Modelling a racing driver Robin Sharp Visiting Professor University of Surrey

Partners Dr Simos Evangelou (Imperial College) Mark Thommyppillai (Imperial College) Robin Gearing (Williams F1)

Published work R. S. Sharp and V. Valtetsiotis, Optimal preview car steering control, ICTAM Selected Papers from 20th Int. Cong. (P. Lugner and K. Hedrick eds), supplement to VSD 35, 2001, 101-117. R. S. Sharp, Driver steering control and a new perspective on car handling qualities, Journal of Mechanical Engineering Science, Proc. I. Mech. E., 219(C8), 2005, 1041-1051. R. S. Sharp, Optimal linear time-invariant preview steering control for motorcycles, The Dynamics of Vehicles on Roads and on Tracks (S. Bruni and G. Mastinu eds), supplement to VSD 44, Taylor and Francis (London), 2006, 329-340. R. S. Sharp, Motorcycle steering control by road preview, Trans. ASME, Journal of Dynamic Systems, Measurement and Control, 129(4), 2007, 373-381. R. S. Sharp, Optimal preview speed-tracking control for motorcycles, Multibody System Dynamics, 18(3), 397-411, 2007. R. S. Sharp, Application of optimal preview control to speed tracking of road vehicles, Journal of Mechanical Engineering Science, Proc. I. Mech. E., Part C, 221(12), 2007, 1571-1578. M. Thommyppillai, S. Evangelou and R. S. Sharp, Car driving at the limit by adaptive linear optimal preview control, Vehicle System Dynamics, in press, 2009.

Objectives Enable manoeuvre-based simulations Understand man-machine interactions Perfect virtual driver able to fully exploit a virtual racecar real-time performance Find best performance Find what limits performance Understand matching of car to circuit

Strategy Specify racing line and speed – (x, y, t) (x, y) gives the racing line, t the speed Track the demand with a high-quality tracking controller Continuously identify the vehicle Modify the t-array and iterate to find limit

Optimal tracking Linear Quadratic Regulator (LQR) control with preview linear constant coefficient plant discrete-time car model road model by shift register (delay line) join vehicle and road through cost function specify weights for performance and control apply LQR software

Close-up of car and road with sampling uT O x y  yr4 yr3 yr2 yr1 road yr0 car current road angle = (yr1-yr0)/(uT) speed, u; time step, T

 Optimal controls from Preview LQR path yr1 K21 shift register state feedback path yr2 K22  steer angle command path yrq K2q K11 car states K12 car state feedback K13 K14

Discrete-time control scheme shift register; n = 14 car linearised for operation near to a trim state xdem xc ydem yc c K2 throttle K1 steer car states + - to cost function + - to cost function

Minimal car model x Mass M; Inertia Iz b a  Fylr  Fylf y  Fyrr Fyrf inertial system b a  Fylr  Fylf y 2w u, constant v  Fyrr Fyrf

K2 (preview) gains for saloon and sports cars Buick Ferrari

The rally car (1)

Tyre-force saturation Saturating nonlinearity of real car Optimal race car control idea Trim states and linearisation for small perturbations Storage and retrieval of gain sets Adaptive control by gain scheduling

car model tyre forces ,

Equilibrium states of front-heavy car decreasing turn radius for fixed speed Axle lateral force / axle weight unique rear slip for given front slip

Optimal preview gain sequences as functions of front axle sideslip ratio Gain value Front tyre side slip angle (Rad) Preview length (s)

Frequency responses Perfect tracking requires: input IC x datum line previous input stored in shift register Perfect tracking requires: unity gain phase lag equal to transport lag For cornering, trim involves circular datum

Controlled car frequency responses

Small perturbations from trim path tangent for cornering trim state IC reference line for straight-running trim state ydem1 ydem2 ydem3 ydem4 ydem3 from curved reference line reference line for cornering trim state ydem4 from curved reference line road path

Tracking runs of simple car at 30m/s (Fixed gain vs. Gain scheduled) 2 2 3 1 1 4 3 4 2 3 1 4 Fixed gain Gain scheduled

Conclusions Optimal preview controls found for cornering trim states Gain scheduling applied to nonlinear tracking problem Effectiveness demonstrated in simple application Rear-heavy car studied similarly Identification and learning work under way