Formation Control Algorithm of Multi-UAVs Based on Alliance

Abstract

Among the key technologies of Multi-Unmanned Aerial Vehicle (UAV) leader–follower formation control, formation reconfiguration technology is an important element to ensure that multiple UAVs can successfully complete their missions in a complex operating environment. This paper investigates the problem of formation reconfiguration due to battlefield mission requirements. Firstly, in response to the mission requirements, the article proposes the Ant Colony Pheromone Partitioning Algorithm to subgroup the formation. Secondly, the paper establishes the alliance for the obtained subgroups. For the problem of no leader within the alliance formed after grouping or reconfiguring, the Information Concentration Competition Mechanism is introduced to flexibly select information leaders. For the problem of the stability of alliance structure problem, the control law of the Improved Artificial Potential Field method is designed, which can effectively form a stable formation to avoid collision of UAVs in the alliance. Thirdly, the Lyapunov approach is employed for convergence analysis. Finally, the simulation results of multi-UAV formation control show that the partitioning algorithm and the competition mechanism proposed can form a stable alliance as well as deal with the no-leader in it, and the improved artificial potential field designed can effectively avoid collision of the alliance and also prove the highly efficient performance of the algorithm in this paper.

1. Introduction

Formation is an orderly movement phenomenon of a large number of autonomous individuals emerging at the group level based on simple local information interactions [1,2,3] and can be applied to the multi-UAV [4,5,6], Multi autonomous Underwater Vehicle [7,8], Auto guided Vehicles [9,10], and so on, including both grouping and subgrouping forms of movement [11]. Among them, grouping behavior is widely used in multiagent systems to achieve cooperative control, such as the assembly [12,13], multi objective monitoring [14,15], reconnaissance [16,17], search and rescue [18,19], attack [20], object detection [21], and so on. The subgrouping behavior has divided the group into small teams to perform the task. It is the interplay of the two behaviors that allow for the complete execution of a task.

In the context of air combat, when multiple UAV formations enter the battlefield to perform tasks, they need to be divided into multiple teams according to the task requirements of the battlefield to perform tasks separately, which can improve the task execution efficiency of the entire formation. Therefore, subgrouping is an important research content of formation reconfiguration. Reference [22] designed a novel swarming control method with multiple virtual leaders, which could solve the problem of subgrouping agent swarming systems with different desired distances and perceived distances, but it requires prior division of each subgroup according to the desired distance. Reference [23] designed control laws for two types of heterogeneous agents with different perceptual ranges, desired distances, and goals to achieve the purpose of the subgroup, but the method still needs to divide the group in advance. References [24,25,26] proposed the self-organized swarm control method based on the information coupling degree, but this type of method requires individuals with certain memory capacity and only achieves the subgrouping behavior under symmetric external stimuli.

After the grouping behavior is completed, the problem of formation assembly and control is another focus in the research of formation reconfiguration. Commonly used formation control techniques are leader–follower [27,28,29], virtual structure [30,31], behavior-based control [32], and consistency algorithm [33]. The leader–follower control method is an excellent intelligent control strategy and is gradually becoming the main method and research direction for solving swarm formation control. The basic idea is to select a UAV as the leader in the formation, with the other UAVs as followers. Reference [34] designed a following linear consistency algorithm based on the leader–follower technique, and the method enabled the follower to track the position, heading, and speed information of the leader, but it required high safety for the leader in the formation, as failure of the leader could make it difficult to maintain the formation. Reference [35] designed a virtual leader consistency algorithm; the algorithm used a virtual leader instead of the leader’s piloting role, which effectively overcame the disadvantage that the formation was difficult to maintain when the leader failed. However, it requires all UAVs to obtain information about the virtual leader, which will lead to a significant increase in computation when the number of formation members increases. Reference [36] designed a distributed UAV formation control method with time delay, and the method used the higher-order consistency theory to solve the distributed formation problem, but it required the system to be a fixed topology communication structure.

In summary, a large amount of literature has been devoted to the grouping and subgrouping behavior of multi-UAVs from various perspectives, but there are some problems:

• The formation needs to be divided into subgroups beforehand when forming new formations in subgroups;
• The member of the formation has a high dependence on the leader, so it is difficult to maintain the formation once the condition of the leader occurs;
• The members of the formation all have to obtain the information about the leader or virtual leader, resulting in a significant increase in computation when the number of members increases.

