Multi-unmanned aerial vehicle flight path planning method based on multi-mode cooperative dung beetle optimizer

Abstract

The invention discloses a multi-unmanned aerial vehicle flight path planning method based on a multi-mode cooperative dung beetle optimizer. The method comprises the steps that 1, a dung beetle optimization algorithm is initialized; 2, updating the position of the dung beetle; 3, mapping position elements exceeding the upper and lower bounds back to the effective search area to obtain a dung beetle position updating result and calculating a population fitness value; 4, updating the position of the dung beetle and calculating a population fitness value; 5, updating the position of the small dung beetle and calculating a population fitness value; step 6, updating the position of dung beetle theft and calculating a population fitness value; 7, comparing population fitness values, and recording a maximum value, a minimum value and corresponding position results; and step 8, iterating to obtain a final position result. Compared with DBO, QHDBO and route planning, the total cost of the method is respectively reduced by 80.78% and 63.8% in a simple scene, and is respectively reduced by 83.96% and 68.89% in a complex scene, and high efficiency and practicability of the method are proved.

Description

Technical Field

[0001] The present invention belongs to the field of unmanned aerial vehicle (UAV) trajectory planning, and in particular relates to a multi-UAV trajectory planning method based on a multimodal collaborative dung beetle optimizer. Background Art

[0002] With unmanned technology [1] With the rapid development of the UAV, UAV plays an increasingly important role in modern warfare. UAV can not only reduce casualties, but also effectively save limited war resources, especially in performing reconnaissance in complex battlefield environments. [2] Its role is particularly prominent in tasks such as attack, electronic jamming, etc. However, as the tasks become more complex, the application of a single UAV in various scenarios is gradually limited by the performance of the UAV itself, thus affecting the efficiency of executing the task. In order to solve the impact of the performance constraints of a single UAV and improve the execution efficiency, scholars have begun to vigorously study the collaborative work of multiple UAVs. [3][4] . The collaboration between multiple UAVs is based on a single UAV. Tasks can be distributed collaboratively by multiple UAVs to solve the limitations of using a single UAV. Collaborative trajectory planning for multiple UAVs is the core issue that needs to be solved when multiple UAVs perform tasks. It usually refers to planning a safe and feasible path from the starting point to the target point. This not only requires considering the factors that a single UAV needs to consider, but also the collaborative relationship between UAVs. The core challenge of collaborative trajectory planning for multiple UAVs lies in how to resolve the contradiction between global optimization and fast and high-quality response under high-dimensional space-time constraints. Its complexity stems from the conflict and balance between the UAV's own performance, the coupling constraints of the three-dimensional unstructured environment, and multi-objective collaborative indicators (including collision risk and task timing).

[0003] In order to solve the problem of multi-UAV trajectory planning, relevant researchers have conducted in-depth exploration and thinking and proposed various classic methods, such as mathematical programming method [5][6] , graph search algorithm [7][8] , spatial sampling algorithm [9]

[10] Potential Field Method

[11]

[12] . Mathematical programming methods such as nonlinear programming and mixed integer linear programming are mature in theory and have high accuracy in trajectory planning solutions in simple scenarios, but if the scale of the problem increases, the calculation time will increase exponentially. The A* algorithm is a widely used graph search algorithm. Its advantage is that the implementation process is relatively simple. However, when faced with planning tasks in large-scale high-dimensional spaces, the algorithm is less efficient and has difficulty in dealing with trajectory planning problems under multiple constraints. As a typical spatial sampling algorithm, RRT does not require discretization of the flight environment and has a fast search speed, but it has shortcomings in finding the optimal route. The artificial potential field method has certain advantages in planning speed, but it is prone to problems such as local oscillations and local minima in large-scale high-dimensional spaces, which may cause the planned routes to be unusable. Therefore, traditional classical algorithms can achieve good trajectory planning results in simple environments, but in complex and highly uncertain battlefield environments, it is difficult to efficiently solve the problem of autonomous trajectory planning for multiple UAVs. In recent years, intelligent optimization algorithms have been widely used.

[13]

[14]

[15] The proposal provides an effective means to solve the multi-UAV trajectory planning problem, such as the famous genetic algorithm

[16] and particle swarm optimization

[17] Genetic algorithm is a type of random search optimization algorithm based on biological evolution and genetics. In route planning, the initial route is genetically encoded, and a fitness function is established based on various constraints. The optimal route is evolved generation by generation through genetic operators such as selection, crossover, and mutation. However, the genetic algorithm has poor local search capabilities in the later stages of the algorithm and suffers from premature convergence.

[18] , there are disadvantages such as long trajectory planning time and low quality of planned routes. Particle swarm optimization is a group intelligence search optimization algorithm that simulates the foraging behavior of bird flocks. In trajectory planning, each particle represents a route, and a fitness function is established to evaluate the quality of the particle route. Based on the information sharing between particles, the particle movement is guided from disorder to order, and the optimal route is finally obtained. However, the algorithm is very easy to fall into local optimality, and the trajectory planning efficiency is low in complex environments.

[19] .

[0004] Xue proposed the dung beetle optimization algorithm in 2022

[20] Because it has the internal differentiation function that other intelligent groups do not have, it is divided into four groups: rolling, foraging, stealing, and breeding. Compared with a single group, it has good fast convergence and the ability to find the global optimal solution, thus having unique advantages in complex optimization problems such as trajectory planning.

[21] The paper demonstrates the trajectory planning effect of the traditional DBO algorithm in a complex mountainous environment. Compared with the classic PSO and SSA algorithms, the route generated by DBO is of higher quality, and the integration of chaos strategy, exponentially decreasing inertia weight and adaptive Couch mutation strategy further improves the algorithm's trajectory planning performance. Although the traditional dung beetle optimization algorithm has a strong global optimization capability, its position update formula for rolling dung beetles and breeding dung beetles depends on the local optimal position, the search direction is too concentrated, and there is a lack of extensive exploration of the solution space. In addition, the parameter adjustment method in the algorithm is simple, and it is impossible to dynamically balance the needs of global exploration and local development according to the iteration stage, resulting in low convergence accuracy and insufficient population coordination ability. In the later stage of the algorithm iteration, the population diversity decreases rapidly, and it is very easy to fall into the local optimal area. In response to the above analysis, many scholars have made in-depth improvements to the classic dung beetle optimization algorithm. Ye M et al.

[22] A multi-strategy improved dung beetle optimization algorithm (MDBO) was proposed. Latin hypercube sampling was used to achieve better population initialization. A new differential change strategy was introduced to enhance the algorithm's ability to avoid local optimality. A strategy combining lens imaging reverse learning and dimension-by-dimension optimization was proposed for optimal solution. The improved algorithm showed excellent performance in both standard benchmark function test sets and engineering application scenarios.

[23] A dung beetle search algorithm (QHDBO) based on quantum computing and multi-strategy hybrid was proposed. The convergence factor and dynamic balance between the number of egg-laying dung beetles and foraging dung beetles were proposed, which enabled the algorithm to focus on global search in the early stages and local exploration in the later stages. Experimental results showed that the improved dung beetle optimization algorithm significantly improved the convergence speed and optimization accuracy, and had good robustness. Qiao L et al.

[24] The Lévy flight strategy and variable spiral strategy were introduced to improve the optimization performance of the Dung Beetle Optimizer (DBO) algorithm. The simulation results show that the improved algorithm is superior to other well-known metaheuristic algorithms. Hu W et al.

