Hybrid golden Jackal and moth flame optimization algorithm based coverage path planning in heterogeneous UAV networks

Abstract

Coverage Path Planning is an important strategy used mainly in junction with the unmanned aerial vehicles (UAVs) such that it can cover the regions of the target with minimized energy consumptions. This coverage path planning specifically with heterogeneous UAVs despite varying capabilities concentrate on the process of flight paths optimization when is employed to cover a target region. In this paper, a coverage path planning strategy using hybrid Golden Jackal and moth flame optimization algorithm (HGJMFOA) is proposed for estimating superior optimal paths that supports the UAVs towards the process of comprehensively covering the generated regions with efficacy. At the initial phase, the regions are generated randomly and the models of UAVs which is formulated using a linear programming model is included for identifying the best point-to-point flight path for each individual UAVs. Then it adopted the merits of HGJMFOA for exploring and exploiting the feasible paths from the source to the destination from which optimal shortest path can be determined with reduced path flight time. It handled the challenges involved during the process of achieving coverage path planning by establishing cooperation between diversified UAVs such that optimal coverage, covering sensor range and superior flight performance is achieved during the application. The simulation experiments of the proposed HGJMFOA approach confirmed minimized task completion time by 21.34%, deviation ratio by 24.56%. and execution time by 34.19%, with randomly generated regions organized for performance evaluation.

Introduction

The rapid advent in the field of artificial intelligent and autonomous control made unmanned aerial Vehicle (UAVs) more ideal and suitable for it to be deployed over diversified number of real-time applications that includes disaster management, object tracking, agricultural monitoring, and so on. These UAVs are designed to assist humans by taking over difficult or repetitive tasks. It is responsible for provisioning an adaptive and efficient way for achieving the activities of civilian and military¹. This UAVs also termed as drones have been widely adopted in most of the real time developments over the last decades, and determined to facilitate more success with the capability of achieving crucial objectives with their inherent merits of modularity cost, mobility, accessibility, and so on. The rapid advent of drones and its exponential increase in its usage motivated the manufacturers in integrating wide number of technologies such that human behaviour detection and Wi-Fi technology systems into the products². Thus, the UAVs in the recent days are widely used in the field of meteorology, search, and rescue, and so on. To accomplish the objectives of UAVs, autonomous decision-making process depending on the dynamic change in the requirements that happens with respect to the time, and area of implementation is essential³. This dynamic decision making can be the tasks related to path planning or selection of trajectory. But an individual UAV face huge challenge in accomplishing the required and essential complicated missions due to its limited storage, communication, and computing potentialities⁴.

In this context, systems with multiple UAVs cooperation have invited tremendous focus for coping up with complexity and increasing scale of real-world applications. These multi-UAV systems due to their potential merits of flexibility and adaptability has evolved into one of the most active hotspots of the research⁵. This multi-UAV systems can possess homogeneous and heterogeneous UAVs. In specific, heterogeneous UAVs generally operates as a group for the objective of improving the system performance and achieving minimized energy consumptions. This operation of heterogeneous UAVs completely depends on how they cooperate with one another and how information is exchanged between them during the process of data dissemination⁶. Heterogeneous UAVs is considered to possesses varying level of potentialities with respect to flight time, region of scanning and load. This significant characteristic of heterogeneous UAVs helps in transforming the search system with fault tolerance and cost effectiveness when they typically used in the large-scale search systems adopted for surveillance or searching. On the other hand, heterogeneous UAVs which possesses different devices are utilized for achieving the task of searching and tracking targets of danger in the military field. This heterogeneous UAVs can also be adopted in the field of agriculture when the crops are cultivated in the regions separated by mountains for the purpose of collecting information associated with the growth of plants and vegetation⁷. Even though heterogeneous UAVs could significantly enhance the degree of flexibility and minimize the cost in diversified domains of applications, they pose the challenge in the process of performing tasks related to optimization performance, reactive decision making and cooperative control and path planning. The path planning of UAVs is defined as the systematic process of identifying the possible and optimal trajectory between the source and destination without collisions. This path planning can be accomplished with efficiency and effectiveness only when the different requirements and limitations related to UAVs and its implementation environment is considered into account. UAVs during this coverage path planning need to handle both static and dynamic types of obstacles but handling these obstacles has maximized probability of resulting in huge physical damage⁸. Thus, path planning needs to be achieved by preventing threats and obstacles, and this is considered as the most important requirements during its process of implementation. This challenge is more crucial to achieve in the systems which especially uses information related to different regions, and coverage path planning associated with heterogeneous UAVs which is responsible for determining the optimal paths between the start and destination scan points. This coverage path planning which needs to scan all regions need to satisfy a large scale of constraints which is even more challenging⁹.

The properties of multi-variable optimization inherently involved during the objective of path planning process is identified to possesses NP-Hard computational complexity. Moreover, the difficulty in adopting heterogeneity components and making them to cooperate with flexibility and adaptability during its implementation in a real time application environment also introduces challenges in establishing the constraints of security and time^10–12. At this juncture, intelligent and efficient heuristic and meta-heuristic optimization algorithms are identified to offer better solutions for the problem of coverage path planning which is determined to possess NP-Hard complexity^13,14. The meta-heuristic optimization algorithms-based coverage path planning is determined to be potent in achieving robustness and efficiency¹⁵. In specific, swarm intelligent meta-heuristic optimization algorithms-based coverage path planning approaches is confirmed to be significant in solving the problem of UAV path planning. These swarm intelligent meta-heuristic optimization algorithms-based coverage path planning are mostly adopted since it facilitates better path quality, and high convergence capability. The system developers started utilizing the support of decision-making tools with respect to the increase in the number of separated regions and heterogeneous UAVs for the process of establishing point-to-point path for each UAVs in a more reasonable manner. Moreover, this decision tools-based path planning helps in accomplishing the objectives through the adoption of highly reconfigurable systems that possesses more intelligence during the process of their inclusion.

A diversified number of bio-inspired approaches of path planning that adopted the merits of ant colony optimization (ACO), Artificial Bee Colony Optimization (ABCO), particle swarm optimization (PSO)¹⁶, Cuckoo search optimization (CSO), Green Anaconda Optimization, and so on are guaranteed to facilitate optimal performance depending on the various requirements that need to be satisfied in the applications over which it is deployed¹⁷. These bio-inspired approaches of path planning that includes evolutionary process is responsible for determining the efficiency related to the solutions of path planning. This path planning process comprises of three phases, such as fitness evaluation, selection, and offspring generation phase. In the phase of fitness evaluation, the constraints related to mission objectives, tasks, vehicle capabilities and dynamic environment are imposed for evaluating the fitness value of the generated possible paths. The second phase of selection identifies the best fitness paths from the possible paths identified during the process of fitness evaluation¹⁸. Finally, the optimized path or trajectory of UAVs are identified for establishing reliable route among the two scan points which need to be covered by the UAVs in the network. In addition, a predominant number of the existing bio-inspired approaches of path planning to the greater extent ignored the collaboration tendencies of UAV swarms and considered only geometrical viewpoint during the implementation process¹⁹. However, the bio-inspired approaches adopted for determining shortest paths suffers from the problem of imbalanced degree that exists between the rate of exploration and exploitation, premature convergence, and poor solution diversity. In this situation, hybrid swarm intelligent meta-heuristic optimization algorithms that provide better trade-off between exploration and exploitation, prevents premature convergence, and confirms maximized solution diversity becomes essential²⁰. Among the hybrid swarm intelligent meta-heuristic optimization algorithms, Golden Jackal Optimization Algorithm which possesses global optimisation capability when integrated with Moth Flame Optimization Algorithm (HSOA) that confirms local optimization potential is determined to satisfy the above-mentioned requirements. In specific, the classical Golden Jackal Optimization Algorithm (GJOA)²¹ was proposed based on the inspiration derived from the group behaviour and cooperation strategy exhibited by the golden jackals in the nature. The major merits of this traditional GJOA algorithm includes, (i) limited number of adjustment parameters, (ii) high stability and (iii) simple implementation. These merits of GJOA made it highly suitable such that numerous researchers and experts used them for solving the problem of real-time applications. It was also used in different applications related to different engineering domains as the single metaheuristic algorithm responsible for determining near optimal solutions in the search space. But as the no free lunch theorem states that “There is no single metaheuristic algorithm which can perform well in the process of solving optimization problems of large-scale”. Even though, GJOA algorithm conformed better competency on par with the existing metaheuristic algorithms of the literature, it still possesses a room of improvement in enhancing its potential in terms of global search attainment and convergence. However, this traditional GJOA algorithm possesses some limitations such as low accuracy of calculation, slow speed of convergence and premature convergence. Hence, these limitations of GJOA need to be addressed by the integration of a metaheuristic algorithm which systematically enhances the rate of convergence and prevents premature convergence with maximized accuracy.

