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Article

Three-Dimensional Path Planning for Post-Disaster Rescue UAV by Integrating Improved Grey Wolf Optimizer and Artificial Potential Field Method

by
Dan Han
1,2,3,
Qizhou Yu
4,*,
Hao Jiang
5,
Yaqing Chen
6,
Xinyu Zhu
1 and
Lifang Wang
2
1
Institute of Electronic and Electrical Engineering, Civil Aviation Flight University of China, Guanghan 618307, China
2
Institute of Electrical Engineering, Chinese Academy of Sciences, Beijing 100190, China
3
University of Chinese Academy of Sciences, Beijing 100049, China
4
Civil Aviation Administration of China, Academy of Flight Technology and Safety, Civil Aviation Flight University of China, Guanghan 618307, China
5
Organization Department of Party Committee, Civil Aviation Flight University of China, Guanghan 618307, China
6
School of Air Traffic Management, Civil Aviation Flight University of China, Guanghan 618307, China
*
Author to whom correspondence should be addressed.
Appl. Sci. 2024, 14(11), 4461; https://doi.org/10.3390/app14114461
Submission received: 18 April 2024 / Revised: 21 May 2024 / Accepted: 21 May 2024 / Published: 23 May 2024
(This article belongs to the Special Issue Advances in Unmanned Aerial Vehicle (UAV) System)
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Abstract

:
The path planning of unmanned aerial vehicles (UAVs) is crucial in UAV search and rescue operations to ensure efficient and safe search activities. However, most existing path planning algorithms are not suitable for post-disaster mountain rescue mission scenarios. Therefore, this paper proposes the IGWO-IAPF algorithm based on the fusion of the improved grey wolf optimizer (GWO) and the improved artificial potential field (APF) algorithm. This algorithm builds upon the grey wolf optimizer and introduces several improvements. Firstly, a nonlinear adjustment strategy for control parameters is proposed to balance the global and local search capabilities of the algorithm. Secondly, an optimized individual position update strategy is employed to coordinate the algorithm’s search ability and reduce the probability of falling into local optima. Additionally, a waypoint attraction force is incorporated into the traditional artificial potential field algorithm based on the force field to fulfill the requirements of three-dimensional path planning and further reduce the probability of falling into local optima. The IGWO is used to generate an initial path, where each point is assigned an attraction force, and then the IAPF is utilized for subsequent path planning. The simulation results demonstrate that the improved IGWO exhibits approximately a 60% improvement in convergence compared to the conventional GWO. Furthermore, the integrated IGWO-IAPF algorithm shows an approximately 10% improvement in path planning effectiveness compared to other traditional algorithms. It possesses characteristics such as shorter flight distance and higher safety, making it suitable for meeting the requirements of post-disaster rescue missions.

