Coupled Optimization of UAV Cluster Path Optimization and Task Assignment on a Mobile Platform

Abstract

This paper focuses on the coupled optimization problem of path optimization and task assignment for UAVs mounted on mobile platforms. Combining the UAV turning angle, minimum direct flight trajectory and other flight characteristics, the path optimization model on the 3D raster map is established with the objectives of shortest flight time and minimum UAV destruction, and the optimal path between the vertices of each mission is derived by using an improved Gray Wolf Optimization algorithm. Combining the takeoff and landing time-window constraints with the range and mission resource constraints, this mission planning model is established with the objective of maximizing the efficiency ratio of mission revenue–UAV damage consumption. Combining the optimal paths between vertices, a complete UAV flight path is formed, which provides a path optimization and goal assignment method for UAV clusters mounted on mobile platforms to perform multiple tasks cooperatively, and its feasibility and effectiveness are verified through simulation experiments.

1. Introduction

Effective task allocation and path optimization can significantly enhance the effectiveness and survivability of UAVs in performing tasks, and there are a large number of related studies on task allocation and path optimization for UAVs. Refs. [1,2,3,4,5,6,7] investigate the path optimization problem of UAVs based on the Adaptive Osprey Optimization algorithm, the Improved Sand Cat swarm algorithm, the Improved Artificial Bee Colony algorithm, etc.; [8,9,10,11,12,13,14,15] focus on efficient target allocation using greedy algorithms, hybrid particle swarm algorithms based on game theory, etc.; [16] investigates the mission planning problem with the constraint of the takeoff and landing time window in the context of shipborne UAV coordinated combat against maritime targets; and [17] proposes a deep reinforcement learning-based method for solving the UAV path planning problem in the presence of multiple charging stations, but the paths of the UAVs in these last two studies are represented by vertices and arcs, and the actual flight trajectories between vertices are not taken into account. Meanwhile, this traditional approach, which solves the task assignment and path optimization separately, fails to fully consider the coupling relationship between the two, which may lead to the problems of path intersection and uneven resource allocation.

In order to solve these problems, several studies have proposed joint optimization methods. Refs. [18,19,20,21,22] propose a multi-UAV cooperative strike strategy by coupling the path optimization of UAVs with the task allocation in a complex environment considering multiple constraints. Refs. [23,24] solve the problem of task allocation and trajectory planning for multiple UAVs striking moving targets on the ground in a 2D environment by predicting the encounter point through UAVs pursuing moving targets. However, there is still a lack of research on coupling and optimizing trajectory planning and task allocation in a 3D raster map environment.

Meanwhile, most of the above studies are applicable to shore-based clustered UAVs, and there are still relatively few studies on the application scenario specificities of UAVs mounted on mobile platforms. Such UAVs face unique challenges, such as the limitations of takeoff and landing platforms and takeoff and landing time due to the complex environment at sea [16], and need to plan the trajectory so that they land at the takeoff and landing points after their moving during the takeoff and landing time window, which makes it impossible to strip the return trajectory out of the trajectory planning study of UAVs mounted on mobile platforms. Refs. [24,25] investigated path planning with time-window constraints considering the effect of weather conditions on UAVs, but the whole process of UAV missions is limited by weather, and the time-window constraints for UAVs mounted on mobile platforms only limit their takeoff and landing phases. Further, it is of great significance to study the task assignment and path optimization of UAVs mounted on mobile platforms in depth.

In view of this, in this paper, a UAV path optimization model with the shortest flight time between vertices and the smallest damage to the UAV as the dual optimization objectives is established on a 3D raster map in the context of coordinated execution of multiple objectives by UAVs mounted on mobile platforms, and the mission planning scheme with the largest efficiency and cost ratio is investigated by combining constraints, such as landing time and space, with the constraints of landing time and space.

2. Multi-Objective-Oriented Mission Planning Problem Description for UAV Clusters Mounted on Mobile Platforms

2.1. Problem Description

A surface mobile platform with $m$ homogeneous UAVs is located in a designated sea area, and the UAVs work together to accomplish $n$ task.

Each UAV carries $N$ tools that can be used to accomplish different tasks. Each UAV is located at the initial position for takeoff, and after takeoff, the surface mobile platform maneuvers along the designated route according to the established task, and can guarantee the UAV takes off and lands in a limited period of time, and the corresponding time allowed for landing is the takeoff and landing window; for simplifying the model, we define the center of the corresponding takeoff and landing route zone as the takeoff and landing point, i.e., the UAVs can arrive at the corresponding takeoff and landing points in the time window, and then complete the landing.

