Robust Coordinated Formation for Multiple Surface Vessels Based on Backstepping Sliding Mode Control

Abstract

We investigate the problem of coordinated formation control for multiple surface vessels in the presence of unknown external disturbances. In order to realize the leaderless coordinated formation and achieve the robustness against unknown external disturbances, a new robust coordinated formation control algorithm based on backstepping sliding mode control is proposed. The proposed coordinated control algorithm is achieved by defining a new switched function using the combination of position tracking error and cross-coupling error. Particularly, the cross-coupling error is defined using velocity tracking error and velocity synchronization error so as to be applicable for sliding mode controller design. Furthermore, the adaptive control law is proposed to estimate unknown disturbances for each vessel. The globally asymptotically stability is proved using the Lyapunov direct method. Finally, the effectiveness of the proposed coordinated formation control algorithm is demonstrated by corresponding simulations.

1. Introduction

Recently, coordination and consensus of multiagent systems have received much attention, and the network-induced constraints are discussed [1]. Many studies on coordination control issues have been widely reported in the existing literature due to the widespread applications, such as multiple mobile robots, spacecraft formation, networked sensors, vehicles coordination [2]. Particulary, the applications of formation control at sea are increasing, which can be found within both civil and military operations. For example, a group of surface vessels travel in formation structures to perform the seabed mapping operations, the collective range of sensor equipment can be maximized, ensuring that larger areas can be covered in shorter time compared with a single vessel. And another example is underway replenishment of vessels which is typically performed by one or more supply vessels lining up at the side of the receiving vessel, after which all vessels strive to maintain equal and constant forward speed and bearing while supplies are being transferred across messenger lines [3]. The complicated operations often cannot be carried out through only one vessel.

In a word, multiple vessels work together to improve performance and reduce fatigue and difficulty for the people involved. What is more, compared with a single vessel, multiple vessels perform the complicated tasks with less time and cost in practical maritime. So this paper will focus on the coordinated formation of multiple surface vessels; And the formation control in this paper aims at the surface vessels, then the robustness to environmental disturbances is highly important when considering the marine and offshore applications. So the study on the robust coordination control algorithm for multiple surface vessels is significative.

With respect to the coordination control issues of multiple surface vessels, abundant studies have been widely reported in the existing literature. There are several typical approaches used to design the coordination formation controller. For example, Lagrangian formation method [4], null-space-based behavioral control [5], nonlinear model predictive control [6], and graph theory [7]. Meanwhile the problem of coordinated path following for multiple vessels also has been discussed in the following references [8–10]. In recent years, the formation control of multiple vessels is researched using the passivity-based control method included in the synchronization control approach [11–13]. A common trait in the aforementioned results is that the coordinated tracking controller is designed by assuming that motions of vessels are disturbance-free. However, surface vessels often encounter external disturbances such as wind, wave, and current. The external environment disturbances are difficult to model well and truly because of the disturbances varying with weather conditions and depth of water area change. So the coordinated formation algorithm for multiple surface vessels should be robust to unknown disturbances. The adaptive control is helpful for solving this problem [14, 15].

This paper develops a coordinated formation control algorithm based on backstepping sliding mode control approach. The sliding mode control possesses the robustness to system uncertainty and external disturbance as a result of the definition of the switched function [16]. And the stabilization for the switched systems are discussed in [17, 18]. For a single surface vessel, the sliding mode control method can achieve robustness to the environment disturbances [19, 20]. Furthermore, the sliding mode control is also used to design the coordinated formation controller as in reference [21, 22]. And the backstepping method can determine the appropriate Lyapunov function systematically and simply, which makes the distributed robust and adaptive redesign implementable. The integration of backstepping and sliding mode control will centralize both the advantages of the two control schemes [23]. Backstepping sliding mode control is used to solve the coordinated formation problem for multi-agent systems in [24, 25]. The aforementioned studies on sliding mode formation control are all based on the leader-follower strategy. However, the leaderless strategy is applicable widely because of most practical task without emphasizing which vessel is important [26]. The cross-coupling synchronization approach is propitious to the leaderless formation controller design. In addition, the synchronization control approach has a simple control structure and convenient implementation capability compared to existing formation control approach. The cross-coupling synchronization error can be convenient for defining switched function in the sliding mode controller design [27]. However, the cross-coupling error of [27] is composed of position tracking error and integral of synchronization error, which cannot be applicable to design sliding mode controller. This motivates that the cross-coupling error is redefined to be applicable to the whole coordinated controller design.

In this paper, a robust coordinated formation controller based on backstepping sliding mode control is proposed for multiple surface vessels. And a new switched function is defined using a new cross-coupling error to achieve leaderless coordination between these vessels. And the cross-coupling error is defined using the velocity error and the synchronization velocity error. This will be applicable to the sliding mode controller design. Furthermore, with respect to the unknown disturbances, the adaptive control law is proposed to improve the robust coordinated formation control algorithm. The rest of this paper is organized as follows. Section 2 introduces the basic notations for the graph theory and establishes the vessel model. Section 3 describes the proposed coordination control algorithm in detail. Section 4 describes robust coordinated tracking controller. The simulation of the proposed coordination algorithm for five vessels is carried out, and the validity of the proposed coordinated control algorithm is demonstrated in Section 5. At last, we draw conclusions in Section 6.

2. Preliminaries

2.1. Notations

In this section, several basic concepts about the directed connected graph are given. A directed graph G consists of the pair (𝜈,𝜀). 𝜈 is a set of vertices; 𝜀∈𝜈² is a set of edges. (𝑖,𝑗)∈𝜀 if information flows from vertex 𝑗 to vertex 𝑖. If any two distinct vertices of a directed graph can be connected through a directed path, then the directed graph is called strongly connected. The adjacent matrix of the directed graph is denoted as 𝐀∈R^𝑛×𝑛, which is defined as diagonal entries 0 and off-diagonal entries 𝑎_𝑖𝑗=1 if (𝑖,𝑗)∈𝜀 and 0 otherwise. And the degree matrix 𝐃∈R^𝑛×𝑛 is defined with off-diagonal entries 0 and diagonal entries 𝑑_𝑖𝑖=∑_𝑗≠𝑖𝑎_𝑖𝑗. Then the Laplacian matrix is obtained as 𝐋=𝐃−𝐀∈R^𝑛×𝑛. That is 𝐋=[𝑙_𝑖𝑗]∈R^𝑛×𝑛, and 𝑙_𝑖𝑖=∑_𝑗≠𝑖𝑎_𝑖𝑗, 𝑙_𝑖𝑗=−𝑎_𝑖𝑗.

