Sliding Mode Observer Design for Automatic Steering of Vehicle

Copyright Iil IFAC Control in Transportation Systems, Braunschweig, Germany, 2000

SLIDING MODE OBSERVER DESIGN FOR AUTOMATIC STEERING OF VEHICLE

ttJ.R Zhang, US.J. Xu, tM. Darouach, and tt A. Rachid ttLSA. 7. rue du Moulin Neu/, 80000 Amiens. France t Box 137. Harbin Institute ofTechnologe. 150001 Harbin. P.R. China tIUT de Longl1~v. 186. rue de Lorraine. 54400 COSNES et ROJ...fAIN, France

Abstract: TIris paper deals with the observer design problem for automatic steering of vehicles. The lateral motion of the vehicles is considered. A sliding mode observer is derived such that the observation errors converge to zero asymptotically in fInite time. The simulation results have sho\\n that the design is very effective. Copyright@ 2000 IFAC Keywords: Sliding Mode, Observers, Automatic Steering, Vehicles

1.

design problem of vehicle-to-vehicle spacing and closing rate for a platoon, then an adaptive observer was proposed to estimate the distance between two vehicles. The estimation of the road friction coefficients using robust adaptive observer was developed by Nishira, et al. (1999). A passivitybased observer has been also proposed by Chen and Tornizuka (1996) to estimate the state variable of vehicle lateral motion based on the measurement of the lateral positions. Saraf and Tomizuka (1997) presented a predictor to estimate the lateral motion states of a vehicle mounted magnetic marker reference frame during sampling interval. Kiencke and Dai~ (1996) proposed a nonlinear observer to estimate the state variables of vehicles, however, no convergence proof is given, while the approach requires exact knowledge of model.

INTRODUCTION

In the past two decades, the automatic steering problem of vehicle has attracted a lot of researchers. A great number of results has been reported (Furukawa, et aI., 1989; Fenton and Mayhan, 1991; Shladover, et al. 1991; Varaiya, 1993; Ackermann, et a1. 1995; Ackermann, 1997 and the references therein). The increasing sophistication of the vehicle electronics package has allowed the development of advanced control algorithms, and whose using will make possible the achievement of safer travel, allow the maintenance of higher speeds and closer intervehicle spacing thus increasing the rate of traffic flow. However, most of the advanced control algorithms require the full-state feedback. In fact, some state variables of vehicles are unavailable to be measured directly or the cost of measuring instruments is too expensive to be used on production vehicles, thus, some means of estimating these variables become imperative. The Kalman filter has been used to estimate roadway curvature and the relative position of vehicles (Litkouhi, et al., 1993). The extended Kalrnan filter has been applied also to estimate the state and the longitudinal and lateral tyre force histories of vehicles (Ray, 1992), the road friction coefficients (Ray, 1995), the tyre force (Huh, et al., 1999), and both the road friction coefficients and the tyre force (Ray, 1997, 1998). Choi and Hedrick (1995) discussed the observer

As well known, Kalman filter and extended Kalrnan filter have the shortcomings that they require perfect system knowledge, and no robustness against modelling errors can be guaranteed (Misawa and Hedrick. 1989). Meanwhile, sliding mode control and estimation algorithms have received much attention recently and have been shown to be extremely effective when applied to nonlinear systems. The sliding mode observer is very attractive due to its ease of design and implementation, allowing the enforcement of finite time convergence of estimated states, as well as its 529

substituting (2) into (1.2) - (1.3), ignoring the twoorder small quantities and rearranging the terms yield

properties of restraining the perturbations in the dynamic model of vehicles. Masmoudi and Hedrick (1992) studied shaft torque estimation problem using sliding mode observers. Breuer et al. (1993) presented a saturation-form observer to estimate the vehicle states and parameters. Krishnaswami and Rizzoni (1995) proposed a sliding mode observer to estimate the vehicle's states, in which it is required both the nonsingularity of a matrix constituted by Lie derivatives of output function and the existence of various order derivatives of input function. By means of a regularization approach Krishnaswami (1998) proposed a sliding mode observer to monitor the vehicle's states, however, there is no convergence proof of observation errors.

.