[25] An improved dung beetle optimization algorithm using a hybrid multi-strategy is proposed. It uses a cubic chaos map to initialize the population, utilizes a cooperative search algorithm to expand the search range of the solution space, introduces T-distribution mutation and differential evolution mutation strategies, provides perturbations to enhance population diversity, and tests and verifies the improved algorithm with a classic intelligent optimization algorithm and its excellent variants at CEC2017. The results show that the proposed algorithm has stronger robustness and optimization capabilities. These improvements have further enhanced the algorithm's global development capabilities to a certain extent, but there are problems with low computational efficiency and poor performance on high-order problems. In addition, the existing methods are not adaptable to the environment, and local optimal problems are difficult to avoid, resulting in difficulty in searching for suitable waypoints in complex terrain to plan high-quality routes. The public literature involved in the above content is as follows:

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[0041] In view of the limitations of the above-mentioned existing methods, in order to efficiently solve the problem of multi-UAV collaborative trajectory planning in complex battlefield environments, the present invention proposes a multi-UAV trajectory planning method based on multimodal collaborative dung beetle optimizer (ARSMDBO for short).

[0042] In order to achieve the above object, the present invention adopts the following technical solutions:

[0043] A multi-UAV trajectory planning method based on a multimodal collaborative dung beetle optimizer specifically includes the following steps:

[0044] Step 1: Initialize the dung beetle optimization algorithm and set algorithm-related parameters to build population fitness;

[0045] Step 2: Update the position of the rolling ball dung beetle and determine whether δ < 0.9 is true. δ = rand(1) represents a random number between 0 and 1. If yes, it means there is no obstacle. The position of the rolling ball dung beetle is updated as follows:

[0046] x i (t+1)=x i (t)+α×k×x i (t-1)+b×Δx,

[0047] Δx=|x i (t)-X w |

[0048] Among them, t is the tth iteration calculation, α is a constant coefficient assigned to -1 or 1, k is a random constant, b is a random constant between (0,1), w is the weight matrix, X w Indicates the worst position of the ball-rolling dung beetle in the current population;

[0049] Otherwise, it means there is an obstacle, and the position of the rolling ball dung beetle is updated as follows:

[0050] x i (t+1)=x i (t)+tan(θ)|x i (t)-x i (t-1)|

[0051] Among them, θ∈(0,π);

[0052] Step 3: Map the position elements exceeding the upper and lower bounds obtained in step 2 back to the valid search area to obtain the updated position result of the rolling ball dung beetle, and calculate the population fitness value of the rolling ball dung beetle;

[0053] Specifically, the specific operations for updating the position result of the rolling ball dung beetle are as follows:

[0054] For s(i) <Lb(i):

[0055] s(i)=Lb(i)+mod(Lb(i)-s(i),Ub(i)-Lb(i))

[0056] For s(i)>Ub(i):

[0057] s(i)=Ub(i)-mod(s(i)-Ub(i),Ub(i)-Lb(i))

[0058] Where s(i) is the i-th element of the individual position vector; Lb(i) is the i-th element of the lower bound vector Lb of the optimization problem; Ub(i) is the i-th element of the upper bound vector Ub of the optimization problem; mod(a,b) is the remainder of a divided by b;

[0059] Step 4, update the position of the dung beetle and calculate the population fitness value of the dung beetle;

[0060] Specifically, the position of the egg ball dung beetle is updated according to the following formula:

[0061] B i (t+1)=X * +b1×(Bi (t)-Lb * )+b2×(B i (t)-Ub * )

[0062] Among them, B i (t) represents the position information of the i-th egg-ball dung beetle at the t-th iteration, b1 and b2 are two independent random vectors of size 1×d, and d is the problem dimension;

[0063] Step 5: Determine whether rand < 0.5. If so, the position of the dung beetle is updated as follows:

[0064] x i (t+1)=x i (t)+C1×(x i (t)-Lb b )+C2×(x i (t)-Ub b )

[0065] Among them, C1 is a random number that obeys the normal distribution; C2 is a random vector in the range of (0,1);

[0066] Otherwise, the position of the dung beetle is updated as follows:

[0067]

[0068] Among them, dim refers to the problem dimension, is the position of the R1th individual in the current iteration, where R1 is a random integer generated from 1 to N;

[0069] Finally, the population fitness value of the small dung beetle is calculated;

[0070] Step 6, update the position of the thieving dung beetle and calculate the population fitness value of the thieving dung beetle;

[0071] Specifically, the position of the stealing dung beetle is updated as follows:

[0072] x i (t+1)=X b +S×g×{|x i (t)-X * |+|x i (t)-X b |}

[0073] Where g is a random vector of size 1×d that follows a normal distribution, and S is a constant;

[0074] Step 7, comparing the population fitness values ​​of the rolling ball dung beetle, the egg ball dung beetle, the small dung beetle, and the stealing dung beetle, recording the maximum population fitness and its corresponding dung beetle population, as well as the minimum population fitness value and its corresponding dung beetle population, as the dung beetle position result of the current iteration number;

[0075] Step 8, determine whether t<iter_max, if yes, set t=t+1 and go to step 2; otherwise, end and take the dung beetle position result of the current iteration number as the final dung beetle position result.

[0076] The technical innovations and beneficial effects of the present invention are as follows:

[0077] 1. Based on the performance of the UAVs themselves and the coordination relationship between them, the cost constraints of multi-UAV trajectory planning are proposed, and a total cost function is constructed to transform the UAV trajectory planning problem into an optimization problem, which simplifies the problem and reduces its complexity.

[0078] 2. Based on the classic dung beetle optimization algorithm, a boundary reflection optimization mechanism is proposed to "reflect" the population beyond the boundary back into the search space, thereby increasing the population's range of movement, improving the algorithm's global search capability, and helping to avoid falling into local optimal solutions. At the same time, an adaptive hybrid exploration strategy is designed. While retaining the advantages of the original dung beetle optimization algorithm, it introduces new update strategies and randomization mechanisms to enhance the algorithm's diversity and stability. Finally, a dynamic adaptive multi-scale mutation strategy is introduced. Through the dual indicators of position and fitness, the population status is accurately assessed. The mutation probability and mutation scale are dynamically adjusted according to the current iteration stage and population distribution. This enables the algorithm to better maintain population diversity at different stages of iteration, thereby planning low-cost, high-quality trajectories.

[0079] The ARSMDBO of the present invention and QHDBO, DBO, BOA

[26] 、PIO

[27] 、BWO

[28] 、COA

[29] Comparisons across different dimensions based on the CEC2017 test function, along with a Wilcoxon rank sum test, demonstrate that the proposed ARSMDBO possesses significantly stronger optimization capabilities. The successful application of the proposed ARSMDBO to the collaborative trajectory planning problem of multiple UAVs in complex battlefield environments also demonstrates its superior ability to solve complex optimization problems. BRIEF DESCRIPTION OF THE DRAWINGS

[0080] Figure 1 Schematic diagram of the boundary reflection optimization mechanism;

[0081] Figure 2 is a flow chart of the method of the present invention;

[0082] Figure 3is the average convergence curve, where (1) to (6) are the average convergence curves corresponding to functions F1, F5, F6, F13, F20, and F30 in the CEC2017 function test set;

[0083] Figure 4 are benchmark scenarios, where (1) is a simple scenario and (2) is a complex scenario;

[0084] Figure 5 It is a three-dimensional track map;

[0085] Figure 6 It is a top view;

[0086] Figure 7 is a side view;

[0087] Figures 5 to 7 Among them, (1) to (3) are the best trajectories in simple scenarios, and (4) to (6) are the best trajectories in complex scenarios;

[0088] Figure 8 The average convergence curves obtained for trajectory planning simulation scenarios, where (1) is a simple scenario and (2) is a complex scenario.

