Compliant Control of Steer-by-Wire Systems

A. Emre Cetin∗, M. Arif Adli†, Duygun Erol Barkana‡, Haluk Kucuk§

∗KaleAltınay Robotik ve Otomasyon, Istanbul, TURKEY

Email: emre.cetin@kalealtinay.com.tr†TUBITAK, Ankara, TURKEY, Email: adli@eng.marmara.edu.tr

‡Yeditepe University, Electrical & Electronics Engineering Department, Istanbul, TURKEY

Email: duygunerol@yeditepe.edu.tr§Marmara University, Mechatronics Department, Istanbul, TURKEY

Email: halukkucuk@marmara.edu.tr

Abstract—Vehicle directional control and driver interactionassemblies have been built to study and implement impedancecontrol strategies on a steer-by-wire system. Adaptive on-lineestimation is used to identify the dynamic parameters of vehicledirectional control and driver interaction assemblies. A nonlinear4 DOF vehicle model, including longitudinal, lateral, yaw andquasi-static roll motions is derived using Newtonian mechanicsto simulate and test the impedance control strategies developed.Impedance control strategy is used to control the dynamiccharacteristics of the steering system for improving the steering-feel and vehicle stability.

Index Terms—steer-by-wire systems, impedance control, compli-ant control, adaptive on-line parameter estimation

I. INTRODUCTION

Nowadays the dynamics of the vehicles are playing an

important role in vehicle development [1]-[4]. As the dynamics

of the vehicles, safety and the comfort of passengers are

getting more important, the steering system is becoming one

of the most important parts of a vehicle [5]-[7]. The steering

system is an important part of the vehicle because it is the part

where the driver interacts with the vehicle to guide it. One

new technology getting great attraction is the steer-by-wire

systems which can be thought of as a subsystem of electric

power assisted steering systems. Various control system have

been developed for steer-by-wire systems.

The control of a steer-by-wire system has been studied on

a hardware-in-the-loop (HIL) system [8]. In [9], a nonlin-

ear tracking controller is presented for steer-by-wire systems

where the tracking errors are asymptotically forced to zero

under parameter uncertainty. The controller is validated on

a HIL simulator. The impedance control strategy has been

used to build a reference model including the driver torque

and the force on rack arising from the tire - road interaction

[10]. The drawback of the impedance controller is that a force

measurement on rack is required in addition to the driver

torque on steering wheel which may be expensive in steer-

by-wire implementations. In [11], a model reference adaptive

controller for steer-by-wire applications has been proposed for

two-wheel steering vehicles which can treat nonlinearities and

uncertainties between the slip angles and lateral forces on tires.

The proposed D* controller is validated on a HIL simulator.

In this study, compliance control of a steer-by-wire system is

developed.

As long as the steering system is a part of a vehicle where

the driver (human) is in interaction with it, it is reasonable

to think that compliance control strategies can be adapted

on electric power-assisted steering systems including steer-

by-wire systems. Compliance control strategies are known

as model based control methods. This means that, the main

requirement of compliance control strategies is that the dy-

namics of the system which will be controlled should be

known to a degree. Therefore, identification of the system

dynamics is necessary in order to implement these control

strategies appropriately. A steer-by-wire model can be obtained

with enough accuracy through the specifications supplied by

the manufacturer or as a result of an identification study.

The identification procedure can be either off-line or on-

line depending on the requirements [12],[13]. In this study,

the dynamics parameters of the steer-by-wire system, i.e.,

driver interaction, and vehicle directional control assemblies

are identified by adaptive on-line parameter estimation method

called output error [14]. In order to validate the compliance

control of a steer-by-wire system, a vehicle model is necessary

to simulate the centering force (steering feel) and forces arising

due to tire-road interaction in order to feed them back to the

driver. Thus, a nonlinear 4 DOF vehicle model is developed.

In this paper, the steering system dynamics, adaptive on-

line parameter estimation, dynamic model of the vehicle and

the controller details are described in Section II. In Section

III, experimental set-up of the steer-by-wire system used in

this work is given. Evaluation results of the steer-by-wire

system system and controllers are given in Section IV. Finally,

conclusion and a discussion about the future work of the

research are provided in Section V.

II. SYSTEM ARCHITECTURE AND CONTROL

The block diagram of the proposed adaptive steering control

system architecture is given in Fig. 1.

