EVS28
KINTEX, Korea, May 3-6, 2015
Optimal Torque Distribution Strategy for a Four
Motorized Wheels Electric Vehicle
Xudong Zhang 1, Dietmar Göhlich 2
Product Development Methods and Mechatronics,
Technical University of Berlin, Germany
Abstract
Electric vehicles (EV) with four motorized wheels allow for an independent and accurate torque control of
each wheel. This offers many possibilities for the development of effective new dynamic stability control
systems. Furthermore, EV should be developed with respect to energy saving. In this paper, an optimal
torque distribution strategy (OTD) for EV with a four motorized wheels is investigated. OTD combines
improved vehicle stability performance with low energy consumption. Objective functions for both vehicle
stability and economy are established with different weighting coefficients. They are determined from the
vehicle body sideslip angle through use of fuzzy logic rules aimed at improving vehicle economy under
stable driving conditions. The solutions to these objective functions are derived under the constraints of
direct yaw moment, desired driving torque and friction coefficient. The effectiveness of this proposed
control approach is tested and confirmed with a suitable simulation model.
Keywords: Electric Vehicles, Vehicle Stability, Vehicle Economy, Torque Distribution
1
Introduction
During past years, due to the declining fossil
resources and polluted environment, various
electric vehicles are developed rapidly [1].
Hereinto, the four motorized wheels electric
vehicle is the fast-growing hotspot, because of its
independent torque control for each wheel [1] [2].
Based on this advantage, a large number of
research projects have been carried out so far
focused on vehicle stability and controllability
improvement. However, a few studies are
involved to explore how to balance the vehicle
performance and energy consumption. It is a
remarkable fact that one of the purposes of the EV
development is global energy crisis.
In the previous research, for a four motorized
wheels EV the desired traction or yaw moment
could be calculated first through LQR or model
following control MFC [3][4], the next problem is
how to distribute it to the actuators.
In this paper, an optimal torque distribution
strategy (OTD) is developed on the basis of
traction control and yaw moment control with
global optimization and fuzzy logic control. And
the effectiveness of OTD is tested in
Matlab/Simulink.
2
System overview
Objective functions for both vehicle stability and
fuel economy are established with different
weighting coefficients, aiming to improve the
vehicle stability and economy performance. The
OTD strategy is able to adjust the weighting
coefficients in real time to meet basically all kinds
of vehicle running conditions. The control
structure is shown the Figure 1.
EVS28 International Electric Vehicle Symposium and Exhibition
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3
ax  u  v  r
a y  v  u  r
Vehicle system modelling
For the four-wheel-drive EV, both the vehicle
dynamic model and tire model are built in
Matlab/Simulink.
β0
Weighting Coefficients for
Objective Functions
β
Vehicle Economy
Objective Function
Desired Driving
Torque
Motor Property
Yaw Moment
Road Condition


