**1. Introduction**

A ship towing system (STS) consists of a tugboat, a towline, and a towed ship [1]. Owing to its powerful transportation ability, the STS plays an increasing role in the development of marine resources, such as oil, natural gas, mineral resource, etc. In the past, due to external environmental disturbances and inherent internal uncertainties, the motion control of the STS was mostly based on experimental works or numerical simulations, rather than theoretical analysis [2]. As a result, an improper control would cause the actual towing trajectory deviate from the target towing route. This may lead to collisions, capsizing, and other safety accidents. As a consequence, it is necessary to investigate the precise motion control of the STS for its safe navigation at sea.

For the STS, it is subject to non-holonomic constraints when the lateral drift motion is small enough to be neglected. In this case, the inter-coupling action generated by the relative motion among tugboat, towline, and towed ship makes the trajectory planning and motion control of the STS especially challenging [3]. In addition, the STS is affected by various factors and its dynamics model is extremely complex, thus imposing challenges to the model of STS. Accordingly, the related research studies mainly focus on the simplified models. For example, in References [4,5], based on the local linearization stability analysis method, the nonlinear dynamics model of the STS was approximated into a linearized model. In Reference [6], the nonlinear dynamics equation of the STS was transformed into a six-dimensional state space model, then the equation was approximated by Taylor series. However, these methods only solve the nonlinear problem of the STS locally. In addition, in Reference [7], the investigation showed that the nonholonomic constraints were destroyed when the hull occurred lateral drift motion. As a result, it is difficult to analyze the motion law of the STS clearly. To overcome this drawback, the relative width of the towed ship should be small. In this case, the STS is not prone to lateral drift so as to ensure the nonholonomic constraints of zero lateral velocity.

In general, STSs could be divided into two types. One is the towed ship without steering capacity, and the other is the towed ship with certain steering ability. To the former, its motion ability is completely depended on the traction of the tugboat, so it is fully passive.

**Citation:** Li, O.; Zhou, Y. Precise Trajectory Tracking Control of Ship Towing Systems via a Dynamical Tracking Target. *Mathematics* **2021**, *9*, 974. https://doi.org/10.3390/ math9090974

Academic Editors: Mikhail Posypkin, Andrey Gorshenin, Vladimir Titarev and Stevan Dubljevic

Received: 18 February 2021 Accepted: 24 April 2021 Published: 27 April 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

To the latter, it has a certain steering ability to achieve steering motion. For the case of the towed ship without steering ability, the dynamics equation of the STS can be derived by conventional method since the nonholonomic constraint is relatively simple. However, the main drawback of such systems is that the towed ship cannot follow the same trajectory as the tugboat during turning movements. In this case, the STS is easy to collide with obstacles. To address this issue, it is necessary to equip the towed ship with a steering assembly, so that it has a certain steering ability. In general, active steering and passive steering are two main steering strategies in practical implementation. Active steering commonly depends on an active control input, and the corresponding nonholonomic constraints become complex. So, it imposes difficulty in deducing the dynamics model [8]. Thus, it is a challenge to design the model-based controllers. In practice applications, the active controller is usually designed by measuring numerous accurate datas, which leads to complicated calculation and expensive cost. In terms of the passive steering method, the rear beam of the towed ship steers passively through a passive steering mechanism, such as the following-up steering. This is helpful to the system lateral stability against rollover.

Since the STS is an underactuated, nonholonomic, and nonlinear system, its motion control is indeed a challenging problem in the control community. The challenge is even harder when the external disturbance or internal uncertainty influence the system. At present, there are mainly two kinds of relevant research methods for the motion control of the STS. On the one hand, extensively studies consider kinematic models only. Usually, advanced control methods, such as model predictive control [9,10], adaptive control [11,12], sliding mode control [13], back-stepping control [14], etc., are used to design speed controllers [15,16] for the STS. According to the kinematics model, the nonlinear adaptive tracking control and nonlinear feedback tracking control methods, together with the path tracking algorithm, are adopted to make the towed ship track the trajectory of the towing boats [17,18]. On the other hand, some studies consider both kinematic and dynamics models at the same time [19,20]. Howeever, the main drawback of these research studies is that they do not make full use of the motion laws, resulting in complex control and insufficient precision. In addition, the problem of inconsistent tracking path between the tugboat and the towed ship cannot be fundamentally solved by only depending on advanced control methods and measurement technologies, which is mainly due to the following two reasons. At first, the steering of the towed ship is not matched with the tugboat, so that the towed ship is easy to deviate from the trajectory of the tugboat. Second, the speed error of the STS at the initial moment is very large, and the accumulated position errors cannot be adjusted. This leads to increasing accumulated position errors, so that the towed ship deviates increasingly from the trajectory of the tugboat. Therefore, it is reasonable to design trajectory tracking controllers by combining the motion laws with its dynamics equation, so that both the tugboat and the towed ship are able to track the same motion path.

In this paper, motivated by the above observations, we aim to seek a novel control strategy to solve the precise tracking control of the STS with two robust torque controllers and a passive steering angle. The major contributions of this paper are summarized as follows.


The remainder of this paper is structured as follows. Section 2 describes the mathematical model of the ship towing system. Section 3 focuses on designing two robust trajectory tracking controllers. Simulation results are reported and discussed in Section 4. Finally, some conclusions are given in Section 5.
