Innovative Solutions for IK:
PROA and Clonal Selection Algorithms Unveiled
AMEL SERAT
Process Engineering Department,
University of Mustapha Stambouli Mascara – UMSM,
El Mamounia Mascara,
ALGERIA
Abstract: - Calculating joint angles for sequential manipulators consists of studying the correlation between
Cartesian and joint variables. The problem-solving technique encounters two main hurdles described as direct
and inverse kinematics. Matrix multiplications usually simplify the direct kinematic problem. However, inverse
kinematic problems are harder as they require solving many nonlinear equations and eliminating variables a lot.
In our work, we introduce two new methods of handling the complicated inverse kinematic problem for robotic
manipulator arms; Poor and Rich Optimization Algorithm and Clonal Selection Algorithm (CSA). These
advanced techniques enhance greatly the estimation of various joints in the arm which makes the solution more
precise and efficient. To demonstrate the effectiveness, robustness, and potential benefits of these approaches
for complicated kinematic problems we present extensive simulation results thereby enabling better
performance of robots.
Key-Words: - Articular angles, Poor and Rich Optimization Algorithm, CSA, manipulator, direct kinematic,
inverse kinematic.
Received: March 5, 2024. Revised: August 29, 2024. Accepted: October 4, 2024. Published: November 5, 2024.
1 Introduction
Robotic arm kinematics can estimate the angles and
positions of a robot arm’s joints for reaching a
specific end-effector position. Inverse kinematics
commonly solves this, [1].
Within different industries inverse kinematic
process has its’ own specifications and challenges,
[2]. In manufacturing automation robot arms are
used for tasks such as welding, packaging,
assembling, or painting.
Their exact articulation ensures precise results
and efficient pick-and-place operations. In medical
rehabilitation, robots are used to assist patients in
physical therapy by tracking movements with great
accuracy; whereas during delicate surgeries where
safety matters most of all, such machines should be
able to estimate each move they make. Among other
service robots that working great when being
articulated accurately there can be mentioned those
designed for disabled persons'care or housekeeping
needs (for example – vacuum cleaners), [3]. For
instance, the ability to estimate articulation
accurately can be used in robotic learning or HRI
research and development since it enhances robots ‘
adaptability and safety during interaction with
humans, [3].
Autonomous robots in agriculture are expected
to manipulate things and explore territories correctly
while maintaining plants that harvest and monitor
crops. In warehousing and logistics robots assist in
sorting packages automatically picking them from
storage places and locating them in designated areas
hence the need for accurate movement estimation
during their operations to enhance productivity, [4].
The entertainment industry as well as media houses
use robotic arms for creating animated characters
that seem humanlike and stage crafts that are
attractive to clients. Finally, in space science, these
machines serve purposes such as maintaining
repairing carrying experiments out on space craft or
even rovers stationed at different planets within our
solar system thus should be articulated with
precision to function well under extreme conditions,
[5].
Depending on the complexity of the robotic
system and the specific application, inverse
kinematics (IK) may be difficult. Inverse kinematics
is challenging due to a variety of factors, [6], [7],
[8], [9]. Solving nonlinear equations which can be
complex and require a lot of computational power is
common in mathematics for IK problems.
Additionally, singularities may cause the robot to
lose degrees of freedom hence making some
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equations unsolvable or infinitely solvable and there
are multiple potential solutions for various
configurations of robots which makes it hard to
select the best one. Robot configuration also
matters; IK becomes complicated in systems with
numerous kinematic chains or several degrees of
freedom (DOF) like humanoid robots, [8], [9].
For instance, IK problems concerning real-time
operations like robotic surgery and interactive
robots are challenging because they require quick
and accurate computation solutions. Moreover, such
problems become more complex when physical
limitations, for example, joint restrictions or
collision avoidance come into play. Numerical
methods offer flexible ways of dealing with these
issues which include the use of the Jacobian inverse
but they may fail due to high computational
costliness caused by their flexibility or lack thereof
as well as non-convergence issues related to
gradient descent while analytical methods provide
exact answers but are often limited to simpler
configurations, [8], [9].
Despite these challenges, advances in methods
and tools (e.g. machine learning strategies,
optimization techniques, or Jacobian inverse), have
significantly improved the controllability of IK for a
diversity of robotic applications. While IK has
challenges in computational, mathematical, and
physical sense there have been a lot of
improvements over time that have improved its
applicability and performance for different
applications, [8], [9].
