![A schematic diagram of the DWMR (resembling the structure described in [20]).](https://www.researchgate.net/publication/389963596/figure/fig1/AS:11431281317230738@1742375904421/A-schematic-diagram-of-the-DWMR-resembling-the-structure-described-in-20_Q320.jpg)
Available via license: CC BY 4.0
Content may be subject to copyright.
Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000.
Digital Object Identifier 10.1109/ACCESS.2023.0322000
Improving Trajectory Tracking of Differential
Wheeled Mobile Robots with Enhanced
GWO-Optimized Back-Stepping and FOPID
Controllers
LI QIANG1,2, HOOI HUNG TANG1and NUR SYAZREEN AHMAD1(Member, IEEE)
1School of Electrical and Electronic Engineering, Universiti Sains Malaysia, 14300 Nibong Tebal, Penang, Malaysia
2Department of Mechanical and Electronic Engineering, Yuncheng University, Yuncheng 044000, China
Corresponding author: Nur Syazreen Ahmad (e-mail: syazreen@usm.my).
’’This work was supported by the Malaysia Ministry of Higher Education under Fundamental Research Grant Scheme with Project Code:
FRGS/1/2024/TK07/USM/02/3.’’
ABSTRACT Improving trajectory tracking in Differential Wheeled Mobile Robots (DWMRs) is vital for
enhancing their effectiveness in various applications, such as autonomous cleaning, mowing, and leader-
following scenarios. These scenarios often involve navigating complex, nonlinear paths, requiring advanced
control strategies for enhanced performance. This work presents the novel integration of a Backstepping
Controller (BSC) and a Fractional-Order Proportional-Integral-Derivative (FOPID) controller within a cas-
cade closed-loop structure for Differential Wheeled Mobile Robots (DWMRs). The proposed BSC-FOPID
controller addresses velocity saturations and nonlinearities, ensuring system stability and precise trajectory
tracking. A key contribution is the enhanced Grey Wolf Optimization strategy, termed GWO-SMA, which
integrates Grey Wolf Optimization (GWO) with Slime Mould Algorithm (SMA). By leveraging opposition
space and optimum cache concepts, GWO-SMA improves fitness optimization in each iteration, enhancing
both exploration and exploitation efficiency. This hybrid approach optimizes controller parameters using
a multi-metric cost function that incorporates Integral Absolute Error (IAE) and Integral Squared Error
(ISE) to minimize long-term steady-state error and enhance responsiveness to larger deviations. Simulations
demonstrate the superior performance of the proposed GWO-SMA algorithm compared to existing opti-
mization techniques, such as Particle Swarm Optimization (PSO), Gazelle Optimization Algorithm (GOA),
and its individual components, GWO and SMA, which have shown strong performance in recent literature
for optimizing PID-type controllers. In addition, simulation results using three distinct reference paths,
i.e. lemniscate, square, and cloverleaf; demonstrate that the GWO-SMA-optimized BSC-FOPID controller
outperforms both adaptive dynamic compensation control (ADCC) and BSC-PID controller in position
and posture tracking accuracy. Specifically, the BSC-FOPID controller achieves significant improvements,
including average reductions of 55.65% in ISE and 38.25% in IAE for position control, as well as 62.12%
and 38.95% improvements in ISE and IAE for posture control, respectively. These improvements highlight
the controller’s enhanced responsiveness and smoother error convergence, particularly during maneuvers
involving sharp curves.
INDEX TERMS Differential wheeled mobile robot, Backstepping control, cascade control,Fractional-order
PID control, Grey Wolf Optimization, Slime Mould Algorithm
I. INTRODUCTION
DIfferential wheeled mobile robots (DWMRs) remain a
popular choice in robotic applications due to their sim-
plicity, efficiency, and effectiveness in navigating complex
environments. Unlike omnidirectional robots that employ
more complex mechanisms for movement, DWMRs utilize
a straightforward design consisting of two independently
driven wheels and a passive caster [1]. This configuration
not only reduces mechanical complexity and manufacturing
costs but also provides reliable performance across a wide
VOLUME 11, 2023 1
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2025.3552312
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
range of applications. The inherent advantages of DWMRs,
such as their ability to execute precise turns and maneuver in
constrained spaces, make them particularly suitable for tasks
that require robust trajectory tracking.
Improving the trajectory tracking performance in DWMRs
is crucial, especially as these robots become increasingly in-
tegrated into various applications. For instance, autonomous
cleaning robots must navigate dynamic environments while
adhering to predefined paths to ensure thorough coverage of
designated areas [2]. Similarly, mowing robots need to follow
intricate patterns across lawns, requiring precise trajectory
tracking to avoid obstacles and ensure efficient operation [3].
In scenarios where robots must follow a leader - be it another
robot or a human - the ability to maintain an accurate tra-
jectory becomes paramount [4], [5]. These interactions often
involve complex, nonlinear paths rather than simple straight
lines, demanding advanced control strategies to enhance per-
formance.
The significance of trajectory tracking performance ex-
tends beyond operational efficiency; it impacts safety, reli-
ability, and user satisfaction. As DWMRs are deployed in
increasingly complex and varied environments, the need for
enhanced tracking capabilities becomes evident [6]. The abil-
ity to follow curves, navigate around obstacles, and adapt to
changing conditions while maintaining stability and accuracy
is essential for achieving the desired functionality in real-
world applications.
To effectively tackle trajectory tracking challenges, there
is a growing trend toward controlling the kinematics and
dynamics of DWMRs using a cascade closed-loop control
approach. [7], [8]. However, in practice, DWMRs operate
under speed and actuator constraints, which introduce non-
linearities into their models [9], [10]. Consequently, a variety
of controllers have been developed, with adaptive control
and robust control emerging as prominent approaches for
managing complex nonlinear control systems. The adaptive
control approach continuously adjusts controller parameters
in response to variations in system dynamics, while robust
controllers are designed to handle uncertainties without alter-
ing controller parameters [11], [12].
The kinematic controller for a DWMR is typically designed
based on the robot’s geometric model to ensure precise trajec-
tory tracking. However, as the complexity of environments
and tasks increases, researchers have introduced advanced
nonlinear control techniques. The work in [13] for instance
introduced a finite-time tracking control method using an
adaptive neural network (NN) approach tailored for practical
DWMR systems with full state constraints. Nevertheless, im-
plementing NN-based methods for real-time control presents
challenges due to the extensive training required and the need
for high-performance computing architectures [14]. Fuzzy
logic controller (FLC) has also been employed to improve
the kinematic control of DWMRs but it can be complex to
design due to the need for extensive rule bases, making tuning
challenging. Additionally, it may struggle to adapt quickly to
rapid changes in the environment, potentially leading to less
precise control performance [15].
Sliding mode control (SMC) is another robust control
method that ensures stability by forcing system states to
converge to a predefined sliding surface; however, backstep-
ping control (BSC) is preferred over FLCs, NNs, and SMC
due to its systematic design approach, which allows for the
derivation of control laws that are explicitly tailored to the
dynamics of the system. BSC employs a recursive strategy to
stabilize the system by progressively designing control laws
for each state variable, ensuring robustness against uncertain-
ties and external disturbances. In [16], the BSC technique
was employed to enhance the performance of the ball and
arc system, particularly in terms of steady-state accuracy and
robustness. Furthermore, this method has proven effective in
addressing the nonlinearities associated with the kinematic
control of Differential Wheeled Mobile Robots (DWMRs),
as demonstrated in [17]–[19].
With regard to the dynamic model of DWMRs, var-
ious schemes have been proposed in the literature. The
study in [20], [21] for instance, introduces an adaptive dy-
namic compensation controller (ADCC) to enhance the linear
and angular velocity performance of DWMRs. The tradi-
tional proportional-integral-derivative (PID) controller is also
widely used due to its simplicity and ease of implementation
[22]. The work in [23] focuses on optimizing PID control to
improve speed tracking performance in DWMRs, while [24]
enhances the PID controller using the H∞control method
to mitigate the effects of parameter variations caused by
fluctuating power supply. However, a key limitation of PID
control is its inability to effectively manage system dynam-
ics and transient responses due to the reliance on integer-
order derivatives, which restricts adaptability to the complex,
nonlinear behaviors often encountered in dynamic environ-
ments [25]. To address these challenges, controllers inte-
grated with fractional-order techniques have been developed
to achieve superior dynamic performance. Notable examples
include the fractional-order extended state observer [26] and
the fractional-order PID (FOPID) controller, both of which
demonstrate exceptional capability in managing system un-
certainties. These advancements significantly improve speed
control and stability, ensuring robust performance across a
broader spectrum of operating conditions [27]. The study
in [28] demonstrates that the NN FOPID controller offers
superior velocity tracking performance compared to the NN
PID controller. Another study in [29] shows the applicabil-
ity of FOPID controllers on a real mobile robotic platform.
The researchers evaluated the performance of the resulting
controller while the robot was subjected to uncertainties and
disturbances in a complex environment.
Nevertheless, the primary challenge in utilizing PID and
FOPID controllers lies in the precise selection of control
parameters. Randomly or manually adjusting these parame-
ters often fails to yield the desired system response, particu-
larly in the presence of nonlinearities and uncertainties [30].
Additionally, when both kinematic and dynamic controllers
are employed, optimizing the control parameters can become
2VOLUME 11, 2023
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2025.3552312
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
computationally taxing, further complicating the tuning pro-
cess. Recently, swarm optimization algorithms have emerged,
drawing inspiration from neurobiological behaviors and nat-
ural systems, such as flocks of birds. These algorithms typi-
cally require only function evaluations rather than derivative
or integral computations [31].
