Containment Control of Underactuated Ships with Environment Disturbances and Parameter Uncertainties

An implementable robust containment control algorithm is proposed for a group of underactuated ships in the presence of hydrodynamic parameter uncertainties and external disturbances.The control objective is to drive all the followers into the convex hull spanned by the virtual leaders, whose state information is available only to a subset of the followers. For this purpose, the ship model is primarily transformed to a strict-feedback form. In the kinematic design, a virtual containment controller, requiring the state information from its neighbors, is presented based on the results obtained from graph theory. In the dynamic design, a robust containment controller is developed through utilizing upper-to-up sliding mode control. In addition, in order to simplify the implementations of the control law, the command filtered backstepping (CFBP) method is introduced to prevent the analytic differentiations of the virtual law from each design step of the backstepping (BP) method. Subsequently, it is well proven that all the tracking errors could converge to and remain small neighborhoods of the equilibriumpoint. Finally, several simulation experiments are conducted to demonstrate the performance of the proposed control algorithm.


Introduction
Over the past few years, a huge and rapidly growing body of research has focused on the cooperative control of the multiple vehicle systems within a motion-control community because of their broad applications, for example, wheeled mobile robot, manipulator, spacecraft, and ship [1][2][3][4].Although an individual vehicle can be used to complete a specific task, some benefits including greater robustness, lower cost, and better performance can be achieved through a group of vehicles working cooperatively.In the marine domain, many applications using cooperative control include various tasks such as environmental monitoring as well as rescue and reconnaissance operations [4][5][6][7].To perform these tasks, an increasing number of studies have focused on the cooperative control of multiple ships.
These studies have included behavior-based method [8], virtual structure method [9][10][11][12][13][14], and leader-follower (L-F) method [4,5,[15][16][17][18][19]. On the basis of null space methods, a formation control strategy has been developed to achieve multiple tasks in [8].In [15], an output synchronization control strategy has been designed for the L-F control of ships.The authors in [9,10] introduce passivity theory as a practical design tool for solving the formation problem, which includes group coordination and path following.A robust cooperative control method has also been applied to form a desired geometric pattern with the aid of a neural network (NN), a backstepping (BP) method, and graph theory [11].In [8][9][10][11]15], the vehicles are assumed to be fully actuated.
In actuality, most ships are underactuated, meaning that they are not actuated in the sway axis [20].Since the dynamics of each underactuated ship do not satisfy the necessary conditions of Brockett for stabilization [21], the cooperative control law for each ship cannot be smooth functions with theirs variables [16].In [12], a decentralized formation control method has been presented to address such an issue by using the nonlinear cascaded system theory and line-of-sight guidance to ensure a straight-line path following for the formations of ships.On the basis of Lyapunov theory, the BP method, and the graph theory, a path-following coordination controller (PFCC) is proposed in [13] and its control strategy also considers time delayed communication among ships.The work of [14] is extended further in [13].In [14], a dynamic surface control (DSC) technique is proposed to estimate the virtual control at each step of the BP method.An additional virtual control is adopted to solve the difficult problems when designing the formation controller for ships in [22].The L-F method for ship has been also reported in [4,5,[16][17][18][19].In [16,17], continuous time-varying cooperative control laws are designed to perform a geometric pattern by using suitable transformations.It is noted that, in [16,17], the yaw velocity is assumed to be nonzero, which is referred to as the "persistent excitation" (PE) condition.Under this condition, a straight line cannot be tracked for underactuated ships.The authors in [18] address the design of the nonlinear model predictive formation controller, where the relative distances and orientations between the follower and the leader can be stabilized.To cope with the uncertainties of the model, two different robust L-F formation control laws have been designed that combine NN with the DSC technique in [4,19].More recently, a robust formation control algorithm is proposed to force ships to maintain the desired orientations and positions relative to one leading vessel, considering the limited magnitude of the control signal in [5].In [4,5,19], the radial basis function NN and adaptive control method can approximate the nonlinear uncertainty systems, which increases the complexity of the online computing.In addition, in all the works on L-F formation control, a common trait is that only one leader exists in the group.
