## 2 Related Works and Problem Statement
### Position Controller Design
\begin{array}{ll}\text{Let }\mathbf{\phi}^{X}=\left[\mathbf{\phi}_{1}^{X},\ldots,\mathbf{\phi}_{N }^{X}\right]^{\mathrm{T}}\text{, }\mathbf{\hat{\phi}}^{X}=\left[\hat{\phi}_{1}^{X},\ldots,\hat{\phi}_{N}^{X}\right] ^{\mathrm{T}}\text{, }\mathbf{\hat{\phi}}^{X}(0)=\left[\hat{\phi}_{1}^{X}(0),\ldots,\hat{\phi}_{N}^{X}(0) \right]^{\mathrm{T}}\text{, }\mathbf{\tilde{\phi}}^{X}=\left[\hat{\phi}_{1}^{X},\ldots,\hat{\phi}_{N}^{X}(0) \right]^{\mathrm{T}}\text{, }\mathbf{\tilde{\phi}}^{X}=\left[\hat{\phi}_{1}^{X},\ldots,\hat{\phi}_{N}^{X} \right]^{\mathrm{T}}\text{, }\mathbf{\tilde{b}}_{d}=\left[\hat{b}_{d1}^{X},\ldots,b_{dN}^{X}\right]^{ \mathrm{T}}\text{, }\mathbf{\tilde{b}}_{d}=\left[\hat{b}_{d1}^{X},\ldots,\hat{b}_{dN}^{X}\right]^{ \mathrm{T}}\text{, and }\overset{\sim}{\mathbf{b}_{d}}(0)\text{=}\left[\hat{b}_{d1}^{X}(0),\ldots,\hat{b}_{ dN}^{X}(0)\right]^{\mathrm{T}}\text{.}\end{array} \]  
Consider the candidate Lyapunov function \(V_{6}\) as follows:  
\[V_{6}=V_{5}+\frac{1}{2\Gamma^{X}}\mathbf{\hat{\phi}}^{\mathrm{XT}}\mathbf{\tilde{\phi}}^{X} +\frac{1}{2\Gamma^{X}_{b}}\mathbf{b}_{d}\overset{\sim X}{\mathbf{b}_{d}} \tag{81}\]The derivative of \(V_{6}\) with respect to time is  
\[\dot{V}_{6} =\mathbf{C}_{2}+\left(\frac{1}{2}+N\rho_{t}k_{p1}\mathbf{s}\right) \delta+\sum\limits_{i=1}^{N}\left\{ \begin{array}{l}z_{i6}^{\mathrm{T}}\Big{(}W _{i}^{X}\circ h_{i}^{X}\Big{)}-\frac{N\dot{\phi}_{i}^{X}}{2\delta}z_{i6}^{ \mathrm{T}}z_{i6}\circ(h_{i}^{X}\circ h_{i}^{X})\\ +\widetilde{\phi}_{i}^{\mathrm{T}}\left(\frac{N}{2\delta}\sum\limits_{j=1}^{3} z_{i6j}^{2}\Big{\|}h_{ij}^{X}\Big{\|}_{2}^{2}\right)\\ +\pi_{i6}^{\mathrm{T}}\left[\delta_{i}-\frac{N\dot{b}_{di}^{X}}{2\delta}z_{i6} \right]+\frac{N}{2\delta}\widetilde{b}_{di}^{X}\|\pi_{i6}\|_{2}^{2}\end{array}  \right\} \tag{82}\] \[-\sum\limits_{i=1}^{N}\left\{M_{\phi_{i}^{X}}\widetilde{\phi}_{i} ^{X}[\hat{\phi}_{i}^{X}-\hat{\phi}_{i}^{X}(0)]+M_{b_{i}^{X}}\widetilde{b}_{di} ^{X}\hat{\phi}_{di}^{X}-\hat{b}_{di}^{X}(0)]\right\}\]  
Let \(\mathbf{C}_{2}=-\Big{(}\mathbf{s}\mathbf{M}_{1}-\frac{1}{2}s\delta\Big{)}z_{4}^{\mathrm{T }}z_{4}-z_{5}^{\mathrm{T}}\Big{[}X^{-1}+\frac{1}{2\delta}\Big{(}1+\mathbf{C}^{T} \mathbf{C}\mathbf{I}\Big{)}\mathbf{I}\mathbf{z}_{5}-z_{6}^{\mathrm{T}}M_{2}\mathbf{z}_{6}\). According to Lemma 4, we have  
\[\mathbf{z}_{i6}^{\mathrm{T}}\Big{(}W_{i}^{X}\circ h_{i}^{X}\Big{)}\leq\frac{N}{2 \delta}\vartheta_{i}^{X}\sum\limits_{j=1}^{3}z_{i6j}^{2}\mathbf{h}_{ij}^{X} \mathbf{h}_{ij}^{X}+\frac{\delta}{2N} \tag{83}\]  
\[\frac{N\hat{\phi}_{i}^{X}}{2\delta}z_{i6}^{\mathrm{T}}\mathbf{z}_{i6}\circ(\mathbf{h}_ {i}^{X}\circ\mathbf{h}_{i}^{X})-\widetilde{\phi}_{i}^{X}\left(\frac{N}{2\delta} \sum\limits_{j=1}^{3}z_{i6j}^{2}\Big{\|}\mathbf{h}_{ij}^{X}\Big{\|}_{2}^{2}\right) =\frac{N}{2\delta}\vartheta_{i}^{X}\sum\limits_{j=1}^{3}z_{i6j}^{2}\mathbf{h}_{ij} ^{X}\mathbf{h}_{ij}^{X} \tag{84}\]  
\[\mathbf{z}_{i6}^{\mathrm{T}}\Bigg{[}\delta_{i}-\frac{N\dot{b}_{di}^{X}}{2\delta} \mathbf{z}_{i6}\Bigg{]}\leq\left(\frac{N}{2\delta}\|\mathbf{z}_{i6}\|_{2}^{2}+\frac{ \delta}{2N}\right)b_{di}^{X}-\frac{N}{2\delta}\hat{b}_{di}^{X}\|\mathbf{z}_{i6}\|_ {2}^{2}=-\frac{N}{2\delta}\widetilde{b}_{di}^{X}\|\mathbf{z}_{i3}\|_{2}^{2}+\frac{ \delta}{2N}b_{di}^{X} \tag{85}\]  
\[\dot{V}_{6}\]  
can be written as  
\[\dot{V}_{6}\leq\mathbf{C}_{2}+\left(\frac{1}{2}+\frac{1}{2}\Big{(}\delta_{N}+b_{d} ^{X}\Big{)}+N\rho_{t}k_{p1}\mathbf{s}\right)\delta-\sum\limits_{i=1}^{N}\left\{ \begin{array}{l}M_{\phi_{i}^{X}}\widetilde{\phi}_{i}^{X}[\hat{\phi}_{i}^{X}- \hat{\phi}_{i}^{X}(0)]\\ +M_{b_{i}^{X}}\widetilde{b}_{di}^{X}[\hat{b}_{di}^{X}-\hat{b}_{di}^{X}(0)] \end{array}