[0089] The present invention is further explained below with reference to the accompanying drawings and embodiments. DETAILED DESCRIPTION

[0090] The research ideas of the present invention are as follows: first, the constraints of the trajectory planning problem are designed; second, the classic dung beetle optimization algorithm is introduced; third, to address its shortcomings, the multi-UAV trajectory planning method based on the multimodal collaborative dung beetle optimizer of the present invention is proposed; the algorithm performance is verified and discussed and analyzed on the CEC2017 function test set; fourth, the improved algorithm is used to solve the multi-UAV collaborative trajectory planning problem in a complex battlefield environment, and the practical applicability of the improved algorithm is further explored.

[0091] 1 Description of the trajectory planning problem

[0092] Formulating accurate trajectory planning evaluation criteria, and comprehensively considering multiple factors to quantitatively evaluate the generated trajectories, thereby quickly screening out the optimal or suboptimal trajectories, can reduce unnecessary waste of computing resources and improve the overall efficiency of trajectory planning. In this invention, by formulating trajectory planning evaluation criteria and constructing a comprehensive cost function, the multi-constrained trajectory planning problem in a complex battlefield environment is converted into a multi-objective optimization problem, and the trajectory corresponding to the minimum cost is calculated as the optimal trajectory.

[30] .

[0093] 1.1 Single Machine Constraints

[0094] 1.1.1 Flight length constraints

[0095] The endurance of a drone has always been the key factor affecting its ability to perform high-intensity and continuous operations. A shorter range can greatly reduce aircraft fuel consumption, thereby reducing costs.

[0096] Flight length cost F Length The specific expression is as follows:

[0097]

[0098] Where n is the number of path nodes on the track line; l i is the straight-line distance from track point i to i+1; L SE From the starting point S(x s ,y s ,z s ) to the target point E(x e ,y e ,z e ) is the Euclidean distance between them.

[0099] 1.1.2 Height Constraint

[0100] In order to ensure the safety and stability of UAV flight, its flight altitude should be kept within a certain range as much as possible. The smoother the fluctuation curve of the flight altitude, the less pressure on the UAV flight control system. In addition, in a stable flight state, the fuel consumption of the UAV will be significantly reduced. In addition, when conducting combat missions such as high-altitude reconnaissance and terrain mapping, it is necessary to meet specific resolution and field of view, which puts forward requirements for flight altitude. Height The specific expressions are as follows:

[0101]

[0102] Among them, h i is the flight altitude of the UAV at track point i; h min and h max is the minimum flight altitude and the maximum flight altitude.

[0103] 1.1.3 Flight Attitude Constraints

[0104] A good flight attitude is the foundation for maintaining stability. A drone's instantaneous changes in direction and its ability to climb and descend during flight are constrained by its own performance and inertia. As shown, for safety and speed control, both its yaw and pitch angles must be limited to a certain range to ensure the drone maintains a stable and safe flight attitude.

[0105] Yaw angle cost F yaw_angle The specific expression is as follows:

[0106]

[0107] Among them, α i Refers to the flight segment i s i+1 With s i+1 s i+2 The yaw angle between max Refers to the maximum yaw angle of the drone; s′ i Refers to s i Projection on the xoy plane, s′ i+1 Refers to s i+1 Projection on the xoy plane, s′ i+2 Refers to s i+2 Projection onto the xoy plane.

[0108] Pitch angle cost F Pitch_angle The specific expression is as follows:

[0109]

[0110] Among them, β i Refers to the flight segment i-1 s i and s i s i+1 The pitch angle between max Refers to the maximum pitch angle of the drone, s′ i Refers to s i Projection on the xoy plane, s′ i+1 Refers to s i+1 Projection on the xoy plane, z i+1 Refers to the voyage point s i+1 The vertical coordinate, z i Refers to the voyage point s i The vertical coordinate of .

[0111] 1.1.4 Minimum track segment constraints

[0112] Due to inertia, changes in a drone's attitude during flight are not instantaneous and typically occur over a steady process. Furthermore, due to communication delays between the drone and the ground control station, there are periods of communication delay during high-speed flight. Therefore, setting a minimum trajectory segment that meets the drone's performance and communication delay constraints is the basis for planning a reasonable route. The specific expression for the minimum trajectory constraint segment is as follows:

[0113]

[0114] Among them, l min is the minimum track segment distance.

[0115] 1.1.5 Enemy Threat Constraints

[0116] In a combat environment, drones may face a variety of threats from the enemy, including radar detection and artillery attacks, which may cause the drone mission to fail. Therefore, when planning the flight path, it is necessary to consider avoiding these potential threats to ensure the smooth execution of the mission.

[31] .

[0117] The success of radar detection mainly depends on the distance between the UAV track point and the radar. The closer the UAV is to the radar center, the greater the threat detected. Radar The specific expression is as follows:

[0118]

[0119] Among them, F Radar is the radar threat cost; d r is the distance between the UAV and the radar threat center; the radar threat center is (x o ,y o ,z o );R r_max is the maximum detection radius of the radar; q is the radar threat level, which is set to 1.2 in the present invention.

[0120] Artillery strikes can directly destroy drones, so it is necessary to avoid areas covered by fire. Artillery threat costs F Artillery Specifically, it can be expressed as:

[0121]

[0122] Among them, F Artillery is the artillery threat cost; d a is the distance between the UAV and the center of the artillery, R a_max It is the maximum killing radius of the artillery.

[0123] 1.1.6 No-fly zone restrictions

[0124] To ensure the safety of both the aircraft and ground personnel during drone missions, they must strictly follow relevant regulations and avoid specific no-fly zones, such as airports, high-voltage power lines, high-rise buildings, and crowded areas. The specific expression for no-fly zone constraints is as follows:

[0125]

[0126] Among them, x min and x max is the upper and lower limits of the x coordinate of the no-fly zone, y min and y max are the upper and lower limits of the y coordinate of the no-fly zone respectively.

[0127] 1.2 Multi-machine collaboration constraints

[0128] When multiple machines work together, they should meet certain specific relationships in time and space to improve the overall coordination of the unmanned swarm and thus efficiently complete the scheduled tasks.

[0129] 1.2.1 Spatial Constraints

[0130] To ensure the safety of multi-UAV coordinated flight and avoid collisions, it is necessary to detect whether they are too close to each other when flying along the planned path. The coordinates of the nodes on each UAV's planned path are compared. If the distance between two UAVs at the same time is less than the safe distance, there is a risk of collision. The specific expression is as follows:

[0131]

[0132] Among them, F space is the space constraint cost, d i is the distance between the track point i of the UAV and other UAVs at the same time; d safe The safe distance for drones.

[0133] 1.2.2 Time Constraints

[0134] When multiple drones work together, it is usually required that each drone arrive at the target point at the same time. However, to avoid complex terrain obstacles, the flight speed of drones varies within a certain range, and the time points at which each drone arrives at the target cannot be exactly the same. Therefore, the present invention stipulates that as long as the time ranges of each drone's arrival at the target point overlap, it can be considered that each drone has reached the target point at the same time. The specific expression is as follows:

[0135]

[0136] Among them, t kmin and t kmax That is, the time range for the kth UAV to reach the target point; L k Refers to the total track length of the kth UAV, v kmin and v kmax Refers to the speed range of the kth UAV.

