# Variable Admittance Control in Sliding Mode for Robust Physical Human-Robot Interaction
## 3 Reachability in VAC
### Reachability in VAC
** A Lyapunov candidate function is selected as  
\[V(\mathbf{\sigma})=\frac{1}{2}\mathbf{\sigma}^{\text{T}}\mathbf{H_{w}}\mathbf{\sigma} \tag{33}\]  
Taking the time derivative of Equation (33) arrives at  
\[\dot{V}(\mathbf{\sigma})=\mathbf{\sigma}^{\text{T}}\mathbf{H_{w}}\dot{\mathbf{\sigma}}+\frac{1 }{2}\mathbf{\sigma}^{\text{T}}\dot{\mathbf{H}_{w}}\mathbf{\sigma} \tag{34}\]  
Assuming \(\mathbf{C_{w}}=\mathbf{C_{w}^{*}}\) that satisfies \(\dot{H_{w}}(\mathbf{x})-2\mathbf{C_{w}^{*}}\big{(}\mathbf{x},\dot{\mathbf{x}}\big{)}\) leads to  
\[\dot{V}(\mathbf{\sigma})=\mathbf{\sigma}^{\text{T}}\mathbf{H_{w}}\dot{\mathbf{\sigma}}+\mathbf{ \sigma}^{\text{T}}\mathbf{C_{w}^{*}}\mathbf{\sigma} \tag{35}\]Substitution of Equation (26) into Equation (35) results in  
\[ \begin{split}\dot{V}&(\mathbf{\sigma})=\mathbf{\sigma}^{\mathbf{T}} \Big{(}\mathbf{H}_{\mathbf{w}}\ddot{\mathbf{x}}-\mathbf{H}_{\mathbf{w}}\ddot{\mathbf{x}}_{\mathbf{s}}+\mathbf{C }_{\mathbf{w}}^{\star}\dot{\mathbf{x}}-\mathbf{C}_{\mathbf{w}}^{\star}\ddot{\mathbf{x}}_{\mathbf{s}} \Big{)}\\ &=\mathbf{\sigma}^{\mathbf{T}}\Big{(}\mathbf{f}_{\mathbf{c}}+\mathbf{f}_{\mathbf{h}}-\mathbf{G }_{\mathbf{w}}-\mathbf{H}_{\mathbf{w}}\ddot{\mathbf{x}}_{\mathbf{s}}-\mathbf{C}_{\mathbf{w}}^{\star}\ddot {\mathbf{x}}_{\mathbf{s}}\Big{)}\end{split}  \tag{36}\]  
where \(\mathbf{f}_{\mathbf{c}}\) is the control input \(\mathbf{u}\). According to \(\dot{\mathbf{\sigma}}=0\) in Equation (27), the equivalent control can be derived as  
\[\mathbf{u}_{\mathbf{eq}}=\hat{\mathbf{H}}_{\mathbf{w}}(\mathbf{x})\ddot{\mathbf{x}}_{\mathbf{s}}+\hat\mathbf{{S}}_{\mathbf{w}}(\mathbf{x},\dot{\mathbf{x}})+\hat{\mathbf{G}}_{\mathbf{w}}(\mathbf{x})-\Delta-\mathbf{f} _{\mathbf{h}} \tag{37}\]  
The approximation of dynamic modeling is assumed to obey the following inequalities. \[ \begin{split}\parallel\triangle\mathbf{H}_{\mathbf{w}}(\mathbf{x})\parallel= \parallel\hat{\mathbf{H}}_{\mathbf{w}}(\mathbf{x})-\mathbf{H}_{\mathbf{w}}(\mathbf{x})\parallel\leq \delta_{H},\\ \parallel\triangle\mathbf{S}_{\mathbf{w}}(\mathbf{x},\dot{\mathbf{x}})\parallel= \parallel\hat{\mathbf{S}}_{\mathbf{w}}(\mathbf{x},\dot{\mathbf{x}})-\mathbf{S}_{\mathbf{w}}(\mathbf{x}, \dot{\mathbf{x}})\parallel\leq\delta_{S},\\ \parallel\triangle\mathbf{G}_{\mathbf{w}}(\mathbf{x})\parallel=\parallel\hat\mathbf{{G}}_{\mathbf{w}}(\mathbf{x})-\mathbf{G}_{\mathbf{w}}(\mathbf{x})\parallel\leq\delta_{G}\end{split}  \tag{38}\]  
where the norm operator represents the Euclidean norm. Analytical solutions of \(\hat{\mathbf{H}}_{\mathbf{w}}(\mathbf{x})\), \(\hat{\mathbf{S}}_{\mathbf{w}}(\mathbf{x})\), and \(\hat{\mathbf{G}}_{\mathbf{w}}(\mathbf{x})\) include kinematic parameters and inertial terms. As long as the modeling uncertainties are bounded, the above assumption is valid. Substituting Equations (37) and (38) into Equation (36) arrives  
\[\dot{V}(\mathbf{\sigma})=\mathbf{\sigma}^{\mathbf{T}}\Big{(}\triangle\mathbf{H}_{\mathbf{w}}\ddot {\mathbf{x}}_{\mathbf{s}}+\triangle\mathbf{S}_{\mathbf{w}}+\triangle\mathbf{G}_{\mathbf{w}}+\mathbf{C}_{ \mathbf{w}}^{\star}\mathbf{\sigma}+\Delta-K_{smc}sign(\mathbf{\sigma})\Big{)} \tag{39}\]  
For an exact dynamic model of the robot, \(\mathbf{C}_{\mathbf{w}}^{\star}\) is bounded as long as the entire system is stable. Moreover, \(\mathbf{\sigma}\) is also bounded and approaches zero. Therefore, it is fair to assume that \(\mathbf{C}_{\mathbf{w}}^{\star}\mathbf{\sigma}\) is bounded, such that it can be modeled as bounded uncertainty. To this end, Equation (39) is reduced to  
\[\dot{V}(\mathbf{\sigma})=\mathbf{\sigma}^{\mathbf{T}}\big{(}\triangle\mathbf{H}_{\mathbf{w}}\ddot {\mathbf{x}}_{\mathbf{s}}+\triangle\mathbf{S}_{\mathbf{w}}+\triangle\mathbf{G}_{\mathbf{w}}+\Delta-K_ {smc}sign(\mathbf{\sigma})\big{)} \tag{40}\]  
As a result, no analytical solution of \(\mathbf{C}_{\mathbf{w}}\) is required in the proof process.