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https://doi.org/10.62909/ejeee.2025.003 Edison Journal for Electrical and Electronics Engineering
Article
Trajectory Tracking Control for Nonholonomic Wheeled Mo-
bile Robots with External Disturbances and Parameter Uncer-
tainties
Nabaa Nasser Ali
1,
*
and Mahmoud Shahi
1,
1
Department of Electricity, College of Engineering, Azad University, Tehran, Iran; nbanasr22@gmail.com
* Correspondence: m.shahi.g@gmail.com
Abstract: The system studied in this research is a nonholonomic wheeled mobile robot (WMR) with
three degrees of freedom (3-DOF). The goal is to design a fast end-sliding mode control for posi-
tioning and stabilizing the system's forward motion and angle. The methodology involves formu-
lating the system's dynamic equations using Newton-Euler equations and then converting them
into state-space form. The designed controller is then applied to the system, and external random
noise disturbances are introduced via MATLAB simulations to evaluate the system's robustness
against uncertainties and disturbances.
Keywords: Non-holonomic Wheeled Mobile Robot, Fast Terminal Sliding Mode Control, Sliding
Surface, Lyapunov Function, Position Control.
1. Introduction
Wheeled mobile robots (WMRs) have gained increasing popularity because of their
amazing mobility and capabilities. Researchers and engineers around the world dedicate
considerable focus on the autonomous supervision and command of these machines.
WMRs are commonly utilized throughout a range of industries, including military ap-
plications space travel, and manufacturing. They are able to function in dangerous set-
tings. and execute tasks independently without the involvement of humans.
On manufacturing lines, self-driving cars equipped with wheels are designed for
rapid and precise movement in different directions. These robots offer cost-saving bene-
fits while maintaining design speed and flexibility. They are typically Having two back
wheels powered by independent motors and a free-moving front wheel, enabling omni-
directional movement. Their robust structure makes them ideal for various applications,
as moving materials and products in factories.
Because there are many unknown parameters, it is difficult to adequately represent
the three-wheeled mobile robot (3WMR) both kinematically and dynamically. It is clas-
sified as an underactuated system, it has less actuators than degrees of freedom, there-
fore. Furthermore, 3WMRs exhibit nonholonomic constraints, preventing them from
moving sideways freely. Therefore, creating control algorithms is a difficult undertaking
for 3WMRs, especially when putting controllers on actual devices because of mechanical
uncertainties and environmental disruptions.
1.1 Related Work
It has been suggested the ABHSMC controller, a novel adaptive controller for
three-wheel mobile robots (3WMR). This controller was created using a collaborative
approach, a combination of a feedback controller and a hierarchical sliding mode con-
Citation: Ali, N. N. and Shahi M.,
Trajectory Tracking Control for
Nonholonomic Wheeled Mobile
Robots with External Disturbances
and Parameter Uncertainties. Edison
Journal for Electrical and Electronics
Engineering, 2025. 3: p. 16-31
Academic Editor: Prof. Dr. Omar
Mohammed Al-Shuja'a
Received: 29 6/2025 /
Revised: 2025/8/1
Accepted: 2025/8/9
Published: 14/8/2025
Copyright: © 2024 by the authors.
Submitted for possible open access
publication under the terms and
conditions of the Creative Commons
Attribution (CC BY) license
(https://creativecommons.org/license
s/by/4.0/).
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2025, Vol.3 17 of 31
troller based on a radial basis function (RBF). It is noteworthy that, the RBF neural net-
work shows a significant ability to calculate the model's uncertainty components and
adjust its parameters in a methodical manner, which leads to the improvement of the
output responses. A new navigation model called ABHSMC The ABHSMC navigation
stack, timed elastic band (TEB) local scheduler, and controller are utilized to efficiently
resolve the obstacle avoidance and tracking problem for a 3-wheel mobile robot. The
Matlab/Simulink platform's implementation of the simulation results show that 3WMRs
are able to adhere to the intended path exactly even when there are disruptions and
modifications to the model. In addition, the reliability of the controller has been verified
in the built Model of self-driving automobile. The experimental findings gained demon-
strate that the suggested navigation structure may successfully guide the real car model
to trace the intended course with a sufficiently small error and simultaneously avoid an
unexpected barrier [1].
