Sliding Mode Control for Trajectory Tracking of Mobile Robot in the ...

International Journal of Control, Automation, and Systems (2009) 7(3):429-435
DOI 10.1007/s12555-009-0312-7
http://www.springer.com/12555
Sliding Mode Control for Trajectory Tracking of Mobile Robot
in the RFID Sensor Space
Jun Ho Lee, Cong Lin, Hoon Lim, and Jang Myung Lee*
Abstract: This paper presents a sliding mode control method for wheeled mobile robots. Because of
the nonlinear and nonholonomic properties, it is difficult to establish an appropriate model of the
mobile robot system for trajectory tracking. A robust control law which is called sliding mode control
is proposed for asymptotically stabilizing the mobile robot to a desired trajectory. The posture of the
mobile robot (including the position and heading direction) is presented and the kinematics equations
are established in the two-dimensional coordinates. According to the kinematics equations, the
controller is designed to find an acceptable control law so that the tracking error will approximate 0 as
the time approaches infinity with an initial error. The RFID sensor space is used to estimate the real
posture of the mobile robot. Simulation and experiment demonstrate the efficacy of the proposed
system for robust tracking of mobile robots.
Keywords: Mobile robot, RFID sensor space, sliding mode, trajectory tracking.
1. INTRODUCTION
There have been many researches on tracking control
of wheeled mobile robot [l,13-15,17,19]. The
stabilization of this kind of robot with restricted mobility
to an equilibrium state is in general quite difficult. The
wheeled mobile robot is a typical nonholonomic system.
Such a system suffers nonlinear and uncertainty
problems, so it can't be stabilized through a fixed
feedback. It implies that methods for linear control
theory cannot be applied to problems of controlling
nonholonomic systems. Due to their richness and
hardness, such nonlinear control problems have
motivated a large number of researches involving various
techniques of automatic control. Another difficulty in
controlling the mobile robots is that in the real world
there are uncertainties in their modeling. Because of this
uncertainty, the trajectory tracking error for the mobile
robot has always been produced and can not be
eliminated.
Until now many methods have been used for trajectory
tracking of mobile robots. PID control becomes unstable
easily when it is affected by the sensor sensitivity [2].
Fuzzy logic control suffers from the slow response time
due to the heavy computation [3]. Feedback linearization
approach is limited by the convergence conditions while
__________
Manuscript received June 30, 2007; Revised June 1, 2008 and
September 27, 2008; accepted January 7, 2009. Recommended by
Sooyong Lee under the direction of Editor Jae-Bok Song. This
work was supported by the Korea Science and Engineering
(KOSEF) grant funded by the Korea government (MOST) (No.
R01-2007-000-10171-0).
Jun Ho Lee, Cong Lin, Hoon Lim, and Jang Myung Lee are
with the School of Electrical Engineering, Pusan National
University, San 30, Jangjeon-dong Kumjung-ku, Busan 609-735,
Korea (e-mails: junho7666@nate.com, jmlee@pusan.ac.kr).
* Corresponding author.
Lyapunov oriented control is difficult to construct a
Lyapunov candidate function. Compared to the
approaches introduced previously, sliding mode control
has many advantages for trajectory tracking of mobile
robot such as fast response, good transient and the
robustness against system uncertainties and external
disturbances. Therefore, it has been researched
extensively [4].
Basically speaking, Sliding Mode Control (SMC)
developed in the 1950’s is a special nonlinear control
strategy with discontinuous property [5,18]. The peculiar
feature of SMC is that the system structure is not fixed
but dynamically changing. By designing switching
functions of state variables or output variables to form
sliding surfaces, SMC can guarantee that when
trajectories reach the surfaces, the switching functions
keep the trajectories on the surfaces to maintain the
desired system dynamics. Due to the above property,
SMC is attractive for many highly nonlinear uncertainty
systems [6,11,12,16].