A. Steering System Dynamics

The steering system is one of the most important parts

of a vehicle since the driver interacts with it to guide the

vehicle. The direction of the vehicle is controlled via servo unit

installed on the steering gear in a similar way as rack mounted

electric power-assisted steering systems. This is the vehicle

2009 IEEE/ASME International Conference on Advanced Intelligent MechatronicsSuntec Convention and Exhibition CenterSingapore, July 14-17, 2009

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Fig. 1. Adaptive Steering Control System Architecture.

directional control assembly composed of steering gear, i.e.,

rack-and-pinion, servo unit and tie-rods connecting rack ends

to wheel assembly. Additionally, in order to feed the centering

force (steering feel) and the forces occurring due to tire road

interaction back to the driver perception, another servo unit

is installed on the steering column which is called the driver

interaction assembly. The driver interaction assembly consists

of steering wheel, steering column and servo unit. The direc-

tion of the vehicle is altered via a dedicated control algorithm

based on steering wheel angular position and the driver torque.

The driver torque is either measured by a torque sensor or

estimated from the known driver interaction assembly dynamic

parameters. The steer-by-wire driver interaction assembly is

illustrated in Fig. 2. 1 DOF system dynamics of the steer-by-

wire driver interaction assembly is given as

JSW θSW + BSW θSW + FSW sgnθSW = iGkAiM + τD (1)

where, JSW is the steering wheel inertia, including the col-

umn, motor and gear-box inertias, i.e., JC , JM , and JG

with respect to column, BSW is the steering wheel viscous

damping, including the column, motor and gear-box viscous

damping, i.e., BC , BM , and BG. θSW , θSW , θSW are the

steering column angular position, velocity and acceleration,

respectively. iG, τD are the servo motor gear box ratio, and

the driver torque, respectively. FSW is the static friction. kA is

the torque constant of the servo motor, and iM is the current

drawn by the motor.

Fig. 2. SBW Driver Interaction Assembly.

The state equations of the driver interaction assembly are

written as

x1 = x2 (2)

x2 = −aSW x2 − fSW sgnx2 + bSW iM + cSW τD (3)

where the state variable x1 is the angular position (θSW ) of the

steering wheel, x2 is the angular velocity (θSW ) of the steering

wheel, the viscous friction aSW is BC/JSW , the static friction

fSW was FSW /JSW and finally the input gains bSW and

cSW are iGkA/JSW , and 1/JSW , respectively. The unknown

parameters aSW , fSW , bSW and cSW in Eqn. 3 are found

using on-line adaptive estimator method which is explained

later in this section.

Steer-by-wire vehicle directional control assembly (rack-

and-pinion) is illustrated in Fig. 3. The experimental setup

is shown in Fig. 5. The 1 DOF system dynamics of the steer-

by-wire vehicle directional control is given as

MRxR+BRxR + FRsgnxR + 2KTRxR = iGkAiM/rP (4)

where MR is the rack mass including the servo motor and gear

box inertias, and BR is the rack viscous friction including

the servo motor and gear box viscous frictions. FR is the

static friction. xR, xR, xR are the rack position, velocity and

acceleration, respectively. KTR is the spring rate connecting

left and right rack ends to the fixture base, rP is the pinion

radius. The state equations of the vehicle directional control

assembly are given as

x3 = x4 (5)

x4 = −aRx4 − fRsgnx4 − 2kTRx3 + bRiM (6)

where the state variable x3 is the position of the rack (xR),x4 is the velocity of the rack (xR). The viscous friction

aR is BR/MR, the static friction fR is FR/MR, spring rate

kTR is KTR/MR and finally the input gain bR is iGkA/MR.

Unknown parameters aR, bR and fR of the vehicle direc-

tional control assembly are estimated using adaptive on-line

estimation techniques. The springs connected to the tie-rods

are disconnected during the estimation. Hence, Eqn. 6 can be

written as

x4 = −aRx4 − fRsgnx4 + bRiM (7)

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Fig. 3. Steer-by-Wire Vehicle Directional Control Assembly.

B. Vehicle Dynamics

This work uses a nonlinear 4 degree-of-freedom (DOF)

vehicle model including the longitudinal, lateral, yaw and

quasi-static roll motions, which is derived using Newtonian

mechanics. The nonlinear vehicle model is used to evaluate

the compliant control of a steer-wire system. Nonlinear Dugoff

tire model is used to estimate tire forces arising due to road-

tire interaction. The vehicle dynamics are not given in this

paper due to space limitations.

