# A Staged Finite-Time Control Strategy for Formation of Underactuated Unmanned Surface Vehicles

Authors: Hui Ye; Xiaofei Yang; Chunxiao Ge; Zhaoping Du
Journal: Complexity
 (2021)

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## Abstract

The formation control issue for a group of underactuated unmanned surface vehicles (USVs) is discussed in the paper, and a staged finite-time control strategy for the USVs is proposed. Firstly, we try to steer each USV to its own starting point in the formation for a limited time, under the initial condition that each of these vehicles is parked at random. To deal with the nonholonomic behavior of the system, the dynamics of the USV is transformed into cascade systems. Then, the finite-time controller is designed for each vehicle based on homogeneity theory. After each USV reaches its own starting point with desired orientation, the model of the vehicle is decomposed into two subsystems under the Serret-Frenet frame. In order to maintain the formation pattern, two finite-time distributed controllers are developed for the surge subsystem and the yaw subsystem, respectively. The settling time for the staged control strategy is limited. Numerical simulations are carried out to illustrate the effectiveness of the proposed formation control strategy.


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## Body

## 1. Introduction

Over the last decade, a great attention has been paid by scholars on unmanned surface vehicles (USVs), which can be used to perform various tasks, such as marine surveillance, oil spill collection, environmental monitoring, bathymetric mapping, and marine floor investigation [1]. These tasks are usually complexity and onerous, which are difficult to do by a single vehicle. Multivehicle cooperation is the inevitable choice, which can not only accomplish the tasks efficiently but also ensure personnel safety, reduce system cost, and provide redundancy against individual failure. Typical cooperative control issues of multiagent systems include consensus, distributed tracking, formation control, containment control, swarming, and aggregation. In this paper, the formation control of multiple USVs is considered.There are two main difficulties in the formation control problem of underactuated USVs, one of which exists in the motion control of each single vehicle. Most existing USVs are typical underactuated systems, whose number of actuators is fewer than their degrees of freedom (DOF). The trajectory tracking control of the underactuated USV is studied in [2, 3], and the virtual target is adopted to generate the reference trajectory. The path following of the underactuated USV is usually addressed by introducing the Serret-Frenet frame and global coordinate transformation [4, 5]. Then, the underactuated problem is converted to a virtual fully actuated problem by designing the virtual control law. For the stabilization of the underactuated USVs, nonsmooth or time-varying cascaded design is usually proposed due to the nonholonomic constraints [6]. Moreover, finite-time controllers are developed in [7, 8] via adding one power integrator technique and terminal sliding mode method, respectively.Another difficulty is to maintain formation pattern of multiple USVs, which will encounter issues of communication and coordination. From a review of the literature, there are mainly four methods used for the formation control of multiple vehicles, including virtual structure method [9], graph theory method [10], behavioral based approach [11], and leader-follower strategy [12, 13]. Among these methods, the leader-follower strategy is most preferred in practice for the formation of USVs because of its high reliability and easily implementation [14]. A robust adaptive formation controller is proposed in [15] for the formation of USVs with uncertainties and ocean disturbances, where neural networks and dynamic surface control technique are used to capture them. In [16], a parameter estimation- and upper bound estimation-based sliding mode control scheme is presented to deal with the model uncertainties and environmental disturbances. The control constraint issues in the formation of underactuated USVs are also considered by many scholars. In [17], a formation tracking controller is developed by using generalized saturation functions to reduce the risk of actuator saturation. The receding horizon control algorithm is adopted in [18] for the formation tracking of a feet of underactuated autonomous underwater vehicles (AUVs) with constrained inputs. In [19], auxiliary dynamic systems and disturbance observers are designed to deal with input saturations and disturbances respectively, and a robust controller is developed for time-varying formation control of underactuated AUVs. The fault tolerant control problem in the leader-follower formation of USVs is considered in [20], while both parametric and nonparametric system uncertainties are discussed.Another popular method for USV formation control is based on the graph theory, which is proposed to settle the consensus problem of multiagents. The literature [21] considers the cooperative control of a group of underactuated USVs using the relative information between neighbor’s information with the aid of results from graph theory and Lyapunov techniques. In [22], a distributed maneuvering controller of multiple USVs is designed in the presence of unknown state constraints. An RNN-based command governor is provided between the kinematic controller and the kinetics controller. A cooperative path maneuvering controller is designed based on a line of sight guidance scheme in [23], where the modular adaptive control method is employed to identify uncertain kinematics and unknown kinetics of each vehicle.Currently, finite-time formation control of multiple USVs become an active research area, since a lot of works have been carried out in both the finite-time control of single USV [24–26] and the finite-time consensus of multiagents [27–29]. In [30], the finite-time formation control of underactuated USVs is investigated with consideration of model uncertainty and environmental disturbance. The reference trajectory of each USV is given first according to the desired geometric pattern. Then, a nonlinear controller with fractional power feedback is designed for each USV, to make sure each vehicle follows its own trajectory in finite time. In [31], the terminal sliding mode approach is used to deal with the finite-time formation of underactuated ships. Specifically, a generalized error which contains topology information of communication is defined so that the method based on graph theory can be used for the nonlinear system.Motivated by the aforementioned works, the finite-time control for single USV and the finite-time cooperative control for multiple USVs are integrated is this paper. A staged finite-time control scheme is proposed for the formation of multiple underactuated USVs. Our major contributions are listed as follows:(1)
To deal with the underactuated nature and achieve the goal of finite-time convergence, we mainly separate the formation control process into two stages. In the first stage, we try to steer each USV to its own starting point in the formation configuration for a limited time. In the second stage, the formation pattern is maintained by using two finite-time cooperative controllers. As a consequence, the resulting control scheme can guarantee finite-time convergence.(2)
In the first stage, the positioning problem of every single underactuated USV is regard as a finite-time stability problem of the homogeneous system. Homogeneous theory is applied in the control design, to make each vehicle get to its own position in the formation pattern.(3)
In the second stage, the model of each USV can be decoupled into two subsystems with a series of integrators: the surge subsystem and the yaw subsystem. Taking communication link topology into account, two finite-time cooperative controllers are designed based on graph theory to maintain the formation pattern.The remainders of the paper are organized as follows. In Section2, preliminaries and problem formulation are introduced. Section 3 presents the main results on the staged control algorithm design, where finite-time control for single USV is designed in Section 3.1 and finite-time cooperative control for formation maintenance is given in Section 3.2. In order to illustrate the effectiveness of the proposed control laws, numerical simulation results are presented in Section 4. Finally, some conclusion remarks are drawn in Section 5.


