## 3 Main Results
### Design of Kinetic Controller  
In this section, a kinetic controller is designed to track the desired approaching angle and the desired surge velocity. Let \(\bar{r}=r-r_{d}\) and \(\tilde{u}_{r}=u_{r}-u_{d}\) be the attitude tracking error and surge velocity tracking error, respectively, where \(r_{d}\) is the virtual control law and \(u_{d}\) is the desired surge velocity. According to Equation (4), the time derivative of \(\tilde{\psi}\) is given by  
\[\dot{\tilde{\psi}}=r-\dot{\psi}_{d} \tag{23}\]  
To stabilize Equation (23), the desired intermediate control law for \(r_{d}\) is chosen as\"  
\[r_{d}=-k_{2}\tilde{\psi}+\dot{\psi}_{d}-U_{r}\phi y_{e} \tag{24}\]  
where \(k_{2}\) is a positive design parameter. From Equation (5), the dynamics of \(\bar{r}\) and \(\tilde{u}_{r}\) are given as follows  
\[ \begin{cases}m_{33}\dot{\tilde{r}}=m_{33}f_{r}+\tau_{r}+\tau_{ur}-m_{33}f_{d}\\ m_{11}\dot{u}_{r}=m_{11}f_{u}+\tau_{u}+\tau_{uu}-m_{11}\dot{u}_{d}\end{cases}  \tag{25}\]  
Since the inertia and damping parameters are unknown, RBFNNs are employed to handle the unknown parts  
\[ \begin{cases}m_{33}f_{r}-m_{33}\dot{r}_{d}=\mathbf{W}_{1}^{*T}\boldsymbol{ \varphi}(\mathbf{z})+\varepsilon_{1}\\ m_{11}f_{u}-m_{11}\dot{u}_{d}=\mathbf{W}_{2}^{*T}\boldsymbol{\varphi}( \mathbf{z})+\varepsilon_{2}\end{cases}  \tag{26}\]where \(\mathbf{W}_{i}^{*}(i=1,2)\) is the ideal constant weight matrix satisfying \(\|\mathbf{W}_{i}^{*}\|\leqslant W_{iM}\); \(\mathbf{\varphi}(\mathbf{z})\) is the radial basis function with \(\mathbf{z}=[u_{r},v_{r},r]^{T}\) being the input vector to the RBFNNs; and \(\varepsilon_{i}(i=1,2)\) is the approximation error with unknown constant upper bound such that \(|\varepsilon_{i}|\leqslant\bar{\varepsilon}_{i}\). Denote the unknown parts as \(m_{33}f_{r}-m_{33}\dot{r}_{d}=f_{u1}\) and \(m_{11}f_{u}-m_{11}\dot{u}_{d}=f_{u2}\). Furthermore, there exists bounded functions \(\delta_{1}\) and \(\delta_{2}\) such that \(|\varepsilon_{1}|+|\tau_{uv}|\leqslant\delta_{1}\), and \(|\varepsilon_{2}|+|\tau_{uv}|\leqslant\delta_{2}\). Therefore, the nominal control inputs are chosen as follows by considering the input saturation  
\[\left\{ \begin{aligned} \tau_{r0}&=-k_{3}\bar{r}-\bar{ \mathbf{\psi}}-\hat{\mathbf{\psi}}_{1}\mathbf{\varphi}(\mathbf{z})-\delta_{1}h(\bar{r})+k_{ \zeta 1}\sigma_{1}\\ \tau_{u0}&=-k_{4}\bar{u}_{r}-\hat{\mathbf{\psi}}_{2}\bm {\varphi}(\mathbf{z})-\delta_{2}h(\bar{u}_{r})+k_{\zeta 2}\sigma_{2}\end{aligned} \right. \tag{27}\]  
with update laws  
\[\left\{ \begin{aligned} \hat{\mathbf{\psi}}_{1}&=\Gamma_{1} \left(\mathbf{\varphi}(\mathbf{z})\bar{r}-\iota_{1}\hat{\mathbf{\psi}}_{1}\right)\\ \hat{\delta}_{1}&=\xi_{1}\left(\bar{r}h(\bar{r})- \lambda_{1}(\delta_{1}-\delta_{1}^{0})\right)\end{aligned} \right. \tag{28}\]  
\[\left\{ \begin{aligned} \hat{\mathbf{\psi}}_{2}&=\Gamma_{2} \left(\mathbf{\varphi}(\mathbf{z})\bar{u}_{r}-\iota_{2}\hat{\mathbf{\psi}}_{2}\right)\\ \hat{\delta}_{2}&=\xi_{2}\left(\bar{u}_{r}h(\bar{u}_ {r})-\lambda_{2}(\delta_{2}-\delta_{2}^{0})\right)\end{aligned} \right. \tag{29}\]  
where \(k_{3}\) and \(k_{4}\) are positive design parameters; \(k_{\zeta 1}\) and \(k_{\zeta 2}\) are positive design parameters; \(\delta_{i}(i=1,2)\) is the estimation of \(\delta_{i}\); \(h(\bar{r})=\tanh\left(\bar{r}/\chi_{r}\right)\), \(h(\bar{u}_{r})=\tanh\left(\bar{u}_{r}/\chi_{u}\right)\), \(\chi_{j}(j=r,u)\) is a positive constant; \(\mathbf{\Gamma}_{i}(i=1,2)\) is a positive define design matrix; \(\lambda_{i}(i=1,2)\) is a small design parameter; \(\delta_{j}^{0}(j=r,u)\) is the prior estimation of \(\delta_{j}\); and \(\sigma_{i}(i=1,2)\) is the state of the auxiliary system. To compensate for the constraint effects of input saturation, auxiliary dynamic systems [28] are given as follows  
\[\dot{\sigma}_{1}=\left\{ \begin{aligned} -k_{\sigma_{1}}\sigma_{1}-\frac{|\bar{r} \Delta\tau_{r}|+0.5\Delta\tau_{r}^{2}}{\sigma_{1}^{2}}\sigma_{1}+\Delta\tau_{r },&|\sigma_{1}|\geqslant\mu_{1}\\ 0,&|\sigma_{1}|<\mu_{1}\end{aligned} \right. \tag{30}\]  
and  
\[\dot{\sigma}_{2}=\left\{