Three-Dimensional Coverage Path Planning for Cooperative Autonomous Underwater Vehicles: A Swarm Migration Genetic Algorithm Approach

Abstract

Cooperative marine exploration tasks involving multiple autonomous underwater vehicles (AUVs) present a complex 3D coverage path planning challenge that has not been fully addressed. To tackle this, we employ an auto-growth strategy to generate interconnected paths, ensuring simultaneous satisfaction of the obstacle avoidance and space coverage requirements. Our approach introduces a novel genetic algorithm designed to achieve equivalent and energy-efficient path allocation among AUVs. The core idea involves defining competing gene swarms to facilitate path migration, corresponding to path allocation actions among AUVs. The fitness function incorporates models for both energy consumption and optimal path connections, resulting in iterations that lead to optimal path assignment among AUVs. This framework for multi-AUV coverage path planning eliminates the need for pre-division of the working space and has proven effective in 3D underwater environments. Numerous experiments validate the proposed method, showcasing its comprehensive advantages in achieving equitable path allocation, minimizing overall energy consumption, and ensuring high computational efficiency. These benefits contribute to the success of multi-AUV cooperation in deep-sea information collection and environmental surveillance.

1. Introduction

The oceans are a treasure trove of underwater resources and hydrographic data. Monitoring water quality at various depths is essential for environmental surveillance, ecological studies, resource discovery, and disaster warning as seen in the methane release from natural gas hydrate extraction [1]. Real-time monitoring helps prevent marine pollution and mitigate geological and meteorological disasters. With growing interest in marine exploration, autonomous underwater vehicles (AUVs) have emerged as efficient tools, offering low energy use, stealth, and maneuverability, and enabling the collection of high-quality, precise data that are difficult to acquire through traditional means [2]. Due to the deep sea’s vastness and complexity, multi-AUV systems have become a leading approach for underwater exploration, enhancing efficiency, coverage, and reducing time costs [3,4].

Finding optimal coverage paths is essential to ensure that multi-AUV systems can explore the entire working area in complex 3D spaces with obstacles [2], a challenge commonly known as the multi-agent coverage path planning (MACPP) problem [5]. Unlike 2D space coverage path planning (CPP) for applications such as unmanned surface vehicles (USVs) scanning, MACPP for AUVs must address three-dimensional spaces with distributed obstacles (usually caused by the irregular sea floor shape) as shown in Figure 1. This task involves multiple requirements: (1) handling complex obstacle avoidance in 3D space, (2) ensuring the coverage of all passable workspaces, (3) minimizing overall energy consumption, and (4) maintaining relatively consistent working paths among agents to optimize the time efficiency of the collective operation. MACPP, which considers only path allocation and spatial coverage, is classified as a non-deterministic polynomial (NP)-hard problem [6]. Considering all these aspects increases the problem’s complexity, which has not yet been fully addressed.

The existing research on CPP problems is summarized in

Table 1. Competence of various methods under multiple requirements.

	3D Workspace with Obstacles	Complete Workspace Coverage	Minimum Energy Consumption	Even Path Assignment
Random coverage approach [7]	✔	✔	✘	✘
zig-zag algorithm [12]	✔	✔	✘	✘
STC [13,14,15]	✘	✔	✘	✔
GBNN [16]	✔	✔	✘	✘
RL [17,18]	✔	✘	✔	✔
SCTG-SMGA (Ours)	✔	✔	✔	✔

. Methods for individual agent CPP include the random coverage approach [7], the zig-zag algorithm for round-trip coverage [8], the spanning tree coverage (STC) algorithm [9], the Glasius Bio-inspired Neural Network (GBNN) [10], and reinforcement learning (RL) [11]. Except for the random method, all these methods have been extended to MACPP applications. An extended idea involves environment representation methods, which divide the workspace into several independent parts according to the number of agents, and then solve the problem using single-agent CPP in each area. The related works, for example, include using space division methods combining zig-zag algorithms [12] or STC [13,14,15]. Another category is agent dynamic interaction methods, where each agent strategically determines its path based on individual goals, interaction with others, and the environment. This includes GBNN [16] and multi-agent reinforcement learning (MARL) [17,18] methods.

Table 1 compares the ability of these methods to address the four requirements of 3D MACPP mentioned in this paper. Overall, no single method provides a complete solution to this problem. The random, zig-zag, and GBNN methods do not optimize energy consumption or ensure evenly assigned paths. While the spanning tree coverage algorithm addresses even path assignment, it lacks the mathematical foundation to extend a 2D spanning tree to 3D. RL is a promising new tool, but due to its lack of mathematical completeness, it cannot guarantee full coverage of the entire workspace.

This article, therefore, introduces a comprehensive solution for MACPP in 3D space. Specifically, it resolves the MACPP problem for AUVs in underwater environments, considering all four cooperative path planning requirements. The innovative contributions of this paper can be summarized as follows:

(1)

We introduce a novel framework for MACPP that comprises two essential steps: autonomous coverage path growth and path allocation. Instead of first dividing the space, we allow short paths to grow and occupy the entire 3D workspace, which is named the snap coverage trails auto-growth (SCTG) algorithm. Then, we address the path evenness and energy optimization problems by optimizing the path allocation and connection among agents. Consequently, the four requirements are fulfilled through these two separate steps.

(2)

We propose a Swarm Migration Genetic Algorithm (SMGA) to solve the path allocation and connection problem. This innovative genetic algorithm introduces the concept of swarms to the traditional GA. The migration operation among swarms of genes provides a mechanism for multi-agent allocation negotiation. Additionally, to adapt to the path planning problem, a unique double chromosome structure is defined to facilitate the exchange of paths among AUVs. The new migration operation achieves efficient gene flow between different swarms with low computational resource consumption, thereby improving the global convergence ability of the algorithm.

(3)

The aforementioned method is applied to facilitate the collaborative exploration undertaken by multiple AUVs within a 3D underwater environment, improving the adaptability and work efficiency of full coverage path planning in underwater environments. Notably, this marks the first MACPP work conducted in 3D space, to the best knowledge of the authors. Experimental results show that the proposed algorithm effectively solves standard problems, including comprehensive indicators such as overall energy consumption, fair path allocation, and computational efficiency.

The remaining sections of this paper are organized as follows: In Section 2, the related works emphasize the novelty of the proposed algorithms concerning the existing body of research. Section 3 presents the problem formulation of multi-AUV 3D coverage path planning in a complex subsea environment. The multi-AUV 3D CPP based on the snap coverage trails auto-growth (SCTG) algorithm, and the SMGA is discussed in Section 4. Section 5 presents the simulation results and performance comparison. Finally, Section 6 concludes the paper with remarks.