## 2. Preliminary

In this section, theories of finite-time stability, homogeneity, and algebraic graph are introduced briefly, and some lemmas are given. Then, the kinematics and dynamics of the underactuated USV are provided, and the task of formation control for multiple USVs is summarized.


### 2.1. Finite-Time Stability and Homogeneity

Both asymptotic stability and finite-time convergence of the closed-loop system are required in our design, which leads to the conception of the finite-time stability.Definition 1.
Consider the system(1)x˙=fx,f0=0,x∈ℝn,where f:ℝn⟶ℝn is continuous. Assume that, for every initial condition x0=x0∈ℝn, system (1) possesses a unique solution xt=ξt,x0 in forward time which is defined on 0,∞. The origin is said to be a finite-time stable equilibrium of (1) if there exists an open neighborhood U of the origin and a function tf:U⟶0,∞ called the settling time function such that the following statements hold:(1)
For everyx0∈U/0, ξt,x0∈U/0,for0≤t<tfx0,ξt,x0=0,fort≥tfx0.(2)
limt⟶tfx0ξt,x0=0.
The origin is said to be a globally finite-time stable equilibrium if it is a finite-time stable equilibrium andU=ℝn.
In general, there are three techniques which can be used to achieve finite-time stability for the feedback stabilization, including homogeneous theory, terminal sliding mode method, and adding one power integrator approach [32]. In this paper, nonsmooth feedback finite-time controllers are designed with the help of the homogeneous system.Definition 2 (see [32]).
Consider the dilation of the formδεγ1,…,γnx1,…,xn=εγ1x1,…,εγnxn, where x1,…,xn are suitable coordinates on ℝn and γ1,…,γn are positive real numbers. A vector field is homogeneous of degree k if and only if the ith component fi is homogeneous of degree k+γi. System (1) is said to be a homogeneous system if its vector field f is homogeneous.
The connection between the finite-time stability and the homogeneous system is concluded in the following two lemmas.Lemma 1 (see [32]).
For system (1), suppose there exists an open neighborhood U of the origin, a C1 positive definite function Vx, and real number ε>0 and 0<α<1, such that V˙x+εVαx is negative semidefinite on U. Then, the origin is a finite-time stable equilibrium of the system.Lemma 2 (see [32]).
Suppose that system (1) is homogeneous of degree k with respect to γ1,…,γn, the origin is a finite-time stable equilibrium of (1) if and only if the origin is an asymptotically stable equilibrium of (1) and k<0.
Consider the following chain of integrators:(2)x˙1=x2,x1t0=x10,x˙2=x3,x2t0=x20,⋮⋮x˙n=u,xnt0=xn0.System (2) is finite-time convergent to the origin if the control input is designed as(3)u=−k1x1α1−k2x2α2−⋯−knxnαn,where the exponents αi∈0,1,i=1,…,n−1, are selected satisfying(4)αi=αi+1αi+22αi+2−αi+1,αn+1=1,αn=α,0<α<1.The itemxiαi is defined as xiαisgnxi, where sgn· is the sign function. The gains ki,i=1,…,n, are selected such that the following matrix is Huiwitz:(5)Ax=010…0001…0⋮⋮⋮⋱⋮000…1−k1−k2−k3…−kn.Define the Lyapunov function candidateVζ=ζTPζ with ζ=x1α1,x2α2,…,xnαnT. P and Q are symmetric positive definite matrices satisfying(6)PAx+AxTP=−Q.Lemma 3 (see [33]).
Consider system (2) together with (3). The state x=x1,x2,…,xnT converges to the origin for a finite time. Moreover, the upper bound of the settling time can be estimated as(7)tfx0≤αVζ01−α/α1−αρ,for all x0=x10,x20,…,xn0T in some open neighborhood U of the origin, where ρ=λminQ/λmaxP. λminQ>0 is the minimum eigenvalue of the matrix Q, and λmaxP>0 is the maximum eigenvalue of the matrix P.


### 2.2. Graph Theory

The communication links topology among the group of NSVs can be represented by a graphG=V,E,A, where V=υ1,υ2,…,υm is the set of vehicles, E represents the set of links between the vehicles (also called edge set), and A is the weighted coefficients of edge set E. In the directed graph, i,j∈E represents the jth vehicle can communicate with ith vehicle, but not conversely. The ith vehicle is called the parent node, and the jth vehicle is also called the child node of the ith vehicle, and they are neighbors. In the undirected graph, i,j∈E denotes the information can be exchanged between the two vehicles. All vehicles are connected with a single undirected path. However, in the directed graph, each vehicle has only one parent node, except for one vehicle, which is called the root. If m vehicles are in the scenario, then the associated adjacency matrix of G can be defined as A=aijm×m∈ℝm×m. aij>0 if and only if there is a link from jth vehicle to ith vehicle; otherwise, aij=0. The Laplacian matrix L=lijm×m∈ℝm×m associated with the matrix A is defined by lii=∑j=1maij and lij=−aij, when i≠j. The Laplacian matrix can be written as L=D−A, where D is the degree matrix [34].Consider a virtual leaderυ0 moving with the group centroid. Define the augmented graph as G¯=V¯,E¯,A¯, where V¯=υ0,υ1,…,υm and E¯=⊆V×V are the new node set and edge set, respectively. Diagonal matrix B=diaga10,a20,…,am0 is called the leader adjacency matrix, where ai0>0 if follower i is connected to the leader; otherwise, ai0=0. In this work, the communication topology verifies the following assumption.Assumption 1.
The graphG associated with the follower is undirected, and in the graph G¯ the leader has directed path to all followers.Lemma 4 (see [35]).
Under Assumption1, L+B is a symmetric positive definite matrix.


