A Bilateral Cooperative Strategy for Swarm Escort under the Attack of Aggressive Swarms

Abstract

With the development of swarm intelligence and low-cost unmanned systems, the offence and defense of a swarm have become essential issues in defense and security technologies. A swarm of drones can be used to attack some high-value units (HVUs), such as bases or fuel tanks. Moreover, some moving HVUs such as cargo ships are also greatly threatened when attacked by a swarm of unmanned surface vehicles. A promising approach to protect a HVU from the attack of an aggressive swarm is to use another low-cost swarm. However, escorting a HVU with a swarm is challenging since defenders must respond to attacks and carry out escorts in a noncentralized manner. It is difficult to balance the above tasks well using the unilateral escort strategy adopted by defenders in previous studies. Therefore, this paper proposes a bilateral cooperative strategy for the swarm escort problem under the attack of aggressive swarms. In this bilateral cooperative strategy, the HVU adaptively select different evasion strategies by inferring the threat level according to the spatial distributions of the defenders and attackers. Meanwhile, the defenders of the swarm take a noncentralized escort algorithm by moving around the HVU in a dual-layer formation. Within each layer, the defenders cluster into several uniformly distributed subswarms to counteract the attackers. Numerical simulations are conducted using different aggressive swarm models to demonstrate the effectiveness of the proposed bilateral cooperative strategy.

1. Introduction

Swarm behavior is common in the natural world, such as when colonies use pheromones to engage with individual profiles within the colony to accomplish the shortest interaction [1,2], and honeybees use individual interactions within the colony to create a balance between flight distance and nectar harvest [3,4,5]. Some studies have found that the emergence of swarming in animals is beneficial for avoiding predators or foraging [6,7] and can compensate for the deficiency of individual abilities, which has inspired artificial swarms. Take the unmanned aerial vehicle (UAV), for example, multiple UAVs can complete more complex and challenging tasks than a single UAV. The use of UAV swarms to attack high-value units (HVUs) such as bases and fuel tanks is on the rise [8,9]. The threat of aggressive swarms is growing across the world and with it the importance of staying ahead in the development of defense systems. Therefore, it has become an urgent issue to develop efficient ways to handle aggressive swarms and to escort the HUVs [10].

1.1. Related Work

Previous studies on escort strategies mainly focus on two issues: offense–defense confrontation and formation control. As for offense–defense confrontations, researchers are usually concerned with target allocation in multi-agent confrontations [11,12,13,14]. The main idea of these studies is to decompose a multi-agent adversarial problem into multiple one-to-one adversarial problems. The one-to-one adversarial problem is often seen as a Target–Attacker–Defender (TAD) game [15], which consists of a target, attacker, and defender. The differential equations are constructed by their geometric relations, and each individual’s desired position then travels to the desired point [16]. Appropriate target allocation strategies significantly influence the defensive validity of such studies. As the swarm size becomes larger, the complexity of the target allocation algorithm increases, leading to difficulties in implementing simple individuals. In comparison, the other type is more concerned with strategy selection. These studies focus on selecting individuals’ strategies and are generally solved using game theory or machine learning approaches [17,18,19,20]. Since the 1990s, studies have been conducted on the application of reinforcement learning (RL) techniques to the solution of multi-agent adversarial games. Littman combined reinforcement learning to solve a two-person stochastic zero-sum game problem [21]. Bowling suggested using multi-agent reinforcement learning (MARL) to express learning in stochastic games [22]. The Multi-Agent Deep Deterministic Policy Gradient (MADDPG) algorithm, developed by Lowe in 2017 and based on the Deep Deterministic Policy Gradient (DDPG), uses a framework of centralized training and decentralized execution, and can be used in hostile circumstances. These techniques, however, have an intractable issue because of their centralized strategy training architecture, which causes the training input to increase exponentially with the number of agents and exerts more of a strain on the agents’ computational capabilities.

The other issue is formation control. Shepherding is an often-discussed formation control algorithm [23,24]. The shepherding model consists of two parts: shepherd agents and herd agents. One or more shepherd agents escort non-cooperative herd agents to the goal area [25], or the defense swarm is treated as multiple shepherd agents, which “escort” attackers to the safety area [26]. However, this problem differs from the escort model in that the HVU has a preset goal rather than being driven by the escort process dynamically. Research on the escort model has been relatively limited compared to the Shepherding model. In the available studies, Jiehong Wu et al. regard the escort swarm and HVU as one heterogeneous swarm [27] and use loss functions to correct the real-time position of escorts to form escort formations to protect the HVU. In that work, escorts and the HVU are unilateral, not bilateral, i.e., only the escorts move according to the HUV but the HUV’s maneuver is independent of the escorts. Moreover, the above research on escorting rarely considers the response of the escort swarm to aggressive swarms. The escort’s unilateral protection effect on the HUV will be diminished when it reacts to the attackers, but the bilateral cooperation between the escort and the HUV can make up for this deficiency and increase the escort efficiency. In this paper, we propose a bilateral cooperative strategy to adjust the inner interactions of the defenders and the reaction of the HVU to counteract aggressive swarms and escort the HUV.

