Enhanced path planning algorithm via hybrid WOA-PSO for differential wheeled mobile robots

Abstract

This study introduces an enhanced algorithm for global path planning of Differential Wheeled Mobile Robots (DWMRs) that merges the Whale Optimization Algorithm (WOA) and Particle Swarm Optimization (PSO). This hybrid strategy, termed HWPSO, is designed to leverage WOA's exploration strength with PSO's efficient exploitation, specifically targeting the challenges of non-holonomic constraints in complex terrains. To validate the effectiveness of the proposed algorithm, its performance is evaluated across five diverse environments and compared against PSO, WOA, and Grey Wolf Optimization which is widely used for mobile robot path planning. Moreover, the comparison broadens to encompass four established environments from the literature where algorithms based on firefly, ant colony, A*, and other PSO variants have previously exhibited optimal performance. Additionally, a new environment is introduced to analyze the efficacy of the proposed approach for path planning for two DWMRs. Simulation results consistently demonstrate the superiority of the proposed HWPSO, manifesting performance improvements of up to 19.3% for path length reduction and up to 12.7% for DWMR travel duration reduction when compared to other methods. This underscores the efficacy of the proposed hybrid approach in achieving enhanced path planning outcomes for DWMRs in diverse scenarios.

Full text

SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL
2024, VOL. 12, NO. 1, 2334301
https://doi.org/10.1080/21642583.2024.2334301
Enhanced path planning algorithm via hybrid WOA-PSO for differential wheeled
mobile robots
Huda Talib Najma,b, Nur Syazreen Ahmadaand Ahmed Sabah Al-Arajic
aSchool of Electrical and Electronic Engineering, Universiti Sains Malaysia, Penang, Malaysia; bBiomedical Engineering Department, University
of Technology, Iraq; cComputer Engineering Department, University of Technology, Iraq
ABSTRACT
This study introduces an enhanced algorithm for global path planning of Differential Wheeled
Mobile Robots (DWMRs) that merges the Whale Optimization Algorithm (WOA) and Particle Swarm
Optimization (PSO). This hybrid strategy, termed HWPSO, is designed to leverage WOA’s exploration
strength with PSO’s efficient exploitation, specifically targeting the challenges of non-holonomic
constraints in complex terrains. To validate the effectiveness of the proposed algorithm, its perfor-
mance is evaluated across five diverse environments and compared against PSO, WOA, and Grey
Wolf Optimization which is widely used for mobile robot path planning. Moreover, the comparison
broadens to encompass four established environments from the literature where algorithms based
on firefly, ant colony, A∗, and other PSO variants have previously exhibited optimal performance.
Additionally, a new environment is introduced to analyze the efficacy of the proposed approach
for path planning for two DWMRs. Simulation results consistently demonstrate the superiority of the
proposed HWPSO, manifesting performance improvements of up to 19.3% for path length reduction
and up to 12.7% for DWMR travel duration reduction when compared to other methods. This under-
scores the efficacy of the proposed hybrid approach in achieving enhanced path planning outcomes
for DWMRs in diverse scenarios.
ARTICLE HISTORY
Received 5 January 2024
Accepted 19 March 2024
KEYWORDS
Differential wheeled mobile
robot; metaheuristic; particle
swarm optimization; path
planning; whale optimization
algorithm
1. Introduction
Autonomous mobile robots represent a category of intel-
ligent machines capable of autonomously performing
designated tasks in challenging environments without
human intervention. They have found increasing deploy-
ment in diverse and unconventional settings, including
hospitals, mining sites, battlefields, and disaster relief
operations (Cuebong Wong et al., 2018; Chen, 2023;Sun
et al., 2020; Liu et al., 2022; Teo et al., 2020). Notably,
the emergence of the COVID- 19 pandemic in 2020
has heightened our recognition of the practical utility
of mobile robots (Banjanovic-Mehmedovic et al., 2021;
Abdel-Basset et al., 2022; Ahmad et al., 2022). These
robots have proven invaluable as they can deliver essen-
tial supplies with- out direct human contact, thereby
reducing the risk of disease transmission and enhancing
operational efficiency.
To address the challenge of safely navigating such
environments, the concept of ‘Path Planning’ has been
established. Path planning involves determining a spe-
cific route for the robot, regardless of its familiarity with
the surroundings (Sanchez-Ibanez et al., 2021; Xie et al.,
2021). It can be classified into global path planning and
CONTACT Nur Syazreen Ahmad syazreen@usm.my
local path planning where they both serve distinct but
complementary purposes, and are crucial for successful
and safe navigation. Global path planning focuses on cre-
ating a path from the start to the destination while consid-
ering the over- all environment and obstacles, identifies a
high-level route that avoids major obstacles, defines way-
points, and navigates through complex environments.
Local path planning, on the other hand, deals with imme-
diate obstacles and short-term adjustments (Kobayashi &
Motoi, 2022). Without a globally planned path, the robots
may get stuck or choose suboptimal routes.
In path planning, paths can be generated using two
distinct approaches: traditional and metaheuristic meth-
ods (Loganathan & Ahmad, 2023). Traditional methods
can be broadly categorized into two groups: sampling-
based and graph-based techniques. Some examples
of the former category include Rapidly-exploring ran-
dom tree, Artificial Potential Field (APF) (Szczepanski
et al., 2022; Ng & Ahmad, 2019), Cell Decomposition
(Salama et al., 2021), and Roadmap Approach (RA) (Yan
et al., 2013). Examples of the latter category encom-
pass methods such as Dijkstra, A∗and D∗methods.
On the other hand, metaheuristic approaches involve
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2H. T. NAJM ET AL.
Figure 1. Path panning methods categorized into traditional and metaheuristic approaches.
bio-inspired algorithms such as fuzzy logic and neural
network (Goay et al., 2018), as well as nature-inspired
algorithms which include genetic algorithms (GA), parti-
cle swarm optimization (PSO) (Das et al., 2016) ant colony
optimization (ACO) (Baoye Song & Xu, 2021; Xiaoxu Wu
et al., 2018) grey wolf optimization (GWO) (Shabeeb &
Alani, 2022), and whale optimization algorithm (WOA)
(Zong et al., 2022). Figure 1provides a visual representa-
tion of these categorized path planning strategies.
The Vertical Cell Decomposition (VCD) is fast for
pathfinding but struggles with obstacles sharing x-
coordinates, leading to the Radial Cell Decomposition
(RCD) for better handling such cases (Salama et al.,
2021). APF excels in dynamic obstacle avoidance with
its simplicity, while RA and visibility graphs risk colli-
sions by navigating close to obstacles. Dijkstra’s method,
though optimal, demands high computation, unlike the
A∗algorithm which efficiently finds the shortest path but
falters with dynamic obstacles (Wu et al., 2022;Lietal.,
2022). D∗adapts well to changing environments, suitable
for local planning.
Metaheuristic approaches for path planning have
become popular due to their adaptability to uncertain
environments using AI algorithms. Fuzzy logic uses rules
for decisions like obstacle avoidance, while neural net-
works apply data-driven learning for similar goals. The
main challenge is choosing the right functions and archi-
tectures, which greatly affect performance (Teo et al.,
2022; Ahmad, 2020; Arrouch et al., 2022). Nature-inspired
algorithms constitute a category of metaheuristic meth-
ods proficient in balancing exploration and exploitation
to facilitate global searches, providing effective solu-
tions for path planning challenges in complex environ-
ments. Notable examples, including the PSO and WOA
algorithms, have found widespread application. How-
ever, due to their restricted search capabilities in intricate
environments, researchers have proposed modifications
to these algorithms, such as expanding their explo-
ration phase or improving their exploitation phase. In
(Li et al., 2023) for instance, a new algorithm called
switching multi-objective whale optimizer was devel-
oped to address the issue of premature convergence in
complex, multi-modal, non-linear decision spaces. These
adjustments are geared towards enhancing adaptability
and overall performance, ensuring the continued effec-
tiveness of nature-inspired algorithms in addressing the
evolving challenges presented by complex environments
(Trivedi et al., 2018; Loganathan & Ahmad, 2024).
Another increasingly popular strategy for enhancing
path planning algorithms involves hybridizing different
techniques to effectively address a broad spectrum of
scenarios (Loganathan & Ahmad, 2024; Wendong Gai
et al., 2018; Garip et al., 2022; Quan et al., 2021; Lazreg &
Benamrane, 2022). In (Kanoon et al., 2022) for instance,
the QOPSO algorithm that combines PSO with Quarter
Orbits (QO) is introduced which allows the robot to steer
using quarter orbit cells. The proposed method signifi-
cantly reduces path length by 29.42% compared to the
Vertical Cell Decomposition VCD algorithm and 24.25%
compared to the RCD algorithm for various maps. In
(Abdul Khaleq & Al-Araji, 2022), a modified chaotic PSO
(MCPSO) is hybridized with firefly algorithm (FA) to form

SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL 3
HFAMCPSO which leads to substantial improvements in
path length, achieving a 43.3% and 25.5% reduction com-
pared to the RCD and the A∗algorithm, respectively,
as well as a 2.3% and 22.7% reduction compared to FA
and GA, respectively. The work in (Yen & Cheng, 2018)
introduces the fuzzy ant colony optimization (FACO)
algorithm, which uses fuzzy control to minimize the itera-
tive learning error of the ACO algorithm for mobile robot
path planning. Simulation results show that the FACO
outperforms the GA, PSO and traditional (ACO), with path
lengths 4.40%, 2.04%, and 6.53% shorter than those algo-
rithms in a simple Z-shaped environment map, and 1.38%
shorter than self-adaptive ACO (SAACO) (Peng et al., 2013)
in a complex environment map.
Despite the strides made in path planning strate-
gies, the direct application to a mobile robot in practice
remains a challenging problem primarily due to motion
constraints (Lin Xu & Cao, 2021; Scharff Willners et al.,
2021; Ahmad, 2023). This challenge is particularly pro-
nounced in the case of differential drive wheeled mobile
robots (DWMRs), which are subject to non-holonomic
constraints. Navigating a DWMR involves addressing the
intricacies of non-holonomic motion, where the robot’s
movement is constrained by the relationship between its
linear and angular velocities. These constraints pose addi-
tional challenges in developing path planning strategies
that are both feasible and efficient for DWMRs.
In this study, we introduce a novel hybrid WOA-
PSO (HWPSO) for the global path planning of DWMRs.
Hybridizing PSO and WOA presents a significant advan-
tage in the global path planning for wheeled mobile
robots by leveraging the complementary strengths of
both algorithms. PSO is renowned for its fast convergence
and ability to quickly find promising areas of the search
space, making it efficient in the early stages of optimiza-
tion. However, it sometimes struggles with local optima
entrapment. On the other hand, WOA, inspired by the
bubble-net hunting strategy of humpback whales, excels
in exploring the search space thoroughly and escaping
local optima due to its unique spiral movement and
encircling prey mechanism. By integrating PSO’s rapid
convergence with WOA’s deep exploration capabilities,
the hybrid approach enhances the robustness and accu-
racy of the path planning process. This synergy not only
improves the efficiency of finding the optimal path, espe-
cially in complex environments with numerous obstacles
and constraints, but also ensures a higher reliability in
reaching the global optimum, thus facilitating more effec-
tive and safer navigation for wheeled mobile robots. To
verify the effectiveness of the proposed algorithm, its per-
formance is evaluated across five diverse environments
and compared against GWO (Shabeeb & Alani, 2022),
PSO and WOA. To underscore its strengths further, the
comparison is extended by including existing environ-
ments from the literature where algorithms such as A∗,
RCD (Salama et al., 2021), QO, QOPSO (Kanoon et al.,
2022), FA, HFAMCPSO (Abdul Khaleq & Al-Araji, 2022),
FACO (Yen & Cheng, 2018), and SAACO (Peng et al.,
2013) have previously demonstrated their optimal per-
formance. Additionally, a new environment introduced
to assess the proposed approach’s effectiveness in path
planning for two DWMRs. The simulation results consis-
tently demonstrate that HWPSO outperforms the other
methods, achieving performance improvements of up to
19.3% for path length and up to 12.7% in terms of DWMR
travel duration.
The subsequent sections of this paper are organized
as follows: In Section 2, we outline the methodology,
encompassing the modeling of the DWMR and the
hybridization of WOA and PSO to create HWPSO, specifi-
cally tailored for DWMR path planning. Section 3 provides
an in-depth presentation and discussion of the simula-
tion results, focusing on the effectiveness of the proposed
algorithm in path planning across five distinct environ-
ments. Finally, in Section 4, we draw conclusions from our
work and propose potential avenues for future research
directions.
2. Differential drive wheeled mobile robot
(DWMR) model
The method of defining the kinematics of a robot that
moves on a surface using wheels is known as DWMR mod-
elling. One of the unique aspects of modelling wheeled
mobile robots is accounting for the nonholonomic lim-
itations imposed by the wheels’ rolling-without-sliding
requirement. This implies that the robot can only move
in the directions indicated by its configuration and wheel
velocities. A DWMR model is a common way to model
non-holonomic wheeled mobile robots. It is composed
of two independent actuated wheels on a common shaft
whose motion is rigidly linked to the robot frame and one
or multiple passively orientable wheel casters which do
not have control and serve for sustenance objectives. In
general, the structure of a DWMR is composed of a cou-
ple of DC motors that operate the DWMR’s left and right
wheels, and a single castor wheel is utilized toward the
front of the cart to stabilize the structure, as illustrated in
Figure 2.
The global coordinate frame is specified as [O,Xaxis,
Yaxis], while qris defined as the posture vector of the
DWMR’s local coordinate, which can be written as qr=
(x,y,θ) where (x,y)is the coordinate of pas depicted in
Fig. 2, and θis the robot’s heading angle with respect
to Xaxis. To demonstrate that the DWMR possesses the
capability for motion and orientation, two fundamental

