A Pseudo-Exponential-Based Artificial Potential Field Method for UAV Cluster Control under Static and Dynamical Obstacles

Abstract

This study presents a novel obstacle evasion method for unmanned aerial vehicle (UAV) clusters in the presence of static and dynamic obstacles. First, a discrete three-dimensional model of the UAV is provided. Second, the proposed improved artificial potential field (APF) is illustrated. In designing the improved scheme, a pseudo-exponential function is fused into the potential field, thus avoiding local extreme points. Frictional resistance is introduced to optimize vibration and maintain stability after reaching the desired endpoints. Meanwhile, the relevant parameters are optimized, and appropriate state limits are defined, thus enhancing the control stability. Third, Lyapunov stability analysis proves that all signals in the closed-loop cluster system are ultimately bounded. Finally, the simulation results demonstrate that the UAV cluster can efficiently reconstruct, form, and maintain formations while avoiding static and dynamical obstacles along with maintaining a safe distance, solving the problem of the local extreme of traditional artificial potential field methods. The proposed scheme is also tested under large-scale multi-UAV scenarios. In conclusion, this study provides valuable insights for engineers working with UAV clusters navigating through formations.

1. Introduction

Due to the swift advancements in aeronautical technology and the increasingly intricate nature of contemporary environments, a single isolated UAV is progressively inadequate in fulfilling the requirements of these intricate scenarios [1]. For example, in civilian applications, a single UAV faces limitations in load capacity, airborne sensors, and communication equipment, hindering its ability to complete tasks as expected, such as supporting agriculture, forestry, environmental conservation, mapping, and aiding in rescues [2]. Hence, multi-UAV clusters, which are able to perform tasks and address complexities in various environments, have become an important topic in the UAV research community [3,4]. The UAV clusters achieve behavioral coordination and adaptability to dynamic environments through information interaction, feedback, motivation, response, and effective control strategies, enabling them to undertake complex, dynamic, and critical tasks [5,6,7,8]. On the one hand, the UAV cluster can fully leverage the advantages of UAVs to improve overall load capacity and information perception processing capabilities. On the other hand, the problems can be solved effectively if a single UAV is attacked or inefficient in task execution. However, with the increasing difficulty of tasks performed by multiple UAVs, the number of UAVs should be increased, and their distribution must be denser in space [9,10], all of which will inevitably bring about potential sudden obstacles that are challenges for flight control and the safety of UAV clusters.

1.1. Related Works

While performing tasks, the UAV should promptly take measures to evade obstacles to ensure safe flight. Scholars have proposed methods to solve intelligent evasion problems. Effective algorithms for obstacle evasion of multi-agent formations include the rapidly exploring random tree algorithm [11,12], leader–follower [13,14], virtual structure method [15,16], behavioral control method [17,18], and APF [19,20]. Among them, the artificial potential field, which is characterized by a simple structure and parameter design, has been widely applied in drone cluster control. It achieves rapid response and intelligent avoidance of sudden changes in the external environment or drone failures during mission execution [21,22]. Li et al. [23] adopted the APF in solving the obstacle avoidance problem for a snake-like robot in water. Tan et al. [24] proposed an APF-based algorithm to realize the collision avoidance and the path following for unmanned surface vehicles operating in an ocean environment. Lin et al. [25] utilized the APF scheme to deal with the path-planning problem for a mobile robot. However, the mentioned studies only considered the algorithm design in 2D, which is an idealized situation compared to practical engineering applications.

Unlike robots and watercraft, UAVs always operate in a 3D environment, in which the volume of obstacles and UAVs must be taken into account. This makes path planning and collision avoidance algorithm design more challenging. To meet the engineering demands, research groups have designed and testified the APF in 3D scenery [26,27]. However, the traditional APF is prone to fall into local optima, leading to obstacle avoidance or formation transformation failures [28]. To solve this problem, research groups have proposed many schemes to escape from the local minimum solution. Zhou et al. [29] fused the APF with a reinforcement learning algorithm to automatically adjust the virtual force of APF. Li et al. [30] designed a dynamical enhanced firework algorithm with APF to handle the minimum problem. There is no doubt that the mentioned intelligent algorithms successfully solved the problem with desired performance. However, such an algorithm design is relatively complicated, and it is difficult to realize in multi-UAV clusters. To find a simpler way, Zhang et al. [31] improved the APF by introducing a slight adjustment to the direction of the repulsive force. Liu et al. [32] integrated a regulatory factor into the obstacle repulsive field of the traditional artificial potential field method, ensuring that both attractive and repulsive forces reduced to zero only at the target point, thus resolving issues of inaccessibility and local optima. Additionally, Wu et al. [33] introduced a disturbance component into the attraction field, which reduced the speed of the UAV, thus escaping from a local minimum. Pan et al. [21] fused a rotating potential function into the APF to guide the UAV out of the local minimum. Szczepanski et al. [34] introduced a vortex potential into the traditional APF, which virtually attracted the UAV away from the extreme point. However, the introduced virtual attracters greatly changed the fundamental characteristic of the APF. Thus, some methods, which are easy to understand and design, are proposed by reconstructing the field function. Pan et al. [35] multiplied the potential field with the absolute distance between the UAV and the target. However, such a method makes the potential field overlarge when the distance is too far. To handle it, Sfeir et al. [36] introduced a Gaussian function into the traditional repulsive potential function. Furthermore, Liu et al. [37] introduced a coefficient into the Gaussian function, thus avoiding being trapped in the local minimum point. Nevertheless, the Gaussian function, which needs index calculation, may occupy too many computational resources. Therefore, it is necessary to design an improved APF algorithm to effectively solve the local extreme problem, along with releasing some calculation burden.