Threat assessment is essential for bilateral cooperative escorts. Threat level, intent, and opportunity are regarded as the three main components of threat assessment according to [28]. Threat assessment entails not only evaluating the threat level and threat intent of the enemy but also considering its limits. Gonsalves P et al. employ the genetic algorithm and fuzzy logic theory to establish a threat assessment model. They use fuzzy logic to analyze the enemy’s threat intent and threat targets, design a genetic algorithm with a fitness function as the threat level, and then use it to generate and evaluate behavior sets [29,30]. Okello N et al. use the Bayesian network to analyze the threat level of an incoming target to the stronghold [31]. Ruining Luo et al. use the analytic hierarchy process (AHP) and information entropy to determine the subjective and objective threat factor weights of low altitude, slow speed, and small targets [32]. In the paper, we will use the Dynamic Bayesian Network (DBN) to construct the threat assessment model. Within the DBN, the HVU evaluates the current threat based on locally perceived information and make strategic selections based on the results. The interaction of the HVU to the escort swarm involves the evaluation of the escort subswarms’ statuses. Since different factors have inconsistent effects on the status evaluation, AHP is used to evaluate their status and select the status-optimal subswarm.

1.2. Contributions

In summary, the main contributions of this paper are to propose a bilateral cooperative strategy for the swarm escort problem under the attack of aggressive swarms, and they are presented as follows:

• To improve the survivability of a HVU, the HVU are guided according to the positions of defenders and attackers. In particular, three evasion strategies are adaptively selected by inferring the threat level.
• A noncentralized escort algorithm is proposed and the defenders in the swarm distribute in two circles centered at the moving HVU, namely dual-layer formation. Within each circle, the defenders cluster into several sub-swarms rather than uniformly distribute over the whole circle.
• Different aggressive swarm models are used to validate the proposed bilateral cooperative strategy through numerical simulations.

The rest of the paper is organized as follows: in Section 2, the bilateral cooperative strategy is presented along with the description of the escort problem. In Section 3, the effectiveness and efficiency of the bilateral cooperative strategy are analyzed from two perspectives: attack patterns and parameter settings. Finally, the conclusions and prospects are drawn in Section 4.

2. Proposed Methods

2.1. Problem Formulation

Let us consider an escort swarm of N individuals escorting a HVU that moves at a constant speed from the start point to the goal. During the escort process, the escort swarm will protect the HVU from the aggressive swarms. An escort individual, i.e., a defender, has state information $s_{D_i}$, consisting of position $x_{D_i}$ and velocity $v_{D_i}$, and the state of the ith defender updates according to

$$\left\{\begin{array}{l}{v}_{D_i}(t+\Delta t)=v_{D_i}(t)+u_{D_i}(t)*\Delta t\\ x_{D_i}(t+\Delta t)=x_{D_i}(t)+v_{D_i}(t+\Delta t)*\Delta t\end{array}\right.$$

(1)

where $\Delta t$ is the simulation step size, u denotes the acceleration. This work considers the social force, which is determined by the interaction rules of the individuals. As for responding to the aggressive swarm, a defender not only cooperates with its neighbors but is also affected by the HVU according to the potential field. Instead of undergoing a passive escort, the HVU can assess threat level and cooperative with the escort swarm by making an active maneuver, which is, namely, the bilateral cooperative strategy.

The essential task of escorting a HVU is to make the HVU reach the goal safely in limited time. In this paper, the escort success rate and travel time are used to evaluate the strategy’s effectiveness. The escort process is defined as successful if the escort swarm eliminates all individuals in the aggressive swarm (no aggressive individual hits the HVU) and the HVU reaching the goal within a preset time. Independent trials will be carried out to sufficiently investigate the effectiveness of the strategy. The success rate is defined as the proportion of successful trails to the total number of trails.

2.2. Aggressive Swarm Model

Responding to aggressive swarm is the basic premise of the escort problem, so another key indicator for escort strategies is the ability to react to swarm attacks with various attack patterns. According to the literature, we generally classify the aggressive swarm into three main categories: non-cooperative swarm attack, single-swarm attack, and multi-subswarm attack.