C. Adaptive On-line Parameter Estimation

In order to estimate the unknown parameters of the vehicle

directional control and driver interaction assemblies on-line,

the output error method is used [14]. Due to the fact that

the parameter estimators developed may be driven unstable by

bounded disturbance, un-modeled dynamics and unconsidered

nonlinearities, a gradient projection is used to eliminate these

improprieties. This method keeps the estimated parameter

values within previously defined bounds in the parameter space

switching the estimator off or on according to the estimated

values.

The state equation of driver interaction assembly has been

given in Eqn. 3. If we substitute the estimated values into this

equation, it becomes:

˙x2 = −aSW x2 − fSW sgnx2 + bSW iM + cSW τD (8)

The estimator equations for the unknown parameters in Eqn.

8 can be written for output error method as:

˙aSW =

{

−γ1εx2 if aSWL ≤ aSW ≤ aSWU

0 otherwise

˙bSW =

{

γ2εiM if bSWL ≤ bSW ≤ bSWU

0 otherwise(9)

˙cSW =

{

γ3ετD if cSWL ≤ cSW ≤ cSWU

0 otherwise

˙fSW =

{

−γ4εsgnx2 if fSWL ≤ fSW ≤ fSWU

0 otherwise

where ε = x2 − x2 is the state estimation error, and aSWL,

aSWU , bSWL, bSWU , cSWL, cSWU , fSWL, fSWU are the

lower and upper bounds for aSW , bSW , cSW and fSW

respectively.

The state equation of the vehicle directional control assem-

bly has been given in Eqn. 6. If we substitute the estimated

values into this equation, it becomes:

˙x4 = −aRx4 − fRsgnx4 + bRiM (10)

In a similar way, the estimator equations for the unknown

parameters in Eqn. 10 can be written for output error method

as:

˙aR =

{

−γ5εx4 if aRL ≤ aR ≤ aRU

0 otherwise

˙bR =

{

γ6εiM if bRL ≤ bR ≤ bRU

0 otherwise(11)

˙fR =

{

−γ7εsgnx4 if fRL ≤ fR ≤ fRU

0 otherwise

where ε = x4− x4 is the state estimation error, and aRL, aRU ,

bRL, bRU , fRL, fRU are the lower and upper bounds for aR,

bR, and fR, respectively. It should be noted that availability

of all state variables of a dynamic system for measurement

does not mean that there is no need for a state observer in

the adaptive control design. As it has been mentioned in the

previous section, the noise disturbance in the measured signals

may degrade the performance of the algorithm and result in

a parameter drift or high-gain instability when the controller

increases gain to counteract the disturbances. Consequently,

adaptation of a state observer to the system will be helpful

to resolve problems which may degrade the performance of

adaptive estimator as well as the stability. In this work the

Luenberger observer is used to observe the state variables to

make the signal better and purified from noise disturbance so

that the adaptive estimation and controller performance will

be improved.

D. Controller

Steer-by-wire systems have the flexibility to implement

any type of feasible control algorithm by modifying the

software. For instance, by selecting proper control input us-

ing impedance control strategy, steer-by-wire systems can be

forced to behave similar to hydraulic steering systems. In

addition to this, it is possible to modify the driver perceived

stiffness of the steering wheel, i.e. the dynamic behavior of

the system. It is possible to make the dynamic response of the

steering wheel a function of other vehicle state variables such

as longitudinal velocity, vehicle yaw rate, etc.

In order to control the steer-by-wire system to behave

similar to hydraulic steering system, the column dynamics of

a conventional hydraulic steering system is written here which

will be used to derive impedance controller for the steer-by-

wire system as,

JC θC + BC(θC − θTB) + KC(θC − θTB) = τD (12)

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where, JC , BC and KC are the steering column inertia,

the steering column viscous damping and the steering col-

umn stiffness, respectively. θC , θC , θC are the steering col-

umn angular position, velocity and acceleration, respectively.

