Robust Coordinated Control Algorithm for Multiple Marine Vessels with External Disturbances

Theproblemof coordinated control formultiplemarine vessels in the presence of external disturbances is considered in this paper. A robust coordinated control algorithm is proposed formultiplemarine vessels.The proposed robust coordinated control algorithm is divided into two parts.The first part develops an extended state observer to estimate the disturbances of marine vessels.The second part presents a robust coordinated control algorithm based on the output of the extended state observer. Furthermore, the robust coordinated control algorithm is designed using the dynamic surface control method. In light of the leader-follower strategy, the trajectory for each vessel is defined according to the desired trajectory of the assigned leader and the relative distance with respect to the leader. The effectiveness of the proposed coordination algorithm is demonstrated by the simulation results.


Introduction
In recent years, coordinated control of multiple vehicles has received increasing attention as an emerging technology [1].Multiple vehicles can perform many complex tasks effectively with less time and lower cost than a single vehicle.And multiple vehicles can accomplish some tasks which cannot be executable by a single one.In order to perform these complicated practical tasks, it is necessary for these vehicles to move collectively as a whole formation.In practice, many relevant applications of coordinated control can be found on the land, in the sea, and in the air [2].For instance, in the operations of underway replenishment by a fleet of surface vessels, it is required that the replenished vehicle should maintain a fixed relative position with respect to the replenishing one, in order to ensure the replenishment operation performed safely and effectively.
The problem of coordinated formation control has been reported in a large number of recent publications.Basic approaches of the coordinated control include leaderfollower approach [3][4][5], behavioral approach [6,7], and virtual structures approach [8,9].In the leader-follower approach, some agents are considered as the leaders, and the rest ones are considered as the followers.The followers will track the leaders, and the leaders will track the predefined desired trajectories.This method is easy to be manipulated and implemented.However, the main criticism of the leaderfollower approach is that it depends heavily on the leader to achieve the goal of the formation task which may be undesirable [3][4][5].In the behavioral approach, the collision avoidance/obstacle avoidance and the target tracking are prescribed for each agent, and the whole formation is achieved by calculating the weight of the relative importance of each behavior.However, it is difficult to analyze the stability of the group behavior using such approach [6,7].In the virtual-structure approach, each member in the formation is considered as a particle embedded in a rigid geometric structure, but the relative applications are limited when the formation structure is time-varying or needs to be frequently reconfigured [8,9].Some advanced approaches including graph theory [10], passivity-based control [11,12], and hybrid control [13] are also used for coordinated control of multiple marine vessels.Most results about the coordinated control problem addressed in the earlier papers are on the assumption that marine vessels are free from environmental disturbances.However, coordinated control for multiple surface vessels encountering exogenous disturbances adds a new level of complexity to the problem.Other advanced methods are proposed to solve the robust coordinated formation control problem, for example, the Lagrangian approach [14], the nonlinear model predictive control [15], the adaptive control [16], and the sliding mode control [17].In addition, the fault tolerant control and the fault diagnosis are studied in references [18][19][20].In particular, the problem of coordinated path following multiple vessels has also been discussed in the related literature studies [21,22].The robustness to environmental disturbances is highly important when performing practical marine and offshore tasks for surface vessels, which is also the concerned issue in this paper.The core of the extended state observer is that the disturbances and the unknown dynamics can be considered as extend state, and then the detailed values can be estimated by designed observer.The correlative applications can be found in literature studies [23][24][25][26].The stability of the extended state observer is analyzed in [27][28][29].The robust coordination control algorithm for multiple surface vessels based on extend state observer and robust control technology is studied in this paper.The designed controller is useful for the practical marine operations.
In this paper, we consider the problem of coordinated formation control of multiple surface vessels in the presence of exogenous disturbances.The coordinated formation controller is proposed by combining the extended state observer and dynamic surface control using the leaderfollower strategy.The extended state observer is developed to estimate the external disturbances of the surface vessels.The coordinated control algorithm is accomplished based on the output of the extended state observer.Furthermore, the trajectory of each vessel is defined using the desired trajectory of the assigned leader and the relative distance with respect to the leader.This paper is organized as follows.In Section 2, the vessel model is established.Section 3 contains a detailed algorithm of the coordination formation control for multiple vessels.Simulation is carried out in Section 4, and we draw conclusions in Section 5.

