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Modelling a racing driver Robin Sharp Visiting Professor University of Surrey

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

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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, 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, 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, R. S. Sharp, Motorcycle steering control by road preview, Trans. ASME, Journal of Dynamic Systems, Measurement and Control, 129(4), 2007, R. S. Sharp, Optimal preview speed-tracking control for motorcycles, Multibody System Dynamics, 18(3), , 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, 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.

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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

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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

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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

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Close-up of car and road with sampling car y O x road y r0 y r1 y r2 y r3 y r4 uT current road angle = (y r1 -y r0 )/(uT) speed, u; time step, T

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K 21 K 22 K 2q car state feedback car states path y r1 path y r2 path y rq steer angle command K 11 K 12 K 13 K 14 Optimal controls from Preview LQR shift register state feedback

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Discrete-time control scheme x dem y dem car linearised for operation near to a trim state K1K1 K2K2 car states xcxc ycyc shift register; n = 14 throttle steer cc + - to cost function + -

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Minimal car model x ab F ylf F yrf F yrr y 0 F ylr Mass M; Inertia I z u, constant v 2w inertial system

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Buick Ferrari K 2 (preview) gains for saloon and sports cars

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The rally car (1)

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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

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car model tyre forces ,,

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

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

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

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Controlled car frequency responses

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y dem4 from curved reference line y dem3 from curved reference line IC reference line for straight-running trim state reference line for cornering trim state road path y dem2 y dem3 y dem4 y dem1 Small perturbations from trim path tangent for cornering trim state

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

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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

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