θTB, θTB, θTB are the torsion-bar angular position, velocity

and acceleration, respectively. τD is the driver torque. The

state equations of the driver interaction assembly have been

given in Eqn. 2 and Eqn. 3. The unknown parameters aSW ,

fSW , bSW and cSW are identified on-line using adaptive on-

line parameter estimation technique. If we select the input iMin Eqn. 3 as,

iM =aSW x2

bSW

+fSW sgnx2

bSW

−BC

JCbSW

(x2 − θTB)

−KC

JCbSW

(x1 − θTB) +τD

bSW

(1

JC

− cSW ) (13)

and substitute into Eqn. 3, the following closed loop system

is obtained

JC x2 + BC(x2 − θTB) + KC(x1 − θTB) = τD (14)

or, substituting x2 = θSW , x2 = θSW , and x1 = θSW , it

becomes,

JC θSW + BC(θSW − θTB) + KC(θSW − θTB) = τD (15)

which is equivalent to Eqn. 12 of hydraulic steering system.

Since there is not a mechanical linkage and torsion-bar avail-

able in steer-by-wire system, torsion-bar and rack-and-pinion

equations of the hydraulic steering system, which are

JTB θTB + BTB

(

θTB − xR/rP

)

+ BC(θTB − θC) + ...

KTB (θTB − xR/rP ) + KC(θTB − θC) = 0 (16)

MRxR + BRxR + BH xR + BTB/rP

(

xR/rP − θTB

)

+ ...

KTB/rP (xR/rP − θTB) + KTR(xR − lSA sin δFL) + ...

KTR(xR − lSA sin δFR) = FH (17)

should be integrated in software. In these equations, JTB, BTB

and KTB are the torsion-bar inertia, the torsion-bar viscous

damping and the torsion-bar stiffness, respectively. BH is

the hydraulic cylinder viscous damping, respectively. KTR

is the tie-rod stiffness connecting left and right rack ends to

the steering arms. lSA is the moment arm length, i.e., the

perpendicular distance between tie-rod end to the king-pin

axis, and FH is the hydraulic assist force. In general, the

hydraulic assist force is a function of the deflection of the

torsion bar. The hydraulic assist force can be calculated using

FH = fH((xR/rP − θTB))PA where, fH is the function of

torsion bar deflection, i.e., the angular difference between the

steering gear pinion (xR/rP ) and the torsion bar (θTB), and

PA is the hydraulic cylinder area.

It is concluded that developed steer-by-wire system in this

work can behave similar to hydraulic steering system using

impedance control [10]. It is assumed that hydraulic steering

system dynamic parameters in Eqn. 12, Eqn. 16 and Eqn.

17 are known. They can even be tuned depending on the

requirements of the steer-by-wire system. Therefore, it is

possible to made a steer-by-wire system imitate a hydraulic

steering system. In addition to this, it is possible to change

the behavior on-line by modifying the coefficients KC , BC ,

and JC of the control algorithm (Eqn. 13) depending either on

the state variables of the vehicle or other requirements such as

the driver preferences like hard or soft driving conditions. The

hydraulic steering system torsion-bar and rack dynamic Eqns.

16 and 17 are solved in software and the rack position is used

as a reference for the vehicle directional control assembly of

steer-by-wire system. As a result, a hardware in the loop steer-

by-wire system is obtained behaving similar with the hydraulic

steering system. Considering that cSW = 1/JSW , it should be

noted that, if JSW > JC or in other words 1/JC > cSW the

last term in Eqn. 13 becomes positive resulting in positive

force feedback. This however does not cause instability until

the difference between JSW and JC does not become larger

than a threshold. In fact, this operation is nothing but changing

the driver gain cSW = 1/JSW in the system with a higher

gain 1/JC . If JC = JSW , the driver torque feedback drops

from the controller. If however JC > JSW , then the last term

becomes negative resulting in negative force feedback.

1) Proportional-Derivative (PD) Control of Vehicle Direc-

tional Control Assembly: During the tests, the torsion bar

and rack Eqns. 16 and 17 of the hydraulic steering system

are integrated in software. As we have vehicle directional

control assembly, it can be controlled to track the position of

the hydraulic rack-and-pinion calculated in software by Eqn.

