Dynamic Recovery and a Resilience Metric for UAV Swarms Under Attack

Abstract

Unmanned Aerial Swarms are attracting widespread interest in fields such as disaster response, environmental monitoring, and agriculture. However, there is still a lack of effective recovery strategies and comprehensive performance metrics for UAV swarms facing communication attacks, especially in capturing dynamic recovery. The aim of this study is to recover the split and disconnected UAV swarm under attacks. A dynamic recovery method is proposed under attacks by establishing the relationship between algebraic connectivity and consensus speed. The proposed recovery method enables each UAV to selectively establish communication links with responsive UAVs based on the proposed recovery method to reduce communication cost, rather than linking with all neighbours within communication range. Based on this, a set of performance indexes is introduced, considering factors such as consensus ability, communication efficiency, mission execution, and resource consumption. Furthermore, a resilience metric is proposed to quantitatively assess the efficiency of recovery and consensus transition, providing a comprehensive measure of the ability to reach consensus after attacks. Simulations utilizing the second-order consensus protocol and dynamics validate that the consensus speed of the proposed recovery method is $18.88\%$ faster than random recovery. The proposed resilience metric captures the change in the time from recovery to new consensus state, and the resilience of the proposed recovery method is $66.99\%$ higher than random recovery.

1. Introduction

Unmanned Aerial Swarms (UAVs) are attracting widespread interest in fields such as disaster response, environmental monitoring, and agriculture [1,2,3]. It is well-known that UAV swarms often work in challenging environments [4,5]. Unanticipated events, including intrusions and attacks, may lead to individual UAVs failing, losing communication, and being unable to continue the mission [6,7,8]. The critical issue for UAV swarms is maintaining function based on a communication topology after disruptions or attacks [9]. Thus, recovery of UAV swarms plays a key role in keeping communication topology [10,11,12].

The recovery process is based on the topology of the UAV swarm [13]. The goal of topology recovery is to maintain effective communication between remaining UAVs [14]. To this end, the recovery methods of UAV swarms have been the focus of extensive research. Redundancy-based approaches are traditionally incorporated into the recovery process [15,16,17]. However, these methods depend on redundant communication links and backup UAVs, which increases the cost. Recently, dynamic adaptive recovery methods have been applied to UAV swarm recovery. The swarm adjusts the communication topology based on global or local states, ensuring fast response to UAV failures or swarm splits. For example, neighbouring UAVs may extend their communication range or alter flight paths to compensate for lost connections. Early works in UAV swarm topological recovery explored distributed algorithms, allowing the swarm to repair broken communication links [18]. Although the adaptive recovery method enables UAV swarms to operate in a self-organized way, the method is not always applicable in dynamic environments [19].

1.1. Related Work

The main goal of UAV swarm recovery is to re-establish and maintain the connectivity of the swarm, ensuring that UAVs communicate and coordinate effectively. Over the years, various strategies have been proposed for UAV swarm recovery. Existing studies can be categorized into topology reconfiguration, optimization-based methods, task-aware recovery, and resilience evaluation.

Some researchers focused on restoring communication links by reconstructing the swarm topology. Zhang et al. [20] developed the recovery problem of leader–follower UAV swarms. Their study provided three recovery strategies to deal with leader disengagement, follower detachment, and new addition scenarios. But their strategies only permit one leader and lack flexibility in dynamic missions. Similarly, Li et al. [21] introduced recovery methods targeting different failure periods but primarily focused on reestablishing communication links without performance considerations. These methods operate in simplified 2D environments and do not assess swarm stability after recovery.

Optimization algorithms have been applied to improve recovery efficiency. Duan et al. [22] adopted a Hybrid Particle Swarm Optimization and Genetic Algorithm (HPSOGA) for multi-UAV formation reconfiguration. They described continuous control input by a piecewise linear strategy for swarm reformation scenarios. Yang et al. [23] applied a modified Artificial Bee Colony (ABC) algorithm to enhance efficiency and safety for the recovery of the UAV swarm. Feng et al. [24] conducted an Adaptive Learning-based Pigeon-Inspired Optimization (ALPIO) algorithm to find the target topology for the recovery of UAV swarms, and the performance of the recovered UAV swarm was superior to other global recovery optimization methods. These methods achieved faster reconnection, but often neglected the stability and convergence speed of the swarm after reconfiguration. They consider idealized communication without considering dynamic environmental interference or 3D kinematics.

More recent works integrate task execution into the recovery process. Liu et al. [25] designed a distributed matching algorithm to solve the recovery problem of a UAV swarm with multiple tasks. Chen et al. [26] utilized a joint optimization model with maintenance and task rescheduling to achieve autonomous recovery. Wei et al. [27] proposed a co-adaptation method for resilience rebound in UAV swarms conducting surveillance missions, where the framework dynamically adjusts formation and task allocation after disruptions to maintain mission continuity. These works highlight the importance of mission-awareness during recovery, but lack evaluation of how quickly and stably the swarm regains coordinated behaviour after damage.

Recent studies discuss the importance of time-based resilience metrics [28,29,30,31,32]. For instance, Cheng et al. [33] proposed a joint reconnaissance resilience framework that measures the mission capability of the UAV swarm under threat. Phadke et al. [34] reviewed UAV swarm resilience requirements and pointed out that the proposed integrated framework enhances the ability of the swarm to withstand disruptions. Feng et al. [35] proposed a recovery importance index to recover the UAV swarm, and the mission convergence area and communication status were involved. However, most existing metrics focus either on topological connectivity or coverage, without fully addressing dynamic performance, such as the time required to reach a new consensus state or the stability of the swarm during recovery.