2.2.1. Non-Cooperative Swarm Attack

The pattern of non-cooperative swarm attack means that individuals in the aggressive swarm only consider collision avoidance without cooperating with each other when attacking the HVU. The main characteristic of this attack pattern is dispersibility, i.e., the attackers tend to move forward to the HVU directly. The formulation of the interaction rules for attackers in the non-cooperative aggressive swarm is shown below [33]:

$${\mathrm{F}}_i^{non-co}={\displaystyle \sum _{j\in N_i}\frac{1}{d_{ij}}{\widehat{d}}_{ij}}$$

(2)

where $N_i$ denotes ith individual’s neighborhood collection in the attacking swarm. $d_{ij}$ is the distance from ith individual to jth neighbor individual. ${\widehat{d}}_{ij}$ represents the unit vector of the position of j with respect to i. Figure 1a shows the non-cooperative attack pattern in the target-free environment.

2.2.2. Single-Swarm Attack

In this pattern, individuals can aggregate due to the interaction rules. The main characteristic of this attack pattern is collectivity, i.e., individuals in the aggressive swarm will aggregate for a saturation attack on the HVU. The formulation of the interaction rules in the single-swarm attack is shown below [33]:

$${\mathrm{F}}_i^{\sin \mathrm{gle}-\mathrm{s}}={\displaystyle \sum _{j\in N_i}(\frac{1}{d_{ij}}-\frac{ls}{{(d_{ij})}^3}){\widehat{d}}_{ij}}$$

(3)

The meaning of the parameters is the same as non-cooperative swarm. Figure 1b shows the single-swarm attack pattern in the target-free environment.

2.2.3. Multi-Subswarm Attack

The aggressive swarm will be divided into some subswarms in this pattern, its characteristic is the combination of the above two attacking patterns. Individuals can aggregate to attack the HVU but do not always remain in a single swarm, which increases the flexibility. The formulation of the interaction rules for a multi-subswarm attacking pattern is shown below [34]:

$${\mathrm{F}}_i^{\mathrm{multi}-\mathrm{s}}={\displaystyle \sum _{j\in N_i}[\frac{1}{d_{ij}}-\frac{ls}{{(d_{ij})}^3}+\exp (-{(d_{ij}-lm)}^2)]{\widehat{d}}_{ij}}$$

(4)

ls and lm are the subswarms’ indicators to adjust spatial distribution and the size of subswarms, respectively. Figure 1c shows the multi-subswarm attacking pattern in the target-free environment.

2.3. Bilateral Cooperative Strategy

Bilateral cooperative strategy consists of two parts: escort swarm cooperative strategy and HVU cooperative strategy. Escort swarm cooperative strategy is based on the social force model, building an escort model and defining meta behavioral rules for defenders to make them self-organized to escort the HVU. In HVU cooperative strategy, the HVU adaptively selects different evasion strategies by inferring the threat level according to the spatial distributions of the defenders and attackers.

2.3.1. Escort Model

In this paper, the self-organizing escort swarm is guided by some meta behavioral rules: position synergy rule, velocity synergy rule, anti-intrusion rule, and escort formation rule. The combined social force is determined by the weighted sum of these behavioral rules:

$$u_{{}_{D_i}}(t)=u_i^{auto}+k^{pos}\ast u_i^{pos}+k^{vel}\ast u_i^{vel}+k^{ain}\ast u_i^{ain}+k^{esc}\ast u_i^{esc}+\eta {\xi }_i$$

(5)

where $u_i^{pos},u_i^{vel},u_i^{ain},u_i^{esc}$ are the vectors indicating the position synergy rule, velocity synergy rule, anti-intrusion rule, and escort formation rule, respectively. $u_{}^{auto}$ denotes a self-driven force that affects the individual speed, $\eta {\xi }_i$ denotes the random noise. $k^{pos}, k^{vel}, k^{ain}, k^{esc}$ denote weights of the corresponding rule.

2.3.2. Formulations of Meta Rules

(a)

Position synergy rule

There is a sensing radius for the defender, and neighboring individuals within the sensing domain can perceive each other’s position and velocity. Based on neighbors’ positions, the defender determines the position synergy force, which describes the attractive or repulsive forces of neighbors to it. The position synergy equation is shown below:

$$u_i^{pos}={\displaystyle \sum _{k\in N_i}\left[(\frac{1}{d_{ik}}-\frac{k_1^{pos}}{d_{ik}{}^3})-e^{-{(d_{ik}-k_2^{pos})}^2}\right]}*{\widehat{d}}_{ik}$$

(6)

where $d_{ik}$ is the distance from the ith defender to kth neighbor individual. ${\widehat{d}}_{ik}$ represents the unit vector of the position of k with respect to i. $N_i$ denotes the ith defender’s neighborhood collection. $k_1^{pos}$ specifies the desired distance between individuals. $k_2^{pos}$ controls the repulsive interaction range, and the repulsive force achieves its maximum when the distance is equal to $k_2^{pos}$. When $d_{ik}$ is smaller than $k_1^{pos}$, the position synergy is portrayed as short-range repulsion, which avoids collision with neighboring individuals; when $d_{ik}$ is slightly larger than $k_1^{pos}$, the position synergy is portrayed as short-range attraction, which enables the defender aggregation with neighboring individuals; when $d_{ik}$ is much larger than $k_1^{pos}$, the position synergy is expressed as medium-range repulsion, which separates the cluster of the defender and neighboring individuals into several smaller groups and achieves its maximum when $d_{ik}$ is equal to $k_2^{pos}$($k_2^{pos}$>>$k_1^{pos}$); when $d_{ik}$ is much larger than $k_2^{pos}$, the position synergy is expressed as long-range attraction, which causes the separated clusters to move close to each other.

