## 2 Related Works and Problem Statement
### Attitude Controller Design
\begin{array}{ll}\dot{V}_{2}&\leq-\bigg{(}K_{1}-\frac{1}{2}\varepsilon\bigg{)}z _{1}^{\mathsf{T}}z_{1}-z_{2}^{\mathsf{T}}\bigg{[}\gamma^{-1}-\frac{1}{2 \varepsilon}\big{(}1+B^{\mathsf{T}}B\big{)}I\bigg{]}z_{2}+\big{(}0.5+N\rho_{M} k_{p}\big{)}\varepsilon\\ &+\sum\limits_{i=1}^{N}z_{i3}^{\mathsf{T}}\bigg{[}z_{i1}+\frac{1}{m_{i}^{33}} \tau_{i}^{\sigma}+W_{i}\circ h_{i}+\varepsilon_{i}\bigg{]}\end{array}  \tag{33}\]  
The attitude control law \(\tau_{i}^{\sigma}\) of each UUSV is designed as follows:  
\[\tau_{i}^{\sigma}=m_{i}^{33}\left(-K_{i2}z_{i3}-\frac{N\dot{\phi}_{i}}{2 \varepsilon}z_{i3}\circ(h_{i}\circ h_{i})-z_{i1}-\frac{N\dot{b}_{di}}{2 \varepsilon}z_{i3}\right) \tag{34}\]  
In \(\tau_{i}^{\sigma},K_{i2}\) is a diagonal matrix, where the diagonal element is a positive constant, and the other elements are 0. Adaptive control law \(\dot{\hat{\phi}}_{i}\) and auxiliary adaptive law \(\dot{\hat{b}}_{di}\) based on MLP are as follows:  
\[\dot{\hat{\phi}}_{i}=\Gamma\Bigg{\{}\frac{N}{2\varepsilon}\sum\limits_{j=1}^{3 }z_{i3,j}^{2}\big{\|}\mathbf{h}_{ij}\big{\|}_{2}^{2}-K_{\phi_{i}}[\hat{\phi}_{i}- \hat{\phi}_{i}(0)]\Bigg{\}} \tag{35}\]  
\[\dot{\hat{b}}_{di}=\Gamma_{b}\Bigg{\{}\frac{N}{2\varepsilon}\big{\|}z_{i3} \big{\|}_{2}^{2}-K_{b_{i}}[\hat{b}_{di}-\hat{b}_{di}(0)]\Bigg{\}} \tag{36}\]  
where \(\phi_{i}=\big{\|}\mathbf{W}_{ij}\big{\|}_{2}^{2}\), \(\phi_{i}>0\). \(\hat{\phi}_{i}\) is the estimation of \(\phi\). The initial value of \(\hat{\phi}_{i}\) is \(\hat{\phi}_{i}(0)\). \(\tilde{\phi}_{i}=\hat{\phi}_{i}-\phi_{i}\) is the estimation error. \(\Gamma>0\) is the adaptive gain. \(K_{\phi_{i}}>0\) is the feedback gain. \(\hat{b}_{di}\) is the estimation of \(b_{di}=\varepsilon_{N}+b_{d}\). \(\tilde{b}_{di}=\hat{b}_{di}-b_{di}\) is the estimation error. \(\Gamma_{b}\) is the positive adaptive gain. \(K_{b_{i}}\) is the positive feedback gain. \(\dot{V}_{2}\) can be written as  
\[ \begin{array}{ll}\dot{V}_{2}&\leq-\bigg{(}K_{1}-\frac{1}{2}\varepsilon I \bigg{)}z_{1}^{\mathsf{T}}z_{1}-z_{2}^{\mathsf{T}}\bigg{[}\gamma^{-1}-\frac{1 }{2\varepsilon}\big{(}1+B^{\mathsf{T}}B\big{)}I\bigg{]}z_{2}\\ &-z_{1}^{\mathsf{T}}K_{2}z_{3}+\big{(}0.5+N\rho_{M}k_{p}\big{)}\varepsilon\\ &+\sum\limits_{i=1}^{N}z_{i3}^{\mathsf{T}}\bigg{[}W_{i}\circ\mathbf{h}_{i}-\frac{N }{2\varepsilon}\hat{\phi}_{i}z_{i3}\circ(\mathbf{h}_{i}\circ\mathbf{h}_{i})-\frac{N}{ 2\varepsilon}\hat{b}_{di}z_{i3}+\varepsilon_{i}\bigg{]}\end{array}  \tag{37}\]  
where \(\mathbf{K}_{2}=\text{diag}\{\mathbf{K}_{2}\}\). For the convenience of subsequent discussion, let \(\mathbf{K}_{\phi}=\text{diag}\{\mathbf{K}_{\phi_{i}}\}\). \(\mathbf{K}_{b}=\text{diag}\{\mathbf{K}_{b_{i}}\}\). **Theorem 1**. : _Under Assumptions 1 and 2, a formation composed of \(N\) UUSVs (10) is considered in a directed graph \(\mathcal{G}\). The proposed attitude control law (35), MLP-based adaptive control law (36), and auxiliary control law (37) for each UUSV guarantee that the formation system is SGUIUB by selecting positive controller parameters \(\mathbf{K}_{0}\), \(\mathbf{K}_{1}\), \(\mathbf{K}_{2}\), \(\mathbf{K}_{\phi}\), \(\mathbf{K}_{b}\), \(\Gamma\), \(\Gamma_{b}\), \(\mathbf{\gamma}\), \(\varepsilon\) under bounded initial conditions, where \(\mathbf{K}_{0}\), \(\mathbf{K}_{1}\), \(\mathbf{\gamma}\) satisfy the following conditions:_  
\[\lambda(\mathbf{K}_{0})_{\text{min}}>N^{2}\rho_{M},\lambda_{\text{min}}(\mathbf{K}_{1})>  
Consider the candidate Lyapunov function \(V_{3}\) as follows:  
\[V_{3}=V_{2}+\frac{1}{2\Gamma}\widetilde{\boldsymbol{\phi}}^{\mathrm{T}} \widetilde{\boldsymbol{\phi}}+\frac{1}{2\Gamma_{b}}\boldsymbol{b}_{d} \boldsymbol{b}_{d} \tag{39}\]  
The derivative of \(V_{3}\) with respect to time is  
\[\dot{V}_{3} =\boldsymbol{B}_{2}+\big{(}0.5+N\rho_{M}k_{p}\big{)}\epsilon+ \sum\limits_{i=1}^{N}\left\{