1. LRV: Laboratoire de Robotique de Versailles 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. LRV: Laboratoire de Robotique de Versailles Outline Problematic for on line estimation Contact models (static & dynamic ones) Vehicle Dynamics an Estimation model Design of a nonlinear robust observer Simulations results Conclusion

3. LRV: Laboratoire de Robotique de Versailles 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  ReRe brakeforce braketorque Introduction Problematic for on line estimation

4. LRV: Laboratoire de Robotique de Versailles Braking and Tractive forces at given Slip Angles vs. Slip Ratio Slip Ratio vs. Lateral Force at given Slip Angles 100 Fx à 50 km/h sol sec MXT 175 R14 -9000 -8000 -7000 -6000 -5000 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 -100-80-60-40-200204040 6080 700 daN 500 daN 300 daN Longitudinal Forces in function of Fz at given Velocity Various intereting Contact Models Exist  ss 2a kipkip 2b k is Braking VxVx VxVx ”still no internal dynamics” Contact models (static or steady state)

5. LRV: Laboratoire de Robotique de Versailles « Coefficient longitudinal » influence of Velocity Longitudinal Models 0102030405060708090100 0 0.5 1 1.5 Glissement (%) Mu µ xmax KxKx µ xbloq Transverse Forces in function of Fz Cannot be reduced to  y (  ”still no internal dynamics”

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

7. LRV: Laboratoire de Robotique de Versailles One-wheel dynamics 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, f w : friction coefficient Slip-Tire force characteristic kinematics relationship of wheel-slip v s represents the slip velocity: v s =v-r  Tire equations

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

9. LRV: Laboratoire de Robotique de Versailles Vehicle dynamics + expression of the 4 forces 4 dynamic equations

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

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

12. LRV: Laboratoire de Robotique de Versailles Convergence analysis The system power is limited, then Forces are bounded, The a priori estimation is also bounded. Then First step : convergence of the sliding surface S is attractive gives Consequently [n] The second step consider the reduced sliding dynamics, x r =(x 3 )

13. LRV: Laboratoire de Robotique de Versailles According to equation ( n) By considering the choice of gain H3>>β we finally obtain the convergence of force estimation: Second step : reduced sliding dynamics, x r =(x 3 ) Convergence analysis Now, let us consider a second Lyapunov function: Note also that the parameters values con also be retrieved

14. LRV: Laboratoire de Robotique de Versailles Simulations The parameters of simlation model ParameterValueUnits M J z F z J f,J r r f,r r 1600 3015 16000 0.7 0.27 Kg Kg.m 2 N Kg.m 2 m 012345678910 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 t(s) Steering Angle rad H 2 = H 3 = Gains and parameters of observer Vehicle Tire/road interface Observer Input x

15. LRV: Laboratoire de Robotique de Versailles 05 1010 11 11.5 12 12.5 13 13.5 14 Vx t(s) m/s Velocities

16. LRV: Laboratoire de Robotique de Versailles Forces

17. LRV: Laboratoire de Robotique de Versailles Conclusion 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.

18. LRV: Laboratoire de Robotique de Versailles 02468 11 12 13 14 15 16 Vx t(s) m/s Steering Angle Velocities 012345678 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 t(s) rad

19. LRV: Laboratoire de Robotique de Versailles Forces

20. LRV: Laboratoire de Robotique de Versailles 0100200300400 0 20 40 60 80 100 trajectory Velocities Steering Angle

21. LRV: Laboratoire de Robotique de Versailles Forces