Cl +C,

v:= ----v+(

bc, -acl

-u)r Mu C u2 +[~ + fg-vr- M (fk, -k 2 )]c5

Mu

(3.1)

. r ''''' b 2 c, +a 2 Cl);] 1 r:=I vu (acl-bc,)-/[A'f,.HU+( z

z

a ~ --[M(vrfg)+(fk, -k 2 )u 2 -c/]v. (3.2) I, These equations can be rewritten in the following state space form x:= Ax+ B(x)c5 (4.1) y:=Cx (4.2)

The present paper discusses the observer design problem for automatic steering of vehicle. The lateral motion of the vehicle is considered. We propose a sliding mode observer which can make the observer errors converge to zero asymptotically in finite time.

where x:= [v ryand

2. DYNAMIC MODEL OF VEHICLE Consider the dynamic model of the vehicle (Alloum, 1994): . 1 v+ar 2 u:=-[T+Mvr-Mfg+cl--c5+u (fk,-k 2 )]

M

u

(1.1) v:=_I_[Tc5-Mur+c l c5-(c l

M

y is the measurement output and matrix C is of appropriate dimension. Separating B(x) as

+c,)~+(bc, -ac/)~]

r:=.!-[aTc5-Mjhur+ac c5-(ac I I I z

u

u (1.2)

-bc)~ ' U

2 2 r -(b c, +a c / )-] u

(1.3)

where, u, v and r are the longitudinal velocity, the lateral velocity and the yaw rate, respectively; M, h and I z are the full mass of the vehicle, the height from center of gravity (CG) to road, and the initial moment around z-axis, respectively; g is the acceleration of gravity force; f is the rotating friction coefficient; a and b are the distances from front and rear tyres to CG respectively; C f and er are the

equations (4.1)-(4.2) become x:= Ax+ Blo + B ovrc5 y:=Cx

(6.1)

(6.2)

Remark J: Advanced automatic steering control generally requires full state feedback. However, some of the states are unavailable to measurement, so an observer is needed. Throughout this paper, we assume that the system (4.1)-(4.2) is observable.

cornering stiffness coefficients of front and rear tyres respectively; 0 is the steering angle; T is the traction and/or braking force; k) and k 2 are the lift and drag coefficients from aerodynamics, respectively.

3. SLIDING MODE OBSERVER DESIGN

Assuming that the longitudinal velocity u is invariant or slowly variant during the lateral control, then one can get, from (1.1), that 2 v+ar T:=-M(vr-fg)-u (fk,-kJ-cl--o, (2)

The observer is taken as

U

530

y=Ci

(7.2) Proof Deriving (12) and substituting (8), (13)-(14) yields

where L is the gain matrix of the observer; K and z are respectively a matrix and a function to be determined. and Defining the error vector as e = x - x subtracting (7.1) from (6. 1) yield

e= Aoe + Boe,8 + Kz

s =Ce =CA oe+CBoe,8 +CKz = CAoe +CBoe,8 - (hs + Esgn(s»

and ss = s(CAoe + CBoe,8) - hs 2

(8)

S Isl(ICAoel

where, and

A o =A-LC

e,

=vr -

4.

T

2

k]=0.005N·s Im\

x(O) =

B - 100.4695] . B - [ - 1 ] I - [ 60.6023 ' 0 - -0.6032

The controller is assumed to have been designed by LQR approach as

Remark 3: The above assumptions are reasonable for the real vehicle. The steering angle of a vehicle is limited, generally it is less than 40 degree (Ackerrnann, et aI., 1995). The error e, is bounded because of observation error e being bounded.

8 where

= -Hi,

H=[-Q.33332.4955J

According to LQR method, choosing Q as identity matrix and R= I, one gets the observer gain matrix L:

Let us consider the sliding surface

L

(12)

Theorem 1: In the observer (7.1)-(7.2), if CK is invertible, and z=-(CKrl(hs+Esgn(S» (13) with h>O, and

to

I].

