AVEC'16
Friction Utilization for Tyre-Road Friction Estimation on
Snow, an Experimental Study
Anton Albinsson, Fredrik Bruzelius, Bengt Jacobson
Department of Applied Mechanics, Chalmers University of Technology, Gothenburg, Sweden
E-mail: [email protected]
Tyre-road friction estimation using effect-based approaches are challenging due to the large tyre excitation needed for
an accurate estimate. The required excitation level varies for different tyres, road surfaces, road conditions and tyre
models used in the estimator. Previous research has shown the required friction utilization on different surfaces.
However, due to the small sample sizes it is hard to draw any general conclusions. This paper investigates the tyre
excitation required to estimate the tyre-road friction coefficient with a generic estimator for 77 different tyres on snow
for four different tyre models and for different levels of measurement noise.
Topics/ Modelling and simulation, state estimation, Active safety and driver assistance systems, Automated driving and
collision avoidance
1. INTRODUCTION
The maximum tyre-road friction limits the
horizontal forces that the tyres can generate. Knowledge
about the tyre-road friction coefficient is thus useful for
active safety systems and driver assistance systems
where the appropriate intervention is dependent on the
maximum available road grip. Vehicles with increasing
levels of autonomous functions, such as autonomous
emergency braking or, in the future, fully autonomous
vehicles, can use this information to adapt intervention
thresholds or vehicle velocity based on the prevailing
road conditions. The information can also be shared
with other drivers or vehicles to warn them of an
approaching low friction area.
The tyre-road friction coefficient is difficult to
estimate during normal driving due to the limited
information about the maximum achievable tyre forces
in the linear tyre region, see [1, 2]. Previous research
have tried to correlate the slip stiffness to the maximum
road friction coefficient to remove the need for large
tyre excitation [3]. However, the relation between the
friction coefficient and the slip stiffness relies on
empirical data and a-priori information about the tyres
that are currently fitted to the vehicle is thus required.
Another approach was evaluated experimentally in [4]
where opposite wheel torques are added to the front and
the rear axle to achieve large tyre excitation during
normal driving, thus making the friction coefficient
available when demanded.
The required utilization to estimate the friction
coefficient for different tyres and road surfaces using a
brush model is investigated in [5]. However, only a few
different tyres are evaluated and it is therefore difficult
to draw any general conclusions.
This study investigates how well the tyre-road
friction coefficient can be estimated by fitting nonlinear tyre models to measurements of 77 different tyres
on a snow surface. Hence, the estimation error that can
be expected without any a-priori information about the
tyre can be evaluated for a large sample size.
Furthermore, the estimation error for different tyre
models are evaluated and compared, thus indicating
which tyre models that are more suitable for friction
estimation on snow. Noise is also added to the input
signals to investigate how different fitting error cost
functions influence the estimation accuracy in the
presence of measurement noise.
2. METHOD
The measurements were taken from [6]. A mobile
test rig was used to produce five to six slip ratio sweeps
up to a fully locked wheel for each test run. The data
was filtered before it was used to evaluate the different
tyre models.
Four different tyre models are evaluated, the brush
model with parabolic pressure distribution, the magic
tyre formula with 4 parameters, Burckhardt tyre model
and Dugoff tyre model. These tyre models are
commonly used for online friction estimation, where the
number of parameters is of a concern due to the limited
number of measurement points.
The tested tyres include new and worn, studded and
non-studded winter tyres. The tyre model parameters are
fitted to the measurements to minimize two different
cost functions, eq. (1) & (2). The cost function in eq. (1)
does not consider the noise in the slip ratio signal and
should thus reasonably be more sensitive to
measurement noise.
