Unmanned aerial vehicle formation obstacle avoidance and trajectory optimization method based on general calculation cooperation

Abstract

The invention discloses an unmanned aerial vehicle formation obstacle avoidance and trajectory optimization method based on general calculation cooperation. The method comprises the following steps: step 1, formation control design among multiple unmanned aerial vehicles: utilizing a leader-follower formation model based on a virtual leader; modeling the unmanned aerial vehicle formation by an undirected graph G = (V, E); and 2, designing a path planning algorithm of the virtual leader: combining an artificial potential field method with an A * algorithm to carry out dynamic path planning. According to the method, the traditional A * algorithm is optimized, and an adaptive path search mechanism is introduced, so that the path search efficiency is remarkably improved. The unmanned aerial vehicles in the formation serve as independent units, and the stability of the formation is maintained while path planning is carried out. The invention provides a new method for unmanned aerial vehicle formation obstacle avoidance and path planning in an obstacle environment.

Description

Technical Field

[0001] The present invention belongs to the field of obstacle avoidance and trajectory optimization of unmanned aerial vehicle formations, and specifically is a method for obstacle avoidance and trajectory optimization of unmanned aerial vehicle formations with comprehensive computing collaboration. The present invention also provides a method for using the system. Background Art

[0002] UAV formation flight relies on the highly autonomous decision-making ability of each UAV and excellent collaborative flight performance. When performing tasks, it is particularly important for UAVs in the formation to avoid obstacles and plan paths, because they form a complex interactive system with other elements of the surrounding environment. This interaction and constraint requires UAVs to not only be able to respond independently, but also to be able to efficiently plan their action paths while maintaining formation coordination to ensure the successful completion of the mission. UAV formations rely on the timeliness and accuracy of information exchange and the real-time calculation of routes by the UAV onboard computers. However, when dealing with the path planning problem of multi-UAV systems, collaborative control between UAVs is particularly critical, with the aim of ensuring the smooth completion of the overall mission and avoiding potential collision risks. Secondly, path planning problems occupy an important position in the fields of automation and drone science, especially in the application of UAVs, where the goal is to plan the optimal or feasible path for UAVs from the starting point to the end point in a given environment. Summary of the invention

[0003] The purpose of the present invention is to overcome the shortcomings of the prior art and provide a method for obstacle avoidance and trajectory optimization of UAV formations with comprehensive computing and coordination. The present invention significantly improves the path search efficiency by optimizing the traditional A* algorithm and introducing an adaptive path search mechanism.

[0004] The objective of the present invention is achieved through the following technical solution: a method for obstacle avoidance and trajectory optimization of a coordinated UAV formation, comprising the following steps:

[0005] Step 1. Formation control design among multiple UAVs: Utilize the leader-follower formation model based on a virtual leader, that is, in the formation, no UAV is designated as the leader, but a virtual UAV with the same motion model as the actual UAV is designated as the leader, and all UAVs in the formation are followers;

[0006] The kinematic model of the drone is expressed as:

[0007]

[0008] In the formula

[0009] are the velocity components of the UAV in the x-axis and y-axis directions, v represents the UAV speed, ψ and They represent the yaw angle and its rate of change over time, respectively, and ω represents the angular velocity;

[0010] The position and heading of the virtual leader are expressed as:

[0011]

[0012] With ΔX i represents the relative distance between UAV i and the virtual leader, then the expected position X of the UAV relative to the virtual leader i,desired It is expressed as:

[0013] X i,desired =X leader +ΔX i

[0014] The status of a virtual leader is determined by sharing the status information between drones, and the formation layout is adjusted accordingly. When receiving the target location information, the formation will take the target point as the destination and start the movement of the virtual leader. During this process, the followers will move freely within the threshold set by the virtual leader.

