Modeling Steering System

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Paper No.: 993076An ASAE Meeting Presentation

UILU-ENG-99-7019

MODELING AND SIMULATION OF AN ELECTROHYDRAULICSTEERING SYSTEM

Hongchu Qiu Qin Zhang John F. Reid Duqiang Wu

Graduate Assistant Assistant Professor Professor Visiting Scholar

Department of Agricultural EngineeringUniversity of Illinois at Urbana-Champaign

1304 West Pennsylvania Avenue

Urbana, IL 61801, USA

Written for Presentation at the 1999 ASAE/CSAE-SCGR Annual International Meeting

Sheraton Center Hall

Toronto, Ontario, CanadaJuly 18 21, 1999.

Summary:

This paper presents a model and its simulation results of an agricultural vehicle electrohydraulic (E/H) steering

system. The dynamic characteristics of an E/H steering system plays an important role in realizing accurate

steering control for an automatically guided agricultural vehicle. E/H steering systems are typically nonlinear with

large deadband, asymmetric flow gain, time delay, hysteresis, and saturation, which result in difficulties in the

system analysis, design, and control. In this paper, a mathematical model has been developed for an E/H steering

system. This model has taken the main components of the E/H steering system, including a PWM driver, an E/H

proportional directional control valve, steering actuating cylinders, steering linkage, and wheel-ground interactions,into consideration. It was then linearized with assumptions of incompressible fluid, the zero lapped valve, and

frictionless. This model has been validated via vehicle steering tests. A third order liner model with considering

the deadband, saturation, and time delay could well approximate the dynamic response within the steering angle

range of 15 to 15 degrees. The validated model was used to estimate the dynamic behaviors of E/H steering

systems in steering controller design. Those parameters are critical for designing high performance steering

controller.

Keywords:

Dynamic model, simulation, nonlinear system, electrohydraulic system, steering system, agricultural vehicles.

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Qiu et al. : Modeling and Simulation of an Electrohydraulic Steering System

2

DESCRIPTION OF ELECTROHYDRAULIC STEERING SYSTEM

This E/H steering system model was developed based on a research platform of agricultural vehicle. This

research platform was based on a Case-IH Magnum18920 MFD agricultural tractor

2, on which the

steering system had been modified by adding a solenoid driven E/H proportional steering control valve for

implementing automatic steering control. The rest of hydraulic components remained unmodified. The

main components of the E/H steering system included a pulse-width-modulus (PWM) E/H valve driver, anE/H steering control valve, two steering actuating cylinders, and the steering linkage (Figure 1).

The E/H steering control valve was a three-position four-port proportional direction control valve. This

control valve controlled both the flow direction and flow rate so that the steering actuators could steer

front wheels to make desired turns. Since the E/H steering system was added in parallel to the existinghandpump steering system, this modified platform could be steered either manually or automatically.

A High Country Tek s (Nevada City, CA) dual-coil PWM driver card2was used to drive solenoids in

implementing valve control. This PWM driver was driven by 12-Volt DC power. A computer-based

controller would provide voltage steering control signals to the PWM driver, which converted steering

signals into duty cycles of DC currents, and then drove corresponding solenoid on the E/H valve to controlflow direction and flow rate.

A linear potentiometer was used as the wheel angle feedback sensor to monitor the steering control

performance. The potentiometer was mounted on the right-side hydraulic steering actuator to measure its

displacement. According to kinematic analysis of the steering linkage, the relationship between wheelangle and actuator displacement could be determined. In this paper, Ackerman steering angle was chosen

as the nominal wheel angle to avoid possible confusion caused by different angles on two front wheels

during turning. Wheel angle conversion model will be discussed in detail below.

The wheel-ground interaction was handled as an external disturbance on steering load. It was affected by

vehicle speed, soil property, tire pressure, and other factors. Therefore, the wheel-ground interaction acted

on wheels as a time-varying load in this model. It is one of the common characteristics for offroadvehicles and results in uncertainty and inconsistency to agricultural vehicle dynamic models.

KINEMATIC ANALYSIS OF STEERING LINKAGE

It is necessary to have kinematic analysis of the steering linkage in order to derive a conversion equation

for converting cylinder displacement reading into steering angle. This relationship provides the feedback

information for the closed-loop control to achieve an accurate tracking and high performance. Moreover,kinematic analysis will also determine the relationship between steering loads and steering actuating

forces.