4H. T. NAJM ET AL.
Figure 2. Schematic of the DWMR.
conditions must be satisfied; achieving pure rolling for
each wheel and ensuring nonslippage for each wheel to
maintain the mobile robot’s lateral velocity at zero which
can be described as
V=˙
xsin θ(t)=˙
ycos θ(t)(1)
where ˙
xand ˙
yare the velocities in Xaxis and Yaxis respec-
tively. So, the kinematic model of the DWMR can be
represented as
x(t)=0.5[νR(t)+νL(t)]cosθ(t)×Ts+x(t−1)(2)
y(t)=0.5[νR(t)+νL(t)] sin θ(t)×Ts+y(t−1)(3)
θ(t)=1
L[νL(t)−νR(t)]×Ts+θ(t−1)(4)
where Lis the distance between the wheels as illustrated
in the figure, Tsis the sampling time, and νR(t)and νL(t)
are the right and left wheels’ linear velocities, respec-
tively. In order to obtain the reference linear velocity, VR,
and angular velocity, WR, for the DWMR, the following
equations can be employed:
VR=˙
x2+˙
y2;WR=¨
yr˙
xr−¨
xr˙
yr
˙
x2
r+˙
y2
r
(5)
where (xr,yr)refers to the reference coordinate. By set-
ting these velocities off, one can then compute the
DWMR’s angular velocities of the right wheel (ωR)and left
wheel (ωL)as follows:
ωR=2VR+LWR
2r;ωL=2VL−LWL
2r(6)
where ris the radius of the wheel.
3. Particle swarm optimization (PSO) and whale
optimization algorithm (WOA)
The concept of PSO draws its inspiration from the col-
lective behavior observed in swarms or flocks of birds
and other social creatures. This computational opti-
mization approach, initially introduced by Eberhart and
Kennedy in 1995 (Eberhart & Kennedy, 1995), operates
as a population-based stochastic optimization technique.
In PSO, a population of potential solutions, referred to as
particles, collaboratively explores the solution space iter-
atively in pursuit of the optimal solution for a given prob-
lem. These particles adapt their positions based on local
and global information, striving to converge toward the
best-known positions within the swarm. The PSO method
begins by initializing the population with random solu-
tions drawn from a multidimensional state space. Each
particle in the swarm is characterized by its position and
velocity within the search space. Two crucial attributes,
namely the local best (pbest) and the global best (gbest ),
guide the adjustment of each particle’s position to esti-
mate its velocity. The pbest represents the best position
a particle has achieved, while the gbest signifies the best
location visited by the entire swarm. The fitness function
of the PSO algorithm plays a pivotal role in determining
the best local and global solutions. The current velocity
and position of individual particles within the swarm can
be expressed as:
vi(k+1)=wvi(k)+c1r1(pbesti(k)−di(k))
+c2r2(gbest(k)−di(k)) (7)
di(k+1)=di(k)+vi(k+1)(8)
where viis the velocity of the i-th particle, diis the
position of the i-th particle, w∈[0, 1] is the inertia
weight, c1is the cognitive parameter, c2is the social
parameter, and r1,r2are random numbers. The veloc-
ity equation in (7) comprises three parts: the inertia term
ensures velocity consistency, the cognitive term steers
the particle toward its best-known position, and the
social term guides it toward the best position within its
neighborhood. These components collaborate to balance
exploration and exploitation. Inertia encourages broad
exploration, while cognitive and social components focus
on promising solutions, aiding in the optimization of
complex problems in PSO. The pseudocode for the PSO
is presented in Algorithm 1.
PSO is a reliable optimization technique with quick
convergence, however, in multi-modal situations, it may
become stuck in a local optimum. To get beyond the
trapping problem, the hybrid metaheuristic technique is
presented. By integrating it with another algorithm, we
enhance its ability to navigate and explore the solution

SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL 5
Algorithm 1 Pseudo-code for the standard PSO
1: Initialize particle population with random positions and velocities
2: Initialize global best position and evaluate fitness
3: Define w,c1,c2,r1and r2
4: While stop condition is not met do
5: for i=1, ...., imax do
6: Update the velocity and velocity limits
7: Update position
8: Evaluate fitness of each particle, f(di)
9: if f(di)<f(pbesti)then
10: pbesti←di
11: end if
12: if f(pbesti)<f(gbest)then
13: gbest ←pbesti
14: end if
15: Update vi(k+1)according to (7)
16: Update di(k+1)according to (8)
17: end for
18: k++
19: end while
space more effectively, thereby increasing the likelihood
of discovering global optimum solutions.
With regard to WOA, it is a relatively recent meta-
heuristic algorithm that draws inspiration from the hunt-
ing strategies of humpback whales. This innovative
approach, introduced by Mirjalili and Lewis in 2016 (Mir-
jalili & Lewis, 2016), has exhibited promising results in
the realm of optimization problem-solving. The core con-
cept behind WOA mimics a pod of whales encircling their
prey, with each individual whale symbolizing a potential
solution. This algorithm comprises several distinct stages,
including encircling, bubble-net attacking, and search-
ing, all crafted to emulate the diverse hunting techniques
observed in whales. By incorporating these characteris-
tics, the algorithm effectively explores the solution space
and converges toward an optimal or near-optimal solu-
tion. The following mathematical model, consisting of a
set of equations, has been devised to emulate the behav-
ior of whales and address optimization problems.
Encircling prey: Let 
Aand 
Cbe coefficient vectors
defined as
−→
A=2.−→
a.−→
r1−−→
a,−→
C=2.−→
r2(9)
where −→
ais decreasing linearly over the course of itera-
tions, and −→
r1,−→
r2∈[0, 1] are random vectors. The model
of encircling behavior employed for updating the posi-
tions of the other whales in the direction of the best
search agent can be written as follows:

D=|
−→
C.−→
X∗(k)−−→
X(k)|
−→
X(k+1)=−→
X∗(k)−−→
A.
D(10)
where −→
X∗refers to the position of the optimal solution,
while −→
Xrefers to the position vector of a candidate
solution.
Bubble-net attacking: Let 
D=|
−→
X∗(k)−−→
X(k)|repre-
sent the distance between the current and the best whale,
which is regarded as the best solution obtained so far.
The spiral equation that underlies the helix-shaped move-
ment pattern of humpback whales for determining the
relative position between the current and the best whale
is as follows:
−→
X(k+1)=
D.ebl.cos(2πl)+−→
X∗(k)(11)
where |l|≤1 is a random number, and bserves as a
constant that characterizes the form of the logarithmic
spiral.
Exploitation: In the WOA algorithm, the primary strat-
egy during the exploitation phase revolves around the
encircling and bubble-net attacking methods. Humpback
whales exhibit two main approaches: they may encir-
cle their prey while continuously adjusting their position
in a spiral manner, or they may swim directly towards
their prey while making positional updates. The WOA
algorithm attributes a 50% probability of reproducing
each of these behaviors, which can be mathematically
described as follows
−→
X(k+1)={Equation (10)if ρ<0.5 (12)
−→
X(k+1)={Equation (11)if ρ≥0.5 (13)
where ρis a random number between 0 and 1.
Exploration (searching for prey): The exploration
phase begins when |A|>1, where the search agent is
forced to move away from the reference whale. The math-
ematical model for this phase can be described as follows:

