Open Journal of Applied Sciences, 2013, 3, 41-46
doi:10.4236/ojapps.2013.32B008 Published Online June 2013 (http://www.scirp.org/journal/ojapps)
Estimation of Longitudinal Tire Force Using
Nonlinearity Observer
Suwat Kuntanapreeda
Department of Mechanical and Aerospace Engineering, Faculty of Engineering,
King Mongkut’s University of Technology North Bangkok, Bangkok, Thailand
Email: suwat@kmutnb.ac.th
Received 2013
ABSTRACT
Tire forces are the major forces propelling the road vehicles. They significantly affect the dynamic behavior of the ve-
hicles. Estimation of the tire forces is essential in vehicle dynamics and control. This paper presents an observer-based
scheme for estimation of the longitudinal tire force of electric vehicles in real time. The observer is based on a
nonlinearity observer method. The pole-placement technique is used for determination of the observer gains. Simulation
results demonstrate that the observer is able to estimate the tire force successfully. The experiments are implemented on
a single-wheel electric vehicle test rig. The test rig comprises an electric motor driven wheel and a free-rolling drum
simulating vehicle-on-road situations. Experimental results confirm the effectiveness of the present scheme.
Keywords: Estimation; Nonlinearity Observer; Tire Force; Traction Control; Electric Vehicles
1. Introduction
Electric vehicles (EVs) have become very attractive for
replacing conventional internal combustion engine vehi-
cles because of environmental and energy problems [1].
The research and development of EVs and hybrid EVs
have been investigated on various topics such as, for
example, propulsion systems [2], power converters [3],
and motion control [4].
Traction control plays an important role in vehicle mo-
tion control because it increases drive efficiency, safety,
and stability. Tire forces are essential in traction control.
They are the vehicular propulsive forces produced by
friction between the rolling wheel and the road surface.
The characteristic of the friction between the wheel and
the road surface is very nonlinear. It mainly depends on
the wheel slip and the tire/road surface condition. In [5],
an approach to estimate the tire-road friction during
normal drive is presented. The approach is based on a
Kalman filter to give estimates of the slip-slope. In [6],
an on-line least-squares method is used to estimate the
parameters concerned with a friction force margin. The
effect of the estimation is evaluated by applying the
method to the breaking control. A slip-based method to
estimate the maximum available tire-road friction during
breaking is developed in [7]. The method is based on the
hypothesis that the slope of the slip curve at the low-slip
region during normal driving can indicate the maximum
friction coefficient. In [8], vehicle-dynamics-based
methods for tire-road friction coefficients estimation are
reviewed. The methods include slip-slope-based, lateral-
ehicle-dynamics-based, and an EKF-based estimation
methods. In [9], three different observers are developed
for the estimation of slip ratios and longitudinal tire
forces. The observers include one that utilizes engine
torque, break torque, and GPS measurements, one that
utilizes torque measurements and an accelerometer, and
one that utilizes GPS measurements and an accelerome-
ter.
This paper presents an observer-based scheme for es-
timation of the longitudinal tire force of EVs using a
nonlinearity observer. Simulation and experimental stud-
ies are used to illustrate the effectiveness of the present
scheme. A single-wheel test rig is used as an experimen-
tal test bench.
2. Nonlinearity Observer
The nonlinearity observer developed in [10,11] is re-
viewed in this section. Consider the following nonlinear
system
()()( (),)()
() ()
tt tt
tt
t

xAxNfx Bu
yCx
(1)
where , , and are, respectively, the state vector,
the control vector, and the output vector, respectively.
, , and are, respectively, the system ma-
trix, the control input matrix, the output matrix, and the
xu
C
y
NA B
Copyright © 2013 SciRes. OJAppS
S. KUNTANAPREEDA
42
nonlinearity matric. is an unknown nonlinear func-
tion, which is estimated by the observer.
()f
(()
()
fx
ˆ
ˆ






x
NH
v
The fundamental idea of the observer is to approxi-
mate by a fictitious system
()f
,) ()
()
tt t
tt
Hv
vVv (2)
By substituting Equation (2) into Equation (1), the ob-
server can be chosen as
ˆˆ
(
ˆv
 


