VRIM: Vehicle Road Interaction Modelling for Estimation of Contact Forces

1. VRIM: Vehicle Road Interaction Modelling for Estimation of Contact Forces N. K. M'SIRDI¹, A. RABHI¹, N. ZBIRI¹ and Y. DELANNE² ¹LRV, FRE 2659 CNRS, Université de Versailles St Quentin, France ² LCPC: Division ESAR; (Nantes) BP 44341 44 Bouguenais cedex

2. Outline • Problematic for on lineestimation • Contact models (static & dynamic ones) • Vehicle Dynamics an Estimation model • Design of a nonlinear robust observer • Simulations results • Conclusion

3. Need of On line Estimation of contact forces The knowledge of the tire/road contact is necessary for vehicle control, road safety, ... Dynamics: Use of the “Relaxation Length” leads to dynamic equation of the longitudinal tire force. Appropriate formulation of the model to permit the on-line estimation of tire forces. Stochastic behaviour (not completely deterministic) Nonstationary processus (time varying) Speed Vx braketorque w Re brakeforce Introduction Problematic for on line estimation

4. 9000 8000 Fx à 50 km/h sol sec MXT 175 R14 Vx 7000 Braking 6000 5000 kip 4000 3000 kis xs 2000 Vx 2b 1000 x 2a 0 -100 -80 -60 -40 -20 0 20 40 60 80 100 -1000 -2000 700 daN -3000 500 daN 300 daN -4000 -5000 -6000 -7000 -8000 -9000 Contact models (static or steady state) Various intereting Contact Models Exist Slip Ratio vs. Lateral Force at given Slip Angles Braking and Tractive forces at given Slip Angles vs. Slip Ratio Longitudinal Forces in function of Fz at given Velocity ”still no internal dynamics”

5. 1.5 µxmax µxbloq 1 Mu 0.5 Influence of Load 6000 pressure : 2.5 bars 7000 N 5000 Kx 5000 N effort Y 4000 3000 3000 N 2000 0 1000 0 10 20 30 40 50 60 70 80 90 100 0 Glissement (%) -10 -8 -6 -4 -2 0 2 4 6 8 10 -1000 -2000 Slip: 0 -3000 carrossage: 0 -4000 -5000 drift -6000 Longitudinal Models « Coefficient longitudinal » influence of Velocity Transverse Forces in function of Fz Cannot be reduced to my(a) ”still no internal dynamics”

6. Friction Models LuGre, Bliman, … Contact Models Assumptions: ponctual, never lost, Stationary pressure distribution, symmetry, perfect rotation, road curvature invariant, … Mechanical Properties PhysicalProperties • - Relaxation length • contact dynamics… • - adhesion/Slipping • Pressure distribution • Stiffness Kx et Ky • Elasticity theory has internal dynamics Pacejka, Fiala , … Dugoff, Sakai, Gim, Guo, Lee, Brush Model • Assume - constant Velocity, slip angle, • invariant Stifness Kx,Ky, Fz constant,… • Uniformity of behaviour

7. One-wheel dynamics One-wheel dynamics Tire equations • Slip-Tire force characteristic • kinematics relationship of wheel-slip • vs represents the slip velocity: vs=v-rw • where • : angular wheel velocity, v : vehicle velocity • F : tire force, T : applied torque • s : wheel-slip • I : wheel inertia, r : Wheel radium, m : vehicle masse • Cx : aerodynamic drag, fw : friction coefficient

8. The wheel-slip can be presented by a first order relaxation length : Modelling of Tire Contact Tire equations • Tire differential equation ( when s<sc, sc is the critical slip) with • Locally we can write Model has internal dynamics Or memory from on state to the next

9. Vehicle dynamics + expression of the 4 forces 4 dynamic equations

10. The model can be written in the state space form Position vector Velocity vector Forces vector With State variable: State space form: Unknown parameters:

11. Input x Vehicle Observer Tire/road interface Robust Observer • Adaptive and robust sliding mode observer design The system is linear with regard to the unknown parameters Adaptive Estimation of Tire forces The dynamics of the estimation errors

12. Convergence analysis • First step : convergence of the sliding surface S is attractive gives The system power is limited, then Forces are bounded, The a priori estimation is also bounded. Then Consequently [n] The second step consider the reduced sliding dynamics, xr=(x3)

13. Convergence analysis • Second step : reduced sliding dynamics, xr=(x3) Now, let us consider a second Lyapunov function: According to equation ( n) By considering the choice of gain H3>>β we finally obtain the convergence of force estimation: Note also that the parameters values con also be retrieved

14. Simulations H2 = 0.2 Steering Angle 0.15 H3 = 0.1 0.05 rad 0 -0.05 Input x Vehicle Observer -0.1 -0.15 Tire/road interface -0.2 0 1 2 3 4 5 6 7 8 9 10 t(s) The parameters of simlation model Gains and parameters of observer

15. Vx 14 13.5 13 12.5 m/s 12 11.5 11 0 5 10 t(s) Velocities

16. Forces

17. An appropriate Model for on line state estimation (can be extended for more than 5 Degres Of Freedom) Robust Observer for on-line tire force estimation (using concept of relaxation length / local linearization) The sliding mode technique is used to be robust with respect to uncertainties on the model, and unknown events (finite time convergence) Possibility to quantify parameters of the tire/road friction. The simulation result illustrate the ability of this approach to give efficient tire force estimation. Conclusion

18. Steering Angle Velocities 0.2 0.15 0.1 0.05 0 rad -0.05 -0.1 -0.15 -0.2 0 1 2 3 4 5 6 7 8 t(s) Vx 16 15 14 m/s 13 12 11 0 2 4 6 8 t(s)

19. Forces

20. trajectory 100 80 60 40 20 0 0 100 200 300 400 Steering Angle Velocities

21. Forces