Formation Control of Multiple Underactuated Surface Vehicles Based on Prescribed-Time Method

This paper investigates the formation control problem of multiple underactuated surface vehicles (USVs). By utilizing input and state transformations, the dynamic model of the USV is converted into an equivalent system consisting of two cascade-connected subsystems. For one of the subsystems, by combining back-stepping method with time-varying scale functions, $C_{1}$ smooth prescribed-time control laws are presented. This approach is proved to stabilize the transformed subsystem in the fixed time and address the formation problem while guaranteeing global exponential stabilization. Simulations are given to demonstrate the effectiveness of the presented method.


I. INTRODUCTION
Formation control of surface vehicles is particularly interesting and has many applications in practice. For example, several autonomous vehicles trace a target and enclose it; several vehicles search a target in a large water area cooperatively, and several vehicles with some sensors on them monitor a large area cooperatively for security, etc. The focus is on designing a control law that can drive a group of vehicles to perform the desired group behavior (e.g. formation, flocking, consensus, rendezvous, and agreement). In the literature, three types of formation control are taken into account for the underactuated surface vehicles. The first type is the formation of tracking with a leader. For this type, intuitively, the vehicles are required to come into the desired formation shape and track the leader [4]- [6]. The second type is the formation following, for which the vehicles are demanded to form a desired geometric pattern and move along a given path [7]. The last type is the formation without a leader, and the control objective is regulating the vehicles to the desired formation shape, not necessarily to track a leader or go along a path [8]. The three types of formation control problems are just like The associate editor coordinating the review of this manuscript and approving it for publication was Norbert Herencsar . trajectory tracking, path following, and stabilization control problems in the conventional control sense [9].
Few works have been done for the third leader-less formation control of underactuated surface vehicles. This is because this problem cannot be solved directly by the tracking control technologies developed for leader-follower formation, or by the approaches developed for the consensus of linear multi-agent. What is more, although the underactuated surface vehicle is subject to nonholonomic constraint, its model cannot be transformed into the standard nonholonomic chained system, so the control schemes designed for the cooperative control of nonholonomic systems [11] cannot be applied here. To solve this problem, a smooth time-varying distributed control law is designed in [3] by using appropriated coordinate transformation and graph theory, which can guarantee that the vehicles converge to the desired geometric pattern with the same orientations and vanishing velocities. For solving the same formation problem, another smooth time-varying distributed control law is proposed in [8] achieving the asymptotic formation of underactuated vehicles with non-diagonal inertia/damping matrices. In [10], the timevarying control laws are proposed to avoid that the communication graph among the vessels is fixed and containing a spanning tree. VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ Above mentioned time-varying control approaches rely on asymptotic methods. Recently, researchers investigated finite/fixed-time control methods for stabilization of USVs such as [15], [16]. Compared with asymptotic control approaches, finite/fixed-time control methods not only have faster convergence speed but also better disturbance rejection properties and robustness [12], [13]. Most finite/fixed-time controllers use fractional power feedback of the form x l p (with p and l being some positive odd integers). In contrast to asymptotic methods, fractional power terms can cause highfrequency oscillations in the system states, i.e. chattering. A number of methods for attenuating chattering have been proposed, such as boundary layer method [17] and highorder sliding-mode method [18]. The boundary layer method includes saturation function and sigmoid function methods. But it can only guarantee the existing condition of the slidingmode outside a small boundary layer around the slidingmode manifold, which will increase the steady-state tracking errors. The high-order sliding-mode method is to hide the discontinuity of control in its higher derivatives.
Fortunately, the prescribed-time control law proposed in [1] can guarantee fixed-time stability of the system without chattering. However, different from trajectory tracking and path following problems, stabilization requires all states converge to zero leads the existence of coupling nonlinear terms, which violates the Brockett necessary condition [21]. So that the USV system cannot be linearized into the strict chain system and hence, the prescribed-time method for the typical nonlinear system cannot be used directly to stabilize USVs. A prescribed-time control scheme of USVs to consider the nonlinear coupling of states might be of great significance.
This paper discusses the exponential formation stabilization of USVs with the prescribed-time approach. After introducing the model transformation in [20], a novel state transformation based on graph theory is proposed to convert the system into a simple cascade form of two subsystems. For the transformed system, we propose the prescribed-time control laws by combining the back-stepping method and scale functions that grow unbounded towards the terminal times. Different from the existing prescribed-time control method, we employ different scale functions for each virtual input respectively to decouple the states of USV. The proposed time-varying control laws are C 1 smooth and proved to guarantee global exponential convergence within a designated time independent of the initial conditions. Compared with the existing fixed/finite time methods of USV stabilization such as [16] and [15], the proposed C 1 smooth approach requires no Sign terms and hence decreases the chattering problem.
The paper is organized as follows. Section II is problem formulation and the objective. Section III gives input and state transformations. Section IV gives the control laws and stability confirmations. Numerical simulations and discussions are given in Section V.