To address the above problems, this paper investigates a formation control algorithm of multi-UAVs based on alliance:

(1)

Adopting the concept of pheromone in the ant colony algorithm and using the principle of closest distance to create subgroups in the formation. This approach solves the problem of the need to divide subgroups in advance.

(2)

Adopting a concentration competition mechanism to select the leader. This method allows flexibility in determining the leader in the alliance and reduces the dependence of the followers on the leader.

(3)

Adopting the improved artificial potential field method to design the control law. This control law can avoid collision within the formation.

The flow chart of the control decision is shown below in Figure 1.

The rest of the paper is organized as follows. Section 2 introduces the multiple UAVs’ system model. Section 3 introduces the multiple UAVs’ formation algorithm based on an alliance algorithm. Section 4 introduces the alliance control based on improved artificial potential. Section 5 illustrates the performance and advantages of the method through simulation. Section 6 is the conclusion.

2. Construction of Multi-UAVs’ System Model

In this section, the basic model is formulated, which can be used to describe the motion state and communication mode of UAVs.

2.1. The Kinematic Model of the UAV

Consider the UAV as a point-mass in two-dimensional space with a position vector and a velocity vector, which can be expressed as:

$$\left\{\begin{array}{l}\dot{x}=v_i\\ {\dot{v}}_i=u_i\end{array}\right.i=1,2,\dots ,N$$

(1)

The speed vector $v_i$ and input control quantity $u_i$ of each UAV need to meet:

$$v_i=\left\{\begin{array}{ll}{v}_i,& ‖v_i‖\le v_{\max }\\ v_{\max }\frac{v_i}{‖v_i‖},& ‖v_i‖>v_{\max } \end{array}\right.$$

(2)

$$u_i=\left\{\begin{array}{ll}{u}_i,& ‖u_i‖\le u_{\max }\\ u_{\max }\frac{u_i}{‖u_i‖},& ‖u_i‖>u_{\max }\end{array}\right.$$

(3)

In the formula, $v_{\max }$ is the maximum speed and $u_{\max }$ is the maximum acceleration.

2.2. The Information Interaction Model of the Multi-UAVs

The information interaction between UAVs in formation, or between formations, is the basis for alliance formation.

• The information interaction mode: adopted the fixed neighbor distance interaction mode. In this mode, UAVs can interact in the formation and have the same communication radius.

 Definition 1 (Neighbor). 

When the distance$d_{ij}$between UAV$i$and UAV$j$satisfies the Equation (4), UAV$j$is said to be the neighbor of UAV$i$, and set$S_i$is called the neighbor set of UAV$i$.

$$S_i=\left\{v_j\in V:d_{ij}=‖x_j-x_i‖\le d_{commu}\right\}$$

(4)

In the Equation (4), $‖x_j-x_i‖$ is the Euclidean distance between UAVs $i$ and $j$ of the fixed coordinate system, abbreviated as $‖d_j{}_i‖$. $d_{commu}$ represents the interaction distance between UAVs with a value of $d_{commu}={\lambda }_1{d}_{expect}$; ${\lambda }_1\ge 1$ and $d_{expect}^{}$ are the desired distances determined by the maximum acceleration, the maximum velocity, the discrete time step ${\delta }_t$, and the UAV wingspan $D_{wing}$, satisfying [37]:

$$d_{expect}\ge D_{wing}+v_{\max }^2/u_{\max }+2v_{\max }{\delta }_t$$

(5)

Figure 2 shows a diagram of the neighbors around the $ua{v}_1$, respectively. The dashed circles are the interaction ranges. It can be seen that the $ua{v}_2$, $ua{v}_3$, and $ua{v}_4$ are all neighbors of the $ua{v}_1$, while $ua{v}_5$ and $ua{v}_6$ are not its neighbors.

• The information interaction content: included broadcasted information about its own speed, position, as well as the selected objective, the subgroup size.
• The information interaction topology chose the algebraic graph theory to describe the topology of UAVs in the formation and used the undirected graph $G=(V,E,A)$ to represent the topological relationship of communication.