All in all, few existing studies combine topology recovery and consensus convergence into a single time-based resilience metric. Most existing metrics evaluate either connectivity or task performance separately and do not measure the time required to reach a stable consensus after an attack. In addition, many approaches assume simple planar motion models and do not include measures for performance after recovery. Few methods consider three-dimensional dynamics with six degrees of freedom, and even fewer quantify the time to reach a new consensus state, which is an essential factor for assessing the realistic resilience of UAV swarms. Therefore, this paper aims to develop a dynamic recovery method for split UAV swarms under attack by linking algebraic connectivity to consensus speed, introducing comprehensive performance indexes and a resilience metric to evaluate recovery and consensus efficiency.

1.2. Contribution and Paper Organization

A critical problem exists in UAV swarms after being attacked—specifically, the swarms split into isolated clusters, resulting in the loss of coordinated operation. Thus, developing a recovery method for maintaining connectivity is still a significant challenge. Moreover, most performance descriptions focus only on communication capability, which is inadequate. Therefore, a comprehensive performance is needed. Furthermore, existing resilience metrics often fail to reflect changes to the recovery algorithm. Hence, this paper proposes a recovery algorithm and designs a resilience metric under attacks. The main contributions include the following:

• A novel dynamic recovery method is proposed that considers maximizing the speed of consensus after an attack. The method involves a disconnected UAV swarm after an attack, dynamic behaviour, and consensus control of the UAV swarm.
• The performance of the UAV swarm is evaluated from the combined perspectives of consensus, communication, mission, and resources. A comprehensive description of the UAV swarm is provided.
• A resilience metric is further proposed, focusing on the time from stopping recovery to a new consensus state. The metric is sensitive to the change in the time from recovery to the new consensus state.

The remainder of this paper is organized as follows. Section 2 defines the problem formulation. Section 3 introduces the proposed recovery method. Section 4 details the performance indexes and resilience metric. Section 5 discusses the experimental results. Finally, Section 6 summarizes the conclusions of this work.

2. Problem Formulation

Definition 1.

(UAV Degree k): For UAV swarms, the degree k of a UAV is defined as the total number of communication links both to and from the UAV, as defined in our previous work [29].

A UAV swarm can be modelled by an undirected graph $\mathcal{G}=\left(\mathcal{V},\mathcal{E}\right)$, where $\mathcal{V}=\left\{v_i|i=0,1,2,\dots ,N\right\}$ denotes the set of UAVs, and $\mathcal{E}=\left\{e_i|i=1,2,\dots ,M\right\}$ represents the set of communication links among UAVs. $\mathcal{E}$ is induced by the adjacency matrix $\mathcal{A}=\left[a_{ij}\right]$. For the i-th UAV $v_i$, if it is within the communication range of UAV $v_j$, the weight $a_{ij}$, which is an element of $\mathcal{A}$, is equal to 1 (i.e., $a_{ij}=1$); otherwise, it is zero. The Laplacian matrix $\mathcal{L}=\left[l_{ij}\right]$ of $\mathcal{G}$ is defined as $\mathcal{L}=\mathcal{D}-\mathcal{A}$, where $\mathcal{D}=diag(k_0,k_1,\dots ,k_N)$ is the diagonal degree matrix and $k_i$ is the degree of the UAV $v_i$ in swarm $\mathcal{G}$. The Laplacian $\mathcal{L}$ of $\mathcal{G}$ can also be represented by $\mathcal{L}=B{B}^{\top }={\sum }_{\ell =1}^M{b}_{\ell }{b}_{\ell }^{\top }$. B denotes incidence matrix of $\mathcal{G}$, with ℓ-th column $b_{\ell }$. The vector $b_{\ell }$ has entries $+1$ and $-1$ in the UAVs connected by the communication link; specifically, $b_{\ell i}=1,b_{\ell j}=-1$, and all other entries are zero. Let the eigenvalues of $\mathcal{L}$ be $0={\lambda }_1\le {\lambda }_2\le \cdots \le {\lambda }_N$. The second-smallest eigenvalue ${\lambda }_2$ is denoted as an algebraic connectivity $\alpha \left(\mathcal{G}\right)$. The normalized eigenvector associated with algebraic connectivity is called the Fiedler vector [36].

Assuming part of the UAVs in the UAV swarm $\mathcal{G}$ are under attack, the attacked UAVs are wirelessly disconnected from the swarm. This means that the corresponding communication links of the attacked UAV did not exist. The severe attacks split the swarm into several clusters. The attacked swarm with multiple clusters can be modelled using an undirected and disconnected graph ${\mathcal{G}}^{\prime }=({\mathcal{V}}^{\prime },{\mathcal{E}}^{\prime })$, where ${\mathcal{V}}^{\prime }=\left\{v_i|i=0,1,2,\dots ,N^{\prime }\right\}$ represents the set of remained UAVs and ${\mathcal{E}}^{\prime }$ represents the set of remained communication links. $L^{\prime }$ is the Laplacian of $G^{\prime }$. The number of zero eigenvalues of $L^{\prime }$ represents the number of split clusters in the attacked swarm.

Definition 2.

(Isolated UAVs): The isolated UAV in the swarm ${\mathcal{G}}^{\prime }$ means that its degree is zero (i.e., $k=0$). For the special case of only one UAV in the cluster, in other words, the UAV degree is 0, which refers to the isolated UAVs. The set of isolated UAVs is $\mathcal{I}=\{v_i\in {\mathcal{V}}^{\prime }|k=0\}$.

Definition 3.

(UAV Cluster): A group of UAVs ${\mathcal{C}}_i\subseteq {\mathcal{V}}^{\prime }$ is called a UAV cluster if, for any $v_i^{\prime },v_j^{\prime }\in {\mathcal{C}}_i$, there exists a communication link between $v_i^{\prime },v_j^{\prime }$.

