Unmanned Aerial Vehicle Path Planning Based on Sparrow-Enhanced African Vulture Optimization Algorithm

Abstract

Drones can improve the efficiency of point-to-point logistics and distribution and reduce labor costs; however, the complex three-dimensional airspace environment poses significant challenges for flight paths. To address this demand, this paper proposes a hybrid algorithm that integrates the Sparrow Search Algorithm (SSA) with the African Vulture Optimization Algorithm (AVOA). Firstly, the algorithm introduces Sobol sequences at the population initialization stage to optimize the initial population; then, we incorporate SSA’s discoverer and vigilant mechanisms to balance exploration and exploitation and enhance global exploration capabilities; finally, multi-guide differencing and dynamic rotation transformation strategies are introduced in the first exploitation phase to enhance the direction of local exploitation by fusing multiple pieces of information; the second exploitation phase achieved a dynamic balance between elite guidance and population diversity through adaptive weight adjustment and enhanced Lévy flight strategy. In this paper, a three-dimensional model is built under a variety of constraints, and SAVOA (Sparrow-Enhanced African Vulture Optimization Algorithm) is compared with a variety of popular algorithms in simulation experiments. SAVOA achieves the optimal path in all scenarios, verifying the efficiency and superiority of the algorithm in UAV logistics path planning.

1. Introduction

In today’s logistics system, the “last mile” has become a distribution problem. As the volume of parcels continues to increase globally, so do the expectations of consumers, who want their packages delivered faster and at a lower cost. To meet this demand, logistics companies need to optimize their distribution routes and ensure that each link is executed accurately and efficiently [1]. On the one hand, during the recent coronavirus epidemic, in order to reduce the risk of virus transmission, some catering companies tried to use drones for food delivery, and this non-contact delivery mode was welcomed and recognized by the majority of consumers [2]. In addition, in the aftermath of earthquakes, floods, typhoons, and other disasters [3], the logistics system needs to have the ability to make a rapid response to ensure that emergency supplies are accurately delivered to the affected area in the shortest possible time. In this case, it is especially critical to efficiently plan a distribution path that is both safe and shortest. On the other hand, traditional logistics and distribution are still based on truck transportation, which is not only slow but also requires high labor costs. Meanwhile, with the improvement in social and economic level, the number of vehicles has increased dramatically, and modern cities are facing problems such as traffic congestion, air pollution, and noise [4]. Drones are an important tool for complex transportation problems due to their small size and flexibility [5]. Many logistics majors such as Google, DHL, and Amazon have long launched pilot programs in order to exploit the potential of drones in logistics applications [6].

Drones not only need to avoid obstacles such as urban buildings and mountainous terrain during flight, but must also consider factors such as airborne exclusion zones, maximum turning angles, and air resistance [7]. The need to improve the efficiency of drones due to their lower load capacity compared to ground vehicles [8]. The core of drone path planning is to provide a flight path that is as short and safe as possible, while at the same time needing to reduce the cost of flight, such as battery energy consumption and airframe wear and tear [9]. In general, path planning problems for UAVs can be categorized into three types: start-goal problems, target-coverage problems, and hybrid problems. The start-goal problem is concerned with planning a shortest path between a starting point and an end point; the goal-coverage problem aims to find an optimal path that completely covers a specified area; and the hybrid problem is a combination of both [10]. This paper explores the planning of a shortest flight path for drones that satisfies multiple constraints under the condition that the starting point and the target point are known. The path needs to meet the constraints of maximum yaw angle, maximum pitch angle, and maximum flight speed based on obstacle avoidance to achieve the feasibility and safety of the path.

For the UAV path-planning problem, researchers have proposed a variety of algorithms. The classic algorithms include the Artificial Potential Field (APF) method [11], Dijkstra’s algorithm [12], A* algorithm [13], and Random-Number Rapid Search (RRT) algorithm [14]. However, the quality of paths planned by these methods tends to be poor when faced with complex environments. Therefore, some scholars have optimized these algorithms. Hao et al. [15] proposed an improved Artificial Potential Field method, which effectively improves the feasibility and smoothness of the path by introducing risk assessment and virtual subgoal mechanism. Chen et al. [16] used Mixed-Integer Linear Programming (MILP) to improve the A* algorithm, expanding the search range and improving the search success rate of UAVs. Yang et al. [17] combined the environmental potential field and RRT to optimize the sampling and expansion strategy, which guided the tree to expand in the direction away from obstacles and close to the target, improving the efficiency of UAV path planning. He et al. [18] proposed a two-layer optimization of A* and DWA, which optimizes the node-expansion model and the DWA evaluation function, extracts the critical path nodes, and achieves the effective tracking of local paths to global paths. Additionally, meta-heuristic algorithms are getting more and more attention from scholars. Traditional representative algorithms include Genetic Algorithm (GA) [19], Particle Swarm Optimization (PSO) [20], Ant Colony Optimization (ACO) [21], Artificial Bee Colony (ABC) [22], etc. New intelligent optimization algorithms emerging in recent years include the Grey Wolf Optimizer Algorithm (GWO) [23], Whale Optimization Algorithm (WOA) [24], Sparrow Search Algorithm (SSA) [25], etc. Although these algorithms have been widely used in UAV path-planning problems, they still suffer from insufficient robustness and tend to fall into local optimization in complex environments.

The African Vulture Optimization Algorithm (AVOA) was proposed in 2021 and is an intelligent optimization algorithm inspired by the foraging and social behavior of African vultures. The algorithm has a relatively simple structure, strong adaptability and search ability, and can show good results in solving a variety of complex optimization problems [26,27]. Abed et al. [28] applied AVOA to robot path planning in static and dynamic environments, and the simulation results showed that AVOA outperforms Adaptive Particle Swarm Optimization (APSO) and Hybrid Fuzzy Wind-Driven Optimization (WDO) algorithms in both environments. Alsirhani et al. [29] utilized AVOA for feature selection in a dataset in a network security intrusion-detection problem, demonstrating the potential of this algorithm for application in smart-grid network security. In the literature [30], Gürses et al. performed cost minimization and created a parametric design of a shell-and-tube heat exchanger (SHTHE) using AVOA and found the optimal solution with all constraints. Kumar et al. [31] proposed an energy-efficient clustering scheme (AVOACS) based on AVOA, which achieves a dynamic balance of node energy consumption by simulating the foraging behavior of vultures, thus enhancing the stability and lifetime of the network. Although AVOA has been used in several fields, the algorithm still has some shortcomings. First, in the early exploration stage, although AVOA added certain exploitation mechanisms to accelerate the convergence speed, this approach weakened the algorithm’s global search ability, which easily led it to fall into a local optimum at a later stage. Secondly, the choice of exploration and exploitation is dependent on the level of starvation and lacks a clear regulatory mechanism, which affects the balance between exploration and exploitation [32].

To solve the above problems, this paper proposes a Sparrow-Enhanced African Vulture Optimization Algorithm (SAVOA) and applies it to UAV path planning in both mountainous and urban environments. First, in the initialization stage of the population, the Sobol sequence is used instead of the traditional random approach to ensure the initial population is more evenly distributed in space, thus effectively improving the quality and diversity of the initial individuals. Second, the discoverer mechanism of SSA is substituted for the exploration phase of AVOA, thus realizing the balance between exploration and exploitation, while the vigilante mechanism is introduced to enhance the ability to jump out of the local optimum. Finally, in the first exploitation stage of the algorithm, multi-guided difference and dynamic rotation transformation strategies are introduced to increase the perturbation of random individuals based on the optimal and suboptimal individuals. Cosine and sinusoidal offsets are utilized to generate jump paths with spiral perturbations to enhance the diversity of local development. In the second exploitation phase, a dynamic balance between elite guidance and population diversity is achieved by combining adaptive weighting coefficients with enhanced Lévy flight, thereby ensuring both individual stability and effective information transfer within the population. SAVOA is applied to the three-dimensional path-planning problem of UAVs, simulation experiments are conducted in two simulated environments, a mountainous area and urban area, and compared with various algorithms including the standard AVOA, and the results show that SAVOA exhibits better optimization performance.

The rest of the paper is structured as follows: Section 2 presents the problem description of UAV path planning, including curve fitting, environment modeling, fitness function, and constraints; Section 3 describes the original AVOA; Section 4 illustrates the improvements to the AVOA; Section 5 shows the results of the experiments in different environments and a specific analysis of these results; and Section 6 briefly describes the improvements in future research directions.

2. Research on UAV Path-Planning Problem

2.1. Problem Assumptions and Description

Within a certain region, it is assumed that the logistics demand point is known, and a rechargeable drone needs to be used to accomplish the parcel delivery task from the distribution point. Due to the performance limitations of drones and the fact that their flight paths are fixed after takeoff and do not accept midway assignments, path planning must be completed prior to flight to ensure a safe and efficient delivery process.