### 2.3. Mathematical Model of Underactuated USV

Two reference coordinate systems are defined to describe the model of the USV, where XEOEYE is the earth-fixed inertial frame and XbObYb is the body-fixed frame, as shown in Figure 1. The model state of the USV in horizontal plane is given by x,y,ψ,u,v,r, where x, y, and ψ denote the position and orientation of the vehicle in the earth-fixed frame, u, v, and r denote the surge, sway, and yaw velocity of the vehicle in the body-fixed frame, respectively.Figure 1
The earth-fixed inertial and the body-fixed frame.Based on the above definition, the kinematic model of the USV can be described as(8)x˙=ucosψ−vsinψ,y˙=usinψ+vcosψ,ψ˙=r.And the dynamic model of the USV can be given by(9)muu˙−mvvr+duu=τu,mvv˙+muur+dvv=0,mrr˙−muvuv+drr=τr,where du,dv,anddr are the hydrodynamic damping of the vehicle and mu,mv,andmr are the inertia of the vehicle including added masses in surge, sway, and yaw. Besides, muv is the hydrodynamic added mass and τu and τr are the surge force and yaw moment, respectively.It can be seen from (9) that the USV can only get the surge force and yaw moment from the thrusters and the rudder in the stern, and there is no sway force from the vessel thruster, so the vehicle is a typical underactuated system.


### 2.4. Formation of Multiple USVs

Consider a group ofm USVs, whose mathematical models are expressed as (8) and (9). For clarity, the subscript “i” (i=1,…,m) is used to distinguish one from another. Such as, the state of the USV i is denoted by xi,yi,ψi,ui,vi,ri. Assume that all the USVs are scattered randomly in the water at any initial states. The goal of this work is to steer these vehicles to achieve the prescribed formation pattern along a given straight line or arc path with a desired speed. Moreover, the settling time is required to be finite. Without loss of generality, the earth-fixed frame is defined with its origin located on the center of the formation at starting point. The XE-axis coincides with the tangent of path with the positive axis pointing to the motion direction of the group, as shown in Figure 2.Figure 2
The reference frame and the formation pattern of the NSVs.A feasible desired trajectory can be described by the statesx0,y0,ψ0,u0,v0,r0 of the group centroid which is moving along the path. The object of the control system can be expressed as follows.In the first stage:(10)limt⟶tf1xit−x0tf1=Δxi,limt⟶tf1yit−y0tf1=Δyi,limt⟶tf1ψit−ψ0tf1=Δψi,limt⟶tf1uit=0,limt⟶tf1vit=0,fori=1,…,m,limt⟶tf1rit=0.In the second stage:(11)limt⟶tf2sit−s0t=Δsi,limt⟶tf2dit−d0t=Δdi,fori=1,…,m,whereΔxi,Δyi,Δψi are the relative position and orientation of USV i to the group centroid, si,di are longitudinal and transversal displacements of USV i with respect to the path, and Δsi,Δdi are the longitudinal and transversal distances between USV i and the group centroid.


## 2.1. Finite-Time Stability and Homogeneity

Both asymptotic stability and finite-time convergence of the closed-loop system are required in our design, which leads to the conception of the finite-time stability.Definition 1.
Consider the system(1)x˙=fx,f0=0,x∈ℝn,where f:ℝn⟶ℝn is continuous. Assume that, for every initial condition x0=x0∈ℝn, system (1) possesses a unique solution xt=ξt,x0 in forward time which is defined on 0,∞. The origin is said to be a finite-time stable equilibrium of (1) if there exists an open neighborhood U of the origin and a function tf:U⟶0,∞ called the settling time function such that the following statements hold:(1)
For everyx0∈U/0, ξt,x0∈U/0,for0≤t<tfx0,ξt,x0=0,fort≥tfx0.(2)
limt⟶tfx0ξt,x0=0.
The origin is said to be a globally finite-time stable equilibrium if it is a finite-time stable equilibrium andU=ℝn.
In general, there are three techniques which can be used to achieve finite-time stability for the feedback stabilization, including homogeneous theory, terminal sliding mode method, and adding one power integrator approach [32]. In this paper, nonsmooth feedback finite-time controllers are designed with the help of the homogeneous system.Definition 2 (see [32]).
Consider the dilation of the formδεγ1,…,γnx1,…,xn=εγ1x1,…,εγnxn, where x1,…,xn are suitable coordinates on ℝn and γ1,…,γn are positive real numbers. A vector field is homogeneous of degree k if and only if the ith component fi is homogeneous of degree k+γi. System (1) is said to be a homogeneous system if its vector field f is homogeneous.
The connection between the finite-time stability and the homogeneous system is concluded in the following two lemmas.Lemma 1 (see [32]).
For system (1), suppose there exists an open neighborhood U of the origin, a C1 positive definite function Vx, and real number ε>0 and 0<α<1, such that V˙x+εVαx is negative semidefinite on U. Then, the origin is a finite-time stable equilibrium of the system.Lemma 2 (see [32]).
Suppose that system (1) is homogeneous of degree k with respect to γ1,…,γn, the origin is a finite-time stable equilibrium of (1) if and only if the origin is an asymptotically stable equilibrium of (1) and k<0.
Consider the following chain of integrators:(2)x˙1=x2,x1t0=x10,x˙2=x3,x2t0=x20,⋮⋮x˙n=u,xnt0=xn0.System (2) is finite-time convergent to the origin if the control input is designed as(3)u=−k1x1α1−k2x2α2−⋯−knxnαn,where the exponents αi∈0,1,i=1,…,n−1, are selected satisfying(4)αi=αi+1αi+22αi+2−αi+1,αn+1=1,αn=α,0<α<1.The itemxiαi is defined as xiαisgnxi, where sgn· is the sign function. The gains ki,i=1,…,n, are selected such that the following matrix is Huiwitz:(5)Ax=010…0001…0⋮⋮⋮⋱⋮000…1−k1−k2−k3…−kn.Define the Lyapunov function candidateVζ=ζTPζ with ζ=x1α1,x2α2,…,xnαnT. P and Q are symmetric positive definite matrices satisfying(6)PAx+AxTP=−Q.Lemma 3 (see [33]).
Consider system (2) together with (3). The state x=x1,x2,…,xnT converges to the origin for a finite time. Moreover, the upper bound of the settling time can be estimated as(7)tfx0≤αVζ01−α/α1−αρ,for all x0=x10,x20,…,xn0T in some open neighborhood U of the origin, where ρ=λminQ/λmaxP. λminQ>0 is the minimum eigenvalue of the matrix Q, and λmaxP>0 is the maximum eigenvalue of the matrix P.