There are two types of obstacle areas that UAVs mounted on mobile platforms face when planning a trajectory; one is no-fly areas, such as islands and mountains and other physical obstacles that are impassable. The other is the threat area, such as the flight punishment area set by the mission, which can be passed, but the passage process must bear a certain destruction cost.

In this paper, a 3D raster map is constructed, and the UAV is required to choose the appropriate takeoff point and landing point, and plan the better mission path, which allows the mission to be completed as early as possible, while the overall mission efficiency ratio is maximized.

2.2. Underlying Assumptions and Explanations

(1)

The UAVs mounted on the mobile platform collaborate to accomplish single-wave missions, the UAVs are not reloaded after landing, a single UAV can perform different missions, and the same mission can be performed by different UAVs.

(2)

The average speed of the UAVs mounted on the mobile platform is a constant value for each segment of the trajectory between the navigation points.

(3)

A UAV on a mobile platform performs tasks instantaneously and does not need to consume additional endurance.

(4)

A UAV on a mobile platform that does not successfully return to the mobile piggyback platform within the airborne endurance time runs out of fuel and crashes.

3. Path Optimization Model Between Flying Vertices of a UAV Mounted on a Mobile Platform

The inter-vertex path of a UAV mounted on a mobile platform consists of line segments connecting multiple navigation points, and finding the optimal path between two vertices is to solve for the coordinates of each navigation point on a 3D raster map.

3.1. Objective Function

3.1.1. Inter-Vertex Time-of-Flight Objective Function

In order for UAVs mounted on mobile platforms to perform their tasks more efficiently and serve the mission requirements, the inter-vertex flight time is therefore minimized, i.e.:

$$\min F_1={\displaystyle \sum _{a=0}^h\frac{\left|\right|P_a{P}_{a+1}\left|\right|}{v}}$$

(1)

$$h=\beta {\displaystyle \frac{S}{d_{\min }}}$$

(2)

In the formula, $P_a=(x_a, y_a, z_a)$ denotes the path node of the $a$th navigation point of the UAV mounted on the mobile platform, $P_{a+1}=(x_{a+1}, y_{a+1}, z_{a+1})$ is the path node of the next navigation point, $\left|\right|P_a{P}_{a+1}\left|\right|$ denotes the Euclidean distance between the two path nodes, $v$ denotes the average speed of the UAV mounted on the mobile platform, $h$ denotes the number of navigation points, $S$ denotes the Euclidean distance between the vertices, $d_{\min }$ denotes the shortest direct flight distance required for the stabilization of the attitude of the UAV mounted on the mobile platform, and $\beta$ denotes the coefficient of selection of the number of navigation points.

3.1.2. Damage Cost Objective Function

The UAVs on the mobile platform, on the path of the mission, will pass through the flight penalty areas set by the mission, and at the same time there are no-fly zones such as islands, reefs, mountains, and so on.

The objective function is to minimize the damage cost of the flight penalty threat to the UAV mounted on the mobile platform throughout the mission, i.e.:

$$\min f_2={\displaystyle \sum _{j=0}^h{\displaystyle \sum _{f=1}^r{T}_o(P_a{P}_{a+1})}}$$

(3)

In the formula, $T_o(P_a{P}_{a+1})$ denotes the damage cost of the flight discipline threat suffered by the UAV mounted on the mobile platform traversing the $o$th mission-defined flight discipline area, and $r$ denotes the number of mission-defined flight discipline area zones.

We assume that every time the UAV trajectory traverses a threat zone, it is subjected to one flight discipline, and the probability of the UAV being destroyed for each flight discipline is $p$, and the number of times the UAV traverses the threat zone is $K$:

$$T_o(P_a{P}_{a+1})=1-{(1-p)}^K$$

(4)

3.1.3. Normalization of the Fitness Function

In order to eliminate the influence of the magnitude between different degrees of adaptation, the flight time adaptation function is standardized, and the time required for the UAV to fly through the Euclidean distance between the starting point and the end point, $l$, is set as the standard flight time, and the flight time adaptation is expressed as the ratio of the difference between the actual flight time and the standard flight time:

$$f_1={\displaystyle \frac{t-t_{s\tan d}}{t_{s\tan d}}}={\displaystyle \frac{{\displaystyle \sum _{j=0}^h(|\left|P_a{P}_{a+1}\right||-l)/v}}{l/v}}={\displaystyle \sum _{j=0}^h\frac{\left|\right|P_a{P}_{a+1}\left|\right|-l}{l}}$$

(5)

Then, the full objective function $F(X_i)$ of the UAV trajectory planning of this UAV mounted on the mobile platform is denoted as:

$$\min F={\omega }_1{f}_1+{\omega }_2{f}_2$$

(6)

Assigning different ${\omega }_i$ to this objective function, the flight time priority or damage cost priority can be set.

