A Study of Collaborative Trajectory Planning Method Based on Starling Swarm Bionic Algorithm for Multi-Unmanned Aerial Vehicle

Abstract

This academic paper addresses the challenges associated with trajectory planning for affordable and light-weight Unmanned Aerial Vehicle (UAV) swarms, despite limited computing resources and extensive cooperation requirements. Specifically, an imitation-based starling cluster cooperative trajectory planning technique is proposed for a fixed-wing model of a six-degree-of-freedom UAV cluster. To achieve this, dynamic trajectory prediction of the rapid random search tree is utilized to generate a track solution adapted to the terrain environment. Additionally, the Dubins aircraft path solution is applied as it is suitable for executing input track commands by the UAV model. Computational simulations on different cluster sizes show the approach can maintain the cluster state while navigating diverse terrains, with the track solution complying with the UAV’s physical model properties.

1. Introduction

With the comprehensive development of UAVs, including updates to communication systems and signal processing technologies [1], as well as continuous improvement of autonomous decision-making capabilities [2], and optimization of cognitive algorithms [3], the application scenarios of UAVs have expanded. A single UAV is no longer sufficient to meet the quality requirements of military and civilian needs. The development of UAVs that can perform extensive and sophisticated tasks such as border management [4], disaster response [5,6], freight distribution [7], and target search [8] is moving towards the use of clusters of UAVs. Compared to single aircraft, UAV clusters outperform them in terms of search range, efficiency, and robustness. Effective cooperation algorithms are essential for enhancing the operational capacity of UAV clusters. Distributed structures can significantly reduce the complexity of the overall control algorithm and enhance the system’s robustness, compensating for the shortage of hardware in UAVs as the number of clusters continues to rise.

Unmanned cluster cooperative systems are designed to integrate the capabilities of each unmanned aerial vehicle (UAV) into a model architecture, achieving increased functionality and efficiency beyond what is achievable by a single aircraft. This enables the perception and processing of the same object from multiple angles. The application of basic UAV cluster systems can significantly reduce the cost of complex and sophisticated aircraft missions, while improving damage tolerance, through the use of effective layout and cooperative operations [9]. In nature, many species, such as bees, ants, and flocks of birds, are capable of accomplishing complex clustering tasks using simple organs and intellect [10]. Algorithmic models of biocluster collaboration have been developed based on the understanding and modeling of biological behavior, and have proven advantageous in various application scenarios [11].

The distributed architecture of bionic clusters comprised of multiple unmanned aerial vehicles (UAVs) endows them with high fault tolerance and robustness, making them suitable for distributed centerless self-organization [12,13]. This architecture allows for a multi-UAV system that is highly compatible and scalable. The UAVs simultaneously perform sub-tasks, enabling the splitting and parallel execution of the overall task, thereby enhancing execution efficiency [14]. Distributed UAV cluster collaboration algorithms draw inspiration from biological models where groups of organisms follow parallel behavioral rules but work together to accomplish complex group tasks such as foraging, defense against enemies, and nesting. These biological groups create a coordinated and orderly state that promptly and consistently responds to external interference patterns [15,16]. By translating the internal principles of biological cluster behavior to cooperative UAV cluster control, it is possible to significantly enhance multi-UAV systems’ autonomous decision-making and intelligent planning in complex situations. The bionic cluster model is highly adaptable, flexible, and effective. However, it is limited by the abilities of individual biological organs that make up the cluster. These limitations include a propensity for the cluster to become trapped in a local optimum solution, lengthy model convergence times, and limited individual intelligence that result in stereotypical performance in autonomous decision-making scenarios. Modern UAVs are much more sophisticated than many living creatures, with exquisite computational and perceptual capacities due to years of development. Consequently, the bionic algorithm used should effectively take advantage of the individual UAV’s capabilities, and improvements and optimizations for UAV operating circumstances can be developed within the bionic-based model framework.

The issue of controlling clusters of UAVs for low-speed, segmented flying duration rotorcraft has been extensively investigated in academic circles. For instance, Bisphagnia University conducted an experiment involving 20 quadrotor UAVs [17], which utilized an optical motion capture system for onboard attitude resolution estimation and control. The system is capable of autonomous operations via an external positioning system and adopts a group organizational structure, wherein centralized planning and control are closely coordinated in small teams. In addition, relatively parallel and independent distributed processing occurs between small teams, allowing the system to use a virtuous cycle to ensure the experiment’s success. Similarly, the University of Southern California’s Preiss team’s indoor dense formation experiment with 49 UAVs mostly relies on the airframe to perform the majority of processing tasks, sensor fusion, and trajectory planning. The formation flight also utilizes the virtual structure method [18]. The Vicsek model, which aims to facilitate coordinated control of clusters in constrained space, is an essential multi-intelligence motion control model. It enabled the orderly construction of 30 rotary-wing UAVs outdoors with a top speed of 8 m/s [19,20]. Furthermore, the US Naval Institute’s Pilot-Follow methodology allowed for the formation flight of 20 fixed-wing UAVs in the fixed-wing sector [21].

Fixed-wing drone clusters are mostly military research experiments in confrontation, pursuit, attack, and other scenarios since they are more frequently deployed for military purposes. The ‘Pixie’ project of the Defense Advanced Research Projects Agency (DARPA) attempts to launch, fly, and land a number of X-61A drones from the air. The Partridge project is comparable. A large-scale fixed-wing UAV cluster flight project is also being worked on by China Electronics Technology Group (CETG), but only a small number of videos ranging from 67 to 200 have been made public. The majority of comparable military projects have the specifics of their work kept secret. The enemy mission must create its own cluster advantage for the UAV cluster to prevail in a conflict, such as the number of advantages, coordination in intelligence advantage, firepower advantage, etc. As a result, intelligent combat is a key area for multi-UAS development, and it consists of four components, of which the UAV cluster is one: the combat model framework, effectiveness analysis, cluster autonomous control decision-making methods, and experimental simulation platform [22,23,24,25]. It is the primary component that controls how the multi-UAS operates. Iterative evolutionary algorithm models affected by the development of artificial intelligence, UAV swarm models [26,27], and bionic swarm models based on behavioral norms inspired by biological swarms can all be used to govern UAV swarms autonomously [28]. The collaborative process is broken down and modeled before the sub-problems are resolved using a number of techniques [29].

