## 2 Problem Formulation  
\(O-X_{0}Y_{0}Z_{0}\), which denotes the Earth-fixed inertial frame, and \(XYZ\), which denotes the body-fixed frame, are revealed in Figure 1.  
The kinematics equation is indicated by  
\[\dot{\xi}=J(\psi)v \tag{1}\]  
where \(\zeta=[x,y,\psi]^{T}\) denotes the position and the heading angle of the USV, respectively. \(v=[u,v,r]^{T}\) denotes surge velocity, sway velocity and yaw rate, respectively. \(J(\psi)\) represents the transition matrix:  
\[J(\psi)=\left[ \begin{array}{ccc}\cos(\psi)&-\sin(\psi)&0\\ \sin(\psi)&\cos(\psi)&0\\ 0&0&1\end{array} \right] \tag{2}\]  
Then, the kinematic equations can be expressed as [35]. \[\left\{ \begin{array}{l}\dot{x}=u\cos\psi-v\sin\psi\\ \dot{y}=u\sin\psi+v\cos\psi\\ \dot{\psi}=r\end{array} \right. \tag{3}\]  
According to Paper [13], the dynamic equations of an underactuated USV model can be represented by  
\[\left\{ \begin{array}{l}\dot{u}=\frac{m_{2}}{m_{11}}v_{T}-\frac{d_{11}}{m_{1 1}}u+\frac{\tau_{u}+d_{u}}{m_{11}}\\ \dot{v}=-\frac{m_{11}}{m_{12}}u-\frac{d_{22}}{m_{22}}v+\frac{d_{2}}{m_{22}}\\ r=\frac{m_{11}-m_{22}}{m_{33}}uv-\frac{d_{33}}{m_{33}}r+\frac{\tau_{u}+d_{t}}{ m_{11}}\end{array} \right. \tag{4}\]  
where \(\tau_{u}\) and \(\tau_{r}\) denote the control input of surge force and the control input of yaw moment, respectively. \(b_{u}\), \(b_{v}\) and \(b_{r}\) represent immeasurable external environmental disturbances owing to waves, wind and ocean current. Figure 1: The Earth-fixed inertial and body-fixed frames. where \(U=\sqrt{u^{2}+v^{2}}\), \(\beta=arc\tan(v,u)\) represents the sideslip angle and \(U_{pp}=\theta\sqrt{x^{2}_{p}+y^{\prime 2}_{p}}\) denotes the virtual reference speed to stabilize \(x_{e}\). A local path parallel reference frame is revealed in Figure 2.  
**Assumption 3**. : _During unmanned surface vessel path following, \(\beta\) is relatively small, \(|\beta|\leq\beta^{*}\) and \(\beta^{*}\) denotes a positive constant number. It is assumed that \(\beta\) changes slowly. \(|\dot{\beta}|\leq C_{\beta}\) and \(C_{\beta}\) denotes a positive constant number. Therefore, \(sin(\beta)=\beta\) and \(cos(\beta)=1\)._  
Rewrite the position error derivatives as the following:  
\[\dot{x}_{e}=U\cos(\psi-\gamma_{p})-U\sin(\psi-\gamma_{p})\beta-U_{pp}+y_{e} \hat{\gamma}_{p} \tag{13}\]  
\[\dot{y}_{e}=U\sin(\psi-\gamma_{p})+U\cos(\psi-\gamma_{p})\beta \tag{14}\]  
Define \(\hat{\beta}\) as an estimated value of \(\beta\). \(\tilde{\beta}=\beta-\hat{\beta}\) denotes the estimation error. Design the Lyapunov function as follows:  
\[V_{g}=\frac{1}{2}x_{e}^{2}+\frac{1}{2}y_{e}^{2}+\frac{1}{2k_{\beta}}\tilde{ \beta}^{2} \tag{15}\]  
where \(k_{\beta}\) is the positive design parameters. Take the time derivative Equation (16) along (14) and (15); then, the following equation is obtained:  
\[\dot{V}_{g}= x_{e}\big{[}U\cos(\psi-\gamma_{p})-U\sin(\psi-\gamma_{p})\beta+ \hat{\gamma}_{p}y_{e}-U_{pp}\big{]}\] \[+y_{e}\big{[}U\sin(\psi-\gamma_{p})+U\cos(\psi-\gamma_{p})\beta- \hat{\gamma}_{p}x_{e}\big{]}+\frac{1}{k_{\beta}}\hat{\beta}(\hat{\beta}-\hat{ \beta}) \tag{16}\]  
IALOS can effectively compute the anticipated yaw angle. In practice, \(\beta\) is often difficult to measure, so the adaptive strategy is employed to estimate the time-varying \(\beta\). **Assumption 4**. : _The reference heading angle can followed perfectly; therefore, \(\psi=\psi_{d}\)._  
Based on (15), the desired heading angle, the sideslip angle adaptive law and the path parameter update law are obtained [16]. \[\psi_{d}=\gamma_{p}-acr\tan(\frac{1}{\Delta}y_{e}+\hat{\beta})) \tag{17}\]  
Figure 2: LOS guidance geometry for curved paths. \[\dot{\beta}=k_{\beta}\left[\frac{Uy_{e}\Delta}{\sqrt{\Delta^{2}+\left(y_{e}+\Delta \hat{\beta}\right)^{2}}}-\hat{\beta}\right] \tag{18}\]  
and  
Figure 2.0  
where \(\Delta\) denotes the positive design parameters. Based on Equation (18), it is found that  
Figure 2.1 Figure 2.2  
where \(\psi_{e}=\psi-\psi_{d}\).