In order to implement sliding mode control on
trajectory tracking of mobile robot, a kinematics model
of the robot needs to be established. The representation
of the kinematic equation of mobile robot for the
trajectory control can be set up in the Cartesian
coordinates [7]. Two sliding surfaces in Cartesian
coordinates are chosen in terms of tracking errors in
position and heading direction, respectively.
To evaluate the real posture of the mobile robot, RFID
sensor space is placed on the floor. An RFID (Radio
Frequency IDentification) technology is essential for a
non-touching recognition system that transmits and
processes the information on events and environments
using a wireless frequency and small chips. The RFID
system can recognize tags within a few centimeters at a
high speed. So in this paper, RFID system is used as a
method of mobile robot localization. Simulation and
© ICROS, KIEE and Springer 2009

430
Jun Ho Lee, Cong Lin, Hoon Lim, and Jang Myung Lee
experiment results show that this control system is
capable of stabilizing the mobile robot to a desired
trajectory.
2. KINEMATICS MODELING OF MOBILE
ROBOT
The mobile robots are increasingly required in
industrial and service robotics, particularly when flexible
motion capabilities are required on reasonably flat
grounds and surfaces. In this research a differential type
wheeled mobile robot is used as a test-bed for the
proposed algorithm. It has two driving wheels and one
passive centered rotatable wheel. The two fixed wheels
are controlled independently by electric motors, and the
passive wheel prevents the robot from tipping over as it
moves on a plane.
The posture of mobile robot can be presented by the
position which is the middle point of the two driving
wheels, and the heading direction θ . Fig. 2 shows the
position of the robot expressed in the X-Y coordinates.
Suppose the posture vector of mobile robot is
T
presented as p = ( xyθ , , ) , here (x, y) denotes the
position of mobile robot and θ is defined as the angle
between the X-coordinate and the heading direction. The
mobile robot’s motion is controlled by the vector
q = (, v ω) T here v is the linear velocity of the robot and
ω is the angular velocity, which are also functions of
time. According to the kinematics, the relationship
between the posture vector p expressed in the X-Y
coordinate and the velocity vector q = (, v ω) T is derived
as:
l
Passive
wheel
Driving wheels
Fig. 1. Overall structure of the mobile robot.
Y
y r
y
x
θ
x r
θ e
θ
r
Fig. 2. Coordinates assignment for a mobile robot.
x e
y
e
X
⎡ ẋ
⎤ ⎡cosθ
0⎤
ṗ
=
⎢
ẏ
⎥
=
⎢
sinθ
0
⎥
⎢ ⎥ ⎢ ⎥
q.
⎢⎣ θ̇
⎥⎦ ⎢⎣ 0 1⎥⎦
This kinematics is common to all kinds of vehicles
which are not omni-directional (For instance, an
automobile, a bicycle, a vehicle with two parallel
independent power wheels - power wheeled steering
system, and a tricycle). The linear velocity, v and
rotational velocity, ω of this kind of vehicle are
controlled by their accelerator and steering wheel,
respectively.
In this control system, two postures are used: the
reference posture, p = ( x , y , θ ) T and the current
T
r
r r r
posture, p = ( xyθ , , ) . A reference posture is a goal
posture of the vehicle and a current posture is its “real”
posture at this moment. An error posture, p e of the two
is defined as difference between the reference posture,
p r , and the current posture, p, which is represented as
T
e e e e
(1)
p ≡ ( x , y , θ ) .
(2)
The error equation of mobile robot can be described as
⎡xe⎤ ⎡ cosθ
sinθ
0⎤⎡xr
− x⎤
p e =
⎢
y
⎥ ⎢
e sinθ
cosθ
0
⎥⎢
yr
y
⎥
⎢ ⎥
=
⎢
−
⎥⎢
−
⎥
. (3)
⎢⎣θe⎥⎦
⎢⎣ 0 0 1⎥⎢ ⎦⎣θr
−θ⎥⎦
The velocity error of the mobile robot can be obtained
as
ṗ
⎡ẋ
e⎤ ⎡ yeω
− v+
vr cosθe⎤
=
⎢
ẏ
⎥ ⎢
x ω v sin θ
⎥
⎢ ⎥
=
⎢
− +
⎥
.
