Differential Speed Steering Control for Four

Differential Speed Steering Control for Four
International Journal of Materials, Mechanics and Manufacturing, Vol. 1, No. 4, November 2013
Differential Speed Steering Control for Four-Wheel
Independent Driving Electric Vehicle
Xiaodong Wu, Member, IACSIT, Min Xu, and Lei Wang

be zero when a spot turn is performed, which is impossible by
traditional steering method.
Abstract—The increasing prospect of electric drive vehicles
has led many different control schemes to apply on the driving
system. In this paper, differential speed steering which do not
use traditional steering mechanism is studied on a
four-wheel-driving vehicle. By comparison with the traditional
Ackerman-steered vehicle, differential-steered method showed
its advantage on the simple structure. Since the steering
performance is just related with the velocities of four wheels, the
strategy of differential speed on each motor is the major
challenge. With the analysis of kinematic model of the vehicle,
the relation between the turning behavior and the wheel
parameters are investigated. A steering speed control method is
proposed to get a steady turning performance during the
acceleration or deceleration. The control strategy is
implemented in a simulation to verify its rationality.
(a) Ackerman-steered vehicle
Index Terms—Differential speed steering, four wheel
independent driving, electric vehicle.
Differential speed steering is widely used in the navigation
of mobile robot [2]-[4]. The steady steering trajectory of the
mobile robot during the steering is not a main consideration,
while the human-driving vehicle should take into account.
Due to identical steering mechanisms, wheeled and tracked
skid-steered vehicles share many properties [5], [6]. Many of
the difficulties associated with modeling and operating in
these kinds of skid-steered vehicles arise from the complex
interaction between wheel and terrain. For Ackerman-steered
vehicles, the wheel motions may often be accurately modeled
by pure rolling, while for differential-steered vehicles, in
general, are modeled by curvilinear motion, the wheels roll
and slide at the same time [7]. This makes it difficult to
develop kinematic and dynamic models, which accurately
describe the motion characteristic. In [8] the comparison of
differential-steering and Ackerman-steering for mobile robot
is discussed. Vehicle can be steered through differential
braking which is to get a speed difference in similar thought
[9].
To obtain the desired turning radius is a difficult issue
during the differential speed steering. Since the steering
mechanism is using to implement the turn motion of traditional vehicle, the change of the wheel speed has no influence
on the performance of turn motion. However, the turn motion
during the differential speed steering is strongly coupled with
the wheel speeds. In this paper, the objective is to analyze the
behavior of differential speed steering applied to a fourwheel-driving electric vehicle when the inner and the outer
wheel are given different speeds. To investigate the steering
performance and behavior, the control model of the differential speed steering is studied. Due to complex tire/ground
interactions and kinematic constraints, high speed driving will
not be considered in this paper. A kinematic model based on
I. INTRODUCTION
With the rapid development of electric vehicles (EVs), it
has been commonly recognized that EVs are inherently more
suitable to realize novel control over conventional internal
combustion engine vehicles (ICEVs). Especially for the EVs
with in-wheel motors installed in each tire can realize
distributed driving system. The use of separate traction
motors at each wheel implies that torque to each drive wheel
can be controlled independently. By making full use of the
excellent control capabilities of electric drive systems, EVs
could not only be clean, but also be able to achieve better
maneuverability that cannot be reached by conventional
ICEVs.
In traditional ICEVs, a steering mechanism is
indispensable for the turn motion. However, by using the
merit of individual wheel traction motors on the electric drive
vehicles, it opens up the possibility of using a differential
speed steering systems [1]. A simple scheme is shown in Fig.
1. Since gearbox, retarder, transmission and steering
mechanism are canceled from the four in-wheel motors
independent drive vehicle, it has flexible layout and more