(b)

Velocity synergy rule

The defender adjusts its velocity by referring to its neighbors’ velocities. In order to react to the surroundings more quickly, the defender chooses the neighboring individual whose orientation change is the greatest in relation to itself and refers to that neighbor’s velocity, as displayed below:

$$k^{*}=\underset{k\in N_i}{\arg \max }\arccos \frac{(x_k(t)-x_{D_i}(t))\cdot (x_k(t-\Delta t)-x_{D_i}(t-\Delta t))}{‖x_k(t)-x_{D_i}(t)‖‖x_k(t-\Delta t)-x_{D_i}(t-\Delta t)‖}$$

(7)

where $a\cdot b$ denotes the dot product of vector a and vector b. The velocity synergy equation is shown below:

$$u_i^{vel}=v_{k^{*}}(t)-v_{D_i}(t)$$

(8)

(c)

Anti-intrusion rule

The defender is locally aware of the attacking individuals and can only perceive their position. In order to protect the HVU, the defender performs “point capture” on the attacking individuals within the sensing domain, i.e., after the attacking individual enters the defender’s capture radius, both the defender and the attacker “die” simultaneously. The magnitude of anti-intrusion force is progressively greater with distance. The anti-intrusion force is designed as:

$$u_i^{ain}={\displaystyle \sum _{k\in T_i}\frac{1}{\sqrt{d_{ik}}}*{\widehat{d}}_{ik}(t)}$$

(9)

where $T_i$ denotes the set of attacking individuals within the sensing domain of the ith defender.

(d)

Escort formation rule

Due to the specificity of the HVU, defenders are globally aware of it and perceive its position. The presence of the escort potential field around the HVU allows the escort swarm to form a dual-layer formation to protect it. The escort potential field consists of three areas: attraction potential field area, repulsion potential field area, and potential-free field area. When a defender is in the attraction potential field area, it is subject to potential field force that drives it towards the HVU; when it is in the repulsion potential field area, the potential field force will drive it away from the HVU; when it is in the potential-free field area, it is not subject to potential field forces. Figure 2 shows the design of the escort potential field.

The potential field force can be stated as:

$$u_i^{esc}=\left\{\begin{array}{l}\sqrt{d_{ip}}\ast {\widehat{d}}_{ip};(d_{ip}\in (d_4^{esc},d_3^{esc})\cup (d_1^{esc},+\infty ))\\ -\frac{1}{d_{ip}}\ast {\widehat{d}}_{ip};(d_{ip}\in (d_3^{esc},d_2^{esc})\cup (0,d_5^{esc}))\end{array}\right.$$

(10)

where $d_{ip}$ is the distance from ith defender to the HVU. ${\widehat{d}}_{ip}$ represents the unit vector of the position of ith defender with respect to the HVU. $d_5^{esc}$ determines the structure of the escort potential field and the remaining parameters are related to it.

$$\left\{\begin{array}{l}{d}_1^{esc}=k1*d_5^{esc}\\ d_2^{esc}=k2*d_5^{esc}\\ d_3^{esc}=k3*d_5^{esc}\\ d_4^{esc}=k4*d_5^{esc}\end{array}\right.$$

(11)

where k1, k2, k3, k4 are potential field coefficients, which indicate the range of the corresponding potential field area, respectively. In this paper, the values of potential field coefficients are taken as k1 = 2, k2 = 1.5, k3 = 1.3, k4 = 1.1.

Figure 2. An illustration of the potential field configuration.

Figure 2. An illustration of the potential field configuration.

2.3.3. HVU Cooperative Strategy

The key concern for escort process is the HVU’s survivability, especially considering aggressive swarms. In practical terms, the HVU (e.g., the carrier, base, etc.) has a relatively stronger ability of perception, i.e., it can perceive the state information (position and velocity) of the escort and aggressive swarms. It is reasonable that the HVU makes an appropriate decision for its maneuver based on the perception information to further improve its survivability. Therefore, in this paper, the cooperative strategy of the HVU consists of two key steps: threat assessment and evasion strategy.