In practical applications, for a group of vehicles multiple leaders might exist.In this case, the control objective is to drive the followers of the group into the convex hull spanned by the leaders, which is called the containment control problem [23].Thus, containment control can be regarded as a special L-F formation control, and the study of containment control is motivated by its possible applications.For example, a group of ships are guided by another group, and some ships are regarded as leaders, which are all equipped with sensors to detect the hazardous obstacles in the marine environment [24].As such, the leaders can form a safe area in which the followers can converge [25].Recent works on the containment control of multiple vehicle systems focus on single integrator systems [25], double integrator systems [26], strictfeedback form systems [23], general linear dynamics [27], and Lagrange dynamics [28].However, these proposed methods cannot be applied directly to an underactuated mechanical system, and the uncertainties from the model might affect the control performance.Accordingly, how to achieve the containment control objective in the presence of uncertain dynamics and external disturbances needs to be investigated further.
For nonlinear systems, various methods have been adopted, and, among these, the BP method can be regarded as a major design tool.However, this method suffers from the "explosion of complexity" problem, which is caused by the repeated derivatives of the virtual control signals [29].To overcome this drawback, the command filtered backstepping (CFBP) method, which introduces, at each design step of the BP method, a command filter to prevent the derivative of nonlinear functions, has also been applied to the strict-feedback form system in [30,31].The CFBP method is regarded as an improved version of standard BP method and is widely used in different fields [32][33][34].However, these works cannot prove the boundedness of the tracking errors, which implies that the stability of the closed-loop system cannot be analyzed quantifiably.In addition, most current CFBP adopted second-order command filter (SOCF), and the firstorder command filter (FOCF) with simpler structure is rarely mentioned.
In conclusion, the containment control of underactuated ships involved the following main difficulties: (1) the underactuated surface ship lacks control input at side, so it belongs to a class of underactuated system; namely, the general nonlinear control theory could not be directly applied; (2) in complex sea cases, ship is often influenced by the external disturbances and parameter uncertainties (EDPU), which makes it complicated to accurately control the motion of ship; (3) the current research results on containment control are applied only in full-actuated system such as single integrator systems, double integrator systems, and Lagrange systems.However, it is not mentioned how to achieve containment control by combining the knowledge of graph theory and the underactuated system, which is one of the research difficulties.In this paper, a robust containment control strategy, inspired by the previous works, which is performed by using the standard BP method, the FOCF technique, the sliding mode control (SMC) method, the Lyapunov stability theory, and the results from graph theory, is developed for a group of ships in the presence of the EDPU.Compared with the existing results, the main features of this paper can be summarized as follows: (1) for the first time, this paper considers the containment problem of underactuated ships; (2) by incorporating the FOCF technique, the BP commands are simplified; (3) different from the previous conclusions about SOCF [30,[32][33][34], the stability of all the closed-loop systems is first analyzed quantifiably; (4) by introducing a polar coordinate, the model is transformed to a strict-feedback form system, and the PE condition from [16,17] is avoided; (5) compared with the results in [4,5,19], the development, using upper-toup SMC to design a robust containment controller, is more practical because the online computation of the uncertainties is avoided.

Notations.
To prepare for the subsequent control design, some notations are standard as below: R × denotes a set of Euclidean  ×  matrices, and R  denotes -dimensional Euclidean space; diag( 1 ,  2 , . . .,   ) denotes a diagonal matrix with entries   ( = 1, 2, . . ., ); the maximum and the minimum eigenvalues of a square matrix are described as  max (⋅) and  min (⋅), respectively; let |⋅| be the absolute value of a scalar; let ‖⋅‖ be the Euclidean norm of a vector; let A ⊗ B be the Kronecker product of matrices A ∈ R × and B ∈ R × ;  fl  denotes that  is defined as .

