## 2 Related Works and Problem Statement
### Attitude Controller Design
Then, the derivative of \(V_{1}\) satisfies  
\[\dot{V}_{1}\leq-\mathbf{z}_{1}^{\mathrm{T}}K_{1}\mathbf{z}_{1}+\mathbf{z}_{1} ^{\mathrm{T}}\mathbf{z}_{2}+\mathbf{z}_{1}^{\mathrm{T}}\mathbf{z}_{3}+N\rho_{ M}k_{p}\varepsilon \tag{23}\]  
Next, the attitude control law \(\tau_{i}^{\sigma}\) of each UUSV is designed. A Lyapunov function is selected as follows:  
\[V_{2}=V_{1}+\frac{1}{2}\mathbf{z}_{2}^{\mathrm{T}}\mathbf{z}_{2}+\frac{1}{2} \mathbf{z}_{3}^{\mathrm{T}}\mathbf{z}_{3} \tag{24}\]  
The derivative of \(V_{2}\) with respect to time is  
\[\dot{V}_{2}=\dot{V}_{1}+\mathbf{z}_{2}^{\mathrm{T}}\dot{\mathbf{z}}_{2}+ \mathbf{z}_{3}^{\mathrm{T}}\dot{\mathbf{z}}_{3} \tag{25}\]  
According to (16), \(\dot{\mathbf{z}}_{2}\) can be represented as follows:  
\[\dot{\mathbf{z}}_{2}=-\gamma^{-1}\mathbf{z}_{2}+\mathbf{K}_{1}\dot{\mathbf{z} }_{1}+\mathbf{K}_{0}\bigg{[}1-\mathrm{tanh}^{\mathrm{T}}\bigg{(}\frac{Nz_{1}} {\varepsilon}\bigg{)}\mathrm{tanh}\bigg{(}\frac{Nz_{1}}{\varepsilon}\bigg{)} \bigg{]}\frac{N\dot{\mathbf{z}}_{1}}{\varepsilon} \tag{26}\]  
Let \(\boldsymbol{B}=K_{1}\dot{\mathbf{z}}_{1}+K_{0}\Big{[}1-\mathrm{tanh}^{\mathrm{ T}}\Big{(}\frac{Nz_{1}}{\varepsilon}\Big{)}\mathrm{tanh}\bigg{(}\frac{Nz_{1}}{ \varepsilon}\Big{)}\Big{]}\frac{Nz_{1}}{\varepsilon}\), \(\boldsymbol{B}\) consist of bounded functions, so there is a positive constant \(\overline{B}\) and \(\left\|\boldsymbol{B}\right\|_{2}\leq\overline{B}\). Equation (25) can be written as  
\[ \begin{array}{ll}\dot{V}_{2}&=\dot{V}_{1}+\mathbf{z}_{2}^{\mathrm{T}}\big{(} -\gamma^{-1}\mathbf{z}_{2}+\boldsymbol{B}\big{)}+\mathbf{z}_{3}^{\mathrm{T}} \dot{\mathbf{z}}_{3}\\ &=-\mathbf{z}_{1}^{\mathrm{T}}K_{1}\mathbf{z}_{1}-\mathbf{z}_{2}^{\mathrm{T}} \gamma^{-1}\mathbf{z}_{2}+\mathbf{z}_{1}^{\mathrm{T}}\mathbf{z}_{2}+\mathbf{z }_{2}^{\mathrm{T}}\mathbf{B}+\mathbf{z}_{3}^{\mathrm{T}}\big{(}\mathbf{z}_{1} +\dot{\mathbf{z}}_{3}\big{)}+N\rho_{M}k_{p}\varepsilon\end{array}  \tag{27}\]  
According to Lemma 4, the following inequalities can be obtained:  
\[z_{1}^{\mathrm{T}}\mathbf{z}_{2}\leq\frac{1}{2}(\varepsilon\mathbf{z}_{1}^{ \mathrm{T}}\mathbf{z}_{1}+\frac{1}{\varepsilon}\mathbf{z}_{2}^{\mathrm{T}} \mathbf{z}_{2}) \tag{28}\]  
\[\mathbf{z}_{2}^{\mathrm{T}}\boldsymbol{B}\leq\frac{1}{2}(\frac{1}{\varepsilon} \mathbf{z}_{2}^{\mathrm{T}}\boldsymbol{B}^{\mathrm{T}}\boldsymbol{B}\mathbf{z} _{2}+\varepsilon) \tag{29}\]  
\(\dot{V}_{2}\) can be written as  
\[ \begin{array}{ll}\dot{V}_{2}&\leq-\Big{(}\boldsymbol{K}_{1}-\frac{1}{2} \varepsilon\boldsymbol{I}\Big{)}\mathbf{z}_{1}^{\mathrm{T}}\mathbf{z}_{1}- \mathbf{z}_{2}^{\mathrm{T}}\Big{[}\gamma^{-1}-\frac{1}{2\varepsilon}\big{(}1+ \boldsymbol{B}^{\mathrm{T}}\boldsymbol{B}\big{)}\boldsymbol{I}\Big{]}\mathbf{z} _{2}\\ &+\big{(}0.5+N\rho_{M}k_{p}\big{)}\varepsilon+\mathbf{z}_{3}^{\mathrm{T}} \big{(}\mathbf{z}_{1}+\dot{\mathbf{z}}_{3}\big{)}\end{array}  \tag{30}\]  
With \(\boldsymbol{B}_{1}=\mathbf{z}_{1}+\dot{\mathbf{z}}_{3}\), \(\boldsymbol{f}_{i}(\cdot)=\boldsymbol{f}_{i}+\gamma_{i}^{-1}\mathbf{z}_{i2}\), the following equation can be obtained:  
\[ \begin{array}{ll}\mathbf{z}_{3}^{\mathrm{T}}\boldsymbol{B}_{1}=\sum_{i=1}^{N }\mathbf{z}_{i3}^{\mathrm{T}}\Bigg{[}\mathbf{z}_{i1}+\frac{1}{m_{i}^{33}} \mathbf{\tau}_{i}^{\sigma}+\boldsymbol{f}_{i}(\cdot)\Bigg{]}\end{array}  \tag{31}\]  
\( \begin{array}{ll}\mathbf{-\sigma}\\ \boldsymbol{f}_{i}\end{array} \) is difficult to be accurate. Thus, the RBF neural network is used to process it. \[\boldsymbol{f}_{i}(\cdot)=\boldsymbol{W}_{i}\circ\boldsymbol{h}_{i}+ \varepsilon_{i} \tag{32}\]  
where \(\boldsymbol{W}_{i}=\big{[}\boldsymbol{W}_{i1}^{\mathrm{T}},\boldsymbol{W}_{i2 }^{\mathrm{T}},\boldsymbol{W}_{i3}^{\mathrm{T}}\big{]}^{\mathrm{T}}\), \(\boldsymbol{h}_{i}=\Big{[}\boldsymbol{h}_{i1}^{\mathrm{T}},\boldsymbol{h}_{i2}^ {\mathrm{T}},\boldsymbol{h}_{i3}^{\mathrm{T}}\Big{]}^{\mathrm{T}}\), and \(\left\|\varepsilon_{i}\right\|\leq\varepsilon_{N}\), \(i=1,\ldots,N\), \(\varepsilon_{N}>0\). \(\dot{V}_{2}\) can be written as  
\[