden-tify two groups: the Eulerian and the Lagrangian approaches. Eulerianapproaches, such as Newton-Euler formalism, uses Newton’s equationsto describe the translational motion and Euler’s equations to describerotational motion of a rigid body. On the other hand, Lagrangian ap-proaches are derived from Hamilton’s Principle (Bruls, 2005; Preumont,2006; Costa, 1992) which states that:

δ

∫ t2

t1

L(q, q)dt = 0 (B.1)

for conservative systems, where the Lagrangian L is defined by L =K − V −W, K is the kinetic energy, V is the potential energy and Wis the deformation energy of the elastic bodies and q ∈ Rn×1 are thegeneralized coordinates.

For system with independent coordinates, i.e. when minimal orrelative coordinates are used to described the system, the equation of

284

B.3. MECHANICAL PRINCIPLE TO GENERATE THE EQUATIONS

motion can be directly derived from Eq. (B.1). Cartesian and finite ele-ment coordinates are redundant and kinematic constraints may appearto restrain the motion. Therefore, the system may satisfy m holonomicconstraints

Φ(q, t) = 0 (B.2)

Non-holonomic constraints, Φ(q, q, t) = 0, are treated in Geradinand Cardona (2001), but are not considered in this description. Thesolution for the constraint problem described by Eqs. B.1 and B.2 isthe following optimization problem:

minq

∫ t2

t1

L(q, q)dt subject to Φ(q, t) = 0 (B.3)

This constrained problem can be restated by the introduction ofLagrange multipliers λ

minq,λ

∫ t2

t1

(L(q, q)− λTΦ(q, t))dt (B.4)

The solution of this optimization problem can be found after inte-grating by parts the following expression:

δ

∫ t2

t1

(L(q, q)− λTΦ(q, t))dt = 0 (B.5)

The solution becomes (Bruls, 2005):

(d

dt

(∂K∂qi

)− ∂K∂qi

+∂W∂qi

+∂V∂qi

+∂Φ∂qi

T

λ

)δqi + δλTΦ = 0 (B.6)

Since the kinetic energy can be described by:

K =12qM(q)q, (B.7)

Eq. (B.4) can be rewritten as:

δqT (M(q)(q)+ggyr(q, q)+gint(q)−gext(q)+BTλ)+δλTΦ = 0 (B.8)

285


where B = ∂Φ/∂q is the matrix of constraint gradient (Jacobian ma-trix), ginti = ∂W/∂qi are the components of the internal elastic forces,gexti = −∂V/∂qi are the components of the external forces and thecomponents of gyroscopic and centrifugal forces ggyri are given by:

ggyri =12

(∂Mij

∂qk+∂Mik

∂qj−∂Mjk

∂qi

)qj qk (B.9)

Since Φ(q, t) = 0, Eq. B.8 can be written in a differential algebraicequation system (DAE):

M(q)q + ggyr(q, q) + gint(q) + BTλ = gext(q) (B.10)Φ(q, t) = 0 (B.11)

Some information regarding the integration methods for multibodysystems can be found in Appendix C.

B.4 Modeling parallel and serial kinematic ma-chines using flexible multibody system

This section is focused on providing all the necessary features to sim-ulate parallel and serial machines in a flexible multibody environment.The modeling of the mechanical system of parallel machines can be per-formed using standard elements available in Oofelie; such as revoluteand spherical joints, and rigid and flexible elements. The modeling ofthe mechanical system of serial machines requires the correct modelingof the translational motion between flexible components. Hereafter,this joint is refereed to as sliding joint and its derivation is describedin Section B.4.4. The sliding joint derivation is based on the derivationof prismatic joints that allow the relative translation between two rigidbodies, while no rotation is allowed.

Some concepts used in flexible multibody should be introduced be-fore the derivation of the sliding joint. Firstly, a short descriptionabout rigid body spherical motion in Subsection B.4.1. Concepts asrotation operator and tangent operator are introduced and are usedduring the sliding joint derivation. Then, concepts about beam kine-matics and displacement gradient measure of deformation are addressedin Subsection B.4.3. These short descriptions can be found in detail inGeradin and Cardona (2001). The derivation of the sliding joint follows

286

B.4. MODELING PARALLEL AND SERIAL KINEMATIC MACHINES USINGFLEXIBLE MULTIBODY SYSTEM

the notation adopted by Geradin and Cardona (2001) when derivingprismatic and revolute joints.

A simplified derivation is presented in the Chapters 6and 7. In thepapers, a simplified notation has been adopted for the sake of brevity.

B.4.1 Parametrization of rigid body spherical motion

Spherical motion corresponds to the rotation of a rigid body about afixed point in a space. The description about spherical motion and itsparametrization described hereafter is based on Geradin and Cardona(2001).

For a point P attached to a rigid body, it is possible to define thematerial coordinates X of the point P, which is the position vector inthe reference frame:

X = [X1 X2 X3]T (B.12)

The spatial coordinates x is defined by the position vector aftertransformation:

x = [x1 x2 x3]T (B.13)

A set of orthogonal base vectors [E1 E2 E3] attached to the bodyin the reference configurations is known as absolute. The same set oforthogonal base vector after transformation [e1 e2 e3] attached tothe current configuration is called body frame. Figure B.2 shows thespherical motion and the definition described before.

The pure rotation can be described by:

x = RX (B.14)

This relation also applies to ei = REi. The rotation operator R isorthogonal, RT = R−1 (Geradin and Cardona, 2001).