Concepts in Graph
Theory.Suppose that a group of under actuated ships interact with each other through a communication network to perform a containment control task.It is natural to model the communication topology among these ships by using graph theory.Without loss of generality, we assume that the individual ship is a node, and the interaction for ships can be described by a directed graph  = {, }, where  = { 1 ,  2 , . . .,   } denotes a set of nodes and  = {(  ,   ) ∈  × } denotes a set of edges with element (  ,   ), which implies that node   can receive information from node   .Here, we say that node   is a neighbor of node   , and the notation   is described as the set of all neighbors of node.Let the adjacency matrix A = {  } ∈ R × be defined such as   = 1 if (  ,   ) ∈  and   = 0 otherwise; Note that we assume   = 0 for all the nodes.The Laplacian matrix L = {  } of the directed graph  is defined such that A directed path from   to   in the graph  is a set of edges: (  ,  +1 ), ( +1 ,  +2 ), . . ., ( −2 ,  −1 ), and ( −1 ,   ), where all the nodes in this path are different [35].
Definition 1.For a group of ships, ship  is said to be a leader if   = 0, and ship  is said to be a follower if   = 1,  = 1, 2, . . ., .
Definition 2. The real set  ∈ R  is said to be convex if, ∀,  ∈ , there exists a point that satisfies (1 − ) +  ∈  for any  ∈ [0, 1].The convex hull for a set of points  = { 1 ,  2 , . . .,   } in  is the minimal convex set including all the points in  and let Co() be the convex hull of .In particular, Co() is defined as follows [36]: 2.3.Underactuated Ship Modeling.Suppose that there are  identical followers, labeled as ship 1 to .According to [37], for the containment control task we can neglect the motions in heave, pitch, and roll; hence, for a ship, the 3-DOF mathematical model can be formulated as (see Figure 1) where Because no control signal is introduced in the sway direction, this model is an underactuated system.Inspired by the work of [38], we define the following polar coordinates to solve this problem: where   = arctan(V  ,   ) is often recognized as side-angle and   and   are recognized as the speed and the course angle of ship .Then, substituting (5) into (3) yields Then, suppose that the group contains  −  leaders in the group, labeled as leaders +1 to .To describe the communication among the followers and leaders in the group, let L denote the Laplacian matrix.As each leader has no neighbors, L can be described by the following: where L 1 ∈ R × and L 2 ∈ R ×(−) .
Assumption 4. For every follower of the group, there exists at least one leader that has a directed path to it [36].7) is available only for the nonzero value of cos   , the condition   > 0 can guarantee that the side-angle satisfies   ̸ = ±0.5;that is, cos   ̸ = 0.In the next part, we will give a certain initial condition for ship  such that   is positive at all times.Remark 8. Using the polar coordinate transformation (5), the underactuated models (3) are transformed to a strictfeedback form of ( 6) and (7).Then, one can apply the standard BP method to design the control input signals   and   , which achieves the containment control objective.