⎢θ̇
⎣ e⎥⎦
⎢⎣
ωr
−ω
⎥⎦
e e e r e
The trajectory tracking problem for the kinematics
model of the mobile robot can be described as finding a
bounded control input q r = ( vr, ωr) T so that given any
initial errors, the system (3) pe = ( xe, ye, θe) T can be
bounded and asymptotically to zero when t →∞
according to this input.
3. DESIGN OF THE SLIDING MODE
CONTROLLER
The process of sliding mode control can be divided
into two steps: 1) The choice of an appropriate sliding
manifold such that if the system trajectory is confined to
lie upon it, then the system exhibit the desired behavior;
2) The determination of a control law which is
discontinuous on the manifold and is capable of forcing
the system trajectory to reach the manifold and remain
on it.
(4)
3.1. Design of switching function
The kinematic equation (3) for mobile robot represents

Sliding Mode Control for Trajectory Tracking of Mobile Robot in the RFID Sensor Space 431
a multiple-input nonlinear system. Therefore the design
of switching function is a difficult problem. According to
[8], to simplify the problem, x e = 0 is chosen as the
first switching surface. When x e = 0, the Lyapunov
candidate function can be defined as
V
y
1 2
= ye
.
(5)
2
By differentiating this equation,
V̇
y = yeẏ
e = ye( − xeω+
vr sin θe)
=−yxω
−vysin(arctan( vy))
e e r e r e
where θ e =− arctan( vy r e)
is chosen as a switching
function candidate.
From Eq. (6), since vy r esin(arctan( vy r e)) ≥ 0, V̇
y ≤ 0.
Proof: ∀x∈R,
x ⊂ ∞
φ ( x) = xsin(arctan x) ≥ 0
(1) when x = 0, φ(0) = 0
(2) x∈(0, ∞), arctan x∈
(0, π / 2)
So : sin(arctan x) ≥0 ⇒ φ( x) > 0
(3) x∈−∞ ( ,0), arctan x∈−
( π / 2,0)
So : sin(arctan x) < 0 ⇒ φ( x) < 0
Notice that when x e converges to 0 and θ e
converges to − arctan( vy r e),
the status of the system,
y e also converges to 0. Therefore the switching
function to be designed should have the status that
x e = 0 and θ e =− arctan( vy r e)
in the sliding mode,
which is asymptotically stable. The switching surface
can be designed as
1
e
⎢
s ⎥
⎣ 2 ⎦ ⎣θe + arctan( vy r e)
⎦
(6)
s ⎡s
⎤ ⎡ x ⎤
= = ⎢ ⎥ .
(7)
The next step is designing a sliding controller, which
can make s 1 → 0 and s 2 → 0 to achieve xe
→ 0
and θe →− arctan( vy r e).
Finally the goal posture of the
mobile robot can be achieved by ye
→ 0 and θe
→ 0.
3.2. Design of the control law
Instead of establishing an analytic expression of a
reaching condition first and then designing a control law
to meet the condition, a reaching law approach is
adopted in this paper. The reaching law approach
describes the condition under which the state move
toward and reach a sliding surface. It directly specifies
the dynamics of the switching function. A kind of
structure which is called constant rate reaching law is
used in this research. In order to reduce the chattering
which is caused by the finite time delays for
computations and limitations of control, the switching
function is replaced by a saturation function. The
reaching law can be defined as
ṡ =−ksat().
s
(8)
This reaching law approach not only establishes the
reaching condition but also specifies the dynamic
characteristics of the system during the reaching phase.
Let α = arctan( vy r e).
Substituting (7) into (8), it can
be derived as
⎡ ẋ
ṡ
= = = ∂ ∂
⎣
⎡ yeω− v+
vr cosθe
⎤
=
⎢
α α
⎥
.
