Lateral Control of a Four-Wheel Steered Vehicle

Copyright © IFAC Intelligent Components for Vehicles, Seville, Spain, 1998

LATERAL CONTROL OF A FOUR-WHEEL STEERED VEHICLE

D. de Bruin and P.P.J. van den Bosch

Measurement and control group, Department of Electrical Engineering Eindhoven University of echnology p.a. Box 513 MB Eindhoven, The Netherlands e-mail: [email protected], Fax: +31402434582, Tel: +31402473795

Abstract: To get insight into the problems connected with the lateral control of vehicles with more than one steerable axis, we designed a lateral controller for a four-wheel steered vehicle. It is shown that under ideal circumstances, the lateral deviation of two points on the vehicle ' s longitudinal axis can be kept zero during cornering with a feed forward compensator only. Since, in reality. circumstances are not ideal, a feedback compensator has been added. Simulations of the total system indicate that with four-wheel steered vehicles a good performance can be achieved. Copyright © 1998lFAC

Key Words: Automotive controL Control system design, Guidance systems, Mobile robots, Position control, Vehicles, Vehicle dynamics

1. INTRODUCTION We participate in a study which has as final goal the design of a double articulated bus with four independently-controllable steering axes and six independently-controllable wheel torques. The total length of the bus is about 25 m and it has a capacity of 160 persons. The bus is equipped with four independently-controllable steering axes to give the bus a tram-like behaviour, so that the bus lane width can be kept small. The requirements are that the bus may not deviate more than 10 cm from the center of the lane during driving at velocities up to 80 km/h and that the bus has to stop within a lateral distance of less than 4 cm from the bus-stop, so that passengers can easily get on and off the bus (Bosch and Hedrikx , 1997). For human drivers, it is impossible to steer such a vehicle satisfactorily. Therefore the bus has to be equipped with a control system. Controlling the 3 carriage system is quite complicated. Therefore a one carriage vehicle with two steerable axes was studied first.

The aim of this study is to investigate whether it is possible to control satisfactorily the lateral deviation of two points on the longitudinal axis of the vehicle, so that as less space as possible is occupied during cornering and under the influence of wind gusts. It is assumed that the position of the sensors for measuring the lateral deviations coincide with the two points along the longitudinal axis that are controlled. This condition can be relaxed. Based on two measured positions the position of the control points can be calculated. In this paper it will be shown that in the ideal case the lateral deviation of both these points can be kept zero during cornering with feed forward of the path curvature when four-wheel steering is applied. Furthermore, in theory the side-slip angle can be kept zero during cornering. For front-wheel steering the design of a feed forward compensator has already been illustrated (Sienel and Ackermann, 1994). However, then it is impossible to keep the side-slip angle zero. The designed feedforward compensator is not applicable in practice since some assumptions have been made that can not be fulfilled in practice, like finite

actuator bandwidth and unknown plant parameters. To overcome this problem a feedback compensator has been added. It will be shown that with this extra feedback compensator good performance can still be achieved under more realistic conditions. The feedback compensator has also to be used for suppressing lateral deviations due to wind gusts .

The lateral velocity at centre of gravity reads YCG=vsin({3+,1cp)==vc{3+,1qJ), where v is the longitudinal velocity. The lateral velocity of senf equals YFYcc+l,(r-vPref), where Prcf is the path curvature at CG. In the same way. the lateral velocity at senr equals y F Y cG-I,(rI'P"f )'

2. MODELlNG THE VEHICLE The model to describe the behavior of the four-wheel steered vehicle is based on the model of Riekert and Schunk (1940) . In this model both the rear wheels and both the front wheels are lumped into two wheels. that are fixed at the centerIine of the vehicle at the rear and at the front respectively. The model of Rieken and Schunk is modified to bring four-wheel steering into account. Figure 1 shows a schematic representation of the model.

reference line

Y l Y

~~, Figf

Xo

Figure 2 Modelfor track following The state-space description of the vehicle then becomes:

Ir

If

/3

Figure 1 Single track model for four-wheel steering

r""

0

v

Y CG

0

[;, =

0 0

r:

