DETECTION OF CRITICAL SITUATIONS FOR LATERAL VEHICLE CONTROL

Transcription

1 DETECTION OF CRITICAL SITUATIONS FOR LATERAL VEHICLE CONTROL N Zbiri, A Rabhi N K M Sirdi Laboratoire des Systèmes Complexes 4 rue du Pelvoux CE 1455 Courcournnes 912 Evry Cedex Laboratoire de Robotique de Versailles LRV 1-12 Avenue de l Europe 7814, Vélizy, France zbiri@lrvuvsqfr msirdi@lrvuvsqfr 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] 21 Lateral model 2 MODELLING 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 J zz the vehicle inertia around gravity center (CG) v velocity of the vehicle at (CG) v y lateral velocity l r, l f distance of front and rear axles from (CG) F xy,f yf are longitudinal and lateral forces

2 δ F front wheel steering angle β sideslip angle ψ yaw rate We consider the following assumptions: 22 Vision system, mesurement M Y F yf Fxf F yr F xr l r V β l f ψ X CG S y s β V ε L 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 m v y = F xf sin(δ F )+F yf cos(δ F )+F yr (1) J z ψ = F xf l f sin(δ F )+F yf l f cos(δ F )+l r F yr v y = v(β + ψ) Assuming equal slip angles on the left and right tires and making small angle approximations leads to ψl f α f = δ f β + v ψl r α r = δ r β + v (2) Using a linear tire model the lateral forces are given by F yf = C f α f F yf = C r α r (3) where C f and C r 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 " # C f a11 a β = 12 βψ + ψ a 21 a mv 22 l f C f δ f (4) J z a 11 = (C f + C r ) mv a 21 = C fl f C r l r J z a 12 = C rl r C f l f mv 2 1(5) a 22 = - C rl 2 r + C f l 2 f J z v (6) Fig 2 Displacement y s 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 + ε L )+l s ψ (7) The angular displacement is obtained from ε L = ψ v R = ψ vw (8) y s is the offset from the centreline at the lookahead distance, ε L the angle between the tangent to the road and the orientation of the vehicle with respect to the road and l s the look-ahead distance at which the measurement is taken, and w is the road curvature The measurement y s is corrupted by a Gaussian white noise with zero mean 23 Complete model of the system Combining the vehicle lateral dynamics and the vision dynamics leads to a single dynamical system of the form x = Ax + Bu + Kw (9) y = Cx where x =[β, ψ, y s,ε L ] T a 11 a 12 a A= 21 a 22 v l s v (1) 1 C f mv l f C f B= J z C= 1 K= (11) v

3 3 OBSERVER FOR THE NOMINAL MODEL An observer [8] is designed for the nominal system (7) described by bx =(A LC)bx + Bu + K bw + Ly (12) by = Cbx Defining the state estimation error e = x bx gives the following dynamic equation: e = ẋ ḃx = A Le + Ke w (13) where : A L = A LC and e w = w bw; A F = λi A L A L = λ a 11 a 12 l 1 a 21 λ a 22 l 2 v l s λ + l 3 v 1 l 4 λ (14) Considering that e w is the error of estimation of the curvature w Innormalsituations,thiserror is less than the road curvature itself which is bounded (depending on road nature) Otherwise, owing to the boundness of w, we can also impose bounds on the estimation bw This estimation depends on the state estimation error This leads to ke w k M kx bxk M kek (with M a positive constant) The state error is therefore sensitive to the change of the road and (C, A) is observable If w satisfies ke w k M kek,wherem is a constant positive matrix then an asymptotic observer can be designed under the following conditions : i) Real (λ i ) <, whereλ i i =1, 2n are the eigenvalues of the matrix A L ii) M< λ min(q) 2λ max(p )kw k P is the positive definite symmetric matrix, solution of the Lyapunov equations Q = (PA L + A T LP ) (15) Where Q is the positive definite and symmetric matrix λ min (Q), λ max (P ) are the minimum eigenvalue of Q and the maximum eigenvalue of P respectively Convergence analysis Consider the following Lyapunov candidate function : V = e T Pe (16) Differentiating V along the trajectory of the system error dynamics yields: V = e T P ė + ėt Pe (17) = e T P (A L e + We w )+(A L e + We w ) T Pe = e T (PA L + A T LP )e +2e T PWe w As Q = (PA L + A T L P ) and ke wk M kek then, we obtain V e T Qe +2λ max (P ) kw k M kek 2 λ min (Q) kek 2 +2λ max (P ) kw k M kek 2 λ min (Q) 2λ max (P ) kw k M kek 2 < (18) To make V< we need M< λ min(q) 2λ max(p )kw k Under condition (i) and (ii), wehave V < e, thene w A numerical application has been done with the following variable values: mu =8; m = 15; Jzz = 2454; v = 2; C r = 575; C f = 575; l r = 14625; l f = 165; l s =1 We have then A L = A LC (19) = (2) Its eigen values are: λ(a L )=[ 31524; i; (21) i; 1864] (22) Choosing Q = I; λ(q) =1 We obtain the matrix P = lyap(a L,Q) (23) = P is symmetric positive definite since it verifies the Sylvester criterion (all minor determinants are positive)