II. PROBLEM STATEMENT A. PRELIMINARIES
Before moving on, we present some definitions of prescribedtime stability.
Definition 1: Consider the system defined bẏ where z ∈ R m is the state vector, f : R + × R m → R m is a nonlinear vector field locally bounded in time. The origin of system (1) is said to be globally uniformly finite-time stable if it is globally asymptotically stable and there exists a locally bounded function T : is an arbitrary solution of the Cauchy problem of (1). The function T is called the settlingtime function.
Definition 2: The origin of system (1) is said to be globally prescribed-time stable if it is globally finite-time stable and the settling-time T is a user-assignable finite constant, i.e.,

B. SYSTEM MODELS
Consider n underactuated surface vehicles [2]. Each surface vehicle has two propellers, which provide the force capable of affecting surge and torque capable of affecting yaw. Following the results in [3], the kinematics of the jth surface vehicle for 1 ≤ j ≤ n can be written as where x j , y j denotes the coordinates of the center of mass of the jth surface vehicle in the earth-fixed frame, ψ j is the orientation of the jth vehicle, and u j , v j , r j are the velocities of the jth vehicle in the surge, sway and yaw, respectively. The dynamics of the jth surface vehicle can be written as: where m i,j (> 0) and d i,j (> 0) for i = 1, 2, 3, j = 1, 2, · · · , n are (effective) inertia and hydrodynamic damping of the jth surface vehicle, respectively. We assume that m i,j and d i,j are constants in this paper. τ 1,j is the surge control force and τ 2,j is the yaw control moment. States x, y, ψ, u, v and r of USVs are measurable. Both τ 1,j and τ 2,j are control inputs to design. For such USVs, an inertial frame and a body-fixed frame are used as depicted in Figure 1.   For the analysis that follows, we assume that: Assumption 1: The environment forces due to wind, currents and waves can be neglected in the model of each underactuated surface vehicle.
Assumption 2: The inertia, added mass and damping matrices are diagonal for each underactuated surface vehicle.

C. COMMUNICATION BETWEEN VEHICLES
During the control, each vehicle knows its own state and the states of some of the other vehicles by communication. If we consider each vehicle as a node, A directed graph contains a directed spanning tree if there exists a directed path from the root to every other node in the graph. For more terminology on graph theory, interested readers may refer to [22], [23].
Assumption 3: The communication topology G = ( , E, A) contains a directed spanning tree.

D. OBJECTIVE
Given a desired geometric pattern p defined by constant vectors p x,j , p y,j T (1 ≤ j ≤ n) noting that rotation by an angle and translation of p do not change its geometric form. We aim to design controllers τ 1,j and τ 2,j for each vehicle based on its relative state information with its neighbors, such that In the formation control problem, (4a) means that the group of surface vehicles converges to the desired geometric pattern P. Equation (4b) means that the desired formation is stationary, centered at the point η 2 , η 3 and that each vehicle converges to the same orientation angle η 4 . The vector η may be predefined or not. In the formation control problem, the control law for vehicle j is required to be designed based on the relative information between vehicle j and vehicle i for i ∈ N j such that the state of each vehicle converges to a stationary point. Noting Brockett's necessary condition for stabilizing a nonholonomic system [21], there does not exist a feedback law for each vehicle which is a smooth function of its own state and the states of its neighbors such that the state of each vehicle converges to a stationary point. So, the defined formation control problem is challenging.

