Transcript Modelling a racing driver

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
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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, 10411051.
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
O
uT
y

yr1
yr0
x
yr2
car
current road angle = (yr1-yr0)/(uT)
speed, u; time step, T
yr3
yr4
road
Optimal controls from Preview LQR
K21
path yr1
path yr2
shift register state feedback
K22
path yrq
K2q

K11
car states
K12
K13
K14
car state feedback
steer angle
command
Discrete-time control scheme
shift register; n = 14
xdem
car linearised for operation
near to a trim state
xc
ydem
yc
c
K2
K1
throttle
steer
car states
+
-
to cost function
+
-
to cost function
Minimal car model
x
0
Mass M; Inertia Iz
b
a

Fylr
2w

y
Fylf
u, constant
v

Fyrr
Fyrf
K2 (preview) gains for saloon and sports cars
preview gain value
0.2
0.1
10 m/s
0
50 m/s
40 m/s
-0.1
q1 = 100, q2 = 0
-0.2
Buick
0
10
0.2
preview gain value
20 m/s 30 m/s
10
0
20
30
40
20 30
40
50
60
70
80
90
100
50
-0.2
Ferrari
-0.4
q1 = 100, q2 = 0
-0.6
0
10
20
30
40
50
60
distance ahead, m
70
80
90
100
attitude, deg -y coordinate, m
0
dotted; car
solid; road
-10
-20
0
15
10
5
0
50
100
150
200
250
300
350
dotted; car
solid; road
0
50
100
150
200
250
300
350
0
50
100
150
200
250
300
350
0
50
100
150
200
250
300
350
20
10
latacc, m/s/s
steer, deg
The rally car (1)
0
8
6
4
2
0
x coordinate, m
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
M (v  ur)  Fylf  Fyrf  Fylr  Fyrr
I z r  a( Fylf  Fyrf )  b( Fylr  Fyrr )
Fy  2D sin(C arctan(B  E(B  arctan(B )))
lateral force, N
4000
3000
2000
Tyre lateral force by Magic Formula
1000
0
0
0.05
0.1
0.15
, lateral slip ratio
0.2
0.25
0.3
Equilibrium states of front-heavy car
decreasing turn radius for fixed speed
Axle lateral force / axle weight
Fy / axle weight
2
1.5
1
0.5
0
Rear axle
Front axle
a < b, understeer only
0
0.05
0.1
0.15
0.2
lateral slip ratio
unique rear slip for given front slip
0.25
0.3
Gain value
Optimal preview gain sequences as
functions of front axle sideslip ratio
Frequency responses
10
8
6
4
input
IC
2
x
0
datum line
-2
-4
previous input stored in shift register
-6
-8
-10
Perfect tracking requires:
0
2
4
6
8
10
12
14
16
18
unity gain
phase lag equal to transport lag
For cornering, trim involves circular
datum
20
Controlled car frequency responses
Bode plot for Front tyre side slip for a speed of 30 m/s
0
-5
-10
Front
Front
Front
Front
-15
-20
-25
-3
10
tyre side slip= 0
tyre side slip= 0.040057
tyre side slip= 0.06009
tyre side slip= 0.38064
-2
10
-1
10
0
10
1
10
Phase plot for Front tyre side slip for a speed of 30 m/s
0
-500
-1000
-1500
-2000
-2500
-3000
-3
10
-2
10
-1
10
Frequency (Hz)
0
10
1
10
Small perturbations from trim
path tangent for cornering trim state
IC
reference line for straight-running trim state
ydem1
ydem2
ydem3
ydem3 from curved reference line
ydem4
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
1
Fixed gain
2
Gain scheduled
3
4
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