efficient driving system. Using four in-wheel motors independently to drive vehicle, the steering of the vehicle is
achieved by the differential speed controls on four wheels.
The minimum turning radius of differential speed steering can
Manuscript received October 15, 2012; revised December 23, 2012. This
work was supported by the National Post-doctoral Science Founda-tion of
China under Grant 108804
X. Wu, M. Xu, and X. Liu are with Institute of Automotive Engineering,
School of Mechanical Engineering, Shanghai Jiaotong University, 800, Rd.
Dongchuan, China (e-mail: xiaodongwu@sjtu.edu.cn, mxu@sjtu.edu.cn,
wanglei19880307@126.com).
DOI: 10.7763/IJMMM.2013.V1.77
(b) Differential-steered vehicle
Fig. 1. Two kinds of steering methods
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International Journal of Materials, Mechanics and Manufacturing, Vol. 1, No. 4, November 2013
driving vehicle is suitable to take a similar control method.
The motion direction of this kind of electric vehicle can be
changed by rotating the left and right side wheels at different
velocities, which is called differential speed steering. This
steering mechanism makes the vehicle mechanically robust
and simple which provides a new way to realize the motion
control of electric vehicle.
the geometrical relationship for vehicle is figured out. Based
on the vehicle model, Steering Speed Control (SSC) is
proposed to keep the turning curve in desired value during
acceleration and deceleration driving. The variable quantity
of left wheels and right wheels are concluded that should
satisfy a proportional relation. Differential speed steering
control strategy for four in-wheel motors independent drive
vehicle is performed to verify rationality of this method by
simulation.
Fig. 3. ODE-based Simulation for the vehicle dynamic.
To analyze the differential speed steering control system, a
simulator for four-wheel independent driving vehicle has
been developed in an Open Dynamics Engine (ODE)
environment. A screenshot is shown in Fig. 3. In the
simulation, the four wheels are driving by independent motor
and keeping the same orientation as the vehicle body. The
vehicle is running on a firm ground surface, and four wheels
are always in contact with the ground surface. The physical
parameters of the simulated vehicle are given in Table I.
Fig. 2. Ackerman-Jeantand steering model.
This paper is organized as follows. Traditional steering
method and the proposed differential speed steering is
compared in Section II. The dynamic model and kinematic
relationship of the differential speed steering is introduced in
Section III. In the Section IV, to get a stable performance of
steering, the control strategy of the proposed model is
discussed. The conclusion and some discussions of future
work are described in Section V.
TABLE I: PHYSICAL PARAMETERS OF THE SIMULATED VEHICLE
II. DIFFERENTIAL SPEED STEERING
A. Traditional Steering Method
According to the Ackerman-Jeantand model [10] of
steering when vehicle is running in a low speed, as be shown
in Fig. 2, we define L is the distance between front and rear
wheel, W is the track width of the vehicle, δ is average steering
angle of two front wheels.
Depending on the above analysis, when the vehicle is
steered with an angle δ, the turning radius R can be expressed
by:
L
(1)
R
tan 
Body Length L:
2.6m
Body Width W:
1.4m
Body Hight H:
1.3m
Radius of wheel:
0.33m
Weight of body mb:
400kg
Weight of wheel mw:
25kg
Total Weight m:
500kg
Friction coefficients:
0.8
To simplify, two wheels on each side of vehicle is assumed
to rotate at the same speed which is similar to the tracked
vehicle. For the differential speed steering, the turning
direction is determined by the side which has higher speed. If
we define Δω = ωR - ωL, the vehicle will turn left when Δω > 0,
and the vehicle will turn right when Δω < 0. To get desired
speed difference, the control strategy can be implemented by
three methods. One is increasing the speed of outer wheel,
another one is decreasing the inner wheel speed, and the third
one is speeding up the outer wheels and slowing down inner
wheels simultaneously. Generally, from safety perspective,
the vehicle speed is required to be slow down or be steady
during the steering motion. So the latter two methods by
decreasing the inner wheel speed will be considered in most
situations. Three turning trajectories under different wheel
speed are shown in Fig. 4. It can be found the turning radius is
