Distributed Swarm Control Algorithm of Multiple Unmanned Surface Vehicles Based on Grouping Method

Abstract

This paper addresses the distributed swarm control problem of multiple unmanned surface vehicles (USVs) in Euclidean space with virtual leader. Firstly, to investigate the topology of the neighborhood relations between vehicles, a new time-variant topology structure is proposed. Secondly, to research the dynamic properties of the group for the case where the number of virtual leader is different, a grouping method based on cosine similarity is proposed. Thirdly, to ensure the high effeciency of information transmission and the reduction of costs, a distributed swarm control algorithm is proposed, which is mainly composed of three parts: gradient descent term, velocity consensus term and navigational feedback term. To analyze the stability of system, the concept of translation framework is introduced. Based on the properties of the Hamiltonian and LaSalle invariance principle, the stability of multiple USVs swarm motion is proved. Finally, simulation results illustrate the effectiveness of the proposed methods.

1. Introduction

Swarming is ubiquition in nature, and it is a form of collective behaviour of multiple interacting agents with a common group objective. Swarm is defined as multiple autonomous agents moving cooperatively to fulfil global objective of a scientific or technological mission. This term is often observed in nature. Each autonomous agent is modelled as a particle and characterised by its position and a function describing its dynamics [1,2,3]. Understanding the mechanisms and operational principles in them can provide useful ideas for developing formation control, distributed cooperative control and coordination of multiple mobile autonomous agents/robots [4].

Over the past several years, the theory and application of multi-agent systems have attracted great attention. Specifically, the multi-agent distributed control not only has received much attention, but also has made significant research progress on account of its extensive applications [5,6,7,8]. Multiple USVs, which is one of the most actively studied topics within the realm of multi-agents systems, generally aims to drive USVs to achieve prescribed constraints on their states [9]. The studies focus on designing appropriate protocols and algorithms so that the multiple USVs can achieve various forms of agreement.

Keeping a swarming group connected is a connectivity problem. In representative literatures [10,11] always establishes group stabilization by assuming that the vehicles are connected through a time-invariant topology. In [12], the vessels are assumed interconnected through a directed topology rather than bidirectional. The connectivity of the group topology relies on the actual distances between vehicles and may be destroyed during group evolution. Efforts should be made to preserve connectivity of the topology while achieving the desired flocking objective. The disadvantage of time-invariant topology is that requiring the number of multiple USVs is fixed. For example, a group vehicles disperse when facing multi-task, and a flock splits when encountering predators, the topology needs to be redesigned.

The second concern is the communication constraints of these strategies. In recent years, a leader-follower system was investigated in many literatures [11,13,14]. A theoretical framework for the design and analysis of flocking algorithm is proposed in [15]. However, these swarming algorithms proposed in thses literatures require substantial communication energy consumptions during the flocking process, it is impossible to apply this algorithm to engineering applications such as distributed sensing in mobile sensor networks. Under the circumstances, the information flow cause substantial increase in the load, which bring enormous pressure on the server. Multiple virtual leaders become another choice, which can reduce the communication loads and realize the information diffluence. Considering that only partial agents can be informed of the virtual leaders, according to the extent of social distancing, a group composed of N mobile agents is divided into multiple different subgroups in [16]. The complex flocking of multiple inertial agents is analyzed via the decomposition approach in [17]. In [18], an algorithm solving the flocking problem for multiple groups of mobile agents in heterogeneous networks is studied. To the best of our knowledge, few results have discussed the method for a group of multiple USVs to flocking in any specified number of groups by cosine similarity method.

The third part focus on the swarm control algorithm. With the deepening research on multi-USVs system, many algorithms have been proposed in literatures [19,20,21,22]. In [19,20], graph theory has been used to investigate the linear consensus problem. Local attractive/repulsive potentials have been used to define the interactive forces between neighboring agents to deal with the separation and cohesion problems in [21,22]. In flocking control, the potential function approach is used to keep a swarming group cohesive and separate. Furthermore, consensus algorithms have been studies extensively in the context of aooperative control of multi-agent systems [23,24,25]. Both the sonsensus algorithm and the potentional function approach obtain information via neighbor-to-neighbor communication. In [26], swarming robots make their motion decisions based on partial information of the entire flocking network. If all members have global information or know the desired trajectory, the cohesive force generated by this global information can keep the flocking group connected. There are also many researches on flocking algorithm. Such as, semi-flocking algorithm [27], local flocking algorithm [28], a individual-based alignment/repulsion algorithm [29], etc. Meanwhile, ref. [11] theoretically proved that provably convergent flocking, formation control, and path following can be integrated in a single control architecture. In the above literature, the formation controller or flocking algorithm use consensus method and distributed approaches for the navigation of multi-USVs. Consensus algorithms and their variants are applied to formation control and swarm control [23]. This means that there is no essential difference between formation control and swarm control algorithm. These algorithms are relevant to the one proposed in this note in the sense that the emergent flocking phenomenon results form the alignment mechanism.

To sum up what is discussed above, firstly, motivated by the importance of network connectivity in multiple USVs, a time-variant topology is designed. It is dependent on proximity nets with interaction range of USVs cluster and $\alpha$-$grids$ with a class of USV satisfying some algebraic constraints. Secondly, to reduce the communication load and realize the information diffluence, a grouping method for a group of multiple USVs to flocking in any specified number of groups based on cosine similarity is proposed. Thirdly, swarming-based algorithms have several advantages that make them suitable for use in sensor management. First, they are completely distributed algorithms. They are highly compatible with the distributed nature of sensor management in sensor networks. Second, in swarming-based algorithms, each vehicle needs to communicate only with its neighbors. Using swarming-based algorithms for sensor management requires only local communication between sensors. Third, in flocking-based algorithms, vehicles apply simple flocking rules, using this type of algorithms for sensor management has low computation overhead for the sensors. Furthermore, they are highly flexible and scalable. To ensure the high effeciency of information transmission and the reduction of costs, a distributed swarm control algorithm is proposed. Furthermore, the concept of translation framework is introduced. The stability of multiple USVs swarm motion is proved by the Hamiltonian and LaSalle invariance principle.

In summary, the main contributions of this paper are summarized as follows.

• To achieve multiple USVs connectivity, a time-variant topology is designed, which is dependent on proximity nets and $\alpha$-$grids$. The connectivity of vehicles are bidirectional.
• To reduce the communication loads and realize the information diffluence, a grouping method for a group of multiple USVs to flocking in any specified number of groups based on cosine similarity is proposed.
• To ensure the high effeciency of information transmission and the reduction of costs, a distributed swarm control algorithm based on gradient descent, velocity consensus term and navigational feedback is proposed.

This paper is organized as follows. A table of notations and some used variables in the paper is presented (

Table 1. Notations and variables used in this paper.

Variable	Definition
$A\backslash B$	Set of elements belonging to $\mathit{A}$ but not belonging to $\mathit{B}$
$\left|·\right|$	Absolute value of a scalar
$∥·∥$	Euclidean norm
${\mathbf{R}}^{n\times n}$	$n\times n$ dimensional Euclidean Space
$diag\left\{a_i\right\}$	A block-diagonal matrix with $a_i$ being the ith diagonal element
${(·)}^{\mathrm{T}}$	Transpose of a matrix
${(·)}^{-1}$	Inverse of a matrix
⊗	Kronecker product of matrix
$A\in {\mathbf{R}}^{n\times n}$	Adjacency matrix defined as $A={\left[a_{ij}\right]}_{n\times n}$ with $a_{ij}=a_{ji}$
$d_i$	Defined as $d_i={\sum }_{j=1}^n{a}_{ij}$
$D\in {\mathbf{R}}^{n\times n}$	Degree matrix defined as $D=diag\{d_1,d_2,\dots ,d_n\}$
$L\in {\mathbf{R}}^{n\times n}$	Laplacian matrix defined as $L=D-A$

). Section 2 describes problem description and some necessary preliminaries. Section 3 designs the model of USVs, virtual leader, and grouping method. Section 4 proposes the distributed swarm control algorithm and proves the stability of the USVs swarming. Section 5 gives the simulation results and comparison results to verify effectiveness of the proposed algortihm. Section 6 concludes this paper.