III. TRANSFORMATION
To facilitate the control law design, we first transform systems (2) and (3) into a suitable form. Using the state transformation [3]: and the control input transformation time derivative of states z 1,j , z 1,j , z 2,j , z 3,j , z 4,j , z 5,j and z 6,j can beż For the transformation, we have the following result according to paper [3]: Lemma 1: By transformations (5) and (6), if (z 2,j , z 3,j , z 5j , z 6j ) andż 5j are bounded, and = 0 means that z 6j exponentially converges to zero. Furthermore, if c 4 = c 5 = 0, then the formation control problem is solved with η = 0.
Hence we can achieve the formation by stabilizing the transformed system (7). Although the state transformation in [3] can simplify discussions, topological relationships between USVs are not considered. To this end, we give the further state transformation based on the graph theory, aiming to achieve the USV formation by the consensus tracking control of states z 1,j , z 2,j , z 4,j and z 5,j and (10) thaṫ For the proposed transformation, we have the following results.
Proof: If states ϑ 2,i , ϑ 3,i , ϑ 3,i , and ϑ 6,i converge to zero in fixed-time, then the constant t s exists such that ∀ t > t s , ||ϑ 2,i ϑ 3,i , ϑ 5,i ϑ 6,i || = 0. According to equations (10), this means for all t ≥ t s , z 2,j = 0, z 3,j = c 4 , z 5,j = c 5 , z 6,j = 0,ż 5,j = 0. (13) Hence we can have that (z 2,j , z 3,j , z 5j , z 6j ) andż 5j are bounded, and By Lemma 1, the proof is completed. Remark 1: Based on the graph theory, we propose the novel transformation. The similar to stabilization of the single USV, the formation system is divided into two chain systems. So that we can achieve the objective by stabilizing the transformed subsystem (12) to simply the control process.

IV. STABILIZATION OF USVs
In this section, we propose prescribed-time control laws to stabilize the transformed system (12) in the fixed-time with time-varying scale functions, which are proved to guarantee the formation exponentially stable.