In the graph $G$, $V=(v_1,v_2,\dots ,v_n)$ is the vertex set and the set UAVs. $E\subseteq V\times V$ is the edge set, representing the communication connection line between UAVs; $A={[a_{ij}]}_{N\times N}$ is the adjacency matrix of $G$; when $(i,j)\in E$, it represents the connection weight between UAV $i$ and UAV $j$.

3. Multiple UAVs’ Formation Algorithm Based on Alliance Algorithm

3.1. Multiple UAVs’ Optimal Alliance Formation Model

In the background of a complex air warfare environment, there are $N$ UAVs $V_i(i=1,2,\dots ,N)$ that need to perform coordinated missions and $M$ stationary objectives $O_k(k=1,2,\dots ,M)$. The number of resources (meaning the number of UAVs in this paper) required at objective $k$ is $R_k^q(q=1,2,\dots ,N)$, which need to meet the formula $R_k^q\le N$ to ensure that the alliance can complete the missions.

In two-dimensional space, the alliance can be described by the expected relative position vector $p_{ij}=p_i-p_j$, and $p_i$ is the expected position of the UAV $V_i$.

 Definition 2 (Stable Alliance). 

If$N$unmanned aerial vehicles in the formation at time$t$simultaneously meet:

$$\underset{t\to \infty }{\lim }{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1,j\ne i}^N‖x_i-x_j-(p_i-p_j)‖}}=0$$

(6)

$$\underset{t\to \infty }{\lim }{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1,j\ne i}^N‖v_i-v_j‖}}=0$$

(7)

It is said that the$N$UAVs can form a stable alliance.

3.1.1. The Formation Process of Multiple UAVs’ Alliance

• Alliance formation proposal:

Once the UAV $V_i$ has searched and confirmed the objective $O_k$, it then evaluates whether it can achieve the task needs. If not, $V_i$ will broadcast the information to its neighbors and recruit an alliance satisfying the requirements to carry out the corresponding missions.

2.

Formation subgroups:

In the formation, the UAVs that do not belong to any of the existing alliances are named Potential Alliance Members (PAM). After receiving the alliance information, the PAMs estimate their Earliest Time of Arrival (ETA) and use the Ant Colony Pheromone Partitioning Algorithm (ACPPA) to divide the formation to participate in the alliance formation.

3.

Alliance formation:

Based on the mission requirements of the objective, the numbers of UAVs involved in the mission are identified. Additionally, then the Information Concentration Competition Mechanism (ICCM) is used to select the leader UAV within the alliance to form the final stability alliance for the mission $O_k$.

3.1.2. The Constraints to Be Met by the Formed Alliance

• Efficiency: to ensure the overall mission time of the formation is the shortest.
• Minimality: to ensure that the alliance is as small as possible. This approach can enable the formation to assign more alliances to perform tasks, which can improve the efficiency of the formation.
• Simultaneity: to ensure the simultaneity of achieving the tasks. This constraint requires the alliance members to reach the objective location at as close to the same time as possible.
• Finiteness: to ensure the alliance can meet the resource requirements of the mission site.

3.1.3. Optimal Alliance Formation Model

$C^k$ denotes the alliance to be formed for the objective $O_k$; ${\mathit{R}}_k^{C^k,q}$ is the resource vector of the alliance (which, in this paper, represents the number of UAVs), satisfying ${\mathit{R}}_k^{C^k,q}\ge {\displaystyle \sum _{V_m\in C^k}{\mathit{R}}_k^{V_m,q}}$, i.e., the resources in the alliance are the sum of all members ${\mathit{R}}_k^{V_m,q}$, where $V_m$ is the member $V_m\in C^k$ in the alliance.

To sum up, the mathematical model of the optimal alliance $C^k$($C^k\subseteq \Lambda$) is:

$$min{T}_{O_k}^{C^k}=\min (\max t_{eta}^{V_m})$$

(8)

$$\min \left|C^k\right|$$

(9)

$$st:{\displaystyle \sum _{V_m\in C^k}{R}_k^{C^k,q}\ge R_k^{V_m,q}}$$

(10)

Equation (8) satisfies the simultaneity of alliance, where $\Lambda$ is the set of all UAVs involved in the formation of the alliance, including the recruiting UAVs and all PAMs; $t_{eta}^{V_m}$ represents the ETA of each alliance member; and $min{T}_{O_k}^C$ is the minimum mission time at the objective $O_k$. Equation (9) satisfies the minimality; Equation (10) satisfies the finiteness, i.e., the sum of resources ${\displaystyle \sum _{V_m\in C^k}{R}_k^{C^k,q}}$ within the alliance should not be less than the number of resources required at the $O_k$.

