## 2 Related Works and Problem Statement
### Position Controller Design
 \tag{72}\]  
With \(\mathbf{C}_{\mathbf{1}}=\mathbf{s}\mathbf{z}_{4}+\dot{z}_{6}\), \(\mathbf{F}_{i}^{\mathrm{X}}(\cdot)=\overset{-X}{f_{i}}+\chi_{i}^{-1}\mathbf{z}_{5}\), the following equation can be obtained:  
\[z_{6}^{\mathrm{T}}\mathbf{C}_{\mathbf{1}}=\sum_{i=1}^{N}z_{i6}^{\mathrm{T}}\bigg{[}s \mathbf{z}_  
The attitude control law \(\tau_{i}^{X}\) of each UUSV is designed as follows:  
\[\tau_{i}^{X}=m_{i}^{11}\left(-M_{i2}z_{i6}-\frac{N\hat{\phi}_{i}^{X}}{2\delta}z_{ i6}\circ(\mathbf{h}_{i}^{X}\circ\mathbf{h}_{i}^{X})-s\mathbf{z}_{i4}-\frac{N\hat{b}_{di}^{X}}{2 \delta}z_{i6}\right) \tag{76}\]  
where \(\mathbf{M}_{i2}\) is a diagonal matrix. Adaptive control law \(\hat{\hat{\phi}}_{i}^{X}\) and auxiliary adaptive law \(\hat{\hat{b}}_{di}^{X}\) based on MLP are as follows:  
\[\hat{\phi}_{i}^{X}=\Gamma^{X}\left\{\frac{3}{2\delta}\sum\limits_{j=1}^{3}\mathbf{ z}_{i6}^{2}\left\|\mathbf{h}_{ij}^{X}\right\|_{2}^{2}-\mathbf{M}_{\hat{\phi}_{i}^{X}}[ \hat{\phi}_{i}^{X}-\hat{\phi}_{i}^{X}(0)]\right\} \tag{77}\]  
\[\hat{\hat{b}}_{di}^{X}=\Gamma_{b}^{X}\left\{\frac{N}{2\delta}\|\mathbf{z}_{i6}\|_{ 2}^{2}-\mathbf{M}_{\hat{b}_{i}^{X}}[\hat{b}_{di}^{X}-\hat{b}_{di}^{X}(0)]\right\} \tag{78}\]  
where \(\phi_{i}^{X}=\left\|\mathbf{W}_{ij}\right\|_{2}^{2}\phi_{i}^{X}>0\). \(\hat{\phi}_{i}^{X}\) is the estimation of \(\phi^{X}\). \(\hat{\phi}_{i}^{X}(0)\) is the initial value of \(\hat{\phi}_{i}^{X}\). \(\widetilde{\phi}_{i}^{X}=\hat{\phi}_{i}^{X}-\phi_{i}^{X}\) is the estimation error. \(\Gamma^{X}>0\) is the adaptive gain. \(\mathbf{M}_{\phi_{i}^{X}}>0\) is the feedback gain. \(\hat{b}_{di}^{X}\) is the estimation of \(b_{di}^{X}=\delta_{N}+b_{d}^{X}\). \(\widetilde{b}_{di}^{X}=\hat{b}_{di}^{X}-b_{di}^{X}\) is the estimation error. \(\Gamma_{b}^{X}\) is the positive adaptive gain. \(\mathbf{M}_{b_{i}^{X}}\) is the positive feedback gain. \(\dot{V}_{5}\) can be written as  
\[ \begin{array}{ll}\dot{V}_{5}&\leq-\left(\mathbf{s}\mathbf{M}_{1}-\frac{1}{2}\mathbf{s} \delta\right)\mathbf{z}_{4}^{\mathrm{T}}\mathbf{z}_{4}-\mathbf{z}_{5}^{\mathrm{T}}\left[ \mathbf{\chi}^{-1}+\frac{1}{2\delta}\left(1+\mathbf{C}^{\mathrm{T}}\mathbf{C}\right)\mathbf{I }\right]\mathbf{z}_{5}-\mathbf{z}_{6}^{\mathrm{T}}\mathbf{M}_{2}\mathbf{z}_{6}\\ &+\left(\frac{1}{2}+N\rho_{\mathrm{I}}k_{p1}\mathbf{s}\right)\delta+\sum\limits_{i =1}^{N}\mathbf{z}_{i6}^{\mathrm{T}}\left[\mathbf{W}_{i}^{X}\circ\mathbf{h}_{i}^{X}-\frac{N \hat{\phi}_{i}^{X}}{2\delta}\mathbf{z}_{i6}\circ(\mathbf{h}_{i}^{X}\circ\mathbf{h}_{i}^{X} )-\frac{N\hat{b}_{di}^{X}}{2\delta}\mathbf{z}_{i6}+\delta_{i}\right]\end{array}  \tag{79}\]  
where \(\mathbf{M}_{2}=\text{diag}\{\mathbf{M}_{i2}\}\). For the convenience of subsequent discussion, let \(\mathbf{M}_{\phi^{X}}\)= diag\(\left\{\mathbf{M}_{\hat{\phi}_{i}^{X}}\right\}\), \(\mathbf{M}_{b^{X}}=\text{diag}\left\{\mathbf{M}_{\hat{b}_{i}^{X}}\right\}\). **Theorem 2**. : _Under assumptions 1 and 2, a formation composed of \(N\) UUISVs (11) is considered in a directed graph \(\mathcal{G}\). The proposed position control law (76), the MLP-based adaptive control law (77), and the auxiliary control law (78) for each UUSV guarantee that the formation system is SGUIUB by selecting positive controller parameters \(\mathbf{M}_{0}\), \(\mathbf{M}_{1}\), \(\mathbf{M}_{2}\), \(\mathbf{M}_{\phi^{X}}\), \(\mathbf{M}_{b^{X}}\), \(\Gamma^{X}\), \(\Gamma^{X}_{b}\), \(\chi\), \(\delta\) under bounded initial conditions, where \(\mathbf{M}_{0}\), \(\mathbf{M}_{1}\), \(\mathbf{\chi}\) satisfy the following conditions:_  
\[\lambda(\mathbf{M}_{0})_{\min}>N^{2}\rho_{\mathrm{I}},\lambda_{\min}(\mathbf{s}\mathbf{M}_{ 1})>\frac{1}{2}\mathbf{s}\delta,\lambda_{\max}(\mathbf{\chi})<\frac{2\delta}{1+\overline {C}^{2}} \tag{80}\]  
\[