On the other hand, Moth Flame Optimization Algorithm (MFOA) was proposed based on the inspiration derived from the navigation behaviour of moths in nature²². In specific, the moths perform a spiral search pattern for selecting and searching around the lights. This classical MFOA algorithm assigned a flame for each moth and, in each iteration the flames’ position is updated. This characteristics of MFOA helped in enhancing the degree of exploration in the search space such that the risk of getting struck into the local point of optimality is completely prevented. It included an adaptive flame quantity approach for balancing the rate of exploration and exploitation in the search space. The best solution identified in each iteration is used for updating the flame sequence such that the moths could reposition themselves depending on the updated flame. This process of updating the flame sequence based on the best solution helps in enhancing the rate of exploration in the search space. Thus, this MFOA is considered to exhibit potential global optimization tendency such that it could be significantly applied to different number of practical applications. To handle these limitations, GJOA and MFOA are integrated using the merits of opposition learning into HGJMFOA algorithm such that well balanced exploration and exploitation rate is achieved in the search space during its application.

In this paper, a Coverage Path Planning Strategy using Hybrid Golden Jackal and Moth Flame Optimization Algorithm (HGJMFOA) is proposed for estimating superior optimal paths that supports the UAVs towards the process of comprehensively covering the generated regions with efficacy. This path planning approach is implemented by considering different scan and flight capabilities. It is also developed as a reliable approach that targeted in the process of UAVs path optimization and region assignment by preventing the problems related to equipment failure, obstacle avoidance and collision.

Major contributions

The proposed HGJMFOA approach is proposed by integrating the merits of GJOA and MFOA such that well-balanced exploration and exploitation rate in the search space is guaranteed during the process of path planning to optimally cover the generated regions. This HGJMFOA approach conformed better coverage of path plan sequentially in a shortest time. It is proposed with the significance of determining best flight paths for UAVs and support in the process of designing multiple regions’ coverage systems. It is contributed as a superior optimal solution to the linear programming model which is formulated for achieving path planning with required constraints which need to be satisfied during its achievement. It is also proposed as a reliable path planning model which considered the impact of distances, and regions’ scan areas into account, for facilitating better global searching capability in the search space.

The remaining section of the paper is organized as follows. Section 2 presents the comprehensive review of the computational intelligent metaheuristic optimization algorithms-based coverage path planning approaches existing is the literature with the pros and cons. Section 3 depicts the detailed view of the proposed HGJMFOA-based optimal path planning mechanism that assists in maximized coverage of UAVs with the primitive of GJOA and HSOA, and its integration which is used for achieving the required objectives. Section 4 demonstrates the simulation results and inferences related to the proposed HGJMFOA compared to the benchmarked path planning approaches with different evaluation metrics required for investigation. Section 5 concludes the paper with major contributions and future scope of enhancement.

Related work

In this section, the computational intelligent coverage path planning metaheuristic optimization algorithms available in the literature is reviewed and the determined pros and cons is presented as follows. The classification of UAV Path Planning Literature is structured into four primary categories: Algorithm Type, UAV Type, Path Planning Objectives, and 2D vs. 3D Path Planning, each of which represents a crucial aspect of UAV navigation and optimization.

The Algorithm Type category classifies research works based on the computational techniques used for UAV path planning. Evolutionary algorithms, such as Genetic Algorithms²³, rely on natural selection mechanisms to iteratively improve flight paths. Hybrid Metaheuristics combine multiple optimization methods, like the integration of chaotic Aquila optimization with Simulated Annealing ²⁴ and Harris Hawk Optimization²⁵, to achieve robust and efficient solutions. Heuristic-Based algorithms ²⁶ use domain-specific rules and approximations, often improving computational efficiency in real-time UAV navigation. Next category is UAV Type, UAV path planning strategies vary depending on whether a Single UAV or Multiple UAVs are involved. Studies focusing on single UAVs ^23,27 optimize the flight path for an isolated drone, often addressing challenges like obstacle avoidance and energy efficiency. Multiple UAV path planning ^28,29 deals with coordination, communication, and collision-free trajectories among multiple drones operating in a shared environment, which is particularly relevant for surveillance, mapping, and delivery applications.

Third category is for Path Planning Objectives, UAVs operate under diverse mission requirements, leading to distinct planning objectives. Shortest Path Optimization^23,30 minimizes travel distance and energy consumption to improve efficiency. Collision Avoidance algorithms^31,26 focus on ensuring safe navigation through complex environments by detecting and avoiding obstacles dynamically. Threat-Aware Planning^25,29 incorporates risk factors such as hostile zones or weather disturbances to generate secure UAV paths, often leveraging adaptive algorithms to re-route drones when necessary. Fourth category is based on 2D vs. 3D Path Planning: UAV path planning can be categorized based on whether navigation is constrained to a 2D plane or involves full 3D movement. 2D path planning^27,32 simplifies computations by focusing on horizontal movements, commonly applied in low-altitude urban environments. In contrast, 3D path planning^28,33 considers altitude variations, enabling UAVs to optimize trajectories in complex airspace, avoiding dynamic obstacles and terrain variations. The detailed review for each paper is detailed further.

Liu et al.²³ contributed a reactive path planning approach that derived the merits of evolutionary algorithm (EADPPM) which addressed the uncertainty issue which is more common during autonomous track planning process. This EADPPM is proposed as a dynamic path planning approach that considered minimized amount of prior knowledge for complex environments. It incorporated the characteristic phenomenon of t-distribution to determined optimal paths that assists multiple UAVs in the randomly generated regions. It addressed the problem of low search efficiency and high computational complexity during the process of determining optimal paths between the source and destination paths in the implementation environment. It was proposed with the capability of dynamically planning the appropriate track with strong degree of robustness properties. It was also proposed as an autonomous path planning approach that mitigated the problem of slow speed during the path calculation and inability of dynamically planning the track which is a major problem in most of the baseline coverage path planning solutions. Flores-Caballero et al.²⁷ proposed a Meta-Heuristic A-Star Method (MHASM) for computing the path such that the path length is minimized by preventing collusions in the implementation environment. This metaheuristic approach determined the firsts set of candidate solutions that suitably adjusted the control paths to identify the path between the source and destination scan points associated with a region. It was proposed as a reliable solution for constrained global optimization problem which considered a single objective function that considered the overall length of the path into account. It considered the impact of obstacles’ collision such that the path shape assignment can be streamlined during the planning process. It was developed as a constrained addressing approach that found out suitable optimized paths by satisfying the properties of non-differentiability, multimodality, and high non-linearity. The comparative investigation of this MHASM method confirmed superior path determination time under dynamic change in generated regions considered for evaluation.

