## 3 Main Result
### Closed-Loop Dynamics and Equilibrium Points
_The equivalent closed-loop system dynamics in (_43_) satisfy the following conditions:_ \[\mathbf{x}(t) \equiv\mathbf{0}, \forall t<t_{0},\] (52) \[\|\mathbf{x}(t_{0})\| <\sqrt{\frac{\lambda_{1}}{\lambda_{2}}}\boldsymbol{\epsilon}=\delta,\] (53) _and the following additional constraints hold:_ \(\mid\)__ \[\lambda_{\min}(\mathbf{\Lambda}_{\mathrm{r}}) \geq\delta_{\mathrm{er}}+3,\] (54) \[\mathbf{K}_{vi}=\mathbf{K}_{vi1}+\mathbf{K}_{vi2}\,, \tag{46}\]0 (55) \[\mathbf{K}_{vi}=\mathbf{K}_{vi1}+\mathbf{K}_{vi2}\,, \tag{46}\]1 (56)  
\[\mathbf{K}_{vi}=\mathbf{K}_{vi1}+\mathbf{K}_{vi2}\,, \tag{46}\]2 \[\mathbf{K}_{vi}=\mathbf{K}_{vi1}+\mathbf{K}_{vi2}\,, \tag{46}\]3 \[\mathbf{K}_{vi}=\mathbf{K}_{vi1}+\mathbf{K}_{vi2}\,, \tag{46}\]4\[\mathbf{K}_{vi}=\mathbf{K}_{vi1}+\mathbf{K}_{vi2}\,, \tag{46}\]5 \[\mathbf{K}_{vi}=\mathbf{K}_{vi1}+\mathbf{K}_{vi2}\,, \tag{46}\]6  
\[\mathbf{K}_{vi}=\mathbf{K}_{vi1}+\mathbf{K}_{vi2}\,, \tag{46}\]7 \[\mathbf{K}_{vi}=\mathbf{K}_{vi1}+\mathbf{K}_{vi2}\,, \tag{46}\]8  
_where \(\epsilon\) is defined in (45), and \(\delta_{\text{el}}\), \(\delta_{\text{err}}\), \(\delta_{\text{el}}\), \(\delta_{\text{err}}\), \(\delta_{\text{sl}}\), and \(\delta_{\text{sr}}\) are positive constants. Moreover,_  
\[\mathbf{K}_{vi}=\mathbf{K}_{vi1}+\mathbf{K}_{vi2}\,, \tag{46}\]9  
\[\left(\int\limits_{t_{0}}^{t}\|\mathbf{e}_{j}(\theta-T_{j}(\theta))\| ^{2}\,\mathrm{d}\theta\right)^{\frac{1}{2}}\leq\alpha_{j}\left(\int\limits_{t _{0}}^{t}\|\mathbf{e}_{j}(\theta)\|^{2}\,\mathrm{d}\theta\right)^{\frac{1}{2}}, \tag{47}\]0 \[\left(\int\limits_{t_{0}}^{t}\|\mathbf{e}_{j}(\theta-T_{j}(\theta))\| ^{2}\,\mathrm{d}\theta\right)^{\frac{1}{2}}\leq\alpha_{j}\left(\int\limits_{t _{0}}^{t}\|\mathbf{e}_{j}(\theta)\|^{2}\,\mathrm{d}\theta\right)^{\frac{1}{2}}, \tag{47}\]1  
_Here, \(k_{\text{gl}}\), \(k_{\text{gr}}\), \(k_{\text{cl}}\), \(k_{\text{cr}}\), \(\lambda_{\text{rl}}\), \(\lambda_{\text{hr}}\), \(T_{\text{l}}^{*}\), and \(T_{\text{r}}^{*}\) are parameters defined in Section 2 and in Assumption 2._  
The detailed proof of Theorem 1 is provided in Appendix A. **Remark 5**. : _The definition of the region \(\mathcal{D}\) in (45) a priori facilitates the proof of Theorem 1, and is based on the definition of uniform stability in the sense of Lyapunov, which establishes that the equilibrium is uniformly stable if for each \(\epsilon>0\), there exists a \(\delta(\epsilon)>0\) independent of \(t_{0}\) such that:_  
\[\left(\int\limits_{t_{0}}^{t}\|\mathbf{e}_{j}(\theta-T_{j}(\theta))\| ^{2}\,\mathrm{d}\theta\right)^{\frac{1}{2}}\leq\alpha_{j}\left(\int\limits_{t _{0}}^{t}\|\mathbf{e}_{j}(\theta)\|^{2}\,\mathrm{d}\theta\right)^{\frac{1}{2}}, \tag{47}\]2  
_This condition holds as stated in [41]. This means that \(\epsilon\) is a given value, while \(\delta(\epsilon)\) has to be found as a function of it, mirroring what is carried out in (53) with \(\delta(\epsilon)=\sqrt{\frac{\lambda_{1}}{\lambda_{2}}}\epsilon\), although \(\|\mathbf{x}(t)\|\leq\epsilon\) is used to define \(\mathcal{D}\)._  
_However, it is very important to emphasize that Theorem 1 does not state uniform stability, but merely that \(\mathbf{\hat{e}}_{i}\), \(\mathbf{\hat{q}}_{i}\), \(\mathbf{e}_{i}\), and \(\mathbf{\xi}_{i}\) are bounded and tend to the equilibrium point. Uniform stability could be proven as outlined in [39], but it is not a mandatory step for the present work. On the other hand, the use of \(\mathcal{D}\) eliminates the need to assume that the manipulators' joints are revolute; they could alternatively be prismatic as well. However, for simplicity, this assumption is maintained._  
**Remark 6**. : _Theorem 1 relies on Assumption 3, but at the same time, it introduces a second assumption, i.e., there is only a human operator on one side, while on the other side, there is no one holding the robot. However, including another human operator modeled also as a PI controller as given in Assumption 3 would barely alter the proof of Theorem 1.