17.1 DOF system dynamics of the vehicle directional control

assembly has been given in Eqn. 4 and the state equations

of the vehicle directional control assembly has been given in

Eqn. 5 and Eqn. 7. A PD controller can be designed to force

the vehicle directional control assembly to track the position

of the hydraulic rack-and-pinion given in Eqn. 17 as

iM = Kv(xR − x4) + Kp(xR − x3) (18)

2) Adaptive Pole Placement Control of Vehicle Directional

Control Assembly: As the vehicle directional control assembly

parameters are estimated on-line, it is reasonable to use model-

based control algorithms. Hence, adaptive pole placement con-

trol algorithm is adapted for the control of vehicle directional

control assembly as

iM =aRxR

bR

+fRsgnxR

bR

+Kv(xR−x4)+Kp(xR−x3) (19)

This method is known as a model-based controller because

the dynamic parameters of the system that is controlled is

used to place the poles of the closed loop system to desired

locations. It is similar with feedforward torque control algo-

rithm used in robotic applications except for one difference. In

feedforward torque control algorithm, viscous friction, static

friction and other system dynamics are compensated by using

reference position and velocity signals, not the actual position

and velocity as in Eqn. 19. The reason for this is to simplify the

calculations since the reference signals are known beforehand.

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III. EXPERIMENTAL SET-UP

A. Driver Interaction Assembly

The driver interaction assembly had been shown in Fig.

4. The brushless servo motor was selected as Baldor brand

BSM63N-275A. The servo controller was Baldor brand BSC

Series 1000/1100. The servo controller translated resolver

signals to encoder signals and gave the current drawn by the

servo motor as an output. An ATI brand Delta 9230-05-1130-

03 torque sensor was installed between the steering wheel and

the column to measure driver torque.

Fig. 4. Driver Interaction Assembly.

B. Vehicle Directional Control Assembly

A Renault mechanical steering gear (rack-and-pinion) was

used for the vehicle directional control assembly (Fig. 5).

The brushless servo motor was selected as Baldor BSM63N-

275A. The servo controller was Baldor BSC Series 1000/1100

which was driven by +/- 10 Volt reference signal. The control

algorithms run on a second PC (client) in real time using

Matlab/Simulink. The PC running the algorithms in real-time

was installed with three data acquisition cards which are

Advantech PCL 812PG MultiLab Analog and Digital I/O Card

(for general purpose digital I/O and analog signal measuring,

i.e., motor velocity and current drawn), Advantech PCI-1720

4-channel Isolated D/A Output Card (to drive the servo motor

controllers) and Advantech PCL-833 3-axis Quadrature En-

coder and Counter Card (to count the encoder signals from

servo controller).

Fig. 5. Vehicle Directional Control Assembly.

IV. RESULTS

Initially, the vehicle model (including steering dynamics)

developed in this work was validated using ADAMS sim-

ulation results of a commercial vehicle model [15]. Sev-

eral sinusoidal steer simulations run with ADAMS and the

simulation results were compared with the outputs of the

vehicle model developed. The comparison of concluded that

the vehicle model used in this work was valid to develop and

test control algorithms for the steer-by-wire systems. Then the

estimated dynamic parameters of the driver interaction and

vehicle directional control assemblies were presented. At the

end evaluation results of the controllers were given.

A. Adaptive On-Line Parameter Estimation

Several tests had been performed to estimate unknown

parameters of the driver interaction and vehicle directional

control assemblies on-line. The tests could be classified ac-

cording to the input types applied to the system as sinusoidal

input (single frequency) (iM = A0sinw0t) (from input 1 to

input 4), ii) sinusoidal input (sum of 2 frequencies) (iM =A0sinw0t+A1sinw1t) (from input 5 to input 6) where wi is

rad/s and Ai is amps. The parameters of these input signals

were listed in Table I and Table II.

Table I: Sinusiodal input (single frequency)

(iM = A0sinw0t)input w0 A0

1 2 0.06

2 2 0.08

3 2 0.1

4 2 0.2Table II: Sinusiodal input (sum of 2 frequencies)

(iM = A0sinw0t + A1sinw1t)input w0 A0 w1 A1

5 2 0.1 0.4 0.2

6 2 1 0.2 1.4

The adaptive on-line estimators that we developed were

parameter-projected, i.e., the bounds of the unknown param-

eters aSW , bSW , and cSW need to be known. However,

at this stage, no information was available about the val-

ues of parameters, except that aSW > 0, bSW > 0, and

cSW > 0. Hence, initially the lower and upper boundaries

of the estimated parameters in Eqn. 9 were set to zero and a

high value, respectively. Inputs were applied to estimate the

unknown parameters of the driver interaction assembly. When

the inputs 1 and 2 were applied to the servo motor controller,

the velocity signals measured from the servo controller were

highly disturbed by noise and hence it was difficult to discern

the signals. This indicated that an observer was required for a

better signal to noise ratio. When input 3 was applied to the

servo motor controller, then the steering wheel started to rotate

continuously and the velocity signal became noticeable. After

a couple of trials using input 4 without driver interference (i.e.,

freely rotating steering wheel), acceptable values of adaptive

gains γ1 and γ2 of the output error method (Eqn. 9) were

determined. It had been observed that the values of aSW and

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bSW approached to 3 and 200 at the limit where t → ∞ for

different values of γ1 and γ2 with different adaptation speeds.