Preliminaries
The vessel model can be divided into two parts: the kinematics and nonlinear dynamics.Generally, only the motion in the horizontal plane is considered for the surface vessel.The elements corresponding to heave, roll, and pitch are neglected.The dynamic model for the ith surface vessel can be represented by the following 3 degrees of freedom (DOF) [30]: where   = [  ,   ,   ]  denotes the north position, east position, and orientation which are decomposed in the Earthfixed reference frame, k  = [  , V  ,   ]  denotes the linear surge velocity, sway velocity, and angular velocity which are decomposed in the body-fixed reference frame.R  (  ) is the transformations matrix from the body-fixed reference frame to the Earth-fixed reference frame, the form of which is as follows: The transformations matrix satisfies R −1  (  ) = R   (  ), for all   .M  denotes the system inertia mass matrix including added mass which is positive definite.C  (k  ) and D  (k  ) denote the Coriolis-centripetal matrix and damping matrix, respectively.The detailed representation of the above three system matrices can be found in reference [30].  = [   V   ]  is the vector of forces and torques input from the thruster system.  is the vector of external environment forces and torques input which is generated by wind, wave, and current.
In order to design the backstepping sliding mode controller, we transform the vessel model as follows: because with   being the angular velocity in the body-fixed reference frame.We can obtain that Taking ( 4) and ( 6) into the vessel dynamic model (2) yields The above equation can be written as where

Coordinated Formation Controller Design
In this section, the controller is designed from two inspects.One is the extended state observer design for each vessel, and the other is the coordinated controller for multiple vessels based on the output of the extended state observer.

Extended State Observer Design.
In this section, we design the extended state observer for each vessel to estimate the disturbances.Let η  = k  , and then (8) can be written as Then we have Let Then the above equation can be rewritten as Here K  is assumed to be unknown.We assume that K  is an extended state.However, K  can be estimated using an extended state observer.Then the disturbances are observed and compensated by the designed controller.
The extended state observer is designed as where And  > 0, 0 <  < 1.
η , k , K are the estimated values of   , k  , K  , respectively.Taking the derivative of z 1 , z 2 , z 3 , respectively, we can obtain that The following assumptions are presumed.
(1) The possibly unknown function K  is continuously differentiable with respect to their variables.| K | ≤  for all  > 0, where  is a positive constant.
Vi rtu al lea de r ( The stability for the extended state observer is analyzed in [24].Then we can obtain η →   , k → k  , K → K  .

Coordinated Controller Design
3.2.1.Formation Setup.This paper considers a fleet of n vessels to perform the desired coordination formation task.Each vessel in the formation is identified by the index set  = [1, 2, . . ., ].The desired formation is established using the leader-follower strategy as shown in Figure 1.Furthermore, the leader is a virtual vessel.If we assume the desired trajectory of the leader vessel is denoted as   , where   = [  (),   (),   ()]  ,   (),   () are sufficiently smooth functions, and   () = arctan( ė  ()/ ṅ  ()).That means the vessel direction is chosen as the tangential vector of the respective desired trajectory.If we define the relative distance between the follower vessel and the leader vessel as l  = [ 0 ,  0 ,  0 ]  , then the desired trajectory of the follower vessel is denoted as   =   + R(  )l  .

