# Event-Sampled Adaptive Neural Course Keeping Control for USVs Using Intermittent Course Data
## 2 Problem Formulation and Preliminaries  
### Problem Formulation  
In the vessel CKC system of USVs, the Nomoto model [7] is usually used to describe the motion process of vessel steering. However, the applicability of Nomoto model is limited, since it is obtained under the condition of small steering angle and low frequency steering. Due to the manipulation process, vessels have the typical nonlinear characteristics [8]. In addition, USVs are characterized by fast steering and large slalom. Therefore, in the control design, it is appropriate to invoke the nonlinear model of Norbbin [9], which can be described as  
\[\ddot{\psi}+hg(\dot{\psi})=b\delta+d \tag{1}\]  
where \(\psi\) is the vessel course; \(h\) and \(b\) are the parameters associated with the manipulation feature, symbiform, loading capacity, and speed; \(\delta\) is the steering angle; \(d\) is the equivalent disturbance; and \(g(\dot{\psi})=a_{1}\dot{\psi}+a_{2}\dot{\psi}^{3}\) is a nonlinear function on the angular turning speed of the vessel. **Remark 1**. : _To better carry out offshore operations, the control scheme of USVs must be highly adaptable and flexible. In addition to considering the control algorithm, the kinematic mathematical model of the USVs should also be considered, and a suitable mathematical model can better describe the motion state of the USVs. The mathematical models commonly used to solve the heading control problem of USVs include the Nomoto model and Northbin model. The Nomoto model is a linear model which is used to describe the motion characteristics of the ship in a straight line, and it is difficult to describe the nonlinear control of the ship in the turning state. In this case, the Northbin model is usually selected. Compared with the Nomoto model, the Northbin model can better describe the steering of USVs._  
Let \(\chi_{1}=\psi\), \(\chi_{2}=\psi\) and \(\tau=\delta\). Furthermore, according to (1), one can obtain  
\[ \begin{cases}\dot{\chi_{1}}=\chi_{2}\\ \dot{\chi_{2}}=f(\chi_{2})+b\tau+d\\ y=\chi_{1}\end{cases}  \tag{2}\]  
where \(f(\chi_{2}=-g(\chi_{2}))\), \(\tau\in R\) is the control input, and \(y\in R\) is the control output, i.e., the ship course \(\psi\). To facilitate the design and analysis, the following assumptions regarding the system (1) are listed. **Assumption 1**. : _The equivalent disturbance \(d\) is bounded, that is, \(|d|\leq\delta\) with \(\delta\) being a constant._  
**Assumption 2**. : _The nonlinear term \(g(\psi)\) and parameter \(b\) are unknown._  
**Assumption 3**. : _The reference course \(y_{r}\) is smooth, and \(\dot{y_{r}}\) and \(\dot{y_{r}}\) are bounded._  
**Remark 2**. : _In this work, the above assumptions are formulated to solve the CKC problem of USVs in a network environment with network resource constraints and discontinuities of course and yaw caused by data transmission. USVs are certainly subject to disturbances from the external environment during actual navigation. Therefore, an equivalent disturbance \(d\) is introduced, which is time-varying and bounded. If the equivalent disturbance \(d\) is not bounded, then it is unreasonable, and a control design method that satisfies this condition does not exist for the time being. From the point of view of engineering practice, the interference to which any USVs are subjected must be bounded, and the control scheme designed under this premise is reasonable. Assumption 2 implies that no prior knowledge of the USV model parameters is required when developing the control scheme. In order to ensure that the ideal trajectory \(y_{r}\) of the USV can be described by a smooth curve, and to ensure the rationality of the control design scheme, Assumption 3 is proposed for this purpose._  
**Remark 3**. : _In practice, due to the modeling technique, ship maneuvering environment, external disturbance, etc., the model parameters are difficult to obtain accurately. In addition, the nonlinear characteristic is inherent in the ship maneuvering model.