⎢
∂ ∂
ωr − ω+ v̇
r + ( − xeω+
vr sin θe)
⎥
⎢⎣
∂vr
∂ye
⎦⎥
e
⎡ṡ
1⎤ ⎡−k1sat( s1)
⎤ ⎢
α α
⎥
⎢
s
⎥ ⎢
2 k2sat( s2)
⎥ ⎢θe vr y ⎥
⎣̇
−
̇
⎦ ⎣ ⎦ + ̇ + ̇e
⎢ ∂vr
∂ye
⎥
The expression of control law can be obtained from
(9).
⎡ yeω+ vr cos θe
+ k1sat( s1)
⎤
⎡v
⎤
q =
⎢
α α
⎥
⎢ ,
ω
⎥ = ∂ ∂
⎣ ⎦
⎢ωr + v̇
r + ( vr sin θe) + k2sat( s2)
⎥
⎢⎣
∂vr
∂ye
⎥⎦
(10)
∂α ye
∂α vr
where =
and =
∂ v
2
2
r 1 + ( vy r e)
∂ y 1 ( )
.
e + vy r e
Using this control law, the posture error of mobile
robot, p = ( x , y , θ ) T asymptotically converges to 0.
e e e e
4. REAL POSTURE EVALUATION USING
RFID SYSTEM
Since the sliding mode controller has been designed,
the real position of mobile robot needs to be evaluated
and feedback to calculate the position error. Because of
the non-holonomic property, the position of mobile robot
is difficult to be obtained through integration of velocity.
Therefore, a localization system based on the RFID
system is introduced in this paper [9,10].
In order to evaluate the robot position using RFID
system, a reference frame needs to be established using
RFID tags. A RFID tag is small and inexpensive and a
robot can be localized fast with this system. Firstly,
RFID tags are placed on the floor in a triangular pattern
to increase the localization accuracy. Since the tag
arrangement on the floor is pre-planned, each tag stores
its absolute position data and sends out when they are
requested. If the origin point is chosen among the tags,
the whole reference frame can be setup. Fig. 3 shows the
tag arrangement and coordinates for the mobile robot.
To communicate with the tags, RFID reader (antenna)
is installed on the bottom of the mobile robot to gather
the position information which is stored by the tag. The
RFID reader antenna receives the signal from the tags
within its effective area. Fig. 4 shows the RFID reader
and tags which are allocated at every 0.05m on the floor.
The size of the epoxy tags is 0.03× 0.03 m. When the
mobile robot passes on the tags, all the tags within the
⎤
⎦
(9)

432
Jun Ho Lee, Cong Lin, Hoon Lim, and Jang Myung Lee
Y
X
Mobile Robot
with RFIDReader
RFIDTag
Fig. 3. Localization based on RFID system.
Fig. 5. Simulation result: sinusoid tracking.
Fig. 4. RFID reader and tag.
circle of radius, r which are under the effective area of
RFID antenna whose size is 0.1× 0.1 m, are activated.
Obviously the distance between antenna and floor is
fixed and cannot be changed, that is the area which is
affected by antenna at the same time cannot be increased.
The RFID reader sequentially gathers the tag
information, since it can recognizes only one tag signal
at a time. In order to receive other tag data within the
effective area of the RFID reader, the tag data previously
read are stored to the memory. Then, the reader receives
the next tag information, and repeats this procedure until
there is no unread tag left within the effective area. After
all the information of tags is stored, the location of the
mobile robot is calculated based on the collected tag data.
At the moment, a new set of tags is going to be selected
for the next step of localization.
5. EXPERIMENT AND RESULT
5.1. Simulation
According to the control law established in Section 3,
the simulation using MATLAB is implemented on the
mobile robot system. The plant to be controlled is the
differential (4) for the mobile robot.
The sinusoid line is used in this simulation to be
considered as the reference trajectory. The linear velocity
is uniform while the angular velocity is sinusoid. That is
when ω r = sin t and v r = 1.0 are required, the rate for
the error of posture instruction pr = ( xr, yr, θr) T can
be written as
x r = vr cos θr,
(11a)
y r = vr sin θr,
(11b)
θ = sin t.