1. Introduction

Unmanned aerial vehicles (UAVs) possess characteristics such as small size, low cost, and high maneuverability, making them capable of replacing humans in penetrating disaster-stricken areas to accomplish certain rescue tasks, thereby effectively assisting overall rescue efforts. During rescue missions, UAVs are equipped with obstacle detection sensors such as laser radar, ultrasonic sensors, or cameras to enable real-time perception of surrounding obstacles such as buildings, trees, and power lines, in order to avoid collisions. By collecting sensor data and using appropriate algorithms, UAVs can generate obstacle-avoidance path planning to ensure safe flight. Furthermore, due to the high complexity of post-disaster environments, rescue UAVs are also equipped with terrain and wind sensors, allowing them to perceive the ever-changing rescue mission scenarios in real-time. With this information, UAVs can plan low-risk flight paths and avoid collisions with ground obstacles. Additionally, they can monitor changes in wind speed and air pressure in the flight environment. The data from these sensors ensure that the path planning algorithm takes into account meteorological conditions, adjusting the flight path to ensure the stability and efficiency of the UAV under different flight altitudes and wind speeds. In the context of disaster response, it is of significant importance to plan a reasonable rescue mission path that effectively avoids obstacles while considering the UAV’s flight characteristics, as this contributes to minimizing disaster losses [1].
Path planning technology is one of the hot research topics in the field of UAVs. In recent years, with the widespread use of UAVs, path planning has become crucial for UAVs to execute missions and avoid obstacles in their work environment. The objective of path planning is to determine a safe and feasible optimal path for UAVs, taking into account the operational feasibility in practical applications. Therefore, enhancing the path optimization capability of UAVs in complex flight environments holds significant research significance, while ensuring both safety and feasibility [2,3,4]. However, the increasingly complex flight environments imply a growing number of path constraints. These UAV path constraints refer to a series of limitations and regulations that must be adhered to during UAV flight to ensure safety, compliance, and effectiveness. Furthermore, path constraints may vary according to different mission scenarios and specific task requirements, adding to the increasing demands placed on UAV path planning algorithms. Although traditional path planning algorithms have reached a high level of maturity, they are not well adapted to the practical flight of UAVs in complex three-dimensional environments [5,6,7,8]. Table 1 provides a comparative analysis of the characteristics of several commonly used planning algorithms. From the table, it can be observed that traditional path planning algorithms such as artificial potential field and velocity vector exhibit high planning efficiency but are limited in their applicability and relatively weak in solving capability. On the other hand, intelligent optimization algorithms like grey wolf optimization have a broader range of applicable scenarios and generally higher solving capabilities, but at the same time, they may sacrifice some efficiency.
The APF method [12,13,14], as one of the traditional algorithms, represents the motion of a UAV in the environment as a virtual force in an artificial gravitational field. The UAV is attracted towards the target point and avoids obstacles through the repulsive force. Although APF has characteristics such as path smoothing and fast planning speed in local path planning, it tends to suffer from problems such as excessive repulsive force and becoming trapped in local minima, resulting in the inability to plan the shortest path or reach the target point. Reference [15] decomposed the repulsive force from obstacles along the path and reconstructed the resultant force, thereby avoiding the occurrence of local minima in the planning process. Reference [16] solved the issues of oscillation and local optima in the APF method by combining advanced avoidance and variable influence range methods.
The GWO algorithm [17,18,19,20,21] is a novel intelligent simulation optimization algorithm inspired by the hunting behavior of grey wolf packs. Compared to other intelligent simulation algorithms, it has advantages such as fewer adjustable parameters, simple structure, and easy implementation, as well as good global search capabilities. However, this method also has drawbacks such as premature convergence, slow convergence speed, and poor accuracy. Currently, domestic and international scholars have conducted in-depth research on the use of the GWO for UAV path planning and have made improvements and optimizations to the original algorithm. Reference [22] introduced reinforcement learning into the GWO to enable individual adaptation based on accumulated performance, allowing UAVs to efficiently obtain smooth paths in three-dimensional complex flight environments. Reference [23] simplified the GWO to accelerate its convergence speed and combined it with the enhanced symbiotic organisms search method to improve development capabilities and enhance population exploration. Reference [24] combined the GWO with the Powell algorithm to address the low optimization accuracy issue of the GWO and improve the effectiveness of UAV path planning. Reference [25] integrated the genetic algorithm and the large-scale neighborhood search algorithm into the GWO algorithm to enhance global search capabilities and further strengthen local search capabilities, thereby improving the algorithm’s solution accuracy and avoiding becoming trapped in local optima during UAV path planning.
Currently, countries worldwide have gradually placed greater emphasis on the practical application of UAVs in post-disaster rescue operations. The utilization of path planning algorithms to enable UAVs to reach rescue destinations more quickly and conduct search and rescue operations more efficiently has become a pressing issue for experts and scholars worldwide. In reference [26], four discrete path planning methods were developed and applied to determine the waypoints that UAVs must follow, thereby improving search efficiency. Reference [27] proposed an enhanced BF (breadth-first) path coverage method, which not only reduced path length but also enhanced the ability to counteract inherent UAV jitter. Reference [28] presented a multi-objective optimization algorithm to allocate tasks and plan paths for a group of UAVs, aiming to minimize task completion time using a genetic algorithm approach. Reference [29] introduced a hybrid algorithm named HC-SAR for UAV path planning, which improved the convergence speed and solution quality by combining heuristic crossover strategies with synthetic aperture radar. Reference [30] designed a path planning method based on sine cosine particle swarm optimization, and linear weight inertia and acceleration coefficients were specifically designed for SCPSO to enhance its performance, escape local minima, and thoroughly explore the search space. While many experts and scholars have conducted research on UAV path planning in post-disaster rescue scenarios, there is limited research specifically focused on post-disaster mountain rescue scenarios. Furthermore, most existing algorithms are not suitable for meeting the requirements of mountain rescue operations, and their efficiency in path planning for mountainous environments is often suboptimal. Therefore, the research presented in this paper aims to address this gap by proposing an innovative algorithm that efficiently caters to UAV post-disaster mountain rescue. By designing an algorithm that can plan efficient and safe flight paths, the proposed approach enables UAVs to effectively navigate around mountainous obstacles and quickly reach rescue points for reconnaissance and aid.
Therefore, the research focus of this paper is defined as UAVs performing post-disaster mountain rescue missions. The UAV’s flight environment is primarily defined as a 10*10 km mountainous terrain, with the main threat being the terrain itself. The task environment is constructed using elevation maps. The task involves the UAV taking off from the starting point (0, 0, 0), traversing mountainous obstacles, and reaching the designated rescue point (100, 100, 100) to initiate the next phase of the rescue mission. In actual post-disaster mountain rescue scenarios, UAVs encounter various complex problems, such as the complexity of the rescue terrain and environment, obstacle recognition and avoidance, operational feasibility, and safety considerations. These challenges impose stringent requirements on the algorithm’s optimization capability and convergence efficiency [31]. Therefore, when UAVs fly in mountainous terrain, it is necessary to consider not only the threats posed by the environment to the UAV but also the performance constraints of the UAV itself and the efficiency of task completion. Therefore, in the context of mountain rescue missions, this paper requires the algorithm to have a fast planning rate and efficient solving capability to meet the requirements of rescue mission scenarios. Based on the analysis in Table 1, it can be observed that the most commonly used UAV-path planning algorithms currently available have certain limitations and are not fully applicable to mountain rescue scenarios. Hence, considering the requirements posed by post-disaster rescue scenarios for UAV path planning algorithms, this paper conducts an analysis and selection process to ultimately choose the APF method and GWO for improvement and integration.
Our research makes three main contributions. Firstly, this paper defines and constructs the task environment by establishing a benchmark terrain model to simulate mountain rescue mission scenarios. In terms of considering the smoothness of the path, the cubic B-spline curve method is used to smooth the path generated by the algorithm. Secondly, addressing the shortcomings of the GWO in terms of optimization performance, long planning time, and the unsuitability of the APF method for three-dimensional task environments, as well as its tendency to get stuck in local optima due to excessive repulsion forces, we propose improvements and integration of the GWO and APF algorithms. Nonlinear adjustment strategies for control parameters are employed in the GWO to balance global and local search capabilities. The update strategy for individual positions is optimized to coordinate the search capability of the algorithm and reduce the probability of falling into local optima. In addition, based on the initial path generated by IGWO, the APF algorithm is improved by assigning route attraction to the path points on the initial path, so that the APF algorithm can adapt to the three-dimensional environment and plan the final path. Finally, we conduct simulation experiments comparing the path planning performance of various algorithms, including the integrated algorithm, in mountainous terrain with obstacles, to validate their effectiveness.
In this paper, the construction method for the UAV post-disaster rescue mission scenario, the fitness function adopted for UAVs, and the path smoothing algorithm used are presented in Section 2. Section 3 provides a detailed description of the innovative hybrid algorithm, IGWO-IAPF, specifically designed to address the proposed mountain rescue mission. Section 4 and Section 5 are dedicated to presenting and discussing the analysis of the simulation results. Finally, Section 6 summarizes the main contributions of the paper.