The steering linkage on the Case-IH Magnum 8920 MFD tractor is a spatial linkage consisting of twoactuating cylinders, a linking rod, two symmetric kingpin axles and front wheels, as shown in Figure 2.

The bars AOa and BOb are turning with the left-side kingpin axle while the bars COc and DOd are

turning with the right-side kingpin axle. aO and bO are fixed points on the left-side kingpin axle while cO

1 CaseIH and Magnum are trade marks of Case Corporation, used with permission.2Mention of a trade name, proprietary product, or specific equipment does not constitute a guarantee or warranty by the

University of Illinois, and does not imply the approval of the named product to the exclusion of other products that may be

suitable.


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and dO are fixed points on the right-side kingpin axle. The linking rod BCconnects the two kingpin

axles. 3AO is the left-side actuator and 4DO is the right-side actuator. The points 3O and 4O are fixed

joints. The kingpin angle is 7to the axiszin the yzplane, which results in the steering system a spatiallinkage. According to mobility analysis of this linkage, this steering linkage is a one-degree-of-freedom

space linkage referred to the frame.

An algebraic analysis method was used to describe the nonlinear geometric relationship for the steeringlinkage. The displacement of the right-side actuator was chosen as the input variable, and wheel angles

and steer arms were theoretically expressed as functions of the actuator displacement. The steering angle,

angular velocity and acceleration of the front wheels could be determined using the developed equations.Assumptions that all components were rigid bodies, and gaps in all joints were negligible were made to

simplify the model.

All other steering system geometric parameters could be determined if the angles of both front wheels

were known. UseR and L , respectively, to represent wheel angles of right-side and left-side wheels

referring to vehicle center line (Fig. 2). The trajectories of points A and B were circles around the left-side kingpin axle. Similarly, points C and D would turn around the right-side kingpin axle. Therefore, theposition of points A, B, C, and D could theoretically be determined by wheel angles using two rotational

transformations and one translation transformation as expressed in Eqs. (1) (4). Since the steering

linkage is a one-degree-of-freedom system, the wheel angles R and L was constrained by the linking

rod (Eq. 5).

+

+

+

=

aAO

AO

aAO

AOLA

AOLA

LL

LL

A

A

A

z

y

x

R

R

z

y

x

a

0

)sin(

)cos(

cossin0

sincos0

001

1

1

(1)

+

+

+

=

b

b

b

BO

BO

BO

BOLB

BOLB

LL

LL

B

B

B

z

y

x

R

R

z

y

x

0

)sin(

)cos(

cossin0

sincos0

001

1

1

(2)

+

+

+

=

c

c

c

CO

CO

CO

CORC

CORC

RR

RR

C

C

C

z

y

x

R

R

z

y

x

0

)sin(

)cos(

cossin0

sincos0

001

2

2

(3)

+

+

+

=

d

d

d

DO

DO

DO

DORD

DORD

RR

RR

D

D

D

z

y

x

R

R

z

y

x

0

)sin(

)cos(

cossin0

sincos0

001

2

2

(4)

BCCBCBCBLzzyyxx =++ 222 )()()( (5)


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Where: L = left-side wheel angle

R = right-side wheel angle

L = left-side kingpin angle

R = right-side kingpin angle = - L

iR = distance to the nearest kingpin axle from point i (i=A, B, C, and D)

0i = initial angle to reference for point i (i=A, B, C, and D)

BCL = link length of tie rod BC

],,[ 000 iii zyx = coordinates on the nearest kingpin axle from point i

With Eqs. (1)-(5), the wheel angels could be numerically determined as a function of the actuator

displacement (Figure 3). This conversion equation was derived from test data using the least square

fitting. The fitted equation of Ackerman angle a (in degree) to the actuator displacement L (in mm) isgiven as Eq. (6).

LLa = 3335.00005.02

(6)

This Ackerman angle curve could be linearized by a straight line near the zero angle and expressed as Eq.

(7), if the displacement L is small (less than 40 mm). The zero angle was defined as the status at whichthe tractor is keeping straight-line tracking.