It has been presented an iterative learning control (ILC) strategy to solve the path
tracking problem of wheeled mobile robots (WMR) based on dynamic model. In the
previous study of trajectory tracking of WMRs, ILC was usually applied to the kinematic
model of WMRs with the assumption that the desired velocity can be tracked immedi-
ately and directly. However, this assumption cannot be realized at all in the real world.
Kinematic and dynamic models of WMR and a new combination of D-type ILC algo-
rithm and dynamic model of WMR with stochastic boundary disturbances are presented.
To analyze the convergence of the algorithm, a convention mapping method is adopted
which shows that a designed controller can completely converge the velocity tracking
errors to zero when the iteration times reach infinity. The simulation results show the
effectiveness of the D-type ILC algorithm in the path tracking problem of WMRs,
demonstrating the effectiveness and robustness of the algorithm under random bound-
ary disturbance conditions. A comparative study between D-type ILC and neural net-
work compound cosine function controller was conducted, which also shows the effec-
tiveness of the ILC strategy [2].
It has been presented the speed and direction angle control of a wheeled mobile
robot based on the sliding mode control technique based on the fractional order adaptive
model. Considering the unique advantages of fractional arithmetic and adaptive mod-
el-based sliding mode PID control method, fractional order and adaptive model-based
sliding mode PID control technique are combined and presented as a controller for the
first time in the literature. effective for real-time control of wheeled mobile robot under
wheeled external robot are proposed. In this proposed method, several critical issues are
considered. First, a kinematic and dynamic model of the wheeled mobile robot is ana-
lyzed according to the uncertainties of the system. Second, fractional-order computations
and model-based PID-type sliding mode control are formulated to realize chatter-free
control, accurate path tracking response, finite-time convergence, and resistance for the
wheeled mobile robot. Finally, an adaptive process is also employed to meet and over-
come the unknown dynamics and uncertain parameters of the system, regardless of prior
information about the uncertainties. The experimental results show that the proposed
controller (fractional-order adaptive model-based sliding mode controller) has accurate
track tracking performance, faster finite-time convergence, and also less speed error un-
der external load when compared to the adaptive model-based PID sliding-mode control.
It can be provided [3].
2. Methodology
Sliding mode controller is one of the most widely used controllers for studying and
applying to nonlinear systems and processes. This acceptance is owing to both its naivety
proportionality and its durability against system unsureness and exterior troubles. The
unruly in its application is the phenomenon of states slipping on the slip surface or gos-
sip. Sign functions are sometimes used in place of smooth functions like saturation func-
tions or hyperbolic tangents in order to mitigate or eradicate this issue. In general, the
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robustness of the sliding mode controller is higher than some controllers such as PID or
LQR. Another advantage is that the control operation is performed on a nonlinear system
and there is no need to linearize and approximate the system. Fast terminal sliding mode
control can also be used. Fast terminal sliding mode control, which is the innovation of
this research, can be used to ensure that the system states are placed on the slip surface in
a short time. The advantages of this controller over the PD sliding mode control are that it
not only increases the speed and accuracy of tracking the desired path, but also grow the
robustness of the system of controls. Lastly, to assess the suggested control plan, a com-
parison between the error outcomes of the simulation of the above sliding mode methods
with MATLAB software in the absence and presence of external disturbances is presented
[4-6].
2.1 State Space Equations
First, we rewrite the dynamic equations governing the mobile robot [2]:
(1)
Equation (1) can be written as follows and used to control the system.
(2)
According to reference [1], the system dynamics equations are transformed as follows:
(3)
We know that
،
،
،
،
،
and
.
Now, considering the following relationships:
(4)
which is the state space vector (in terms of the robot's position and velocity) and the
torque vector (control input).
Therefore, we rewrite the set of equations (3) in the following form:
(5)
So that
،
،
،
،
و
and have the following definitions.:
(6)
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2.2 Control input design [7]
Here we will design the control input for two displacement states & & the
front orientation angle of the mobile robot. The control diagram of the system is as
follows in Figures 2.1 and 2.2:
Figure 2.1. A model of a three-wheeled mobile robot
Figure 2.2. The control diagram of the system
Now, considering the values of the parameters, the initial state vector and the de-
sired positions of the robot as displayed in the Table 1 below:
Table 1. Parameters considered for the simulation and command of the mobile, wheeled
robot
Robot parameters
Values
Units
10
Kg
818/3
Kg/m^2
07/0
M
19/0
M
Favorable conditions
]
Initial conditions of the robot
First, we will design the controller for the angle. Considering the control method,
which in this study is the fast terminal sliding mode control method, we first consider a
nonlinear sliding surface called so that
the value of
and
therefore, the tracking of the angle and angular velocity is done. Therefore, we de-
fine the slip surface as equation (7):
(7)
So that
and
are positive parameters and and are also integers.
and
is, which is
the desired state or reference of the robot orientation
angle according to Table 1.
