Distributed Cooperative Multi-Target Search for an Autonomous Underwater Vehicle Swarm in Unknown 3D Underwater Environments

Abstract

This paper investigates the problem of multi-target search by an Autonomous Underwater Vehicle (AUV) swarm in unknown three-dimensional (3D) underwater environments with obstacles under limited communication conditions. To address this problem, a distributed cooperative search framework is proposed. Within this framework, an adaptive dual-state search mechanism driven by a target response function is designed. This mechanism enables the swarm to transition between independent large-scale roaming search and precise cooperative search. On this basis, a multi-target search method is developed by integrating a virtual force model, motion-constrained 3D Particle Swarm Optimization (PSO), and a sectional 3D tangent-plane obstacle-avoidance method. Simulation results demonstrate the effectiveness and engineering feasibility of the proposed framework. Under the conditions of unknown terrains and communication limits, the AUV swarm can adaptively execute state transitions, safely avoid 3D obstacles, and complete multi-target search tasks. Specifically, as the swarm size increases from 30 to 60 AUVs, the mean number of iterations drops from 432.97 to 269.73, while the total energy consumption expectedly rises from 11.79 × 10⁴ to 15.51 × 10⁴, reflecting a well-balanced trade-off between efficiency and cost. This study provides a practical distributed control reference for AUV swarms in complex communication-constrained underwater scenarios.

1. Introduction

AUVs are unmanned mobile platforms that can navigate underwater and perform sensing, inspection, search, and data collection tasks without real-time human control. They are important tools for marine exploration and underwater missions [1]. Driven by the development of applications such as ocean resource exploration [2], underwater rescue [3], marine environmental monitoring [4], underwater sensor data collection [5], and coverage optimization in underwater sensor networks [6], AUV-based target search and cooperative operation have received increasing attention. Limited by its sensing range and operational efficiency, a single-AUV system can hardly satisfy the practical requirements of multi-target search in complex underwater environments. By contrast, through distributed cooperation, an AUV swarm can expand search coverage, improve target detection, and reduce the impact of individual platform failures on the overall mission. Therefore, AUV swarms have become an important technical approach for addressing underwater multi-target search problems [7].

In recent years, extensive research has been devoted to AUV target search. From the perspective of algorithmic implementation, existing approaches can be broadly divided into Bio-inspired Neural Networks (BNNs), reinforcement learning (RL), heuristic optimization algorithms, and other methods.

BNN methods have been widely studied in AUV target search. Cao et al. [8,9,10] applied the Glasius BNN (GBNN) to 3D underwater environments, where dynamic propagation among neurons was used to realize target search and obstacle avoidance, and further combined it with a self-organizing map to optimize task allocation. Song et al. [11] verified the real-time obstacle-avoidance capability of the GBNN in two-dimensional environments with dynamic obstacles and achieved multi-target search. Li et al. [12] reformulated the GBNN update process as a convolution operation and improved global information propagation through mean pooling and multi-step convolution. Sun et al. [13] combined fuzzy control with a BNN to improve trajectory smoothness and navigation safety. These methods do not require offline training. Instead, their parameters are preset and can adapt online to environmental changes. However, further improvements are still needed in complex marine environment modeling and computational complexity control.

RL methods have provided new adaptive decision-making tools for AUV target search. Wang et al. [14] incorporated spatiotemporal information into the multi-agent Deep Deterministic Policy Gradient algorithm and constructed an exploration map to reduce repeated search. Cai et al. [15] combined fuzzy clustering with an RL based on policy iteration and achieved efficient cooperative search over large areas based on regional importance. Liu et al. [16] proposed autonomous collaborative search learning, which improved cooperative decision-making for multiple AUVs through an information fusion mechanism. Hou et al. [17] combined the Age of Information mechanism with multi-agent reinforcement learning to investigate a heterogeneous unmanned-system cooperative optimization method for AUV target search in complex underwater environments. Masmitja et al. [18] proposed an underwater target localization method that combines Deep RL with Least Squares and Particle Filter estimation to achieve range-only target localization using AUVs and Autonomous Surface Vehicles. However, existing RL methods still face several limitations, including low sample efficiency, poor training stability, limited generalization ability, and insufficient consideration of underwater communication constraints.

Heuristic optimization algorithms have also been widely applied to AUV target search. Yan et al. [19] proposed an improved Ant Colony Optimization algorithm with fuzzy logic and a dynamic pheromone evaporation mechanism, which improved the efficiency of AUV search path planning. Ni et al. [20] developed a three-stage cooperative search strategy based on an improved Dolphin Swarm Algorithm, which enhanced the search efficiency and environmental adaptability of multiple AUVs in unknown 3D environments. Li et al. [21] addressed target search in a heterogeneous AUV–USV system and proposed a local dynamic predictive control framework combined with the Lévy flight method to improve space exploration capability and system robustness. Under local prior information, Li et al. [22] proposed a two-layer search strategy based on the Grey Wolf-enhanced Remora Optimization Algorithm, which improved the efficiency and stability of cooperative search by multiple AUVs. You et al. [23] combined Voronoi diagrams with a biological competition model to allocate task regions for multiple AUVs and further proposed an improved Particle Swarm Optimization method for search path planning within subregions. Although existing heuristic methods perform well in task allocation and path optimization, they still lack a unified framework for task allocation and safe obstacle avoidance in multi-target search by AUV swarms in unknown three-dimensional obstacle environments.

In addition, some researchers have explored AUV target search from other algorithmic perspectives. For complete-coverage path planning, Yao et al. [24] proposed a hierarchical search method that combines region decomposition with adaptive elliptical spiral coverage for efficient search. Zhang et al. [25] adopted K-means+ for dynamic cooperative partitioning and used the Optimal Allocation Minimum Spanning Tree to generate smooth backtrack-free paths. For region partitioning and task allocation, Ling et al. [26] performed multi-AUV task region division based on Minimum Spanning Tree clustering. Wang et al. [27] formulated multi-AUV task allocation as a capacitated vehicle routing problem and used the SCIP solver to obtain energy-constrained solutions. Ling et al. [28] further improved multi-AUV cooperative search efficiency by combining Minimum-Spanning-Tree-based target clustering with path optimization. For adaptive search in unknown environments, Li et al. [29] used forward-looking sonar information to guide multiple AUVs to search unfinished subregions and proposed a multi-AUV adaptive predictive search algorithm for unknown 3D environments, achieving target search and localization. In terms of heterogeneous cooperation and encirclement, Xue et al. [30] proposed a USV–AAV heterogeneous cooperative search algorithm based on time-series constraints, while Li et al. [31] adopted distributed dynamic predictive control to realize cooperative search and encirclement. Although these methods have advanced AUV target search, they still face limitations in terms of their environmental adaptability, communication constraints, and cooperative capability in 3D scenarios.

