Mission Planning for UAV Swarm with Aircraft Carrier Delivery: A Decoupled Framework

Abstract

Due to the limited endurance of UAVs, especially in scenarios involving large areas and dense target nodes, it is challenging for multiple UAVs to complete diverse tasks while ensuring timely execution. Toward this, we propose a cross-platform system consisting of an aircraft carrier (AC) and multiple UAVs, which makes unified task planning for included heterogeneous platforms to maximize the efficiency of the entire combat system. The carrier-based UAV swarm mission planning problem is formulated to minimize completion time and resource utilization, taking into account large-scale targets, multi-type tasks, and multi-obstacle environments. Since the problem is complex, we design a decoupled framework to simplify the solution by decomposing it into two levels: upper-level AC path planning and bottom-level multi-UAV cooperative mission planning. At the upper level, a drop point determination method and a discrete genetic algorithm incorporating improved A* (DGAIIA) are proposed to plan the AC’s path in the presence of no-fly zones and radar threats. At the bottom level, an improved differential evolution algorithm with a market mechanism (IDEMM) is proposed to minimize task completion time and maximize UAV utilization. Specifically, a dual-switching search strategy and a neighborhood-first buying-and-selling mechanism are developed to improve the search efficiency of the IDEMM. Simulation results validate the effectiveness of both the DGAIIA and IDEMM. An animation of the simulation results is available at simulation section.

1. Introduction

1.1. Background

Due to the advantage of high mobility and flexible deployment, unmanned aerial vehicles (UAVs) have become a promising tool in both military and civilian fields, especially for reconnaissance, mobile relays, disaster rescue, and environmental surveillance [1,2,3,4] over the past two decades. Furthermore, technological advancements have driven the miniaturization of UAVs, making it increasingly common for multiple UAVs to cooperate as a team. Team collaboration is to enable the operational effectiveness of UAV swarms to surpass that of a single UAV. Compared to a single UAV with limited payload capacity and onboard fuel, UAV swarms are not only capable of performing a variety of tasks but also significantly enhance the overall system’s execution efficiency through mutual complementarity of capabilities and coordinated actions. However, the generally short endurance of UAVs makes it difficult to meet the requirements of long-duration, large-scale operations. This not only limits the operational range of UAVs but also affects their sustained combat capability in complex battlefield environments.

As a paradigm of mobile combat platforms, aircraft carriers (ACs) possess potent integrated combat effectiveness and comprehensive defense systems. They are capable of carrying a variety of ACs to carry out multiple missions such as long-range precision strikes and sea blockades, while also being equipped with diversified combat units. Given the AC’s notable advantages in long-distance transportation and heavy-lift capabilities, integrating it into UAV swarm operations can effectively address the issue of limited UAV endurance. This model can fully leverage the combat advantages of ACs and the flexible characteristics of UAVs, achieving complementary strengths. Therefore, integrating UAV platforms with AC systems is crucial. We aim to develop an innovative collaboration model to enhance operational efficiency and effectiveness through this integration.

In this innovative collaborative operation mode, the AC plays a crucial role in transporting UAVs from airports to preset drop points (DPs), where the UAVs are released to execute predetermined tasks. Crucially, the AC proceeds directly to transport other UAVs to the next specified DP without awaiting completion of deployed UAV tasks, repeating the process until all DPs have been visited. Simultaneously, deployed UAVs maneuver flexibly from the DPs to designated target locations, complete their tasks, and then return to the DPs to stand by for recovery. Compared to traditional modes, this proposed mode significantly enhances the operational effectiveness of UAV swarms. Leveraging the AC as a transport platform, UAV swarms gain the capabilities of long-distance, low-cost, multi-wave, and hierarchical mission execution. Figure 1 visually contrasts the traditional mode with the innovative mode. Notably, the use of ACs for transportation can greatly reduce the flight distance of UAVs, especially in large-scale scenarios. Furthermore, while the deployed UAVs are executing their assigned tasks, the AC continues with its transportation duties. This parallel operation mode further significantly enhances the efficiency of task execution, as illustrated in Figure 1b.

By employing the cooperative mode, the AC can serve as both a transportation station and a launch center for UAVs. The AC transports UAVs to designated positions for task execution, and energy refueling for UAVs can be performed while they are on board the AC. In this cooperative mode, the AC becomes an important platform for enhancing UAV combat capabilities. It can rapidly deploy UAVs to various operational areas, improving mission flexibility and response speed while also expanding the operational range.

As far as we know, most existing research in UAV mission planning focuses on small-scale scenarios, considering the limited endurance of UAVs [5,6,7,8,9,10,11,12,13,14]. These small-scale mission planning methods are usually designed based on specific task environments and constraints, while large-scale mission planning involves more target points and more complex interactions. Therefore, existing mission planning methods typically cannot be scalable to handle large-scale problems. Meanwhile, there are notable studies focusing on the coupling of UAVs to other platforms [15,16,17,18,19,20,21,22], particularly in conjunction with trucks. In [18,19,20,21,22], researchers have addressed the vehicle routing problem by using a truck and multi-UAV to provide delivery and pickup services in rural areas. The truck, carrying several UAVs, fulfills customer requests along its route. UAVs are launched independently to perform sub-routes at the depot and at any node visited by the truck. In each sub-route, UAVs are assigned to service one or multiple customers. The truck is allowed to take a trip from the depot, while each UAV is allowed to make a single trip from its launch location. Consequently, the truck–UAV delivery problem is essentially a new variant of the traveling salesman problem.

The aforementioned truck–UAV cooperative system represents the collaboration between heterogeneous platforms. It shares some operational similarities with the AC and UAV collaboration system that we propose. However, significant disparities emerge between the two systems when considering obstacles, radar threats, diverse tasks, and time window constraints. On the one hand, the systems operate in different environments. The truck–UAV collaboration system assumes a benign environment, characterized by linear connections between access nodes and the absence of threats. In contrast, the carrier–UAV collaboration system operates in a complex environment with no-fly zones and radar threats. The linear trajectory of the AC between the DPs may introduce a risk of mission failure. On the other hand, their task types are markedly different. While the truck–UAV system typically involves a single task, the AC and UAV collaboration system involves multiple heterogeneous tasks such as reconnaissance and attack. Towards this, our proposed system must require task prioritization and time window constraints. Consequently, due to the significant differences between truck–UAV and carrier–UAV systems, the algorithms developed for the truck–UAV system [18,19,20,21,22] are not suitable for the scenarios studied in this paper.

Motivated by this, we propose the carrier-based UAV swarm mission planning system, aimed at efficiently executing multi-type tasks in complex and large-scale environments, including those with radar threats and no-fly zones. However, the carrier-based UAV swarm mission planning problem faces several challenges, such as determining optimal DPs for UAVs, devising secure trajectories for the AC, and developing an effective allocation scheme for multiple UAVs.

1.2. Literature Review

In the AC delivery model we propose, there are two core issues: one is the path planning of the AC, and the other is the task allocation of multi-UAVs at the DPs. These two issues are not only crucial to the successful implementation of this model but also of great significance to traditional combat planning, as they play important roles in optimizing execution efficiency and resource allocation. Therefore, we will focus on reviewing the extensive research conducted around these topics.

Given the significant similarity between UAV path planning and AC path planning, and considering the abundant research findings accumulated in the former, we have thus focused our review on UAV path planning. UAV path planning algorithms can be broadly categorized into three types: sampling-based methods, search-based methods, and optimization-based methods [23,24,25]. Among them, the A* algorithm, as a type of search-based method, has been widely adopted due to its ease of implementation, high search efficiency, and broad applicability. As a result, the design of the A* algorithm is increasingly receiving attention, and various improvement measures have been proposed to enhance its efficiency and effectiveness. For instance, Trovato et al. [26] proposed an improved differential A* algorithm aimed at improving node expansion efficiency and convergence speed. Compared with the conventional A* algorithm, this improved version offers better reconfigurability in real time. Xie et al. [27] proposed a multi-directional A* algorithm capable of generating paths resembling real-world scenarios. To address the limitations of the conventional multi-directional A* algorithm in expansion efficiency, Wu et al. [28] presented a two-way adaptive A* algorithm, which significantly improved both expansion efficiency and path smoothness through directional search strategies. Additionally, Song et al. [29] developed a new path smoothing algorithm to improve the performance of generated paths. Building upon this work, Hawa et al. [30] developed an A* algorithm designed for environments with complex-shaped no-fly zones.