2.2. Polynomial Interpolation Fitting Curves

During the planning process, a relatively smooth flight trajectory can be generated by adding control points including the start point $W\left(0\right)=(x_0,y_0,z_0)$ and the end point $W(m+1) = (x_{m+1}, y_{m+1}, z_{m+1})$, with the intermediate point $W\left(q\right) = (x_q,y_q,z_q)(q = \mathrm{1,2}, \dots , m)$, and the total number of control points is $N=m+2$.

Assign the parameters by equally spacing the control points on $\tau \in \left[0, 1\right]$:

$${\tau }_i={\displaystyle \frac{i}{N-1}},\hspace{1em}i=0, 1, 2,\dots ,N-1$$

(1)

Obtain the corresponding control points:

$$W_{\left({\tau }_i\right)}=W\left(i\right),\hspace{1em}\forall i\in \{\mathrm{0,1},2,\dots ,N-1\}$$

(2)

Then $N-1$ interpolating polynomials are constructed independently for each coordinate component:

$$\left[\begin{array}{c}W(q{{)}_{\left(\tau \right)}}_x\\ W(q{)}_{\left(\tau \right)y}\\ W(q{{)}_{\left(\tau \right)}}_z\end{array}\right]={\sum }_{q=0}^{N-1}\left[\begin{array}{c}{\alpha }_q\\ {\beta }_q\\ {\gamma }_q\end{array}\right]{\tau }^q$$

(3)

where the coefficients ${\alpha }_q,{\beta }_q,{\gamma }_q$ can be determined by solving the Vandermonde system of equations:

$$\left[\begin{array}{ccccc}1& {\tau }_0& {\tau }_0^2& \dots & {\tau }_0^{N-1}\\ 1& {\tau }_1& {\tau }_1^2& \dots & {\tau }_1^{N-1}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& {\tau }_{N-1}& {\tau }_{N-1}^2& \dots & {\tau }_{N-1}^{N-1}\end{array}\right]\left[\begin{array}{c}{\alpha }_0\\ {\alpha }_1\\ ⋮\\ {\alpha }_{N-1}\end{array}\right]=\left[\begin{array}{c}{x}_0\\ x_1\\ ⋮\\ x_{N-1}\end{array}\right]$$

(4)

The coefficients of the $y$ and $z$ components can be derived identically. Then, $k$ points are sampled uniformly in the range $\tau \in [0, 1]$:

$${\tilde{\tau }}_j={\displaystyle \frac{j}{k-1}},\hspace{1em}j=\mathrm{0,1},\dots ,k-1$$

(5)

$k$ points to generate continuous paths:

$$W_{\left({\tilde{\tau }}_j\right)}=\left({\sum }_{\mu =0}^{N-1}{\alpha }_{\mu }{\tilde{\tau }}_j^{\mu },\hspace{0.33em}{\sum }_{\mu =0}^{N-1}{\beta }_{\mu }{\tilde{\tau }}_j^{\mu },\hspace{0.33em}{\sum }_{\mu =0}^{N-1}{\gamma }_{\mu }{\tilde{\tau }}_j^{\mu }\right)$$

(6)

The set of path nodes $\tilde{W}$ for the final curve is:

$$\tilde{W}=\left\{W_{\left({\tilde{\tau }}_j\right)}\mid j=0, 1,\dots ,k-1\right\}$$

(7)

In this paper, the number of intermediate points $m=3$, control points $N=5$, and the number of polynomials is 4. Retaining the display of 5 control points, a smooth curve using polynomial interpolation can be obtained, as shown in Figure 1.

2.3. Environmental Modeling

2.3.1. Mountain Environment Model

Mountainous environments have a lot of undulating terrain and are more complex, which puts higher demands on the UAV’s path planning. This paper uses the method of Gaussian peak function superposition to construct three-dimensional mountain terrain simulation maps [33]; the mathematical model is:

$$z_M(x,y)={\sum }_{i=1}^M{H}_i\mathit{exp}\left[-{\left({\displaystyle \frac{x-x_i}{x_{si}}}\right)}^2-{\left({\displaystyle \frac{y-y_i}{y_{si}}}\right)}^2\right]$$

(8)

$$center=(x_i,y_i),height=z_{M\left(i\right)},range=(x_{si},{y_s}_i)$$

(9)

where $M$ is the total number of peaks; $(x_i,y_i)$ are the center coordinates of peaks in the $x-y$ plane; $z_{M\left(i\right)}$ indicate the height of the peak; $H_i$ are terrain parameters, used to control the height of the peaks; $x_{si}$ and $y_{si}$ denote the attenuation coefficients in the x-axis and y-axis directions, respectively, which are used to adjust the range of the peaks in different directions.

2.3.2. Urban Environment Model

In urban terrain, the shape of buildings is usually highly irregular. To simplify the modeling process and highlight the obstacle avoidance feature in path planning, this paper adopts a rectangular body to abstractly simulate buildings in the city:

$$b_i=\left\{(x,y,z)\mid b_{x\left(i\right)}\le x\le b_{x\left(i\right)}+w_i,b_{y\left(i\right)}\le y\le b_{y\left(i\right)}+d_i,0\le z\le b_{h\left(i\right)}\right\}$$

(10)

The set of buildings is:

$$b={\bigcup }_{i=1}^{n_c}{b}_i$$

(11)

In the expression, $b_i$ corresponds to the area range of the $i$ rectangular building; $b_{x\left(i\right)}$ and $b_{y\left(i\right)}$ represent the coordinates of the lower left corner $x$ and $y$ of the $i$ building, respectively; $w_i$, $d_i$, and $b_{h\left(i\right)}$ denote the extension (length, width, and height) of the $i$ building’s lower-left coordinates in the $x$, $y$, and $z$ directions, respectively; $n_c$ is the number of buildings and $b$ is the set of all buildings.

2.3.3. Exclusion Zone Model

In actual flights, drones are often subject to restrictions from special areas such as the military and government. Here, these restrictions are set as spherical no-fly zones. The sphere can be expressed by the following equation:

$$s_i=\left\{\right(x,y,z\left)\right|(x-c_{x\left(i\right)}{)}^2+(y-c_{y\left(i\right)}{)}^2+(z-c_{z\left(i\right)}{)}^2\le (r_{\left(i\right)}{)}^2\}$$

(12)

The set of spheres is:

$$s={\bigcup }_{i=1}^{n_s}{s}_i$$

(13)

In the above two formulas,$s_i$ is the range of the region of the $i$ sphere, $(c_{x\left(i\right)},c_{y\left(i\right)},c_{z\left(i\right)})$ are the center coordinates (sphere centers) of the $i$ sphere, and $r_i$ represents the radius of the $i$ sphere; $n_s$ denotes the total number of spheres, and $s$ denotes the set formed by all the spheres.

2.3.4. Wind-Field Model

Wind speed and direction can have an impact on UAV energy consumption, flight stability, and path accuracy. Therefore, the effects of wind need to be taken into account when performing UAV path planning, which may change speed, altitude, or direction of the UAV [34]. This paper adds a wind-field model to the three-dimensional map, and the wind vector $\overrightarrow{w}=(w_x,w_y,w_z)$ is [35,36]:

$$\left\{\begin{array}{l}{w}_x=v_w\cdot \mathit{sin}\theta \\ w_y=v_w\cdot \mathit{cos}\theta \\ w_z=v_w\cdot e^{-\left({\displaystyle \frac{(x-x_1{)}^2+(y-y_1{)}^2}{(r_1{)}^2}}\right)}\left[1-\left({\displaystyle \frac{(x-x_1{)}^2+(y-y_1{)}^2}{(r_1{)}^2}}\right)\right]\end{array}\right.$$

(14)

The relative airspeed and ground speed of the drone were [37]:

$${\overrightarrow{v}}_a=\left(\begin{array}{c}{v}_a\mathit{cos}\psi \mathit{cos}\gamma \\ v_a\mathit{sin}\psi \mathit{cos}\gamma \\ -v_a\mathit{sin}\gamma \end{array}\right)$$

(15)

$$\overrightarrow{r}={\overrightarrow{v}}_a+\overrightarrow{w}$$

(16)

where $w_x,w_y,w_z$ denote the components of the wind field in the $x,y,z$ directions; $v_w$ and $v_a$ are the wind speed at the center of the updraft ${(x}_1,y_1)$ and the initial speed of the UAV, respectively; $r_1$ is the radius of influence of the airflow center; $\theta$ is the angle between the point $(x,y)$ and the line connecting the center of the airflow ${(x}_1,y_1)$ relative to the north direction; ${\overrightarrow{v}}_a$ and $\overrightarrow{r}$ are the relative airspeed and ground-speed vectors of the UAV, respectively; $\psi$ and $\gamma$ are the yaw and pitch angles. Figure 2 shows the mountain model and the city model with wind field, respectively. The number of peaks or buildings in this paper is chosen to be 20 and 25 (i.e., the number of peaks is taken to be 20 and 25, and the number of buildings in the city is taken to be 20 and 25, for a total of four terrains), and this number is determined in the initialization phase. The heights of the peaks or buildings are randomly generated, and the terrain parameters are recorded after generation to repeat the experiment. Only mountainous and urban terrain with a peak or building count of 25 is shown in Figure 2.

