## 2 Related Works and Problem Statement
### Position Controller Design
\right\} \tag{86}\]  
According to the following two inequalities,  
\[\widetilde{\phi}_{i}^{X}[\hat{\phi}_{i}^{X}-\hat{\phi}_{i}^{X}(0)]\geq\frac{1 }{2}\widetilde{\phi}_{i}^{X2}-\frac{1}{2}\Big{[}\phi_{i}^{X}-\hat{\phi}_{i}^{X} (0)\Big{]}^{2} \tag{87}\]  
\[\widetilde{b}_{di}^{X}[\hat{b}_{di}^{X}-\hat{b}_{di}^{X}(0)]\geq\frac{1}{2} \widetilde{b}_{di}^{X2}-\frac{1}{2}\Big{[}b_{di}^{X}-\hat{b}_{di}^{X}(0)\Big{]} ^{2} \tag{88}\]  
Further, it can be obtained  
\[\dot{V}_{6} \leq\mathbf{C}_{2}+\left(\frac{1}{2}+\frac{1}{2}\big{(}\delta_{N}+b_{ d}^{X}\big{)}+N\rho_{t}k_{p1}\mathbf{s}\right)\delta-\frac{1}{2}\sum\limits_{j=1}^{N} \left\{M_{\phi_{i}^{X}}\widetilde{\phi}_{i}^{X2}+M_{b_{i}^{X}}\widetilde{b}_{ di}^{X2}\right\} \tag{89}\] \[+\frac{1}{2}\sum\limits_{i=1}^{N}\left\{M_{\phi_{i}^{X}}\big{[} \phi_{i}^{X}-\hat{\phi}_{i}^{X}(0)\big{]}^{2}+M_{b_{i}^{X}}\Big{[}b_{di}^{X}- \hat{b}_{di}^{X}(0)\big{]}^{2}\right\}\] \[=-\bigg{(}\mathbf{s}\mathbf{M}_{1}-\frac{1}{2}s\delta\Big{)}\mathbf{z}_{4}^{ \mathrm{T}}\mathbf{z}_{4}-\mathbf{z}_{5}^{\mathrm{T}}\bigg{[}\mathbf{x}^{-1}+\frac{1}{2 \delta}\Big{(}1+\mathbf{C}^{T}\mathbf{C}\mathbf{I}\Big{)}\mathbf{I}\mathbf{z}_{5}-z_{6}^{\mathrm{T }}M_{2}\mathbf{z}_{6}\] \[-\frac{1}{2}\widetilde{\mathbf{\phi}}_{X}^{\mathrm{T}}\mathbf{M}_{\phi^{X}} \widetilde{\mathbf{\phi}}^{X}-\frac{1}{2}\mathbf{b}_{d}\ \ M_{\phi^{X}}\mathbf{b}_{d}+\frac{1}{2}\big{[}\mathbf{\phi}^{X}-\hat{\mathbf{\phi}}^{X}(0 )\big{]}^{\mathrm{T}}\mathbf{M}_{\phi^{X}}\big{[}\mathbf{\phi}^{X}-\hat{\mathbf{\phi}}^{X}(0 )\big{]}\] \[+\frac{1}{2}\left[ \begin{array}{cc}-X&-\\ \mathbf{b}_{d}&-\mathbf{b}_{d}&(0)\end{array} \right]^{\mathrm{T}}\mathbf{M}_{\mathrm{b}^{X}} \left[ \begin{array}{cc}-X&-\\ \mathbf{b}_{d}&-\mathbf{b}_{d}&(0)\end{array} \right]\] \[+\bigg{(}\frac{1}{2}+\frac{1}{2}\big{(}\delta_{N}+b_{d}^{X}\big{)}+N \rho_{t}k_{p1}\mathbf{s}\bigg{)}\delta\]If  