 





x
L
xA B
uyC
L
0V 0
v
)
x
(3)
where x and v are the observer gain matrices that
must be chosen such that the observer is asymptotically
stable. In this paper, the pole-placement technique is
used to obtain the observer gains. When
L L
HI
and
are chosen, the observer is reduced to a propor-
tional-integral (PI) observer [11]
V0
ˆˆˆ ˆ
ˆ
xv
v
dt
dt
 

xAxB (yCx)L(yCx)
vL x)
ˆ
uL
(yC
ˆ()f
(4)
and the estimated nonlinearity is given by
ˆ().ttv
(5)
3. Single-Wheel Test Rig
The single-wheel test rig used in the experiments is
shown in Figure 1. It consists of a drum set, a wheel set
and a measurement/control unit. The role of the drum set
is to let the wheel to behave like rolling on the road. The
diameter and width of the drum are approximately 1.0
and 0.3 meters, respective. The wheel set consists of a
tire, a 4-inch-rim wheel, a brushed DC motor, and load-
ing masses. The wheel is directly driven by the motor
thought a rigid shaft. The measurement/control unit con-
sists of a signal condition circuit and a PC computer. The
computer is installed with a 12-bit analog/digital inter-
face board. The reader is referred to [12] for more details
of the test rig.
The mathematical model of the wheel set can be sim-
ply written as
1
11101
T
in
dK K
TE
K
J
CKVFr
dt RR

 


 (6)
where 1
is the rotational speed of the wheel, in
V is
the input voltage applied to the motor,
F
is the tire
force, 1 is the outer radius of the tire, 1
r
J
is the total
inertia of the rotating part including the wheel set, 1 is
the equivalent rotational damping constant, T
C
K
is the
torque constant of the motor,
K
is the back-EMF con-
stant of the motor, is the resistance of the motor
windings, and
R
0
K
is the effective gain of the motor
drive.
The mathematical model of the rotating drum can be
expressed as
2
222
d
2
J
CF
dt
r (7)
where 2
is the rotational speed of the drum, 2 is the
outer radius of the drum, 2
r
J
is the total inertia of the
drum, and is the equivalent rotational damping con-
stant.
2
C
The tire force
F
can be expressed as
()
F
N
(8)
where N is the normal force between the wheel and the
drum, ()
is the friction (or adhesion) coefficient,
is the driving slip ratio. The slip ratio is defined as
11
1
11
,
rv
r
0
(9)
where is the vehicle velocity. Here, since the motion
of the vehicle velocity is simulated by the rotation of the
drum, is substituted by the circumferential velocity of
the drum, i.e.,
v
v
22
vr
(10)
The parameters of the model are determined by direct
measurements and simulation tuning by comparing with
real experimental data. The first set of the parameters are
the mechanical parameters whose values are determined
directly from measurements. The values are summarized
in Table 1.
Figure 1. Photograph of the single-wheel test rig.
Table 1. Mechanical parameters of the test rig.
Symbol Value
1
J 0.0098 km m2
2
J 24.95 km m2
1
r0.13125 m
2
r 0.5 m
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S. KUNTANAPREEDA 43
By substituting the values from Table 1 into Equations
(6) and (7), it yields
1
1
0.0098 0.13125
in
dabV F
dt
  (11)
and
2
22
24.95 0.5
dC
dt F
 (12)
where 1
TE
KK
aC
R




and 0
T
K
b
R
K
. The values
of , , and 2 are experimentally determined by
measuring the step responses of the speeds of the wheel
and the drum. Figure 2 shows an example of the step
response. The initial velocities of the wheel and the drum
are zero. There is also a step change of in at time =
500 sec. After conducting several simulation tuning, the
approximated values of the parameters are found to be
, , and . In simulations,
a
0.
()
b
0392
C
0.47
V
a04b20.0C
is set as
22
2
() pp
p

(13)

with 0.3
p
and 0.9
p
are, respectively, the slip
ratio and the friction coefficient when the maximum fric-
tion occurred.
Simulation results of the same operating condition
done in the experiment are shown in Figure 3. The re-
sults confirm the validness of the simulation model. Note
that there is high-slip behavior occurred in the first 50
sec. The simulation model is also able to detect this be-
havior, with some small errors.
4. Observer Design
To design an observer, it is assumed that the output is
12
[T
]
. Note that, in actual vehicles, 1
can be eas-
ily obtained from an ABS speed sensor and 2
, which
represents the vehicle velocity, can be computed from the
velocity of a non-driven wheel or GPS measurements.
Equations (11) and (12) can be combined and written
in the form of Equation (1) as
40 12.448
00 0.20
10
01
F


 





xx
yx
u
]
(14)
where 12
[
T
x. Here,
F
is the nonlinearity
estimated by the observer. From Equation (3) and let
and , the observer can be expressed as
HI V0
4 012.448100
ˆˆˆ
00 0.20()
010
00 00