(a)

Threat assessment

Threat assessment is the premise of evasion of the HVU. In the paper, we use Dynamic Bayesian Network (DBN) to conduct threat assessment. Bayesian Networks (BN) [35] are also called as Belief Networks. Based on probabilistic inference theory, the conditional probability relationship between variables is described as a directed acyclic graph. The DBN is an extension in time of the static BN. The DBN consists of two parts: network structure and conditional probability. In this work, we consider these three factors for threat assessment: distance to the nearest attacker, cooperative strength with regard to escort swarm, and HVU heading declination, which are designed as nodes of the DBN, as shown in Figure 3.

As for the distance to the nearest attacker (D_attack), we use three different distance levels, i.e., D_attack1, D_attack2, and D_attack3 to indicate close, medium, and far, respectively. As for the cooperative strength (N_sup), it is determined by the number of defenders and attackers in the most core escort domain (the domain determined by $d_5^{esc}$ as shown in Figure 2). We use N_sup1, N_sup2, and N_sup3 to represent that the number of escort individuals is less than that of attacking individuals, similar (within two), and more than, respectively. HVU heading declination (β) represents the angle between the velocity direction of the nearest attacking individual and the line from the attacking individual to the HVU. β1, β2, and β3 are used to represent small, medium, and big angle value, respectively.

The D_attack and β reflect the intent of the nearest attacking individual with regard to the HVU. Both are positively correlated with the threat level. In other words, the attacker is close or the HVU heading declination is small means a higher probability of a high threat level for the HVU. N_sup reflect the available power of the escort swarm that can be used to counteract the attackers. A higher N_sup level will possibly lead to a lower threat level for the HVU.

Based on the structure (see Figure 3) and conditional probabilities (see

Table 1. Dynamic Bayesian Network conditional probabilities for HVU threat assessment.

Conditional Probability	Variable	Threat Level (TL)
		L	M	H
P(D_attack|TL)	D_attack1	0.02	0.10	0.83
	D_attack2	0.40	0.58	0.10
	D_attack3	0.58	0.32	0.07
P(N_sup|TL)	N_sup1	0.15	0.27	0.58
	N_sup2	0.35	0.50	0.27
	N_sup3	0.50	0.23	0.15
P(β|TL)	β1	0.20	0.35	0.50
	β2	0.37	0.45	0.35
	β3	0.43	0.20	0.15

), we can combine such three factors to evaluate the threat level and the corresponding probability. The output of the network (i.e., the probabilities of threat levels) at the previous step is used as the prior probability of the root node at the coming step. As a result, using on the constructed DBN, the threat assessment can be updated step by step. We use the Junction Tree Propagation Algorithm [36] to make inference of the DBN.

(b)

Evasion strategy

In this paper, the HVU can make decisions on evasion strategy according to the threat assessment. The threat level with the highest probability will be regarded as the current threat level. When the threat level is low, the HVU has not perceived the attacking swarm, or the attacking swarm does not pose significant threat to the HVU. The HVU is suggested to maintain its original move state. When the threat level is medium, the HVU needs to cooperate with the escort swarm to avoid the aggressive swarm. When the HVU is under high threat, it needs to escape immediately. To achieve the above evasion process, we have designed the corresponding strategies for three different threat levels as shown below.

Strategy for low threat level. The HVU’s task is to not only avoid being attacked but also to reach the goal as soon as possible. Thus, when the HVU is under the low threat level, it will move towards the goal without considering swarm attacks. The velocity of the HVU adjusts according to:

$$v_p=\frac{x_{goal}-x_p}{‖x_{goal}-x_p‖}$$

(12)

where $x_{goal}$, $x_p$ denote the coordination of the goal and HVU, respectively.

Strategy for medium threat level. As for the medium threat level, the HVU needs to cooperate with the escort swarm. To escort the HVU and counteract the attackers, the defenders will move around the HVU in a dual-layer multi-swarm manner, as mentioned in Section 2.3.1 and Section 2.3.2. From the HVU side, it needs to move cooperatively with the escort swarm to obtain better protection, which will be described in detail in this section.

First, the HVU will identify the escort subswarms. In this paper, we introduce the Density-Based Spatial Clustering of Applications with Noise (DBSCAN) algorithm [37] to cluster the escort swarm into several subswarms. The internal density of a subswarm is controlled by two parameters according to the DBSCAN algorithm: the radius of the neighborhood and the minimum number of a subswarm.

Second, the HVU evaluates each subswarm’s status. To achieve this, the following criteria will be taken into account: distance to the subswarm center, distance from the subswarm to attackers, the subswarm size, and distance from the subswarm to the nearest attacker. In this paper, the geometric center will be used to represent the spatial position of a given swarm. Additionally, each criterion has inconsistent effects on subswarm status. Therefore, we use the Analytic Hierarchy Process (AHP) [32] to build a status evaluation hierarchy model, as shown in Figure 4.