2. Research on UAV Post-Disaster Path Planning Problem

2.1. Problem Description

During emergency rescue missions, UAVs need to determine the optimal flight path, through path planning, in order to reach the accident site quickly and efficiently. Due to the limitations of UAV performance, it is necessary to plan the UAV’s path to ensure the avoidance of obstacles and flight risks. Additionally, suitable landing areas need to be selected based on the specific conditions of the accident site to facilitate safe landing and provide rescue services [32].
This study utilizes a digital elevation model (DEM) to construct the simulation environment. The elevation refers to the description of the ups and downs of the terrain. The elevation model is commonly used in three-dimensional environments and is also known as a three-dimensional elevation model. The digital elevation model represents the digitalized representation of the terrain undulations using corresponding terrain data. It is a collection of plane environment coordinates x , y and elevation data, describing the environmental distribution in the area. Real-world terrain models are a part of the digital elevation model, from which the environmental distribution can be derived [33]. Through the digital elevation model, we can obtain more relevant factors such as terrain, landforms, and spatial distribution, including altitude, central coordinates, slope, and slope variation rate, compared to a general digital terrain model.
The information required for UAV path planning needs to be extracted from the terrain model, and effective terrain modeling can significantly improve the efficiency of path planning. This paper considers factors such as the original terrain and obstacle areas and models the environment based on mountainous terrain. The primary flight mission of the UAV is to avoid mountainous obstacles and find the optimal path to reach the target for rescue purposes. The established baseline terrain model [34] is as follows:
c e n t e r = x i , y i h e i g h t = z i r a n g e = a i , b i , c i
Z ( x , y ) = ∑ i = 1 n h i exp − x − x i x s i 2 − y − y i y s i 2
x s i = a i y s i = b i
In the equation: x i , y i represent the point coordinates of the center of obstacle peaks in the three-dimensional environment, z i represents the height of the obstacle, r a n g e represents the extent of the peak’s expansion, a , b , c are user-defined constants that control the range and steepness of the mountain peaks. By adjusting the values of various parameters, different terrain morphologies can be obtained; h i is a terrain parameter that controls the height, x s i and y s i correspond to the attenuation along the x and y axes, respectively, for the i peak, controlling the slope, n represents the total number of peaks. The environmental model is shown in Figure 1.
The path planning nodes for the UAV in this three-dimensional environment can be represented by coordinates x i , y i , z i . The starting and ending points of the path are set as point (0, 0, 0) and point (100, 100, 100), respectively. Multiple obstacles are placed in the UAV operating environment, and the detailed parameter information is provided in Table 2. Therefore, the UAV path planning task can be described as finding the optimal route from the takeoff point that avoids all obstacles and reaches the target point.

2.2. Fitness Function

During the research of UAV path planning using optimization algorithms, the quality of the planned path can be measured by setting a fitness function. In this paper, considering the actual flight conditions of the UAV and safety concerns, a fitness function is constructed by taking into account three aspects: path length, obstacle threat, and ground collision cost. A lower fitness value indicates a better generated path.
  • Cost of path length
This study selects the minimization of path length as a criterion to determine the optimality of the path [35]. As UAVs are controlled by ground control stations, the flight path C i of the UAV is represented as a list composed of n waypoints, where each waypoint corresponds to a path node in the path planning search map. Assuming there are i flight paths and j path nodes, and each node’s coordinates can be represented as P i j x i j , y i j , z i j , the calculation of the cost T 1 of path length can be expressed as:
T 1 ( C i ) = ∑ j = 1 n − 1 P i , j P i , j + 1 →
2.
Navigation altitude cost
The smoothness of the path is determined by the UAV’s stable flight. A stable flight altitude not only enables the planning of smooth flight paths but also helps save fuel, alleviating the burden on the UAV control system to some extent [36]. Furthermore, in practical UAV applications, the flight altitude is often restricted between two given extremes. It is necessary to cruise at an appropriate altitude to meet requirements such as search operations and payload delivery. Assuming the minimum and maximum altitudes are h min and h max , respectively, the cost T 2 of flight altitude can be expressed as:
T 2 C i = ∑ j = 1 n H i j
H i j = h min − h i j ,   h i j < h min 0 ,   h min ≤ h i j ≤ h max h i j − h max ,   h i j > h max
In the equation, h represents the flight altitude relative to the ground.
3.
Ground collision risk
During the operation of UAVs, any issues such as explosions can affect the safety of the life and property of ground personnel. Therefore, it is necessary to use ground collision risk T 3 to determine the impact of UAV accidents on ground personnel. The objective function is as follows:
T 3 = N r D
In the equation, N r represents the number of injured personnel after the UAV collides with the ground, and D is the fatality rate resulting from the collision. For N r and D , they can be obtained using Equations (7), (8), (9) and (10) respectively.
N r = A r ρ
A r = W U A V L U A V 2 + H U A V 2
In the equation, ρ represents the population density, and A r represents the coverage area of the accident, which is determined by the UAV’s wingspan W U A V , length L U A V , and height H U A V .
This article evaluates the impact mortality rate d based on the logistic growth variable model proposed by Dalamagkidis [37]. When the coordinates of the UAV are x i , y i , z i , its expression is:
D = 1 1 + α r / β r | β r / E r | 1 / ( 4 P s )
E r = 1 2 M + Q v 2 + M + Q g z i
In the equation, P s is the shielding coefficient of the environment, with a value range of 0 , 1 , indicating the degree of exposure of the ground personnel; α is the required impact energy when the shielding coefficient is equal to 0.5 and the mortality rate reaches 50%; β is the threshold of impact energy required when the shielding coefficient is approximately equal to 0 and causes death; E r is the kinetic energy generated when the UAV collides with the ground; M is the weight of the UAV; v is the operating speed of the UAV; Q is the maximum carrying capacity of the UAV; z i is the height at which the UAV experiences a system malfunction; g is the gravitational acceleration.
The fitness function for the path, obtained by integrating the costs of path optimality, obstacle threat, and flight altitude, is given as follows:
T C i = b 1 T 1 C i + b 2 T 2 C i + b 3 T 3 C i b 1 + b 2 + b 3 = 1
In the equation, b 1 , b 2 , b 3 is a weight coefficient ranging from 0 to 1. The magnitude of each weight represents the importance of the corresponding flight cost. By adjusting the weight coefficients, the path requirements for different application needs can be met. For example, in rescue missions where UAVs need to maintain stable flight at a certain altitude, the weight coefficient b 2 for the flight altitude cost can be appropriately increased.

2.3. Path Smoothing Algorithm Based on Cubic B-Spline Curves

In order to ensure smooth and flyable paths while reducing computational time, this study employs cubic B-spline curves for path smoothing. The smoothing technique described in reference [38] is utilized, which primarily involves adding control points to effectively avoid obstacles and generate smooth flight paths.