2. Problem Description and Preliminaries

2.1. Problem Description

Multiple USVs system is a complex system. In this paper, we investigate the dynamic properties of the group for the case where the number of virtual leader is different and the topology of the neighborhood relations between vehicles is dynamic. Under the premise of learning preparatory knowledge, a grouping method based on cosine similarity is proposed. It is grouped according to the number of virtual leaders. Then, a disributed swarm control algorithm is propsoed. It is mainly composed of gradient descent term, velocity consensus term and navigational feedback term. The structure of the proposed grouping method and distributed swarm control algorithm for multiple USVs system is shown in Figure 1.

2.2. Algebraic Graph Theory

A directed graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ consists of a vertex set $\mathcal{V}=\{1,2,\dots ,n+1\}$ and the set of edges $\mathcal{E}\subseteq \{\mathcal{V}\times \mathcal{V}\backslash (i,i\left)\right\}$. The graph is in general directed and has no self-loops, it means that $\mathcal{E}\subseteq \left\{\right(i,j):i,j\in \mathcal{V},j\ne i\}$.

The adjacency matrix $A={\left[a_{ij}\right]}_{n\times n}$ of a graph is called to be undirected if $(i,j)\in \mathcal{E}$$\iff (j,i)\in \mathcal{E}$ (i.e., $a_{ij}=a_{ji}=1$); otherwise, A is directed. The adjacency matrix A is a matrix with nonzero elements, in which $a_{ij}\ne 0$$\iff (i,j)\in \mathcal{E}$. For an undirected graph $\mathcal{G}$, The adjacency matrix A is symmetric, $A^{\mathrm{T}}=A$. The Laplacian matrix L associates with the graph $\mathcal{G}$ is define as $L=D-A$ where $D=diag\{d_1,d_2,\dots ,d_n\}$ with $d_i={\sum }_{j=1}^n{a}_{ij}$. Laplacian matrix L always has a right eigenvector of $1_n={(1,1,\dots ,1)}^{\mathrm{T}}$ associated with eigenvalue ${\lambda }_1=0$. It satisfies the following assumption.

Assumption 1.

In the graph theory, there is at least one path from the virtual leader navigator to each vehicle.

2.3. Proximity Nets

In low-speed applications, the damping can be assumed to be linear. Its quadratic velocity term can be ignored. Ship models are usually reduced-order models for control of the horizontal palne motions (surge, sway and yaw).

The set of neighbors of ith USV is described as

$$\begin{array}{c}\hfill N_i=\{j\in \mathcal{V}:(i,j)\in \mathcal{E}\}=\{j\in \mathcal{V}:a_{ij}\ne 0\}.\end{array}$$

(1)

Define the interaction radius $r>0$, it is the interaction range between any two USVs. As shown in Figure 2, an open-loop circle neighborhood is defined as [30]

$$\begin{array}{c}\hfill N_i=\{j\in \mathcal{V}:\parallel X_j-X_i\parallel <r\},\end{array}$$

(2)

where $∥·∥$ represents the Euclidean norm in ${\mathbf{R}}^2$, $X_i:={(x_i,y_i,{\psi }_i)}^{\mathrm{T}}\in {\mathbf{R}}^3$ denotes the position and yaw angle of ith USV for all $i\in \mathcal{V}$, $X_i$ is the center, r is the radius.

The set of edges ${\mathcal{E}}_X$ is denoted as

$$\begin{array}{c}\hfill {\mathcal{E}}_X=\{(i,j)\in \mathcal{V}\times \mathcal{V}:\parallel X_j-X_i\parallel <r,i\ne j\}.\end{array}$$

(3)

The vector $X=col(X_1,X_2,\dots ,X_n)\in {\mathbf{R}}^{3\times n}$ is called the configuration of all USVs of the graph $\mathcal{G}$. A framework is a pair $(\mathcal{G},X)$ consists of a graph $\mathcal{G}$ and the configuration X of its USVs. The proximity net $\mathcal{G}\left(X\right)=(\mathcal{V},{\mathcal{E}}_X)$ can be defined by $\mathcal{V}$ and ${\mathcal{E}}_X$. The framework $\left(\mathcal{G}\right(X),X)$ is called a proximity structure.

Assumption 2.

The interaction radius r of all USVs is the same. The topology of the USV wireless sensor network is a proximity net $\mathcal{G}$ with a interaction range r. All proximity nets are bidirectional graphs.

2.4. Geometric Structure: $\alpha$-$grids$

In order to describe the movement process of multiple USVs swarm more clearly, we use $grid$ to represent the geometry of all USVs swarm.

First, we find a set of USVs with the following properties form all USVs, where each USV is equally distanced form all of its neighbors on a proximity net $\mathcal{G}\left(X\right)$. This kind of USV can be described as a set of solutions satisfying the following algebraic constraints:

$$\begin{array}{c}\hfill \parallel X_j-X_i\parallel =d,\forall j\in N_i\left(X\right).\end{array}$$

(4)

To describe the set of solutions $X^{\prime }$ of another class different form X, which are very close to the set of solutions satisfying Formula (4), the set of inequalities is shown as

$$\begin{array}{c}\hfill -\delta \le \parallel X_j-X_i\parallel -d\le \delta ,\forall (i,j)\in {\mathcal{E}}_X,\end{array}$$

(5)

and refer to these solutions satisfying the above inequalities constraints as quasi $\alpha$-$grid$. The $\alpha$-$grid$ and quasi $\alpha$-$grid$ are shown in Figure 3 and Figure 4, respectively.

Next, to quantitatively describe the different degrees of configuration X and $\alpha$-$grid$, we introduce a variable: deviation energy function (DEF). It is defined as

$$\begin{array}{c}\hfill E\left(X\right)=\frac{1}{\left(\right|{\mathcal{E}}_X|+1)}\sum _{i=1}^n\sum _{j\in N_i}\omega (\parallel X_j-X_i\parallel -d),\end{array}$$

(6)

where $\omega \left(z\right)=z^2$ is called potential energy function (PEF).

For the swarm with n USVs, the DEF can be regarded as a nonsmooth PEF. The $\alpha$-$grids$ are the solutions with PEF equal to zero. Moreover, the $\alpha$-$grids$ are also the global minimum value of the $E\left(x\right)$.

For the quasi $\alpha$-$grid$ X with an side length uncertainty of $\delta$, the deviation energy function is defined as

$$\begin{array}{c}\hfill E\left(x\right)\le \frac{|{\mathcal{E}}_x|}{\left(\right|{\mathcal{E}}_x|+1)}{\delta }^2\le {\delta }^2={\epsilon }^2{d}^2,\epsilon \ll 1.\end{array}$$

(7)

2.5. Nonnegative Mapping Function: $\lambda$-Norm

In order to construct the smooth collective potential energy function (CPEF) of the USVs swarm and the planar adjacency matrix of proximity structure, we define a concept of nonnegative mapping function: $\lambda$-norm.

The $\lambda$-norm of a vector is defined as

$$\begin{array}{c}\hfill {\parallel z\parallel }_{\lambda }=\frac{1}{\sigma }[\sqrt{1+{\sigma \parallel z\parallel }^2}-1],\end{array}$$

(8)

where $\sigma >0$ is a constant, $\lambda$-norm is a map ${\mathbf{R}}^3\to {\mathbf{R}}_{\ge 0}$.