3.2. Ant Colony Pheromone Partitioning Algorithm

In the classical ant colony algorithm [38,39,40], ants select the next node by calculating the selection probability $P$ based on residual and heuristic information:

$$P_{ij}^k(t)=\frac{{[{\tau }_{ij}(t)]}^{\alpha }·{[{\eta }_{ij}(t)]}^{\beta }}{{\displaystyle \sum _{k\in {\alpha }_k}{[{\tau }_{ik}(t)]}^{\alpha }{[{\eta }_{ik}(t)]}^{\beta }}}$$

(11)

In this formula, ${\tau }_l$ is the residual pheromone on the path $l$; ${\eta }_l=\frac{1}{d_l}$ is the heuristic function indicating the expected degree of ants from node to node; $d_l$ is the Euclidean distance of the path $l$; $\alpha$ is the pheromone heuristic factor. The larger this value, the more likely the ants choose the path that most ants walk though. $\beta$ is the visibility heuristic factor. The larger this value, the more likely the ants choose the node that is the nearest; ${\alpha }_k$ is the set of next nodes that ants can choose.

On the basis of the pheromone theory, the ACPPA includes the following steps:

• Firstly, referring to the selection probability of the ant colony algorithm, the probability formula is designed for the selection of alliance $k$ by UAV $V_i$ at time $t$.

The selection probability is designed as:

$$p\_selectio{n}_{i,k}=\frac{{({\tau }_{i,k})}^{\alpha }{({\eta }_{i,k})}^{\beta }}{{\displaystyle \sum _{k=1}^K{({\tau }_{i,k})}^{\alpha }{({\eta }_{i,k})}^{\beta }}}$$

(12)

${\tau }_{ik}={\tau }_k-n_{ik}$ is the residual information to ensure that the formed alliance satisfies the demand; ${\tau }_k$ is the desired alliance size, which is not larger than the initial formation size; $n_{ik}$ is the number of members in the neighborhood of UAV $V_i$ who choose the alliance. It can be seen from the equation that ${\tau }_{ik}$ decreases as the $n_{ik}$ increases; ${\eta }_{ik}=\frac{1}{d_{ik}}$ is the heuristic factor to ensure that the alliance is divided according to the closest distance as much as possible under the condition of basically satisfying the alliance size requirement. $\alpha$ and $\beta$ have the same meaning as the corresponding parameters in Formula (12), respectively, reflecting the importance of residual information items and heuristic information items.

2.

Secondly, referring to the path selection strategy of ants, the PAMs within the formation are divided.

3.

Finally, referring to the selection probability formula, the subgrouping is implemented within the formation by the roulette method.

Forming the alliance combination is shown in Figure 3.

The number of alliances is determined based on the number of mission points. The Algorithm 1 Ant Colony Pheromone Partitioning Algorithm flows as below:

Algorithm 1: Ant Colony Pheromone Partitioning Algorithm
Input:	$\mathrm{Number} \mathrm{of} \mathrm{alliances} M_k$
1	$sum=0$$, k=1,k\in M_k$
2	$P_{random}\in (0,1)$$, p\_selectio{n}_{i,k}$
3	$sum\leftarrow sum+p\_selectio{n}_{i,k}$
4	$\mathrm{If} P_{random}<sum, V_i \mathrm{join} \mathrm{the} \mathrm{alliance} k$
	$\mathrm{Else} k\leftarrow k+1$, go to step3

3.3. Information Concentration Competition Mechanism

When the subgrouping is completed, the leader within the alliance needs to be selected. This leader is responsible for the information interaction within and between the alliances. The other followers of the alliance receive the communication information from the information leader. This approach to using leader communications can effectively reduce the communication pressure, while at any moment allowing all UAVs within the alliance to obtain the information directly or indirectly at the fastest speed. This article uses ICCM to choose leaders.