D=|
−→
C.−→
Xrand(k)−−→
X(k)|
−→
X(k+1)=−→
Xrand(k)−−→
A.
D(14)
where −→
Xrand represents a randomly chosen position vec-
tor from the current population. The pseudo code for
WOA is depicted in Algorithm 2.
4. Proposed hybrid WOA-PSO for DWMR path
planning
The fusion of WOA and PSO algorithms represent a pow-
erful tool for tackling challenging optimization problems.
This innovative hybrid approach harnesses the strengths
of both algorithms, leading to enhanced search capabili-
ties and improved convergence efficiency. PSO is known
to have limitations when dealing with complex, higher-
order design challenges, primarily due to its inherent ten-
dency to operate within a narrow search region caused
by the constant inertia weight. To address this limitation,

6H. T. NAJM ET AL.
Algorithm 2 Pseudo-code for the standard WOA
1: Initialize whale population with random positions, −→
Xi
2: Evaluate fitness of each search agent
3: Find the best search agent, −→
X∗
4: While stop condition is not met do
5: for i=1, ...., imax do
6: Update a,A,C,pandl
7: if p<0.5 then
8: If |A|<1then Update −→
X(k+1)based on (10)
9: else
10: Update −→
X(k+1)based on (14)
11: end if
12: else
13: Update −→
X(k+1)based on (11)
14: end if
15: end for
16: Check the fitness of each −→
Xi
17: Update −→
X∗if there is a better solution
18: k++
19: end while
20: Return −→
X∗
the Hybrid PSO and WOA, or HWPSO, is introduced, cap-
italizing on the favorable attributes of both PSO and
WOA.
The proposed HWPSO algorithm mitigates the risk of
getting trapped in local optima by strategically com-
bining the explorative strengths of the WOA with the
exploitative efficiency of PSO. WOA’s exploration phase
utilizes a logarithmic spiral approach to navigate through
and beyond local optima, covering extensive search
areas. Meanwhile, PSO’s role is refined to fine-tune the
search around promising areas identified by WOA. This
synergy not only accelerates convergence towards opti-
mal solutions but also diversifies the search process, effec-
tively reducing the likelihood of premature convergence
in complex search landscapes. The hybrid algorithm gains
from the diversity-enhancing exploration capabilities of
WOA while retaining the exploitation prowess of PSO by
incorporating ‘pbest’ from WOA into the velocity update
equation. This collaboration aims to strike the optimal
balance between exploration and exploitation, resulting
in quicker convergence rates and superior performance
across various optimization challenges. Equation (15)
illustrates the updated velocity update equation pro-
posed by this approach
vi(k+1)=wvi(k)+c1r1(Whale∗(k)−di(k))
+c2r2(gbest(k)−di(k)) (15)
where ‘Whale∗’ refers to the best solution. The other com-
ponents of the PSO and WOA methods are unaltered to
guarantee preservation the essential mechanics of the
two techniques. In the context of DWMR path planning,
the fitness of a candidate solution is evaluated using a
cost function that is based on path length which is also
subject to penalty imposed in collision avoidance. The
collision of the robot with obstacles is denoted as viola-
tion, νL, and is directly proportional to the cost function.
If the robot’s path intersects any obstacle at any point
on the map, νLwill assume a value determined by (18).
This will elevate the cost function’s value, indicating an
unsuitable solution and steering away from the global
minima.
d(i)=(x−xobs(i))2+(y−yobs (i))2(16)
∅(i)=⎧
⎪
⎪
⎨
⎪
⎪
⎩
0if d(i)
Robs(i)≥1
1−d(i)
Robs(i)if d(i)
Robs(i)<1
(17)
vL=1
no
no

i=1
∅(i)(18)
Let the path length be written as
fρ=
N

i=1(x(i+1)−x(i))2+(y(i+1)−y(i))2(19)
where Nis the total number of populations or nodes in the
path. The cost function can then be developed as follows:
f(.)=fρ(1+μvL)(20)
where μwhich is set to 100 denotes the penalty imposed
for the violations. The penalty is crucial for emphasizing
the severity of violations (obstacle collisions). It directly
impacts the cost function, discouraging paths that inter-
sect with obstacles. This mechanism is vital for guid-
ing the optimization process toward more feasible, safe
paths by penalizing collisions heavily, thus balancing
exploration and exploitation in complex decision spaces.
Figure 3depicts the flowchart of the suggested hybrid
algorithm.
It is also essential to highlight that the DWMR is con-
strained by a maximum angular velocity of 3rad/sand
a maximum linear velocity of 0.5m/s. To compute the
DWMR travel duration, the total time required to traverse
the path can be simply written as
tf=
nw−1

j=1(x(i+1)−x(i))2+(y(i+1)−y(i))2
v
+
nw−1

j=1
θj+1−θj
ω(21)
where vrepresents the linear velocity of the DWMR, ω
signifies the angular velocity of the DWMR,and(xj,yj,θj)
and (xj+1,yj+1,θj+1)denote the coordinates of adjacent
waypoints.

SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL 7
Figure 3. The proposed HWPSO algorithm.
It is worth to highlight that our research focuses
on solving real-world robot path planning challenges,
emphasizing practical rather than theoretical aspects
Figure 4. Comparison of the proposed HWPSO against PSO, WOA and GWO in terms of shortest route and convergence curve for
Environment 1.
(Trivedi et al., 2018). We point out that the complex
nature and specific constraints of robot path planning,
detailed in (16)-(20), are not fully captured by stan-
dard benchmarks. Therefore, we use evaluation metrics
directly related to path planning, such as path length
and avoiding obstacles, which are more pertinent to our
research goals than typical benchmark metrics.
5. Results and discussions
The dimensions of the DWMR model used in this work
are L=0.39m,r=0.075m, and the sampling time, Ts,
is set to 0.1s. The maximum linear velocity of the robot
is set to 0.5m/s, while the maximum angular velocity of
the robot is set to 2rad/s. In order to demonstrate the
effectiveness of the proposed HWPSO, the path planning
algorithm with the DWMR is evaluated via MATLAB sim-
ulation across five different cluttered environments with
black shapes representing obstacles or barriers and white
areas representing free space. Among these, the first four
environments (i.e. Environments 1 to 4) are sourced from
existing literature where algorithms such as QO, QOPSO,
FA, HFAMCPSO, FACO, SAACO, A∗and RCD have previ-
ously demonstrated their best performance, while the last
one (i.e. Environment 5) is introduced for the first time
to evaluate the efficiency of the proposed method for
path planning of two DWMRs. With regard to GWO, PSO,
WOA, and HWPSO algorithms, the number of populations
and maximum iteration number are set to 100 and 25
respectively.
Environment 1 which measures 7mby 7mas illustrated
in Figure 4(a) is employed from a recent study in (Kanoon

8H. T. NAJM ET AL.
Figure 5. Illustration on the DWMR’s trajectory via the proposed HWPSO compared with GWO, PSO and WOA in Environment 1 based
on left and right wheels’ angular velocities as well as travel duration, tf.
et al., 2022). In this environment, the path generated by
the proposed HWPSO, depicted by the black line, is ini-
tially compared against those generated by its individual
components, i.e. PSO and WOA which are represented by
the orange and blue lines respectively. The DWMR starts
at the red circle which marks the initial position, and con-
cludes at the green diamond which marks the goal posi-
tion. The corresponding convergence curves are shown
in Figure 4(b) where it can be observed that HWPSO
surpasses GWO, PSO, and WOA in terms of finding the
shortest path and achieving faster convergence. Figure 5
illustrates the corresponding DWMR’s trajectory based on
the left and right wheels’ angular velocities as well as the
time for the robot to traverse the path via HWPSO, PSO,
WOA and GWO algorithms. As shown in the figure, apart
from a reduced travel duration, the proposed algorithm
produces a notably smoother velocity profile which is
crucial for ensuring the robot’s movement is seamless
in practical applications. The corresponding numerical
results are recorded in Table 1which also includes the
outcomes from (Kanoon et al., 2022). The table reveals
that the QOPSO algorithm results in a shorter path com-
pared to GWO, PSO and WOA. Nevertheless, the proposed
HWPSO outperforms QOPSO by not only generating the
shortest path but also exhibiting the fastest convergence,
requiring only 9 iterations to reach the steady-state value.
Tab le 1. Numerical results for Environment 1.
Method Optimal Path Length (cm) Best Iteration
QO (Kanoon et al., 2022) 659.4 N/A
QOPSO (Kanoon et al., 2022) 657.1 12
GWO (Shabeeb & Alani, 2022) 921 17
PSO 679.9 20
WOA 658 10
HWPSO 634.4 9
The outcomes for Environment 2 are showcased in
Figures 6–9and the corresponding quantitative perfor-
mance is documented in Table 2which also includes the
findings from (Abdul Khaleq & Al-Araji, 2022). Examin-
ing Figures 6and 7, it’s clear that the proposed HWPSO
exhibits superior performance in both path length and
travel duration compared to GWO, PSO and WOA. The
travel duration in particular is distinctively shorter with
the proposed method, clocking in at only 22.4s, in con-
trast to GWO, WOA and PSO, which demand 25.66s,
32.63s and 30.74s, respectively, as depicted in Figure 7.
This discrepancy is primarily attributed to the smoother
trajectory illustrated in Figure 6(a). In comparison to the
findings in (Abdul Khaleq & Al-Araji, 2022) presented in
Table 2where HFAMCPSO demonstrates the best per-
formance in terms of path length, WOA yields a shorter
path but is surpassed by the superior performance of the
proposed HWPSO.

SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL 9
Figure 6. Comparison of the proposed HWPSO against PSO, WOA and GWO in terms of shortest route and convergence curve for
Environment 2.
Figure 7. Illustration on the DWMR’s trajectory via the proposed HWPSO compared with GWO, PSO and WOA in Environment 2 based
on left and right wheels’ angular velocities as well as travel duration, tf.
The results for Environment 3 are displayed in Figures 8
and 9, and the corresponding quantitative performance
is detailed in Table 3, which also incorporates the out-
comes from (Yen & Cheng, 2018) and (Peng et al., 2013).
By examining Figures 8and 9, it becomes evident that
the proposed HWPSO outperforms PSO and WOA in both
path length and travel duration. Notably, the travel dura-
tion is significantly shorter with the proposed method,
registering at only 99.85s, in contrast to WOA and PSO,
which require 116.6s and 111.6s, respectively, as depicted
in Figure 9. Comparing the results in Table 3with the find-
ings from (Yen & Cheng, 2018) and (Peng et al., 2013), the