2.4. Fitness Function

In UAV path planning, the fitness function is used to comprehensively measure the degree of merit of a flight path. In this paper, the UAV’s flight distance, flight-altitude standard deviation, flight time, and stability (including yaw angle and pitch angle) are used as comprehensive evaluation indexes to construct the fitness function, and the lower the fitness indicates the higher quality of the flight path.

2.4.1. Flight-Distance Cost

The battery consumption of the drone is closely related to the flight distance; the shorter the path, the less power is used, and the more effectively the drone can complete logistics and distribution tasks per unit of power, which to a certain extent can reduce operating costs. In this paper, the flight path of the drone consists of $k$ nodes. By calculating the Euclidean distance between two adjacent nodes and accumulating them, the flight-distance cost function $f_1$ is obtained:

$$f_1={\sum }_{j=1}^{k-1}\sqrt{(x_{j+1}-x_j{)}^2+(y_{j+1}-y_j{)}^2+(z_{j+1}-z_j{)}^2}$$

(17)

In this function, $k$ represents the total number of path nodes including the start and end points; $(x_j,y_j,z_j)$ and $(x_{j+1},y_{j+1},z_{j+1})$ represent the coordinates of the $j$ and $j+1$ node, respectively.

2.4.2. High Standard Deviation Cost

Altitude fluctuations of UAVs can have an impact on flight stability and energy consumption, and if there are large altitude ups and downs in the path, it will not only increase the regulation burden of the flight-control system but also may cause energy waste and reduce flight efficiency. This paper introduces the altitude standard deviation as an evaluation index; the smaller the altitude standard deviation is, the smoother the altitude change during flight is represented, and its model $f_2$ is:

$$\left\{\begin{array}{l}\stackrel{-}{z}={\displaystyle \frac{1}{k}}{\sum }_{i=1}^k{z}_i\\ f_2=\sqrt{{\displaystyle \frac{1}{k}}{\sum }_{i=1}^k(z_i-\stackrel{-}{z}{)}^2}\end{array}\right.$$

(18)

In the equation, $k$ is the number of height values (total number of path nodes); $z_i$ is the height value of the $i$ node; $\stackrel{-}{z}$ is the average height of all the nodes; and $f_2$ is the standard deviation of the final obtained path height.

2.4.3. Flight-Time Cost

Flight time is an important metric in drone logistics and distribution missions. The variation in wind speed and direction affects the actual flight speed of the UAV, and the flight speed vector has been given in Equation (16), which is $\overrightarrow{r}$. The time cost function $f_3$ is:

$$\left\{\begin{array}{l}{v}_i=\Vert \overrightarrow{r}\Vert =\sqrt{r_{x,i}^2+r_{y,i}^2+r_{z,i}^2}\\ \stackrel{-}{v}={\displaystyle \frac{1}{k}}{\sum }_{i=1}^k{v}_i\\ f_3={\displaystyle \frac{L}{\stackrel{-}{v}}}={\displaystyle \frac{f_1}{mean\left(v_i\right)}}\end{array}\right.$$

(19)

where $k$ denotes the total number of path nodes; $r_{x,i},r_{y,i},r_{z,i}$ are the components of $\overrightarrow{r}$ in the $x,y,z$ directions, respectively; $v_i$ is the modulus of the $i$ node’s ground speed in Equation (16), which is the magnitude of the UAV’s speed under the influence of wind; $\stackrel{-}{v}$ is the average flight speed over the full path; $f_3$ is the time required to fly the path.

2.4.4. Stability Cost

The stability of the UAV’s flight path is very important, as large fluctuations during flight can easily cause an unstable attitude and increase the wear and tear of the equipment, thus shortening the life of the UAV. The smoothness of a path largely reflects its stability, which can be measured by yaw and pitch angles; the smaller the angle, the more stable the path. Figure 3 shows the flight path from the path node $W_{j-1}(x_{j-1},y_{j-1},z_{j-1})$ to the path node $W_{j+1}(x_{j+1},y_{j+1},z_{j+1})$ in a 3D coordinate system, and $W_{j-1}^{\prime },W_j^{\prime },W_{j+1}^{\prime }$ are the projection points of $W_{j-1},W_j,W_{j+1}$ in the $X-Y$ plane, respectively. The angle ${\psi }_j$ formed by the intersection of the extended line of line segment $W_{j-1}^{\prime }{W}_j^{\prime }$ and line segment $W_j^{\prime }{W}_{j+1}^{\prime }$ is the yaw angle, which is calculated as follows:

$${\psi }_j=\mathit{arctan}\left({\displaystyle \frac{‖\overrightarrow{W_{j-1}^{\prime }{W}_j^{\prime }}\times \overrightarrow{W_j^{\prime }{W}_{j+1}^{\prime }}‖}{\overrightarrow{W_{j-1}^{\prime }{W}_j^{\prime }}\cdot \overrightarrow{W_j^{\prime }{W}_{j+1}^{\prime }}}}\right)$$

(20)

The path point $W_{j+1}^{″}$ is the projection of onto the plane where the line segment $W_{j-1}{W}_j$ is located. The angle ${\gamma }_j$ formed by line segment $W_{j+1}$ and line segment $W_j{W}_{j+1}^{″}$ is the pitch angle, which is calculated as follows:

$${\gamma }_j=\mathit{arctan}\left({\displaystyle \frac{z_j-z_{j-1}}{‖\overrightarrow{W_{j-1}^{\prime }{W}_j^{\prime }}‖}}\right)$$

(21)

The smoothness of the paths in this paper is measured by computing the second-order difference in the angular change (the acceleration of the angular change):

$$\left\{\begin{array}{c}{f}_4^{\prime }={\sum }_{j=2}^{k-1}\left|{\psi }_{j+1}-2{\psi }_j+{\psi }_{j-1}\right|\\ f_4^{\prime \prime }={\sum }_{j=2}^{k-1}\left|{\gamma }_{j+1}-2{\gamma }_j+{\gamma }_{j-1}\right|\end{array}\right.$$

(22)

$k$ is the total number of path nodes, $f_4^{\prime }$ and $f_4^{″}$ and are the smoothing of yaw and pitch angles, respectively. The total smoothness is:

$$f_4=f_4^{\prime }+f_4^{″}$$

(23)

To summarize, the expression for the fitness function $F_t$ for each path is:

$$F_t={\sum }_{i=1}^4{l}_i{f}_i$$

(24)

$l$ is the weight coefficient and satisfies $l={\sum }_{i=1}^4{l}_i=1$; in this paper, we take $l_1=0.6, l_2=0.2, l_3=0.1, l_4=0.1$.

2.5. Restrictive Condition

UAVs are subject to multiple constraints in actual flight, and two types of constraints are considered in this paper when performing path planning. On the one hand, there is the impact of environmental factors on flight trajectories, including mountainous terrain, urban buildings and flight exclusion zones, which UAVs need to avoid when flying. On the other hand, the limitations of the UAV’s own performance must be considered, including maximum flight speed, maximum yaw angle, and pitch angle.

2.5.1. Constraints of Environmental Factors

To avoid collisions when flying a UAV in a complex environment, the flight altitude should be higher than the height of the obstacles, which is modeled as:

$$h_{\mathrm{drone}}(x,y)>h_{\mathrm{terrain}}(x,y)+sd$$

(25)

where $h_{\mathrm{drone}}(x,y)$ denotes the flight height of the UAV at position $(x,y)$, $h_{\mathrm{terrain}}(x,y)$ is the height of the obstacle at the corresponding position, and $sd$ is the set minimum safety gap for ensuring a certain safety distance.

2.5.2. Self-Performance Constraints

The UAV itself has certain limitations in its structural and dynamic performance, and the maximum velocity $v_{aMax}$, maximum yaw angle ${\psi }_{Max}$, and maximum pitch angle ${\gamma }_{Max}$ that it can withstand also need to be taken into account and need to satisfy the following inequalities:

$$\left\{\begin{array}{c}{v}_{a\left(i\right)}\le v_{aMax}\\ {\psi }_{\left(i\right)}\le {\psi }_{Max}\\ {\gamma }_{\left(i\right)}\le {\gamma }_{Max}\end{array}\right.$$

(26)

$v_{a\left(i\right),}{\psi }_{\left(i\right)},{\gamma }_{\left(i\right)}$ are the velocity, yaw angle, and pitch-angle magnitude of the $i$ path node, respectively. The fitness function $F_t$ is penalized when a collision of the path with an obstacle is detected or when the flight parameters exceed the UAV limits:

$$F_t=F_t\times 1000$$

(27)

3. Standard African Vulture Optimization Algorithm

The AVOA balances the algorithm’s global search performance and local exploitation performance through the feeding behavior of vultures at different levels of hunger, which can be divided into four steps.