Ait-Saadi et al.²⁴ contributed a Hybrid Simulated Annealing and Chaotic Aquilla Optimization Algorithm (HSACAOA) for facilitating optimal path planning to ensure Collison-free, smooth and short path between the two points considered for flight path determination. It specifically adopted the characteristics of simulated annealing for establishing better balance between the exploration and exploitation rate. It was tested with three different chaotic maps under different number of threats that may crumble the process of determining the path planning process between the source and destination points. The results of this HSACAOA-based coverage path planning mechanism ensured minimized path cost and fitness value on par with the baseline approaches used for evaluation. Chen et al.³¹ proposed an Ant Colony Optimization Algorithm-based Coverage Path Planning Mechanism (ACOACPPM) for determining superior and optimal paths to the UAVs for the purpose for fully enveloping the comprehensive number of regions in a more efficient manner. It initially adopted the UAVs models and constructed the regions such that linear programming-based problem can be modelled for establishing the best point-to-point path of flight between each of the individual UAVs. It then adopted the characteristic properties of ants foraging behaviour for estimating the shortest flight path between the source and destination scan points in the network. It was implemented for seeking optimal solutions approximately for minimizing the tasks’ time consumptions in the corporative search system. The simulation results conducted for ACOACPPM using different random generated regions confirmed better time completion time, minimized execution time and deviation ratio compared to the benchmarked approaches used for evaluation.

Aslan et al.³² proposed a multi-Population-based Immune Plasma Algorithm (MPIPA) for planning the path for the UAVs with maximized optimality such that the regions which need to be covered is enveloped with the shortest span of time. It adopted the remarkable properties of MPIPA for identifying the fuel potential paths that helps in achieving more possible candidate solutions which can be identified in least amount of time. This MPIPA algorithm explore and exploited the regions with maximized degree of efficiency such that premature convergence is prevented with maximized solution diversity. It was also identified to be more potential in exploring the greedy candidate solutions which formed the basis behind the determination of optimal paths estimated between the source and destination paths. Zhang et al.³⁰ implemented an Improved Artificial Fish Swarm Algorithm-based Path Planning Mechanism (AFSAPPM) for assisting the UAVs with well-established paths that helps in achieving the required tasks in a more efficient manner. This path planning mechanism used an improved AFSA algorithm for preventing the problem related to local optimization susceptibility and non-uniform population distribution. This AFSAPPM scheme handled the challenges related to slow convergence rate such that the UAVs can tackle the problem of path planning in a more effective manner. It also handled the problem of local optimization which combines global search potentialities with rapid rate of convergence. The simulation results of AFSAPPM confirmed high global convergence potential with high speed compared to the baseline approaches used for comparison.

Xu et al.²⁶ contributed a Random Tree Star Metaheuristic Algorithm (RTSMA) that determines better path quality and convergence speed for achieving better path planning support for UAVs with maximized efficiency. This path planning approach which included the merits of bias sampling and asymptotic optimality aided in better exploration of search space to support the flight of heterogeneous UAVs in the network. The RTSMA adopted for path planning possessed a large area of random sampling which converges systematically with better optimality. This improvement of adopted RTSMA is mainly due to the incorporation of artificial potential field for identifying the scope of greedy expansion which systematically improves the efficiency of the search process. This path planning approach obtained the initial path and updated the cost of the path depending on the generated random tree. It determined sampling points and rejected high-cost nodes depending on the estimation of the current cost path and heuristic cost which is responsible for speeding up the rate of convergence. The results of this RTSMA approach confirmed better results in terms of convergence rate, speed in determining initial paths and inflection points and, path cost compared to the baseline approaches used for comparison. Tan et al.²⁸ proposed a Game Theory Model Using PSO (GTMPSO) for finding the flight paths in the complex environments that possesses obstacles since achieving Nash Equilibrium under complex environments is more challenging. This GTMPSO approach handled the challenges that are more common in complex environments during the process of determining optimal flight paths. It established the balance between the UAVs exploitation and exploration capabilities by adopting a game theory-based Nash Equilibrium. This path planning approach is specifically developed for simulating the UAVs flight environment which comprises of obstacles that considered the coefficient of group awareness and self-awareness. The adopted PSO played an important role in better simulation of the UAVs flight situations. This approach was identified to be one of the optimal schemes which helped multiple number of multiple UAVs to navigate well in the presence of obstacles with well balance between the tendencies of local and global search. The simulation experiments of this GTMPSO approach on an average confirmed better flight time, flight distance and convergence time compared to the traditional approaches used for evaluation.

You et al.²⁵ proposed a UAV planning approach using multi-strategy enhanced Harris Hawk Optimization Algorithm (MSEHHOA) for improving the accuracy and rapidness in determining the possible paths that the UAVs travel to accomplish a specific task. This MSEHHO approach initially used the merits of double adaptive weight methodology which improved the speed of path planning with improved convergence accuracy. It was proposed with the objective of balancing the trade-off between the rate of exploration and exploitation with sustained solution diversity through the employment of dimension learning-based hunting methodology. It also integrated the merits of dung beetle optimization algorithm for preventing the algorithm of path planning from being struck into the local point of optimality. The results of this MSEHHOA method confirmed better performance in terms of stability, speed of convergence and ability of optimization determination. Liu et al.²⁹ proposed a PSO-based spherical vector-based adaptive evolutionary PSO algorithm for better path planning of UAVs in the network. This PSO-based path planning specifically used the merits of reverse learning and tent map for proper guiding of the paths that the UAVs need to follow for better accomplishment of tasks. It was proposed with the capability of distributing the initial solutions and enhancing the diversity of the solutions in the search space. It also used the merits of adaptive and dynamically non-linear factors that helped well in balancing the trade-off between the rate of exploration and exploitation. It further used an adaptive acceleration approach which aided in avoiding local optima during the process of UAVs path planning. Moreover, the optimization capabilities were improved by the evolutionary programming methodology. The results of this PSO-based confirmed better scalability, stability, convergence and solution accuracy during the process of path planning.

Hu et al.³³ proposed an ACO and Said-Ball curve-based UAVs’ path planning approach for guiding the UAVs for accomplishing the task whish need to be achieved in the network. This ACO-SBC approach used the primitive principles of intelligent metaheuristic algorithms for continuous monitoring the movement of UAVs in the network. It used the merits of adaptive foraging strategy which optimizes the potential of ACO towards the process of balancing the phases of exploration and exploitation in the search space. It further used the merits of multi-stage stochastic strategy for expanding the range of exploration facilitated by ACO in the search space. It furthermore used the strategy of aggregation mutation for the purpose of introducing dynamics of ACO at one level, and improved diversity on the other level. The results of this ACO-SBC approach confirmed minimized communication overhead and collision avoidance during the process of path planning. In addition, Tables 1 and 2 presents the review of the existing swarm intelligent metaheuristic optimization algorithms-based path planning mechanisms with the merits and limitations.

Table 1.