 v  ar 

1    arctan
 u  1 Tr 


2 



 v  br 

 2   arctan 
 u  1 Tr 


2 



 v  ar 

 3    arctan
 u  1 Tr 


2 



 v  br 

 4   arctan
 u  1 Tr 


2 

Optimal Solutions
T1,T2,T3,T4
Figure 1: Control structure
3.1
(5)
Besides, the rotational dynamic equations for each
motorized wheel would be
  Tmi  beta  Fti  r  Fzi  d (6)
Jm  w
The slip angle for each tire could be described by,
Vehicle Stability
Objective Function
Constraints
(4)
Vehicle dynamics model
The vehicle model consists of the longitudinal
motion, lateral motion, yaw motion, and rotational
motion of the four wheels, which leads to a
vehicle model with seven degrees of freedom, as
shown in Figure 2.
(7)
(8)
(9)
(10)
It is very essential to take into account the load
transfer due to the longitudinal and lateral
acceleration for a better analysis about the vehicle
characteristics. So the normal load expression for
each wheel could be written as
b
h
b
 max  may
2l
2l
l
b
h
a
Fz 2  mg  max  may
2l
2l
l
b
h
b
Fz 3  mg  max  may
2l
2l
l
b
h
a
Fz 4  mg  max  may
2l
2l
l
Fz1  mg
Figure 2: Schematic of vehicle dynamic motion
According to Figure 2, the vehicle motion
equations can be expressed as the following
equations.
Longitudinal and lateral motions along the X and
Y-axis:
m  ax 
4
F
i 1 4
xi
m  ay 
1
 CD Af u 2
2
(1)
4
 Fyi
(2)
i 1 4
Rotational motions of yaw about Z-axis:
I z  r  Fx1  Fx 2  
B
B
 Fx 3  Fx 4  
2
2
 Fy1  Fy 3  a  Fy 2  Fy 4  b
These acceleration terms could be calculated as
(3)
3.2
h
B
h
B
h
B
h
B
(11)
(12)
(13)
(14)
Tire model
The tire lateral force and tire slip angle depend
linearly on each of them only if the lateral
acceleration is lower than 0.4g and slip angle is
less than 5 degrees. However, any situation that is
beyond this range will lead to a serious nonlinear
tire characteristic. Therefore, an accurate tire
model is necessary for the dynamic analysis. So
this paper applies the well-known Pacejka Magic
Formula tire model because of its high precision
EVS28 International Electric Vehicle Symposium and Exhibition
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and confidence [5]. The “Magic Formula” is
given as:
y=Dsin(Carctan{Bx-E[Bx-arctan(Bx)]}) (15)
Where y is the lateral or longitudinal force, x is
the slip angle or slip rate, B, C, D and E are the
stiffness factor, the shape factor, the peak value
and the curvature factor respectively [6]. The tire
longitudinal and lateral force characteristics used
in this paper are depicted in Figure 3 and Figure 4.
0.9
0.8
0.6
0.8
0.4
Ft/Fz
0.2
0.7
0
0.6
-0.2
-0.4
0.5
-0.6
-0.8
0.4
-1
-1
-0.5
0
Longitudinal slip rate [%]
0.5
maximum tire/road friction coefficient
1
1
1
Figure 3: Tire longitudinal force characteristics
4
4.1
Optimal torque distribution
strategy design
Vehicle stability objective function
On the basis of the Kamm Circle concept [7] [8],
which represents the friction limit for the rolling
wheel transmitting longitudinal and lateral forces
at the same time, the following objective function
is chosen,
min J s 
Fti2  Fsi2
 2 2
i 1 4 Fzi  ui
4
(16)
This vehicle stability objective function is the sum
of the squared normalized tire’s resultant
indicating the tire actual load, which means the
lower Js the greater potential the tire has to cope
with the possible contingency [9].
The objective function should satisfy the
following equality constraints and inequality
constraints, the desired driving torque Td , yaw
moment Myaw from upper control layer, the motor
properties and the road condition.
4
s.t.
1
0.9
0.8
0.6
0.8
Fs/Fz
0.4
0.2
0.7
0
0.6
-0.2
-0.4
0.5
-0.6
-0.8
0.4
-1
-40
-30
-20
-10
0
10
20
30
maximum tire/road friction coefficient
1
40
slip angle [degree]
Figure 4: Tire lateral force characteristics
It can be seen that in pure driving case the
longitudinal force is the function of normal force,
the road friction coefficient and longitudinal slip
while in pure cornering case the lateral force is
the function of normal force, the road friction
coefficient and slip angle. Besides, the tire
characteristics are quite different on different
roads. The longitudinal force reaches a maximum
value at the slip rate from 3% to 17%, and then it
decreases with slip rate increasing. Likewise,
lateral force attains its maximum at the slip angle
of around 3° to 10°, and then slowly decreases for
higher slip angles. What’s more, the tire
performance rapidly deteriorates on the low
adhesion road, which may well lead to vehicle
instability. This is one of the situations our control
strategy tries to avoid.
 T  beta  T
i 1 4
i
d
(17)
 T1  T2 T3  T4 