2. Related Works

The CPP problem has received extensive research attention over several decades; it involves generating an optimal path to cover a specific area of interest [19]. In the context of a multi-AUV system, the problem of MACPP can be defined as the generation of a set of paths that can cover all free 3D space and avoid obstacles at the same time, and the paths are to be assigned equally among the agents. The MACPP algorithms in previous studies are typically classified into two categories: environment representation methods and the agent dynamic interaction method [5].

2.1. Environment Representation Methods

Environment representation methods involve initially dividing the work area into several independent parts based on the number of agents, and then solving the problem by using a single-agent CPP in each respective area [20].

In the division of task areas, some researchers have used the Voronoi algorithm [21,22,23] and an improved precise cell decomposition [24] method to divide the entire region into multiple subregions, allowing the agents to complete the coverage task in the assigned subregions, respectively. A similar rectangle decomposition algorithm [25] is proposed for enhancing the speed of region segmentation. However, these partitioning algorithms do not achieve a distribution of the task area on average. The divide areas for multi-robot coverage path planning (DARP) algorithm [26] is proposed as an approach for achieving average partitioning. By continuously iterating the values of the grid evaluation matrix, the algorithm can gradually achieve the goal of achieving an average distribution of regions among multiple agents. However, it is important to note that the different initial positions of the agents significantly impact the speed and effectiveness of the partitioning process.

After partitioning the space, it is essential to strategically plan the path for covering each distinct area block. A large number of researchers have applied the spanning tree coverage (STC) method [9,13,14,15,27] to single-agent CPP after delineating regions. A minimum STC algorithm is then employed to minimize conflicts among the agents [28]. However, this algorithm falls short in achieving the complete coverage of areas such as holes or similar unidirectionally connected spaces. As another widely used algorithm for intra-area coverage, the GBNN algorithm is a biologically inspired neural network CPP algorithm that generates coverage paths by adjusting neuron activity values [2,29,30]. However, when agents encounter consecutive obstacles, a path planned by a human template is necessary to prevent agents from falling into a dead zone.

Environment representation methods ensure comprehensive coverage of the working space and prevent path repetition by constraining agents’ movements to distinct subregions. However, challenges arise due to variations in agent numbers, initial positions, and the intricate distribution of obstacles, making it difficult to achieve consistently reasonable partitioning results. Finding a universally applicable division strategy for all conditions becomes especially challenging, and this difficulty escalates when addressing scenarios in 3D space.

2.2. Agent Dynamic Interaction Methods

The agent dynamic interaction methods are characterized by the real-time coordination of multiple agents, wherein each agent strategically determines its continuous working path based on individual goals and environmental conditions. This continuous path planning enables agents to avoid conflicts and collisions, promoting cooperative and efficient task execution. Some researchers have made enhancements to the GBNN algorithm by incorporating considerations for multiple agents regarding each other as dynamic obstacles [31,32,33]. This allows for the automatic generation of coverage paths within the working area based on the neural network activity values derived from the global map.

In recent years, there has been growing interest in combining collaborative coverage planning problems with reinforcement learning to achieve new breakthroughs [11,17,34]. Some researchers utilize a dual guidance mechanism for area coverage path planning [18]. An approach involves training multiple agents to interact with the target environment using the convolutional neural network (CNN) algorithm. This training allows the agents to achieve the overall area coverage requirement while working together and avoiding conflicts [35]. Reinforcement learning-based path planning algorithms offer advantages such as high adaptability, robustness, and improved performance in uncertain environments. However, these methods typically require a substantial amount of high-quality data for training. Additionally, selecting an appropriate model structure and tuning algorithm hyperparameters significantly increase the complexity of algorithm development and debugging.

In the GBNN algorithm, agents adhere to rule-based behavior, while agents in reinforcement methods acquire appropriate behavior through extensive data learning. Both approaches typically exhibit high execution efficiency. However, reinforcement methods demand substantial training efforts and may exhibit suboptimal performance when applied to environments that differ significantly from their training datasets. In contrast, GBNN, relying on straightforward interactive rules, falls short of meeting the combined requirements, including the evenness of the energy consumption distribution and an overall short working time, in intricate environments.

The framework introduced in this paper shares the inherent logic more akin to that of agent dynamic interaction methods. However, as illustrated in Figure 2, instead of consolidating path generation and allocation into a single step, our approach addresses the path coverage issue through an initial path auto-growth method. Subsequently, path segments are allocated in a second step through negotiation among agents using a newly proposed Swarm Migration Genetic Algorithm. This decouples the MACPP problem, simplifying coverage path generation in 3D space. The optimization, featuring combined objectives such as complete area coverage, high computational efficiency, strong adaptability to complex environments, and improved balance in path assignment and energy consumption, can thus be achieved. The detailed approach for these two steps will be elucidated in Section 4.1 and Section 4.2, respectively.

3. Multi-AUV CPP Problem Formulation

3.1. Problem Statement

The underwater mission environment is a three-dimensional space denoted by ℜ, which is discretized into a finite number of $(x,y,z)$ coordinates:

$$\Re =\left\{\mathrm{x},y,z:x\in \left[0,l\right],y\in \left[0,w\right],z\in \left[-h,0\right]\right\}$$

(1)

where $(x,y,z)$ denotes the discrete coordinates, and $(l,w,h)$ represents the length, width, and depth range of the working space, respectively. The obstacles are distributed in a known set O of cells, and the area to be covered is denoted as $\Pi =\Re -O$.

Assumption 1.

All AUVs covered in this paper are isomorphic agents with an identical detection depth of $d_p$, detection width of $d_w$, and turning radius of $d_r$ during the coverage mission.

The impact of layering and gridding, following Assumption 1 within a working space, is depicted in Figure 3. The inter-layer spacing is denoted as $d_p$, and the grid size of the horizontal layers $d_l$ is determined by the maximum of $d_w$ and $d_r$. Obstacle areas are illustrated in blue cells, while areas to be covered are depicted in flesh-colored cells. Irregularly shaped obstacles are approximated as external cuboids, which may classify some free spaces around the obstacle as impassable. However, this method aligns with the common strategy of expanding obstacle boundaries and map simplification found in many existing studies [2,36].

Assumption 2.

The kinematics and dynamics of the AUVs are designed to accommodate right-angle turns with the selected $d_l$ and $d_p$ values.

