We investigate the problem of finite-time cooperative tracking for multiple surface vessels in the presence of external disturbances. A robust finite-time cooperative tracking algorithm based on terminal sliding-mode control is proposed for multiple surface vessels. In light of the leader-follower strategy, a virtual leader vessel is defined to provide reference point for other surface vessels to form the desired formation. Specifically, the proposed algorithm only requires the communication topology among the surface vessels to be a directed graph with a directed spanning tree. The robustness is achieved by compensating the upper bound of external disturbance in the control input, and the global finite-time stability is proved by Lyapunov stability theory. Finally, the effectiveness of the proposed finite-time cooperative tracking control algorithm is demonstrated by simulation results.

With the rapid development of marine technology, the cooperative motion control for multiple vessels has received increasing attention during the last decades. The cooperative formation of multiple vessels has become popular for military and commercial applications. For example, coast patrol requires multiple vessels to perform cooperative tracking operation while maintaining a desired formation pattern. During winter, the tanker must be escorted by icebreakers, which requires the tanker to keep a fixed distance to the icebreakers. Besides, underway replenishment is performed by coordinating one or more supply vessels and the receiving vessel such that all vessels maintain the desired relative distances and hold the equal course and forward speed. These complicated operations of multiple vessels are carried out by moving collectively as a whole formation. Compared with individual vessel, cooperative operations of multiple vessels have higher operational efficiency, larger serve areas, better fault-tolerant property, and stronger robustness [

With respect to the cooperative control issues, formation control as a special case, a large number of studies have been widely reported in existing publications. The formation strategies mainly include leader-follower strategy, virtual structures strategy, and behavioral strategy [

When multiple agents are to be coordinated to perform complicated task, information exchange between them is a necessary condition. In order to accomplish cooperative tracking operations, both position and velocity information need to be shared. In practice, the communication topology among these agents might be directed as a result of the external disturbances. That means one agent might receive the information from neighbors but cannot send his own information to the neighbors. Under directed communication topologies, Ren had studied the consensus tracking algorithm for multiagent with single-integrator kinematics [

For marine control, finite-time control is quite desirable when considering the huge inertia of the surface vessels. Compared to asymptotic stability control, the convergence rate of finite-time control is faster, and the system with finite-time convergence has better disturbance rejection properties and robustness against uncertainties [

In this paper, the problem of robust cooperative tracking control for multiple surface vessels is considered, and the communication topology among these surface vessels is directed graph which has a directed spanning tree. The finite-time cooperative tracking control algorithm is designed using the terminal sliding-mode control method, and the desired formation configuration is achieved using the virtual leader-follower strategy. The rest of this paper is organized as follows. In Section

With respect to the surface vessels, only the motions on the surge, sway, and yaw are considered. If we define the generalized position and orientation which are expressed in the inertial reference frame as

In order to design the tracking controller for surface vessels in the sequel, the expression of vessel model can be transformed as

Inertia mass matrix

where

which means it is skew symmetric;

In order to model the information transmit relationship between the group of surface vessels, several basic concepts of directed graph are given here [

Let one vertex represent one vessel in the group and the edges represent information exchange links by available directed communication; then the communication relationship between the group of vessels is described by a directed graph. Specially, in this paper we consider the communication topology as a directed graph with a directed spanning tree; that is, the digraph has at least one vertex with a directed path to all other vertexes.

Define the Kronecker product of two matrices

Given a variable vector

Let the Laplacian matrix of a directed graph G be defined as

For the non-Lipschitz system

In this section, we will design the finite-time cooperative tracking controller based on terminal sliding-mode control. Here we consider

The desired formation pattern among the surface vessels is established based on the leader-follower strategy. The leader vessel is virtual and it is labeled by 0. Then the communication topology among all the vessels (include the virtual leader) is described by a directed graph

Consider the following.

We assume that the position of the virtual leader vessel is denoted as

The virtual vessel is free to external disturbances, so the leader vessel model in the inertial reference frame can be written as

We assume that the position of the virtual vessel and its velocity are available to its neighbors only and the control force input

Define the relative position error of the formation reference point for the

Consider the vessel with the nonlinear model as in (

The Laplacian matrix of the communication graph among these surface vessels is

If we define

The control input vector of all these vessels can be written as

Substituting the control input (

Let

Substituting the control input (

However, on this new terminal sliding-mode surface, that is,

Define the Lyapunov function as

In this section, simulation results are presented to evaluate the performance of the proposed finite-time cooperative formation control algorithm. We consider four surface vessels to perform the cooperative tracking task. For detailed system parameters matrices of vessel mathematic model, we can refer to the literature [

The information exchange topology among all the vessels.

From the above information exchange topology graph, we can know that the adjacent matrix of the graph is as follows:

Then the Laplacian matrix of the information exchange topology graph of the practical vessels can be written as

The initial conditions are

With the proposed finite-time cooperative tracking control law, the dynamic trajectory of each vessel is shown in Figure

The dynamic trajectory of each vessel.

The heading consensus for these vessels.

The surge velocities consensus of the vessels.

The sway velocities consensus of the vessels.

The angular velocities consensus of the vessels.

Based on the above simulation results, we can know that the cooperative tracking task of multiple surface vessels is achieved by the proposed finite-time cooperative control algorithm. That means that these surface vessels can form the desired formation and perform the cooperative tracking as a whole formation in finite time. Overall, the proposed finite-time cooperative tracking control algorithm for multiple surface vessels is effective and satisfactory.

In this paper, the finite-time cooperative tracking control scheme for multiple surface vessels has been proposed. The cooperative formation is achieved by defining the formation reference point of each vessel based on the virtual leader-follower strategy. Furthermore, the communication topology among these vessels (include the virtual leader) is only the directed graph with a directed spanning tree. The cooperative tracking control scheme is designed using the terminal sliding-mode control approach which requires defining a nonlinear sliding variable function. In addition, the robustness against the external disturbances is achieved by compensating for the upper bound in the control input. It is proved that the cooperative tracking with desired formation can be achieved in finite time. Finally, the effectiveness of the proposed finite-time cooperative tracking control algorithm is validated by the simulation results.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to acknowledge the support of the National Technology Momentous Special Program of China (2011ZX05027-002), High Technology of ships Research Program of China (Z12SJENA0011), the National Natural Science Fund of China (NSFC51209056), and the Basic Research Business Particular Item Fund of the Central High School of China (HEUCF041405).