By deriving from (7) we arrive at the following relation:
(8)
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Given that the slip the state variables are used to define the surface equation. and
the desired state, the system will converge to the desired state on the slip surface. So, it
can be said that the following relations hold on the slip surface:
(9)
Based on According to sliding mode theory, the control input has the following
definition:
(10)
So that
the control component is equivalent and keeps the states on the slip
surface
and the switching component is responsible for directing the states towards
the slip surface, which is actually responsible for stabilizing the system and is determined
with the help of the Lyapunov stability criterion. To acquire the comparable control el-
ement, it is enough to put
and also take the help of equation (5):
(11)
Now evaluating the potential Lyapunov function as a positive definite function in
the manner described below
(12)
This function's derivative needs to be negative definite for asymptotic stability:
Using relation (11), and substituting into equation (13), we have:
(14)
Consequently, a good option for the switching control legislation to meet the require-
ment (16) is the following relation:
(15)
Where
and
are positive real values. Substituting from equation (15) into equation
(14), we have:
By combining equations (11) and (15) and substituting into relation (10), the general ro-
bust control equation for the fast terminal sliding mode to be applied to the wheeled
mobile robot to track the desired reference path will be obtained:
(17)
Now we will design the controller for the displacements and . Since we want to con-
trol these two states with one input (
). Therefore, first we consider two nonlinear slip
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surfaces named
and
such that with the value of
and
and
with the value of
and
. Therefore, linear position and velocity
tracking is performed. For this purpose, we define the slip surfaces as equation (18):
(18)
So that
up
to the positive parameters and and are also integers. And
and
, which is
the desired state or reference of the
robot movement and
the desired state or reference of the robot movement ac-
cording to Table 1. Now, considering that we have only one control input in this part, we
define a slip surface as a combination of
and
:
(19)
so that and are positive parameters. To acquire the comparable control element, it is suf-
ficient to set and also use relations (5) and (18):
(20)
Which
by solving the above equation we have:
(21)
Now evaluating the potential Lyapunov function as a positive definite function in the
manner described below:
(22)
Using relation (20) and the fact that the derivative for asymptotic stability must be
negative definite, we have:
(23)
Using relation (21), and substituting into equation (23), we have:
( 42 )
Consequently, a good option for the switching control legislation to meet the require-
ment (16) is the following relation
( 52 )
where
and
are positive real values. Substituting from equation (25) into equation
(24), we have:
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( 62 )
By combining equations (21) and (25) and substituting into relation (10), the general ro-
bust control equation for fast terminal sliding mode to be applied to a wheeled mobile
robot to track the desired linear position and velocity reference path will be obtained:
( 72 )
3. Results and Discussion
3.1. Software simulation with MATLAB software
3.1.1. Fast terminal sliding mode control with sign function
Now, considering the simulation time
and sampling time constant
and the parameter values according to Table 1 are. Taking into account the control
input that was created and their application in Matlab software, the graphs shown below
are taken out [8, 9]:
In addition, to evaluate the performance of the control strategy used, in each graph
its performance is compared with the classical sliding mode PD control, as shown in
Figure 3.1. It is observed that the position and Ɵ angle tracking for PD sliding mode
control and the proposed method are well performed, but the position tracking for PD
sliding mode control has diverged from 15 seconds onwards, while the fast terminal
sliding mode control has performed all tracking well, as shown in Figure 3.2
Figure 3.1. Tracking the paths related to the positions and
angles of a mobile, wheeled robot with the sign function
Figure 3.2. Robot tracking on the screen with the
mark function
It is observed that the sinusoidal path tracking in the two-dimensional plane for PD
sliding mode control deviates from the path after some time, while for fast terminal
sliding mode control it is accompanied by some persistent error but always remains
convergent. In Figure 3.3, it is observed that in the control effort diagrams, the unwanted
oscillations at the first input for the PD sliding mode are very large. In Figure 3.4, it is
observed that in the slip surface diagrams, the unwanted oscillations at the first slip sur-
face for the PD sliding mode are very large.
