Resilient Formation Reconfiguration for Leader–Follower Multi-UAVs

Abstract

Among existing studies on formation reconfiguration for multiple unmanned aerial vehicles (multi-UAVs), the majority are conducted on the assumption that the swarm scale is stationary. In fact, because of emergencies, such as communication malfunctions, physical destruction, and mission alteration, the scale of the multi-UAVs can fluctuate. In these cases, the achievements of formation reconfiguration for fixed-scale multi-UAVs are no longer applicable. As such, in this article, the formation reconfiguration problem of leader–follower multi-UAVs is investigated with a variable swarm scale taken into consideration. First, a streamlined topological structure is designed on the basis of the parity of the vertex numbers. Then, three formation reconfiguration strategies corresponding to the scenarios covering leader disengagement, follower detachment, and new member additions are developed with the aim of reducing the frequency of connection changes. Moreover, in terms of the leader election link of the leader disengagement scenario, a knowledge-based performance assessment model for UAVs is constructed with the help of the hierarchical belief rule base (BRB). Finally, the proposed formation reconfiguration strategies for leader disengagement, new member addition, and follower disengagement are demonstrated through simulations. The connection retention rate (CRR) for swarm communication topology under the three formation reconfiguration strategies can reach 67%, 90%, and 100%, respectively.

1. Introduction

Compared with individual intelligence [1], the power contained in swarm intelligence has been increasingly utilized in recent years [2,3,4,5,6,7]. Accordingly, the unmanned aerial vehicle (UAV) swarm, as one modern distributed system, has been extensively applied to numerous fields. For instance, multiple unmanned aerial vehicles (multi-UAVs) have broad uses in military actions, such as surveillance, reconnaissance, and strikes, as well as some civilian scenarios including performance and rescue [8]. It should be noted that the majority of the tasks mentioned above are executed on the basis of the formation control of the multi-UAVs, which has a tremendous influence on the performance of the whole distributed system.

Formation control for multi-UAVs refers to the fact that a group of UAVs can form and maintain a specific geometry in order to collaboratively accomplish a series of goals and can adapt to changes in the external environment by adjusting the geometry [9,10,11]. In general, formation control can be divided into three phases: the initial formation phase, the formation maintenance phase, and the formation reconfiguration phase–the last phase has the most complexity and challenges. To the best of our knowledge, the majority of the existing research on formation reconfiguration for multi-UAVs is based on the assumption of a fixed swarm size. Nonetheless, in actual flights, it is common for some members to disengage from the swarm due to accidents such as communication failure or physical damage. Meanwhile, it is also not uncommon for UAVs to leave or join the formation for mission changes. Under the aforementioned circumstances where the number of members is dynamic, multi-UAVs of good robustness are required to reconfigure the formation in a timely manner in accordance with the required tactics. Therefore, it is of practical significance to investigate the problems of formation reconfiguration for multi-UAVs under the condition of a dynamic swarm scale.

For simplicity and good control, the leader–follower architecture is widely applied to the formation control of multi-UAVs [12,13,14]. However, the architecture is dependent on the leader so that the entire swarm is more inclined to collapse or become out of control if the leader has a communication failure or is unexpectedly shot down by the enemy [15]. When such accidents occur, adopting appropriate policies to select a new leader for the impaired multi-UAVs is required so that the expected geometry can be reformed. A hasty countermeasure of randomly appointing a member as the new leader can bring about disastrous consequences. The kinematic-information-based leader election mechanism proposed in [16] can be an effective method. Compared to single source information, electing the leader with the help of the information from multiple sources can be another feasible solution, which can synthesize the kinematic information and the non-kinematic information of each UAV to make decisions with more accuracy and reliability.

As an uncertainty inference approach, the evidential reasoning (ER) rule can effectively fuse qualitative knowledge and quantitative data from multiple information sources in an explicable way. In comparison, the ER rule avoids the harsh a priori probability conditions of Bayesian inference and overcomes the defect of Dempster–Shafer evidence theory that cannot cope with conflicting evidence. With the superior ability to represent unknowns and uncertainties, the ER rule has been widely used to deal with the problems of multi-attribute decision analysis (MADA) [17,18]. With the development of evidence theory, the belief rule base (BRB), an expert system capable of tolerating uncertainties, was created [19,20]. The system contains a group of belief rules with some additional parameters determined by expert knowledge and uses the algorithm based on the ER rule as the core inference engine. This allows for good interpretability when using the BRB method for both the model structure and inference process. Until now, BRB has been applied to many fields successfully, including fault diagnosis [21,22,23,24], performance assessment [25], and risk analysis [26]. Therefore, it is reasonable to introduce the BRB technique into the leader election link of the formation reconfiguration strategy for leader disengagement.

Motivated by the above observations and analyses, the research on formation reconfiguration for leader–follower multi-UAVs triggered by changes to the swarm scale are carried out in this article. In addition, a knowledge-based performance assessment model for UAVs is established for electing a new leader. The main contributions of this article can be summarized as:

(1) Focused on three scenarios of the formation reconfiguration problem for leader–follower multi-UAVs with a changeable swarm scale—namely, leader disengagement, follower detachment, and new member addition—three reconfiguration strategies are developed, which include most of the existing studies [27,28] on fixed swarm scales as special cases.

(2) With respect to the communication topology adjustment link of the formation reconfiguration for leader–follower multi-UAVs with a dynamic swarm scale, a node–number-based strategy is proposed with the aim of minimizing the frequency of connection changes and making the formation reconfiguration process more efficient and resilient, especially in the case of large swarm scales.

(3) Targeting the leader election link of the formation reconfiguration strategy for leader disengagement, a hierarchical-BRB-based performance assessment model is established with two kinematic pivotal performance metrics and two non-kinematic pivotal performance metrics as the inputs and the final performance assessment score as the output.

The rest of the article is arranged as follows. Related works are discussed in Section 2. In Section 3, the kinematic models of the leader–follower multi-UAVs and the main objectives of the article are clarified. In Section 4, the construction of the hierarchical BRB performance assessment model for UAVs is detailed. The formation control protocol and three formation reconfiguration strategies are outlined in Section 5, and their effectiveness is demonstrated in Section 6. The article is concluded in Section 7, which also proposes some directions for future studies.

2. Related Works

With the aim of improving the safety and autonomy of multi-rotor UAV swarm flight, a trajectory generation approach based on B-spline and its corresponding adaptive optimization approach were proposed [29]. In order to boost the rapidity and precision of task processing for UAV swarms, a modified rational clustering method (RCM) algorithm was developed with the help of intensive task neighbor information exchange and state learning [30].

Aiming at the problems of formation and reconfiguration for the vertical take-off and landing multi-UAVs with dynamic topologies, the authors in [31] designed a cascade-formation control law based on the artificial potential function. The authors in [32] devised a time-optimal heuristic formation reconfiguration approach for multi-UAVs based on the gradient algorithm. Based on the assumption that the role of each swarm member is alterable, the authors in [33] developed a formation reconstruction policy with the help of weighted graph matching, by which the total amount of member movement during topology adjustment can be reduced.

Aiming at the UAV formation moving target tracking problem, a fusion model was constructed with the help of a receding horizon control (RHC) algorithm and standoff algorithm [34], and compared with the general RHC algorithm, more satisfactory results were achieved in real-time collaborative trajectory planning and real-time obstacle avoidance. Nonetheless, the validity of the fusion model is on the premise that the swarm communication topology is fully connected and the swarm scale is fixed. With the expansion of the swarm scale, the complexity of the swarm communication can increase greatly, thus making the swarm communication topology reconfiguration more challenging. In [35], an intelligent formation controller based on the RHC algorithm, adaptive hybrid particle swarm optimization and differential evolution (AHPSODE) algorithm was proposed for a multi-rotor UAV swarm with a fixed scale, by which the formation reconfiguration can be achieved under the condition of minimizing the total movement distance and reducing the computation cost.

In response to the changes in the requirements of battlefield missions, an alliance-based formation control framework was established for leader–follower multi-UAVs [16], where the ant colony pheromone partitioning algorithm (ACPPA) was adopted to divide the original leader–follower multi-UAVs into different subgroups. In terms of the leaderless subgroups, the information concentration competition mechanism (ICCM) was introduced to achieve flexible leader election. However, only the kinematic information of each UAV was utilized in this leader election scheme.