[0015] The virtual leader drone formation is introduced into the undirected graph as a real drone, and is modeled by the undirected graph G = (V, E), denoted as G(V, E); where V = {1, 2, ..., N} is the vertex set, N is the number of drones, and V is the set of drones; the vertex v represents the position vector p v =[x v ,y v ,z v ]; w vn is the edge connecting vertex v and vertex n, representing the weight of the edge between the two drones;

[0016] According to the undirected graph G, we get the adjacency matrix A and degree matrix D, and further get the Laplace matrix as follows:

[0017] L=DA

[0018] According to the Laplace matrix L, the normalized Laplace matrix is ​​obtained as follows:

[0019]

[0020] Where I represents the identity matrix;

[0021] In order to achieve the desired group formation, a comparison is made between the current formation shape and the desired shape, and the formation is quantified. The specific formula is as follows:

[0022]

[0023] In the formula is the symmetric normalized Laplace matrix of the current formation, is the symmetric normalized Laplace matrix of the desired formation shape, and the Frobenius norm ||·|| F Used for distance measurement; tr{} represents the trace of the matrix, that is, the sum of the diagonal elements of the matrix;

[0024] For drone v, use its n adjacent edges {e v1 ,e v2 ,...,e vn} to form the weight vector w v =[w v1 , w v2 , ..., w vn ] T ; By the chain rule, f is relative to p v The gradient of is written as:

[0025]

[0026] Where w v is a weight vector consisting of the weights of the adjacent edges of drone v, is the gradient of the similarity measure f with respect to the weight vector; this formula is used to calculate the position of the drone v, and how the similarity measure will change; if the formation shape is different from the expected shape, this gradient is used to decide how to move the drone v to reduce this difference until the difference between the current formation shape and the expected shape meets the preset requirements;

[0027] Step 2: Design of the path planning algorithm for the virtual leader, including the following steps:

[0028] Step 2-1: Use the traditional A* algorithm to plan the path for the drone;

[0029] Step 2-2, path optimization processing: using the dynamic programming characteristics of the Floyd algorithm, solve the multi-source shortest path problem in the weighted graph to form an adaptive A* algorithm;

[0030] Step 2-3, local obstacle avoidance algorithm: optimize the local obstacle avoidance of the path optimized in step 2-2 by using the artificial potential field method.

[0031] The beneficial effects of the present invention are as follows: the present invention models the UAV formation problem as an undirected graph model and utilizes an improved virtual leader-follower mechanism. In this model, each UAV can communicate bidirectionally with other UAVs to form a highly connected network structure. This not only ensures the formation-keeping capability of the formation, but also enhances the collision avoidance capability. By introducing a position consistency index, the stability of the UAV formation and the ability to avoid collisions are effectively indicated. Secondly, in combination with the A* algorithm and the artificial potential field method, the present invention proposes an innovative path planning scheme to improve the efficiency and real-time performance of path planning. By optimizing the traditional A* algorithm and introducing an adaptive path search mechanism, the path search efficiency is significantly improved. At the same time, in view of the redundant point problem in the path and the smoothing requirements of the UAV trajectory, redundant point processing and B-spline smoothing processing are respectively performed to ensure the optimization of the path and the smoothness of the flight, and combined with the artificial potential field method for real-time obstacle avoidance. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] Figure 1 A flowchart of the obstacle avoidance and trajectory optimization method for UAV formation of the present invention;

[0033] Figure 2 A schematic diagram of a UAV formation scenario of the present invention;

[0034] Figure 3 It is a schematic diagram of the simulation of the UAV formation of the present invention;

[0035] Figure 4 This is a real-scene test diagram of the UAV formation of the present invention;

[0036] Figure 5 It is the obstacle-free time-unmanned vehicle spacing diagram of the present invention;

[0037] Figure 6 It is the distance diagram between drones when there are obstacles in the present invention;

[0038] Figure 7 This is a diagram showing the results of the formation obstacle avoidance algorithm of the present invention. DETAILED DESCRIPTION

[0039] The technical solution of the present invention is further described below in conjunction with the accompanying drawings.

[0040] like Figure 1 As shown, a method for obstacle avoidance and trajectory optimization of a UAV formation based on comprehensive computing and coordination of the present invention comprises the following steps:

[0041] Step 1. Formation control design among multiple UAVs: Utilize the leader-follower formation model based on a virtual leader, that is, in the formation, no UAV is designated as the leader, but a virtual UAV with the same motion model as the actual UAV is designated as the leader, and all UAVs in the formation are followers;

[0042] The kinematic model of the drone is expressed as:

[0043]

[0044] In the formula

[0045] are the velocity components of the UAV in the x-axis and y-axis directions, v represents the UAV speed, ψ and They represent the yaw angle and its rate of change over time, respectively, and ω represents the angular velocity;

[0046] The position and heading of the virtual leader are expressed as:

[0047]

[0048] With ΔX i represents the relative distance between UAV i and the virtual leader, then the expected position X of the UAV relative to the virtual leader i,desired It is expressed as:

[0049] X i,desired =X leader +ΔX i

[0050] In the mutual communication between drones, they determine the status of a virtual leader by sharing each other's status information, and adjust their formation layout accordingly; when receiving the target location information, the formation will take the target point as the destination and start the movement of the virtual leader; in this process, the followers will move freely within the threshold range set by the virtual leader, and the threshold range is set according to the environment.