The gradient ${\lambda }_{\sigma }\left(z\right)=\nabla {\parallel z\parallel }_{\lambda }$ is given as

$$\begin{array}{c}\hfill {\lambda }_{\sigma }\left(z\right)=\frac{z}{\sqrt{1+{\sigma \parallel z\parallel }^2}}=\frac{z}{1+{\sigma \parallel z\parallel }_{\lambda }}.\end{array}$$

(9)

The constant $\sigma$ of the $\lambda$-norm remains fixed in this paper. The mapping ${\parallel z\parallel }_{\lambda }$ is differentiable everywhere, but it is not differentiable at $z=0$. Since the model of multi-USVs swarm is discrete, we can construct a smooth continuous CPEF of USVs swarm by using the property of $\lambda$-norm.

The scalar function ${\rho }_h\left(z\right)$ is a convex function, which has the following properties: it is continuous and smooth in the interval $(0,1)$. In this paper, we use convex function to construct smooth PEF with finite cut-off and smooth adjacency matrix. The convex function is as follows [15]:

$${\rho }_h\left(z\right)=\left\{\begin{array}{cc}1,\hfill & z\in [0,h)\hfill \\ \frac{1}{2}[1+cos\left(\frac{z-h}{1-h}\pi \right)],\hfill & z\in [h,1]\hfill \\ 0,\hfill & otherwise\hfill \end{array}\right.,$$

(10)

where $h\in (0,1)$.

Through the convex function ${\rho }_h\left(z\right)$, we define a concept of planar adjacency matrix $A\left(X\right)$. The elements in $A\left(X\right)$ are defined as

$$\begin{array}{c}\hfill a_{ij}\left(X\right)={\rho }_h\left(\frac{\parallel X_j-X_i{\parallel }_{\lambda }}{r_{\alpha }}\right)\in [0,1],j\ne i,\end{array}$$

(11)

where $r_{\alpha }={\parallel r\parallel }_{\lambda }$ and $a_{ii}\left(X\right)=0$ for all i and X.

2.6. Collective Potential Energy Function

The collective potential energy function of multiple USVs swarm is a nonnegative function V, which is a mapping: ${\mathbf{R}}^{3n}\to {\mathbf{R}}_{\ge 0}$. In this paper, the CPEF is a smooth form of the DEF with the PEF that has a finite cut-off. The most common way to construct a CPEF is to multiply a convex function by using PEF.

First, define a smooth attractive/repulsive potential function $\omega \left(z\right)$: ${\mathbf{R}}_{\ge 0}\to {\mathbf{R}}_{\ge 0}$. It has the following properties: when $z=d$, it has a global minimum, and it has a finite cut-off r.

Next, define the function $\phi \left(X\right)$ is defined as:

$$\begin{array}{c}\hfill \phi \left(X\right)=\frac{1}{2}\sum _i^n\sum _{j\ne i}^n\omega (\parallel X_j-X_i\parallel ),\end{array}$$

(12)

where the $\phi \left(X\right)$ is not differentiable at the singular configuration where two different nodes coincide. In order to solve this problem, we introduce $\lambda$-norm into Formula (4), and its algebraic constraints can be written as

$$\begin{array}{c}\hfill \parallel X_j-X_i{\parallel }_{\lambda }=d_{\alpha },\forall j\in N_i\left(X\right),\end{array}$$

(13)

where $d_{\alpha }={\parallel d\parallel }_{\lambda }$.

The end, according to Formulas (12) and (13), the smooth CPEF of multiple USVs swarm is obtained as

$$\begin{array}{cc}\hfill V\left(X\right)& =\frac{1}{2}\sum _i^n\sum _{j\ne i}^n{\omega }_{\alpha }(\parallel X_j-X_i{\parallel }_{\lambda }),\hfill \end{array}$$

(14)

where

$$\begin{array}{cc}\hfill {\omega }_{\alpha }\left(z\right)& ={\int }_{d_{\alpha }}^z{\varphi }_{\alpha }\left(s\right)ds,\hfill \\ \hfill {\varphi }_{\alpha }\left(z\right)& ={\rho }_h(z/r_{\alpha })\varphi (z-d_{\alpha }),\hfill \\ \hfill \varphi \left(z\right)& =\frac{1}{2}[(a_1+a_2){\lambda }_1(z+a_3)+(a_1-a_2)],\hfill \end{array}$$

(15)

and the ${\lambda }_1\left(z\right)=z/\sqrt{1+z^2}$, and the parameters in $\varphi \left(z\right)$ are: $0<a_1\le a_2$, $a_3=|a_1-a_2|/\sqrt{4a_1{a}_2}$, so as to sure $\varphi \left(0\right)=0$.

Through the above analysis, it can be concluded that the CPEF $V\left(X\right)$ has the following properties: it has a finite cut-off at $r_{\alpha }={\parallel r\parallel }_{\lambda }$; when $z=d_{\alpha }$, it has a global minimum.

3. Modeling of USV, Virtual Leader, and Grouping Method

3.1. Modeling of USV

In low-speed applications, the damping can be assumed to be linear. Its quadratic velocity term can be ignored. Ship models are usually reduced-order models for control of the horizontal palne motions (surge, sway and yaw). The 3 degrees of freedom (DOFs) kinematic equations of the USV can be expressed in vector form as [31]

$$\begin{array}{cc}\hfill & \dot{\eta }=J\left(\eta \right)\upsilon ,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & M\dot{\upsilon }+D\left(\upsilon \right)\upsilon =\tau +{\tau }^{\ast },\hfill \end{array}$$

(17)

where $J\left(\eta \right):=R\left(\psi \right)$ is the rotation matrix, it is given as

$$\begin{array}{c}\hfill R\left(\psi \right)=\left[\begin{array}{ccc}cos\left(\psi \right)& -sin\left(\psi \right)& 0\\ sin\left(\psi \right)& cos\left(\psi \right)& 0\\ 0& 0& 1\end{array}\right],\end{array}$$

(18)

with the properties: $\parallel R\left(\psi \right)\parallel =1$ and $R^{\mathrm{T}}\left(\psi \right)R\left(\psi \right)=I_{3\times 3}$. $\eta :={[x,y,\psi ]}^{\mathrm{T}}$ is the position and yaw angle in the earth-fixed frame $X_E{O}_E{Y}_E$ (see Figure 5). $\upsilon :={[u,v,r]}^{\mathrm{T}}$ is the velocity vector in the body-fixed frame $X_B{O}_B{Y}_B$, where u, v and r represent the surge, sway and yaw angular velocities, respectively. The system inertia matrix M is positive definite and constant, where $M=M^{\mathrm{T}}\in {\mathbf{R}}^{3\times 3}$ and $\dot{M}=0$. The damping matrix D is also symmetric and positive define, that is $D\in {\mathbf{R}}^{3\times 3}$. $\tau :={[{\tau }_1,{\tau }_2,{\tau }_3]}^{\mathrm{T}}$ is the control input, which is produced by the propellers. ${\tau }^{\ast }:={\tau }_{wind}+{\tau }_{wave}+{\tau }_{ocean}$ is total environment disturbance by the wind, waves, and ocean currents, respectively.

Through the derivation of the above formula, we can rewritten the Formulas (16) and (17) as

$$\left\{\begin{array}{c}\dot{X}=\theta ,\hfill \\ \dot{\theta }=U,\hfill \end{array}\right.$$

(19)

where $X=\eta$, $\theta =\dot{\eta }$, $U=-M_{\eta }^{-1}\left(\eta \right)D_{\eta }(\upsilon ,\eta )\dot{\eta }+M_{\eta }^{-1}\left(\eta \right)[{\tau }_{\eta }+{\tau }_{\eta }^{\ast }]$.