A rigid body undergoing translation and rotation simultaneouslycan be described by

xp = x0 + RXp (B.15)

where xp and Xp are the spatial and material positions of the point P,respectively. The vector x0 describes the spatial position of the origin.

The velocity of point P in spatial coordinates is obtained by

287


E2

E3

E1

e2

e3

e1

n

f

P(X , X , X )1 2 3

P’(x , x , x )1 2 3

Figure B.2: Spherical motion

vp = xp = RXp = RRTxp = ωxp (B.16)

where ω is a skew symmetric matrix, defined by ω = RRT . Thisrelation is similar to v = ωx = ω × x. The vector part of ω providesthe spatial expression of the angular velocity:

ω = vect(RRT ) (B.17)

In a similar way the velocity vector of the point P may also betransformed to material coordinates:

V = RTv = RT RX (B.18)

It is possible to define Ω = RT R that can be interpreted as thematrix of material angular velocities. The vector is obtained by

Ω = vect(RT R) (B.19)

The associated virtual displacement of Eq. B.15 is obtained throughvariation of this expression

δx = δRX (B.20)

288


Substituting Eq. B.15 in Eq. B.20, the spatial matrix of infinitesi-mal rotations, δθ, can be defined as:

δx = δRRTx = δθx (B.21)

δθ = δRRT (B.22)

And restating that RT = R−1, and therefore RRT = I, the mate-rial matrix of infinitesimal rotations, δΘ, can be defined:

δx = RRT δRX = RδΘX (B.23)

δΘ = RT δR (B.24)

The spatial and the material matrices of infinitesimal rotations arerelated by

δΘ = RT δθR (B.25)

And the associated vectors, δθ = vect(δθ) and δΘ = vect(δΘ), arerelated by

δΘ = RT δθ (B.26)

The parametrization of spherical motion usually results from thechoice of an independent set of three parameters aT = [a1 a2 a3] todescribe the rotational operator:

R = R(a) (B.27)

Among the choice of rotation parameters, it is possible to highlightthe Cartesian rotation vector, Rodrigues parameters and Euler param-eters. A full description of these choices is presented in Geradin andCardona (2001). Hereafter, the Cartesian is briefly introduced. Someconcepts related with this description are used to formulate the slidingjoints in Subsection B.4.4.

The Cartesian rotation vector is defined as the vector which has thedirection of the rotation axis and the length equal to the amplitude ofthe rotation (see Fig. B.2):

Ψ = nφ (B.28)

289


The rotation operation can be correlated with the Cartesian rota-tion vector, via trigonometric relations (Geradin and Cardona, 2001),by:

R(Ψ) = I +sin‖Ψ‖‖Ψ‖

Ψ +1− cos‖Ψ‖‖Ψ‖2

ΨΨ (B.29)

or in exponential form R(Ψ) = exp(Ψ). The rotation operator admitsthe series expansion:

R(Ψ) = I + Ψ +12!

Ψ2 +13!

Ψ3 + . . .+1n!

Ψn (B.30)

Analyzing the rigid body motion and the Cartesian rotation deriva-tives, the parameterized expressions of the material and angular veloc-ities can be derived yielding

Ω = T(Ψ)Ψ (B.31)

ω = TT (Ψ)Ψ (B.32)

where T(Ψ) is the tangent operator and is described by

T(Ψ) = I +cos‖Ψ‖ − 1‖Ψ‖2

Ψ +(

1− sin‖Ψ‖‖Ψ‖

) ΨΨ‖Ψ‖2

(B.33)

The tangent operator admits the series expansion

T(Ψ) = I− 12!

Ψ +13!

Ψ2 + . . .+(−1)n

(n+ 1)!Ψn (B.34)

Truncated series expansions may be used to simplify the computa-tion.

The material and spatial rotation increments of rotation can alsobe described in terms of tangent rotation replacing the time-derivativeby the variation operator:

δΘ = T(Ψ)δΨ (B.35)

δθ = TT (Ψ)δΨ (B.36)

290


The geometric description of spherical motion can be described interms of successive elementary rotations about the coordinate axes.Several textbooks provide a description of finite rotation transfor-mation in terms of Euler angles and Bryant angles (Shabana, 2005;Geradin and Cardona, 2001). Describing this procedures to gener-ate independent parameters to express the spatial transformation fromspatial frame to material frame is out of scope of this thesis.

B.4.2 Prismatic Joint: an example

The prismatic joint allows the translational movement between tworigid bodies (see Fig. B.3a). The description of the prismatic joint issimilar to the one found in Geradin and Cardona (2001). The maindifference is that in the reference, the constraints related to the orienta-tions are imposed directly via Boolean identification, and in this work,these constraints are fully described. This description is performed asan introduction for the derivation of the sliding joint in Section B.4.4.

Figure B.3(a) shows the local frames of two rigid bodies connectedby a translational joint. The triads µ1,µ2,µ3 and ξ1, ξ2, ξ3 are orthog-

A,

µ

µ

µ

1

2

3

’

’

’

B

B ξ1’

2’ξ

ξ3’

A

µ

µ

µ

1

2

3

’

’

’

C

ζ

ζ

ζ

1

2

3

’

’

’

η

,

,

,

(b)(a)

Figure B.3: (a) Translational joint and (b) Sliding joint

291


onal unit vectors attached to nodes A and B at the reference configura-tion. The triads µ′1,µ

′2,µ3

′ and ξ′1, ξ′2, ξ3

′ are obtained by the mappingfrom the reference into the actual configuration through the rotationaloperators RA and RB:

ξ′i = RAξi µ′i = RBµi (B.37)