[0051] The virtual leader drone formation in the above content is introduced into the undirected graph as a real drone, and the drone formation is modeled by the undirected graph G = (V, E), denoted as G(V, E); the point set V cannot be an empty set, while the edge set E can be an empty set. V = {1, 2, ..., N} is the vertex set, N is the number of drones, and V is the set of drones. In the undirected graph G, the vertex v represents the position vector p v =[x v ,y v ,z v ]. vn ∈E is the edge connecting vertex v∈V and vertex n∈V, which represents the weight of the edge between the two drones. The drones are connected in both directions, and each drone can communicate with other drones, so the graph G is a complete undirected graph. Each edge of the graph G is associated with a non-negative number as a weight;

[0052] w vn =‖pv -p n ‖ 2 (v,n)∈E

[0053] Where w vn is the weight of the edge between drone v and drone n in the graph, p v and p n are the position vectors of drones v and n respectively, ||·|| represents the Euclidean norm, that is, the square of the straight-line distance between two points; E represents the set of all edges in the formation graph.

[0054] According to the undirected graph G, we get the adjacency matrix A and degree matrix D, and further get the Laplace matrix as follows:

[0055] L=DA

[0056] According to the Laplace matrix L, the normalized Laplace matrix is ​​obtained as follows:

[0057]

[0058] Where I represents the identity matrix;

[0059] As a graph representation matrix, Laplacian contains information about the graph structure. In order to achieve the desired group formation, a comparison between the current formation shape and the desired shape is performed, and the formation is quantized. The specific formula is as follows:

[0060]

[0061] In the formula is the symmetric normalized Laplace matrix of the current formation, is the symmetric normalized Laplace matrix of the desired formation shape, and the Frobenius norm ||·|| F Used for distance measurement; tr{} represents the trace of the matrix, which is the sum of the diagonal elements of the matrix; the Laplacian matrix can capture the information of the formation structure, so this metric can well reflect the preservation of the formation shape. The Frobenius norm provides a way to measure the size of the matrix difference, while the trace of the matrix can quantify this difference into a single scalar value.

[0062] The indicators are analytically distinguishable based on the location of each drone. For a drone v, use its n adjacent edges {e v1 ,e v2 ,...,e vn} to form the weight vector w v =[w v1 , w v2 , ..., w vn ] T; By the chain rule, f is relative to p v The gradient of is written as:

[0063]

[0064] Where w v is a weight vector consisting of the weights of the adjacent edges of drone v, is the gradient of the similarity measure f with respect to the weight vector; this formula is used to calculate the position of the drone v, and how the similarity measure will change. If the formation shape is different from the expected shape, this gradient can be used to determine how to move the drone v to reduce this difference until the difference between the current formation shape and the expected shape meets the preset requirements. Then the gradient can be written as:

[0065]

[0066] Each component of the gradient vector reflects the sensitivity of the formation similarity measure to the weight. Adjusting the distance between the drone and other drones changes the overall shape similarity of the formation. The cooperative controller designed for the multi-drone system can meet the stability condition, and the error between the expected distance and the actual distance is asymptotically stable, that is, the drone group can achieve the desired formation.

[0067] By implementing a frame synchronization communication mechanism between drones, a detailed analysis of the drone group formation matrix is ​​performed to accurately construct and implement an expected drone formation pattern; the matrix is ​​constructed through communication based on the position of the follower drones, and the position of the followers is determined by the position of the virtual leader; Figure 2 It is a virtual leader-follower formation, with the virtual leader in the middle. A matrix is ​​built through communication based on the positions of the three follower drones, and the positions of the followers are determined by the position of the virtual leader.