The objective of the bionic swarm model is to revisit the cooperative group behavior mechanism of biological swarms and abstractly extrapolate it into the self-organizing behavior regulations of UAV swarms. Because multi-UAV swarms and bird swarms are similar in nature, different flock models of birds are frequently used in UAV swarm systems. For instance, the Beynolds-proposed BIRD model [30], which simulates bird flock behavior and exhibits the consistency of a flock of birds, is based on three fundamental rules: separation, aggregation, and coherence. The drones’ perceptive capabilities and behavior norms are set up by Gaudiano’s team using a non-central decision-making control mechanism [31], which also enables distributed drone cluster search for tracking and striking targets. The Allison team develops a probabilistic state transition model for autonomous behavior decision-making at the macro level of individual UAVs using the Probabilistic Finite State Machine (PFSM) method based on a decentralized hybrid architecture for multi-UAV systems to adapt to dynamic environments [32]. The team of UAV swarms developed a synchronous control approach to execute multi-point attacks on chosen targets, which showcases the swarm intelligence of UAV swarms [33]. The model’s simulation results demonstrate durability and reliability, this tactic demonstrates the concept of scalability.

Pheromone algorithms, artificial immune algorithms, particle swarm algorithms, simulated annealing algorithms, genetic algorithms, and other creative algorithms are used to iteratively improve the model of the UAV cluster [34,35,36,37]. The cluster architecture is carefully optimized based on artificial intelligence. Using an overlay of pheromone-tagged maps, Sauter’s team suggested a completely distributed digital pheromone algorithm that can effectively and simultaneously manage large-scale intelligent body systems for path planning [38,39], cooperative reconnaissance, and target tracking. The US Naval Institute discusses linear and particle swarm optimization algorithms for dealing with handling adversary issues [40], respectively. The findings of particle swarm algorithms cause more mistakes due to variances in UAV motion and particle motion, and linear algorithms are more in line with the performance characteristics of UAVs, but this approach offers a promising solution to deal with the cluster solving of adversary challenges. Ernest examined an improved tactical decision processing-learning system based on genetic fuzzy trees to address the UAV cluster combat game challenge [41]. The approach breaks down the problem analysis into smaller, more manageable chunks, and dealing with those first. The method then uses genetic algorithms to solve the remaining issues and learn–understand–adapt–memory problems. It exhibits strong robustness and fault tolerance in challenging, uncertain battlefield environments, has excellent applicability, and can be used in a wide range of combat scenarios.

In this thesis, we present a dynamic path planning algorithm for complex UAV flight spaces and construct a cluster bionic algorithm framework to address the needs of multi-UAV cluster cooperative flight, path planning, and cluster control in complex adversarial environments. The starling swarm bionic algorithm alone is not effective for UAV clustering due to its lack of anticipation of subsequent movements, large variation in UAV control, and interferences between UAVs. To address these limitations, we optimize the algorithm by adding prediction based on the cluster as a whole and combining it with a six-degree-of-freedom UAV model. This makes the algorithm more applicable to UAVs and closer to real-world operations. The addition of a prediction link improves flight efficiency and reduces interference and loss among individuals. Combining the starling swarm bionic algorithm with the RRT algorithm enhances the cluster’s autonomous decision-making capability and enables it to cope with increasingly complex terrains.

In response, the thesis uses fixed-wing UAVs as the research subject, focuses on the flight process of multi-UAV systems, revolves around the issue of finding workable solutions for maintaining adaptable and effective clustering patterns in complex environments, and incorporates the use of tools such as the Bionic Eustachian Flock algorithm, artificial intelligence models, and control system theory to complete the research on the Bionic Cluster Synergy algorithm.

This paper is divided into four parts. The first part describes the principle and algorithm structure of the modified starling bionic algorithm for UAV clusters. The second part discusses the UAV path planning strategy based on the RRT algorithm. The third part presents the algorithm framework and implementation process of the overall clustering system, along with cluster flight path simulation results. Concluding remarks are given in the final part.

2. Starling-Mimicking Flock Algorithm

This model presents a strategy for simulating intelligent group synergy that was motivated by starling swarming behavior [42]. It builds a flock synergy algorithm from four aspects: local perception, locomotor behavior, safety avoidance, and adaptive evolution through simple behavioral rules of interaction between agents, to achieve a description of the entire group movement in the group synergy problem. This algorithm uses the concept of centerless self-organization to find the optimal solution from the information of its nearest neighbors. In the solution space, the evolution from disordered behavior to ordered behavior is described.

To discuss the synergy problem, which describes the co-movement structure of a population, we set the cluster size of agents to $Na$, $agen{t}_i$ and $agen{t}_j$ denote the target agent i and the surrounding agents j, respectively, where $i,j\in (1,2,3,\dots ,Na)$, and $i\ne j$. The intelligent body $agen{t}_i$ learns by perceiving the surrounding near-neighbors $agen{t}_j$ and adjusts its own motion behavior to achieve a certain safe distance from the surrounding intelligent bodies, and after a period of time $\Delta t$ is able to satisfy:

$$\left\{\begin{array}{c}\underset{t\to t+\Delta t}{\lim }∥\frac{1}{Na}{\displaystyle \sum _{i=1}^{Na}}\frac{v_i}{∥v_i∥}∥\to 1\hfill \\ \underset{t\to t+\Delta t}{\lim }{\displaystyle \sum _{i=1}^{Na}}{\displaystyle \sum _{j=1}^{Na}}∥v_i-v_j∥\to 0\hfill \\ \underset{t\to t+\Delta t}{\lim }{\displaystyle \sum _{i=1}^{Na}}{\displaystyle \sum _{j=1}^{Ni}}∥p_i-p_j∥<disN\hfill \end{array}\right.$$

(1)

where $v_i$ is the velocity of the ith rack of agent, $p_i$ is the position of the ith rack of agent, $Ni$ is the set of agent close to the $agen{t}_i$, and $disN$ is a threshold value for the cluster distance.