La = 3129.0 (7)

Relationships of steer arms to the right-side actuator displacement could be derived similarly and areshown in Eqs. (8) and (9). The plot of this relationship is shown in Figure 4.

94.1651329.00027.0

10104108

2

3648511

+

+=

LL

LLLLr

(8)

63.1651356.00017.0

10103102

2

354759

++

=

LL

LLLLl (9)

Equation (10) gives the calibration between the linear potentiometer output in volts and the displacement

of the right-side actuator in mm.

605.100365.43 += VL (10)

where: a = Ackermann angle and

L = displacement of right-side actuator

lL = steer arm of left-side actuator

rL = steer arm of right-side actuator.

MODELING OF STEERING SYSTEM

The E/H steering system was modeled by several subsystems, including an E/H valve subsystem, an

orifice flow control subsystem, and a steering linkage subsystem. The E/H valve subsystem includes a


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PWM driver, two solenoids, and the steering control valve. Since the orifice flow plays a very important

role in the proportional flow control, therefore, it is discussed separately.

E/H Valve Subsystem

In controlling an E/H proportional valve, a PWM driver is commonly used to convert the voltage controlsignal into a modulated current to energize solenoid drivers, which will then generate a magnetic force topush the valve spool. The spool controls the size of valve orifices to govern the flow rate passing the

valve so that the speed of hydraulic steering actuators can proportionally be adjusted. The E/H valve

subsystem is shown in Figure 5.

The current to the solenoid drivers is modulated by a PWM driver (Eq. 11). The solenoids will thengenerate magnetic forces proportionally to the current to push the spool (Eq. 12). With the consideration

of Coulomb friction, the displacement of the valve spool can be determined by the Eq. (13).

R

V

dt

dii =+ (11)

iBF p= (12)

dt

xdm

dt

dxckx

dt

dxsignFF f

2

)( +++= (13)

Where: pB = solenoid coefficient

fF = Coulomb friction force that changes direction as the velocity changes its sign

V= applied voltage to the PWM

= coil time constant (L/R)i = coil currentc = proportional damping coefficient

k = spool spring stiffness

L = coil inductancem = spool and solenoid total mass

R = coil resistance including the inductance resistance.

x = spool displacement

Solenoid forces can be simplified as a linear function of the input voltage to PWM driver (Eq. 14).Therefore, dynamics behaviors of the spool can be expressed by Eq. (15).

uBF t= (14)

dt

xdm

dt

dxckx

dt

dxsignFuBF ft

2

)( ++== (15)

Where: tB = total factor coefficient including PWM, coil and solenoid,

u = input voltage.

Ori f ice Flow Control Subsystem

The volumetric flow rate can be expressed by the orifice equation (Anderson, 1988).


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PxACQ q

=

2)( (16)

Where Q = volumetric flow rate

A(x) = restricted area based on spool position,x

qC = flow coefficient

P = pressure differential= fluid mass density

Equation (16) is a nonlinear equation and can be linearized using a Taylor series expansion. It is chosen a

steady state operating point, ( 0X , 0lP ), as the expansion reference point to linearize the nonlinear

equation.

)(),(

)(),(

),( 000

000

00 ll

l

lll PP

P

PXQXX

X

PXQPXQQ

+

+= (17)

Equation (17) can further be written approximately as a linear homogenous equation as shown in Eq. (18)

if the initials ( 0X , 0lP ) equals to zero. Eq. (18) is often used to determine the flow rate passing the

steering control valve.

lvv PRXKQ = (18)

where,

X

PXQK lv

=

),( 00 (19)

l

lv

P

PXQR

=),( 00 (20)

Steering L inkage Subsystem

According to mobility analysis, the steering linkage subsystem is a one degree-of-freedom (DOF) spatiallinkage referred to the frame. Two single-rod double-action steering cylinders drive the steering linkage

to implement the steering operation. The total active steering torque is determined by actuating forces and

steer arms. All the moving parts in the steering system contribute the inertial mass. Moreover, suchinertia is often neglected because the tractor chassis moves in steering operation.

Lagrange s equations are used to establish the equivalent dynamic equations for this one DOF steering

system. Using Ackerman steer angle, a , as the generalized coordinates, all the other moving parts could

be expressed as functions of the Ackerman steer angle, )( aif . The total equivalent inertial moment is

noted as eqJ . Equations. (21) and (22) describe the dynamic behaviors of steering linkage subsystem.