In a prismatic joint, the orientation of the triads should be the sameduring the motion yielding the following constraints

Φ1 = µ′T1 ξ′3 = µT1 RT

ARBξ3 = 0Φ2 = µ′T2 ξ

′3 = µT2 RT

ARBξ3 = 0Φ3 = µ′T1 ξ

′2 = µT1 RT

ARBξ2 = 0(B.38)

The variation of these constraints is performed in order to obtainthe Jacobian matrix (matrix of constraint gradient). The variation ofΦ1 is

δΦ1 = (δRAµ1)TRBξ3 + (RAµ1)T δRBξ3 (B.39)

From the definition of the the spatial and material matrices of in-finitesimal rotations and applying the skew-symmetric matrices prop-erty uv = −vu, this variation can be rewritten as

δΦ1 = (RATAδΨA)T µ′1ξ′3 − (RBTBδΨB)T µ′1ξ

′3 (B.40)

where TA = T(ΨA) and TB = T(ΨB) are tangent operators.The variation of constraints Φ2 and Φ3 can be performed in a

similar way. Moreover, two translational constraints should be added

Φ4 = µ′T1 (xB − xA) = (RAµ1)T (xB − xA) = 0Φ5 = µ′T2 (xB − xA) = (RAµ2)T (xB − xA) = 0

(B.41)

The variation of constraint Φ4 is

δΦ4 = (RAδΘA)T (µ′1(xB − xA)) == −µ′1δxA + µ′1δxb++µ′1(xB − xA)RATA

(B.42)

In this way, this joint removes 5 degrees-of-freedom. Consideringthe varied coordinates δqT = [δxTa δxTb δΨT

a δΨTb ], the Jacobian

matrix is finally described by

292


B =

0 0 (µ′1ξ

′3)RATA −(µ′1ξ

′3)RBTB

0 0 (µ′2ξ′3)RATA −(µ′2ξ

′3)RBTB

0 0 (µ′1ξ′2)RATA −(µ′1ξ

′2)RBTB

−µ′T1 µ′T1 µ′1(xB − xA)RATA 0−µ′T2 µ′T2 µ′2(xB − xA)RATA 0

(B.43)

B.4.3 The Elastic Beam: geometry and deformation

In order to derive a sliding joint which allows the translational move-ment between rigid and flexible bodies, the appropriate description ofthe flexible members is required. Beam elements are the focus of thiswork and are treated hereafter. Details can be found in Geradin andCardona (2001).

According to Fig. B.4, the reference configuration can be definedby

X = ηE1 +X2E2 +X3E3 (B.44)

where η is the parameter that determines the reference position alongthe beam neutral axis. In a concise form X = X0+Y, where X0 = ηE1

and Y = X2E2 +X3E3.

x

x

x

3

2

1

E1

E2

E3

η

η

X2

X1

X3

e2

e1

e3

x0

Figure B.4: Description of the beam kinematics (Geradin andCardona, 2001)

293


After the deformation, the position of the same point can be writtenas

x = x0 +X2e2 +X3e3 = x0 + RY (B.45)

where x0 = X0 + u0 and u0 is the displacement of the cross sectionfrom its reference position.

The most convenient way of expressing the beam deformation isby computing the position gradients with respect to the parameter ηbefore and after the deformation, and express their difference in thematerial frame:

D(η,X2, X3) = RT dxdη− dXdη

(B.46)

Differentiating Eqs. B.44 and B.45 and substituting in Eq. B.46,the material deformation measure yields:

D(η,X2, X3) = RT

(dx0

dη− e1

)+ RT dR

dηY (B.47)

The second term of Eq. B.47 describes the deformations involvingrotation of the cross section and can be interpreted as the materialmeasure of the curvature:

K = RT dRdη

(B.48)

This term represents the rotation gradient along the neutral axisand its correspondent axial vector is K(η) = vect(K).

When adopting the cartesian rotation vector as the spherical motionparametrization, the rotation operator R(Ψ) can be described by Eq.B.29, the tangent operator T(Ψ) by Eq. B.33 and the curvature vectorK(Ψ) by

K(Ψ) = T(Ψ)dΨdη

(B.49)

B.4.4 Sliding joints

This formulation can be extended to consider the connection betweenrigid and flexible bodies representing a sliding joint. The location ofthe rigid body sliding over a flexible beam can be represented by aparameter η which determines the location of the rigid body A on the

294


flexible body (Figs. B.3b and B.5). In this case, the triad ξ′1, ξ′2, ξ3

′

is related to the actual connection between the bodies. The positionx(η) and the Cartesian rotation vector Ψ(η), related to the actualconnection, can be calculated by the following discretization:

x(η) =2∑i=1

Ni(η)xi

Ψ(η) =2∑i=1

Ni(η)Ψi

(B.50)

where Ni(η) is the shape function used in the discretization of the beamelements and the parameter η determines a location between the nodes1 and 2 whose the position vectors are x1 and x2 and the Cartesianrotation vectors Ψ1 and Ψ2, respectively (see Fig. B.5).