When the target equation is met, the entire group will continue to travel in a somewhat compact formation structure with essentially the same speed and direction, displaying an overall ordered movement condition. In order to establish a macroscopically ordered movement behavior of the intelligent body cluster under distributed and centerless control settings, each intelligent body interacts with the surrounding neighboring agent bodies. This is the essence of starling group synergy. Because of the shorter interaction distance and higher interaction frequency, it is a more dependable and secure flight procedure. The clustering algorithm is specifically a cluster collaboration algorithm based on behavior.

2.1. A Model of Starling Cluster Behavior

Each individual starling interacts with its immediate neighbors, which are often seven to eight in number but will hereafter be referred to as seven, in a localized and frequent manner. According to Cavagna and Cimarelli’s research [43], because they are situated at a set distance from the starling’s perceptual range, the closest seven to eight neighbors within that range are typically identified as nearest neighbors. The closest neighbor is not always chosen as the closest person since starlings rely on visual perception to assess the position and status of other people, but rather the closest observable unobstructed companion inside the field of view. Two starlings will not be able to communicate with one another if they are hidden by other items. As a result, while the entire cluster may exhibit a variety of behaviors, those of the individual and its seven nearby neighbors are essentially homogeneous and similar. This is illustrated in Figure 1, where the blue circle represents the perceived range and the brown circle represents the perceived individual.

Five agents behaviors—gathering behavior, dispersal behavior, avoidance escape behavior, return behavior, and free flight exploration behavior—are designed in this paper based on behavioral observation of a flock of starlings and the flight situation applicable to UAVs, combined with the $RRT$ algorithm in the latter section.

• Gathering behavior (Figure 2a):
  When $agen{t}_i$ performs the gathering behavior, set the current position of $agen{t}_i$ to $p_i$, retrieve other agents in the sensing range $R_p$, and select the seven nearest neighbors $agen{t}_j$. A central location C is determined according to the location $p_j$ of each nearest neighbor object, and $d_{ic}$ denotes the distance between $agen{t}_i$ and the centroid C of the nearest neighbor, if $d_{ic}>R_{maxl}$ which is beyond the farthest range $R_{maxl}$ of keeping distance, the centroid C is calculated as:

  $$C=\frac{1}{7}\sum _n^i{p}_n$$

  (2)
  When the agent performs gathering behavior, $agen{t}_i$ flies in the direction of the nearest neighbor centroid C as shown in Equation (3) The average velocity of the nearest neighbor cluster $v_{neighbo{r}_i}$ is the average of the seven nearest neighbors:

  $$v_{neighbo{r}_i}=\frac{1}{7}\sum _n^7{v}_n$$

  (3)
  The acceleration $a_i$ of the agent body $agen{t}_i$ is calculated as shown in Equation (4):

  $$a_i=K_d(d_{ic}-R_{maxl})+K_v(v_{neighbo{r}_i}-v_i)$$

  (4)

  where $K_d$ and $K_v$ are the displacement deviation and velocity deviation weight parameters, respectively.
  The values of $K_d$ and $K_v$ determine the sensitivity of the UAV to the clustering state. These values are adjusted based on the movement speed of each UAV in the cluster and the expected distance between individuals. A larger value for $K_d$ makes the UAV more sensitive to distance and increases the amplitude of its clustering behavior. The value of $K_v$ smooths the clustering behavior and constrains the amplitude of the UAV’s actions, resulting in a smoother clustering process.
• Dispersal behavior (Figure 2b):
  When $agen{t}_i$ performs dispersion behavior, $d_{ic}$ denotes the distance between $agen{t}_i$ and the nearest neighbor center point C, if $d_{ic}<R_{minl}$ that is, it enters the nearest range $R_{minl}$ of keeping distance, when the agent body performs dispersion behavior, $agen{t}_i$ faces away from the nearest neighbor center point C in the direction, the acceleration $a_i$ of the agent body $agen{t}_i$ is calculated as shown in Equation (4):

  $$a_i=\frac{K_d}{(R_{minl}-d_{ic})}$$

  (5)

  where $K_d$ is the displacement deviation.
  Figure 2. (a) Gathering behavior. (b) Dispersal behavior.

  Figure 2. (a) Gathering behavior. (b) Dispersal behavior.
• Return behavior:
  After conducting specific experimental tests, it was found that it is less efficient to return to the cluster by gathering behavior when multiple agents are far away from more than the cluster because of exploring the environment, so a returning behavior is defined, $C_a$ denotes the center of the cluster, $d_{ia}$ denotes the distance from the agent to the center of the cluster, and when $d_{ia}>R_a$, it means the agent has left the cluster. Performing regression behavior, $R_a$ denotes the cluster activity range, which is a dynamic variable. When the cluster is in a safe flight space $R_a$ is at a smaller value, constraining the cluster to fly as a compact whole, and when the cluster is in a complex environment or disturbed by invasion $R_a$ is at a larger value, enabling the cluster to have a relatively large margin of space to escape and evade, shown in the Figure 3. When none of the agents in the whole cluster is disturbed by the external environment and it lasts for a period of time, the environment at that time is defined as a safe space. If the disturbance information is received by one agent body, then the information will be quickly propagated to the whole through individual local information interactions, canceling the definition of safe space. The acceleration $a_i$ of the regression behavior is calculated as follows:

  $$a_i=K_{dr}(d_{ia}-R_a)+(K_{vr}{v}_{neighbo{r}_i}-v_i)$$

  (6)

  where $K_{dr}$ and $K_{vr}$ are the displacement deviation weight parameter and velocity proportional adjustment parameter, respectively.
  When traversing certain terrains, the cluster may split into two clusters. In such cases, it is important for the two clusters to quickly reconstitute as a whole. This can be achieved by adjusting the $K_{dr}$ and $K_{vr}$ parameters. Increasing the value of $K_{dr}$ results in a greater acceleration of cluster regression. Increasing the value of $K_{vr}$ increases the desired speed of cluster regression, allowing for faster cluster regression even in challenging terrains such as narrow mountains.
  Figure 3. Flight space transformation diagram.