2.3.1. Smoothing of Path Using Basis Functions and Control Points

By providing spatial vertices P i ( i = 0 , 1 , ⋯ , m + n ) , n -degree curve segments can be obtained.
P ( t ) = ∑ i = 0 n P i F i , k ( t )
In the equation, P i represents the curve equation corresponding to the i control point, and F i , k t represents the k order B-spline basis function. Since the value of k represents the smoothness of the curve, a higher value of k results in a smoother curve but also increases the computational complexity. The parameter t represents the nodes used to divide the curve into segments and control the shape of the curve. To balance smoothness and complexity, this study selects k = 3 , resulting in the cubic B-spline basis function as follows:
F i ,   k ( t ) = 1 k ! ∑ m = 0 k − i ( − 1 ) j C k + 1 j ( t + k − m − j ) k
The set of points obtained by the algorithm calculation is referred to as the path planning points M . Assuming there are n + 1 ordered spatial pose vectors V i ( i = 0 , 1 , ⋯ , n ) in M , the linear combination of k + 1 adjacent position vectors can be used to derive the curve equation for the i segment of k + 1 adjacent control points as follows:
P i ( x ) = ∑ j = 0 k F i , k ( x ) V i + j
In the equation, P i ( x ) represents the i B-spline curve function. x is the data point. By substituting Equation (13) into Equation (15), the matrix form of the curve is obtained as follows:
P i x = 1 6 1 x x 2 x 3 1 4 1 0 − 3 0 3 0 3 − 6 3 0 − 1 3 − 3 1 V i − 1 V i V i + 1 V i + 2

2.3.2. Path Smoothing Basis Functions and Control Points

In Equation (15), V i − 1 , V i , V i + 1 , V i + 2 represents a set of control points. In UAV path planning, it is necessary to determine the positional values of the desired points on the path, which are obtained by solving the inverse kinematics problem of the UAV motion. If planning a path based on known positional values, the control points need to be determined. Let P represent the positional values, and the following conditions are obtained:
P i − 1 ( 1 ) = P i ( 0 ) = P i = 1 6 ( V i − 1 + 4 V i + V i + 1 ) i = 1 , 2 , ⋯ , m − 1
In Equation (16), there are m + 2 unknowns but only m equations. Therefore, the following constraint conditions are added to the endpoints of the global path point set:
V 1 = V 0 V m + 1 = V m
A unique set of V i − 1 , V i , V i + 1 , V i + 2 can be determined by given boundary conditions. The B-spline curve determined by Equation (15) is represented as follows:
P i x = 1 6 − x 3 + 3 x 2 − 3 x + 1 V i − 1 + 1 6 − x 3 + 6 x 2 + 4 V i + 1 6 − 3 x 3 + 3 x 2 − 3 x + 1 V i + 1 + 1 6 x 3 V i + 2
In Equation (18), controlled by four control points V i − 1 , V i , V i + 1 , V i + 2 , the coordinates of the control points are represented by X i , Y i , Z i ( i = 1 , 2 , ⋯ , n ) . Among them, the parameter x takes all values uniformly from 0 to 1 with an interval of 0.01, which yields the i segment of the B-spline curve. The coordinates X i , Y i , Z i of any point on the curve are represented as follows:
X t = ∑ i = 0 m X i F i , k t Y t = ∑ i = 0 m Y i F i , k t Z t = ∑ i = 0 m Z i F i , k t
Finally, based on Equations (18) and (19), the connecting points of the cubic B-spline curve can be obtained. To reduce computational complexity, redundant connecting points are removed, and the remaining connecting points are used to generate the smooth effect of the cubic B-spline curve, as shown in Figure 2.

3. Description of the Integrated Improved Algorithm

In this paper, an integrated algorithm, namely the improved grey wolf optimization-improved artificial potential field (IGWO-IAPF), is proposed to solve the problem of UAV post-disaster rescue path planning. Firstly, the IGWO is employed for initial path planning. Then, the IAPF is applied for secondary optimization to enhance the feasibility and efficiency of the planned path, resulting in an efficient three-dimensional path.

3.1. Grey Wolf Optimizer

The grey wolf optimizer simulates the leadership hierarchy and predation mechanism within a grey wolf pack. The wolf pack is divided into four hierarchical levels, Represented by α , β , δ and ω , respectively. The α wolf represents the leader of the pack, responsible for leading the entire pack in hunting and decision-making. The β wolf assists the α wolf in decision-making and provides auxiliary commands. The δ wolves obey the commands of the α and β wolves, respectively, and perform tasks such as scouting and reconnaissance. The ω wolves are considered as the lowest hierarchy and engage in activities around the α , β or δ wolves. In the mathematical simulation, the α wolf represents the best solution in the algorithm, the β wolf represents a suboptimal solution, the δ wolf represents the third best solution with relatively poor fitness, and the remaining solutions are considered as δ wolves.
During the hunting process of the wolf pack, three steps can be distinguished: encircling the prey, chasing the prey, and attacking the prey. In the grey wolf optimizer, the following formula can be used to update the position of the wolves, achieving the encircling of the prey:
D → = C → × X P → t − X → t
X → t + 1 = X p → t − A → × D →
In the equation, D → represents the distance between an individual wolf and the prey. X → and X p → are the position vectors of the wolf and the prey, respectively. t denotes the iteration number. A → and C → are synergy vectors determined by coefficients, and their calculation formula is as follows:
A → = 2 a → × r 1 → − a →
C → = 2 r 2 →
In the equation, r 1 → and r 2 → are random vectors within [0, 1]. a → is the convergence factor, which linearly decreases from 2 to 0 during the iteration process.
During the chasing process, since the α , β and δ wolves are the closest to the prey, the ω wolf pack follows the guidance of the α , β and δ wolves to move, thus achieving the encirclement of the prey. In the mathematical model, this can be expressed as follows:
D α → = C 1 → × X α → − X → D β → = C 2 → × X β → − X → D δ → = C 3 → × X δ → − X →
X 1 → = X α → − A 1 × D α → X 2 → = X β → − A 2 × D β → X 3 → = X δ → − A 3 × D δ →
X → t + 1 = X 1 → + X 2 → + X 3 → 3
where D α → , D β → , D δ → represent the distances between the α , β and δ wolves and the rest of the wolf pack, respectively. X 1 → , X 2 → , X 3 → represent the current positions of the α , β and δ wolves, respectively. X → is the final position of the ω wolf pack.

