## 3 Main Results
### Stability Analysis  
In this section, the main theorem of the path following controller is presented. Theorem 4.1: _Consider the USV model in Equations (4) and (5) in the presence of uncertainties and unknown external time-varying disturbances under input saturation, and suppose that Assumptions 1 and 2 are satisfied, under the guidance law in Equation (11) along with the finite-time currents observer in Equation (12). The given path is parameterized by \(\theta\) with the update laws in Equation (21), and the desired velocity \(u_{d}\) is given as well. The control laws in Equation (27) together with the adaptive laws in Equations (28) and (29) and the auxiliary systems in Equations (30) and (31) are incorporated to assist in handling input saturation, guaranteeing that all tracking error signals are locally UIUB._  
Consider the following Lyapunov function  
\[V=\frac{1}{2}x_{e}^{2}+\frac{1}{2}y_{e}^{2}+\frac{1}{2}\tilde{\psi}^{2}+\frac{1 }{2}m_{33}\tilde{r}^{2}+\frac{1}{2}m_{11}\tilde{u}_{r}^{2}+\frac{1}{2}\sum_{i=1 }^{2}\sigma_{i}^{2}+\frac{1}{2}\sum_{i=1}^{2}\tilde{\mathbf{W}}_{i}^{T}\mathbf{ \Gamma}_{i}^{-1}\tilde{\mathbf{W}}_{i}+\frac{1}{2\xi_{i}}\sum_{i=1}^{2}\delta_{ i}^{2} \tag{33}\]  
Taking the time derivative of Equation (33) along with Equations (22)-(24) and (32) yields  
\[ \begin{split}\dot{V}&\leqslant-k_{1}x_{e}^{2}- \frac{U_{r}}{\sqrt{\left(y_{e}+\alpha_{y}\right)^{2}+\Delta^{2}}}y_{e}^{2}-k_{ 2}\tilde{\psi}^{2}-k_{3}\tilde{r}^{2}-\tilde{\mathbf{W}}_{1}\boldsymbol{ \varphi}(\mathbf{z})\tilde{r}-\delta_{1}h(\tilde{r})\tilde{r}\\ &\quad+k_{\xi 1}\sigma_{1}\tilde{r}+\delta_{1}\tilde{r}+\Delta\tau_{r} \tilde{r}-k_{4}\tilde{u}_{r}^{2}-\tilde{\mathbf{W}}_{2}\boldsymbol{\varphi}( \mathbf{z})\tilde{u}_{r}-\delta_{2}h(\tilde{u}_{r})\tilde{u}_{r}+k_{\zeta 2} \sigma_{2}\tilde{u}_{r}+\delta_{2}\tilde{u}_{r}+\Delta\tau_{u}\tilde{u}_{r}\\ &\quad+\tilde{\theta}_{x}x_{e}+\tilde{\theta}_{y}y_{e}+\sum_{i=1 }^{2}\sigma_{i}\dot{\sigma}_{i}+\sum_{i=1}^{2}\tilde{\mathbf{W}}_{i}^{T} \mathbf{\Gamma}_{i}^{-1}\dot{\mathbf{W}}_{i}+\frac{1}{\xi_{i}}\sum_{i=1}^{2} \delta_{i}\dot{\delta}_{i}^{\dot{\delta}}\end{split}  \tag{34}\]  
(1) When \(|\sigma_{i}|\geqslant\mu_{i}(i=1,2)\), according to Equations (27) to (30) and [PERSON]'s equalities \(k_{\xi 1}\sigma_{1}\tilde{r}\leqslant\frac{1}{2}\tilde{r}^{2}+\frac{1}{2}k_{ \zeta 1}^{2}\sigma_{1}^{2}\), \(\sigma_{1}\Delta\tau_{r}\leqslant\frac{1}{2}\sigma_{1}^{2}+\frac{1}{2}\Delta \tau_{r}^{2}\), \(k_{\zeta 2}\sigma_{2}\tilde{u}_{r}\leqslant\frac{1}{2}\tilde{u}_{r}^{2}+\frac{1}{2}k_{ \zeta 2}^{2}\sigma_{2}^{2}\), \(\sigma_{2}\Delta\tau_{u}\leqslant\frac{1}{2}\sigma_{2}^{2}+\frac{1}{2}\Delta \tau_{u}^{2}\), \(\tilde{\theta}_{x}x_{e}\leqslant\frac{1}{2}\tilde{\theta}^{2}+\frac{1}{2}x_{e}^ {2}\), and \(\tilde{\theta}_{y}y_{e}\leqslant\frac{k_{1}}{2}\tilde{\theta}^{2}+\frac{1}{2k _{1}}y_{e}^{2}\), we have  
\[ \begin{split}\dot{V}&\leqslant-\left(k_{1}-\frac{1 }{2}\right)x_{e}^{2}-\left(\frac{U_{r}}{\sqrt{y_{e}^{2}+\Delta^{2}}}-\frac{1}{2 k_{1}}\right)y_{e}^{2}-k_{2}\tilde{\psi}^{2}-\left(k_{\sigma 1}-\frac{1}{2}-\frac{1}{2}k_{ \zeta 1}^{2}\right)\sigma_{1}^{2}-\left(k_{3}-\frac{1}{2}\right)\tilde{r}^{2} \\ &\quad-\left(k_{\sigma 2}-\frac{1}{2}-\frac{1}{2}k_{\zeta 2}^{2} \right)\sigma_{2}^{2}-\left(k_{4}-\frac{1}{2}\right)\tilde{u}_{r}^{2}-\sum_{i=1 }^{2}\iota_{i}\tilde{\mathbf{W}}_{i}^{T}\tilde{\mathbf{W}}_{i}-\sum_{i=1}^{2} \lambda_{i}\tilde{\delta}_{i}\left(\delta_{i}-\delta_{i}^{0}\right)\\ &\quad+\tilde{r}\left(\delta_{1}-\delta_{1}h\left(\tilde{r}\right) \right)+\tilde{\delta}_{1}h\left(\tilde{r}\right)\tilde{r}+\tilde{u}_{r}\left( \delta_{2}-\delta_{2}h\left(\tilde{u}_{r}\right)\right)+\tilde{\delta}_{2}h \left(\tilde{u}_{r}\right)\tilde{u}_{r}+\frac{1+k_{1}}{2}\theta^{2}\end{split}