Trajectory tracking control for underactuated surface vessels using neural network

LIU Yang(刘杨)¹, GUO Chen(郭晨)²

(1. School of Electronic and Information Engineering, Dalian Jiaotong University, Dalian 116028, China;

2. College of Information Science and Technology, Dalian Maritime University, Dalian 116026, China)

Abstract: Aiming at the trajectory tracking control of underactuated vessels with parameters uncertain and external disturbance problems, a stable adaptive neural network control method is proposed. Based on the diffeomorphism transformation, the new tracking variables are given. First, the kinematics controller is designed to track the new variables and the velocity command is proposed. Second, the dynamics controller using neural network method is designed to make the real velocity of the underactuated surface vessel reach the desired velocity command. All the parameters’ update laws are selected in terms of Lyapunov functions, which guarantee the closed-loop systems to be uniformly ultimately bounded (UUB). Finally, a numerical simulation illustrates the effectiveness of the proposed control method.

Key words: underactuated vessel; trajectory tracking; nonlinear system; neural network

CLC number: TP273                  Document code: A             Article ID: 1672-7207(2011)S1-0032-04

1 Introduction

The past few decades have witnessed an increased research effort in the area of motion control of underactuated surface vessels. A typical motion control problem is trajectory tracking control^[1]. Which is demand to design of control laws that force a vessel to reach and follow a time parameterized reference. There is another definition of the trajectory tracking that is allowed to reach the geometric path with an associated timing law. The trajectory tracking problem for the underactuated surface vessel is especially challenged because most of these systems are not fully feedback linearizable and nonholonomic constraints, in general, it can’t be solved by smooth time-invariant state feedback or continuous control^[2]. In Refs. [3-4], the trajectory tracking control of nonlinear underactuated surface vessels has been studied. But the yaw velocity was needed to be nonzero. And the problem was solved in Ref. [5], that the yaw velocity is not nonzero. In Ref. [6], the vessel’s dynamics include nonlinear off-diagonal damping terms. In this paper, we needn’t assumption that the mass and damping matrices of the vessel to be diagonal, because the uncertain nonlinear off-diagonal terms can be estimated by neural network. Using the diffeomorphism transformation, new tracking variables are given. In kinematics loop, based on the new tracking errors, the reference yaw angle and the reference speed are designed. In dynamics loop, an adaptive neural network controller is proposed, and all the parameters’ update laws are selected in terms of Lyapunov functions, which guarantee the closed-loop systems to be UUB.

2 Ship model and problem descriptions

We consider the underactuated surface vessel that can be described by the 3 degrees of freedom (DOF) model^[1]:

          (1)

where (x, y) is the position of the vessel;  is the orientation in the earth-fixed frame; ; u, v and r are the velocities in the surge, sway and yaw directions, respectively; ,  and  are unknown constant vectors with known dimensions, ,  and  are known smooth function vectors;  is the surge force and  is the yaw force; g_u and g_r are nonzero positive constant coefficients; b_u, b_v and b_r are unstructured uncertainties which include external disturbance and measurement noise, etc.

Define the reference trajectory as (x_d(t), y_d(t), ψ_d(t)). In general, there is a deviation angle between the orientation of the vessel and tangent of the reference trajectory when external disturbance exists. So we consider the dynamic position tracking problems that reach the geometric path with an associated timing law.

3 Controller design

First, we define the new errors that x_e=x-x_d, y_e=y-y_d and,, where .

Lemma 1: (z₁, z₂) is the diffeomorphism of (x_e, y_e).

Proof: Define , we can obtain

Then we know that at the point (x_e, y_e)=(0, 0), the relative degree is 2. So (z₁, z₂)is the diffeomorphism of (x_e, y_e).

3.1 Kinematic controller design

Derive the variable z₁ and z₂, and  ,  . Where ;  when ,  when ;  , .

The new variables   is proposed. The state (1) can be written as

        (2)

            (3)

Define the candidate Lyapunov function for the state variable (2) such that . If we propose the desirable velocity variable and desirable orientation as ,  . The time derivative of V₁ becomes  where k₁＞0, k₂＞0.

3.2 Dynamics controller design

We propose a new variable . And the tracking errors here proposed as , .

Remark 1:  denotes n-dimensional Euclidean space, and  denotes a set of  Euclidean matrices.