In this paper, the communication topology between these vessels is described by a strongly connected graph. Each vertex in the graph represents a vessel in the group. The edges represent information exchange links by available communication.

Define the Kronecker product of two matrices 𝐀∈R^𝑚×𝑛 and 𝐁∈R^𝑝×𝑞 as 𝑎𝐀⊗𝐁=¹¹𝐁⋯𝑎_1𝑛𝐁𝑎⋮⋱⋮_𝑚1𝐁⋯𝑎_𝑚𝑛𝐁. (1)

Theorem 1 (see [28]). Let 𝑥=0 be an equilibrium point for 𝑥̇=𝑓(𝑡,𝑥), and let 𝐷⊂R^𝑛 be a domain containing 𝑥=0. Let 𝑉∶[0,∞)×𝐷→R be a continuously differentiable function such that 𝑊₁(𝑥)≤𝑉(𝑡,𝑥)≤𝑊₂(𝑥),𝜕𝑉+𝜕𝑡𝜕𝑉𝑓()𝜕𝑥𝑡,𝑥≤𝑊₃(𝑥) (2) for all 𝑡≥0 and for all 𝑥⊂𝐷, where 𝑊₁(𝑥), 𝑊₂(𝑥), and 𝑊₃(𝑥) are continuous positive definite functions on 𝐷. Then, 𝑥=0 is uniformly asymptotically stable. If 𝐷=R^𝑛 and 𝑊₁(𝑥) is radially unbounded, then 𝑥=0 is globally uniformly asymptotically stable.

Lemma 2 (Barbalat's lemma [28]). Let 𝜙∶R→R be a uniformly continuous function on [0,∞). Suppose that lim_𝑡→∞∫^𝑡₀𝜙(𝜏)𝑑𝜏 exists and is finite. Then, 𝜙(𝑡)→0 as 𝑡→∞.

2.2. Mathematical Vessel Model

The vessel model can be divided into two parts: the kinematics and nonlinear dynamics. Generally, only the motion in the horizontal plane is considered for the surface vessel. The elements corresponding to heave, roll, and pitch are neglected. The dynamic model for the 𝑖th surface vessel can be represented by the following 3 degrees of freedom (DOF) [29]: 𝜂̇_𝑖=𝐑_𝑖𝜓_𝑖𝐯_𝑖,𝐌_𝑖𝐯̇_𝑖+𝐂_𝑖𝐯_𝑖𝐯_𝑖+𝐃_𝑖𝐯_𝑖𝐯_𝑖=𝜏_𝑖+𝜏_𝑑𝑖, (3)(4) where 𝜂_𝑖=[𝑛_𝑖,𝑒_𝑖,𝜓_𝑖]^𝑇 denotes the north position, east position, and orientation which are decomposed in the earth-fixed reference frame, and 𝐯_𝑖=[𝑢_𝑖,𝑣_𝑖,𝑟_𝑖]^𝑇 denotes the linear surge velocity, sway velocity and angular velocity, which are decomposed in the body-fixed reference frame. 𝐑_𝑖(𝜓_𝑖) is the transformations matrix from the body-fixed reference frame to the earth-fixed reference frame, the form of which is as follows: 𝐑_𝑖𝜓_𝑖=cos𝜓_𝑖−sin𝜓_𝑖0sin𝜓_𝑖cos𝜓_𝑖0001. (5) Then we can know that the transformations matrix satisfies the following properties as 𝐑_𝑖⁻¹(𝜓_𝑖)=𝐑^𝑇_𝑖(𝜓_𝑖), for all 𝜓_𝑖. 𝐌_𝑖 denotes the system inertia mass matrix including added mass which is positive definite. 𝐂_𝑖(𝐯_𝑖) and 𝐃_𝑖(𝐯_𝑖) denote the Corioliscentripetal matrix and damping matrix, respectively. The detailed representation of the previous three-system matrix can be found in [29]. 𝜏_𝑖=𝜏_𝑢𝑖𝜏_𝑣𝑖𝜏_𝑟𝑖^𝑇 is the vector of forces and torques input from the thruster system. 𝜏_𝑑𝑖 is the vector of external environment forces and torques input which are generating by wind, wave and current.