The Table 1 enumerates developments in the
use of neural networks and evolutionary algorithms
for inverse kinematics solving as of late. [10],
proposed a hybrid strategy that fuses genetic
algorithms (GA) and neural networks. This is
illustrated in the following table highlights new
works based on neural networks and evolutionary
algorithms applied to the IK problem. [10],
suggested the use of GA along with neural networks
but the approach was a mixed strategy. This method
minimizes the likelihood of end-effector faults with
very small accuracy by creating an initial population
for the GA by three off-Elman neural networks. The
studies were performed on a six-axis serial robot
platform.
Another method named semantic niching
technique that is adaptive in its nature was proposed
in [11] and has employed a local search-based
quasi-Newton algorithm in combination with
niching genetic algorithm. It added significantly to
the findings of the simulation, whatever the
particular system under examination had not been
stated yet. The self-adaptive mutation rate in the
genetic algorithm was proposed in [12], along with
combining sequential mutation genetic algorithm
with extreme learning machine. The ELM was used
first to calculate the first IK solution and then GA
with its basic steps WHERE used for further
optimization. Stanford MT-ARM robotic
manipulator with six degrees of freedom (DOFs)
was used to implement and test this approach with
improved performance in terms of increased speed
of processing and the ability to give as many and as
accurate Inverse Kinematics solutions as needed.
Table 1. Overview of Research on IK Solutions for
Robotic Manipulators
Author
Approach
Key
Techniques
Results
[10]
A
hybridapproach
using neural
networks and
GA
Three Elman
neural
networks for
the initial
population
Achieved
high precision
in end-
effector error
minimization
[11]
Adaptive
niching
strategy
Niching
genetic
algorithm,
quasi-
Newton
algorithm
Improved
precision and
resolution of
simulation
results
[12]
Sequential
mutation
genetic
algorithm
combined with
extreme-
learning
machine
Extreme-
learning
machine for
preliminary
IK solution,
simple GA
for
optimization
Improved
computational
time without
reducing the
accuracy of
IK solutions
[13]
Continuous
genetic
algorithm
Continuous
GA
operators for
initialization,
crossover,
mutation
Smoothened
joint space
while
maintaining
the accurate
Cartesian
path
[13], employed a continuous genetic algorithm
and used the CGA operators for initializing,
selection, crossover, and mutation. In their
experiment, they applied their method on a 3R
planar manipulator and proved that their approach
was advantageous at the Least Squared cost function
since it smoothed the joint space and kept the
Cartesian route accurate without losing the
Cartesian path whereas proved continuous GA as
effective to generate exact IK solutions.
The so-called inverse kinematics (IK) problem
has to be solved in the field of robotics in order to
enable the precise and adaptable motion of robotic
arms and manipulators. IK issues present
considerable challenges because they often involve
complex and non-linear relationships, and there are
multiple solution possibilities with varying
optimalities. Such problems characteristically
possess vast solution spaces that are irregular and
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complex for classical techniques to manage. On the
other hand, new approaches for solving these
problems appeared with the main group of newly
emerged optimization algorithms as the “Poor and
Rich Optimization Algorithm” PROA) [14], [15],
[16], [17] and the “Clonal Selection Algorithm”
(CSA) [18], [19], [20], [21], [22], [23].
Because of its two-methodology method [14],
the application of the PROA on the IK is also
particularly appealing because it resembles the
dynamic requirements of IK solutions for robotics.
Taking the use of the social terms “rich” and “poor”
groups, PROA provides a clear divide between more
harm-oriented means of optimization [14], [15],
[16], [17], and more beneficially-oriented ones. The
population is basically split into two subpopulations
by this algorithm: In terms of local exploitation, it is
referred to as the “rich,” which exploits the potential
places utilizing the local search, and the “poor,”
which tries to search an area within the entire
solution space by using the global search. Like the
discussed tiered approach that uses different forms
of the memory system to ensure there is progressive
learning or development while also ensuring that
there is a fast search through the complex solution
spaces that are characteristic of most robotic joint
configurations. This functionality is rather beneficial
for real-time iterative apps where the speed of
convergence and performance increment of even a
few percent means a lot, [14], [15],[16], [17].