Multi-UAV collaborative mission planning is a complex process that allocates resources based on the specific requirements of the task, the execution capabilities of the UAVs, and the characteristics of the target objects, with the core objective of maximizing operational effectiveness. In ref. [31], the author introduces a multi-UAV collaborative mission planning system and employs various algorithms to solve it. Wu et al. [11] and Zhu et al. [13] considered UAV path length as a key factor in their evaluation functions and developed a multi-UAV mission planning model accordingly. Moreover, to meet the requirement of multi-attack in practical operations, Yu et al. [12] explored multitasking scenarios. They devised a mechanism to resolve deadlocks; however, the implementation of this mechanism has led to an increase in computational complexity, thereby hindering the rapid resolution of the problem. Luo et al. [14] studied multi-type task assignment problems for multi-functional and heterogeneous UAVs. Although their problem does not suffer from deadlock, the model does not account for UAV constraints like ammunition load and flight time. In addition, the heterogeneous UAVs investigated may not fully represent real-world scenarios in UAV swarm cooperative mission planning. In this regard, Chen et al. [9] and Jiang et al. [10] studied mission planning for homogeneous UAVs, proposing relevant solution algorithms with theoretical and practical implications. However, their approaches face limitations when applied to large-scale mission planning problems.

1.3. Motivation and Contribution

In this paper, we develop the carrier-based UAV swarm mission planning system. This system is designed to transport UAVs to combat area using AC in environments where radar threats and no-fly zones exist, and assign UAVs to execute attack and assessment tasks against targets. Compared to the traditional single-platform execution mode, our proposed operational mode significantly enhances the overall execution efficiency and operational coverage of the combat system, particularly when dealing with numerous targets. According to our knowledge, there has been no relevant research addressing the problem of collaborative planning between UAVs and ACs in large-scale and complex combat scenarios.

The carrier-based UAV swarm mission planning problem encompasses threat sources at various detection ranges, diversified combat platforms, and multiple types of tasks, undoubtedly posing significant challenges in finding solutions. To effectively tackle large-scale carrier-based UAV swarm mission planning problems, we propose a decoupled framework, which divides the planning of carrier-based UAV swarm mission planning into two phases: the upper-level path planning for AC and the lower-level coordinated mission planning for multi-UAV. In the upper-level planning, the AC is responsible for transporting UAVs to each combat sub-area. The challenge at this phase lies in how to plan the AC’s path, ensuring it can effectively avoid no-fly zones and radar threats while minimizing flight costs. Given that each sub-area is visited only once by the AC, its path planning problem is somewhat similar to the traveling salesman problem with collision avoidance constraints. However, it is important to note that in the scenario discussed in this paper, the DPs for UAVs in each sub-area have not been determined. In other words, each destination that the traveling salesman needs to reach is unknown, which constitutes a significant difference between the two problems. Therefore, this paper also necessitates decision-making on the UAV deployment points in each sub-zone to enhance the efficiency of UAV missions. In bottom-level planning, our goal is to minimize time and UAV resources while considering various constraints such as the type of tasks executed, sequence requirements, time window restrictions, UAV flight time, and payload capacity.

The main contributions of this paper are summarized as follows:

• We design a decoupled planning framework that decomposes the carrier-based UAV swarm mission planning problem into a two-level planning process. This decoupled architecture enhances the adaptability and extensibility of the planning process, making it capable of addressing coordinated planning in complex large-scale scenarios. Additionally, to the best of our knowledge, this paper is the first to propose and investigate the carrier-based UAV swarm mission planning problem.
• For the upper-level problem, we develop a selection method based on the principles of distance and threat minimization to address the DP determination problem. Then, we propose a discrete genetic algorithm incorporating an improved A* algorithm (DGAIIA) to plan the global path of AC. In the DGAIIA method, a variant of the single traveler problem is formulated, which considers radar threats and no-fly zones.
• For the bottom-level problem, we establish the multi-objective optimization model for cooperative mission planning of multiple UAVs, with time cost and UAV utilization as the optimization objective. To solve this problem, we propose an improved differential evolution algorithm with a market mechanism (IDEMM), which features a dual switching search strategy and a neighborhood-first buying-and-selling mechanism.
• Simulation experiments demonstrate that the carrier-based UAV swarm mission planning problem can be efficiently solved by the DGAIIA and IDEMM algorithms. Furthermore, the IDEMM algorithm is superior to other benchmark algorithms in terms of convergence performance. Our proposed framework can be easily extended to other mission scenarios involving coordinated planning of multiple types of platforms.

1.4. Organization

The rest of the paper is organized as follows. Section 2 introduces the basic definition of the carrier-based UAV swarm mission planning, the path planning model for AC, and the multi-UAV cooperative mission planning model at each drop point. In Section 3, we present the algorithm design for solving the carrier-based UAV swarm mission planning. The results of numerical experiments are provided in Section 4. Finally, Section 5 concludes the paper and discusses future research directions.

2. System Model

2.1. An Overview of the Carrier-Based UAV Swarm Mission Planning

In the carrier-based UAV swarm mission planning system, UAVs serve as the task execution platforms, while the AC functions as the transport and logistical support platform for these UAVs. Based on the characteristics of the proposed planning method, the entire operational area is divided into several sub-areas. The AC is responsible for transporting UAVs to each sub-area, where the specific locations for deploying UAVs by the AC are termed as DPs. To ensure the maximization of subsequent UAV mission execution efficiency, the positions of these deployment points need to be optimized. Furthermore, in complex environments featuring no-fly zones and radar threats, the flight paths for AC need also be planned, and UAVs need to be allocated rationally to achieve the minimization of flight costs and the maximization of utilization rates.

We consider the carrier-based UAV swarm mission planning system in a two-dimensional plane, which is composed of one AC, one control center $v_0$, K isomorphic UAVs with attack and assessment capabilities, denoted by $\mathcal{U}=\left\{u_1,u_2,\cdots ,u_K\right\}$, G radars, represented as $\mathcal{R}=\left\{r_1,r_2,\cdots ,r_G\right\}$, L no-fly zones, denoted by $\mathcal{O}=\left\{o_1,o_2,\cdots ,o_L\right\}$, and M targets, represented as $\mathcal{T}=\left\{t_1,t_2,\cdots ,t_M\right\}$. The position of radar $r_g\in \mathcal{R}$ is expressed as ${\mathit{s}}_g=\left[x_g,y_g\right]$. Let $\mathcal{S}=\left\{{\mathit{s}}_g\left|r_g\in \mathcal{R}\right.\right\}$ denote the set of positions of radars. There are attack and assessment tasks for each target, which are represented by $k_1$ and $k_2$, respectively. The set of task types is denoted as $\mathcal{K}=\left\{k_1,k_2\right\}$.