XXuL
y
X
(15)
where 12
ˆˆ
ˆˆ
[]
T
F

X.
The pole-placement technique is used here to obtain
the observer gain L. The desired poles in continu-
ous-time domain are simply chosen to be .
1, 2, 3 
In implementations, the observer is implemented in a
digital computer. The sampling period is set to be 0.5
second. Thus, the corresponding discrete-time desired
poles are 0.60 . Using the pole place-
ment technique, it results the following observer gain
65,0.3679, 0.2231
0.1612 0.1309
0.0021 0.7766
0.0859 0.0232

L (16)
To verify the effectiveness of the observer, simulations
are conducted in MATLAB/Simulink. The simulation
results are shown in Figure 4. In Figure 4(a), the results
show that the estimated values follow the true values
very well. Figure 4(b) displays the responses for only
the first 100 sec, where the system started from zero ve-
locity. The results show that the observer estimates the
tire force effectively even though the high slip occurred.
The convergent time is less than 100 sec. The responses
during the input voltage in stepping up are shown in
Figure 4(c). The estimates track the change effectively.
In summary, the simulation results indicate that the ob-
server is able to estimate the tire force successfully.
V
0100 200 300 400 500 600 700 800900 1000
0
50
100
Ti me( sec)
Wheel's speed(rad/sec)
0100 200 300 400 500 600 700 800900 1000
0
10
20
30
Ti me( sec)
Drum's speed(rad/sec)
Figure 2. Step response of the experimental test rig.
0100 200300 400 500600 700 800900 1000
0
50
100
Time(sec)
Wheel's speed(rad/sec)
0100 200300 400 500600 700 800900 1000
0
10
20
30
Time(sec)
Drum's speed(rad/sec)
Figure 3. Step response of the simulation model.
Copyright © 2013 SciRes. OJAppS
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0100 200300 400500 600 700800 9001000
0
50
100
150
Time(sec)
Wheel's speed(rad/sec)
True value
Estimated value
0100 200300 400500 600 700800 9001000
-10
0
10
20
30
Time(sec)
Drum's speed(rad/sec)
True value
Estimated value
0100 200300 400500 600 700800 9001000
-5
0
5
10
15
Time(sec)
Tire force(N)
True value
Estimated value
(a)
010 2030 4050 60 70 80 90 100
0
20
40
60
80
Time(sec)
Wheel's speed(rad/sec)
True value
Estimated value
010 2030 4050 60 70 80 90 100
-10
0
10
20
Time(sec)
Drum's speed(rad/sec)
True value
Estimated value
010 2030 4050 60 70 80 90 100
-5
0
5
10
15
Time(sec)
Tire force(N)
True value
Estimated value
(b)
500 505 510515 520 525 530 535540 545550
70
80
90
100
110
Time(sec)
Wheel's speed(rad/sec)
True value
Estimated value
500 505 510515 520 525 530 535540 545550
20
22
24
26
Time(sec)
Drum's speed(rad/sec)
True value
Estimated value
500 505 510515 520 525 530 535540 545550
-5
0
5
10
Time(sec)
Tire force(N)
True value
Estimated value
(c)
Figure 4. Simulation results: (a) time = 0 - 1000 sec, (b) time
= 0 - 100 sec, (c) time = 500 - 550 sec.
5. Experimental Results
The sampling rate of 0.5 sec is used in all experiments.
The same operation condition used in the simulations is
repeated in the experiments. The experimental results are
shown in Figure 5. They indicate that the observer is
successfully implemented. Note that, in experiments, the
true value of the tire force is unknown because it is not
measurable. Since the experimental responses are very
similar to those in the simulations, this implies that the
estimates of the tire force is convincing.
6. Conclusions
Observer-based estimation of tire forces for road vehicles
is studied in this paper. The observer is based on the
nonlinearity observer developed in [10,11]. The pole-
placement technique is used for determination of the ob-
server gains. Both simulation studies and experiments are
conducted in the paper. In the experiments, a single- wheel
test rig is used as a testing platform. Both simulation and
Copyright © 2013 SciRes. OJAppS
S. KUNTANAPREEDA 45
0100 200300 400500 600 700800 900 1000
0
50
100
150