Particle Swarm Optimization (PSO) [32] is one of the
widely utilized algorithms for optimizing parameters in
robotics and control systems [33]. For instance, the work
presented in [34] employs PSO to enhance the path planning
of a DWMR while adhering to specified velocity constraints.
Similarly, in [35], PSO is utilized to optimize a PID con-
troller for automatic generation control, specifically targeting
inter-area oscillation suppression within a unified two-area
hydro-thermal deregulated power system. In [36], the PSO-
optimized PID has also shown a superior performance in
robot trajectory tracking compared to PSO-optimized FLC.
Another commonly employed optimization technique is the
Grey Wolf Optimizer (GWO) [37]. In [38], GWO is applied
to parameterize an adaptive fractional-order parallel fuzzy
PID controller, while in [39], it is used to optimize the PID
controller of a brushless DC motor. The study in [40] applies
GWO to optimize both FOPID and Linear Quadratic Regula-
tor (LQR) controllers for use in a DWMR.
A comparative analysis has been conducted in [41] to eval-
uate various optimization algorithms for tuning a PID con-
troller which is designed to regulate the speed of a DC motor
for maintaining the terminal output of an automatic voltage
regulator system. The simulation results indicate that the
Slime Mould Algorithm (SMA) [42] outperforms several al-
ternatives, including GWO, Harris Hawks Optimization [43],
and Atom Search Optimization [44]. Additionally, in [45],
the SMA-optimized cascade PD-PI controller demonstrated
enhanced frequency stability in power systems compared to
controllers optimized using the Golden Search Optimization
Algorithm [46]. The superior performance of SMA can be
attributed to its unique mechanism of mimicking the forag-
ing behavior of slime moulds, which allows it to efficiently
explore the solution space while avoiding local optima. This
adaptability enables SMA to effectively balance exploration
and exploitation during the optimization process, resulting
in more accurate and robust controller tuning. Furthermore,
in [47], the Gazelle Optimizer Algorithm (GOA) [48] is
employed to enhance the performance of a FOPID controller
for speed regulation of micromotors, demonstrating superior
performance compared to the GWO-optimized FOPID con-
trollers. This further highlights the importance of choosing
the right optimization algorithm to achieve optimal controller
performance in dynamic systems.
Nevertheless, integrating a kinematic controller with a PID
or FOPID controller for the trajectory tracking of DWMRs
introduces increased complexity in optimization due to the
additional parameters involved. This complexity is further
exacerbated when nonlinearities, such as speed constraints
inherent in practical robot models, are considered, leading to
a higher computational burden. To the best of the authors’
knowledge, the optimization of such integrated systems re-
mains relatively underexplored in the literature. In contrast,
existing approaches like ADCC primarily focus on position
tracking while neglecting posture tracking, despite both being
critical for the effective operation of DWMRs. The primary
contributions of this work are summarized as follows:
1) Development of GWO-SMA with a tailored multi-
metric cost function for DWMR: We introduce a
novel enhanced GWO strategy, termed GWO-SMA,
which incorporates a tailored multi-metric cost function
specifically designed for DWMR trajectory tracking.
This approach combines Integral Absolute Error (IAE)
and Integral Squared Error (ISE) to balance steady-state
accuracy and rapid error correction, ensuring precise
position and posture tracking. Additionally, the GWO-
SMA method leverages opposition space concepts and
an optimum cache mechanism to enhance fitness op-
timization in each iteration, improving both explo-
ration and exploitation efficiency compared to individ-
ual algorithms. Simulation results demonstrate that the
proposed GWO-SMA algorithm outperforms existing
control optimization techniques, including PSO [36],
GWO [39], SMA [45], and GOA [47], in terms of
trajectory tracking error.
2) Optimization of BSC and FOPID controller for a
DWMR with speed constraints: This work integrates
BSC and FOPID controller (i.e. BSC-FOPID) within
a cascade closed-loop control structure to enhance
the trajectory tracking of a DWMR that is subject to
nonlinearities resulting from speed constraints. Sim-
ulations were conducted using three distinct refer-
ence paths-lemniscate, square, and cloverleaf-to evalu-
ate the controller’s performance. The proposed GWO-
SMA-optimized BSC-FOPID demonstrates superior
trajectory tracking compared to the ADCC approach
across all tested paths. Furthermore, when bench-
marked against the GWO-SMA-optimized BSC-PID
controller, the BSC-FOPID achieves significant im-
provements, including at least a 55% reduction in ISE
and a 38% reduction in IAE for position and posture
tracking.
The remainder of the article is organized as follows: Sec-
tion II outlines the general model of the DWMR based
on kinematic and dynamic principles, deriving the velocity-
based dynamic model of the robot along with the reference
trajectory generation and the tracking control system, which
includes BSC and FOPID controllers, as well as the proposed
enhanced GWO algorithm. Section III presents the simula-
tion results and discusses the findings. Finally, the work is
concluded in Section IV.
II. METHODOLOGY
A. DWMR VELOCITY-BASED DYNAMIC MODEL
A DWMR consists of two independently driven wheels at the
rear and one unpowered steering wheel to maintain balance,
VOLUME 11, 2023 3
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2025.3552312
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
as illustrated in Fig. 1. In this figure, Bdenotes the center of
the wheel baseline, Grepresents the center of mass, and b
signifies the distance between Gand B. The point of interest,
denoted as h= [x,y]T, resides in the XY plane and is essential
for trajectory tracking.
Figure 1: A schematic diagram of the DWMR (resembling
the structure described in [20]).
Additionally, arepresents the distance from point hto B,Ψ
denotes the robot’s posture, measured as an angular deviation
from the x-axis to the dashed line hB, and ωrepresents
the robot’s angular velocity. The parameter dindicates the
distance between the robot’s wheels. The mathematical model
of the DWMR is defined as [20]:
˙
x
˙
y
˙
Ψ
˙
u
˙ω
=
ucos Ψ −aωsin Ψ
usin Ψ + aωcos Ψ
ω
θ3
θ1ω2−θ4
θ1u
−θ5
θ2uω−θ6
θ2ω
+
0 0
0 0
0 0
1
θ10
01
θ2
uref
ωref
(1)
where uand ωrepresent the outputs of linear velocity and
angular velocity of the robot, respectively. Similarly, uref and
ωref denote velocity inputs of DWMR system. It is worth
to note that the dynamic model of the DWMR includes the
steering passive wheel. In (1), θ= [θ1, . . . , θ6]Trepresents
the inner parameters of the robot model, constituting a vector
of identified parameters. The entire robot can be described by
the following kinematic and dynamic models [20]:
Kinematic model:
˙η=J(η)ς(2)
Dynamical model:
˙ς=D(ς) + n(ς)τ(3)
where η= [x,y,Ψ]Trepresents the position of the robot,
while ς= [u, ω]Tand τ= [uref , ωref ]Tdenote the output
and input of the dynamic model, respectively. The parameters
J,Dand nare defined as follows:
J(η) =
cos Ψ −asin Ψ
sin Ψ acos Ψ
0 1
D(ς) = "θ3
θ1ω2−θ4
θ1u
−θ5
θ2uω−θ6
θ2ω#,n(ς) = 1
θ10
01
θ2
The DWMR’s kinematic model and velocity model men-
tioned above, while generally reflecting both motion and
speed characteristics of mobile robots under ideal conditions,
are limited in practical motion by mechanical constraints.
Therefore, The inputs of linear velocity and angular veloc-
ity of the vehicle for dynamic model should be reasonably
confined within the specified range as follows:
|u(t)| ≤ umax ,|ω(t)| ≤ ωmax (4)
where, umax and ωmax represent the maximum linear velocity
and angular velocity of the mobile robot output, respectively.
B. TRACKING CONTROL SYSTEM
In this study, a cascade closed-loop control structure is pro-
posed for controlling the dynamic behaviours of the DWMR
as illustrated in Fig. 2. The inner loop employs FOPID con-
trollers for the robot’s dynamic model, enabling it to track
the robot’s velocities as commanded by the BSC denoted as
cς. Given the current velocity of the robot ς= [u, ω]T, the
inner loop controller aims to determine an appropriate control
action τ= [uref , ωref ]T, such that limt→∞ ς(t) = cς(t).
The outer loop, comprising the BSC and the mobile robot’s
kinematic model, is responsible for the position control sub-
system. This subsystem ensures the tracking stabilization of
the mobile robot’s position and posture. When the reference
trajectories for the robot are set as rd= [xd,yd,Ψd]T, and
the current trajectory is η= [x,y,Ψ]T, the position controller
seeks to find an appropriate output cς= [uc, ωc]Tsuch that
limt→∞ η(t) = rd(t).
It is important to note that the saturation blocks in Fig. 2
constrain the velocities from the FOPID controllers, consid-
ering the mechanical properties and dynamic model of the
mobile robot. This ensures that the robot tracks the reference
trajectories in a practical and reasonable manner.
1) Reference Trajectory
Generally, the reference trajectory of a virtual mobile robot
mainly concentrates on the desired coordinate positions in
XY plane. The studies in [17], [20] only explored the position
of the mobile robot as it moved along different shapes during
tracking, it’s necessary to incorporate the desired angular
posture as a variable in subsequent analyses. In order to obtain
the desired velocity signals, the inverse kinematic transfor-
mation is needed based on the kinematic model of DWMR.