0 5 10 15 20 25
0
5
10
Time (Seconds)
x (m)
Tracking performance of x position
0 5 10 15 20 25
-2
0
2
Time (Seconds)
y (m)
Tracking performance of y position
0 5 10 15 20 25
-1
0
1
Time (Seconds)
teta(rad)
Tracking performance of orientation
Reference
smc
FTsmc
Reference
smc
FTsmc
Reference
smc
FTsmc
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Figure 3.3. Control efforts related to tracking reference
paths with the sign function
Figure 3.4. Slip surfaces related to tracking reference paths
with the sign function
In Figures 3.5 and 3.6, it can be seen that the path tracking in the three-dimensional
plane for PD sliding mode control deviates from the path after traveling a certain dis-
tance, while for fast terminal sliding mode control, it is accompanied by some persistent
error but always remains convergent.
3.1.2 Hyperbolic tangent function control of fast terminal sliding modes
As observed, fluctuations were observed, especially in the control effort diagram, as
chattering. One of the commonly proposed strategies to eliminate or reduce these fluc-
tuations is to replace the hyperbolic tangent or saturation function in the control input
relation [10, 11]:
ℎ
(28)
And
ℎ
(29)
Now run the simulation again with the same values as in the previous section and the
following graphs are obtained:
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Figure 3.5. Overall tracking path in 3D space with the sign
function
Figure 3.6. Tracking paths related to positions and an-
gles of a wheeled mobile robot with the hyperbolic
tangent function
It is observed that the tracking of the positions of , and angles for the PD
sliding mode control and the proposed method are well performed. It is observed that
the sinusoidal path tracking in the two-dimensional plane for PD sliding mode control is
tracked with persistent error, while for terminal sliding mode control it converges
quickly after some time without error. In Figures 3.7 and 3.8, it is observed that in the
control effort diagrams, the amplitude of the second input for the proposed method is
greater than that of the PD sliding mode, which can be a weakness. Similarly, some
momentary changes are seen in the first input. In Figure 3.9, it is observed that in the slip
surface diagrams, some momentary changes are seen in the first slip surface in the pro-
posed method.
Figure 3.7. Robot tracking in a plane with the hyperbolic
tangent function
Figure 3.8. Control efforts related to tracking reference
paths with the hyperbolic tan-gent
0 1 2 3 4 5 6 7
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
x(m)
y(m)
curve Robot direction
Reference
smc
FTsmc
0 5 10 15 20 25
-50
0
50
100
Time (Seconds)
u1(N.m)
Control input 1= T1-T2
0 5 10 15 20 25
-5000
0
5000
10000
Time (Seconds)
u2(N.m)
Control input 2= T1+T2
smc
FTsmc
smc
FTsmc
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Figure 3.9 - Slip surfaces related to tracking reference paths
with the hyperbolic tangent function
Figure 3.10 - Overall path traced in 3D space with the
hyperbolic tangent function
In Figure 3.10, it can be seen that the overall path of movement traveled is always
followed with some error in the three-dimensional plane for PD sliding mode control, but
it is established error-free for rapid terminal sliding mode control. According to diagram
of Figure 3.11, the following point can be explained: Lyapunov's stability principle, which
states that if a state variable is increasing, its derivative must be decreasing for stability
(first and second cases), and if a state variable is decreasing, its derivative must be
increasing for stability (third case), which is significant.
3.1.3 Fast terminal sliding mode control in the presence of random noise
Now again, considering the simulation time
and sampling time
constant
and the parameter values according to Table 1,
and
and and . Considering the control input that was created and how they
are used in Matlab program, the graphics that follow are taken from:
We introduce an external disturbance in the form of random noise with an amplitude of
0/7 N-m at the input to the system, which can be considered to include an infinite number
of frequencies. Then, We look at the suggested controller's resistance to this shock as well
as PD's sliding mode control as shown in Figure 3.12 [12, 13].