Based on the above review, existing studies still have several limitations. First, some methods rely on global information, prior environmental knowledge, or centralized coordination, which may be difficult to obtain in unknown underwater environments with limited communication. Second, many studies focus on a single aspect of the problem, such as task allocation, path optimization, coverage planning, or obstacle avoidance, while the integration of cooperative multi-target search and safe navigation in a unified distributed framework remains insufficient. Third, the cooperative response of AUV swarms to multiple detected targets has not been adequately addressed, especially under local communication and three-dimensional motion constraints.

To address the limitations identified above, particularly the reliance on global information and the insufficient consideration of multi-target response under constrained communication conditions, this paper investigates a distributed collaborative search framework for AUV swarms operating in unknown 3D underwater environments with obstacles. The proposed framework enables AUVs to coordinate search activities using local sensing and communication information, while dynamically adjusting their search behaviors according to target-response information. Specifically, AUVs can switch between independent exploration and target-oriented cooperation, allowing vehicles to conduct cooperative and precise target searches while continuing to explore the environment. This design aims to support effective distributed multi-target search while satisfying navigation safety requirements in complex underwater environments. The main contributions of this paper are summarized as follows:

⮚

A distributed collaborative search framework is established for multi-target search by AUV swarms in unknown 3D underwater obstacle environments under limited communication conditions.

⮚

An adaptive dual-state search mechanism is designed, where the roaming state enables rapid exploration of the task area to detect targets, and the cooperative state focuses on precise target search. Driven by a target response function, AUVs seamlessly transition between these two states to ensure efficient multi-target search.

⮚

A motion-constrained 3D search strategy is developed by integrating virtual-force-based swarm interaction, PSO-based search updating, and tangent-plane obstacle avoidance to jointly address search efficiency, coordination, and navigation safety.

⮚

Simulation studies in unknown 3D obstacle environments demonstrate the effectiveness of the proposed framework in terms of adaptive state switching, multi-target search, and obstacle avoidance under limited communication conditions.

The remaining content of this paper is organized as follows. Section 2 introduces the multi-target search model. In Section 3, the search method is proposed. In Section 4, the method is verified by simulation. The conclusions are given in Section 5.

2. Modeling of Seafloor Target Search in Unknown Environments

The search task is performed in a 1000 m × 1000 m × 200 m underwater environment, with a horizontal area of 1000 m × 1000 m and a vertical range of 200 m. The seafloor elevation difference is less than 200 m and the seabed slope changes continuously, implying that no cliffs are present. The task entities include AUVs, targets, and obstacles. The AUV swarm is denoted by

$$\mathit{U}=\left\{{auv}_i|i=1,2,\dots ,n_u;{30\le n}_u\le 100\right\}$$

(1)

where ${\mathrm{a}\mathrm{u}\mathrm{v}}_i$ denotes the $i$-th AUV. The target set is denoted by

$$\mathit{T}=\left\{{tar}_j|j=1,2,\dots ,n_T;n_T\ge 1\right\}$$

(2)

By discretizing the seafloor, the obstacle set is obtained as

$$\mathit{S}=\left\{{obs}_k|k=1,2,\dots ,n_s;n_s\ge 1\right\}$$

(3)

At time $t$, the positions of ${auv}_i$, ${tar}_j$, and ${obs}_k$ are denoted by ${\mathit{a}\mathit{u}\mathit{v}}_i\left(t\right)$, ${\mathit{t}\mathit{a}\mathit{r}}_{\mathit{j}}$, and ${\mathit{o}\mathit{b}\mathit{s}}_k$, respectively, and the velocity of ${auv}_i$ is denoted by ${\mathit{V}}_i\left(t\right)$. The Euclidean distances among entities are defined as follows: the distance between two AUVs is

$${dv}_{i_1,i_2|t}=‖{\mathit{a}\mathit{u}\mathit{v}}_{{\mathit{i}}_{\mathbf{1}}}(\mathit{t})-{\mathit{a}\mathit{u}\mathit{v}}_{{\mathit{i}}_{\mathbf{2}}}(\mathit{t})‖,$$

(4)

the distance between an AUV and a target is

$${dvt}_{i,j|t}=‖{\mathit{a}\mathit{u}\mathit{v}}_{\mathit{i}}(\mathit{t})-{\mathit{t}\mathit{a}\mathit{r}}_{\mathit{j}}‖,$$

(5)

and the distance between an AUV and an obstacle is

$${dvs}_{i,k|t}=‖{\mathit{a}\mathit{u}\mathit{v}}_{\mathit{i}}\left(\mathit{t}\right)-{\mathit{o}\mathit{b}\mathit{s}}_{\mathit{k}}‖.$$

(6)

Within the mission environment, the target search task can be described as follows. Given a target-reaching threshold $d_0$, if there exists an AUV such that

$${dvt}_{i,j|t}<d_0.$$

(7)

then target ${\mathrm{t}\mathrm{a}\mathrm{r}}_j$ is regarded as successfully found. The search mission is complete when all targets have been found. In practice, $d_0$ is calibrated based on the effective range of onboard target-verification sensors (e.g., optical cameras or other close-range observation devices).

The AUVs participating in the search are homogeneous. The velocity of ${\mathrm{a}\mathrm{u}\mathrm{v}}_i$, denoted by $V_i\left(t\right),$ satisfies $0\le V_i\left(t\right)\le V_m$. Each AUV can be located at any obstacle-free position within the mission environment. Let $d_{\mathrm{c}\mathrm{o}\mathrm{m}}$, $d_{\mathrm{o}\mathrm{b}\mathrm{s}}$, and $d_{\mathrm{t}\mathrm{a}\mathrm{r}}$ denote the maximum communication range, obstacle sensing range, and target detection range, respectively. The communication between AUVs is limited by the maximum communication range $d_{\mathrm{c}\mathrm{o}\mathrm{m}}$. Two AUVs can communicate only when

$${dv}_{i_1,i_2|t}\le d_{com}.$$

(8)

In addition, an AUV can detect a target when $d_{vt(i,j\mid t)}\le d_{\mathrm{t}\mathrm{a}\mathrm{r}}$ and can perceive an obstacle when $d_{vs(i,k\mid t)}\le d_{\mathrm{o}\mathrm{b}\mathrm{s}}$. In practice, $d_{com}$ is determined by the capability of acoustic modems, while $d_{\mathrm{t}\mathrm{a}\mathrm{r}}$ and $d_{\mathrm{o}\mathrm{b}\mathrm{s}}$ are calibrated based on the specifications of onboard detection and obstacle-avoidance sensors.