The process can be induced as follows with the condition
[xd,yd,˙
xd,˙
yd]known first.
ud
ωd=cos (Ψd)−asin (Ψd)
sin (Ψd)acos (Ψd)−1˙
xd
˙
yd
=cos(Ψd) sin(Ψd)
−sin(Ψd)
a
cos(Ψd)
a˙
xd
˙
yd(5)
Ψd=Zωd(t)dt (6)
4VOLUME 11, 2023
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2025.3552312
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
Figure 2: Overview of the cascade closed-loop control structure.
where, ˙
xd,˙
ydare the derivatives of desired coordinate from
virtual mobile robot in x axis and y axis.
In summary, the reference trajectory signals not only pro-
vide the information of desired position coordinate and an-
gular posture, but also contain the desired linear velocity and
angular velocity, which can be symbolized as:
rd= [xd,yd,Ψd,˙
xd,˙
yd,ud, ωd]
2) Trajectory controller based on backstepping technique
BSC is a nonlinear system control technique grounded in
Lyapunov theory. This method decomposes the nonlinear sys-
tem into several subsystems by introducing virtual variables,
allowing the construction of control laws from Lyapunov
functions. For the position and posture of DWMR, error
transformation and error dynamic system from [49] can be
expressed by (7) and (8):
ex
ey
eΨ
=
cos (Ψ) sin (Ψ) 0
−sin (Ψ) cos (Ψ) 0
0 0 1
xd−x
yd−y
Ψd−Ψ
(7)
˙
ex
˙
ey
˙
eΨ
=
ωey−u+udcos(eΨ)
udsin(eΨ)−ωex
ωd−ω
(8)
In order to ensure the error system of kinematic control stable,
the Lyapunov candidate function is constructed as follows:
L=e2
x+e2
y
2+1−cos (eΨ)
k2
which is positive. The time derivative of Lis,
˙
L=ex˙
ex+ey˙
ey+˙
eΨ
k2
sin(eΨ)
=ex(−u+udcos(eΨ)) + udsin(eΨ)ey+ωd−ω
k2ud
(9)
Define two virtual variables u1=−u+udcos(eΨ)and u2=
ey+ωd−ω
k2ud, select the error feedback appropriately for u1,u2,
and obtain the control law of system (8) as follows:
u1=−k1ex
u2=−k3
k2
sin(eΨ)(10)
Substituting Eq. (10) into Eq. (9), the derivative of Lcan be
further simplified as:
˙
L=−k1e2
x−k3
k2
udsin2(eΨ)(11)
Obviously, ˙
L≤0and is uniformly continuous at the con-
ditions that k1,k2,k3are all positive for ud>0, and Lya-
punov function Ltends to be positive value and the norm
of vector [ex,ey,eΨ]Tis bounded. According to Barbalat’s
Lemma [50], it is immediate that error variable exand eΨ
converge asymptotically to zero. Since ex→0and eΨ→0,
along with the uniform continuity of ˙
exand ˙
eΨin Eq. (8), it
follows that ω→ωdand ey→0for ud>0. Therefore, the
tracking errors converge to zero globally.
3) Velocity controller based on FOPID technique
In this section, two FOPID controllers are designed for the
dynamic control of the mobile robot, addressing the distinct
dynamics of linear and angular velocities. The scheme of the
FOPID controllers is depicted in Fig. 3.
Figure 3: Internal structure of the velocity control loop.
FOPID controllers stem from fractional calculus. In control
system problems, the transfer function of the FOPID con-
troller is written as:
C(s) = Kp+Ki1
sλ
+Kdsµ(12)
A FOPID controller comprises five tuning parameters
[Kp,Ki,Kd]and [λ, µ]. Notably, if λ=µ= 1, the FOPID
controller simplifies to a classical PID controller. There is a
relationship between the FOPID controller and the Oustaloup
filter via fractional-order operators. The Oustaloup filter is
used to implement fractional-order operators in the frequency
domain, providing a good approximation within a specified
frequency range (ωb, ωh)and order N. The Oustaloup’s re-
cursive filter is presented by:
sα=K
N
Y
k=−N
s+ω′
k
s+wk
(13)
VOLUME 11, 2023 5
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2025.3552312
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
where K=ωα
h,
ωk=ωbωh
ωb
k+N+1
2(1+α)
2N+1
;ω′
k=ωbωh
ωb
k+N+1
2(1−α)
2N+1
.
For a fractional operator, Nis the number of zeros and poles,
(2N+ 1) is the order of the approximation. In the mobile
robot application, 3rd order of the filter with the frequency
range 10−2,102is chosen.
Based on the above analysis and the control structure
shown in Fig. 3. The transfer functions of the linear velocity
controller and the angular velocity controller based on FOPID
for the mobile robot can be represented by the following
equations [51]:
f(s) = ux(s)
eu(s)=Kp1+Ki1(1
s)λ1+Kd1sµ1;(14)
g(s) = ωx(s)
eω(s)=Kp2+Ki2(1
s)λ2+Kd2sµ2;(15)
According to the controller outputs in the above equations,
the velocity saturation components can be limited by the
following piecewise function:
uy=φu(ux) =
−umax ux<−umax
ux−umax ≤ux≤umax
umax ux>umax
(16)
ωy=φω(ωx) =
−ωmax ωx<−ωmax
ωx−ωmax ≤ωx≤ωmax
ωmax ωx> ωmax
(17)
After implementing the FOPID velocity controllers under
velocity constraints, the DWMR closed-loop system of dy-
namic model can be formed and expressed by the state-space
equations.
˙
eu
˙
eω="˙
uc−θ3
θ1(ωc−eω)2+θ4
θ1(uc−eu)−uref
θ1
˙ωc+θ5
θ2(ωc−eω)(uc−eu) + θ6
θ2(ωc−eω)−ωref
θ2#
(18)
where eu=uc−u,eω=ωc−ωare the state variables of the
closed-loop velocity system. The vector cς= [uc, ωc]Trepre-
sents the desired velocity vector in the inner loop. Regarding
the saturation constraints, the maximum velocity limits used
in this study were set to umax = 0.75 m/s and ωmax = 1.745
rad/s. The main objective in designing the FOPID controllers
is to determine the parameter sets [Kp1,Ki1, λ1,Kd1, µ1]for
linear velocity and [Kp2,Ki2, λ2,Kd2, µ2]for angular velocity,
ensuring the stability of the mobile robot’s velocity.
Therefore, it is essential to optimize the parameters of
both the kinematic and velocity controllers to ensure that the
robot can track the desired trajectory as closely as possible.
To address this issue, we propose a new enhanced GWO
strategy, termed GWO-SMA, in which the cost function is
specifically tailored for the trajectory tracking of DWMRs in
the following section.
C. GWO-SMA: ENHANC ED GWO ALGORITHM
The GWO and SMA are two meta-heuristic optimization
algorithms commonly employed in engineering problem-
solving. However, they operate on different search mecha-
nisms. SMA is inspired by the oscillation behavior of slime
mould in nature, while GWO draws inspiration from the
social structure and hunting patterns of grey wolves. Although
both are effective, GWO often exhibits sensitivity to parame-
ter settings and may prematurely converge. Conversely, SMA
strikes a balance between exploring new solution spaces and
exploiting known promising regions. In the context of trajec-
tory tracking for a DWMR, especially considering actuator
saturation, the optimization task becomes more complex. This
study aims to address the limitations of individual algorithms
by combining GWO and SMA into a hybridized optimization
approach for enhanced efficiency. The subsequent sections
will elaborate on the conventional GWO and SMA methods,
followed by an explanation of the proposed hybridization
technique.
1) Grey wolf optimizer
GWO was firstly invented in 2014 by Seyedali Mirjialili [37],
and it is widely used in science and engineering for optimiza-
tion. Within the GWO framework, each wolf embodies a posi-
tion point in a search space of D dimension, especially three
best wolves α,βand δobtained so far are saved in current
iteration and used for updating the positions of wolves group.
Take the i-th wolf
Xifor example, the renewing position is
searched by the following formulas:
−→
Dα=
−→
C1·−→
Xα−−→
Xi,−→
X1=−→
Xα−−→
A1·−→
Dα(19)
−→
Dβ=
−→
C2·−→
Xβ−−→
Xi,−→
X2=−→
Xβ−−→
A2·−→
Dβ(20)
−→
Dδ=
−→
C3·−→
Xδ−−→
Xi,−→
X3=−→
Xδ−−→
A3·−→
Dδ(21)
−→
Xi(t+ 1) =
−→
X1+−→
X2+−→
X3
3(22)
where, tindicates the current iteration. −→
Dα,β,δ and −→
X1,2,3
are intermediate vectors of wolf, which show the behaviour
of encircling prey. −→
Ais a sign of exploitation for wolf during
attacking prey, it is determined by the random value in the
interval [−a,a],ais linearly decreased from 2 to 0 over the
course of iterations. |A|>1shows the wolf search for the
prey while as |A|<1force the wolf to attack towards the prey.
−→
Cimprove the ability of wolf exploration during searching
the prey, it dynamically adjusts the the distance between the
prey and wolf by rand weights for prey. The conventional
parameters are provided by:
−→
A= 2−→
a−→
r1−→
a(23)
−→
C= 2−→
r2(24)
r1,r2represent two uniformly distributed random numbers
in [0,1]. The pseudo code of the standard GWO is shown in
Algorithm 1.