Assuming that the USVs swarm is composed of n USVs, the motion state equation of the ith USV is described as

$$\left\{\begin{array}{c}\dot{X_i}={\theta }_i,\hfill \\ \dot{{\theta }_i}=U_i,\hfill \end{array}\right.$$

(20)

where $X_i={\eta }_i\in {\mathbf{R}}^{3n}$ ($i=1,2,\dots ,n$) is the position and yaw angle vectors, ${\theta }_i={\dot{\eta }}_i\in {\mathbf{R}}^{3n}$ is the velocity vector, $U_i=-{M_i}_{{\eta }_i}^{-1}\left({\eta }_i\right){D_i}_{{\eta }_i}({\upsilon }_i,{\eta }_i)\dot{{\eta }_i}+{M_i}_{{\eta }_i}^{-1}\left({\eta }_i\right)[{{\tau }_i}_{{\eta }_i}+{{\tau }_i}_{{\eta }_i}^{\ast }]\in {\mathbf{R}}^{3n}$ is the control input acting on the ith USV.

3.2. Modeling of Virtual Leader: $\gamma$-USV

To get better effect of navigational feedback term in USVs swarm algorithm (see Formula (28) for details), the motion state of virtual leader also conforms to Formula (20). Therefore, the motion state of virtual leader ($\gamma$-USV, $\gamma =1,\dots ,m$) is as follows [32,33]:

$$\left\{\begin{array}{c}\dot{X_{\gamma }}={\theta }_{\gamma },\hfill \\ \dot{{\theta }_{\gamma }}=U_{\gamma },\hfill \end{array}\right.$$

(21)

where $X_{\gamma }$, ${\theta }_{\gamma }$, and $U_{\gamma }$$\in {\mathbf{R}}^3$ are the position and yaw angle vector, velocity vector, and control input of the virtual leader, respectively. $(X_i\left(0\right),{\theta }_i\left(0\right))=(X_d,{\theta }_d)$. A static USV has a fixed static that is equal to $(X_d,{\theta }_d)$ for all time. The design of $U_{\gamma }$ for virtual leader is part of tracking control design for a group of USVs. For example, the choice of $U_{\gamma }\equiv 0$ leads to that the virtual leader moves along a straight line with a desired velocity ${\theta }_{\gamma }$. Based on expression of $u_i^{\gamma }$ (see Formula (29)), a secondary objective of an $\alpha$-$grid$ is to track a dynamic/static virtual leader ($\gamma$-USV).

3.3. Grouping Method

In order to expand the search scope of multiple USVs cluster, it is not enough to increase the number of USVs. By increasing the number of virtual leaders, multi-USV can be grouped and searched from different directions, so as to expand the search scope. Under the circumstances, a multi-USV swarm grouping method based on cosine similarity is proposed.

Suppose that all USVs are in a rectangular region, the coordinates of the lower left corner of the rectangular is $Q(x_q,y_q)$, and the ith USV initial positions are $T_i(x_i\left(0\right),y_i\left(0\right))$, as shown in Figure 6. Connect point Q and $T_i$ to form a line segment $Q{T}_i$. The angle ${\theta }_i$ is formed by the line segment $Q{T}_i$ and the positive direction of the x axis. Counter clockwise is positive. Tangent angle ${\theta }_i$ is as follows:

$${\theta }_i=atan((y_i\left(0\right)-y_q)/(x_i\left(0\right)-x_q)),$$

(22)

where $i=1,2,\dots ,n$.

The design process of this method is as follows.

Step 1. According to Formula (22), the tangent angle of the ith USV is ${\theta }_1$, ${\theta }_2$,..., ${\theta }_i$,..., and ${\theta }_n$, respectively. $i=1,2,\dots ,n$.

Step 2. Sort them by the order of tangent angle. The ordered array satisfies the following condition:

$${\theta }_{k_1}\le {\theta }_{k_2}\le \cdots \le {\theta }_{k_n},$$

(23)

The index $k_i$ of ${\theta }_{k_i}$ (i = 1,..., n) represents the ith number in an ordered array from the step 2.

Step 3. USVs were grouped according to the number of virtual leaders. If we divide n by m, the quotient is W and the remainder is c. The value W meets the following condition

$$W=\lfloor n/m\rfloor ,$$

(24)

where $\lfloor ·\rfloor$ denotes rounding down number.

There are following cases to explain.

1 If $c=0$, all USVs are evenly distributed, and the number of USVs in each group is equal. The subscripts of $\gamma$th group USV satisfy the following conditions:

$$k_{(\gamma -1)\ast W+1},k_{(\gamma -1)\ast W+2},\dots ,k_{(\gamma -1)\ast W+W},$$

(25)

where $\gamma =1,2,\dots ,m$.

2 If $c\ne 0$, the subscripts of $\gamma$th group USV are as follows:

(1) For the $(m-c)$ groups, the subscripts of USV are as follows:

$$k_{(\gamma -1)\ast W+1},k_{(\gamma -1)\ast W+2},\dots ,k_{(\gamma -1)\ast W+W},$$

(26)

where $\gamma =1,2,\dots ,m-c$.

(2) For the rest of c groups, the subscripts of USV are as follows:

$$k_{(\gamma -1)\ast (W+1)+1},k_{(\gamma -1)\ast (W+1)+2},\dots ,k_{(\gamma -1)\ast (W+1)+W+1},$$

(27)

where $\gamma =m-c+1,\dots ,m$.

4. USVs Distributed Swarm Control Algorithm

In this section, the distributed swarm control problem for multiple unmanned surface vehicles is addressed by grouping method, distributed swarm control algorithm and other control technology. To accurately estimate environmental disturbances and guarantee estimation error converges in finite time, a finite time disturbance observer is designed. In order to show the control algorithm more intuitively, the schematic diagram is presented in Figure 7.

4.1. USVs Distributed Swarm Control Algorithm

Based on the flocking algorithm proposed in [15], heading angle and yaw angular velocity are considered in the flocking algorithm. In this case, a distributed swarm control algorithm (DSCA) is proposed.

In USVs swarm, a control input $u_i$ is used to denote the input signal of the ith USV, which consists of three terms. The input signal $u_i$ is as follows:

$$\begin{array}{c}\hfill u_i=f_i^g+f_i^v+f_i^{\gamma },\end{array}$$

(28)

where $f_i^g$, $f_i^v$ and $f_i^{\gamma }$ are gradient-based term, velocity consensus term and navigational control law, respectively. The specific expressions are as follows:

$$\begin{array}{cc}\hfill f_i^g=& p_i\sum _{j\in N_i}{\varphi }_{\alpha }(\parallel X_j-X_i{\parallel }_{\lambda })n_{ij},\hfill \\ \hfill f_i^v=& p_i\sum _{j\in N_i}{a}_{ij}\left(X\right)({\theta }_j-{\theta }_i),\hfill \\ \hfill f_i^{\gamma }=& u_i^{\gamma }=f_i^{\gamma }(X_i,{\theta }_i,X_{\gamma },{\theta }_{\gamma }),\hfill \\ \hfill =& p_i(-b_1(X_i-X_{\gamma })-b_2({\theta }_i-{\theta }_{\gamma })),(b_1,b_2>0),\hfill \end{array}$$

(29)

where $p_i=1$ if ith USV is informed; otherwise, $p_i=0$. $n_{ij}={\lambda }_{\sigma }(X_j-X_i)=((X_j-X_i)/$$\sqrt{1+\sigma \parallel X_j-X_i{\parallel }^2})$ is a vector along the line connecting $X_j$ to $X_i$. $\sigma \in (0,1)$ is a fixed parameter of the $\lambda$-norm. the pair $(X_{\gamma },{\theta }_{\gamma })$ is the state of the virtual leader. The parameters $b_1>0$ and $b_2>0$ represents the navigational feedback weight coefficients of X and $\theta$, respectively.