The targets are assumed to be stationary and located within a central 800 m × 800 m × 200 m region of the mission environment, with a vertical distance of 0–20 m from the seafloor. During the search process, each AUV can continuously detect surrounding target signals through onboard sensors. The target signal is related to $d_{vt(i,j\mid t)}$ through an environment-disturbed function, which is referred to as the target response function and is given in Equation (9):

$$I_{i,j}(t)=\left\{\begin{array}{cc}{\displaystyle \frac{mQ}{{dvt}_{i,|t}^2}}+\eta & {dvt}_{i,j|t}\le d_{tar}\\ 0& {dvt}_{i,j|t}>d_{tar}.\end{array}\right.$$

(9)

where $I_{i,j}(t)$ is the target signal detected by ${auv}_i$ at time $t$, $\eta$ represents the stochastic ambient noise, $m$ denotes the environmental attenuation coefficient with $0<m<1$, and $Q$ denotes the constant signal power of the target. The term $d_{vt(i,j\mid t)}$ denotes the actual AUV–target distance in the simulation environment, which is introduced only for target signal modeling and cannot be directly obtained by the AUV. This formulation replicates the physical characteristics of underwater sensory signals decaying over distance and suffering from environmental interference, thereby providing a realistic simulation environment for algorithm validation [32].

The seafloor surface is discretized into stationary obstacle points with a spacing of 1 m, and the position of each obstacle point is expressed as ${\mathit{o}\mathit{b}\mathit{s}}_{\mathit{k}}=[{xs}_k,{ys}_k,{zs}_k]$.

When operating in the three-dimensional mission environment, an AUV can obtain its own position and velocity through onboard sensors, communicate with other AUVs within its communication range, and detect obstacle information within its sensing range. Accordingly, the state of the AUV is defined as follows:

$$\left\{\begin{array}{l}{\mathit{a}\mathit{u}\mathit{v}}_{\mathit{i}}(t)=\left[{xu}_{i|t},{yu}_{i|t},{zu}_{i|t}\right]\\ {\mathit{V}}_{\mathit{i}}(t)=\left[\dot{{xu}_{i|t}},\dot{{yu}_{i|t}},\dot{{zu}_{i|t}}\right]\\ \dot{{xu}_{i|t}}={\displaystyle \frac{d{xu}_{i|t}}{dt}}=‖{\mathit{V}}_{\mathit{i}}\left(t\right)‖\mathrm{c}\mathrm{o}\mathrm{s}\theta \mathrm{s}\mathrm{i}\mathrm{n}\varphi \\ \dot{{yu}_{i|t}}={\displaystyle \frac{d{yu}_{i|t}}{dt}}=‖{\mathit{V}}_{\mathit{i}}\left(t\right)‖\mathrm{c}\mathrm{o}\mathrm{s}\theta \mathrm{c}\mathrm{o}\mathrm{s}\varphi \\ \dot{{zu}_{i|t}}={\displaystyle \frac{d{yu}_{i|t}}{dt}}=‖{\mathit{V}}_{\mathit{i}}\left(t\right)‖\mathrm{s}\mathrm{i}\mathrm{n}\theta \end{array}\right.$$

(10)

where ${xu}_{i|t}$, ${yu}_{i|t}$ and ${zu}_{i|t}$ denote the coordinates of the $i$-th AUV at time $t$ in the global Cartesian coordinate system $xoyz$, respectively. ${\mathit{V}}_{\mathit{i}}\left(t\right)$ denotes the velocity of the $i$-th AUV at time $t$; $\varphi$ denotes the yaw angle, and $\theta$ denotes the pitch angle.

3. Proposed Method

3.1. Response-Threshold-Based Multi-Target Task Allocation

To improve the coordination and efficiency of target search by an AUV swarm, three states are defined for the AUVs: roaming search, cooperative search, and a task-completed state.

As shown in Figure 1, when no target information is available, an AUV remains in the roaming search state, in which the AUVs repel and disperse from one another at the maximum speed to rapidly explore the task environment. Once a target signal is detected, a Response-Threshold-based Multi-Target Task Allocation (RT-MTTA) model is employed to form sub-alliances. The AUVs in the same sub-alliance then search for the corresponding target, and their states switch to the cooperative search state. When the distance between an AUV and a target is smaller than a given threshold, i.e., $d_{vt(i,j\mid t)}<d_0$, the target is regarded as found. The AUV then dissolves the sub-alliance, broadcasts that the target has been found, and continues searching for other targets. The detailed execution of the RT-MTTA is presented in the remainder of this subsection.

During the search process, task allocation mainly depends on how an AUV autonomously organizes, selects between different targets, and decides whether to switch tasks before completing the selected one. Specifically, the AUV first detects the target response values within its sensing range through onboard sensors. If multiple target signals are detected, the response probability of the AUV to each target is calculated using a response probability evaluation model, after which a roulette-wheel selection algorithm is employed to determine the search target. The response probability evaluation is given as follows:

$$p(i,j)={\displaystyle \frac{I_{i,j}^2(t)}{{\displaystyle {\sum }_{k=1}^m}{I}_{i,k|t}^2}},\forall j,k=\{1,2,3,\cdots ,m\}$$

(11)

where $I_{i,j}^2(t)$ denotes the signal of target ${tar}_j$ detected by AUV ${\mathrm{a}\mathrm{u}\mathrm{v}}_i$ at time $t$. If the AUV detects signals from $m$ targets, then $p(i,j)$ denotes the probability that ${auv}_i$ responds to the stimulus of target ${tar}_j$. The roulette-wheel selection is formulated as follows:

$$P(i,j)={\sum }_{k=1}^j{p}_{ik},j=1,2,\cdots ,m$$

(12)

When $P(i,j-1)<r_a<P(i,j)$, AUV ${auv}_i$ selects target ${tar}_j$ for cooperative search, where $r_a~U(0,1)$. This mechanism executes probabilistic selection based on target signal intensities, avoiding a greedy search for localized individual targets and thereby maintaining search diversity.

An AUV can obtain target information in two ways. The AUV that directly detects target signals through onboard sensors is referred to as a Type-I AUV. The AUV that fails to detect target signals but indirectly acquires target information through communication with other AUVs is referred to as a Type-II AUV. If the target information obtained by Type-I and Type-II AUVs corresponds to the same target, they can jointly participate in the corresponding search task. When multiple AUVs are assigned to the same target, they form a sub-alliance for coordinated search.

In the 3D mission environment, uneven AUV resource allocation may occur during sub-alliance formation, such as multiple AUVs being assigned to the same target while only a few AUVs are assigned to another target. To address this issue, a closed-loop regulation mechanism is introduced. After the initial sub-alliance task allocation, the resource allocation of each sub-alliance is re-evaluated. If the number of members in a sub-alliance reaches the upper limit $N_m$, only the top $N_m$ AUVs are retained, while the remaining AUVs withdraw from the alliance and either select another target or switch to the roaming search. If the number of members in a sub-alliance is below $N_m$, the alliance can recruit nearby AUVs to perform their task.

