1. Modelling a racing driver
Robin Sharp
Visiting Professor
University of Surrey

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

3. 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), , 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,
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.

4. 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

5. 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

6. 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

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

8.  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

9. 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

10. 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

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

12. The rally car (1)

14. 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

15. car model tyre forces ,

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

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

18. 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

19. Controlled car frequency responses

20. 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

21. 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

22. 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