We assume that AUVs can follow the planned paths through right-angle turns in Assumption 2. Similar assumptions were used in previous studies [37,38,39], which tended to prioritize traversability and path length during the path planning phase. To account for kinematic constraints, we consider the AUV turning capability (i.e., the turning radius in all directions) when determining the size of the mapping grid. This ensures that the planned paths align closely with the motion capabilities of the AUV.

Assumption 3.

The battery capacity carried by the AUVs is sufficient to support each round-trip mission from the initial location to the designated operation area. That is, the batteries are able to support the need of the AUVs to continuously consume electrical energy.

As shown in Assumption 3, we assume that the AUVs carry enough battery capacity at a time to support a round-trip mission from the initial location to the designated operational area and reach the scheduled resupply point after completing the tasks.

In the initial phase, our objective is to identify a set of disjoint paths, denoted as snap coverage trails Q, capable of fully covering the unoccupied region within the 3D space as described in Definition 1. Subsequently, these trails must be assigned to the AUVs. Assuming a specific assignment strategy yields M trail groups $Q_i,i=1,\dots ,M$; the next step involves determining the optimal sequence for connecting all trails within $Q_i$ to construct a continuous operational path $P_i$ as described in Definition 2.

Definition 1.

Snap coverage trails Q are a set of short paths that fully cover the free region Π.

$$\left\{\begin{array}{c}Q=\left[Q_1,Q_2,\cdots ,Q_M\right]\hfill \\ Q_1\cup Q_2\cup \cdots \cup Q_M=\Pi \hfill \end{array}\right.$$

(2)

where M is the total number of snap coverage trails Q, and $Q_M$ is the M-th trail.

Definition 2.

The planned path $P_i$ for the i-th AUV is defined as a sequence of snap coverage trails Q with designated endpoint orders. The trails Q are connected following the designated order shown in Equation (3):

$$P_i=\left[\pm Q_{x_1},\pm Q_{x_2},\cdots \pm Q_{x_{n_i}}\right],\forall i\in \left\{1,2,\cdots ,M_r\right\}$$

(3)

where $Q_{xj}\in Q$,$\forall j\in \left\{1,2,\cdots ,n_i\right\}$ represents a trail having the order of endpoints as $\left\{S_1^x,S_2^x\right\}$ with the sign +, and $\left\{S_2^x,S_1^x\right\}$ with the sign −, $n_i$ is the number of trails allocated to the i-th AUV, and $M_r$ is the number of AUVs.

Based on the aforementioned definition, the MACPP can be reformulated as a doublefold problem: (1) generate a trail set Q to comprehensively cover all unoccupied space, (2) allocate trails to distinguish AUVs as $Q_i,i=1,\dots ,M$, and establish the optimal connectivity sequence within $Q_i$ to construct a continuous path $P_i$ for each AUV. This paper addresses the first problem through a path auto-generation algorithm in Section 4.1, while the second problem is resolved by tackling a constraint multi-objective optimization problem in Section 4.2.

3.2. AUV Energy Consumption Model

The motion of an AUV during task execution comprises three primary modes: horizontal linear movement, vertical jumping, and horizontal steering.

The horizontal linear motion is depicted in Figure 4a, and the energy consumption $f_{a_i}$ of the i-th AUV moving from $J_1$ to $J_2$ is calculated by the following equation [40]:

$$f_{a_i}={\displaystyle \frac{1}{2}}\rho C_F\left({\alpha }_o\right)S_x{V}_o^2{d}_{x_i}+{\displaystyle \frac{1}{2}}\rho C_F\left({\alpha }_o\right)S_y{V}_o^2{d}_{y_i}$$

(4)

where ${\displaystyle \frac{1}{2}}\rho C_F\left({\alpha }_o\right)S_x{V}_o^2$ and ${\displaystyle \frac{1}{2}}\rho C_F\left({\alpha }_o\right)S_y{V}_o^2$ denote the drag force on the i-th AUV in the x and y directions, respectively. $\rho$ denotes the density of seawater; $C_F\left({\alpha }_o\right)$ represents the resistance coefficient at the angle ${\alpha }_c$ between the drag force and the direction of motion; $S_x$ and $S_y$ indicate the force-bearing length in the x and y directions, respectively; $V_o$ is the velocity of the ocean; $d_{x_i}$ represents the distance sailed by the i-th AUV along the x-axis; and $d_{y_i}$ represents the distance sailed by the i-th AUV along the y-axis.

The vertical jumping motion is shown in Figure 4b, and the energy consumption of the i-th AUV floating up or sinking from $H_1$ to $H_2$ is calculated by the following Equation [39,40,41]:

$$f_{b_i}={\lambda }_p\left({\displaystyle \frac{1}{2}}\rho C_F\left({\beta }_o\right)S_z{V}_o^2{d}_{z_i}\pm G_B{d}_{z_i}\right)$$

(5)

where ${\lambda }_p$ indicates the energy consumption correction factor of the AUV in different depth scenarios; the first term denotes the drag force on the i-th AUV in the z direction, and the second represents the amount of change in the gravitational potential energy. $C_F\left({\beta }_o\right)$ represents the resistance coefficient at the angle ${\beta }_c$ between the drag force and the direction of motion; $S_z$ indicates the force-bearing length in the z directions; $G_B$ denotes a combination of the gravity and buoyancy; and $d_{z_i}$ represents the distance sailed by the i-th AUV along the z-axis.

The common turning styles mainly include “Flat-shaped”, “U-shaped”, “Bulb-shaped”, and “Hook-shaped” [42]. As illustrated in Figure 4c, we only consider the “U-shaped” turnings in this paper. The following equation is the energy consumption of the turning process for the i-th AUV:

$$f_{c_i}=L_T{N}_{T_i}·\left({\displaystyle \frac{\pi d_l}{2}}+{\displaystyle \frac{d_l}{tan\psi }}\right)$$

(6)

where $\psi$ is the angle between the guide direction and the boundary, $L_T$ is the single turning path length, and $N_{T_i}$ is the number of turns of the i-th AUV.

4. Methodology

4.1. Auto-Growth of the Snap Coverage Trails

Snap coverage trails Q are short paths with a combined complete coverage of the 3D free space. The term “short” is relative to the total continuous path length required for the AUV to accomplish its mission. Considering the complex obstacle distributions in the underwater environment, we design a snap coverage trails auto-growth (SCTG) algorithm for efficient trail generation as depicted at the top of the framework figure in Figure 2. The underwater three-dimensional space is layered according to the detection depth $d_p$, and the trails growth in the k-th layer is shown in Figure 5. Firstly, the lower-left grid is selected as the starting point of the first trail growth. The growth direction of each trail is determined by comparing the values of $T_x$ and $T_y$, where $T_x$ represents the maximum generatable length in the x-axis direction and $T_y$ represents the maximum generatable length in the y-axis direction. If the starting point has $T_x>T_y$, the trail grows along the x-axis direction. If the starting point has $T_x\le T_y$, the trail grows along the y-axis direction.