Step one: Divide each generation of vultures into three groups. After the population initialization is complete, enter the iterative loop. In the loop, the fitness of all individuals in each generation is evaluated. The individual with the best fitness value is divided into the first group, the individual with the second-best fitness value is divided into the second group, and the remaining individuals become the third group. The individuals in the third group will move toward the positions of these two groups of vultures according to Formula (28) in order to complete the positional updating.

$$R(i)=\left\{\begin{array}{cc}BestVultur{e}_1& p_i=L_1\\ BestVultur{e}_2& p_i=L_2\end{array}\right.$$

(28)

$$p_i={\displaystyle \frac{F_i}{{\sum }_{i=1}^p{F}_i}}$$

(29)

In Equation (28), the probability of an individual moving closer to the optimal vulture in each group is regulated by two control parameters, $L_1$ and $L_2$, both of which take values in the range $[0, 1]$ and satisfy $L_1+L_2=1$. $L_1$ and $L_2$ were determined by a roulette-wheel mechanism, and in Equation (29), $F_i$ represents the fitness of the first and second groups of vultures; $p$ denotes the total number of vultures in these two groups.

Step two: Starvation rate of vultures. When vultures have sufficient energy, they can fly longer distances in search of food and enter a global exploration phase; when they are low on energy, they reduce their range and rely on stronger individuals in the vicinity, thus entering an exploitation phase. This behavior can be mathematically modeled using the following equation:

$$T=h\times \left({sin}^w\left({\displaystyle \frac{\pi }{2}}\times {\displaystyle \frac{iter}{{iter}_{Max}}}\right)+\mathit{cos}\left({\displaystyle \frac{\pi }{2}}\times {\displaystyle \frac{iter}{{iter}_{Max}}}\right)-1\right)$$

(30)

$$F=(2\times ran{d}_1+1)\times z\times \left(1-{\displaystyle \frac{iter}{ite{r}_{Max}}}\right)+T$$

(31)

where $T$ is a tuning parameter used to prevent falling into a local optimal solution; the range of the random variable $h$ is −2 to 2; the parameter $w$ is an initially set fixed constant used to regulate the switching of the algorithm between the exploration and development phases; $iter$ and $ite{r}_{Max}$ are the current and maximum number of iterations; $F$ represents the vulture’s starvation rate; $ran{d}_1$ is a random number between 0 and 1; and $z$ is a random number between −1 and 1 that is dynamically updated with the number of iterations. When $\left|F\right| \ge 1$, vultures tend to look for food in a wider search space, at which point the AVOA enters the exploration phase; when $\left|F\right|<1$, the vultures are unable to fly long distances due to physical limitations, at which time they prefer to search locally in the vicinity and the algorithm enters the exploitation phase.

Step three: Exploration phase. Vultures usually need to spend long periods of time scrutinizing a wide area and flying long distances in search of food. In AVOA, this behavior can be simulated by two exploration strategies that enable a global search of different areas. The choice of strategy can be adjusted by a control parameter $P_1$, which is fixed and ranges from 0 to 1. In this paper, it is set to 0.6. The equations are as follows:

$$P(i+1)=\left\{\begin{array}{cc}Equation \left(33\right)& P_1\ge ran{d}_{P_1}\\ Equation \left(35\right)& P_1<ran{d}_{P_1}\end{array}\right.$$

(32)

$$P(i+1)=R(i)-D(i)\times F$$

(33)

$$D(i)=|X\times R(i)-P(i)|$$

(34)

$$P(i+1)=R(i)-F+ran{d}_2\times \left((ub-lb)\times ran{d}_3+lb\right)$$

(35)

where $P(i+1)$ represents the vulture position in the next iteration; Equations (33) and (34) describe the first exploration strategy, where $X$ is used to enhance the randomness of an individual’s movement and changes dynamically in each iteration in the range $[0, 2]$, which can be obtained by the equation $X = 2 \times rand$; both $rand$ and $ran{d}_{P_1}$ are random numbers between 0 and 1. Equation (35) shows the location-update formula for the second exploration strategy, $ub$ and $lb$ represent the upper and lower boundaries of the map, respectively; $ran{d}_2$ and $ran{d}_3$ are both random numbers between 0 and 1.

Step four: Exploitation phase. The exploitation phase of AVOA also consists of two sub-phases, each corresponding to a different search strategy. The AVOA enters the first exploitation phase when $0.5\le \left|F\right|<1$ is satisfied. In this stage, the behavior of vultures is modeled by two strategies: food competition and rotational flight, and the choice of strategy is controlled by the initial parameter $P_2$, which ranges from $[0, 1]$ and is set to 0.4 in this paper; $ran{d}_{P_2}$ is a random number between $[0, 1]$; the position-update equations are:

$$P(i+1)=\left\{\begin{array}{cc}\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\left(37\right)& P_2\ge ran{d}_{P_2}\\ \mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\left(41\right)& P_2<ran{d}_{P_2}\end{array}\right.$$

(36)

$$P(i+1)=D(i)\times \left(F+ran{d}_4\right)-d(t)$$

(37)

$$d\left(t\right)=R\left(i\right)-P\left(i\right)$$

(38)

$$S_1=R(i)\times \left({\displaystyle \frac{ran{d}_5\times P_i}{2\pi }}\right)\times \mathit{cos}(P(i))$$

(39)

$$S_2=R(i)\times \left({\displaystyle \frac{ran{d}_6\times P_i}{2\pi }}\right)\times \mathit{sin}(P(i))$$

(40)

$$P(i+1)=R(i)-\left(S_1+S_2\right)$$

(41)

When $\left|F\right|<0.5$ is satisfied, the AVOA enters the second exploitation phase, which simulates the behavior of vultures gathering around food or fighting fiercely for it. The initial phase of the algorithm generates the decision parameter $P_3$, and when $ran{d}_{P_3}\le P_3$, the algorithm goes to the vulture rallying food-source strategy; otherwise, it uses the intense siege-fighting strategy, where $ran{d}_{P_3}\in (0, 1)$.

$$P(i+1)=\left\{\begin{array}{cc}\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\left(45\right)& P_3\ge ran{d}_{P_3}\\ \mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\left(46\right)& P_3<ran{d}_{P_3}\end{array}\right.$$

(42)

$$A_1=BestVultur{e}_1(i)-{\displaystyle \frac{BestVultur{e}_1(i)\times P(i)}{BestVultur{e}_1(i)-P(i{)}^2}}\times F$$

(43)

$$A_2=BestVultur{e}_2(i)-{\displaystyle \frac{BestVultur{e}_2(i)\times P(i)}{BestVultur{e}_2(i)-P(i{)}^2}}\times F$$

(44)

$$P(i+1)={\displaystyle \frac{A_1+A_2}{2}}$$

(45)

$$P(i+1)=R(i)-\left|d\right(t\left)\right|\times F\times Levy\left(d\right)$$

(46)

$$Levy(d)=0.01\times {\displaystyle \frac{u}{|v{|}^{\frac{1}{{\beta }_u}}}}$$

(47)

$$u={\left({\displaystyle \frac{\Gamma (1+{\beta }_u)\times \mathit{sin}\left(\frac{\pi {\beta }_u}{2}\right)}{\Gamma \left(\frac{1+{\beta }_u}{2}\right)\times {\beta }_u\times 2^{\left(\frac{{\beta }_u-1}{2}\right)}}}\right)}^{{\displaystyle \frac{1}{{\beta }_u}}}$$

(48)

$v$ is a random number between $[0, 1]$; ${\beta }_u = 1.5$; control parameter $P_3=0.6$; $Levy\left(d\right)$ denotes Lévy flight.

4. Description of the SAVOA

4.1. Population Initialization Strategy Based on Sobol Sequence

The initial population with good distribution is the basis of the metaheuristic algorithms [38]. The standard African vulture algorithm uses a random approach to population initialization, which can lead to an uneven distribution of generated individuals and affect the performance of the algorithm. This paper introduces the Sobol sequence as a mechanism for population initialization, which is a low-discrepancy random sequence that possesses high uniform coverage in multidimensional space.

Define the range of the search space as $[l{b}_j,u{b}_j]$ and the initial position of the population can be defined as:

$$X_{i,j}=l{b}_j+S_{i,j}\times (u{b}_j-l{b}_j),\hspace{1em}i=1, 2,\dots ,pop;\hspace{1em}j=1, 2,\dots ,d$$

(49)

where $l{b}_j$ and $u{b}_j$ represent the lower and upper boundaries of the $j$ dimension, respectively; $S_{i,j}$ is the value of the $i$ sample in the $j$ dimension generated by the Sobol sequence, which ranges from $[0,1]$; $pop$ and $d$ are population size and spatial dimension, respectively. Figure 4 shows the distribution comparison between random initialization and Sobol sequence initialization when generating a population of 500 individuals in two-dimensional space. It can be seen that the randomized approach is prone to uneven distribution and aggregation, whereas the individuals generated by the Sobol sequence cover the entire search space more evenly.