From the Ackermann-Jeantand model, the kinematic
geometrical relation of the vehicle parameters during the
steering are expressed. The speed distributive relationship is
achieved by balancing the force of the four wheels. It can be
found that the turning radius of the vehicle is mainly
influenced by the steering angle of the front wheels, and has
no relationship with the advance speed of the vehicle.
B. Differential Speed Steering
Steering by differential speeds between the left and right
wheels is widely used in the motion control of mobile robots.
Benefited from the in-wheel-motor, four-wheel independent
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International Journal of Materials, Mechanics and Manufacturing, Vol. 1, No. 4, November 2013
m and I are the mass and moment of inertia of the vehicle. W
and L are the width and length of the vehicle.
increased corresponding to the decrease of the speed
difference Δω.
mx  FLfx  FRfx  FLrx  FRxx  m o y
0.5
Δω=0.15
Δω=0.2
Δω=0.25
0
my  FLfy  FRfy  FLry  FRxy  m o x
Io  ( FLfx  FRfx  FLrx  FRxx )
-0.5
-1
( FLfy  FRfy  FLry  FRxy )
-1.5
-2.5
-1
-0.5
0
0.5
1
(2)
L
2
In Fig. 5, signs such as β denote the angle between the body
axis and the moving direction of the vehicle. It is also can be
taking as the sideslip angle of the tire which refers to the angle
formed by the traveling direction and the surface of revolution
of the tire. If the wheel radius is defined as r, the longitudinal
and lateral force to the tire expressed in equation (2) is
expanded in the following equations:
-2
-3
-1.5
W
2
1.5
Fig. 4. Trajectory of the four-wheel-driving vehicle under different speed
between left and right wheels (ωR = 0.3).
J   T  F x r
III. MODELING OF VEHICLE STEERING
F y  C 
(3)
FRfx
where T is the driving/braking torque of the motor, Jω is wheel
inertia, ω is wheel angular velocity, Cα is the cornering power
of the individual tires.
FRfy
x
FLfx
V
β
Fy
B. Steering Kinematic
ICR denote the instantaneous center of rotation of the
vehicle body in Fig. 5. Based on the previous study, it is
believed the ICR is laid on a line parallel to the y-axis [2], [7],
[8]. In the local body coordinate, the longitudinal location
ICRx and the lateral location ICRy satisfies the following
constraints:
L
Lf
o
ICR
y
.
ψo
FRrx
FRry
Fx
Lr
Y
O
voy
ICRx   .
FLry
X
o
W
v
ICRy  .ox
(4)
o
Fig. 5. Schematic of the kinematics model for a skid-steered vehicle.
where vox and voy are the velocity of the mass center of vehicle
body in longitudinal direction and lateral direction.
Define vR and vL be center linear velocities of right and left
vehicle wheels shown in Fig. 6. The radius of the turn can be
calculated from similar triangles:
A kinematic model of a differential-steered vehicle which
maps the wheel velocities to the vehicle velocities is very
important during the development of control strategy. To
mathematically analyze the dynamic models of the
differential speed steering, a schematic of four-wheel-driving
vehicle moving at constant velocity about an ICR
(instantaneous centers of rotation) is shown in Fig. 5. The
local coordinate frame, which is attached to the body center of
gravity, is denoted by L(x,y,z), where x is the lateral
coordinate, and y is the longitudinal coordinate.
vL  vR
2
.
vL  vR
o 
W
W vL  vR
ICRy 
2 vL  vR
vox 
A. Vehicle Dynamic
The mass center of the vehicle is assumed at the geometric
center of the body frame. The 3-DOF dynamic equations in
the body frame L(x,y,z) can be written in equation (2), where
the Fx and Fy, indicate the longitudinal force and the lateral
force. In the suffixes to each force F, the first letter refers to
the left or right and the second to the front or rear. For
example, FLf refers to the force acting on the left front wheel.
(5)
The instantaneous turning radius of the vehicle can be
expressed by
Ro  ICRx2  ICRy2
(6)
In the local body coordinate, using equation (4) and
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International Journal of Materials, Mechanics and Manufacturing, Vol. 1, No. 4, November 2013
calculate from this complex nonlinear relationship.
To figure out the relation between the turning radius and
the rotation speed of wheels, we introduce a nondimensional
variable γ as the ratio of sum and difference of left-side and
right-side wheel rotating velocities. It can be expressed as
equation (5), the kinematic relationship between the whole
vehicle and the wheel speed can be summarized as:
vR
x
vL

