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An Extended Adaptive Kalman Filter for Real-time State
Estimation of Vehicle Handling Dynamics
Running header: An EA Kalman Filter for Handling Dynamics
M. C. BEST
*
, T. J. GORDON
*
and P. J. DIXON
*
SUMMARY
This paper considers a method for estimating vehicle handling dynamic states in real-time,
using a reduced sensor set; the information is essential for vehicle handling stability control and
is also valuable in chassis design evaluation. An extended (nonlinear) Kalman filter is designed
to estimate the rapidly varying handling state vector. This employs a low order (4 DOF)
handling model which is augmented to include adaptive states (cornering stiffnesses) to
compensate for tyre force nonlinearities. The adaptation is driven by steer-induced variations
in the longitudinal vehicle acceleration.
The observer is compared with an equivalent linear, model-invariant Kalman filter. Both filters
are designed and tested against data from a high order source model which simulates six
degrees of freedom for the vehicle body, and employs a combined-slip Pacejka tyre model. A
performance comparison is presented, which shows promising results for the extended filter,
given a sensor set comprising three accelerometers only. The study also presents an insight
into the effect of correlated error sources in this application, and it concludes with a discussion
of the new observer’s practical viability.
Keywords : Vehicle Handling Dynamics, Kalman Filter, State Observer
NOTATION
* Department of Aeronautical and Automotive Engineering, Loughborough University, Loughborough,
Leics. LE11 3TU, U.K
Dynamic Variables
r yaw rate (rad/s)
v sideslip velocity (m/s)
u forward velocity (m/s)
w wheel rotation velocity referred to tyre
contact patch (m/s)
p roll rate (rad/s)
φ
roll angle (rad)
δ
(front) wheel steer angle (rad)
C
αf
C
αr
front, rear cornering stiffness (N/rad)
S
1
– S
5
acceleration sensors (m/s
2
)
F
lat
F
long
lateral, longitudinal components of front
tyre force (N)
y sensor set including measurement noise
(m/s2)
x
s
state vector – s superscript denotes
source model data.
^ caret denotes estimated signal
Model parameters
M vehicle mass : 830 Kg
I
xx
roll moment of inertia : 500 Kgm
2
I
zz
yaw moment of inertia : 1600 Kgm
2
I
xz
yaw/roll product of inertia : 30 Kgm
2
h height of CG above roll centre : 0.25 m
d height of sensor above roll centre : 1.0 m
b CG to front axle distance : 1.06 m
c CG to rear axle distance : 1.53 m
K
f
K
r
front, rear roll stiffness :
K
f
+ K
r
= 40 kN/rad (identified)
B
f
B
r
front, rear roll damping :
B
f
+ B
r
= 1.5 kNs/rad (identified)
g gravitational acceleration
µ
longitudinal tyre friction coefficient :
5000 Ns/m
ρ
measurement noise amplification factor
λ
adaptation rate coefficient
Kalman Filter
ω
process noise
υ
measurement noise
A,B,C,D discrete-time linear observer model
K Optimal gain matrix
P state error covariance matrix
R measurement noise covariance matrix
Q process noise covariance matrix
S measurement / process noise cross
covariance matrix
f(x) nonlinear system model
h(x) nonlinear output model
F(x) system model Jacobian
H(x) output model Jacobian
1. INTRODUCTION
Recent research interest in vehicle stability has resulted in a number of proposed strategies for
yaw rate, attitude and cornering control for vehicles under extreme steer manouevres and with
uncertain road friction. A number of schemes have also been introduced in production vehicles
– most notably by Mercedes-Benz.
Although much effort has gone into the controller designs, relatively few studies have been
published in observer design. Also, while simple strategies have been suggested which require
only the directly measureable yaw rate, there is an increasing need for a more complete
handling state vector (eg yaw rate, sideslip velocity, roll rate and roll angle) to be available
[1,2,3]. This data set is not directly measurable, and sideslip velocity in particular is extremely
difficult to reconstruct without precise knowledge of the dynamic tyre characteristic.
Two recent examples which do consider the wider state vector are Kiencke and Daiβ [4],
which examines pole-placed observers in both linear and nonlinear forms – though these are
stationary – and Ray [5], which develops an extended adaptive Kalman filter. This latter study
is similar to the work presented here, but it relies on a higher-order model and larger sensor set
including measurement of applied wheel torques.
The Kalman filter is a popular candidate for observers of this type as it utilises available model
information along with the measurement set to maximise accuracy. However, to ensure
practical real-time operation, the model must remain low-order and time-invariant, whilst
retaining a useful level of accuracy; this is a significant constraint given the complex tyre force
variations which prevail. Some form of nominal model adaptation, or more formal on-line
identification seems inevitable in conjunction with any practically useful Kalman filter.