Considering Eqs. B.35, B.36 and B.49, the discretized materialrotation, δΘ, and the discretized expression of the curvatures, K(η),become:

δΘ(η) = T(Ψ)2∑i=1

Ni(η)δΨi (B.51)

δK(η) = T(Ψ)2∑i=1

N ′i(η)Ψi (B.52)

Considering Eq. B.50, the rotation matrix can be described by:

R(η) = R(Ψ(η)) = exp(Ψ(η)) (B.53)

Moreover, considering the expansion Eq. B.30, the discretized ro-tation matrix can be calculated by:

δR(η) = R(Ψ(η))K(η)δη + R(Ψ(η))δΘ(η) (B.54)

The inclusion of the parameter η adds a degree-of-freedom thatshould be removed by an extra constraint. Therefore, the set of con-straints to correctly represent a sliding joint is

295


Φ1 = µ′T1 ξ′3 = µ1RT

AR(η)ξ3

Φ2 = µ′T2 ξ′3 = µ2RT

AR(η)ξ3

Φ3 = µ′T1 ξ′2 = µ1RT

AR(η)ξ2

Φ4 = µ′T1 (x(η)− xA) = (RAµ1)T (x(η)− xA)Φ5 = µ′T2 (x(η)− xA) = (RAµ2)T (x(η)− xA)Φ6 = µ′T3 (x(η)− xA) = (RAµ3)T (x(η)− xA)

(B.55)

The variations of the constraints Φ1 and Φ4 yield, respectively

δΦ1 = (δRAµ1)TR(η)ξ3 + (RAµ1)T δR(η)ξ3 == (δRAµ1)TR(η)ξ3 + (RAµ1)TR(η)K(η)ξ3δη + (RAµ1)TR(η)Θ(η)ξ3

(B.56)

δΦ4 = (RAδΘA)T (µ′1(x(η)− xA)) + (RAµ1)T (δx(η)− δxA)(B.57)

The variation of the constraints Φ2, Φ3, Φ5 and Φ6 can be per-formed in the same way.

Considering that the body A is sliding along a beam between thenodes B and C (see Fig B.5) and adopting that x(η) = (1−η)xB+ηxCand Ψ(η) = (1 − η)ΨB + ηΨC , the Jacobian matrix can be de-scribed by Eq. B.58 according to the varied coordinates δqT =[δxTa δxTb δxTc δΨT

a δΨTb δΨT

c δη]

Node η

Rigid Body

flexible beam

Node 1

Node 2Node 1

Node 2

L

Figure B.5: Sliding joint scheme

296

B.5. CONCLUSIONS

B =

0 0 0 (µ′1R(η)ξ3)RATA . . .0 0 0 (µ′2R(η)ξ3)RATA . . .0 0 0 (µ′1R(η)ξ2)RATA . . .−µ′T1 µ′T1 (1− η) µ′T1 η µ′1(x(η)− xA)TRATA . . .−µ′T2 µ′T2 (1− η) µ′T2 η µ′2(x(η)− xA)TRATA . . .−µ′T3 µ′T3 (1− η) µ′T3 η µ′3(x(η)− xA)TRATA . . .

. . . µ′T1 ξ′3(1− η)R(η)T(η) µ′T1 ξ

′3ηR(η)T(η) . . .

. . . µ′T2 ξ′3(1− η)R(η)T(η) µ′T2 ξ

′3ηR(η)T(η) . . .

. . . µ′T1 ξ′2(1− η)R(η)T(η) µ′T1 ξ

′2ηR(η)T(η) . . .

. . . 0 0 . . .

. . . 0 0 . . .

. . . 0 0 . . .

. . . µT1 ξ′3K(η)

. . . µT2 ξ′3K(η)

. . . µT1 ξ′2K(η)

. . . µ′T1 (xC − xB)

. . . µ′T2 (xC − xB)

. . . µ′T3 (xC − xB)

(B.58)

This is a time-varying problem where η determines the locationof contact between the bodies for each instant. Considering the pick-and-place robot as test-case, the relative motion between the flexiblebeam and the linear motor can be implemented using Eq. B.58, wherexB, xC , xA, their related rotation matrices, tangent operators andcurvatures are updated according to the parameter η.

B.5 Conclusions

A non-exhaustive overview about flexible multibody theory has beendescribed. These concepts and techniques are going to be appliedthroughout the thesis, especially, in Chapters 5, 6, 7 and 8.

Moreover, a specific joint has been derived to enable the modeling ofserial machines: the sliding joint. Its derivation has been based on thederivation of the prismatic joint. A simplified derivation is presentedin the Chapters 6 and 7.

297

Appendix C

Review onTime-Integration methods

This appendix summarizes the time integration techniques applied inthe multibody field.

C.1 Integration methods for mechatronic sim-ulation

Differential-Algebraic Equations (DAEs) may appear during the mod-eling process of a multibody system (see Schiehlen (1997)). In general,DAEs can be described by the general form:

f(x,dx

dt, y, t) = 0 (C.1)

where x ∈ Rn are variables for which derivatives are present (differ-ential variables), y ∈ Rm are variables for which no derivatives arepresent (algebraic variables) and t ∈ R is an independent variable(usually time).

An actively controlled multibody systems with mechatronic com-ponents may also require a representation by differential-algebraicalequations.

Ordinary Differential Equation (ODE) can be described by Eq. C.2,in the case that Eq. C.1 can be re-written in an explicit form.

dx

dt= f(x, t) = 0 (C.2)

299

C Review on Time-Integration methods

In same cases, the algebraic constraint can be described explicitlyyielding semi-explicit DAE:

f(x,dx

dt, y, t) = 0

g(x, y, t) = 0(C.3)

The time integration of these equations may become an issue sincetheir numerical treatment is difficult. The problem of the numericalintegration of DAEs is due to the inherent instability caused by theindex-3 found for mechanical systems (Schiehlen, 1997). Therefore, itrequires sophisticated implicit integration algorithms. The differentialindex of a DAE system as defined in Eq. C.3 is the number of timesthat the equations should be differentiated to obtain a set of ODEs.