• UAV information concentration:

Defining the UAV Information Concentration $W_i$ concept:

$$W_i={\sigma }_1{z}_i+{\sigma }_2{g}_i^{-1}$$

(13)

${\sigma }_1$ and ${\sigma }_2$ are positive constants; $z_i$ is the number of nodes, and its units are the number of links; $g_i$ is the number of the guided links, and its units are the same as above.

2.

Guided links number:

 Definition 3 (Guided Links Number). 

The number of links that the information transmission needs to pass through.

For example, in Figure 4a, the blue UAV is the leader within the alliance, and the followers receive information from it, assuming that the time required for a message to pass through a communication link is $t$. At the time $T$, it takes $t$ for the information leader to transmit the message to $ua{v}_1$ and $ua{v}_4$, and $3t$ to transmit it to $ua{v}_3$. Since the information transmission is simultaneous, the maximum transmission time is $3t$, i.e., the maximum number of links that the leader can pass through unilaterally is 3, meaning that the guided links number is 3. Similarly, the guided links number in Figure 4b is 2. It is obvious that the information can be transmitted to the whole alliance in a shorter time than required in Figure 4b.

3.

Competition strategy within the alliance

UAVs in the alliance calculate their own information concentration and send it to the recruiting UAV. The UAV with the largest information concentration competes to become the leader.

The leader–follower alliance has higher alliance-controlling accuracy and more simple communication network, but once the leader fails, the alliance is difficult to maintain. The ICCM can be a good solution to this situation, selecting the leaders of the alliance and keeping it on track.

4. Alliance Control Based on the Improved Artificial Potential

4.1. Collision Avoidance Potential Field Construction in the Alliance

During the formation of the alliance, the possible collision between UAVs due to flight is a problem to be considered. The solution to this problem in this paper is to build an artificial potential field within the alliance. If the UAVs in the alliance are regarded as moving obstacles, the obstacle avoidance method based on artificial potential field can be used to achieve collision avoidance.

The potential field between UAVs acts in the neighborhood, that is, the UAVs need to be neighbors to each other, as beyond the neighborhood the distance between the UAVs is so far that the effect of the potential field is negligible.

To construct a smooth potential function, convex function (14) is used to define the adjacency matrix between UAVs $A(x)=[a_{ij}(x)]$:

$$f_{\mathrm{convex}}(\xi )=\left\{\begin{array}{ll}1,& \xi \in [0,h)\\ \frac{1}{2}(1+\cos (\pi \frac{\xi -h}{1-h})),& \xi \in [h,1]\\ 0,& \mathrm{others}\end{array}\right.$$

(14)

where $h$ is the parameter that controls the position of the bulge, $h\in (0,1)$.

Therefore, the elements in the adjacency matrix are:

$$a_{ij}(x)=\left\{\begin{array}{ll}{f}_{convex}(\frac{‖x_j-x_i‖}{d_{expect}}),& i\ne j\\ 0,& i=j\end{array}\right.$$

(15)

Then, the potential function is as follows:

$$U(d_{ij})=\frac{1}{2}a\cdot {(\frac{1}{d_{ij}}-\frac{1}{d_{expect}})}^2\cdot \overline{e}(d_{ij},d_{expect})$$

(16)

where $a>0$ is the gain coefficient of repulsion force; $\overline{e}(X,Y)$ is the judgment function. When $X>Y$, $\overline{e}(X,Y)=0$, and, vice versa, $\overline{e}(X,Y)=1$. That is, when the distance $d_{ij}$ between UAVs is greater than the expected distance $d_{expect}$, the repulsion force is 0. Then, the repulsion force on UAV $V_i$ is:

$$f_{ij}^{all}=-\nabla U(d_{ij})={\displaystyle \sum _{j=1}^{n}b(\frac{1}{d_{ij}}-\frac{1}{d_{expect}})\cdot \frac{1}{d_{ij}^2}\cdot {\eta }_{ij}\cdot \overline{e}(d_{ij},d_{expect})}$$

(17)

${\alpha }_1$ is the distance influence factor between UAVs. ${\chi }_{ij}$ is the distance normalization function:

$${\chi }_{ij}=\frac{x_j-x_i}{\sqrt{{‖x_j-x_i‖}^2}}$$

(18)

4.2. Design of Alliance Control Law

In this section, the consistency theory is used to achieve the autonomous formation control of the alliance members, and the desired stable formation under the alliance control law was defined in Section 2.2.