Then, through a comprehensive assessment of the four criteria, the pairwise comparison matrix $X_0$, which depicts the ratio of importance among the criteria, is illustrated in

Table 2. The pairwise comparison table of criteria.

B	B1	B2	B3	B4
B1	1	1/2	3	4
B2	2	1	4	5
B3	1/3	1/4	1	2
B4	1/4	1/5	1/2	1

. For example, B1 is three times more important than B3, and B2 is twice as important as B1.

The eigenvector of $X_0$ can be calculated as the weight vector of four criteria, i.e.,${\omega }_0$. The subswarms are compared based on one criterion, $Bi$(I = 1,2,3,4), to obtain a pairwise comparison matrix $X_i$, as in Table 2, the eigenvector ${\omega }_i$ of $X_i$ is the weight vector of $Bi$ for the subswarms. The weight matrix of the four criteria for the subswarms can then be derived:

$$W={[{\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4]}^T$$

(13)

The subswarm’s status ranking weight vector V can be obtained as follows:

$$V={\omega }_0{}^{T}W$$

(14)

Thus, the subswarm with the highest ranking can be inferred from V as the status-optimal subswarm.

Finally, the HVU moves towards the status-optimal subswarm, and the velocity of the HVU is defined as:

$$v_p=\frac{x_{c^{*}}-x_p}{‖x_{c^{*}}-x_p‖}$$

(15)

Here, $x_{c^{*}}$ is the position of the status-optimal subswarm’s center.

Strategy for high threat level. When the threat level is high, the HVU needs to escape. The escape direction is determined by the position of all attacking individuals. The formulation is shown below:

$$v_p=\frac{{\displaystyle \sum _{j\in T_p}{\widehat{d}}_{jp}/d_{jp}^2}}{‖{\displaystyle \sum _{j\in T_p}{\widehat{d}}_{jp}/d_{jp}^2}‖}$$

(16)

where $T_p$ denotes a collection of attacking individuals in the HVU’s perceptual domain. $d_{jp}$ is the distance from the HVU to jth attacking individual. ${\widehat{d}}_{jp}$ represents the unit vector of the position of an attacking individual with respect to the HVU.

3. Results

In this section, we conduct a series of numerical simulations to validate the proposed algorithm under different aggressive swarms. The HVU’s initial position is (0,0) and its speed is set to a constant 1 m/s. The escort swarm is initially placed apart away rather than around the HVU. Unless otherwise stated, the settings of the following numerical simulations are given in

Table 3. Parameters setting.

Parameter	Description	Value
HVU
$R_p^{sen}$	Perception radius of the HVU	10 m
$‖v_p‖$	Constant speed	1 m/s
Escort swarm
${‖v_D‖}_{\max }$	The maximum speed of defenders	2 m/s
$d_5^{esc}$	Escort potential field radius	5 m
$k^{pos}$, $k^{vel}$,$k^{ain}$,$k^{esc}$ Simulation & environment	Weights of interaction rules	5, 2, 10, 20
$\Delta t$	Time interval	1 s
$\eta$	The amplitude of random noise	0.1
Maxsimstep	Maximum simulation steps	1500
${‖v_A‖}_{\max }$	The maximum speed of attackers	3 m/s

.

3.1. Escort without Swarm Attacks

The most essential issue in the self-organized escort problem is whether the escort swarm can adaptively form an escort formation to protect the HVU. The first series of simulations is to validate the dual-layer escort formations without aggressive swarms. In the absence of attackers, the HVU’s threat level is always low. Thus, the HVU only needs to be considered for the task of reaching the goal successfully. Figure 5A shows the escort trajectories of different number of defenders. It is observed that escort individuals can form a dual-layer escort formation around the HVU. It can also be found that escort individuals form multiple subswarms which are uniformly distributed around the HVU. Such behavior is achieved according to simple rules proposed in this work. From the practical perspective, the dual-layer uniformly distributed subswarms can counteract the attacking swarms from any direction with various fleet size.

Moreover, the escort task, i.e., simulation time, is not over when the HVU reaches the goal, the escort formation will vortex around the HVU, as shown in Figure 5B. We use the angular momentum to measure the degree of the orderliness of the swarm during the vortex motion:

$$m_{swarm}=\frac{1}{N}\left|{\displaystyle \sum _{i=1}^N{r}_{ip}(t)\times v_{D_i}(t)}\right|$$

(17)

where $r_{ip}$ denotes the position vector from the ith defender to the HVU. We can find that the transition time from escort to vortex becomes longer as the escort size increases. The vortex trajectories with different size of escort swarms are shown in Figure 5C. It can be seen that the escort individuals begin to vortex around the HVU after it reaches the goal, and the orderliness of the vortex steadily improves. This emergent vortex behavior may be a useful base (static HVU) protection. The vortex formation, like a school of fish, is helpful for the defenders to perceive the threats from various directions in time.