Next the Luenberger observer was adapted into the output error

method. Then input 4 was applied with γ1 and γ2 as 0.05 and

100. Estimated aSW and bSW parameters were shown in Fig.

6 and Fig. 7, respectively. As information about the values of

aSW and bSW were available, the initial values were selected

as aSW (0) = 3 and bSW (0) = 200. The measured steering

wheel velocity x2 was plotted together with the estimated

velocity x2 in Fig. 8. It could be seen from Fig. 8 that the

estimated steering wheel velocity followed the measured signal

perfectly at low velocities. Note that during the estimation, it

was assumed that there was no friction (fSW = 0).

Fig. 6. Estimated aSW with Output Error Method and Luenberger Observer(γ1 = 0.05, γ2 = 100, Input 4).

Fig. 7. Estimated bSW with Output Error Method and Luenberger Observer(γ1 = 0.05, γ2 = 100, Input 4).

Fig. 8. Measured/Observed State Variables x2 vs. x2 Using Output ErrorMethod and Luenberger Observer (γ1 = 0.05, γ2 = 100, Input 4).

When the static friction is considered, the inputs 1 and 2

indicated the level of reference input at which the steering

wheel started to rotate freely. It was observed that the level of

0.06 Volts at the reference input of the servo controller initiated

the steering wheel rotation without driver interaction. Lower

values than this could not generate smooth, recognizable

motion. Hence, this was recorded approximately the level

where the servo motor torque compensated the static friction

available in the system and started the motion. Again using

Input 4, the estimated fSW is plotted in Fig. 9. aSW and bSW

were similar as shown in Fig. 6 and Fig. 7. Hence, due to

space limitations, they are not plotted.

Fig. 9. Estimated fSW with output error method and Luenberger observer(γ1 = 0.05, γ2 = 100, γ4 = 0.05, Input 4).

Up to this point, the driver interaction assembly parameters

were identified without driver interaction, i.e., the steering

wheel was rotated freely. However, note that there was an

additional parameter in the system equation which was the

driver gain represented by cSW and equal to 1/JSW . Thus,

there was a need to determine the driver gain cSW .

x2 = −aSW x2 − fSW sgnx2 + bSW iM + cSW τD (20)

In order to estimate the driver gain cSW , the test results with

subject (driver) interaction were used. The subject was directed

to steer the steering wheel sinusoidally without taking the input

applied to the system under consideration. cSW results from

input 5 and input 6 in Fig. 10 and 11.

Fig. 10. Estimated cSW with Output Error Method and Luenberger Observer(γ1 = 0.001, γ2 = 10, γ3 = 10, γ4 = 0.005, Input 5).

Fig. 11. Estimated cSW with Output Error Method and Luenberger Observer(γ1 = 0.001, γ2 = 10, γ3 = 10, γ4 = 0.005, Input 6).

In order to estimate the vehicle direction control assembly

parameters, similar tests had been performed. The parameters

641

of the input signal was selected iM = A0sinw0t (Input

1) where w0 = 1.5, A0 = 1.5. The observer gains were

determined as k1 = 600 and k2 = 80000 by trial and error. The

initial values of the unknown parameters aR(0), fR(0), and

bR(0) were very important for the estimation, therefore after

some trials the initial values were selected as aR(0) = 7.5,

bR(0) = 1, and fR(0) = 1. The estimated parameter values

aR, bR, and fR, where the adaptive gains were selected as

γ1 = 50, γ2 = 10, and γ3 = 10, were given in Fig. 12, 13,

and 14.

Fig. 12. Estimated aR with Output Error Method and Luenberger Observer(γ5 = 50, γ6 = 10, γ7 = 10).

Fig. 13. Estimated bR with Output Error Method and Luenberger Observer(γ5 = 50, γ6 = 10, γ7 = 10).

Fig. 14. Estimated fR with Output Error Method and Luenberger Observer(γ5 = 50, γ6 = 10, γ7 = 10).