Figure 3.11 Phase plane of system state variables with
hyperbolic tangent function
Figure 3.12 Random noise introduced into the robot system
Considering the control input that was with the hyperbolic tangent function and
their application in MATLAB software, the graphs listed below are once more extracted:
0 5 10 15 20 25
-0.2
-0.1
0
0.1
0.2
Time (Seconds)
S1
Sliding Surface for Teta orientation
0 5 10 15 20 25
-300
-200
-100
0
100
Time (Seconds)
S2
Sliding Surface for x-y positions
smc
FTsmc
smc
FTsmc
0
2
4
60.2
0.4
0.6
0.8
1
1.2
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
x(m)
curve Robot direction
y(m)
Teta(rad)
Reference
smc
FTsmc
0 5 10 15 20 25
-3
-2
-1
0
1
2
3
time(Seconds)
Disturbance
External Disturbance
n(t)
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(30)
It is observed that the tracking of positions and angles is well done for the
suggested approach and the PD sliding mode control , but the tracking of position is
well done for the proposed method while it is deviated in the PD sliding mode control as
shown in Figures 3.13 and 3.14.
Figure 3.13. Tracking the paths related to the posi-
tions and angles of a mobile, wheeled robot with ran-
dom noise.
Figure 3.14. Robot Tracking in a Plane with Random Noise
It is observed that the sinusoidal path tracking in a two-dimensional plane for PD
sliding mode control is tracked with persistent error while for terminal sliding mode
control it converges quickly with less error. According to Figure 3.15, it is observed that
the path tracking for PD sliding mode control is tracked with a higher error while for fast
terminal sliding mode control it converges with a lower error.
Figure 3.15. Tracking error for reference paths with ran-
dom noise
Figure 3.16. Overall tracking path in 3D space with hy-
perbolic tangent function
In Figure 3.16, the overall motion path can be seen, which shows that tracking the
path in the 3D plane for PD sliding mode control is always followed with some error,
while for fast terminal sliding mode control it is accompanied by less error. It is observed
that the tracking of linear
and
angular velocitie s
for PD sliding mode control is
performed with larger fluctuations than the proposed method.
0 5 10 15 20 25
0
5
10
Time (Seconds)
x (m)
Tracking performance of x position
0 5 10 15 20 25
0
0.5
1
1.5
Time (Seconds)
y (m)
Tracking performance of y position
0 5 10 15 20 25
-1
0
1
Time (Seconds)
teta(rad)
Tracking performance of orientation
Reference
smc
FTsmc
Reference
smc
FTsmc
Reference
smc
FTsmc
0 1 2 3 4 5 6 7
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
x(m)
y(m)
curve Robot direction
Reference
smc
FTsmc
23.5 24 24.5 25
-0.15
-0.1
-0.05
0
0.05
Time (Seconds)
Error of x(m)
0 5 10 15 20 25
-1
0
1
Time (Seconds)
Error of y(m)
24.91 24.92 24.93 24.94 24.95 24.96 24.97 24.98 24.99 25
-1
0
1
2
x 10
-4
Time (Seconds)
Error of Teta(rad)
smc
FTsmc
smc
FTsmc
smc
FTsmc
0
2
4
6
8
0
0.5
1
1.5
-1
-0.5
0
0.5
1
x(m)
curve Robot direction
y(m)
Teta(rad)
Reference
smc
FTsmc
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3.1.4 Sliding Mode for the Fast Terminal Defend Against Parametric Variability
Taking the simulation duration into account once more
and
sampling time constant
and the parameter values according to Table
2, and and are. In this case, even up to 50% changes are made in
parameters such as arm length, arm mass and inertia moment, but the robustness of
the proposed controller as well as PD sliding mode control is very good and no
noticeable change was observed, of course, the range of required control effort
increases significantly [14, 15]:
Table 2. Parameters considered for simulation and control of a wheeled mobile robot
with 50% uncertainty applied to the parameters
Robot parameters
Values
Units
(5/0 +1*)10
Kg
(5/0 +1 *)818/3
Kg/m^2
(5/0 +1 *)07/0
M
(5/0 +1*) 19/0
M
To evaluate the performance of the applied control strategy, its performance graph is
compared with the sliding mode PD control as shown in Figurs 3.18:
Figure 3.17. Tracking the paths of linear and angular velocities
of a wheeled mobile robot with random noise
Figure 3.18. Control efforts related to tracking refer-
ence paths with 50% parametric uncertainty applied
In Figures 3.17 and 3.18, it is observed that the amount of input amplitude changes
at the beginning of time is higher for the proposed method, which is due to the need for
faster convergence of the state variables to the desired state of position and velocity. The
range of changes of the first control effort is between Nm and
Nm for the second input, which is a large increase and this means that the system
tolerates this level of uncertainty, but it will bear a large cost.