Typically, all UAVs fly with a pre-defined safety distance and the same velocity, thereby forming a coordinated formation. However, when communication links among UAVs are disrupted due to an attack, the swarm is split into multiple clusters. In this case, the safety distance and velocity can only be maintained in each cluster, while inter-cluster coordination fails due to the lack of communication between clusters. This may lead to the downtime of the entire swarm system. To address this issue, this work proposes an algorithm for the rapid recovery of the communication network. Once inter-cluster communication is recovered, the swarm can re-establish coordination among clusters, enabling the entire swarm to maintain the desired inter-UAV safety distances. A summary of the main symbols is provided in

Table 1. Summary of notations.

Symbol	Description
$\mathcal{G}$, ${\mathcal{G}}^{\prime }$, ${\mathcal{G}}^r$	Swarm before attack, after attack, and at recovery start with added links
${\mathcal{E}}^{\prime }$, ${\mathcal{E}}^s$, ${\mathcal{E}}^r$, ${\mathcal{E}}^C$	Communication links in ${\mathcal{G}}^{\prime }$, links added to ${\mathcal{G}}^{\prime }$ at recovery start, links in ${\mathcal{G}}^r$, and all possible links in a complete graph of ${\mathcal{G}}^{\prime }$ excluding existing links
N, $N^{\prime }$	Number of UAVs in the swarm before and after attack
$\alpha \left(\mathcal{G}\right)$	Algebraic connectivity of swarm $\mathcal{G}$
A, $a_{ij}$	Adjacency matrix of swarm with element $a_{ij}$ indicating edge between i and j
B, $b_l$	Incidence matrix of UAV swarm with l-th edge vector column $b_l$
k	UAV degree
${\mathcal{C}}_i$	UAV cluster
$\mathcal{I}$	Set of isolated UAVs, special case of UAV cluster ${\mathcal{C}}_i$
$x_i,y_i,z_i$	Position of UAV i in 3D space
$V_i$	Velocity of UAV i
${\gamma }_i,{\chi }_i,{\varphi }_i$	Pitch, yaw and roll angles of UAV i
$T_i$, $D_i$, $L_i$	The thrust, drag force and lift of UAV i
$u_i$, $u_i^{pre}$, $u_i^{rec}$	Virtual input of UAV i: current, before recovery, and when recovery start
${\kappa }_r$	A binary value represents the signal of recovery
t	Time variable
${\mathcal{N}}_i$	The number of neighbours of UAV i
${\varphi }_{\alpha }$	A smooth pairwise potential function
$n_{ij}$	The vector along the communication link $e_{ij}$
$\rho h(·)$	A bump function.
R	Communication range of the UAVs
F, $f_i$	Normalized Fiedler eigenvector according to $\alpha \left({\mathcal{G}}^r\right)$, $f_i$ is the ith element of F
$n_i$	The number of UAVs in the UAV cluster ${\mathcal{C}}_i$

.

3. Proposed Recovery Method

3.1. Swarm Dynamics and Consensus Control

A swarm of fixed-wing UAVs is considered in this study. The i-th fixed-wing UAV in the swarm has dynamics of

$$\begin{array}{cc}\hfill & {\dot{x}}_i=V_{i}cos{\gamma }_{i}cos{\chi }_i\hfill \\ \hfill & {\dot{y}}_i=V_{i}cos{\gamma }_{i}sin{\chi }_i\hfill \\ \hfill & {\dot{z}}_i=V_{i}sin{\gamma }_i\hfill \\ \hfill & \dot{V_i}={\displaystyle \frac{T_i-D_i}{m_i}}-sin{\gamma }_i\hfill \\ \hfill & \dot{{\gamma }_i}={\displaystyle \frac{L_{i}cos{\varphi }_i-m_{i}gcos{\gamma }_i}{m_i{V}_i}}\hfill \\ \hfill & \dot{{\chi }_i}={\displaystyle \frac{L_{i}sin{\varphi }_i}{m_i{V}_{i}cos{\gamma }_i}}\hfill \end{array}$$

(1)

where $x_i, y_i, z_i$ are the position along three dimensions [37]. $V_i$ is the ground velocity of the i-th UAV. ${\gamma }_i$ and ${\chi }_i$ are the pitch and yaw angles. ${\varphi }_i$ is the roll angle. $T_i$ and $D_i$ denote the thrust and drag force of the i-th UAV. $L_i$ is the lift. g represents the gravitational acceleration. $m_i$ is the mass of the fixed-wing UAV.

In this work, feedback linearization is applied to transform the nonlinear dynamics of the fixed-wing UAV to second-order dynamics for more straightforward swarm control designing [38,39]. The second-order dynamic of the UAV is given as

$$\left\{\begin{array}{c}{\ddot{x}}_i=u_{xi}\hfill \\ {\ddot{y}}_i=u_{yi}\hfill \\ {\ddot{z}}_i=u_{zi}\hfill \end{array}\right.$$

(2)

where $u_{xi}, u_{yi}, \mathrm{and} u_{zi}$ are the virtual control inputs that adjust the acceleration of the i-th UAV. The virtual input $u_i$ through the entire mission process is proposed as follows:

$$\begin{array}{cc}\hfill u_i& =\left[\begin{array}{c}{u}_{xi}\\ u_{yi}\\ u_{zi}\end{array}\right]\hfill \\ \hfill u_i& =(1-{\kappa }_r)u_i^{pre}+{\kappa }_r{u}_i^{rec}\hfill \end{array}$$

(3)

where $u_i$ is composed of $u_{xi}, u_{yi}, \mathrm{and} u_{zi}$. $u_i$ consists of two parts, $u_i^{pre}$ before recovery and $u_i^{rec}$ according to the proposed recovery algorithm. ${\kappa }_r$ is a binary value that represents the signal of recovery, as described in Equation (4).

$${\kappa }_r=\left\{\begin{array}{c}1,\mathrm{if}\mathrm{recovery}\mathrm{has}\mathrm{started}\hfill \\ 0,\mathrm{otherwise}\hfill \end{array}\right.$$

(4)

$u_i^{pre}$ is the virtual input before starting recovery. The $u_i^{pre}$ minimizes the differences between communication UAVs, thus achieving consensus across the swarm [40]. $u_i^{pre}$ is introduced in Equation (5).

