Matt Best (Loughborough University)

Tyre and Vehicle Model Identification using Identifying

Kalman Filters

Matthew C Best

Department of Aeronautical and Automotive Engineering

Loughborough University, UK

Matt Best (Loughborough University)

Background

• Kalman filters are used widely for state estimation, but rely on good, suitably simple vehicle / tyre models

• System identification provides a way to optimise simple model accuracy

• The Extended Kalman filter can include parameter adaptation, but design covariance matrices still need to be chosen

• If the full state set is measurable, the Extended Kalman filter can be re-configured for system identification directly

Matt Best (Loughborough University)

References[1] Best M.C., Gordon T.J. and Dixon P.J., ‘An Extended Adaptive Kalman Filter for Real-time State Estimation of Vehicle Handling Dynamics,’ Vehicle System Dynamics : International Journal of Vehicle Mechanics and Mobility, Vol 34, No 1, pp 57-75, 2000

[2] Best M.C. and Gordon T.J., ‘Combined State and Parameter Estimation of Vehicle Handling Dynamics’ proceedings from the 5th International Symposium on Advanced Vehicle Control (AVEC), Ann Arbor, USA, August 2000, pp 429-436

[3] Best, M.C., ‘Parametric Identification of Vehicle Handling using an Extended Kalman Filter’, International Journal of Vehicle Autonomous Systems, Vol 5, No 3 / 4, 2007, pp 256 - 273

[4] Best, M. C., Newton, A. P., and Tuplin, S. ‘The identifying extendedKalman filter: parametric system identification of a vehicle handlingmodel’. Proc. Instn Mech. Engrs, Part K: J. Multi-body Dynamics, 2007,221(O1), 87-98

[5] Best, M. C., ‘Identifying Tyre Models Directly from Vehicle Test Data using an Extended Kalman Filter’. Vehicle System Dynamics : International Journal of Vehicle Mechanics and Mobility, forthcoming

Matt Best (Loughborough University)

Extended Kalman Filter for State Estimation

, ,k k k k k x f x u θ ω , ,k k k k k y h x u θ υ

, , k kE E k ω 0 υ 0

, , T Ti j i jE E i j ω ω 0 υ υ 0

, , T T Tk k k k k k k k kE E E Q ω ω S ω υ 0 R υ υ

ˆ

ˆ

, ,ˆ

, ,ˆ

k

k

k kk

k kk

x x

x x

f x u θF x

x

h x u θH x

x

Assuming :

From Model and gradients

And Estimating :

Real-time state estimate : 1ˆ ˆ k k x x

Matt Best (Loughborough University)

Combined State & Parameter Estimation [1],[2]

, ,k k k k k y h x u θ υ

Can extend the Model to include Parameters :

( )

( )

, ,k k k k kk

k

x

θ

x f x u θ ωz

0 ωθ

RS

SQT

But design covariance is not much easier, and still fixed :

Matt Best (Loughborough University)

Real-time combined x and Result [1]

Source modelEAKF, = 5x10-4

EAKF, = 5x10-3

0,1 ,

00

00

00

, 5x5

I

ISIS

QIIQ

( )

( )

, ,k k k k kk

k

x

θ

x f x u θ ωz

0 ωθ

Matt Best (Loughborough University)

Adaptive combined x and Result [2]

Mass centre fore/aft position (m 0.3)

Mass (kg 0.3)

Yaw inertia, Izz (kgm2 0.3)

Roll inertia, Ixx (kgm2 0.3)

Matt Best (Loughborough University)

System Identification or Parameter Estimation ?

If k kx y Eg by combined GPS / inertia instrumentation of test vehicle

1 1 1, ,k k k k k y h y u θ υ

k kθ ωReplace state set completely. Model (h) only appears in y :

then f = 0, F = 0

yk = [ r, v ]T or [ r, v, , p ]T or [ r, v, ay]T

Matt Best (Loughborough University)

Dedicated System Identification [3],[4],[5]

Error sequences now known explicitly

k kθ ω

1 1 1, ,k k k k k y h y u θ υ

1

1 ˆ ˆ( )k k kT ω θ θ

1ˆ, ,k k k k k υ y h y u θ

ˆ

, ,ˆ

k

k kk

θ θ

h x u θH θ

θ

21 (1 ) T

k k k k Q Q ω ω

1 (1 ) Tk k k k S S ω υ

1 (1 ) Tk k k k R R υ υ

1ˆ ˆ ˆ, ,k k k k k k k k kT

1θ θ K S R y h y u θ

/1 Te

0 < < 1 Acceptable change in params (fast / slow)