3.2. Restrictive Conditions

3.2.1. No-Fly Zone Restrictions

UAVs mounted on mobile platforms may encounter no-fly zones such as mountains on their mission paths and crash immediately after crossing them; so, to ensure the safety of UAV flights, it is necessary to maintain a safe distance from the no-fly zones:

$${\Omega }_a\cap V_c=Ø, \forall a\in [0,h]$$

(7)

In the formula, ${\Omega }_a$ denotes the set of all points on the track segment $P_a{P}_{a+1}$, and $V_c$ denotes the set of all points in the no-fly zone.

3.2.2. Flight Turn Angle Constraints

Due to the constraints on the maneuverability of UAVs mounted on mobile platforms, to ensure the flyability of UAVs, their flight turning angles are constrained and divided into turning declination constraints in the horizontal direction and turning inclination constraints in the vertical direction:

$${\theta }_{P_a{P}_{a+1},P_{a+1}{P}_{a+2}}=\arctan ({\displaystyle \frac{(y_{a+2}-y_{a+1})-(y_{a+1}-y_a)}{(x_{a+2}-x_{a+1})-(x_{a+1}-x_a)}})\in [{\theta }_{\min },{\theta }_{\max }]$$

(8)

$$\begin{array}{cc}{\phi }_{P_a{P}_{a+1},P_{a+1}{P}_{a+2}}& =\arctan ({\displaystyle \frac{(z_{a+2}-z_{a+1})-(z_{a+1}-z_a)}{\sqrt{{((x_{a+2}-x_{a+1})-(x_{a+1}-x_a))}^2+{((y_{a+2}-y_{a+1})-(y_{a+1}-y_a))}^2}}})\hfill \\ & \in [{\alpha }_{\min },{\alpha }_{\max }]\hfill \end{array}$$

(9)

In the formula, ${\theta }_{P_a{P}_{a+1},P_{a+1}{P}_{a+2}}$ denotes the horizontal declination of the two segments of the track between the $a$th navigational point and the $a+2$th navigational point, and ${\phi }_{P_a{P}_{a+1},P_{a+1}{P}_{a+2}}$ denotes the vertical inclination of the two segments of the track between the $a$th navigational point and the $a+2$th navigational point.

3.2.3. Flight Level Constraint

UAVs mounted on mobile platforms face flight scenarios mostly over the ocean, where the climate environment is complex and changeable, and they face unpredictable factors such as waves, which may cause a certain degree of obstruction to the mission execution of the UAVs, and at the same time, the constraints on the maneuverability of the UAVs themselves impose a constraint on the flight altitude of the UAVs mounted on the mobile platform:

$$z_a\in [h_{safe},h_{\max }]$$

(10)

In the formula, $h_{safe}$ denotes the minimum safe distance to avoid the influence of waves, and $h_{\max }$ denotes the maximum altitude that a UAV mounted on a mobile platform can fly to.

3.2.4. Minimum Trajectory Constraint

In order to ensure that UAVs mounted on mobile platforms can be operated in a stable and safe manner while performing their tasks, so that they can maintain a sufficiently straight flight distance before changing its flight attitude, a constraint is imposed on the shortest length of the trajectory to avoid the instability and potential risks caused by frequent turns:

$$\left|\right|P_a{P}_{a+1}\left|\right|\ge d_{\min }, \forall a\in [0,h]$$

(11)

In the formula, $d_{\min }$ denotes the shortest direct flight distance required to stabilize the attitude of a UAV mounted on a mobile platform.

3.3. Trajectory Planning Algorithm Based on Improved GWO

3.3.1. Track Crossing Threat Area Detection

In this paper, the flight disciplinary area zone and no-fly zone set by the task is set as a spherical model, and the trajectory crossing threat zone detection is divided into the following steps in the algorithm:

(1)

Judge whether the points are in the spherical threat zone.

(2)

Judge whether the distance between the midpoint of the line segment and the center of the danger zone is less than the radius.