[0137] The build time constraint costs are as follows:

[0138]

[0139] Among them, t c is the instruction time for the kth UAV to reach the target destination.

[0140] 1.3 Overall evaluation criteria function

[0141] The linear weighted method is used to add the constraint conditions into the objective function in the form of penalty functions to construct the overall evaluation standard function. The specific expression is as follows:

[0142] F sum =w·[F length +F Height +F yaw_angle +F Pitch_angle +F path segment +F Radar +F Artillery +F flight ban +F space +F time ] T (17)

[0143] Among them, w is the weight matrix.

[0144] 2ARSMDBO algorithm

[0145] The ARSMDBO algorithm is improved by introducing three strategies, namely boundary reflection optimization mechanism, adaptive hybrid exploration strategy and dynamic adaptive multi-scale mutation, on the framework of DBO algorithm (dung beetle optimization algorithm), to solve the problems of simple boundary processing, easy to fall into local optimality and insufficient maintenance of population diversity in the DBO algorithm.

[0146] 2.1 Basic Principles of DBO Algorithm

[0147] The dung beetle optimization algorithm generates the initial population in a randomized manner, and the position of each dung beetle corresponds to a solution to the optimization problem. For the d-dimensional optimization problem, the position of the i-th dung beetle is represented by x i =[x i,1 ,x i,2 ,…,x i,d ], the dung beetle population of size N can be expressed as X = [x1, x2,…, x N ] T , as shown in formula (18).

[0148]

[0149] The dung beetle's ball rolling behavior can be divided into two modes: obstacle-free mode and obstacle-supported mode. In the absence of obstacles, the dung beetle uses the sun to navigate and keep the dung ball rolling in a straight line. The intensity of the light affects the dung beetle's path, and the position of the rolling ball is updated as shown in formula (19).

[0150]

[0151] Among them, t is the tth iteration calculation, α is a constant coefficient assigned to -1 or 1, k is a random constant, b is a random constant between (0,1), w is the weight matrix, X w Indicates the worst position of the ball-rolling dung beetle in the current population;

[0152] When encountering an obstacle, the dung beetle uses dancing to reorient itself to obtain a new route. The position update method of the dancing behavior is shown in formula (20).

[0153] x i (t+1)=x i (t)+tan(θ)|x i (t)-x i (t-1)| (20)

[0154] Where,θ∈(0,π).

[0155] During reproduction, female dung beetles roll their dung balls to a safe location suitable for laying eggs and hide them there, thereby providing a suitable habitat for their offspring. Inspired by this, the original authors proposed a boundary selection strategy to simulate the location where female dung beetles lay eggs, as shown in formula (21).

[0156]

[0157] Among them, Lb * ,Ub * Indicates the lower and upper bounds of the spawning area; X * Represents the current local optimal solution; R = 1-t / T max ,T max represents the maximum number of iterations; Lb and Ub represent the lower and upper bounds of the optimization problem.

[0158] The area where the eggs are laid is dynamically adjusted with the number of iterations, so the position of the egg-laying dung beetle is also dynamic during the iteration process. As shown in formula (22), the egg-laying dung beetle is strictly limited to the egg-laying area.

[0159] B i (t+1)=X * +b1×(B i (t)-Lb * )+b2×(B i (t)-Ub * ) (twenty two)

[0160] Among them, B i (t) represents the position information of the i-th egg-ball dung beetle at the t-th iteration; b1 and b2 are two independent random vectors of size 1×d, where d is the problem dimension.

[0161] During foraging behavior, some mature dung beetles emerge from the ground to look for food, and the optimal foraging area for young dung beetles is dynamically updated, as shown in formula (23).

[0162]

[0163] Among them, Xb represents the global optimal position, Lb b ,Ub b They represent the lower and upper bounds of the optimal foraging area, respectively.

[0164] The position update of the dung beetle is shown in formula (24).

[0165] x i (t+1)=x i (t)+C1×(x i (t)-Lb b )+C2×(x i (t)-Ub b ) (twenty four)

[0166] Where C1 is a random number that follows a normal distribution; C2 is a random vector in the range of (0,1).

[0167] Some dung beetles will steal the dung balls of other dung beetles. The position update of the stealing behavior is shown in formula (25).

[0168] x i (t+1)=X b +S×g×{|x i (t)-X * |+|x i (t)-X b |} (25)

[0169] Where g is a random vector of size 1×d that follows a normal distribution, and S is a constant.

[0170] 2.2 Improvement Strategy

[0171] Although the Dung Beetle Optimizer (DBO) performs well in terms of convergence speed, it has shortcomings such as insufficient maintenance of population diversity during the iteration process, easy to fall into local optimality, and simple and crude boundary processing, which restrict the performance of the algorithm. Therefore, the present invention proposes the following three strategies to improve the algorithm's capabilities.

[0172] 2.2.1 Boundary reflection optimization mechanism

[0173] In traditional boundary processing, variables that exceed the boundary are usually directly set to the boundary value, that is, truncated. Although this method is simple, it will lose the original movement direction and position information of the variable. Because the variable is forced to be restricted to the boundary, it cannot continue to explore the area outside the boundary, which may cause the algorithm to fall into a local optimal solution. Frequent truncation may interfere with the convergence process of the algorithm, resulting in a slow convergence speed. Based on the above analysis, in order to ensure the flexibility of enhanced boundary processing, a boundary reflection optimization mechanism is designed to "reflect" the variables that exceed the boundary back into the search space, rather than simply setting them to the boundary value. The range of movement of the variable can be increased, thereby improving the global search capability of the algorithm and helping to avoid falling into a local optimal solution.

[0174] The details are as follows:

[0175] For s(i) <Lb(i):

[0176] s(i)=Lb(i)+mod(Lb(i)-s(i),Ub(i)-Lb(i)) (26)

[0177] For s(i)>Ub(i):

[0178] s(i)=Ub(i)-mod(s(i)-Ub(i),Ub(i)-Lb(i)) (27)

[0179] Where s(i) is the i-th element of the individual position vector; Lb(i) is the i-th element of the lower bound vector Lb of the optimization problem; Ub(i) is the i-th element of the upper bound vector Ub of the optimization problem; and mod(a,b) is the remainder of dividing a by b.

[0180] By optimizing the reflection, such as Figure 1 As shown, each element s(i) of the individual position vector s is ensured to be between the defined upper and lower bounds Lb(i) and Ub(i), while retaining the motion information of the variable, thus avoiding the problems caused by direct truncation.

[0181] 2.2.2 Adaptive Hybrid Exploration Strategy

[0182] In the study of optimization algorithms, finding the ideal balance between global and local search has always been a key focus. Traditional optimization algorithms are prone to falling into local optima during iteration, especially when faced with complex multimodal optimization problems. Against this backdrop, this paper proposes an adaptive hybrid exploration strategy. While retaining the advantages of the original dung beetle optimization algorithm, this strategy introduces a new update strategy and randomization mechanism to enhance the algorithm's global search capabilities while also improving its diversity and stability. This is specifically demonstrated as follows:

[0183] For the individuals of the dung beetle foraging population, the original update method is adopted with a probability of 50%, and the new update method is adopted with a probability of 50%. The new update method is as follows:

[0184] Randomly select 2 to 5 individuals and calculate the mean X of their positions pmean :

[0185]

[0186] Where p is the number of randomly selected individuals, which is set to 2 to 5 in the present invention, p(k) is the index of the randomly selected individual, and X p(k) is the position of the kth individual among the p selected individuals.