Command Filter.
To overcome the problem of "explosion of complexity" which occurs in a strict-feedback form [39], a FOCF is introduced to eliminate the analytic differentiation of the virtual control signal in the standard BP design procedure [30].The standard state space form of FOCF is presented as where the FOCF parameter   is a constant and satisfies   > 0;  and ż are estimates of the input  and its derivative, respectively; the initial conditions of states  and  −  are  3 and  4 ; the following lemma proves the input-to-state stability property of FOCF which will be used in Section 3 to analyze the stability of the closed-loop system.
Lemma 9. Consider the FOCF described by (11).Assume that the positive constants  1 and  2 exist, such that Then, the following properties of the FOCF can be given: (1)  1 is bounded and satisfies (2) ż 1 is bounded and satisfies Proof.Define the tracking error q =  − .By using (11), one can obtain Consider a Lyapunov function candidate (LFC) Then, the time derivative of ( 15) is Using inequality (16), one obtains Thus, for all  ≥ 0, Because |()| ⩽  1 , it follows that  is bounded and satisfies From ( 11) and ( 18), one obtains That is,  1 and ż 1 are bounded for all  ≥ 0.
Remark 10.For a given input , the larger value of the FOCF parameter   will decrease the estimated error  − , which leads to better final estimate accuracy.
Remark 11.To address the problem of excessive estimation error in the initial time, we assume that the FOCF initial values are (0) = (0) in this paper.Combined with ( 11) and ( 18), the following inequalities hold for all  ≥ 0: 2.5.Control Formulation.This article aims to design a containment control scheme such that all the followers in the group move into the convex hull formed by the leaders [28,40]; that is, where With this previous notation, the control problem under study can be formulated as below.
Control Objective.Consider the model given by ( 6) and ( 7) under Assumptions 3-5, and the objective of this work is to design a robust cooperative controller for each ship on the basis of its local states and the information from neighbors and a portion of the leaders such that lim where   is a positive constant, p  () ∈ R 2 denotes the actual position of follower , and p  () ∈ R 2 denotes the desired position of follower  which satisfies where Remark 14.Because of the movement of the ship affected by the ocean environment and the filtering error induced by FOCF, the tracking error p  − p  cannot converge to zero but can be uniformly ultimately bounded by   , which will be discussed in detail in the next part.

Kinematic Loop Design
Step 1. First, to prepare for the design, the following error variables are introduced: where   and   denote the virtual control laws of   and   , respectively.Then, it is convenient to expand (6) into where It is obvious that   is the virtual control law for (27) which will be specified later.To solve the control problem (24),   based on its local states and the information from neighbors and a portion of the leaders is designed as below: where   > 0 is a constant.To make the presentation clearer, the following notations are used: Then, the following lemma is presented.
Proof.This proof is divided into three parts.
(3) Then, with the aid of the triangle inequality, the norm of   can be expressed as follows: where  3 is a constant.Clearly, if condition (31) holds, it would imply the strict positiveness of the norm of   for all  ≥ 0. This calculation completes this proof.
Remark 16.To facilitate the analysis of the properties of the proposed containment control law (29), the tracking error variable V  −   is omitted, which will be considered in the following BP design procedure.
At this point, one should also notice that the design of this control system will utilize the virtual control of   and   in the following design steps.To facilitate the computation of the virtual control, let   in (29) be expressed as where Then, combing with ( 28), (29), and (39), the expressions of   and   can be given by Mathematical Problems in Engineering Lemma 17.For   and   described by (41), if   in (36) satisfies condition (31) Proof.From ( 28),   is bounded by Differentiating both sides of the first equation of (41) gives According to the expression of   in (41),   is bounded by As inequality (45), α  satisfies the following: This calculation completes this proof.
Step 2. In this work, the BP method is applied to the strictfeedback nonlinear system of ( 6) and (7).However, it uses the derivatives of the virtual control such as   and   , which increases the complexity of calculations and needs the acceleration information from its neighbors.To avoid the problems described previously, FOCF is introduced to estimate each virtual control law and its derivative [30,43].
To facilitate the expressions of the variables involved in filtered BP method, a particular notation is introduced in this paper.For example, let   go through FOCF depicted in Figure 2 With (48) applied, the kinematic model ( 6) can be rewritten as where p  , V  ,   , Φ 1 (⋅), Φ 2 (⋅), and Φ 3 (⋅) are defined as (28), Then, substituting (29) into (50) and multiplying both sides of (50) with   give Clearly, if the last two terms converge to zero, s  exponentially converges to zero.To prepare for the following step, define the compensating tracking error vector as where   ,   , and   denote the compensating signals of   ,   , and   , respectively.In order to remove the effect of Mathematical Problems in Engineering the nonlinear term V   −   in (51), a compensating signal is selected as where the initial conditions are   = [0; 0] and   =  0 exp(−  ).From (51), (52), and (53), one obtains noting that the last nonlinear term of (54) will be compensated in the next step.Then, considering the first LFC Taking the time derivative of  1 along the solution of (54) gives Step 3. At this step,   will be considered as a control input to stabilize the tracking error   .With notation (48), the kinematic model for   can be rewritten as below: where   is the virtual control for   .To stabilize the tracking error   in (48) and to compensate the last term for (56), the virtual control law   is designed as where   > 0 is a constant.Then, from (34) and (35), it follows that Select the compensating signal for   as where   (0) = 0.By the definition of   in (52) and ( 59) and (60), the compensating tracking error   can be expressed as Consider the second LFC: whose time derivative along (56) and ( 61) is It should be noted that the last two terms of (63) will be compensated in the dynamic design.