3
.
2 Performance evaluation criteria
To assess the controllers' performance in this investigation, We make use of the
following error metrics, which are the average of the sum of the absolute values of the
errors [16, 17]:
(31)
23.026 23.028 23.03 23.032 23.034 23.036
-4
-2
0
2
4
Time (Seconds)
Vx-dot (m/s
2
)
Tracking performance of Vdot -acceleration
16.126 16.128 16.13 16.132 16.134 16.136
-4
-2
0
2
4
Time (Seconds)
Vy-dot (m/s
2
)
Tracking performance of Vdot -acceleration
6.194 6.1945 6.195 6.1955 6.196
-1
0
1
2
Time (Seconds)
Omega-dot(rad/s
2
)
Tracking performance of Omegadot- angular acceleration
Reference
smc
FTsmc
Reference
smc
FTsmc
Reference
smc
FTsmc
0 5 10 15 20 25
-50
0
50
100
150
Time (Seconds)
u1(N.m)
Control input 1= T1-T2
0 5 10 15 20 25
-5000
0
5000
10000
15000
Time (Seconds)
u2(N.m)
Control input 2= T1+T2
smc
FTsmc
smc
FTsmc
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3.2.1. Performance evaluation in normal mode
Now, we will qualitatively and quantitatively examine the graphs and Table 2 of the
system performance in the mode where the saturation function is used, for the tracking
error of the robot's position and linear velocity, orientation angle, and angular velocity:
Figure 3.19 Position and status error diagram in normal
mode
Figure 3.20. Speed error diagram in normal mode.
By observing and examining Figures 3.19 and 3.20, it can be said that the position
and velocity tracking error tends to zero and accurate position and velocity tracking is
well done. Now, for greater accuracy, the performance quantity of these two controllers
is given in Table 3 for the average values of the absolute value integral of the position and
velocity error in the time interval of 25 seconds for the two controllers:
Table 3. Average values of the absolute value integral of the position and velocity errors
for the fast terminal sliding mode and PD sliding mode controllers
Speed error
Status error
Average integral of absolute value of position and
velocity error
0754/0
0335/0
FTSMC
0725/0
0635/0
PDSMC
According to Table 3, it is observed that the average integral value of the absolute
value of the position tracking When compared to the PD sliding mode control, the error
in the rapid terminal sliding mode control is cut in half. However, the average integral
value of the absolute value of the speed tracking error in the proposed method is slightly
higher. The value of the suggested approach is displayed in this Table.
3
.
2.2. Performance evaluation in the case of random noise perturbation.
Now, we will qualitatively and quantitatively examine the graphs and Table of the
system performance again in the case where the saturation function and random noise
disturbance with a height of 0.7 Nm are introduced into the system, for the tracking error
of the robot's position and angular velocity [15, 18]:
0 5 10 15 20 25
-2
0
2
Time (Seconds)
Error of x(m)
0 5 10 15 20 25
-1
0
1
Time (Seconds)
Error of y(m)
0 5 10 15 20 25
-0.05
0
0.05
Time (Seconds)
Error of Teta(rad)
smc
FTsmc
smc
FTsmc
smc
FTsmc
0 5 10 15 20 25
-50
0
50
Time (Seconds)
Error of xdot(m/s)
0 5 10 15 20 25
-50
0
50
Time (Seconds)
Error of ydot(m/s)
0 5 10 15 20 25
-0.2
0
0.2
Time (Seconds)
Error of Tetadot(rad/s)
smc
FTsmc
smc
FTsmc
smc
FTsmc
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Figure 3.21 - Position and status error diagram in the
presence of random noise.
Figure 3.22 Velocity error diagram in the presence of
random noise.