UAVs frequently operate in complex environments, where the presence of dynamical threats poses the risk of collisions. However, many approaches tend to ignore this scenario, often resorting to basic obstacle avoidance strategies and assuming an uncomplicated environment. Aiming to address the dynamical obstacle evasion, Yao et al. [38] proposed a model predictive control algorithm combining the interfered fluid dynamical system and the Lyapunov guidance vector field to realize path planning and obstacle avoiding for a UAV in 3D. Huang et al. [39] designed an ant colony optimization-based APF scheme for the UAV path-planning problem in static and moving obstacles. Lindqvist et al. [40] developed a nonlinear model predictive control scheme for the tracking control of UAVs under dynamical threats. However, most existing work focuses solely on obstacle evasion by individual UAVs, overlooking algorithm verification within UAV clusters, which requires consideration of the complex interactions among UAVs. Additionally, the mentioned formation control problems are established on a continuous-time model, assuming that all states of the agents are continuous. However, as microcomputers have become an overwhelming topic in UAV engineering applications [41,42], digital systems and controllers are being adopted more frequently, where continuous information cannot be directly processed. Consequently, there is an urgent need to develop an effective discretized control algorithm that ensures the reliability and survivability of the discrete-time UAV clusters in environments with static obstacles and dynamic threats.

1.2. Main Contributions

Motivated by the extensive and varied literature mentioned above, we intend, in this study, to develop an obstacle evasion algorithm for a discrete-time 3D UAV cluster under both static obstacles and dynamic threats. An improved APF scheme is presented, incorporating a pseudo-exponential function to avoid local extremes, frictional resistance to optimize vibration and maintain stability, and optimized parameters with defined velocity and acceleration limits to enhance control stability. Simulations demonstrate that the UAV cluster can efficiently form and maintain formations while avoiding obstacles and maintaining a safe distance, addressing the limitations of traditional APF methods. The scheme is also validated in large-scale multi-UAV scenarios, providing valuable insights for engineers working with UAV formation control.

The main contributions of this paper are as follows:

First, compared to those of 2D [23,24,25] and single isolated UAV [34,35,36], the proposed improved APF algorithm is tested in a 3D multi-UAV cluster under discrete-time calculation. Such an environment is considerably more complex, and the mutual effects between the UAVs need further consideration.

Second, a pseudo-exponential function is fused into the traditional APF scheme to address its inherent local extreme problem. Unlike the schemes in references [21,33,34], an extra potential field is not introduced, thus retaining the original components of the APF. Compared to the Gaussian-function-based APF [36,37], which needs complicated exponential calculation, the proposed method has the advantage of calculation efficiency.

Third, we consider the dynamic threat, which is an actual condition in flight environments. In contrast with the works on APF verification for a single UAV under moving threats [38,39,40], our research addresses the path planning and obstacle evasion for a large-scale multi-UAV cluster involving complex internal interactions. And, we validate the effectiveness and robustness of our scheme in the presence of static obstacles and moving threats.

The rest of this work is organized as follows: Section 2 presents an overview of the mathematical model for UAVs. Section 3 details the design process for the artificial potential field and the improvement scheme, and it includes a theoretical stability analysis. In Section 4, comprehensive simulation results demonstrate the effectiveness and robustness of the proposed control scheme. Finally, Section 5 provides conclusions and discusses future work.

2. System Modeling and Problem Formulation

This study focuses on improving the artificial potential field for intelligent evasion control of UAV clusters. The objective of intelligent evasion control is to enable the clusters to autonomously navigate around dynamic and static obstacles while maintaining their formations or swiftly forming new ones. The UAV cluster, consisting of N UAVs, allows each UAV to obtain real-time position information and to coordinate with other UAVs. The scenario assumes that the task area of the UAV cluster is rectangular, and each UAV takes off from a starting point and flies at a constant velocity. The UAVs must avoid all obstacles throughout the flight to reach their target points safely and to avoid crashes involving any single UAV.