[0187] Update individual position x i (t):

[0188]

[0189] Among them, dim refers to the problem dimension, is the position of the R1th individual in the current iteration, where R1 is a random integer generated from 1 to N.

[0190] By introducing new update strategies and randomness mechanisms, the algorithm achieves a better balance between global search and local search, thereby improving the optimization performance and stability of the algorithm.

[0191] 2.2.3 Dynamic Adaptive Multi-Scale Mutation Strategy

[0192] Despite its impressive global search capabilities, the Dung Beetle Optimizer (DBO) algorithm struggles to maintain population diversity during iterations. As iterations progress, the positions of individuals in the population gradually converge toward the global optimal solution. While this helps improve the algorithm's convergence rate, it also leads to a gradual decrease in population diversity. This reduction in diversity can cause the algorithm to become trapped in local optimal solutions, especially when faced with complex multimodal optimization problems. Furthermore, fixed-scale mutation strategies may not effectively balance the needs of global and local search across different iterations, further limiting the algorithm's performance.

[0193] To overcome these shortcomings, it is particularly important to introduce a dynamic adaptive multi-scale mutation strategy. By dynamically adjusting the mutation probability and mutation scale, this strategy can better maintain population diversity during the iteration process, while flexibly balancing the capabilities of global and local search according to the current iteration stage and population distribution. The details are as follows:

[0194] First, calculate the diversity index:

[0195] First calculate the position diversity:

[0196]

[0197] Then calculate the fitness diversity:

[0198]

[0199] Among them, fit i is the current fitness value of the i-th population;

[0200] Then weighted to get a new global diversity:

[0201] RR=0.5·(0.5·p d +0.5·f d ) (33)

[0202] The adaptive mutation probability is then calculated by weighting the global diversity:

[0203]

[0204] Among them, RR max is the maximum value of RR, iter_max refers to the maximum number of iterations, and t is the current number of iterations;

[0205] Conduct extensive exploration in the early stages and detailed development in the later stages. max , σ min are two constants, σ max =1.0,

[0206] σ min =0.1, used in formula (36);

[0207] Then perform multi-scale mutation operation:

[0208] x i (t+1)=x i (t)+σ·(Ub-Lb)·randn·(2·rand(1,dim)-1) (35)

[0209]

[0210] Then conduct search enhancement strategy:

[0211] x i (t+1)=x i (t)+0.01·(X b -x i ) (37)

[0212] Using the dual indicators of position and fitness, the population state is accurately assessed, and the population state automatically adjusts the intensity of disturbances. Large-scale mutations avoid local optimality, while small-scale fine-tuning improves convergence accuracy. Multi-scale mutations are then performed to increase the probability of the algorithm escaping the local optimality.

[0213] 2.3 Algorithm Flow

[0214] Based on the above analysis, the present invention provides a multi-UAV trajectory planning method based on a multimodal collaborative dung beetle optimizer. The process is as follows: Figure 2 As shown, the specific steps include:

[0215] Step 1: Initialize the dung beetle optimization algorithm and set algorithm-related parameters to build population fitness.

[0216] Preferably, the calculation formula of population fitness is as follows:

[0217] F sum =w·[F length +F Height +F yaw_angle +F Pitch_angle +F path segment +F Radar +F Artillery +F flight ban +F space +F time ] T Among them, w is the weight matrix, F Length is the flight length cost, F Height is the height cost, F yaw_angle is the yaw angle cost, F Pitch_angle is the pitch angle cost, F path segment is the minimum track constraint, F Radar is the radar detection cost, F Artillery is the artillery threat cost, F flight ban F is the no-fly zone constraint, space is the space constraint cost, F time The time constraint cost.

[0218] Step 2: Update the position of the rolling ball dung beetle and determine whether δ < 0.9 is true. δ = rand(1) represents a random number between 0 and 1. If yes, it means there is no obstacle. The position of the rolling ball dung beetle is updated as follows:

[0219] x i (t+1)=x i (t)+α×k×x i (t-1)+b×Δx,

[0220] Δx=|x i (t)-X w |

[0221] Among them, t is the tth iteration calculation, α is a constant coefficient assigned to -1 or 1, k is a random constant, b is a random constant between (0,1), w is the weight matrix, X w Indicates the worst position of the ball-rolling dung beetle in the current population;

[0222] Otherwise, it means there is an obstacle, and the position of the rolling ball dung beetle is updated as follows:

[0223] x i (t+1)=x i (t)+tan(θ)|x i (t)-x i (t-1)|

[0224] Among them, θ∈(0,π);

[0225] Step 3: Map the position elements exceeding the upper and lower bounds obtained in step 2 back to the effective search area to obtain the updated position result of the rolling ball dung beetle, and calculate the population fitness value of the rolling ball dung beetle.

[0226] Specifically, the specific operations for updating the position result of the rolling ball dung beetle are as follows:

[0227] For s(i) <Lb(i):

[0228] s(i)=Lb(i)+mod(Lb(i)-s(i),Ub(i)-Lb(i))

[0229] For s(i)>Ub(i):

[0230] s(i)=Ub(i)-mod(s(i)-Ub(i),Ub(i)-Lb(i))

[0231] Where s(i) is the i-th element of the individual position vector; Lb(i) is the i-th element of the lower bound vector Lb of the optimization problem; Ub(i) is the i-th element of the upper bound vector Ub of the optimization problem; and mod(a,b) is the remainder of dividing a by b.

[0232] Step 4: Update the position of the dung beetle and calculate the population fitness value of the dung beetle. Specifically, the position of the dung beetle is updated according to the following formula:

[0233] B i (t+1)=X * +b1×(B i (t)-Lb * )+b2×(B i (t)-Ub * )

[0234] Among them, B i (t) represents the position information of the i-th egg-shaped dung beetle at the t-th iteration, b1 and b2 are two independent random vectors of size 1×d, and d is the problem dimension.

[0235] Step 5: Determine whether rand < 0.5. If so, the position of the dung beetle is updated as follows:

[0236] x i (t+1)=x i (t)+C1×(x i (t)-Lb b )+C2×(x i (t)-Ub b )

[0237] Among them, C1 is a random number that obeys the normal distribution; C2 is a random vector in the range of (0,1);

[0238] Otherwise, the position of the dung beetle is updated as follows:

[0239]

[0240] Among them, dim refers to the problem dimension, is the position of the R1th individual in the current iteration, where R1 is a random integer generated from 1 to N;

[0241] Finally, calculate the population fitness value of the dung beetle.

[0242] Step 6: Update the position of the thieving dung beetle and calculate the population fitness value of the thieving dung beetle. Specifically, the position of the thieving dung beetle is updated according to the following formula:

[0243] x i (t+1)=X b +S×g×{|x i (t)-X * |+|x i (t)-X b |}

[0244] Where g is a random vector of size 1×d that follows a normal distribution, and S is a constant.

[0245] Step 7: Compare the population fitness values ​​of the rolling ball dung beetle, egg ball dung beetle, small dung beetle, and stealing dung beetle, record the maximum population fitness and its corresponding dung beetle population, as well as the minimum population fitness value and its corresponding dung beetle population, as the dung beetle position result of the current iteration number.

[0246] Step 8, determine whether t<iter_max, if yes, set t=t+1 and go to step 2; otherwise, end and take the dung beetle position result of the current iteration number as the final dung beetle position result.