[0068] Step 2: Design of the path planning algorithm for the virtual leader. In the application scenario of path planning, in view of the complex requirements of the drone in global path planning and real-time obstacle avoidance, and considering the limitations of the A* algorithm in real-time obstacle avoidance, the present invention proposes an innovative path optimization solution combining the improved A* algorithm and the artificial potential field method, with the focus on improving the efficiency and real-time responsiveness of path planning through computational optimization. Accordingly, the present invention proposes an ASAP-A* algorithm to enhance the obstacle avoidance capability and path efficiency during the calculation process, including the following steps:

[0069] Step 2-1, use the traditional A* algorithm to plan the path for the UAV; the traditional A* algorithm is a heuristic function, and there are many search nodes and large turning angles in the path acquisition. Different search strategies will also lead to different results of the UAV path planning. By changing its evaluation function, the global path planning time of the UAV can be effectively reduced. When the UAV is far away from the target point, increase the coefficient of the actual loss function g(n). When the distance between the UAV and the target point gradually decreases, reduce the coefficient of g(n), and gradually increase the coefficient of the estimation function h(n). This can effectively reduce the search space and avoid too much invalid search near the starting point. And while maintaining the optimal path, the UAV can effectively reduce the computational complexity, reduce the planning time, and adapt to different flight phases, thereby improving the navigation performance of the UAV in complex environments. The specific formula is as follows:

[0070]

[0071] Where ρ(x,x g ) is the Euclidean distance from the current point to the target point, ρ(x s ,x g ) is the Euclidean distance from the starting point to the target point; it can be seen from the formula that when the UAV moves toward the target position, the proportion of g(n) in F(n) gradually decreases, which can make the UAV quickly move away from the starting position. When the UAV moves toward the target position, the proportion of h(n) in F(n) gradually increases, which can make the UAV quickly move toward the unknown target.

[0072] Step 2-2, path optimization processing: When using the A* algorithm for path planning, it is often encountered that the route moves close to the edge of the obstacle, resulting in many unnecessary turning points and redundant points in the path. The existence of these points increases the length of the route and reduces the smoothness of the path. In order to optimize this problem, the present invention introduces the idea of ​​the Floyd algorithm to improve the path generated by the A* algorithm. By removing these unnecessary redundant points and turning points, the total length of the route can be effectively shortened and the route can be made smoother, thereby improving the efficiency and quality of path planning.

[0073] The present invention uses the dynamic programming characteristics of the Floyd algorithm to solve the multi-source shortest path problem in the weighted graph and form an adaptive A* algorithm; the Floyd algorithm is combined with the A* algorithm to perform initial generation and subsequent simplification of the obstacle avoidance route. The path from the starting point to the target point is significantly shortened, effectively improving the performance of route planning.

[0074] The path planned by the conventional A* algorithm is generally planned along obstacles, which will result in many turning points and large turning angles, and does not make use of flight. The route planned by the conventional A* algorithm is starting point → A → B → C → D → E → target point. It moves along the obstacles at each step. It can be clearly seen that the inflection points of the middle points A, B, D, and E are redundant. There are both redundant inflection points and redundant points, so the path planning is unreasonable. Through the Floyd algorithm change, the starting point → C → target point route, this path planning is more reasonable, and the turning points and redundant points are reduced. However, there are also shortcomings. The turning angle of the turning point C is too large, which is not suitable for the smooth path of multiple drones.

[0075] After deleting redundant points and turning points, the remaining turning points still have too large turning angles, so the path is smoothed. B-spline curves generate smooth curves through control points and control polygons. When smoothing multiple line segments of convex polygons, the local path can be changed, but the overall path will not be changed.

[0076]

[0077] Where P(t) represents any point on the B-spline curve, P i is the control point, N i,k (t) is the basis function, which depends on the knot vector and the degree k of the curve, and n is the number of control points. Basis function N i,k The recursive definition of (t) is as follows:

[0078]

[0079] The specific process of the adaptive A* algorithm is shown in the following table.

[0080]

[0081]

[0082] Step 2-3, local obstacle avoidance algorithm: Although the A* algorithm performs well in global route planning and static environment obstacle avoidance, there is still room for improvement in local obstacle avoidance in dynamic environments. Currently, most algorithms are integrated with other algorithms for dynamic path planning. The artificial potential field method has a simple model, high real-time performance, high environmental adaptability, and small load on the drone. However, it also has the disadvantage of being easily trapped in the local optimal solution. Therefore, the artificial potential field method is used to optimize the path optimized in step 2-2 for local obstacle avoidance; the above shortcomings can be effectively avoided.