3.2. Artificial Potential Field Algorithm

The artificial potential field method treats the environment in which the unmanned aerial vehicle operates as an artificial force field. The target exerts an attractive force on the UAV, creating a gravitational potential field, while obstacles generate repulsive forces, forming a repulsive potential field. The UAV moves towards the target under the combined effects of attraction and repulsion forces.
The gravitational potential field between the target point and the unmanned aerial vehicle is defined as:
U a t t X = 1 2 K a t t x − x G 2
In the equation, K a t t is the position gain coefficient, x represents the position of the unmanned aerial vehicle, and x G represents the position of the target point. The attraction force between the UAV and the target point is the negative gradient of U a t t X and can be expressed as:
F a t t X = − g r a d K a t t = − K a t t x − x G
The repulsive potential field generated by obstacles on the unmanned aerial vehicle can be expressed as:
U r e p X = 1 2 K r e p 1 x − x 0 − 1 ρ 0 2 , x − x 0 ≤ ρ 0 0 , x − x 0 > ρ 0
In the equation, K r e p is the repulsion gain coefficient, x − x 0 represents the distance between the UAV and the obstacle, and ρ 0 represents the influence distance of the obstacle. The repulsion force between the UAV and the obstacle is the negative gradient of F r e p X and can be expressed as:
F r e p X = − g r a d K r e p = K r e p 1 x − x 0 − 1 ρ 0 1 x − x 0 2 , x − x 0 ≤ ρ 0 0 , x − x 0 > ρ 0
Once the attraction force from the target point to the unmanned aerial vehicle and the repulsive forces from various obstacles on the UAV are determined, the resultant force acting on the UAV can be obtained.
F a l l = F a t t + ∑ i = 1 n F r e p _ i
In a two-dimensional environment, due to the constraints of the task environment dimensions, the APF typically follows a more direct path to reach the target point. However, in a three-dimensional environment, with the increase in dimensions, obstacles may have larger volumes or heights, while the influence range of the force field is limited. When the UAV is far from the obstacles, the repulsive force may not be sufficient to completely avoid the obstacles, resulting in the occurrence of a path crossing through obstacles. Moreover, due to the complexity of the shape and distribution of the obstacles, traditional force field functions may not accurately describe the influence of obstacles, leading to more possible flight directions and paths during the planning process, which increases the number of local minima. Therefore, the UAV may become trapped in a local minimum due to the characteristics of the force field, unable to find a better path. Figure 3 shows the effect of the traditional APF algorithm for three-dimensional path planning. It can be clearly seen that during path planning with the APF, the route crosses over the peaks.

3.3. IGWO-IAPF Algorithm Description

In order to comprehensively improve the accuracy, stability, and convergence speed of the algorithm’s path optimization, and to avoid getting trapped in local minima, improvements have been made to the traditional GWO. These improvements include enhancing the distance control parameter and incorporating probabilistic mutation. Additionally, the traditional APF method has been modified by introducing initial path attraction and modifying the traditional force field. Finally, the two improved algorithms are integrated together.

3.3.1. Improvement of Distance Control Parameters

From Equation (22), it can be observed that the parameter A balances the exploration and exploitation capabilities of the GWO. When A > 1 , the wolf population tends to expand the search range to find more suitable prey, which corresponds to the algorithm’s global search capability. When A < 1 , the wolf population tends to narrow down the search range, surrounding the prey from various directions and launching attacks, corresponding to the algorithm’s local search capability. The value of A is influenced by the parameter a , which linearly decreases from 2 to 0 during the iteration process. However, the problem solved by the grey wolf optimizer is a nonlinear optimization process, and linearly decreasing a cannot fully represent this process. Therefore, it is necessary to redesign the parameter a .
This study proposes a dynamically changing parameter a based on the distance ratio. During the iteration process, the distances d i and the average distance d a between all wolf individuals and the α wolf (the best individual) are calculated. The ratio d i − s i t e of the distance d i to d a represents the relative proximity of an individual’s current position. A larger value of d i − s i t e indicates a greater relative distance to the individual’s position, and a larger α value is assigned to wolves with relatively distant positions to enhance global search capability. Conversely, a smaller value of d i − s i t e indicates a closer relative position of the individual, and a smaller α value is assigned to wolves with relatively close positions to enhance local search capability. Different values of α are assigned based on the different d i − s i t e values of each individual in each iteration, aiming to balance the local search and global search capabilities of the algorithm.
d i = ∑ d = 1 D ( X i d − X a d ) 2
d a = ∑ i N d i N
d i − s i t e = d i d a
a = 2 − 2 × t t max λ × d i − s i t e
In the equation, d i represents the distance between the i wolf and the α wolf. X i denotes the position of the i wolf, while X a represents the position of the α wolf. d a represents the average distance in the current iteration, and d i − s i t e denotes the relative distance of the i wolf. t max corresponds to the maximum number of iterations, and λ is a positive constant.

3.3.2. Probability Variation

To enhance the ability of the GWO algorithm to escape local optima, a mutation operation from genetic algorithms is introduced. After the first iteration of the algorithm, N wolf individuals are generated. In order to increase the number of excellent wolf individuals in the next iteration and prevent them from getting trapped in local optima, mutation is applied to the wolves with a relative position d i − s i t e less than 1. The fitness of the mutated individuals is then compared with the original individuals, and the superior individuals are retained.
X n e w t = X i t + r a n d
In the equation: X n e w t represents a newly generated individual after mutation, where X n e w t is the i individual, and r a n d is a random vector composed of random numbers in the range (−1, 1) for each dimension.

3.3.3. Improvements to the APF Algorithm

Due to the issue of excessive repulsive forces that can prevent the UAV from reaching the target point, improvements have been made to the APF algorithm in the context of path planning. In this regard, the GWO is utilized for multiple iterations to optimize and generate an initial path. The initial path is then assigned an attractive force, which is strongest at the point closest to the current position of the UAV. This approach enhances the attractive force and mitigates the problem of excessive repulsive forces. The attractive force along the path can be defined as follows:
U a t t 2 X = 1 2 K _ l i n e x − x p 2
F a t t 2 X = − g r a d K _ l i n e = − K _ l i n e x − x p
In the equation, where K _ l i n e represents the path attraction gain coefficient, and x G denotes the position of the point on the initial path closest to the current position of the UAV.
After calculating the line attraction force obtained by the UAV, the line attraction force is incorporated into the total force experienced by the UAV, resulting in a new formula for the total force:
F a l l = F a t t 1 + F a t t 2 + ∑ i = 1 n F r e p _ i
Guided by the direction of the total force, the UAV navigates towards the target point. When approaching obstacles, the UAV experiences repulsive forces from the obstacles, leading to avoidance behavior.
Due to the continuous proximity and distancing of obstacles during the UAV’s flight, the influence distance of obstacles has a significant impact on the range and magnitude of the force field experienced by the UAV. Figure 4, Figure 5 and Figure 6 illustrate the comparison of various parameters of the IAPF influenced by obstacles. In conducting algorithmic simulation experiments, the flexibility to adaptively adjust the gain coefficients based on the experimental environment is crucial to avoid the algorithm becoming trapped in local optima.