Substituting (29) into (28), the expression (28) is written as

$$\begin{array}{cc}\hfill u_i=& p_i(\sum _{j\in N_i}{\varphi }_{\alpha }(\parallel X_j-X_i{\parallel }_{\lambda })n_{ij}+\sum _{j\in N_i}{a}_{ij}\left(X\right)({\theta }_j-{\theta }_i)\hfill \\ \hfill & -b_1(X_i-X_{\gamma })-b_2({\theta }_i-{\theta }_{\gamma })),(b_1,b_2>0).\hfill \end{array}$$

(30)

4.2. USVs Swarm Dynamics

From the Formulas (20) and (30), the collective dynamics equation of the USVs swarm is as follows:

$$\left\{\begin{array}{c}\dot{X_i}={\theta }_i,\hfill \\ \dot{{\theta }_i}=-\nabla V\left(X_i\right)-\widehat{L}\left(X_i\right){\theta }_i+f_i^{\gamma }(X_i,{\theta }_i,X_{\gamma },{\theta }_{\gamma }),\hfill \end{array}\right.$$

(31)

where $V\left(X\right)$ is a smooth CPEF (see Formula (14)), $\widehat{L}\left(X\right)$ is the 3-D Laplacian of the proximity net $\mathcal{G}\left(X\right)$. $\widehat{L}=L\otimes I_m$. The $\widehat{L}$ has the following property:

$$z^{\mathrm{T}}\widehat{L}z=\frac{1}{2}\sum _{(i,j)\in \mathcal{E}}{a}_{ij}{\parallel z_j-z_i\parallel }^2,z\in {\mathbf{R}}^{3n},$$

(32)

where $z=col(z_1,z_2,\dots ,z_n)$ and $z_i\in {\mathbf{R}}^{3n}$ for all i.

When $f_i^{\gamma }\equiv 0$, the USVs swarm dynamics whose motion form is Formula (31) is a energy consumption system with Hamiltonian (see the Formula (41) for details):

$$H_{\xi }(X,\theta )=E_{\xi }\left(X\right)+\frac{1}{2}\sum _{i=1}^n{\parallel {\theta }_i\parallel }^2.$$

(33)

Consider that a translational frame is centered at $X_c$, i.e., the center of mass of all USVs, let $Ave\left(z\right)=(1/n){\sum }_{i=1}^n{z}_i$ denote the average of the $z_i$, $z=col(z_1,z_2,\dots ,z_n)$. Let $X_c=Ave\left(X\right)$ and ${\theta }_c=Ave\left(\theta \right)$ denote the position and yaw angle vector and velocity vector of the origin of the translational frame, respectively. Therefore, ${\dot{X}}_c\left(t\right)={\theta }_c\left(t\right)$ and ${\dot{\theta }}_c\left(t\right)=Ave\left(u\left(t\right)\right)$.

Lemma 1.

([15]) Suppose that the navigational feedback $f_{\gamma }(X,\theta ):=f_i^{\gamma }(X,\theta ,X_{\gamma },{\theta }_{\gamma })$ is linear, i.e., there exists a decomposition of $f_{\gamma }(X,\theta )$ in the following form:

$$f_i^{\gamma }(X,\theta ,X_{\gamma },{\theta }_{\gamma })=g(X,\theta )+1_n\otimes h(X_c,{\theta }_c,X_{\gamma },{\theta }_{\gamma }),$$

(34)

where $g(X,\theta )=-p_i{b}_{1}X-p_i{b}_2\theta$, $1_n={(1,1,\dots ,1)}^{\mathrm{T}}$, $h(X_c,{\theta }_c,X_{\gamma },{\theta }_{\gamma })=-p_i{b}_1(X_c-X_{\gamma })-p_i{b}_2({\theta }_c-{\theta }_{\gamma })$, and the pair $(X_{\gamma },{\theta }_{\gamma })$ is the state of the virtual leader.

Substituting (34) to (31), the expression (31) can be consisted of n second-order systems

$$\left\{\begin{array}{c}\dot{X}=\theta \hfill \\ \dot{\theta }=-\nabla V\left(X\right)-\widehat{L}\left(X\right)\theta +g(X,\theta )\hfill \end{array}\right.$$

(35)

and one second-order systems

$$\left\{\begin{array}{c}\dot{X_c}={\theta }_c\hfill \\ \dot{{\theta }_c}=h(X_c,{\theta }_c,X_{\gamma },{\theta }_{\gamma }).\hfill \end{array}\right.$$

(36)

4.3. Stability Analysis

According to the Lemma 1, we can decompose the stable USVs swarm motion into the combination of the following two forms of stability properties: (1) the stability of certain equilibria of the structural dynamics, and (2) the stability of the expected equilibrium points in translational dynamics. Then, the stability of USVs swarm is equivalent to the stability of these two forms of systems.

According to the Formula (31), the structural dynamics of the multiple USVs cluster has the following forms:

$$\left\{\begin{array}{c}\dot{X}=\theta ,\hfill \\ \dot{\theta }=-\nabla E_{\xi }\left(X\right)-D_d\left(X\right)\theta ,\hfill \end{array}\right.$$

(37)

where

$$\begin{array}{cc}\hfill E_{\xi }\left(X\right)& =V\left(X\right)+\xi J\left(X\right),\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill J\left(X\right)& =\frac{1}{2}\sum _{i=1}^n{\parallel X_i\parallel }^2,\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill D_d\left(X\right)& =b_2{I}_3+\widehat{L}\left(X\right),\hfill \end{array}$$

(40)

where $E_{\lambda }\left(X\right)$ denotes the aggregate potential energy function (APEF), $J\left(X\right)$ is the moment of inertia of all USVs, $\xi =b_1>0$ is the navigational feedback parameter, $D_d\left(X\right)$ is a positive definite matrix, $b_2>0$.

Before proving the stability of the system, we introduce a concept: the structural Hamiltonian of the system [15]. Hamiltonian is the sum of the potential energy of all particles plus the kinetic energy of the particles related to the system. In this paper, its expression is as follows:

$$\begin{array}{c}\hfill H_{\xi }(X,\theta )=E_{\xi }\left(X\right)+K\left(\theta \right),\end{array}$$

(41)

where $K\left(\theta \right)=(1/2){\sum }_i{\parallel {\theta }_i\parallel }^2$ is the kinetic energy function (KEF). According to Formulas (14), (38) and (39), the expression (41) is written as

$$\begin{array}{cc}\hfill H_{\xi }(X,\theta )=& \frac{1}{2}\sum _i^n\sum _{j\ne i}^n{\omega }_{\alpha }(\parallel X_j-X_i{\parallel }_{\lambda })+\frac{1}{2}\sum _i^n\xi \parallel X_i{\parallel }^2+\frac{1}{2}\sum _i^n{\parallel {\theta }_i\parallel }^2\hfill \\ \hfill =& \frac{1}{2}\sum _i^n(\sum _{j\ne i}^n{\omega }_{\alpha }(\parallel X_j-X_i{\parallel }_{\lambda })+\xi \parallel X_i{\parallel }^2+\parallel {\theta }_i{\parallel }^2).\hfill \end{array}$$

(42)

We also need to introduce two other concepts: cohesion of the USV group and cluster.

Definition 1.

Cohesion of the USV group. If there exists a circle with $X_c\left(t\right)=Ave\left(X\left(t\right)\right)$ as the center and $R>0$ as the radius, which contains all the USV for all time $t\in [t_0,t_f]$, (i.e., $\exists R>0:\parallel X\left(t\right)\parallel \le R,\forall t\in [t_0,t_f]$), we called the USV group is cohension.

Definition 2.

Swarms. If the proximity net $\mathcal{G}\left(X\right(t\left)\right)$ is connected over $[t_0,t_f)$$(t_f>t_0)$, the group of α-$grids$ is called a swarm over the interval $[t_0,t_f)$.

Lemma 2.

The local minima of CPEF $V\left(X\right)$ is a α-$grid$, and vice versa.

Proof. 