The impassible region ${\Omega }_k$ is the union of the k-th layer already grown region $A_{cov}$ and the obstacle region $O_k$, denoted as ${\Omega }_k=A_{cov}\bigcup O_k$. For each new generated trail $Q_{new}$, its corresponding region needs to be stored in $A_{cov}$, and the range of region ${\Omega }_k$ is updated. The current generated trail $Q_{new}$ is checked to determine if it fully covers the grid column where the starting point is located. If the column is not fully covered, the first uncovered grid in that column along the y-axis is selected as the new starting point. If the column is fully covered, the first uncovered grid in the next column along the y-axis is selected as the new starting point. The growth direction of the new trails follows the same scheme as mentioned earlier, where the longest side is selected from the x and y axes as the growth direction. When $A_{cov}$ in each layer of space equals the entire region of free grid cells, the trails stop growing autonomously, and $\Omega$ stops being updated. This marks the completion of the overall growth of snap coverage trails Q in the underwater multi-layer space. The SCTG algorithm is formally presented as shown in Algorithm 1.

Algorithm 1: Snap coverage trails Q auto-growth algorithm.

4.2. SMGA for Trail Assignment

The remaining problem to be solved is to assign trails to the AUVs from the original set of snap coverage trails Q. The SMGA as illustrated in the lower part of the framework figure in Figure 2 is proposed for solving this path allocation and ordering problem, with the optimal connection described in Section 4.2.2. The specific pseudocode for SMGA has been provided in Algorithm 2.

Algorithm 2: Swarm migration genetic algorithm.

4.2.1. Chromosomes Design

As illustrated in Figure 6, the basic idea for SMGA is to design a chromosome structure in swarms, and each swarm represents the path cluster for an AUV. The sequence of the genes in a swarm stands for the path connection sequence for the specific AUV.

Correspondingly, each chromosome includes two parts $Chrom\_p$ and $Chrom\_r$. The former includes $M_r$ swarms denoted as $Chrom\_p_1,\dots ,Chrom\_p_{M_r}$, where $Chrom\_p^i$ is the trail index sequence for the i-th AUV. $Chrom\_r$ has the same length as $Chrom\_p$, and it records the AUV index for each trail in the current chromosome.

A population for the GA consists of multiple chromosomes and can be represented by:

$$Pop=\left[\begin{array}{cc}Chrom\_p^1& Chrom\_r^1\\ Chrom\_p^2& Chrom\_r^2\\ ⋮& ⋮\\ Chrom\_p^N& Chrom\_r^N\end{array}\right]$$

(7)

$$Chrom\_p^i=[Chrom\_p_1^i,\cdots ,Chrom\_p_{M_r}^i],\forall i\in \left\{1,2,\cdots ,N\right\}$$

(8)

where $Pop$ is the overall population, $Chrom\_p^i$ is the trail index, $Chrom\_r^i$ is the arrangement code of the AUV index corresponding to $Chrom\_p^i$, and N is the number of individuals in the population.

As a result, each row in $Pop$ corresponds to a double-layer chromosome, and individual chromosomes represent information in two aspects: (1) the assignment of the trails Q to different AUVs in the second layer in $Chrom\_r^i$, and (2) the sequence of the trails Q belonging to an AUV in $Chrom\_p^i$.

4.2.2. Fitness Function

The fitness function is a crucial metric for evaluating the quality of chromosomes for all swarms. Essentially, it defines the cost function for the multi-AUV CPP problem in Section 3.1. Based on the problem formulation, the fitness function should include the path assignment unevenness and the total energy consumption. During path allocation, the allocation results among AUVs and the system’s overall energy consumption are optimized based on the fitness function, iteratively. When the fitness function converges to a minimum value, it indicates that the allocation evenness and energy consumption have reached optimal levels, resulting in a cooperative win for the agents.

The balance of the snap coverage trails $Q_i$ allocation is represented by their variations in paths $P_i$. For a given chromosome, the sequence of the trails in one swarm is determined. However, the endpoint connection sequence is not fixed, and we need to obtain the optimal connectivity for the planned paths $P_i$ of each AUV. As shown in Figure 7, the connection sequence of the endpoints between two neighbor trails can affect the total path length.

We construct a dynamic programming problem to find the best connection method of the endpoints for a given trail sequence. The state transition equations are defined in Equations (9) and (10). The two states for the p-th trail connection are defined as $d[1,p]$ and $d[2,p]$. They represent the travel cost of connecting $S_1^p$ or $S_2^p$. $dis(·,·)$ is the distance between two endpoints. With the state equations defined above, dynamic programming can be applied to find the endpoint sequence with the smallest travel cost, which can be calculated by Equation (11):

$$d[1,p]=min(dis(S_1^p,S_1^{p-1})+d[2,p-1],dis(S_1^p,S_2^{p-1})+d[1,p-1])$$

(9)

$$d[2,p]=min(dis(S_2^p,S_1^{p-1})+d[2,p-1],dis(S_2^p,S_2^{p-1})+d[1,p-1])$$

(10)

$$\left|P_i\right|=min(d[1,n_i],d[2,n_i])$$

(11)

where $n_i$ is the number of trails allocated to the i-th AUV, and $\left|P_i\right|$ represents the path length for each AUV.

4.2.3. Swarm Migration Operations

Then, the constraints for the optimization problem in Section 3.1 are given as Equation (12). The first condition ensures that no trail is assigned to more than one AUV, and the second condition ensures that all the trails Q are assigned:

$$\left\{\begin{array}{c}{P}_i\cap P_j=\varnothing ,\forall i,j\in \left\{1,2,\cdots ,M_r\right\}\hfill \\ P_1\cup P_2\cup \cdots \cup P_{M_r}=Q\hfill \end{array}\right.$$

(12)

The optimization problem’s cost function accounts for two key practical considerations. First, it aims to conserve the operational energy of the AUVs. Second, it seeks to achieve a balanced distribution of the working path among the AUVs. This not only reduces their collaborative working time but also ensures the optimal designation of the trail connection sequence. It is formulated by Equation (13):

$$F={\omega }_1·\sum _{i=1}^{M_r}{E}_{c_i}+{\omega }_2·\sqrt{{\displaystyle \frac{1}{M_r}}\sum _{i=1}^{M_r}{\left(\left|P_i\right|-{\displaystyle \frac{1}{M_r}}\sum _{i=1}^{M_r}\left|P_i\right|\right)}^2}$$

(13)

where ${\omega }_i$ are the weight factors. The first term is the total energy consumption, and the second term evaluates the path length inequality among the AUVs. $E_{c_i}$ and $\left|P_i\right|$ can be calculated from Equations (11) and (14), respectively.

$$E_{c_i}=f_{a_i}+f_{b_i}+f_{c_i},\forall i\in \left\{1,2,\cdots ,M_r\right\}$$

(14)

where $E_{c_i}$ represents the task energy consumption of the i-th AUV, and $f_{a_i}$, $f_{b_i}$, and $f_{c_i}$ are calculated by Equations (4)–(6), respectively.

