## 1 Introduction
According to engineering practice, the ship model has the characteristics of nonlinearity and uncertainty due to the continuous change of operating conditions. To deal with the uncertainty of the model problem, neural network (back propagation (BP) neural network and radial basis function (RBF) neural network) [18; 19] or fuzzy logic [20; 21], which have the ability of universal approximation, are usually introduced into the control system. However, fuzzy logic requires more expert prior knowledge, so the neural network is used more widely in the field of universal approximation. In [22], a new novel neural network adaptive controller was developed for cooperative path following of marine vessels. Although the neural network can solve the problem of the unknown dynamic model, as a multi-layer neural network, it will undoubtedly increase the amount of computation of the control system. Besides, in practical engineering, there is a very important factor that needs to be taken into consideration, that is the input saturation of the actuator. Input saturation is easy to ignore for the path following design since the commanded control inputs calculated by the path-following control laws are possibly constrained by the maximum outputs that the actuator can produce. If input saturation is not taken into consideration, it is very likely to reduce system performance and even lead to instability of the entire path following. In [23], an auxiliary design system was introduced to analyze the effect of input constraints, and its states were used for adaptive tracking control design. In the presence of unknown time-varying disturbances and input saturation, an auxiliary dynamic system, a disturbance observer and a dynamic surface control (DSC) technique were used to design a robust nonlinear control law for a fully-driven supply ship [24]. In [25], a path-following controller for USV was developed by combining the neural network and DSC technique subject to input saturation. Motivated by the above-mentioned observations, the goal of this article is that based on the establishment of the padded propulsion USV model, the ALOS algorithm with a time-varying \(\Delta\) is proposed to provide guidance. Besides, a path-following controller, which is proposed by using the backstepping method, the neural shunting model, the neural network minimum parameter learning method and the auxiliary dynamic system, is developed for USV without knowing the exact information of the model structure and the time-varying external disturbances. The main contributions of this note can be summarized as follows:  
1. Based on force analysis and MMG (Ship Manoeuvring Mathematical Model Group)separation modeling theory, the padded propulsion USV is proven to be an underactuated system. 2. An improved LOS algorithm is employed as a navigation strategy for USV, which means that it not only ensures the expected compensation effect, but also avoids the use of expensive sensor equipment. 3. A novel neural shunting model is adopted to deal with the \"explosion of complexity\", which can reduce the computational complexity of the control system. 4. Model uncertainties and time-varying external disturbances are estimated by the neural network minimum parameter learning method. Compared with RBF and BP, the neural network minimum parameter learning method has a smaller amount of computation. 5. The auxiliary dynamic system is introduced to prevent the input saturation problem, which is closer to practical engineering. The rest of the paper is organized as follows. Section 2 describes the process of modeling. In Section 3, ALOS with a time-varying \(\Delta\) guidance algorithm is proposed. In Section 4, the path following controller is designed. The stability of the closed-loop system is demonstrated in Section 5. In Section 6, numerical simulations are given to prove the correctness of the strategy proposed in this note. Finally, Section 7 summarizes the full text.