## 1 Introduction
Combining an adaptive fuzzy backstepping technique with Nussbaum approach, the researchers in [13] proposed a novel Nussbaum-based adaptive fuzzy control scheme for trajectory tracking of a USV in the presence of complex unknown nonlinearities and fully unknown dynamics, and tracking errors can converge to an arbitrarily small neighborhood of zero using this method. In [14], the authors proposed an estimator-based backstepping controller to control the unmanned surface ship to follow a desired trajectory considering external disturbance and system uncertainty, and the estimator was used to precisely estimate external disturbance and system uncertainties, which guaranteed that the desired trajectory can be followed with an exponential rate of convergence. A novel model based backstepping controller was designed in [15] for trajectory tracking control of underactuated USV, in which the well-known persistent exciting conditions of yaw velocity was completely relaxed; the USV could track an arbitrary trajectory, including a circle, straight-line and generally curved trajectories. In [16], an adaptive robust controller proposed by hierarchical sliding mode and a neural network was used for trajectory tracking and stabilization of underactuated surface vessels. From the above review we can see that previous research about the trajectory tracking problem of USVs have motivated researchers to proposed different control methods, such as SMC, backstepping technique, [PERSON]'s direct method, robust control method and hybrid control technology. However, in [2; 3; 4; 5; 6; 7; 8; 10; 15], though different traditional methods were conducted, many of them didn't consider the parameter uncertainties or disturbances. In [9], the author considered disturbance rejection, but control inputs were not analyzed. In [11; 12; 13; 14; 16], novel strategies were proposed considering parameter uncertainties or disturbances, and they were effective for the trajectory tracking of USVs. However, in most of the recent studies, USVs or ships are required to follow a specified path or track a predefined position strictly. Unlike the specified path or trajectory tracking of the previous researches, in this study, the proposed strategy is conducted to control the USV to track a moving regional trajectory rather than move along a pre-defined trajectory strictly. We know that, in practice, some missions of USVs or ships do not require a strict point or a strictly particular path. For example, a USV sailing through a narrow waterway can be considered sailing through a region rather than a particular path. A USV or ship sailing into a port can be regarded as sailing into a region rather than a strict point. Thus, when the USV or ship sails in the open sea, a strict trajectory tracking is unnecessary. For the trajectory tracking problem, one can consider control of a USV to track a common moving target region instead of a moving point. Then, the USV will track a region trajectory generated by the moving region. We call this problem a non-strict trajectory tracking. Figure 1 shows the concept of non-strict trajectory tracking. The USV can move into any place of the moving region trajectory rather than strictly moving along the pre-defined trajectory. [PERSON] et al. [17; 18] proposed a new region reaching concept for a robot manipulator and adopted similar concepts for an underwater vehicle in 2004 [19]. The desired region was specified by an objective function \(f(\delta X)\leq 0\), where \(\delta X\) is the error in position leading to the region control (RC) problem. [PERSON] [20] proposed a new region-reaching controller for an underwater vehicle mounted with a manipulator based on RC. [PERSON] [21] used RC to control a swarm of robots to establish any arbitrary complex formation. Region-reaching tasks save energy and result in fast motion [18]. Region control is a good method for non-strict trajectory tracking. Some related works have been conducted in recent years. [PERSON] et al.