To facilitate the motion model, we only consider lateral motion and ignore factors such as aerodynamic performance and flight status. The UAV is considered as a mass point on a three-dimensional plane, and the task area is divided into x, y and z directions [3,5]. Hence, the discrete-time second-order motion model of a single UAV in the cluster can be expressed through the following equation:

$$\{\begin{array}{cc}\begin{array}{c}{X}_i\left(k+1\right)=X_i\left(k\right)+v_i\left(k\right)T\\ v_i\left(k+1\right)=v_i\left(k\right)+u_i\left(k\right)T\end{array}& i=1,2,\mathrm{\dots },N\end{array},$$

(1)

where $k\in {\mathbb{Z}}^{+}$ is the sample instant; $T\in ℝ$ is the sample period; $X_i\left(k\right)={\left[x_i\left(k\right),y_i\left(k\right),z_i\left(k\right)\right]}^{\mathrm{T}}$ represents the position vector of UAV i at k moment; $v_i\left(k\right)={\left[v_{ix}\left(k\right),v_{iy}\left(k\right),v_{iz}\left(k\right)\right]}^{\mathrm{T}}$ means velocity vector of UAV i at k moment; similarly, $u_i\left(k\right)={\left[u_{ix}\left(k\right),u_{iy}\left(k\right),u_{iz}\left(k\right)\right]}^{\mathrm{T}}$ is the acceleration of UAV i at k instant.

In addition, for the safe operation of the UAV cluster, the following constraints need to be considered.

(1)

Distance

Limited by the size and deceleration distance, a minimum safety distance should be maintained between UAVs to avoid collisions with adjacent UAVs in the cluster. Any two UAVs (UAV i and UAV j) can fly safely outside this range, that is, $\left|X_i\left(k\right)-X_j\left(k\right)\right|\ge {\rho }_0(i\ne j)$, and ${\rho }_0$ is the influence distance of the obstacle.

(2)

Velocity

Due to their limited performance, UAVs usually have a maximum velocity. UAVs can fly safely at velocities within a specified range, that is, $\left|v_i\left(k\right)\right|\le v_{max}$.

(3)

Acceleration

After receiving instructions, UAVs cannot change velocity and direction immediately. Therefore, the acceleration limit should conform to $\left|u_i\left(k\right)\right|\le a_{max}$.

In the following calculation, we replace the variables such as $X_i\left(k\right)$ to $X_i$ for simplicity.

3. Modeling of Cluster Intelligent Evasion Based on Artificial Potential Field

3.1. Traditional Artificial Potential Field

First proposed by Khatib [19] in 1986, the artificial potential field method is used to solve the collision between intelligent objects and obstacles. A virtual force method guides intelligent objects toward reduced potential energy by constructing a potential function that covers the entire field. The basic idea is to construct an artificial potential field made up of attractive and repulsive forces. The target point attracts the UAV, while obstacles and other UAVs repel it. With the combination of the above potential fields, the movement of a single UAV is controlled by the resultant force. Thus, the UAV can achieve path planning and obstacle evasion, and a schematic diagram of the working process is shown in Figure 1.

An artificial potential field is defined by a potential function. This potential function is differentiable, and the value of the function at a point in space represents the strength of the potential field at that point. Under the combined action of potential field functions $U_{att}$, $U_{rep}$ and $U_{ij}$, the resultant force applied to the UAV will drive it away from obstacles and approach the target. The sum of gravitational potential and repulsive potential is expressed by the function $U\left(X_i\right)$ of the UAV at the point $X_i=\left(x_i,y_i,z_i\right)$. Thus, we obtain the whole potential field function as:

$$U\left(X_i\right)=U_{att}\left(X_i\right)+U_{rep}\left(X_i\right)+U_{ij}\left(X_i\right).$$

(2)

In practice, the simplest and most common method to construct a gravitational potential field is to calculate the square of the Euclidean distance from each point in free space to the target endpoint and then multiply this value by a scaling factor, known as the gravitational gain. The gravitational potential field function is illustrated in (3):

$$U_{att}\left(X_i\right)={\displaystyle \frac{1}{2}}{K}_{att}{\rho }^2\left(X_i,X_i^G\right).$$

(3)

where $X_i^G=\left(x_i^G,y_i^G,z_i^G\right)$ is the coordinates of the target point for UAV i, and $\rho \left(X_i,X_i^G\right)$ represents the distance between the UAV i and its target point.

