# Adaptive Trajectory Tracking Control for
## 1 Introduction  
With the continuous advancement of science and technology, the modern ship system is moving toward a systematic, intelligent and unmanned direction. In recent years, a new type of ship research project has been developed by combining the achievements of small speedboats and advanced control technologies, like the Unmanned Surface Vehicles (USVs) [1; 2]. Following the trend of ship systems, USVs has become a unique research field, which attracts more and more institutions and researchers to conduct relevant research [3; 4]. Generally, trajectory tracking is an important topic in the field of USV research, and it is the guarantee for many applications in the future, such as formation operations, and so on [5; 6]. However, a very realistic and urgent problem to be solved is that the overwhelming majority of USVs with propellers and rudders belong to an underactuated system, while without any actuators for direct control of sway motion [7; 8; 9]. Therefore, how to design robust trajectory tracking controllers under the condition of underactuated characteristics for USVs is a very worthwhile problem. Many researchers have made outstanding contributions to cope with the trajectory tracking problem for underactuated ships [10; 11; 12; 13]. In [14], a unified backstepping design methodology is hired to solve the stabilisation and tracking problems for an underactuated ship with only two propellers. [PERSON] proposes a modular way that cascaded kinematic and dynamic linearizations can be achieved similarly as in the backstepping method. Based on this, under the premise of inputand velocity constraints, dynamic surface control (DSC) technology is used to design a trajectory tracking control strategy for an underactuated ship [15]. For an underactuated USV, the experimental implementation of two trajectory tracking control algorithms: a cascade of proportional-derivative controllers and a nonlinear controller are proposed by [PERSON] et al. through the backstepping control technology [16]. In [17; 18], for an underactuated USV, the switching control and modified backstepping methods are used to design the trajectory tracking controller, which can track a straight line track or fixed point. However, in actual navigation, besides the external disturbances caused by wind, waves and currents, the ship's quality, speed and moment of inertia will also change, which will lead to some changes or unknown dynamics in the model of ship. In [19], the backstepping method is hired to design the USV trajectory tracking controller. At the same time, a disturbance observer is employed to compensate for the time-varying external disturbances on line. The sliding mode algorithms have strong robustness, which are often introduced into the motion control of various nonlinear systems. Hierarchical sliding mode-based trajectory tracking controller is designed to allow the underactuated surface vessels to track a pre-specified trajectory [20]. Besides, in [21], the sliding mode control algorithm is introduced into the design of a trajectory tracking controller for underactuated USV, but it does not consider the effects of unknown dynamics and time-varing disturbances. In practical engineering application, when dealing with model structure unknown or unknown dynamics problems, the universal approximation ability of neural network [22; 23] or fuzzy logic [24; 25; 26] is widely used. Fuzzy logic needs experts' prior knowledge, which often increases the design difficulty of controller. So the multilayer neural network represented by radial basis function (RBF) neural network is the most widely used method to deal with the unknown dynamics problem. Paper [27] proposes a single layer neural network to cope with the completely unknown vehicle dynamics for an autonomous surface vehicle (ASV) to track a predetermined trajectory.