\[ \begin{array}{ll}\beta^{X}&=\frac{1}{2}\big{[}\mathbf{\phi}^{X}-\mathbf{\phi}^{X}(0) \big{]}^{\mathrm{T}}\mathbf{M}_{\mathbf{\phi}^{X}}\big{[}\mathbf{\phi}^{X}-\mathbf{\phi}^{X}(0) \big{]}+\frac{1}{2}\Bigg{[}\begin{matrix}-X&\stackrel{{\mathbf{\alpha}^{X}} }{{-}}\\ \mathbf{b}_{d}-\mathbf{b}_{d}&(0)\end{matrix}\Bigg{]}^{\mathrm{T}}\mathbf{M}_{\mathbf{b}^{X}} \begin{bmatrix}-X&\stackrel{{\mathbf{\alpha}^{X}}}{{-}}\\ \mathbf{b}_{d}&-\mathbf{b}_{d}&(0)\end{bmatrix}\\ &+\Big{(}\frac{1}{2}+\frac{1}{2}\big{(}\delta_{N}+b_{d}^{X}\big{)}+N\rho_{ \mathrm{I}}k_{p1}\mathbf{s}\Big{)}\delta\end{array}  \tag{90}\]  
we have  
\[ \begin{array}{ll}\dot{V}_{6}&\leq-\Big{(}\lambda_{\min}(\mathbf{sM}_{1})-\frac{1 }{2}\mathbf{s}\delta\Big{)}\mathbf{z}_{4}^{\mathrm{T}}\mathbf{z}_{4}-\Big{(}\lambda_{\min }\big{(}\mathbf{\chi}^{-1}\big{)}-\frac{1}{2\delta}\Big{(}1+\overline{\mathcal{C} }^{2}\Big{)}\Big{)}\mathbf{z}_{5}^{\mathrm{T}}\mathbf{z}_{5}\\ &-\lambda_{\min}(\mathbf{M}_{2})\mathbf{z}_{6}^{\mathrm{T}}\mathbf{z}_{6}-\frac{1}{2} \lambda_{\min}\Big{(}\mathbf{M}_{\mathbf{\phi}^{X}}\Big{)}\mathbf{\tilde{\phi}}^{\mathrm{ X}}\mathbf{\tilde{\phi}}^{\mathrm{X}}-\frac{1}{2}\lambda_{\min}(\mathbf{M}_{\mathbf{\phi}^{X}}) \mathbf{b}_{d}&\mathbf{b}_{d}+\mathbf{\beta}^{X}\end{array}  \tag{91}\]  
If  
\[\lambda(\mathbf{M}_{0})_{\min}>N^{2}\rho_{\mathrm{I}},\lambda_{\min}(\mathbf{sM}_{1})> \frac{1}{2}\mathbf{s}\delta,\lambda_{\max}(\mathbf{\chi})<\frac{2\delta}{1+\overline{ \mathcal{C}}^{2}} \tag{92}\]  
\(\dot{V}_{6}\) can be written as  
\[\dot{V}_{6}\leq-\alpha^{X}V_{6}+\mathbf{\beta}^{X} \tag{93}\]  
where  
\[\alpha^{X}=\min\left\{ \begin{array}{ll}[2\lambda_{\min}(\mathbf{sM}_{1})-\mathbf{s} \delta],\Big{[}2\lambda_{\min}\big{(}\mathbf{\chi}^{-1}\big{)}-\frac{1}{\delta} \Big{(}1+\overline{\mathcal{C}}^{2}\Big{)}\Big{]}\\,2\lambda_{\min}(\mathbf{M}_{2}),\Gamma^{X}\lambda_{\min}\Big{(}\mathbf{M}_{\mathbf{\phi}^{ X}}\Big{)},\Gamma^{X}_{\mathbf{b}}\lambda_{\min}(\mathbf{M}_{\mathbf{\phi}^{X}})\end{array} \right\} \tag{94}\]  
According to Lemma 5, we can conclude that all position subsystems are SGUUB.