4.2. SSA Convergence Strategy

The standard AVOA enters either the exploration or exploitation phase determined by the starvation rate $F$. The calculation of the starvation rate in turn relies on parameters such as $w$, $z$, and $h$, which are mainly fixed or randomly generated, resulting in a strong randomness in the switch between exploration and exploitation. In order to enhance the balance between exploration and exploitation, this paper introduces the SSA’s discoverer mechanism.

Discoverers are primarily responsible for exploring areas of greater food abundance and have a larger foraging search area. To improve the global search capability of the algorithm in the early stage, the paper removes the division mechanism of AVOA that regulates the exploration and exploitation based on the starvation rate and replaces the exploration stage of AVOA with the discoverer mechanism in SSA. Its position-update equation is:

$$X_{i,j}^{t+1}=\left\{\begin{array}{cc}{X}_{i,j}^t\cdot \mathit{exp}\left({\displaystyle \frac{-i}{\alpha \cdot ite{r}_{Max}}}\right)& R_2<ST\\ X_{i,j}^t+Q\cdot L& R_2\ge ST\end{array}\right.$$

(50)

where $X_{i,j}^t$ represents the position of the $i$ sparrow in the $j$ dimension at the $t$ iteration, where $j=\mathrm{1,2},3,...,d$; $\alpha$ is a random number in the range $(0, 1]$; $ite{r}_{Max}$ is the maximum number of iterations; the values of $Q$ are randomly generated and conform to a normal distribution; $L$ is a $1 \times d$ matrix with all elements 1; $R_2$ and $ST$ are the warning and safety thresholds, with ranges of $[0, 1]$ and $[0.5, 1]$, respectively.

Also, to improve the algorithm’s ability to jump out of the local optimum, the vigilante mechanism is added at the end. The core is to activate some individuals as “vigilantes” to perturb the search space directionally when the population is trapped in a local optimum, to break the current local convergence. Its position-update equation is:

$$X_{i,j}^{t+1}=\left\{\begin{array}{cc}{X}_{best}^t+\beta \cdot \left|X_{i,j}^t-X_{best}^t\right|& f_i>f_g\\ X_{i,j}^t+K\cdot {\displaystyle \frac{\left|X_{i,j}^t-X_{worst}^t\right|}{(f_i-f_w)+\epsilon }}& f_i=f_g\end{array}\right.$$

(51)

where $X_{best}$ is the global optimal position; $\beta$ is the step-size adjustment factor, which is a normally distributed random number with mean 0 and variance 1; $K$ is a $[-1, 1]$ random number used to introduce directional perturbations; $f_i$ is the fitness value of the current sparrow individual; $f_g$ and $f_w$ represent the globally optimal global worst fitness; ε is a very small constant used to avoid the molecule being zero. The proportions $p_d$ of discoverers and $p_v$ of vigilantes in this paper are 0.2 and 0.15.

4.3. Multi-Guides Differential and Dynamic Rotation Transforms

The dynamic decay factor $\delta$ is first introduced in the exploitation phase:

$$\delta =1-{\left({\displaystyle \frac{iter}{ite{r}_{Max}}}\right)}^2$$

(52)

In the first exploitation phase, when $P_2\ge ran{d}_{P_2}$, to avoid over-reliance on a single leader individual, the positional differentials of the three bootstraps (optimal, suboptimal, and random individuals) are combined to strengthen the local exploitation capability while retaining some global search potential. The difference $\Delta d$ is given by:

$$\left\{\begin{array}{l}{\eta }_1=U-0.5,\hspace{1em}U~U(\mathrm{0,1}{)}^d\\ \Delta d=0.6\left(BestVultur{e}_1\right(i)-P(i\left)\right)\\ +0.3\left(BestVultur{e}_2\right(i)-P(i\left)\right)\\ +0.1\cdot {\eta }_1\cdot (ub-lb)\end{array}\right.$$

(53)

where ${\eta }_1$ is a randomized perturbation vector for adding small perturbations; $\mathcal{U}(\mathrm{0,1}{)}^d$ is a Gaussian distribution; $BestVultur{e}_1\left(i\right)$ and $BestVultur{e}_2\left(i\right)$ are the optimal and suboptimal solutions, respectively; $P\left(i\right)$ represents the location of the current individual; $ub$ and $lb$ denote the upper and lower boundaries, and $d$ is the spatial dimension; the coefficients for the three guides are 0.6, 0.3, and 0.1.

When $P_2<ran{d}_{P_2}$, a rotational perturbation strategy that utilizes sinusoidal and cosinusoidal offset generation can produce periodic perturbation jumps to avoid overexploitation. The strategy generates a perturbation amplitude control factor $\xi$ and a randomized perturbation template ${\eta }_2$:

$$\left\{\begin{array}{l}\xi =(0.8+0.4\cdot rand)\cdot (1+2\cdot |F\left|\right)\cdot \delta \\ {\eta }_2=U+0.5,\hspace{1em}U~U(\mathrm{0,1}{)}^d\end{array}\right.$$

(54)

$\xi$ is used to prevent poor convergence due to too large a jump or too small a perturbation; ${\eta }_2$ is used as the coefficient template of the random perturbation factor to scale the individual update amount; $\delta$ is the dynamic attenuation factor in Equation (52).

In summary, the positional equation for the first phase of SAVOA’s exploitation is updated to:

$$P^{\prime }(i+1)=\left\{\begin{array}{cc}\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\left(56\right)& P_2\ge ran{d}_{P_2}\\ \mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\left(61\right)& P_2<ran{d}_{P_2}\end{array}\right.$$

(55)

$$P^{\prime }(i+1)=D^{\prime }\left(i\right)\times \left(F+0.3\times ran{d}_4\right)-d^{\prime }\left(t\right)$$

(56)

$$D^{\prime }\left(i\right)=|X^{\prime }\times R(i)-P(i\left)\right|$$

(57)

$$d^{\prime }\left(t\right)=R\left(i\right)-P\left(i\right)-\Delta d$$

(58)

$${S_1}^{\prime }=R(i)\cdot \left({\displaystyle \frac{ran{d}_5\times P_i}{2\pi }}\right)\cdot \mathit{cos}(2\cdot \pi \cdot ran{d}_{\theta })\cdot \xi \cdot {\eta }_2$$

(59)

$${S_2}^{\prime }=R(i)\times \left({\displaystyle \frac{ran{d}_6\times P_i}{2\pi }}\right)\times \mathit{sin}(2\cdot \pi \cdot ran{d}_{\theta })\cdot \xi \cdot {\eta }_2$$

(60)

$$P^{\prime }(i+1)=R\left(i\right)-\left({S_1}^{\prime }+{S_2}^{\prime }\right)$$

(61)

where the range of $X$ is changed to $[0, 1.5]$; $d^{\prime }\left(t\right)$ requires the subtraction of the difference $\Delta d$ from $d\left(t\right)$; $ran{d}_{\theta }$ is a random number between 0 and 1; $P^{\prime }(i+1)$ denotes the vulture position in the next iteration.

4.4. Adaptive Weighting Factors and Enhanced Lévy Flight

In the second exploitation phase, a random tiny value is added when $ran{d}_{P_3}\le P_3$:

$$ϵ=10^{-6}+10^{-5}\cdot ran{d}_{ϵ}$$

(62)

Meanwhile, the denominators of the two guiding objectives $A_1$ and $A_2$ change to:

$$\left\{\begin{array}{c}de{n}_1=BestVultur{e}_1\left(i\right)-P(i{)}^2+ϵ\cdot \left(\left|P\right(i\left)\right|+0.1\cdot (ub-lb)\right)\\ de{n}_2=BestVultur{e}_2\left(i\right)-P(i{)}^2+ϵ\cdot \left(\left|P\right(i\left)\right|+0.1\cdot (ub-lb)\right)\end{array}\right.$$

(63)

$ran{d}_{ϵ}$ is a random number between $[0, 1]$; $ϵ$ is used to carry out the stabilization of the values to avoid too small a denominator which would lead to too large a gradient and thus destroy convergence; $de{n}_1$ and $de{n}_2$ utilize the nonlinear coupling between the current individual and the superior individual to construct an offset bootstrap mechanism that differentiates more by diversity than linearity.