4 4 SYSTEM DETECTION Computing the steady state of equation (4), leads to the steady state yaw rate described by ψ δ f = v (24) l mv2 (l f C f l r C r ) l f C r The actual curvature radius R, wherer ψ = v can be given by where l = l f + l r R = l mv 2 (l f C f l r C r ) l f C r (25) δ f So the actual curvature tracked by the vehicle is w = R 1 (26) The fault detection system is built on the decision whether the vehicle is in an over steering or under steering case The residue used for the detection is given by the difference between the estimated curvature bw obtained from the vision system equations (8) and the actual value w obtained from the relation (21) An abrupt change of the steering modifies the average of the residue which should be close to zero Among the tests for averages, we kept (WSAT) which takes into account the passed evolution of the tested variable For every instant we have to define two thresholds A(k) and B(k) According to the value of the likelihood ratio with respect to the previous two thresholds, we then make either decision H o, if the vehicle is in over steering case or H 1, if it is in the under steering case The test decision is described by: (k) A(k) Decision hypothesis by B(k) (k) Decision hypothesis H 1 is valid A(k) (k) B(k) Neutral situation, go to the next observation A and B control the errors linked to the decision procedure Under hypothesis H o, the deviation k has the probability density f(e/h o ) while under H 1 it becomes f(e/h 1 ) The likelihood ratio is defined by V = f(e/h 1) f(e/h ) (27) Wald has proposed to calculate the thresholds H o and H 1 by linking them to the probability of false alarms P F and the probability of no detection P ND Once these probabilities are imposed, the thresholds are given by A = PND 1 P F B = 1 PND P F For independent Gaussian observations of standard deviation σ and mean µ under H o and µ 1 under H 1, the test becomes with S o = km + S o <S(k) <km+ S l (28) M = µ 1 µ 2 σ2 µ 1 µ ln A(k) (29) S 1 = σ2 µ 1 µ ln B(k) Intheinequation,thecumulatedsum kx S(k)= e j µ depends on k, which intersects j=1 the curves representing the right and left side of the inequalities But these curves are straight lines (showing upper and lower limits) having the same slope µ 1 µ 2 and the y axis intersects respectively at σ 2 σ µ 1 µ ln A(k) and at 2 µ 1 µ ln B(k) After ach detection S(k) is initialised at zero 5 SIMULATION The detection test consists in creating separately two deviations of the steering angle At the 18th iteration, we increase it and at the 3th iteration we decrease it to simulate respectively an over steering and under steering case Figure 3 shows the form of the steering anglethe estimation of the state variables given by the observer are represented in figures 4, 5, 6,and 7 The estimation of the road trajectory obtained from (8) is shown in figure 8 and has the same profile than the steering angle The curves in figure 9 shows the deviation between bw and and the test results A detection delay appears (arround 2 samples delay), however, we can remark that this test does not generate a false alarm Figure 1, represent the two upper and lower bounds lines In fact we observe parts of lines This comes from setting the cumulated sum to zero after each decision A comparison of the signal at the two fixed thresholds (lower and

5 upper) would give the worse results with repeated false alarms We notice that the cumulated sum increases and decreases randomly However at the 18th iteration, this sum increases more rapidly (instant where the mean increase) and intercept the upper line which means that the vehicle is in an over steering situation (rad/s) (rad) Fig 5 yaw rate Fig 3 steering angle S Obsr (rad) (rad) Fig 6 deviation of the cap Fig 4 sideslip angle (m) 4 3 After detecting whether the over steering situation persists this sum will increase again very rapidly to the upper line which it is not the case here Then, we have introduced a deviation in the other side at the 3th iteration and we see that the sum decrease and intercept the lower line 2 samples later 6 CONCLUSION In this paper we have presented a methodology of detecting under over steering situation for a vehicle This approach is based on state estimation and statistics tests The obtained results are very Fig 7 lateral displacement good The aim here is that only one sensor is used and no knowledge on the path road is needed The model used is linear and the curvature estimation depends on a derivative signal which may limit the method in presence of high measurement noise A further work will enhance the robustness and will extend the approach to nonlinear models

6 (1/m) Fig 8 estimation of road trajectory [4] EDDickmanns and BDMysliwetz " Recursive 3-D road and relative ego-state estimation " IEEE Transaction on PAMI, 14(2): , February 1992 [5] A Wald, 1947 Sequentiel analysis, John Wiley Sons New york [6] A El Hadri " Modélisation de véhicules, observation d état et estimation des forces pneumatiques : Application au contrôle longitudinal " Thèse présentée à l Université de Versailles Saint Quentin en Yvelines [7] HPeng and MTomizuka (199) " Vehicle lateral control for highway automation " In Proceeding of the American Control Conference pp San Diego, USA [8] Philippe de Larminat 1996, Hermès Automatique, commande des systèmes linéaires 2 e édition revue et augmentée Fig 9 the observation of the residue and the test results Fig 1 Test of wald REFERENCES [1] UOzguner, KAyelioglu, and CHatipoglu " An analytical study of vehicle steering control " In Proceedings of the 4th IEE Conference on Control Applications, pages , 1995 [2] DRaviv and MHerman " A non reconstruction approach for road following " In Proceedings of the SPIE, editor, Intelligent Robots and Computer Vision, pages 2-12,1991 [3] JKosecka RBlasi,CJTaylor, and JMalik A Vision-Based Lateral Control of Vehicules Proceding Intelligent Transportation Systems Conference, Boston, 1997