Likewise, the CSA [18], [19], [20], [21], [22],
[23] offers a robust background for tackling the IK
challenge because it relies on immunological clonal
choice principles. This method works as a form of
an ‘antibody’ that is aimed at the afflicted ‘antigen’
of the target end effector site through mimicking the
Biológical evolution processes of the B cells that
affords affinity maturation and selection. Finally, as
with CSA, it starts with a wide range of initial
solutions and focuses on locating and making copies
of high-performing solutions and using
hypermutation, [19]. This selection and mutation
course is most acceptable and aligned to the
complicated situation of IK where there are several
joint configurations that could achieve the intended
goal with different levels of effectiveness, [18],
[19], [20], [21], [22], [23]. CSA is actually very
successful when robotic systems are required to
remix from one pre-established task to another as
CSA has the ability to learn and develop new
approaches, [22].
With improved applicability offered by using
PROA and CSA, scientists and engineers may be
able to use effective techniques towards enriching
the versatility and feasibility of robots in solving
problems that require kinematic inverse solutions.
These are cutting-edge approaches to robotic
interface that create way to advanced robotic
systems that are even more sensitive and adaptive to
the environment and at the same time create new
frontiers of robotic automation.
The rest of the paper is arranged as follows: The
rest of the paper is arranged as follows:
• Part 2: Robotic Arm, this section then goes deeper
into describing the PUMA 560 robotic arm
manipulator in detail its features and size, and lastly
stresses the need to find a proper solution to the IK
problem to enable accurate and adaptive control of
it.
• Part 3: Poor Optimization Algorithm (PROA), let
us discuss about that technique and implementation
of the Poor Optimization Algorithm. In this section,
the primary emphasis is made on the fact that PROA
has specific benefits and unique characteristics,
which deal with the enhancement of the quality of
the IK solution through the application of the
developed approach based on the use of complex
arrays of stimuli.
• Part 4: that part considers more in detail the Clonal
Selection Algorithm, exploring the ideas and
solutions behind it. This paper explores the
application of methodologies borrowed from the
field of immunology to solve the IK problem in the
case of CSA with the objective of establishing the
applicability and effectiveness of the concept.
• Part 5: PROA and CSA simulation experiment on
the PUMA 560 robotic arm. In this part, the
simulation outcomes acquired using the PROA and
CSA models are described. We compare the
accuracy of both algorithms and then give an
analysis of the findings which have further
implications for the field of robotics.
• Part 6: Recommendation, the last segment of a
report provides a summary of critical observations
that may have been derived from the inquiry.
Finally, it accredits the contributions of the study to
the advanced optimization techniques in robotics
and suggests direction for further research in
enhancing the prospect and applicability of these
methodologies.
2 PUMA 560 Arm
Robotics is an organized method of command and
execution to enforce the desired task by combining
mechanical, electrical, and computational
technology. It begins with the initial stage of
sensation of the environment followed by an
analysis of the sensations by specific algorithms to
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produce a set of instructions for the movement of
the motor.
These are instructions issued to guide and
instruct mechanical parts into accomplishing work
that has been anticipated, and as such you look at
the mechanical components to ensure that they heed
the instructions given to them.
However, the scope of the system is within the
reach of the robot’s manipulator, which is usually
mounted on a rigid element, for example, the floor
or ceiling, and consists of several links and joints;
the working area is sometimes called the workspace.
The end-effector denotes the tool mounted on the
terminal link of the manipulator, which performs the
specific task. End-effectors are tools and products
such as scalpels, graspers, and needles that are used
during surgery.
Fig. 1: PUMA 560 arm
In a PUMA560 type robot, six rotational joints
are represented by variable Q1 to Q6 as depicted in
Figure 1, and in order to make the system work, one
has to find these angles q1, q2...q6 for the robotic
arm ("end-effector") relative to the End-Effector
point M(x,y,z) in a Cartesian plane and they can be
inserted into the manipulator equation, thus the
equations obtained are finished. After the joint
angles were determined, they made the output of the
joint parameters Q which are represented by the
vector Q (Q1, Q2, Q3,..., Q6).
The process of finding the joint parameters,
which are represented by the vector Q (Q1, Q2,
Q3,..., Q6) is as follows:
Enter the coordinate of the M point in Cartesian
space
Apply an optimization method f that minimizes
norm(f(Q) – M) = 0.