  Figure 3. Flight space transformation diagram.
• Avoidance escape behavior:
  The avoidance escape behavior is the highest priority behavior, which is triggered when an individual agent is disturbed by external factors. The individual agent that perceives the disturbance will randomly choose a direction to avoid the disturbing individual and start to escape, continue accelerating to the maximum speed $v_{max}$ during the escape process, and stop the behavior when the disturbing individual is outside the perception range for a period of time, and at the same time, the agent will synchronize the escape information to the surrounding neighbors. The agent that receives the synchronized escape information but does not perceive the interfering individual will imitate and learn from the agent at the outermost part of the current perception range $R_p$ and will no longer stay in close formation, but will change into a spread formation away from the center of its immediate neighbors C. The escape acceleration is calculated as:

  $$a_i=\frac{K_a}{∥p_d-p_i∥}$$

  (7)

  where $K_a$ denotes the escape acceleration parameter and $p_d$ denotes the position of the interfering individual.
  The value of $K_a$ determines the amplitude of the cluster’s evasive escape behavior. Setting a larger value, based on the acceleration capability of the UAVs, allows the cluster to reach its maximum escape speed faster and avoid danger in a timely manner.
• Free flight exploration behavior:
  Free flight exploration behavior is the lowest priority intelligent body behavior. When the agent body keeps in the appropriate position interval $[R_{minl},R_{maxl}]$ in the near-neighbor group, the starling will make a comparative judgment of its own state and the surrounding near-neighbor individuals, and adjust its flight state. When the flight position estimated by the agent body based on its own state is better than other individuals, in this paper this judgment criterion is the distance $d_{aim}$ from the target position, the intelligent body chooses to explore the flight and randomly selects a direction to fly. When the agent body’s flight position is not the best among its immediate neighbors, the intelligent body executes a learning flight strategy to adjust its position by selecting a learning object. The better the current flight position, the higher the probability of being selected as the learning object, and the probability $l{p}_i$ is shown in the Formula (8):

  $$l{p}_i=\frac{(1/d_{j,a})}{{\displaystyle \sum _j^7}{d}_{j,a}}$$

  (8)

  where $d_{j,a}$ is the distance of each nearest neighbor individual $agen{t}_j$ from the target position. According to the calculated probability, one nearest neighbor UAV is selected as the learning object, and the learning flight policy update speed is given by:

  $$a_i=K_l(v_j-v_i)+K_{rand}\Delta v_{rand}$$

  (9)

  where $K_l$ is the learning flight weight parameter, the larger this parameter is the faster the rate of cluster convergence, $K_{rand}$ is the weight function of randomly explored paths, which determines the intensity of exploring feasible paths at each iteration, and $\Delta v_{rand}$ is the increment of randomly chosen velocity and heading.
  By interchanging the above five behaviors under different conditions, it is possible to achieve a cluster of agents to keep in a certain formation for flight missions.

2.2. Drone Half-Spring Model

A semi-spring model algorithm is utilized to achieve companion and obstacle avoidance for the cluster’s individual UAVs in order to ensure that they can successfully finish their mission without the risk of collision.

$$\begin{array}{c}\dot{v}s.=-K_{spring}(R_{spring}-d_{i,j})+K_{v}s(v_i-v_j)\hfill \\ R_{spring}=\left\{\begin{array}{c}{R}_e{d}_{i,j}<R_e\hfill \\ R_m{d}_{i,j}>R_m\hfill \\ d_{i,j}else\hfill \end{array}\right.\hfill \end{array}$$

(10)

where $K_{spring}$ denotes the spring elastic recovery coefficient, $K_{v}s$ denotes the kinetic energy weight coefficient, $d_{i,j}$ denotes the distance between two UAVs $agen{t}_i$ and $agen{t}_j$, $R_e$ and $R_m$ denote the repulsion radius and holding radius, respectively, and positive values represent the acceleration direction for the UAV $agen{t}_i$ pointing to the $agen{t}_j$ direction.

This study sets characteristics for UAVs such as perception radius, rejection radius, and holding radius based on the aforementioned methodology, as shown in Figure 4. It also defines a collision radius to determine whether a collision between UAVs has occurred.

• perception radius:
  Each $agent$ can perceive the maximum radius of the search, that is, the perception radius $R_p$
• holding radius:
  An appropriate distance between each $agent$ and its neighbors is used to constrain the tightness of the cluster. The drones within this range and the current drone do not interfere with each other to perform various actions, that is, maintain the radius $R_m$.
• rejection radius:
  In order to avoid collisions, UAVs need to receive early warning of upcoming dangers, so a repelling range is set within the surrounding range of UAVs, and the repelling radius $R_e$ is the minimum distance that can be maintained between UAVs.
• collision radius:
  The collision radius is the range in which the UAV cannot successfully complete the avoidance behavior for the encountered object, which is defined as the collision radius $R_{collision}$.

The near-neighbor UAV uses an attraction strategy when it is both inside the sensing range and beyond the holding range. To do this, the acceleration caused by the UAV $agen{t}_j$ pulling on $agen{t}_i$ is estimated using Equation (10). A holding strategy is applied to the drone after it is in holding range to synchronize the speeds of the two sides. The acceleration caused by the pressure created when a drone enters the present drone’s repulsion range is calculated using Equation (10). The UAV may keep a safe range of motion with its close neighbors by utilizing the three “repel–hold–attract” techniques.

The formula for the UAV update speed at each step can be determined by modeling the UAV swarm in this chapter in terms of the UAV swarm and the individual:

$${{\dot{v}}^k}_i=K_{ai}{a}_i+K_{vi}\sum {\dot{v}}_i$$

(11)

The speed update reference values among them are $a_i$ and ${\dot{v}}_i$, which were computed using the half-spring model and the starling behavior, respectively. The weight parameter governing starling swarm behavior is represented by $K_{ai}$, and the weight parameter governing a single UAV’s safe flying is represented by $K_{vi}$. The two weight parameters can be modified to alter the UAV swarm’s operating mode. Cluster convergence and divergence will be more adaptable and sensitive to interference with individual perception while focusing on swarm activity. The manner in which people react to disturbances seems to be fairly slow.