6VOLUME 11, 2023
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2025.3552312
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
Algorithm 1 Pseudo code for GWO
1: Initialize a, A and C
2: for i=1,...,N do
3: Randomly initialize Xi
4: end for
5: Xα=the best search agent
6: Xβ=the second best search agent
7: Xδ=the third best search agent
8: while t<Tdo
9: Calculate the fitness of all search agents
10: Update Xα,Xβ, and Xδ
11: Update a,A, and C
12: for i=1,...,N do
13: Update the position of current search agent by
equations (19)-(22)
14: end for
15: t=t+ 1
16: end while
17: Return Xα
2) Slime mould algorithm (SMA)
The SMA’s concept revolves around adaptive weighting, sim-
ulating the slime mould’s process of navigating towards food
sources with a delicate balance of exploration and exploita-
tion [42]. This adaptive process is mathematically captured
through the vector W:
−→
W(SmellIndex(i)) =
1 + r·log bF−S(i)
bF−wF + 1,conditions
1−r·log bF−S(i)
bF−wF + 1,others
(25)
SmellIndex =sort(S)(26)
where S(i)represents the fitness of the i-th agent, wF and bF
denote the worst and optimal fitness obtained in the current
iteration, and SmellIndex is an array of sorted fitness values.
The algorithm progresses through two stages. First, agents
adjust their positions based on the adaptive weights, seeking
optimal paths towards food sources. Second, agents wrap
around food sources and update their positions, ensuring
efficient exploration and exploitation. The position update
equation is defined as follows:
−→
X(t+ 1) =
ϵ·(UB −LB) + LB, ϵ < z
−→
Xb(t) + −→
vb ·−→
W·−→
XA(t)−−→
XB(t)r<p
−→
vc ·−→
X(t),r≥p
(27)
where Xbdenotes the location with the highest concentration,
XAand
XBrepresent two randomly selected agents, LB and UB
are the search range boundaries, and zand pare selective con-
ditions determining the transition of X(t), which determine
the transition of X(t),ϵand rare random numbers from [0,1].
The value of −→
vb vary randomly between [−a,a]and decrease
to zero as the iterations increase, the value of −→
vc has the same
properties as −→
vb, but the boundary is limited by [−1,1]. The
parameters pand aare defined as follows:
p= tanh |S(i)−DF|(28)
a=arctanh −t
T+ 1(29)
where, DF is the best fitness calculated in all iterations, and
Trepresent the maximum iteration number. The pseudo code
of the standard SMA is shown in Algorithm 2.
Algorithm 2 Pseudo code for SMA
1: Initialize the agents of slime mould Xi(i= 1,2, ..., N),
the maximum iteration T, dimension and boundaries.
2: while t<Tdo
3: Fitness evaluation
4: Update Xb,wF,bF
5: Update the weight
Wby (25)-(26)
6: for i=1,...,N do
7: Calculate parameters p,
vb,
vc by (28)-(29)
8: Update search agent Xiposition by (27)
9: end for
10: t=t+ 1
11: end while
12: Return bestFitness Xb
3) Proposed hybrid GWO-SMA for controller optimization
In order to improve the exploration of hybrid algorithm, the
search space Ωwas divided into Ω1and Ω2, the GWO try
to solve the best three solutions of α, β, and δin Ω1, while
the SMA endeavour to obtain the best fitness Xbin Ω2in
consecutive iterations. The agent X∗
iin Ω2can be expressed
by by the following equation.
X∗
i=UB +LB −Xi,Xi∈Ω1,X∗
i∈Ω2(30)
A globally optimum cache was introduced in the hybrid
algorithm. The cache consists of two structures, one is the best
fitness and position from αwolf, the other is the best food cost
and position which comes from SMA during every iteration.
Then the two fitness values make a comparison at the last
process, only the solution of lower fitness update the value of
alpha wolf or the best food position of slime mould. Adopting
this mechanism ensures that in each round of iteration, the
solution of the entire algorithm is optimal and the best agent is
active in the whole searching space, thus avoids the algorithm
to be stuck in a local minimum.
In order to improve the exploitation of new algorithm,
SMA is implemented in Ω2at the same time. The best fitness
Xbwill be recorded for each iteration. However, there are dif-
ferences in terms of complexities of GWO, SMA and hybrid
GWO-SMA. The total complexity of GWO can be roughly
estimated using the (31), while the total complexity of SMA
is shown in (32). The parallel hybrid algorithm mainly consist
of subsequent parts: fitness evaluation , sorting, weigh update
and location update for every iteration, as well as initialization
executed once at the start. D denotes the dimensions of func-
VOLUME 11, 2023 7
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2025.3552312
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
tions, and T represents the maximum number of iterations, N
has the number of agents. For the hybrid algorithm in every
iteration, the computational complexities of weight update
and location update are the same, which is O(N∗D), while
the computation complexity of fitness evaluation and sorting
is 2N+NlogN . Therefore, the total complexity of hybrid
optimization based on GWO-SMA, is represent by (33)
approximately.
O(GWO) = N·T·D(31)
O(SMA) = N·(2 + T·(1 + logN + 2 ·D)) (32)
O(GWO −SMA) = T·N(2 + logN + 2D) + N(33)
The flow chart of proposed hybrid algorithm is presented in
Fig. 4.
To assess the performance of the hybrid GWO-SMA for
the DWMR trajectory tracking, a cost function F(35), is
proposed by incorporating both IAE and ISE error metrics
for position and posture, denoted as IAEXY ,IAEΨ,ISEXY , and
ISEΨ, respectively. By incorporating both IAE and ISE in
the cost function, we leverage their complementary strengths:
IAE effectively minimizes long-term steady-state error, while
ISE enhances responsiveness to larger deviations [29], [52].
This multi-metric approach enables the DWMR to achieve
more accurate position and posture tracking, ensuring a bal-
anced trade-off between precision and stability. The specific
definitions of these metrics are as follows:
IAEXY =Ztf
0
|ex(t)|+|ey(t)|dt
IAEΨ=Ztf
0
|eΨ(t)|dt
ISEXY =Ztf
0e2
x(t)+e2
y(t)dt
ISEΨ=Ztf
0e2
Ψ(t)dt
(34)
where tf represents the total simulation time. The cost func-
tion is then formulated as
F=w1IAEXY +w2IAEΨ+w3ISEXY +w4ISEΨ(35)
where, w1=w3= 0.4,w2=w4= 0.1are the weights
used during the optimization. These weights are configured to
prioritize the robot’s position in the XY plane over its posture,
while still maintaining consideration for the latter.
In this work, a hierarchical optimization method is pro-
posed, where the first phase focuses on optimizing the BSC
parameters using only the kinematic model of the DWMR.
In the second phase, the optimized BSC parameters are then
employed to optimize the dynamic controller (i.e. FOPID)
based on the dynamic model of the DWMR. This approach
offers several advantages; it simplifies the optimization pro-
cess by decoupling the kinematic and dynamic complexities,
enabling efficient parameter tuning in a step-wise manner.
By optimizing the simpler kinematic model first, the method
Figure 4: Flow chart of proposed GWO-SMA algorithm for
BSC-FOPID optimization.
improves convergence and narrows the search space for dy-
namic control, leading to more accurate and reliable overall
system performance. Additionally, the modular design allows
each phase to be optimized independently, making the process
more adaptable and efficient.
To validate the effectiveness of the proposed hybrid method
for BSC-FOPID optimization, it is benchmarked against
PSO [36], GWO [39], SMA [45], and GOA [47], which
have demonstrated strong performance in recent literature for
optimizing PID-type parameters. Table 1 presents the control
parameters for each algorithm. The maximum number of
iterations and population size are uniformly set to 20 and 60
respectively. The parameters of the DWMR system are the
same as in reference [20], where a= 0.2m, and
θ= [0.2604,0.2509,−0.000499,0.9965,0.00263,1.0768] .
(36)
The actual position of the DWMR in the global coordinate
system is η(0) = [0.2,0,0]. A lemniscate trajectory is se-
lected for the initial control parameter optimization, which
can be mathematically described by
xd=sin(2ωdt),yd=sin(ωdt)(37)
where ωd= 0.25 rad/s and t∈[0,30]. The results for this
8VOLUME 11, 2023
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2025.3552312
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
part are presented in Section III-A.
Table 1: Parameter Setting for Each Optimization Algo-
rithm.
Algorithm Parameters
PSO ω∈[0.4,2],c1=c2= 2
GWO a∈[0,2]
SMA z= 0.03
GOA s∈[0,1],µ∈[−1,1],PSRs = 0.34
GWO-SMA s∈[0,1],PSRs = 0.34
To demonstrate the generalizability of the optimized con-
troller and ensure an unbiased evaluation, two additional
trajectories; square, and cloverleaf; are introduced for vali-
dation. These trajectories can be mathematically described as
follows:
•Square:
xdyd=
[L
T(t−k(t)T),0] k(t)=0
[L,L
T(t−k(t)T)] k(t)=1
[L−L
T(t−k(t)T),L]k(t)=2
[0,L−L
T(t−k(t)T)] k(t)=3
(38)
•Cloverleaf [53]:
˙
xd= 0.052cos(0.13t)+0.156cos(0.39t),(39)
˙
yd=−0.052sin(0.13t)+0.156sin(0.39t),t∈[0,50]
(40)
Additionally, the optimized BSC-FOPID is compared
against the optimized BSC-PID and the ADCC, the latter
of which has shown promising results in DWMR trajectory
tracking [20]. The ADCC is a speed control strategy based on
the dynamic model, which enhances the tracking performance
of the DWMR’s linear and angular velocities in response to
the position controller’s reference signals. The results for this
part are presented in Section III-B.
