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Table 2. An illustration of the number of nodes, distance, and time spent traversing with the map to each waypoint
Model Nodes Distance Time spent ( )
Zhang’s model before node reduction 242 271.1 2.25
Zhang’s model after node reduction 24 253.4 0.66
Proposed model 38 277.7 0.40

Figure 10. Illustration of the path created from the other models [43] . (A) It depicts the path created by Zhang et al.’s model by the green lines
(redrawn by Zhang et al., 2021 [43] ). (B) It represents the proposed method point order and traversed path. The point order is illustrated by
the violet arrows, while the orange path represents the robot path. Safety-aware roads are depicted by the blue dashed lines. The waypoints
are illustrated by the violet circles.

models. The Simulated Annealing (SA) algorithm, Grey Wolf Optimization (GWO) algorithm, Ant Colony
Optimization(ACO)algorithm, GeneticAlgorithms(GA),ImperialistCompetitiveAlgorithm(ICA),andSelf-
Organizing Maps (SOM) were chosen as the heuristic-based algorithms used in the comparison studies. The
ICAalgorithmisabiologicallyinspiredalgorithmbythehuman, whichsimulatesthesocial-politicalprocessof
imperialism and imperialistic competition. The SOM algorithm is similar to a typical artificial neural network
algorithm, except it utilizes a competitive learning process instead of backpropagation that utilizes gradient
descent.

Heuristic-based algorithms have similar attributes; due to this feature, the same parameters can be used to
construct a stable comparison study for our proposed IPSO algorithm. The conducted comparison studies
focus on six key attributes such as: min length ( ), average length ( ), length standard deviation ( ), min
time( ), averagetime( ), andtimestandarddeviation( ). Thevariancebetweeneachalgorithmcanbeseenby
assessing each parameter. The above analyses show how effective the IPSO model can generate the minimum
overall global trajectory in Table 1. The global trajectory generated by the compared algorithms is notably
larger than the IPSO model. However, regarding the time aspect, the IPSO model was unable to achieve the
shortesttime. ThesignificanceoftheproposedmodelcanbeseenintheSTDevaluationparameter. Theresults
of the comparison studies more than show the validity and performance of the proposed model to discover
the optimal waypoint visiting sequence.

6.2. Model comparison studies
The compared models were developed to address the issues of multi-waypoint navigation and mapping in
various applications. Each model uses some variation of a global navigation system in combination with an
obstacle avoidance technique. The models were selected based on their map configuration and overall effi-
ciency in solving the multi-waypoint navigation problem. Our comparison studies analyze the number of
nodes, the trajectories produced, and the total time to fulfill the fastest route.