3.3.4. The Steps of IGWO-APF Fusion Algorithm

Based on the content presented in Section 2.1, Section 2.2 and Section 2.3, the algorithmic flowchart of the proposed method is illustrated in Figure 7. The detailed description of the fusion algorithm is as follows:
Step 1: Compute the influence distance p o between the UAV and obstacles to determine if the UAV is located within a relatively free space away from obstacles. This is important because in practical planning scenarios, it is common to encounter situations where the UAV is far from obstacles. Based on the fast planning capability of the IAPF algorithm, the distance d i s between the UAV and obstacles is calculated. If d i s > p o , the IAPF is directly chosen for path planning to reduce the planning time. If d i s < p o , the IGWO is used for initial path planning of the UAV.
Step 2: Perform population initialization for the GWO algorithm. Generate the first generation of wolves using a random distribution following a normal distribution, and conduct collision detection (flag judgment). If flag = 0, it indicates that the path does not intersect with any obstacles, signifying that this path can be selected. If flag = 1, it implies that the path intersects with obstacles, thus increasing the fitness value. Finally, update the global best solution.
Step 3: Generate path points for all grey wolves based on the given formula. Calculate the cost values for all grey wolves, and identify the three wolves with the lowest costs as the α , β , δ wolves. Update the convergence factor using the formula, and then calculate A using another formula.
Step 4: Determine if t is greater than T. If t is less than T, increment t by 1 and return to Step 2. Otherwise, the algorithm iteration concludes, and the global optimal value and the optimal path are outputted as the results of the algorithm.
Step 5: After a certain number of iterations of the GWO algorithm, preserve the path points generated by the GWO algorithm. Set this path as the initial path for the UAV. Assign an attractive force to the path point g i , which is the closest to the UAV’s position, to provide a planning direction using the APF method. Calculate the force field experienced by the UAV at point x i − 1 using the IAPF algorithm and plan the position of the next path point x i , as shown in Figure 8.
Step 6: Compare the fitness values of g i and x i . Determine if x i > g i . If x i > g i , retain x i as the final path point for the UAV. If x i < g i , select four points g i , g i + 1 , g i + 2 , g i + 3 , assign attractive forces to each of them, update the force field, replan the position of point x i , , and compare it again with g i . If x i ′ > g i , retain x i ′ as the final path point for the UAV. If x i ′ < g i , update g i to x i , as shown in Figure 9.
Step 7: Iterate in a loop until x n is reached, indicating that the UAV has reached the target point. If the distance d i s between the UAV and obstacles becomes less than p o during the mission, the subsequent path planning is conducted using the IAPF algorithm until the UAV reaches the destination point.

4. Results

4.1. Experimental Setup

Based on the modeling research and algorithm design mentioned above, to validate the effectiveness of the proposed algorithm in this paper, simulations were conducted using Matlab 2022a. Matlab was employed to simulate the operation of unmanned aerial vehicles (UAVs) in a post-disaster mountain rescue scenario. The experiments were conducted using an Intel(R) Core(TM) i9-13900HX CPU @ 2.2 GHz processor, 32 GB of RAM, and a 1 TB hard drive.
To closely resemble the actual flight process of the UAV in post-disaster rescue scenarios, the specific parameter settings in this study are as follows:
(1)
The path planning task space is 10,000 m × 10,000 m × 10,000 m, and the corresponding parameters for setting up the benchmark terrain obstacle peaks are provided in Table 2.
(2)
The coordinates for the initial takeoff and landing point are set at (0, 0), and the coordinates for the final takeoff and landing point are set at (100, 100, 100). The maximum range of the UAV is set to 5000 m.
(3)
The crowd density at ground impact ρ was set to 0.02 persons/ m 2 .
(4)
The parameter settings for the IGWO algorithm are presented in Table 3.
(5)
The parameter settings for the IAPF algorithm are presented in Table 4.
(6)
The fitness weights b 1 , b 2 , b 3 are set to 0.5, 0.2, and 0.3, respectively.

4.2. Comparison of Different Algorithms

In order to verify the convergence of the IGWO, based on Equations (1)–(3), this paper utilizes the random function to randomly define the center point x i , y i and height z i of obstacle peaks, the expansion range r a n g e of the peaks, as well as the number of peaks n . Ten instances of mountainous obstacle environments were randomly generated, and for each randomly generated task environment, the convergence of the IGWO algorithm, GWO algorithm, particle swarm optimization (PSO) algorithm, genetic algorithms (GA), and wolf pack algorithm (WPA) were simulated and computed. Each environment underwent 20 simulation experiments, and the average results of each algorithm were obtained. The fitness iteration curves for each algorithm are shown in Figure 10, with the number of iterations on the x-axis and the best fitness on the y-axis.
From Figure 10, it can be observed that after 50 iterations, the IGWO algorithm achieves a better optimal fitness value compared to the GWO algorithm, PSO algorithm, GA, and WPA. The optimal fitness obtained by the IGWO algorithm is approximately 2239.36.

4.3. Initial Path Planning Effect of IGWO Algorithm

Due to the fact that the fusion algorithm in non-free space path planning is based on the point closest to the UAV on the path generated by the IGWO algorithm for the next step computation, it is essential for the IGWO algorithm to consistently generate high-quality paths under different starting points. In this paper, three points (1, 1, 1), (12, 28, 30), and (20, 40, 61) in the task environment are selected as the starting planning points of the IGWO algorithm to validate the effectiveness of the IGWO planning from various aspects. As shown in Figure 11, it depicts the effect of the initial path planning by the IGWO-IAPF fusion algorithm.

4.4. Comparison of Path Planning Results

The IGWO-IAPF algorithm was employed for path planning simulation experiments in the task environment designed in Figure 1. Additionally, the GWO algorithm, PSO algorithm, GA algorithm, and WPA algorithm were also included in the task environment for path planning simulation experiments, which were compared with the IGWO-IAPF algorithm. During the computation process, all four algorithms, except for the IGWO-IAPF algorithm, generated path effect maps after 20 iterations. Table 3 summarizes the data results obtained from running the different algorithms 30 times in the task environment scenario. Figure 12 illustrates the path effect generated by the IGWO algorithm in the task environment, while Figure 13 presents a comparison of the path planning results of the five algorithms.