In order to design the backstepping sliding mode controller, we transform the vessel model as following. Equation (3) can be transformed as 𝐯_𝑖=𝐑_𝑖⁻¹𝜓_𝑖𝜂̇_𝑖. (6) By differentiating 𝐑_𝑖(𝜓_𝑖), we have 𝐑̇_𝑖(𝜓_𝑖)=𝐑_𝑖(𝜓_𝑖)𝑆, where 𝑆=0−𝑟_𝑖0𝑟_𝑖00000=−𝑆^𝑇, (7) where 𝑟_𝑖 is the angular velocity in the body-fixed reference frame. Take the derivative of (6), we can obtain that: 𝐯̇_𝑖=𝐑̇_𝑖⁻¹𝜓_𝑖𝜂̇_𝑖+𝐑_𝑖⁻¹𝜓_𝑖𝜂̈_𝑖=−𝐑_𝑖⁻¹𝜓_𝑖𝐑̇_𝑖𝜓_𝑖𝐑_𝑖⁻¹𝜓_𝑖𝜂̇_𝑖+𝐑_𝑖⁻¹𝜓_𝑖𝜂̈_𝑖. (8) Taking (6) and (8) into the vessel dynamic model (4) can yield 𝐌_𝑖𝐑_𝑖⁻¹𝜓_𝑖𝜂̈_𝑖−𝐌_𝑖𝐑_𝑖⁻¹𝜓_𝑖𝐑̇_𝑖𝜓_𝑖𝐑_𝑖⁻¹𝜓_𝑖𝜂̇_𝑖+𝐂_𝑖𝐯_𝑖𝐑_𝑖⁻¹𝜓_𝑖𝜂̇_𝑖+𝐃_𝑖𝐯_𝑖𝐑_𝑖⁻¹𝜓_𝑖𝜂̇_𝑖=𝜏_𝑖+𝜏_𝑑𝑖. (9) The above equation can be written as 𝐌_𝑛𝑖𝜂_𝑖𝜂̈_𝑖+𝐂_𝑛𝑖𝜂_𝑖,𝜂̇_𝑖𝜂̇_𝑖+𝐃_𝑛𝑖𝜂_𝑖,𝜂̇_𝑖𝜂̇_𝑖=𝜏_𝑖+𝜏_𝑑𝑖, (10) where 𝐌_𝑛𝑖𝜂_𝑖=𝐌_𝑖𝐑_𝑖⁻¹𝜓_𝑖;𝐂_𝑛𝑖𝜂_𝑖,𝜂̇_𝑖=𝐂_𝑖𝐯_𝑖−𝐌_𝑖𝐑_𝑖⁻¹𝜓_𝑖𝐑̇_𝑖𝜓_𝑖𝐑_𝑖⁻¹𝜓_𝑖;𝐃_𝑛𝑖𝜂_𝑖,𝜂̇_𝑖=𝐃_𝑖𝐯_𝑖𝐑_𝑖⁻¹𝜓_𝑖. (11)

3. Coordinated Formation Controller Design

In this section, we will design the coordinated trajectory tracking controller for multiple surface vessels based on backstepping sliding mode control method. And we assume that all the vessels are disturbance-free. The detailed procedure of controller design is as follows.

3.1. Formation Setup

This paper considers a fleet of 𝑛 vessels to perform the desired coordination formation task. And each vessel in the formation is identified by the index set 𝐼=[1,2,…,𝑛]. As in [12], the desired formation is established by defining the formation reference vector for each vessel, which is denoted as 𝐥_𝑖=[𝑥_0𝑖,𝑦_0𝑖,𝜓_0𝑖]^𝑇, where 𝑥_0𝑖, 𝑦_0𝑖, 𝜓_0𝑖 are constant, respectively, and −𝜋≤𝜓_0𝑖≤𝜋. Then the formation reference point for each vessel is given by: 𝐱_𝑖=𝜂_𝑖+𝐑_𝑖(𝜓_𝑖−𝜓_0𝑖)𝐥_𝑖. If we assume the desired trajectory of the formation point is denoted as 𝜂_𝑑, where 𝜂_𝑑=[𝑛_𝑑(𝑡),𝑒_𝑑(𝑡),𝜓_𝑑(𝑡)]^𝑇, 𝑛_𝑑(𝑡), 𝑒_𝑑(𝑡) are sufficiently smooth functions, and 𝜓_𝑑(𝑡)=arctan(𝑒_𝑑̇(𝑡)/𝑛̇_𝑑(𝑡)). That means that the vessel direction is chosen as the tangential vector of the respective desired trajectory. We can know that the coordinated formation is achieved if all the formation reference points of the group of vessels are synchronized; that is 𝐱₁=⋯=𝐱_𝑖=⋯=𝐱_𝑛=𝜂_𝑑.

3.2. Formation Controller Design

The proposed coordinated formation controller for multiple surface vessels is designed using the backstepping sliding mode control approach, and a new switched function is defined to accomplish the sliding mode controller.

Define the position tracking error of the formation reference point for each vessel as 𝐳_1𝑖=𝐱_𝑖−𝜂_𝑑=𝜂_𝑖+𝐑_𝑖𝜓_𝑖−𝜓_0𝑖𝐥_𝑖−𝜂_𝑑. (12) If we define the new variable 𝐟_𝑖 as 𝐟_𝑖=𝐑_𝑖(𝜓_𝑖−𝜓_0𝑖)𝐥_𝑖, take the derivative of 𝐳_1𝑖, we can obtain that 𝐳̇_1𝑖=𝜂̇_𝑖+𝐟_𝑖̇−𝜂̇_𝑑. (13) Define the following stabilizing function for each vessel as 𝛼_𝑖=𝜆_𝑖𝐳_1𝑖, (14) where 𝜆_𝑖∈R^3×3 is a diagonal positive definite matrix.