  B  beta  M yaw (18)
r 
 r
(19)
Ti  Tmotor
T1  T3  T2  T4
(20)
Fti2  Fsi2  Fzi  ui
(21)
Where Fzi is the normal load, ui is the road friction
coefficient, Ti is the torque output of each motor,
beta is the reducer ratio, r is the tire rolling radius,
B is the wheels track, Tmotor is the motor
maximum torque, Fti is the tire longitudinal force
and Fsi is the tire lateral .
4.2
Economy objective function
The relationship among the motor’s torque, RPM
and efficiency is shown in Figure 3. It can be seen
that the motor efficiency is quite distinguishing in
different working regions and especially when it
works in the low speed or low torque output
region, the motor has poor efficiency [10].
EVS28 International Electric Vehicle Symposium and Exhibition
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0.94
3
0.9
0.94
0.97
0.96
T1  T3  T f  p  Td
5
0.9
0.95
0.91
0.91
0.9
0.89
9
0.8
0.880.87
0.86
0.88
0.87
0.86
0.85
0.84
0.83
1500
2000
2
0.8
0.81
0.8
0.79
0.78
0.770.76 0.75
0.74
0.73
0.85
0.84
0.83
0.82
0.81
0.8
0.79
0.78
0.77
2500
3000
3500
4000
RPM [r/min]
Figure 5: Motor efficiency map
For example when the vehicle is running at high
speed and the driving torque is distributed equally
to the four wheels in the most common way,
based on Figure 5, for one single motor it must be
poorly efficient because of its low torque output.
Oppositely, if we use front-wheel-drive (FWD) or
rear-wheel-drive (RWD) instead of four-wheeldrive (4WD), then the motor torque output will
gain about twice as the original. That means the
motor works more efficiently.
According to the above qualitative analysis, an
economy objective function is established without
regard to regenerative braking.
min J e 
4
4
niTi
 P    T , n 
i 14
i
i 14 i
i


 p  T
1  p   Td  (28)
d
Je  2  n  


  p  Td , n   1  p   Td , n 


 
  2
2

In this paper the sequential quadratic
programming (SQP) method is applied to solve
the economy objective function. However, the
solving process contains a large number of
interpolation and iteration. Online calculation
cannot ensure system’s real-time performance.
Therefore, p is solved with the above method
in advance shown in the following graph.
1
 T  beta  T
(23)
T1  T3  T2  T4
(24)
i 1 4
𝑝∈[0.5, 1] is proportional distribution coefficient.
Then the final economy objective function is
obtained,
i
4
i
d
u F 