Define a $\epsilon$-neighborhood of X as

$$\begin{array}{c}\hfill {\mathcal{N}}_{\epsilon }\left(X\right)=\{X^{\prime }\in {\mathbf{R}}^{3n}:\parallel X_i^{\prime }-X_i\parallel <\epsilon ,\forall i\in \mathcal{V}\}\end{array}$$

(43)

where $X^{\prime }=col(X_1^{\prime },X_2^{\prime },\dots ,X_n^{\prime })$. If there exists a $\epsilon$-neighborhood ${\mathcal{N}}_{\epsilon }\left(X\right)$ of $X^{\ast }$ such that $V\left(X\right)\ge V\left(X^{\ast }\right)$ for all $X\in {\mathcal{N}}_{\epsilon }\left(X\right)$, we called the configuration $X^{\ast }$ is a local minima of $V\left(X\right)$.

The $V\left(X\right)$ can be decomposed into two terms: a graph-induced PEF by the proximity net $\mathcal{G}\left(X\right)$ and an integer factor of $h_0={\omega }_{\alpha }\left(r_{\alpha }\right)$, the expression of $V\left(X\right)$ is as follows:

$$\begin{array}{c}\hfill V\left(X\right)=V_{\mathcal{G}}\left(X\right)+k\left(X\right)h_0,\end{array}$$

(44)

where

$$\begin{array}{cc}\hfill V_{\mathcal{G}}\left(X\right)& =\frac{1}{2}\sum _{(i,j)\in \mathcal{E}\left(X\right)}{\omega }_{\alpha }(\parallel X_j-X_i{\parallel }_{\lambda }),\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill k\left(X\right)& =\frac{1}{2}(n(n-1)-|\mathcal{E}\left(X\right)\left|\right)\in \mathcal{I}\hfill \end{array}$$

(46)

with $\mathcal{I}=\{0,1,2,\dots ,n(n-1\left)\right\}$. For a fixed $k\in \mathcal{I}$, we have

$$\begin{array}{c}\hfill V\left(X\right)=V_{\mathcal{G}}\left(X\right)+k{h}_0\ge k{h}_0.\end{array}$$

(47)

Any configuration $X^{\ast }$ that achieves the equality in Formula (46) is a local minima of $V\left(X\right)$ and satisfies $V_{\mathcal{G}}^{\ast }\left(X\right)=0$. ${\omega }_{\alpha }\left(z\right)=0$ if and only if $z=d_{\alpha }$. Therefore, the configuration $X^{\ast }$ satisfy the following constraints:

$$\begin{array}{cc}\hfill \parallel X_j^{\ast }-X_i^{\ast }{\parallel }_{\lambda }=& d_{\alpha }\hfill \\ \hfill ⟹\parallel X_j^{\ast }-X_i^{\ast }\parallel =& d,\forall j\in N_i\left(X\right).\hfill \end{array}$$

(48)

It means that $X^{\ast }$ is a $\alpha$-$grid$. The proof of the converse is rather similar and is omitted.

This completes the proof. □

Remark 1.

The environmental disturbances consist of low-frequency part and high-frequency part. In this paper, only the low-frequency part is considered during the control process.Regardless of the rotation and translation of $X_i$, the CPEF $V\left(X\right)$ remains invariant. Therefore, the $V\left(X\right)$ has no isolated local minimum.

Theorem 1.

Consider that a group of USVs adopt USVs swarm algorithm of Formula (30) with $b_1,b_2>0$ and satisfy structural dynamics of Formula (37). Let ${\Omega }_c=\{(X^{\mathrm{T}},{\theta }^{\mathrm{T}})\in {\mathbf{R}}^{6n}:=H_{\xi }(X,\theta )\le c\}$ be a level-set of the Hamiltonian $H_{\xi }(X,\theta )$ of Formula (41) such that for any solution starting in ${\Omega }_c$. Suppose that their initial inertia $J\left(X\right(0\left)\right)$ and the kinetic energy $K\left(\theta \right(0\left)\right)$ are finite. Then, the following statements hold. (i) The group of USVs remain cohesive for all $t\ge 0$. (ii) Almost every solution of Formula (37) asymptotically converges to an equilibrium point $(X_{\lambda }^{\ast },0)$, where $X_{\lambda }^{\ast }$ is a local minimum of function $E_{\xi }\left(X\right)$. (iii) All USVs move asymptotically at the same speed. (iv) It is supposed that the initial structural energy of USVs swarm system is less than $(k+1)c^{\ast }$ with $c^{\ast }=M{\omega }_{\alpha }\left(0\right)$ and $k\in Z_{+}$. Then, at most k distinct pairs of different α-grids may collide. When $k=0$, there is no interaction collision in the USVs swarm system.

Proof. 

According to the symmetry of $\omega \left(z\right)=z^2$ and adjacency matrix $A\left(t\right)$, we can get the following equation:

$$\begin{array}{c}\hfill \frac{\partial {\omega }_{\alpha }(\parallel X_{ji}{\parallel }_{\lambda })}{\partial X_{ji}}=\frac{\partial {\omega }_{\alpha }(\parallel X_{ji}{\parallel }_{\lambda })}{\partial X_j}=\frac{\partial {\omega }_{\alpha }(\parallel X_{ji}{\parallel }_{\lambda })}{\partial X_i}\end{array}$$

(49)

with ${\omega }_{\alpha }(\parallel X_{ji}{\parallel }_{\lambda })={\omega }_{\alpha }(\parallel X_j-X_i{\parallel }_{\lambda })={\omega }_{\alpha }(\parallel X_i-X_j{\parallel }_{\lambda })$.

Due to the USVs swarm is an energy consuming system, the time derivative of Formula (41) is obtained as

$$\begin{array}{cc}\hfill {\dot{H}}_{\xi }(X,\theta )& =-{\theta }^{\mathrm{T}}(b_2{I}_3+\widehat{L}\left(X\right))\theta \hfill \\ \hfill & =-b_2\left({\theta }^{\mathrm{T}}\theta \right)-{\theta }^{\mathrm{T}}\widehat{L}\left(X\right)\theta \le 0.\hfill \end{array}$$

(50)

According to the Formula (50), $H_{\xi }(X,\theta )$ is a monotonically decreasing function for all $(X,\theta )$, such that

$$\begin{array}{c}\hfill H_{\xi }(X\left(t\right),\theta \left(t\right))\le H_0:=H_{\xi }(X\left(0\right),\theta \left(0\right))<+\infty ,\end{array}$$

(51)

where

$$\begin{array}{cc}\hfill H_0:=& H_{\xi }(X\left(0\right),\theta \left(0\right))\hfill \\ \hfill =& V\left(X\right(0\left)\right)+\xi J\left(X\right(0\left)\right)+K\left(\theta \right(0\left)\right).\hfill \end{array}$$

(52)

According to the Formulas (51) and (52) and the assumption that initial inertia $J\left(X\right(0\left)\right)$ and the kinetic energy $K\left(\theta \right(0\left)\right)$ are finite, we have

$$\begin{array}{cc}\hfill E_{\xi }\left(X\left(t\right)\right)& \le H_0,\forall t\ge 0\hfill \\ \hfill E_{\xi }\left(X\left(t\right)\right)& =V\left(X\left(t\right)\right)+\frac{\xi }{2}{X}^{\mathrm{T}}\left(t\right)X\left(t\right).\forall \xi >0\hfill \end{array}$$

(53)

From the Formula (53), the following equation is obtained:

$$\begin{array}{cc}\hfill & V\left(X\left(t\right)\right)+\frac{\xi }{2}{X}^{\mathrm{T}}\left(t\right)X\left(t\right)\le H_0\hfill \\ \hfill ⟹& X^{\mathrm{T}}\left(t\right)X\left(t\right)\le \frac{2(H_0-V\left(X\left(t\right)\right))}{\xi }\hfill \\ \hfill ⟹& X^{\mathrm{T}}\left(t\right)X\left(t\right)\le \frac{2H_0}{\xi }.\forall t\ge 0\hfill \end{array}$$

(54)

According to the same proof process, yields

$$\begin{array}{cc}\hfill & K\left(\theta \left(t\right)\right)=\frac{1}{2}\sum _i^n{\parallel {\theta }_i\left(t\right)\parallel }^2\le H_0,\forall t\ge 0\hfill \\ \hfill ⟹& {\theta }^{\mathrm{T}}\left(t\right)\theta \left(t\right)\le 2H_0.\forall t\ge 0\hfill \end{array}$$

(55)

According to the Formulas (50)–(55), we can draw the following conclusion: the USVs swarm can achieve aggregation for all $t\ge 0$, and keep in a circle neighborhood with $X_c$ as the center and $R=\sqrt{2H_0/\xi }$ as the radius.