III. RESULTS AND DISCUSSIONS
This section presents the results for control parameter opti-
mization of BSC-PID and BSC-FOPID (Section III-A), as
well as results comparing these optimized controllers against
the ADCC (Section III-B).
A. CONTROL PARAMETER OPTIMIZATION AND
EVALUATION
Table 2 presents the upper and lower limits imposed during
the control parameter optimization. As discussed in the pre-
vious section, a hierarchical optimization method is employed
where the first phase involves optimizing the BSC control
parameters, i.e. k1,k2,and k3as described in (10). To ensure
the reliability of the results, each algorithm was executed 20
times. Table 3 presents the statistical results based on the
fitness values. It can be observed that the proposed GWO-
SMA algorithm outperforms the rest in terms of mean, min-
imum, maximum, median, and variance values, followed by
SMA, GWO, and GOA, with PSO yielding the least favorable
2 4 6 8 10 12 14 16 18 20
Iterations
75
80
85
90
95
100
105
Fitness Value
PSO
GWO
SMA
GOA
Proposed
(a)
2 4 6 8 10 12 14 16 18 20
Iterations
50
60
70
80
90
100
Fitness Value
PSO
GWO
SMA
GOA
Proposed
(b)
Figure 5: Comparison of the convergence curves of the
proposed hybrid GWO-SMA algorithm with other algo-
rithms for (a) optimization of PID controllers; and (b)
optimization of FOPID controllers.
performance. Based on these results, the best-performing
BSC structure for each algorithm was selected for the second
optimization phase, with the parameters displayed in Table 4.
The numerical results for the second phase, which fo-
cuses on optimizing the velocity controllers, are presented
in Table 5. The proposed GWO-SMA algorithm consistently
demonstrates superior performance across both controller
types. For the BSC-PID controller, GWO-SMA achieved
the lowest minimum value of 78.0092 and the smallest de-
viation of 0.1812, indicating high stability and consistent
performance across multiple trials. Similarly, for the BSC-
FOPID controller, GWO-SMA exhibited outstanding results
with a minimum value of 44.4077 and a deviation of 2.9533,
showcasing improved accuracy and reliability compared to
other optimization methods. These findings underscore the
effectiveness of the GWO-SMA hybrid approach in attaining
optimal control performance with minimal variation, making
it a robust solution for velocity control in the tested scenarios.
Furthermore, it is evident that the GWO-SMA-optimized
VOLUME 11, 2023 9
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2025.3552312
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
BSC-FOPID controller provides more accurate and consistent
control, outperforming the traditional PID controller across
multiple runs. Table 6 records the FOPID parameters corre-
sponding to the best solutions obtained from each algorithm.
Fig. 5 illustrates the convergence curves of the proposed
hybrid GWO-SMA in comparison to other algorithms. It
specifically demonstrates the performance of the hybrid al-
gorithm in optimizing two types of controllers: (a) PID con-
trollers and (b) FOPID controllers. The convergence curves
demonstrate the superior efficiency and effectiveness of the
GWO-SMA algorithm, as evidenced by its lower final fit-
ness values for both types of controllers. Notably, FOPID
controllers generally achieve lower fitness values compared
to PID controllers, highlighting their enhanced optimization
potential. This underscores the GWO-SMA algorithm’s ca-
pability for improved performance in controller optimization
tasks.
B. PERFOR MANCE EVALUATION BET WEEN DIFFER ENT
CONTROL SCHEMES
This section evaluates the performance and generalizability of
the proposed GWO-SMA-optimized BSC-FOPID controller
by comparing it with other DWMR control schemes, specif-
ically the GWO-SMA-optimized BSC-PID and ADCC [21].
To facilitate this evaluation, apart from the lemniscate path,
additional trajectories as defined in (38) and (39) are em-
ployed. These additional trajectories are not part of the initial
control optimization process, providing a rigorous test of
the controller’s adaptability and effectiveness across diverse
path-following scenarios.
1) Lemniscate Trajectory
The results in Fig.6 illustrate the trajectory tracking per-
formance of the DWMR using various control schemes:
ADCC, GWO-SMA-optimized BSC-PID, and GWO-SMA-
optimized BSC-FOPID for the lemniscate reference path.
The ADCC controller exhibits significant deviation from the
reference trajectory, as depicted in Fig. 6(a), while its corre-
sponding posture error response, shown in Fig. 6(b), is only
marginally worse than that of the BSC-FOPID. This deviation
primarily arises from the fact that the ADCC controller is
designed to address the robot’s dynamic behavior rather than
complex kinematic paths. When the trajectory involves con-
tinuous curves, as seen in practical applications, the kinematic
limitations of ADCC lead to degraded tracking performance.
On the other hand, both the BSC-PID and BSC-FOPID
controllers significantly enhance tracking accuracy by lever-
aging their backstepping structure. Among the two, the BSC-
FOPID controller demonstrates superior performance, par-
ticularly when handling sharp turns, as it maintains closer
-1 -0.5 0 0.5 1
x (m)
-1
-0.5
0
0.5
1
1.5
y (m)
Start
End
Reference
ADCC
BSC-PID
BSC-FOPID
Start
End
(a)
0 5 10 15 20 25
Time (s)
-0.5
0
0.5
e (rad)
ADCC
BSC-PID
BSC-FOPID
(b)
Figure 6: Trajectory tracking performances of the DWMR
with different control schemes for the lemniscate path in
terms of (a) position; and (b) posture error response.
adherence to the reference path throughout the trajectory. This
advantage can be attributed to the fractional-order compo-
nents, which provide finer control by balancing the trade-off
between responsiveness and robustness. Further insights can
be drawn from the posture error response in Fig. 6(b), where
the BSC-FOPID controller exhibits smoother and more stable
error convergence compared to both the BSC-PID and ADCC
controllers. This improved stability indicates the controller’s
enhanced precision and adaptability to varying conditions,
highlighting its suitability for complex trajectory tracking
tasks.
To further investigate the differences between BSC-PID
and BSC-FOPID, their linear and angular velocity com-
mands—before (i.e., uxand ωx) and after (i.e., uyand ωy)
passing through the saturation or speed constraint blocks-
along with the corresponding velocity error responses, are
recorded in Fig. 7. The figures reveal that the saturation
Table 2: Parameter settings of the BSC-PID and BSC-FOPID for the optimization process.
Controller k1k2k3kp1ki1λ1kd1µ1kp2ki2λ2kd2µ2
BSC-PID [0, 30] [0, 30] [0, 30] [0, 30] [0, 30] 1 [0, 30] 1 [0, 30] [0, 30] 1 [0, 30] 1
BSC-FOPID [0, 30] [0, 30] [0, 30] [0, 30] [0, 30] [0.01, 1] [0, 30] [0.01, 1] [0, 30] [0, 30] [0.01, 1] [0, 30] [0.01, 1]
10 VOLUME 11, 2023
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2025.3552312
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
(a)
(b)
(c)
Figure 7: Comparison between BSC-PID and BSC-FOPID
performance for the lemniscate path in terms of (a) an-
gular velocity command; (b) linear velocity command; and
(c) velocity error response.
Table 3: Statistical comparison of various optimization
algorithm for position controllers.
Metric PSO GWO SMA GOA GWO-SMA
Max 5.0826 5.0031 4.9846 4.9859 4.9764
Min 4.9521 4.9219 4.9209 4.9239 4.9203
Median 5.0397 4.9311 4.9268 4.9463 4.9216
Mean 5.0345 4.9379 4.9344 4.9497 4.9266
Variance 0.0370 0.0191 0.0187 0.0180 0.0140
Table 4: Optimal BSC parameters for each algorithm.
Algorithm k1k2k3F
PSO [36] 1.5496 25.6676 4.5779 4.9521
GWO [39] 1.5360 24.8424 4.5498 4.9219
SMA [41] 1.3536 25.1842 4.5885 4.9209
GOA [47] 1.5353 24.5607 4.4454 4.9239
GWO-SMA 1.5342 24.9959 4.5807 4.9203
blocks effectively suppress oscillations, as seen in Fig.7(a)
and Fig. 7(b). However, the BSC-FOPID controller not only
achieves faster suppression of oscillations but also minimizes
both linear and angular velocity errors more rapidly than the
BSC-PID controller, as illustrated in Fig. 7(c). This faster
error convergence underscores the superior adaptability and
control precision of the BSC-FOPID approach, making it a
more reliable choice for complex trajectories.
2) Square Trajectory
Fig. 8 presents the trajectory tracking performance of the
DWMR for a square reference path using the same control
schemes. As with the lemniscate path, the ADCC controller
exhibits significant deviation from the reference trajectory
and slow convergence as shown in subfigure (a). This devi-
ation is expected, as the ADCC controller primarily focuses
on compensating for the robot’s dynamic behavior, making it
less effective when encountering abrupt directional changes.
However, it demonstrates slightly better posture tracking
compared to the BSC-PID and BSC-FOPID controllers, as
indicated in subfigure (b). In contrast, both the BSC-PID
and BSC-FOPID controllers continue to exhibit superior tra-
jectory tracking, with the BSC-FOPID controller showing
the closest adherence to the square path, especially dur-
ing sharp turns where precise control becomes increasingly
challenging. This superior performance can be attributed to
Table 5: Statistical comparison of various optimization
algorithm for velocity controller.