From the aspect of formation reconfiguration for multi-UAVs with a cut-down scale, compared with swarm communication reconfiguration, the authors in [36] placed more emphasis on swarm geometry reconfiguration. In [36], a framework of UAV formation reconfiguration based on system resilience was constructed. The concept of resilience was defined to quantitatively evaluate the disaster-tolerant capability of multi-UAVs. Then, a parameter optimization model was established on the basis of system resilience. Meanwhile, an improved adaptive learning-based pigeon-inspired optimization (ALPIO) algorithm was designed to achieve formation reconfiguration of multi-UAVs after random attacks with maximal system resilience, reduced reconfiguration time and expanded coverage area.

Compared with existing works, the innovation of the article is clarified in

Table 1. The innovation of the article compared with existing works.

Aspect	Existing Works	The Article
Reconfiguration	Geometry [9,16,36]	Communication
Swarm scale	Static [16,34,35]	Dynamic
Leader election	Single-source information [16]	Multi-sources information

.

3. Problem Formulation

Consider the following state equation for the ith follower of the leader–follower multi-UAVs:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left[\begin{array}{c}{\dot{V}}_i\\ {\dot{\chi }}_i\\ {\dot{\varphi }}_i\\ {\dot{x}}_i\\ {\dot{y}}_i\\ {\dot{z}}_i\end{array}\right]& =\left[\begin{array}{c}-gsin{\chi }_i\\ \frac{-gcos{\chi }_i}{V_i}\\ 0\\ V_{i}cos{\chi }_{i}cos{\varphi }_i\\ V_{i}cos{\chi }_{i}sin{\varphi }_i\\ V_{i}sin{\chi }_i\end{array}\right]+\left[\begin{array}{ccc}1& 0& 0\\ 0& \frac{1}{V_i}& 0\\ 0& 0& \frac{1}{V_{i}cos{\chi }_i}\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{c}{a}_{1i}\\ a_{2i}\\ a_{3i}\end{array}\right],i=1,\dots ,N.\hfill \end{array}\end{array}$$

(1)

Therein, N is the amount of the followers. $V_i$, ${\chi }_i$, ${\varphi }_i$ stand for the linear velocity, the pitch angle and the yaw angle of the follower. $q_i={\left[\begin{array}{ccc}{x}_i& y_i& z_i\end{array}\right]}^T$ stands for the 3D coordinate of the follower. $a_i={\left[\begin{array}{ccc}{a}_{1i}& a_{2i}& a_{3i}\end{array}\right]}^T$ is the control input consisting of the horizontal acceleration, the vertical acceleration and the angular acceleration of the follower as [34]. In practice, the horizontal acceleration $a_{1i}$ and the vertical acceleration $a_{2i}$ can be obtained by decomposing the available linear acceleration while the angular acceleration $a_{3i}$ can be obtained directly. g stands for the gravitational acceleration constant. Denote $p_i={\left[\begin{array}{ccc}{\dot{x}}_i& {\dot{y}}_i& {\dot{z}}_i\end{array}\right]}^T$. From (1), one can obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\ddot{q}}_i& ={\dot{p}}_i\hfill \\ \hfill & =\left[\begin{array}{c}{\dot{V}}_{i}cos{\chi }_{i}cos{\varphi }_i-V_{i}sin{\chi }_i{\dot{\chi }}_{i}cos{\varphi }_i-V_{i}cos{\chi }_{i}sin{\varphi }_i{\dot{\varphi }}_i\\ {\dot{V}}_{i}cos{\chi }_{i}sin{\varphi }_i-V_{i}sin{\chi }_i{\dot{\chi }}_{i}sin{\varphi }_i+V_{i}cos{\chi }_{i}cos{\varphi }_i{\dot{\varphi }}_i\\ {\dot{V}}_{i}sin{\chi }_i+V_i{\chi }_i{\dot{\chi }}_i\end{array}\right].\hfill \end{array}\end{array}$$

(2)

Substitute ${\dot{V}}_i=-gsin{\chi }_i+a_{1i}$, ${\dot{\chi }}_i=\frac{-gcos{\chi }_i}{V_i}+\frac{1}{V_i}{a}_{2i}$, ${\dot{\varphi }}_i=\frac{1}{V_{i}cos{\chi }_i}{a}_{3i}$, and one can obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\ddot{q}}_i& ={\dot{p}}_i\hfill \\ \hfill & =\left[\begin{array}{c}0\\ 0\\ -g\end{array}\right]+\left[\begin{array}{ccc}cos{\chi }_{i}cos{\varphi }_i& -sin{\chi }_{i}cos{\varphi }_i& -sin{\varphi }_i\\ cos{\chi }_{i}sin{\varphi }_i& -sin{\chi }_{i}sin{\varphi }_i& cos{\varphi }_i\\ sin{\chi }_i& cos{\chi }_i& 0\end{array}\right]\left[\begin{array}{c}{a}_{1i}\\ a_{2i}\\ a_{3i}\end{array}\right].\hfill \end{array}\end{array}$$

(3)

Denote $a_i=A_i^{-1}(u_i-b)$, where $A_i=\left[\begin{array}{ccc}cos{\chi }_{i}cos{\varphi }_i& -sin{\chi }_{i}cos{\varphi }_i& -sin{\varphi }_i\\ cos{\chi }_{i}sin{\varphi }_i& -sin{\chi }_{i}sin{\varphi }_i& cos{\varphi }_i\\ sin{\chi }_i& cos{\chi }_i& 0\end{array}\right]$, $b={\left[\begin{array}{ccc}0& 0& -g\end{array}\right]}^T$ and $u_i={\left[\begin{array}{ccc}{u}_{xi}& u_{yi}& u_{zi}\end{array}\right]}^T$ is the virtual control input described by

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_{xi}=a_{1i}cos{\chi }_{i}cos{\varphi }_i-a_{2i}sin{\chi }_{i}cos{\varphi }_i-a_{3i}sin{\varphi }_i\hfill \\ u_{yi}=a_{1i}cos{\chi }_{i}sin{\varphi }_i-a_{2i}sin{\chi }_{i}sin{\varphi }_i+a_{3i}cos{\varphi }_i\hfill \\ u_{zi}=a_{1i}sin{\chi }_i-a_{2i}cos{\chi }_i-g\hfill \end{array}\right.,\end{array}$$

(4)

where $u_{xi}$, $u_{yi}$, $u_{zi}$ are the accelerations on the x-axis, the y-axis and the z-axis of the follower. Thus, the kinematics of the ith follower in the leader–follower multi-UAVs can be further represented as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{q}}_i=p_i\hfill \\ {\dot{p}}_i=u_i\hfill \end{array}\right. ,i=1,\dots ,N.\end{array}$$

(5)

Next, model the kinematics of the leader in the leader–follower multi-UAVs as

$$\begin{array}{c}\hfill q_0\left(t\right)=q_0\left(t_0\right)+p_0·(t-t_0),t\ge t_0\ge 0,\end{array}$$

(6)

where $q_0\left(t\right)$ denotes the 3D coordinate of the leader at time instant t while $p_0$ refers to the constant linear velocity of the leader.

Remark 1. 

It is well known that UAVs have slower translational mechanics and faster rotational dynamics. Therefore, the flight controller of the UAV adopts a cascade structure, in which the outer-loop controller focuses on position and speed while the inner-loop controller focuses on attitude stability. The article focuses on the formation reconfiguration of multi-UAVs, where UAVs can be regarded as point-mass systems. Therefore, the article only considers the outer-loop controller of the UAV kinematics model and does not consider the dynamic model attitude controller of each UAV.

The communication topology of the leader–follower multi-UAVs are illustrated in Figure 1, where the leader is denoted by node 0 and the followers are associated with the rest of the nodes, and satisfies the following assumptions.

Assumption A1. 

The scale of the leader–follower multi-UAVs is no less than five, with one UAV as the leader.

Assumption A2. 

Only the followers numbered one and two can receive the signals from the leader.

Assumption A3. 

Followers with adjacent odd (even) numbers are neighbors to each other.

Remark 2. 

With the expanding scale of the leader–follower multi-UAVs, some problems are emerging gradually. On the one hand, the communication topology becomes increasingly complicated, which can further make the design of a formation controller more difficult. On the other hand, in terms of each follower, the plethora of neighbors can place more of the burden on topology reconfiguration. In light of these, the topological structure displayed in Figure 1 with minimal edges is devised based on the parity of vertex numbers, which can make the communication inside the multi-UAVs more effective and the formation reconfiguration more resilient.