## 2.2. Graph Theory

The communication links topology among the group of NSVs can be represented by a graphG=V,E,A, where V=υ1,υ2,…,υm is the set of vehicles, E represents the set of links between the vehicles (also called edge set), and A is the weighted coefficients of edge set E. In the directed graph, i,j∈E represents the jth vehicle can communicate with ith vehicle, but not conversely. The ith vehicle is called the parent node, and the jth vehicle is also called the child node of the ith vehicle, and they are neighbors. In the undirected graph, i,j∈E denotes the information can be exchanged between the two vehicles. All vehicles are connected with a single undirected path. However, in the directed graph, each vehicle has only one parent node, except for one vehicle, which is called the root. If m vehicles are in the scenario, then the associated adjacency matrix of G can be defined as A=aijm×m∈ℝm×m. aij>0 if and only if there is a link from jth vehicle to ith vehicle; otherwise, aij=0. The Laplacian matrix L=lijm×m∈ℝm×m associated with the matrix A is defined by lii=∑j=1maij and lij=−aij, when i≠j. The Laplacian matrix can be written as L=D−A, where D is the degree matrix [34].Consider a virtual leaderυ0 moving with the group centroid. Define the augmented graph as G¯=V¯,E¯,A¯, where V¯=υ0,υ1,…,υm and E¯=⊆V×V are the new node set and edge set, respectively. Diagonal matrix B=diaga10,a20,…,am0 is called the leader adjacency matrix, where ai0>0 if follower i is connected to the leader; otherwise, ai0=0. In this work, the communication topology verifies the following assumption.Assumption 1.
The graphG associated with the follower is undirected, and in the graph G¯ the leader has directed path to all followers.Lemma 4 (see [35]).
Under Assumption1, L+B is a symmetric positive definite matrix.


## 2.3. Mathematical Model of Underactuated USV

Two reference coordinate systems are defined to describe the model of the USV, where XEOEYE is the earth-fixed inertial frame and XbObYb is the body-fixed frame, as shown in Figure 1. The model state of the USV in horizontal plane is given by x,y,ψ,u,v,r, where x, y, and ψ denote the position and orientation of the vehicle in the earth-fixed frame, u, v, and r denote the surge, sway, and yaw velocity of the vehicle in the body-fixed frame, respectively.Figure 1
The earth-fixed inertial and the body-fixed frame.Based on the above definition, the kinematic model of the USV can be described as(8)x˙=ucosψ−vsinψ,y˙=usinψ+vcosψ,ψ˙=r.And the dynamic model of the USV can be given by(9)muu˙−mvvr+duu=τu,mvv˙+muur+dvv=0,mrr˙−muvuv+drr=τr,where du,dv,anddr are the hydrodynamic damping of the vehicle and mu,mv,andmr are the inertia of the vehicle including added masses in surge, sway, and yaw. Besides, muv is the hydrodynamic added mass and τu and τr are the surge force and yaw moment, respectively.It can be seen from (9) that the USV can only get the surge force and yaw moment from the thrusters and the rudder in the stern, and there is no sway force from the vessel thruster, so the vehicle is a typical underactuated system.


## 2.4. Formation of Multiple USVs

Consider a group ofm USVs, whose mathematical models are expressed as (8) and (9). For clarity, the subscript “i” (i=1,…,m) is used to distinguish one from another. Such as, the state of the USV i is denoted by xi,yi,ψi,ui,vi,ri. Assume that all the USVs are scattered randomly in the water at any initial states. The goal of this work is to steer these vehicles to achieve the prescribed formation pattern along a given straight line or arc path with a desired speed. Moreover, the settling time is required to be finite. Without loss of generality, the earth-fixed frame is defined with its origin located on the center of the formation at starting point. The XE-axis coincides with the tangent of path with the positive axis pointing to the motion direction of the group, as shown in Figure 2.Figure 2
The reference frame and the formation pattern of the NSVs.A feasible desired trajectory can be described by the statesx0,y0,ψ0,u0,v0,r0 of the group centroid which is moving along the path. The object of the control system can be expressed as follows.In the first stage:(10)limt⟶tf1xit−x0tf1=Δxi,limt⟶tf1yit−y0tf1=Δyi,limt⟶tf1ψit−ψ0tf1=Δψi,limt⟶tf1uit=0,limt⟶tf1vit=0,fori=1,…,m,limt⟶tf1rit=0.In the second stage:(11)limt⟶tf2sit−s0t=Δsi,limt⟶tf2dit−d0t=Δdi,fori=1,…,m,whereΔxi,Δyi,Δψi are the relative position and orientation of USV i to the group centroid, si,di are longitudinal and transversal displacements of USV i with respect to the path, and Δsi,Δdi are the longitudinal and transversal distances between USV i and the group centroid.


## 3. Staged Control Design for the Formation of USVs

For formation control of the USVs, vehicles should be forced to achieve the desired formation firstly and then maintain the geometric configuration. Consequently, the proposed control strategy includes two stages. In the first stage, each vehicle is steered to its own position in the formation pattern within a finite time. After that, the dynamics of the underactuated USV can be transformed into two integrator systems. In the second stage, two finite-time cooperative controllers are employed to maintain the formation shape and guarantee the stabilities of the systems.