Review of the existing swarm intelligent metaheuristic optimization algorithms-based path planning mechanisms

Author	Mechanism used	Merits
Liu et al.23	Evolutionary algorithm based dynamic path planning Mechanism	It addressed the problem of low search efficiency and high computational complexity during the process of determining optimal paths between the source and destination paths in the implementation environment
Flores-Caballero et al.27	Meta-heuristic A-Star method	This metaheuristic approach determined the firsts set of candidate solutions that suitably adjusted the control paths to identify the path between the source and destination scan points associated with a region
Ait-Saadi et al.24	Hybrid Simulated Annealing and Chaotic Aquilla Optimization Algorithm (HSACAOA)-based coverage path planning mechanism	It ensured Collison-free, smooth and short path between the source and destination points, and it specifically adopted the characteristics of simulated annealing for establishing better balance between the exploration and exploitation rate
Chen et al.31	Ant Colony Optimization Algorithm-based Coverage Path Planning Mechanism (ACOACPPM)	It initially adopted the UAVs models and constructed the regions such that linear programming-based problem can be modelled for establishing the best point-to-point path of flight between each of the individual UAVs
Aslan et al.32	Multi-Population-based Immune Plasma Algorithm (MPIPA)	It adopted the remarkable properties of MPIPA for identifying the fuel potential paths that helps in achieving more possible candidate solutions which can be identified in least amount of time
Zhang et al.30	An Improved Artificial fish swarm algorithm-based path planning mechanism (AFSAPPM)	This path planning mechanism used an improved AFSA algorithm for preventing the problem related to local optimization susceptibility and non-uniform population distribution
Xu et al.26	Random Tree Star Metaheuristic Algorithm (RTSMA)	This path planning approach which included the merits of bias sampling and asymptotic optimality aided in better exploration of search space to support the flight of heterogeneous UAVs in the network
Tan et al.28	Game Theory Model Using PSO (GTMPSO) for optimal path planning	It handled the challenges that are more common in complex environments during the process of determining optimal flight paths, and established better balance between the UAVs exploitation and exploration capabilities by adopting a game theory-based Nash Equilibrium
You et al.25	UAV planning approach using multi-strategy enhanced Harris Hawk Optimization Algorithm (MSEHHOA)	This MSEHHO approach initially used the merits of double adaptive weight methodology which improved the speed of path planning with improved convergence accuracy It was proposed with the objective of balancing the trade-off between the rate of exploration and exploitation with sustained solution diversity through the employment of dimension learning-based hunting methodology
Liu et al.29	PSO-based spherical vector-based adaptive evolutionary PSO algorithm	This PSO-based path planning specifically used the merits of reverse learning and tent map for proper guiding of the paths that the UAVs need to follow for better accomplishment of tasks It was proposed with the capability of distributing the initial solutions and enhancing the diversity of the solutions in the search space
Hu et al.33	ACO and Said-Ball curve-based UAVs’ path planning approach	This ACO-SBC approach used the primitive principles of intelligent metaheuristic algorithms for continuous monitoring the movement of UAVs in the network It used the merits of adaptive foraging strategy which optimizes the potential of ACO towards the process of balancing the phases of exploration and exploitation in the search space

Table 2.

Limitations of the reviewed existing swarm intelligent metaheuristic optimization algorithms-based path planning mechanisms

Author	Mechanism used	Limitations
Liu et al.23	Evolutionary algorithm based dynamic path planning Mechanism	The path determination cost and computational complexity needs significant improvement since the time used for covering all regions need to be speeding up
Flores-Caballero et al.27	Meta-heuristic A-Star method	The fine tuning of parameters used for fitness function evaluation needs potential improvement, since the weighting coefficient in complex scenarios are not reactive
Ait-Saadi et al.24	Hybrid Simulated Annealing and Chaotic Aquilla Optimization Algorithm (HSACAOA)-based coverage path planning mechanism	The uncertainty involved in determining the candidate solutions need to be handled with maximized balance such that multiple regions are enveloped with the shortest time of flight
Chen et al.31	Ant Colony Optimization Algorithm-based Coverage Path Planning Mechanism (ACOACPPM)	The linear programming model needs improvement in satisfying the constraints essential in-flight path optimization process
Aslan et al.32	Multi-Population-based Immune Plasma Algorithm (MPIPA)	The concept of multi-population even through helps in determining potential number of candidate solutions, still need to maximize its solution diversity and prevent premature convergence
Zhang et al.30	An Improved Artificial fish swarm algorithm-based path planning mechanism (AFSAPPM)	The performance offered by the included algorithm for path planning needs improvement in determining optimal solutions independent to the number of candidate solutions determined in the search space
Xu et al.26	Random Tree Star Metaheuristic Algorithm (RTSMA)	The sampling method used for exploration need to be improved, and balance between exploitation and exploration needs significant enhancement
Tan et al.28	Game Theory Model Using PSO (GTMPSO) for optimal path planning	The process of achieving Nash equilibrium depending on the number of players included for path planning may incur a comparatively high time and computational complexity
You et al.25	UAV planning approach using multi-strategy enhanced Harris Hawk Optimization Algorithm (MSEHHOA)	The time involved during the process of path planning was comparatively high, and hence there exists a scope of enhancement in reducing the time of UAVs flight
Liu et al.29	PSO-based spherical vector-based adaptive evolutionary PSO algorithm	The balance established between the rate of exploitation and exploration during the process of path planning is marginal and needs high degree of accuracy in UAVs flight for target achievement
Hu et al.33	ACO and Said-Ball curve-based UAVs’ path planning approach	The convergence speed of this path planning algorithm is moderate and hence needs significant improvement in achieving the required degree of tasks

Extract of the literature

The review of the existing swarm intelligent metaheuristic optimization algorithms-based path planning mechanisms^] conducted helped in determining the shortcomings^34,35 which is listed as follows. Initially, it is identified that most of the approaches concentrated on geometry points-based path planning rather than considering the local and global search which forms the foundation for supporting the UAVs during their flight. Then most of the UAVs path planning approaches mainly used a single swarm intelligent metaheuristic optimization algorithm which often possesses issues related to imbalance between the rate of exploration and exploitation, poor solution diversity, and probability of the solutions from being struck into local point of optimality. It is confirmed that the parameters considered for fitness function evaluation in some of the path planning schemes was not comprehensive, and some were not ideal for realizing the situations in the real time environment. It is also realized that the problem of coverage path planning of UAVs over multiple regions is less studied to the best of the knowledge, since a greater number of constraints need to be explored in process of optimization. It is also visualized that a comparatively large number of path planning approaches paid attention to the process of optimizing multiple number of autonomous UAVs over a single region, rather than covering maximized number of regions with least amount of time. On the other hand, even though the existing coverage path planning approaches solve the problem in establishing the path of UAVs efficiently, they mainly targeted on applications that possessed only one region of interest, and which offered restricted range of applications. In addition, it is also concluded that the results of the existing coverage path planning approaches cannot be directly employed over the coverage systems of multiple regions which need to traverse comprehensively by heterogeneous UAVs in the network.

Proposed path planning

In this section, the models, fitness function, and the process of hybridizing GJOA and MFOA for achieving coverage path planning is presented in a more detailed way.

System models

Consider UAVs, which fly to Mission Regions (MRs), for executing complete Reconnaissance Mission (RM). represents UAV phase, where and represents the initial horizontal and vertical co-ordinates, and is the maximum speed of flight. Diverse factors like battery capacity, flying environment and payload have an impact on . However, these parameters are overlooked, as research completely focusses on problem related to Coverage Path Planning (CPP). It is assumed that information is at hand. Distance amid 2 areas represented as matrix, where is the length of flight from to . In case and are identical, . On the contrary, length may be computed by coordinates.

Task region (TR) model

UAVs include reconnaissance sensors (visual) for investigation of ground target. It is assumed that line of device sight and attitude of UAV’s flight is not dependent and may not have an impact on one another. When obstacles are not considered, detection range aligns with Visual Range (VR) of UAV obtained by intersecting visual cone with ground. In case location of detected target, orientation of detection device, factors of field detection as well as surface feature are identified, Line-of-Sight (LoS) propagation can be used for finding the VR for Task Region (TR). A polygonal model is constructed in terms of surface features of TR. Primarily, TR centre is considered as centre of sphere, surrounding surface features in a hemisphere called visual hemisphere. Then, the associated location relationship amid detection device as well as target is examined by employing LoS method of propagation. Obstruction conditions levied by surface features are considered and congested parts in view are eliminated offering identifiable range of TR. Eventually by applying this method, polygonal models may be generated for all TRs. Depending on formerly defined modelling method; RM is assumed to be complete by UAV when it crosses TR, through any point in VR.