[0083] The artificial potential field method is divided into gravitational field and repulsive field. The algorithm regards the target and obstacle as objects that have gravitational and repulsive forces on the drone respectively. The drone moves along the combined force of gravitational and repulsive forces. The movement in this potential field is like finding a path from the starting point to the lowest point of the target point in a potential energy with ups and downs; the total potential energy U of the drone in the environment is the target potential energy U attr and obstacle potential energy U rep The sum is:

[0084] U=U attr +U rep

[0085] Where U attr is the gravitational potential energy, U rep is the repulsive potential energy;

[0086] The attractive potential energy is used to describe the driving force of the drone moving towards the target point. The specific formula is as follows:

[0087]

[0088] Where q is the current position of the drone, q goal is the target position, ζ is the attractive potential energy coefficient, which is used to adjust the size of the attractive force, and ‖·‖ is the Euclidean distance from the drone position to the target position;

[0089] Repulsive potential energy is used to describe the repulsive effect of obstacles on drones. The formula is as follows:

[0090]

[0091] Where U rep is the repulsive potential energy, ρ(q) is the shortest distance from the current position q of the drone to the nearest obstacle; ρ0 is the distance of the repulsive force, that is, the repulsive force will take effect only when the distance between the drone and the obstacle is less than ρ0; η is a positive proportional constant used to adjust the strength of the repulsive force;

[0092] The direction of motion of the drone in the potential field is determined by the negative gradient of the total potential energy, that is:

[0093]

[0094] F indicates the direction of total potential energy reduction, and the drone should move in this direction to reach the target position while avoiding collision with obstacles;

[0095] Step 2-4, in combination with steps 2-1, 2-2, and 2-3, the trajectory of the drone formation is optimized using the ASAP-A* algorithm. The A* algorithm itself cannot avoid dynamic obstacles. On the other hand, although the artificial potential field algorithm can perform dynamic obstacle avoidance, its planning results show path oscillation compared with the A* algorithm. Therefore, the present invention designs the ASAP-A* algorithm, and the specific process of the algorithm is shown in the following table.

[0096]

[0097]

[0098] The simulation environment of this embodiment is based on a computer equipped with a processor AMD Ryzen 7 5800H with RadeonGraphics 3.20GHz, a graphics card NVIDIA GeForce GTX3060, and a memory of DDR16G, and this experiment is run on the Ubuntu 20.04 operating system. In order to ensure the safety of UAV formation flight and reduce the risk of crashes, while simulating the real physical environment as much as possible, this embodiment aims to narrow the difference between the simulation environment and the on-site flight test. To this end, the Gazebo robot simulation software was used to simulate the flight test. This simulation used three P450 model drones to conduct simulation tests in a dense forest scene, such as Figure 2 As shown. During the simulation process, the flight altitude of the drone was set to 1.5m to ensure the accuracy and reliability of the test. This embodiment uses the ROS+Gazebo+RViz joint simulation platform. Figure 3 It is a schematic diagram of the simulation of the UAV formation of the present invention; Figure 4 This is a real-scene test diagram of the UAV formation of the present invention; Figure 5 It is the obstacle-free time-unmanned vehicle spacing diagram of the present invention; Figure 6 It is the distance diagram between drones when there are obstacles in the present invention; Figure 7 This is the result diagram of the formation obstacle avoidance algorithm of the present invention; it can be seen from the figure that the drone obstacle avoidance and trajectory optimization algorithm designed by the present invention can effectively solve the drone path optimization problem in obstacle scenarios and realize stable and flexible formation flight.

[0099] Those skilled in the art will appreciate that the embodiments described herein are intended to help readers understand the principles of the present invention, and should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific variations and combinations that do not deviate from the essence of the present invention based on the technical revelations disclosed by the present invention, and these variations and combinations are still within the protection scope of the present invention.