4.2.4. Population Initialization

SMGA exhibits a higher diversity of chromosomes and faces greater challenges in achieving convergence compared to the single-population genetic algorithm. Therefore, in this paper, we use an enhanced K-means algorithm to assign trails for the initialization of the swarms. The coordinates of endpoints for all trails Q can be obtained through the work in Section 4.1. With these endpoint coordinates, the distance matrix between different endpoints can be calculated. In accordance with Figure 7, 4 different ways to connect two trails that do not include the starting point are identified. The sum of the distances between the 4 endpoints is taken, and by averaging them, the average distance between the two trails can be obtained. By iterating this process for all trails Q, the overall average distance matrix for trails can be calculated. To obtain a relatively good initial population chromosome using the K-means algorithm, the average distance matrix and $M_r$ can be used as inputs. The K-means algorithm can be applied to cluster the trails Q based on their distance matrix, and the resulting clusters can be used to initialize the chromosomes of the population. This way, a relatively good initial population chromosome for further genetic operations can be obtained.

Some nature species exhibit behavior for individuals moving from one swarm to another. For instance, in certain bird species, individuals may leave their original flock and join another flock to seek better feeding opportunities or find a more suitable breeding partner. Inspired by such behavior, we propose a swarm migration algorithm that mimics this behavior. Genetic operations in SMGA mainly include operations such as selection, variation, migration, and crossover. The selection and variation operations are similar to traditional GA and are not elaborated here. The migration and crossover operations involve swarm interactions and are explained in detail below.

Migration encourages individuals to move among swarms. For the MACPP problem, it corresponds to changing the assignment of some trails from one AUV to another. As illustrated in Figure 6, the first step in the migration operation is to find the swarms that need to be operated. In this article, we compare the total path lengths for each swarm. The one with the longest path is considered too crowded, and some individuals should migrate to the swarm with the shortest path length. In the second step, some individuals are selected for migration. The migrants are chosen such that their length is closest in the swarm to the difference between the longest and average path lengths.

Crossover is the basic genetic operation of the traditional GA and is widely used in single swarm operations. However, in SMGA, the interactions between chromosomes of different swarms must be considered. As shown in Figure 8, a division operation is performed on the chromosomes of different individuals performing crossovers. Two parent chromosomes are copied and the selected swarm is exchanged at the crossover point. Gene conflicts are handled between the remaining swarms by filling in missing or duplicate entities in order. If the offspring chromosomes need to add new entities to complete, we follow the principle of maintaining the swarm and add the entities to the original swarm to which they belong as shown in chromosome $Cld2$ in Figure 8 with (6,3).

5. Experimental Results and Discussion

In the ensuing sections, we undertake three categories of experiments: the 3D MACPP performance evaluation, ablation experiments, and comparative experiments. For the 3D MACPP tests, we utilize two distinct three-dimensional underwater maps to assess the efficacy of our SCTG-SMGA algorithm. Following that, three sets of ablation experiments are carried out to assess the performance variations of the SCTG-SMGA algorithm under different configurations of critical parameters. Lastly, we benchmark the performance of the SCTG-SMGA algorithm against two previous MACPP methods. The methodologies are executed in Matlab R2022b software on a computer equipped with an Intel^®Xeon^® 8124 M 3.00 GHz CPU and 256 GB of memory.

5.1. Performance Validation Experiments

The relevant experimental parameters for the 3D MACPP tests are detailed in

Table 2. Experimental parameters.

Parameter	Value	Parameter	Value
$\rho$	$1.00\times {10}^3$ kg/m3	$S_y$	$1.00\times {10}^1$ m
$C_F\left({\alpha }_o\right)$	$3.00\times {10}^{-1}$	$S_z$	$1.50\times {10}^0$ m
$C_F\left({\beta }_o\right)$	$5.00\times {10}^{-1}$	$V_o$	$2.00\times {10}^{-1}$ m/s
$S_x$	$5.00\times {10}^{-1}$ m	$d_l$	$1.30\times {10}^2$ m
${\omega }_1$	$5.00\times {10}^{-6}$	${\omega }_2$	$1.00\times {10}^{-4}$
$G_B$	$5.00\times {10}^4$ N	$d_p$	$2.00\times {10}^1$ m

. The values of the energy consumption parameters, including $\rho$, $C_F\left({\alpha }_o\right)$, $C_F\left({\beta }_o\right)$, $S_x$, $S_y$, $S_z$, $V_o$, and $G_B$ are sourced from the literature [40,41]. The AUV optimal detection ranges $d_l$ and $d_p$ are defined based on the findings presented in [43]. The values of ${\omega }_1$ and ${\omega }_2$ were determined through sensitivity analyses, and fine-tuning efforts were applied to strike a balance between path allocation and total energy consumption. We seek to minimize the overall energy consumption while maintaining the balance of assigning paths among the agents, so the weight of ${\omega }_1$ is generally taken to be higher than that of ${\omega }_2$.

As depicted in Figure 9a,b, our study employs two distinct 3D underwater maps to evaluate the SCTG-SMGA algorithm. First, 3D Map1 presents a complex environment with numerous obstacles, while 3D Map2 features a more scattered small obstacle scenario with more passable spaces. Figure 10a,b showcase the SCTG-SMGA algorithm’s three-dimensional coverage path planning outcomes for scenarios with 3 and 6 agents in 3D Map1, and Figure 10c,d depict the corresponding results in 3D Map2. There are a total of 6 vertical layers in both maps. With 3 agents, we observe the interlacement of paths by different agents within the same layer, ensuring a balanced distribution of paths. However, with 6 agents, the algorithm tends to assign one agent to each layer to prevent inter-layer jumping, thus minimizing the overall energy consumption.