To make UAVs avoid obstacles in the environment, a repulsive potential field is also needed to construct a particular potential field that is closer to the obstacle, especially when the repulsive force is greater. The following function usually constructs it:

$$U_m^{rep}\left(X_i\right)={\displaystyle \frac{1}{2}}{K}_{rep}{\left[{\displaystyle \frac{1}{\rho \left(X_i,X_m^o\right)}}-{\displaystyle \frac{1}{{\rho }_0}}\right]}^2, m=1,\cdots ,M,$$

(4)

where $K_{rep}$ is repulsion gain; $X_m^o=\left(x_m^o,y_m^o,z_m^o\right)$ represents the coordinate of the obstacle; $\rho \left(X_i,X_m^o\right)$ means the distance between the current position of the UAV i and the obstacle m; ${\rho }_0$ is the influence distance of the obstacle. The repulsion effect shall only occur if the distance from the track point X to the nearest obstacle is greater than ${\rho }_0$.

In the working environment of UAVs, both dynamical threats and static obstacles should be considered. To address such obstacles, we establish the following equation:

$$\{\begin{array}{cc}\begin{array}{c}{X}_m^o\left(k+1\right)=X_m^o\left(k\right)+v_m^o\left(k\right)T\\ v_m^o\left(k+1\right)=v_m^o\left(k\right)+u_m^o\left(k\right)T\end{array}& m=1,\cdots ,M\end{array},$$

(5)

where $k\in {\mathbb{Z}}^{+}$ is the sample instant; $T\in ℝ$ is the sample period; $X_m^o\left(k\right)={\left[x_m^o\left(k\right),y_m^o\left(k\right),z_m^o\left(k\right)\right]}^{\mathrm{T}}$ represents the position vector of obstacle m at k moment. Similarly, $v_m^o\left(k\right)$ and $u_m^o\left(k\right)$ denote the velocity and acceleration vectors.

Additionally, there are interaction relationships between the UAVs $V_i$ and $V_j$, where the potential function is defined as

$$U_{ij}\left(X_i,X_j\right)=K_u\left(d_{ij}\left(X_i,X_j\right)+{\rho }_0^2/d_{ij}\left(X_i,X_j\right)\right),$$

(6)

where $K_u$ denotes a positive parameter, $d_{ij}\left(X_i,X_j\right)=\Vert X_i-X_j\Vert$ represents the distance between $V_i$ and $V_j$, and ${\rho }_0$ is the minimum distance of between the UAVs.

The gravitational and repulsive forces received by UAV X are obtained by calculating the negative gradient of the UAV position according to the gravitational, repulsive, and interaction potential field functions, respectively:

$$F_{att}=-\nabla U_{att}\left(X_i\right)=-K_{att}\rho \left(X_i,X_i^G\right),$$

(7)

$$F_m^{rep}=-\nabla U_m^{rep}\left(X_i\right)=\{\begin{array}{c}\begin{array}{cc}\left\{\begin{array}{l}\left[{\displaystyle \frac{1}{\rho \left(X_i,X_m^o\right)}}-{\displaystyle \frac{1}{{\rho }_0}}\right]{\displaystyle \frac{-K_{rep}}{{\rho }^2\left(X_i,X_m^o\right)}}\cdot \\ {\displaystyle \frac{\partial \rho \left(X_i,X_m^o\right)}{\partial X_i}}\end{array}\right\}& \rho \left(X_i,X_m^o\right)\le {\rho }_0,\\ 0& \rho \left(X_i,X_m^o\right)>{\rho }_0,\end{array}\end{array}$$

(8)

$$\begin{array}{ll}{F}_{ij}& =-{\nabla }_{X_i}{U}_{ij}\left(X_i,X_j\right)\\ & =-K_u\left[1-{\rho }_0^2/d_{ij}\left(X_i,X_j\right)\right]{\nabla }_{X_i}{d}_{ij}\left(X_i,X_j\right).\end{array}$$

(9)

3.2. Improved Artificial Potential Field

The traditional artificial potential field is a simple algorithm with good real-time performance. However, a local extreme can quickly arise when the gravitational and repulsive forces acting on a UAV at a certain point are equal in magnitude but opposite in direction, causing the UAV to oscillate around the local extreme point. Alternatively, when obstacles are near the target point, the repulsive force will be significant, and the gravitation is relatively small, making it difficult to reach the target point. The greater the number of UAVs, the more likely it is to have these problems.