Controller Design. Define the first dynamic surface as
Taking the derivative of the first surface, with (13), we obtain Mathematical Problems in Engineering Define a virtual velocity k  as follows: where Λ 1 is a positive definite matrix.With this definition, if k  = k , then Ṡ 1 = −Λ 1 S 1 .Then S 1 → 0 with a convergence rate determined by the choice of Λ 1 .Because of the definition of S 1 , this will also guarantee that η →   →   .k  is passed through a first order filter in order to avoid the problem existed in the backstepping scheme: where T is a diagonal matrix of the filter time constants which are chosen to be as small as possible.Because k → k  , then we can define the second sliding surface as where k  is the estimated value of k  ; take the derivative of k  , then we have Taking the derivative of the second surface, with (13), we have We consider the following Lyapunov function candidate: We take the time derivative of ( 25): So we choose the control input as The control input of the vessel is   = M  (  )u  .So the control force input  is selected as Theorem 1.Consider the vessel with the nonlinear model as in (1), (2), and (8), with the control law (27), and then one can guarantee that the vessels approach the desired trajectory ultimately while holding the desired formation structure.
Proof.With the definition of the second surface, (19) can be rewrite as Define the estimated error of the first order filter as Taking the derivative of S 3 yields With ( 19) and ( 21), we obtain Define the Lyapunov function as Differentiating the above equation yields If we define the maximum of ( 1 , z 1 , , , ż 1 , η  , Ṡ 1 ) is  max , we can know that S  1 S 1 + S  2 MS 2 + S  3 S 3 ≤ 2,  is a positive constant.Then we can obtain that   ≤ .Let Λ 1 = K  =  0 +2 3 , and the filter time constant can be chosen as T = ( 3 + (  max  max /2 3 ) +  0 ) −1 ; then we can obtain that  If we choose  0 > /2, then V  < 0. We can guarantee that S 2 → 0. This implies k → k  → k  , in turn, S 1 → 0 and η →   →   .

Simulation Results
In this section, experimental simulations are carried out to evaluate the effectiveness of the proposed coordinated formation control algorithm.The detailed parameters of the vessel are presented in the literature [11].At the beginning, the proposed extended state observer of one vessel is evaluated by the simulation.Similarly, the performance of the extended state observer of other vessels is achieved.Compare with the existing literature studies, we let the initial position of the vessel is  = [0, 0, 0]  .The vessel moves in a beeline northward.assume that the vessels encounter the wind, wave, and current.The wind is assumed to be fixed direction and  The simulation results are shown in Figures 6 to 10. Figure 6 shows the movements for these vessels in the plane.The heading change curve of each vessel is shown in Figure 7. Figures 8, 9, and 10 show the surge velocity, the sway velocity, and the angular velocity of each vessel during the coordinated control process, respectively.We can see that these vessels realized the coordinated tracking task from Figures 6 and 7. From Figures 8, 9, and 10, the velocities of these vessels achieve consensus as a whole, and the velocities cannot achieve consensus absolutely when the vessels move to the inflexion of the curves.With the analysis of the simulation results, we can conclude that these vessels can accomplish coordinated trajectory tracking task while keeping the desired formation.It means that the proposed coordination control algorithm is effective.

Conclusion
This paper has proposed a new robust coordinated formation control algorithm for multiple surface vessels in the presence of external environmental disturbances.The proposed coordinated formation controller for these vessels is designed by combining the extended state observer and the dynamic surface control together.The extended state observer is designed to estimate the external disturbances of the surface vessels.The coordinated formation is realized based on the leader-follower strategy.The desired trajectory of each vessel is defined using the desired trajectory of the assigned leader and relative distance with respect to the leader.The controller is designed based on the output of the extended state observer and using the dynamic surface control method.The proposed coordinated controller is robust to the external disturbances.Finally, the effectiveness of the proposed robust coordination control algorithm is demonstrated by the simulation results.

Figure 1 :
Figure 1: The formation of these vessels.

Figure 2 :
Figure 2: The movement of the vessel in the plane.

Figure 3 :
Figure 3: The practical value and estimated value of the positions.

Figure 4 :Figure 5 :
Figure 4: The practical value and estimated value of the velocities.

Figure 10 :
Figure 10: Angular velocity change of the vessels.