A preliminary study by the authors [6] illustrates the problem; a bicycle handling model was
used as the basis for the filter;
() () ()
ttt
δ
Γ+Φ= xx
&
with
()
[]
T
vrt =x and
=Γ
−−−+−
+−−−
=Φ
M
C
I
bC
Mu
CC
Mu
MucCbC
uI
cCbC
uI
CcCb
f
zz
f
rfrf
zz
rf
zz
rf
α
α
αααα
αααα
2
22
(1)
where the cornering stiffnesses C
αf
and C
αr
were adapted by a simple recursive least-squares
method. Both Kalman filter and adaptation were driven using lateral acceleration
measurements, eg at the mass centre,
() () ()
turtvty +=
&&
(2)
Figure 1 is typical of the findings of that study – although relatively fast adaptation of the C
α
can be achieved and the dynamic variations are reasonably well tracked, the model order is too
low. As a result, significant errors develop at very low frequencies in estimates
f
C
α
ˆ
,
r
C
α
ˆ
and
v
ˆ
which can not be corrected without direct measurement of the sideslip. Manipulation of
equations (1) and (2) shows that the errors in adapted cornering stiffness (
α
C
ˆ
- C
α
) can be
explained by steady-state sideslip error (
v
ˆ
- v) according to :
−−
−−
=
−
−
=
brvu
brvu
CC
crv
crv
CC
rrff
ˆ
ˆ
,
ˆ
ˆ
δ
δ
αααα
.(3)
0 2 4 6 8 10
−0.2
0
0.2
0.4
Yaw velocity, r (rad/s)
0 2 4 6 8 10
50
100
150
200
C
α f
(kN/rad)
0 2 4 6 8 10
−1
−0.5
0
0.5
Time (secs)
Sideslip velocity, v (m/s)
0 2 4 6 8 10
50
100
150
200
Time (secs)
C
α r
(kN/rad)
Figure 1 : Illustration of adaptive Kalman filter based on a ‘bicycle’ handling model
Similar manipulation of alternative extended models shows that inclusion of roll, pitch and or
vertical dynamics do not overcome the problem of equation (3), and neither will extension of
the tyre model, unless one resorts to the unreasonable assumption of an accurate tyre model
without adaptation.
In this study we retain the simple linear type C
α
model in the Kalman filter, but extend it to
include longitudinal dynamics, utilising the small perturbations that steering manouevres induce
in forward acceleration, to decouple C
α
f
from C
α
r
and v. The philosophy is that steady-state
errors can be reduced by adapting the model to closely track a forward acceleration sensor.
Source model
Adaptive observer
Further enhancements in this simulation study include the use of model nonlinearities in the
filter – an extended Kalman filter is prescribed; this has the additional advantage that the
adapted parameters (C
α
f
and C
α
r
) can be included in the state vector, so adaptation is implicit
within the filter design. Practical viability for production vehicles is also considered
throughout the study, in terms of computing requirements, and particularly in the prescription
of an inexpensive, reduced sensor set.
Section 2 describes the simulation models, Section 3 details the design algorithm for an
extended adaptive filter, and specifies a comparator algorithm. A verification experiment on
the influence of correlated noise sources is then given in Section 4. After presentation and
discussion of the results in Section 5, a summary of issues for practical viability is provided in
Section 6.
2. SIMULATION
2.1 Modelling
The study is carried out using a reference source model to provide ‘true’ state trajectories,
sensor measurements and also data for simple parameter identifications for the Kalman filter
model. This source model simulates full order motion of a rigid vehicle body, with
independent suspension freedoms, though vertical suspension modes are suppressed through
the assumption of inertialess wheels. A Pacejka tyre model is implemented in both longitudinal
and lateral axes, incorporating a friction limiting ellipse.
The precise model equations are omitted here, for brevity, but also – as we will show in
Section 4 – because the detail and even to some extent the accuracy of the source data is of
secondary importance; the study should reveal similar results for any suitably formulated high
order model, or indeed for an actual test vehicle.
For the purposes of the study, we will restrict attention to simulations where the longitudinal
dynamic is affected only by steer. Thus the source model is initiated at a constant speed,
25m/s, and the (front wheel) drive torque is maintained at a level just sufficient to overcome
simulated aerodynamic and rolling friction forces.
This provides a suitable isolated test case for the adaptive observer, and it allows a simplified
longitudinal tyre force model to be used in the observer model (though we shall see in Section
5.1 that more representative models could readily be accommodated within the proposed
observer design). Figure 2 shows a vehicle plan view, with front axle forces detailed; using
small-angle approximations these can be modelled in the SAE axis system as :
()
+
−−
=
−=
+≈
−≈
δ
µ
δ
δ
α
u
rbv
CF
uwF
FFF
FFF
flat
long
latlongy
latlongx
where,
(4)