Define the velocity error as 𝐳_2𝑖; the form of which is 𝐳_2𝑖=𝐳̇_1𝑖+𝛼_𝑖=𝐳̇_1𝑖+𝜆_𝑖𝐳_1𝑖. (15) Then we can obtain 𝐳_2𝑖=𝜂̇_𝑖+𝐟_𝑖̇−𝜂̇_𝑑+𝜆_𝑖𝐳_1𝑖. (16) With the form of vessel model, then we have 𝐌_𝑛𝑖𝜂_𝑖𝐳̇_2𝑖=𝐌_𝑛𝑖𝜂_𝑖𝜂̈_𝑖+𝐟_𝑖̈−𝜂̈_𝑑+𝜆_𝑖𝐳_1𝑖=𝜏_𝑖+𝜏_𝑑𝑖−𝐂_𝑛𝑖𝜂_𝑖,𝜂̇_𝑖𝜂̇_𝑖−𝐃_𝑛𝑖𝜂_𝑖,𝜂̇_𝑖𝜂̇_𝑖+𝐌_𝑛𝑖𝜂_𝑖𝐟_𝑖̈−𝐌_𝑛𝑖𝜂_𝑖𝜂̈_𝑑+𝐌_𝑛𝑖𝜂_𝑖𝜆_𝑖𝐳_1𝑖. (17) In light of (16), we can obtain that 𝜂̇_𝑖=𝜂̇_𝑑−𝐟_𝑖̇+𝐳_2𝑖−𝜆_𝑖𝐳_1𝑖. (18) Then (17) can be calculated as follows 𝐌_𝑛𝑖𝜂_𝑖𝐳̇_2𝑖=𝜏_𝑖+𝜏_𝑑𝑖−𝐂_𝑛𝜂_𝑖,𝜂̇_𝑖+𝐃_𝑛𝜂_𝑖,𝜂̇_𝑖𝜂̇_𝑑−𝐟_𝑖̇−𝐂_𝑛𝑖𝜂_𝑖,𝜂̇_𝑖+𝐃_𝑛𝑖𝜂_𝑖,𝜂̇_𝑖𝐳_2𝑖−𝜆_𝑖𝐳_1𝑖−𝐌_𝑛𝑖𝜂_𝑖𝜂̈_𝑑−𝐟̈+𝐌_𝑛𝑖𝜂_𝑖𝜆_𝑖𝐳̇_1𝑖. (19) For representing conveniently, we define 𝐳₁=[𝑧^𝑇₁₁𝑧^𝑇₁₂⋯𝑧^𝑇_1𝑛]^𝑇; 𝐳₂=[𝑧^𝑇₂₁𝑧^𝑇₂₂⋯𝑧^𝑇_2𝑛]^𝑇; 𝜂=[𝜂^𝑇₁𝜂^𝑇₂⋯𝜂^𝑇_𝑛]^𝑇; 𝐟=[𝐟^𝑇₁𝐟^𝑇₂⋯𝐟^𝑇_𝑛]^𝑇; 𝜏=[𝜏^𝑇₁𝜏^𝑇₂⋯𝜏^𝑇_𝑛]^𝑇; 𝜏_𝑑=[𝜏^𝑇_𝑑1𝜏^𝑇_𝑑2⋯𝜏^𝑇_𝑑𝑛]^𝑇; 𝜂_𝐷=[𝜂^𝑇_𝑑𝜂^𝑇_𝑑⋯𝜂^𝑇_𝑑]^𝑇; 𝜆=diag𝜆₁𝜆₂⋯𝜆_𝑛,𝐌_𝑛𝜂=diag𝐌_𝑛1𝜂₁⋯𝐌_𝑛𝑛𝜂_𝑛,𝐂_𝑛𝜂,𝜂̇=diag𝐂_𝑛1𝜂₁,𝜂̇₁⋯𝐂_𝑛𝑛𝜂_𝑛,𝜂̇_𝑛,𝐃_𝑛𝜂,𝜂̇=diag𝐃_𝑛1𝜂₁,𝜂̇₁⋯𝐃_𝑛𝑛𝜂_𝑛,𝜂̇_𝑛. (20) So the error dynamics of multiple surface vessels can be written in terms of matrix and vector: 𝐳̇₁=𝐳₂−𝜆𝐳₁,𝐳̇₂=𝐌_𝑛⁻¹𝜂𝜏+𝜏_𝑑−𝐌_𝑛𝜂𝜂̈_𝐷−𝐟̈−𝐂_𝑛𝜂,𝜂̇+𝐃_𝑛𝜂,𝜂̇𝜂̇_𝐷−𝐟̇−𝐂_𝑛𝜂,𝜂̇+𝐃_𝑛𝜂,𝜂̇𝐳₂−𝜆𝐳₁+𝐌_𝑛𝜂𝜆𝐳̇₁. (21)(22) Define the synchronization velocity error vector for these vessels as 𝜀=𝐋⊗𝐈₃𝐳₂, (23) where 𝐋∈R^𝑛×𝑛 is the Laplacian matrix of the communication topology graph for these vessels.

Define the cross-coupling error using the velocity error and the synchronization velocity error; the form of which is 𝐞=𝐳₂+𝛾𝜀=𝐳₂+𝛾𝐋⊗𝐈₃𝐳₂=𝐈_3𝑛+𝛾𝐋⊗𝐈₃𝐳₂, (24) where 𝛾∈R^3𝑛×3𝑛 is a diagonal positive definite matrix.

The switched function is defined using the position tracking error and the cross-coupling error. The detailed form is as follows 𝐬=𝐤𝐳₁+𝐞=𝐤𝐳₁+𝐈_3𝑛+𝛾𝐋⊗𝐈₃𝐳₂, (25) where 𝐤∈R^3𝑛×3𝑛 is a diagonal positive definite matrix.

If we assume these vessels are disturbance-free; that is 𝝉_𝑑=0, then the control input can be chosen as follows 𝜏=𝜏_eq+𝜏_sw, (26) where 𝜏_eq is the equivalent control input, and 𝜏_sw is the switch control part of the backstepping sliding mode control input. The detailed expressions are as follows: 𝜏_eq=𝐌_𝑛𝜂𝜂̈_𝐷−𝐟̈−𝐌_𝑛𝜂𝜆𝐳̇₁+𝐂_𝑛𝜂,𝜂̇+𝐃_𝑛𝜂,𝜂̇𝜂̇_𝐷−𝐟̇+𝐂_𝑛𝜂,𝜂̇+𝐃_𝑛𝜂,𝜂̇𝐳₂−𝜆𝐳₁−𝐌_𝑛𝜂𝐈_3𝑛+𝛾𝐋⊗I₃⁻¹𝐤𝐳₂−𝜆𝐳₁,𝜏_sw=−𝐌_𝑛𝐈_3𝑛+𝛾𝐋⊗I₃⁻¹𝐏𝐬+𝛽sign(𝐬), (27) where 𝐏∈R^3𝑛×3𝑛, 𝛽∈R^3𝑛×3𝑛 are a diagonal positive definite matrix, respectively. And sign(⋅) is sign function. And sign(𝐬)=sign(𝐬^𝑇₁)sign(𝐬^𝑇₂)⋯sign(𝐬^𝑇_𝑛)^𝑇.