The measured and estimated rack velocity, i.e., the state

variable x4 and x4 were plotted in Fig. 15. The rack velocity

plot was limited to 9 seconds to interpret the difference easily.

B. Controller

In order to show the effect of the value of JC , 3 different

values for JC were selected as 0.02, 0.08 and 0.16. BC and

KC are kept constant. The driver was directed to steer the

vehicle sinusoidally to the same steering wheel angle in all

three cases, i.e. the steering wheel angle and angular velocities

Fig. 15. Measured/Observed State Variables x4 vs. x4 Using Output ErrorMethod and Luenberger Observer (γ5 = 50, γ6 = 10, γ7 = 10).

Fig. 16. Driver torque.

Fig. 17. Steering wheel position.

are nearly same in all three experiments. The driver torque and

steering wheel position were shown respectively in Fig. 16, 17.

In Fig. 16, it could be seen that increasing the column

inertia JC , the steering wheel resistance to the driver was

decreasing due to the fact that the column stiffness and viscous

friction was decreasing with the increasing column inertia

according to the second and third terms in the controller

Eqn. 13. Increasing the column inertia from 0.02 to 0.08,

the positive force feedback was switching to negative force

feedback. Further increasing the inertia from 0.08 to 0.16increased the negative force feedback according to the forth

term in the controller Eqn. 13. However, this increase in the

negative feedback did not cause and increase in the stiffness

of the steering wheel perceived by the driver. To the contrary,

the stiffness of the steering wheel perceived by the driver

decreased due to the fact that the decrease in the stiffness

and viscous friction of the system was more pronounced than

the increase in the negative feedback.

1) PD Control of Vehicle Directional Control Assembly:

A PD controller was designed with Kv = 10 and Kp = 1500

642

to force the vehicle directional control assembly to track the

position of the hydraulic rack-and-pinion given in Eqn. 17. The

positions of hydraulic steering rack-and-pinion and vehicle

directional control assembly were plotted in Fig. 18.

Fig. 18. SBW vehicle directional control assembly rack position vs. hydraulicsteering system rack position (Kp=1500, Kv=10).

2) Adaptive Pole Placement Control of Vehicle Directional

Control Assembly: The positions of hydraulic steering rack-

and-pinion and vehicle directional control assembly were

plotted in Fig. 19 when gains were selected as Kv = 10 and

Kp = 1500.

Fig. 19. SBW vehicle directional control assembly rack position vs. hydraulicsteering system rack position (Kp=1500, Kv=10).

Since the damping affect and the static friction was excluded

from the system dynamics by the first term in controller Eqn.

19, then the PD controller input was left for the disturbing

effects which ensured that the tracking performance improved.

V. CONCLUSION AND DISCUSSION

In this study, compliance control of a steer-by-wire system

was studied. In order to study the behavior of steer-by-

wire systems and adaptive compliance control strategies, two

experimental setups were designed and produced which were

driver interaction and vehicle directional control assembly. The

main requirement of compliance control strategies was that the

dynamics of the system should be known to a degree. Hence,

identification of the system dynamics was necessary in order to

implement this control strategy appropriately. In this study, the

dynamics parameters of the steer-by-wire system, i.e., driver

interaction, and vehicle directional control assemblies were

identified by adaptive on-line parameter estimation output

error method. In order to validate the controller developed for

the steer-by-wire system, a nonlinear 4 DOF vehicle model

was derived using Newtonian mechanics.

It has been concluded that steer-by-wire systems were

suitable to implement compliance control strategies appropri-

ately. By selecting proper control parameters using impedance

control strategy, steer-by-wire system can be forced to behave

similar with hydraulic steering systems with the following

advantages; i) it was possible to modify the driver-perceived

stiffness of the steering wheel, even make it a function of other

vehicle state variables such as longitudinal velocity, yaw rate,

etc.; and ii) it was possible to modify the dynamic parameters

of the hydraulic steering system which was used as a reference

model for the impedance controller. This was only possible

with a real mechanical construction to a degree by changing

the material and dimensions. However, in steer-by-wire sys-

tems, this was possible by modifying the coefficients in soft-

ware using impedance controller. Additionally, by estimating

dynamic parameters of vehicle directional control assembly,

the dynamic effects of the system can be compensated by the

adaptive pole placement controller and PD controller input

can be left only for disturbing effects. It was concluded that

this feed-forward loop decreased the position-error. Hence,

a model based controller was suitable for a better tracking

performance.

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