Definition 1 

(Leader–follower formation problem). Under the designed control input $u_i,i=1,\dots ,N$, $\underset{t\to \infty }{lim}{q}_0-q_i={\Delta }_i$ and $\underset{t\to \infty }{lim}{p}_0-p_i=0$ for any initial condition, where ${\Delta }_i,i=1,\dots ,N$ represents the desired distance of the ith follower with respect to the leader.

Definition 2 

(Neighbor). In a topology, node j is a neighbor of node i ($i,j=0,\dots ,N$, $i\ne j$) in the sense that node i can receive and use the information of node j as a control input. The neighbor set of node i is denoted by ${\mathcal{N}}_i$. When node i and node j can communicate in both directions, node i and node j are neighbors of each other.

Remark 3. 

In practice, each UAV within the swarm is associated with a unique number in the swarm communication topology, which can be changed based on real-time situations. Each UAV stores the numbers of all its neighbors. Furthermore, each UAV only sends its state information to other UAVs with proper numbers in the swarm communication topology and receives state information from UAVs with proper numbers in the swarm communication topology. Updating the number of each UAV in the swarm communication topology and subsequently updating the numbers of each UAV’s neighbors are integral parts of the communication topology reconfiguration that the article focuses on.

The core target of the article is to design formation reconfiguration strategies for leader–follower multi-UAVs in terms of three scenarios, including the disengagement of the leader, the disengagement of the follower and the addition of new members.

4. Construction of the Hierarchical BRB Performance Assessment Model for UAVs

In this section, the basics of BRB are mentioned first. Then, the hierarchical architecture of the BRB performance assessment model is described. Subsequently, the construction details are elaborated from the perspectives of performance metrics, belief rules and inference mechanism, respectively.

4.1. Basics on BRB

As mentioned in Section 1, BRB is an optimizable expert system with several belief rules as the knowledge base and the algorithm based on the evidential reasoning (ER) rule as the inference machine. In general, the kth belief rule, $k=1,\dots ,L$, is of the following structure:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & Rul{e}_k:\hfill \\ \hfill & If{X}_{1}is{A}_{k,1}\wedge \cdots \wedge X_{F}is{A}_{k,F},\hfill \\ \hfill & Then\{(D_1,{\beta }_{k,1}),\dots ,(D_P,{\beta }_{k,P})\},\hfill \\ \hfill & Withruleweight{\theta }_k,antecedentattributeweights{\delta }_1,\dots ,{\delta }_F.\hfill \end{array}\end{array}$$

(7)

Therein, $X_f,f=1,\dots ,F$, is the antecedent attribute. $A_{k,f}\in A_f,k=1,\dots ,L,f=1,\dots ,F$, is the referential value of $X_f$ in the kth rule while $A_f,f=1,\dots ,F$, is the referential value set of $X_f$. Note that when $f,f=1,2$ is determined, $A_{k,f},k=1,\dots ,L$ can be the same. Denote $J_f$ the size of $A_f$, that is, $J_f=\left|A_f\right|,f=1,\dots ,F$. The amount of belief rules $L=\prod _{f=1}^F{J}_f$. $D_p,p=1,\dots ,P$, is the consequent attribute and ${\beta }_{k,p},k=1,\dots ,L,p=1,\dots ,P$, is the belief degree of $D_p$ in the kth rule. In terms of the kth belief rule, $k=1,\dots ,L$, the summation of the belief degree of the consequent attribute satisfy $\sum _{p=1}^P{\beta }_{k,p}\le 1$. The kth belief rule is complete in the sense that

$$\begin{array}{c}\hfill \sum _{p=1}^P{\beta }_{k,p}=1,k=1,\dots ,L,\end{array}$$

(8)

otherwise the belief rule is incomplete. Note that the completeness of belief rules is not mandatory, depending on the degree of certainty of expert knowledge. For more basics about BRB, refer to [37] and references therein.

4.2. Hierarchical Architecture Illustration

As exhibited in Figure 2, two layers, the basic layer and the aggregation layer, are encompassed in the designed hierarchical architecture. Therein, the basic layer consists of two BRBs, where $BR{B}_{11}$ is used to assess the kinematic performance for the ith follower, $i=1,\dots ,N$, using the designed kinematic pivotal performance metrics ${\mathcal{M}}_{11}^i\left(t\right)$ and ${\mathcal{M}}_{12}^i\left(t\right)$ while $BR{B}_{12}$ is used to assess the non-kinematic performance for the ith follower using the designed non-kinematic pivotal performance metrics ${\mathcal{M}}_{21}^i\left(t\right)$ and ${\mathcal{M}}_{22}^i$. The outputs of the BRBs in the basic layer for the ith follower ${\gamma }_1^i\left(t\right)$ and ${\gamma }_2^i\left(t\right)$ will be taken as the inputs of $BR{B}_{21}$ in the aggregation layer to obtain the final performance assessment result $Z_i\left(t\right)$ for the ith follower.

Remark 4. 

As clearly portrayed in Figure 2, the hierarchical BRB performance assessment model for UAVs is composed of two layers. In the first layer, namely, the basic layer, $BR{B}_{11}$ and $BR{B}_{12}$ are incorporated. Here, subscripts ‘11’ and ‘12’ indicate the 1st BRB and the 2nd BRB in the first layer, respectively. There is only $BR{B}_{21}$ in the second layer, namely, the aggregation layer. Likewise, the subscript ‘21’ indicates the 1st BRB in the 2nd layer. With respect to all parameters of $BR{B}_{11}$, $BR{B}_{12}$ and $BR{B}_{21}$, ‘11’, ‘12’, and ‘21’ are attached as the subscript or superscript accordingly.

4.3. Pivotal Performance Metrics Design

Prior to comprehensively assessing the performance for each follower, some pivotal performance metrics are required to be designed as the inputs to the basic layer of the hierarchical BRB performance assessment model for UAVs. First, based on the kinematics of the followers (5), two time-varying kinematic pivotal performance metrics are designed as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\mathcal{M}}_{11}^i\left(t\right)=\frac{1}{|{\mathcal{N}}_i|}\sum _{j\in {\mathcal{N}}_i}\parallel (q_j\left(t\right)-q_i\left(t\right))-({\Delta }_j-{\Delta }_i)\parallel ,\hfill \\ \hfill & {\mathcal{M}}_{12}^i\left(t\right)=\frac{1}{|{\mathcal{N}}_i|}\sum _{j\in {\mathcal{N}}_i}\parallel p_j\left(t\right)-p_i\left(t\right)\parallel ,i=1,\dots ,N,\hfill \end{array}\end{array}$$

(9)

where $|{\mathcal{N}}_i|$ denotes the neighbor amount of the ith follower and $\parallel ·\parallel$ stands for the Euclidean norm. The two metrics indicate the average error between the ith follower and its neighbors in terms of the position and the linear velocity, respectively, while both reflect the distributed state consensus degree of the ith follower. It should be noted that the closer the two metrics are to zero, the better the performance of the follower can be. Then, oriented to non-kinematics, two pivotal performance metrics are determined for each follower as well. One of them is the residual energy, which can be abstracted into the following monotonically decreasing continuous function $\Psi (·)$

$$\begin{array}{c}\hfill {\mathcal{M}}_{21}^i\left(t\right)={\Psi }_i\left(q_i\left(t\right)\right),t\in \left[0,+\infty \right),i=1,\dots ,N.\end{array}$$

(10)

Here, one supposes that the more energy that remains, the better performance of the follower. Furthermore, the equipment value is taken as another non-kinematic pivotal performance metric satisfying

$$\begin{array}{c}\hfill {\mathcal{M}}_{22}^i\in \Phi ,i=1,\dots ,N,\end{array}$$

(11)

where $\Phi$ is a non-empty finite set composed of elements representing the equipment value of the followers. Note that the equipment value is fixed for each follower and the assumption that the higher equipment value means the better performance is made.