### 3.1. Finite-Time Control for Single USV

In this stage, each vehicle is controlled separately to reach its initial position in the formation. We hope that this process could be completed in a limited time. For the underactuated USVi, its desired pose vector to the group centroid is Δxi,Δyi,Δψi. We can transform the set point tracking problem of USV i to the stabilization problem by defining the following new state variables:(12)x¯i=xi−Δxi,y¯i=yi−Δyi,ψ¯i=ψi−Δψi.Then, the kinematic equation of USVi can be rewritten as(13)x¯.i=uicosψ¯i−visinψ¯i,y¯.i=uisinψ¯i+vicosψ¯i,ψ¯.i=ri.Since the mathematical model of the underactuated USV is a typical nonholonomic system, the following global coordinate transformation is employed before control design [36]:(14)z1i=x¯icosψ¯i+y¯isinψ¯i,z2i=−x¯isinψ¯i+y¯icosψ¯i+mvdvvi,z3i=ψ¯i,z4i=−mudvui−z1i,z5i=vi,z6i=ri,τ1i=dudv−1ui−z2iz6i−τuidv,τ2i=muvmruivi−drmrri+τrimr,where τ1i,τ2i represents the new input transformation pattern. Thus, system (13) together with (9) can be transformed into(15)z˙2i=z4iz6i,z˙3i=z6i,z˙4i=τ1i,z˙6i=τ2i,(16)z˙1i=−dvmuz1i−dvmuz4i+z2iz6i−mvdvz5iz6i,z˙5i=−dvmvz5i+dvmvz6iz1i+z4i.Remark 1 (see [36]).
The system stabilization problem in (13) and (9) is equal to systems (15) and (16). If system (16) is global asymptotic convergent within finite time, the overall system of (15) and (16) is global asymptotic convergent.Remark 2.
Asτui,τri and τ1i,τ2i can be converted to each other, just τ1i,τ2i need to be designed to make variables z1i,z2i,z3i,z4i,z5i,andz6i asymptotically converge to zero, which means that the tuple xi,yi,ψi,ui,vi,ri converges to Δxi,Δyi,Δψi,0,0,0. That is to say, the USV has arrived to its initial place in the formation pattern with desired orientation.
In view of the above remarks, our objective has shifted from set point tracking for each USV to simply stabilizing system (15). Note that this system does not satisfy Brockett’s necessary condition. Hence, we divide system (15) into two cascade subsystems and try to provide a discontinuous control law. Suppose z6i is not zero at the initial time, such as z6i0≠0. The first cascade subsystem is given as the follows:(17)z˙2i=z4i′,z˙4i′=τ1i′,where z4i′=z6i0z4iandτ1i′=z6i0τ1i. The finite-time control law is designed as(18)τ1i′=−k1iz2iα1−k2iz4iα2,τ2i=0,fort∈0,t1i,where the exponents α1,α2 are selected as α1=α2/2−α2,0<α2<1. The gains k1iandk2i are selected to satisfy(19)P1i01−k1i−k2i+01−k1i−k2iTP1i=−Q1i.
Thus,t1i is the settling time which can be calculated by(20)t1i=α2ζ1iT0P1iζ1i01−α2/α21−α2ρ1i,where ζ1i0=z2i0,z4i0T and ρ1i=λminQ1i/λmaxP1i. According to Lemma 3, the feedback control law (18) presents a homogeneous closed-loop system with negative degree of homogeneity. Thus, the states z2i and z4i of system (17) reach to zero in finite time; then, τ2i would not affect system (17) any more. Moreover, τ2i can be designed independently for the second cascade system:(21)z˙3i=z6i,z˙6i=τ2i.
After timet1i, we have(22)τ1i′=0,τ2i=−k3iz3iα3−k4iz6iα4,fort∈t1i,t1i+t2i,where the exponents α3andα4 are selected as α3=α4/2−α4,0<α4<1. The gains k3iandk4i are selected to satisfy(23)P2i01−k3i−k4i+01−k3i−k4iTP2i=−Q2i.
Thus,t2i is the settling time which can be calculated by(24)t2i=α4ζ2iTt1iP2iζ2it1i1−α4/α41−α4ρ2i,where ζ1it1i=z3it1i,z6it1iT and ρ2i=λminQ2i/λmaxP2i. By using Lemma 3 again, the control law (22) makes the states z3i and z6i of system (21) converge to zero in finite time.
After that the states of (15) converge to zero, system (16) is also asymptotically stabilized according to Remark 1. Actually, system (16) is simplified to first-order systems, whose settling time periods are 3mu/dv and 3mv/dv. We let(25)t3i=max3mudv,3mvdv.
So, the final convergence time of all USVs is calculated as(26)tf1=maxit1i+t2i+t3i.
That is to say, all USVs can arrive to their positions in the formation pattern after timetf1.