(3)

Determine whether there is an intersection point between the line segment and the sphere.

3.3.2. Overview of the GWO Algorithm

Gray Wolf Optimization (GWO) is a group-intelligence-based optimization algorithm that simulates the social hierarchy and hunting behavior of gray wolves [26]. In the algorithm, the wolf pack consists of α-, β-, and δ-ranked wolves, where the α wolf is the leader with the strongest leadership and hunting skills. The algorithm searches for the optimal solution by simulating the wolf pack’s behavior of tracking, attacking and rounding up prey.

In the field of modern optimization, the Gray Wolf Optimization (GWO) algorithm has received widespread attention for its simplicity and effectiveness. However, traditional GWO algorithms may encounter the problems of slow convergence and premature convergence when dealing with complex problems. In order to overcome these limitations, this paper employs an improved multiple-swarm Gray Wolf Optimization algorithm (MP-GWO) [27].

3.3.3. Design of Multiple-Pack Gray Wolf Optimization Algorithm (MP-GWO)

In the traditional GWO algorithm, the whole wolf pack is searched as a whole, which may lead to a lack of diversity in the search process, especially when the solution space is large or the problem is more complex. In order to improve the search ability and adaptability of the algorithm, this paper proposes a multiple pack strategy, which divides the wolf pack into multiple sub-populations, each of which independently performs search and optimization [24]. The core framework of the MP-GWO algorithm includes the following steps:

• Initialization: initial wolf packs are randomly generated and clustered to form multiple sub-populations.
• Sub-population search: each sub-population is searched independently, and the position of the wolf pack is updated by simulating the hunting behavior of the wolf pack.
• Dynamic updating: a dynamic updating mechanism is introduced to dynamically adjust the search strategy according to the change of the wolf pack’s adaptation.
• Merging and reorganization: during the iterative process, sub-populations are merged periodically to enhance the diversity of the population and avoid premature convergence.

4. Modeling Multi-Objective-Oriented Mission Planning for Unmanned Aerial Vehicles on Mobile Platforms

The path of a UAV mounted on a mobile platform consists of a start point, a target point, a drop point and an arc connecting the relevant vertices, and the essence of solving its planning problem is to find an optimal or near-optimal sequence of vertices and arcs that satisfies the various constraints and optimization objectives in the graph structure constructed by the vertices and arcs.

4.1. Description of Decision Variables

The matching relationship between UAVs mounted on mobile platforms and targets is expressed in terms of 0–1 decision variables, i.e.:

(1)

Execution task decision variable $x_{ij}^k$:

$$x_{ij}^k=\left\{\begin{array}{l}1\\ 0\end{array}\right.\begin{array}{l}The kth UAV flies from vertex i to vertex j\\ Other\end{array}$$

(2)

Missile allocation decision variable $y_i^k$:

$$y_i^k=\left\{0,1,2,\dots ,N\right\} Number of mission tools assigned to the ith vertex by the kth UAV$$

4.2. Objective Function

When assigning objectives to the UAV mounted on the mobile platform, we have to take into account both the benefits obtained after the completion of the mission and the damage to the UAV mounted on the mobile platform during the execution of the mission, so we select the maximum mission cost-effectiveness ratio as the optimization objective of the objective assignment model.

4.2.1. Objective Function of Mission Benefit

Different targets have different values, and the completion probability of the UAVs mounted on the mobile platform to perform tasks with different targets varies, and the purpose is to select the tasks that satisfy the UAV resource constraints so that the overall task gain is the highest, i.e.:

$$\max G={\displaystyle \sum _{b=1}^n(1-{\displaystyle \prod _{k=1}^m{(1-p_b)}^{y_b^k}})V_b}$$

(12)

In the formula, $V_b$ denotes the value of the $b$th target task, $p_b$ denotes the completion probability of the $b$th target task performed by each task tool, $n$ denotes the number of target vertices, and $m$ denotes the number of UAVs.

4.2.2. Damage Cost Objective Function

UAVs assigned to carry out different target tasks and mounted on mobile platforms suffer different levels of damage, which the optimization aims to minimize, i.e.:

$$\min J={\displaystyle \sum _{k=1}^m(1-{\displaystyle \prod _{i=1}^{n+c+d}{\displaystyle \prod _{j=1}^{n+c+d}(1-x_{ij}^k{p}_{ij}}}}))V$$

(13)

In the formula, $p_{ij}$ denotes the damage probability on the way from the $i$th vertex to the $j$th vertex, $V$ denotes the value of the UAV on the mobile platform, $c$ denotes the number of takeoff vertices, and $d$ denotes the number of landing vertices.

