1. Tyre and Vehicle Model Identification using Identifying Kalman Filters
Matthew C Best
Department of Aeronautical and Automotive Engineering
Loughborough University, UK

2. 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

3. 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
[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
[4] Best, M. C., Newton, A. P., and Tuplin, S. ‘The identifying extended Kalman filter: parametric system identification of a vehicle handling model’. 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

4. Extended Kalman Filter for State Estimation
From Model
and gradients
Assuming :
And Estimating :
Eqn not given at bottom – result of Ricatti equation. Q, S and R are typically NOT VARYING with k
Real-time state estimate :

5. Combined State & Parameter Estimation [1],[2]
Can extend the Model to
include Parameters :
But design covariance is not much easier, and still fixed :
Point out flaws with this approach AS LISTED IN PAPER, aswell as apparent from design covariances

6. Real-time combined x and q Result [1]
Point out flaws with this approach AS LISTED IN PAPER, aswell as apparent from design covariances
Source model
EAKF, l = 5x10-4
EAKF, l = 5x10-3

7. Adaptive combined x and q Result [2]
Mass (kg ´ )
Yaw inertia, Izz (kgm2 ´ )
Roll inertia, Ixx (kgm2 ´ )
Mass centre fore/aft position (m ´ 0.3)
Point out flaws with this approach AS LISTED IN PAPER, aswell as apparent from design covariances

8. System Identification or Parameter Estimation ?
Eg by combined GPS / inertia instrumentation of test vehicle
Replace state set completely. Model (h) only appears in y :
then
f = 0, F = 0
R and v is minimal y set in order to generate slip angles and hence forces. Others can be added but not necessary – model complexity can also be increased.
yk = [ r, v ]T or [ r, v, f, p ]T or [ r, v, ay]T

9. Dedicated System Identification [3],[4],[5]
Error sequences now known explicitly
Speed of convergence
And now noise matrices are known explicitly and hence can be time-varying to pick up the REAL correlation information, and aren’t subject to multi-parameter ‘design’ decisions of earlier covariance matrices set as ‘design’. Now only got two, scalar design parameters.
0 < l < 1
Acceptable change in params (fast / slow)

10. 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 :

11. 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 steer
Validation data : ‘free drive’ around proving ground, varying speed & steer
steer, d, (rad)
yaw rate, r, (rad/s)
sideslip, v, (m/s)

12. Parameter and Cost Convergence
q0 qopt P G C E S a h C G
Parameter value & cost a P h E
Iteration number

13. Open loop state accuracy : Identification test
Sideslip velocity, v
Yaw angular velocity, r
Typical time history (s)
Source data
Original model
Optimised model
Roll angular velocity, p

14. Open loop state accuracy : Validation test
Sideslip velocity, v
Lateral acceleration, ay
Section of time history (s)
Source data
Original model
Optimised model
Roll angular velocity, p

15. Identified Tyre Model : Normalised Plot
front
rear
Normalised lateral force
x : left
+ : right
Normalised slip

16. Fast parameter adaptation : Friction tracking [5]
Algorithm changes :
: was 0.01
now 0.1
t : was 350
now 0.01
q = [G]
GPS position track : Proving ground wet handling track

17. Friction tracking result
Source data (estimated from GPS trace and proving ground data
Friction coefficient
IEKF estimate
Time (s)

18. Friction tracking : open loop state variation
Source data
Invariant model
m varying model
Sideslip velocity, v
Time (s)

19. 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