The member selection priorities are defined as follows. Type-I AUVs have higher priority than Type-II AUVs. If the priority is the same, the AUV with the stronger target signal is preferred. If the number of Type-I AUVs is smaller than $N_m$, the remaining members are selected from Type-II AUVs by giving priority to those closer to their communicating AUV. If multiple Type-II AUVs have the same distance to their communicating AUV, the one with the stronger signal is preferred. The detailed rules are listed in

Table 1. Priority of members in sub-alliance $U_8$ of the AUV swarm at $t=36$.

Index	AUV	Type	Target Signal	Distance to Communicating AUV	Priority
1	${auv}_6$	II	-	76.97	4
2	${auv}_8$	II	-	33.08	3
3	${auv}_{10}$	II	-	93.73	6
4	${auv}_{11}$	II	-	27.42	2
5	${auv}_{17}$	II	-	86.53	5
6	${auv}_{26}$	I	1.55	-	1

.

The AUV swarm must not only avoid all obstacles but also search for all targets. To accomplish the target-search tasks efficiently, AUVs can form sub-alliances based on detected target signals and communication with neighboring AUVs, so that multiple AUVs jointly search for the same target. In Table 1, an example of member priority is given for sub-alliance $U_8$. In this case, AUV ${auv}_{26}$ detects the signal of target ${tar}_8$ and then is classified as a Type-I AUV. Since the number of Type-I AUVs is smaller than $U_8$, ${auv}_{26}$ recruits neighboring AUVs within its communication range. After receiving the recruitment message from ${auv}_{26}$, ${auv}_{66}$, ${auv}_8$, ${auv}_{10}$, ${auv}_{11}$, and ${auv}_{17}$ join the search task for target ${tar}_8$, thereby forming sub-alliance $U_8$. According to the member-selection priority rules, the priority order in sub-alliance $U_8$ is ${auv}_{26}>{auv}_6>{uav}_8>{auv}_{10}>{auv}_{11}>{auv}_{17}$, and these members jointly participate in the search task for target ${tar}_8$.

3.2. Roaming Search Based on the Virtual Force Model

When no target signal is detected, the AUVs conduct roaming search to explore the mission area more efficiently. A virtual force model is adopted, in which repulsive forces are generated when the inter-AUV distance is smaller than $min(d_{com},d_{tar})$ to promote rapid dispersion of the AUV swarm. For computational simplicity, each AUV is assumed to be affected only by the two nearest neighboring AUVs.

Assume that the nearest neighboring AUV of ${auv}_i$ is ${auv}_{i_1}$. Their positions at time $t$ are given by ${\mathit{a}\mathit{u}\mathit{v}}_{\mathit{i}}\left(t\right)=\left[{xu}_{i|t},{yu}_{i|t},{zu}_{i|t}\right]$ and ${\mathit{u}\mathit{a}\mathit{v}}_{{\mathit{i}}_{\mathbf{1}}}\left(t\right)=\left[{xu}_{i_1|t},{yu}_{i_1|t},{zu}_{i_1|t}\right]$, respectively, with ${dv}_{i,i_2|t}\le \mathrm{m}\mathrm{i}\mathrm{n}(d_{com},d_{tar})$. The repulsive force exerted by ${auv}_{i_1}$ on ${auv}_i$ is calculated by Equation (13), and its direction is from the former to the latter:

$${\mathit{f}}_{i,i_1}(t)={\displaystyle \frac{c\cdot l_m^2}{{{dv}_{i,i_1|t}}^3}}\left[\left({xu}_{i|t}-{xu}_{i_1|t}\right),\left({yu}_{i|t}-{yu}_{i_1|t}\right),\left({zu}_{i|t}-{zu}_{i_1|t}\right)\right]$$

(13)

where ${\mathit{f}}_{i,i_1}(t)$ denotes the repulsive force exerted by ${\mathrm{a}\mathrm{u}\mathrm{v}}_{i_1}$ on ${\mathrm{a}\mathrm{u}\mathrm{v}}_i$ at time $t$. $l_m$ denotes the obstacle-avoidance enhancement distance, and $c$ is a coefficient for path optimization.

If the two nearest neighboring AUVs of ${auv}_i$ both satisfy ${dv}_{i,|t}\le min(d_{com},d_{tar}),$ the virtual forces acting on ${\mathrm{a}\mathrm{u}\mathrm{v}}_i$ are as illustrated in Figure 2. In the figure, ${xu}_{i_1|t}>{xu}_{i|t}>{xu}_{i_2|t},{yu}_{i_1|t}>{yu}_{i|t}>{yu}_{i_2|t}$, and ${zu}_{i|t}>{zu}_{i_2|t}>{zu}_{i_1|t}$. The virtual forces ${\mathit{f}}_{\mathit{i},{\mathit{i}}_{\mathbf{1}}}(\mathit{t})$ and ${\mathit{f}}_{\mathit{i},{\mathit{i}}_{\mathbf{2}}}(\mathit{t})$ are both calculated according to Equation (13). The resultant virtual force acting on ${auv}_i$ is given by the vector sum

$${\mathit{f}}_{\mathit{i}}(\mathit{t})={\mathit{f}}_{\mathit{i},{\mathit{i}}_{\mathbf{1}}}(\mathit{t})+{\mathit{f}}_{\mathit{i},{\mathit{i}}_{\mathbf{2}}}(\mathit{t})$$

(14)

For an AUV in the roaming search state, its velocity is directed along the resultant virtual force. Accordingly, the desired velocity at the next step is given by Equation (15).

$${\mathit{V}\mathit{e}}_{\mathit{i}}(t+1)=Vm{\displaystyle \frac{{\mathit{f}}_i(t)}{‖{\mathit{f}}_i(t)‖}}$$

(15)

3.3. Cooperative Search Based on Motion-Constrained 3D Particle Swarm Optimization

An AUV swarm is a typical distributed system. Since a mapping exists between an AUV swarm and a particle swarm, Particle Swarm Optimization can be introduced into the AUV cooperative search decision. By further considering the motion constraints and communication limitations of AUVs, a motion-constrained particle swarm model is constructed to compute the desired velocity ${\mathit{V}\mathit{e}}_i\left(t+1\right)$ at the next step, as described below:

$$\left\{\begin{array}{l}{\mathit{V}\mathit{p}}_{\mathit{i}}(\mathit{t}+\mathbf{1})=\omega {\mathit{V}}_{\mathit{i}}(\mathit{t})+c_1{r}_1({{\mathit{a}\mathit{u}\mathit{v}}_{\mathit{i}}}^{\mathbf{\ast }}(\mathit{t})-{\mathit{a}\mathit{u}\mathit{v}}_{\mathit{i}}(\mathit{t}))+c_2{r}_2({{\mathit{g}}_{\mathit{i}}}^{\mathbf{\ast }}(\mathit{t})-{\mathit{a}\mathit{u}\mathit{v}}_{\mathit{i}}(\mathit{t}))\\ {\mathit{V}\mathit{e}}_{\mathit{i}}\left(\mathit{t}+\mathbf{1}\right)={\mathit{V}}_{\mathit{i}}\left(\mathit{t}\right)+\left({\mathit{V}\mathit{p}}_{\mathit{i}}\left(\mathit{t}+\mathbf{1}\right)-{\mathit{V}}_{\mathit{i}}\left(\mathit{t}\right)\right)·\lambda \end{array}\right.$$