4.4. Belief Rule Definition

In terms of $BR{B}_{11}$ in the basic layer of the hierarchical BRB performance assessment model for UAVs, the kth belief rule can be defined as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & Rul{e}_k^{11}:\hfill \\ \hfill & If{X}_{11}\left(t\right)is{A}_{11,k,1}\wedge X_{12}\left(t\right)is{A}_{11,k,2},\hfill \\ \hfill & Then\{(D_1^{11},{\beta }_{k,1}^{11}),\dots ,(D_{P_{11}}^{11},{\beta }_{k,P_{11}}^{11})\},\hfill \\ \hfill & Withruleweight{\theta }_k^{11},antecedentattributeweights{\delta }_1^{11},{\delta }_2^{11}.\hfill \end{array}\end{array}$$

(12)

Therein, $X_{11}\left(t\right)$ and $X_{12}\left(t\right)$ are the antecedent attributes of $BR{B}_{11}$ corresponding to the kinematic pivotal performance metrics ${\mathcal{M}}_{11}^i\left(t\right)$ and ${\mathcal{M}}_{12}^i\left(t\right)$, $i=1,\dots ,N$, respectively. $A_{11,k,f}\in A_{11,f},k=1,\dots ,L_{11},f=1,2$ refers to the referential value of the fth antecedent attribute in the kth rule. Note that $A_{11,1,f},\dots ,A_{11,L_{11},f}$ are not different from each other. $A_{11,f}=\{A_{11,f}^1,\dots ,A_{11,f}^{J_{11,f}}\}$ is the corresponding referential value set with $J_{11,f}=\left|A_{11,f}\right|$, where $|·|$ denotes the amount of the elements contained in the set and the sequence $A_{11,f}^1,\dots ,A_{11,f}^{J_{11,f}}$ is assumed to be increasing, which means $A_{11,f}^1,\dots ,A_{11,f}^{J_{11,f}}$ are different from each other. $L_{11}$ signifies the amount of the belief rules in $BR{B}_{11}$ and is calculated by $L_{11}=\prod _{f=1}^2{J}_{11,f}$. ${\beta }_{k,p}^{11},k=1,\dots ,L_{11},p=1,\dots ,P_{11}$, is the belief degree of the pth assessment grade $D_p^{11}$ in the kth rule. Here, one assumes that each belief rule of $BR{B}_{11}$ is complete, that is,

$$\begin{array}{c}\hfill \sum _{p=1}^{P_{11}}{\beta }_{k,p}^{11}=1,k=1,\dots ,L_{11},\end{array}$$

(13)

while the weight of the belief rules and the antecedent attributes are normalized, namely,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & 0\le {\theta }_k^{11}\le 1,k=1,\dots ,L_{11},\hfill \\ & 0\le {\delta }_f^{11}\le 1,f=1,2.\hfill \end{array}\end{array}$$

(14)

The kth belief rule of $BR{B}_{12}$ in the basic layer of the hierarchical BRB performance assessment model for UAVs can be represented as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & Rul{e}_k^{12}:\hfill \\ \hfill & If{X}_{21}\left(t\right)is{A}_{12,k,1}\wedge X_{22}is{A}_{12,k,2},\hfill \\ \hfill & Then\{(D_1^{12},{\beta }_{k,1}^{12}),\dots ,(D_{P_{12}}^{12},{\beta }_{k,P_{12}}^{12})\},\hfill \\ \hfill & Withruleweight{\theta }_k^{12},antecedentattributeweights{\delta }_1^{12},{\delta }_2^{12},\hfill \end{array}\end{array}$$

(15)

where $X_{21}\left(t\right)$ and $X_{22}$ are the antecedent attributes of $BR{B}_{12}$ corresponding to the non-kinematic pivotal performance metrics ${\mathcal{M}}_{21}^i\left(t\right)$ and ${\mathcal{M}}_{22}^i$, $i=1,\dots ,N$, respectively. $A_{12,k,f}\in A_{12,f},k=1,\dots ,L_{12},f=1,2$, is the referential value of the fth antecedent attribute in the kth rule of $BR{B}_{12}$ while $A_{12,f}=\{A_{12,f}^1,\dots ,A_{12,f}^{J_{12,f}}\}$ is the corresponding referential value set with $J_{12,f}=\left|A_{12,f}\right|$. It is supposed that the sequence $A_{12,f}^1,\dots ,A_{12,f}^{J_{12,f}}$ is increasing, that is, $A_{12,f}^1,\dots ,A_{12,f}^{J_{12,f}}$ are different from each other. By contrast, $A_{12,1,f},\dots ,A_{12,L_{12},f}$ are not different from each other. ${\beta }_{k,p}^{12},k=1,\dots ,L_{12},p=1,\dots ,P_{12}$, represents the belief degree of the pth assessment grade $D_p^{12}$ in the kth rule. With respect to $BR{B}_{12}$, the following constraints are also specified:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \sum _{p=1}^{P_{12}}{\beta }_{k,p}^{12}=1,k=1,\dots ,L_{12},\hfill \\ \hfill & 0\le {\theta }_k^{12}\le 1,k=1,\dots ,L_{12},\hfill \\ & 0\le {\delta }_f^{12}\le 1,f=1,2.\hfill \end{array}\end{array}$$

(16)

In terms of the $BR{B}_{21}$ in the aggregation layer of the hierarchical BRB performance assessment model for UAVs, the kth belief rule can be described as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & Rul{e}_k^{21}:\hfill \\ \hfill & If{Y}_1\left(t\right)is{A}_{21,k,1}\wedge Y_2\left(t\right)is{A}_{21,k,2},\hfill \\ \hfill & Then\{(D_1^{21},{\beta }_{k,1}^{21}),\dots ,(D_{P_{21}}^{21},{\beta }_{k,P_{21}}^{21})\},\hfill \\ \hfill & Withruleweight{\theta }_k^{21},antecedentattributeweights{\delta }_1^{21},{\delta }_2^{21}.\hfill \end{array}\end{array}$$

(17)

Therein, $Y_1\left(t\right)$ and $Y_2\left(t\right)$ are the antecedent attributes of $BR{B}_{21}$ corresponding to the outputs of the basic layer for the ith follower ${\gamma }_1^i\left(t\right)$ and ${\gamma }_2^i\left(t\right)$, $i=1,\dots ,N$, which will be articulated in Section 4.5. The referential value of the fth antecedent attribute in the kth rule is denoted by $A_{21,k,f}\in A_{21,f},k=1,\dots ,L_{21},f=1,2$, where $A_{21,f}=\{A_{21,f}^1,\dots ,A_{21,f}^{J_{21,f}}\}$ is the corresponding referential value set with $J_{21,f}=\left|A_{21,f}\right|$ and the increasing sequence $A_{21,f}^1,\dots ,A_{21,f}^{J_{21,f}}$ assumed. Note that $A_{21,f}^1,\dots ,A_{21,f}^{J_{21,f}}$ are different from each other while $A_{21,1,f},\dots ,A_{21,L_{21},f}$ are not different from each other. ${\beta }_{k,p}^{21},k=1,\dots ,L_{21},p=1,\dots ,P_{21}$, denotes the belief degree of the pth assessment grade $D_p^{21}$ in the kth rule. Moreover, the following conditions with respect to the rule completeness and the weight normalization are also satisfied by $BR{B}_{21}$ in the aggregation layer

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \sum _{p=1}^{P_{21}}{\beta }_{k,p}^{21}=1,k=1,\dots ,L_{21},\hfill \\ \hfill & 0\le {\theta }_k^{21}\le 1,k=1,\dots ,L_{21},\hfill \\ & 0\le {\delta }_f^{21}\le 1,f=1,2.\hfill \end{array}\end{array}$$

(18)

4.5. Inference Process Description

Inside $BR{B}_{11}$ in the basic layer, the fth antecedent attribute, $f=1,2$, will be transformed into the following belief distribution:

$$\begin{array}{c}\hfill \{(A_{11,f}^1,{\alpha }_{11,f}^1\left(t\right)),\dots ,(A_{11,f}^{J_{11,f}},{\alpha }_{11,f}^{J_{11,f}}\left(t\right))\}.\end{array}$$

(19)

Therein, ${\alpha }_{11,f}^j\left(t\right)$, $f=1,2$, $j=1,\dots ,J_{11,f}$, is the belief degree of the jth referential value with respect to the fth antecedent attribute of $BR{B}_{11}$ in the basic layer, which can be calculated according to the flowchart displayed in Figure 3.