According to the characteristics of the carrier-based mission planning, all targets will be divided into N clusters $\left(1\le N\le M\right)$, specifically, when there is only one target in each cluster, i.e., $N=M$, or when all targets belong to a single cluster, i.e., $N=1$. If target $t_m$ belongs to n-th cluster, ${\zeta }_{m,n}=1$; otherwise, ${\zeta }_{m,n}=0$. The set of targets belonging to the n-th cluster is represented by ${\mathcal{T}}_n=\left\{t_m\left|{\zeta }_{m,n}=1,t_m\in \mathcal{T}\right.\right\}$. Since there is a DP in each cluster, there are a total of N DPs that need to be optimized, denoted by $\mathcal{V}=\left\{v_1,v_2,\cdots ,v_N\right\}$. The position of DP $v_n$ is expressed as ${\mathit{w}}_n=\left[x_n,y_n\right]$. Let $\mathcal{W}=\left\{{\mathit{w}}_n\left|v_n\in \mathcal{V}\right.\right\}$ denote the set of DP position. Let $\tilde{\mathcal{V}}=\left\{v_0,\mathcal{V}\right\}$ denote the set of the control center and all DPs. As shown in Figure 2, the AC transports UAVs from the control center $v_0$, sequentially visiting all DPs, and ultimately returning to $v_0$. At each DP, the AC will deploy some UAVs to carry out tasks. These deployed UAVs will take off from the DP and, upon completion of their tasks, return to the same DP to await recovery. Let $\mathcal{U}=\left\{u_1,u_2,\cdots ,u_{K_N}\right\}$ denote the set of deployed UAVs at the DP $v_n\in \mathcal{V}$, $K_n$ is the number of deployed UAVs at the DP $v_n\in \mathcal{V}$, and ${\displaystyle \sum _{n=1}^N}{K}_n=K$. Note that the recovery task of UAVs has been extensively studied, but it is not the focus of this paper [32,33,34,35]. The speed of AC and UAV $u_k$ is expressed as $V_c$ and $V_u$, respectively. Therefore, the trajectory of AC passes through the DPs and the control center $v_0$, while the trajectories of the UAV pass through the target points and the DPs. If the AC flies from DP $v_n$ to DP $v_{\tilde{n}}$, ${\lambda }_{n,\tilde{n}}=1\left(v_n,v_{\tilde{n}}\in \tilde{\mathcal{V}},v_n\ne v_{\tilde{n}}\right)$; otherwise, ${\lambda }_{n,\tilde{n}}=0$. Let ${\eta }_{k,m,p}$ denote the association variable between UAV $u_k$ and task $k_p$ of target $t_m$, where ${\eta }_{k,m,p}=1$ indicates that UAV $u_k$ is assigned to perform task $k_p$ of target $t_m$; otherwise, ${\eta }_{k,m,p}=0$. The set of tasks performed by UAV $u_k$ is represented by ${\mathcal{K}}_k=\left\{\left(t_m,k_p\right)\left|{\eta }_{k,m,p}=1,t_m\in \mathcal{T},k_p\in \mathcal{K}\right.\right\}$. Let $\mathit{\zeta }$, $\mathit{\lambda }$ and $\mathit{\eta }$ denote the set of variables $\left\{{\zeta }_{m,n}\right\}$, $\left\{{\lambda }_{n,\tilde{n}}\right\}$ and $\left\{{\eta }_{k,m,p}\right\}$, respectively.

Based on the above discussion, we propose a decoupled mechanism to simplify problem analysis and model construction. As shown in Figure 3, the carrier-based UAV swarm mission planning is decomposed into two levels, that is, the upper-level AC path planning problem and the bottom-level multi-UAV cooperative mission planning problem.

2.2. The Path Planning Model of AC

We consider radars and no-fly zones in the carrier-based UAV swarm mission planning system. According to [36], when the AC enters the radar threat zone, it faces the risk of detection, and the closer it is to the radar source, the greater the degree of threat. The distance $d_{r_g,v_n}$ between the radar $r_g$ and the DP $v_n$ is given by

$$d_{r_g,v_n}=∥{\mathit{s}}_g-{\mathit{w}}_n∥=\sqrt{{\left(x_g-x_n\right)}^2+{\left(y_g-y_n\right)}^2}.$$

(1)

Based on the threat model in [36], the radar threat at DP $v_n$ is expressed as

$$\mathcal{H}\left(v_n\right)=\sum _{g=1}^G{\displaystyle \frac{1}{{\left(d_{r_g,v_n}\right)}^4}}{I}_{\left(d_{r_g,v_n}\le R_g^{\det }\right)},$$

(2)

where $d_{r_g,v_n}$ as shown in Equation (1), $R_g^{\det }$ is the detection radius of the radar $r_g$, and $I_{\left(d_{r_g,v_n}\le R_g^{\det }\right)}$ is the indicator function, which is expressed as

$$I_{\left(d_{r_g,v_n}\le R_g^{\det }\right)}=\left\{\begin{array}{c}1,d_{r_g,v_n}\le R_g^{\det },\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

(3)

Thus, the radar threat of the AC flying from the DP $v_n$ to another DP $v_{\tilde{n}}$ is derived as follows:

$$\mathcal{H}\left(l_{n,\tilde{n}}\right)=\sum _{i=1}^{N_{n\tilde{n}}}\mathcal{H}\left(p_{n\tilde{n},i}\right),$$

(4)

where $l_{n,\tilde{n}}$ is the flight path of the AC from DP $v_n$ to another DP $v_{\tilde{n}}$, which is made up of a series of flight path points $\left\{p_{n\tilde{n},0},p_{n\tilde{n},1},p_{n\tilde{n},2},\dots ,p_{n\tilde{n},N_{n\tilde{n}}}\right\}$. $N_{n\tilde{n}}$ is the number of path points in the flight path $l_{n,\tilde{n}}$. Obviously, $p_{n\tilde{n},0}=v_n$, $p_{n\tilde{n},N_{n\tilde{n}}}=v_{\tilde{n}}$. $\mathcal{H}\left(p_{n\tilde{n},i}\right)$ denotes the radar threat at point $p_{n\tilde{n},i}$. Let $\overline{d}\left(l_{n,\tilde{n}}\right)$ denote the distance length of the flight path $l_{n,\tilde{n}}$, which is expressed as

$$\overline{d}\left(l_{n,\tilde{n}}\right)=\sum _{i=1}^{N_{n\tilde{n}}-1}{d}_{p_{n\tilde{n},i},p_{n\tilde{n},i+1}}=\sum _{i=1}^{N_{n\tilde{n}}-1}\sqrt{{\left(x_{p_{n\tilde{n},i}}-x_{p_{n\tilde{n},i+1}}\right)}^2+{\left(y_{p_{n\tilde{n},i}}-y_{p_{n\tilde{n},i+1}}\right)}^2},$$

(5)

where $x_{p_{n\tilde{n},i}}$, $x_{p_{n\tilde{n},i+1}}$, $y_{p_{n\tilde{n},i}}$ and $y_{p_{n\tilde{n},i+1}}$ are the x-axis and y-axis coordinates of points $p_{n\tilde{n},i}$ and $p_{n\tilde{n},i+1}$, respectively.

Let $\mathcal{L}=\left\{l_{n,\tilde{n}}\left|v_n,v_{\tilde{n}}\in \mathcal{V},v_n\ne v_{\tilde{n}}\right.\right\}$ denote the set of flight paths of the AC. Therefore, in order to ensure that the AC does not collide with the no-fly zone, the following constraint needs to be met:

$$\mathcal{L}\cap \mathcal{O}=\varnothing .$$

(6)

Since the AC only passes through each DP once when transporting UAVs, there is only one path for the AC to reach each DP and only one path departing from each DP. This constraint is represented by

$$\sum _{n=0}^N{\lambda }_{n,\tilde{n}}=1,\sum _{\tilde{n}=0}^N{\lambda }_{n,\tilde{n}}=1,v_n,v_{\tilde{n}}\in \tilde{\mathcal{V}},$$

(7)

$$\sum _{\tilde{n}=0}^N\sum _{n=0}^N{\lambda }_{n,\tilde{n}}=N,v_n,v_{\tilde{n}}\in \tilde{\mathcal{V}}.$$

(8)