A. PRESCRIBED TIME THEORY
Consider the following systeṁ where z = [z 1 , . . . , z n ] T is the state, τ ∈ R is the control input and f (z, t) , g (z, t) are possibly non-vanishing. Then we have the following two lemmas, whose proofs are in the reference [1].
Lemma 4 (Inverse Transformation): Given the transformation x(t) → w(t) given by w (t) = µ m+1 1 P(µ 1 )x, in (32), the inverse transformation w(t) → z(t) is given by where σ (t) = T −t T is a monotonically decreasing linear function with the properties that σ (0) = 1 and σ (T ) = 0, which means, in particular, that regulation is achieved in prescribed time T , the inverse matrix Q(v) = P (µ 1 ) −1 is a lower triangular matrix having elements q i,j given by Furthermore, q = sup v∈(0,1] |Q (σ )|. Denote where and where k n−1 an appropriately chosen coefficient vector so that the polynomial s n−1 + k n−1 s n−2 + · · · + k 1 and the matrix are both Hurwitz. Now we replace the state w n by the new variable z as This then results inṙ where e n−1 = [0, · · · , 0, 1] T ∈ R n−1 . Before proceeding, we note that the linear system (31) is ISS.
The derivative of the new state (30) iṡ which, by substitution ofẇ n = w n+1 ,ẋ n = x n+1 , and then (A.1), and writing out the k = 0 term from the sum, yieldṡ with In the following lemma, the quantity L 0 is expressed in terms of w. Lemma 5 (Rewriting L 0 ): The quantity L 0 is expressed as where l 0 (v) = [l 0,1 , l 0,2 , · · · , l 0,n ], and for j = 1, 2, · · · , n, with Furthermore, l 0 (σ ) is bounded. Lemma 6: The system (16) with the controller has a globally fixed-time asymptotically stable equilibrium at the origin, with a prescribed convergence time T , and there exist M , δ > 0 such that for all t ∈ [0, T ). Furthermore, the control τ remains bounded over [0, T ) and, if f (z, t) is vanishing at z = 0, τ also converges to zero as t → T .
Proof: If control laws τ 1,j and τ 2,j ensure that e 3,j , e 5,j and e 6,j converge to zero before time T 1 , then we can derive that Due to K > 1, α 3,j and α 6,j are continuous at the point t = T 1 . If the control law 2,j ensures that e 3,j and e 6,j can converge to zero before time T 2 < T 1 , then in the time interval [T 0 , T 2 ], we can have that According to the dynamics of subsystem [ϑ 2,j , ϑ 3,j , ϑ 5,j , ϑ 6,j ], the time derivative of ϑ 2,j can be stated aṡ If the control law 1,j ensures e 5,j can converge to zero before time T 0 , in interval [T 0 , +∞], This means thaṫ In time interval [0, T 1 ], According to Binomial Theorem, we can have ∀ t ∈ [0, T 1 ], Therefore, after calculations, where 1 can be presented as: According to the above equations, we can have ϑ 2,j (t = T 1 ) = 0. Since in time interval [T 1 , +∞],θ 2,j = 0, we can conclude that ∀ t > T 1 , ϑ 2,j = 0.
(ii). The expression of α 5,j can be stated as Then the (K − 2)th order derivative of α 5,j and α 6,j are . Hence α 5,j and α 6,j are K − 2 smooth. The proof is completed.
Remark 2: Compared with conventional time-varying functions, instead of terms sin(t) or cos t, we introduce the scale function 1 T −t to provide the excitation conditions. So that the subsystem (12) can be stabilized to the origin before the fixed-time T 2 . Note that different from previous fixed-time approaches, the proposed prescribed-time control laws contain no Sign terms. Hence the chattering can be decreased.
Remark 3: The parameter K is to increase the smoothness of α 5,j and α 6,j , which can affect the smoothness of control laws. Since we need control laws are C 1 smooth, the parameter K requires to be equal or greater than 3.
Remark 4: Since the existence of nonlinear termθ 2,j = α 5,i α 6,j and the requirement that all states converge to zero, the system cannot be linearized and stabilized by prescribedtime methods for linear systems. Hence, this paper introduces different scale functions for the virtual inputs α 5,j and α 6,j , to decouple the nonlinear term by designing terminal times respectively.
According to Theorem 1, based on the back-stepping method, virtual inputs α 5,j and α 6,j can stabilize the subsystem [ϑ 2,j , ϑ 3,j ] in the prescribed-time and ensure that α 5,j and α 6,j are K − 2 smooth. Next, we design the actual control inputs 1,j and 2,j such that ϑ 2,j , ϑ 3,j , e 3,j , e 5,j and e 6,j converge to zero in the prescribed-time. The dynamics of [e 3 , e 5 , e 6 ] can be stated as: e 3,j = e 6,j ,ė 6,j = 2,j −α 6,j , (57a) With Lemma 6, we are now in a position to present the prescribed-time control scheme Here a,j and b,j can be presented as: Theorem 2: Consider the system (12), control laws 1,j and 2,j in (60) and (61) ensure following properties hold.
(ii). The time derivative of a,j and b,j can be presented as: By Lemma 6, for all t ∈ [0, T 0 ), states e 3,j , e 5,j e 6,j satisfy that Therefore, we can have that According to the above equations, we can have that˙ a,j and˙ b,j are continuous implying that a,j and b,j are C 1 smooth. Additionally, by Theorem 1, we know α 5,j and α 6,j are C 2 smooth, meaningα 5,j andα 6,j are C 1 smooth. Hence inputs 1,j and 2,j are C 1 smooth. Remark 5: Although switching points exist in the control process, we can ensure the C 1 smoothness by designing parameters, such as K . So that the derivatives on the left and right sides of the switch points are equal.
It is interesting to note that, in contrast to most existing finite-time control methods, the proposed fixed-time control scheme, as shown in (60) and (61), is built not only upon regular (rather than fractional power) state feedback but also upon time-varying (rather than constant) gain. It is such a structural feature that renders the convergence time not only finite but also user-assignable. Also, with the time-varying gain, the proposed control avoids excessively large initial driving force as encountered in many high and constant gainbased control methods, because the initial value of the timevarying gain here can be set as small as desired, rendering the initial control effort small. It should be mentioned that the nature of the time-varying gain as involved in the control scheme, although calling for gain updating according to the given simple analytical algorithm, does not cause any technical difficulty for implementation because the computation involved in updating the gain is even simpler than those involved in updating the parameters in traditional adaptive control.
This paper emphasizes to use C 1 smooth time-varying scaling functions achieving exponential formation control of USVs, based on which the closed system always has certain robustness against uncertainties and disturbances. Hence, due to space limitations, we focus on designing timevarying scale functions instead of disturbance/uncertainty rejection approaches. In addition, some previous robust and adaptive methods of nonlinear systems in [24]- [26] can be adopted to solve the disturbance and uncertainty problems. Besides, the neural network technology is combined with back-stepping or adaptive methods to achieve tracking control of nonlinear systems (e.g. [27]- [29]). However, due to differences of kinematics, both kinds of nonlinear approaches can not be utilized directly here. An improved back-stepping control law based on above methods can be considered as an alternative for formation control of USVs in the future.