Claims

1. A method for obstacle avoidance and trajectory optimization of UAV formation based on comprehensive computing and coordination, characterized in that: The following steps are involved: Step 1. Formation control design among multiple UAVs: Utilize the leader-follower formation model based on a virtual leader, that is, in the formation, no UAV is designated as the leader, but a virtual UAV with the same motion model as the actual UAV is designated as the leader, and all UAVs in the formation are followers; The kinematic model of the drone is expressed as: In the formula are the velocity components of the UAV in the x-axis and y-axis directions, v represents the UAV speed, ψ and They represent the yaw angle and its rate of change over time, ω represents the angular velocity; The position and heading of the virtual leader are expressed as: With ΔX i represents the relative distance between UAV i and the virtual leader, then the expected position X of the UAV relative to the virtual leader i,desired It is expressed as: X i,desired =X leader +ΔX i The status of a virtual leader is determined by sharing the status information between drones, and the formation layout is adjusted accordingly. When receiving the target location information, the formation will take the target point as the destination and start the movement of the virtual leader. During this process, the followers will move freely within the threshold set by the virtual leader. The virtual leader drone formation is introduced into the undirected graph as a real drone, and is modeled by the undirected graph G = (V, E), denoted as G(V, E); where V = {1, 2, ..., N} is the vertex set, N is the number of drones, and V is the set of drones; the vertex v represents the position vector p v =[x v ,y v ,z v ]; w vn is the edge connecting vertex v and vertex n, representing the weight of the edge between the two drones; According to the undirected graph G, we get the adjacency matrix A and degree matrix D, and further get the Laplace matrix as follows: L=DA According to the Laplace matrix L, the normalized Laplace matrix is ​​obtained as follows: Where I represents the identity matrix; In order to achieve the desired group formation, a comparison is made between the current formation shape and the desired shape, and the formation is quantified. The specific formula is as follows: In the formula is the symmetric normalized Laplace matrix of the current formation, is the symmetric normalized Laplace matrix of the desired formation shape, and the Frobenius norm ||·|| F Used for distance measurement; tr{} represents the trace of the matrix, that is, the sum of the diagonal elements of the matrix; For drone v, use its n adjacent edges {e v1 ,e v2 ,...,e vn } to form the weight vector w v =[w v1 , w v2 , ..., w vn ] T ; By the chain rule, f is relative to p v The gradient of is written as: Where w v is a weight vector consisting of the weights of the adjacent edges of drone v, is the gradient of the similarity measure f with respect to the weight vector; This formula is used to calculate the position of drone v and how the similarity measure will change; if the formation shape is different from the expected shape, this gradient is used to determine how to move drone v to reduce the difference until the difference between the current formation shape and the expected shape meets the preset requirements; Step 2: Design of the path planning algorithm for the virtual leader, including the following steps: Step 2-1: Use the traditional A* algorithm to plan the path for the drone; Step 2-2, path optimization processing: using the dynamic programming characteristics of the Floyd algorithm, solve the multi-source shortest path problem in the weighted graph to form an adaptive A* algorithm; Step 2-3, local obstacle avoidance algorithm: optimize the local obstacle avoidance of the path optimized in step 2-2 by using the artificial potential field method.

2. The method for obstacle avoidance and trajectory optimization of a UAV formation based on comprehensive computing and coordination according to claim 1, characterized in that: The specific implementation method of steps 2-3 is as follows: the artificial potential field method is divided into gravitational field and repulsive field. The target and obstacle are regarded as objects with gravitational and repulsive forces on the drone respectively. The drone moves along the combined force of gravitational and repulsive forces. The movement in this potential field is like finding a path from the starting point to the lowest point of the target point in a potential energy with ups and downs; the total potential energy U of the drone in the environment is the target potential energy U attr and obstacle potential energy U rep The sum is: U=U attr +U rep Where U attr is the gravitational potential energy, U rep is the repulsive potential energy; The attractive potential energy is used to describe the driving force of the drone moving towards the target point. The specific formula is as follows: Where q is the current position of the drone, q goal is the target position, ζ is the attractive potential energy coefficient, which is used to adjust the size of the attractive force, and ‖·‖ is the Euclidean distance from the drone position to the target position; Repulsive potential energy is used to describe the repulsive effect of obstacles on drones. The formula is as follows: Where U rep is the repulsive potential energy, ρ(q) is the shortest distance from the current position q of the drone to the nearest obstacle; ρ0 is the distance of the repulsive force, that is, the repulsive force will take effect only when the distance between the drone and the obstacle is less than ρ0; η is a positive proportionality constant used to adjust the strength of the repulsive force; The direction of motion of the drone in the potential field is determined by the negative gradient of the total potential energy, that is: F indicates the direction in which the total potential energy decreases, and the drone should move in this direction.

Images