A = [-5.9259 -30.0000]. o -4.9979'

(11)

converges

[~:~]; i(O) = [_0~~2]; C = [0

After a simple calculation based on (4.3), one gets

i.e., there is a positive scale a such that

e

k 2 =o.41 N.s 2 1m 2 ;

the initial conditions are: a.=5; 00=40°; ~O)=O; ~0.6;

is bounded,

0

SIMULATION RESULTS

cJ =J35000N/rad; c =105000N/rad ;

(10)

error

Elsl

f=0.02; a=J.05m; b=1.35m; Iz=2350kg ·m 2 ;

Assumption 2: The error e is bounded, i.e., there is a positive scale p such that

pllC4 1 +8oaICBol,

2

Consider the model of vehicle given by (1.l)-(1.3), where, u=30m/sec;M=J350kg; h=o.5m; g = 9.8lm/sec 2 ;

(9)

E = the observation asymptotically.

2

The proof is complete.

Assumption I: The steering angle is bounded, i.e., there is a positive scale 8 0 such that

s=Ce

Elsl

s-hs 2 <0, if s;tO, S;tO.

To design a sliding mode observer, we need the following assumptions:

vr -vr

-

S

Remark 2: The task of the observer design is to choose L, K and z such that the observer error e converges to zero asymptotically in limited time. For this end, L must be chosen to ensure the matrixA-LC stable. The gain matrix L can be chosen using various ready approaches, for example, standard pole placement or LQR approaches. K and z need to be designed carefully so that the term Boe, can be restrained.

Assumption 3: The error e, =

+ ICBoe, 81) - hs

2

IsJ(IICAollllell +ICBolle, 1181) - hs = Isl(IICAol~~1I + ICBolle, 1181) - hs - (plICA o1 + 8 oa lCB oI) Isl

vr.

Ilellsp.

EI~

-

0:

-0.2696] then A [ 0.0991' , 0

0:

[-5.9259 -29.7304] 0 -5.0970

.

The sliding mode surface is So: [0 IJ e Based on theorem 1, choosing proper matrix K, one gets the sliding mode observer.

(14) zero

The simulation results based on the observer are shown by Fig.I-Fig.6.

53\

0.2

,....,(C-,.------------~I

eJ o I '--.~-~~e2

I,

0.8

-0.2 -0.4 -0.6 -0.8

o

0.5

1.5

1

oL.i.._ _- ' - - _ - - - '_ _- ' -_ _

Fig. I. Observation errors ej and e2

0.6

1.44

1.45

1.46

1.47

__

~_~

1.48

1.49

1.5

Fig.5 The chattering of e J

---~-------------,

'--1

OA~ 02

-3

5

X rl

10

-~--~-------------,

1

o

rr-~------------i

,I

o

-0.2 -OA

~

~

0.5

1

L-'

o

I

_

---ll

_

1.5

Fig.2 The history of steering angle 0 -5

1

_ _~_ _~_ _~ _ ~

L...!.~

0.05

,----------~-----,

0.06

0.07

0.08

0.09

_

0.1

Fig.6 The chattering of e2

V

0.5'

\

v

///

The simulation results show that the observation errors eJ and e2 converge to zero after 0.8 sec and 0.045 sec. respectively, even if there exists the chattering due to the switching effects, the amplitudes of errors are very small, whose magnitudes are 10-4 and 10'3 respectively. The chattering can be eliminated by using a boundary layer controller, however, the asymptotic stability will be lost. The proposed observer is effective for estimating lateral velocity of vehicle.

°1\~P~'·-~ -0.5

~",!.,

I· ,

-11'----~--~-o 0.5 1 1.5 Fig.3 The convergent history of vand v

5.

::[ I'

In this paper, the observer design problem for automatic steering of vehicle has been investigated. A sliding mode observer has been proposed to estimate the lateral velocity of the vehicles, which can make the observer errors converge to zero asymptotically in fInite time. Simulation results showed that the proposed observer is effective for estimating lateral velocity of vehicle.