When a UAV reaches the target point and is on a collision path with an obstacle, it may be unable to reach the target. This is because it is pushed by the repulsive force generated by the obstacle, causing it to move at the same velocity and in the same direction as the obstacle is approaching, as illustrated in Figure 2a. If the path of the UAV to the target point coincides with the obstacle’s path away from the target point, and the directions of the repulsive force and gravitational force on the UAV are aligned, the UAV will move in the direction of the resultant force. The UAV cannot reach the target point once it reaches the equilibrium point where the gravitational and repulsive forces balance each other, as shown in Figure 2b.

To solve such a problem, Krogh et al. [43] multiplied the $U_{rep}\left(X_i\right)$ by the absolute distance between the UAV and the target. However, this method makes the potential field overlarge when the distance is too far. To handle it, Sfeir et al. [36] introduced a Gaussian function into the traditional repulsive potential function, which is shown in Equation (10). Furthermore, Liu et al. [37] introduced a coefficient into Equation (10), thus avoiding being trapped in the local minimum point.

$$G\left(X_i\right)=1-\exp \left(-{\displaystyle \frac{{\left(X_i-X_i^G\right)}^2}{R^2}}\right).$$

(10)

Remark 1.

Considering that real-time computational resources are limited due to the inherent restrictions of the communicational and computational abilities of the hardware, it is necessary to construct a new function, whose calculation process is relatively easy. Inspired by the above discussion, we intend to design a function f satisfying $f(\cdot )\to 1$ when the UAV the far away from the target, and $f(\cdot )\to 0$ when the UAV is close the target. Therefore, we propose the following pseudo-exponential function:

$$P\left(X_i\right)=\left(1-{\displaystyle \frac{1}{1+\beta {\left(X_i-X_i^G\right)}^2}}\right),$$

(11)

where $\beta$ is a positive adjustable parameter, $X_i-X_i^G$ denotes the distance between the UAV and the target.

Substituting Equation (11) into Equation (4) yields:

$$U_m^{irep}\left(X_i\right)={\displaystyle \frac{K_{rep}}{2}}{\left[{\displaystyle \frac{1}{\rho \left(X_i,X_m^o\right)}}-{\displaystyle \frac{1}{{\rho }_0}}\right]}^2\left(1-{\displaystyle \frac{1}{1+\beta {\left(X_i-X_i^G\right)}^2}}\right).$$

(12)

When the UAV is far away from the target, the repulsive potential is approximately equal to the traditional repulsive potential, making the shape of the field unchanged. If the UAV approaches the target, the whole potential field tends to be zero, making the UAV reach the stable state.

According to Equations (8) and (12), we further deduce the gradient of $U_m^{irep}\left(X_i\right)$, and then obtain the corresponding repulsive force:

$$F_m^{irep}=-\nabla U_m^{irep}\left(X_i\right)=\{\begin{array}{cc}{F}_m^{irep1}+F_m^{irep2}& \rho \left(X_i,X_m^o\right)\le {\rho }_0,\\ 0& \rho \left(X_i,X_m^o\right)>{\rho }_0.\end{array}$$

(13)

And the repulsive force consists of two parts:

$$\begin{array}{l}{F}_m^{irep1}=-K_{rep}\left[{\displaystyle \frac{1}{\rho \left(X_i,X_m^o\right)}}-{\displaystyle \frac{1}{{\rho }_0}}\right]{\displaystyle \frac{1}{{\rho }^2\left(X_i,X_m^o\right)}}\left(1-{\displaystyle \frac{1}{1+\beta {\left(X_i-X_i^G\right)}^2}}\right){\displaystyle \frac{\partial \rho \left(X_i,X_m^o\right)}{\partial X_i}}\\ F_m^{irep2}=K_{rep}{\left[{\displaystyle \frac{1}{\rho \left(X_i,X_m^o\right)}}-{\displaystyle \frac{1}{{\rho }_0}}\right]}^2{\displaystyle \frac{\beta \left(X_i-X_i^G\right)}{{\left[1+\beta {\left(X_i-X_i^G\right)}^2\right]}^2}}{\displaystyle \frac{\partial \rho \left(X_i,X_i^G\right)}{\partial X_i}}.\end{array}$$

(14)

In addition, to make the improved algorithm more stable, the algorithm is optimized to reduce vibration after reaching the target point, to limit acceleration, to manage velocity saturation levels, and to refine parameter selection.