Controller Statistics PSO GWO SMA GOA GWO-SMA
BSC-PID
Max∗80.6821 78.9279 80.2932 79.6844 78.8869
Min∗78.0753 78.2757 78.1861 78.3887 78.0092
Median∗78.6621 78.5939 78.4910 78.8878 78.3504
Mean∗78.7297 78.6216 78.5689 78.9370 78.3584
Deviation∗0.5098 0.1933 0.4619 0.3276 0.1812
BSC-FOPID
Max 64.9780 85.0551 64.3667 59.0236 55.6728
Min 47.6662 45.4945 47.0879 46.1770 44.4077
Median 53.3309 55.0098 53.5231 53.2342 52.2939
Mean 53.6018 59.6594 53.5568 52.3888 51.1963
Deviation 3.4682 11.4948 3.7634 3.7522 2.9533
VOLUME 11, 2023 11
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2025.3552312
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
Table 6: Optimized FOPID controllers for the DWMR via various algorithms.
Algorithm kp1ki1λ1kd1µ1kp2ki2λ2kd2µ2
PSO [35] 8.0201 0.5938 1.0000 30.0000 1.0000 30.0000 0.0100 1.0000 23.8625 1.0000
GWO [39] 1.0314 22.8409 0.1523 30.0000 0.3619 30.0000 0.0477 1.0000 24.5442 0.9793
SMA [45] 16.7424 0.5945 1.0000 29.9957 1.0000 0.2891 1.0000 0.0172 17.2801 0.8580
GOA [47] 0.9496 5.6550 0.4729 27.5296 0.5549 0.2968 1.0000 0.2179 30.0000 0.9606
GWO-SMA (proposed) 6.3722 0.5398 1.0000 29.5024 0.9483 0.2277 26.9509 0.0613 26.7139 0.9665
-0.5 0 0.5 1 1.5 2 2.5 3 3.5
x (m)
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
y (m)
Start
End
Reference
ADCC
BSC-PID
BSC-FOPID
Start
End
(a)
0 5 10 15 20 25 30 35 40
Time (s)
-0.5
0
0.5
1
e (rad)
ADCC BSC-PID BSC-FOPID
(b)
Figure 8: Trajectory tracking performances of the DWMR
with different control schemes for the square path in terms
of (a) position; and (b) posture error response.
the fractional-order terms in the FOPID controller, which
enhance the system’s ability to adapt to abrupt changes by
providing finer control over both transient and steady-state
responses.
The enhanced performance of these controllers relative to
ADCC can also be linked to the weighting strategy employed
in the cost function, as detailed in (35). Specifically, the pos-
ture error is assigned a lower weight than the position error,
prioritizing trajectory adherence over posture stabilization.
This strategy proves particularly effective for paths with sharp
turns, enabling the controllers to minimize position errors
more aggressively while maintaining acceptable posture ac-
curacy.
Further insights can be drawn from Fig. 9, which illustrates
the corresponding velocity responses. A trend similar to that
observed in Fig. 7 emerges, where the BSC-FOPID controller
effectively suppresses velocity errors more rapidly than the
BSC-PID controller. Subfigures (a) and (b) show that both
angular and linear velocity commands exhibit less fluctuation
under the BSC-FOPID controller, highlighting its superior
ability to manage control effort while ensuring smoother tran-
sitions at cornering points. This capability not only improves
tracking accuracy but also reduces the risk of instability due
to aggressive control inputs.
Moreover, the velocity error responses in Fig. 9(c) fur-
ther underscore the advantage of the BSC-FOPID controller.
Compared to the BSC-PID controller, the FOPID-based ap-
proach achieves faster error convergence, ensuring that the
robot quickly returns to the desired velocity profile after
encountering sharp turns. This rapid error minimization high-
lights the robustness and adaptability of the BSC-FOPID con-
troller, making it particularly suitable for complex trajectories
with abrupt path changes.
3) Cloverleaf Trajectory
Fig. 10 illustrates the trajectory tracking performance of the
DWMR for the cloverleaf reference path using the ADCC,
BSC-PID, and BSC-FOPID control schemes. As observed
with previous path results, the ADCC controller exhibits sig-
nificant deviation from the reference trajectory, particularly
during the tight turns of the cloverleaf path, as shown in sub-
figure (a). This limitation arises from the ADCC controller’s
primary focus on compensating for dynamic behavior, with-
out sufficient adaptability to complex, continuous curves.
Although the posture error, depicted in subfigure (b), remains
relatively stable, the position tracking performance is notably
compromised.
In contrast, both the BSC-PID and BSC-FOPID controllers
demonstrate superior tracking accuracy. The BSC-FOPID
controller, in particular, maintains closer adherence to the
reference trajectory, especially during the intricate curved
segments. This enhanced performance is attributed to the
fractional-order components of the FOPID structure, which
provide more flexible tuning and better disturbance rejection
compared to the integer-order BSC-PID.
Fig. 11 further supports this observation by comparing
the velocity command and error responses for the BSC-PID
and BSC-FOPID controllers. Subfigures (a) and (b) reveal
that the saturation blocks effectively suppress oscillations
in both angular and linear velocity commands. Notably, the
BSC-FOPID controller achieves smoother velocity transi-
tions with fewer fluctuations, particularly during the sharp
12 VOLUME 11, 2023
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2025.3552312
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
(a)
(b)
(c)
Figure 9: Comparison between BSC-PID and BSC-FOPID
performance for the square path in terms (a) angular
velocity command; (b) linear velocity command; and (c)
velocity error response.
turns, as evidenced by the zoomed-in insets. Moreover, sub-
figure (c) highlights the velocity error responses, where the
BSC-FOPID controller consistently minimizes both linear
and angular velocity errors more rapidly than the BSC-PID
controller. This faster error convergence reflects the FOPID’s
ability to adapt to varying conditions and maintain stability
under dynamic path changes.
Overall, the results confirm that the BSC-FOPID controller
outperforms both the ADCC and BSC-PID controllers, not
only in trajectory adherence but also in maintaining smoother
and more stable velocity control, making it particularly effec-
tive for complex paths like the cloverleaf.
C. SUMMARY
The results across all three trajectory scenarios-lemniscate,
square, and cloverleaf paths demonstrate the superior perfor-
mance of the GWO-SMA-optimized BSC-FOPID controller
compared to both the ADCC and BSC-PID controllers. The
ADCC consistently underperformed, especially when navi-
gating complex curves, as evidenced by its suboptimal track-
ing performance on both lemniscate and cloverleaf paths.
While the BSC-PID controller improved tracking accuracy,
it struggled to maintain precise trajectory adherence in more
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
x (m)
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
y (m)
Start
End
Reference
ADCC
BSC-PID
BSC-FOPID
Start
End
(a)
0 10 20 30 40 50
Time (s)
-1
-0.5
0
0.5
e (rad)
ADCC
BSC-PID
BSC-FOPID
(b)
Figure 10: Trajectory tracking performances of the DWMR
with different control schemes for the cloverleaf path in
terms of (a) position; and (b) posture error response.
VOLUME 11, 2023 13
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2025.3552312
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
demanding path segments, such as tight corners and rapid
directional changes. In contrast, the BSC-FOPID controller
consistently showed better responsiveness, smoother error
(a)
(b)
(c)
Figure 11: Comparison between BSC-PID and BSC-FOPID
performance for the cloverleaf path in terms of (a) angular
velocity command; (b) linear velocity command; and (c)
velocity error response.
convergence, and superior tracking accuracy, particularly
in complex and dynamic conditions. Table 7 records the
position and posture tracking performance of the DWMR
with GWO-SMA-optimized BSC-FOPID and BSC-PID con-
trollers across the three scenarios in terms of ISE and IAE.
The average improvement in ISE is notably higher than in
IAE, with 55.65% and 38.25% improvements in position con-
trol and 62.12% and 38.95% in angular control, respectively.
The greater improvement in ISE compared to IAE can be
attributed to the fact that ISE penalizes larger errors more
heavily due to the squaring of the error term. This means that
the BSC-FOPID controller is especially effective in reducing
larger deviations from the reference trajectory, which are
disproportionately more impactful on ISE than IAE. Thus,
while both error metrics improve, the BSC-FOPID controller
excels in minimizing the most significant deviations, leading
to higher ISE reductions.
IV. CONCLUSION AND FUTURE WORK
In conclusion, this work presents significant advancements in
the trajectory tracking of DWMR through the integration of a
BSC and a FOPID controller within a cascade closed-loop
control structure. The introduction of the enhanced GWO
strategy, GWO-SMA, specifically tailored for DWMR tra-
jectory tracking, further elevates performance by optimizing
the cost function and improving exploration and exploitation
efficiencies. The GWO-SMA-optimized BSC-FOPID con-
troller provides enhanced responsiveness and smoother error
convergence, particularly during maneuvers involving sharp
curves. Performance metrics reveal average improvements of
55.65% in ISE and 38.25% in IAE for position control, along
with 62.12% and 38.95% improvements in ISE and IAE for
posture control. These results highlight the effectiveness of
the proposed BSC-FOPID controller in minimizing signifi-
cant deviations from the reference trajectory.
Several promising directions for future research can be
explored to extend the contributions of this study. First, the
applicability of the GWO-SMA optimization strategy could
be expanded to other types of mobile robots or robotic sys-
tems to evaluate its versatility and effectiveness across diverse
operational scenarios. This may involve refining weighting
Table 7: Performance evaluation in terms of position and
posture errors across the three reference paths.