Speed of convergence

Matt Best (Loughborough University)

Tyre Model Identification [3],[4]

Objective : Identify a single, load dependent tyre model, directly from the vehicle data

Parameters : • Pacejka curve B, C, D, E (P, G, C, E here) • Steering compliance S• CG long. and vert. position, a, h

3dof model :r

v

p

y1k k kT h y y

Matt Best (Loughborough University)

Vehicle Testing [4]

OXTS RT3200 GPS / inertia measurement (6dof body kinematics)Steering input from CAN, 3.5l Jaguar XJ8 Test Vehicle

100Hz sampling rate, data & Kalman Filter

Identification data : Four constant speeds, increasing step steerValidation data : ‘free drive’ around proving ground, varying speed & steer

0 10 20 30 40 50 60 70-1

-0.5

0

0.5

1

1.5

steer, , (rad)yaw rate, r, (rad/s)sideslip, v, (m/s)

Matt Best (Loughborough University)

Parameter and Cost Convergence

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

-1

0

1

2

3

4

5

Iteration number

Par

amet

er v

alue

& c

ost

P

GC

E

S

a

h

0 opt

P 0.9 1.07

G 1.0 1.96

C 1.4 1.78

E -0.2 -2.29

S 2.0 4.52

a 1.5 1.69

h 0.5 0.41

Matt Best (Loughborough University)

Open loop state accuracy : Identification test

Typical time history (s)

Sid

esl

ip v

elo

city

, v

S

290 300 310 320 330 340 350-4

-3

-2

-1

0

1

2

3

80 85 90 95 100 105 110

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

180 185 190 195 200

-0.2

-0.1

0

0.1

0.2

0.3

Rol

l ang

ula

r ve

loci

ty,

p

Yaw

ang

ula

r ve

loci

ty,

r

Source dataOriginal modelOptimised model

Matt Best (Loughborough University)

80 85 90 95 100 105 110-2

-1

0

1

2

80 85 90 95 100 105 110-10

-5

0

5

10

Open loop state accuracy : Validation test

Section of time history (s)

Sid

esl

ip v

elo

city

, v

Rol

l ang

ula

r ve

loci

ty,

p

Late

ral a

ccel

era

tion,

ay

Source dataOriginal modelOptimised model

80 85 90 95 100 105 110

-0.2

-0.1

0

0.1

0.2

0.3

Matt Best (Loughborough University)

Identified Tyre Model : Normalised Plot

-3 -2 -1 0 1 2 3-1.5

-1

-0.5

0

0.5

1

1.5

front

rear

x : left

+ : right

Normalised slip

Nor

mal

ised

late

ral f

orc

e

Matt Best (Loughborough University)

400 500 600 700 800 900 1000

-700

-600

-500

-400

-300

-200

Fast parameter adaptation : Friction tracking [5]

Algorithm changes :

GPS position track : Proving ground wet handling track

: was 0.01 now 0.1

: was 350 now 0.01

= [G]

Matt Best (Loughborough University)

0 50 100 150 200 250 300 350 400 4500

0.2

0.4

0.6

0.8

1

1.2

Friction tracking resultF

rictio

n c

oeff

icie

nt

Time (s)

Source data (estimated from GPS trace and proving ground data

IEKF estimate

Matt Best (Loughborough University)

100 110 120 130 140 150 160 170 180 190 200

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Friction tracking : open loop state variation

Time (s)

Sid

esl

ip v

elo

city

, v

Source dataInvariant model varying model

Matt Best (Loughborough University)

Conclusions

• A simple model structure can provide very good state tracking

• Combined state and parameter estimation can be effective, but needs accurate design covariance, and care with conditioning

• A ‘reasonable’ model set-up can estimate yaw rate and lateral acceleration fairly easily

• Optimisation of the tyre model improves these states, and dramatically improves sideslip velocity prediction

• A single tyre model can be identified and validated, adding credibility to the sideslip velocity measurement accuracy claims of the GPS / inertia instrumentation manufacturers.

• Fast adaptation is successful and can track rapid friction variation