𝑁𝑁
𝐽𝐽𝑓𝑓 = �(𝑓𝑓𝑚𝑚 [𝑖𝑖] − 𝑓𝑓𝑑𝑑 [𝑖𝑖])2
𝑖𝑖=1
(1)
AVEC'16
𝑁𝑁
2)
𝐽𝐽𝑠𝑠 = �(𝑓𝑓𝑚𝑚 [𝑖𝑖] − 𝑓𝑓𝑑𝑑 [𝑖𝑖])
𝑖𝑖=1
𝜕𝜕𝑓𝑓
+
|
(𝜎𝜎 [𝑖𝑖] − 𝜎𝜎𝑑𝑑 [𝑖𝑖])2
𝜕𝜕𝜎𝜎𝑑𝑑 𝜎𝜎𝑑𝑑 [𝑖𝑖] 𝑚𝑚
(2)
4. DISCUSSION & CONCLUSION
where 𝑓𝑓𝑚𝑚 is the normalized force from the tyre
model at the measured slip ratio, 𝑓𝑓𝑑𝑑 is the measured
normalized force, 𝜎𝜎𝑑𝑑 is the measured slip ratio and 𝜎𝜎𝑚𝑚
is the slip ratio at the measured normalized force
obtained from an inverse tyre model. Eq. 2 is inspired
by the total least square approach but has been
simplified so that the terms in the cost function is
available directly from the tyre model.
The parameters of the tyre models were found using
the gradient based optimization function fmincon with
the interior-point algorithm in matlab. Constraints were
added to the tyre parameters in order to obtain
reasonable tyre characteristics.
3. PRIMARY RESULTS
The mean of the tyre-road friction coefficient estimate,
normalized with the measured maximum friction
coefficient, for the different tyre models and for
different utilization levels is shown in figure 1.
Friction estimation [µest/µmax]
2.5
Brush Model Parabolic
MTF C=1
MTF C=Free
Dugoff
BurckHardt
Current Utilization
2
1.5
1
0.5
0
0
0.2
0.6
0.4
Utilized friction [µ/µmax]
0.8
1
a)
Brush Model Parabolic
MTF C=1
MTF C=Free
Dugoff
BurckHardt
Current Utilization
Friction estimation [µest/µmax]
4
3
2
added the force and the slip signal and minimizing force
error. c) White Gaussian noise added to the force and
the slip signal and minimizing force & slip error (eq. 2).
When noise is present only minimizing the force
error, eq. 1., is not sufficient since the friction
coefficient estimate tends to follow the current
utilization at low tyre utilizations (0 to 20% utilization).
The additional slip term in the alternative cost function,
eq. 2, increases the error at larger tyre utilizations but
creates a more distinguished separation between low
and high friction at low tyre utilizations, compare figure
1b) with figure 1c). Although the actual friction
coefficient is unknown for small tyre forces, the
separation makes is possible to distinguish between
high- and low-friction surfaces at low excitation levels.
The Magic tyre formula together with the Dugoff tyre
model shows good promise to be able to separate
between low and high friction at low excitation and to
achieve accurate friction estimation at larger excitation.
In the full paper the minimum and maximum friction
estimates will be presented and discussed. The tyre
models will be evaluated versus the raw measurement
data to investigate if the same trends can be observed
with real measurement noise. The consequences of
adding a penalty on the slip error in the cost function
will be further discussed. The estimation error for
different utilization levels will also be presented and
discussed with respect to the utilization required to
achieve a given estimation error.
REFERENCES
[1]
[2]
1
0
-1
0
0.2
0.6
0.4
Utilized friction [µ/µmax]
0.8
1
[3]
b)
Friction estimation [µest/µmax]
6
Brush Model Parabolic
MTF C=1
MTF C=Free
Dugoff
BurckHardt
Current Utilization
5
4
3
[4]
2
1
[5]
0
-1
0
0.2
0.6
0.4
Utilized friction [µ/µmax]
0.8
1
c)
Fig 1. Mean normalized friction estimation as a
function of utilized friction with standard deviation
shown as error bars. The current friction utilization is
plotted as a dotted black line. a) Noise free input signals
and minimizing force error. b) White Gaussian noise
[6]
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Coefficient of Friction between Tire and Road
Based on Vehicle State Measurements," PhD
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