Keeping the weighting coefficient $w_t$ between 0.25 and 0.85 allows for adaptive regulation of equilibrium:

$$w_t=clip\left(0.65+0.25\cdot \mathit{tan}h(3F),\hspace{0.33em}0.25,\hspace{0.33em}0.85\right)$$

(64)

The use of the hyperbolic tangent function $\mathit{tan}h(3F)$ allows for a nonlinear smooth change in the weight curve, ensuring a natural transition in the middle; instead of direct averaging, the adaptive weighting coefficient $w_t$ allows for greater flexibility by allowing the current individual to be biased in the direction of a more optimal bootstrap but without being overly dependent on one superior individual. On the basis of maintaining excellent individual guidance, Gaussian perturbation noise $\sigma$ is added to maintain the diversity of local search:

$$\sigma =0.05\cdot (ub-lb)\cdot \mathcal{N}\left(\mathrm{0,1}\right)\cdot \delta$$

(65)

where $\mathcal{N}\left(\mathrm{0,1}\right)$ is the Gaussian distribution; $ub$ and $lb$ are the upper and lower boundaries; and $\delta$ is the dynamic decay factor.

Reinforcement of Lévy flight when $ran{d}_{P_3}>P_3$:

$$\left\{\begin{array}{l}scale=1.2-0.5\cdot \left({\displaystyle \frac{iter}{ite{r}_{\mathrm{Max}}}}\right)\\ \varphi =1+0.15\cdot sign\left(Bestvultur{e}_1\left(i\right)-P\left(i\right)\right)\end{array}\right.$$

(66)

$scale$ denotes the perturbation-scaling factor, whose value decreases with the number of iterations. It is used to control the overall scaling of the $Levy$ step, with a larger perturbation in the early stage to favor exploration and a smaller perturbation in the later stage to favor fine search; $\varphi$ is a scaling factor used to enhance positive jumps or fine tune negative jumps to form an asymmetric search mechanism, and $sign$ is a directional indicator. In summary, the positional formula for the second exploitation phase of the SAVOA is updated as:

$$P^{\prime }(i+1)=\left\{\begin{array}{cc}Equation\left(70\right)& P_3\ge ran{d}_{P_3}\\ Equation\left(71\right)& P_3<ran{d}_{P_3}\end{array}\right.$$

(67)

$${A_1}^{\prime }=Bestvultur{e}_1(i)-{\displaystyle \frac{Bestvultur{e}_1(i)\times P\left(i\right)}{de{n}_1}}\times F$$

(68)

$${A_2}^{\prime }=Bestvultur{e}_2(i)-{\displaystyle \frac{Bestvultur{e}_2(i)\times P(i)}{de{n}_2}}\times F$$

(69)

$$P^{\prime }(i+1)=w_t\cdot {A_1}^{\prime }+(1-w_t)\cdot {A_2}^{\prime }\cdot \sigma$$

(70)

$$P^{\prime }(i+1)=R\left(i\right)-\left|d\right(t\left)\right|\cdot F\cdot scale\cdot Levy\left(d\right)\cdot \varphi$$

(71)

The pseudo-code and flowchart of SAVOA are Algorithm 1 and Figure 5.

Algorithm 1. Pseudo-code for SAVOA

$pop$: size of the population; $pd$: the number of the producers; $pv$: the number of vultures who perceive the danger; Generating first-generation populations using Sobol sequences while $iter<{iter}_{Max}$ do Find the best and the second-best individuals;     for $i=1:pd$         Updating the location of vultures using Equation (50);         Calculate the fitness of vultures;     end for     for $i=pd+1:pop$         Calculate the starvation rate $F$ using Equations (30) and (31);         if $\left|F\right|\ge 0.5$ then             Updating the position using Equation (55);         else             Updating the position using Equation (67);         end if             Calculate the fitness of vultures;     end for     for $i=1:pv$         Updating the location of vultures using Equation (51);         Calculate the fitness of vultures;     end for $iter=iter+1$; end while return Vultures with the lowest overall costs

5. Results

5.1. Experimental Environment

In order to verify the performance of SAVOA in UAV path planning, this paper carries out comparative simulation experiments of six algorithms, namely SAVOA, AVOA, SSA, ABC, WOA, and GA, in mountainous environments and urban environments, respectively, with two scenarios for each environment. Also, to avoid the serendipity of the algorithm and to verify the stability of the algorithm, 12 repetitions of the experiment are performed in each scenario. All experiments were performed in MatlabR2024a with Windows 10, an Intel(R) Core (TM) i5-10300H processor, and an NVIDIA GeForce GTX 1650 Ti (4 GB) graphics card. The parameters in this paper are specified as follows:

• Table 1. Path-planning parameters.

  Parameter	Value	Unit
  Map range	(500, 500, 400)	m
  Starting point	(5, 5, 5)	m
  Ending point	(480, 480, 300)	m
  Wind-field center	(0, 0, 0)	m
  $v_w$	8	m/s
  $\theta$	Calculations based on path points	degree
  $v_a$	8	m/s
  $v_{aMax}$	23	m/s
  ${\psi }_{Max}$	45	degree
  ${\gamma }_{Max}$	60	degree

  records the map parameters, the UAV’s own performance parameters, and the wind-field parameters.
• The mountain model and the city model have two scenarios each, with 20 and 25 mountains or buildings, respectively. The specific terrain parameters are shown in

  Table 2. Terrain parameters.

  	Mountainous1			Urban1
  Number of Terrains	Center Coordinate	Height	Range	Lower-Left Coordinate	Length, Width, Height
  20	[287.13, 213.17]	194.24	[44.61, 44.61]	[167, 212]	[25, 27, 296]
  	[220.33, 267.09]	147.98	[21.89, 21.89]	[415, 132]	[29, 30, 170]
  	[176.52, 361.81]	376.50	[24.46, 24.46]	[310, 36]	[23, 24, 288]
  	[159.80, 251.00]	129.55	[30.53, 30.53]	[245, 441]	[23, 28, 268]
  	[400.63, 225.58]	398.25	[45.41, 45.41]	[330, 431]	[28, 29, 126]
  	[381.27, 315.55]	336.44	[31.85, 31.85]	[304, 332]	[23, 32, 202]
  	[212.60, 417.70]	193.70	[36.84, 36.84]	[392, 299]	[29, 28, 192]
  	[234.86, 151.39]	213.90	[41.69, 41.69]	[426, 215]	[30, 31, 165]
  	[305.14, 104.33]	280.19	[28.78, 28.78]	[327, 250]	[27, 26, 247]
  	[307.79, 296.05]	217.89	[38.10, 38.10]	[333, 135]	[32, 25, 130]
  	[122.14, 133.60]	211.61	[43.94, 43.94]	[45, 323]	[24, 24, 206]
  	[390.61, 403.49]	134.71	[25.41, 25.41]	[132, 137]	[25, 32, 160]
  	[100.60, 270.32]	352.88	[33.36, 33.36]	[127, 304]	[23, 29, 298]
  	[185.65, 79.55]	162.52	[21.75, 21.75]	[392, 74]	[28, 30, 133]
  	[306.62, 397.52]	262.34	[36.80, 36.80]	[250, 346]	[26, 27, 224]
  	[84.86, 406.10]	286.21	[22.46, 22.46]	[397, 431]	[30, 25, 276]
  	[235.38, 349.01]	286.95	[20.78, 20.78]	[51, 47]	[26, 24, 318]
  	[91.82, 343.22]	203.71	[41.42, 41.42]	[228, 131]	[27, 30, 217]
  	[389.50, 134.50]	140.15	[33.91, 33.91]	[46, 166]	[30, 31, 270]
  	[337.28, 174.07]	214.65	[37.74, 37.74]	[224, 230]	[23, 25, 334]
  	Mountainous2			Urban2
  25	[403.27, 359.92]	129.63	[27.35, 27.35]	[412, 353]	[23, 28, 314]
  	[224.92, 400.70]	204.54	[39.74, 39.74]	[46, 163]	[25, 27, 330]
  	[213.24, 320.39]	173.35	[21.82, 21.82]	[58, 427]	[31, 24, 212]
  	[296.68, 76.46]	270.48	[17.51, 17.51]	[174, 154]	[25, 23, 232]
  	[310.17, 348.24]	243.25	[26.18, 26.18]	[34, 220]	[23, 32, 196]
  	[192.87, 97.22]	327.47	[17.64, 17.64]	[256, 52]	[28, 31, 209]
  	[295.47, 177.03]	399.90	[25.07, 25.07]	[331, 444]	[32, 23, 305]
  	[119.10, 105.49]	199.56	[35.73, 35.73]	[300, 209]	[26, 24, 134]
  	[79.71, 307.94]	266.84	[20.29, 20.29]	[255, 338]	[26, 25, 161]
  	[173.89, 260.91]	371.96	[18.98, 18.98]	[83, 316]	[23, 30, 123]
  	[131.65, 363.93]	378.60	[35.35, 35.35]	[181, 329]	[25, 27, 357]
  	[114.35, 235.63]	218.12	[36.09, 36.09]	[160, 68]	[24, 26, 267]
  	[407.51, 251.52]	322.87	[16.44, 16.44]	[356, 318]	[23, 26, 288]
  	[352.72, 116.96]	334.02	[17.09, 17.09]	[123, 214]	[28, 23, 356]
  	[228.10, 137.21]	333.24	[22.42, 22.42]	[223, 142]	[24, 30, 206]
  	[317.64, 287.60]	207.80	[20.54, 20.54]	[141, 384]	[24, 31, 340]
  	[99.47, 169.70]	303.52	[19.33, 19.33]	[217, 265]	[29, 29, 206]
  	[237.93, 238.95]	209.25	[23.79, 23.79]	[80, 58]	[25, 26, 251]
  	[346.21, 190.18]	157.34	[34.94, 34.94]	[439, 77]	[23, 30, 335]
  	[89.31, 404.45]	385.51	[22.61, 22.61]	[428, 432]	[24, 29, 148]
  	[397.21, 81.05]	121.88	[27.08, 27.08]	[319, 56]	[25, 27, 233]
  	[354.52, 424.85]	326.06	[30.50, 30.50]	[427, 163]	[24, 29, 355]
  	[137.97, 422.35]	131.05	[17.18, 17.18]	[391, 247]	[24, 31, 289]
  	[186.90, 166.03]	180.83	[23.63, 23.63]	[328, 171]	[24, 29, 173]
  	[300.20, 232.11]	355.47	[30.70, 30.70]	[248, 406]	[26, 25, 265]