(11c)
r
Fig. 6. Tracking error of x[m], y[m], and θ [degree].
The posture instruction can be obtained through
solving the differential (4). Let k 1 = k 2 = 6.0 and the
initial error of the mobile robot is set to be (x, y, θ ) =
(0.5, 1.8, 0). The simulation results are shown in Figs. 5
and 6. Fig. 5 shows the trajectory tracking result for the
sinusoid line. The actual trajectory approaches to the
reference line quickly. Fig. 6 shows the tracking error in
the direction of x, y, and θ , respectively.
According to the simulation results for the sinusoid
trajectory tracking, the movement can remain a stable
status although the system generates error. That is, the
posture of the robot converges to the desired trajectory.
This proves the validity of the sliding control algorithm
theoretically.
5.2. Real experiment
To demonstrate the effectiveness and applicability of
the proposed method, a real-time control system is
implanted for the mobile robot. In the experiment, a
mobile robot with an antenna fixed on the bottom moves
on the RFID tags. Fig. 7 shows the picture of the robot
which is used in the experiment. It has the same structure
as Fig. 1, with two driving wheels and one passive wheel.
The diameter of the robot is 230 mm and the radius of
driving wheel is 20 cm. The driving wheels are driven by
motors with the maximum permissible speed of 8,200
rpm. The motor and the driving wheel are connected by a
timing belt. The RFID tags are arranged on the floor in
the triangular pattern.
The control board of the mobile robot consists of the

Sliding Mode Control for Trajectory Tracking of Mobile Robot in the RFID Sensor Space 433
Fig. 7. Mobile robot used in the experiment.
Fig. 9. Straight line tracking with initial error (0.1 m,
0.3 m, − π /2).
Fig. 8. Schematic diagram of the control system.
main controller and motor controller. The main
controller of the robot is dsPIC30F6014, which is
running at 32MHz. It is used to communicate with host
computer and motor controller. It receives the velocity
instruction from the host computer and calculates the
velocity distribution on the right and left motors,
respectively, and then sends the data through SPI
communication to the auxiliary motor controller,
dsPIC4012. The motor controller generates PWM signal
with different duty cycles according to the velocity
instruction.
Fig. 8 shows the whole schematic diagram of the
trajectory tracking system for the mobile robot. Because
of the complexity of the calculation process, the sliding
controller is carried out in the main computer running at
the frequency of 1.86MHz. The software for implementing
the algorithm is developed in Visual C++ 6.0. After
the path has been set up, the sliding mode controller
generates the real velocity instruction. The dsPIC
controller can generate the PWM signal to control the
velocity of the mobile robot so that the mobile robot
moves according to the instruction. The RFID sensor
space evaluates the posture of the robot and feedback the
information to the host computer until the posture error is
minimized.
In order to validate the applicability of the proposed
control scheme, the mobile robot was required to track
reference trajectories. The mobile robots move on RFID
tags which is set to be 3 m by 3 m according to the
reference trajectory. The real position of the mobile robot
is feedback to the mobile robot every 0.25 second.
The experimental results for the straight-line tracking
are shown in Figs. 9-12. Fig. 9 is the result where there
are relatively small initial posture errors of 0.1 m, 0.3 m,
and -1.57 radian, for xe, y e,
and θ e,
respectively. Fig.
10 shows the trajectory tracking error for each axis.
Fig. 10. Tracking error of x[m], y[m], and θ [degree].
Fig. 11 is the result in which the robot started tracking
with large initial errors of 0.5 m, 0.75 cm and -2.1 radian.
As it is shown, the mobile robot eventually approaches
the reference trajectory with asymptotic stability within
0.5 second to 3% error bound. In particular, as shown in
Fig. 11, if the robot is located at a position with large
initial posture errors, there exists a transition time during
which the tracking error of the heading direction is
diminished. Fig. 12 shows the trajectory tracking error
from which it is recognized that the tracking speed is
slower within 1.2 second to 3% error bound than that of
the first situation.