Position Posture
Path Controller ISE IAE ISE IAE
Lemniscate
BSC-PID 19.441 130.335 20.568 99.701
BSC-FOPID 8.391 78.365 2.376 32.382
Improvement (%) 56.84 39.87 88.45 67.52
Square
BSC-PID 655.132 1037.000 201.305 460.799
BSC-FOPID 30.589 272.159 148.574 386.966
Improvement (%) 95.33 73.76 26.19 16.02
Cloverleaf
BSC-PID 26.669 172.242 80.487 249.758
BSC-FOPID 22.726 170.319 22.754 166.568
Improvement (%) 14.79 1.12 71.73 33.31
Average Improvement (%) 55.65 38.25 62.12 38.95
14 VOLUME 11, 2023
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2025.3552312
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
factors or exploring alternative cost function designs to en-
hance performance in more complex and challenging en-
vironments. Furthermore, comparing the GWO-SMA strat-
egy with other optimization algorithms, such as the Whale
Optimization Algorithm [54] and the Spotted Hyena Opti-
mizer [55], which have been employed for different types
of controllers, could provide valuable insights into relative
performance and potential areas for improvement. Such com-
parisons would help establish benchmarks for optimization
techniques in robotic control applications. Another promising
avenue is the integration of machine learning techniques to
detect and compensate for wheel slips in real-time. This
capability would significantly enhance the system’s adapt-
ability and performance in dynamic environments charac-
terized by unpredictable changes and disturbances. Finally,
while this study is validated through simulations, future work
will focus on real-time implementation to further validate
the proposed controller’s effectiveness. This transition will
involve addressing practical challenges such as real-time pro-
cessing constraints, sensor noise, communication delays, and
dynamic uncertainties in physical environments. Overcoming
these challenges will be critical to ensuring the controller’s
robustness and reliability in real-world applications.
CONFLICT OF INTEREST
The authors declare no known competing financial interests
or personal relationships that could have appeared to influ-
ence the work reported in this paper.
References
[1] L. Tagliavini, G. Colucci, A. Botta, P. Cavallone, L. Baglieri, and
G. Quaglia, ‘‘Wheeled mobile robots: State of the art overview and kine-
matic comparison among three omnidirectional locomotion strategies,’’
Journal of Intelligent & Robotic Systems, vol. 106, pp. 1–18, 10 2022.
[2] T. Asafa, T. Afonja, E. Olaniyan, and H.Alade, ‘‘Development of a vacuum
cleaner robot,’’ Alexandria Engineering Journal, vol. 57, no. 4, pp. 2911–
2920, 2018.
[3] I. Daniyan, V. Balogun, A. Adeodu, B. Oladapo, J. K. Peter, and K. Mpofu,
‘‘Development and performance evaluation of a robot for lawn mowing,’’
Procedia Manufacturing, vol. 49, pp. 42–48, 2020, proceedings of the
8th International Conference on Through-Life Engineering Services –
TESConf 2019.
[4] B. Wen and J. Huang, ‘‘Leader-following formation tracking control of
nonholonomic mobile robots considering collision avoidance: A system
transformation approach,’’ Applied Sciences, vol. 12, no. 24, 2022.
[5] J. H. Teo, N. S. Ahmad, and P. Goh, ‘‘Visual stimuli-based dynamic
commands with intelligent control for reactive bci applications,’’ IEEE
Sensors Journal, vol. 22, no. 2, pp. 1435–1448, 2022.
[6] P. Y. Leong and N. S. Ahmad, ‘‘Exploring autonomous load-carrying
mobile robots in indoor settings: A comprehensive review,’’ IEEE Access,
p. (in press), 2024.
[7] O. W. Abdulwahhab and N. H. Abbas, ‘‘Design and stability analysis of
a fractional order state feedback controller for trajectory tracking of a
differential drive robot,’’ International Journal of Control, Automation and
Systems, vol. 16, p. 2, 2023.
[8] H. Xie, J. Zheng, R. Chai, and H. T. Nguyen, ‘‘Robust tracking control of
a differential drive wheeled mobile robot using fast nonsingular terminal
sliding mode,’’ Computers & Electrical Engineering, vol. 96, p. 107488,
2021.
[9] A. Loganathan and N. S. Ahmad, ‘‘A hybrid hho-avoa for path planning of
a differential wheeled mobile robot in static and dynamic environments,’’
IEEE Access, vol. 12, pp. 25 967–25 979, 2024.
[10] N. S. Ahmad, J. H. Teo, and P. Goh, ‘‘Gaussian process for a single-channel
eeg decoder with inconspicuous stimuli and eyeblinks,’’ Computers, Ma-
terials & Continua, vol. 73, no. 1, p. 611–628, 2022.
[11] L. Xin, Q. Wang, J. She, and Y. Li, ‘‘Robust adaptive tracking control of
wheeled mobile robot,’’ Robotics and Autonomous Systems, vol. 78, pp.
36–48, 2016.
[12] S. A. A. Syed Mubarak Ali, N. S. Ahmad, and P. Goh, ‘‘Flex sensor
compensator via hammerstein–wiener modeling approach for improved
dynamic goniometry and constrained control of a bionic hand,’’ Sensors,
vol. 19, no. 18, 2019.
[13] S. Li, Q. Wang, L. Ding, X. An, H. Gao, Y. Hou, and Z. Deng, ‘‘Adaptive
nn-based finite-time tracking control for wheeled mobile robots with time-
varying full state constraints,’’ Neurocomputing, vol. 403, pp. 421–430,
2020.
[14] C. H. Goay, P. Goh, N. S. Ahmad, and M. Ain, ‘‘Eye-height/width predic-
tion using artificial neural networks from s-parameters with vector fitting,’’
Journal of Engineering Science and Technology, vol. 13, pp. 625–639, 03
2018.
[15] H. H. Tang and N. S. Ahmad, ‘‘Fuzzy logic approach for controlling
uncertain and nonlinear systems: a comprehensive review of applications
and advances,’’ Systems Science & Control Engineering, vol. 12, no. 1, p.
2394429, 2024.
[16] A. J. Humaidi and M. R. Hameed, ‘‘Design and performance investigation
of block-backstepping algorithms for ball and arc system,’’ in 2017 IEEE
International Conference on Power, Control, Signals and Instrumentation
Engineering (ICPCSI). IEEE, 2017, pp. 325–332.
[17] L. Xu, J. Du, B. Song, and M. Cao, ‘‘A combined backstepping and
fractional-order pid controller to trajectory tracking of mobile robots,’’
Systems Science & Control Engineering, vol. 10, no. 1, pp. 134–141, 2022.
[18] P. T. H. Sen, N. Q. Minh, and P. X. Minh, ‘‘A new tracking control algorithm
for a wheeled mobile robot based on backstepping and hierarchical sliding
mode techniques,’’ in 2019 First International Symposium on Instrumen-
tation, Control, Artificial Intelligence, and Robotics (ICA-SYMP). IEEE,
2019, pp. 25–28.
[19] M. Yue, F. Tang, B. Liu, and B. Yao, ‘‘Trajectory-tracking control of a
nonholonomic mobile robot: Backstepping kinematics into dynamics with
uncertain disturbances,’’ Applied Artificial Intelligence, vol. 26, no. 10, pp.
952–966, 2012.
[20] F. N. Martins, M. Sarcinelli-Filho, and R. Carelli, ‘‘A velocity-based
dynamic model and its properties for differential drive mobile robots,’’
Journal of Intelligent & Robotic Systems, vol. 85, pp. 277–292, 2017.
[21] N. A. Martins and D. W. Bertol, Wheeled Mobile Robot Control: Theory,
Simulation, and Experimentation. Springer Nature, 2021, vol. 380.
[22] S. Y. Chan, N. S. Ahmad, and W. Ismail, ‘‘Anti-windup compensator for
improved tracking performance of differential drivemobile robot,’’ in 2017
IEEE International Systems Engineering Symposium (ISSE), 2017, pp. 1–
5.
[23] H. Khan, S. Khatoon, P. Gaur, M. Abbas, C. A. Saleel, and S. A. Khan,
‘‘Speed control of wheeled mobile robot by nature-inspired social spider
algorithm-based pid controller,’’ Processes, vol. 11, no. 4, p. 1202, 2023.
[24] N. S. Ahmad, ‘‘Robust H∞-Fuzzy Logic Control for Enhanced Tracking
Performance of a Wheeled Mobile Robot in the Presence of Uncertain
Nonlinear Perturbations,’’ Sensors, vol. 20, no. 13, p. 7673, 2020.
[25] N. S. Ahmad, J. Carrasco, and W. P. Heath, ‘‘LMI searches for discrete-
time Zames-Falb multipliers,’’ in 52nd IEEE Conference on Decision and
Control, 2013, pp. 5258–5263.
[26] A. F. Hasan, A. J. Humaidi, A. S. M. Al-Obaidi, A. T. Azar, I. K. Ibraheem,
A. Q. Al-Dujaili, A. K. Al-Mhdawi, and F. A. Abdulmajeed, ‘‘Fractional
order extended state observer enhances the performance of controlled tri-
copter uav based on active disturbance rejection control,’’ in Mobile Robot:
Motion Control and Path Planning. Springer, 2023, pp. 439–487.