Dynamic Loop
Design.This section aims to design the robust control inputs   and   which make the compensated tracking errors   and   asymptotically stable.
Step 1 (surge control design).The definition of   in (52) is chosen as the sliding surface: Differentiating both sides of (64) along (7), one can obtain   dynamics as To ensure that   is always equal to zero, the equivalent control input   can be obtained by solving Ė  = 0 without the parameter uncertainties and external disturbances.Thus,   can be chosen as where "̂" denotes the estimated ship parameters.It could be easily found that the equivalent control input cannot guarantee   = 0 when the EDPU are considered; to eliminate the effect of the uncertainties of the control system and to ensure the convergence of   to zero, an additional control input   is introduced: where  2 is a positive constant and   is the control parameter, determined from the following bounds of the ship parameters in (7) Moreover, to compensate for the last term of (63), a compensated control input   is set to be Combined with the previously mentioned control input, the actual yaw control input   can be chosen as To determine   , consider the LFC for   as whose time derivative along (65) and ( 70) is Considering the third LFC  3 =  2 +   and taking the time derivative of  3 along (63) give Step 2 (yaw control design).Similar to Step 1, a sliding surface for   is introduced: Taking the time derivative of   along ( 7) results in Similar to the previous step, the equivalent control input   can be designed as where "̂" denotes the estimated ship parameters.To compensate for the last term of inequality (74) and reject the uncertainties, the actual surge control input can be chosen as where  1 > 0 is a constant;   can be determined by the following bounds of the ship parameters in (7): and, to determine   , we consider the LFC for   as whose time derivative along (76) and ( 78) is According to (80) and ( 82), computing the first-order derivative of   results in (84) Substituting ( 84) into (83), one obtains Consider the overall LFC  4 =  3 +   ; take the time derivative of  4 along ( 74) and (85) gives (86)