3.2. Escort against Swarm Attacks

When encountering swarm attacks, the HVU needs to not only reach the goal but also survive the swarm attacks. Figure 6A shows the snapshots of the 50 escort individuals protecting the HVU under the attack of 30 aggressive individuals with different attacking models, i.e., non-cooperative swarm, single swarm and multi-subswarm attacking models (see Section 2.2). At the beginning, the threat, i.e., the attacking swarm, is far away from the HVU and the escort swarm, the HVU moves towards the goal point and the escort swarm is forming a dual-layer escort formation around the HVU. Later, when encountering the aggressive swarm, the defenders will intercept and eliminate the aggressive individuals, meanwhile the HVU continually evaluates the threat and selects the best evasion strategy. After the aggressive individuals have been eliminated, the escort swarm reverts to a stable dual-layer escort formation.

As for the evasion of the HVU, Figure 6B shows the trajectories of escort individuals and HVU during the whole escort process. It can be seen that the HVU is almost always in the center of the escort swam. Moreover, the trajectory of the HVU indicates that when moving towards the goal the HVU can adaptively make its decision on maneuver for the purpose of evasion, which validates the effectiveness of the proposed bilateral cooperative strategy. Figure 6C shows the inside view of the threat assessment of the HVU. It can be seen that the duration of high threat in the single-swarm attack case is uninterrupted and long. This is because the single-swarm attack is likely to perform a saturation attack in a specific direction, which has been proven to be an effective military tactic. Interestingly, the total duration of high threat is longest despite a small interruption in the non-cooperative swarm attack case. Obviously, the non-cooperative swarm is also very threatening because the attacking direction of each individual does not necessary align with the others. Attacking from diverse directions can also significantly improve the effectiveness. In comparison, the multi-subswarm attack causes the shortest duration of high threat level. This is because this attack model is superior neither in saturation attack in a specific direction nor in the diverse attacking directions.

To investigate the impacts of different swarm attack models, we show the escort success rate and HVU travel time in Figure 7. It can be seen in Figure 7A that the success rate under the single-swarm attack decreases the most with the increase in the attacking swarm’s size, which implies an advantage of the single-direction saturation attack. Interestingly, the decline in success rate is not significant when attacking by a non-cooperative swarm (diverse attacking direction). This reflects the effectiveness of the dual-layer ring formation of the escort swarm to some extent. As for the travel time, the HVU requires the longest time duration to reach the goal when attacked by a non-cooperative swarm. This reveals that the HVU adjusts its direction more frequently when the attacking direction is more diverse, which matches the first subplot in Figure 6C.

3.3. Sensitivity to Parameters

To gain a deep insight into the proposed bilateral cooperative escort method, in this section we will conduct sensitivity analysis of the parameters from the perspective of the HVU and escort swarm, respectively. Without loss of generality, in the following experiments we draw conclusions based on 100 independent runs, the non-cooperative aggressive swarm is used to investigate the success rate of the escort process.

First, we investigate the HVU’s perception range which is set to 8, 10, 12, 14, and 16, respectively. According to the statistical results shown in Figure 8A, the perception range is helpful for maintaining a high success rate. It is intuitive that a short perception range of the HVU means little information (the information about defenders and attackers) that the HVU can obtain to make evasion decision. Such a situation becomes worse when counteracting a larger size of aggressive swarm, see the curve with perception range 8.

Second, as for the other side in the bilateral cooperative escort, i.e., the escort swarm, the basic radius of the escort potential field (i.e., $d_5^{esc}$ in Figure 2) is studied. The $d_5^{esc}$ determines the scale of the dual-layer escort formation of the defenders. The statistical results shown in Figure 8B indicate that there is an optimal $d_5^{esc}$ given an escort swarm size (50 in this experiment). It can be seen that the smallest and largest $d_5^{esc}$ lead to the worst two success rate curves, while the best escort performance is obtained when $d_5^{esc}$ is equal to 5. This is because a large escort potential field helps the defenders find the attackers early and counteract them in a large scale. However, when given the size of escort swarm, this may also lead to a decrease in spatial density, which is harmful to protecting the HVU. Therefore, the value of $d_5^{esc}$ should be carefully selected when applying the proposed bilateral cooperative escort method.

Lastly, it is wort studying the impact of swarm size of both defenders and attackers in the defense-and-offence process. To this end, we set the size of the aggressive swarm to 10, 20, and 30, respectively, to investigate the success rate of the escort process with various size (from 30 to 50) of the escort swarm. As shown in Figure 8C, the success rate can keep above 0.9 if the escort swarm size is larger that of the aggressive swarm by at least 10. In addition, the ratio of aggressive swarm size to escort swarm size significantly impacts the success rate, i.e., a high ratio leads to a low success rate. Interestingly, when the size difference is larger than 20, the success rate always remains at a high level. In contrast, when the size difference is smaller than 10, the success rate dramatically decreases with the increase in aggressive swarm size.