Moreover, the integration algorithm may face ’stiff’ problems thatcan be defined by a system which its eigenfrequencies are distributedover a broad frequency range. This can be caused by the physicalproperties of the system or by the numerical techniques required tospatial discretization. Stiffness affects the speed at which the equationsof motion can be integrated, causing certain integrators to proceed veryslowly with small time steps, even though the solution found is verysmooth and well behaved.

Some remarks about time integration for multibody system andintegrated environment for mechatronic systems are stated in the fol-lowing sections.

C.1.1 Integration methods for multibody dynamics

Most of commercial available multibody software are based on Carte-sian coordinates. Software as LMS V.Lab Motion and ADAMS, theequations of motion are derived using Newton-Euler formalism and theconstraints are applied using Lagrange multipliers. This approach leadsto a set of semi-explicit DAEs.

Table C.1 shows an overview of the integration algorithms availablein LMS V.Lab Motion. ADAMS preferably offers Backward Differenti-ation Formula (BDF) which is an implicit integrator inherently stablefor stiff systems.

A singlestep solver requires only the solution at the immediatelypreceding time point. A multistep solver, on the other hand, requiresthe solutions at several preceding time points to compute the currentsolution.

300

C.1. INTEGRATION METHODS FOR MECHATRONIC SIMULATION

Acronym Name Type Usage

PECEPredict explicit discontinuous

Evaluate multistep and non-stiffCorrect, Evaluate systems

BDFBackward implicit smooth and

Differentiation multistep stiffFormula systems

RKexplicit extremely

Runge-Kutta singlestep discontinoussystems

Table C.1: Integration Methods for Multibody Systems

The basic idea of implicit methods is the development of the alge-braic relationship, such as the trapezoidal rule, between the state ofthe system at different time instances. Then, assuming that the sys-tem state is known at the required previous time, the equations aresolved to yield the state of the system at the current time. On the con-trary, explicit methods are based on explicit equations which providethe state of the system at the current time as a function of the systemstate at the required previous time.

Explicit methods are simpler to implement. However, a disadvan-tage of explicit methods is their numerical instability under inappropri-ately chosen integration time step. In general, stiff large scale systemsrequire relatively small time steps to remain stable, resulting in longexecution times. Adjustable time step algorithms have been imple-mented. However, the speed improvement depends on the problem be-ing solved, and in many cases no improvement is possible (Solodovniket al., 1998).

Besides its computational complexity, implicit methods are uncon-ditionally stable. Therefore, implicit methods may be the best optionfor stiff equations of motion.

C.1.1.1 Integration methods for integrated environment - Aco-simulation scheme

Some commercial multibody software alow a connection with Mat-lab/Simulink in order to simulate controllers, for instance Virtual.LabMotion and ADAMS. Aiming this, some connection elements, such as

301


actuators and sensors, are available to make the interface between theenvironments (LMS.International, 2006). Generally, the solution of thisintegrated environment can be performed in two ways (Adams, 2001;LMS.International, 2006):

1. Co-simulation mode : In this case, the integrators on both sides(controls package and multibody environment) are running inparallel. They exchange data (via connection elements) at a spec-ified step-size.

2. Block evaluation mode : In this mode, the controls package inte-grates all of the equations.

When the co-simulation mode is chosen, any of the integrationmethods in Matlab can be used. However, the integration algorithm ofthe multibody environment should be the BDF solver (implicit solver).The reason for not electing an explicit solver is that the same perfor-mance is achieved when using co-simulation or block evaluation modes.The co-simulation provides the best performance when a large stiffmechanism model is being simulated (LMS.International, 2006), how-ever not the most accurate results (Adams, 2001). The sampling rate,the duration at which information is exchanged between the environ-ments, has great effects in the co-simulation mode. It should be atleast more than twice the highest interested frequency on the mechan-ical side. A different choice can cause aliasing deteriorating the resultssince it generates a false (alias) frequency. Reasonable smooth resultsare obtained when the sample frequency lies between 20 and 40 timesthe frequency of interest.

Selecting an integrator on the controls system side is really model-dependent. In Matlab, the most used time-integration algorithms areode45 (Runge-Kutta), ode113, ode15s (the Gear method) and ode23tb.The results may differ depending on which integrator is used. The bestintegrator is probably the BCS Gear integrator, which works very wellfor numerically stiff models but it is not available in Matlab/Simulink.

C.1.2 Integration methods for flexible multibody dy-namics

According to the augmented Lagrangian method (Geradin and Car-dona, 2001), the dynamic equations and the constraint equations can

302


be described by Eqs. C.4, where k and p are the scaling and penaltyfactors.

Mq + BT (pΦ + kλ) = gkΦ(q) = 0

(C.4)

where q denotes the generalized coordinates vector, M the mass ma-trix, Φ the constraint vector, B the constraint gradient vector, λ theLagrange multipliers associated with the constraints and g the vector ofapparent forces representing the internal, external and complementaryinertia forces.

Due to the spatial discretization required by the FE approach, flex-ible multibody system may be consisted by a sparse system of DAEs.Besides, the solution algorithms being based on sparse matrix algebra,the flexibility causes the appearance of high frequencies. This charac-teristic yield the motion equations to be stiff, which motivates the useof implicit time integration according to Geradin and Cardona (2001).

Therefore, the algorithms used in multibody system, as Runge-Kutta algorithm, can not be applied. Moreover, BDF may introducean excess of artificial damping in the frequency range of interesting(Geradin and Cardona, 2001).