The free body diagram of the equivalent steering system is shown in Figure 6.

=+ iaeqaeq MCJ (21)


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Lfcabba

Lrfcbbaai

LysignFAPAP

LysignFAPAPM

+

=))((

))((

(22)Where eqJ = total equivalent inertial moment for the steering system

eqC = steering system total equivalent damping

a = Ackermann steer angle second derivativea = derivative of the Ackermann steer anglesa = Ackermann steer angle

LM = total equivalent resistant moment (steering load)

aP = pressure in chamber a

bP = pressure in chamber b

aA = effective area in chamber a

bA = effective area in chamber b

fcF = Coulomb friction force in actuator piston

y = displacement of the right actuator.

Simpl if ication Assumptions and Model L ineari zation

As a flow control system, the output of the steering system is the angular velocity of front wheelscorresponding to the steering control input u. Model linearization is desirable to simplify the above

nonlinear equations of the steering system. Assumptions, including frictionless, no hydraulic leakage, and

incompressible fluid, were made in the linearization. From Eq. (7), the displacement of right-side

actuator, L , had a linear relationship to Ackerman angle, a . Assume that a zero lapped valve and zerosteering load were used, this steering system could be simplified as a third order model for velocity

control, or a fourth order model for position control.

)2)(1()(

)()(

22

nn

pssTs

K

sU

ssG

+++=

= (23)

)2)(1()(

)()(

22

nn

pssTss

K

sU

ssG

+++=

= (24)

Where = wheel angle velocity

= wheel angleU = input signal

T = time constant for the steering linkage subsystem

= damping ratio for the valve subsystem

n = natural frequency for the valve subsystem

K = overall gain


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Generally, the natural frequency of the valve, n is much higher than that of the steering system (1/T).

Therefore, this steering control system could be further simplified as a first order model (Eq. 25) for

velocity control, or a second order model (Eq. 26) for position control.

1)(

)()( +=

= Ts

K

sU

ssGp (25)

)1()(

)()(

+=

=

Tss

K

sU

ssGp (26)

SIMULATION AND VERIFICATION

From the developed model, dynamic behaviors of this E/H steering system could be simulated. In thisresearch, a SIMULINK model (Figure 7) has been developed for the simulation of the E/H steering

system. The simulation model in SIMULINK was based on a third order linearized model of this E/Hsteering system, which had considered the deadband, asymmetric flow gain, and saturation, with or

without considering the time delay. The output variable of the simulation model was the turning rate ofAckerman steering angle for this vehicle.

The E/H steering system model was validated through comparing simulation results with field test results

in terms of angular velocity of the vehicle Ackerman steering angle. In the field test, the steering control

signals were generated by the computer-based controller, and delivered to the PWM driver to control thesteering control valve in the research tractor platform. Figures 8 and 9 show the comparison of system

dynamic responses to different step input steering control signals from both the simulation and field tests.

The simulation model was developed based on a third order linearized E/H steering system model without

consideration of time delay. Figure 8 shows system dynamic responses obtained from both the simulationand field tests corresponding to exciting steering signals of 5 and 4 volts for a right-turn. Figure 9 shows

the similar results with exciting signals of -5 and -4 volts to make a left-turn. Results from both cases

indicated that the model was capable of providing system dynamic response evaluation with fare accuracy.However, there was some time delay from this steering system as the test results showed in Figures 8 and

9. The time delay was varying with the levels of input signal, and was estimated about 0.1 second for this

particular steering system.

When taking consideration of the time delay in the third order linearized E/H steering model, it could

improve the accuracy in simulating dynamic responses significantly. Figures 10 and 11 show thecomparison of system responses of both the simulation results and the test results from the field test results

with the exciting steering signals as showed in Figures 8 and 9. In both cases, the calculated system

responses matched to the test data very well after adding the time delay into the model. It could beconcluded that it is necessary to include the time delay in a third-order linearized model for accurate

dynamic responses simulation for an E/H steering system.