$$\begin{array}{cc}\hfill u_i^{pre}& =-\sum _{j\in N_i}{\varphi }_{\alpha }(\parallel \left[\begin{array}{c}{x}_j\\ y_j\\ z_j\end{array}\right]-\left[\begin{array}{c}{x}_i\\ y_i\\ z_i\end{array}\right]{\parallel }_2)n_{ij}-\sum _{j\in N_i}\rho h(\parallel \left[\begin{array}{c}{x}_j\\ y_j\\ z_j\end{array}\right]-\left[\begin{array}{c}{x}_i\\ y_i\\ z_i\end{array}\right]{\parallel }_2/R)(\left[\begin{array}{c}\dot{x_j}\\ \dot{y_j}\\ \dot{z_j}\end{array}\right]-\left[\begin{array}{c}\dot{x_i}\\ \dot{y_i}\\ \dot{z_i}\end{array}\right])\hfill \end{array}$$

(5)

where ${\mathcal{N}}_i$ represents the set of neighbours of UAV i. ${\varphi }_{\alpha }$ is a smooth pairwise potential function. $n_{ij}$ is the vector along the communication link $e_{ij}$. $\rho h(·)$ is a bump function [40]. R is the communication range of the UAVs. Particularly, $u_i^{rec}$ is the virtual input of the UAV swarm once the recovery has started. $u_i^{rec}$ is proposed and described in the following section.

3.2. Swarm Recovery Strategy

Indeed, the recovery process is based on the swarm ${\mathcal{G}}^{\prime }$ with several clusters. From Equation (5), the corrupted communication link means that $a_{ij}$ is equal to zero. Thus, the isolated UAV or the UAV in each cluster can only compute the control input $u_i$ locally, without considering neighbouring UAVs. The proposed recovery method reconstructs $a_{ij}$, leading to the connection between clusters or isolated UAVs and enhancing the UAV swarm connectivity. Then, the control input $u_i^{rec}$ is calculated based on the reconstructed wireless communication links, namely the recovered $a_{ij}$.

The recovery of the swarm ${\mathcal{G}}^{\prime }$ is denoted by ${\mathcal{G}}^r=({\mathcal{V}}^{\prime },{\mathcal{E}}^r)$. ${\mathcal{E}}^r={\mathcal{E}}^{\prime }\cup {\mathcal{E}}^s$. ${\mathcal{E}}^s$ is a set of m added communication links which is selected from candidate communication links ${\mathcal{E}}^C$ on ${\mathcal{V}}^{\prime }$, ${\mathcal{E}}^{\prime }\cap {\mathcal{E}}^C=\varnothing$. ${\mathcal{E}}^s$ is added to ${\mathcal{G}}^{\prime }$ for the greatest increase in algebraic connectivity of ${\mathcal{G}}^r$. Algebraic connectivity of the UAV swarm influences the speed of convergence, since higher connectivity generally leads to faster consensus. The second-smallest eigenvalue of the Laplacian matrix ${\mathcal{L}}^r$ of the recovery swarm ${\mathcal{G}}^r$ is the algebraic connectivity $\alpha \left({\mathcal{G}}^r\right)$. A large $\alpha \left({\mathcal{G}}^r\right)$ indicates that the recovery swarm has stronger connectivity, faster consensus convergence, and higher resilience. The recovery of the UAV swarm can be described as follows:

$$\begin{array}{cc}\hfill maximize& \alpha \left({\mathcal{G}}^r\right)\hfill \\ \hfill subjectto& {\mathcal{G}}^r=({\mathcal{V}}^{\prime },{\mathcal{E}}^r)\hfill \\ \hfill & {\mathcal{E}}^r={\mathcal{E}}^{\prime }\cup {\mathcal{E}}^s\hfill \\ \hfill & |{\mathcal{E}}^s|=m\hfill \\ \hfill & {\mathcal{E}}^s\subseteq {\mathcal{E}}^C\hfill \end{array}$$

(6)

where the optimization variable is the subset ${\mathcal{E}}^s$ of recovery communication links. A communication link ℓ is added at each step. However, the recovery process is computationally intensive for UAV swarms. The problem of Equation (6) can be solved by a heuristic method other than exhaustive search. The inequality

$$\alpha ({\mathcal{G}}^r+H)\le \alpha \left({\mathcal{G}}^r\right)+Tr\left(Hh{h}^{\top }\right)$$

(7)

holds when $\alpha \left({\mathcal{G}}^r\right)$ is simple and h is a unit-norm eigenvector associated with $\alpha \left({\mathcal{G}}^r\right)$ [41]. H is any symmetric matrix. Thus, for isolated $\alpha$ (${\lambda }_1\left(\mathcal{L}\right)<{\lambda }_2\left(\mathcal{L}\right)<{\lambda }_3\left(\mathcal{L}\right)$), the supergradient of $\alpha \left({\mathcal{G}}^r\right)$ is

$$f_{\ell }^{\prime }\left(\alpha \left({\mathcal{G}}^r\right)\right)=F^{\top }{f}_{\ell }^{\prime }\left(\mathcal{L}\right)F$$

(8)

F is normalized Fiedler eigenvector according to $\alpha \left({\mathcal{G}}^r\right)$, $F=[f_1,f_2,\cdots ,f_N]$. Considering $f_{\ell }^{\prime }\left(\alpha \left(L\right)\right)=b_{\ell }{b}_{\ell }^{\top }$, the supergradient of $\alpha \left({\mathcal{G}}^r\right)$ is

$$\begin{array}{cc}\hfill f_{\ell }^{\prime }\left(\alpha \left({\mathcal{G}}^r\right)\right)& =F^{\top }{b}_{\ell }{b}_{\ell }^{\top }F={\left(b_{\ell }^{\top }F\right)}^2\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill =({\left[\begin{array}{c}0\\ 1\\ ⋮\\ -1\\ 0\end{array}\right]}^{\top }\left[\begin{array}{c}{f}_1\\ f_i\\ ⋮\\ f_j\\ f_N\end{array}\right]{)}^2={(f_i-f_j)}^2& \hfill \end{array}$$