Bruls (2005) states that integration algorithms as constraint elim-ination or index reduction do not fulfil the original constraints at thedisplacement level. Moreover, integration algorithms that rely on theLagrangian structure of the mechanical system are not suitable since itis not always possible to describe control systems equation using thisapproach. Bruls (2005) concluded that improved Newmark algorithms,such as Hilber-Hughes-Taylor (HHT) and generalized-α methods, areappropriated for flexible multibody system.

Hereafter, the improved Newmark algorithms are revised. Thisoverview is based on the original Newmark scheme and its generalized-α extension described by Bruls and Golinval (2006).

The Taylor expansion of the displacement and velocity based ongeneralized coordinates can be described by:

qn+1 = qn + hqn +h2

2qn +

h3

6...qn + . . .

qn+1 = qn + hqn +h2

2qn + . . .

(C.5)

where h is the time-step size. Newmark truncated these equation in

303


the following form:

qn+1 = qn + hqn +h2

2qn + βh3...qn

qn+1 = qn + hqn + γh2qn(C.6)

Assuming that the acceleration is linear within the time step, Eq.C.7 can be written.

...qn+1 =qn+1 − qn

h(C.7)

Substituting Eq. C.7 into Eq. C.6 produces the Newmark equationsin the standard form:

qn+1 = qn + hqn +(1

2− β

)h2qn + βh2qn+1

qn+1 = qn + (1− γ)hqn + γhqn+1

(C.8)

where γ and β are numerical parameters. Choosing γ = 1/2 andβ = 1/4 leads to an unconditionally stable integration operator ofsecond-order accuracy. The average constant acceleration scheme canbe modified to introduce numerical damping. This numerical coeffi-cient may vary within the unconditional stability limit (Geradin andCardona, 2001):

γ =12

+ α β =14

(γ +

12

)2α > 0 (C.9)

where α is the numerical damping.However, more numerical damping should be introduced without

degrading the order of accuracy. The generalized-α method fulfill thisrequirement. The Newmark formulae, Eq. C.8, is applied to a modifiedsystem equation which considers both time steps (Bruls, 2005):

(1− αm)(Mq)n+1 + αm(Mq)n + (1− αm)g∗n+1 + αmg∗n = 0(1− αf )k(Φ)n+1 + αfk(Φ)n = 0

(C.10)where g∗ = Φq

T (pΦ + kλ − g, αm and αf are numerical parameters.The HHT algoritnm is obtained for αm = 0 and αf ∈ [0, 1/3]. Thegeneralized-α method is second-order accurate if

γ =12

+ αfm (C.11)

304


and high frequency dissipation is maximized if

β =14

(1 + αfm)2 (C.12)

where αfm = αf − αm. In summary, this method is second-orderaccurate and unconditionally stable for αm < αf ≤ 1/2 and γ = 1/2−αm+αf (Geradin and Cardona, 2001). Details about this method andthe stability analysis are described in Bruls (2005).

C.1.3 Integration methods for integrated environment

The explicit state-space form is widely used in the control field and isdescribed by Eq. C.13.

x = f(u,x, t)y = h(u,x, t) (C.13)

where x is the state-space variables vector, u is the sensor measurementvector, y is the generalized forces vector, f are the differential equationsof the dynamical states and h are the algebraic output equations (Fig.2.6).

The inclusion of the state-space representation in an integrated en-vironment as a flexible multibody package may be a challenge due tothe strongly coupling effect. Bruls (2005) has proved that an exten-sion of the generalized-α can be applied to integrate the applied tostate-space equations.

A stabilized scheme for the integration of the state-space is sug-gested Bruls (2005):

xn+1 = xn + h(1− θ)xn + hθxn+1 (C.14)

To increase the accuracy, a similar strategy applied to integrate theflexible multibody model was adopted when dealing with an isolatedstate-space system. Therefore, the discretized state equations can bedescribed by:

(1− δm)xn+1 + δmxn − (1− δf )fn+1 − δf fn = 0(1− δf )(yn+1 − hn+1) + δf (yn − hn) = 0 (C.15)

where δm and δf are numerical parameters. This scheme is second-order accurate for θ = 1/2 + (δf − δm).

305


When combining the mechanical equations and the state equations,some variable modification was performed in order to uniform the treat-ment of the state and displacement variable. Therefore, variables z areintroduced, where z = x. Moreover, the numerical parameters arechosen to be δm = αm, δf = αf and θ = γ.

Besides that, an observer equation will be defined foe each accel-eration measurement qi. In this way the output equation becomesy = h(u,x, t) + Lq, where L is a matrix that defines the localizationswhere the accelerations should be calculated.

The discretized equations of motion of an active system can bedescribed by Eq. C.16. Details about this method its stability analysisare described in Bruls (2005).

(1− αm)(Mq)n+1 + αm(Mq)n + (1− αm)g∗n+1 + αmg∗n = 0(1− αf )k(Φ)n+1 + αfk(Φ)n = 0

(1− αm)zn+1 + αmzn − (1− αf )fn+1 − αf fn = 0−(1− αm)Lqn+1 − αmLqn + (1− αf )(yn+1 − hn+1)+

+αf (yn − hn) = 0

qn+1 = qn + hqn +(1

2− β

)h2qn + βh2qn+1

zn+1 = zn + hzn +(1

2− β

)h2zn + βh2zn+1

qn+1 = qn + (1− γ)hqn + γhqn+1

zn+1 = zn + (1− γ)hzn + γhzn+1

(C.16)

C.2 Conclusions

This appendix have summarized the time integration techniques re-garding co-simulation schemes and integrated simulation employed inthe multibody field.