Operators’ intelligent and skillful decisions are
necessary for the tele-operation of a mobile robot when
there are many scattered obstacles. Among the sensors
used for environment recognition, the camera is the most
popular and powerful. However there are several
limitations in the camera-based tele-operation of a
mobile robot. For example, shadowed or curved area
cannot be viewed using a narrow view-angle camera,
especially in an environment with bad illumination and

434
Jun Ho Lee, Cong Lin, Hoon Lim, and Jang Myung Lee
Fig. 11. Straight line tracking with initial error (0.5 m,
0.75 m, − 2 π /3).
(a) Small initial error (0.1 m, 0.3 m, − π / 2).
(b) Large initial error (0.5 m, 0.75 m, − 2 π /3).
Fig. 13. Straight line tracking result with PID controller.
Fig. 12. Tracking error of x[m], y[m], and θ [degree].
several obstacles. Therefore, it is necessary to have other
sensory information for reliable tele-operations. In this
study, sixteen ultrasonic sensors are attached around a
mobile robot in a ring pattern to measure the distances to
the obstacles, and a collision vector is introduced as a
new tool for obstacle avoidance, which is defined as the
normal vector from an obstacle to the mobile robot.
Based on this collision vector, a virtual reflection force is
generated to avoid the obstacles and then the reflection
force is transferred to the operator who is holding the
joystick used to control the mobile robot. Based on this
reflection force, the operator can control the mobile robot
more smoothly and safely. For this bi-directional teleoperation,
a master joystick system using a hall sensor
was designed to eliminate the nonlinear region which
exists in a general joystick with two motors and
potentiometers. The effectiveness of the collision vector
and force reflection joystick is verified by comparing
two vision-based tele-operation experiments, with and
without force reflection.
In order to prove the superiority of the sliding control,
The PID controller is implemented for the desired
trajectory of a straight line. As shown in Fig. 13, the
error is much bigger than that of the proposed algorithm.
Furthermore the tracking speed is slow relatively. It takes
3.4 and 4.2 seconds to be within 5% of the desired
trajectory for small and large initial errors, respectively.
Those two factors demonstrate the superiority of the
proposed sliding mode controller.
6. CONCLUSION
This proposed sliding mode control algorithm consists
of design of switching function and design of the control
law based on reaching law approach. According to the
simulation and experimental results, the proposed sliding
mode control is an important method to deal with the
system which has uncertainties and nonlinearities. This
proposed algorithm demonstrates a good tracking
performance. In spite of large initial error, the robot
posture converges to the desired trajectory. The
comparison to PID control shows the superiority of this
algorithm in trajectory tracking performance. The
adoption of RFID sensor space solves the problem that
the real position of mobile robot is difficult to be
obtained due to its non-holonomic property and provides

Sliding Mode Control for Trajectory Tracking of Mobile Robot in the RFID Sensor Space 435
accurate position information. It is confirmed that the
proposed sliding mode control law combined with RFID
system can be successfully used for the purpose of
trajectory tracking of the non-holonomic mobile robot in
the presence of disturbances.
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intelligent control.
Jun Ho Lee received the M.S degree in
Mechanical Engineering from Pusan
National University. His research
interests include factory automation and
sliding mode control.
Cong Lin received the B.S. degree in
Electrical Engineering from Jilin
University and the M.S degree in
Electrical Engineering from Pusan
National University. His research
interests include neural network and
sliding mode control.
Hoon Lim is currently a M.S student in
Electrical Engineering of Pusan National
University. His research interests include
mobile manipulator and sliding mode
control.
Jang Myung Lee received the B.S. and
M.S degrees in Electronics Engineering
from Seoul National University, Korea.
He received the Ph.D. degree in
Computer from the University of
Southern California, Los Angeles. Now,
he is a Professor in Pusan National
University. His research interests include
integrated manufacturing systems and