In addition, Fig. 1 presents the overall view of the proposed HGJMFOA-based path planning scheme that helps in determining the route between the source and destination points in the network.

Fig. 1.

Overall view of the proposed HGJMFOA-based path planning scheme

Precise formulation

CPP is framed as reduction of objective function (Overall distance covered by UAVs for accomplishing mission) when bound by linear constraints (Energy consumption restrictions of UAV). Once the regions are completely covered, Flight Path (FP) of UAV is subject to ensuing constraints:

• Once a UAV is chosen for RM, it starts from primary location and returns once tasks are finished.

• To guarantee that every MR has no duplications or exclusions, only one UAV in MR performs reconnaissance.

• The quantity of UAVs must not be more than total number of MRs.

• UAV returns to initial position before energy is totally drained.

• : Turning Radius (TR) of UAV’s flight should have a minimum value.

is the UAV’s flight path, where is Boolean depicts those flies from to . In case UAV flies from to directly, ; Else, Further suggests that UAV should take off as well as land from initial location only once. Initial position is shown as virtual region (). In addition, and represent whether starts flying or lands in initial location .

1

Then is the only one aircraft can reach and leave MR.

2

Further indicates that quantity of UAVs involved in diverse tasks is not more than number of MRs.

3

Furthermore shows that UAV should return to before its energy is depleted. On-board energy limiting its journey is . Hence necessitates that the total distance flown on mission by UAV is less than or equal to its journey.

4

In addition, represents restrictions of TR. Contrary to rotorcraft that is capable of hovering before turning, UAVs with fixed-wing demand a circular arc for executing the action. Minimum radius of circle where arc is present is determined, guaranteeing the drone to effectively complete turning behaviour.

Minimum value of TR restricts length of Flight Segment (FS). When UAVs enter and exit a particular FS through arc, length of FS should be adequate to include at least 2 circular arcs. With decrease in length, UAVs will be unable to turn normally. In case starting and ending waypoints of FS () are and , then constraint on length of shortest segment is given by Eq. (5)

5

where, represents lower bound.

Further, minimum TR imposes limitations on angle between neighbouring segments. In case angle is very small, drone cannot finish present segment before moving to next one. Then angle amid and should be more than minimum. Limitations on least arc radian is given by Eq. (6)

6

where represents upper bound.

The above aspects are combined and is given by Eq. (7)

7

Flight Distance (FD) computed through multi-UAV formation for performing tasks of regional reconnaissance which is the cost of longest distance cost for characterising performance of execution of Coverage Task (CT). and represents the FD of a UAV as well as UAV formation.

8

9

The CPP problem focuses on determining ideal FP sequence for formation of multi-UAV such that total FD of the CTs is the least. represents objective function with constraints to . The exact model is a linear one as presented in Eq. (10)

10

As this model gives a solution to CPP problem, it is essential to explore complete solution space involving considerable time. With increase in number of UAVs and regions, solution time becomes unrealistic, leading to unacceptable efficiency. To handle this restriction, improved Fuzzy C-Means (FCM) algorithm is used for clustering TR. Then, Improved IGJOA is used for assigning and planning paths more efficiently.

Golden Jackal optimization algorithm (GJO)

Golden Jackal Optimization (GJO) is designed for optimization³⁶ depending on the inspiration derived from hunting behaviour of GJ pairs, leveraging methods to optimize updates of prey locations. Hunting of GJ involves the following steps that includes, the detecting and tracing prey, enclosing and inspiring prey till it stops movement, attacking and seizing prey. Alike other meta-heuristic algorithms, population is initialised. A collection of matrices represented as is produced using Eq. (11) and arbitrarily positioned in search space. The matrices are uniformly distributed as presented in Eq. (12).

11

12

Where and are the lower and upper bounds of variables, represents random vector which is uniform in range . and signify population size of prey and variable dimension respectively. signify dimension of prey. During the phase of exploration phase (), mathematical model of GJ hunting is given by Eqs. (13) and (14)

13

14

where presents the current iteration, is the prey’s position vector, and are the positions of male and female jackals. and represent new locations of male as well as female jackals conforming to prey. In this case, is the prey’s escape energy as defined in Eqs. (15–17).

15

16

17

Where

- Reduction in prey energy, - Initial phase of . - Random number within range [0, 1]. - a constant set with value 1.5. -maximum amount of iterations, -present amount of iterations.

Then in Eqs. (13) and (14) is used for determining distance amid jackal as well as prey. Depending on avoidance energy of prey, jackal’s present location will lead to an increase or decrease in distance. is the random vector determined using Levy Flight (LF) function.

18

19

where, and are arbitrary values within range [0,1], represents a constant (1.5). Average of Eqs. (13) and (14) is involved in the updating of prey location.

20

When jackals harass the prey, gets reduced. Jackals encircling and consuming prey is represented mathematically as shown below ().

21

22

In addition, Algorithm 1 presents the flow process of GJOA adopted for achieving dynamic UAV path planning process.

Algorithm 1.

Steps included into GJOA for dynamic UAV pathplanning process.

Multi-strategy fusion-based improved GJOA

This Multi-Strategy Fusion-Based Improved GJOA was proposed for addressing the challenges that includes reduced convergence speed, insufficient exploration as well as exploitation efficiency and inclination to fall into spaces with local extreme values, specifically when handling difficult and optimization problems with high dimensions. In specific, the possible enhancements introduced into GJOA for addressing its limitations includes Tent Chaotic Mapping (TCM), Opposition-Based Learning (OBL), Dynamic Inertia Weight and Logarithmic Spiral Search-based Location Update Rule. Initially, tent mapping is used for attaining even distribution of the initial population in the search space. Then opposition learning mechanism is used for achieving population replacement such that population diversity with search space expansion is guaranteed for the objective of preventing the solutions from being struck into the local optima. Further it used a logarithmic spiral path strategy into MFOA for searching the solutions in the search space. In addition, adaptive dynamic inertia weights are used for balancing the trade-off between the rate of exploration and exploitation in the search space.

Tent chaotic mapping (TCM)

Initially, population in GJO is produced at random, diversity as well as rationality of distribution may not be certain. This has an impact on algorithm’s optimization performance as well as search efficiency. Chaotic Sequences (CSs) which possess features like randomness, sensitivity and ergodicity for initial values³⁷ are extensively employed in optimization of search space. Among diverse chaotic mappings, TM offers better results as it is simple and capable of producing evenly distributed outcomes. The mapping shows exceptional ergodicity and may be easily linked with other algorithms.

Random behaviour of initial population is leveraged while including TCM³⁸ to produce CSs as well as performing population initialisation. This method guarantees that initial solutions are evenly distributed in solution space, enabling highly complete search within search space and improving convergence accuracy as well as optimization performance of algorithm. of CS produced by TCM is given by Eq. (23)

23

where, setting initially aims at arbitrarily generating value of solution vector in range given by random vector. This enhances probability that primary population strictly lines up with real situation. Population is got by inverse mapping of regularly distributed CS after TM by using Eq. (24).

24

In contrast to random scheme, TCM distribution is comparatively even in feasible domain. For comparing distributions of stochastic scheme and TM methods, Lyapunov index is introduced to determine mean exponential divergence. Lyapunov index of TM is taken as 0.603 > 0. It enters a chaotic phase signifying increased divergence level in contrast to random scheme. Hence, TCM is used for initialising the population during initial phase of algorithm. This guarantees that the positions of the individuals are evenly spread in search space. This aids in improving solving capability of algorithm. Moreover, Algorithm 2 demonstrates the process of TCM initialization included into the path planning process.