Ti  min  Tmotor , i zi 
r  beta 

(27)
(22)
The objective function should satisfy the
following equality constraints and inequality
constraints, the desired driving torque Td, the
motor properties, the road condition and a few
assumptions.
s.t.
(26)
T2  T4  Tr  1  p   Td
u
n
r
0.
9
0.93
0.92
0.93
Distribution coefficient
1000
0.9 0.
6 95
0.94
0.82
0.81
0.8
0.79
0.78
0.77
500
0.9
7
96
0.
91
0. 2
9
0.
0.9
0.89
0.88
0.87
0.86
20
10
0.
91
0.7
3
0.8
6
0 0
0 .75 .74
0. .76
77
0.
78
0.7
9
0.8
0.8 0.81
0.83
4 0.82
0
0.8
0.8.88
5
9
0.9
0.6
0.66 5
0.67
0.68
0.69
0.7
0.71
0.72
0.7
3
8
0.7
09.8
0.83 0.81
0.84 0.
82
0.85
30
5
0.9
92
0.
0.92
0. 0.7
0 75 4
0.7.76
7
0.7
Torque (N*m)
40
87
0.
94
0.
0.91
0.92
50
0.93
9
0.93
60
0.91
0.92
0.880.89
70
0.88
0.89
0.
88
0. 9
0.8 0.9
The assumptions are justified, since economy
optimal control will be activated only when the
controller determines the car runs stable.
The assumptions could be explained as the
following equations,
0.87
84 85
0. 0.
0.8
6
0.8
7
80
0.86
8
0.
0.
83
0.
0. 75
0.776
0.77
8
0.7
9
0.6
9
0
0. .7
0.771
2
0.6
0.676
0.68
90
0.
00. 56
0..5587
0 0.59
0 .6 6
0.60.6.621
43
100
0.9
0.8
0.7
0.6
0.5
0
100
0
200
Torque [N*m]
1000
2000
300
3000
400
(25)
Where Fzi is the normal load, ui is the friction
coefficient, Ti is the torque output of each motor,
beta is the reducer ratio, r is the tire rolling radius,
Tmotor is the motor maximum torque, ni is the
wheel rotational speed, and  i is the efficiency
function of each motor.
In order to simplify the modelling, a few
assumptions are used here.
1) The slip rate and rotational speed difference
of each wheel is quite small and hence, the
wheel rotational speed n equals u/r.
2) The output torque is identical if the motors
are on the same axle.
4000
RPM [r/min]
Figure 6: Proportional distribution coefficient as a
function of motor speed and torque output
4.3
Weighting Coefficient distribution
Figure 7, 8 show that when the steering angle of
front wheels is changed from -0.12 rad to 0.12 rad,
the characteristic between yaw moment, lateral
force and vehicle body sideslip angle respectively,
indicating that when the vehicle body sideslip
angle is big enough, both the yaw moment and the
lateral force will be approaching a constant, which
means vehicle loses its further steering ability
[11].
EVS28 International Electric Vehicle Symposium and Exhibition
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the top thread of vehicle lateral acceleration is
also used as stability criterion [12],
a y _ max    8 m s 2
(28)
4
1.5
x 10
Yaw moment [N*m]
1
|β/βmax|, |ay/ay_max| and w(β) are the inputs and
output variable of the fuzzy logic controller
respectively. They all have five linguistic values,
described as Z (zero), PS (positive small), PM
(positive middle), PB (positive big), P (positive).
The fuzzy rule base and input-output-state relation
are shown in Table 1 and Figure 9.
0.12 rad
0.5
0
-0.5
-0.12 rad
-1
Table 1: Fuzzy rule-base
-1.5
-0.2
-0.1
0
0.1
0.2
w(β)
0.3
Body sideslip angle [rad]
Z
PS
PM
PB
P
Figure 7: Body sideslip angle vs. yaw moment
4
1.5
|ay/ay_max|
x 10
PS
PS
PS
PM
PM
PB
|β/βmax|
PM
PM
PM
PM
PB
PB
PB
PB
PB
PB
P
P
P
P
P
P
P
P
-0.12 rad
0.5
1
0
0.8
w(beta)
Lateral force [N]
1
Z
Z
PS
PS
PM
PM
-0.5
-1
0.12 rad
0.6
0.4
0.2
0
1
-1.5
-0.2
-0.1
0
0.1
0.2
0.3
0.5
Body sideslip angle [rad]
ay/aym
0
0
0.2
0.6
0.4
0.8
1
beta/betam
Figure 8: Body sideslip angle vs. lateral force
Therefore, in order to improve vehicle economy
in the premise of stable driving, vehicle body
sideslip angle β is used to build the connection
between the two objective functions for stability
and energy saving.
max J  w J s  1  w J e
(26)
w(β)∈[0,1] is the weighting coefficient. When β
is relatively small, due to its steady running w
should be reduced to increase economy objective
function weighting coefficient. Conversely when
β is relatively big, w should be increased to ensure
the vehicle stability.
Here the top thread of vehicle body sideslip angle
is introduced [12],
 max  10  7 
2
vCoG
40 m s 2
Figure 9: Input-output-state relation
5
Simulation in Matlab/Simulink
The effectiveness evaluation of the proposed
torque distribution strategy is presented from two
aspects. One emphasizes on the vehicle stability.
The other focuses on the vehicle economy. The
simulation parameters are shown in the following
table.
(27)
Where vCoG is the speed of the centre of gravity.
Besides, another parameter that can describe
