Applied Ocean Research 36 (2012) 1–11 Contents lists available at SciVerse ScienceDirect Applied Ocean Research journal homepage: www.elsevier.com/locate/apor Ship’s tracking control based on nonlinear time series model , Jun Wu a , Hui Peng a,∗ , Kohei Ohtsu b , Genshiro Kitagawa c , Tohru Itoh d a School of Information Science & Engineering, Central South University, Changsha, Hunan 410083, China b Tokyo University of Marine Science and Technology, 2-1-6 Etchujima, Koto-ku, Tokyo 135-8533, Japan c Research Organization of Information and Systems, 4-3-13 Toranomon, Minato-ku, Tokyo 105-0001, Japan d Advanced Simulation Technology of Mechanics R&D, Co. Ltd., 2-3-13 Minami, Wako-shi, Saitama 351-0104, Japan a r t i c l e i n f o Article history: Received 1 May 2011 Received in revised form 11 January 2012 Accepted 12 January 2012 Available online 15 February 2012 Keywords: Radial basis function (RBF) AutoRegressive model with eXogenous variable (ARX) Ship tracking control Model predictive control (MPC) Real-time control results a b s t r a c t For actualization of ship tracking control along a desired path with a constant velocity, a hybrid model is proposed to represent the ship’s tracking dynamic behavior. Firstly, a single-input single-output nonlin- ear time series model is built for characterizing the responses between the ship’s heading angle deviation and its rudder angle. To represent nonlinearity of the ship motion, the rolling angle is used as the model index to make the model parameters vary with the ship sailing states considering the yaw-heel-effect. The nonlinear time series model is identified offline by using previously observed real data. Then, a state-space model combined with the relationship between the heading angle deviations and the cross track errors is proposed to represent the tracking dynamic behavior. On the basis of the identified state- space type tracking motion model, a predictive controller is designed to steer the ship sailing forward with the constant velocity along the predefined reference path. The effectiveness of the nonlinear time series model-based method for the tracking control proposed in this paper is demonstrated by simulation studies and actual experiments. © 2012 Elsevier Ltd. All rights reserved. 1. Introduction The principal role of the classical autopilot of a ship is mainly concerned with keeping its heading angle to a desired course angle. However, nowadays by making use of the position feedback signals obtained from a GPS navigation device, a ship guidance system, namely a tracking control system, can be designed to make the ship track a desired trajectory. The first and simple way is to change the ship’s course to keep its trajectory following the desired trajectory. The principles in many commercial base tracking systems follow this type of control [1]. However, it does not directly consider the rudder motion. Therefore, the rudder motion might make useless large angle in many cases.  This document is a collaborative effort.  This work was supported by the Institute of Statistical Mathematics, Tokyo, Japan, and Tokyo University of Marine Science and Technology, Japan and by the International Science & Technology Cooperation Program of China under Grant 2011DFA10440, the Key International Cooperation Programs of China’s Hunan Provincial Science & Technology Department under Grant 2009WK2009 and the National Natural Science Foundation of China under Grant 70921001, and also received the financial support from Japan Radio Company. ∗ Corresponding author. Tel.: +86 138 08429700; fax: +86 731 88830642. E-mail addresses: Jun.WU@csu.edu.cn, jun.wu@qq.com, jun.vvu@gmail.com (J. Wu), huipeng@mail.csu.edu.cn (H. Peng), ohtsu@kaiyodai.ac.jp (K. Ohtsu), kitagawa@rois.ac.jp (G. Kitagawa), itoh@astom.co.jp (T. Itoh). The recent important concern for ship owners is to save fuels as much as possible. In order to realize it, the captain determines the optimum route using weather routing technique considering the ship’s resistance due to wind, wave and current. Namely, in real navigation, a ship must track along the selected route as precise as possible by efficient rudder control. This is an important role of the ship’s tracking system. Holzh ¨ uter [2] presented a linear-quadratic Gaussian (LQG) approach for high-precision track control of a ship, which can avoid useless rudder motion by imposing an appropri- ate penalty to the rudder motion in the cost function. However, the ship’s model in the paper contains many hydro-dynamical param- eters which were assumed to be known or need to be estimated by troublesome maneuvering tank tests. Kvam et al. [3] and Fukuda et al. [4] also proposed this type of tracking control based on the Bryson and Ho’s time varying control theory. However, these con- trollers did not consider the rolling motion induced by the rudder motion and did not give a general method of tracking to a circular route. There have been many researches on ship tracking control based on ship physical models, which revealed very good tracking con- trol performances. Pettersen and Nijmeijer [5] and Lefeber et al. [6] studied a complete state-tracking problem for an underactuated ship that has only surge control force and yaw control moment. The ship dynamics was described by a simplified physical model based on some assumptions. Using the same physical model, Jiang [7] designed two systematic tracking controllers with the aid of Lyapunov’s direct method. Moreover, it shows how to render the 0141-1187/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.apor.2012.01.004