Due to the unpredictability of battlefield environments, the path length has a significant impact on the safety and mission completion time of AC. The shorter the path, the faster the mission is completed, and the less risk AC faces. Thus, path length is one of the important indicators for evaluating the quality of a path. Additionally, in the presence of radar, radar threats cannot be overlooked. Once AC is detected by radar, the mission is prone to failure. However, most existing works solely use path length as the evaluation criterion [37], without considering the potential radar threats encountered during flight. Therefore, we consider optimizing the DP association ${\lambda }_{n,\tilde{n}}$ and the path $\mathcal{L}$ of AC in order to minimize the path length and radar threat for AC while satisfying the collision avoidance constraints and the unique access constraint of DP. The optimization problem is formulated as follows:

$$\begin{array}{cc}\hfill \left(\mathbf{P}\mathbf{1}\right):& \underset{\left\{{\lambda }_{n,\tilde{n}},\mathcal{L}\right\}}{\max }\sum _{{\displaystyle \genfrac{}{}{0pt}{}{v_n,v_{\tilde{n}}\in \mathcal{V}\hfill }{v_n\ne v_{\tilde{n}}\hfill }}}\left(\overline{d}\left(l_{n,\tilde{n}}\right)+\mathcal{H}\left(l_{n,\tilde{n}}\right)\right){\lambda }_{n,\tilde{n}}\hfill \\ & \mathrm{s}.\mathrm{t}. \left(6\right),\left(7\right),\left(8\right).\hfill \end{array}$$

(9)

2.3. The Task Assignment Model of UAVs

At each DP, the UAVs deployed are assigned both attack and assessment tasks. Despite variations in the number of UAVs and targets at different DPs, the principle of allocation remains consistent. Based on this, we can adopt a standardized model to describe the task allocation for all DPs. Next, we will discuss the task allocation problem at DP $v_n$ as an example. Let $M_n$ denote the number of targets at DP $v_n$, and ${\displaystyle \sum _{n=1}^N}{M}_n=M$.

In the process of task allocation, while assigning a separate UAV to each target can significantly enhance task execution efficiency, this practice leads to inadequate utilization of UAV resources, resulting in resource wastage. Furthermore, this approach increases the number of UAVs required for deployment, which in turn necessitates opening larger hatch doors on the AC during launch, thereby enlarging the radar cross-section and significantly increasing the risk of detection for the AC. However, reducing the number of deployed UAVs would correspondingly increase the workload for each UAV, prolonging the execution time and even failing to complete the mission due to the limited endurance of the UAV. In view of this, we must optimize the number of UAVs deployed $K_n$ at each DP.

Since each target needs to be attacked once, and the number of ammunition required for each attack by a UAV is $\delta$, with the number of ammunition each UAV carried is ${\sigma }_{\max }$, the minimum number of UAVs required to fulfill all attack tasks can be expressed as

$$K_{n,\min }=⌈{\displaystyle \frac{\delta \times M_n}{{\sigma }_{\max }}}⌉,$$

(10)

where ⌈⌉ is the integer up sign.

To improve the efficiency of task execution, the number of deployed UAVs needs to be greater than $K_{n,\min }$. However, in order to ensure the full use of UAVs, the number of deployed UAVs cannot be too high. Thus, we set the maximum number of deployed UAVs as

$$K_{n,\max }=K_{n,\min }+⌈{\displaystyle \frac{\delta \times M_n}{\beta }}⌉,$$

(11)

where $\beta$ denotes the scaling factor for the number of deployed UAVs.

Each target must undergo both attack and assessment, where the assessment task aims to measure the degree of damage to the target and therefore can only be conducted after the attack task is completed. Given that the smoke generated from ammunition during the attack process can interfere with the assessment results, there must be at least a time interval $T_{gap}$ after the end of the attack task before the assessment task can be initiated. Let $T_{t_m,k_1}$ be the time when the target $t_m\in {\mathcal{T}}_n$ is attacked and $T_{t_m,k_2}$ be the time when the target $t_m$ is evaluated. The task execution time constraint is expressed as

$$T_{t_m,k_1}+T_{gap}\le T_{t_m,k_2},\forall t_m\in {\mathcal{T}}_n.$$

(12)

In this paper, aiming to achieve the trade-off between execution and utilization efficiency, we define the utility function as the weighted sum of execution time of the mission and the effective utilization of UAV. Let $\mathcal{G}$ denote the utility function, which is expressed as

$$\mathcal{G}=\left(\underset{u_k\in {\mathcal{U}}_n}{\max }\left(T_{u_k}\right)\right)+\sum _{u_k\in {\mathcal{U}}_n}\left({\alpha }_1{\displaystyle \frac{T_{\max }-T_{u_k}}{T_{\max }}}+{\alpha }_2{\displaystyle \frac{{\sigma }_{\max }-\delta \times N_{u_k}^{attack}}{{\sigma }_{\max }}}\right).$$

(13)

In the utility function, the first part denotes the execution time of the mission, where $T_{u_k}$ is the execution time of the UAV. The second part denotes the UAV utilization rate, which is measured by the remaining ammunition and remaining flight time. $T_{\max }$ is the maximum endurance of UAV, and $N_{u_k}^{attack}$ is the number of attack tasks carried out by UAVs $u_k$. ${\alpha }_1$ and ${\alpha }_2$ denote the penalty factor for remaining flight time and remaining ammunition, respectively.

Hence, the multi-UAV cooperative mission planning problem at DP $v_n\left(n=1,2,\dots N\right)$ is formulated as

$$\begin{array}{c}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\begin{array}{ccc}\left(\mathbf{P}\mathbf{2}\right):\hfill & \underset{\left\{K_n,{\eta }_{k,m,p}\right\}}{\max }& \mathcal{G}\hfill \\ & \mathrm{s}.\mathrm{t}.& \left(12\right),\hfill \end{array}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\displaystyle \sum _{u_k\in {\mathcal{U}}_n}}{\eta }_{k,m,p}=1,\forall t_m\in {\mathcal{T}}_n,\forall k_p\in \mathcal{K},\hfill & \hfill \left(14\mathrm{a}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\displaystyle \sum _{u_k\in {\mathcal{U}}_n}}{\displaystyle \sum _{t_m\in \mathcal{T}}}{\displaystyle \sum _{k_p\in \mathcal{K}}}{\eta }_{k,m,p}=2M_n,\hfill & \hfill \left(14\mathrm{b}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{T}_{u_k}\le T_{\max },\forall u_k\in {\mathcal{U}}_n,\hfill & \hfill \left(14\mathrm{c}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\delta \times N_{u_k}^{attack}\le {\sigma }_{\max },\forall u_k\in {\mathcal{U}}_n,\hfill & \hfill \left(14\mathrm{d}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{K}_{n,\min }\le K_n\le K_{n,\max },\hfill & \hfill \left(14\mathrm{e}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\eta }_{k,m,p}\in \left\{0,1\right\},\forall u_k\in {\mathcal{U}}_n,\forall t_m\in {\mathcal{T}}_n,\forall k_p\in \mathcal{K}.\hfill & \hfill \left(14\mathrm{f}\right)\end{array}$$

Problem (P2) is a multi-objective optimization problem in which execution time minimization and UAV utilization maximization are considered. Constraints (14a) and (14b) specify that each task is executed only once. Constraints (14c) and (14d) are introduced to limit the flight time and the ammunition load of UAVs, respectively. Constraints (14e) is the restriction on the number of UAVs deployed.

However, multi-UAV systems inevitably encounter dynamic task scheduling problems during missions, such as emergency task insertion, sudden UAV failures or damages. These issues are of great practical significance and have been extensively studied in the literature [38,39,40]. Among existing distributed methods, contract net protocols are an effective solution for dynamic task reallocation, typically achieved through mechanisms such as buy–sell contracts, swap contracts, and replacement contracts.

3. Algorithm Design

In this section, we address problems (P1) and (P2) in three steps based on the decomposition mechanism proposed in Section 2. Firstly, to reduce the radar threat at the DPs and the flight time of UAVs to execute tasks, the position of the DPs and the target–DP association are optimized using the proposed clustering algorithm. Secondly, the trajectory of AC is optimized using an improved discrete genetic algorithm incorporating improved A* (DGAIIA). Lastly, the task allocation problem at each DP is optimized using an improved differential evolution algorithm with market mechanism (IDEMM).

3.1. DP Determination Combined with Distance Clustering

The number of clusters is determined by the number of DPs; thus, we divide M targets into N clusters. Clustering based on distance can generate clusters of arbitrary shapes and process relatively large datasets. Thus, we design the following three steps to search for the optimal position of DPs.