5. Discussion

5.1. IGWO Algorithm Planning Efficiency

From Figure 10, it can be observed that the fitness value of the improved IGWO algorithm is 2239.36, while the fitness value of the traditional GWO algorithm is 2371.93, indicating an improvement of approximately 5% in fitness. The IGWO algorithm tends to converge by the 10th iteration, whereas the traditional GWO algorithm requires around 30 iterations to start converging. The improved IGWO algorithm exhibits an approximately 60% improvement in convergence compared to the traditional GWO algorithm. When compared to other path planning algorithms, the IGWO algorithm shows significant improvements in both optimization efficiency and effectiveness. In comparison, the IGWO algorithm shows significant improvements in both optimization efficiency and effectiveness compared to the GWO algorithm. Moreover, the IGWO algorithm outperforms the PSO algorithm, GA algorithm, and WPA algorithm in terms of optimization effectiveness. Additionally, the convergence efficiency of the IGWO algorithm is remarkable. Compared to the PSO algorithm, although the PSO algorithm tends to converge after the 5th iteration, the fitness value is nearly 100 higher than that of the IGWO algorithm. Furthermore, compared to the other two algorithms, GA and WPA, the IGWO algorithm tends to converge after the 10th iteration, while the GA algorithm takes 25 iterations to converge, and the WPA algorithm starts converging at the 21st iteration. This indicates that the IGWO algorithm possesses stronger global optimization capabilities. Additionally, while the IGWO algorithm achieves fast convergence, it also maintains superior search accuracy in the later stages. This results in higher path efficiency and lower path cost in the planning conducted by the IGWO algorithm.
Considering the fitness iteration curve in Figure 10, it can be observed that the IGWO algorithm tends to converge after 10 iterations. In order to enhance the planning efficiency of the fusion algorithm IGWO-IAPF, this study sets the iteration count for the initial path planning generated by the IGWO algorithm, as mentioned in Section 3.3.4 and Steps 2–10. This approach ensures both the high efficiency of the initial path planning by the IGWO algorithm and fast planning speed during the subsequent global path planning process.

5.2. Analysis of Path Planning Effectiveness

From Figure 11, it can be observed that the IGWO algorithm successfully completes the path planning task from three different starting points. However, in Figure 11a, it should be noted that the paths generated by the IGWO algorithm are too close to the mountainous obstacles from (12, 14, 19) to (12, 21, 25). This proximity increases the risk of the UAV colliding with the mountains, potentially leading to a crash. Additionally, the path planned at (15, 23, 40) exhibits considerable twists and turns, indicating lower path smoothness. Therefore, the use of the APF algorithm is necessary for secondary path planning to address these challenges in proximity to obstacles and path smoothness.
According to Table 5, it can be observed that the IGWO-IAPF algorithm consistently achieves superior optimal fitness values and average optimal fitness values compared to the GWO, PSO, GA, and WPA algorithms. This indicates that the IGWO-IAPF algorithm consistently finds the optimal path compared to other algorithms. The standard deviation of the IGWO algorithm is significantly lower than that of other algorithms, indicating lower curve dispersion and improved stability in path optimization.
The planning time of the IGWO-IAPF algorithm is relatively good compared to several other algorithms, being only around 6% slower than the PSO algorithm. However, due to the adoption of a quadratic programming mode during the planning process, the IGWO-IAPF algorithm has a higher computational complexity compared to the PSO algorithm. Nevertheless, the IGWO-IAPF algorithm demonstrates a relatively better planning performance. Therefore, this study believes that the improved IGWO-IAPF algorithm is capable of rapidly conducting effective path planning and improving planning efficiency.
From Figure 12, it can be observed that the IGWO-IAPF algorithm achieves favorable results in UAV path planning. The legend and textual labels in the figure provide corresponding parameter descriptions. The algorithmic steps of the IGWO-IAPF algorithm proposed in this paper are shown in the box in the figure. The fusion algorithm consists of three stages during the path planning process: the first and third stages involve fast path planning using the IAPF algorithm, while the second stage represents the path planning phase of the IGWO-IAPF algorithm. In the first and third stages, when the distance between the UAV and obstacles is greater than p 0 , the algorithm directly utilizes the IAPF algorithm for fast planning. However, when the distance between the UAV and obstacles is less than p 0 , the fusion algorithm is employed for planning. From the figure, it can be observed that although the planned flight path comes close to the peaks of obstacles, it consistently maintains a safe distance.
From Figure 13, it can be observed that both the IGWO-IAPF algorithm and the other four algorithms successfully plan paths that avoid obstacle areas, allowing the UAV to reach the target point smoothly. However, the paths planned by the traditional GWO algorithm are too far from the obstacle at (80, 35, 62), resulting in significantly reduced flight efficiency. On the other hand, the WPA algorithm plans a path that is too close to the obstacle at (80, 35, 62), which increases the risk of collision during actual operation. Additionally, the paths planned by the four comparative algorithms exhibit significant turning angles, which have a considerable impact on the flight range and path smoothness. In contrast, the path planned by the IGWO-IAPF algorithm maintains a straight flight after the waypoint at (41, 28, 65), resulting in better obstacle avoidance and higher path smoothness compared to the other four algorithms. Furthermore, the path length generated by the IGWO-IAPF algorithm is relatively shorter with less redundancy, resulting in better path quality.

6. Conclusions

In this study, the problem of UAV path planning in mountainous post-disaster rescue scenarios is addressed by considering surrounding obstacles, weather conditions, and real-time sensor-perceived terrain features. To tackle this problem, this paper makes the following three contributions:
(1)
A comparative analysis of existing commonly used path planning algorithms was conducted, and the GWO algorithm and the APF algorithm were selected as the foundational algorithms for UAV path planning in post-disaster mountain rescue scenarios. A simulation environment was constructed using elevation maps, and a third-order B-spline curve was employed to achieve smooth trajectory generation for the UAV paths.
(2)
The traditional GWO algorithm was improved by employing a nonlinear adjustment strategy for control parameters to balance the global and local search capabilities. Additionally, the update strategy for individual positions in the GWO algorithm was optimized to enhance the coordination of search capabilities and reduce the likelihood of falling into local optima. The traditional APF algorithm was also enhanced by introducing attractive forces along the planned flight paths to address the issue of becoming trapped in local optima when the attractive and repulsive forces are equal. These improvements overcome the limitations of the APF algorithm in three-dimensional path planning tasks.
(3)
In this paper, a fusion algorithm that combines the IGWO algorithm with the IAPF method is proposed. The initial paths generated by the IGWO algorithm provide the basis for the IAPF algorithm to incorporate both the attractive forces along the flight paths and the planning directions.
The simulation results demonstrate that compared to the traditional GWO algorithm, the IGWO-APF algorithm achieves faster convergence to optimal paths. The planned paths effectively avoid obstacles, resulting in lower path planning costs and higher path quality. Compared to the PSO, GA, and WPA algorithms, the fusion algorithm also exhibits higher planning efficiency and superior planning effectiveness. The effectiveness and superiority of the IGWO-APF algorithm in solving the three-dimensional UAV path planning problem in post-disaster rescue missions are confirmed by the experiments.
However, this study still has some limitations in the simulation experiments. For instance, it did not consider the real-time obstacle avoidance problem posed by dynamic obstacles such as birds in mountainous environments. Further improvements are necessary to make the UAV path planning more practical and realistic. Additionally, this study only conducted simulation-based experiments without deploying the algorithms on actual UAVs for real-flight experiments. Therefore, in future work, the algorithms should be deployed and tested in real-world scenarios to further validate their performance.