/'

0.2 ~(//;: 1',

I'

o ~ V-~----------I

-0.2 LI---~----~---

o

CONCLUSION

r

0.5

1

1.5

Fig.4 The convergent history of ;. and r

532

Systems. Proceedings ofthe 13 th IFAC, pp. 273278. Saraf, S. and M. Tomizuka (1997). Slip Angle Estimation for Vehicles on Automated Highways. Proceedings of the American Control Conference, pp. 1588-1592. Kiencke, U. and A. Dai~ (1996). Observation of Lateral Vehicle Dynamics. Proceedings of the 13/h IFAC, pp. 7-10. Misawa, EA and 1.K. Hedrick (1989). Nonlinear Observers-A State-of-the-Art Survey. Journal ofDynamic Systems, Measurement, and Control, Vol.11I, Sep.I989, pp. 344-352. Masmoudi, RA. and J.K. Hedrick (1992). Estimation of vehicle Shaft Torque Using Nonlinear Observers. Journal of Dynamic Systems, Measurement, and Control, Vol. 114, pp. 394-400. Breuer, W., Friedrich Pfeiffer, and Bemd Gebler (1993). State- and Parameter-Estimation for Four-Wheel-Drive Passenger-Cars. Proceedings ofEuropean Control Conftrence, pp. 992-997. Krishnaswami, V. and G. Rizzoni (1995). Vehicle Steering System State Estimation Using Slidin~ Mode Observers. Proceedings of the 34/ Conference on Decision and Control, pp. 33913396. Krishnaswami, V. (1998). A Regularization Approach to Robust Variable Structure Observer Design Applied to Vehicle Parameter and State Estimation. Proceedings of the American Control Conference, pp. 2258-2262. Alloum, A. (1994). Modelisation et commande dynamique dlun vehicule pour la securite de conduite. Ph. D thesis, U. T C. Compiegne, France.

REFERENCES Furukawa, Y, N.Yuhara, S.Sano, H.Takeda and YMatsushita (1989). A Review of Four-Wheel Steering Studies from the Viewpoint of Vehicle Dynamics and Control. Vehicle System Dynamics, Vo1.18, pp. 151-186. Fenton, R.E. and RI. Mayhan (1991). Automated Highway Studies at Ohio State University- An Overview. IEEE Transaction on Vehicular Technology. Vol.VT-40, No.I, pp. 100-113. Shladover, S., et al. (1991). Automatic Vehicle Control Developments in the PATH Program. IEEE Transactions on Vehicular Technology, Vo1.40, No.I, pp. 114-130. Varaiya, P. (1993). Smart Cars on Smart Roads: Problems of Control. IEEE Transactions on Automatic Control, Vo1.38, No.2, pp.l95-207. Ackermann, 1. (1997). Robust Control Prevents Car Skidding. IEEE Control System, June, pp.23-31. Ackermann, 1., et al. (1995). Linear and Nonlinear Controller Design for Robust Automatic Steering. IEEE Transactions on Control Systems Technology, YoU, No.I, pp. 132-143. Litkouhi, RB., A1lan YLee and Douglas B.Craig (1993). Estimator and Controller Design for Lane Trak, a Vision-Based Automatic Vehicles Steering System. Proceedings of the Conference on Decision and Control, pp.18681873. Ray, L.R (1992). Nonlinear Estimation of Vehicle State and Tire Force. Proceedings of the American Control Conference, pp. 526-530. Ray, L.R. (1995). Real-time Detemtination of Road Coefficient of Friction for IVHS and Advanced Vehicle Control. Proceedings of the American Control Conference, pp. 2133-2137. Hub, K., C.Seo, 1. Kim, and D. Hong (1999). Active Steering Control Based on The Estimated Tire Forces. Proceedings of the American Control Conference, pp. 729-733. Ray, L.R (1997). Nonlinear Tire Force Estimation and Road Friction Identification: Simulation and Experiments. Automatica, Vo1.33, No.IO, pp.1819-1833. Ray, L.R (1998). Experimental Detemtination of Tire Forces and Road Friction. Proceedings of the American Control Conference, pp. 18431847. Choi, S.B. and J.K.Hedrick (1995). Vehicle Longitudinal Control Using an Adaptive Observer for Automated Highway Systems. Control Proceedings of the American Conference, pp. 3106-3110. Nishira. H.. T. Kawabe. and S. Shin (1999). Road Friction Estimation Using Adaptive Observer with Periodical a-modification. Proceedings of the 1999 IEEE International Conference on Control Applications, pp. 663-667. Chen, C. and M. Tomizuka (1996). Passivity-Based Nonlinear Observer for Lateral Control of Tractor-Trailer Vehicles in Automated Highway

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