Stability Analysis of Closed-Loop
System.This section consists of four steps.
Step 2 (boundedness of   ).The definition of   in (52) and the definition of   in (48) can be rewritten as Then, the bounds of   ,    , and   give It is obviously concluded that the strict positiveness of   can be guaranteed if   satisfies the condition Thus, |  | < 0.5; that is, cos   ̸ = 0, for all  > 0. Because the determinant of   in (78) is given by the term cos   , condition (91) can guarantee that   are well defined for all  > 0.
which implies the constant  5 > 0 exists which satisfies , where  5 > 0 is a constant.Then, consider the LFC as where and   ,   , and   satisfy With equality (93), one obtains It could easily be found from equality (96) that ‖  ‖ ≤ √ 2 for all  ≥ 0.
Step 4 (implementation of the control objective).By noting that, with the boundedness of E  ,   , the signal s  in ( 30) is bounded by  + √ 2, define the vector as Then, the upper bound for the norm of S() can be obtained as below: Because L 1 is invertible, equality (98) means that Thus, ‖p  − p  ‖ is bounded by  max for all  ≥ 0, where p  is defined in (25).Subsequently, the containment control objective ( 24) is solved.
Remark 21.In [16,17,44], the yaw velocity is required to be nonzero, which means that a straight-line reference cannot be tracked for ships.However, the PE condition is very restrictive from a practical point of view.By using a polar coordinate transformation, the proposed control strategy for ships can track not only straight-line paths but curved-line paths as well.
Remark 22.It is important to note that the traditional DSC method estimates the value of the virtual control law and its derivative only through first-order filter and fails to adjust its estimate through feedback.By contrast, FOCF method introduces compensation tracking error signal on the basis of traditional DSC method, realizes the real-time correction of virtual control law, and greatly improves the accuracy of the estimation.
Remark 23.The authors in [45] also adopt CFBP method to design the trajectory coordinated control strategy and use SOCF to estimate the virtual control law in the BP design procedure online.However, SOCF belongs to a class of secondorder systems, and it is more tedious to implement than FOCF.It should be mentioned that there are many deduction errors in [45].In addition, the authors in [45] adopt the adaptive method to estimate the hydrodynamic parameters online in the dynamic design, but it is difficult to implement in practice.Therefore, in the paper, we refer to the design structure framework of literature [45], use first-order filter to estimate the virtual control laws, solve the containment control objective, and analyze the stability of the closed-loop system by using the input-output boundedness property of FOCF.Moreover, in the dynamic design, upper-to-up sliding mode method is used to design the actual control inputs of ship.Rather than online estimation of hydrodynamic parameters, the method only needs to use the boundedness of hydrodynamic parameters and external disturbances and has the advantages of simple structure and strong robustness.
From (100), the time derivative of  5 can be obtained as Because  11 ,  V ,   , and V  are bounded as shown in Section 3.2, one can obtain where That is to say, the sway velocity V  is uniformly ultimately bounded and will eventually converge to the invariant set

Simulation Results
To verify the effectiveness of the proposed robust cooperative scheme, a networked group that consists of the following five ships with three virtual leaders is considered.The communication topology among the ships is shown in Figure 3 with  = {1, 2, . . ., 5} and  = {6,7, 8}.It can be noted from Figure 3 that the network topology has a directed spanning tree.Assume that all the following ships have the identical structure and that mathematical model parameters are presented in Table 1.To satisfy conditions ( 31), (91), and (96), the cooperative control parameters for each following ship are chosen as follows: and the FOCF parameter is selected as   = 10.Next, four experiments will be presented to verify the effectiveness of the proposed algorithms.4-7.Also, the total simulation time is 200 s. Figure 3 shows the motion traces of the followers.As shown in Figure 4, the five followers converge to the convex hull formed by those of the moving leaders. Figure 5 demonstrates that the norms of the tracking error for each follower converge asymptotically to zero as Figure 6: Control inputs for each follower.
→ ∞, which reveals that the containment control objective (24) is solved.Figure 6 shows the actual control inputs of each follower.
It is obvious from Figure 7 that FOCF can accurately estimate virtual control laws.The simulation results are depicted in Figures 8-11, which exhibit almost the same control performance as those given in Section 4.1.Figure 8 shows the motion traces of the five followers, from which it is evident that the followers also converge to the convex hull formed by those of the moving leaders.The norms of the tracking errors are plotted in Figure 9, and it reveals that the tracking error for each follower are bounded to a small neighborhood of the origin as  → ∞.It is clearly seen from Figures 5 and 9 that the proposed controller yields slightly worse performance than the previous section due to the EDPU.However, the tracking errors are bounded, which means that the tracking errors for each follower are GUUB as proved in Theorem 20. Figure 10 shows the actual control inputs applied to each follower under the EDPU, whereas Figure 11 shows the time evolution of the FOCF estimated errors.Therefore, from the simulation results in this section, it is obvious to conclude that the proposed containment control strategy is also practical when each follower is exposed to the EDPU.Also, it is clear that the proposed containment control strategy for each follower can track straight lines without assuming the PE condition.