Remark 3. The control input vector for each vessel which is disturbance-free can be written as 𝜏_𝑖=𝜏^𝑖_eq+𝜏^𝑖_sw. (28) Define 𝐘=𝐈_3𝑛+𝛾𝐋⊗I₃⁻¹=𝑌_𝑖𝑗∈R^3×3∈R^3𝑛×3𝑛,𝛾=diag𝛾₁𝛾₂⋯𝛾_𝑛,𝐤=diag𝐤₁𝐤₂⋯𝐤_𝑛,𝐏=diag𝐏₁𝐏₂⋯𝐏_𝑛,𝛽=diag𝛽₁𝛽₂⋯𝛽_𝑛. (29) The detailed expressions of the control input for each vessel are as follows: 𝜏^𝑖_eq=𝐌_𝑛𝑖𝜂_𝑖𝜂̈_𝑑−𝐟_𝑖̈−𝐌_𝑛𝑖𝜂_𝑖𝜆𝐳̇_1𝑖+𝐂_𝑛𝑖𝜂_𝑖,𝜂̇_𝑖+𝐃_𝑛𝑖𝜂_𝑖,𝜂̇_𝑖𝜂̇_𝑑−𝐟_𝑖̇+𝐂_𝑛𝑖𝜂_𝑖,𝜂̇_𝑖+𝐃_𝑛𝑖𝜂_𝑖,𝜂̇_𝑖𝐳_𝟐𝐢−𝜆_𝑖𝐳_1𝑖−𝐌_𝑛𝑖𝜂_𝑖^𝑛_𝑗=1𝑌_𝑖𝑗𝑘_𝑗𝐳_2𝑗−𝜆_𝑗𝐳_1𝑗𝜏^𝑖_sw=−𝐌_𝑛𝑖𝜂_𝑖^𝑛_𝑗=1𝑌_𝑖𝑗𝐏_𝑗𝐬_𝑗+𝛽_𝑗sign𝐬_𝑗, (30) where the switched function is defined as 𝐬_𝑗=𝐤_𝑗𝐳_1𝑗+𝐳_2𝑗+𝛾_𝑗^𝑛_𝑖=1𝑎_𝑗𝑖𝐳_2𝑖−𝐳_2𝑗, (31) where 𝐀={𝑎_𝑗𝑖}∈R^𝑛×𝑛 is the adjacency matrix for the communication topology graph. 𝑎_𝑖𝑖=0 and 𝑎_𝑗𝑖=1 if information flow from vessel 𝑖 to vessel 𝑗 and 0 otherwise, for all 𝑗≠𝑖.

Theorem 4. Consider a group of 𝑛 vessels to perform the coordination formation task; the vessel model is described as (3) and (4), and we assume that these vessels are disturbance-free. With the distributed coordinated formation control law (26) and (27), the following conditions are satisfied;(i)all the matrices 𝜆, 𝐏, and 𝐤 are diagonal positive definite;(ii)(𝜆Γ^𝑇+𝐤)𝐏Γ−(1/4)𝐈_3𝑛>0, where Γ=𝐈_3𝑛+𝛾(𝐋⊗𝐈₃);(iii)the matrix 𝛾 is diagonal positive definite and small enough. Then the position tracking error, the velocity error and their synchronization error, approach zero asymptoticallyl; that is, these vessels can realize the coordination formation asymptotically.