V
L
vox
ICRx
Ro
.
ψo
(11)
From the experimental results, we find that the turning
radius has a linear relation with the parameter γ. A group of
the experimental data shown in Fig. 8 has demonstrated such
an observation. Here we set the right wheels rotating with a
fixed speed, and the left wheel will change its speed from the
zero to the same value as the right one. Therefore, we can use
equation (12) to evaluate desired turning radius.
voy
o
ICR
y
R  L
R  L
ICRy
W
R  K  K
Fig. 6. Simplified kinematic model of the differential speed steering.
1 1




2
2
 vox  

ICRx   vL 
   ICRx
v


 
 oy  
W   vR 
.   W

 o
1
1



W
W


R  L
R  L
(12)
where K is a proportionality coefficient which can be
calculated by a curve fit of the experimental data. If ωR = ωL,
then R = ∞, and the vehicle is moving forward directly. If ωR =
-ωL, then R = 0, since the vehicle is performing in-place
rotating about its center without any translational motion.
Since the turning radius during the differential speed
steering is strongly coupled with the wheel speeds on both
sides, the steering performance is sensitive to the real-time
velocity of vehicle. During the steering, the acceleration or
deceleration of vehicle with simple speed variation will make
an oversteering or understeering. To keep stable turning
performance, the change of the speed in the both right-side
and left-side wheels should satisfy a proportional relationship.
If we define the change of the wheel speed on both side as Δx
and Δy, to keep the turning radius during differential speed
steering, the following equation should be satisfied:
(7)
Wheel slip plays a critical role in the behavior of the
differential speed steering. When the vehicle turns, the
longitudinal slip of the wheels on both sides will be different.
For the outer wheel, the wheel rotate speed is greater than the
wheel line speed due to acceleration, and for the inner wheel,
the wheel rotate speed is smaller than the wheel line speed due
to deceleration. The longitudinal slip will be greater than zero
in the inner wheel while be less than zero in the outer wheel.
To obtain the desired turning radius and the yaw rate, the
difference between the outer and inner speed must be
controlled in real time.
R  x  L  y R  L

R  x  L  y R  L
(13)
10
14
ωR = 0.3
ωR = 0.4
ωR = 0.5
ωR = 0.3 rad, ωL = 0.0 ~ 0.3 rad
ωR = 0.4 rad, ωL = 0.0 ~ 0.4 rad
ωR = 0.5 rad, ωL = 0.0 ~ 0.5 rad
12
10
R (m)
5
R (m)
8
6
4
0
0
0.1
0.2
0.3
0.4
2
ωR - ωL (rad)
Fig. 7. Relation between the turning radius and wheel speed.
0
0
4
8
12
16
20
γ
Fig. 8. Relation between the turning radius and wheel speed parameter γ.
After simplification, if the change of the wheel speed Δx
and Δy satisfy the proportional relationship in equation (14),
the turning radius can keep the same. To facilitate the
following discussion, we call this Steering Speed Control
(SSC).
IV. CONTROL FOR DIFFERENTIAL SPEED STEERING
From the analysis of the kinematic relation during the
differential speed steering, the turning radius has nonlinear
relation with the speed difference between the wheels on both
sides. Here a group of turning experiment under different
initial speed is carried out. The results are shown in Fig. 7. It
can be found that desired turning radius is difficult to
x R