Curve Tracking with EDPU.
In this section, the simulation experiments are presented, which concern the robustness properties of the containment control law to the EDPU.The initial states of the followers and the curved predefined trajectories of leaders are the same as the previous section.These simulation results are demonstrated in Figures 16-19.Figure 16 shows the actual trajectories of the followers.The norms of the tracking errors are plotted in Figure 17.As seen from Figures 16 and 17, the containment control laws can force the followers to converge into the triangular region in the presence of the EDPU, and the norms of the tracking  errors are all less than 0.75 m.In Figure 18, the control forces   and   are shown.Figure 19 illustrates the responses of the FOCF estimated errors.In summary, from Figures 4-19, we can see that the proposed containment control law in this paper provided acceptable results as proven in Theorem 20 and all the tracking errors of the closed-loop system converge to a small neighborhood of the origin as time approaches infinity.

Conclusions
(1) Because the sway force is unavailable, the underactuated model is transformed to a strict-feedback form by introducing a polar coordinate transformation.In the kinematic loop design, a virtual controller is designed on the basis of the results from graph theory so as to achieve the control objective.(2) To avoid the problem of "explosion of complexity" from the standard BP method, FOCF is introduced to estimate the virtual control laws and their derivatives.In addition, a compensating signal is introduced to compensate the nonlinear terms in BP design.
(3) Under the consideration that each follower is exposed to the EDPU, the actual control inputs are presented by using the SMC method in the dynamic loop design, which can guarantee that all the signals of the closed-loop system are GUUB and all the tracking errors eventually converge to a small neighborhood of the equilibrium point.
(4) Future works will be required to account for communication constraint among ships, such as packets dropouts and constant or time-varying communication delays as well as switching communication topology.

Lemma 12 .
Under Assumption 4, all the eigenvalues of L 1 have positive real parts.Furthermore, each of −L −1 1 L 2 is nonnegative, and all the row sums of this matrix are equal to 1 [41].Remark 13.Let X  () fl [  +1 (), . . .,    ()]  be the positions of the dynamic leaders.From Definition 2 and Assumption 4, Lemma 12 implies that every point of −(L −1 1 L 2 ⊗I 2 )X  () is in the convex hull spanned by the leaders X  ().

Figure 4 :
Figure 4: Motion trace of the group.

Figure 8 :
Figure 8: Motion trace of the group.

Figure 10 :
Figure 10: Control inputs for each

Figure 12 :
Figure 12: Motion trace of the group.

Figure 16 :
Figure 16: Motion trace of the group.
where (  ,   ) and   denote the position and heading angle of the ship  in the inertial reference frame {};   , V  , and   denote surge, sway, and yaw velocities expressed in the vessel-fixed reference frame {}, respectively;   ,  V , and   describe ocean environment disturbances;   and   are used to describe actual control inputs;   ,  2 ,  V ,  V2 ,  V ,   ,  2 , and  V denote hydrodynamic damping parameters;  11 ,  22 , and  33 denote ship inertia including added mass.
to obtain    and α   , which represent the estimated values of   and α  , respectively.Similarly, we can also define the variables    , α   ,    , and α   .For the sake of the design work, the following tracking errors are defined as   =   −    ,   =   −    ,   =   −    .Remark 18.It is noted the boundedness of   , α  ,   , and α  ensures the boundedness of    , α  According to Lemma 17,   α  ,   , and α  are bounded by  1 ,  1 ,  2 , and  2 , respectively, and then the constants  3 ,  3 ,  4 , and  4 exist and satisfy ≤  4 .
Remark 19.Notice that, with   being bounded by 0.5, the constants  1 and 2 exist, such that | β  | ≤  1 , | β  | ≤  2 all the time.Step 3 (boundedness of   ).Prior to analyzing the boundedness   the boundedness of   ,    should be discussed first.From the definition of   in (58) and the Young equality, one obtains ≤  4 +

Table 1 :
Parameters of ships.