The raw data containing path lengths, energy consumption, and algorithm execution time are presented in

Table 3. Experimental data of SCTG-SMGA algorithm in 3D maps.

Type	Agents	SCTG-SMGA algorithm
		Path length of each agent (m)						Average length (m)	Algorithm execution time (s)
3D Map1	3	$1.82\times {10}^5$	$1.81\times {10}^5$	$1.82\times {10}^5$	-	-	-	$1.82\times {10}^5$	65.78
	6	$9.26\times {10}^4$	$8.96\times {10}^4$	$9.24\times {10}^4$	$9.13\times {10}^4$	$9.07\times {10}^4$	$8.85\times {10}^4$	$9.08\times {10}^4$	122.52
3D Map2	3	$1.85\times {10}^5$	$1.90\times {10}^5$	$1.90\times {10}^5$	-	-	-	$1.88\times {10}^5$	71.87
	6	$9.63\times {10}^4$	$9.70\times {10}^4$	$9.37\times {10}^4$	$9.50\times {10}^4$	$9.11\times {10}^4$	$9.20\times {10}^4$	$9.42\times {10}^4$	126.61
Type	Agents	Energy consumption of each agent (kJ)						Average energy (kJ)	Total energy consumption (kJ)
3D Map1	3	$4.26\times {10}^7$	$4.34\times {10}^7$	$4.32\times {10}^7$	-	-	-	$4.31\times {10}^7$	$1.29\times {10}^8$
	6	$2.19\times {10}^7$	$2.21\times {10}^7$	$2.17\times {10}^7$	$2.20\times {10}^7$	$2.11\times {10}^7$	$2.13\times {10}^7$	$2.17\times {10}^7$	$1.30\times {10}^8$
3D Map2	3	$4.51\times {10}^7$	$4.42\times {10}^7$	$4.45\times {10}^7$	-	-	-	$4.46\times {10}^7$	$1.34\times {10}^8$
	6	$2.23\times {10}^7$	$2.20\times {10}^7$	$2.19\times {10}^7$	$2.20\times {10}^7$	$2.21\times {10}^7$	$2.24\times {10}^7$	$2.21\times {10}^7$	$1.33\times {10}^8$

. From the average values, path allocation for 3 agents is evenly distributed in both maps, but the balance diminishes with 6 agents due to the algorithm favoring minimizing total energy cost by avoiding inter-layer jumps. Total energy consumptions are comparable for both 3 and 6 agent scenarios. Map2 requires slightly more energy for full exploration due to its larger open spaces. The converge time costs for 3-agent cases are 53.69% and 56.76% of the 6-agent cases for Maps 1 and 2, respectively, suggesting a nearly proportional relationship between the number of agents and algorithm running time. In summary, the SCTG-SMGA successfully accomplished its role in achieving balanced and efficient path planning among AUVs within intricate 3D environments.

5.2. Ablation Experiments

This study conducts three sets of ablation experiments to evaluate how the performance of SCTG-SMGA varies with changes in the parameters. The experiments MACPP with 3 agents are carried out under different conditions, including variations in depth coefficients (${\lambda }_p$), the presence or absence of swarm clustering initialization, and different numbers of migration paths.

The parameter ${\lambda }_p$ denotes the cost associated with inter-layer jumping. MACPP results for the two 3D maps with ${\lambda }_p$ values of 50, 500, and 5000 are depicted in Figure 11 and

Table 4. Experimental data under different ${\lambda }_p$ in 3D maps.

Type	${\mathit{\lambda }}_{\mathit{p}}$	Path length of each agent (m)			Average length (m)	Algorithm converge time (s)
3D Map1	50	$1.81\times {10}^5$	$1.82\times {10}^5$	$1.82\times {10}^5$	$1.82\times {10}^5$	66.71
	500	$1.81\times {10}^5$	$1.82\times {10}^5$	$1.82\times {10}^5$	$1.82\times {10}^5$	65.98
	5000	$1.85\times {10}^5$	$1.78\times {10}^5$	$1.82\times {10}^5$	$1.82\times {10}^5$	64.65
3D Map2	50	$1.91\times {10}^5$	$1.85\times {10}^5$	$1.89\times {10}^5$	$1.88\times {10}^5$	61.98
	500	$1.88\times {10}^5$	$1.91\times {10}^5$	$1.87\times {10}^5$	$1.88\times {10}^5$	63.62
	5000	$1.88\times {10}^5$	$1.91\times {10}^5$	$1.86\times {10}^5$	$1.88\times {10}^5$	59.34
Type	${\lambda }_p$	Energy consumption of each agent (kJ)			Average energy (kJ)	Total energy consumption (kJ)
3D Map1	50	$4.30\times {10}^7$	$4.32\times {10}^7$	$4.36\times {10}^7$	$4.33\times {10}^7$	$1.30\times {10}^8$
	500	$1.34\times {10}^8$	$1.35\times {10}^8$	$1.34\times {10}^8$	$1.35\times {10}^8$	$4.04\times {10}^8$
	5000	$6.36\times {10}^8$	$6.13\times {10}^8$	$6.25\times {10}^8$	$6.25\times {10}^8$	$1.87\times {10}^9$
3D Map2	50	$4.49\times {10}^7$	$4.37\times {10}^7$	$4.49\times {10}^7$	$4.45\times {10}^7$	$1.33\times {10}^8$
	500	$1.37\times {10}^8$	$1.46\times {10}^8$	$1.38\times {10}^8$	$1.41\times {10}^8$	$4.22\times {10}^8$
	5000	$6.46\times {10}^8$	$6.51\times {10}^8$	$6.40\times {10}^8$	$6.46\times {10}^8$	$1.94\times {10}^9$

. As ${\lambda }_p$ increases, the SCTG-SMGA algorithm tends to reduce inter-layer jumps to prevent a significant increase in the total energy costs. Consequently, with ${\lambda }_p$ assigned a very large value of 5000, the algorithm allocates two neighboring layers for each AUV to avoid any unnecessary jumps.

The MACPP for 3D Map1 with and without swarm clustering initialization (as in Section 4.2.4) and the corresponding convergence curves are depicted in Figure 12. Convergence of the SMGA is considered achieved when the difference in F between iterations is less than 0.08 of its own value. When swarm clustering is initialized, the algorithm converges at the 1043th iteration with a fitness value of 123.67. Without swarm clustering initialization, convergence occurs at the 1508th iteration with a fitness value of 158.53. The final fitness value with clustering initialization is 78.01% of the convergence value without initialization clustering, with a convergence speed that is 30.84% higher. Similarly, the results of three-dimensional coverage path planning are presented for 3D Map2 with and without swarm clustering initialization in Figure 13. The trends align with the performance for 3D Map1, achieving a fitness value of 67.96% and 45.60% fewer converging iterations when incorporating clustering initialization. The initialization considerably decreases the number of iterations to converge and improves the final performance.