(1)

Vibration optimization

Frictional resistance is fused into the algorithm to reduce the oscillation amplitude of the UAV after reaching the target point, shorten oscillation time, and make it stable:

$$F_{fri}=-K_{fri}{v}_i,$$

(15)

where $K_{fri}$ indicates the coefficient of frictional resistance. Therefore, the resultant force of a UAV refers to the vector sum of the gravitational force generated by the target point, the repulsive force of obstacles, the interaction forces among the UAVs, and frictional resistance:

$$F_{total}=F_{att}+F_m^{irep}+{\displaystyle \sum _{i=1,j\ne i}^N{F}_{ij}}+F_{fri}.$$

(16)

(2)

State constraints of the system

According to Equation (16), the theoretical acceleration of UAV can be obtained as:

$$a_i^k=F_{total}/\psi ,i=1,\cdots ,N,F_{fri}=-K_{fri}{v}_i,$$

(17)

where $\psi$ is the mass of UAV. Acceleration should be limited, with an absolute value not greater than the maximum acceleration $a_{\max }$, to ensure UAVs are continuous and stable in posture change. As a result, the actual acceleration can be obtained as:

$$a_i=\{\begin{array}{ll}{a}_i^k& a_i^k\le a_{\max }\\ a_{\max }& a_i^k>a_{\max }\end{array}, i=1,\cdots ,N, F_{fri}=-K_{fri}{v}_i.$$

(18)

Furthermore, the actual velocity of the UAV should be constrained in

$$v_i=\{\begin{array}{ll}{v}_i^k& v_i^k\le v_{\max },\\ v_{\max }& v_i^k>v_{\max }.\end{array}$$

(19)

(3)

Parameters adjusting

Parameters include the minimum safety distance ${\rho }_0$, gravitational coefficient $K_{att}$, and coefficient of repulsive force $K_{rep}$. In particular, the minimum safety distance is decided by the size, velocity, and GPS positioning accuracy of the UAV. The coefficients for gravitational and repulsive forces significantly influence the flight trajectory. For instance, if the repulsion coefficient is too small or the gravitational coefficient is too large, UAVs may fail to evade obstacles using the traditional artificial potential field method. Otherwise, unnecessary trajectory curvature will occur, or UAVs fail to converge to the target point. In summary, an essential step for this algorithm is to select appropriate gravitational and repulsive gain coefficients. Considering the distance from the UAV to the obstacle and the obstacle’s radius of influence, we conducted several simulations to adjust these coefficients and selected a more appropriate set of data conducive to avoiding obstacles. In the meantime, an area of action with limited repulsive force was set to reduce unnecessary turning maneuvers of trajectory.

According to the above discussions, the maximum velocity and acceleration of a UAV are $v_{\max }$ and $a_{\max }$. Assuming that the size of each UAV body is c, positioning accuracy of GPS is d, and s means the shortest distance for UAV to decelerate from the maximum velocity to zero, then we have:

$$s={\displaystyle \frac{v_{\max }^2}{2a_{\max }}}$$

(20)

To ensure the safe operation of the UAV cluster, we set the rational distance constraint. Hence, the minimum safety distance ${\rho }_0$ between two UAVs can be further given as:

$${\rho }_0=2d+2s+c$$

(21)

Next, a gain coefficient is determined. In this study, the coefficients of repulsive force and gravitation must satisfy two conditions:

First, the resultant force received by UAVs should be in the same direction as the gravity to make UAVs continuously approach the target point. It can be seen from Equations (7) and (11) that the gravitational force is opposing and the repulsive force is positive, so the resultant force should also be less than zero. Second, to ensure the safety of each UAV to the greatest extent and avoid a collision, the coefficient of repulsive force should be significant based on the first condition satisfied.

$$F_{att}+F_m^{irep}<0$$

(22)

This can be substituted into

$$\begin{gathered} -K_{att}\rho \left(X_i,X_i^G\right)-K_{rep}\left[{\displaystyle \frac{1}{\rho \left(X_i,X_m^o\right)}}-{\displaystyle \frac{1}{{\rho }_0}}\right]{\displaystyle \frac{1}{{\rho }^2\left(X_i,X_m^o\right)}}\cdot \\ \left(1-{\displaystyle \frac{1}{1+\beta {\left(X_i-X_i^G\right)}^2}}\right){\displaystyle \frac{\partial \rho \left(X_i,X_m^o\right)}{\partial X_i}}+ \\ {K}_{rep}{\left[{\displaystyle \frac{1}{\rho \left(X_i,X_m^o\right)}}-{\displaystyle \frac{1}{{\rho }_0}}\right]}^2{\displaystyle \frac{\beta \left(X_i-X_i^G\right)}{{\left[1+\beta {\left(X_i-X_i^G\right)}^2\right]}^2}}{\displaystyle \frac{\partial \rho \left(X_i,X_i^G\right)}{\partial X_i}}<0 \end{gathered}$$

(23)

Considering that $K_{att}$ and $K_{rep}$ are both greater than 0, we can deduce the following inequality.