(1)

Target Clustering

Let $c_n\left(n=1,2,\dots ,N\right)$ denote the n-th cluster center. According to the proximity principle, each target will be distributed to one of the clusters, which is expressed as

$$\left\{t_m\in {\mathcal{T}}_n\left|\underset{n=1,2,\dots ,N}{\min }∥t_m-c_n∥\right.\right\},$$

(15)

where $∥t_m-c_n∥$ is the Euclidean distance between target $t_m$ and cluster center $c_n$.

(2)

Optimization of the Cluster Center

Cluster centers $c_n$ are optimized to search DPs that take the least time to execute the mission. According to the principle of distance minimization, the problem of optimizing cluster center can be expressed as

$$\underset{t_m\in {\mathcal{T}}_n}{\min }sd\left(t_m\right)=\sum _{t_{\tilde{m}}\in \left({\mathcal{T}}_n\setminus t_m\right)}∥t_{\tilde{m}}-t_m∥.$$

(16)

In Equation (16), $sd\left(t_m\right)$ represents the sum of the distances from all other target points to $t_m$ when $t_m$ serves as the center of the n-th cluster. We can observe that the cluster center $c_n$ is also the target $t_m$. Compared with other points in the cluster, the optimized cluster center has the shortest sum of distances to other points in the cluster.

(3)

Determination of the DPs

Although the center of the cluster offers the lowest flight cost for the UAV to execute its mission, the radar threat at this location may be significant. Therefore, we select the location with the minimum radar threat as DP within the circle centered at $c_n$ with radius $R_e$. The problem of DP determination can be expressed as

$$\underset{p_e,∥p_e-c_n∥\le R_e}{\min }\mathcal{H}\left(p_e\right),$$

(17)

where $p_e$ is the point within the circle centered at $c_n$ with radius $R_e$, and $\mathcal{H}$() is the function for calculating radar threats, as shown in Equation (2).

The details of the whole process are shown in Algorithm 1.

Algorithm 1 The DP Determination Algorithm

Input: The position of targets and the number of DPs. Output: The classification of target and the position of DPs. 1: Initialize cluster center $c_1,c_2,\dots ,c_N$. ****************************Clustering of targets **************************** 2: For $t_m\in \mathcal{T}$ do 3:   Cluster targets according to Equation (15). 4: End For **********************Optimization of the Cluster Center********************** 5: For $n=1,2,\cdots ,N$ do 6:     Calculate $sd\left(c_n\right)$. 7:     For $\forall t_{\tilde{m}}\in \left({\mathcal{T}}_n\setminus c_n\right)$ do 8:        Calculate $sd\left(t_m\right)$. 9:        If $sd\left(c_n\right)\le sd\left(t_m\right)$ do 10:           $c_n$ unchanged. 11:      Else 12:           $c_n=t_m$. 13:      End If 14:     End For 15: End For ****************************Determination of DPs*************************** 16: For $n=1,2,\dots ,N$ do 17:   The DP is determined according to Equation (17). 18: End For 19: Return ${\mathcal{T}}_n$, $v_n\left(n=1,2,\dots ,N\right)$.

3.2. AC Path Planning Combined with Improved Genetic and A* Algorithm

We design the DGAIIA algorithm to solve the AC’s trajectory optimization problem (P1), which minimizes flight costs and radar threats by optimizing the DP association decision ${\lambda }_{n,\tilde{n}}$ and the flight path $\mathcal{L}$ between DP $v_n$ and DP $v_{\tilde{n}}$. Firstly, we optimize the flight paths between any two DPs. Subsequently, based on the established path evaluation criteria, we optimize the order in which the AC visits the DPs.

Based on the idea of the A* algorithm, we define a heuristic function $\mathcal{F}$ to calculate the path cost from the start node $v_n$ to the end node $v_{\tilde{n}}$ via an expandable node $p_{n\tilde{n},i}$, to search the optimal flight path $\mathcal{L}$ between DP $v_n$ and DP $v_{\tilde{n}}$, which is expressed as

$$\mathcal{F}\left(p_{n\tilde{n},i}\right)=\mathcal{G}\left(p_{n\tilde{n},i}\right)+\mathcal{B}\left(p_{n\tilde{n},i}\right),$$

(18)

where $\mathcal{G}\left(p_{n\tilde{n},i}\right)$ and $\mathcal{B}\left(p_{n\tilde{n},i}\right)$ represent the evaluation function of flight time and the radar threat, respectively. $\mathcal{G}\left(p_{n\tilde{n},i}\right)$ includes the actual flight time from the start node $v_n$ to the current node $p_{n\tilde{n},i}$, as well as the predicted time from the current node $p_{n\tilde{n},i}$ to the end node $v_{\tilde{n}}$, which is expressed as

$$\mathcal{G}\left(p_{n\tilde{n},i}\right)={\displaystyle \frac{\left({\displaystyle \sum _{j=0}^{i-1}}{d}_{p_{n\tilde{n},j},p_{n\tilde{n},j+1}}+\left|x_{p_{n\tilde{n},i}}-x_{\tilde{n}}\right|+\left|y_{p_{n\tilde{n},i}}-y_{\tilde{n}}\right|\right)}{V_c}}.$$

(19)

In Equation (19), ${\displaystyle \sum _{j=0}^{i-1}}{d}_{p_{n\tilde{n},j},p_{n\tilde{n},j+1}}$ is the Euclidean distance from the start node $v_n$ to the current node $p_{n\tilde{n},i}$; $\left|x_{p_{n\tilde{n},i}}-x_{\tilde{n}}\right|+\left|y_{p_{n\tilde{n},i}}-y_{\tilde{n}}\right|$ is the Manhattan distance from the current node $p_{n\tilde{n},i}$ to the end node $v_{\tilde{n}}$.

The performance of the optimal path is influenced by the heuristic functions. Although the heuristic functions with Euclidean distance explore a broader space, they occupy more memory space, and the square root operation of Euclidean distance also increases the computational load. In contrast, heuristic functions with Manhattan distance can traverse relatively fewer redundant nodes, enabling more precise spatial exploration. This type of heuristic function not only improves the efficiency of the algorithm but also enhances the quality of the path. Therefore, this paper adopts a heuristic function with Manhattan distance to calculate the predicted distance, as shown in Equation (19).

Similarly, $\mathcal{B}\left(p_{n\tilde{n},i}\right)$ consists of the actual radar threat from the start node $v_n$ to the current node $p_{n\tilde{n},i}$, and the predicted radar threat from the current node $p_{n\tilde{n},i}$ to the end node $v_{\tilde{n}}$, which is expressed as

$$\mathcal{B}\left(p_{n\tilde{n},i}\right)={\theta }_1\times \sum _{j=1}^i\sum _{g=1}^G{\displaystyle \frac{1}{{\left(d_{r_g,p_{n\tilde{n},j}}\right)}^4}}{I}_{\left(d_{r_g,p_{n\tilde{n},j}}\le R_g^{\det }\right)}+{\theta }_2\times N_{p_{n\tilde{n},i}}^r,$$

(20)

where ${\theta }_1$ denotes the weight of the actual radar threat, ${\theta }_2$ denotes the weight of the predicted part, and $N_{p_{n\tilde{n},i}}^r$ denotes the number of radars included in the rectangular formed by the current node $p_{n\tilde{n},i}$ and the end node $v_{\tilde{n}}$.

According to the path evaluation criteria (18), the DP association ${\lambda }_{n,\tilde{n}}$ on the path of AC is optimized by the improved genetic algorithm we designed. First, each individual is encoded by a chromosome, which has one row and $N+2$ columns. Since the AC departs from the airport and eventually returns to it, the first coded value and the last coded value are $v_0$, and the remaining encoded value are the numbers of DPs. Hence, an individual $I_i$ is evaluated by calculating $\mathcal{E}\left(I_i\right)={\displaystyle \sum _{v_n\in \tilde{\mathcal{V}}}}{\displaystyle \sum _{v_{\tilde{n}}\in \tilde{\mathcal{V}}}}{\lambda }_{n,\tilde{n}}\mathcal{F}\left(p_{n\tilde{n},\tilde{n}}\right)$, where $\mathcal{F}\left(p_{n\tilde{n},\tilde{n}}\right)$ as shown in (18). $\mathcal{E}\left(I_i\right)$ represents the path cost of AC, it consists of the flight time and the radar threat.