10 H. T. NAJM ET AL.
Figure 8. Comparison of the proposed HWPSO against PSO, WOA and GWO in terms of shortest route and convergence curve for
Environment 3.
Figure 9. Illustration on the DWMR’s trajectory via the proposed HWPSO compared with GWO, PSO and WOA in Environment 3 based
on left and right wheels’ angular velocities as well as travel duration, tf.
GWO, WOA and PSO exhibit inferior performance. How-
ever, a notable improvement is observed with the pro-
posed hybridization of WOA and PSO, leading to a reduc-
tion in both path length and the number of iterations
required to reach the optimal value.
Environment 4 is sourced from a recent work in
(Salama et al., 2021), depicting a corridor setting within
a building, which measures 10mby 10m. In (Salama et al.,
2021), the A∗and RCD algorithms have exhibited supe-
rior performance compared to the VCD which encounters
challenges in finding the optimal path when applied to
environments with obstacles having vertices that share
the same x-coordinates. In the work, two scenarios are
presented; Scenario 1 (S1) where the starting position is

SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL 11
Tab le 2. Numerical results for Environment 2.
Method
Optimal Path
Length (cm)
Best
Iteration
CPSO (Abdul Khaleq & Al-Araji, 2022) 739.168 60
MCPSO (Abdul Khaleq & Al-Araji, 2022) 738.507 58
FA (Abdul Khaleq & Al-Araji, 2022) 743.555 65
HFAMCPSO (Abdul Khaleq & Al-Araji, 2022) 737.399 50
GWO (Shabeeb & Alani, 2022) 730 80
PSO (Kanoon et al., 2022) 751.7 45
WOA 672.88 20
HWPSO 653.1 26
Tab le 3. Numerical results for Environment 3.
Method
Optimal Path
Length (m)
Best
Iteration
FACO (Yen & Cheng, 2018) 29.3 100
SAACO (Peng et al., 2013) 29.7 100
GWO (Shabeeb & Alani, 2022)37.8 50
PSO 34.5 50
WOA 33.1 50
HWPSO 29.07 50
at (0, 0), and the goal position is at (6.9,5.7), and Scenario
2 (S2), where the starting position is at (0, 0),andthe
goal position is at (8.9,7.8). Via simulations, the A∗method
outperforms the RCD method by a considerable margin.
However, with the applications of the GWO, WOA, PSO,
and HWPSO algorithms, a substantial reduction in path
lengths is attainable, as depicted in Figure 10, Figure 12,
and Table 4. Among these three, HWPSO produces the
shortest paths in both scenarios. Figures 11 and 13 illus-
trate the resulting trajectories of the HWPSO for S1 and S2,
respectively, where it can be observed that the proposed
Tab le 4. Numerical results for Environment 4 (Scenarios 1 and 2).
Optimal Path Length (m)
Method Environment 4 (S1) Environment 4 (S2)
A∗(Salama et al., 2021) 12.62 16.87
RCD (Salama et al., 2021) 17.44 25.36
GWO (Shabeeb & Alani, 2022) 10.1 12.9
PSO 10.4 13.90
WOA 9.9 13.90
HWPSO 9.38 12.33
HWPSO not only leads to a smaller variation in the veloc-
ity profiles but also achieves a shorter travel duration
particularly when the path is longer.
Environment 5 which measures 6.3mby 6.3mas
depicted in Figure 14(a) is introduced in this work to
evaluate the performance of the proposed HWPSO in
path planning of two DMWRs (denoted as DWMR1 and
DWMR2 in the figure). The corresponding convergence
curves are shown in Figure 14(b,c) with the numeri-
cal results detailed in Table 5. Based on the data pre-
sented in both the figures and the table, it is evident that
HWPSO surpasses the other two methods in terms of find-
ing the shortest path and achieving faster convergence,
thus exhibiting a consistent performance for path plan-
ning of multiple DWMRs. Specifically, for DWMR1, HWPSO
attains the best cost at 5.72mwith only 10 iterations,
whereas GWO, PSO and WOA require 37, 18 and 15 iter-
ations, respectively, to reach slightly higher best costs
than HWPSO. In the case of DWMR2, HWPSO exhibits
notably faster convergence with a significantly lower best
cost, approximately 2.5 m less than the second-best cost.
Figure 15 illustrates the corresponding DWMRs’ trajecto-
ries based on the left and right wheels’ angular velocities
Figure 10. Comparison of the proposed HWPSO against PSO, WOA and GWO in terms of shortest route and convergence curve for
Environment 4 (Scenario 1).

12 H. T. NAJM ET AL.
Figure 11. Illustration on the DWMR’s trajectory via the proposed HWPSO compared with GWO, PSO and WOA in Environment 4
(Scenario 1) based on left and right wheels’ angular velocities as well as travel duration, tf.
Figure 12. Comparison of the proposed HWPSO against PSO, WOA and GWO in terms of shortest route and convergence curve for
Environment 4 (Scenario 2).
as well as the time for the robots to traverse the path via
HWPSO, GWO, PSO and WOA. As illustrated in the figure,
apart from achieving a shorter travel duration, the pro-
posed algorithm generates a distinctly smoother veloc-
ity profile. This feature is vital for ensuring the seamless
movement of the robots in practical applications.
Table 6provides a summary of the average optimal
path lengths, and travel durations (tf) for the HWPSO
using the proposed HWPSO method, in comparison
to GWO, PSO and WOA across all environments. The
table additionally quantifies the percentage improve-
ment achieved by HWPSO compared to the second-best

SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL 13
Figure 13. Illustration on the DWMR’s trajectory via the proposed HWPSO compared with GWO, PSO and WOA in Environment 4
(Scenario 2) based on left and right wheels’ angular velocities as well as travel duration, tf.
Figure 14. Comparison of the proposed HWPSO against PSO, WOA and GWO in terms of shortest route and convergence curve for
Environment 5.