$$\begin{array}{l}{\displaystyle \frac{K_{att}}{K_{rep}}}<{\displaystyle \frac{{\rho }_0-\rho \left(X_i,X_m^o\right)}{\rho \left(X_i,X_i^G\right){\rho }^3\left(X_i,X_m^o\right){\rho }_0}}\left(1-{\displaystyle \frac{1}{1+\beta {\left(X_i-X_i^G\right)}^2}}\right)\\ -{\displaystyle \frac{\left[{\rho }^2\left(X_i,X_m^o\right)-2\rho \left(X_i,X_m^o\right){\rho }_0+{\rho }_0^2\right]\beta \left(X_i-X_i^G\right)}{\rho \left(X_i,X_i^G\right){\rho }^2\left(X_i,X_m^o\right){\rho }_0^2{\left[1+\beta {\left(X_i-X_i^G\right)}^2\right]}^2}}\end{array}$$

(24)

The approximate range of gravitational and repulsive coefficients can be determined in light of the specific UAV working environment and the minimum safety distance. Then, following working time requirements, the appropriate values of $K_{rep}$ and $K_{att}$ are set.

3.3. Stability Analysis

Theorem 1.

For the UAV system described in Equation (1), construct the control law as Equation (16). If all coefficients are designed appropriately, all signals of the closed-loop system will be bounded [21].

Proof. 

Choose the Lyapunov-like function.

$$V\left(X\right)={\displaystyle \sum _{i=1}^N\left[U_{att}\left(X_i\right)+{\displaystyle \sum _{m=1}^M{F}_m^{irep}}+{\displaystyle \sum _{i=1,j\ne i}^N{U}_{ij}\left(X_i,X_j\right)}+\frac{1}{2}{X}_i^{\mathrm{T}}{X}_i\right]}$$

(25)

Since Equations (3), (6) and (12) are all non-negative, $V\left(X\right)\ge 0$ holds.

Calculating the derivation of Equation (25), it yields:

$$\dot{V}\left(X\right)={\displaystyle \sum _{i=1}^N{\dot{X}}_i^{\mathrm{T}}\left[{\nabla }_{X_i}{U}_{att}\left(X_i\right)+{\displaystyle \sum _{m=1}^M{U}_m^{irep}\left(X_i\right)}+{\displaystyle \sum _{i=1,j\ne i}^N{\nabla }_{X_i}{U}_{ij}\left(X_i,X_j\right)}+{\ddot{X}}_i\right]}$$

(26)

Equation (26) can be further deduced as

$$\dot{V}\left(X\right)={\displaystyle \sum _{i=1}^N{\dot{X}}_i^{\mathrm{T}}\left[-F_{att}-F_m^{irep}-{\displaystyle \sum _{i=1,j\ne i}^N{F}_{ij}}+{\ddot{X}}_i\right]}$$

(27)

Substituting the whole control force (16) into the UAVs system (1), we can obtain

$${\ddot{X}}_i=F_{att}+F_m^{irep}+{\displaystyle \sum _{i=1,j\ne i}^N{F}_{ij}}-K_{fri}{\dot{X}}_i$$

(28)

Based on Equations (27) and (28), we can finally obtain

$$\dot{V}\left(X\right)=-K_{fri}{\displaystyle \sum _{i=1}^N{\dot{X}}_i^{\mathrm{T}}}{\dot{X}}_i\le 0$$

(29)

Therefore, the Lyapunov condition is satisfied, and we can draw the conclusion that all signals are bounded and the proof of Theorem 1 is complete. □

4. Obstacle Avoidance Simulation Experiment and Result Analysis

The improved intelligent evasion method is implemented on a personal computer using the MATLAB® 2018b platform, with the following configuration: Intel Core i7-13700F CPU @ 2.10 GHz; 32 GBRAM; Windows 10 Professional Operating System. In the simulation of the following cases, the necessary parameters can be seen in

Table 1. Parameters for simulations.

Explanations	Value
Simulation step size T	0.02 s
Safety distance ${\rho }_0$	2 m
Maximum flight velocity $v_{\max }$	2 m/s
Maximum acceleration $a_{\max }$	9.8 m/s2
Initial velocity $v_0$	0 m/s
Gravitational coefficient $K_{att}$	10
Repulsive force coefficient $K_{rep}$	120
Designed parameter of the pseudo-exponential function $\beta$	50

. The initial position and target position could be determined according to the required formations, and the pseudocode of the algorithm is shown in Algorithm 1.

Figure 3 illustrates the shape of the artificial potential field. It highlights the issue of local minima in the traditional APF scheme, where UAVs are easily trapped at local minimum points, preventing them from reaching the target.