Ti me(s ec )
Wheel's speed(rad/sec)
True value
Estimated value
0100 200300 400500 600 700800 900 1000
0
10
20
30
Ti me(s ec )
Drum's speed(rad/sec)
True value
Estimated value
0100 200300 400500 600 700800 900 1000
-10
0
10
20
Ti me(s ec )
Tire force(N)
Estimated value
(a)
0102030405060 708090 100
0
20
40
60
80
Time ( sec)
Wheel's speed(rad/sec)
True value
Estimated value
0102030405060 708090 100
0
5
10
15
20
Time ( sec)
Drum's speed(rad/sec)
True value
Estimated value
0102030405060 708090 100
0
5
10
15
20
Time ( sec)
Tire force(N)
Estimated value
(b)
500 505 510 515 520525 530 535540 545 550
70
80
90
100
Time (sec)
Wheel's speed(rad/sec)
True value
Estimated value
500 505 510 515 520525 530 535540 545 550
18
20
22
24
26
Time (sec)
Drum's speed(rad/sec)
True value
Estimated value
500 505 510 515 520525 530 535540 545 550
0
2
4
6
8
Time (sec)
Tire force(N)
Estimated value
(c)
Figure 5. Experimental results: (a) time = 0 - 1000 sec, (b)
time = 0 - 100 sec, (c) time = 500 - 550 sec.
experimental results show the effectiveness of the present
observer.
7. Acknowledgements
The author gratefully acknowledges the support provided
by Science and Technology Research Institute, King
Mongkut’s University of Technology North Bangkok .
REFERENCES
[1] C. C. Chan, “The State of the Art of Electric, Hybrid, and
Fule Cel Vehicles,” Proceedings of the IEEE, Vol. 95, No.
4, 2007, pp. 704-718. doi:10.1109/JPROC.2007.892489
[2] K. T. Chau, C. C. Chan and C. Liu, “Overview of Per-
manent-Magnet Brushless Drives for Electric and Hybrid
Electric Vehicles,” IEEE Transactions on Industrial Elec-
tronics, Vol. 55, No. 6, 2008, pp. 2246-2257.
doi:10.1109/TIE.2008.918403
[3] M. J. Hoeijmakers and J. A. Ferreira, “The Electric Vari-
able Transmission,” IEEE Transactions on Industry Ap-
plications, Vol. 42, No. 4, 2006, pp. 1092-1093.
doi:10.1109/TIA.2006.877736
[4] Y. Hori, “Future Vehicle Driven by Electricity and Con-
trol-Research on Four-Wheel-Motored ‘UOT Electric
March II’,” IEEE Transactions on Industrial Electronics,
Vol. 51, No. 5, 2004, pp. 954-962.
Copyright © 2013 SciRes. OJAppS
S. KUNTANAPREEDA
Copyright © 2013 SciRes. OJAppS
46
doi:10.1109/TIE.2004.834944
[5] F. Gustafsson, “Slip-Based Tire-Road Friction Estima-
tion,” Automatica, Vol. 33, No. 6, 1997, pp. 1087-1099.
doi:10.1016/S0005-1098(97)00003-4
[6] E. Ono, K. Asano, M. Sugai, S. Ito, M. Yamamoto, M.
Sawada and Y. Yasui, “Estimation of Automotive Tire
Force Chraracteristics Using Wheel Velocity,” Control
Engineering Practice, Vol. 11, 2003, pp. 1361-1370.
doi:10.1016/S0967-0661(03)00073-X
[7] S. Müller, M. Uchanski and K. Hedrick, “Estimation of
the Maximum Tire-Road Friction Coefficient,” ASME J.
Dynamic Systems, Measurement, and Control, Vol. 125,
2003, pp. 607-617. doi:10.1115/1.1636773
[8] R. Rajamani, D. Piyabongkarn, J. Y. Lew, K. Yi and G.
Phanomchoeng, “Tire-Road Friction-Coefficient Estima-
tion,” IEEE Control Systems Magazine, Vol. 30, No. 4,
2010, pp. 54-69. doi:10.1109/MCS.2010.937006
[9] R. Rajamani, G. Phanomchoeng, D. Piyabongkarn and J.
Y. Lew, “Algorithms for Real-Time Estimation of Indi-
vidual Wheel Tire-Road Friction Coefficients,”
IEEE/ASME Trans. Mechatronics, Vol. 17, No. 6, 2012,
pp.1183-1195.
[10] P. C. Müller, “Estimation and Compensation of Nonlin-
earities,” Proceedings of the 1st Asian Control Confer-
ence, Tokyo, 1994, Vol. II, pp. 641-644.
[11] D. Söffker, T. J. Yu and P. C. Müller, “State Estimation
of Dynamical Systems with Nonlinearities by Using Pro-
Portional-Integral Observer,” International Journal of
Systems Science, Vol. 26, 1995, pp. 1571-1582.
doi:10.1109/MCS.2010.937006
[12] A. Kraithaisri, S. Kuntanapreeda and S. Koetniyom, “De-
velopment of A Single-Wheel Test Rig for Traction Dy-
namics and Control of Electric Vehicles,” Proceedings of
the 2012 Hong Kong International Conference on Engi-
neering and Applied Science, 2012, pp. 360-367.