## Appendix A
This facilitates the derivation of a well-structured inequality for \(\dot{V}_{i}(\mathbf{x})\), which is central to analyzing the system's stability. The positive definite function \(V_{i}(\mathbf{x})\) is selected based on standard Lyapunov candidatefunctions, ensuring that its gradient satisfies the conditions for stability as outlined in Theorem 1. This approach guarantees that the stability proof aligns with the principles of Lyapunov theory while maintaining the robustness required for the system's dynamics. Using Property 2, the following relationship is established:  
\[-\mathbf{s}_{i}^{\mathrm{T}}\mathbf{K}_{\mathrm{viz}}\mathbf{s}_{i}=-\mathbf{\varepsilon}_{i}^{ \mathrm{T}}\mathbf{K}_{\mathrm{viz}}\mathbf{\dot{e}}_{i}-2\mathbf{\varepsilon}_{i}^{\mathrm{ T}}\mathbf{K}_{\mathrm{viz}}\mathbf{\Lambda}_{i}\Big{(}\mathbf{B}_{i}\mathbf{e}_{i}-\mathbf{ \varepsilon}_{j}\Big{)}-\Big{(}\mathbf{B}_{i}\mathbf{e}_{i}-\mathbf{\varepsilon}_{j}\Big{)} ^{\mathrm{T}}\mathbf{\Lambda}_{i}\mathbf{K}_{\mathrm{viz}}\mathbf{\Lambda}_{i}\Big{(}\mathbf{B }_{i}\mathbf{e}_{i}-\mathbf{\varepsilon}_{j}\Big{)}, \tag{10}\]  
derived from (29) and (46). Substituting (10) into the derivative of (11), we obtain:  
\[\dot{V}_{i}(\mathbf{x})\leq -\lambda_{\min}(\mathbf{\Lambda}_{i})\lambda_{\min}(\mathbf{B}_{i})\|\mathbf{ e}_{i}\|^{2}+\|\mathbf{e}_{i}\|\|\mathbf{s}_{i}\|+\lambda_{\max}(\mathbf{\Lambda}_{i})\| \mathbf{e}_{i}\|\|\mathbf{\dot{e}}_{j}\|-\lambda_{\min}(\mathbf{K}_{\mathrm{viz}})\|\mathbf{s }_{i}\|^{2}\] \[-\lambda_{\min}(\mathbf{K}_{\mathrm{viz}})\|\mathbf{\dot{e}}_{i}\|^{2}+k_ {\mathrm{g}i}\|\|\mathbf{e}_{i}\|+\lambda_{\mathrm{H}i}\lambda_{\max}(\mathbf{\Lambda} _{i})\|\mathbf{s}_{i}\|\Big{(}\lambda_{\max}(\mathbf{B}_{i})\|\mathbf{\dot{e}}_{i}\|+\|\mathbf{ \dot{e}}_{j}\|\Big{)}\] \[+\lambda_{\max}(\mathbf{\Lambda}_{i})k_{\mathrm{c}i}\|\mathbf{s}_{i}\|\| \mathbf{\dot{e}}_{i}\|\Big{(}\lambda_{\max}(\mathbf{B}_{i})\|\mathbf{e}_{i}\|+\|\mathbf{\dot{e} }_{j}\|\Big{)}+\lambda_{\max}(\mathbf{K}_{\mathrm{viz}})\|\mathbf{s}_{i}\|\|\mathbf{\dot{e} }_{j}\|\] \[\mathbf{x} =\mathbf{0} \Rightarrow \hat{\mathbf{e}}_{i}=\mathbf{0},\,\hat{\mathbf{q}}_{i}=\mathbf{0},\]0  
Next, assuming \(\|\mathbf{x}(t_{0})\|<\epsilon\), where \(\epsilon\) is defined in (45), and considering that \(T_{j}(t)\geq 0\) for \(j=l,r\) (by Assumption 2), the state variables remain bounded within a region \(\mathcal{D}\). Substituting these bounds into (11), the following refined inequality is obtained:  
\[\mathbf{x} =\mathbf{0} \Rightarrow \hat{\mathbf{e}}_{i}=\mathbf{0},\,\hat{\mathbf{q}}_{i}=\mathbf{0},\]1 \[\mathbf{x} =\mathbf{0} \Rightarrow \hat{\mathbf{e}}_{i}=\mathbf{0},\,\hat{\mathbf{q}}_{i}=\mathbf{0},\]2 \[\mathbf{x} =\mathbf{0} \Rightarrow \hat{\mathbf{e}}_{i}=\mathbf{0},\,\hat{\mathbf{q}}_{i}=\mathbf{0},\]3 \[\mathbf{x} =\mathbf{0} \Rightarrow \hat{\mathbf{e}}_{i}=\mathbf{0},\,\hat{\mathbf{q}}_{i}=\mathbf{0},\]4  
Next, the inequality (12) refines (11) by incorporating the assumption that \(\|\mathbf{x}(t_{0})\|<\epsilon\). This ensures that \(\mathbf{x}(t)\) remains within \(\mathcal{D}\), maintaining the boundedness of terms such as \(\|\mathbf{\dot{e}}_{j}\|\) and \(\|\mathbf{\dot{e}}_{j}\|\). These bounds are critical in demonstrating that \(\dot{V}_{i}(\mathbf{x})\) remains negative definite for all \(\mathbf{x}\
eq\mathbf{0}\). Finally, since \(\dot{V}_{i}(\mathbf{x})<0\), it follows that \(V_{i}(\mathbf{x})\) is a strictly decreasing function. Combined with the positive definiteness of \(V_{i}(\mathbf{x})\), this guarantees that \(\mathbf{x}(t)\) asymptotically approaches \(\mathbf{0}\) as \(t\to\infty\), completing the proof of stability for this subsystem. 3. Since any of the robots can be considered as local and the other one as remote, without loss of generality, it is assumed that \(\mathbf{A}_{\mathrm{r}}=\mathbf{O}\) and \(\mathbf{B}_{\mathrm{r}}=\mathbf{I}\), according to (30), while \(\mathbf{A}_{1}>\mathbf{O}\).