(16)

where ${\mathit{a}\mathit{u}\mathit{v}}_i\left(t\right)$ and ${\mathit{V}}_i\left(t\right)$ denote the position vector and velocity vector of AUV ${auv}_i$ at step $t$, respectively; ${\mathit{V}\mathit{p}}_{\mathit{i}}\left(\mathit{t}+\mathbf{1}\right)$ is the velocity obtained directly by particle swarm iteration; and ${\mathit{V}\mathit{e}}_{\mathit{i}}(\mathit{t}+\mathbf{1})$ is the desired velocity vector of AUV ${auv}_i$ at the next step. The parameter $\lambda$ is introduced to account for the inertia of AUV motion, while $\Delta t$ represents the discretization of continuous time and can also be regarded as a step size. $c_1$ and $c_2$ are the individual cognitive coefficient and social cognitive coefficient, respectively; $r_1$ and $r_2$ are random variables in the interval (0, 1); and $\omega$ is the inertia weight. ${{\mathit{a}\mathit{u}\mathit{v}}_i}^{*}\left(t\right)$ denotes the best position experienced by AUV ${auv}_i$ since joining the current sub-alliance, and ${{\mathit{g}}_{\mathit{i}}}^{\mathbf{\ast }}\left(\mathit{t}\right)$ denotes the best position visited by the alliance up to step $t$. The number of particles corresponds to the number of AUVs within the current sub-alliance.

3.4. Three-Dimensional Tangent-Plane Obstacle Avoidance

Target search for near-bottom objects in a 3D seafloor environment is similar to target search in a 2D environment. The seafloor surface can be regarded as a curved extension of a 2D search plane. If an AUV moves along the seafloor surface while searching, near-bottom target search can be achieved. Since the seafloor is a curved surface, the AUV is assumed to move along its local tangent plane.

The seafloor within the obstacle-sensing range of an AUV is discretized into obstacle points. An example is shown in Figure 3, where Figure 3a presents the detected seafloor terrain and Figure 3b its discretized representation. The discrete points are treated as the obstacle set, and the Euclidean distance between the AUV and an obstacle is given by Equation (17).

$${dvs}_{i,k|t}=‖{\mathit{a}\mathit{u}\mathit{v}}_{\mathit{i}}\left(\mathit{t}\right)-{\mathit{o}\mathit{b}\mathit{s}}_{\mathit{k}}‖=\sqrt{{\left({xu}_{i|t}-{xs}_k\right)}^2+{\left({yu}_{i|t}-{ys}_k\right)}^2+{\left({zu}_{i|t}-{zs}_k\right)}^2}$$

(17)

For obstacle avoidance, the AUV considers the nearest obstacle point and two neighboring obstacle points that are not collinear with it. In Equation (18), ${obs}_{k_0}$ denotes the obstacle point nearest to AUV ${auv}_i$.

$${dvs}_{i,k_0|t}=\underset{{\mathit{obs}}_k\in \mathit{S}}{\min }\left({dvs}_{i,k|t}\right)$$

(18)

Based on ${obs}_{k_0}$, two additional obstacle points, ${obs}_{k_1}$ and ${obs}_{k_2}$, are selected to satisfy Equation (19). Since ${obs}_{k_0}$, ${obs}_{k_1}$, and ${obs}_{k_2}$ are not collinear, they determine a plane ${fl}_{i|t}$. To avoid obstacles while searching for near-bottom targets, the AUV moves parallel to this plane.

$$\left\{\begin{array}{l}{\mathit{o}\mathit{b}\mathit{s}}_{{\mathit{k}}_{\mathbf{0}}}=\left[{xs}_{k_0},{ys}_{k_0},{zs}_{k_0}\right]\\ {\mathit{o}\mathit{b}\mathit{s}}_{{\mathit{k}}_{\mathbf{1}}}=[{xs}_{k_1},{ys}_{k_1},{zs}_{k_1}]\\ {\mathit{o}\mathit{b}\mathit{s}}_{{\mathit{k}}_{\mathbf{2}}}=[{xs}_{k_2},{ys}_{k_2},{zs}_{k_2}]\\ {xs}_{k_1}={xs}_{k_0}-\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}({xs}_{k_0}-{xu}_{i|t})\\ {ys}_{k_1}={ys}_{k_0}\\ {xs}_{k_2}={xs}_{k_0}\\ {yo}_{k_2}={ys}_{k_0}-\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}({ys}_{k_0}-{yu}_{i|t})\end{array}\right.$$

(19)

The plane ${fl}_{i|t}$ is translated to a new plane ${fl}_{i|t^{\prime }}$ passing through ${\mathit{u}\mathit{a}\mathit{v}}_{\mathit{i}}\left(\mathit{t}\right)$. A local coordinate system ${\mathit{a}\mathit{u}\mathit{v}}_{\mathit{i}\mathit{t}}xyz$ is then established with ${\mathit{u}\mathit{a}\mathit{v}}_{\mathit{i}}\left(\mathit{t}\right)$ as the origin, under which ${\mathit{a}\mathit{u}\mathit{v}}_{\mathit{i}}\left(\mathit{t}\right)$, ${\mathit{o}\mathit{b}\mathit{s}}_{{\mathit{k}}_{\mathbf{0}}}$, ${\mathit{o}\mathit{b}\mathit{s}}_{{\mathit{k}}_{\mathbf{1}}}$, and ${\mathit{o}\mathit{b}\mathit{s}}_{{\mathit{k}}_{\mathbf{2}}}$ are expressed as in Equation (20).

$$\left\{\begin{array}{l}{\mathit{a}\mathit{u}\mathit{v}}_{\mathit{i}}^{\mathbf{\prime }}\left(\mathit{t}\right)=\left[0,0,0\right]\\ {\mathit{o}\mathit{b}\mathit{s}}_{{\mathit{k}}_{\mathbf{0}}}^{\mathbf{\prime }}=\left[\left({xs}_{k_0}-{xu}_{i|t}\right),\left({ys}_{k_0}-{yu}_{i|t}\right),\left({zs}_{k_0}-{zu}_{i|t}\right)\right]\\ {\mathit{o}\mathit{b}\mathit{s}}_{{\mathit{k}}_{\mathbf{1}}}^{\mathbf{\prime }}=\left[\left({xs}_{k_1}-{xu}_{i|t}\right),\left({ys}_{k_1}-{yu}_{i|t}\right),\left({zs}_{k_1}-{zu}_{i|t}\right)\right]\\ {\mathit{o}\mathit{b}\mathit{s}}_{{\mathit{k}}_{\mathbf{2}}}^{\mathbf{\prime }}=\left[\left({xs}_{k_2}-{xu}_{i|t}\right),\left({ys}_{k_2}-{yu}_{i|t}\right),\left({zs}_{k_2}-{zu}_{i|t}\right)\right]\end{array}\right.$$