Denote ${\alpha }_{11,f}\left(t\right)={[{\alpha }_{11,f}^1\left(t\right),\dots ,{\alpha }_{11,f}^{J_{11,f}}\left(t\right)]}^T,f=1,2$, and then, the antecedent attribute matching degree matrix $M_{11,a}\in {\mathbb{R}}^{L_{11}\times 2}$ of $X_{11}\left(t\right)$ and $X_{12}\left(t\right)$ can be obtained by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill M_{11,a}\left(t\right)& =\left[M_{11,a1}\left(t\right),M_{11,a2}\left(t\right)\right],\hfill \\ \hfill M_{11,a1}\left(t\right)& ={\alpha }_{11,1}\left(t\right)\otimes {\mathbf{1}}_{J_{11,2}}\hfill \\ \hfill & ={\left[m_{11,a1}^1\left(t\right),\dots ,m_{11,a1}^{L_{11}}\left(t\right)\right]}^T,\hfill \\ \hfill M_{11,a2}\left(t\right)& ={\mathbf{1}}_{J_{11,1}}\otimes {\alpha }_{11,2}\left(t\right)\hfill \\ \hfill & ={\left[m_{11,a2}^1\left(t\right),\dots ,m_{11,a2}^{L_{11}}\left(t\right)\right]}^T.\hfill \end{array}\end{array}$$

(20)

Therein, $m_{11,af}^k\left(t\right)$, $f=1,2$, $k=1,\dots ,L_{11}$ denotes the matching degree between the fth input and the fth antecedent attribute of the kth belief rule, ⊗ denotes the Kronecker product of matrices while ${\mathbf{1}}_{J_{11,f}}\in {\mathbb{R}}^{J_{11,f}},f=1,2$, denotes the column vector with all entries 1. Subsequently, one can obtain the following rule matching degree matrix:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill M_{11,r}\left(t\right)& ={\overline{M}}_{11,a1}\left(t\right)\odot {\overline{M}}_{11,a2}\left(t\right)\hfill \\ \hfill & ={\left[m_{11,r}^1\left(t\right),\dots ,m_{11,r}^{L_{11}}\left(t\right)\right]}^T,\hfill \end{array}\end{array}$$

(21)

where ⊙ denotes the Hadamard product of matrices and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\overline{M}}_{11,a1}\left(t\right)=\left[\begin{array}{c}{\left(m_{11,a1}^1\left(t\right)\right)}^{{\overline{\delta }}_1^{11}}\\ ⋮\\ {\left(m_{11,a1}^{L_{11}}\left(t\right)\right)}^{{\overline{\delta }}_1^{11}}\end{array}\right],\\ \hfill {\overline{M}}_{11,a2}\left(t\right)=\left[\begin{array}{c}{\left(m_{11,a2}^1\left(t\right)\right)}^{{\overline{\delta }}_2^{11}}\\ ⋮\\ {\left(m_{11,a2}^{L_{11}}\left(t\right)\right)}^{{\overline{\delta }}_2^{11}}\end{array}\right],\\ \hfill {\overline{\delta }}_f^{11}=\frac{{\delta }_f^{11}}{\underset{f=1,2}{max}\left\{{\delta }_f^{11}\right\}},f=1,2.\end{array}\end{array}$$

(22)

Based on the rule matching degree matrix (21), the activation weight of the kth belief rule of $BR{B}_{11}$, $k=1,\dots ,L_{11}$, can be yielded by

$$\begin{array}{c}\hfill {\omega }_k^{11}\left(t\right)=\frac{{\theta }_k^{11}{m}_{11,r}^k\left(t\right)}{{\sum }_{l=1}^{L_{11}}{\theta }_l^{11}{m}_{11,r}^l\left(t\right)}.\end{array}$$

(23)

Then, all activated belief rules will be fused using the following analytical ER algorithm as [38]:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\beta }_p^{11}\left(t\right)=\frac{\mu \left(t\right)(\prod _{k=1}^{L_{11}}({\omega }_k^{11}\left(t\right){\beta }_{k,p}^{11}+1-{\omega }_k^{11}\left(t\right)\sum _{p=1}^{P_{11}}{\beta }_{k,p}^{11})-\prod _{k=1}^{L_{11}}(1-{\omega }_k^{11}\left(t\right)\sum _{p=1}^{P_{11}}{\beta }_{k,p}^{11})}{1-\mu \left(t\right)\prod _{k=1}^{L_{11}}(1-{\omega }_k^{11}\left(t\right))},\end{array}\end{array}$$

(24)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mu \left(t\right)=\frac{1}{\sum _{p=1}^{P_{11}}\prod _{k=1}^{L_{11}}({\omega }_k^{11}\left(t\right){\beta }_{k,p}^{11}+1-{\omega }_k^{11}\left(t\right)\sum _{p=1}^{P_{11}}{\beta }_{k,p}^{11})-(P_{11}-1)\prod _{k=1}^{L_{11}}(1-{\omega }_k^{11}\left(t\right)\sum _{p=1}^{P_{11}}{\beta }_{k,p}^{11})}.\end{array}\end{array}$$

(25)

Therein, $\mu \left(t\right)$ is the correction factor and ${\beta }_p^{11}\left(t\right),p=1,\dots ,P_{11}$, denotes the aggregated belief degree of the pth assessment grade $D_p^{11}$. Couple $(D_p^{11},{\beta }_p^{11}\left(t\right)),p=1,\dots ,P_{11}$, can constitute the following assessment grade belief distribution:

$$\begin{array}{c}\hfill \{(D_p^{11},{\beta }_p^{11}\left(t\right)),p=1,\dots ,P_{11}\}.\end{array}$$

(26)

Then, the output of $BR{B}_{11}$ in the basic layer of the hierarchical BRB performance assessment model for UAVs in terms of the ith follower, $i=1,\dots ,N$, can be obtained by the following utility conversion:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\gamma }_1^i\left(t\right)& =BR{B}_{11}({\mathcal{M}}_{11}^i\left(t\right),{\mathcal{M}}_{12}^i\left(t\right))\hfill \\ \hfill & =\sum _{p=1}^{P_{11}}{D}_p^{11}{\beta }_p^{11}\left(t\right).\hfill \end{array}\end{array}$$

(27)

Similarly, the output ${\gamma }_2^i\left(t\right)$ of $BR{B}_{12}$ in the basic layer of the hierarchical BRB performance assessment model for UAVs in terms of the ith follower, $i=1,\dots ,N$, can be obtained by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\gamma }_2^i\left(t\right)& =BR{B}_{12}({\mathcal{M}}_{21}^i\left(t\right),{\mathcal{M}}_{22}^i)\hfill \\ \hfill & =\sum _{p=1}^{P_{12}}{D}_p^{12}{\beta }_p^{12}\left(t\right).\hfill \end{array}\end{array}$$

(28)

Afterwards, $BR{B}_{21}$ in the aggregation layer will further fuse the outputs ${\gamma }_1^i\left(t\right)$ and ${\gamma }_2^i\left(t\right)$ of the basic layer in terms of the ith follower, $i=1,\dots ,N$, to obtain the ultimate performance assessment result of the ith follower $Z_i\left(t\right)$, which can be summarized as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill Z_i\left(t\right)& =BR{B}_{21}({\gamma }_1^i\left(t\right),{\gamma }_2^i\left(t\right))\hfill \\ \hfill & =\sum _{p=1}^{P_{21}}{D}_p^{21}{\beta }_p^{21}\left(t\right).\hfill \end{array}\end{array}$$

(29)

5. Articulation of Formation Reconfiguration Strategies for Multi-UAVs with Variable Swarm Scale

In this section, the formation controller is introduced in the first place, based on which three formation reconfiguration strategies in terms of three scenarios leading to swarm scale change, that is, leader disengagement, follower disengagement and new member addition, are developed, respectively.

5.1. Introduction of Formation Control Protocol

In accordance with Definition 1, a consensus-based formation control protocol is introduced from [39] to accomplish formation control of the leader–follower multi-UAVs (5) and (6):

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u_i=& -\sum _{j\in {\mathcal{N}}_i}{K}_1[(q_i-q_0+{\Delta }_i)-(q_j-q_0+{\Delta }_j)]-\sum _{j\in {\mathcal{N}}_i}{K}_2(p_i-p_j)\hfill \\ \hfill & -K_3(q_i-q_0+{\Delta }_i)-K_4(p_i-p_0),i=1,\dots ,N.\hfill \end{array}\end{array}$$

(30)

Therein, $K_1,K_2$ are positive definite matrices, while $K_3,K_4$ are positive semi-definite matrices in the sense that the ith follower can receive the signals from the leader and are null matrices otherwise. ${\mathcal{N}}_i$ denotes the neighbor set of the ith follower. ${\Delta }_i$ stands for the desired relative position of the ith follower with respect to the leader, which satisfies ${\Delta }_i=q_0-q_i$.

Remark 5. 

