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

car

yO x

roadyr0

yr1yr2

yr3yr4

uT

current road angle = (yr1-yr0)/(uT)

speed, u; time step, T

K21

K22

K2q

car state feedbackcar states

path yr1

path yr2

path yrq

steer angle

command

K11

K12

K13

K14

Optimal controls from Preview LQR

shift register state feedback

Discrete-time control scheme

xdem

ydem

car linearised for operation near to a trim state

K1

K2

car states

xc

yc

shift register; n = 14

throttlesteer

c

+- to cost function

+- to cost function

Minimal car modelx

ab

Fylf

FyrfFyrr

y

0

Fylr

Mass M; Inertia Iz

u, constant

v

2w

inertial system

0 10 20 30 40 50 60 70 80 90 100

-0.2

-0.1

0

0.1

0.2

prev

iew

gai

n va

lue

10 m/s20 m/s 30 m/s 40 m/s 50 m/s

q1 = 100, q2 = 0

0 10 20 30 40 50 60 70 80 90 100

-0.6

-0.4

-0.2

0

0.2

distance ahead, m

prev

iew

gai

n va

lue 10

20 30 4050

q1 = 100, q2 = 0

Buick

Ferrari

K2 (preview) gains for saloon and sports cars

0 50 100 150 200 250 300 350

-20

-10

0-y

coo

rdin

ate,

m

dotted; carsolid; road

0 50 100 150 200 250 300 350

05

1015

attit

ude,

deg

dotted; carsolid; road

0 50 100 150 200 250 300 350

0

10

20

stee

r, d

eg

0 50 100 150 200 250 300 350

02468

x coordinate, m

lata

cc,

m/s

/s

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

)()( yrrylryrfylfz FFbFFarI

0 0.05 0.1 0.15 0.2 0.25 0.30

1000

2000

3000

4000

lateral slip ratio

late

ral f

orc

e, N

Tyre lateral force by Magic Formula

yrrylryrfylf FFFFurvM )(

)))arctan((arctan(sin(2 BBEBCDFy

,

Equilibrium states of front-heavy car

0 0.05 0.1 0.15 0.2 0.25 0.30

0.5

1

1.5

2

lateral slip ratio

Fy

/ axl

e w

eig

ht

a < b, understeer onlyRear axle

Front axle

unique rear slip for given front slip

Axle lateral force / axle weight

decreasing turn radius for fixed speed

Front tyre side slip angle (Rad)Preview length (s)

Optimal preview gain sequences as functions of front axle sideslip ratio

Frequency responses

0 2 4 6 8 10 12 14 16 18 20-10

-8

-6

-4

-2

0

2

4

6

8

10

IC

xinput

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

Controlled car frequency responses

10-3

10-2

10-1

100

101

-25

-20

-15

-10

-5

0

Bode plot for Front tyre side slip for a speed of 30 m/s

10-3

10-2

10-1

100

101

-3000

-2500

-2000

-1500

-1000

-500

0Phase plot for Front tyre side slip for a speed of 30 m/s

Frequency (Hz)

Front tyre side slip= 0Front tyre side slip= 0.040057Front tyre side slip= 0.06009Front tyre side slip= 0.38064

ydem4 from curved reference line

ydem3 from curved reference line

IC

reference line for straight-running trim state

reference line for cornering trim state

road path

ydem2

ydem3

ydem4

ydem1

Small perturbations from trimpath tangent for cornering trim state

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

1

1

1

2

3

4

2

2

3

3

4

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