### 3.2. Cooperative Control for Formation Maintenance

The goal of the second stage is to maintain the formation pattern by means of finite-time distributed control. It is assumed that only local distributed state information would be utilized by each vehicle to make its own surge velocity and yaw angular consistent with others. Consider a virtual leader moving with the group centroid, labeled 0. The communication graphG¯ among the n vehicles and the virtual leader satisfies Assumption 1.After the first stage, the group of USVs has come into a prespecified geometric pattern with desired yaw angular. A reference pathΓ0 is previously specified for the centroid of the group which must be followed by the group of the USVs, as shown in Figure 3. The kinematic model of each vehicle can be rewritten according to the Serret–Frenet frame. Suppose that there is a virtual S-F frame XtO0Xn moving along the path Γ0 for the virtual vehicle 0. The kinematics of vehicle i expressed in new state coordinates is [37](27)d˙i=uisinφi+vicosφi,φ˙i=ri−cs˙i,s˙i=11−cdiuicosφi−visinφi,where φi=ψi−Δψi−ψ0 and c is the curvature of the path which is constant for the straight line or arc. Since each vehicle is on its own path and tangent to it at the initial moment, we assume that di≈Δdi,vi≈0 and sinφi≈φi,cosφi≈1. We define(28)r¯i=ri−cs˙i,hi=11−cdi,τ3i=1muτui−duui,τ4i=1mrτri−drri−chiτ3i.Figure 3
The virtual leader and the Serret–Frenet frame.Then, the dynamics of vehiclei can be decoupled into two subsystems: the surge subsystem(29)s˙i=hiui,u˙i=τ3i,and the yaw subsystem(30)d˙i=uiφi,φ˙i=r¯i,r¯.i=τ4i.Suppose thathi,i=1,…m are not zero. The cascade subsystem (29) can be written as(31)s˙i=u1i′,u˙1i′=τ3i′,where ui′=hiuiandτ3i′=hiτ3i. The finite-time distributed protocols based on the relative position and relative velocity are designed as(32)τ3i′=−k5i∑j=1maijsi−sj−Δsi−Δsj+ai0si−Δsi−s0α5−k6i∑j=0maijui′−uj′+ai0ui′−u0′α6,fort∈tf1,∞,where the exponents α5andα6 are selected as α5=α6/2−α6,0<α6<1. The gains k5iandk6i are selected such that s2+k6is+k5i are Hurwitz polynomials.Similarly,ui,i=1,…m are assumed to change slowly and not zero. The cascade system (30) can be written as(33)d˙i=φi′,φ˙i′=r¯i′,r¯.i′=τ4i′,where φi′=uiφ′,r¯i′=uir¯i,andτ4i′=uiτ4i. The finite-time distributed protocols based on the relative position and relative velocity are designed as(34)τ4i′=−k7i∑j=1maijdi−dj−Δdi−Δdj+ai0di−Δdi−d0α7−k8i∑j=1maijφi′−φj′+ai0φi′−φ0′α8−k9i∑j=1maijr¯i′−r¯j′+ai0r¯i′−r¯0′α9,fort∈tf1,∞,where α7,α8,andα9 are selected as α7=α8α9/2α9−α8andα8=α9/2−α9,0<α9<1. The gains k7i,k8i,andk9i are selected such that s3+k9is2+k8is+k7i are Hurwitz polynomials.The main result is given by the following theorem.Theorem 1.
Under Assumption1, protocols (32) and (34) can guide system (31) and (33) to realize finite-time consensus, respectively.Proof.
For systems (33) and (34), define the error states as(35)ei,1=∑j=1maijdi−dj−Δdi−Δdj+ai0di−Δdi−d0,ei,2=∑j=1maijφi′−φj′+ai0φi′−φ0′,ei,3=∑j=1maijr¯i′−r¯j′+ai0r¯i′−r¯0′.
Denotinge1=e1,1,e2,1,…em,1T,e2=e1,2,e2,2,…em,2T,ande3=e1,3,e2,3,…em,3T, it is straightforward to express the error dynamics as(36)e˙1=e2,e˙2=e3,e˙3=L+B−K7e1a7−K8e2a8−K9e3a9,where K7=diagk71,k72,…,k7m,K8=diagk81,k82,…,k8m,K9=diagk91,k92,…,k9m, and ejα,j=1,2,3, denotes e1,jα,e2,jα,…,em,jαT.
Let(37)Ae=0I000I−K7−K8−K9,M=IIL+B,A¯e=MAe=0I000I−L+BK7−L+BK8−L+BK9.
Consider a Lyapunov function candidateVe=eTPee, where e=e1T,e2T,e3TTandPe=PeT>0. It is obviously that Ae is a Hurwitz satisfying the Lyapunov equation PeAe+AeTPe=−Qe, where Qe=QeT>0. Since M is a symmetric positive definite matrix, the Lyapunov equation P¯eA¯e+A¯eTP¯e=−Qe holds with P¯e=PeM−1>0, which means that V¯e=eTP¯ee is a Lyapunov function for the linear system e˙=A¯ee.
From the proof of Theorem1 in [33], we can see that Vζe=ζeTP¯eζe is a Lyapunov function for system (36), where ζe=e1α7T,e2α8T,e3α9TT. So, system (36) is asymptotically stable. Moreover, it can be verified that system (36) is a homogeneous system of degree α9−1/α9<0. Hence, by Lemma 2, system (36) is finite-time convergent to the origin. It follows that the finite-time consensus problem of system (33) with protocol (34) is solved.
The finite-time consensus of system (31) with protocol (32) can be proved similarly. Theorem 1 is proven.