The formation reconfiguration process of a UAV swarm under dynamic swarm scale can be divided into two phases: swarm communication topology reconfiguration and swarm geometry reconfiguration. The article focuses on the former. In fact, protocol (30) is a universal consensus-based control protocol whose effectiveness has been well demonstrated in existing works, such as [39]. Therefore, the proof regarding protocol (30) is omitted from the article.

5.2. Design of Formation Reconfiguration Strategy for Multi-UAVs under Leader Disengagement

With respect to the leader–follower control architecture, the leader is critical to the stable flight of the multi-UAVs. Once the command capability of the leader is lost for malfunction or destruction, the multi-UAVs can be at risk of losing control and collapse. As such, in the case of leader departure, the hierarchical BRB performance assessment model for UAVs in Section 4 is utilized to select a new leader. Then, update the swarm communication topology and the desired distance of each follower with respect to the leader. Finally, reform the intended shape under the formation control protocol. The formation reconfiguration strategy for leader disengagement can be further summarized as the following steps, with the renewal of the swarm communication topology illustrated in Figure 4.

Step 1. Select a new leader based on the hierarchical BRB performance assessment model for UAVs, which is noted as NEW_LEADER.

Step 2. If NEW_LEADER has two neighbors, disconnect the bidirectional connection between NEW_LEADER and the neighbor with smaller number, and then, establish a bidirectional connection between the node and the last node of the path starting with NEW_LEADER.

Step 3. Change the bidirectional connection between NEW_LEADER and the current neighbor to a directed connection incident on NEW_LEADER.

Step 4. If the current number of NEW_LEADER is odd, establish a unidirectional connection from NEW_LEADER to node 2, otherwise establish a unidirectional connection from NEW_LEADER to node 1.

Step 5. If the difference in length between the two paths starting from NEW_LEADER is 2, disconnect the bidirectional connection between the end node of the longer path and its neighbor, and then, establish a bidirectional connection between the node and the end node of the shorter path.

Step 6. Update the amount of followers inside the multi-UAVs by N = N−1.

Step 7. Update the number of nodes from 0 to N.

Step 8. Update the desired distance of nodes 1 to N with respect to node 0, and then reform the desired geometry under the formation control law (30).

5.3. Design of Formation Reconfiguration Strategy for Multi-UAVs under Follower Disengagement

During the formation flight of the leader–follower multi-UAVs, followers can suffer from accidents as well, including malfunction, battle damage and mission change, which leads to a reduced swarm size. In the case of follower departure, the first step for reconfiguring the formation is to restore the swarm communication topology for the smoothness of information interaction within the multi-UAVs. Then, renew the parameters of the multi-UAVs, including the amount of followers, the desired distance of each follower with respect to the leader and the number for each swarm member. Finally, the impaired multi-UAVs reform the intended geometry under the formation control law (30). The formation reconfiguration scheme in this case can be further summarized as the following steps with the restoring process of the swarm communication topology illustrated in Figure 5.

Step 1. Determine if there are nodes that cannot be reached by node 0. If yes, find the node with the smallest number that node 0 cannot access and enter Step 2, otherwise go to Step 3.

Step 2. Determine whether the disengaging follower is node 1 or 2. If yes, establish a directed connection from node 0 to the node found in Step 1. If not, establish a bidirectional connection between the node found in Step 1 and the node whose number is two less than that of the disengaging follower.

Step 3. Determine whether the length of the two paths starting at node 0 are the same. Disconnect the bidirectional connection between the terminating node of the longer path and its neighbor and establish a bidirectional connection between the node and the terminating node of the shorter path in the sense that the difference in length between the two paths is 2.

Step 4. Update the amount of the followers inside the multi-UAVs, namely, N = N − 1.

Step 5. Update the number of each follower from 1 to N.

Step 6. Update the desired distance of each follower with respect to the leader and reform the intended formation under the control law (30).

5.4. Design of Formation Reconfiguration Strategy for Multi-UAVs under New Member Addition

The formation reconfiguration strategy for leader–follower multi-UAVs in the case of an increasing swarm scale covers the update of follower amount, the renewal of the swarm communication topology, the parameter configuration of the new member and the reconstruction of the intended geometry, which can be further summarized as the following steps with the renewing process of the swarm communication topology illustrated in Figure 6.

Step 1. Update the amount of the followers inside the multi-UAVs by N = N + 1.

Step 2. Set the number of the new member to N.

Step 3. Establish a bidirectional connection between node N and node N − 2.

Step 4. Set the desired distance of node N with respect to node 0.

Step 5. Reform the expected geometry under the formation control law (30).

6. Simulation Verification

Suppose that 11 UAVs, UAV i, $i=0,\dots ,10$, are incorporated in the initial leader–follower multi-UAVs with UAV 0 as the leader. The default linear velocity is in meters per second and the default displacement is in meters. In terms of UAV 0, the constant linear velocity $p_0={[4,4,4]}^T$ and the initial Cartesian coordinate $q_0\left(0\right)={[100,100,100]}^T$. In terms of UAV i, $i=1,\dots ,10$, the desired distance with respect to the leader ${\Delta }_i=i\kappa$ where $\kappa ={[20,0,20]}^T$ in the sense that i is odd and $\kappa ={[0,20,20]}^T$ otherwise. The initial position $q_i\left(0\right)={[q_{i1}\left(0\right),q_{i2}\left(0\right),q_{i3}\left(0\right)]}^T$, $i=1,\dots ,10$, where $q_{i1}\left(0\right)\in [0,100]$, $q_{i2}\left(0\right)\in [0,100]$ and $q_{i3}\left(0\right)=0$. The initial linear velocity $p_i\left(0\right)={[p_{i1}\left(0\right),p_{i2}\left(0\right),p_{i3}\left(0\right)]}^T$, where $p_{i1}\left(0\right)\in [0,0.3]$, $p_{i2}\left(0\right)\in [0,0.3]$ and $p_{i3}\left(0\right)\in [0,0.3]$. In terms of the constant matrices in the formation control protocol (30), $K_1=I_3$, $K_2=0.6I_3$, $K_3=I_3$, $K_4=0.8I_3$, with $I_3$ representing the 3D identity matrix. The simulation is conducted in MATLAB R2021a and lasts 240 s, where UAV 11 joins the multi-UAVs at the 60th s, UAV 0 disengages from the multi-UAVs at the 120th s and UAV 9 disengages from the multi-UAVs at the 180th s.

6.1. Phases of Forming Up and Formation Maintenance

When $t\in \left[0,60\right)$, the leader–follower multi-UAVs can achieve the anticipant geometry and keep flying in the geometry under the formation control law (30), whose trajectories are portrayed in Figure 7. The position trace error of the followers can converge to zero within 15 s, as shown in Figure 8. In addition, the convergence of linear velocity, pitch angle and yaw angle for UAV i, $i=1,\dots ,10$, can also be made as displayed in Figure 9, Figure 10 and Figure 11.

6.2. Formation Reconfiguration under New Member Addition

At the 60th second, UAV 11 joins the multi-UAVs midway. Based on the strategy developed in Section 5.4, the leader–follower multi-UAVs composed of UAV i, $i=0,\dots ,11$, can reform the intended geometry, as portrayed in Figure 12 and Figure 13. Meanwhile, the linear velocity, the pitch angle and the yaw angle for UAV i, $i=1,\dots ,11$, can becoma convergent as displayed in Figure 14, Figure 15 and Figure 16, respectively.

6.3. Formation Reconfiguration under Leader Disengagement

At the 120th second, UAV 0 breaks away from the leader–follower multi-UAVs by accident. According to the strategy developed in Section 5.2, the hierarchical BRB performance assessment model for UAVs needs to be established to elect a new leader for the impaired multi-UAVs.