Proof. Define the first Lyapunov function as 𝑉₁=12𝐳^𝑇₁𝐳₁. (32) Differentiating 𝑉₁ with respect to time, then we have 𝑉̇₁=−𝐳^𝑇₁𝜆𝐳₁+𝐳^𝑇₁𝐳₂. (33) Define the second Lyapunov function as 𝑉₂=𝑉₁+12𝐬^𝑇𝐬. (34) Take the time derivative of the above equation as 𝑉̇₂=𝑉̇₁+𝐬^𝑇𝐬̇=𝑉̇₁+𝐬^𝑇𝐤𝐳̇₁+𝐬^𝑇𝐈_3𝑛+𝛾𝐋⊗I₃𝐳̇₂=𝐳^𝑇₁𝐳₂−𝐳^𝑇₁𝜆𝐳₁+𝐬^𝑇𝐤𝐳₂−𝜆𝐳₁+𝐬^𝑇𝐈_3𝑛+𝛾𝐋⊗I₃𝐌_𝑛⁻¹𝜂×𝜏−𝐌_𝑛𝜂𝜂̈_𝐷+𝐌_𝑛𝜂𝐟̇+𝐌_𝑛𝜂𝜆𝐳̇₁−𝐂_𝑛𝜂,𝜂̇+𝐃_𝑛𝜂,𝜂̇𝜂̇_𝐷−𝐟̇−𝐂_𝑛𝜂,𝜂̇+𝐃_𝑛𝜂,𝜂̇𝐳₂−𝜆𝐳₁. (35) With the control input (26) and (27), we can obtain that 𝑉̇₂=−𝐳^𝑇₁𝜆𝐳₁+𝐳^𝑇₁𝐳₂−𝐬^𝑇𝐏𝐬+𝛽sign(𝐬)=−𝐳^𝑇₁𝜆𝐳₁+𝐳^𝑇₁𝐳₂−𝐬^𝑇𝐏𝐬−𝐬^𝑇𝐏𝛽sign(𝐬)=−𝐳^𝑇₁𝜆𝐳₁+𝐳^𝑇₁𝐳₂−𝐬^𝑇𝐏𝐬−^3𝑛_𝑟=1𝑝_𝑟𝛽_𝑟𝑠_𝑟, (36) where |𝑠_𝑟| denotes the absolute value of the variable of 𝑠_𝑟.
If we define Γ=𝐈_3𝑛+𝛾(𝐋⊗𝐈₃), then we choose matrix 𝐐 as 𝐐=𝜆+𝐤^𝑇𝐏𝐤𝐤^𝑇1𝐏Γ−2𝐈_3𝑛Γ^𝑇𝐏^𝑇1𝐤−2𝐈_3𝑛Γ^𝑇𝐏Γ. (37) If we define 𝐳=[𝑧^𝑇₁,𝑧^𝑇₂]^𝑇, then we can obtain that 𝐳^𝑇𝐐𝐳=𝐳^𝑇₁𝐳^𝑇₂𝜆+𝐤^𝑇𝐏𝐤𝐤^𝑇1𝐏Γ−2𝐈_3𝑛Γ^𝑇1𝐏𝐤−2𝐈_3𝑛Γ^𝑇𝐳𝐏Γ𝐳¹₂=𝐳^𝑇₁𝜆+𝐳^𝑇₁𝐤^𝑇𝐏𝐤+𝐳^𝑇₂Γ^𝑇1𝐏𝐤−2𝐳^𝑇₂𝐈_3𝑛×𝐳^𝑇₁𝐤^𝑇1𝐏Γ−2𝐳^𝑇₁𝐈_3𝑛+𝐳^𝑇₂Γ^𝑇𝐳𝐏Γ¹𝐳₂=𝐳^𝑇₁𝜆𝐳₁+𝐳^𝑇₁𝐤^𝑇𝐏𝐤𝐳₁−12𝐳^𝑇₂𝐳₁+𝐳^𝑇₂Γ^𝑇𝐏𝐤𝐳₁+𝐳^𝑇₁𝐤^𝑇𝐏Γ𝐳₂−12𝐳^𝑇₁𝐳₂+𝐳^𝑇₂Γ^𝑇𝐏Γ𝐳₂=𝐳^𝑇₁𝜆𝐳₁−𝐳^𝑇₂𝐳₁+𝐬^𝑇𝐏𝐬. (38) Then (36) can be written as 𝑉̇₂=−𝐳^𝑇𝐐𝐳−^3𝑛_𝑟=1𝑝_𝑟𝛽_𝑟𝑠_𝑟≤0. (39) If we choose the matrix 𝛾 as small enough, then Γ will be positive definite. All the matrices 𝜆, 𝐏, and 𝐤 are diagonal positive definite, if they are chosen to satisfy (𝜆Γ^𝑇+𝐤)𝐏Γ−(1/4)𝐈_3𝑛>0; then we can guarantee that the matrix 𝐐 is positive definite. If we define 𝑊(𝑡)=𝐳^𝑇𝐐𝐳+∑^3𝑛_𝑟=1𝑝_𝑟𝛽_𝑟|𝑠_𝑟|, it is obvious that the bounded limit of 𝑊(𝑡) exists. From the front definition, we can know that 𝐳₁, 𝐳₂, and 𝐬 are bounded. With (16), we can know that 𝐳̇₁ is bounded. In terms of (22), (26), and (27), 𝐳̇₂ is bounded. Then 𝐳 and 𝐳̇ are bounded. Due to the definition of 𝐬, 𝐬 and 𝐬̇ are bounded. Then we can get that 𝑊̇(𝑡) is also bounded; then 𝑊(𝑡) is uniformly continuous. With Barbalats lemma, 𝑊(𝑡)→0 as 𝑡→∞. Then 𝐳₁→0, 𝐳₂→0, 𝐬→0 as 𝑡→∞. With (23), we can get 𝜀→0. Then the synchronization error for each vessel is 𝜀_𝑖=^𝑛_𝑗=1𝑎_𝑖𝑗𝐳_2𝑗−𝐳_2𝑖⟶0. (40) Because 𝑎_𝑖𝑗 is constant, then we can obtain that 𝐳_2𝑖(𝑡)−𝐳_2𝑗(𝑡)→0.
Though the following calculation 𝐳_2𝑖(𝑡)−𝐳_2𝑗(𝑡)=𝐳̇_1𝑖+𝜆_𝑖𝐳_1𝑖−𝐳̇_1𝑗−𝜆_𝑖𝐳_1𝑗=𝐱̇_𝑖−𝜂̇_𝑑+𝜆_𝑖𝐱_𝑖−𝜂_𝑑−𝐱̇_𝑗−𝜂̇_𝑑−𝜆_𝑖𝐱_𝑗−𝜂_𝑑=𝐱̇_𝑖−𝐱̇_𝑗+𝜆_𝑖𝐱_𝑖−𝐱_𝑗=𝐞̇_𝑖𝑗+𝜆_𝑖𝐞_𝑖𝑗, (41) where 𝐞_𝑖𝑗=𝐱_𝑖−𝐱_𝑗. From the above equation, we can know that it is a linear exponential system with the input as 𝐳_2𝑖(𝑡)−𝐳_2𝑗(𝑡), and 𝐳_2𝑖(𝑡)−𝐳_2𝑗(𝑡)→0, and 𝐞_𝑖𝑗 is bounded; then we can obtain that lim_𝑡→∞‖𝐞_𝑖𝑗‖=0; that is 𝐱_𝑖−𝐱_𝑗→0. According to 𝐳_1𝑖(𝑡)→0, then 𝐱_𝑖−𝜂_𝑑→0. Finally, we can know that 𝐱₁=⋯=𝐱_𝑖=⋯=𝐱_𝑛=𝜂_𝑑. So the group of vessels achieve the coordinated tracking while holding the desired formation.

4. Robust Formation Controller Design

In this section, we will design the robust coordinated controller for multiple vessels in the presence of external disturbances. This section will be divided into two parts. The first part is that the upper bounded of the disturbance is known in advance. The second part is that the upper bounded is unknown. For the second part, the adaptive control law is designed to estimate the external disturbances. In this section, for a vector 𝐱∈R^𝑛×1, the absolute value of the vector is denoted as |𝐱|=[|𝑥₁|,…,|𝑥_𝑛|]^𝑇.

