## 2 Related Works and Problem Statement
### Attitude Controller Design
\begin{array}{l}\boldsymbol{z}_{i3}^{\mathrm{T}} (\boldsymbol{W}_{i}\circ\boldsymbol{h}_{i})-\frac{N}{2\epsilon}\hat{\phi}_{i} \boldsymbol{z}_{i3}^{\mathrm{T}}\boldsymbol{z}_{i3}\circ(\boldsymbol{h}_{i} \circ h_{i})\\ +\hat{\phi}_{i}\left(\frac{N}{2\epsilon}\sum\limits_{j=1}^{3}\boldsymbol{z}_{i3,j}^{2}\|\boldsymbol{h}_{ij}\|_{2}^{2}\right)\\ +\boldsymbol{z}_{i3}^{\mathrm{T}}\Big{[}\epsilon_{i}-\frac{N}{2\epsilon} \hat{b}_{di}\boldsymbol{z}_{i3}\Big{]}+\frac{N}{2\epsilon}\tilde{b}_{di}\| \boldsymbol{z}_{i3}\|_{2}^{2}\end{array} \right\} \tag{40}\] \[-\sum\limits_{i=1}^{N}\left\{K_{\hat{\phi}_{i}}\widehat{\phi}_{i} [\hat{\phi}_{i}-\hat{\phi}_{i}(0)\right]+K_{b}\tilde{b}_{di}\Big{[}\hat{b}_{di }-\hat{b}_{di}(0)\Big{]}\right\}\]  
Let \(\boldsymbol{B}_{2}=-\left(K_{1}-\frac{1}{2}\epsilon\boldsymbol{I}\right) \boldsymbol{z}_{1}^{\mathrm{T}}\boldsymbol{z}_{1}-\boldsymbol{z}_{2}^{ \mathrm{T}}\Big{[}\boldsymbol{\gamma}^{-1}-\frac{1}{2\epsilon}\big{(}1+ \boldsymbol{B}^{\mathrm{T}}\boldsymbol{B}\big{)}\boldsymbol{I}\Big{]} \boldsymbol{z}_{2}-\boldsymbol{z}_{3}^{\mathrm{T}}\boldsymbol{K}_{2} \boldsymbol{z}_{3}\). According to Lemma 4, we have  
\[z_{i3,j}\boldsymbol{W}_{ij}^{\mathrm{T}}\boldsymbol{h}_{ij}\leq\frac{N}{2 \epsilon}\phi_{i}\boldsymbol{z}_{i3,j}^{2}\boldsymbol{h}_{ij}^{\mathrm{T}} \boldsymbol{h}_{ij}+\frac{\epsilon}{2N} \tag{41}\]  
It can then be concluded that  
\[z_{i3}^{\mathrm{T}}(\boldsymbol{W}_{i}\circ\boldsymbol{h}_{i})=\sum\limits_{j =1}^{3}z_{i3,j}\boldsymbol{W}_{ij}^{\mathrm{T}}\boldsymbol{h}_{ij}\leq\frac{ N}{2\epsilon}\phi_{i}\sum\limits_{j=1}^{3}\boldsymbol{z}_{i3,j}^{2} \boldsymbol{h}_{ij}^{\mathrm{T}}\boldsymbol{h}_{ij}+\frac{\epsilon}{2N} \tag{42}\]  
\[\frac{N}{2\epsilon}\hat{\phi}_{i}\boldsymbol{z}_{i3}^{\mathrm{T}}\boldsymbol{ z}_{i3}\circ(\boldsymbol{h}_{i}\circ\boldsymbol{h}_{i})-\frac{N}{2\epsilon} \widetilde{\phi}_{i}\left(\sum\limits_{j=1}^{3}z_{i3,j}^{2}\|\boldsymbol{h}_{ ij}\|_{2}^{2}\right)=\frac{N}{2\epsilon}\phi_{i}\sum\limits_{j=1}^{3}z_{i3,j}^{ 2}\boldsymbol{h}_{ij}^{\mathrm{T}}\boldsymbol{h}_{ij} \tag{43}\]  
\[z_{i3}^{\mathrm{T}}\bigg{[}\epsilon_{i}-\frac{N}{2\epsilon}\hat{b}_{di} \boldsymbol{z}_{i3}\bigg{]}\leq\|\boldsymbol{z}_{i3}\|_{2}b_{di}-\frac{N}{2 \epsilon}\hat{b}_{di}\|\boldsymbol{z}_{i3}\|_{2}^{2} \tag{44}\]  
\[z_{i3}^{\mathrm{T}}\Big{[}\epsilon_{i}-\frac{N}{2\epsilon}\hat{b}_{di} \boldsymbol{z}_{i3}\Big{]} \leq\Big{(}\frac{N}{2\epsilon}\|\boldsymbol{z}_{i3}\|_{2}^{2}+ \frac{\epsilon}{2N}\Big{)}b_{di}-\frac{N}{2\epsilon}\hat{b}_{di}\| \boldsymbol{z}_{i3}\|_{2}^{2} \tag{45}\] \[=-\frac{N}{2\epsilon}\widetilde{b}_{di}\|\boldsymbol{z}_{i3}\|_{2} ^{2}+\frac{\epsilon}{2N}b_{di}\]  
\(\dot{V}_{3}\) can be written as  
\[\dot{V}_{3}\leq\boldsymbol{B}_{2}+\bigg{[}\frac{1}{2}+N\rho_{M}k_{p}+\frac{1}{2 }(\epsilon_{N}+b_{d})\bigg{]}\epsilon-\sum\limits_{i=1}^{N}\left\{ \begin{array}[ ]{l}K_{\hat{\phi}_{i}}\widehat{\phi}_{i}[\hat{\phi}_{i}-\hat{\phi}_{i}(0)]\\ +K_{b_{i}}\widetilde{b}_{di}\Big{[}\hat{b}_{di}-\hat{b}_{di}(0)\Big{]}\end{array} \right\} \tag{46}\]  
According to the following two inequalities,  
\[\widetilde{\phi}_{i}[\hat{\phi}_{i}-\hat{\phi}_{i}(0)]\geq\frac{1}{2} \widetilde{\phi}_{i}^{2}-\frac{1}{2}\big{[}\phi_{i}-\hat{\phi}_{i}(0)\big{]}^{2} \tag{47}\]  
\[\widetilde{b}_{di}\Big{[}\hat{b}_{di}-\hat{b}_{di}(0)\Big{]}\geq\frac{1}{2} \widetilde{b}_{di}^{2}-\frac{1}{2}\Big{[}b_{di}-\hat{b}_{di}(0)\Big{]}^{2} \tag{48}\]it can further be obtained as  
\[\dot{V}_{3}\leq\mathbf{B}_{2} +\bigg{[}\frac{1}{2}+N\rho_{M}k_{p}+\frac{1}{2}(\epsilon_{N}+b_{d}) \bigg{]}\varepsilon-\frac{1}{2}\sum\limits_{i=1}^{N}\Big{\{}K_{\phi_{i}}\widehat {\phi}_{i}^{2}+K_{b_{i}}\widehat{b}_{di}^{2}\Big{\}} \tag{49}\] \[+\frac{1}{2}\Bigg{[} \begin{matrix}-\overset{\Lambda}{\mathbf{b}_{d}- \mathbf{b}_{d}(0)}\end{matrix} \Bigg{]}^{\mathrm{T}}\mathbf{K}_{b} \begin{bmatrix}- \overset{\Lambda}{\mathbf{b}_{d}-\mathbf{b}_{d}(0)}\end{bmatrix}