6.3.1. Model Inputs Determination

In terms of UAV i, $i=1,\dots ,11$, the observed value of the four pivotal performance metrics devised in Section 4.3 at the 120th second needs to be obtained as the inputs to the basic layer of the hierarchical BRB performance assessment model for UAVs. Here, the residual power function in (10) is determined as

$$\begin{array}{c}\hfill {\Psi }_i\left(t\right)=\left\{\begin{array}{cc}1500-0.0001\parallel q_i\left(t\right)-q_i\left(0\right)\parallel ,\hfill & i=1,3,5,7,9,\hfill \\ 2000-0.0001\parallel q_i\left(t\right)-q_i\left(0\right)\parallel ,\hfill & i=2,4,6,8,10,\hfill \\ 1500-0.0001\parallel q_i\left(t\right)-q_i\left(60\right)\parallel ,\hfill & i=11.\hfill \end{array}\right.\end{array}$$

Additionally, the equipment value set $\Phi$ in (11) is determined as

$$\begin{array}{c}\hfill \Phi =\{\mathbb{A},\mathbb{B},\mathbb{C}\},\end{array}$$

where $\mathbb{A}=30$, $\mathbb{B}=20$ and $\mathbb{C}=10$. The equipment value for UAV i is as follows:

$$\begin{array}{c}\hfill {\mathcal{M}}_{22}^i=\left\{\begin{array}{cc}\mathbb{C},\hfill & ifi=1,4,7,10,\hfill \\ \mathbb{B},\hfill & ifi=2,5,8,11,\hfill \\ \mathbb{A},\hfill & ifi=3,6,9.\hfill \end{array}\right.\end{array}$$

6.3.2. Performance Assessment Model Construction

In terms of $BR{B}_{11}$ in the basic layer, the referential information of the two time-varying antecedent attributes $X_{11}\left(t\right)$ and $X_{12}\left(t\right)$ is determined by analyzing the observed value of the kinematic pivotal performance metrics ${\mathcal{M}}_{11}^i\left(t\right)$ and ${\mathcal{M}}_{12}^i\left(t\right)$ of UAV i, $i=1,\dots ,11$, at the 120th second, which are displayed in

Table 2. The referential information of the antecedent attribute $X_{11}\left(t\right)$ of $BR{B}_{11}$ in the basic layer.

Referential point	$\mathbb{A}$	$\mathbb{B}$	$\mathbb{C}$
Referential value	28	33	38

and

Table 3. The referential information of the antecedent attribute $X_{12}\left(t\right)$ of $BR{B}_{11}$ in the basic layer.

Referential point	$\mathbb{A}$	$\mathbb{B}$	$\mathbb{C}$
Referential value	0	0.0005	0.001

, respectively. The assessment grades of $BR{B}_{11}$ in the basic layer are given in

Table 4. The assessment grades of $BR{B}_{11}$ in the basic layer.

Referential point	$\mathbb{D}$	$\mathbb{C}$	$\mathbb{B}$	$\mathbb{A}$
Referential value	20	40	60	80

. The belief degree of each assessment grade in each rule is assigned based on empirical knowledge, as shown in

Table 5. The consequent belief distribution table of $BR{B}_{11}$ in the basic layer.

Rule Number	Antecedent Attributes		Consequent Belief Distribution
	${\mathit{X}}_{\mathbf{11}}\left(\mathit{t}\right)$	${\mathit{X}}_{\mathbf{12}}\left(\mathit{t}\right)$	{$\mathbb{D},\mathbb{C},\mathbb{B},\mathbb{A}$}
1	$\mathbb{A}$	$\mathbb{A}$	{0, 0.1, 0.1, 0.8}
2	$\mathbb{A}$	$\mathbb{B}$	{0, 0.2, 0.5, 0.3}
3	$\mathbb{A}$	$\mathbb{C}$	{0.2, 0.3, 0.3, 0.2}
4	$\mathbb{B}$	$\mathbb{A}$	{0, 0.2, 0.5, 0.3}
5	$\mathbb{B}$	$\mathbb{B}$	{0.1, 0.45, 0.45, 0}
6	$\mathbb{B}$	$\mathbb{C}$	{0.4, 0.4, 0.2, 0}
7	$\mathbb{C}$	$\mathbb{A}$	{0.2, 0.3, 0.3, 0.2}
8	$\mathbb{C}$	$\mathbb{B}$	{0.4, 0.4, 0.2, 0}
9	$\mathbb{C}$	$\mathbb{C}$	{0.6, 0.3, 0.1, 0}

. The antecedent attribute weight ${\delta }_f^{11}$, $f=1,2$, and the rule weight ${\theta }_k^{11}$, $k=1,\dots ,9$, are all set to 1 for the equal importance.

In terms of $BR{B}_{12}$ in the basic layer, the referential information of the time-varying antecedent attribute $X_{21}\left(t\right)$ and the constant antecedent attribute $X_{22}$ is configured on the basis of the observed value of the non-kinematic pivotal performance metrics ${\mathcal{M}}_{21}^i\left(t\right)$ and ${\mathcal{M}}_{22}^i$ of UAV i, $i=1,\dots ,11$, at the 120th second, as shown in

Table 6. The referential information of the antecedent attribute $X_{21}\left(t\right)$ of $BR{B}_{12}$ in the basic layer.

Referential point	$\mathbb{C}$	$\mathbb{B}$	$\mathbb{A}$
Referential value	500	1000	1500

and

Table 7. The referential information of the antecedent attribute $X_{22}$ of $BR{B}_{12}$ in the basic layer.

Referential point	$\mathbb{C}$	$\mathbb{B}$	$\mathbb{A}$
Referential value	10	20	30

, respectively. The assessment grades and the experience-based consequent belief distribution table are given in

Table 8. The assessment grades of $BR{B}_{12}$ in the basic layer.

Referential point	$\mathbb{D}$	$\mathbb{C}$	$\mathbb{B}$	$\mathbb{A}$
Referential value	20	40	60	80

and

Table 9. The consequent belief distribution table of $BR{B}_{12}$ in the basic layer.

Rule Number	Antecedent Attributes		Consequent Belief Distribution
	${\mathit{X}}_{\mathbf{21}}\left(\mathit{t}\right)$	${\mathit{X}}_{\mathbf{22}}$	{$\mathbb{D},\mathbb{C},\mathbb{B},\mathbb{A}$}
1	$\mathbb{C}$	$\mathbb{C}$	{0.6, 0.3, 0.1, 0}
2	$\mathbb{C}$	$\mathbb{B}$	{0.4, 0.4, 0.2, 0}
3	$\mathbb{C}$	$\mathbb{A}$	{0.2, 0.3, 0.3, 0.2}
4	$\mathbb{B}$	$\mathbb{C}$	{0.4, 0.4, 0.2, 0}
5	$\mathbb{B}$	$\mathbb{B}$	{0.1, 0.45, 0.45, 0}
6	$\mathbb{B}$	$\mathbb{A}$	{0, 0.2, 0.5, 0.3}
7	$\mathbb{A}$	$\mathbb{C}$	{0.2, 0.3, 0.3, 0.2}
8	$\mathbb{A}$	$\mathbb{B}$	{0, 0.2, 0.5, 0.3}
9	$\mathbb{A}$	$\mathbb{A}$	{0, 0.1, 0.1, 0.8}

. The antecedent attribute weight ${\delta }_f^{12}=1$, $f=1,2$, and the rule weight ${\theta }_k^{12}=1$, $k=1,\dots ,9$.

In terms of $BR{B}_{21}$ in the aggregation layer, the referential points and the corresponding value of the time-varying antecedent attributes $Y_1\left(t\right)$ and $Y_2\left(t\right)$ are determined by the analysis of the outputs of the basic layer for UAV i, $i=1,\dots ,11$, that is, ${\gamma }_1^i\left(t\right)$ and ${\gamma }_2^i\left(t\right)$, at the 120th second, which are exhibited in

Table 10. The referential information of the antecedent attribute $Y_1\left(t\right)$ of $BR{B}_{21}$ in the aggregation layer.

Referential point	$\mathbb{C}$	$\mathbb{B}$	$\mathbb{A}$
Referential value	20	50	80

and

Table 11. The referential information of the antecedent attribute $Y_2\left(t\right)$ of $BR{B}_{21}$ in the aggregation layer.

Referential point	$\mathbb{C}$	$\mathbb{B}$	$\mathbb{A}$
Referential value	20	50	80

, respectively. The configuration of the assessment grades is given in

Table 12. The assessment grades of $BR{B}_{21}$ in the aggregation layer.

Referential point	$\mathbb{D}$	$\mathbb{C}$	$\mathbb{B}$	$\mathbb{A}$
Referential value	20	40	60	80

and the consequent belief distribution corresponding to each combination of the referential points of the antecedent attributes $Y_1\left(t\right)$ and $Y_2\left(t\right)$ is exhibited in

Table 13. The consequent belief distribution table of $BR{B}_{21}$ in the aggregation layer.