Author Contributions

Conceptualization, D.H. and Q.Y.; methodology, D.H.; software, D.H. and Q.Y.; validation, Q.Y., H.J. and Y.C.; formal analysis, H.J. and X.Z.; investigation, D.H. and Q.Y.; resources, Q.Y. and D.H.; data curation, H.J., Y.C. and X.Z.; writing—original draft preparation, D.H. and Q.Y.; writing—review and editing, D.H., Q.Y. and L.W.; visualization, D.H.; supervision, Q.Y., H.J., Y.C. and L.W.; project administration, D.H. and Q.Y.; funding acquisition, H.J. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the Research Center for Scientific Development of Higher Education Institutions, Ministry of Education, grant number 2022 IT 177; Independent research project of the Key Laboratory of Flight Techniques and Flight Safety, grant number FZ2021ZZ04.

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. The environmental model.
Figure 1. The environmental model.
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Figure 2. Effect of cubic B-spline curve.
Figure 2. Effect of cubic B-spline curve.
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Figure 3. APF planning rendering.
Figure 3. APF planning rendering.
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Figure 4. Comparison chart of the impact of obstacles on the original algorithm parameters.
Figure 4. Comparison chart of the impact of obstacles on the original algorithm parameters.
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Figure 5. Diagram of the impact of obstacles on the attraction of airlines.
Figure 5. Diagram of the impact of obstacles on the attraction of airlines.
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Figure 6. Diagram of the impact of obstacles on the total force field.
Figure 6. Diagram of the impact of obstacles on the total force field.
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Figure 7. Fusion algorithm flowchart.
Figure 7. Fusion algorithm flowchart.
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Figure 8. Schematic diagram of the second planning of IAPF algorithm.
Figure 8. Schematic diagram of the second planning of IAPF algorithm.
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Figure 9. Schematic diagram of the third planning of IAPF algorithm.
Figure 9. Schematic diagram of the third planning of IAPF algorithm.
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Figure 10. Algorithm fitness iteration curve.
Figure 10. Algorithm fitness iteration curve.
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Figure 11. Initial path diagrams.
Figure 11. Initial path diagrams.
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Figure 12. IGWO-IAPF algorithm path planning rendering.
Figure 12. IGWO-IAPF algorithm path planning rendering.
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Figure 13. Comparison diagram of path planning.
Figure 13. Comparison diagram of path planning.
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Table 1. Comparison of path planning algorithms.
Table 1. Comparison of path planning algorithms.
Algorithm NameSolving RequirementsApplicable ScenariosEfficiencySolving Ability
Artificial potential field methodLow2DHighWeak
Velocity vector method [9]Low2DHighGeneral
Genetic algorithm [10]High2D/3DLowStrong
Particle swarm optimization [11]General2D/3DHighGeneral
Grey wolf optimizerGeneral2D/3DGeneralStrong
Table 2. Obstacle parameter settings.
Table 2. Obstacle parameter settings.
NumberCenterHeightRange
1(20, 20)45(10, 10, 5)
2(39, 29)45
3(30, 30)50
4(16, 40)55
5(61, 43)61
6(81, 51)60
7(45, 55)60
8(36, 61)50
9(70, 70)60
10(88, 92)65
Table 3. IGWO parameter settings.
Table 3. IGWO parameter settings.
ParameterMeaningValue
N The number of grey wolf populations20
T Maximum number of iterations10
N _ length The movement step size for each sampling0.5
Table 4. IAPF parameter settings.
Table 4. IAPF parameter settings.
ParameterMeaningValue
K a t t Gravitational gain coefficient0.05
K _ l i n e Gravity gain coefficient of flight route500
K r e p Repulsive gain coefficient100
ρ 0 Distance affected by obstacles60
Table 5. Comparison results of algorithm multiple runs.
Table 5. Comparison results of algorithm multiple runs.
AlgorithmOptimal Fitness ValueAverage Optimal Fitness ValueStandard DeviationAlgorithm Planning Time/s
IGWO-IAPF1583.681607.3923.5253.06
GWO1827.691890.71158.98126.37
PSO1683.291694.52103.4146.58
GA1709.161733.2353.6262.14
WPA1699.041723.4189.2755.42
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MDPI and ACS Style

Han, D.; Yu, Q.; Jiang, H.; Chen, Y.; Zhu, X.; Wang, L. Three-Dimensional Path Planning for Post-Disaster Rescue UAV by Integrating Improved Grey Wolf Optimizer and Artificial Potential Field Method. Appl. Sci. 2024, 14, 4461. https://doi.org/10.3390/app14114461

AMA Style

Han D, Yu Q, Jiang H, Chen Y, Zhu X, Wang L. Three-Dimensional Path Planning for Post-Disaster Rescue UAV by Integrating Improved Grey Wolf Optimizer and Artificial Potential Field Method. Applied Sciences. 2024; 14(11):4461. https://doi.org/10.3390/app14114461

Chicago/Turabian Style

Han, Dan, Qizhou Yu, Hao Jiang, Yaqing Chen, Xinyu Zhu, and Lifang Wang. 2024. "Three-Dimensional Path Planning for Post-Disaster Rescue UAV by Integrating Improved Grey Wolf Optimizer and Artificial Potential Field Method" Applied Sciences 14, no. 11: 4461. https://doi.org/10.3390/app14114461

APA Style

Han, D., Yu, Q., Jiang, H., Chen, Y., Zhu, X., & Wang, L. (2024). Three-Dimensional Path Planning for Post-Disaster Rescue UAV by Integrating Improved Grey Wolf Optimizer and Artificial Potential Field Method. Applied Sciences, 14(11), 4461. https://doi.org/10.3390/app14114461

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