[0247] In particular, the following process is also included between step 6 and step 7:

[0248] Step a, calculate the location diversity of all dung beetle populations:

[0249]

[0250] Where D is the problem dimension; x i,j represents the position of the i-th population in the j-th dimension, represents the average position of all populations in the jth dimension;

[0251] Step b, calculate the fitness diversity of all dung beetle populations:

[0252]

[0253] Among them, fit i is the current fitness value of the i-th population; N is the population size;

[0254] Step c: weighted global diversity under the current number of iterations:

[0255] RR=0.5·(0.5·p d +0.5·f d )

[0256] Step d, calculate the adaptive mutation probability by weighted global diversity:

[0257]

[0258] Among them, iter_max refers to the maximum number of iterations, t is the current number of iterations; RR max is the maximum value of RR for all iterations;

[0259] Step e: Determine whether rand() < P mut If yes, update the dung beetle position according to the following formula; otherwise, go directly to step 7:

[0260] x i (t+1)=x i (t)+σ·(Ub-Lb)·randn·(2·rand(1,dim)-1)

[0261]

[0262] Step f, update the dung beetle position obtained in step e again according to the following formula:

[0263] x i (t+1)=x i(t)+0.01·(X b -x i ).

[0264] 2.4 Complexity Analysis

[0265] The ARSMDBO algorithm of the present invention is based on the logical framework of the original DBO algorithm and is improved by incorporating three different strategies. In order to verify the effectiveness of the improved algorithm, its complexity needs to be evaluated. Assuming that the number of dung beetle populations is N, the dimension of the optimization space is D, and the maximum number of algorithm iterations is iter_max, the time complexity of the classic dung beetle optimization algorithm is T DBO =PO(D×N×iter_max)=O(N). The introduced boundary reflection optimization mechanism nests an additional loop. Assuming that its loop running time is t1, its time complexity is T1=O(N×D×t1)=O(N); for the update of the individual positions of foraging dung beetles in the population, an adaptive hybrid exploration strategy is added. Only a new mutation method is added, and no new loop is introduced. The time complexity is T2=O(N×D×iter_max)=O(N); In the later stage of the algorithm, a dynamic adaptive multi-scale mutation strategy is added, which traverses all populations. Assuming that its loop running time is t3, its time complexity is T3=O(N×D×t3)=O(N), so the total time complexity is T ARSMDBO =T1+T2+T3=O(N), which is on the same order of magnitude as the DBO algorithm. This shows that the improved ARSMDBO algorithm has a higher optimization capability without increasing the time complexity of the algorithm, thus maintaining the operation efficiency.

[0266] 2.5 Benchmark Function Testing

[0267] This paper selects six typical intelligent optimization algorithms, QHDBO, DBO, BOA, PIO, BWO, and COA, for comparison. The input parameter settings of the algorithms are shown in Table 2. The parameter settings of the original literature of these algorithms are adopted. To reduce the influence of algorithm randomness, each algorithm is run 30 times separately, and the test dimensions are 30, 50, and 100 dimensions. All simulations in this paper are carried out on a Windows 11 platform equipped with a 13th generation Intel Core processor with a main frequency of 2.20GHz. TM i9-13900HX processor. In addition, MATLAB 2024a software was used for the simulation.

[0268] Table 2 Algorithm parameters

[0269]

[0270] The present invention uses the CEC2017 function test set to evaluate the algorithm, which is an effective method for evaluating algorithm performance and verifying its ability to solve complex performance. The specific details are shown in Table 3. This function test set has a total of 29 functions (it should be noted that F2 has been removed), of which F1 and F3 are unimodal functions with only one global optimal solution and no local optimal solution. Therefore, they are mainly used to test the convergence efficiency of the optimization algorithm in an undisturbed environment. F4 to F10 are basic multimodal functions, with rotation and offset added to the function, and the main evaluation algorithm is the trade-off strategy between extensive search and fine convergence. F11-F20 are hybrid functions, each of which is composed of at least three CEC2017 benchmark functions that are rotated or translated and assigned different weights. This provides a challenge close to the real scene for the optimization algorithm and tests the comprehensive ability of the algorithm in a complex environment. F21-F30 are composition functions that simulate the multi-factor coupling and dynamic changes in real optimization problems through multi-level structures and nonlinear transformations, proposing the ultimate challenge to the robustness, adaptability and global coordination ability of the algorithm. The following is an analysis and discussion of the test results of the algorithm on the four types of functions, and the numerical results are recorded in Tables 4 to 7. The significant values ​​are represented in bold. The mean and standard deviation of the fitness values ​​are used to evaluate the search accuracy and stability of the algorithm. The average convergence curve is shown in Table 4. Figure 3 As shown, it is used to observe the convergence speed and accuracy of the algorithm.

[0271] Table 3 CEC2017 test function details (N represents the number of functions)

[0272]

[0273] Table 4 Unimodal function test results

[0274]

[0275]

[0276] As shown in Table 4, ARSMDBO exhibits excellent global optimization capabilities on unimodal functions. Except for some cases where its stability is slightly lower than that of other high-performance algorithms, ARSMDBO has the best average optimization results in both low-dimensional and high-dimensional experiments, and is far superior to the classic DBO algorithm and the excellent DBO variant algorithm QHDBO. This reflects the superiority of the boundary reflection optimization mechanism in retaining variable position information, which enhances the algorithm's ability to escape from local optimality and gradually approach the global optimality.

[0277] Table 5 Basic multimodal function test results

[0278]

[0279]

[0280] Simple multimodal problems F4-F10, due to the addition of rotations and offsets to the functions, place a greater test on the algorithm's ability to balance local escape and global exploration than unimodal functions. The experimental results, shown in Table 5, demonstrate that ARSMDBO outperforms the other six compared algorithms in finding the optimal solution. In particular, ARSMDBO achieves far superior convergence accuracy on F4 and demonstrates strong robustness. On F6, ARSMDBO's stability is slightly lower than that of the other algorithms, but its continuous exploration of the global optimum demonstrates its strong ability to escape local optima, ultimately leading to the discovery of a superior solution. These results further validate the superiority of the ARSMDBO algorithm in solving complex problems.

[0281] Table 6 Hybrid function test results

[0282]

[0283]

[0284]

[0285] ARSMDBO demonstrated significant performance in experimental studies on mixed problems F11-F20, as shown in Table 6. ARSMDBO achieved significantly better mean and standard deviation results than the comparison algorithms in both low- and high-dimensional experiments on F11 to F15, F18, and F19, demonstrating the effectiveness and competitiveness of its diverse search strategy. Clearly, ARSMDBO's standard deviation was slightly weaker than that of several other well-known algorithms on F16, F17, and F20. However, the algorithm maintains population diversity by dynamically adjusting the mutation probability and scale in the later stages of the search, thereby enhancing its ability to explore the optimal value. Consequently, it ultimately achieved superior optimization results.

[0286] Table 7 Composition function test results

[0287]

[0288]

[0289] ARSMDBO demonstrates strong competitiveness in solving complex problems using functions F21-F29. As shown in Table 7, across all dimensions, ARSMDBO outperforms the other compared algorithms on average. This is particularly true for F30. Due to the introduction of an adaptive hybrid exploration strategy, which adds different position update strategies during the iteration process, ARSMDBO has more tools to explore the optimal solution, ultimately outperforming both the average and standard deviation metrics. These findings clearly highlight the unique advantages and adaptability of ARSMDBO in solving complex problems, enabling more efficient exploration and optimization of complex search spaces.