To show the effectiveness of the proposed improved APF algorithm, we demonstrate our scheme with traditional APF under the parallel flying of UAVs, and the corresponding results are shown in Figure 4. From Figure 4a, we can see that UAV5 cannot reach the target point, which means it is caught in a local minimum point. In Figure 4b, UAV5 successfully escapes from the local minimum and achieves obstacle evasion, which validates the feasibility of the proposed improved APF scheme.

Algorithm 1: Pseudo-exponential based Artificial Potential Field Method

Meanwhile, we demonstrate the control performance of the proposed scheme in the presence of three static obstacles. Figure 5 illustrates the evasion process of the UAV cluster. It can be seen that all UAVs successfully evade the static obstacles without any collision. Additionally, they all reach and stay in the predefined target points.

From Figure 6, it is evident that throughout the evasion process, the distances between the UAVs and the obstacles consistently exceed the minimum required distance. Moreover, Figure 7 indicates that each UAV maintains a safe distance from other UAVs. As a result, the UAV cluster successfully completes path planning and obstacle evasion while minimizing the risk of coming too close to the minimum safe distance under the implemented control algorithm.

The operational environment for UAVs is inherently complex, necessitating their ability to navigate around static obstacles, evade moving threats, and track targets concurrently. Hence, we further test the path-planning and obstacle evasion performance under static obstacles and moving threats. In this case, two moving threats traverse through a cluster of nine UAVs, and the whole dynamical process is shown in Figure 8. We observe that UAV8 successfully evades the dynamical threat (red cross). Meanwhile, when a threat (magenta cross) enters the cluster, UAV4 changes the path immediately and avoids the collision. Additionally, UAV5 first passes a static obstacle and then evades the threat. The remaining UAVs effectively evade static obstacles, enabling all of them to ultimately reach their target destinations. The specific distances between the UAVs and the obstacles are illustrated in Figure 9 and Figure 10. It is evident that these distances consistently exceed the safety thresholds. According to Figure 11, the safety criteria of the UAV cluster are never violated. Consequently, the proposed algorithm demonstrates effective control performance in this dynamic and challenging environment.

Moreover, to verify the robustness, we further test our scheme under four static obstacles and two dynamical threats, achieving a change in formation. Figure 12 shows that UAVs 1, 2, 4, and 9 successfully bypass the static obstacles. Meanwhile, UAVs 5, 6, 7, and 9 evade the dynamical threats (red cross) without any collision. Moreover, the whole cluster transitions from a rectangular formation to a cross-shaped formation and stays in the desired position. Then, we present Figure 13, from which we can observe that the distances between the UAVs and the obstacles meet the prescribed safety constraints. Additionally, in Figure 14, it is evident that UAVs 5, 6, 7, and 9 avoid the dynamic threats as they approach while maintaining safe distances. In Figure 15, all UAVs can escape from the extreme distance immediately and work safely throughout the entire flying process, indicating that the collision avoidance task is successful.

Additionally, we evaluate the performance of this method in large-scale multi-UAV (unmanned aerial vehicle) systems. Consequently, we designed a control system for a cluster consisting of 100 UAVs. Figure 16 demonstrates that all UAVs successfully reached their target positions under the control scheme, and the subfigures confirm that no collisions occurred within the cluster. To quantitatively assess the obstacle avoidance effectiveness, we created Figure 17, which shows the distances between the UAVs and the obstacles. It is clear that all UAVs maintain a safe separation as they approach the threshold distance.

Furthermore, the inter-UAV distances are illustrated in Figure 18, indicating that all UAVs adhered to the safety criteria throughout the entire motion process. Hence, we can conclude that the proposed method can achieve obstacle avoidance for at least 1 to 100 drones.

5. Conclusions

Focusing on the research of intelligent obstacle avoidance and path planning for a distributed UAV cluster, this study developed an improved artificial potential field algorithm. First, a mathematical model of a discrete-time 3D UAV was established. Then, to address the inherent local minima issue in the traditional artificial potential field method, a pseudo-exponential function was introduced. Concurrently, measures, such as vibration optimization, parameter tuning, and state constraints, are implemented to ensure the UAV cluster adeptly navigates dynamic obstacles, maintains a safe distance of 2 meters, and successfully reaches the target area in complex environments. Stability analysis was conducted using Lyapunov stability theory. Additionally, numerical simulation results showed that the problem of local minima was effectively resolved. The proposed scheme achieves path-planning, obstacle evasion, and formation changes under both static obstacles and dynamic threats for a cluster with nine UAVs. Meanwhile, the obstacle avoidance effectiveness of this method is verified under a cluster of 100 UAVs. Thus, the effectiveness and robustness of the designed algorithm were demonstrated. In our future work, we aim to implement the designed algorithm for path planning and obstacle evasion on a practical UAV cluster platform.