2.3. Overall Flow of the Algorithm to Mimic a Flock of Starlings

Figure 5 depicts a flowchart of the starling cluster algorithm. The algorithm in this part is predicated on the existence of a drone cluster in the space, where N is the size of the cluster. The drones in this cluster are represented by $agen{t}_1$, $agen{t}_2$, $agen{t}_3$, …, and $agen{t}_N$. Each drone $agen{t}_i$ has a specific flight speed $v_i$. Each drone’s initial position is unique, out of range of other drones’ collisions, and its initial speed and direction are both randomly generated. Each UAV also has a memory function that allows it to estimate and evaluate its own state using a value function. Before receiving the information from these drones as the parameter $a_i$ of the cluster speed update method, one of the drones, denoted by $agen{t}_i$, searches for other drones within the current perception radius and chooses seven of them as neighbors, designated by $agen{t}_j$. Check the collision radius, the repulsion radius, the presence of UAVs and other obstructive individuals, and the presence of neighboring UAVs inside the maintenance radius and outside the perception radius, and update the parameters $v_{ij}$ of the individual speed algorithm based on this data. Through the use of the speed update equation, state update equation, and position update equation, the UAV dynamically modifies the associated speed, state, and position after a unit of flying time. One cycle is finished after the aforementioned steps have been repeated once for each UAV in the cluster. The drone group’s task completion is estimated using the value judgment function, and the consistency of the cluster state is determined using the cluster description function.

3. Rrt UAV Path Planning

3.1. Rrt Path Planning

LaValle and Kuffner first introduced the fast random search tree algorithm ($RRT$) as a method for quickly searching nonconvex high-dimensional areas by constructing a tree structure that randomly fills the area. The technique, which has advantages for coping with obstructions and differential spatial limitations, is built by taking random sample increments from the search space. It has been extensively utilized for a variety of motion robot path planning issues.

A directed graph’s unique search tree structure is called a fast random search tree. Directed line segments link the child nodes to the parent nodes. One node is connected to another, from the root node to create the structure of the search tree; each child node corresponds to a parent node, and a parent node corresponds to n child nodes. Both connected line segments and nodes have distinct physical meanings in the $RRT$ algorithm, with connected lines denoting a path with direction and distance and nodes denoting particular location state data. The cost of each node may be determined and the path structure can be optimized by predicting the path between two nodes.

The path planning problem in this section is described as the search for a continuous feasible path solution in a metric space X from the starting point $x_{init}$ to the target region $X_{goal}\in X$ or the target location $x_{goal}\in X$, where there are various types of no-fly regions $X_{obt}\in X$ in the space that must be avoided, where the intelligent body is not aware of the expression of each no-fly. The limits of the development space X will be included if they are developed and the random exploration tree can be constructed in all $X_{free}$ areas, which are defined as all regions of the metric space X excluding the no-fly zone $X_{obt}$. The workflow of the fast random search tree is shown in Figure 6.

3.2. Path Smoothing

The path solution obtained by the fast probe random search tree algorithm in the preceding section has the issue of low efficiency; there are many paths that are consumed by each other, wasting agent energy as well as the lengthy action time to complete the path, and the path solution is smoothed for this issue.

Connect the third node $x_3$ on the path to form a new path $C_{13}$ from the starting point $x_{init}$. If the new path $C_{13}$ in $X_{free}$, connect the next node $x_4$. Repeat the previous step until the path $C_{1n}\notin X_{free}$ is connected to a specific node $x_n$. To find a new alternative path, connect from $x_{n-1}$ to $x_{n+1}$, and carry on with this optimization procedure until this path comes to an end. The optimization’s outcome is depicted in Figure 7. On the basis of the original path, remove inefficient nodes and create a new, shorter path. This technique enhances the RRT algorithm’s ability to find paths, and it can additionally be applied to the UAV’s three-dimensional space path.

3.3. Spatial Path Planning Algorithm

The $RRT$ algorithm must be extended from the two-dimensional plane to three-dimensional space and be able to detect collisions in three-dimensional space in order to be used for UAV cluster trajectory planning. This section of the algorithm uses discrete path sampling first, followed by the use of sampling points on the map to detect collisions. This effectively reduces the algorithm’s computing power requirements and accelerates operation; however, if the sampling frequency is too low, the path may pass through a barrier without a collision being detected, which is known as the mode phenomenon. In order to keep the experimental findings readable, invisible barriers were placed at the map’s edge to limit the pathways to a specific visual range, and for the purposes of the later UAV path planning requirements, the paths could not be lower than the height of the ground. Additionally, the possible locations for the flight path corner are constrained to the selected range of random points.

The random search tree algorithm in 3D terrain first selects a random location x in the location space reachable per unit interval time of the defined UAV model, finds the node $v^{\ast }$ closest to the point, plans the new node $v^{+}$, no collision can be generated in the path $(x^{\ast },x^{+})$, add the new path to the path graph, and when there is no obstacle in the path between the new node and the target node $x_e$, adds the target node to the path, and ends the algorithm. The 3D $RRT$ algorithm can handle the complicated static environment and can obtain a workable solution for the UAV flight path. The algorithm operates as illustrated in Figure 8, where the red path in the picture is the UAV path predicted by the algorithm.

The starling clustering algorithm lacks path planning for the entire goal, reacts immediately to the environment, and so is less effective at solving the resulting path. In order to account for the global aspect of the clustering technique and the path flight efficiency, this part uses the fast random search tree algorithm. The rapid random search tree algorithm is still in a better position even though it can plan a workable solution that can handle complex paths. The following section evaluates the shortest path for the UAV based on this section, modifying the attitude throughout the flight and arriving at the intended position and attitude in the shortest amount of time in order to better adapt the path planning to the UAV flying state.

4. Cluster Path Planning Method That Resembles a Starling Flock

4.1. Dubins Aircraft Path

The automobile path, with a height component added, serves as the foundation for the Dubins aircraft path. According to Chitsaz’s method [44], the path of the Dubins aircraft is separated into three categories based on the altitude difference between the beginning point and the ending position, the length of the Dubins car’s path, and the maximum output pitch angle $\left(\overline{\gamma }\right)$ of the aircraft [45]. Further incorporated are many circumstances, including low, medium, and high ascents. The Formula (12) yields the Dubins aircraft’s minimum turning radius:

$$R_{\min }=\frac{V^2}{g}\tan \overline{\varphi }$$

(12)

$\overline{\varphi }$ is the maximum tilt angle of the drone, V is the Aircraft airspeed, g is the Gravitational acceleration.