The migration number among swarms in the SMGA is the final parameter under examination. In Figure 14, the MACPP results for 3D Map1 with migration path numbers 1 and 5 are depicted in (a) and (b), respectively, with corresponding convergence curves in (c) and (d). Similarly, Figure 15 shows the MACPP results for 3D Map2 with migration path numbers 1 and 5 in (a) and (b), with corresponding convergence curves in (c) and (d). For both maps, the impact of elevating the migration number from 1 to 5 is uniform: it expedites the convergence speed by 61.79% and 29.10%, respectively, concurrently elevating the final fitness value by 6.06% and 9.6%, respectively. This aligns with the expectation that a higher migration number accelerates the allocation of entities among different swarms. However, it also compromises some capacity for nuanced adjustments, resulting in a slightly larger final fitness value.

5.3. Comparative Experiments

Due to the absence of existing research on MACPP methods specifically tailored for 3D environments, comparative experiments must be conducted in 2D complex maps. We have selected GBNN [29] and DARP-STC [26] algorithms for comparison. GBNN is an innovative bio-inspired CPP algorithm recognized for its rapid computational speed. DARP-STC is widely employed in 2D CPP due to its superior multi-agent balanced allocation. As illustrated in Figure 16, this paper presents three sets of comparative experiments for the algorithm, involving non-concave 2D maps with varying obstacle distributions, including hollow shapes and narrow connections.

The coverage path generation outcomes for the SCTG-SMGA, GBNN, and DARP-STC algorithms with 3, 4, and 5 agents are depicted in Figure 17, Figure 18 and Figure 19, respectively. The paths generated by the SCTG-SMGA algorithm demonstrate a more evenly distributed coverage compared to the GBNN algorithm and notably involve fewer turns compared to the DARP-STC algorithm.

Quantitative results concerning the path length, number of turns, and energy consumption for the three algorithms are summarized in

Table 5. Comparison of experimental data under three algorithms.

Type	Agents	Length standard deviation (m)			Total Length (m)			Number of turns
		SCTG-SMGA	GBNN	DARP-STC	SCTG-SMGA	GBNN	DARP-STC	SCTG-SMGA	GBNN	DARP- STC
2D Map1	3	368	39051	245	$\mathbf{2.71}\times {\mathbf{10}}^{\mathbf{5}}$	$2.92\times {10}^5$	$\mathbf{2.71}\times {\mathbf{10}}^{\mathbf{5}}$	154	176	261
	4	306	41083	225	$2.72\times {10}^5$	$2.86\times {10}^5$	$\mathbf{2.71}\times {\mathbf{10}}^{\mathbf{5}}$	158	168	290
	5	659	25671	208	$2.72\times {10}^5$	$2.88\times {10}^5$	$\mathbf{2.71}\times {\mathbf{10}}^{\mathbf{5}}$	154	160	357
2D Map2	3	324	69676	0	$2.49\times {10}^5$	$2.55\times {10}^5$	$\mathbf{2.48}\times {\mathbf{10}}^{\mathbf{5}}$	130	134	203
	4	225	67941	225	$\mathbf{2.48}\times {\mathbf{10}}^{\mathbf{5}}$	$2.55\times {10}^5$	$\mathbf{2.48}\times {\mathbf{10}}^{\mathbf{5}}$	126	132	295
	5	416	31453	255	$\mathbf{2.48}\times {\mathbf{10}}^{\mathbf{5}}$	$2.48\times {10}^5$	$\mathbf{2.48}\times {\mathbf{10}}^{\mathbf{5}}$	133	140	251
2D Map3	3	738	40602	0	$3.20\times {10}^5$	$3.21\times {10}^5$	$\mathbf{3.18}\times {\mathbf{10}}^{\mathbf{5}}$	138	132	270
	4	294	37618	0	$3.20\times {10}^5$	$3.18\times {10}^5$	$\mathbf{3.18}\times {\mathbf{10}}^{\mathbf{5}}$	134	136	276
	5	874	36274	255	$3.21\times {10}^5$	$3.22\times {10}^5$	$\mathbf{3.18}\times {\mathbf{10}}^{\mathbf{5}}$	130	128	310
Type	Agents	Energy standard deviation (kJ)			Total energy consumption (kJ)			Algorithm execution time (s)
		SCTG-SMGA	GBNN	DARP-STC	SCTG-SMGA	GBNN	DARP-STC	SCTG-SMGA	GBNN	DARP- STC
2D Map1	3	$\mathbf{2.77}\times {\mathbf{10}}^{\mathbf{5}}$	$7.54\times {10}^6$	$3.63\times {10}^5$	$\mathbf{5.46}\times {\mathbf{10}}^{\mathbf{7}}$	$5.91\times {10}^7$	$5.80\times {10}^7$	6.55	6.84	11.43
	4	$\mathbf{2.07}\times {\mathbf{10}}^{\mathbf{5}}$	$8.31\times {10}^6$	$4.10\times {10}^5$	$\mathbf{5.49}\times {\mathbf{10}}^{\mathbf{7}}$	$5.78\times {10}^7$	$5.89\times {10}^7$	18.40	19.25	26.12
	5	$\mathbf{1.36}\times {\mathbf{10}}^{\mathbf{5}}$	$5.07\times {10}^6$	$5.45\times {10}^5$	$\mathbf{5.48}\times {\mathbf{10}}^{\mathbf{7}}$	$5.79\times {10}^7$	$6.10\times {10}^7$	25.49	27.75	39.15
2D Map2	3	$\mathbf{1.77}\times {\mathbf{10}}^{\mathbf{5}}$	$1.21\times {10}^7$	$4.14\times {10}^5$	$\mathbf{4.98}\times {\mathbf{10}}^{\mathbf{7}}$	$5.10\times {10}^7$	$5.19\times {10}^7$	14.81	9.48	20.91
	4	$\mathbf{2.79}\times {\mathbf{10}}^{\mathbf{5}}$	$1.32\times {10}^7$	$7.53\times {10}^5$	$\mathbf{4.95}\times {\mathbf{10}}^{\mathbf{7}}$	$5.10\times {10}^7$	$5.49\times {10}^7$	20.32	20.16	60.89
	5	$\mathbf{2.49}\times {\mathbf{10}}^{\mathbf{5}}$	$5.99\times {10}^6$	$3.94\times {10}^5$	$\mathbf{4.97}\times {\mathbf{10}}^{\mathbf{7}}$	$4.99\times {10}^7$	$5.35\times {10}^7$	25.18	29.57	35.39
2D Map3	3	$\mathbf{4.03}\times {\mathbf{10}}^{\mathbf{5}}$	$6.52\times {10}^6$	$6.50\times {10}^5$	$\mathbf{6.31}\times {\mathbf{10}}^{\mathbf{7}}$	$\mathbf{6.31}\times {\mathbf{10}}^{\mathbf{7}}$	$6.69\times {10}^7$	8.95	6.99	14.89
	4	$3.20\times {10}^5$	$6.62\times {10}^6$	$\mathbf{2.03}\times {\mathbf{10}}^{\mathbf{5}}$	$6.29\times {10}^7$	$\mathbf{6.26}\times {\mathbf{10}}^{\mathbf{7}}$	$6.71\times {10}^7$	12.67	12.37	18.23
	5	$4.79\times {10}^5$	$6.49\times {10}^6$	$\mathbf{1.01}\times {\mathbf{10}}^{\mathbf{5}}$	$\mathbf{6.30}\times {\mathbf{10}}^{\mathbf{7}}$	$6.31\times {10}^7$	$6.82\times {10}^7$	15.08	17.69	22.67