Therefore, conclusion (i) is valid.

From to the Formula (50), $H_{\xi }(X,\theta )$ is a monotonically decreasing function for all $(X,\theta )$. Hence, $H_{\xi }(X,\theta )\le c$ for all $t\ge 0$ that implies the ${\Omega }_c=\{(X^{\mathrm{T}},{\theta }^{\mathrm{T}})\in {\mathbf{R}}^{6n}:=H_{\xi }(X,\theta )\le c\}$ is an invariant set.

From the LaSalle’s invariance principle [34], all the solutions of Formula (41) starting in ${\Omega }_c$, will converge to the largest invariant set in S. The expression of S is as follows:

$$\begin{array}{c}\hfill S=\{(X^{\mathrm{T}},{\theta }^{\mathrm{T}})\in {\mathbf{R}}^{6n}:={\dot{H}}_{\xi }(X,\theta )=0\}.\end{array}$$

(56)

Thus, all the solutions are in the set ${\Omega }_c$. Any solution $\left(X\right(t),\theta (t\left)\right)$ of the USVs swarm algorithm is uniquely mapped to a solution $\left(X\right(t),\theta (t\left)\right)$ of the multiple USVs system.

According to the Formula (55), the velocity will be bounded as

$$\begin{array}{cc}\hfill & K\left(\theta \left(t\right)\right)=\frac{1}{2}\sum _i^n{\parallel {\theta }_i\left(t\right)\parallel }^2\le H_0\le c.\forall t\ge 0\hfill \\ \hfill ⟹& {\theta }^{\mathrm{T}}\left(t\right)\theta \left(t\right)\le 2H_0\le 2c.\forall t\ge 0\hfill \end{array}$$

(57)

This ensures that the upper bound of uniform convergence of velocity is $\theta \left(t\right)\le \sqrt{2c}$.

The Formulas (53)–(57) guarantee that the solution of the USVs swarm is bounded. Let $z=col(X,\theta )$, we have

$$\begin{array}{cc}\hfill {\parallel z\left(t\right)\parallel }^2& =X^{\mathrm{T}}\left(t\right)X\left(t\right)+{\theta }^{\mathrm{T}}\left(t\right)\theta \left(t\right)\hfill \\ \hfill & \le 2(\frac{1}{\xi }+1)H_0\le R^2+2c:=C<+\infty ,\hfill \end{array}$$

(58)

where $C>0$ is a constant.

From the Formula (50), we draw the following conclusions: if $\theta =0$, ${\dot{H}}_{\xi }(X,\theta )=0$; otherwise, ${\dot{H}}_{\xi }(X,\theta )<0$. Therefore, the USVs swarm are always in motion for all $t\ge 0$. Thus, the velocities of the group of USVs match in the moving frame, or ${\theta }_1={\theta }_2=\cdots ={\theta }_n$. When ${\sum }_i{\theta }_i=0$, we can get ${\theta }_i=0$$(i=1,2,\dots ,n)$. In the reference frame, the velocities of USVs are asymptotically consistent. Meanwhile, $X_1=X_2=\cdots =X_n$. Thus, $\nabla E_{\xi }\left(X_{\xi }^{\ast }\right)=0$. Therefore, almost every solution of the multiple USVs system asymptotically converges to the equilibrium point $z_{\xi }^{\ast }=(X_{\xi }^{\ast },0)$, where $X_{\xi }^{\ast }$ is a local minima of $E_{\xi }\left(X\right)$. According to the Lemma 2, the local minima of CPEF $V\left(X\right)$ is a $\alpha$-$grid$. These prove that (ii) is holds.

We now prove (iii) of Theorem 1. Through the above proof of the (i) and (ii), we can draw the following conclusion: the velocities of all USVs are asymptotically consistent. Thus, the velocity of each USV asymptotically consistent in the reference frame.

In order to prove (iv), we use the counter evidence method to prove.

According to the information in the (iv), we can conclude that the following inequality holds:

$$\begin{array}{c}\hfill H_0<(k+1)c^{\ast }=(k+1){\omega }_{\alpha }\left(0\right),k\in Z_{+}.\end{array}$$

(59)

Thus, there are at most k distinct pairs of USVs that collide at a given time $t_1\ge 0$.

Next, we make the following assumption: there must be at least $k+1$ distinct pairs of USVs collide with each other at a given time $t_1$. Therefore, we get the following conclusion: the Hamiltonian $H_0$ of the USVs swarm is at least $(k+1){\omega }_{\alpha }\left(0\right)$ at a given time $t=t_1$. Then, the following inequality holds:

$$\begin{array}{cc}\hfill & H_0\ge (k+1){\omega }_{\alpha }\left(0\right)\hfill \\ \hfill & (k+1)c^{\ast }=(k+1){\omega }_{\alpha }\left(0\right)\hfill \\ \hfill ⟹& H_0\ge (k+1)c^{\ast }.\hfill \end{array}$$

(60)

Obviously, the Formulas (59) and (60) are contradictory. Therefore, our hypothesis is not tenable. Therefore, no more than k distinct pairs of the USVs can possibly collide at any time $t>0$.

Next, we prove that: there is no collision between USV when $k=0$. When $k=0$, the Hamiltonian $H_0<c^{\ast }={\omega }_{\alpha }\left(0\right)$. Assuming that there exists two USV colliding at $t=t_1$, the two USVs are labeled as $n_1$th USV and $n_2$th USV, respectively. i.e., $X_{n1}\left(t_1\right)=X_{n2}\left(t_1\right)$. From the Formula (44), we can get the following results:

$$\begin{array}{cc}\hfill V\left(X\right(t\left)\right)=& \frac{1}{2}\sum _i^n\sum _{i\ne j}^n{\omega }_{\alpha }(\parallel X_j-X_i{\parallel }_{\lambda })\hfill \\ \hfill =& \frac{1}{2}\sum _{i\in \mathcal{V}\backslash \{n1,n2\}}^n\sum _{j\in \mathcal{V}\backslash \{i,n1,n2\}}^n{\omega }_{\alpha }(\parallel X_j-X_i{\parallel }_{\lambda })+{\omega }_{\alpha }(\parallel X_{n1}\left(t\right)-X_{n2}\left(t\right){\parallel }_{\lambda })\hfill \\ \hfill \ge & {\omega }_{\alpha }(\parallel X_{n1}\left(t\right)-X_{n2}\left(t\right){\parallel }_{\lambda }).\hfill \end{array}$$

(61)

Substituting $t=t_1$ into Formula (61), the following inequality holds:

$$\begin{array}{c}\hfill V\left(X\left(t_1\right)\right)\ge {\omega }_{\alpha }\left(0\right)=c^{\ast }.\end{array}$$

(62)

However, from the Formula (33), the inequality is shown as follows:

$$\begin{array}{c}\hfill V\left(X\right(t\left)\right)\le H\left(X\right(t),\theta (t\left)\right)-K\left(\theta \right(t\left)\right)\\ \hfill ⟹V\left(X\left(t\right)\right)\le H(X\left(t\right),\theta \left(t\right))\le c<c^{\ast }.\end{array}$$

(63)

Obviously, the Formulas (62) and (63) are contradictory. Therefore, our hypothesis is untenable. Thus, no two USVs collide with $k=0$ at any time $t\ge 0$.