(10)

The zero eigenvalues of L have multiplicity s when ${\mathcal{G}}^r$ is disconnected. In this case, the swarm has s UAV clusters. It is worth mentioning that following the change in $\alpha \left({\mathcal{G}}^r\right)$ is complicated because of repeated eigenvalues of zero when ${\mathcal{G}}^r$ is still disconnected. The added communication links can be a perturbation of ${\mathcal{G}}^r$. The fractional power p of the added communication links to eigenvalues of $\mathcal{L}$ is $p^{{\displaystyle \frac{1}{J}}}$, where J denotes the order of the Jordan submatrix. This means that even a newly added communication link can affect the eigenvalues of $\mathcal{L}$. In addition, let $F^1,\dots ,F^s$ be the vectors according to the multiple eigenvalue zero. $F^1={\left[\begin{array}{ccccc}{\displaystyle \frac{1}{n_1}}& {\displaystyle \frac{1}{n_1}}& {\displaystyle \frac{1}{n_1}}& 0& 0\end{array}\right]}^{\top }$, …, $F^s={\left[\begin{array}{ccccc}0& 0& 0& {\displaystyle \frac{1}{n_s}}& {\displaystyle \frac{1}{n_s}}\end{array}\right]}^{\top }$. $n_1$ and $n_s$ are the number of UAVs in the first and last UAV cluster, respectively. ${(f_i-f_j)}^2>0$ only when the added communication link ℓ connects two UAV clusters. Therefore, the recovery strategy should always link UAVs from different UAV clusters. In general, the increase in algebraic connectivity $\alpha \left({\mathcal{G}}^r\right)$ can be approximated by Equation (10).

Accordingly, the proposed recovery method aims to solve the optimization problem with a heuristic method. The heuristic recovery method enables each UAV to selectively establish connections with responsive UAVs to reduce communication cost rather than linking with all neighbours within communication range. When communication fails, UAVs assess the reconnection criteria, prioritizing the maximization of algebraic connectivity. The heuristic method is divided into three main steps. At first, the isolated UAV is considered. It starts to find the largest connected clustered UAVs ${\mathcal{C}}_i$ within the communication range. If no connected ${\mathcal{C}}_i$ is found, it increases the transmit power and communication range until one is identified. It links the isolated UAV and the found ${\mathcal{C}}_i$ and adds communication links until all isolated UAVs are connected. Hence, there exists a swarm without isolated UAVs.

The second key step deals with the existing UAV clusters. If the UAV swarm is still disconnected and composed of several connected clustered UAVs ${\mathcal{C}}_i$, the recovery process is continued for each UAV cluster. Next, it adds a communication link between the i-th UAV in the largest UAV cluster and the j-th UAV in the smallest UAV cluster, where both the i-th and j-th UAVs have the highest UAV degree. The process continues until all UAVs are connected.

Finally, the UAV swarm is connected, and communication links are added until the difference in algebraic connectivity between the current swarm and the updated swarm is less than a threshold $ϵ$. The threshold $ϵ$ is set to ${10}^{-3}$ for balancing precision and practicality in the recovery process.

A small threshold helps recovery without adding too many communication links, thereby avoiding unnecessary computational costs. Overall, the improvement in algebraic connectivity is significant, and the proposed recovery process is completed. The proposed recovery method is described in detail in Algorithm 1.

Algorithm 1 Swarm recovery strategy.

1: Input: Disconnected swarm ${\mathcal{G}}^{\prime }$, Adjacency matrix $A^{\prime }$ of ${\mathcal{G}}^{\prime }$, threshold $ϵ={10}^{-3}$ 2: Output: Recovered UAV swarm ${\mathcal{G}}^r$, Adjacency matrix $A^r$ of ${\mathcal{G}}^r$ 3: for each isolated UAV $\mathcal{I}$ do 4:    Find the largest connected connected clustered UAVs ${\mathcal{C}}_i$ within communication range 5:    if ${\mathcal{C}}_i$ is not found then 6:       Increase communication range until ${\mathcal{C}}_i$ is found 7:    end if 8:    Add a communication link between $\mathcal{I}$ and ${\mathcal{C}}_i$ 9:    Update the adjacency matrix $A^r$ 10: end for 11: while the swarm ${\mathcal{G}}^r$ is not connected do 12:    Identify the largest ${\mathcal{C}}_{max}$ and smallest UAV cluster ${\mathcal{C}}_{min}$ 13:    Add a link between UAVs with the highest degree in $i\in {\mathcal{C}}_{max}$ and UAV $j\in {\mathcal{C}}_{min}$ 14:    Update the adjacency matrix $A^r$ 15: end while 16: while Algebraic connectivity improvement $\Delta \alpha \left({\mathcal{G}}^r\right)>ϵ$ do 17:    Add communication links according to Equation (10) 18:    Update the adjacency matrix $A^r$ 19: end while

At the same time, due to the proposed recovery strategy, new communication links are added. Hence, the adjacency matrix and Laplacian matrix are updated. The recovery consensus protocol $u_i^{rec}$ is designed as

$$\begin{array}{c}\hfill u_i^{rec}=-\sum _{j\in N_i}{a}_{ij}^r[\varphi (\parallel \left[\begin{array}{c}{x}_j\\ y_j\\ z_j\end{array}\right]-\left[\begin{array}{c}{x}_i\\ y_i\\ z_i\end{array}\right]{\parallel }_2-d_e)n_{ij}+(\left[\begin{array}{c}\dot{x_j}\\ \dot{y_j}\\ \dot{z_j}\end{array}\right]-\left[\begin{array}{c}\dot{x_i}\\ \dot{y_i}\\ \dot{z_i}\end{array}\right])]\end{array}$$