If the external disturbance for each vessel is bounded and the upper bounded satisfy |𝜏_𝑑𝑖|≤𝐹_𝑖, where 𝐹_𝑖∈R^3×1 is a positive constant vector, and for multiple vessels, the vector form is |𝜏_𝑑|≤𝐹. If we choose the control input as 𝜏=𝜏_eq+𝜏_sw+𝜏_en, (42) where 𝜏_eq and 𝜏_sw are the same with the definition in the front section and 𝜏_en is the control input to compensate for the external disturbances, the detailed representation is 𝜏_en=−𝐹₁sign𝑠₁⋯𝐹_3𝑛sign𝑠_3𝑛^𝑇=−𝐅. (43) For each vessel, the control input to compensate the external disturbance is as follows 𝜏^𝑖_en=−𝐅_𝑖∈R^3×1. (44)

Theorem 5. Consider 𝑛 vessels to perform the coordination formation task, the vessel model is described as (3) and (4). with the distributed coordinated formation control law (42), (27), and (43), and the conditions in Theorem 4 are satisfied, then the position tracking error, the velocity tracking error and their synchronization error, are asymptotically stable; that is, these vessels can realize the coordination formation asymptotically.

Proof. The proof procedure in this section is similar to the front section. The main difference is that the external disturbances are considered in this section.
Define the same Lyapunov function with the front section as 𝑉₂=𝑉₁+12𝐬^𝑇𝐬. (45) Taking the time derivative of the above equation as 𝑉̇₂=𝑉̇₁+𝐬^𝑇𝐬̇=𝑉̇₁+𝐬^𝑇𝐤𝐳̇₁+𝐬^𝑇𝐈_3𝑛+𝛾𝐋⊗I₃𝐳̇₂=𝐳^𝑇₁𝐳₂−𝐳^𝑇₁𝜆𝐳₁+𝐬^𝑇𝐤𝐳₂−𝜆𝐳₁+𝐬^𝑇𝐈_3𝑛+𝛾𝐋⊗I₃𝐌_𝑛⁻¹𝜂×𝜏+𝜏_𝑑−𝐌_𝑛𝜂𝜂̈_𝐷+𝐌_𝑛𝜂𝐟̇+𝐌_𝑛𝜂𝜆𝐳̇₁−𝐂_𝑛𝜂,𝜂̇+𝐃_𝑛𝜂,𝜂̇𝜂̇_𝐷−𝐟̇−𝐂_𝑛𝜂,𝜂̇+𝐃_𝑛𝜂,𝜂̇𝐳₂−𝜆𝐳₁; (46) with the control input (42), (27), and (43) yield: 𝑉̇₂=−𝐳^𝑇₁𝜆𝐳₁+𝐳^𝑇₁𝐳₂−𝐬^𝑇𝐏𝐬+𝛽sign(𝑠)+𝐬^𝑇𝐈_3𝑛+𝛾𝐋⊗I₃𝐌_𝑛⁻¹𝜂𝜏_𝑑−𝐅≤−𝐳^𝑇₁𝜆𝐳₁+𝐳^𝑇₁𝐳₂−𝐬^𝑇𝐏𝐬−^3𝑛_𝑟=1𝑝_𝑟𝛽_𝑟𝑠_𝑟+|𝐬|𝐈_3𝑛+𝛾𝐋⊗I₃𝐌_𝑛⁻¹𝜂𝜏_𝑑−𝐅≤−𝐳^𝑇₁𝜆𝐳₁+𝐳^𝑇₁𝐳₂−𝐬^𝑇𝐏𝐬−^3𝑛_𝑟=1𝑝_𝑟𝛽_𝑟𝑠_𝑟. (47) From the above inequality, we can see that the same results with the front section are achieved; then we can prove that the position tracking error, velocity tracking error, and their synchronization error are asymptotically stable according to the proof procedure in the front section.

If the external disturbances are unknown in advance, and we assume that the external disturbances vary slowly; that is 𝜏̇_𝑑=0, then we adopt the adaptive control to estimate the disturbances. Then control law also includes three parts, which is written as 𝜏=𝜏_eq+𝜏_sw+𝜏_en, (48) where 𝜏_en=−𝜏_𝑑, (49) and 𝜏_𝑑 is the estimate value of the external disturbances. The adaptive control law is chosen as follows 𝜏̇_𝑑=𝜎𝐌_𝑛𝐈_3𝑛+𝛾𝐋⊗I₃⁻¹𝐬, (50) where 𝜎 is a positive real number.

Theorem 6. Consider a group of 𝑛 vessels to perform the coordination formation task, the vessel model is described as (3) and (4). With the coordinated formation control law (42), (27), and (49), and the adaptive control law (50) and the conditions in the Theorem 4 are satisfied, then the position tracking error, the velocity tracking error, and their synchronization error are asymptotically stable; that is these vessels can realize the coordination formation asymptotically.

Proof. Define the new Lyapunov function as 𝑉₃=𝑉₂+1𝜏2𝜎^𝑇𝑑𝜏_𝑑, (51) where the definition of 𝑉₂ is same as the front one, and 𝜏_𝑑 is defined as the estimated error, the form of which is 𝜏_𝑑=𝜏_𝑑−𝜏_𝑑, where 𝜏_𝑑 is the estimated value of the disturbances.
Take the time derivative of the previous equation as 𝑉̇₃=𝑉̇₁+𝐬^𝑇𝐬̇+1𝜎𝜏^𝑇_𝑑̇𝜏_𝑑=𝐳^𝑇₁𝐳₂−𝐳^𝑇₁𝜆𝐳₁+𝐬^𝑇𝐤𝐳̇₁+𝐬^𝑇𝐈_3𝑛+𝛾𝐋⊗I₃𝐳̇₂−1𝜎𝜏^𝑇_𝑑𝜏̇_𝑑=𝐳^𝑇₁𝐳₂−𝐳^𝑇₁𝜆𝐳₁+𝐬^𝑇𝐤𝐳₂−𝜆𝐳₁+𝐬^𝑇𝐈_3𝑛+𝛾𝐋⊗I₃𝐌_𝑛⁻¹𝜂×𝜏+𝜏_𝑑−𝐌_𝑛𝜂𝜂̈_𝑑+𝐌_𝑛𝜂𝐟̇+𝐌_𝑛𝜂𝜆𝐳̇₁−𝐂_𝑛𝜂,𝜂̇+𝐃_𝑛𝜂,𝜂̇𝜂̇_𝑑−𝐟̇−𝐂_𝑛𝜂,𝜂̇+𝐃_𝑛𝜂,𝜂̇𝐳₂−𝜆𝐳₁−1𝜎𝜏^𝑇_𝑑𝜏̇_𝑑=−𝐳^𝑇₁𝜆𝐳₁+𝐳^𝑇₁𝐳₂−𝐬^𝑇𝐏(𝐬)−1𝐬+𝛽sign𝜎𝜏^𝑇_𝑑𝜏̇_𝑑−𝜎𝐌_𝑛𝜂𝐈_3𝑛+𝛾𝐋⊗I₃⁻¹𝐬=−𝐳^𝑇₁𝜆𝐳₁+𝐳^𝑇₁𝐳₂−𝐬^𝑇𝐏𝐬+𝛽sign(𝐬)=−𝐳^𝑇₁𝜆𝐳₁+𝐳^𝑇₁𝐳₂−𝐬^𝑇𝐏𝐬−^3𝑛_𝑟=1𝑝_𝑟𝛽_𝑟𝑠_𝑟. (52) From the above equality, we can see that the same results with the front section are achieved; then we can prove that the position tracking error, velocity tracking error and their synchronization error, are asymptotically stable according to the proof procedure in the front section.