(20)

In the local coordinate system ${\mathit{a}\mathit{u}\mathit{v}}_{\mathit{i}\mathit{t}}xyz$, the equation of the translated plane ${fl}_{i|t^{\prime }}$ can be assumed as

$$ax+by+cz=0$$

(21)

Then, the normal vector of the plane is given by ${\mathit{n}\mathit{l}}_{\mathit{i}|\mathit{t}}=[a,b,c]$, and two vectors in the plane are

$$\left\{\begin{array}{l}{\mathit{p}}_{\mathbf{1},\mathit{i}|\mathit{t}}=\left[\left({xs}_{k_1}-{xs}_{k_0}\right),\left({{ys}_{k_1}-ys}_{k_0}\right),\left({{zs}_{k_1}-zs}_{k_0}\right)\right]\\ {\mathit{p}}_{\mathbf{2},\mathit{i}|\mathit{t}}=\left[\left({{xs}_{k_2}-xs}_{k_0}\right),\left({ys}_{k_2}-{ys}_{k_0}\right),\left({zs}_{k_2}-{zs}_{k_0}\right)\right]\end{array}\right.$$

(22)

From the above equations, the normal vector can be obtained as

$${\mathit{n}\mathit{l}}_{\mathit{i}|\mathit{t}}=\left[a,b,c\right]=\left[\begin{array}{ccc}[1,0,0]& [0,1,0]& [0,0,1]\\ ({xs}_{k_1}-{xs}_{k_0})& ({{ys}_{k_1}-ys}_{k_0})& ({{zs}_{k_1}-zs}_{k_0})\\ ({{xs}_{k_2}-xs}_{k_0})& ({ys}_{k_2}-{ys}_{k_0})& ({zs}_{k_2}-{zs}_{k_0})\end{array}\right]$$

(23)

$$\left\{\begin{array}{l}a=({{ys}_{k_1}-ys}_{k_0})({zs}_{k_2}-{zs}_{k_0})-({ys}_{k_2}-{ys}_{k_0})({{zs}_{k_1}-zs}_{k_0})\\ b=({{zs}_{k_1}-zs}_{k_0})({{xs}_{k_2}-xs}_{k_0})-({zs}_{k_2}-{zs}_{k_0})({xs}_{k_1}-{xs}_{k_0})\\ c=({xs}_{k_1}-{xs}_{k_0})({ys}_{k_2}-{ys}_{k_0})-({{xs}_{k_2}-xs}_{k_0})({{ys}_{k_1}-ys}_{k_0})\end{array}\right.$$

(24)

When $\underset{{\mathit{obs}}_k\in \mathit{S}}{\min }\left({dvs}_{i,k|t}\right)\le l_n$, the AUV starts obstacle avoidance; when $\underset{{\mathit{obs}}_k\in \mathit{S}}{\min }\left({dvs}_{i,k|t}\right)\le l_m$, the AUV enters an enhanced obstacle-avoidance mode, where $l_m<l_n$. Let the obstacle-avoidance velocity generated by the 3D tangent-plane obstacle-avoidance algorithm be

$${\mathit{V}\mathit{o}}_{\mathit{i}}\left(t+1\right)=[{\dot{xo}}_{i\left|\right(t+1)},{\dot{yo}}_{i\left|\right(t+1)},{\dot{zo}}_{i\left|\right(t+1)}]$$

(25)

and let ${\mathit{V}\mathit{e}}_{\mathit{i}}\left(\mathit{t}+\mathbf{1}\right)=[{\dot{xe}}_{i\left|\right(t+1)},{\dot{ye}}_{i\left|\right(t+1)},{\dot{ze}}_{i\left|\right(t+1)}]$. The obstacle-avoidance process under the algorithm is divided into two cases.

(1)

$l_m<\underset{{\mathit{obs}}_k\in \mathit{S}}{\min }\left({dvs}_{i,k|t}\right)\le l_n$

$$\left\{\begin{array}{c}{\dot{xo}}_{i\left|\right(t+1)}={\dot{xe}}_{i\left|\right(t+1)} \\ {\dot{yo}}_{i\left|\right(t+1)}={\dot{ye}}_{i\left|\right(t+1)} \\ {\dot{zo}}_{i\left|\right(t+1)}=-{\displaystyle \frac{a {\dot{xo}}_{i\left|\right(\mathit{t}+\mathbf{1})}+b {\dot{yo}}_{i\left|\right(\mathit{t}+\mathbf{1})}}{c}}\end{array}\right.$$

(26)

(2)

$\underset{{\mathit{obs}}_k\in \mathit{S}}{\min }\left({dvs}_{i,k|t}\right)<l_m$

In the coordinate system $oxyz$, the normal vector of plane ${fl}_{i|t}$ pointing toward AUV ${auv}_i$, denoted by ${\mathbf{n}}_{l,i}^{\prime }(t)$, can be obtained according to Equation (27).

$${\mathit{n}\mathit{l}}_{\mathit{i}|\mathit{t}}^{\mathbf{\prime }}=\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\left({\mathit{p}}_{\mathbf{1},\mathit{i}|\mathit{t}}·({\mathit{u}\mathit{a}\mathit{v}}_{\mathit{i}|\mathit{t}}-{\mathit{o}\mathit{b}\mathit{s}}_{{\mathit{k}}_{\mathbf{0}}})\right){\mathit{n}\mathit{l}}_{\mathit{i}|\mathit{t}}$$

(27)

Plane ${fl}_{i|t}$ repels AUV ${\mathrm{a}\mathrm{u}\mathrm{v}}_i$ with virtual force ${\mathit{f}}_{\mathit{i},\mathit{n}\mathit{l}|\mathit{t}}$.

$${\mathit{V}\mathit{o}}_{\mathit{i}}\left(t+1\right)={\mathit{f}}_{\mathit{i},\mathit{n}\mathit{l}|\mathit{t}}={\displaystyle \frac{c\cdot l_m^2}{{‖{\mathit{n}\mathit{l}}_{\mathit{i}|\mathit{t}}^{\mathbf{\prime }}‖}^3}}{\mathit{n}\mathit{l}}_{\mathit{i}|\mathit{t}}^{\mathbf{\prime }}$$

(28)

The values of $l_n$ and $l_m$ vary with the AUV state. Specifically, $l_n=0.8d_{tar}$ and $l_m=V_m$ in the roaming search state, while $l_n=V_m$ and $l_m=0.5V_m$ in the cooperative search state.