Rule Number	Antecedent Attributes		Consequent Belief Distribution
	${\mathit{Y}}_{\mathbf{1}}\left(\mathit{t}\right)$	${\mathit{Y}}_{\mathbf{2}}\left(\mathit{t}\right)$	{$\mathbb{D},\mathbb{C},\mathbb{B},\mathbb{A}$}
1	$\mathbb{C}$	$\mathbb{C}$	{0.6, 0.3, 0.1, 0}
2	$\mathbb{C}$	$\mathbb{B}$	{0.4, 0.4, 0.2, 0}
3	$\mathbb{C}$	$\mathbb{A}$	{0.2, 0.3, 0.3, 0.2}
4	$\mathbb{B}$	$\mathbb{C}$	{0.4, 0.4, 0.2, 0}
5	$\mathbb{B}$	$\mathbb{B}$	{0.1, 0.45, 0.45, 0}
6	$\mathbb{B}$	$\mathbb{A}$	{0, 0.2, 0.5, 0.3}
7	$\mathbb{A}$	$\mathbb{C}$	{0.2, 0.3, 0.3, 0.2}
8	$\mathbb{A}$	$\mathbb{B}$	{0, 0.2, 0.5, 0.3}
9	$\mathbb{A}$	$\mathbb{A}$	{0, 0.1, 0.1, 0.8}

. Here, the antecedent attribute weight ${\delta }_f^{21}=1$, $f=1,2$, and the rule weight ${\theta }_k^{21}=1$, $k=1,\dots ,9$.

At this point, the hierarchical BRB performance assessment model for UAVs is established, by which the kinematic performance assessment result, the non-kinematic performance assessment result and the final performance assessment result for UAV i, $i=1,\dots ,11$, at the 120$th$ second are obtained, as shown in Figure 17.

Remark 6. 

In the article, model parameter optimization is not considered for the hierarchical BRB performance assessment model for UAVs. Once all model parameters have been determined with the help of empirical knowledge, the model outputs are determined based on the given model inputs. Nonetheless, when model parameter optimization is considered, the stability of BRB can also be guaranteed, that is, the output error of BRB can converge to zero, which has been demonstrated in [40].

6.3.3. Formation Reconfiguration for the Multi-UAVs with the New Leader

In the leader election link, UAV 2 turns out to be the new leader by the highest final performance score as displayed in Figure 17. Complete the remaining steps of the strategy developed in Section 5.2 in turn and then, the multi-UAVs can reform the intended formation under the leadership of UAV 2, as depicted in Figure 18 and Figure 19. Moreover, the linear velocity, the pitch angle and the yaw angle of each follower are portrayed in Figure 20, Figure 21 and Figure 22, respectively.

Remark 7. 

It should be noted that, in the article, ‘UAV i’ is just an identifier for each UAV, which has nothing to do with the number in the swarm communication topology. For instance, the number of the original leader ‘UAV 0’ in the swarm communication topology is 0, whereas after its disengagement, ‘UAV 2’ turns to be the new leader of the swarm, which means at the moment, the UAV identified by ‘UAV 2’ corresponds to the node numbered 0 in the swarm communication topology. At the same time, the numbers of other followers in the swarm communication topology can change correspondingly. On the other hand, as aforementioned, the desired distance for each follower with respect to the leader is associated with the number in the swarm communication topology rather than the number contained in the identifier. Therefore, as demonstrated in Figure 18, under the leadership of ‘UAV 2’, obvious changes occur for the movement of the followers whose numbers in the swarm communication topology change, including ‘UAV 1’, ‘UAV 3’, ‘UAV 5’, ‘UAV 7’, ‘UAV 9’ and ‘UAV 11’, while the movement trends of the followers ‘UAV 4’, ‘UAV 6’, ‘UAV 8’, and ‘UAV 10’ remain unchanged for their unaltered numbers in the swarm communication topology.

6.4. Formation Reconfiguration under Follower Detachment

At the 180th second, UAV 9 detaches itself from the leader–follower multi-UAVs. With the help of the strategy developed in Section 5.3, the multi-UAVs can form the expected geometry once again, as portrayed in Figure 23 and Figure 24. After the disengagement of UAV 9, with the aim of achieving the communication topology reconfiguration with minimal edge changes, all followers’ numbers in the swarm communication topology alter, further contributing to the change of each follower’s desired distance with respect to UAV 2. These can well explain the change of the movement trend for each follower in Figure 23. Meanwhile, the linear velocity, the pitch angle and the yaw angle for each follower can achieve consistency as plotted in Figure 25, Figure 26 and Figure 27, respectively.

6.5. Summary and Evaluation

In the end, the full flight trajectories of the leader–follower multi-UAVs when $t\in \left[0,240\right)$ are portrayed in Figure 28, where the reconfiguration triggering events including the addition of UAV 11, the disengagement of UAV 0 and the disengagement of UAV 9 are labeled. The correspondence between the identifier and the number in the swarm communication topology for each member of the multi-UAVs is displayed in

Table 14. Correspondence between the identifier and the number in the swarm communication topology for each member of the multi-UAVs.

Triggering Event	0	1	2	3	4	5
Initial correspondence	UAV 0	UAV 1	UAV 2	UAV 3	UAV 4	UAV 5
Addition of UAV 11	UAV 0	UAV 1	UAV 2	UAV 3	UAV 4	UAV 5
Disengagement of UAV 0	UAV 2	UAV 1	UAV 4	UAV 3	UAV 6	UAV 5
Disengagement of UAV 9	UAV 2	UAV 4	UAV 1	UAV 6	UAV 3	UAV 8
Triggering Event	6	7	8	9	10	11
Initial correspondence	UAV 6	UAV 7	UAV 8	UAV 9	UAV 10
Addition of UAV 11	UAV 6	UAV 7	UAV 8	UAV 9	UAV 10	UAV 11
Disengagement of UAV 0	UAV 8	UAV 7	UAV 10	UAV 9	UAV 11
Disengagement of UAV 9	UAV 5	UAV 10	UAV 7	UAV 11

.

Furthermore, in order to quantitatively evaluate the proposed formation reconfiguration strategies under the condition of dynamic swarm scale from the perspective of swarm communication topology, the concept of connection retention rate (CRR) is introduced, whose formula can be described as

$$\begin{array}{c}\hfill CRR=\frac{C_{total}-C_{altered}}{C_{total}},\end{array}$$

(31)

where $C_{total}$ denotes the amount of unidirectional connections contained in the post-reconfiguration swarm communication topology and $C_{altered}$ denotes the amount of altered unidirectional connections during the reconfiguration of the swarm communication topology. It should be noted that a bidirectional connection is treated as two unidirectional connections. Under the specific formation reconfiguration triggering event, the closer CRR is to 1, the less impact the reconfiguration strategy has on the stability of the swarm communication topology. The information related to CRR for the swarm communication topology using the reconfiguration strategies developed in Section 5.2–Section 5.4 is displayed in

Table 15. Connection retention rate (CRR) for the swarm communication topology using the proposed reconfiguration strategies.

Triggering Event	${\mathit{C}}_{\mathit{altered}}$	${\mathit{C}}_{\mathit{total}}$	CRR
Addition of UAV 11	2	20	90%
Disengagement of UAV 0	6	18	67%
Disengagement of UAV 9	0	16	100%

.

7. Conclusions

The majority of the existing formation reconfiguration studies are carried out under the assumption that the swarm scale is fixed. By contrast, in the article, the problems of formation reconfiguration for leader–follower multi-UAVs under three conditions of dynamic swarm scale are investigated with the corresponding formation reconfiguration strategies developed. In terms of the formation reconfiguration under leader disengagement, the hierarchical BRB performance assessment model for UAVs is constructed with four designed pivotal performance metrics for each UAV as the inputs and the ultimate performance assessment result for each UAV as the output. Furthermore, connection retention rate (CRR) is introduced to quantitatively analyze the influence of the proposed formation reconfiguration strategies for changing swarm scale on the stability of the swarm communication topology. In simulation, CRR for the swarm communication topology under the formation reconfiguration strategies for leader disengagement, new member addition and follower disengagement can reach 67%, 90% and 100%, respectively.

In the article, it is assumed that there is only one leader for an UAV swarm and only two communication methods are permitted, that is, leader-to-follower and follower-to-follower. The control architecture involving multiple leaders and how to design the communication mechanism under this architecture, including leader-to-leader, leader-to-follower and follower-to-follower, will be further studied. Moreover, in the article, the main focus was put on how to reconfigure the formation when the scale of the multi-UAVs is altered and how to fuse the multiple sources of information of followers to elect a leader for a leaderless swarm, where inter-vehicle collision avoidance during formation reconfiguration is not incorporated. This challenge needs to be considered in our future works.