306

Curriculum Vitae

Personal data

Maıra Martins da SilvaAddress: Rua Lucio Rodrigues 100

13572-040 Sao Carlos - SP, BrazilE-mail: [email protected] and date of birth: Sao Carlos, April 19th 1979Nationality: Brazilian

Education

• 2005-2009: Ph.D. student at the Department of Mechanical En-gineering, Katholieke Universiteit Leuven, Belgium; funded bythe CAPES (2005-2008) and by the K.U. Leuven Research Coun-cil (2008-2009).

• 2004-2005: Pre-doctoral student at the Department of Mechan-ical Engineering, Katholieke Universiteit Leuven, Belgium.

• 2002-2004: Master student at the Department of MechanicalEngineering, University of Sao Paulo, Brazil; funded by the CNPq(2002-2004), Master thesis: Handling analysis of a light commer-cial vehicle considering the frame flexibility.

• 1997-2001: degree in Mechanical Engineering, specializationMechatronics (awarded as the best undergraduate student ofthe Engineering School of Sao Carlos), University of Sao Paulo,Brazil.

307

Curriculum Vitae

Working Experience

• 2002-2004: Mechanical Engineer at Multicorpos S.A -MSC.Software Brazil

• 2002: Trainee at DaimlerChrysler AG - Stuttgart

• 2001: Trainee at Eaton Transmissions Brazil

308

List of publications

International peer reviewed journal articles

[1] M.M. da Silva, J. Swevers, J. De Caigny, O. Bruls, M. Miche-lin, C. Baradat, O. Tempier, W. Desmet and H. Van Brussel,“Computer-Aided Integrated Design of Parallel Kinematic Ma-chines”, Mechanism and Machine Theory, submitted, 2009.

[2] M.M. da Silva, O. Bruls, W. Desmet and H. Van Brussel, “In-tegrated Structure and Control design for Mechatronic Systemswith Configuration-Dependent Dynamics”, Mechatronics, sub-mitted, 2008.

[3] M.M. da Silva, O. Bruls, S. Swevers, W. Desmet and H. VanBrussel, “Computer-Aided Integrated Design for Machines withVarying Dynamics”, Mechanism and Machine Theory, pp. 1-22,10.1016/j.mechmachtheory.2009.02.006.

[4] M.M. da Silva, W. Desmet and H. Van Brussel, “Design of Mecha-tronic Systems with Configuration-Dependent Dynamics: Simu-lation and Optimization”, IEEE/ASME Transactions on Mecha-tronics, vol. 13, no. 6, pp. 638-646, 2008.

[5] L.P.R. de Oliveira, M.M. da Silva, P. Sas, W. Desmet and H. VanBrussel, “Concurrent Mechatronic Design Approach for ActiveControl of Cavity Noise”, Journal of Sound and Vibration (JSV),vol. 314, no. 3-5, pp. 507-525, 2008.

[6] M.M. da Silva and A. Costa Neto, “Handling Analysis of a LightCommercial Vehicle Considering the Frame Flexibility”, Interna-tional Review of Mechanical Engineering (IREME), vol. 1, pp.332-339, 2007.

309


[7] H. Van der Auweraer, K. Janssens, L. de Oliveira, M. da Silvaand W. Desmet, “Virtual Prototyping for Sound Quality Designof Automobiles”, Sound and Vibration (SV), vol. 41, no. 4, pp.26-30, 2007.

Articles in books

[1] M.M. da Silva , O. Bruls, B. Paijmans, W. Desmet and H. VanBrussel , “Computer-aided integrated design for mechatronic sys-tems with varying dynamics”, Motion and Vibration Control - Se-lected Papers from MOVIC 2008, SPRINGER, pp. 53-62, ISBM978-1-4020-9437-8, 2008.

Full papers in proceedings of international con-ferences

[1] M.M. da Silva , O. Bruls, B. Paijmans, W. Desmet and H. VanBrussel , “Computer-aided integrated design for mechatronic sys-tems with varying structural flexibilities”, Proceedings of The9th International Conference on Motion and Vibration Control- MOVIC 2008, Munich (Germany), pp. 1-10, September 15-18,2008.

[2] F. Haase, M. Kauba, D. Mayer , H. Van der Auweraer, P. Gaj-datsy, L.P.R. de Oliveira, M. da Silva, P. Sas and A. Deraemaeker,“Active Vibration Control of an Automotive Firewall for InteriorNoise Reduction”, Proceedings of Adaptronic Congress, Berlin(Germany), pp. 1-10, May 20-22, 2008.

[3] T. Cardone, R. d’Ippolito, S. Donders, H. Van der Auweraer,L.P.R. de Oliveira and M. da Silva , “Robustness Assessmentand Optimization of a Simplified Smart Structure Model forStructural Acoustic Control”, Proceedings of Leuven Symposiumon Applied Mechanics in Engineering, LSAME.08, Leuven (Bel-gium), pp. 1-6, March 31 - Appril 2, 2008.

[4] M.M. da Silva , W. Desmet and H. Van Brussel , “Designof mechatronic systems with configuration-dependent dynam-ics: simulation and optimization”, Proceedings of the 2007

310


IEEE/ASME International Conference on Advanced IntelligentMechatronics - AIM 2007, Zurich (Switzerland), pp. 1-6,September 4-7, 2007.

[5] M.M. da Silva , W. Desmet and H. Van Brussel, “Towards a con-current optimization of mechatronic systems with configuration-dependent dynamics”, Proceedings of the 12th IFToMM WorldCongress - IFtoMM 2007, Besancon (France), pp. 1-6, June 18-21, 2007.