## 3.1. Finite-Time Control for Single USV

In this stage, each vehicle is controlled separately to reach its initial position in the formation. We hope that this process could be completed in a limited time. For the underactuated USVi, its desired pose vector to the group centroid is Δxi,Δyi,Δψi. We can transform the set point tracking problem of USV i to the stabilization problem by defining the following new state variables:(12)x¯i=xi−Δxi,y¯i=yi−Δyi,ψ¯i=ψi−Δψi.Then, the kinematic equation of USVi can be rewritten as(13)x¯.i=uicosψ¯i−visinψ¯i,y¯.i=uisinψ¯i+vicosψ¯i,ψ¯.i=ri.Since the mathematical model of the underactuated USV is a typical nonholonomic system, the following global coordinate transformation is employed before control design [36]:(14)z1i=x¯icosψ¯i+y¯isinψ¯i,z2i=−x¯isinψ¯i+y¯icosψ¯i+mvdvvi,z3i=ψ¯i,z4i=−mudvui−z1i,z5i=vi,z6i=ri,τ1i=dudv−1ui−z2iz6i−τuidv,τ2i=muvmruivi−drmrri+τrimr,where τ1i,τ2i represents the new input transformation pattern. Thus, system (13) together with (9) can be transformed into(15)z˙2i=z4iz6i,z˙3i=z6i,z˙4i=τ1i,z˙6i=τ2i,(16)z˙1i=−dvmuz1i−dvmuz4i+z2iz6i−mvdvz5iz6i,z˙5i=−dvmvz5i+dvmvz6iz1i+z4i.Remark 1 (see [36]).
The system stabilization problem in (13) and (9) is equal to systems (15) and (16). If system (16) is global asymptotic convergent within finite time, the overall system of (15) and (16) is global asymptotic convergent.Remark 2.
Asτui,τri and τ1i,τ2i can be converted to each other, just τ1i,τ2i need to be designed to make variables z1i,z2i,z3i,z4i,z5i,andz6i asymptotically converge to zero, which means that the tuple xi,yi,ψi,ui,vi,ri converges to Δxi,Δyi,Δψi,0,0,0. That is to say, the USV has arrived to its initial place in the formation pattern with desired orientation.
In view of the above remarks, our objective has shifted from set point tracking for each USV to simply stabilizing system (15). Note that this system does not satisfy Brockett’s necessary condition. Hence, we divide system (15) into two cascade subsystems and try to provide a discontinuous control law. Suppose z6i is not zero at the initial time, such as z6i0≠0. The first cascade subsystem is given as the follows:(17)z˙2i=z4i′,z˙4i′=τ1i′,where z4i′=z6i0z4iandτ1i′=z6i0τ1i. The finite-time control law is designed as(18)τ1i′=−k1iz2iα1−k2iz4iα2,τ2i=0,fort∈0,t1i,where the exponents α1,α2 are selected as α1=α2/2−α2,0<α2<1. The gains k1iandk2i are selected to satisfy(19)P1i01−k1i−k2i+01−k1i−k2iTP1i=−Q1i.
Thus,t1i is the settling time which can be calculated by(20)t1i=α2ζ1iT0P1iζ1i01−α2/α21−α2ρ1i,where ζ1i0=z2i0,z4i0T and ρ1i=λminQ1i/λmaxP1i. According to Lemma 3, the feedback control law (18) presents a homogeneous closed-loop system with negative degree of homogeneity. Thus, the states z2i and z4i of system (17) reach to zero in finite time; then, τ2i would not affect system (17) any more. Moreover, τ2i can be designed independently for the second cascade system:(21)z˙3i=z6i,z˙6i=τ2i.
After timet1i, we have(22)τ1i′=0,τ2i=−k3iz3iα3−k4iz6iα4,fort∈t1i,t1i+t2i,where the exponents α3andα4 are selected as α3=α4/2−α4,0<α4<1. The gains k3iandk4i are selected to satisfy(23)P2i01−k3i−k4i+01−k3i−k4iTP2i=−Q2i.
Thus,t2i is the settling time which can be calculated by(24)t2i=α4ζ2iTt1iP2iζ2it1i1−α4/α41−α4ρ2i,where ζ1it1i=z3it1i,z6it1iT and ρ2i=λminQ2i/λmaxP2i. By using Lemma 3 again, the control law (22) makes the states z3i and z6i of system (21) converge to zero in finite time.
After that the states of (15) converge to zero, system (16) is also asymptotically stabilized according to Remark 1. Actually, system (16) is simplified to first-order systems, whose settling time periods are 3mu/dv and 3mv/dv. We let(25)t3i=max3mudv,3mvdv.
So, the final convergence time of all USVs is calculated as(26)tf1=maxit1i+t2i+t3i.
That is to say, all USVs can arrive to their positions in the formation pattern after timetf1.


## 3.2. Cooperative Control for Formation Maintenance

The goal of the second stage is to maintain the formation pattern by means of finite-time distributed control. It is assumed that only local distributed state information would be utilized by each vehicle to make its own surge velocity and yaw angular consistent with others. Consider a virtual leader moving with the group centroid, labeled 0. The communication graphG¯ among the n vehicles and the virtual leader satisfies Assumption 1.After the first stage, the group of USVs has come into a prespecified geometric pattern with desired yaw angular. A reference pathΓ0 is previously specified for the centroid of the group which must be followed by the group of the USVs, as shown in Figure 3. The kinematic model of each vehicle can be rewritten according to the Serret–Frenet frame. Suppose that there is a virtual S-F frame XtO0Xn moving along the path Γ0 for the virtual vehicle 0. The kinematics of vehicle i expressed in new state coordinates is [37](27)d˙i=uisinφi+vicosφi,φ˙i=ri−cs˙i,s˙i=11−cdiuicosφi−visinφi,where φi=ψi−Δψi−ψ0 and c is the curvature of the path which is constant for the straight line or arc. Since each vehicle is on its own path and tangent to it at the initial moment, we assume that di≈Δdi,vi≈0 and sinφi≈φi,cosφi≈1. We define(28)r¯i=ri−cs˙i,hi=11−cdi,τ3i=1muτui−duui,τ4i=1mrτri−drri−chiτ3i.Figure 3
The virtual leader and the Serret–Frenet frame.Then, the dynamics of vehiclei can be decoupled into two subsystems: the surge subsystem(29)s˙i=hiui,u˙i=τ3i,and the yaw subsystem(30)d˙i=uiφi,φ˙i=r¯i,r¯.i=τ4i.Suppose thathi,i=1,…m are not zero. The cascade subsystem (29) can be written as(31)s˙i=u1i′,u˙1i′=τ3i′,where ui′=hiuiandτ3i′=hiτ3i. The finite-time distributed protocols based on the relative position and relative velocity are designed as(32)τ3i′=−k5i∑j=1maijsi−sj−Δsi−Δsj+ai0si−Δsi−s0α5−k6i∑j=0maijui′−uj′+ai0ui′−u0′α6,fort∈tf1,∞,where the exponents α5andα6 are selected as α5=α6/2−α6,0<α6<1. The gains k5iandk6i are selected such that s2+k6is+k5i are Hurwitz polynomials.Similarly,ui,i=1,…m are assumed to change slowly and not zero. The cascade system (30) can be written as(33)d˙i=φi′,φ˙i′=r¯i′,r¯.i′=τ4i′,where φi′=uiφ′,r¯i′=uir¯i,andτ4i′=uiτ4i. The finite-time distributed protocols based on the relative position and relative velocity are designed as(34)τ4i′=−k7i∑j=1maijdi−dj−Δdi−Δdj+ai0di−Δdi−d0α7−k8i∑j=1maijφi′−φj′+ai0φi′−φ0′α8−k9i∑j=1maijr¯i′−r¯j′+ai0r¯i′−r¯0′α9,fort∈tf1,∞,where α7,α8,andα9 are selected as α7=α8α9/2α9−α8andα8=α9/2−α9,0<α9<1. The gains k7i,k8i,andk9i are selected such that s3+k9is2+k8is+k7i are Hurwitz polynomials.The main result is given by the following theorem.Theorem 1.
Under Assumption1, protocols (32) and (34) can guide system (31) and (33) to realize finite-time consensus, respectively.Proof.
For systems (33) and (34), define the error states as(35)ei,1=∑j=1maijdi−dj−Δdi−Δdj+ai0di−Δdi−d0,ei,2=∑j=1maijφi′−φj′+ai0φi′−φ0′,ei,3=∑j=1maijr¯i′−r¯j′+ai0r¯i′−r¯0′.
Denotinge1=e1,1,e2,1,…em,1T,e2=e1,2,e2,2,…em,2T,ande3=e1,3,e2,3,…em,3T, it is straightforward to express the error dynamics as(36)e˙1=e2,e˙2=e3,e˙3=L+B−K7e1a7−K8e2a8−K9e3a9,where K7=diagk71,k72,…,k7m,K8=diagk81,k82,…,k8m,K9=diagk91,k92,…,k9m, and ejα,j=1,2,3, denotes e1,jα,e2,jα,…,em,jαT.
Let(37)Ae=0I000I−K7−K8−K9,M=IIL+B,A¯e=MAe=0I000I−L+BK7−L+BK8−L+BK9.
Consider a Lyapunov function candidateVe=eTPee, where e=e1T,e2T,e3TTandPe=PeT>0. It is obviously that Ae is a Hurwitz satisfying the Lyapunov equation PeAe+AeTPe=−Qe, where Qe=QeT>0. Since M is a symmetric positive definite matrix, the Lyapunov equation P¯eA¯e+A¯eTP¯e=−Qe holds with P¯e=PeM−1>0, which means that V¯e=eTP¯ee is a Lyapunov function for the linear system e˙=A¯ee.
From the proof of Theorem1 in [33], we can see that Vζe=ζeTP¯eζe is a Lyapunov function for system (36), where ζe=e1α7T,e2α8T,e3α9TT. So, system (36) is asymptotically stable. Moreover, it can be verified that system (36) is a homogeneous system of degree α9−1/α9<0. Hence, by Lemma 2, system (36) is finite-time convergent to the origin. It follows that the finite-time consensus problem of system (33) with protocol (34) is solved.
The finite-time consensus of system (31) with protocol (32) can be proved similarly. Theorem 1 is proven.