DISCUSSION AND CONCLUSIONS

This paper analyzed an agricultural vehicle E/H steering system by modeling this system. A third-order

linearized model, with assumptions of zero lapped valve, frictionless, and no leakage, has been developed

for E/H steering system analysis. This model used Ackerman angle and steering actuator displacement to


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describe the dynamic behaviors in steering rate control. Vehicle steering field tests, with different step

steering rate control signals to steer the vehicle Ackerman angle within -19 to +19 degrees, have beenconducted to validate the model. Results obtained from the field test indicated that the third-order

linearized model could satisfactorily approximate on system dynamic responses without considering time

delay effects for a nonlinear E/H steering system. Simulation accuracy could be greatly improved if the

time delay was considered in this third-order model. When the absolute value of the steering angle wasgreater than 15 degrees, the linearized model could result in some large errors compared with test results.

It could be concluded that a third-order linearized E/H steering model with time delay in consideration was

capable of describing the dynamics responses satisfactorily for a nonlinear E/H steering system when thesteering angle was in the range of 15 to +15 degrees.

Steering loads could be expressed as a disturbance to the system in this model. The steering load does

affect the system deadband and time delay significantly. In this linearized model, a zero steering load was

assumed. However, it should take steering load into consideration when developing a high performancemodel. System deadband was found varying with steering load. It is primarily resulted from the overlap

of valve spool, the spring preload, the friction, and the variation in steering load. The deadband could be

compensated by properly identifying its range, and design a proper compensation gain.

The causes for time delay were much more complicated and harder to handle than the deadband. There

were several factors resulting in time delay. System deadband and steering loads were among the oneshave the most significant effects. Valve configuration and system design specifications were the other

major effects on system time delay. Introducing a time delay factor to the simulation model, and a

response prediction function to the control system could compensate the effect of time delay effectively.

ACKNOWLEDGEMENTS

This project was partially supported by USDA Hatch Funds, Illinois Council on Food and Agricultural

Research (98-069 AE), and Case Corporation. All the mentioned supports are gratefully acknowledged.

REFERENCES

Anderson, W. 1988. Controlling Electrohydraulic Systems, Marcel Dekker, Inc.

Backe, W.1993. The present and future of fluid power. Proceeding Instn. Mech. Engrs., Part I: Journal

of Systems and Control Engineering, 207:193-212.

DeSantis, R.M. 1997. Modeling and path-tracking for a load-haul-dump mining vehicle. Transactions ofthe ASME,119: 40-47

Fitzsimons, P.M., and J.J. Palazzolo. 1996. Part I: Modeling of one-degree-of-freedom active hydraulic

mount, Journal of Dynamic Systems, Measurement, and Control, 118: 439-442.Heck, B.S., and A.A. Ferri. 1996. Model reduction of coulomb friction damped systems using singular

perturbation theory, Transactions of the ASME, 118: 84-91.

Krus, P., K. Weddfelt, and J. Palmgerg. 1994. Fast pipeline models for simulation of hydraulic systems.Transactions of the ASME, 116: 132-136.

Manring, N.D. and R.E. Johnson. 1996. Modeling and designing a variable-displacement open-loop pump,

Journal of Dynamic Systems, Measurement, and Control, 118: 267-271.


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Sepehri, N., F. Sassani, P. D. Lawrence, and A. Ghasempoor. 1996. Simulation and experimental studies

of gear backlash and stick-slip friction in hydraulic excavator swing motion, Journal of DynamicSystems, Measurement, and Control, 118: 463-467.

Vaughan, N.D. and J.B. Gamble. 1996. The modeling and simulation of a proportional solenoid valve,

Transactions of the ASME, 118: 120-125.

Wu, D. Q. Zhang, and J.F. Reid. 1998. Dynamic steering simulator of wheel type tractors,ASAE Paper983115. ASAE.


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Ob

O a

AB

C

D

O 3 O 4

O c

O d

O 1

O 2

zx

yo

Figure 2. Schematic diagram of the steering linkage on Case-IH Magnum

8920 MFD agricultural tractor.

PWM E/H Valve Actuatorsu i Q aSteering

Linkage

Hydraulic

System

P

F

Figure 1. Block diagram of E/H steering system for an agricultural tractor.