  .
• The number of exclusion zones is 10 for all four scenarios; see

  Table 3. Spherical exclusion zone parameters.

  Mountainous1		Urban1		Mountainous2		Urban2
  Center	Radius	Center	Radius	Center	Radius	Center	Radius
  [160, 247, 348]	28	[127, 469, 302]	29	[81, 226, 254]	26	[292, 29, 302]	21
  [259, 210, 250]	25	[113, 390, 330]	27	[167, 161, 253]	26	[25, 124, 203]	20
  [474, 46, 204]	24	[323, 306, 171]	22	[445, 186, 162]	21	[125, 136, 211]	20
  [136, 435, 203]	20	[260, 26, 359]	24	[304, 39, 319]	30	[334, 276, 338]	26
  [41, 79, 338]	21	[188, 90, 306]	26	[224, 253, 257]	29	[453, 293, 289]	22
  [132, 391, 315]	28	[33, 394, 200]	20	[327, 386, 320]	20	[364, 237, 199]	23
  [386, 64, 253]	25	[296, 131, 323]	28	[52, 102, 279]	26	[192, 223, 306]	27
  [205, 226, 283]	26	[457, 452, 204]	20	[183, 458, 336]	26	[292, 129, 282]	26
  [424, 459, 348]	20	[46, 260, 192]	22	[381, 212, 247]	26	[306, 301, 217]	21
  [237, 117, 338]	27	[330, 192, 329]	26	[205, 145, 359]	21	[39, 128, 243]	20

  for specific parameters.

5.2. Comparison of Different Algorithms for Path Planning

This paper uniformly set the population size $pop$ to 40 for the six algorithms; the maximum number of iterations $ite{r}_{Max}$ is set to 200; the number of intermediate points m is 3; the safe distance $sd$ is 2; the weight coefficients of the fitness function are $0.6, 0.2, 0.1, 0.1$ for $l_1, l_2,{ l}_3, l_4$, respectively.

Table 4. Algorithmic parameters.

Algorithmic	Parameter	Value
SAVOA	Warning parameter $ST$	0.8
	Percentage of discoverers $p_d$	0.2
	Percentage of vigilantes $p_v$	0.15
	Control parameter $P_2$	0.4
	Control parameter $P_3$	0.6
	Control parameter $w$	2.5
AVOA [39]	Control parameter $P_1$	0.6
	Control parameter $P_2$	0.4
	Control parameter $P_3$	0.6
	Control parameter $w$	2.5
SSA [40]	Warning parameter $ST$	0.8
	Percentage of discoverers $P_d$	0.2
	Percentage of vigilantes $P_v$	0.15
ABC [41]	Number of honeys harvesting bees $N_e$	$0.2\times pop$
	Judgment threshold $limit$	5
WOA [42]	Random number between 0 and 1	$r$
	Convergence factor a	$a=2-2·iter/{iter}_{Max}$
	Control parameter $A$	$A=2a·r-a$
GA [43]	Probability of selection $P_s$	0.5
	Crossover probability $P_c$	0.8
	Probability of mutation $P_m$	0.2

lists the parameter settings for each algorithm.

Based on the parameters of Table 4, these six algorithms are simulated for UAV path planning in mountainous environments and urban environments.

In GA, the selection probability $P_s$ refers to the selection pressure applied during the parental selection stage, which affects the likelihood of individuals with higher fitness being selected for reproduction. In this study, we used a roulette-wheel selection method, which selects individuals probabilistically based on their relative fitness values. The higher the fitness, the greater the probability of being selected. In addition, the crossover operation uses a one-point crossover method, which selects a single crossover point on the chromosomes of the parent individuals and then generates offspring by exchanging the sub-sequences after that point.

5.2.1. Path Planning in Mountainous Environments

Figure 6 and Figure 7 show representative results from 1 of 12 experiments comparing the UAV flight-path performance of the six algorithms in mountainous environments, while Figure 8 gives graphs of the fitness of the six algorithms in two different mountainous environments as a function of the number of iterations.

5.2.2. Path Planning in the Urban Environment

Figure 9 and Figure 10 show the results of 1 representative experiment selected from 12 experiments comparing the performance of the six algorithms for UAV flight paths in urban environments. Figure 11 gives the graphs of the fitness of the six algorithms in two different urban environments with the number of iterations.

All the above images are representative results from 1 of the 12 experiments. The results show that the SAVOA outperforms AVOA, SSA, WOA, GA, and ABC algorithms in both mountainous and urban environments and is able to plan paths that are close to the shortest distance between the start and end points. Meanwhile, according to the iteration graph, the optimal fitness of SAVOA is also at the lowest level and has the best convergence performance. Since the fitness values in this paper are based on weighted results, there is no fixed theoretical optimum, but the shortest path is fixed and can have a reference value: $F_{t\_theoretical}=l_1{f}_1=440.203$.

5.3. Results of Multiple Repeat Experiments

5.3.1. Validation of the Validity of Algorithms

In order to verify the performance of SAVOA more comprehensively, four performance metrics are introduced in this paper: the best fitness value, the average fitness value, the flight time of the UAV, and smoothness. These four indicators are the average results of 12 replicated experiments, and according to Figure 12 and Figure 13 it can be seen that SAVOA exhibits the lowest values in all indicators.

From

Table 5. Mean fitness and standard deviation for 12 experiments.

	Mountainous1		Mountainous2		Urban1		Urban2
	Mean	Std	Mean	Std	Mean	Std	Mean	Std
SAVOA	468.754	25.927	468.181	24.275	466.796	24.024	467.715	29.487
AVOA	499.513	51.747	510.897	53.555	509.129	43.598	530.630	60.249
SSA	569.460	36.992	598.256	45.568	589.180	34.399	581.370	42.766
ABC	495.496	36.656	501.567	45.855	503.765	29.808	510.041	43.887
WOA	528.749	61.233	587.686	55.121	521.323	41.307	566.806	49.879
GA	553.432	9.552	561.809	13.173	534.067	7.069	556.463	10.071

, it can be seen that SAVOA has the best average fitness regardless of the environment, and it has the second lowest standard deviation after GA (which has a smaller standard deviation due to the fact that GA often falls into a local optimum very quickly).

5.3.2. Stability Verification of Algorithms

Figure 14 and Figure 15 show the changes in the optimal fitness of the six algorithms in four environments over 12 repetitions, and it is easy to see that the optimal fitness of SAVOA is not only the lowest, but also has the smallest fluctuation in change.

5.4. Comparative Experimental Analysis of Path Planning

5.4.1. Experimental Analysis in Mountainous Environments

Figure 6 and Figure 7 are schematic illustrations of the results from 1 of the 12 replicate trials in the two mountain scenarios. In the mountainous region 1 scenario, the paths planned by SSA and WOA make substantial turns during flight, which not only greatly increases the flight distance, but also poses a serious threat to the flight safety of the UAV; the GA path flies at a higher altitude, avoiding obstacles but at the same time increasing energy consumption; although the paths generated by AVOA and ABC are relatively smooth, the flight paths are still not optimal; SAVOA is able to find a path that is both smooth and relatively short. For the variation in fitness, according to Figure 8a, SSA starts to converge at generation 32, but it has the highest fitness value of 603.063; WOA falls into local optimality at generation 69 with an fitness value of 585.779; the GA underwent three convergences, at generations 2, 26, and 168, with a final fitness value of 534.291; ABC falls into a local optimum at generation 33 with a fitness of 486.8; AVOA converges multiple times and is the least adapted algorithm besides SAVOA at 481.708; SAVOA gradually converged at the 31st generation, with slight changes afterwards, and the final fitness was 464.328, with the best convergence performance and optimization ability.