(11)

$$\begin{array}{cc}\hfill \varphi \left(q\right)& ={\displaystyle \frac{1}{2}}[(a+b){\sigma }_1(q-d_e+c)+(a-b)]\hfill \\ \hfill {\sigma }_1(q-d_e+c)& ={\displaystyle \frac{q-d_e+c}{\sqrt{1+{(q-d_e+c)}^2}}}\hfill \\ \hfill c& ={\displaystyle \frac{|a-b|}{\sqrt{4ab}}}\hfill \\ \hfill n_{ij}& =(\left[\begin{array}{c}{x}_j\\ y_j\\ z_j\end{array}\right]-\left[\begin{array}{c}{x}_i\\ y_i\\ z_i\end{array}\right])/\sqrt{1+ϵ\parallel \left[\begin{array}{c}{x}_j\\ y_j\\ z_j\end{array}\right]-\left[\begin{array}{c}{x}_i\\ y_i\\ z_i\end{array}\right]{\parallel }^2}\hfill \end{array}$$

(12)

where $a_{ij}^r$ is the element of the adjacency matrix $A^r$ after recovery. $d_e$ is the expected distance between the UAVs. $\varphi \left(q\right)$ is a collective potential. Most importantly, the communication range of UAVs could be changed during the recovery process. $n_{ij}$ gives the gradient of distance. The constants a and b are equal to 10.

Remark 1. 

Moreover, for fixed-wing UAVs, the real control inputs are roll angle, lift, and thrust. Thus, the virtual control input $u_i$ should be transformed to the real control inputs, which are described as Equation (13) [37].

$$\left\{\begin{array}{c}{\varphi }_i={tan}^{-1}\left({\displaystyle \frac{u_{yi}cos{\chi }_i-u_{xi}sin{\chi }_i}{(u_{zi}+g)cos{\gamma }_i-\eta sin{\gamma }_i}}\right)\hfill \\ L_i=m_i{\displaystyle \frac{(u_{zi}+g)cos{\gamma }_i-\eta sin{\gamma }_i}{cos{\varphi }_i}}\hfill \\ T_i=m_i[(u_{zi}+g)sin{\gamma }_i+\eta cos{\gamma }_i]+D_i\hfill \end{array}\right.$$

(13)

where $\eta =(u_{xi}cos{\chi }_i+u_{yi}sin{\chi }_i)$.

4. Proposed Swarm Performance Indexes and Resilience Metric

4.1. Proposed Swarm Performance Indexes

The performance of UAV swarms can be described from four perspectives: consensus ability, communication ability, mission ability, and resource ability. Consensus ability denotes that each UAV of the UAV swarm achieves the same acceleration. Communication ability includes transmitting messages between UAVs. Mission ability means the completion of missions. Resource ability includes energy management, for example, of the battery. Therefore, the performance of UAV swarms can be described as a weighted sum of these four abilities.

However, it is hard to decide the weight of these four abilities. Thus, the analytic hierarchy process (AHP) is performed to calculate the weight [42]. Furthermore, the four abilities are measured by four different indexes, including variant of acceleration ($id{x}_1$), interaction between UAVs ($id{x}_2$), completion of missions ($id{x}_3$), and energy management ($id{x}_4$). The notations involved in the proposed swarm performance indexes and resilience metric are listed in

Table 2. Summary of notations of proposed performance metric and resilience metric.

Symbol	Description
$id{x}_1$, $id{x}_2$, $id{x}_3$, $id{x}_4$	Variant of acceleration, interaction between UAVs, completion of missions, energy management
$y\left(t\right)$	The performance of UAV swarms
$y_r$	The new stable performance after recovery
$y_b$	The $\zeta$ percentage of its initial performance before attack. It emphasizes mission success by ensuring the swarm maintains a sufficient level of required functionality
$t_{\alpha }$	Attack time
$t_r^{\ast }$	The time complete recovery
$t_s$	The time to the new consensus state
$t_f$	The time to the end of the simulation
$Rs$	The proposed resilience metric
E	The ratio of buffer time and time to consensus after recovery
$\tau$	A decreasing function that measures the increase in the time it takes to reach the new consensus state
$S_R$	The ratio of the performance of the UAV swarm after reaching the new consensus to $y_b$
$t_r-t_{\alpha }$	Buffer time: the time after attack time $t_{\alpha }$ but before recovery time $t_r$. It allows the UAV swarm to absorb the impact of the attack and prepare for recovery.
$t_s-t_r^{\ast }$	The transition time measured as the time interval from the end of the recovery action to the point where the states of all remaining UAVs converge within a predifined threshold

. Compare the four indexes with each other to get the judgement matrix. The judgement is shown in

Table 3. Importance of the indexes.

Indexes	${\mathit{idx}}_1$	${\mathit{idx}}_2$	${\mathit{idx}}_3$	${\mathit{idx}}_4$
$id{x}_1$	1	3	4	7
$id{x}_2$	0.333	1	2	6
$id{x}_3$	0.25	0.5	1	4
$id{x}_4$	0.143	0.167	0.25	1

. The weight is $w=(0.54447, 0.25434, 0.15123, 0.04995)$. The consistency index (CI) is $0.039$. The random index (RI) is $0.882$. Thus, the consistency ratio (CR) is $CI/RI=0.044<0.1$, and the judgement matrix can pass the consistency test.