y L
358
(14)
International Journal of Materials, Mechanics and Manufacturing, Vol. 1, No. 4, November 2013
REFERENCES
In Fig. 9, we show the steering trajectories of simulated
vehicle with SSC and without SSC. From the black circle, the
speed of vehicle is changed for acceleration or deceleration
until it reach white circle. An identical variable quantity (Δx =
Δy = 0.3rad) has been added on each wheel for the
acceleration trajectory. Meanwhile, for the braking trajectory
each wheel has an identical decrease of the speed (Δx = Δy =
-0.3rad). From the differential speed steering without SSC, it
can be found that the vehicle is oversteering or understeering
during the acceleration and deceleration. While the change of
the wheel velocity is based on the ratio relation in equation
(14), the vehicle can moving in the desired steering circle
steadily.
L. Zhai and S. Dong. “Electronic Differential Speed Steering Control
for Four In-wheel Motors Independent Drive Vehicle,” in Proc. the 8th
World Congress on Intelligent Control and Automation, Taipei, China,
pp. 780-783, 2011
[2] J. Yi, H. Wang, J. Zhang, D. Song, S. Jayasuriya and J. Liu. “Kinematic
Modeling and Analysis of Skid-Steered Mobile Robots With
Applications to Low-Cost Inertial-Measurement-Unit-Based Motion
Estimation,” IEEE Transactions on Robotics, vol. 25, no. 5, pp.
1087-1097, 2009
[3] W. Yu, O. Chuy, E. G. Collins, and P. Hollis, “Analysis and
Experimental Verification for Dynamic Modeling of A Skid-Steered
Wheeled Vehicle,” IEEE Transactions on Robotics, vol. 26, no. 2, pp.
340-353, 2010
[4] K. Kozlowski and D. Pazderski, “Modeling and control of a 4-wheel
skid-steering mobile robot,” Int. J. Appl. Math. Comput. Sci., vol. 14,
no. 4, pp. 477-496, 2004
[5] J. Y. Wong, Theory of Ground Vehicles, 3rd ed. New York: Wiley,
2001.
[6] K. R. Weiss, “Skid-steering,” Auto. Eng., pp. 22-25, 1971.
[7] J. Y. Wong and C. F. Chiang, “A general theory for skid steering of
tracked vehicles on firm ground,” Proceedings of the Institution of
Mechanical Engineers, Part D: Journal of Automobile Engineering,
vol. 215, no. 3, pp. 343-355, 2001
[8] B. Shamah, “Experimental Comparison of Skid Steering Vs. Explicit
Steering for a Wheeled Mobile Robot,” Master Thesis, The Robotics
Institute, Carnegie Mellon University, Pittsburgh Pennsylvania, USA,
1999.
[9] T. Pilutti, G. Ulsoy, and D. Hrovat. “Vehicle Steering Intervention
Through Differential Braking,” in Proc. American Control
Conference, Washington, USA, pp. 1667-1671, 1995.
[10] J. Ackermann and W. Sienel, “Robust yaw damping of cars with front
and rear wheel steering.” IEEE Transactions on Control Systems
Technology, vol. 1, no. 1, pp. 15-20, 1993.
[1]
1
Accelerate
Trajectory
Without SSC
-4
Brake Trajectory
Without SSC
Turn Trajectory
With SSC
-9
-10
-8
-6
-4
-2
0
2
Fig. 9. Steering trajectories of the vehicle with SSC and without SSC.
Xiaodong Wu received the B.Eng. and M.Eng. degree
in Mechanical & Electronic Engineering Science from
China University of Petroleum, Beijing in 2008, and
the Ph.D. degree from Ritsumeikan University, Japan
in 2010, for research on the bio-inspired control
system of the snake-like robots. He is currently
working as a Postdoctoral Researcher in the
Department of Mechanical Engineering at Shanghai
Jiao Tong University. His research interests include
dynamic control of electric vehicles, bio-inspired robots, and automotive
electronics. He is a member of IEEE, IACSIT.
V. CONCLUSIONS AND FUTURE WORKS
A. Conclusions
By utilizing the benefits of the independent motor driving
electric vehicle, differential speed steering is studied to
replace traditional mechanical steering in this paper. When
the inner and the outer wheel are given different speeds, the
steering behavior of the four-wheel-driving electric vehicle is
analyzed in an ODE-based simulation platform. To solve the
problem of changed turning radius during the velocity
alteration, the kinematic model of steering based on the
geometrical relationship for vehicle is figured out. Based on
the vehicle model, Steering Speed Control (SSC) is proposed
to keep the turning curve in desired value during acceleration
and deceleration driving. The simulated results also verify the
rationality of control strategy for differential speed steering in
four in-wheel motors independent drive vehicle.
Min Xu received the Bachelor's degree of the Marine
Powertrain Engineering, from Shanghai Jiao Tong
University in 1983, and M.Eng. and Dr.Eng. degree in
Transport Phenomena Engineering from University of
Hiroshima in 1987 and 1991. He worked as
postdoctoral researcher from 1991 to 1992, and as a
research staff from 1992 to 1995 in Carnegie Mellon
University. Since 1995 he worked in GM, Delphi,
Ford, Visteon respectively until 2003. From 2003 to
2006, he joined to Chery Automobile Company and worked as Vice
President. He is currently working as Assistant President, Director of
Institute of Automotive Engineering in Shanghai Jiao Tong University. His
research interests include spray and combustion, combustion system for
internal combustion engines, design and development for vehicle engines
and vehicle integration and development. He is Deputy Director of National
Engineering Laboratory for Automotive Electronic Control Technology. He
also is an expert committee for “863” Mega Project of Energy-efficient and
New Energy Vehicle. He is a SAE Fellow, and serves many societies and
conferences as committee members and chairs.
B. Future Works
During the differential speed steering with four wheels, the
minimum turning radius can be zero for a pivot steering. But
greater power and torque are required as a greater sideslip is
encountered. Wheel slip plays a critical role in kinematic and
dynamic modeling of the steering. Understanding the effect
slip of the differential-steering, we can optimize the torque
control of the motor to decrease the power consumption and
tyre wear. In the future work, the wheel/ground interactions
which directly provide traction and braking forces that affect
the steering stability and turning radius will be studied.
Wang Lei received the Bachelor's degree in the
Department of Dynamic & Mechanical Engineering
from Wuhan University in 2011. He is currenty
pursueing M.S. degree in the Department of
Mechanical Engineering at Shanghai Jiao Tong
University. His research interests include embedded
control system development, and vehicle dynamic
research.
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