Then, the objective function $E$ of the mission planning model of the UAV mounted on the mobile platform is denoted as:

$$\max L={\displaystyle \frac{G}{J}}$$

(14)

4.3. Constraints

4.3.1. Continuity Constraint

In order to avoid crashes of UAVs on mobile platforms due to fuel depletion during missions, it is necessary to ensure that the range of the UAVs is less than their endurance range:

$${\displaystyle \sum _{i=1}^{n+c+d}{\displaystyle \sum _{j=1}^{n+c+d}{x}_{ij}^k}}{t}_{ij}\le S,\forall k\in [1,m]$$

(15)

In the formula, $S$ denotes the range distance of a single UAV.

4.3.2. Start and End Vertex Constraints

In UAV path optimization when mounted on a mobile platform, a start vertex and an end vertex must be selected at the takeoff and landing points, i.e.:

$${\displaystyle \sum _{i\in V_{start}^k}{\displaystyle \sum _{j\in V_k}{x}_{ij}^k}}=1, \forall k\in [1,m]$$

(16)

$${\displaystyle \sum _{i\in V_{end}^k}{\displaystyle \sum _{j\in V_k}{x}_{ij}^k}}=1, \forall k\in [1,m]$$

(17)

In the formula, $V_k$ denotes the set of all mission vertices, $V_{s\tan d}^k$ denotes the set of takeoff vertices of the UAV $k$ mounted on the mobile platform, and $V_{land}^k$ denotes the set of landing vertices of the UAV $k$ mounted on the mobile platform.

4.3.3. Takeoff and Landing Time-Window Constraints

Due to the specificity of the takeoff and landing conditions of the UAV mounted on the mobile platform, its delivery platform sails along the mission route in accordance with the established task arrangement, and arrives at the suitable takeoff and landing sea area, within the time window $t_{s\tan d}$, the takeoff and landing task of the UAV mounted on the mobile platform is open, and for the purpose of simplifying the model, the center of the sea area is taken as the takeoff and landing point, i.e., the UAV arrives at the center of the sea area within the corresponding time window, which means that the UAV can be finished landing:

$$\inf t_{land}^k={\displaystyle \sum _{i=1}^{n+c+d}{\displaystyle \sum _{j=1}^{n+c+d}{x}_{ij}^k}}{t}_{ij}+\inf t_{start}^k={\displaystyle \sum _{i=1}^{n+c+d}{\displaystyle \sum _{j=1}^{n+c+d}{x}_{ij}^k}}{t}_{ij}+\inf t_s$$

(18)

$$\sup t_{land}^k={\displaystyle \sum _{i=1}^{n+c+d}{\displaystyle \sum _{j=1}^{n+c+d}{x}_{ij}^k}}{t}_{ij}+\sup t_{start}^k={\displaystyle \sum _{i=1}^{n+c+d}{\displaystyle \sum _{j=1}^{n+c+d}{x}_{ij}^k}}{t}_{ij}+\sup t_s$$

(19)

$$t_{land}^k\cap t_l\ne Ø$$

(20)

In the formula, $t_{start}^k$ denotes the takeoff time interval of the UAV $k$ mounted on the mobile platform, $t_{land}^k$ denotes the landing time interval of the UAV $k$ mounted on the mobile platform, $t_s$ denotes the takeoff time window of the takeoff point $s$, and $t_l$ denotes the landing time window of the landing point $l$.

4.3.4. Path Vertex Constraints

Each UAV mounted on the mobile platform carries a total of $N$ mission tools and can select at most $N$ mission vertices, so the number of its path vertices is constrained to be:

$${\displaystyle \sum _{i\in V_k}{\displaystyle \sum _{j\in V_k}{x}_{ij}^k}}\le N+2,\forall k\in [1,m]$$

(21)

4.3.5. Path Continuity Constraint

For any two consecutive vertices $i$ and $j$ of a UAV mounted on a mobile platform, if the UAV flies from a vertex to a vertex, then it must be ensured that the vertex must be the successor vertex of some previous vertex of the UAV, i.e.:

$$x_{ij}^k\le {\displaystyle \sum _{q\in V_k}{x}_{qi}^k}, \forall k\in [1,m], \forall i\in V_k-V_{start}^k, \forall j\in V_k-V_{end}^k$$

(22)