In the mountainous region 2 scenario, the SSA, WOA, and GA turn angles are extremely large, which can lead to very unstable UAV flight; the AVOA has a medium turn angle, but flies a long distance; in contrast, SAVOA plans flight paths that are close to the shortest distance with the lowest fitness. From Figure 8b, it can be seen that SSA has the highest fitness of 775.521; this is followed by GA, which falls into the first local optimum at generation 6 and the second local optimum at generation 73, with a fitness of 568.374; WOA reached convergence at the 17th generation, but its fitness was relatively high, at 524.655; AVOA nears convergence at generation 67, with slight changes afterward and a final fitness of 506.123. ABC’s fitness decreases sharply at generation 9 and then falls to a local optimum at generation 48 with a fitness of 488.051; SAVOA finds the near-optimal path in generation 34, followed by fine-tuning, and the final fitness is 460.661, which is significantly better than the other five algorithms.

5.4.2. Experimental Analysis in Urban Environments

Figure 9 and Figure 10 illustrate sample results from two urban scenarios, while Figure 11 presents the fitness evolution curve under the urban environment. In the city 1 scenario, the paths generated by SSA, ABC, and WOA all show large turns and are not suitable for UAV flight; GA path smoothness is not high, but the higher flight path reduces economy; although AVOA outperforms the first four algorithms in overall performance, it still fails to find the optimal path. In terms of fitness change, SSA has a rapid decrease in fitness in the 25th generation, reaching the first convergence state, followed by a second convergence in the 149th generation, with a final fitness of 588.687; GA falls into local optimality three times, in generations 2, 153, and 190, with a fitness of 545.625; ABC fell into local optimality several times, the last time in generation 124, with a fitness of 516.886; WOA converged several times, the last time in generation 158, with a final fitness of 501.689; although AVOA converged relatively early at the 39th generation, its fitness did not reach the optimum, ending at 482.18. SAVOA reaches convergence at generation 43 and is gradually fine-tuned later, with a final fitness of 461.04.

In the city 2 scenario, the paths generated by AVOA, SSA, ABC, and WOA all have sharp turns with high fluctuation ups and downs; although the GA path is relatively smooth, its higher flight altitude causes the drone to take a longer detour. According to the fitness variation graph, WOA has the highest fitness value, converging at the 56th generation with a final value of 609.014. SSA’s fitness decreases sharply in the 27th generation and then converges in the 85th generation, but its fitness is as high as 543.75; AVOA converged at generation 46, but experienced significant fluctuations starting at generation 94 and continued to vary thereafter, with a fitness value of 501.002. SAVOA began converging at generation 64; although slight variations occurred afterward, they were minimal, resulting in a final fitness value of 457.691.

5.5. Effectiveness Analysis of SAVOA Path Planning

5.5.1. Effectiveness Analysis of Mountainous Terrain

Figure 12 shows the average of the four-performance metrics for GA, WOA, ABC, SSA, AVOA, and SAVOA in the mountainous environments. In mountainous 1 scenario, for optimal fitness, SAVOA is 15.48%, 9.86%, 4.73%, 17.19%, and 4.68% less than GA, WOA, ABC, SSA, and AVOA, respectively; on average fitness was reduced by 15.30%, 11.35%, 5.40%, 17.68%, and 6.16%, respectively; for flight time, SAVOA decreased by 23.89%, 22.36%, 19.70%, 31.55%, and 19.56%, respectively; in terms of smoothness, SAVOA is 86.98%, 92.39%, 72.71%, 94.23%, and 68.01% lower than the other five algorithms.

In the mountainous 2 scenario, the optimal fitness of SAVOA was reduced by 16.87%, 19.67%, 5.79%, 20.88%, and 6.39% compared to GA, WOA, ABC, SSA, and AVOA; the average fitness was reduced by 16.67%, 20.33%, 6.66%, 21.74%, and 8.36%; in terms of flight time, SAVOA was reduced by 32.41%, 39.27%, 19.15%, 38.33%, and 23.20%, respectively, while smoothness was reduced by 92.42%, 96.34%, 90.72%, 97.76%, and 89.07%.

5.5.2. Effectiveness Analysis of Urban Terrain

Figure 13 shows the average values of the four-performance metrics for GA, WOA, ABC, SSA, AVOA, and SAVOA in the urban environments. In the city 1 scenario, SAVOA’s four performance metrics remained at their lowest values, where the optimal fitness decreased by 13.04%, 10.19%, 6.94%, 20.11%, and 6.34% over GA, WOA, ABC, SSA, and AVOA, respectively; for average fitness, SAVOA was reduced by 12.60%, 10.46%, 7.34%, 20.77%, and 8.31%, respectively; above the flight time, the reduction was 25.92%, 27.01%, 22.93%, 34.39%, and 24.21%, respectively; and the smoothness was 93.73%, 94.75%, 91.24%, 98.84%, and 94.34%.

For the city 2 scenario, the optimal fitness of SAVOA decreased by 16.97%, 17.22%, 8.17%, 18.40%, and 9.57%; the average fitness decreased by 15.95%, 17.48%, 8.30%, 19.55%, and 11.86%; the flight time decreased by 33.73%, 34.12%, 32.47%, 37.10%, and 26.73%; in terms of smoothness, SAVOA was reduced by 93.33%, 99.00%, 97.07%, 99.04%, and 98.19%, respectively.

According to the above analysis, SAVOA outperforms the other five algorithms in all four metrics. In the four scenarios, its average fitness is only 1.37%, 1.64%, 1.28%, and 2.10%, higher than the optimal fitness, respectively, which is a small difference, which shows that SAVOA converges faster while achieving the lowest fitness. In addition, it having the least flight time and the best smoothing also indicates the efficiency and safety of the algorithm for UAV path planning.

5.6. Stability Analysis of SAVOA Path Planning

According to Figure 14 and Figure 15, in 12 repeated experiments, the highest values of the optimal fitness of SAVOA, AVOA, SSA, ABC, WOA, and GA in the mountainous region 1 scenario were 463.799, 496.220, 739.612, 501.345, 592.912, and 633.083, respectively; the minimum values of optimal fitness were 460.729, 474.182, 486.660, 473.403, 473.589, and 479.604; the standard deviations of optimal fitness were 0.852, 5.871, 73.349, 7.249, 39.825, and 49.885.

In the mountainous region 2 scenario, the highest values of optimal fitness for the six algorithms are 462.106, 511.813, 775.521, 503.359, 751.339, and 587.792, respectively; the minimum values of optimal fitness were 459.800, 474.635, 512.939, 474.443, 473.261, and 523.344; the standard deviations were 0.674, 13.792, 82.232, 10.203, 86.826, and 20.586.

The highest values of optimal fitness for SAVOA, AVOA, SSA, ABC, WOA, and GA were 461.579, 560.919, 654.730, 531.821, 549.571, and 580.526, respectively, for the city 1 scenario; the lowest values were 460.392, 472.498, 506.979, 471.515, 475.985, and 491.416; the standard deviations were 0.385, 25.300, 39.284, 18.873, 19.500, and 27.187.

The highest values of optimal fitness in the city 2 scenario were 458.763, 589.909, 737.192, 542.138, 675.592, and 620.824, respectively; the lowest values were 457.216, 478.428, 481.252, 475.628, 495.523, and 489.958; and the standard deviations were 0.509, 28.505, 68.256, 21.159, 50.948, and 36.761, respectively.

In summary, the optimal fitness of SAVOA in all four environments fluctuates slightly above and below 460, the standard deviation is not only the smallest but also is much better than other algorithms, and the fitness value is the most stable, which is an experimental result that fully demonstrates the excellent stability of SAVOA in the global path-planning task.

6. Conclusions

Global path planning is the foundation of the UAV logistics and transportation system; in order to plan an efficient flight path, this paper firstly establishes two three-dimensional models of mountainous area and city, and constructs a multi-evaluation objective function by combining constraints and mission requirements. Secondly, the standard AVOA principle is analyzed, and then the improvement strategies of SAVOA in the initial, exploration, and exploitation phases are presented. Finally, SAVOA is compared with five algorithms in simulation experiments in four environments, and the results show that SAVOA not only outperforms the standard AVOA in various indexes, but also outperforms the traditional algorithms ABC and GA, as well as the new intelligent algorithms SSA and WOA which have emerged in recent years, and demonstrates a better optimization ability and stronger robustness.

However, only the flight path of a single UAV is considered in this paper. In subsequent studies, how to ensure efficient collaboration among multiple UAVs to accomplish the delivery task will be investigated in depth, which not only requires optimization of the algorithm to avoid mutual collisions among UAVs but also considers the recognition and adaptive adjustment of obstacles in various dynamic environments.