[6] H. Van der Auweraer, K. Janssens, A. Vecchio, L.P.R. de Oliveira,M.M. da Silva and W. Desmet, “Optimisation of ANVC SystemsDesign Using an Integrated Simulation Approach”, Proceedingsof Adaptronic Congress, Gottingen (Germany), pp. 1-10, May23-24, 2007.

[7] M. da Silva, W. Desmet and H. Van Brussel, “Simulation ofsystems parametrically dependent on the spatial configuration:a mechatronic approach”, Proceedings of the XII InternationalSymposium on Dynamic Problems of Mechanics - DINAME 2007,Ilhabela (Brazil), pp. 1-10, February 26 - March 2, 2007.

[8] R. Cunha, M. da Silva and A. Costa Neto, “Modeling and qual-itative analysis of tire wear for steady state cornering maneu-vers”, Proceedings of the XII International Symposium on Dy-namic Problems of Mechanics - DINAME 2007, Ilhabela (Brazil),pp. 1-10, February 26 - March 2, 2007.

[9] H. Van der Auweraer, P. Mas, L.P.R. de Oliveira, M.M. da Silvaand W. Desmet, “Simulation-Based Optimization of an ActiveNoise and Vibration Control Solution”, Proceedings of IMACXXV, Orlando, (Florida, USA), pp. 1-10, February 19-22, 2007.

[10] H. Van der Auweraer, P. Mas, P. Segaert, L.P.R. de Oliveira,M.M. da Silva and W. Desmet, “CAE-based Design of ActiveNoise Control Solutions”, Proceedings of 10th Symposium on In-ternational Automotive Technology - SIAT 2007, Pune (India),SAE Paper No. 2007-26-032, January 17-20, 2007.

[11] H. Van der Auweraer, K. Janssens, L. de Oliveira, M. da Silvaand W. Desmet, “Sound synthesis approach for sound quality

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design of ANVC applications”, Proceedings of the 35th Interna-tional Congress and Exposition on Noise Control Engineering -INTERNOISE 2006, Honolulu (Hawai, USA), pp. 1-10, Decem-ber 3-6, 2006.

[12] H. Van der Auweraer, L. de Oliveira, M. da Silva, S. Herold, J.Mohring and A. Deraemaeker, “A virtual prototyping approachto the design of smart structures applications”, Proceedings of theInternational Conference on Noise and Vibration Engineering -ISMA 2006, Leuven (Belgium), pp. 273-284, September 18-20,2006.

[13] M. Da Silva, O. Bruls, B. Paijmans, W. Desmet and H. Van Brus-sel, “Concurrent simulation of mechatronic systems with variablemechanical configuration”, Proceedings of the International Con-ference on Noise and Vibration Engineering - ISMA 2006, Leuven(Belgium), pp. 69-79, September 18-20, 2006.

[14] H. Van der Auweraer, S. Herold, J. Mohring, L. de Oliveira, M.da Silva and G. Pinte, “Virtual prototyping of active noise andvibration solutions”, Proceedings of the Transportation ResearchArena - TRA 2006, Goteborg (Sweden), pp. 1-10, June 12-16,2006.

[15] H. Van der Auweraer, S. Herold, J. Mohring, L. de Oliveira, M. daSilva and A. Deraemaeker, “CAE approach to the design of smartstructures applications”, Proceedings of the 6th European Confer-ence on Noise Control - EURONOISE 2006, Tampere (Finland),pp. 1-6, May 10 - June 1, 2006.

[16] M.M. da Silva, A. Costa Neto, “Multibody handling analysis ofa light commercial truck considering frame flexibility”, Proceed-ings of the XI International Symposium on Dynamic Problemsof Mechanics - DINAME 2005, Ouro Preto (Brazil), pp. 1-10,February 28 - March 4, 2005.

[17] L.P.R. Oliveira, M.M. da Silva and P.S. Varoto, “ExperimentalModal Analysis Approaches for High Quality Tests”, Proceedingsof SAE Congress Brazil, Sao Paulo (Brazil), SAE Number 251PE, November 1-2, 2003.

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[18] M.M. da Silva, L.P.R. Oliveira, L.G.S. Ericsson, A. Costa Netoand P.S. Varoto, “An experimental investigation on the modalcharacteristics of an off-road competition vehicle chassis”, Pro-ceedings of SAE Congress Brazil, Sao Paulo (Brazil), pp. 1-10,November 1-2, 2003.

[19] M.M. da Silva, L.P.R. Oliveira, L.G.S. Ericsson, A. Costa Netoand P.S. Varoto, “An experimental investigation on the modalcharacteristics of an off-road competition vehicle chassis”, Pro-ceedings of the XXIV CILAMCE - CILAMCE 2003, Ouro Preto(Brazil), pp. 1-10, October 29-31, 2003.

Abstracts in proceedings of international con-ferences

[1] M. Da Silva, O. Bruls, W. Desmet and H. Van Brussel,“Computer-aided integrated design for mechatronic systemswith configuration-dependent dynamics”, Book of Abstracts ofACOMEN 2008, Liege (Belgium), pp. 1, May 26-28, 2008.

[2] H. Van der Auweraer, S. Herold, J. Mohring, L. de Oliveira, M.da Silva and G. Pinte, “Optimal integrated design procedurefor ASAC”, Book of Abstracts Engineered Adaptive Structures,Maiori (Italy), pp. 1, June 18-23, 2006.

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