It must also be ensured that the vertex must be the antecedent vertex of some subsequent vertex of the UAV, i.e.:

$$x_{ij}^k\le {\displaystyle \sum _{e\in V_k}{x}_{je}^k}, \forall k\in [1,m], \forall i\in V_k-V_{end}^k, \forall j\in V_k-V_{end}^k$$

(23)

To avoid UAVs mounted on mobile platforms staying at the same vertex:

$${\displaystyle \sum _{i\in V_k}{x}_{ii}^k}=0, \forall k\in [1,m]$$

(24)

5. Simulation Results and Analysis

Taking four UAVs mounted on the mobile platform to execute 10 tasks collaboratively as an example, a fixed task execution position is set for each maritime target, which is the target vertex. The main parameter information settings in the simulation are shown in

Table 1. Position information of each task vertex.

	Coordinate		Coordinate
$v_1$	(0, 200, 0)	$v_8$	(775, 825, 0)
$v_2$	(50, 100, 0)	$v_9$	(825, 825, 0)
$v_3$	(100, 67, 0)	$v_{10}$	(830, 830, 0)
$v_4$	(200, 0, 0)	$v_{11}$	(500, 700, 0)
$v_5$	(800, 800, 0)	$v_{12}$	(500, 725, 0)
$v_6$	(750, 750, 0)	$v_{13}$	(200, 850, 0)
$v_7$	(825, 775, 0)	$v_{14}$	(225, 850, 0)

,

Table 2. Location information for each no-fly zone.

	Coordinates of the Center Point	Radius
$o_1$	(400, 200, 0)	100
$o_2$	(750, 150, 0)	100
$o_3$	(550, 650, 0)	50
$o_4$	(200, 650, 0)	50

,

Table 3. Information on the location of the flight disciplinary area zones set by the missions.

	Coordinates of the Center Point	Radius
$r_1$	(600, 600, 70)	100
$r_2$	(800, 600, 70)	50
$r_3$	(600, 800, 70)	50
$r_4$	(150, 500, 40)	30
$r_5$	(400, 700, 40)	30
$r_6$	(400, 500, 40)	30
$r_7$	(800, 800, 0)	10
$r_8$	(750, 750, 0)	20
$r_9$	(825, 775, 0)	10
$r_{10}$	(775, 825, 0)	10
$r_{11}$	(825, 825, 0)	10
$r_{12}$	(830, 830, 0)	5
$r_{13}$	(500, 700, 0)	10
$r_{14}$	(500, 725, 0)	5
$r_{15}$	(200, 850, 0)	10
$r_{16}$	(225, 850, 0)	5

,

Table 4. Value attributes of each target vertex.

$v_5$	$v_6$	$v_7$	$v_8$	$v_9$
300	150	100	100	100
$v_{10}$	$v_{11}$	$v_{12}$	$v_{13}$	$v_{14}$
50	100	50	100	50

,

Table 5. Completion probability of each task tool for each goal vertex.

$v_5$	$v_6$	$v_7$	$v_8$	$v_9$
0.4	0.5	0.6	0.6	0.6
$v_{10}$	$v_{11}$	$v_{12}$	$v_{13}$	$v_{14}$
0.8	0.6	0.8	0.6	0.8

and

Table 6. Opening time of each take-off and landing window.

$v_1$	$v_2$	$v_3$	$v_4$
[0, 3800]	[9400, 14,200]	[15,900, 20,600]	[23,600, 29,800]

. The value of the UAV is 50, the endurance time is 19,600 s, the speed is 0.3 Ma and each UAV carries two mission tools; $v_1~v_4$ are the take-off and landing points of the UAV carried on the mobile platform, and $v_5~v_{14}$ are the attack target vertices.

The composed simulation map is shown in Figure 1.

5.1. Trajectory Planning for a Single UAV Mounted on a Mobile Platform

To plan the UAV trajectory of a single UAV mounted on a mobile platform, UAV No. 1 is taken from the take-off and landing point $v_1$, arrives at the target vertices $v_7$ and $v_{11}$, and then returns to the take-off and landing point $v_3$ as an example, and the traditional Gray Wolf Optimization algorithm and the improved two-population Gray Wolf Optimization algorithm are respectively used for simulation and comparison experiments; a schematic comparison of the UAV trajectories is shown in Figure 2, and a comparison of the adaptation convergence curves of the two algorithms is shown in Figure 3.