[0290] Figure 3 The average convergence curves of each algorithm are shown. In comparison, ARSMDBO can quickly approach the global optimal value, reflecting the algorithm's powerful ability to balance local exploration and global development.

[0291] Table 8 Rank Sum Test

[0292]

[0293]

[0294] In addition, in order to more intuitively illustrate the difference between ARSMDBO and other algorithms, the Wilcoxon rank sum test was used

[32]

[33] Statistical analysis of the experimental data was performed. The results of 30 independent optimization runs of ARSMDBO were compared with those of 30 independent optimization runs of QHDBO, DBO, BOA, PIO, BWO, and COA, using a Wilcoxon rank sum test at a significance level of 5%. All algorithms were set to have a dimension of 30, a population size of 30, and 500 iterations. A "<" indicates when ARSMDBO outperforms the other algorithms. A "=" indicates when it is comparable to the other algorithms, and a ">" indicates when its performance is worse than the other algorithms. The test results are shown in Table 8, where significant values ​​are indicated in bold. On all 29 test functions, the probability values ​​of ARSMDBO's optimization results compared to the optimization results of the other six algorithms were all less than 0.05, indicating that ARSMDBO's optimization results were significantly different from those of the other comparison algorithms and that ARSMDBO was significantly better. This statistically demonstrates the superiority of the ARSMDBO algorithm.

[0295] Through the above analysis, we can conclude that ARSMDBO achieved the best results in some statistical indicators, including mean, standard deviation and convergence speed, which shows a stronger optimization ability than other high-performance algorithms; and ARSMDBO is closest to the optimal value compared with other algorithms in multiple different types of functions, which shows that it is more likely to find and converge to the optimal global solution, and has a stronger ability to solve complex problems than other algorithms; compared with DBO and QHDBO, ARSMDBO has achieved a significant improvement in convergence speed and accuracy, showing a stronger global development and local escape ability, reflecting the effectiveness of the improvement strategy.

[0296] 3 Collaborative trajectory planning simulation and discussion

[0297] 3.1 Algorithm and UAV parameter settings

[0298] To better demonstrate the superiority of the proposed algorithm, this section selects DBO and QHDBO as comparison algorithms. By setting the starting and target points and adjusting information about threats in the environment, trajectory planning simulations were performed in two environments of varying complexity. The algorithm population size N was set to 30, the number of iterations was 200, and the number of waypoints n was 12. Each algorithm was independently run 30 times, and the optimal value was taken as the final result to avoid the influence of randomness. Because drone flight is subject to its own performance constraints, this section sets relevant performance parameters based on real-world conditions, as shown in Table 9.

[0299] Table 9 Constraint parameter settings

[0300]

[0301] The experiment sets up three UAVs for trajectory collaborative planning. The coordination time of the three UAVs is set to 60s, 40s and 30s respectively, and the weight matrix w = [0.05, 0.05, 0.7, 0.7, 0.7, 2, 2, 0.9, 0.7, 0.7]. The flight trajectory of a UAV is usually constructed by connecting n-2 waypoints. However, due to the performance constraints of the UAV, the trajectory constructed by simply connecting each waypoint with a simple straight line cannot meet the conditions for safe flight of UAVs in reality. Therefore, this paper introduces the B-spline curve method

[34]

[35] Smooth optimal track.

[0302] 3.2 Simulation environment construction

[0303] When UAVs are performing combat missions such as material distribution, reconnaissance and strike, and high-altitude broadcasting, they need to plan their routes in advance based on the current terrain obstacles and known obstacles, electronic interference zones and other no-fly zones. The quality of the established three-dimensional model determines whether it can reflect the actual combat environment, UAV performance and combat mission requirements. This invention uses the real digital elevation model (DEM) map obtained from the LiDAR sensor.

[36] On this basis, various dangerous areas that UAVs need to avoid when performing combat missions are added. The two together form a three-dimensional space model. The parameters related to the space model are shown in Table 10. According to the number of threats, the following are formed: Figure 4 Two benchmark scenarios, complex and simple, are shown.

[0304] Table 10 Spatial model parameters

[0305]

[0306]

[0307] 3.3 Trajectory planning simulation and analysis

[0308] In order to better analyze the trajectory of multi-aircraft collaborative planning, this section observes the optimal trajectory from three perspectives: 3D panoramic view, top view, and side view based on two benchmark scenarios, such as Figures 5 to 7 As shown in the figure, (1) to (3) in each figure are the optimal trajectories in simple scenarios, and (4) to (6) are the optimal trajectories in complex scenarios. In addition, the average cost, optimal cost and standard deviation of each algorithm running 30 times independently to find the optimal trajectory are also counted.

[0309] Figure 5 The three algorithms were shown to find the optimal trajectory after 30 independent runs in two scenarios of varying complexity. As can be seen from the three-dimensional plot, all three algorithms generated routes that satisfied the constraints, effectively guiding the drone to circumvent various environmental threats and reach the target point within the preset time. In comparison, ARSMDBO generated a shorter and smoother trajectory. In particular, in complex scenarios, DBO and QHDBO both experienced varying degrees of detours and sudden changes in trajectory direction, which contradicted the inertial laws of high-speed flight, demonstrating ARSMDBO's superior optimization capabilities.

[0310] from Figure 6It can be observed that DBO and QHDBO have weak global optimization capabilities. They usually plan routes by significantly increasing the trajectory. The fluctuation of the track planning points is large, resulting in more and larger turning angles, and this situation is particularly obvious in complex scenarios. The overhead view of the ARSMDBO algorithm trajectory can more clearly feel that the track point fluctuations are smaller, the track is less tortuous, the turning angle changes are smaller, and the curve is smoother. The path length of the track is also the shortest among these algorithms, indicating that the ARSMDBO algorithm overcomes the result of falling into local optimality, and therefore the track quality is the highest.

[0311] Figure 7 Observing the collaboratively planned trajectories from a side view, in simple scenarios, the three algorithms have comparable planning effects, with no noticeable difference in the routes. However, in complex scenarios, the trajectories planned by DBO and QHDBO show significant and large variations in altitude, with instantaneous pitch inflection points. This indicates that the two traditional intelligent optimization algorithms, DBO and QHDBO, are stuck in a local optimal solution and cannot exit, resulting in a relatively low-quality planned trajectory. The trajectory planned by ARSMDBO, on the other hand, has smaller altitude variations and less tortuosity, making it more suitable for drone flight.

[0312] Depend on Figure 8 As shown in Table 11, the total cost of the trajectory planned by ARSMDBO is lower and converges faster at the same starting and ending points. The optimal values, average values, and standard deviations calculated for 30 times show that the ARSMDBO algorithm outperforms the unoptimized DBO algorithm and the excellent DBO variant QHDBO in both optimal and average values, and exhibits greater stability. In simple scenarios, the average total cost of ARSMDBO is 80.78% lower than that of DBO and QHDBO, respectively, and the average cost is 63.8%. In complex scenarios, the advantages of the proposed algorithm are even more pronounced, with average total costs decreasing by 83.96% and 68.89% compared to DBO and QHDBO, respectively. This indicates that ARSMDBO reduces costs by at least 60% compared to other algorithms, significantly improving trajectory planning efficiency.