(1)

The Selection Operation

The roulette wheel method is used to select the best individual, and its selection probability is calculated as follows. First, the estimated value of each individual is subtracted from the largest estimated value, i.e., $\Delta \left(I_i\right)=\mathcal{E}\left(I_i\right)-\underset{I_i\in \mathcal{I}}{\max }\mathcal{E}\left(I_i\right)$, where $\mathcal{I}$ is the set of all individuals. The bigger $\Delta \left(I_i\right)$ is, the better individual $I_i$ is. Accordingly, the selection probability $p_i^{sel}$ of the i-th individual $I_i$ is shown as

$$\sum _{j=0}^{i-1}{\displaystyle \frac{\Delta \left(I_j\right)}{{\displaystyle \sum _{I_j\in \mathcal{I}}}\Delta \left(I_j\right)}}\le p_i^{sel}<\sum _{j=0}^i{\displaystyle \frac{\Delta \left(I_j\right)}{{\displaystyle \sum _{I_j\in \mathcal{I}}}\Delta \left(I_j\right)}}.$$

(21)

(2)

The Crossover Operation

To enrich the diversity of individuals, the crossover operation is carried out as follows. A contiguous fragment containing $\rho$ encoded values (except the first coded value and the last coded value) is taken from parent 1, and then each coded value (except the first coded value and the last coded value) of parent 2 is checked in turn. If the coded value being checked appears in the intercepted fragment, the coded value in the intercepted fragment will be assigned to the coded value being checked in parent 2. Otherwise, the coded value being checked in parent 2 is skipped and the next coded value is checked. This process is repeated until all coded values of parent 2 have been checked.

An example of the crossover operation is shown in Figure 4. A contiguous fragment containing $\rho =4$ encoded values is intercepted from parent 1. It can be observed that the first coded value of parent 2 is 2, which appears in the intercepted fragment, thus updating the first coded value of parent 2 to the first coded value of the intercepted fragment. The second coded value of parent 2 also appears in the intercepted fragment, thus the second coded value of parent 2 is also updated. The third coded value of parent 2 does not appear in the intercepted fragment; this position is skipped and the next position is checked.

(3)

The Mutation Operation

To avoid falling into local optimality, the mutation operation is performed on the crossover individual, which is implemented as follows. Two random integers $r_1,r_2\in \left[1,N\right]$ are generated, and then the coding value in the $r_1$-th position is exchanged with that in the $r_2$-th position.

The DGAIIA is designed by incorporating the path planning method between two DPs (improved A* algorithm) into the DPs’ travePSOl order optimization method (improved genetic algorithm), and its pseudo-code is shown in Algorithm 2.

3.3. The Multi-UAV Cooperative Mission Planning Combined with Improved Differential Evolution Algorithm

To solve the task assignment problem (P2) of multi-UAV, we design the IDEMM algorithm. We optimize the number of deployed UAVs $K_n$ and the task assignment decision ${\eta }_{k,m,p}$ to minimize execution time and maximize UAV utilization.

Algorithm 2 The AC path planning algorithm

Input: Location of DPs, radars, and no-fly zones; the number of DPs N; population size ${\mathcal{N}}_{DGAIIA}$; maximum iteration ${\mathcal{M}}_{DGAIIA}$; mutation rate $P_{mut}$. Output: The DP association decision ${\lambda }_{n,\tilde{n}}$, and the flight path $\mathcal{L}$ of AC. 1: Initialize the population based on the improved A* algorithm. 2: For $i=1,2,\cdots ,{\mathcal{M}}_{DGAIIA}$ do 3:   Let $n=0$. 4:   While $n\le {\mathcal{N}}_{DGAIIA}$do 5:       Execute the selection operation. 6:       Execute the crossover operation. 7:       If $rand\le P_{mut}$ do 8:           Execute the mutation operation. 9:       End If 10:     $n=n+1$. 11:   End While 12:   Record the optimal individual $O_i$ in the i-th generation. 13: End For 14: Return ${\lambda }_{n,\tilde{n}}^{*},{\mathcal{L}}^{*}$.

The differential evolution (DE) algorithm has advantages in solving the multi-UAV cooperative task assignment problem due to its simple calculation and strong searching ability [41,42]. However, as the number of iterations increases, the fundamental DE diminishes population diversity and prematurely converges to the local minima. To address these issues, we propose an improved DE algorithm, which includes the first variation operator based on the dual switching search strategy and the second variation operator based on the neighborhood-first buying-and-selling mechanism. The flowchart of the IDEMM algorithm for solving the multi-UAV cooperative mission planning is shown in Figure 5.

(1)

Task Assignment Scheme Coding

Multi-dimensional chromosome coding can effectively present multiple types of data information. Since the task assignment scheme is related to the target, task types, and UAVs, the multi-dimensional chromosome matrix coding can clearly represent the assignment relationship between them. The task assignment scheme coding is shown in Figure 6; each column shows the assignment of the UAV, and there are $2\times M_n$ columns in total. Take the first column as an example, it indicates that the UAV $u_{j_1}^n$ is assigned to execute the attack task $k_1$ of target $t_1^n$.

(2)

The First Variation Operator

As shown in Figure 5, we propose the dual search strategy mechanism in the variation operator to update the population, which includes the local search strategy based on the auction mechanism and the large-scale search strategy based on the crossover operation. These two strategies manage to maintain population diversity while ensuring the speed of convergence.

The mission completion time is determined by the UAV with the longest execution time, it will decrease by optimizing the tasks performed by the UAV with the longest execution time. Thus, we optimize the tasks performed by the UAV with the longest execution time through the auction idea. In this process, the UAV with the longest execution time is selected as the auction center and posts the auction task to other UAVs. UAVs that are not capable of performing auction tasks do not participate in the auction activity, while other UAVs will calculate the task completion time for adding the auction task to their task list.

The bid price for UAVs is set as the difference between the task completion time required to add the auction task to their task list and the task completion time of the auction center. Therefore, a lower bid price for a UAV indicates that the new scheme has an advantage, meaning that it can complete the task faster compared to the existing scheme. Ultimately, the auction center will select the UAV with the lowest bid price to execute the auction task. The auction mechanism is illustrated in Figure 7.

The local search technique may fail when the bid prices of all UAVs are positive. To tackle this issue, a large-scale search strategy based on the crossover operation is proposed, as shown in Figure 8. The details of the operation are listed as follows.

Multi-UAV crossover: Multiple tasks are selected by the random method, and UAVs performing selected tasks are exchanged under the ammunition constraint. For example, there are three tasks selected in Figure 8: $k_1$ of target $t_4^n$, $k_1$ of target $t_1^n$, and $k_2$ of target $t_5^n$. The crossover operation will assign UAV $u_{j_1}^n$ to $k_1$ of target $t_1^n$, UAV $u_{j_2}^n$ to $k_2$ of target $t_5^n$, and UAV $u_{j_3}^n$ to $k_1$ of target $t_1^n$.

Target crossover: Two targets are randomly selected for crossover under the constraint of task sequence. As shown in Figure 8, two targets, $t_1^n$ and $t_3^n$, are selected, and then the coding positions of the two targets are swapped.

(3)

The Second Variation Operator

To enhance the optimization ability of the proposed algorithm, we design a buying-and-selling strategy based on neighborhood-first. Both the mission completion time and the utilization efficiency of UAVs is determined by the number of deployed UAVs. Thus, the proposed strategy is designed to optimize the number of deployed UAVs, and it mainly removes UAVs that only perform one task. It is carried out as follows:

Step 1: Calculate the number of tasks $N_{t,j}$ performed by UAV $u_j^n\left(1\le j\le K_n\right)$.