## 4. Numerical Simulation and Results

To illustrate the effectiveness of the proposed control law, a simulation study based on the model is presented in [15]. The same model parameters are chosen as in [38] and given as follows: mu=25.8, mv=33.8, mr=2.76, muv=6.2, du=12+2.5u, dv=17+4.5v, and dr=0.5+0.1r.Without loss of generality, in our simulation scene, we assume that there are only 3 USVs in the formation. Consider the fact that not all the follower vehicles can receive information from each other, but all of them can directly communicate with the “virtual leader,” as depicted in Figure4. In addition, their initial conditions and the desired positions and orientations of each USV in the first stage are listed in Table 1.Figure 4
The communication links relationship in the formation.Table 1
Initial and desired states of the USVs.
Initial statesUSV1: [−30, 60,π/3, 0, 0,−π/18]USV2: [-20,-30,π/2, 0, 0,−π/18]USV3: [50,-50,−π/4, 0, 0, π/18]Desired states in formationUSV1: [0, 10, 0, 0, 0, 0]USV2: [6.3, 0.2,π/50, 0, 0, 0]USV3: [0, −10, 0, 0, 0, 0]According to the setting of aforementioned scenario, the adjacent matrixA, the degree matrix D, the Laplace matrix L, and the leader adjacency matrix B can be deduced as follows:(38)A=010101010,D=100020001,L=1−10−12−10−11,B=100010001.The expected path of the group centroid is given as a circle with radius 100 m. Thus, the expected paths for USV1, USV2, and USV3 are concentric circles with radius 90 m, 100 m, and 110 m, respectively. According to the design guideline, the control parameters are chosen as listed in Tables2 and 3.Table 2
The gains of the controllers.
Gainsk1k2k3k4k5k6k7k8k9USV150.50.110.50.5555USV250.50.110.50.5555USV350.50.110.50.5555Table 3
The exponents of the controllers.
Exponentsα1α2α3α4α5α6α7α8α9USV1.810.901.810.900.200.330.250.330.50The final convergence timetf1 of the first stage for all USVs is selected as 150s, which is more than the sum of t1i, t2i, and t3i. In the second stage, we assume that there is a virtual leader moving along the path with a fixed velocity ut=2m/s.The simulation results are shown in Figures5–8. Figure 5 shows the formation trajectory of the USVs under the proposed control scheme. It can be seen that each of the USVs could arrive at their own starting points in formation first and then could track the desired paths and maintain the triangular pattern well.Figure 5
Trajectory of the USVs.Figure 6
Curves of states with respect to time.
(a)(b)(c)(d)(e)(f)Figure 7
Curves of control efforts with respect to time.
(a)(b)Figure 8
Curves of state errorsdi−d0 and si−s0 with respect to time.
(a)(b)Figure6 shows the curves of the states and control inputs of the USVs. It can be observed that, in the first stage, the positions of the USVs converge to the expected values within about 40 seconds, before the first switch of the control algorithm. And then the orientations of the USV converge to the expected values within about 140 seconds, which is also before the second switching time of the control algorithm. The control inputs of the USVs are depicted in Figure 7. In the first stage, the exponents α2andα4 have great influences on the convergence time. A good choice of them can greatly reduce the settling time of stage one. In the second stage, the feedback gains of the yaw subsystem k7i,k8i,andk9i are usually selected to more than the gains of the surge subsystem k5iandk6i, to make the USVs on their paths.Figure8 shows the errors of the longitudinal displacement and the transversal displacement between the follower vehicles and the leader vehicle. The cooperative control starts working after which every USV gets to its own start points. Therefore, the time starts at 150 seconds and the errors of the displacements are constant, which means that the formation of the USV is kept well.


## 5. Conclusion

A formation control strategy based on finite-time control and leader-follower theory is proposed in the paper to achieve formation maneuver control for USVs. The formation control process is divided into two stages. In the first stage, all the USVs would be steered to their starting points in the formation pattern in finite time. Then, in the second stage, all the vehicles would be controlled to maintain the formation pattern based on finite-time distributed cooperative control. By using tools from homogeneous theory, it is theoretically shown that the proposed protocol can guarantee finite-time converge and consensus, respectively. Numerical simulation is made to illustrate effectiveness of our strategy.



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*Source: 6695221-2021-01-13.xml*