3.4. Validation of Real-World Settings

To confirm the adaptability of the bilateral cooperative strategy to the actual escort scenario, in this section we will consider some constraints and disruptions in the real world. In this paper, we take the drone escort scenario as an example, consider the individual constraints and environmental perturbations in this scenario. The escort scenario is set as 8 km × 8 km and there is a no-fly zone in this scenario. The individual constraints are mainly kinematic constraints, which include maximum velocity, maximum acceleration and maximum turning angular velocity limits. Environmental perturbations include ambient noise and measurement noise, where measurement noise can be caused by various sources, such as wind flow during escort or individual perception accuracy error. Here, we introduce Gaussian noise to simulate the effect of environmental perturbations on the turning angle and speed. The constraints and perturbation parameters of the escort individuals are given in

Table 4. Drone settings.

Parameters	Description	Value/Unit
${‖v_D‖}_{\max }$	Maximum speed	30 m/s
${‖u‖}_{\max }$	Maximum acceleration	10 m/s2
${\omega }_{\max }$	Maximum turning angular velocity	0.6 rad/s
$n_v$	Speed noise	N (0.34,0.9)
$n_{\omega }$	Angular velocity noise	N (−0.4,1.0)

. The attacker’s maximum speed is greater than defender’s (${‖v_A‖}_{\max }$=32 m/s), and other constraints are consistent with the defender. The HVU’s speed is constant, and its value is 20 m/s, i.e., $‖v_P‖$=20 m/s.

In addition, according to the actual scenario, the relevant parameters of the bilateral cooperative strategy are also adjusted. The perception radius of the HVU is $R_p^{sen}$ = 1000 (m) and the escort potential field radius is $d_5^{esc}$=400 (m).

Based on the above constraints and adjustments, the escort simulation is carried out with 50 escort individuals and 30 aggressive individuals with non-cooperative attack patterns. Figure 9a shows the trajectories of the escort individuals and HVU during the whole escort process. The trajectory of the escort individuals (light gray lines) indicates that under the influence of individual constraints and environmental perturbations, the escort individuals can still spontaneously and stably form a dual-layer escort formation. Additionally, compared to Figure 6B, it can be found that the constraint causes the individuals not to show sharp turning and there is a turning radius. Figure 9b shows the snapshots of the 50 escort individuals protecting the HVU under the attack of 30 aggressive individuals with non-cooperative attack patterns. It can be seen that although the constraints have some influence on the escort process, the escort individuals can still spontaneously form a stable dual-layer escort formation around the HVU and intercept and eliminate the aggressive individuals.

To further validate the effectiveness of the escort strategy in a real-world setting, we perform 100 independent experiments and obtain the escort effectiveness, as in Figure 10. Compared with Figure 7, it can be found that, similar to the unconstrained ideal scenario, an increase in the size of the aggressive individuals makes the escort success rate decrease and the travel time become longer. In addition, after considering the real-world setting, the tendency of the defense effectiveness to degrade as the attack size increases is more apparent. However, overall, the escort strategy is effective in real-world settings for responding to swarm attacks at different attack sizes. Therefore, it can be concluded that the bilateral escort strategy can be well adapted to real escort scenarios.

4. Conclusions

The offence and defense of a swarm have been essential issues for defense technology and security. Escort a moving HVU is challenging since the defenders should move around the HVU in a noncentralized manner and the HVU should also move adaptively according to the defenders and attackers to obtain a high survival rate. Obviously, this is a bilateral cooperative escort process which should be designed from the perspective of both the escort swarm and the HVU.

In this paper, we proposed a bilateral cooperative escort strategy for a moving HVU guarded by a noncentralized escort swarm. In particular, the escort swarm always move around the HVU in a dual-layer circle formation. In addition, the escort swarm is adaptively clustered into several uniformly distributed subswarms along the corresponding circles. Moreover, a threat assessment based on AHP method is proposed to make the threat level apparent so as to make decision on evasion for survival. Numerical simulations were conducted using different aggressive swarm models, i.e., non-cooperative swarm, single-swarm, and multi-subswarm attacks. The effectiveness of the proposed bilateral cooperative escort strategy is validated. The sensitivity of the parameters from the perspective of HVU and escort swarm is also analyzed, and the value of $d_5^{esc}$ is more sensitive than the other two parameters. The effectiveness of the escort strategy in a real-world setting is also validated. In the future work, the dynamics of the HVU, escort platforms and aggressive platforms will be considered and adaptive parameter adjusting strategies will also be investigated for the field applications. Moreover, the use of bilateral cooperative escort strategy will be considered for physical platforms, such as drone formations.