Step 2: If $N_{t,j}=1$, this task is referred to as $a_{exe}$, and then proceed to Step 3, or otherwise, terminate the process.

Step 3: A buying-and-selling activity will be initiated by $u_j^n$, its sales task $a_{exe}$ or purchase task $a_{neigh}$, which represents the other tasks with the same target. For example, the neighboring task of $k_1$ is $k_2$. Note that the task $a_{exe}$ is preferentially sold to UAVs that execute the neighboring task of $a_{exe}$.

Step 4: The UAV $u_j^n$ calculates the gain $b_i$ obtained by purchasing $a_{neigh}$ and the gain $b_n$ obtained by selling $a_{exe}$ to the UAV that executes the neighboring task of $a_{exe}$. If $b_n>b_i>0$, the UAV $u_j^n$ sells the task $a_{exe}$. If $b_i>b_n>0$, $u_j^n$ purchases $a_{neigh}$. Otherwise, we calculate the gain obtained by selling $a_{exe}$ to other UAVs. Note that the gain refers to the improved performance of the new scheme compared to the original scheme.

4. Simulation and Analysis

4.1. Environment Settings

The simulation environment was Lenovo YOGA Book, Windows 11, and Inter Core i5-1135G7, and the program was implemented in Python 3.6.

We assume that the scenario contains an airport at (0 m, 0 m), 80 targets, 11 no-fly zones, and 10 radars. The targets are unevenly distributed in a square area of 1000 km × 1000 km. The distribution of targets, no-fly zones, radars, and airport is shown in Figure 9, in which the no-fly zones are depicted in black and their expansion is depicted in gray. The major simulation settings are shown in

Table 1. Simulations settings.

Symbol	Meaning	Value
N	The number of DPs	8
${\sigma }_{\max }$	The maximum ammunition load of UAV	3
$\delta$	The ammunition number needed for the attack task	1
$T_{\max }$	The maximum flight time of UAV (h)	10
$R_g^{\det }$	The detection radius of the radar (km)	80
$V_u$	The flight speed of UAV (km/h)	100
$T_{gap}$	The time interval between tasks (h)	0.1
$R_e$	The search radius of the DP (km)	50

. The parameters of the DGAIIA and the IDEMM algorithm are shown in

Table 2. Parameters of the DGAIIA.

Parameter	Meaning	Value
${\mathcal{N}}_{DGAIIA}$	The population size	50
${\mathcal{M}}_{DGAIIA}$	The maximum iteration	400
$P_{mut}$	The variation rate	0.1
${\theta }_1$	The weight of actual radar threat	$1\times {10}^7$
${\theta }_2$	The weight of predicted radar threat	10

and

Table 3. Parameters of the IDEMM.

Parameter	Meaning	Value
${\mathcal{N}}_{IDEMM}$	The population size	50
${\mathcal{M}}_{IDEMM}$	The maximum iterations	400
$\beta$	The scaling factor for the number of dropped UAVs	3
${\alpha }_1$	The penalty factor of remaining flight time	0.8
${\alpha }_2$	The penalty factor of remaining ammunition	0.2

, respectively.

4.2. The Location Determination of DPs

In this section, the DP optimization algorithm is demonstrated. First, the targets are divided into 8 clusters according to the first step and second step of Algorithm 1, and the numbers of targets for each cluster are 5, 9, 14, 13, 6, 9, 14, and 10, respectively. As shown in Figure 10, the targets of each cluster are marked with a different color. Then, the optimal DP is searched in the drawn circle with a radius of 50 km centered at the center of each cluster according to the third step of Algorithm 1, and the optimal DP is marked with a black cross. It can be observed that the center of the drawn circle is selected as the DP when there are no radars in the drawn circle; otherwise, the position of DP will be far away from the radar. This is because the DP needs to stay as close to the targets as possible for reducing the flight cost of UAVs, while also considering staying away from the radar to avoid detection. Therefore, the proposed DP determination algorithm can effectively avoid radar threats and reduce the flight cost of UAVs. The locations of the optimal DPs are presented in

Table 4. The locations of DPs.

Number	Location (km)	Number	Location (km)
$v_1$	(772,729)	$v_2$	(467,891)
$v_3$	(890,195)	$v_4$	(154,290)
$v_5$	(847,640)	$v_6$	(283,811)
$v_7$	(189,639)	$v_8$	(491,155)

.

To investigate the impact of varying the number of DPs N on system performance, we provide the path lengths of AC, the total number of deployed UAVs, and the total task execution time with different N in

Table 5. Performance of the proposed algorithm with varying numbers of DPs.

	AC Path Length (km)	Deployed UAV Number	Execution Time (h)
N = 6	2966.94	54	7.9
N = 8	3128.13	51	8.3
N = 10	3208.97	49	9.6

. When N is small, each DP corresponds to a larger target cluster covering a wider area. Due to the limited flight radius and payload capacity of UAVs, long-distance targets require separate UAV deployment for task execution, resulting in lower UAV utilization and consequently higher UAV deployment numbers. As N increases, the AC path length extends since it needs to traverse more DPs. This prolonged transportation path increases AC travel time, thereby lengthening total mission duration. In summary, selecting N involves a trade-off between deployed UAV numbers and mission execution time.

4.3. The AC Path Planning

(1)

The Path Planning between any two DPs

In this section, the performance of the proposed A* algorithm is verified by numerical results. We present the optimal paths from DP $v_6$ to DP $v_3$, from DP $v_5$ to DP $v_8$, and from DP $v_0$ to DP $v_6$, as shown in Figure 11. It is observed that the optimal paths avoid the radar threat and no-fly zones, indicating the feasibility and reliability of the improved A* algorithm. Furthermore, we also investigate the impact of radar detection radius on the optimal path, as shown in Figure 11. As the radar detection radius increases, the optimal path of the AC moves farther from the radar, resulting in a longer flight distance. Additionally, we conduct a comparison between the proposed A* algorithm and the classical rapidly exploring random tree (RRT) algorithm [43], as presented in

Table 6. Comparison between RRT and the proposed algorithm.

	RRT		Proposed Algorithm
	Runtime (s)	Path Length (km)	Runtime (s)	Path Length (km)
$v_6\to v_3$	19.94	1094.76	5.18	950.31
$v_5\to v_8$	10.09	764.93	1.58	684.74
$v_0\to v_6$	16.61	1147.26	3.95	941.68

. The results demonstrate that the proposed algorithm outperforms RRT in all tested scenarios. Specifically, for the path from DP $v_6$ to DP $v_3$, the proposed algorithm reduces runtime from 19.94 s to 5.18 s and shortens path length from 1094.76 km to 950.31 km. Consistent improvements are observed in other tested scenarios, with significant reductions in runtime and substantial improvements in path optimality. Therefore, for obstacle avoidance tasks, the proposed algorithm significantly enhances computational efficiency and path optimality compared to the classical RRT algorithm, while ensuring path safety.

(2)

The Overall Path Planning of AC

In this section, through numerical results, we validate our proposed DGAIIA algorithm for flight cost and radar threat minimization, and we also confirm the feasibility of our proposed scheme for solving the overall path of the AC in the carrier-based UAV swarm mission planning. We present the optimal path of the AC in Figure 12, and an animation of the optimal paths can be viewed on the website. As shown in Figure 12, the AC does not revisit DPs, and the optimal visiting order is $v_0\to v_4\to v_7\to v_6\to v_2\to v_1\to v_5\to v_3\to v_8\to v_0$.

Next, we study the influence of the weight of actual radar threat ${\theta }_1$ on the optimal path. As shown in Figure 13, with the increase in the weight ${\theta }_1$, the optimal paths move further away from radar threats. When the weight of actual radar threat is set as ${\theta }_1=1\times {10}^8$, the optimal path is farthest from radars.

