# Variable Admittance Control in Sliding Mode for Robust Physical Human-Robot Interaction
## 3 Reachability in VAC
### Reachability in VAC
Applying inequality property of the inner product and norm (i.e., \(<\mathbf{a},\mathbf{b}>\leq\parallel\mathbf{a}\parallel\parallel\mathbf{b}\parallel\)) and the property of \(\parallel\mathbf{a}\cdot\mathbf{b}\parallel\leq\parallel\mathbf{a}\parallel\parallel\mathbf{b}\parallel\) to Equation (40), one gets  
\[\dot{V}\leq\parallel\mathbf{\sigma}^{\mathbf{T}}\parallel\big{(}\delta_{H}\parallel \ddot{\mathbf{x}}_{\mathbf{s}}\parallel+\delta_{S}+\delta_{G}+\delta-K_{smc}\big{)} \tag{41}\]  
Substituting Equation (31) into Equation (40), the well-known \(\eta\)-reachability condition [33], \(\dot{V}\leq-\eta\parallel\mathbf{\sigma}\parallel\), is satisfied. For variable admittance system, the extreme values of admittance parameters are assumed to be known. Then, the following inequalities are assumed:  
\[ \begin{split}\frac{1}{\lambda(\mathbf{M}_{\mathbf{m}ax})}\leq& \parallel\mathbf{M}_{\mathbf{d}}(\mathbf{t})^{-1}\parallel\leq\frac{1}{\lambda(\mathbf{M}_{ \mathbf{min}})},\\ \lambda(\mathbf{D}_{\mathbf{min}})\leq&\parallel\mathbf{D}_{\mathbf{d}}( \mathbf{t})\parallel\leq\lambda(\mathbf{D}_{\mathbf{max}}),\\ \lambda(\mathbf{K}_{\mathbf{min}})\leq&\parallel\mathbf{K}_{\mathbf{d}}( \mathbf{t})\parallel\leq\lambda(\mathbf{K}_{\mathbf{max}})\end{split}  \tag{42}\]  
Taking the Euclidean norm of Equation (A7) (refer to Appendix A) with the definition of \(\mathbf{A}=\ddot{\mathbf{x}}_{\mathbf{r}}+\mathbf{M}_{\mathbf{d}}^{-1}\mathbf{f}_{\mathbf{h}}\) and\(\mathbf{B}=\mathbf{M}_{\mathbf{d}}^{-1}\mathbf{D}_{\mathbf{d}}\big{(}\ddot{\mathbf{x}}-\dot{\mathbf{x}}_{\mathbf{r}}\big{)}+\mathbf{M}_{\mathbf{d}} ^{-1}\mathbf{K}_{\mathbf{d}}(\mathbf{x}-\mathbf{x}_{\mathbf{r}})\), and applying the inequality property \(\parallel\mathbf{A}-\mathbf{B}\parallel\geq\parallel\parallel\mathbf{A}\parallel- \parallel\mathbf{B}\parallel\) results in  
\[\parallel\ddot{\mathbf{x}}_{\mathbf{s}}\parallel\geq\parallel\parallel\mathbf{A}\parallel- \parallel\mathbf{B}\parallel\parallel \tag{43}\]  
Simplifying the above equation gives  
\[\parallel\ddot{\mathbf{x}}_{\mathbf{s}}\parallel\leq\parallel\mathbf{B}\parallel-\parallel \mathbf{A}\parallel \tag{44}\]By substituting \(\mathbf{A}\) and \(\mathbf{B}\) in the above equation, one obtains  
\[\parallel\ddot{\mathbf{x}}_{\text{s}}\parallel\leq\frac{\lambda(\mathbf{D_{max}})}{ \lambda(\mathbf{M_{min}})}\parallel\left(\dot{\mathbf{x}}-\dot{\mathbf{x}}_{\text{r}} \right)\parallel+\frac{\lambda(\mathbf{K_{max}})}{\lambda(\mathbf{M_{min}})}\parallel \mathbf{x}-\mathbf{x}_{\text{r}}\parallel-\frac{1}{\lambda(\mathbf{M_{min}})}\parallel \mathbf{f_{h}}\parallel-\parallel\ddot{\mathbf{x}}_{\text{r}}\parallel \tag{45}\]  
and  
\[\parallel\ddot{\mathbf{x}}_{\text{s}}\parallel\geq\frac{1}{\lambda(\mathbf{M_{min}}) }\parallel\mathbf{f_{h}}\parallel+\parallel\ddot{\mathbf{x}}_{\text{r}}\parallel- \frac{\lambda(\mathbf{D_{max}})}{\lambda(\mathbf{M_{min}})}\parallel\left(\dot{\mathbf{x}} -\dot{\mathbf{x}}_{\text{r}}\right)\parallel-\frac{\lambda(\mathbf{K_{max}})}{\lambda (\mathbf{M_{min}})}\parallel\mathbf{x}-\mathbf{x}_{\text{r}}\parallel \tag{46}\]  
Then, a more conservative bound in terms of system stability, i.e., Equation (32), is obtained that further proves the reachability condition. \(\square\)  
In the following discussion, a relatively large modeling error in the mass of each link is intentionally introduced to investigate system robustness and stability. Analytical expressions for \(\mathbf{H_{w}}\), \(\mathbf{S_{w}}\), and \(\mathbf{G_{w}}\) incorporated the mass of every link in the manipulator. Subsequently, the threshold values in Equation (38) were determined to maintain consistency in position and velocity while varying the mass values. In both simulation and experimental validation, a higher sliding mode gain was employed to ensure system stability and responsiveness in the face of other uncertainties. It is crucial to note that in real-world scenarios, the sliding mode gains and other control parameters must be meticulously calibrated, considering the unique demands of the given task.