3 Poor and Rich Optimization
Algorithm
The Poor and Rich Optimization Algorithm (PROA)
which is a multi-population optimization technique
based on socioeconomic concepts relies on [14],
[15], [16], [17] references. It has implementation of
a stratification-based algorithm that divides the
population into rich and poor classes. The rich
group includes subjects with higher fitness values
and the poor one comprises those with lower fitness
values. PROA modifies solutions iteratively using
different tactics for each group. For affluent
subpopulations, intensification prevails as its
guiding principle, the algorithm gives preference to
exploitation by carrying out small changes in
solutions to gain better ones locally, [15].
Conversely, the underprivileged subpopulation
approach focuses on diversity; making big
changes/manipulations to explore new regions in the
search space, [15].
PROA picks a method that uses the average of
the top, middle, and bottom picks from the rich
group, [16]. This approach guides the less rich
group toward better choices. By mixing these ways,
PROA keeps a good balance between trying new
things and sticking with what works, [16]. It also
adds steps for changing, combining, and ordering
groups to keep variety and make sure of ending up
with the best answers. PROA's way of doing things
from many angles helps it solve tough problems
well, [16], [17].
Fig. 2: PROA flowchart
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This is just a simple explanation [14], [15], [16],
[17] of PROA in the form displayed by Figure 2,
[17]:
Start by employing random candidates to
form a deprived early population.
By combining those found through
searching the neighborhoods and those
found randomly you create a starting
'middle class'.
Assess each candidate solution in terms of
its appropriateness for both rich and poor
communities.
Let the number of candidates to be moved
from rich to poor be determined by a
method that is based on probability.
According to the number of iterations that
were completed, re-distribute the ratio of
rich and poor people at different points in
time.
Diversify the search process by modifying
random solutions in both groups.
Continue steps 3 to 8 until a stop criterion is
met (e.g., maximum number of iterations or
finding a feasible solution).
4 Clonal Selection Algorithm
Artificial Immune Systems (AIS) fall in the
category of computer simulations of the immune
system which help immunologists do
difficult/impractical research that is not possible to
solve using other approaches, predict the future, and
run simulations and trial runs. Another name for this
topic is computational immunology, [18], [19], [20],
[21], [22], [23]
This branch is growing very fast at present and
its goal is to develop computer models that simulate
the mechanisms of mammalian immunity. These
systems are focused on the ability of a body to
recognize foreign substances, and antigens, and
destroy cancer cells without damaging normal
human cells, [18], [19], [20], [21], [22], [23].
Immune Network, Clonal Selection, and
Negative Selection are among the important
algorithms in this area, [18].
The comprehension of immunological responses
to antigens is based on the principle of clonal
selection. It conveys the idea that immune cells will
proliferate and be selected (recruited) only if they
can recognize and bind to antigens, instead of
immune cells that do not associate with antigens,
[18].
Based on the above, it can be summarized thus:
• cloning immune cells which are likely to
mutate with a possibility of mutations, or
somatic hypermutation, [18], [19], [20], [21],
[22], [23].
• The elimination of newly generated
lymphocytes with self-reactive receptors and,
development and maturation of naive cells
that respond to antigens.
Our algorithm extends the Clonal Selection
paradigm when it is made clear that only antibodies
with the highest affinity for antigens are picked to
proliferate. In effect, our approach combines the
principles of clonal selection and function
approximation. Below is the algorithm [23],
together with the flowchart shown in Figure 3:
1. The initial settings
Establish Base Population: Generate random
population within given limits.
2. Main Loop: Perform operations until
reaching the point where the stop is
required.
• Calculate Affinity: Get the affinity of
each individual in the population.
• Choose Top Individuals: Select the best
individuals depending on affinity.
• Create People for Cloning Based on
Clone Rate: Some people were selected
for cloning using the clone rate
criterion.
• New Affinities From Hypermutation Of
Clones Are Produced: The clones have
new affinities through hypermutation.
• Updatting population: this is made by
selecting the best individuals from
(Original And Cloned Populations) to
keep up with population change
• Introduce Randomness (to Ensure
Genetic Diversity): Form and pick out
random new people to maintain genetic
diversity.
• Go to the next iteration by incrementing
the loop counter.
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Fig. 3: Clonal selection flowchart
5 Experimentation and Results
A Pentium 4 with 1.8 GHz CPU was used to
simulate the PUMA560 robotic arm on the machine.