The control law of the UAV within the alliance is designed as:

$$u_i(t)=\underset{\mathrm{Location} \mathrm{coordination} \mathrm{items}}{\underbrace{{\lambda }_1{\displaystyle \sum _{i\in S_i}^N{a}_{ij}}(x_i-x_j-(p_i-p_j))}}+\underset{\mathrm{Velocity} \mathrm{coordination} \mathrm{items}}{\underbrace{v_i-v_j}}+\underset{\mathrm{Anti}-\mathrm{collision} \mathrm{potential} \mathrm{field}}{\underbrace{{\lambda }_2{\displaystyle \sum _{\in S_i}^N{\alpha }_1(\frac{1}{d_{ij}}-\frac{1}{d_{expect}})\cdot \frac{1}{d_{ij}^2}\cdot {\chi }_{ij}\cdot \overline{e}(d_{ij},d_{expect})}}}$$

(19)

Among them, ${\lambda }_1$ and ${\lambda }_2$ are the positive constants; $u_i$ denotes the control input of UAV $V_i$; $a_{ij}$ represents the acceleration vector of UAV; $v_i$ and $x_i$ denote the velocity and position vectors, respectively; $p_i$ and $p_j$ are the desired relative position.

${\overline{x}}_i(t)=x_i(t)-x_0(t)-p_i$, ${\overline{x}}_j(t)=x_j(t)-x_0(t)-p_j$, and ${\overline{v}}_i=v_i-v_j$; the control law can be transformed into:

$$u_i(t)={\lambda }_1{\displaystyle \sum _{j\in S_i}^N{a}_{ij}}(t)({\overline{x}}_j-{\overline{x}}_i)-{\overline{v}}_i-{\lambda }_2{\displaystyle \sum _{j\in S_i}^N{\alpha }_1(\frac{1}{d_{ij}}-\frac{1}{d_{expect}})\cdot \frac{1}{d_{ij}^2}\cdot {\chi }_{ij}\cdot \overline{e}(d_{ij},d_{expect})}$$

(20)

4.3. Analysis of Alliance Stability

For the control law of a multi-UAV alliance system under the action of an intra-alliance obstacle avoidance potential field, the stable convergence of the action of the control law (20) is to be analyzed.

First, the following theorems are introduced:

 Theorem 1 (Stability). 

Consider the multi-UAVs’ alliance system (1) which consists of UAVs and has a connected communication topology. Suppose that the total energy of the alliance is a finite value$E(t)$, namely,$E(t)\le E_0$. If the system under the action of an alliance control law (20), then the multi-UAVs’ alliance system will eventually form the desired formation and ensure that there is no collision between UAVs.

 Proof (Stability). 

For the multi-UAVs system, the following Lyapunov [37] equation is constructed:

$$E(t)={\displaystyle \sum _{i=1}^n(\frac{{\lambda }_1}{2}{\displaystyle \sum _{j\in S_i}^N{a}_{ij}}(t){({\overline{x}}_i-{\overline{x}}_j)}^2+\frac{1}{2}{\overline{v}}_i^T\overline{v}+{\lambda }_2{\displaystyle \sum _{j\in S_i}^{N}U(d_{ij})})}$$

(21)

In the equation, $E(t)$ is a positive semidefinite function. Calculate $\dot{E}(t)$ and bring (20) into:

$$\begin{array}{ll}\dot{E}\left(t\right)& =\sum _{i=1}^n({\lambda }_1\sum _{j\in S_i}^N{a}_{ij}\left(t\right)({\overline{x}}_i-{\overline{x}}_j)·{\overline{v}}_i+{\overline{v}}_i^T\dot{\overline{v}}+{\lambda }_2\sum _{j\in S_i}^N\nabla U\left(d_{ij}\right)·\overline{v})\\ & =\sum _{i=1}^n\begin{array}{l}({\lambda }_1\sum _{j\in S_i}^N{a}_{ij}\left(t\right)({\overline{x}}_i-{\overline{x}}_j)·{\overline{v}}_i+{\lambda }_2\sum _{j\in S_i}^N\nabla U\left(d_{ij}\right)·{\overline{v}}_i+{\overline{v}}_i^T({\lambda }_1\sum _{j\in S_i}^N{a}_{ij}\left(t\right)({\overline{x}}_j-{\overline{x}}_i)\\ -{\overline{v}}_i-{\lambda }_2\sum _{j\in S_i}^N\nabla U\left(d_{ij}\right)\left)\right)\end{array}\end{array}$$