5. Simulation Results

In this section, experimental simulations are carried out to evaluate the effectiveness of the proposed coordinated formation control algorithm. Five marine vessels are considered to perform the coordinated tracking task. Detailed parameters of these vessels are presented in [12]. The proposed algorithm has achieved the leaderless coordination based on the graph theory. In this experiment, the topology graph of communication among these five vessels is chosen as Figure 1.

Figure 1 

The communication topology graph of these vessels.

And the Laplacian matrix of the communication topology graph is as follows 𝐿=1000−1−130−1−10−110000−110000−11. (53)

The initial positions of the five vessels are 𝜂₁=[−80800−7𝜋/30]^𝑇, 𝜂₂=[−30800−𝜋/3]^𝑇, 𝜂₃=[−100780−𝜋/4]^𝑇, 𝜂₄=[20830−𝜋/2]^𝑇, and 𝜂₅=[−100800−𝜋/4]^𝑇, respectively. In order to evaluate the performance of the coordinated tracking, the desired formation pattern of the coordinated formation controller is described by 𝐥₁=[000]^𝑇, 𝐥₂=[0500]^𝑇, 𝐥₃=[0−500]^𝑇, 𝐥₄=[01000]^𝑇, and 𝐥₅=[0−1000]^𝑇. The desired trajectory for all the formation points is chosen as 𝜂_𝑑(𝑡)=[𝑛_𝑑𝑒_𝑑𝜓_𝑑]^𝑇, and the detailed forms are 𝑛_𝑑=𝑡, 𝑒_𝑑=800−800sin(𝑡/400), and 𝜓_𝑑=arctan(𝑒_𝑑̇/𝑛̇_𝑑).

Although the proposed coordination algorithm is robust to unknown external disturbance, the disturbance model in the simulation can be chosen as a fixed term instead of the unknown disturbance. We assume that the vessels encounter the wind, wave, and current. And the wind is assumed to be fixed direction and fixed velocity; then the disturbance of wind is a constant; the wave and current is assumed to be the sine wave with a fixed frequency at one time. So the external disturbances can be chosen as 𝜏_𝑑𝑖=10⁶∗0.05sin𝜋𝑡100+0.02,0.03sin𝜋𝑡,1000.01sin𝜋𝑡100+0.01^𝑇(𝑁). (54)

The control parameters of the coordinated formation controller are chosen as 𝜆_𝑖=diag0.05,0.05,0.2, 𝛾_𝑖=diag1,1,1, 𝐏_𝑖=diag0.3,0.3,0.15, and 𝛽_𝑖=diag1,1,0.5.

The simulation results are shown from Figure 2 to Figure 9. Figure 2 shows the movements for these vessels in the plane. The heading change curve of each vessel is shown in Figure 3. Figures 4, 5, and 6 show the surge velocity, the sway velocity, and the angular velocity of each vessel during the coordinated control process, respectively. Figures 7, 8, and 9 show the surge force, sway forces, and the heave torques applied to the vessels, respectively.

Figure 2 

The formation trajectories in horizontal plane.

Figure 3 

The heading curve of each vessel.

Figure 4 

The surge velocity curve of each vessel.

Figure 5 

The sway velocity curve of each vessel.

Figure 6 

The heave velocity curve of each vessel.

Figure 7 

The surge force of each vessel.

Figure 8 

The sway force of each vessel.

Figure 9 

The heave force of each vessel.

We can see that these vessels realized the coordinated tracking task from Figures 2 and 3. From Figures 4, 5, and 6, the velocities of these vessels achieve consensus as a whole, and the velocities cannot achieve consensus when the vessels move to the inflexion of the respective path curve. The phenomenon of the surge velocity is obvious in the experimental simulations due to the vessels move along the tangent directions of the desired trajectory. This phenomenon appears as a result of vessel speed regulated to maintain the desired formation pattern at the inflexion of the respective path curve. From Figures 7, 8, and 9, we know that the forces and torques for the group of 𝑛 vessels approach consensus, in a way. This is the result of that these vessels are uniform in these experimental simulations. With the analysis of the simulation results, we can conclude that these vessels can accomplish coordinated trajectory tracking task while keeping the desired formation. It means that the proposed coordination control algorithm is effective.

6. Conclusion

This paper has proposed a new backstepping sliding mode coordinated formation control algorithm for multiple surface vessels in the presence of external environmental disturbances. The proposed coordinated formation controller for these vessels is designed by defining a new switched function. And a new cross-coupling error is defined using the velocity error and velocity synchronization error to be applicable for the backstepping sliding mode controller. In addition, the adaptive control law is also designed to compensate for the external disturbances and then achieve the robustness. Finally, the effectiveness of the proposed coordination control algorithm is demonstrated by experimental simulations.