vehicle steering stability is lateral acceleration ay.
It is interpreted as the slope of β here. Similarly,
EVS28 International Electric Vehicle Symposium and Exhibition
Table 2: Vehicle parameters
Vehicle mass
Vehicle inertia about Z axis
Distance of c.g. from front axle
Distance of c.g. from rear axle
Frontal projected area
Wheels track
air resistance coefficient
Reducer ratio
Reducer efficiency
Tire effective rolling radius
1280 kg
2460 kg·m²
1.2 m
1.3 m
2 m2
1.5 m
0.35
3.5
0.9
0.3 m
5
Air density
Height of the sprung mass c.g.
Wheel rotational inertia
Economy Evaluation
friction coefficient and a wet earth road with 0.55
as the maximum tire/road friction coefficient.
Vehicle velocity is 40 km/h. The sinusoidal signal
is taken as a steering angle input shown in Figure
11.
0.2
This economy evaluation is according to the
NEDC (New European Driving Cycle) shown in
Figure 10. The even torque distribution and front
wheels torque distribution are chosen to compare
with optimal torque distribution strategy. Figure
12, 13 and 14 shows the simulation results.
25
0.15
Steering angle [rad]
5.1
1.1 kg/m3
0.35 m
2.22 kg·m²
0.1
0.05
0
-0.05
-0.1
20
Speed [m/s]
-0.15
-0.2
0
15
3
4
5
Figure 11: Sinusoidal steering angle input
5
200
400
600
800
1000
1200
time [s]
Figure 10: New European driving cycle
Figure 12 shows that for even distribution and
front wheel drive, they have their own suitable
working area. However, the optimal torque
distribution performs lowest power consumption
in whole time history. Besides, as it’s seen in the
Figure 13, the optimal torque distribution also
achieves lower thermal loss than the other two
strategies. Figure 14 illustrates that OTD strategy
can make motors work more efficiently, which
proves the effectiveness of OTD.
Table 3 lists the results of economy improvement
and equivalent weight reduction. It can indicate
that compared with even distribution strategy and
front wheel drive, the OTD deceases 3.584 % and
1.992 % energy consumption respectively. If we
use weight to measure the energy saving effect, it
means 47 kg and 26 kg weight reduction
equivalently.
5.2
2
time [s]
10
0
0
1
Stability Evaluation
Most accidents on the road are caused by the
sudden lane changing situation aiming at avoiding
a previous vehicle or other obstacles. Here the
single lane changing maneuver is used to test the
performance of the control system.
The lane change test is implemented on both a dry
concrete road with 0.8 as the maximum tire/road
From the Figure 15 and 16, it can be seen that not
only on a dry concrete road but also on a wet
earth road, with OTD control the vehicle’s yaw
rate and route can almost perfectly follow the
desired value. Oppositely, without control,
comparatively large deviation will appear in the
yaw rate response and movement path when the
car runs on a low friction road. Besides, on the
dry road the error between the actual sideslip
angle and desired sideslip angle is quite small
shown in Figure 15(c), which illustrates that the
vehicle runs stable. Therefore, the weighting
coefficient w is also relatively small. The driving
torque for each motor is mainly determined by
economy objective function Je. In contrast, if
there is a big difference between the actual
sideslip angle and desired sideslip angle shown in
Figure 16(c), which will increase the weighting
coefficient w, then the motor’s torque is mainly
determined by stability objective function Js.
6
Conclusions
A four motorized wheels EV nonlinear dynamic
model has been built in this paper. Based on
stability and economy objective functions a novel
optimal torque distribution strategy is proposed.
Fuzzy logic control is used here to allocate the
weighting coefficient between the two objective
functions. Several simulation experiments have
been implemented in Matlab/Simulink to verify
the strategy’s effectiveness. The simulation results
indicate that this optimal torque distribution
strategy is able to enhance vehicle stability and
improve fuel economy effectively.
EVS28 International Electric Vehicle Symposium and Exhibition
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Power [kW]
20
15
10
5
0
0
200
400
600
time [s]
equal distribution
800
front-wheel drive
1000
1200
1000
1200
1000
1200
1000
1200
optimal distribution
Thermal power [kW]
Figure 12: Power consumption
4
3
2
1
0
0
200
400
600
time [s]
Average allocation
800
Front-wheel drive
Optimal allocation
Figure 13: Thermal power consumption
Efficiency