(22)

Collating the equation gives:

$$\dot{E}(t)={\displaystyle \sum _{i=1}^n(-{\overline{v}}_i^T{\overline{v}}_i)}$$

(23)

From Equation (23), it can be seen that $\dot{E}(t)\le 0$, that is, $E(t)$ is a nonincreasing function. Consider set $\mathrm{\Omega }=\left\{({\overline{d}}_{ij},{\overline{v}}_i)|E(t)\le E_0\right\}$, where ${\overline{d}}_{ij}={\overline{x}}_i-{\overline{x}}_j$ is bounded because:

$${\overline{v}}_i^T{\overline{v}}_i\le {\displaystyle \sum _{i=1}^n({\overline{v}}_i^T{\overline{v}}_i)}\le 2E\le 2E_0$$

(24)

Therefore, if $‖{\overline{v}}_i‖\le \sqrt{2E_0}$, namely, ${\overline{v}}_i$ is bounded, then set $\mathrm{\Omega }$ is a compact. According to the LaSalle invariant set principle, if the initial solution of the system is in $\mathrm{\Omega }$, their trajectories will converge to the maximum invariant set $\overline{\mathrm{\Omega }}=\left\{({\overline{d}}_{ij},{\overline{v}}_i)\in \mathrm{\Omega }|\dot{E}(t)=0\right\}$.

According to Formula (23), if $\dot{E}(t)=0$, then ${\overline{v}}_i=0$, so there is $v_i(t)-v_j(t)=0$ in $\overline{\mathrm{\Omega }}$.

In addition,

$$\frac{d({\overline{v}}_i(t))}{dt}=2{\overline{v}}_i^T{\overline{v}}_i(t)=0$$

(25)

From the formula, it can be seen that ${\overline{v}}_i(t)\equiv c$; therefore, it has ${\overline{x}}_i(t)-{\overline{x}}_j(t)=0$, namely $x_j(t)-x_i(t)=p_{ij}$.

According to definition 2 (stability), multi-UAVs’ alliance can form the expected stable formation structure.

The rebuttal method can prove that the members of the multi-UAVs’ alliance will not collide during flight.

Suppose UAV $V_i$ and UAV $V_j$ collide at time $t_1>0$ in the alliance, that is, meeting:

$$x_i(t_1)=x_j(t_1),(i\ne j,i,j\in (1,2,\dots ,n))$$

(26)

Then, the potential field between the two UAVs is:

$${\displaystyle \sum _{i=1}^n{\lambda }_2[{\displaystyle \sum _{j\in S_i}^{n}U({\overline{d}}_{ij})}]}={\displaystyle \sum _{i=1}^n{\lambda }_2[{\displaystyle \sum _{j\in S_i}^n\frac{1}{2}a\cdot {(\frac{1}{d_{ij}}-\frac{1}{d_{expect}})}^2]}}$$

(27)

Through Equations (26) and (27):

$$\underset{t\to t_1}{\lim }\frac{1}{2}{\displaystyle \sum _{i=1}^n{\lambda }_2[{\displaystyle \sum _{j\in S_i}^{n}U({\overline{d}}_{ij})]}]}=+\infty$$

(28)

According to the Lyapunov equation:

$${\displaystyle \sum _{i=1}^n[\frac{1}{2}{\lambda }_2{\displaystyle \sum _{j\in S_i}^{n}U({\overline{d}}_{ij})]}]}=E-{\displaystyle \sum _{i=1}^n[\frac{1}{2}{\lambda }_1{\displaystyle \sum _{j\in S_i}^n{a}_{ij}(t){({\overline{x}}_i-{\overline{x}}_j)}^2-\frac{1}{2}{\overline{v}}_i^T{\overline{v}}_i]}]}\le E\le E_0$$

(29)

That is:

$$\underset{t\to t_1}{\lim }\frac{1}{2}{\displaystyle \sum _{i=1}^n{\lambda }_2[{\displaystyle \sum _{j\in S_i}^{n}U({\overline{d}}_{ij})]}]}\le E_0$$

(30)

Since $E_0$ is bounded, there is a contradiction between Formulas (28) and (30), that is, during the flight process, the alliance members can ensure safe flight without collision.