The system configuration consists of one 40 GB
hard drive and 1 GB RAM, with Matlab 7 as the
simulation environment.
The Poor and Rich Optimization Algorithm
(PROA) and the Clonal Selection Algorithm (CSA),
as well as the application of these methods, have
resulted in great improvement in the performance
and flexibility of robotic arms and manipulators to
cope with the difficulties of inverse kinematics (IK).
Here, we will show the results of applying the two
optimization methods to tackle the nonlinear and
complex characteristics of IK issues and a
comparison with the Wavelet Network method's
2006 findings, [24].
To improve the performance and reflexivity of
robotic arms and manipulators the poor and rich
optimization algorithm (PROA) and the clonal
selection algorithm (CSA) are the biggest
breakthroughs that have been seen in the aforesaid
problems of IK difficulties. This section is dedicated
to demonstrating the application of these two novel
methods of optimization in solving the nonlinear
and intricate nature of IK problems as well as to
discuss the insights from our own implementation
experiences and the comparison as well as the
Wavelet Network method's 2006 findings, [24].
The Table 2 gives the values of the mean square
error (MSE) by the wavelet network method at the
six different configurations (Q1 to Q6). The errors
scaled down by a factor of 10-3, signify that Q3
(0.029) is the position with the least error, and that
Q1 (0.276) is the one with the biggest error.
Table 2. MSE with Wavelet Networks
Angles
Error (*10-3)
Q1
0.276
Q2
0.144
Q3
0.029
Q4
0.259
Q5
0.198
Q6
0.151
This variability in error reflects the
effectiveness and accuracy level of the Wavelet
Network method in solving inverse kinematics
problems for different configurations.
The Artificial Immune System (AIS) algorithm's
parameters are listed in the Table 2. The number of
generations was set to 400, with a mutation
probability of 0.001, and the parameter β was set to
0.1. These characteristics were crucial in
determining the AIS's behavior and performance
during the optimization phase as shown in Table 3.
Another tuned parameter is the Mutation
probability, which was set to 0.001, with a mutation
rate of 5% for each gene and the β was set to 0.3.
These characteristics were the ones that significantly
influenced the behavior and the performance of the
random elements of the AIS at the optimization
phase.
Table 3. CSA parameters
parameter
Values
Generation Number
400
Mutation Probability
0.001
β
0.1 and 0.3
Table 4 is a detailed, vivid, and informative
table presenting the Mean Square Error (MSE)
values of six different robotic joint angles (the first
through the sixth) after the Clonal Selection
Algorithm (CSA) has been applied to them. The
values are presented as coefficients of 10-3 and thus
indicate great deviation between the performance of
the different links or variances across joint angles.
One way to interpret it is: that a figure of 0.002
indicates a negligible error, therefore, the
corresponding joint Q6 assembly is practically
error-free. On the flip side, the highest error
produced at joint Q2 was 0.4, which is enough to
show that the method might benefit from more
development.
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Table 4. MSE with CSA
Angles
Error (*10-3)
Q1
0.05
Q2
0.4
Q3
0.009
Q4
0.02
Q5
0.04
Q6
0.002
As a whole, these researches show that the CSA
is a powerful and flexible approach in dealing with
the inverse kinematics problems, which are the most
challenging part of the activity on the robotic
system.
The data not only verifies the CSA technique's
reliability but also brings to the fore some of the
details underpinning its operational subtleties and
the possibilities for optimization in real-world
robotic applications.
Table 5. MSE With PROA
Angles
Error (*10-3)
Q1
0.03
Q2
0.054
Q3
0.00015
Q4
0.014
Q5
0.098
Q6
0.003
A summary is given in the Table 5 of the Mean
Square Error (MSE) values obtained using the New
and Better Optimization Algorithm (NOA) for the
six different robotic joint angles (Q1 to Q6). These
figures, which are scaled ata factor of 10-3, show
algorithm's ability to attain accurate performance in
the different establishments. In the most error-free
case, Q3 is displaying a tiny 0.00015, thus
underlining the flawless performance of the
algorithm in a certain environment. On the other
hand, Q5 exhibits the biggest error of 0.098 which
indicates that the robot can be improved its
performance by some tuning of the algorithm. From
our side, the upcoming part explores the robustness
of the PROA method. It also discusses the possible
pitfalls of this method if it is not correctly used for
accurate control and adaptation in difficult robotic
kinematic situations.