Furthermore, the performance of UAV swarms can be set as:

$$\begin{array}{cc}\hfill y\left(t\right)& =w\ast {[id{x}_1,id{x}_2,id{x}_3,idx4]}^T\hfill \\ \hfill & =0.54447\ast id{x}_1+0.25434\ast id{x}_2+0.15123\ast id{x}_3+0.04995\ast id{x}_4\hfill \end{array}$$

(14)

$id{x}_1=1-var{v}_{norm}$, $var{v}_{norm}=(varv-var{v}_{min})/(var{v}_{max}-var{v}_{min})$ is the variant of acceleration. $id{x}_2=msgs/msg{s}_0$ is calculated by the total messages between UAVs, and $msg{s}_0$ is all the messages at $t=0$. $id{x}_3$ is calculated by the coverage ratio of the UAV swarm to the initial coverage. This means part of the area is monitored by at least two UAVs. As shown in Figure 1, two UAVs are overlapped if the distance d between them is smaller than $\sqrt{3}R$ under the hexagonal approximation of two overlapped footprints [43].

Thus, the double convergence rate of the UAV swarm is the ratio of the number of two UAVs that are overlapped to the total number of UAVs. $id{x}_4$ is calculated by the ratio of the remaining battery level to the initial battery level. The consumption of the battery increases as the transmission power increases. It is worth noting that all four indexes are normalized to the range of $[0, 1]$.

4.2. Proposed Resilience Metric

The resilience metric proposed in this section is directly derived from the core objectives and operational characteristics of Algorithm 1, aiming to precisely quantify its performance in terms of recovery speed and topological integrity. As shown in Figure 2, the performance of the UAV swarm is composed of five parts: normal operation, buffer, recovery, recovery to consensus, and new consensus state. $y_t$ is the performance of the UAV swarm, and $y_r$ is the new stable performance after recovery. It is worth mentioning that $y_b$ is the $\zeta$ percentage of its initial performance before attack. It is vital to measure the performance of a UAV swarm above $\zeta$ of its initial capability, as it emphasizes mission success by ensuring the swarm maintains a sufficient level of required functionality. $t_r^{\ast }$ is the time for complete recovery. $t_s$ is the time to the new consensus state. $t_f$ is the time to the end of the simulation.

The proposed resilience measurement is expressed as:

$$\begin{array}{cc}\hfill & Rs=E\ast \tau \ast S_R\hfill \\ \hfill & E={\displaystyle \frac{t_r-t_{\alpha }}{t_s-t_r^{\ast }}}\hfill \\ \hfill & \tau =t_f\ast [1-{(t_s-t_r^{\ast })}^a]\hfill \\ \hfill & S_R={\displaystyle \frac{y_r-y_b}{y_b}}\hfill \\ \hfill & y_r={\displaystyle \frac{{\sum }_{t_s}^{t_f}{y}_t}{t_f-t_s}}\hfill \\ \hfill & y_b=\zeta {\displaystyle \frac{{\sum }_0^{t_{\alpha }}{y}_t}{t_{\alpha }-0}}\hfill \end{array}$$

(15)

where E is the ratio of buffer time and time to consensus after recovery. Buffer time represents the time after attack time $t_{\alpha }$ but before recovery time $t_r$. It allows the UAV swarm to absorb the impact of the attack and prepare for recovery. $\tau$ is a decreasing function that measures the increase in the time it takes to reach the new consensus state. $t_s-t_r^{\ast }$ is the transition time measured as the time interval from the end of the recovery action to the point where the states of all remaining UAVs converge within a predefined threshold. $S_R$ is the ratio of the performance of the UAV swarm after reaching the new consensus to $y_b$.

5. Simulation

5.1. Configuration

In this study, the ground station serves as the controller, overseeing a functioning UAV swarm. The UAVs utilize distributed algorithms for local control and decision-making. However, two UAVs in the swarm communicate with the ground station to ensure that their local information is consistent with the global swarm. This communication enables the ground station to maintain global connectivity and provides real-time updates to the global Laplacian matrix. The scenario of the UAV swarm is shown in Figure 3.

The UAV swarm is supposed to be attacked and completely disrupted when performing disaster monitoring missions. Simulations are conducted based on Python and NetworkX, which model UAV dynamics, communication, and environmental interactions in 3D. The dynamics of the UAV are based on Equations (1) and (2), ensuring realistic movement. The UAV swarm performs a mission with randomly distributed UAV positions in a defined area. $N^{\prime }$ UAVs in the swarm are attacked and removed based on the highest UAV degree k. Then, corresponding communication links are removed. The proposed recovery method is applied to the swarm after buffer time $\beta$ [21,44].

The UAV swarm is distributed within an area of 500 m × 500 m × 100 m at an altitude between 800 m and 90 m. The communication frequency of each UAV is 100 Hz. The communication range R is 250 m. The simulations are performed using Python 3.7.2 and NetworkX 2.6.3 on Ubuntu 18.04.6, with an Intel i7-10700F processor (Intel, Santa Clara, CA, USA) and 16 GB RAM. The dynamics of the UAV swarm are simulated, including the normal operation, attacks, buffer, recovery process, and consensus, as shown in Figure 4. Figure 4a shows the UAV swarm operating normally at $t=0$ s. There are ten UAVs in the swarm. Then, five UAVs in the swarm are attacked and disrupted at $t=2$ s.

All the attacked UAVs and their corresponding communication links are removed from the swarm. After that, it should be noted that the UAVs may connect to others during the buffer time. Furthermore, as detailed in Figure 4c, the UAV swarm starts to recover via the proposed method after a buffer time at $t=4$ s. $x, y, z$ are the three-dimensional axes. One communication link is added to the UAV swarm after each communication instance. The recovery process is shown in Figure 4c–f. Finally, the recovery process is completed at $t=4.03$ s.

The comparison of the proposed recovery method with the random recovery method is shown in Figure 5. The difference between the two methods is that the random recovery method randomly adds a communication link without considering the algebraic connectivity. Similarly, the random recovery method ends when the difference in algebraic connectivity between the current swarm and the updated swarm is less than a threshold $ϵ$. The value of $ϵ$ was determined empirically. The smaller values lead to adding more communication links and costs without a significant improvement in recovery. On the other hand, larger values did not recover well. Therefore, the threshold $ϵ$ is set as ${10}^{-3}$. Thus, four communication links are added in the proposed recovery method, while two are added in the random recovery. It is apparent that the proposed recovery method has a higher $\alpha \left({\mathcal{G}}^r\right)$ than the random recovery method.