I,

0 -I,

y,

o

0

0

tHP

In this figure: CG= the center of gravity of the front-wheel steering angle [rad] Or the rear-wheel steering angle [rad] f3 side slip angle at the CG [rad] r vehicle yaw rate [rad/s] Ff lateral force generated by the front tire [N] Fr = lateral force generated by the rear tire [N] If distance from the CG to front axis [m] I, distance from the CG to rear axis [m] I = vehicle wheel base [m] V = velocity of the vehicle at the CG [mJs]

Q oo

21

~F,l+

0 0 0

an Q

t,,(p

~1~H~

o

t:J.1{J

b"

b'2

0

b21

boo

0

0 0

0

0

0

- v

0 0

0 0 0

-T'] Mv

8

~

p,:/

o

id

0 0 - vi., vi,

o 8, o p,,/ o id

Tl

(1)

with al/=- j.Jf.er-cr)!Mv a2l=j.J(c,1,-cff)/J blJ=J.iCf IMv b2J =J.iCJtIJ

For automatic tracking the model has to be extended, as described by Ackermann et. al. (1993). The model of Ackermann has been modified to make automatic tracking of two points at the centerline of the vehicle, instead of one point, possible. Also a modification has been made to bring disturbance forces into account. Figure 2 shows the modified model for automatic tracking. Here, YCG is the lateral deviation of the center of mass with respect to the reference line. Yf and Yr are the lateral deviations of the sensor at the front (sent) and the sensor at the rear (senr) respectively . Further cp is the vehicle yaw angle with respect to the earth-fixed coordinate frame (Xo,Jo) , CPr is the angle between Xo and the tangent to the path and ,1CP=qJ-qJ,. (xv,Yv) is a vehicle-fixed coordinate frame. Finally fd is a disturbance force due to wind acting on the CG.

aJ2=-1 +j.J(c,1r-cJt)/Mv a22=-j.J(c,1/+cflhlJv b J2=J.iCrIMv b22=- J.iC,1 r IJ

2

In (I), it is assumed that the longitudinal velocity v is constant. The parameters j.J, cf and C r describe the behavior of the tires. cf and Cr are the cornering stiffnesses of the front and rear tire respectively and j.J is the road adhesion coefficient which equals I for dry road and 0.1 for an icy road. For the compensator design in this paper neither this parameter variation, nor the variations of other parameters will be considered.

3.

FEEDFORWARD COMPENSATOR DESIGN

Ideal path tracking means YFYF,1q;=O. It can be shown that this can be accomplished by making y" Yf

2

and ,1cp equal to zero. Following (1). ,1qJ=r- vP ref, so for ,1cp equal to zero r has to be equal to vP rel With this, y F v{3+ls(r-vPref)=v{3. In the same way y ,=v{3 . Thus when it is possible to steer the vehicle so that {3 equals zero and r equals vPref in all circumstances, Yf and Y r equal zero in all circumstances. Thi s in turn means that the lateral deviation stays zero at the front and rear sensor.

sator with a high bandwidth lowpass filter, so that (4) becomes

Now it will be shown that with four-wheel steering the requirements stated above can be met under the assumptions that vPref is known (a condition that can be met when e.g. the discrete marker scheme di scussed in (Zhang and Parson, 1990; Asaoka and Ueda. 1996) is used) , that the actuators have an infinite bandwidth and that there are no parameter variations . {3 and r can be written in the laplace domain as:

where T has to be sufficiently small.

a ( s) b(s) [3 ( s) = - O,(s) + - o,(s) n ( s) n(s)

° _ , ( s) -

4. FEEDBACK COMPENSATOR DESIGN By deriving the feedforward law (5 ) It was assumed that the actuators have infinite bandwidth. In reality the bandwidth can be as low as 5Hz (Guldner et. aI. , 1996; Guldner et. aI., 1997), so the actuator really disturbs the ideal conditions. Furthermore it was assumed that the parameters like M , I , v and /-l don't change during dri ving, but in reality these parameters change. Moreover there can be a timing error between the real curvature and Pref which causes lateral deviations too. So in practice the feed forward compensator (5) is not sufficient. An extra feedback compensator has to be added to suppress disturbances and uncertainty.