When the altitude difference $\Delta h=\left|x_{de}-x_{ds}\right|$ between the starting point and the ending point satisfies the following formula, it is a low climb situation:

$$\left|x_{de}-x_{ds}\right|⩽L_{car}\left(R_{\min }\right)\tan \overline{\gamma }$$

(13)

where $L_{car}\left(R_{\min }\right)$ is the minimum Dubins car path, and the right-hand side of the equation is the maximum altitude increment for flight climb at the limiting flight pitch angle $\overline{\gamma }$. $\Delta h$ satisfies the Equation (14) for the medium climbing case:

$$L_{car}\left(R_{\min }\right)\tan \overline{\gamma }\left|x_{de}-x_{ds}\right|⩽[L_{car}\left(R_{\min }\right)+2\pi R_{\min }]\tan \overline{\gamma }$$

(14)

The high climb case applies when at a climb height even greater than the minimum path plus a section to complete the circling $2\pi R_{\min }$ ascent:

$$R_{\min }=[\frac{V^2}{g}+2\pi R_{\min }]\tan \overline{\varphi }$$

(15)

The Dubins aircraft trajectories are produced for each of the three scenarios in the following three subsections:

• Low Climbing Dobins Path:
  The height increase between the start and end points can be achieved by increasing the flight path to satisfy the necessary pitch angle within the limitations, provided that the required climb height does not exceed the minimum path’s limited climb height. The optimal route can be determined using the following equation:

  $${\gamma }^{\ast }={\tan }^{-1}(\frac{\left|x_{de}-x_{ds}\right|}{L_{car}\left(R_{\min }\right)})$$

  (16)
  The formula below determines the length of the shortest flight path.

  $$L_{air}(R_{\min },{\gamma }^{\ast })=\frac{L_{car}\left(R_{\min }\right)}{\cos {\gamma }^{\ast }}$$

  (17)
  The steep climb of the Dubins flight route is based on the minimum path of the car with the inclusion of the optimal flight pitch angle ${\gamma }^{\ast }$, together with the pitch angles ${\psi }_s$ for the start point attitude and the pitch angles ${\psi }_e$ for the termination point attitude.
  A Dubins airplane path with a medium height difference of 75 m and a Dubins automobile route distance of 185 m are depicted in Figure 9.
  Figure 9. (a) Low-altitude climb Dobbins aircraft path. (b) Top view of the path.

  Figure 9. (a) Low-altitude climb Dobbins aircraft path. (b) Top view of the path.
• Mid Climb Dobins Path:
  In the case of a medium climb, the climb height is both greater than the model’s maximum height restriction at the minimum car path distance, is insufficient to add a climb to the initial path for a full climb spiral, and is also higher than the termination point without adjusting the proper pitch angle. This means that the pitch angle can reach the termination point under the model restriction, and only a section of an arc can be added to the path. In order to increase the path length in this paper so that the climb height with pitch angle $\gamma =\pm \overline{\gamma }$ is exactly $X_{de}-X_{ds}$, an additional section of path $\gamma =sign(x_{de}-x_{ds})\overline{\gamma }$ must be inserted. As indicated in Figure 10a, the arc is added after the starting helix if the final height is greater than the starting height. In Figure 10b, the arc is inserted before the end helix if the final height is lower than the starting height.
  Figure 10. (a) Arc insertion method one. (b) Arc insertion method two.

  Figure 10. (a) Arc insertion method one. (b) Arc insertion method two.
  The intermediate arc is parameterized as shown in Figure 11 to demonstrate how the Dubins path can be obtained for modest climb heights, where the expression for $z_i$ is given by Equation:

  $$z_i=c_s+R\left(\phi \right)(z_s-c_s)$$

  (18)
  On the basis of the discovered position $(z_i,{\psi }_s+\phi )$, a typical Dubins aircraft path is then plotted to the endpoint. The path is a distance from:

  $$L\left(\phi \right)=\phi R_{\min }+L_{car}(z_i,{\psi }_s+\phi ,z_e,{\psi }_e)$$

  (19)
  The pitch angle ${\phi }^{\ast }$ is determined using the binary search algorithm, which produces:

  $$L\left({\phi }^{\ast }\right)=tan\overline{\gamma }=\left|z_{de}-z_{ds}\right|$$

  (20)
  The following equation yields the corresponding Dubins aircraft path distance:

  $$L_{air}=\frac{L\left({\phi }^{\ast }\right)}{\cos \overline{\gamma }}.$$

  (21)
  The parameters required for modeling the planning of the Dubins aircraft path for the medium climb case are shown in Figure 11. The intermediate parameters introduced are $c_i,{\psi }_i,{\lambda }_i,w_i,q_i$, so the parameters of the Dubins aircraft path can be defined as:

  $$D_{air}=(R,\gamma ,c_s,{\psi }_s,{\lambda }_s,w_s,q_s,c_s,{\psi }_i,{\lambda }_i,w_i,q_i,w_l,q_l,c_e,{\psi }_e,{\lambda }_e,w_e,q_e).$$

  (22)
  Figure 11. Illustration of the criteria for the medium-height climbing Dobbins path.

  Figure 11. Illustration of the criteria for the medium-height climbing Dobbins path.
  A Dubins airplane path with a medium altitude variation of 175 m and a Dubins automobile route distance of 185 m are depicted in Figure 12.
  Figure 12. (a) Mid-altitude climbing Dobins aircraft path. (b) Top view of the path.

