1. Optimization of Global Chassis Control Variables Josip Kasać*, Joško Deur*, Branko Novaković*, Matthew Hancock**, Francis Assadian** * University of Zagreb, Faculty of Mech. Eng. & Naval Arch., Zagreb, Croatia (e-mail: josip.kasac@fsb.hr, josko.deur@fsb.hr, branko.novakovic@fsb.hr). ** Jaguar Cars Ltd, Whitley Engineering Centre, Coventry, UK (e-mail: fassadia@ford.com, mhancoc1@jaguar.com).

2. 2 Introduction Introduction of new actuators - active rear steering (ARS), active torque vectoring differential (TVD), active limited-slip differential (ALSD), offers new possibilities of improving active vehicle stability and performance However, the control system becomes more complex (Global Chassis Control = GCC), which calls for application of advanced controller optimization methods Benefits of using the nonlinear open-loop optimization: –assessment on the degree of GCC improvement achieved by introducing different actuators; –gaining an insight on how the state controller can be extended by feedforward and/or gain scheduling actions to improve the performance. In this paper a gradient-based algorithm for optimal control of nonlinear multivariable systems with control and state vectors constraints is proposed GCC application - double lane change maneuver executed by using control actions of active rear steering and active rear differential actuators.

3. 3 Optimal control problem formulation Find the control vector u(t) that minimizes the cost function  subject to the nonlinear MIMO dynamics process equations Euler time- dicretization  subject to control & state vector inequality and equality constraints  with initial and final conditions of the state vector Time- dicretization

4. 4 Extending the cost function with constraints-related terms Basic cost function defined above Penalty for final state condition Penalty for inequality constraints Penalty for equality constraints Final problem formulation Weighting factors

5. 5 Comparison with nonlinear programming based algorithms Penalty functions: Nonlinear programming approach: Plant equation constraints :  Advantage vs. Nonlinear Programming based algorithms: Process equations constraints (ODE) are not included in the total cost function as equality constraints  The control and state vectors are treated as dependent variables, thus leading to backward in time structure of algorithm  similar to BPTT algorithm from NN

6. 6 Exact gradient calculation Implicit but exact calculation of cost function gradient - chain rule for ordered derivatives BPTT algorithm – time generalisation of BP algorithm

7. 7 Backward-in-time structure of the algorithm

8. 8 Modified gradient algorithm - convergence speed-up  a “sliding-mode”-based modification of the gradient algorithm  provides a stronger influence of the gradient near the optimal solution, and better convergence  The gradient algorithm with the constant convergence coefficient  and a linear gradient is characterized by a slow convergence.  Small value of the gradient near the optimal solution is the main reason for the slow convergence.

9. 9 Definition of vehicle dynamics quantities 1 2 3 4 t x t y f  0  f T x y z f T i T r T U V r CoG r T  r  2/t2/t b c l c T  Active Front Steering Power Plant Central Differential Rear Differential Active Rear Steering State Variables

10. 10 1. State-Space Subsystem 1.1 Longitudinal, lateral, and yaw DOF U, V - longitudinal and lateral velocity, r - yaw rate, X, Y - vehicle position in the inertial system ψ - yaw angle F xi, F yi, - longitudinal and lateral forces M - vehicle mass, I zz - vehicle moment of inertia, b - distance from the front axle to the CoG, c - distance from the rear axle to the CoG, t - track

11. 11 1.3 Delayed total lateral force (needed to calculate the lateral tire load shift): 1.4 The actuator dynamics: - rear wheel steering angle, - rear differential torque shift, - actuator time constants. 1.2 The wheel rotational dynamics  j - rotational speed of the i-th wheel, F xti - longitudinal force of the i-th tire, T i - torque at the i-th wheel, I wi - wheel moment of inertia, R - effective tire radius.

12. 12 2. Longitudinal Slip Subsystem 3. Lateral Slip Subsystem 4. Tire Load Subsystem 5. Tire Subsystem l - wheelbase h g - CoG height μ - tire friction coefficient B, C, D - tire model parameters

13. 13 6. Rear Active Differential Subsystem Active limited-slip differential (ALSD): Torque vectoring differential (TVD): ΔT r - differential torque shift control variable, T i - input torque (driveline torque) and T b - braking torque

14. 14 GCC optimization problem formulation Nonlinear vehicle dynamics (process) description  Control variables (to be optimized):  r (ARS) and  T r (TVD/ALSD)  Other inputs (driver’s inputs):  f  State variables: U, V, r,  i (i = 1,...,4), , X, Y Cost functions definitions  Path following (in external coordinates):  Control effort penalty:  Different constraints implemented:  control variable limit:  vehicle side slip angle limit:  boundary condition on Y and dY / dt : Reference trajectory

15. 15 Example: Double line change maneuver (22 m/s,  =1) Reference trajectory for next optimizations  Front wheel steering optimization results for asphalt road (  = 1)

16. 16  Optimization results for ARS+TVD control and  = 0.6

17. 17  Optimization results for ARS control and  = 0.6

18. 18  Optimization results for TVD control and  = 0.6

19. 19  Optimization results for ALSD control and  = 0.6

20. 20 Optimization results for different actuators (  =0.6 )  ARS and TVD gives comparable results; no advantage of combined ARS/TVD (except for lower control effort); ALSD less effective due to lack of oversteer generation ARS+TVD ARS TVD ALSD

21. 21 Optimization results for different actuators (  =0.3 )  At low-  surface the lateral optimizer limits lateral acceleration to stabilize vehicle; as a result trajectory tracking is worsen ARS+TVD ARS TVD ALSD

22. 22 Conclusions A back-propagation-through-time (BPTT) exact gradient method for optimal control has been applied for control variable optimization in Global Chassis Control (GCC) systems. The BPTT optimization approach is proven to be numerically robust, precise (control variables are optimized in 5000 time points), and rather computationally efficient Recent algorithm improvement: –numerical Jacobians calculation –implementation of higher-order Adams methods The future work will be directed towards: –use of more accurate tire model –introduction of a driver model for closed-loop maneuvers –model extension with roll, pitch, and heave dynamics –implementation of different gradient methods for convergence speed-up