## 2 Related Works and Problem Statement
### Position Controller Design
\begin{array}{l}\dot{V}_{4} \leq z_{4}^{\rm T}\mathcal{R}^{\rm T}\big{(}\overline{\mathcal{L} }\otimes I\big{)}(z_{5}+z_{6})-z_{4}^{\rm T}\mathcal{R}^{\rm T}\big{(} \overline{\mathcal{L}}\otimes I\big{)}\mathbf{M}_{1}z_{4}\\ +\|z_{4}\|_{\infty}\kappa^{\rm T}\big{(}\overline{\mathcal{L}}\otimes I_{1} \big{)}N\rho_{U}-\mathbf{M}_{0}z_{4}^{\rm T}\mathcal{R}^{\rm T}\big{(}\overline{ \mathcal{L}}\otimes I\big{)}\text{tanh}\bigg{(}\frac{Nz_{4}}{\delta}\bigg{)}\\ \leq z_{4}^{\rm T}\mathcal{R}^{\rm T}\big{(}\overline{\mathcal{L}}\otimes I \big{)}(z_{5}+z_{6})-z_{4}^{\rm T}\mathcal{R}^{\rm T}\big{(}\overline{ \mathcal{L}}\otimes I\big{)}\text{\bf\bf\Lambda}_{1}z_{4}+N\rho_{U}k_{P1} \delta\mathcal{R}^{\rm T}\big{(}\overline{\mathcal{L}}\otimes I_{1}\big{)}\\ +N^{2}\rho_{U}z_{4}^{\rm T}\mathcal{R}^{\rm T}\big{(}\overline{\mathcal{L}} \otimes I_{1}\big{)}\text{tanh}\bigg{(}\frac{Nz_{4}}{\delta}\bigg{)}-M_{0}z_{4 }^{\rm T}\mathcal{R}^{\rm T}\big{(}\overline{\mathcal{L}}\otimes I\big{)} \text{tanh}\bigg{(}\frac{Nz_{4}}{\delta}\bigg{)}\end{array}  \tag{63}\]  
Let \(\lambda(\mathbf{M}_{0})_{\rm min}>N^{2}\rho_{U}\), and we can further conclude that \(N^{2}\rho_{U}z_{4}^{\rm T}\mathcal{R}^{\rm T}\big{(}\overline{\mathcal{L}} \otimes I_{1}\big{)}\text{tanh}\ln\big{(}\frac{Nz_{4}}{\delta}\big{)}-M_{0}z_{4 }^{\rm T}\mathcal{R}^{\rm T}\big{(}\overline{\mathcal{L}}\otimes I\big{)} \text{tanh}\bigg{(}\frac{Nz_{4}}{\delta}\bigg{)}<0\). Further, we have  
\[\dot{V}_{4}\leq-z_{4}^{\rm T}\mathcal{R}^{\rm T}\big{(}\overline{\mathcal{L}} \otimes I\big{)}\mathbf{M}_{1}z_{4}+z_{4}^{\rm T}\mathcal{R}^{\rm T}\big{(} \overline{\mathcal{L}}\otimes I\big{)}(z_{5}+z_{6})+N\rho_{U}k_{P1}\delta \mathcal{R}^{\rm T}\big{(}\overline{\mathcal{L}}\otimes I_{1}\big{)} \tag{64}\]Let \(\mathbf{s}=\mathcal{R}^{\mathrm{T}}(\overline{\mathcal{L}}\otimes\mathbf{I})\). We have  
\[\dot{V}_{4}\leq-z_{4}^{\mathrm{T}}\mathbf{s}M_{1}z_{4}+z_{4}^{\mathrm{T}}\mathbf{s}(z_{ 5}+z_{6})+N\rho_{U}k_{p1}\delta\mathbf{s} \tag{65}\]  
Next, the attitude control law \(\mathbf{\tau}_{i}^{X}\) of each UUSV is designed. And a Lyapunov function is selected as follows:  
\[V_{5}=V_{4}+\frac{1}{2}z_{5}^{\mathrm{T}}z_{5}+\frac{1}{2}z_{6}^{\mathrm{T}}z_ {6} \tag{66}\]  
The derivative of \(V_{5}\) with respect to time is  
\[\dot{V}_{5}=\dot{V}_{4}+z_{5}^{\mathrm{T}}\dot{z}_{5}+z_{6}^{\mathrm{T}}\dot{z }_{6} \tag{67}\]  
where \(\dot{z}_{5}\) can be represented as follows:  
\[\dot{z}_{5}=-\chi^{-1}z_{5}+M_{1}\dot{z}_{4}+M_{0}\bigg{[}1-\tanh^{\mathrm{T}} \bigg{(}\frac{N\mathbf{z}_{4}}{\delta}\bigg{)}\tanh\bigg{(}\frac{N\mathbf{z}_{4}}{ \delta}\bigg{)}\bigg{]}\frac{N\dot{z}_{4}}{\delta} \tag{68}\]  
Let \(\mathbf{C}=M_{1}\dot{z}_{4}+M_{0}\bigg{[}1-\tanh^{\mathrm{T}}\bigg{(}\frac{N\mathbf{z }_{4}}{\delta}\bigg{)}\tanh\bigg{(}\frac{N\mathbf{z}_{4}}{\delta}\bigg{)}\bigg{]} \frac{N\dot{z}_{4}}{\delta}\). And there exists a positive constant \(\overline{\mathcal{C}}\) that satisfies \(\left\|\mathbf{C}\right\|_{2}\leq\overline{\mathcal{C}}\). Then, (67) can be written as  
\[\dot{V}_{5}=-z_{4}^{\mathrm{T}}\mathbf{s}M_{1}\mathbf{z}_{4}+s\mathbf{z}_{4}^{\mathrm{T}}z _{5}+z_{6}^{\mathrm{T}}(\mathbf{s}\mathbf{z}_{4}+\dot{z}_{6})+z_{5}^{\mathrm{T}}\Big{(} -\chi^{-1}z_{5}+\mathbf{C}\Big{)}+N\rho_{U}k_{p1}\delta\mathbf{s} \tag{69}\]  
According to Lemma 4, the following inequalities can be obtained:  
\[z_{4}^{\mathrm{T}}z_{5}\leq\frac{1}{2}(\delta z_{4}^{\mathrm{T}}\dot{z}_{4}+ \frac{1}{\delta}z_{5}^{\mathrm{T}}z_{5}) \tag{70}\]  
\[z_{5}^{\mathrm{T}}\mathbf{C}\leq\frac{1}{2}\bigg{(}\frac{1}{\delta}z_{5}^{\mathrm{ T}}\mathbf{C}^{\mathrm{T}}\mathbf{C}z_{5}+\delta\bigg{)} \tag{71}\]  
\(\dot{V}_{5}\) can be written as  
\[ \begin{array}{rl}\dot{V}_{5}&\leq-\Big{(}\mathbf{s}M_{1}-\frac{1}{2}\mathbf{s}\delta \Big{)}z_{4}^{\mathrm{T}}z_{4}-z_{5}^{\mathrm{T}}\Big{[}\chi^{-1}+\frac{1}{2 \delta}\Big{(}1+\mathbf{C}^{\mathrm{T}}\mathbf{C}\Big{)}\mathbf{I}\Big{]}z_{5}\\ &+\Big{(}\frac{1}{2}+N\rho_{U}k_{p1}\mathbf{s}\Big{)}\delta+z_{6}^{\mathrm{T}}\big{(} \mathbf{s}\mathbf{z}_{4}+\dot{z}_{6}\big{)}\end{array}