A comprehensive comparison of three
optimization strategies: PROA, CSA, and Wavelet
Networks employed in robotic arm joint rotations
(Q1 to Q6) provides an uneven playing field of
algorithmic effectiveness.
Let me give you an analysis here:
• Q1: The lowest error (0.03*10-3) of PROA
exhibits justification of PROA as best among
the rest and thus more accurate. This earned
PROA first place among the accuracy
requirements that are as difficult as the first
example.
• Q2: PROA made a significant success in Q2
with a huge descent in MSE (0.054*10-3) while
CSA had a bigger error (0.4*10-3). As such,
PROA proves to be a useful tool for faster
optimization under Q2 conditions.
• Q3: PROA is more effective in comparison to
both CSA and Wavelet Networks with an
almost zero margin of error (0.00015*10-3),
thereby expressing excellent accuracy and
sensitivity in the precision calibration settings
on the robot.
• Q4: PROA has the MSE reduction factor which
is the greatest, illustrating the strong capacity of
the PROA even with the most intricate and
convoluted situations, and as a result, the
greatest error reduction to 0.014*10-3 in all joint
configurations.
• Q5: The two algorithms have the same error
rates -0.04*10-3 and PROA (0.098*10-3) had
errors, too. These algorithms have lower errors
than the Wavelet Networks, whereas CSA has
the lowest MSE, showing that it is more
promising in joint Q5.
• Q6: Both CSA (0.002 *10-3) and PROA
(0.003*10-3) exhibit minimum mistakes and
high efficiency isthe most efficient, with CSA
in particular coming out ahead of PROA in this
case.
What can be gathered, PROA consistently
betters the rivalries in the main but on the other
hand, it is extremely low in MSE and the best in
precision and optimization. This is particularly
evident in scenarios that need high levels of
accuracy, such as Q1, Q2, Q3, and Q4. CSA is also
an excellent support of the specific arrangements
namely Q5 and Q6, where it is slightly ahead of
PROA, to indicate its potential for some distinct
purposes. This research proves the significance of
picking the algorithms according to the operational
requirements of the robotic system to have
exceptional performance and be adaptable.
6 Conclusion
PROA, CSA, and Wavelet Networks, are the three
optimization algorithms used in this study. For
solving the inverse kinematics of the robot arm, we
are using a PUMA 560 robotic arm manipulator.
This study was primarily concerned with decreasing
the Mean Square Error over the whole joint (Q1–
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Q6) set upto identify the method that offers
precision and the least use of energy.
In most cases, it is notable that the use of the
Poor and Rich Optimization Algorithm (PROA) is
more favorable to the other methods, synchronically
in Q1, Q2, Q3, and Q4. This is the true test
according to the competitiveness of PROA in the set
of movements that are complex and fluctuating, the
realization of PROA as the stand-out choice for the
applications requiring high precision and excellent
performance is clear.
The Clonal Selection Algorithm (CSA), though
it has some restrictions in Q2, is very effective in
Q5 and Q6. Its outstanding performance in the
creation of these specific joints shows its ability to
shape a variable tool that would adapt and respond
dynamically and thus can be used purposefully
wherever flexibility and specificity are paramount.
Fortunately, Wavelet Networks, among other
algorithms that have become obsolete, still provide
some foundational reasoning when studying the
evolution and development of the algorithmic
strategy over the years. The obvious performance
lag hints at the remarkable developments in the
domain of robotics inverse kinematics.
Therefore, the main stress of the article lies in
the role of the algorithm in the robotic system
design and implementation.
This study's results are a piece of helping for
knowing the best driving digital systems for specific
tasks, but they also contribute to robotics part by
increasing our understanding of the fact that
different optimization techniques can be used to
improve the adaptability and efficiency of the
robotic systems.
Progress made in the robotics field will often be
directed by the conclusions of this research since we
will be able to execute efforts to figure out why
some of the robots are capable of adapting to
changes faster and more accurately than others.
Acknowledgement:
We are grateful to all of the writers whose research
and contributions have been cited in this work.
Declaration of Generative AI and AI-assisted
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author, Amel Serat, was responsible for all
aspects of the research presented in this article,
including the formulation of the problem, the design
and execution of the simulation, data analysis, and
interpretation of results.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
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