14 H. T. NAJM ET AL.
Figure 15. Illustration on the DWMR1’s and DWMR2’s trajecto-
ries via the proposed HWPSO compared with GWO, PSO and WOA
in Environment 5 based on left and right wheels’ angular velocities
as well as travel duration, tf.
Tab le 5. Numerical results for Environment 5.
DWMR Method
Optimal Path
Length (m)
Best
Iteration
1GWO (Shabeeb & Alani, 2022)5.89 37
PSO 5.828 18
WOA 5.773 15
HWPSO 5.72 10
2GWO (Shabeeb & Alani, 2022)7.98 20
PSO 7.879 34
WOA 7.33 24
HWPSO 4.86 14
algorithm in each category. From the numerical results, it
is evident that the HWPSO effectively addresses the lim-
itations of its individual components, i.e. WOA and PSO,
which tend to get trapped in local minima during path
optimization in complex environments. A further analy-
sis of the table reveals that, while WOA outperforms PSO
in determining optimal paths, the reverse is observed in
terms of tf, where PSO yields a shorter duration for nearly
each environment. Thus, for a DWMR which is subject
to non-holonomic constraints, a shorter path does not
necessarily translate to a shorter travel duration. Never-
theless, HWPSO consistently outperforms the rest in both
performance evaluations. The percentage improvement
ranges from 2.94% to 19.26% for path length and from
2.98% to 12.7% for tf.
6. Conclusion
In conclusion, this study presents HWPSO which is a novel
hybrid algorithm integrating WOA and PSO for the global
path planning of DWMRs. HWPSO strategically leverages
the exploration capabilities of WOA and the exploitation
efficiency of PSO to provide a well-balanced and adapt-
able solution for enhancing DWMR path planning. The
integration of WOA and PSO elements aims to address
challenges associated with non-holonomic constraints,
thereby improving the algorithm’s performance in nav-
igating complex environments. The performance eval-
uation across diverse environments, compared against
GWO, PSO and WOA, as well as established algorithms
from the literature, has demonstrated HWPSO’s consis-
tent superiority.
In future work, we aim to refine the HWPSO algorithm
further, enhancing its adaptability for complex, dynamic
environments, particularly for local path planning. This
includes optimizing its parameters and testing its effec-
tiveness across various real-world scenarios and robotic
platforms. The focus will also extend to assessing the
algorithm’s robustness and versatility in practical applica-
tions. HWPSO presents a promising framework for DWMR
path planning, offering substantial potential for ongoing
advancements and application in a broader context.
Nomenclature
Symbols/Acronyms Definitions
PSO Particle Swarm Optimization
WOA Whale Optimization Algorithm
HWPSO Hybrid WOA-PSO
GWO Grey Wolf Optimization
DWMR Differential wheeled mobile
robot
ωL,ωRLeft wheel’s angular velocity, right
wheel’s angular velocity
tfTravel duration of the robot
QO Quarter orbits
QOPSO Quarter orbits PSO

SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL 15
Tab le 6. Comparison of the average optimal path length and DWMR’s travel duration, tf, using the proposed HWPSO method versus its
GWO, PSO and WOA across all environments.
Environment
Performance metric Method 1 2 3 4 5
GWO (Shabeeb & Alani, 2022) 9.21 7.30 37.8 22.9 13.87
PSO 6.80 7.52 34.50 24.30 13.71
Path length (m) WOA 6.58 6.73 33.10 22.50 13.10
HWPSO 6.34 6.53 29.07 21.71 10.58
Improvement (%) 3.59 2.94 12.18 3.51 19.26
GWO (Shabeeb & Alani, 2022) 36.6 25.66 132.7 80.68 43.58
PSO 23.18 30.74 111.64 79.14 38.64
WOA 24.59 32.63 116.61 85.62 39.93
tf(s)HWPSO 21.93 22.40 99.85 76.78 36.91
Improvement (%) 5.40 12.70 10.56 2.98 4.48
CPSO, MCPSO Chaotic, modified chaotic PSO
FA Firefly algorithm
HFAMCPSO Hybrid FA with MCPSO
FACO Fuzzy ant colony optimization
SAACO Self-adaptive ant colony
optimization
RCD Radial cell decomposition
Disclosure statement
No potential conflict of interest was reported by the author(s).
Funding
Funding for this project was provided by the Malaysia Ministry
of Higher Education through the Fundamental Research Grant
Scheme with Project Code: FRGS/1/2021/TK0/USM/02/18.
Declaration of competing interest
The authors declare that they have no known compet-
ing financial interests or personal relationships that could
have appeared to influence the work reported in this
paper.
Authors’ contributions
Data curation, H.T.N.; Design, H.T.N; Formal analysis,
H.T.N. and N.S.A; Funding acquisition, H.T.N and N.S.A.;
Investigation, H.T.N and A.S.A; Methodology, H.T.N and
A.S.A; Project administration, N.S.A.; Software, H.T.N and
A.S.A; Supervision, N.S.A. and A.S.A; Validation, N.S.A. and
A.S.A.; Writing – original draft, H.T.N; Writing – review &
editing, N.S.A.
Data availability statement
Derived data supporting the findings of this study are available
from the corresponding author, N.S.A, upon reasonable request
following the publication of the paper.
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