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Quadratic:

y = -0.0005x2- 0.3335x

R2= 1

Linear:

y = -0.3192x

R

2

= 0.9869

-40

-30

-20

-10

0

10

20

30

40

-120 -90 -60 -30 0 30 60 90 120

Displacement of right-sided cylinder, mm

Ackermannangle,

degree

Ackermann Angle [Degrees]

Poly. (Ackermann Angle [Degrees])

Linear (Ackermann Angle [Degrees])

Figure 3. Ackerman angle versus the displacement of right-side steeringactuator. The Ackerman angle curve was fitted by a quadratic

polynomial.

120

130

140

150

160

170

-150 -100 -50 0 50 100

Right-side Cylinder Displacement, mm

Ste

eringarmsoffrontwheels,mm

Effective Steer Arm of Left-side Cylinder

Effective Steer Arm of Right-side Cylinder

Figure 4. Steer arms of front wheel versus the displacement of

steering actuator.


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L r

L L

a

M f

J e q

C e q

F a

Fb

Figure 6. Free body diagram of the steering system. Fa and Fb

are , respectively, the actuating forces of left-side and

right-side actuators.

P W M S o l e n o i d S p o o l

ui F em x

Figure 5. Block diagram of the E/H valve subsystem.


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Zero-Order

Hold

Transport

Delay

94

s+12

Transfer Fcn1

2500

*0.7*50s+2

Transfer Fcn

angle

To Workspace2input

To Workspace1

output

To Workspace

Switch

Step

1

Slider

Gain

Signal

Generator

Scope2

Scope1Scope

Saturation1

SaturationProduct

s

1

Integrator

Dead Zone

1.18

Constant1

1

Constant

Figure 7. Simulation model using a third-order transfer function with

considering system deadband, saturation, and asymmetric flow

rate. Time delay can be either being considered or not in this

model.


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-5

0

5

10

15

20

25

30

35

40

1 1.1 1.2 1.3 1.4 1.5 1.6

Time, sec.

Stee

ringanglechangerate,

deg/s

Vel_R5 Vel_R4 Vel_SIM_R5 Vel_SIM_R4

Figure 8. Comparison of dynamic responses of field test and simulation results with a third-

order model without considering the time delay. Vel_R5 and Vel_R4 are,

respectively, the field test with step steering rate inputs of 5 and 4 volts.Vel_SIM_R5 and Vel_SIM_R4 are, respectively, simulation results corresponding

to the same step inputs. The steering actuators make right-turn.


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-50

-40

-30

-20

-10

0

10

1 1.1 1.2 1.3 1.4 1.5 1.6

Time, sec.

Steering

anglechangerate,

deg/s

Vel_L5 Vel_L4 Vel_SIM_L4 Vel_SIM_L5

Figure 9. Comparison of dynamic responses of field test and simulation results with a third-order model without considering the time delay. Vel_L5 and Vel_L4 are,respectively, the field test with step steering rate inputs of 5 and 4 volts.

Vel_SIM_L5 and Vel_SIM_L4 are, respectively, simulation results corresponding to

the same step inputs. The steering actuators make left-turn.


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-5

0

5

10

15

20

25

30

35

40

1 1.1 1.2 1.3 1.4 1.5 1.6

Time, sec.

Steeringang

lechangerate,

deg/s

Vel_R5 Vel_R4 Vel_R5_SIM Vel_R4_SIM

Figure 10. Comparison of dynamic responses of field test and simulation results with a third-

order model with considering the time delay. Vel_R5 and Vel_R4 are,

respectively, the field test with step steering rate inputs of 5 and 4 volts.

Vel_SIM_R5 and Vel_SIM_R4 are, respectively, simulation results corresponding

to the same step inputs. The steering actuators make right-turn.


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-50

-40

-30

-20

-10

0

10

1 1.1 1.2 1.3 1.4 1.5 1.6

Time, sec.

Steeringanglechangerate,

deg/s

Vel_L5 Vel_L4 Vel_L5_SIM Vel_L4_SIM

Figure 11. Comparison of dynamic responses of field test and simulation results with athird-order model with considering the time delay. Vel_L5 and Vel_L4 are,

respectively, the field test with step steering rate inputs of 5 and 4 volts.Vel_SIM_L5 and Vel_SIM_L4 are, respectively, simulation results

corresponding to the same step inputs. The steering actuators make left-turn.

Modeling Steering System

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