DETECTION OF CRITICAL SITUATIONS FOR LATERAL VEHICLE CONTROL N. Zbiri ∗,∗∗ A. Rabhi ∗∗ N. K. M’Sirdi ∗∗ ∗

Laboratoire des Systèmes Complexes 40 rue du Pelvoux CE 1455 Courcournnes 91020 Evry Cedex ∗∗ Laboratoire de Robotique de Versailles LRV 10-12 Avenue de l’Europe 78140, Vélizy, France [email protected] [email protected]

Abstract: In this paper, we propose and develop a "look-ahead" system for detection of over steering or under steering situations. The approach is based on the combination of an observer and a vision system which estimates the road curvature and a change detection procedure. We use an average sequential test to detect the abnormal situations. Simulation results shows the efficiency of the method. Keywords: On line observers and estimation, faults and changes detection, sequential test, vehicle lateral control, assistance, driving.

1. INTRODUCTION Recently, safety and drivability of vehicles is of increasing interest. Today’s advanced driver assistance systems are widely used in modern vehicles. Their objective is to assist the driver by preventing any unstable or unpredictable vehicle behavior. One critical situation which has received a particular attention is the deviation of the vehicle from its trajectory. Different reference systems have been examined for detecting the lateral vehicle motion. Most existing approaches to automatic steering can be classified into “look-ahead” [1 − 3], and “look-down” [4] systems according to the point of measurement of lateral displacement. The advantage of “look-ahead” systems is their ability to replicate human driving by measuring lateral displacement. The objective of this work is to design a “lookahead” system detection for over steering or under steering situations. The methodology developed is based on vehicle state and trajectory estimation. The detection of the deviation of the vehicle from

its trajectory is then obtained using the Wald Sequential Average Test (WSA) [5].

2. MODELLING 2.1 Lateral model The dynamics of a single track vehicle can be described by a detailed 16-DOF [6] non linear model. In other systems, it is possible to decouple the longitudinal and lateral dynamics [7]. The kinematical behaviour of the vehicle can be in this application approximated by the bicycle model, see figure 1. In this paper we use the following notation : m total mass of the vehicle Jzz the vehicle inertia around gravity center (CG) v velocity of the vehicle at (CG) vy lateral velocity lr , lf distance of front and rear axles from (CG) Fxy , Fyf are longitudinal and lateral forces

δ F front wheel steering angle β sideslip angle ψ yaw rate We consider the following assumptions:

2.2 Vision system, mesurement. M

Y Fyf

Fxf

Fyr

V

V

X

β

ψ

β

CG S

F xr

εL

ys

lf lr

Fig. 1. Bicycle model - the two front wheels turn slightly differentially - the steering angles are small - a linear tire model is used. Newton law’s applied to the gravity centre lead to equations .

mvy = Fxf sin(δ F ) + Fyf cos(δ F ) + Fyr

Fig. 2. Displacement ys shows a vehicle runing along the roadway Figure 2 shows a vehicle runing along roadway. The equations describing the evolution of the measurements extracted from image, caused by the motion of the car and changes in the road geometry, are as follows: y s = v(β + εL ) + ls ψ

..

Jz ψ = −Fxf lf sin(δ F ) + Fyf lf cos(δ F ) + lr Fyr .

vy = v(β + ψ)

Assuming equal slip angles on the left and right tires and making small angle approximations leads to .

ψlf αf = δ f − β + (2) v . ψlr αr = δ r − β + v Using a linear tire model the lateral forces are given by Fyf = Cf αf Fyf = Cr αr

(3)

"

.

β .. ψ

#

=

∙

a11 a12 a21 a22

−(Cf + Cr ) a11 = mv Cf lf − Cr lr a21 = Jz

¸∙

⎤

Cf ¸ β ⎢ ⎥ mv . + ⎣ l C ⎦ δf f f ψ Jz a12 a22

(4)

Cr lr − Cf lf = −1(5) mv 2 Cr lr2 + Cf lf2 =(6) Jz v

(7)

The angular displacement is obtained from . . v . εL = ψ − = ψ − vw R

(8)

ys is the offset from the centreline at the lookahead distance, εL the angle between the tangent to the road and the orientation of the vehicle with respect to the road and ls the look-ahead distance at which the measurement is taken, and w is the road curvature. The measurement ys is corrupted by a Gaussian white noise with zero mean.

2.3 Complete model of the system Combining the vehicle lateral dynamics and the vision dynamics leads to a single dynamical system of the form

where Cf and Cr are the front and rear cornering stuffiness, respectively. Substituting the forces into 1 of motion and making small angle approximations on the steering angle yields ⎡

.

.

(1)

.

x = Ax + Bu + Kw

(9)

y = Cx .

where x = [β, ψ, ys , εL ]T ⎡ ⎡ C f ⎢ l mv ⎢ f Cf B=⎢ ⎣ Jz 0 0

a11 a12 a a ⎢ A=⎣ 21 22 v ls 0 1

⎤

0 0 0 0

⎤

0 0⎥ v⎦ 0

(10)

⎡ ⎤ 0 ⎥ ⎥ C=£ 0 0 1 0 ¤ K=⎢ 0 ⎥ ⎥ ⎣ 0 ⎦ ⎦ −v

(11)

3. OBSERVER FOR THE NOMINAL MODEL An observer [8] is designed for the nominal system (7) described by

V = eT P e

(16)

Differentiating V along the trajectory of the system error dynamics yields:

.

x b = (A − LC)b x + Bu + K w b + Ly

(12)

yb = C x b

.

(13)

b AF = λI − where : AL = A−LC and ew = w − w; AL ⎡

l1 λ − a11 −a12 ⎢ −a21 λ − a22 l2 AL = ⎢ ⎣ −v −ls λ + l3 0 −1 l4

⎤

0 0 ⎥ ⎥ −v ⎦ λ

(14)

= eT (P AL + ATL P )e + 2eT P W ew As Q = −(P AL + ATL P ) and kew k ≤ M kek then, we obtain .

2

V ≤ −eT Qe + 2λmax (P ) kW k M kek

≤ −λmin (Q) kek2 + 2λmax (P ) kW k M kek2

≤ −λmin (Q) − 2λmax (P ) kW k M kek < 0(18) .

To make V < 0 we need M