By observing and examining the Figures 3.21 and 3.22, it can be said that the effect of
this disturbance with the desired amplitude has a negligible effect on the position
tracking and tracking speed in the fast terminal sliding mode control, while its effects on
the position tracking part on the PD sliding mode control are much greater. It is also
observed that the effect of noise on the speed graphs is less. Now, for greater accuracy,
the performance quantity of these two controllers is given in Table 4 for the average
values of the absolute value integral of the position and speed error in the time interval of
25 seconds for the two controllers:
Table 4. Average values of the absolute magnitude integral of the position and velocity
errors for fast terminal sliding mode and PD sliding mode controllers due to random
noise disturbance
Speed error
Status error
Average integral of absolute value of position and
velocity error
1003/0
3955/0
FTSMC
1722/0
9127/0
PDSMC
According to Table 4, it is observed that the average integral value of the absolute
value of the tracking error in the fast terminal sliding mode control is reduced to about
one third compared to the PD sliding mode control. In addition, unlike the noiseless case
of Table 4, the average integral value of the absolute value of the tracking error in the
proposed method is less than that of the PD sliding mode control. Of course, in both
methods, the magnitude of the tracking error increases significantly compared to the
conventional case, which limits the controllers.
3.3 Discusion
Finally, by investigating some graphs, it can be seen that chattering effects are more
effective in the derivative of states (for example, control inputs). which according to the
relation of derivation in the Fourier transform, which indicates that the act of time deri-
vation means the linear amplification of system frequencies. It can be interpreted as
meaning that higher frequencies get more amplification.
The non-holonomic wheeled mobile robot system is a system in which the number
of inputs is less than the number of controlled state variables, which are also called un-
der-excitation systems. From the control point of view, when the dynamic equations of
non-holonomic systems are written in the state space form, they have fewer inputs than
the number of state variables.
0 5 10 15 20 25
-2
0
2
Time (Seconds)
Error of x(m)
0 5 10 15 20 25
-4
-2
0
Time (Seconds)
Error of y(m)
0 5 10 15 20 25
-0.05
0
0.05
Time (Seconds)
Error of Teta(rad)
smc
FTsmc
smc
FTsmc
smc
FTsmc
0 5 10 15 20 25
-50
0
50
Time (Seconds)
Error of xdot(m/s)
0 5 10 15 20 25
-50
0
50
Time (Seconds)
Error of ydot(m/s)
0 5 10 15 20 25
-0.2
0
0.2
Time (Seconds)
Error of Tetadot(rad/s)
smc
FTsmc
smc
FTsmc
smc
FTsmc
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For this reason, the issue of stabilizing them, or in other words, designing a regula-
tor for them, is a problem for which a precise solution has not been proposed so far.
Although it is proven that non-holonomic systems can be controllable despite having
fewer inputs than the number of state variables, it is proved that finding continuous in-
puts that are only a function of the error of the state variables for stabilization These
systems are impossible.
This problem makes the control system design for non-holonomic systems very
difficult. As seen, the fast terminal sliding mode control, which is a non-linear controller,
is well suited to the control of this non-linear robot.
In addition, although the control system has good robustness against the external
disturbances and parametric uncertainties, at the same time, the error values and the
range of the control inputs indicate the limitation of the amount of imposed external
disturbances or the percentage of allowed parametric uncertainty.
4. Conclusions
The non-holonomic wheeled mobile robot system is a system in which the number of
inputs is less than the number of controlled state variables, which are also called un-
der-excitation systems. From the control point of view, when the dynamic equations of
non-holonomic systems are written in state space form, they have fewer inputs than the
number of state variables. For this reason, the issue of stabilizing them, or in other words,
designing a regulator for them, is a problem for which no exact solution has been pro-
posed so far. Although it is proved that non-holonomic systems can be controllable de-
spite having fewer inputs than the number of state variables, it is proved that finding
continuous inputs that are only a function of the error of the state variables for stabiliza-
tion These systems are impossible. This problem makes the design of the control system
for non-holonomic systems very difficult. But as seen, the fast terminal sliding mode
control, which is itself a non-linear controller, is a good response to the control of this
non-linear robot. In addition, although the control system has good resistance against
external disturbances and parametric uncertainties, at the same time, the error values and
the range of the control input indicate the limit of the amount of imposed external dis-
turbances or the percentage of allowed parametric uncertainty. In the end, it is reminded
again that the sliding mode is among the most often used controllers. for managing
non-linear processes and systems. This acceptance is due to its relative simplicity and
resistance to external disruptions and system uncertainty. The issue that the field is fac-
ing of its application is the occurrence of the phenomenon of sliding modes on the surface
that slides, which is especially noticeable in systems that have fast dynamics. As seen, to
solve or reduce this problem, replacing the hyperbolic tangent function instead of the
sign function in the switching control part was used as a suitable option, so that this
combination softened the behavior and reduced system vibrations.
Conflicts of Interest: “The authors declare no conflict of interest.”
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