Algorithm 2.

Initialization of TCM.

Opposition-based learning (OBL)

OBL³⁹ is efficient in improving numerous meta-heuristic algorithms. OBL determines conflicting solution of solution during assessment, offering an added chance to determine universally optimal solution. This method aids in assessing the feasible and inverse solutions. Outstanding individuals are chosen from inverse and present population for generating a fresh population, thus improving population diversity and increasing algorithm’s search space. OBL-based population replacement scheme aims at enhancing diversity of population and improving algorithm’s search efficiency. Solutions of better quality are to be preserved when substituting solutions of inferior quality, leading to exploration of wider solution space. It improves initial solution quality and prevents algorithm from converging to local optimum and decreases its inclination to blind optimization. In this context the current population is expressed as . OB-based opposite population, is determined through Eq. (25)

25

Fresh population represented as is got by merging present population ‘’ with ‘’ produced through OL. Once fitness of every individual in population is assessed and ranked, individuals with best fitness (top ‘’) are chosen as preliminary population. Then Algorithm 3 is presented for highlighting the strategy of population replacement guaranteed by the proposed OBL-based UAV path planning process.

Algorithm 3.

Population replacement scheme based on OBL.

Combine and into .

Determine prey’s fitness in fresh population.

Select top fit individuals () as .

Dynamic inertia weight and logarithmic spiral search-based location update rule

Moth-Flame Optimization Algorithm (MFOA) is a lateral positioning as well as navigation-based optimization algorithm²². It is based on moths’ navigation behaviour, where they select and search round about lights in a spiral pattern. Every moth is allocated a flame, and flame’s location is updated in every iteration⁴⁰. This method improves search space exploration and decreases risk of falling into local optima. Furthermore, algorithm includes dynamic flame quantity scheme for obtaining a balance amid exploration as well as exploitation⁴¹. In every iteration, flame sequence is modified depending on best solution, and moths relocate themselves based on updated flame, thus enhancing performance of exploration of search space⁴². MFOA offers better global optimization abilities and is simple to understand. Its location update rule is based on LSS pattern:

26

where is the moth, denotes flame, represents a constant which defines shape of LS and is the random number within range . specifies immediacy of ensuing moth location to flame. is the extreme location from flame. highlights the closest location to flame. To improve exploitation ability of algorithm, represents random number within range , where denotes convergence factor. shows a linear decrease from to ‘’. This guarantees that conforming flame is used more precisely depending on quantity of iterations.

27

To stop algorithm from extreme exploitation of local optimal solution and ensuring usage of information related to present optimal position, Dynamic Inertia Weight (DIW), a transition mode is included in the study. DIW inspired by Cosine curve is shown in Eq. (28). By including this method, population diversity is maintained while facilitating quicker convergence to global optimal solution as well as enhanced accuracy of convergence.

28

where is the quantity of iterations, represents a factor for weight adjustment. During exploration, lesser weights are essential for facilitating individuals to comprehensively search space. With progress in exploitation, IW increases progressively. This method maximizes usage of present optimal position and guarantees convergence accuracy during development. By combining LS path and DIW with optimization method of Golden Retriever (GR), rules for position updating of female as well as male GRs during exploitation are given by Eqs. (29) and (30)

29

30

To determine whether new prey location is retained for ensuing iteration, greedy method is used.

31

Position updating rule in optimization algorithm uses double LS curve rule of updating, drawing motivation from encirclement approach used by GJ for hunting. The GJ surrounds the prey by using a spiral curve, slowly reducing physical strength of prey when contracting adjoining region, eventually seizing its target. GJOA includes double LS curve updating rule to simulate encirclement approach. This enhances performance of algorithm in searching search space to degree and closely aligning with real-life hunting nature of GJ.

Hybrid OL-improved MFO and GJO (OMGJO)

Integrating TCM initialization, OBL along with double helix curve’s location update rule with DIW improves capabilities of GJOA. It enhances diversity of population, increases search space as well as improves solution quality. This is favourable for algorithm to perform a complete search by efficiently evading local optima and guaranteeing accuracy of convergence during exploitation. During initialization, TCM produces initial solution, and OBL approach produces opposite solution. Values of Fitness Function (FF) of populations are compared, and better solution is reserved. OBL approach is used for finding inverse solution of present feasible solution. It maintains optimal solution using values of FF. During exploitation phase, locations of both female and male jackals are determined using new rules of updating. Figure 2 and Algorithm 4 specifies the steps included into the process of hybridizing MFOA and GJOA using the benefits of OL.

Fig. 2.

OMGJO-based path planning mechanisms for heterogeneous UAVs

Algorithm 4.

OMGJO algorithm determined using MFOA and GJOA using OL.

Proposed IGJOA -based path planning mechanism (HGJMFOA)

IGJOA is used for establishing order of access within clustering region, eventually obtaining an ideal FP^43,44. This IGJOA includes 2 phases: initial phase including TCM initialization, OBL along with double helix curve’s location update rule, and DIW improving the capabilities of GJOA.

Phase 1

The golden jackals termed as search agents () are set in a search space of ‘d’ dimensions, the location and speed are determined with iterations as and . They are determined after every iteration.

32

33

Where, is a parameter of TCM which ranges between 01, is an arbitrary number in the range [0, 1]. represents the current iteration. represents global ideal location before iterations.

Phase 2

This phase is entered when reaches maximum iteration . Set centre of remaining golden jackals as centroid at each iteration. Determine and based on Eqs. (33), (34).

34

35

36

Fitness Function (FF) is an indicator used for assessing whether individuals have reached ideal level. In FCM algorithm, least value of objective function signifies the best outcome. Hence, FF is given by,

37

It is obvious that FF is proportional directly to value of Fuzzy Clustering (FC) objective function. A smaller value corresponds to least value of fitness representing nearness to optimal solution. Likewise, DIW stops functioning, and search of IGJOA stops when ‘ ’ reaches maximum iteration in present phase.

Simulation results and discussion

In this section, the performance evaluation results of the proposed HGJMFOA technique and the benchmarked path planning approaches are presented with respect to number of regions generated and the number of UAVs used for accomplishing the objective of covering the required area of interest. The benchmarked coverage path planning approaches used for evaluation are Ant Colony Optimization Algorithm-based Coverage Path Planning Mechanism (ACOACPPM)³¹, multi-population-based Immune Plasma Algorithm (MPIPA)³², Improved Artificial fish swarm algorithm-based path planning mechanism (IAFSAPPM)³⁰ and Game Theory Model Using PSO (GTMPSO) for optimal path planning²⁸, respectively. The benchmarked approaches such as ACOACPPM, MPIPA, IAFSAPPM and GTMPSO approaches are compared with the proposed HGJMFOA technique since all these mechanisms were contributed to the existing literature for addressing the similar kind of path planning problem which is solved by the proposed mechanism. Thus, the proposed HGJMFOA technique and the baseline approaches target on the process of optimizing the flight paths and reduce the time of task completion during the operation of UAVs. The parameter settings used for implementing the proposed HGJMFOA technique follows the same used for implementing the benchmarked approaches. The proposed HGJMFOA technique and the baseline approaches are implemented using C + + language which is installed over the Microsoft Visual Studio 2015 version. The machine used for implementation comprises of 64u-bit Windows 8.1 operating system and Intel(R) Core (TM) i5-3320 M CPU 2.60 GHz, 4.00 GB RAM. This simulation experiments are conducted using the evaluation parameter of execution time, task completion time and deviation ratio.

Evaluation of the proposed HGJMFOA using execution time with different regions and constant UAVs

In this initial part of analysis, the execution time incurred by the proposed HGJMFOA, and the baseline path planning approaches are compared under varying randomly generated regions by assigning the UAVs as constant.