To sum up, under the control law (20), the multi-UAVs’ alliance system can form the expected formation and avoid collisions between members.

Thus, the theorem (stability) is proved. □

5. Simulation Verification and Analysis

In order to verify the correctness of the designed algorithm as well as the model, a case of five UAVs’ formations on the mission was executed.

According to the given model parameters (

Table 1. Model parameters.

Parameter	Value	Parameter	Value	Parameter	Value
$ua{v}_1/\mathrm{km}$	(0, 0)	$a$	5.0	${\delta }_t/\mathrm{s}$	1
$ua{v}_2/\mathrm{km}$	(0, 100)	$b$	4.0	$d_{commu}/\mathrm{km}$	$1.2d_{expect}$
$ua{v}_3/\mathrm{km}$	(0, 200)	$D_{wing}/\mathrm{m}$	6	$d_{expect}/\mathrm{km}$	290
$ua{v}_3/\mathrm{km}$	(100, 0)	$\alpha$	6	${\lambda }_1$	5
$ua{v}_5/\mathrm{km}$	(200, 0)	$\beta$	5	${\lambda }_2$	5
$\mathrm{h}$	0.2	$v(\mathrm{km}/\mathrm{h})$	$[40,60]$

), an example verification was conducted. Figure 4 shows the motion trajectories of five UAVs using the alliance formation algorithm. In the pictures, the blue drones denote the leader and the black dashed boxes denote two mission places.

It can be seen from Figure 5a that the V-shaped UAV formation carried out the subgrouping behavior, according to the demand of the mission places, to form two alliances and fly to the mission locations to execute the mission, respectively; Figure 5b shows the two alliances executing the mission at the task zone, respectively; in the above alliance, the middle UAV is the leader, and in the below alliance, the upper drone is the leader; Figure 5c shows that the alliances’ detachment leaves the task zone and group again to form a unified formation; Figure 5d shows that all drones carried out grouping behavior to form a zigzag formation.

The change in the number of alliances per moment in the formation and the total number of communications between UAVs are shown in Figure 6 and Figure 7. During the simulation, the UAV receives a message from a friendly aircraft in the domain once per moment as one communication. It can be seen that when the alliance tends to be stable, the number of inter-UAV communications also tends to be stable.

Figure 8 and Figure 9 show the speed and yaw angle change curves of the five UAVs. It can be seen from Figure 7 that the UAVs can adjust their speed within the controllable range according to the mission situation so that the formation speed eventually converges, and all UAVs will fly stably at 45 km/h; Figure 8 shows that the UAVs are able to maintain a consistent yaw angle after forming a stable formation.

6. Conclusions

This paper discusses the formation reconfiguration control problem under the switching topology network. It uses the Ant Colony Pheromone Partitioning Algorithm to deal with the problem of partitioning due to task requirements of objective position, presents an Information Concentration Competition Mechanism to handle the problem of no-leader at the new alliance, and proposes a control law based on artificial potential field to solve the problem of alliance stability. The simulation results show that the algorithm can achieve a stability alliance reconfiguration under the guidance of mission objectives, find the leader of the new alliance, and reduce the communication pressure. The formation control algorithm of multi-UAVs based on alliance is still a key point of interest for the future.

Formation reconfiguration control of multi-UAVs is a very complex problem. Despite the above-mentioned research results, this paper inevitably has the following shortcoming:

In the article, the communication state between UAVs is ideal and does not consider actual situations such as communication delay and packet loss. Therefore, designing more optimized algorithms to tackle the control of multi-UAVs under nonideal conditions such as interference or instability in the communication data chain will be the main work in the future. In addition, the artificial potential field imposed on the alliance can effectively solve the problem of collision avoidance between UAVs. However, the problem that the APF method is easy to fall into local minimum is not fully considered, which will be another main direction of future work.