3.5. AUV Velocity and Position Update

Once the desired velocity is obtained, the AUV starts obstacle avoidance. The next-step velocity ${\mathit{V}}_i(t+1)$ is then determined according to the search state, $\underset{{\mathit{obs}}_k\in \mathit{S}}{\min }\left({dvs}_{i,k|t}\right),$ and the desired velocity ${\mathit{V}\mathit{e}}_i(t+1)$. The velocity update is divided into six cases:

(1)

AUV ${auv}_i$ is in the roaming search state, and $\underset{{\mathit{obs}}_k\in \mathit{S}}{\min }\left({dvs}_{i,k|t}\right)<Vm$.

According to Equations (27) and (28), ${\mathit{V}\mathit{o}}_{\mathit{i}}\left(t+1\right)$ is obtained, and ${\mathit{V}}_i(t+1)$ is then given by Equation (29).

$$\left\{\begin{array}{cc}\hfill \left[\dot{x},\dot{y},\dot{z}\right]& ={\mathit{V}\mathit{e}}_{\mathit{i}}\left(\mathit{t}+\mathbf{1}\right)+{\mathit{V}\mathit{o}}_{\mathit{i}}\left(t+1\right)\hfill \\ \hfill {\mathit{V}}_{\mathit{i}}\left(\mathit{t}+\mathbf{1}\right)& =\left[\dot{x},\dot{y},\dot{z}\right]{\displaystyle \frac{Vm}{\sqrt{{\dot{x}}^2+{\dot{y}}^2+{\dot{z}}^2}}}\hfill \end{array}\right.$$

(29)

(2)

AUV ${auv}_i$ is in the roaming search state, and $Vm<\underset{{\mathit{obs}}_k\in \mathit{S}}{\min }\left({dvs}_{i,k|t}\right)<0.8d_{tar}$.

According to Equation (26), ${\mathit{V}\mathit{o}}_{\mathit{i}}\left(t+1\right)$ is obtained, and ${\mathit{V}}_i(t+1)$ is then given by Equation (30).

$${\mathit{V}}_{\mathit{i}}(\mathit{t}+\mathbf{1})={\mathit{V}\mathit{o}}_{\mathit{i}}\left(t+1\right){\displaystyle \frac{V_m}{‖{\mathit{V}\mathit{o}}_{\mathit{i}}\left(t+1\right)‖}}$$

(30)

(3)

AUV ${auv}_i$ is in the roaming search state, and $\underset{{\mathit{obs}}_k\in \mathit{S}}{\min }\left({dvs}_{i,k|t}\right)>0.8d_{tar}$.

${\mathit{V}}_i(t+1)$ is given by Equation (31).

$$\left\{\begin{array}{l}{\dot{xu}}_{i\left|\right(t+1)}={\dot{xe}}_{i\left|\right(t+1)}\\ {\dot{yu}}_{i\left|\right(t+1)}={\dot{ye}}_{i|\left(t+1\right)}\\ \dot{{zu}_{i\left|\right(t+1)}}=-\left|{\dot{ze}}_{i\left|\right(t+1)}\right|\end{array}\right.$$

(31)

(4)

AUV ${auv}_i$ is in the cooperative search state, and $\underset{{\mathit{obs}}_k\in \mathit{S}}{\min }\left({dvs}_{i,k|t}\right)<0.5V_m$.

According to Equations (27) and (28), ${\mathit{V}\mathit{o}}_{\mathit{i}}\left(t+1\right)$ is obtained, and ${\mathit{V}}_i(t+1)$ is then given by Equation (32).

$$\begin{array}{c}{\mathit{V}}_{\mathit{i}}\left(t+1\right)=\\ \left\{\begin{array}{ccc}\left({\mathit{V}\mathit{e}}_{\mathit{i}}\left(t+1\right)+{\mathit{V}\mathit{o}}_{\mathit{i}}\left(t+1\right)\right){\displaystyle \frac{Vm}{‖{\mathit{V}\mathit{e}}_{\mathit{i}}\left(\mathit{t}+\mathbf{1}\right)+{\mathit{V}\mathit{o}}_{\mathit{i}}\left(t+1\right)‖}}& ,\hfill & ‖{\mathit{V}\mathit{e}}_{\mathit{i}}\left(t+1\right)+{\mathit{V}\mathit{o}}_{\mathit{i}}\left(t+1\right)‖>Vm\\ {\mathit{V}\mathit{e}}_{\mathit{i}}\left(t+1\right)+{\mathit{V}\mathit{o}}_{\mathit{i}}\left(t+1\right)& ,\hfill & ‖{\mathit{V}\mathit{e}}_{\mathit{i}}\left(t+1\right)+{\mathit{V}\mathit{o}}_{\mathit{i}}\left(t+1\right)‖\le Vm\end{array}\right.\end{array}$$

(32)

(5)

AUV ${auv}_i$ is in the cooperative search state, and $0.5V_m<\underset{{\mathit{obs}}_k\in \mathit{S}}{\min }\left({dvs}_{i,k|t}\right)<V_m$.

According to Equation (26), ${\mathit{V}\mathit{o}}_{\mathit{i}}\left(t+1\right)$ is obtained, and ${\mathit{V}}_i\left(t+1\right)$ is then given by Equation (32).

(6)

AUV ${auv}_i$ is in the cooperative search state, and $\underset{{\mathit{obs}}_k\in \mathit{S}}{\min }\left({dvs}_{i,k|t}\right)>V_m$.

${\mathit{V}}_i(t+1)$ is given by Equation (31).

After ${\mathit{V}}_i(t+1)$ is obtained, the AUV position is updated accordingly.

$${\mathit{u}\mathit{a}\mathit{v}}_{\mathit{i}}\left(t+1\right)={\mathit{u}\mathit{a}\mathit{v}}_{\mathit{i}}\left(t\right)+{\mathit{V}}_{\mathit{i}}\left(t+1\right)$$

(33)

Based on the above analysis, the flowchart and pseudocode of the multi-target search algorithm are shown in Figure 4 and Algorithm 1, respectively. The convergence of the overall search algorithm primarily relies on the convergence properties of the PSO algorithm [33], while being secondarily supported by the rapid and complete area exploration capabilities of the roaming search phase. Once target signals are detected, some AUVs transition from roaming search to the convergent cooperative phase, ensuring stable target search.

Algorithm 1 Multi-Target Search by AUV Swarms
1	Initialization
2	while not all targets found do
3	All AUVs detect target signals
4	Execute Response-Threshold-based Multi-Target Task Allocation
5	for each AUV do
6	if AUV performs collaborative search in sub-alliances then
7	Calculate ${\mathit{V}\mathit{e}}_{\mathit{i}}\left(\mathit{t}+\mathbf{1}\right)$ according to Equation (16)
8	else
9	Calculate ${\mathit{V}\mathit{e}}_{\mathit{i}}\left(\mathit{t}+\mathbf{1}\right)$ according to Equation (15)
10	end if
11	Calculate ${\mathit{V}}_{\mathit{i}}\left(t+1\right)$ by 3D tangent-plane obstacle avoidance
12	Update AUV position by Equation (33)
13	if ${dvt}_{i,j|t}<d_0$  then
14	Declare target as found
15	end if
16	end for
17	end while

4. Simulation Results and Analysis

The simulation parameters were set with reference to the literature and practical search requirements, as listed in

Table 2. Values of parameters.