The number of UAVs which is considered during the process of evaluation is 10, 20, 30 and 40, respectively. When the regions are increased the exploration and exploitation increases systematically with a constant number of UAVs available in the network for achieving the flight between the source and destination points of scan. This increase in the search space increases the execution time generally during the implementation of the proposed HGJMFOA scheme and the benchmarked approaches, but the proposed approach handled the problems of path planning in a remarkable manner such that execution time is considerably reduced as specified in Figs. 3, 4, 5 and 6. The results confirmed that the proposed HGJMFOA scheme under different obstacles handles its impact by determining the paths through which the UAVs can flight from the source and destination scan points within minimized amount of time. When the number of regions is increased, the length between the source and destination points has the probability of being increase since the flight path which need to be eventually traversed also gets potentially increased.

Fig. 3.

Execution time under different number of regions (number of UAVs-10)

Fig. 4.

Execution time under different number of regions (number of UAVs-20)

Fig. 5.

Execution time under different number of regions (number of UAVs-30)

Fig. 6.

Execution time under different number of regions (number of UAVs-40)

However, the proposed HGJMFOA scheme handled the increase in the regions by adopting a dynamic exploitation rate which performs better local optimization at one level, and at the same time it determined candidate solutions which possessed better solution diversity. Thus, the results of the HGJMFOA scheme with varying randomly generated regions ensured a comparatively minimized execution time by 19.84%, better than the baseline approaches used for comparative investigation.

Evaluation of the proposed HGJMFOA using execution time with different UAVs and constant regions

In this section, the proposed HGJMFOA and the benchmarked path planning approaches evaluated using execution time with different number of UAVs (keeping the number of regions as constant).

The number of regions which considered during the process of evaluation is 10, 25, 40 and 50, respectively. When the number of UAVs is increased with constant number of regions, the execution time gets systematically increased due to the time incurred during the determination of optimal flight over the implementation environment. From Figs. 7, 8, 9 and 10, it is also determined that the proposed HGJMFOA scheme performed well by reducing the execution time since it exploited and explored the feasible candidate solution remarkably such that different possible paths within a considerable amount of time is comparatively better than the baseline approaches. It is also confirmed that the proposed HGJMFOA scheme incurred least amount of time in determining the feasible solution which is helpful for seeking the paths over the multiple number of regions considered for evaluation. When the number of regions are increased, the performance of the proposed HGJMFOA scheme still is visualized to exhibit the same pattern in terms of execution time, This capability of the proposed HGJMFOA scheme aided in sustaining reduced time of execution due to reasons that includes, (i) it reactive tendency in adjusting the path plan depending on the constraints which are employed, (ii) the number of constraints which are contextually derived in the deployment environment, and (iii) algorithmic tendency in preventing the candidate solution from being struck into local point of optimality. Thus, the results of the HGJMFOA scheme under different number of UAVs on an average confirmed minimized execution time by 21.38%, better than the baseline approaches used for comparative investigation. On the other hand, the baseline ASFAPPM approach exhibited unusual behaviour since the synergic method used for coverage path planning was sub-optimal which still possesses the possibility of significant improvement. This baseline ASFAPPM approach initially struggled with minimized number of UAVs and regions, but however tackled the situation during path planning to the required level when the number of regions and UAVs scaled up the required degree in the network.

Fig. 7.

Execution time under different number of UAVs (number of regions-10)

Fig. 8.

Execution time under different number of UAVs (number of regions-25)

Fig. 9.

Execution time under different number of UAVs (number of regions-40)

Fig. 10.

Execution time under different number of UAVs (number of regions-50)

Performance evaluation of the proposed HGJMFOA-based path planning using deviation ratio with different number of regions and UAVs

In this section, the proposed HGJMFOA and the baseline path planning approaches (inexact methods) are evaluated using deviation ratio with different number of regions and UAVs. This deviation ratio used for comparative evaluation is defined as a ratio between the relative difference identified between the times of task completion incurred by the inexact methods and task completion time incurred by the proposed HGJMFOA approach (exact method). Figure 11 demonstrates the deviation rate realized in the implementation environment during the application of the proposed HGJMFOA approach and the benchmarked path planning approaches with different number of UAVs by keeping the number of regions as 50. It is visualized the deviation ratio of the proposed HGJMFOA approach and the benchmarked path planning approaches under different number of UAVs gets decreased with respective increase in the number of regions, However, the proposed HGJMFOA approach exhibits minimized deviation rate than the benchmarked path planning approaches since it adopted GJOA and MFOA for exploring and exploiting the search space well such that unnecessary points of scan in the network is maximally prevented. It is also proposed with the capability of dynamically adjusted the path movement depending on the factors that are contextually determining during the flight of the UAVs for achieving its core objective. This potentiality of the proposed HGJMFOA approach comparatively helped in minimizing the deviation rate by 13.21%, 15.98%, 18.34% and 21.89%, better than benchmarked path planning approaches used for comparative investigation.

Fig. 11.

Deviation ratio under different number of UAVs (number of regions-50)

Figure 12 depicts the plots of deviation rate attained during the implementation of the proposed HGJMFOA approach and the benchmarked path planning approaches with different number of regions by keeping the number of UAVs as 40. It is seen that the deviation rate of the proposed HGJMFOA approach gets declined with increase in the number of regions. However, the proposed HGJMFOA approach compared to the baseline path planning approaches performed well since it shows minimized deviation rate with increase in the number of regions at any point of time. It is also determined that when the number of regions increases in the implementation environment, the need for determining required path with the available number of UAVs is required, and more and more optimal paths need to be determined such that deviation ratio during the initial phase is not increased. It is also identified that the solution space needs to determine a greater number of feasible solutions from which optimal solutions could be determined by imposing the required number of constraints during the flight paths confirmation. This capability of the proposed HGJMFOA approach under different number of regions comparatively helped in minimizing the deviation rate by 12.64%, 14.86%, 17.12% and 19.56%, better than benchmarked path planning approaches used for comparative investigation.

Fig. 12.

Deviation ratio under different number of regions (number of UAVs-40)

In addition, Tables 3 and 4 presents the comparative results of the proposed HGJMFOA approach and the baseline approaches using deviation ratio with different number of UAVs and regions.

Table 3.

Deviation ratio (in %) with different number of UAVs

Number of UAVs	Deviation ratio (in %)
	Proposed HGJMFOA	GTMPSO	MPIPA	AFSAPPM	ACOACPPM
4	6.82	7.38	7.52	7.64	7.88
8	6.64	7.22	7.41	7.56	7.76
12	6.23	7.11	7.32	7.51	7.62
16	5.88	7.02	7.19	7.42	7.54
20	5.65	6.84	7.04	7.31	7.48
24	5.44	6.75	6.98	7.22	7.41
28	5.32	6.61	6.92	7.28	7.35
32	5.19	6.44	6.89	7.24	7.31
36	5.04	6.38	6.84	7.21	7.29
40	4.86	6.31	6.82	7.38	7.28

Table 4.

Deviation ratio (in %) with different number of regions

Number of regions	Deviation ratio (in %)
	Proposed HGJMFOA	GTMPSO	MPIPA	AFSAPPM	ACOACPPM
200	12	15	18	25	28
400	17	21	25	28	32
600	21	27	32	32	39
800	24	32	37	37	43
1000	28	36	42	42	48
1200	37	43	49	49	57
1400	48	55	58	58	67
1600	56	62	66	66	76
1800	64	68	73	73	82
2000	72	73	79	79	85

In addition, Tables 5 and 6 presents the comparative results of the proposed HGJMFOA approach and the baseline approaches using computation time for path planning (seconds) with different number of UAVs and regions.

Table 5.