This completes the proof. □

5. Simulation Results

In simulations, the model of surface ship Cybership II is used in [35]. The time-varying environmental disturbances are modeled as the white Gaussian noise. Numerical simulation and comparison results are given to show the effectiveness of the proposed DSCA. We consider twenty follower vehicles ($n=20$) under different number virtual leader ($\gamma$ USV), i.e., case 1, case 2 and comparison group, respectively.

The parameters of DSCA are choose as follows. The sensing radius is choose at $r=5m$, the desired distance $d=3m$, $\lambda =0.1$ for the $\lambda$-norm, $h=0.2$ for the bump function ${\rho }_h\left(z\right)$. After a lot of simulation testing and verification, the navigation feedback gain are set as $b_1=0.01$ and $b_2=0.15$, respectively.

5.1. Case 1: 20 USVs and a Virtual Leader

In this subsection, the simulation results are given to verify the performance of proposed DSCA of the USVs with a virtual leader.

The initial positions of 20 USVs and a virtual leader are randomly distribute in the range of abscissa and ordinate [−60 m, 60 m] × [−80 m, 60 m], and heading angle of the USVs is also randomly distributed in [$-\pi rad$, $\pi rad$]. The initial parameters of the virtual leader are $X_{\gamma }\left(0\right)={[x\left(0\right)m,y\left(0\right)m,\psi \left(0\right)rad]}^{\mathrm{T}}$, where $\left(x\right(0),y(0\left)\right)$ denotes the position of the initial moment, and $\psi \left(0\right)=\pi /4$, and ${\theta }_{\gamma }\left(0\right)={[2m/s,2m/s,0m/s^2]}^{\mathrm{T}}$.

The simulation results are shown in Figure 8 and Figure 9. Figure 8 shows the topology structure of 20 USVs and a virtual leader at different times. As shown in Figure 8a, it represnets the position of 20 USVs and a virtual leader at the initial time. Figure 8b shows that 19 USVs are clustered at 30$s$. Figure 8c represents that 20 USVs are clustered at 40$s$. Figure 8d represents that 20 USVs can always maintain cluster navigation. During sail, the topology structure composed of multiple USVs remained stable. Figure 9a represents the trajectories of 20 USVs and a virtual leader. Figure 9b shows the positions and headings (X). Figure 9c represents the tracking errors. Figure 9d shows the USVs speeds and velocities mismatch. The simulation results show that the group of USVs remain cohesive and all USVs move asymptotically at the same speed.

Since the initial conditions (x, y positions, heading and velocity) of the USVs are different, there are small deviations from vessel to vessel. Apparently, the topology show that no collisions occur during the whole swarming process. Through the comparison of Figure 8b–d, it can be seen that the topology structure is time-varying. Quasi $\alpha$-$grid$ is unstable. After a long enough time, quasi $\alpha$-$grid$ will eventually form $\alpha$-$grid$. It shows the effective of the proposed time-variant topology structure. Through the simulation results, it can be found that under the proposed DSCA, the positions converge uniformly without collision, the heading errors tend to zero, and velocities consensus are also achieved.

5.2. Case 2: 20 USVs and Three Virtual Leaders

In this subsection, the simulation results are given to verify the performance of proposed DCSA with multi-USV under three virtual leaders.

The initial positions of the 20 USVs are distribute in the range of abscissa and ordinate [−100 m, 50 m] × [−100 m, 50 m], and heading angle of the USVs is also randomly distributed in [$-\pi rad$, $\pi rad$].

Different parameters are set for the initial state of the three virtual leaders. The parameters of the first virtual leader are $X_{\gamma 1}\left(0\right)={[-20m,-80m,\pi /12rad]}^{\mathrm{T}}$ and ${\theta }_{\gamma 1}\left(0\right)={[1.5m/s,1.5m/s,0m/s^2]}^{\mathrm{T}}$. The parameters of the second virtual leader are $X_{\gamma 2}\left(0\right)={[-20m,-20m,\pi /4rad]}^{\mathrm{T}}$ and ${\theta }_{\gamma 2}\left(0\right)={[2m/s,2m/s,0m/s^2]}^{\mathrm{T}}$. The parameters of the third virtual leader are $X_{\gamma 3}\left(0\right)={[-80m,10m,5\pi /12rad]}^{\mathrm{T}}$ and ${\theta }_{\gamma 3}\left(0\right)={[2.5m/s,2.5m/s,0m/s^2]}^{\mathrm{T}}$.

Figure 10a represents the trajectories of 20 USVs and three virtual leaders. Figure 10b shows the positions and headings (X). Figure 10c represents the tracking errors. Figure 10d shows the USVs speeds and velocities mismatch. The simulation results show that the grouping method based on cosine similarity is effectiveness, the group of USVs remain cohesive and all USVs move asymptotically at the same speed.

5.3. Comparison Group

The comparison group is carried out by using a distributed convex optimization proposed in [36].

The controller is designed by using only local interaction information rather than global bidirectional topology, and the case of multi-USV grouping is not considered in [36]. In simulations, the control inputs and environment distribution of two methods are both same. The simulation results are shown in Figure 11 and Figure 12.

Figure 11 shows the topology structure of 20 USVs and a virtual leader at different times. As shown in Figure 11a, it represnets the position of 20 USVs and a virtual leader at the initial time. Figure 11b shows that 12 USVs are clustered at 30 s. Figure 11c represents that 17 USVs are clustered at 50 s. Figure 11d represents that 20 USVs are clustered at 100 s. Figure 12a represents the trajectories of 20 USVs and a virtual leader. Figure 12b shows the positions and headings (X). Figure 12c represents the tracking errors. Figure 12d shows the USVs speeds and velocities mismatch.

After 50 experiments, the number of USVs achieving clusters by using defferent algorithms are shown in

Table 2. The number of USVs achieving clusters via different algorithms ($n=20$).

	t = 0 s	t = 30 s	t = 50 s	t = 100 s
distributed swarm control algorithm	0	19	20	20
distributed convex optimization	0	12	17	20

. Through the comparative analysis of the data in Table 2, it is shown that the number of USV obtained by method distributed swarm control algorithm is more than that obtained by distributed convex optimization. It can be proved that the effectiveness of the proposed algorithm.

In order to study the influence of environmental disturbances on the algorithm, we use a finite time disturbance observer to estimate the environmental disturbances. The comparative simulation is carried out by using a modular design approach proposed in [37].

The simulation result is shown in Figure 13. Taking the first USV as an example, Figure 13 depicts that the time-varying environmental disturbances value and estimates obtained by using modular design approach and finite time disturbance observer.

Through the analysis of Figure 13, it is shown that the more accurate extimation of the environmantal disturbances is obtained by using finite time disturbance observer than modular design approach. This means that the proposed control scheme is robust against disturbances.

6. Conclusions

In this paper, we investigate the dynamic properties of the group for the case where the number of virtual leader is different and the topology of the neighborhood relations between vehicles is dynamic. Firstly, to achieve multiple USVs connectivity, a new time-variant topology structure is proposed, which is based on proximity nets and $\alpha$-$grids$. Secondly, to reduce the communication load and realize the information diffluence, a grouping method based on cosine similarity is proposed. Thirdly, to ensure the high effeciency of information transmission and the reduction of costs, a distributed swarm control algorithm is proposed. To analyze the stability of system, the concept of translation framework is introduced. The stability of multiple USVs system is proved by using the Hamiltonian and LaSalle invariance principle. Simulation results show the effectiveness of the proposed methods. However, this paper does not consider the problem of static/dynamic obstacles. These problems will be studied in the next step.