(2)

where ).I.C ,

2 /JC ,(/JC,l , l - l , M v )

Mv

Jv - M

).I.C,

b( s) = - s +

Mv

).I.C, l ,

c( s) = - - s J

,

/JC , (/JC , l , l+l, Mv

2

We designed a feedback compensator with H~ techniques (Zhou, K. et.a!., 1996), since with these techniques it is easy to put constraints on output and actuator signals. Figure 3 shows the augmented plant used for the ~ design .

)

,

Jv- M /-l 2c,c , l

+ ---'-

(3)

JvM

-).I.C , I ,

/-l 2c, c, 1

J

JvM

()_ )

[) _ - n( s)a (s)v I ,Cs) - b(s)c(s) _ a(s)d (s) Ts + I pul es)

c(s) d es) res ) = - o,(s) + - o,( s) n (s) n(s)

a ( s) =--s+

n(s)b(s)v 1 Pu , (s) b(s)c(s) - a (s) d (s) Ts + 1

d es) = - - - s - ----'-

n( s)

=s

2

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/-l( M (c r I; + c , l ~ ) + J (c, + JMv

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c, )s +

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1 --1; _-4~':"'.

vehicle

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I

I I & ' 1 1/S ~ 1

~~ ____ _

l_ ~

-

I

"

Since there are two inputs, both the required conditions 13=0 and r=O. can be met. These conditions are met when:

n (s) b(s)v ,(s) = b(s )c( s) _ a( s)d (s / u, ( s)

';--l - B~-;consq I

~

+---~--~---J-M-v'~--~~--~-~

° °

,------, I

+

~....c:..:..:"'---r--1Cjf-----'_ \~ const~1

I

6.cp Yf . - LP

:=J

LP

H

I II Yr~

" I!i!! I i

~.-J

I I~ _

_

_

-l

Figure 3 Augmented plant used for f eedback compensator design

(4)

n(s)a(s)v , (s) = b ( s)c(s) _ a( s)d ( s/u, ( s)

In this figure , LO stands for lowpass weighting filter, HP stands for highpass weighting filter and const stands for constant weighting filter. We will not consider these filters in detail, since this is too space consuming. The control inputs of the augmented plant are the time derivatives of the steering angles . These are chosen as inputs to make it possible to put constraints on

So in de ideal case, when Pref is known, both Yf, Yr and can be kept zero when feed forward compensation (4) is applied. The relative degree of the feed forward compensator equals -I. The relative degree can be made zero by mUltiplying the feed forward compen,1cp

3

them. The time derivatives are integrated to get the steering angles as input for the vehicle. With the implementation of the controller these integrators are added to the feedback compensator. It was assumed that all states where available for feedback. In practice, some states have to be estimated with an observer (Farrelly and Wellstead, 1996), but this extension was not considered here. The requirements on the actuator inputs were that they may not exceed 0.7 [rad] and that the time derivatives of them may not exceed 0.45 [rad/s]. The final control scheme is shown in figure 4 .

0.01

r--~------~--'---

0.005

o ·0.005 . ·0. 01 ·0.015 ' - - " - - - ' - - - ' - - - - ' - - - ' - - - ' - - - - ' o 0.5 1.5 2 2.5 3 timers]

(a) lateral deviation [m] 0.08 r - - - - - - - - - - - - - -

Figure 4 Final control scheme 0.5

1.5 timers]

2

2.5

3

(b) steering angles [rad]

5. SIMULATIONS For the feedforward- and feedback compensator design a vehicle with IF5 [m], IF5 [m], CFCF300000 [N/rad], M=10000 [kg] and J=83000 [kgm 2] has been used. The two sensors were place symmetrical with respect to the CG at a distance of 2.5 m from it. Since parameter variations where not considered in the compensator design, only the simulation results for the vehicle with nominal parameters will be discussed. The simulations where carried out with the linear vehicle (1), with taking actuator saturation (i 8 j I <0.7 [rad] and 18 1<0.45 [rad/s]) into account. The next simulations where carried out: • entering a curve of 250 [m] radius • riding straight ahead with a disturbing lateral wind force starting at t=l s and ending at t=4 [s]. Both simulation where carried out at v=20 [mls].

timers]