  Figure 12. (a) Mid-altitude climbing Dobins aircraft path. (b) Top view of the path.
• High Climbing Dobins Path:
  States within the model limitations cannot achieve height increments for starting or ending locations with significant height variations relative to the car path. To achieve the height increase, the Dubins car path can be extended by adding a specified number of spiral revolutions and increasing the turning radius at the beginning or end of the path. The minimum number of spiral revolutions required, if any, is the lowest integer k that satisfies the following conditions:

  $$(L_{car}\left(R_{\min }\right)+2\pi k{R}_{\min })\tan \overline{\gamma }⩽\left|x_{de}-x_{ds}\right|⩽(L_{car}\left(R_{\min }\right)+2\pi (k+1)R_{\min })\tan \overline{\gamma }$$

  (23)
  The formula for k is given by

  $$k=⌊\frac{1}{2\pi k{R}_{\min }}(\frac{\left|x_{de}-x_{ds}\right|}{\tan \overline{\gamma }}-L_{car}\left(R_{\min }\right))⌋$$

  (24)
  The formula $⌊x⌋$ represents a function that rounds x down to the nearest integer and then increases the radius of the beginning and ending spiral to $R^{\ast }$, achieving:

  $$(L_{car}(R\ast )+2\pi k{R}^{\ast })\tan \overline{\gamma }=\left|x_{de}-x_{ds}\right|.$$

  (25)
  To create a query in $R^{\ast }$ that answers Equation (25), use a tiling search. The resulting path, which is the ideal Dobins plane path, has the following formula for path length:

  $$L_{air}(R^{\ast },\overline{\gamma })=\frac{L_{car}\left(R^{\ast }\right)}{\cos \overline{\gamma }}$$

  (26)
  The path of the automobile is 185 m, the height difference is 400 m, and Figure 13 shows the course of the high-climbing Dubins aircraft. The minimum radius $R_{\min }$ is swapped out for $R^{\ast }$ when defining the high-rise Dubins aircraft path, which otherwise shares the same specifications as the Dubins automobile path. Increase the optimal flight path angle $\pm \overline{\gamma }$, the starting point flight pitch angle ${\psi }_S$, the terminal point flight pitch angle ${\psi }_e$, and the quantity of spiral spin k following the initial spiral.
  Figure 13. (a) High-altitude climbing Dobins aircraft path. (b) Top view of the path.

  Figure 13. (a) High-altitude climbing Dobins aircraft path. (b) Top view of the path.

This section explains how Dubins vehicle routes can be used for aircraft. Planning outcomes for the UAV path can be improved by switching the RRT algorithm’s path generation approach to the Dubins aircraft path.

4.2. Cluster Path Planning Method That Resembles a Starling Flock

According to the starling cluster cooperative method described in Section 2, the UAV cluster can move in space while maintaining a specific formation. However, the algorithm focuses on maintaining the cluster flight state and ensuring the safe flight of each UAV. To adapt the current UAV input commands based on prediction results with locally feasible solutions, dynamic prediction must first anticipate future behavior based on the current situation and gathered environmental information. This enables effective flight in challenging terrain and reduces the likelihood of getting stuck. The dynamic prediction process is divided into two phases: the first adjusts the system’s position and state within the cluster; the second uses the RRT algorithm to solve the path for the immediate environment based on these adjustments. The final step uses a value function to evaluate the path solution and determine a locally optimal path based on known cluster and environmental data. For each exploration of a new node, the starling clustering algorithm’s learning behavior generates a moving path and state update that is added as a set of branches to the value function estimation and final path solution selection. This increases the efficiency of obtaining feasible solutions. The value estimation function J is defined as follows:

$$J=\sum _{k=1}^{k_p-1}{L}_{Dubins}(x_i^k,x_i^{k+1})+K_a{L}_{Dubins}(x_i^{k_p},x_{aim}).$$

(27)

where $x_i^k$ denotes the kth prediction state of the ith UAV, $k_p$ denotes the number of prediction iterations, $L_{Dubins}(x^1,x^2)$ denotes the Dubins aircraft path distance between two nodes $x^1,x^2$, and $K_a$ is the target distance weight function. Increasing $K_a$ can speed up the speed of getting feasible solutions, but it also tends to lead to the space of no solutions.

The final outcome for the entire prediction process is specified as:

$${{\rm Y}}_p={{\rm Y}}_b({x^1}_i,{x^2}_i,\dots ,{x^{k_s}}_i,)+{{\rm Y}}_{rrt}({x^{s+1}}_i,{x^{s+2}}_i,\dots ,{x^{k_p}}_i,)$$

(28)

where ${{\rm Y}}_b$ denotes cluster-location-adjusted path prediction and ${{\rm Y}}_{rrt}$ denotes path planning prediction for 3D fast search trees.

$${{\rm Y}}_b=update\_starling({X^k}_i,k_s)$$

(29)

Equation (29) denotes position-adjusted path generation, ${X^k}_i$ is the set of all UAV state quantities within the sensing range of $agen{t}_i$, $k_s$ is the number of iterations required to enter the free exploration flight, and $starling\left(\right)$ the state update equation of the starling clustering algorithm.

$${{\rm Y}}_{rrt}=K_{learn}\left(starling({X^k}_i,(k_p-k_s))\right)+K_r\left(RRTpath({X^k}_i,(k_p-k_s))\right)$$

(30)

Equation (30) denotes the path solution generated by location path planning, $RRTpath\left(\right)$ is a three-dimensional spatial fast random search tree algorithm, $K_{learn},K_r$ is the corresponding weight function, $K_r$ the larger the predicted solution space per iteration is larger.

$${{\rm Y}}_i={{\rm Y}}_p{L}_{Dubins}({x^1}_i,{x^2}_i,\dots ,{x^k}_i)<v_i$$

(31)

${{\rm Y}}_i$ is the execution path of this iteration.

The longer the time in the non-secure environment, the more the $K_{learn}$ weight parameter varies with time:

$$\left\{\begin{array}{c}{K}_{learn}=K_{learn}-t(K_{learn}/t_e)t⩽t_e\hfill \\ K_{learn}=0else\hfill \end{array}\right.$$

(32)

In summary, the wider the solution space is for each iteration, the longer the UAV cluster is imprisoned in a complex environment.

As shown in Figure 14, the starling clustering algorithm’s structure is maintained, with a higher priority given to maintaining cluster positions in flight dynamics prediction than to path planning. The priority is adjusted to correct cluster positions in one iteration time increment according to path prediction planning, with the remaining time used for locally optimal path execution. When the number of predicted steps exceeds a certain threshold, it is likely to predict planning to the appropriate cluster position, especially after several iterations. We added a three-dimensional RRT to the clustering algorithm’s free exploration flight behavior.