From the comparison of simulation results, it can be seen that the algorithm’s optimization search efficiency is higher after the introduction of two-species parallel computation, and at the same time, the improved algorithm can more closely approximate the optimal solution, and it achieves a lower fitness value more quickly.

Analyzed from the perspective of stability, the convergence curve of the improved algorithm is relatively smooth with less fluctuation. This indicates that its performance is stable under different operating conditions and reduces the deviation of the results caused by random factors. In contrast, the curve of the traditional Gray Wolf algorithm fluctuates more, reflecting the instability of the search process.

On the basis of this experiment, experiments with the Particle Swarm Optimization Algorithm and the Cuckoo Search Algorithm were added to compare the path planning effect of the related methods and prove the superiority of the improved Gray Wolf Optimization algorithm.

Figure 4 visualizes the path planning results for the UAV using the various algorithms on a simulated map model; as shown in Figure 5, the improved Gray Wolf Optimization algorithm demonstrates path planning results that will be superior to the simple heuristic algorithm. Several types of methods completely avoided the obstacles in the map, but in terms of flight time metrics, the improved Gray Wolf Optimization algorithm can better match the relevant environment to achieve a faster mission and reduce the cost of UAV flight.

5.2. UAV Cluster Mission Planning Mounted on a Mobile Platform

In this paper, the priority of flight time and damage cost is set to be the same, i.e., ${\omega }_1={\omega }_2=0.5$, and combined with the path optimization model, the simulation obtains the time consumed for flying between vertices and the probability of being destroyed for UAVs flying between vertices, and the results are shown in

Table 7. Time spent flying between mission vertices.

	$v_1$	$v_2$	$v_3$	$v_4$	$v_5$	$v_6$	$v_7$
$v_1$	-	1151.27	1764.65	3052.21	10,278.54	9497.61	10,206.44
$v_2$	-	-	602.05	1962.20	11,093.95	11,118.85	11,434.75
$v_3$	-	-	-	1345.55	12,003.44	10,333.20	11,308.20
$v_4$	-	-	-	-	10,445.67	10,019.47	10,636.18
$v_5$	-	-	-	-	-	914.76	589.63
$v_6$	-	-	-	-	-	-	945.86
$v_7$	-	-	-	-	-	-	-
$v_8$	-	-	-	-	-	-	-
$v_9$	-	-	-	-	-	-	-
$v_{10}$	-	-	-	-	-	-	-
$v_{11}$	-	-	-	-	-	-	-
$v_{12}$	-	-	-	-	-	-	-
$v_{13}$	-	-	-	-	-	-	-
$v_{14}$	-	-	-	-	-	-	-
	$v_8$	$v_9$	$v_{10}$	$v_{11}$	$v_{12}$	$v_{13}$	$v_{14}$
$v_1$	10,069.32	12,585.24	10,724.01	7406.71	7456.93	7069.78	6963.98
$v_2$	11,001.53	10,957.14	13,182.25	7569.92	9310.38	8254.26	8262.84
$v_3$	11,450.98	12,061.30	11,475.61	7609.01	7842.51	7986.65	8057.77
$v_4$	10,876.55	11,273.50	11,229.18	7608.60	8800.76	8618.56	8759.38
$v_5$	368.50	353.06	421.42	4397.88	4712.22	6347.37	6365.75
$v_6$	1218.60	1621.24	1744.31	2851.06	3971.10	6393.08	6281.76
$v_7$	1230.67	670.70	773.87	4877.26	3903.62	6509.98	6187.45
$v_8$	-	525.99	556.99	4139.09	5465.82	6106.80	5705.21
$v_9$	-	-	69.34	4295.84	4323.40	6591.09	6137.45
$v_{10}$	-	-	-	4409.17	4839.63	6590.20	6278.59
$v_{11}$	-	-	-	-	245.13	4864.30	3674.09
$v_{12}$	-	-	-	-	-	3935.71	4363.17
$v_{13}$	-	-	-	-	-	-	247.03
$v_{14}$	-	-	-	-	-	-	-

and

Table 8. Destruction probability during UAV navigation between mission vertices.