Define χ as a variable, then  denotes Euclidean norm.  is the absolute value of χ.denotes the estimation of χ.is the corresponding estimation error, denotes the minimum singular value of the matrix .

RBF neural networks are used to approximate the nonlinear uncertain functions in dynamics loop, considering the on-line approximation of the uncertain terms by linearly parameterized approximation models. The RBF neural networks are of the general form, whereis a vector of regulated weights and a vector of RBF’s. It is shown that given a smooth function , where  is a compact subset of , m is an appropriate integer, there exists an RBF vector  and a weight vector  such that  . Here ε is called network reconstruction error, it can also be considered as uncertainty due to modeling errors. The unknown parameters w will be estimated on-line. Let θ be the parameter estimation value of w, and develop adaptive laws for updating these parameters. So the uncertain terms can be written as , ,  , . We take the control input , and parameters’ update laws as

 (4)

  (5)

       (6)

       (7)

     (8)

     (9)

    (10)

     (11)

where , are strictly positive definite matrix; ＞0, ＞0, ＞0, ＞0, ＞0; w_ui0 and w_ri0 are design parameters; i=f, g. And the smooth functionsatisfies that  and , ＞0, ^[7].

4 Stability analysis

Theorem 1: Consider an underactuated vessel described by the dynamic system (1). If we chose the control law as Eqs. (4) and (5), and parameters’ estimation laws as Eqs. (6)-(11), then all the signals in the closed-loop system are satisfied as UUB.

Proof: Define the following candidate Lyapunov function:

  (12)

Differentiating Eq. (22) and substituting Eqs. (14)- (21) into it, finally we have

      (13)

   (14)

where .

Therefore, the entire signals written in Eq. (12) satisfy as UUB. Define  , . So the subsystems  and  are cascade systems. Based on the analysis above, we know that the signals listed in  as being global uniformly asymptotically stable, and the signals listed in  as UUB, then the signals listed in the  as UUB.

5 Simulations

The underactuated surface vessel model which used in Ref. [7] is adopted, and the reference trajectory is selected as:,, ,  , u_d=5,  when t≤200, r_d=0.4 when 200＜t＜310, r_d=0 when t＞310. Take the parameters as: k₁=0.01, k₂=0.5, k₃=0.4, k₄=4, k₅=1, , ,  , . Initial conditions are as follows: , , , ,  , ,  , h=6,   .

The numerical simulation results are shown in Fig.1, which shows the trajectory tracking results, and position and orientation tracking errors, and velocity tracking errors. From the results we can easily know that all the signals tracking errors are UUB.

Fig.1 Trajectory tracing and tracking errors

6 Conclusions

A stable adaptive neural control method is proposed for the underactuated surface vessels. Lyapunov theory guarantees all the signals in closed-loop systems to be UUB. The simulation shows the underactuated surface vessel can track the desired trajectory even with parameters uncertain and external disturbances.

References

[1] Fossen T I. Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[M]. Trondheim, Norway: Marine Cybernetics, 2002: 17-113.

[2] Reyhanoglu M, van der Schaft A J, McClamroch N H, et al. Dynamics and control of a class of underactuated mechanical systems[J]. IEEE Transactions on Automatic Control, 1999, 44(9): 1663-1671.

[3] Jiang Z P. Global tracking control of underactuated ships by Lyapunov’s direct method[J]. Automatica, 2002, 38(2): 301-309.

[4] Pettersen K Y, Nijmeijer H. Underactuated ship tracking control: theory and experiments[J]. International Journal of Control, 2001, 74(14): 1435-1446.

[5] Do K D, Jiang Z P, Pan J. Underactuated ship global tracking under relaxed conditions[J]. IEEE Transactions on Automatic Control, 2002, 47(9): 1529-1536.

[6] Do K D, Pan J. Global tracking control of underactuated ships with nonzero off-diagonal terms in their system matrices[J]. Automatica, 2005, 41(1): 87-95.

[7] Li J H, Lee P M, Jun B H. Point-to-point navigation of underactuated ships[J]. Automatica, 2008, 44(12): 3201-3205.

(Edited by CHEN Can-hua)

Received date: 2011-04-15; Accepted date: 2011-06-15

Foundation items: Project (61074053) supported by the National Natural Science Foundation of China

Corresponding author: LIU Yang(1981-), PhD, Lecturer; Tel: +86-13942076915; E-mail: muxiaoyi123@sina.com