Compared to the algorithm flow in Section 2 Figure 5, this algorithm incorporates the RRT method into the free exploration flight behavior and adds a path planning method while maintaining the overall structure of the starling bionic algorithm. This improves the algorithm’s ability to handle complex environments. Additionally, subsequent path prediction is added in each iteration to improve the cluster’s flight efficiency and reduce interference and loss among individuals.

The UAVs are initially distributed in space [0 100, 0 100, 50 150] at a safe distance, with initial velocities randomly selected between [40, 100] and heading angles set to 0. Figure 15 displays the algorithm’s test results in MATLAB, Each line represents the trajectory of a drone. The main parameters of the simulation experiment included a perception range of 260 m, a starling nearest neighbor holding distance interval of [35, 70], an individual safety distance holding interval of [30, 60], a cluster holding distance extension of 50 m after triggering the danger space signal, and a desired cluster flight speed of 50 m/s.

We conducted experiments in MATLAB using different terrains to test the method’s application in various terrain situations. As shown in Figure 16, a small obstacle was added to the cluster path in terrain 2, and the clusters can be seen travelling around it from different directions and regrouping. In Figure 17, the obstacle’s size was increased in terrain 3 and the cluster travelled around it from the right side. The two different obstacle avoidance methods arise from whether the UAV cluster’s internal structure is disrupted. When the UAV cluster is dispersed to a certain distance, the algorithm regroups the cluster to achieve obstacle avoidance as a whole.

Figure 18, Figure 19 and Figure 20 show the cluster flight of 12, 16, and 20 UAVs, respectively. It can be observed that the clusters are able to traverse various terrains while still maintaining a certain cluster structure. In the simulation environment, the volume of obstacles was increased to observe the state of UAV clusters traversing narrow terrain. When encountering narrow terrain, the clusters changed into a narrow formation for flight. The dynamic predictive path planning-based starling cluster collaborative algorithm can meet the requirements of various environments while maintaining its original scalability.

This section will extract the data from the algorithm process for the experimental findings in Figure 15 and explain them in greater detail in order to further examine the state changes of the complete cluster during the cluster flight. Figure 21 depicts the drone’s course change first.

The positive direction of pitch angle is deflected downward on the positive x-axis and the positive direction of yaw angle is deflected on the positive x-axis toward the positive y-axis, both expressed in magnitude. Figure 21 samples the cluster flight process at a frequency of 1 Hz. The yaw angle changes in the range of $[-1,3]$ and finally converges to the range of $[0,0.2]$, and the pitch angle changes in the range of $[-0.5,0.3]$ and finally converges to the interval of $[-0.1,0]$. This clearly shows that the UAV cluster has achieved essentially the same heading after crossing the complex terrain. The fluctuation of the heading’s second half results from the UAV’s dominant position in the clustering algorithm’s free flight exploration behavior.

Similar to this, the speed range when flying over difficult terrain is $[15,95]$, and finally all fall on the speed line of 50 m/s. The cluster operates at a constant speed. Sample their position information at four time nodes (5 s, 10 s, 50 s, and 80 s, respectively) to better examine changes in the status of the cluster structure (Figure 22 and Figure 23):

We demonstrate the effectiveness of the cluster cooperative control method using the track-following UAS design suggested by Beard and McLain. In this design, the path manager configures the flight path between two attitudes and positions based on the trajectory command. The vector field guidance then generates speed, altitude, and heading commands for transmission to the fixed-wing UAV’s autopilot. The UAV’s sensors provide information to the state estimator, which generates the necessary state estimates. In this study, a fixed-wing UAV mathematical model based on the control framework of the appeal architecture and the UAV model is utilized in MATLAB to test the viability of the proposed method. The input frequency of the command operates the system in this study from 10 Hz to 50 Hz, as indicated in Figure 24, and the results are shown here in the form of 1 Hz for viewing purposes.

The following trajectory of the model and the algorithm planning trajectory experience a significant error in the early stages due to the UAV position being randomly generated, as seen in Figure 25 from the overhead path. However, after a period of model parameter adaptation, it is then able to follow the algorithm planning trajectory more accurately.

At altitude, the algorithm produced a larger following error in the early part of the algorithm and a smaller following error in the later section, as shown in Figure 26. However, since the goal of this paper is to design a cluster collaboration algorithm rather than concentrating on the UAV autopilot control system, the model and control system are sufficient for algorithm testing and can show that the algorithm has good robustness and can execute the UAV model in the procedure. Even in the face of significant variations in the UAV model’s execution, the algorithm is strong enough to finish the task and maintain safe flying.

The algorithm can effectively maintain the cluster flight pattern of UAVs, keep the dispersed formation in the environment where pathfinding is necessary, solve the path more efficiently, tighten the cluster range in a safe environment, and form a tighter cluster pattern flight, according to comparisons between the heading and velocity change curves and the terrain map. A path planning scheme based on a fast search tree algorithm with dynamic prediction is added to improve the applicability of the algorithm to complex environments and the efficiency of the algorithm itself to maintain the flight pattern of the starling cluster in Europe. The starling cluster collaborative algorithm framework is applied to the UAV cluster, and the algorithm is targeted and optimized according to the characteristics of the UAV.

5. Conclusions

This work examines a clustering control algorithm for complex operating scenarios using fixed-wing UAV clustering. A bionic clustering algorithm framework is constructed to achieve consistent clustering flight of multiple UAVs by combining the behavior of starling clusters with a semi-spring model of individual UAVs. The bionic model includes a three-dimensional fast random search tree algorithm based on dynamic prediction and modifies the path calculation to Dubins aircraft paths. The algorithm improves cluster convergence and collaboration efficiency using the Starling bionic algorithm framework, enhances energy utilization, and increases the cluster’s ability to process environmental information. A three-dimensional flight environment is created for algorithm verification using MATLAB. Experimental data analysis shows that the algorithm enables UAV clusters to navigate complex environments while maintaining a specific cluster structure.