(c) side-slip angle [rad] Figure 5 Simulation results for entering a corner

1

Figure 6 shows the simulation results for riding straight ahead (PrerO) with a wind gust starting at t=l s and ending at t=4 s. The disturbing force due to the wind was taken equal to 10000 N . For a vehicle with these dimensions this is equivalent with about windforce 7. In figure 6-a it can be seen that after a peak of about 1 cm the steady state deviations of Yt and Yr are very small during the wind gust, and both are the same, which could be expected since four-wheel steering is applied. After the wind gust has disappeared, the steady state error returns to zero again. Figure 6-b shows that the steering angles are the same, which could be expected too. Finally, figure 6-c shows the side-slip angle, which has a small peak upwards and a

Figure 5 shows the simulation results for entering a curve. Figure 5-a shows that Yt is max. about 0.015 [m] and both Yt and Yr equal zero in steady state. Also {3 equals zero in steady-state, shown in figure 5-c, which was expected from the feedforward compensator design. The deviations are not zero for t=1.5 [s] since the actuator has no infinite bandwidth. Especially the limitation on the rate of turn poses the constraint. Figure 5-b shows the steering angles occurring during cornering. Remarkable is that both the steering angles have the same sign. This is due to the fact that the lateral acceleration of the vehicle is rather high .

4

with four-wheel steering and that with four-wheel steering a very low value for the side slip angle can be achieved during cornering.

small peak downwards during the start of the gust and the end of the gust respectively.

0.01

7. FUTURE RESEARCH

0005

In this work we didn't consider variations of vehicle mass and tire parameters. Furthermore, we didn' t consider velocity changes. Future work has to bring parameter variations and changes of velocity into account. When this work gives satisfactory results the model and controller will be extended to a fourwheel steered vehicel with a semi trailer with a steered axis.

0 -0 .005 -0. 01

0

2

3 4 time rs]

5

6

(a) lateral deviation [m] 0 .01

0

REFERENCES Asoaka, A. and Ueda, S.(1996). An experimental study of a magnetic sensor in an automated highway system, Proceedings of the 1996 IEEE intelligent vehicle symposium, pp.373-378 Ackermann, J.(1993), Robust control, systems with uncertain physical parameters, Springer- verlag London Bosch, P.PJ. van den and Hendrix, W.H.A. (1997) Control of the lateral and longitudinal position of a bus ,lFAC symposium on transportation sys terns, Chania, Greece, 1997, pp 385-390 Farrelly,1. and Well stead, P. , (1996), Estimation of vehicle lateral velocity, Proceedings of the IEEE international conference on control appli cations, pp.552-557 Guldner et. al. (1996), Analysis of automatic steering control for highway vehicles with look-down lat eral reference systems. Vehicle system dynamics, 26, pp 243-269 Guldner et. al. (1997) , Robust control design for automatic steering based on feedback of front and tail lateral displacement, Proceedings of the 1997 European Control Conference Riekert, P. and Schunck, T.E., (1940) Zur fahr mechanik des gummibereiften kraftfahrzeugs ingenieur-archiv, band 1940, pp. 210-223 Sienel, W. and Ackermann, J (1994), Automatic steering of vehicles with reference angular veloc ity feedback, Proceedings of the 1994 American Control Conference Zhang W . and Parson R.E . (1990) . An intelligent roadway reference system for vehicle lateral guidance/control, Proceedings of the American control conference, pp 281-286 Zhou, K., Koyle, J.c. and Glover K. , (1996) , Robust and Optimal Control, Prentice Hall, New Jersey

-0. 01

-002 0

2

3 4 time Is]

5

6

(b) steering angles [rad] x 10.3 3 2

o -1

-2 -3

'----'-----~--'----'---'--

o

2

4 3 time Is]

5

6

(c) side-slip angle [rad] Figure 6 Simulation results for wind a gust

6. CONCLUSIONS We designed a controller for controlling the lateral deviation of two points on the longitudinal axis of a four-wheel steered vehicle so that the vehicle occupies as less road space as possible during cornering. This controller consists of a compensator that feeds forward the curvature of the road. A feedback compensator compensates deviations that the feed forward compensator can not compensate due to non-idealities and for suppression of disturbances due to lateral wind gusts. This paper shows that it is possible to control two points on the vehicle longitudinal aXIs

5