[0313] Table 11 Total cost of trajectory planning

[0314]

[0315]

[0316] 4 Conclusion

[0317] In order to solve the problem of multi-UAV collaborative trajectory planning in complex environments, the present invention proposes a trajectory planning method based on an adaptive reflective multimodal collaborative dung beetle optimization algorithm, and achieves excellent results in simulation experiments. In battlefield environments, due to the rugged terrain and complex and changeable threats, multi-UAV collaboration also needs to consider coordination relationships, and has high requirements for the safety and stability of planned trajectories. Traditional intelligent optimization algorithms are very easy to fall into local optimality and have slow convergence speed, making it difficult to quickly generate high-quality trajectories. To meet the above challenges, the present invention makes several improvements to the classic DBO algorithm: (1) Designing a boundary reflection optimization mechanism to retain the original position information of the variables and increase the range of movement of the variables, thereby improving the global search capability of the algorithm; (2) Introducing an adaptive hybrid exploration strategy, adding a new population position update strategy and randomness mechanism, and enhancing the diversity and stability of the algorithm; (3) Proposing a dynamic adaptive multi-scale variation, in the iterative process, according to the current iteration stage and population distribution, flexibly balances the capabilities of global search and local search.

[0318] In simulation experiments, the proposed algorithm and six well-known comparative algorithms were tested using 29 functions from CEC2017. The improved algorithm's convergence speed and optimization capabilities were verified. The comparison revealed significant differences between ARSMDBO and the other algorithms, with superior performance across multiple metrics. In a battlefield multi-aircraft collaborative trajectory planning simulation, ARSMDBO, DBO, and QHDBO were used in two benchmark scenarios. Results from 30 independent runs showed that ARSMDBO reduced total flight costs by more than 60% compared to the classic DBO and its superior variant, QHDBO. The resulting trajectory was also of higher quality, better meeting the UAV's flight performance constraints.

[0319] In the next step of research, we will further explore the trajectory planning effect of the algorithm deployed on real drones, and further seek the optimization space of the algorithm based on the characteristics of the variable obstacles and threats in the actual battlefield environment, so as to improve its applicability in large-scale multi-machine collaborative trajectory planning scenarios.

Claims

1. A multi-UAV trajectory planning method based on a multimodal collaborative dung beetle optimizer, characterized in that: The specific steps include: Step 1: Initialize the dung beetle optimization algorithm and set algorithm-related parameters to build population fitness; Step 2: Update the position of the rolling ball dung beetle and determine whether δ < 0.9 is true. δ = rand(1) represents a random number between 0 and 1. If yes, it means there is no obstacle. The position of the rolling ball dung beetle is updated as follows: x i (t+1)=x i (t)+α×k×x i (t-1)+b×Δx, Δx=|x i (t)-X w | Among them, t is the tth iteration calculation, α is a constant coefficient assigned to -1 or 1, k is a random constant, b is a random constant between (0,1), w is the weight matrix, X w Indicates the worst position of the ball-rolling dung beetle in the current population; Otherwise, it means there is an obstacle, and the position of the rolling dung beetle is updated as follows: x i (t+1)=x i (t)+tan(θ)|x i (t)-x i (t-1)| Among them, θ∈(0,π); Step 3: Map the position elements exceeding the upper and lower bounds obtained in step 2 back to the effective search area to obtain the updated position result of the rolling ball dung beetle, and calculate the population fitness value of the rolling ball dung beetle; Specifically, the specific operations for updating the position result of the rolling ball dung beetle are as follows: For s(i) <Lb(i): s(i)=Lb(i)+mod(Lb(i)-s(i),Ub(i)-Lb(i)) For s(i)>Ub(i): s(i)=Ub(i)-mod(s(i)-Ub(i),Ub(i)-Lb(i)) Where s(i) is the i-th element of the individual position vector; Lb(i) is the i-th element of the lower bound vector Lb of the optimization problem; Ub(i) is the i-th element of the upper bound vector Ub of the optimization problem; mod(a,b) is the remainder of a divided by b; Step 4, update the position of the dung beetle and calculate the population fitness value of the dung beetle; Specifically, the position of the egg ball dung beetle is updated according to the following formula: B i (t+1)=X * +b1×(B i (t)-Lb * )+b2×(B i (t)-Ub * ) Among them, B i (t) represents the position information of the i-th egg-ball dung beetle at the t-th iteration, b1 and b2 are two independent random vectors of size 1×d, and d is the problem dimension; Step 5: Determine whether rand < 0.

5. If so, the position of the dung beetle is updated as follows: x i (t+1)=x i (t)+C1×(x i (t)-Lb b )+C2×(x i (t)-Ub b ) Among them, C1 is a random number that obeys the normal distribution; C2 is a random vector in the range of (0,1); Otherwise, the position of the dung beetle is updated as follows: Among them, dim refers to the problem dimension, is the position of the R1th individual in the current iteration, where R1 is a random integer generated from 1 to N; Finally, the population fitness value of the dung beetle is calculated; Step 6, update the position of the thieving dung beetle and calculate the population fitness value of the thieving dung beetle; Specifically, the position of the stealing dung beetle is updated as follows: x i (t+1)=X b +S×g×{|x i (t)-X * |+|x i (t)-X b |} Where g is a random vector of size 1×d that follows a normal distribution, and S is a constant; Step 7, comparing the population fitness values ​​of the rolling ball dung beetle, the egg ball dung beetle, the small dung beetle, and the stealing dung beetle, recording the maximum population fitness and its corresponding dung beetle population, as well as the minimum population fitness value and its corresponding dung beetle population, as the dung beetle position result of the current iteration number; Step 8, determine whether t<iter_max, if yes, set t=t+1 and go to step 2; otherwise, end and take the dung beetle position result of the current iteration number as the final dung beetle position result.

2. The multi-UAV trajectory planning method based on the multimodal collaborative dung beetle optimizer according to claim 1, characterized in that: In step 1, the calculation formula of population fitness is as follows: F sum =w·[F length +F Height +F yaw_angle +F Pitch_angle +F path segment +F Radar +F Artillery +F flight ban +F space +F time ] T Among them, w is the weight matrix, F Length is the flight length cost, F Height is the height cost, F yaw_angle is the yaw angle cost, F Pitch_angle is the pitch angle cost, F pathsegment is the minimum track constraint, F Radar is the radar detection cost, F Artillery is the artillery threat cost, F flightban F is the no-fly zone constraint, space is the space constraint cost, F time The time constraint cost.

3. The multi-UAV trajectory planning method based on the multimodal collaborative dung beetle optimizer according to claim 1, characterized in that: The following process is also included between step 6 and step 7: Step a, calculate the location diversity of all dung beetle populations: Where D is the problem dimension; x i,j represents the position of the i-th population in the j-th dimension, represents the average position of all populations in the jth dimension; Step b, calculate the fitness diversity of all dung beetle populations: Among them, fit i is the current fitness value of the i-th population; N is the population size; Step c: weighted global diversity under the current number of iterations: RR=0.5·(0.5·p d +0.5·f d ) Step d, calculate the adaptive mutation probability by weighted global diversity: Among them, iter_max refers to the maximum number of iterations, t is the current number of iterations; RR max is the maximum value of RR for all iterations; Step e: Determine whether rand() < P mut If yes, update the dung beetle position according to the following formula; otherwise, go directly to step 7: x i (t+1)=x i (t)+σ·(Ub-Lb)·randn·(2·rand(1,dim)-1) Step f, update the dung beetle position obtained in step e again according to the following formula: x i (t+1)=x i (t)+0.01·(X b -x i )。 4. The multi-UAV trajectory planning method based on the multimodal collaborative dung beetle optimizer according to claim 3, characterized in that: Weight matrix w = [0.05, 0.05, 0.7, 0.7, 0.7, 2, 2, 0.9, 0.7, 0.7].

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