Computation time for path planning (seconds) with different number of UAVs

Number of UAVs	Computation time for path planning (in seconds)
	Proposed HGJMFOA	GTMPSO	MPIPA	AFSAPPM	ACOACPPM
4	1.86	1.94	2.05	2.23	2.35
8	1.95	2.08	2.16	2.39	2.46
12	2.14	2.21	2.32	2.43	2.54
16	2.29	2.32	2.45	2.54	2.68
20	2.38	2.44	2.56	2.65	2.78
24	2.49	2.56	2.65	2.77	2.89
28	2.56	2.88	2.96	2.84	2.95
32	2.67	2.94	3.12	3.24	3.35
36	2.79	3.12	3.25	3.45	3.56
40	2.84	3.25	3.38	3.56	3.69

Table 6.

Computation time for path planning (seconds) with different number of regions

Number of regions	Computation time for path planning (in s)
	Proposed HGJMFOA	GTMPSO	MPIPA	AFSAPPM	ACOACPPM
200	3.56	3.78	3.86	3.97	4.04
400	3.64	3.89	3.95	4.04	4.11
600	3.78	3.97	4.12	4.12	4.21
800	3.89	4.04	4.24	4.18	4.38
1000	4.12	4.21	4,68	4.24	4.46
1200	4.26	4.39	4.76	4.29	4.56
1400	4.45	4.54	4.84	4.32	4.64
1600	4.58	4.69	4.99	4.39	4.72
1800	4.76	4.86	5.08	4.46	4.78
2000	4.84	4.98	5.14	4.54	5.12

The above-mentioned results confirmed that the proposed HGJMFOA confirmed minimized computational time compared to the baseline approaches since it used better optimization process independent to the number of UAVs and regions considered in the network.

Statistical validation

In this section, the statistical validation of the proposed scheme and the baseline approaches are presented using descriptive analysis, Homogeneous Subset Test and Tukey’s Hinges percentile distribution conducted for path planning time is presented. In specific, Table 7 presents the descriptive analysis of the proposed HGJMFOA scheme, and the baseline approaches determined based on path planning time. This descriptive analysis confirmed that the mean of the path planning time confirmed the maximized value of 1.864 which is comparatively better than the benchmarked approaches used for comparison.

Table 7.

Descriptive analysis with respect to mean, standard deviation and standard error using path planning time

Algorithms compared	N	Mean	Standard deviation	Standard error
HSACAOA	30	− 0.018	0.006	0.001
MSEHHOA	30	1.586	0.031	0.005
ACOACPPM	30	− 0.007	0.027	0.000
AFSAPPM	30	− 0.013	0.004	0.000
MPIPA	30	0.085	0.063	0.013
GTMPSO	30	− 0.019	0.001	0.000
Proposed HGJMFOA	30	1.864	0.408	0.078
Total	210	3.478	0.54	0.097

Then Figs. 13 and 14 presents the plots of standard deviation and standard error of the proposed scheme and the benchmarked approaches under different number of iterations (keeping regions-2000 as constant). These plots confirmed that standard deviation and standard error determined with respect to path planning time is comparatively lower than the baseline approaches since is confirmed maximized coverage of regions which significantly minimized the path planning time to the expected level.

Fig. 13.

Standard deviation under different iterations (number of regions-2000 as constant)

Fig. 14.

Standard error under different iterations (number of regions-2000 as constant)

Then Table 8 demonstrates the homogeneous subset test results which revealed that the proposed HGJMFOA scheme outperformed the baseline algorithms which confirmed a mean value of 11.654 at 1.000 level of significance and included into the group 3. At the same time, the bassline GTMPSO approach conformed a mean performance and included into the gropu-2 at the significance level of 1.000. On the other hand, the ither baseline algorithm confirmed similar performance in terms of path planning time and hence they are included into the same group-1. Thos result also identified that the baseline approaches exhibited the worst performance in terms of path planning time.

Table 8.

Homogeneous subset test for path planning time

Algorithms	N	Subset for alpha = 0.05
		1	2	3
HSACAOA	30	− 0.016
MSEHHOA	30	− 0.011
ACOACPPM	30	− 0.008
AFSAPPM	30	− 0.009
MPIPA	30	− 0.083
GTMPSO	30		1.512
Proposed HGJMFOA	30			1.816
Significance		0.382	1.000	1.000

In addition, Table 9 presents the percentile distribution for the path planning time with respect to Tukey’s Hinges percentile distribution. This Table 9 presented the 1st. 2nd and 3rd quartile with respect to 25 percentile, 50 percentile (median) and 75 percentile distribution.

Table 9.

Tukey’s hinges percentile distribution for path planning time

Algorithms	Percentiles
	5	10	25	50	75	90	95
HSACAOA			− 0.022	− 0.023	− 0.013
MSEHHOA			`1.516	1.516	1.516
ACOACPPM			− 0.013	− 0.012	− 0.008
AFSAPPM			− 0.014	− 0.011	− 0.009
MPIPA			− 0.014	0.086	− 0.148
GTMPSO			− 0.016	− 0.017	− 0.0118
Proposed HGJMFOA			1.128	2.112	2.134

These results confirmed that the median value (50 percentile = 2.112) for the proposed HGJMFOA scheme is the best with the interquartile range of 0.896.

Conclusions

The proposed HGJMFOA scheme confirmed better coverage path planning mechanism by determining superior optimal paths between the source and destination scan points which eventually assist the UAVs towards the efficient process of comprehensively covering all the regions of interest. It initially constructed the regions and UAVs models with a linear programming model which is formulated for accurate determination of best point-to-point flight path for each individual UAVs. Then it adopted the merits of HGJMFOA for obtaining shortest path between the source and destination points in the region of interest to determine optimal solutions approximately with minimized tasks time consumption in the cooperative search systems. The results confirmed that the path planning time under different UAVs and regions is minimized compared to the baseline approaches used for comparison. The proposed approach under different number of regions comparatively minimized the deviation rate by 12.64%, 14.86%, 17.12% and 19.56% than the benchmarked path planning approaches used for the comparative investigation. The statistical validation achieved using descriptive analysis (mean, standard deviation, standard error), Homogeneous Subset Test and Tukey’s Hinges percentile distribution confirmed that the proposed HGJMFOA scheme confirmed better path planning time.

Limitations of the study

This proposed strategy faces the challenge when the obstacles existing in the region of target is comparatively high since it hurdles the process of enveloping the entire coverage area with the support of UAVs. Energy management is the major concern which is not highly focused during the implementation of the proposed strategy use for path planning of UAVs which possesses restricted amount of energy during the process of.

Future scope of enhancement

The future research can include fuzzy set theory-based multi-criteria decision-making algorithm for determining the effective path planning of UAVs to coverage the network in an efficient manner. The proposed path planning algorithm can be extended with maximized convergence speed of the algorithm such that it can be adopted for simulation of more realistic UAV flight scenario. In addition, the applicability of the proposed path planning algorithm can be scaled up to different types of applications. In specific, multicriteria decision making technique such as VIKOR-DEMATEL approach can used for determining the weights of the factors and the contribution of the impactful factors towards the process of network coverage path planning process. Moreover, Delaunay Triangulation method can be adopted for determining the optimal number of UAVs which may be used for covering the required regions of target in an optimal manner.

Author contributions

K.K.: Writing—Original Draft, Methodology, Conceptualization, Investigation, Experiments, Software, Data curation, Validation, Review & Editing. C.B: Supervision, Conceptualization, Methodology, Data curation, Validation, Writing—Review & Editing. R.P.: Conceptualization, Methodology, Data curation, Formal analysis, Validation, Writing—Review & Editing.

Data availability

All data generated or analyzed during this study are included in this article.

Declarations

Competing interests

The authors declare no competing interests.