	$v_1$	$v_2$	$v_3$	$v_4$	$v_5$	$v_6$	$v_7$
$v_1$	-	-	-	-	0.2	0.2	0.2
$v_2$	-	-	-	-	0.2	0.2	0.2
$v_3$	-	-	-	-	0.2	0.2	0.2
$v_4$	-	-	-	-	0.2	0.2	0.2
$v_5$	-	-	-	-	-	0.2	0.2
$v_6$	-	-	-	-	-	-	0.2
$v_7$	-	-	-	-	-	-	-
$v_8$	-	-	-	-	-	-	-
$v_9$	-	-	-	-	-	-	-
$v_{10}$	-	-	-	-	-	-	-
$v_{11}$	-	-	-	-	-	-	-
$v_{12}$	-	-	-	-	-	-	-
$v_{13}$	-	-	-	-	-	-	-
$v_{14}$	-	-	-	-	-	-	-
	$v_8$	$v_9$	$v_{10}$	$v_{11}$	$v_{12}$	$v_{13}$	$v_{14}$
$v_1$	0.2	0.2	0.2	0.2	0.2	0.2	0.2
$v_2$	0.2	0.2	0.2	0.2	0.2	0.2	0.2
$v_3$	0.2	0.2	0.2	0.2	0.2	0.2	0.2
$v_4$	0.2	0.2	0.2	0.2	0.2	0.2	0.2
$v_5$	0.2	0.36	0.2	0.2	0.2	0.2	0.2
$v_6$	0.2	0.2	0.2	0.2	0.36	0.2	0.2
$v_7$	0.2	0.2	0.2	0.2	0.2	0.2	0.2
$v_8$	-	0.2	0.2	0.2	0.36	0.2	0.2
$v_9$	-	-	0.2	0.2	0.2	0.2	0.2
$v_{10}$	-	-	-	0.36	0.36	0.36	0.36
$v_{11}$	-	-	-	-	0.2	0.2	0.2
$v_{12}$	-	-	-	-	-	0.2	0.2
$v_{13}$	-	-	-	-	-	-	0.2
$v_{14}$	-	-	-	-	-	-	-

.

Combined with the mission planning model, the simulation using the annealing algorithm results in a mission planning scheme that maximizes the overall mission efficiency ratio of a UAV cluster mounted on the mobile platform, for which the number of mission tools assigned to each UAV for each target vertex and the paths of the UAVs moving between the mission vertices are shown in

Table 9. Number of mission tools assigned to each target vertex by each UAV.

	$u_1$	$u_2$	$u_3$	$u_4$
$v_5$	0	0	0	0
$v_6$	2	0	0	1
$v_7$	0	0	0	1
$v_8$	0	0	0	0
$v_9$	0	1	0	0
$v_{10}$	0	0	0	0
$v_{11}$	0	0	0	0
$v_{12}$	0	0	2	0
$v_{13}$	0	1	0	0
$v_{14}$	0	0	0	0

and

Table 10. UAV movement paths between mission vertices.

	Take-Off Vertex	Take-Off Moment	Mission Vertex 1	Mission Vertex 2	Landing Vertex	Landing Moment	Objective Function Value
$u_1$	1	0	6		2	12,756	3.4769
$u_2$	3	15,900	13	9	4	29,764
$u_3$	2	14,160	12		4	24,600
$u_4$	3	17,900	6	7	4	28,906

.

According to the task allocation scheme, the path optimization model is used to obtain the overall path optimization scheme for a UAV cluster mounted on the mobile platform, and the simulation trajectory planning results are shown in Figure 6.

6. Conclusions

In this paper, we mainly study the mission planning problem of UAVs on mobile take-off and landing platforms to perform multiple tasks in a complex marine environment. First, the optimal paths between UAV flight vertices on a 3D raster map are planned based on a path optimization model, and an improved two-population Gray Wolf Optimization algorithm is proposed to speed up the path finding process. Subsequently, a task planning model is constructed with several specific constraints in this context, and a UAV cluster on the mobile platform is planned with the help of the annealing algorithm and the path pre-planning data, which gives a task sequence that fits the optimization objective and provides a smooth and feasible flight trajectory for each UAV mounted on the mobile platform.

In this study, we proposed a corresponding solution for the UAV application of mobile platforms in marine environments; however, we also clearly recognize the limitations of the study. On the one hand, the current framework is more specific to maritime mobile platforms, and in future research we will strive to expand its application scenarios to adapt to diverse environments, such as land and aviation, and enhance its versatility and flexibility by optimizing the model structure and parameter settings. On the other hand, we will increase our research efforts on the effectiveness of the method in large-scale UAV clusters and complex environments. By improving the algorithms, introducing more advanced cooperative strategies, and conducting more comprehensive simulations and practical tests, we will ensure that our method can still function efficiently and stably in large-scale clusters and complex and changing environments, so as to provide a more solid theoretical and practical foundation for the wide application of UAV technology.