Parameters	Values	Parameters	Values
$n_u$	30~60	$N_m$	6
$n_T$	10	$m$	0.1
$Vm$	10	$Q$	105
$r_{com}$	300	$c_1$	1
$r_{obs}$	100	$c_2$	1.2
$r_{tar}$	100	$\omega$	0.5
$d_0$	10	$\lambda$	0.1

[32,33,34,35].

Taking $n_u=30$ as an example, the multi-target search process is illustrated. In Figure 5a, the seafloor terrain for target search is presented. Figure 5b shows the top view of the search area at $t=1,$ where the AUVs are initially distributed in the $100\times 100$ region at the lower left corner, while the targets are located in the central $800\times 800$ region.

In Figure 5c, ${auv}_{26}$ detects the signal from target ${tar}_8,$ and AUVs ${auv}_6$, ${auv}_8$, ${auv}_{10}$, ${auv}_{11}$, and ${auv}_{17}$ are recruited into the sub-alliance.

In Figure 5d, from $t=36$ to $t=77$, ${auv}_3$, ${auv}_6$, ${auv}_8$, ${auv}_{10}$, ${auv}_{11}$, ${auv}_{17}$, ${auv}_{18}$, ${auv}_{20}$, ${auv}_{26}$, and ${auv}_{30}$ participate in the sub-alliance to cooperative search for ${tar}_8$ at different times, with some AUVs joining or leaving during the process. Eventually, ${auv}_{30}$ finds ${tar}_8$ at $t=77$.

In Figure 5e, ${tar}_2$ is found through cooperative search at $t=222$. Up to this time, eight targets, ${tar}_1$, ${tar}_2$, ${tar}_4$, ${tar}_6$, ${tar}_7$, ${tar}_8$, ${tar}_9$, and ${tar}_{10}$, have been found.

In Figure 5f, from $t=1$ to $t=222$, 11 AUVs, namely ${auv}_6$, ${auv}_7$, ${auv}_{13}$, ${auv}_{14}$, ${auv}_{19}$, ${auv}_{20}$, ${auv}_{21}$, ${auv}_{23}$, ${auv}_{26}$, ${auv}_{27}$, and ${auv}_{29}$, are involved in the cooperative search for ${tar}_2$ at different times, with dynamic joining and leaving during the process. Before that, multiple AUVs had already participated in the cooperative search for other targets.

In Figure 5g, all 10 targets are found by the AUV swarm at $t=292$. Figure 5h presents the path of all AUVs during the search process, from which it can be seen that the AUVs successfully avoided the seafloor.

The number of AUVs was set to $n_u=30, 40, 50,$ and 60, while the number of targets was fixed at $n_T=10$. For each setting, 30 repeated trials were conducted, and the results are summarized in

Table 3. Search steps and energy consumption for $n_T=10$ under different AUV swarm sizes.

	Steps				Energy Consumption ($\mathbf{\times }{\mathbf{10}}^{\mathbf{4}}$)
$n_u$	$n_u=30$	$n_u=40$	$n_u=50$	$n_u=60$	$n_u=30$	$n_u=40$	$n_u=50$	$n_u=60$
Results of 30 trials	366	248	295	231	9.97	9.56	13.91	13.52
	326	327	259	240	8.83	12.38	12.47	13.74
	409	323	263	256	11.43	12.10	12.65	14.73
	383	332	300	282	10.65	12.51	14.43	16.23
	324	255	245	219	9.02	9.59	11.82	12.48
	472	392	245	224	12.98	14.55	11.80	12.93
	365	284	285	295	10.03	10.69	13.68	17.12
	359	364	237	211	9.95	13.59	11.46	12.30
	467	271	310	223	12.57	10.26	14.76	12.95
	603	239	254	263	14.90	9.21	12.00	15.21
	406	297	239	251	10.95	11.01	11.36	14.50
	443	276	248	348	11.90	10.32	11.55	20.05
	413	357	298	800	11.23	13.41	14.42	44.37
	501	281	292	213	13.92	10.44	13.89	12.32
	672	306	258	256	18.45	11.54	12.46	14.64
	348	346	329	246	9.51	13.01	15.44	14.34
	402	341	300	273	11.05	12.71	14.37	15.99
	413	466	279	244	11.22	17.42	13.23	14.20
	588	255	335	223	15.72	9.66	15.87	12.98
	348	333	323	248	9.72	12.49	15.643	14.14
	338	267	260	247	9.37	9.97	12.47	14.19
	422	324	256	251	10.86	12.12	12.29	14.38
	757	273	244	258	20.19	10.31	11.63	14.97
	357	325	226	285	9.97	12.19	10.82	16.46
	425	313	276	234	11.76	12.09	13.26	13.62
	376	326	242	210	9.90	12.03	11.75	12.18
	481	313	270	236	13.27	11.15	12.88	13.78
	407	261	234	228	11.38	9.85	11.16	12.95
	418	287	269	372	11.53	11.02	12.93	21.18
	400	483	237	225	11.15	17.62	11.30	13.00
Mean	432.97	315.50	270.27	269.73	11.78	11.83	12.92	15.51

. It can be seen that, as the number of AUVs increases, the target search task is completed more quickly, while the energy consumption also increases. These results indicate that an AUV swarm can substantially improve the efficiency of multi-target search.

5. Conclusions

This paper addressed the problem of multi-target search by an AUV swarm in unknown 3D underwater environments with limited communication. A dual-state search mechanism, including cooperative and roaming, was established, and an AUV-swarm-based search method was developed by integrating a virtual force model, motion-constrained 3D Particle Swarm Optimization, and a 3D tangent-plane obstacle-avoidance strategy. The proposed method enables AUVs to switch adaptively between roaming search and cooperative search, form sub-alliances for different targets, and maintain safe motion near the seafloor.

Simulation results verified that the proposed method can effectively accomplish multi-target search in unknown 3D underwater obstacle environments. The results also showed that increasing the swarm size can reduce the search time, although with higher energy consumption. Overall, the proposed method provides a feasible solution for AUV-swarm-based multi-target search with communication constraints.

However, this study primarily focused on multi-target search by AUV swarms in unknown 3D underwater obstacle environments under limited communication conditions, while factors such as heterogeneous swarms, communication uncertainties, and ocean current disturbances were not fully considered. Future work will primarily focus on searching for moving targets, extending to heterogeneous swarms, addressing time-varying ocean currents and communication uncertainties, and conducting real-world experimental validation.