[27] A. Tepljakov,B. B. Alagoz, C. Yeroglu, E. Gonzalez, S. H. HosseinNia, and
E. Petlenkov, ‘‘Fopid controllers and their industrial applications: A survey
of recent results,’’ IFAC-PapersOnLine, vol. 51, no. 4, pp. 25–30, 2018, 3rd
IFAC Conference on Advances in Proportional-Integral-Derivative Control
PID 2018.
[28] A. M. Abed, Z. N. Rashid, F. Abedi, S. R. M. Zeebaree, M. A. Sahib,
A. J. M. Jawad, G. A. R. Ibraheem, R. A. Maher, A. I. Abdulkareem,
I. K. Ibraheem, A. T. Azar, and A. Al-khaykan, ‘‘Trajectory tracking of dif-
ferential drive mobile robots using fractional-order proportional-integral-
derivative controller design tuned by an enhanced fruit fly optimization,’’
Measurement and Control, vol. 55, no. 3-4, pp. 209–226, 2022.
[29] M. Gheisarnejad and M. H. Khooban, ‘‘An intelligent non-integer pid
controller-based deep reinforcement learning: Implementation and experi-
VOLUME 11, 2023 15
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2025.3552312
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
mental results,’’ IEEE Transactions on Industrial Electronics, vol. 68, no. 4,
pp. 3609–3618, 2020.
[30] N. S. Ahmad, ‘‘Modeling and hybrid pso-woa-based intelligent pid and
state-feedback control for ball and beam systems,’’ IEEE Access, vol. 11,
p. 137866–137880, 2023.
[31] A. Loganathan and N. S. Ahmad, ‘‘A systematic review on recent advances
in autonomous mobile robot navigation,’’ Engineering Science and Tech-
nology, an International Journal, vol. 40, p. 101343, 2023.
[32] R. Mendes, J. Kennedy, and J. Neves, ‘‘The fully informed particle swarm:
simpler, maybe better,’’ IEEE Transactions on Evolutionary Computation,
vol. 8, no. 3, pp. 204–210, 2004.
[33] H. H. Tang and N. S. Ahmad, ‘‘Enhanced fuzzy logic control for active
suspension systems via hybrid water wave and particle swarm optimiza-
tion,’’ International Journal of Control, Automation, and Systems, vol. 23,
pp. 560–571, 2025.
[34] H. T. Najm, N. S. Ahmad, and A. S. Al-Araji, ‘‘Enhanced path planning
algorithm via hybrid woa-pso for differential wheeled mobile robots,’’
Systems Science & Control Engineering, vol. 12, no. 1, p. 2334301, 2024.
[35] Y. Bhateshvar, K. C. Vora, H. Mathur, and R. Bansal, ‘‘A comparison
on pso optimized pid controller for inter-area oscillation control in an
interconnected power system,’’ Technology and Economics of Smart Grids
and Sustainable Energy, vol. 7, pp. 1–14, 02 2022.
[36] E. Aner, M. Awad, and O. Shehata, ‘‘Performance evaluation of pso-pid and
pso-flc for continuum robot’s developed modeling and control,’’ Scientific
Reports, vol. 14, p. 18, 01 2024.
[37] S. Mirjalili, S. M. Mirjalili, and A. Lewis, ‘‘Grey wolf optimizer,’’ Ad-
vances in Engineering Software, vol. 69, pp. 46–61, 2014.
[38] K. Singhal, V. Kumar, and K. P. S. Rana, ‘‘Robust trajectory tracking con-
trol of non-holonomic wheeled mobile robots using an adaptive fractional
order parallel fuzzy pid controller,’’ Journal of the Franklin Institute, vol.
359, no. 9, pp. 4160–4215, 2022.
[39] P. Dutta and S. K. Nayak, ‘‘Grey wolf optimizer based pid controller
for speed control of bldc motor,’’ Journal of Electrical Engineering &
Technology, vol. 16, pp. 955–961, 2021.
[40] R. M. Hussein, ‘‘Design a new hybrid controller based on an improved
version of grey wolf optimization for trajectory tracking of wheeled mobile
robot,’’ FME Transactions, vol. 51, no. 2, pp. 140–148, 2023.
[41] D. Izci and S. Ekinci, ‘‘Comparative performance analysis of slime
mould algorithm for efficient design of proportional-integral-derivative
controller,’’ Electrica, vol. 21, pp. 151–159, 01 2021.
[42] S. Li, H. Chen, M. Wang, A. A. Heidari, and S. Mirjalili, ‘‘Slime mould
algorithm: A new method for stochastic optimization,’’ Future generation
computer systems, vol. 111, pp. 300–323, 2020.
[43] A. A. Heidari, S. Mirjalili, H. Faris, I. Aljarah, M. Mafarja, and H. Chen,
‘‘Harris hawks optimization: Algorithm and applications,’’ Future Gener-
ation Computer Systems, vol. 97, pp. 849–872, 2019.
[44] W. Zhao, L. Wang, and Z. Zhang, ‘‘Atom search optimization and its
application to solve a hydrogeologic parameter estimation problem,’’
Knowledge-Based Systems, vol. 163, pp. 283–304, 2019.
[45] S. Abid, A. El-Rifaie, M. Elshahed, A. Ginidi, A. Shaheen, G. Moustaffa,
and M. Tolba, ‘‘Development of slime mold optimizer with application for
tuning cascaded pd-pi controller to enhance frequency stability in power
systems,’’ Mathematics, vol. 11, 04 2023.
[46] M. Noroozi, H. Mohammadi, E. Efatinasab, A. Lashgari, M. Eslami, and
B. Khan, ‘‘Golden search optimization algorithm,’’ IEEE Access, vol. 10,
pp. 37 515–37532, 2022.
[47] D. Izci and S. Ekinci, ‘‘Fractional order controller design via gazelle opti-
mizer for efficient speed regulation of micromotors,’’ e-Prime - Advances
in Electrical Engineering, Electronics and Energy, vol. 6, p. 100295, 2023.
[48] O. Agushaka, A. Ezugwu, and L. Abualigah, ‘‘Gazelle optimization algo-
rithm: a novel nature-inspired metaheuristic optimizer,’’ Neural Computing
and Applications, vol. 35, pp. 1–33, 10 2022.
[49] H. Ye and S. Wang, ‘‘Trajectory tracking control for nonholonomic
wheeled mobile robots with external disturbances and parameter uncertain-
ties,’’ International Journal of Control, Automation and Systems, vol. 18,
no. 12, pp. 3015–3022, 2020.
[50] M. HOU, G. Duan, and M. Guo, ‘‘New versions of barbalat’s lemma with
applications,’’ Journal of Control Theory and Applications, vol. 8, pp. 545–
547, 11 2010.
[51] A. Tepljakov, B. B. Alagoz, C. Yeroglu, E. Gonzalez, S. H. HosseinNia,
and E. Petlenkov, ‘‘Fopid controllers and their industrial applications: A
survey of recent results11this study is based upon works from cost action
ca15225, a network supported by cost (european cooperation in science
and technology).’’ IFAC-PapersOnLine, vol. 51, no. 4, pp. 25–30, 2018, 3rd
IFAC Conference on Advances in Proportional-Integral-Derivative Control
PID 2018.
[52] R. Sharma, K. Rana, and V. Kumar, ‘‘Performance analysis of fractional or-
der fuzzy pid controllers applied to a robotic manipulator,’’ Expert Systems
with Applications, vol. 41, no. 9, pp. 4274–4289, 2014.
[53] L. Yan, B. Ma, and Y. Jia, ‘‘Trajectory tracking control of nonholonomic
wheeled mobile robots using only measurements for position and velocity,’’
Automatica, vol. 159, p. 111374, 2024.
[54] Z. A. Waheed and A. J. Humaidi, ‘‘Design of optimal sliding mode con-
trol of elbow wearable exoskeleton system based on whale optimization
algorithm,’’ Journal Européen des Systèmes Automatisés, vol. 55, no. 4,
pp. 459–466, 2022.
[55] A. F. Hasan, N. Al-Shamaa, S. S. Husain, A. J. Humaidi, and A. Al-dujaili,
‘‘Spotted hyena optimizer enhances the performance of fractional-order pd
controller for tri-copter drone,’’ International Review of Applied Sciences
and Engineering, vol. 15, no. 1, pp. 82–94, 2024.
LI QIANG received the B.S. from Yuncheng Uni-
versity, Shanxi, China in 2006, and M.S. degrees
from Taiyuan University of Technology, Shanxi in
2010. He is currently pursuing the Ph.D degree
in the School of Electrical and Electronic Engi-
neering, Universiti Sains Malaysia. His research
interests include the design of robust controller and
applications of different mobile robots.
HOOI HUNG TANG received the B.Eng. degree
in Mechatronics Engineering from Universiti Sains
Malaysia (USM) in 2023. He is currently taking his
Ph.D. with the School of Electrical and Electronics
Engineering, USM. His research interests include
nonlinear control, adaptive control, and model op-
timization.
NUR SYAZREEN AHMAD (Member, IEEE) re-
ceived her B.Eng degree in Electrical and Elec-
tronic Engineering from the University of Manch-
ester, United Kingdom, and her Ph.D. degree in
Control Systems from the same university. She is
currently an Associate Professor at the School of
Electrical and Electronic Engineering, Universiti
Sains Malaysia (USM), specializing in intelligent
control, sensor networks, and mobile robotics. Her
current research focuses on autonomous systems,
emphasizing intelligent optimization strategies and AI-driven methods for
motion control, multi-modal sensing, indoor navigation, and human-robot
interaction.
16 VOLUME 11, 2023
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2025.3552312
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