C. CONTROL ALGORITHM
To sum up, with the main results, we give the following algorithm for the implementation of the presented method and results in USVs.
(ii). Control laws τ 1,j and τ 2,j are C 1 smooth. Proof: (i). According to input transformation (6) and Theorem 2, control laws (65a) and (1) ensure the fixed-time stability of the subsystem (12). By Lemma 2, this indicates the exponential stability of system (11) and (12). So that the formation control problem is solved based on the results of Lemma 1.
(ii). By Theorem 2, we can have that 1,j and 2,j are C 1 smooth, which can be developed to inputs τ 1,j and τ 2,j .   To help readers grasp the essence of our formation control methodology, the structure of proposed algorithm can be described as five steps: 1). control parameter selection; 2). model parameter measurement; 3) differential homeomorphism transformation; 4). virtual controller design; 5). real controller design. Details can be seen in Table 2 and Figure 2.
Remark 6: Generally, time-varying control laws cannot guarantee the exponential stability of USVs. To this end, some special terms such as e κt are used [3]. Differently, this paper solves this problem by fixed-time stabilize the   subsystem (12). Compared to the previous fixed/finite-time approaches [15], [16], the presented control laws are C 1 smooth.
For the explosion of terms in back-stepping method, some estimation approaches have been utilized, such as the command filtered technology. However, since it always causes filter errors in the closed system and our control laws have no or few explosion of terms, the command filtered backstepping technology is not used here.

A. FORMATION STABILITY
In this section, the effectiveness of the proposed control laws is shown by simulation. Consider three identical underactuated surface vehicles with the model parameters: m 1,j = 25.8, m 2,j = 33.8, m 3,j = 2.76, d 1,j = 12, d 2,j = 17, and d 3,j = 0.5 for j = 1, 2, 3. To illustrate the formation stability of USVs by our method. Different initial conditions of three systems are considered as two cases, which are presented in Table 3 2]. Choose the control parameters as T 0 = 0.5, T 1 = 1.5, T 2 = 3, K = 3, κ = 1.
Assume the communication digraph G among the surface vessels is fixed and as in Figure 3, the cooperative control laws can be obtained by Corollary 1. Without loss of generality, we consider two cases of initial conditions, whose details are stated in Table 3. Figures 4(a) and 4(b) show the path of each vehicle, which converge to the desired geometric pattern. The simulation also shows that in both two cases, each [ψ j , u j , v j , r j ] converges to a constant vector, where velocities u j , v j and r j converge to zero.   In Figures 5(a) and 5(b), responses of inputs illustrate that proposed control laws have no chattering problems.

B. COMPARISON 1) EFFECTS OF INITIAL CONDITIONS
To illustrate the effectiveness of our method under different initial conditions, we have the comparisons of states ϑ 2,j between our method and finite-time control laws of [15] in Cases 1 and 2. Note that the convergence of states ϑ 2,j mean that the stability of the transformed system (12), where j = 1, 2, 3.
Figures 6(a)-7(b) show that in both two cases, the state ϑ 2,j can be converged to zero before 3s by our approach, whose settling time is designed as T 2 . However, with different initial conditions, the finite-time control method has different settling times. This illustrates that by prescribed-time control laws, the state ϑ 2,j can be converged in the fixed-time, which is independent of initial conditions.

2) SMOOTHNESS COMPARISON
To evaluate the smoothness of our method, we consider the comparison of the input trajectories with the fixed-time control method in [16] under initial condition Case 1 in Figure 8. Obviously, due to the existence of Sign function, the conventional fixed-time control law can cause chattering. By introducing a time-varying scale function, the proposed prescribed-time control laws are C 1 smooth and have better smoothness on trajectories. This implies the proposed control laws relatively easy to implement in engineering due to the smoothness.

VI. CONCLUSION
In this paper, based on a global diffeomorphism transformation, by introducing time-varying scale functions, the prescribed-time control laws are proposed to stabilize the transformed system in the fixed-time. Then the control laws are proved to ensure the exponential formation stabilization of USVs.
As actuator saturation due to mechanical constraints may have significant impacts on the system transient behavior and even stability, stabilization of USVs subject to actuator saturation is still an open problem. Some intelligent control methods of nonlinear systems can.
TINGTING YANG was born in Heilongjiang, China. She received the B.S. and Ph.D. degrees from Dalian Maritime University, in 2012 and 2020, respectively. Her current research interests include formation control of underwater vehicles, disturbance observers, and fault tolerant control.
DONG LI received the master's degree from Dalian Maritime University, in 2015. He mainly engaged in machine vision, electrical control, and robotics research.