To reveal a useful and fundamental trade-off between flight cost and radar threat of the optimal path, we provide the values of path length $J_1={\displaystyle \sum _{v_n,v_{\tilde{n}}\in \mathcal{V},v_n\ne v_{\tilde{n}}}}\overline{d}\left(l_{n,\tilde{n}}\right){\lambda }_{n,\tilde{n}}$ and radar threat $J_2={\theta }_1\times {\displaystyle \sum _{j=1}^i}{\displaystyle \sum _{g=1}^G}{\displaystyle \frac{1}{{\left(d_{r_{g,}{p}_{n\tilde{n},j}}\right)}^4}}{I}_{\left(d_{r_{g,}{p}_{n\tilde{n},j}}\le R_g^{\det }\right)}$, as well as the actual radar threat $J_3={\displaystyle \frac{J_2}{{\theta }_1}}$ with different ${\theta }_1$ in

Table 7. $J_1$, $J_2$, and $J_3$ of the optimal paths with different ${\theta }_1$.

${\mathit{\theta }}_1$	${\mathit{J}}_1$	${\mathit{J}}_2$	${\mathit{J}}_3$
$1\times {10}^5$	3105.81	33.10	$3.31\times {10}^{-4}$
$1\times {10}^6$	3110.54	91.43	$9.14\times {10}^{-5}$
$1\times {10}^7$	3162.46	250.63	$2.51\times {10}^{-5}$
$1\times {10}^8$	3341.18	907.58	$9.08\times {10}^{-6}$

. As the weight ${\theta }_1$ increases from $1\times {10}^5$ to $1\times {10}^7$, the increase in $J_1$ and $J_2$ is minimal. However, when ${\theta }_1$ increases from $1\times {10}^7$ to $1\times {10}^8$, the increase in $J_1$ and $J_2$ is quite significant, though the reduction in actual radar threats is negligible. It can be seen that while a sufficiently large weight ${\theta }_1$ can effectively avoid radar detection, the length of the AC’s flight path also increases. Therefore, the weight is set to ${\theta }_1=1\times {10}^8$ in this paper.

4.4. The Multi-UAV Cooperative Mission Planning

In this section, we validate the feasibility and effectiveness of the IDEMM algorithm. Specifically, the proposed algorithm is applied to solve the task assignment problem at DP $v_3$, and a detailed analysis is discussed. Readers interested in the task assignment results at other DPs can obtain them by visiting the website we have created (available at bilibili.com/video/BV1GN4y1N7eg, accessed on 7 January 2025). We present the optimal task assignment scheme obtained by the proposed algorithm in

Table 8. The optimal task assignment scheme at DP $v_3$.

UAV	Task List	Execution Time (h)
$u_1$	$(t_{61},k_1),(t_{20},k_1),(t_{20},k_2),(t_{68},k_2)$	7.9
$u_2$	$(t_{19},k_1),(t_{19},k_2),(t_{46},k_1),(t_{46},k_2),(t_{55},k_1),(t_{55},k_2)$	6.5
$u_3$	$(t_{32},k_1),(t_{32},k_2),(t_{27},k_1),(t_{27},k_2),(t_{68},k_1),(t_{68},k_2)$	7.6
$u_4$	$(t_{72},k_1),(t_{36},k_1),(t_{23},k_2),(t_{40},k_1)$	7.4
$u_5$	$(t_{72},k_2),(t_{23},k_1),(t_{61},k_2),(t_{21},k_1),(t_{21},k_2)$	7.1
$u_6$	$(t_{36},k_2),(t_{14},k_1),(t_{14},k_2)$	7.4

, as well as the sequence and timing of task execution for each UAV in Figure 14.

It is observed from Table 8 and Figure 14 that the optimal assignment scheme satisfies the ammunition load constraint, the flight time limit, and the task sequence requirement. Taking UAV $u_3$ as an example, it departs from DP $v_3$, first flying to target $t_{32}$, then sequentially reaching targets $t_{27}$, $t_{68}$, and $t_{40}$, and finally returning to DP $v_3$. Additionally, UAV $u_3$ first performs the attack task $k_1$ on target $t_{32}$, and then carries out the assessment task $k_2$ on target $t_{32}$. The execution order for the remaining targets is the same. UAV $u_3$ performs a total of six tasks, including three attack tasks, with an execution time of 7.6 h. Moreover, it can be observed from Figure 14 that the execution time of each UAV is similar and close to its maximum endurance time. Thus, the above analysis indicates that the proposed algorithm can efficiently utilize UAVs and quickly complete tasks under multiple constraints.

To demonstrate the performance of our proposed algorithm, we compare it with three other algorithms: particle swarm optimization (PSO) [17], the DE algorithm [44], and the improved differential evolution algorithm with first variation operator (IDEFV). IDEFV is used to validate the optimization capability of the second mutation operator of IDEMM, which is designed by introducing the first mutation operator of IDEMM into the basic DE algorithm. The population size and maximum number of iterations for PSO, DE, IDEFV, and IDEMM are uniformly set to 50 and 400, respectively. We run each algorithm 50 times and calculate the average value of these 50 times to obtain the experimental results.

Figure 15 shows the convergence behavior of the different algorithms, where J is the objective function of problem (P2). We can observe that IDEMM converges the fastest, followed by IDEFV, DE, and PSO. Thus, we conclude that the convergence performance of IDEMM is better than that of other schemes. In Figure 16, we show the distribution of the optimal objective values obtained by different algorithms. The optimal objective values obtained by PSO and DE are concentrated at higher values, indicating that PSO and DE become trapped in local optima too early, leading to poorer optimization efficiency. In Figure 15 and Figure 16, IDEFV demonstrates faster convergence speed and stronger optimization performance due to the introduction of the first mutation operator, but its ability to escape local optima is relatively weak. Compared to other three algorithms, IDEMM generally achieves lower objective values and its median values are farther from the lower edge values, indicating that IDEMM is less likely to become trapped in local optima. Furthermore, to quantify the stability of four algorithms,

Table 9. The mean and Std of the optimal values obtained by four algorithms.

	PSO	DE	IDEFV	IDEMM
Std	0.35	0.24	0.25	0.23
Mean	10.22	10.39	9.58	9.05

presents the mean and standard deviations (Stds) of the optimal objective function values obtained from 50 independent runs. The results indicate that IDEMM achieves the lowest Std and a significantly smaller mean than other algorithms. This demonstrates that IDEMM not only provides higher solution accuracy but also exhibits stronger stability and robustness for this problem. In summary, we can conclude that the proposed algorithm has superior optimization capabilities.

5. Conclusions

In this paper, we address the problem of cross-platform collaborative operations against large-scale targets. We develop a two-level framework to decompose this problem into two sub-problems: the upper-level AC path planning problem and the bottom-level multi-UAV cooperative mission planning problem. For the upper-level problem, we propose the DGAIIA algorithm, which plans the AC’s path according to the position of optimal DPs. The optimal path indicates that DGAIIA effectively minimizes both flight cost and radar threat. For the bottom-level problem, we introduce the IDEMM algorithm, which employs a dual switching search strategy and a neighborhood-first approach combined with a buying-and-selling mechanism to tackle the multi-UAV task assignment problem with multiple constraints. Finally, we conduct simulations to demonstrate the effectiveness of our proposed design.

Due to space limitations, several important issues in cross-platform collaborative operations remain unaddressed and require further investigation. We outline the following concrete directions for future work:

(1) Handling Unknown DP Numbers: Our current approach assumes a predetermined number of DPs. To address this limitation, future work should explore integrating unsupervised clustering algorithms, such as adaptive K-means, to automatically determine both the optimal number and locations of DPs without prior knowledge [45]. Developing robust clustering algorithms in this combat context remains a substantial challenge.

(2) Robustness and Adaptability in Dynamic Environments: The complex and unpredictable battlefield necessitates enhanced adaptability. Future research should focus on the following:

a. Real-time Path Adaptation: Develop algorithms with response capabilities, enabling the AC and UAVs to dynamically modify their paths in real-time using sensor data (e.g., to avoid moving obstacles or sudden threats detected mid-mission) [46].

b. Dynamic Re-planning under Uncertainty: Future work could investigate online partially observable Markov decision process planners or robust model predictive control frameworks to handle inherent uncertainties (e.g., intermittent communication, sudden target appearances, or UAV failures) [47,48].