After recovery, the UAV swarm continues to operate under consensus. The velocity of the UAV swarm by the proposed recovery method and random recovery can be shown in Figure 6. During $t=0\sim 2$ s, ten UAVs operate under normal conditions with different velocities. When $t=2$ s, five UAVs—UAV1, UAV2, UAV4, UAV7 and UAV8—are attacked and removed. Therefore, the velocities of these UAVs are not shown in the figure after $t=2$ s. Accordingly, the remained UAV clusters operate for a while during the buffer time. Then, the UAV cluster starts to recover via the proposed method, and its velocity drops in an instant. It could be inferred that the newly added communication links prevent the UAVs from becoming chaotic. After recovery, the UAV clusters are connected and able to reach consensus in the new state. The consensus is achieved at $t=66.88$ s, which is much quicker than the random recovery method $t=82.45$ s. It is clear that the consensus speed of the proposed method is $18.88\%$ faster than random recovery. Overall, the proposed recovery method is more efficient than the random recovery method for increasing the consensus speed.

Furthermore, the scalability of the proposed recovery method is evaluated by increasing the swarm size to $N=50$. A severe attack is simulated, where 25 UAVs (i.e., $50\%$) of the swarm are removed at $t_{\alpha }=2$ s. As described in Figure 7, the proposed recovery method maintains fast convergence at $t=137.05$ s, while the random recovery method takes longer, converging at $t=146.35$ s. Compared to the case with $N=10$, the convergence takes more time due to the increased number of UAVs.

The performance of the UAV swarm based on the proposed performance indexes is shown in Figure 8. The total simulation time $t_f$ is 100 s. The inset of Figure 8 shows a zoom of the normal operation, attack, and buffer time. The UAV swarm is attacked at $t=2$ s, and the performance of the UAV swarm degrades sharply. The buffer time is 2 s; during this period, the swarm absorbs the attack and prepares for recovery. After that, the proposed recovery method is performed until the algebraic connectivity between the current swarm and the updated swarm is less than a threshold $ϵ$. Furthermore, the swarm continues to perform in order to reach a new consensus state.

The resilience of the UAV swarm under attack is compared by the proposed resilience metric and two other resilience metrics in Refs. [28,45]; the results are shown in

Table 4. Resilience of the UAV swarm under attack.

Method	Resilience in Ref. [28]	Resilience in Ref. [45]	Proposed Resilience
Proposed Recovery	1.0957	0.1416	0.7750
Random Recovery	1.6593	0.1968	0.4641

. It is apparent that the proposed recovery method has a higher resilience than the random recovery method measured by the proposed resilience metric. The proposed resilience metric is sensitive to the time from recovery to the new consensus state. Generally, the resilience should be more resilient with less time from recovery to a new consensus state. It should, however, be noted that the resilience metrics in Refs. [28,45] are both larger in the random recovery. However, they can not capture the change in time from recovery to the new consensus state. Moreover, the proposed resilience metric captures the change in the time from recovery to new consensus state, and the resilience of the proposed recovery method is $66.99\%$ higher than random recovery.

In addition, an intermittent attack scenario is considered as shown by the red dashed lines in Figure 9. Five UAVs are attacked and removed from each attack at $t=2$ s, $t=12$ s, $t=22$ s, $t=32$ s, $t=42$ s, and $t=52$ s. Hence, the proposed recovery method is re-triggered after each attack. Despite the intermittent attacks, the swarm regains connectivity and reaches consensus at $t=133.54$ s. It is apparent that the proposed recovery is also efficient in recovering from intermittent attacks.

Figure 10 shows the change in the performance of UAV swarm $y\left(t\right)$ during the same scenario. The performance drops sharply following each attack, then recovers over time. The stepwise drops reflect the attack times, while the rises indicate the ability to recover. The zoomed-in view of the early stage highlights the rapid initial response and the progressive improvement of the state of the UAV swarm. The resilience of the swarm measured by the proposed method is $0.7205$, which indicates a high level of resilience of the UAV swarm under intermittent attacks.

5.2. Discussion

Our results confirm that the proposed recovery method makes the UAV swarm connect faster. The proposed resilience metric specifically evaluated the recovery time until convergence. While the resilience metric effectively evaluated the proposed recovery method, more realistic factors should be involved. The communication of the UAV swarm assumes fixed range and stable links, but does not consider the communication capacity and delay of the communication links, which may introduce bias to the dynamic.

Future work should explore larger swarms with more UAVs, such as 100 or more, operating in complex environments. In addition, the influence of communication range should be examined, as it is an important factor affecting swarm connectivity and resilience. Future studies could also investigate recovery parameters from the resilience metric to further improve recovery efficiency.

6. Conclusions

In summary, a new dynamic recovery method, performance indexes, and a new resilience metric are proposed for the UAV swarm, which can increase the speed of consensus and thus improve the resilience of the UAV swarm under attack. Depending on the relationship between algebraic connectivity and consensus speed, the proposed dynamic recovery method can be more efficient than the random recovery method. For a realistic description of the UAV swarm, four aspects should be used, including the ability to reach consensus, communication, mission, and resource. To capture the change in time from recovery to a new consensus state, a resilience metric is proposed. Simulations conducted on the second-order consensus protocol and a six-degree-of-freedom (6-DOF) dynamic UAV swarm have shown the resilience of the UAV swarm under attack by the proposed method. Simulation results show that the resilience of the proposed recovery method is 66.99% higher than random recovery. The proposed recovery methods and resilience evaluation improve the real-world deployment of UAV swarms by enhancing their ability to quickly recover from attacks. This ensures more stable, autonomous operation in dynamic, uncertain environments, reducing the need for manual intervention and increasing mission reliability.