Note: Bold numbers represent the optimal data in that case.

. The SCTG-SMGA algorithm offers substantial advantages in minimizing the overall energy consumption of tasks. Across all map scenarios, SCTG-SMGA exhibits the lowest total energy consumption (with marginal differences of 0.5% in 2D Map3, slightly trailing the GBNN algorithm). Its average task energy consumption is 97.38% and 93% of the other two methods. Despite not explicitly optimizing for energy consumption evenness, the proposed algorithm achieves the minimum standard deviation in energy distribution among agents in the majority of scenarios.

When multi-AUVs collaborate to perform coverage tasks, the balanced allocation paths among the agents are maintained under the premise of minimizing the overall energy consumption so as to achieve maximization of the overall efficiency of the multi-AUV system. Achieving minimal energy use with a relatively balanced path distribution is particularly crucial for underwater missions where the AUVs have limited onboard energy resources. In terms of path length, SCTG-SMGA maintains a favorable balance among AUVs, with an average standard deviation of 467 m, representing only 0.017% of the total path length. DARP-STC demonstrates exceptional performance in evenly allocating the path to different AUVs, with a path length standard deviation of only 157 m. However, this achievement comes at the cost of significantly higher energy consumption, introducing a total energy 7.04% higher than SCTG-SMGA due to a large number of turning actions. On the other hand, GBNN exhibits a path length standard deviation as large as $3.89\times {10}^5$ m, which could result in unbalanced workloads and low overall working efficiency during multi-AUV cooperation. Overall, SCTG-SMGA demonstrates impressive performance in both path allocation evenness and total energy savings, making it a compelling choice as a planner for multi-AUV cooperation.

In terms of computational time costs, SCTG-SMGA exhibits the most efficient performance, outperforming in five out of nine tests. Its time costs display a nearly linear correlation with the number of agents. Known for swift execution, GBNN remains highly competitive with our SCTG-SMGA method, showing only a modest 1.80% increase in the average time cost. Conversely, DARP-STC not only has significantly longer time consumption than the other two but also exhibits heightened sensitivity to both map configurations and initial agent conditions. The longest runtime, 60.89 s, is recorded on 2D Map2 for four agents, despite not having the largest agent count in the tests. In summary, SCTG-SMGA achieves commendable performance with minimal time cost in most cases.

Overall, experiments demonstrate that SCTG-SMGA achieves a better balance between path allocation and energy optimization. On the contrary, the optimization objectives of GBNN and DARP-STC align with the energy consumption and path allocation evenness factors in this study, respectively. While GBNN often achieves optimal energy consumption, its path allocation is highly inefficient, potentially leading to system inefficiencies. Conversely, DARP-STC ensures even path distribution but results in numerous turns, increasing energy consumption. Additionally, SCTG-SMGA generally outperforms in algorithmic running time. However, due to the stochastic nature of genetic algorithms, SCTG-SMGA may yield different results when run multiple times. Typically, it provides good planning results, but in the event of an anomaly where the algorithm converges to a local minimum, it may be necessary to adjust key GA parameters to achieve better outcomes.

6. Conclusions

This paper presents a comprehensive solution for the MACPP problem in the challenging marine domain of 3D underwater space, specifically focusing on AUV applications. The introduced framework, consisting of autonomous coverage path growth and path allocation, adeptly addresses the intricate challenges posed by multi-agent cooperative path planning in complex underwater environments. The SMGA is introduced as an innovative approach, showcasing its effectiveness in optimizing both the allocation and intelligent connectivity of multi-agent coverage paths. The algorithm’s incorporation of a novel migration operation, along with path clustering in the initialization phase, contributes to improved convergence capabilities.

The experimental application of the proposed method to the cooperative exploration task of multiple AUVs in a 3D underwater environment demonstrates its effectiveness in considering comprehensive metrics, including overall energy consumption, path allocation balance, and computational efficiency. The results indicate a significant reduction in task energy consumption when compared with previous works, highlighting the algorithm’s efficiency under varying map complexities. Furthermore, ablation experiments are conducted to explore the algorithm’s sensitivity to parameter variations, shedding light on the impact of depth coefficients, swarm clustering initialization, and the number of migration paths. These experiments provide valuable insights into the algorithm’s robustness and performance under different conditions. Comparative experiments with existing methods, including GBNN and DARP-STC, showcase the superiority of the proposed SCTG-SMGA algorithm in achieving a balanced path allocation among AUVs, minimizing energy consumption, and demonstrating competitive computational efficiency. However, due to the stochastic nature of genetic algorithms, SCTG-SMGA may yield different results when run multiple times. Typically, it provides good planning results, but in the event of an anomaly where the algorithm converges to a local minimum, it may be necessary to adjust key GA parameters to achieve better outcomes.

In summary, this paper contributes to the field of multi-agent cooperative path planning in 3D underwater environments. In addition, the proposed framework and algorithm offer a practical and efficient solution to challenges beyond multi-AUV underwater missions, such as more general task allocation problems. In future work, the proposed framework and algorithm can be extended to address broader challenges beyond multi-AUV underwater missions. For instance, with a modified fitness model and path generation mechanism that interacts with the environment, this method could enable rapid path replanning for clusters of unmanned systems in dynamic environments.