Efficiency [%]
1
0.9
0.8
0.7
0
200
400
600
time [s]
equal distribution
800
front-wheel drive
optimal distribution
Improvement Rate
Improvement rate [%]
15
10
5
0
0
200
400
600
time [s]
equal distribution & optimal distribution
800
front-wheel drive & optimal distribution
Figure 14: Efficiency and its improvement rate
Table 3: Improvement analysis among different distribution strategies
Cycle Stage
General
Speed
High Speed
Whole
Cycle
Energy Consumption (kJ)
OTD*
*
ED
1406.36
1373.35
FWD* 1431.29
*
ED
1658.66
1581.82
FWD* 1583.95
*
ED
3065.02
1373.35
FWD* 3015.24
Energy Saving
2.347 %
4.048 %
4.632 %
0.134 %
3.584 %
1.992 %
Equivalent Weight
Reduction
30.91 kg
53.04 kg
60.69 kg
3.8 kg
47.02 kg
26.11 kg
* ED: even distribution, FWD: front wheel drive, OTD: optimal torque distribution strategy
EVS28 International Electric Vehicle Symposium and Exhibition
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0.5
0.4
0.4
0.3
0.3
Yaw rate [rad/s]
Yaw rate [rad/s]
0.5
0.2
0.1
0
0.2
0.1
0
-0.1
-0.1
-0.2
-0.2
-0.3
-0.3
-0.4
-0.4
-0.5
-0.5
0
1
2
3
4
5
0
time [s]
desired response
with control
1
2
desired response
without control
(a): Yaw rate response
3
4
with control
5
without control
(a): Yaw rate response
5
5
4.5
4.5
4
4
3.5
3.5
y postion [m]
y postion [m]
time [s]
3
2.5
2
1.5
3
2.5
2
1.5
1
1
0.5
0.5
0
0
10
20
30
40
0
0
50
x postion [m]
desired path
10
20
30
40
50
x postion [m]
path with control
path without control
desired path
(b): X-Y Movement
path with control
path without control
(b): X-Y Movement
0.04
0.03
0.03
0.02
Sideslip angle [rad]
Sideslip angle [rad]
0.02
0.01
0
-0.01
0.01
0
-0.01
-0.02
-0.02
-0.03
-0.03
0
1
2
3
4
5
time [s]
desired response
-0.04
0
1
(c): Sideslip angle response
time [s]
3
4
5
actual response
(c): Sideslip angle response
80
Torque output of each motor [Nm]
80
Torque output of each motor [Nm]
2
desired response
actual response
60
40
20
0
-20
-40
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
60
40
20
0
-20
-40
-60
-80
0.5
1
front left motor
rear left motor
front right motor
rear right motor
1.5
2
2.5
3
3.5
4
front left motor
rear left motor
front right motor
(d): Torque output of each motor
(d): Torque output of each motor
Figure 15: on a dry concrete road
Figure 16: on a wet earth road
EVS28 International Electric Vehicle Symposium and Exhibition
4.5
5
time [s]
time [s]
rear right motor
8
126, No.4, 2004,pp:753-763
Acknowledgments
The authors would like to thank CSC for
providing a scholarship as the financial support
for the first author to pursue his Ph.D. degree at
TU Berlin.
[10]
YU Zhuo-ping, ZHANG Li-jun, Xiong Lu,
Optimized torque distribution control to achieve
higher fuel economy of 4WD electric vehicle with
four in-wheel motors, Journal of Tongji University
(Natural Science), Vol. 33, No. 10, Oct 2005, pp:
1356-1361
[11]
Shibahata Y, Shimada K, Tomari T,
Improvement of vehicle maneuverability by direct
yaw moment control, Vehicle System Dynamics,
Vol. 22, No. 5-6,1993, pp: 465-481.
[12]
Kienecke U, Nielsen L, Automotive control
systems, ISBN 3-540-66922-1, Berlin, Springer,
2000
References
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Xudong Zhang received the B.E. and
M.E. in School of Mechanical and
Vehicle Engineering of Beijing
Institute of Technology. Currently he
is a Ph.D. candidate in Product
Development
Methods
and
Mechatronics at Technical University
of Berlin. His research focuses on
vehicle dynamic.
Dietmar Göhlich received his Ph.D.
degree in Mechanical Engineering at
Georgia Institute of Technology.
Currently he is the professor and the
leader of Product Development
Methods and Mechatronics at
Technical University of Berlin. His
research interests include electrics
vehicle and related areas, product
development methods, concept design
and CAD/CAE technology.
[7] M. Gafvert and J. Svendenius. Construction of
novel semi-empirical tire models for combined
braking and cornering. Technical Report ISRN
LUTFD2/TFRT--7606--SE,
Department
of
Automatic Control, Lund University, Sweden,
April 2003
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in-wheel motors - mythsand realities, Proceedings
of the 10th International Symposium on
Advanced Vehicle Control (AVEC 10), 2010, pp
261-266
[9] Mokhiamar O, Abe M, Simultaneous optimal
distribution of lateral and longitudinal tire forces
for the model following control, Journal of
dynamic systems, measurement, and control, Vol.
EVS28 International Electric Vehicle Symposium and Exhibition
9