# Variable Admittance Control in Sliding Mode for Robust Physical Human-Robot Interaction
## 3 Reachability in VAC
### Reachability in VAC  
In general, there are two steps involved in SMC design: defining a sliding surface to satisfy desired dynamic behavior and designing a controller to drive system states to stay on the sliding surface. As seen in the previous section, selecting a sliding surface for  
Figure 3: SMC with the new sliding surface being applied: (**a**) displays the same force profile in Figure 1. Admittance response, \(\mathbf{x}_{\mathbf{c}}\), does not explicitly appear in the controller; nonetheless, it is tracked by actual states accurately. In comparing (**b**) to Figure 1e, an improvement is observed that suggests the efficiency of the proposed method. Moreover, the same sliding surface works well for both human-leading and human-following cases as appeared in (**c**). admittance control was completed. This section aims to prove the sufficient condition (or reachability condition) for ensuring sliding mode in VAC. Previous studies have proven the reachability of conventional SMC in the sense of Lyapunov [35; 36; 37]. However, the proof is based on a 2-DoF system where the analytical expression of Coriolis and centrifugal matrix can be uniquely determined, such that the skew-symmetric property of \(\dot{H_{w}}-2\mathbf{C_{w}}\) is valid in this case. For a higher DoF system, \(\mathbf{C_{w}}\) has infinite solutions. Only a specific definition of \(\mathbf{C_{w}^{*}}\) (Chapter 9 in [33]) satisfies \(\dot{H_{w}}=\mathbf{C_{w}^{*}}+\mathbf{C_{w}^{T}}\), that is  
\[C_{ij}=\frac{1}{2}\dot{H_{ij}}+\frac{1}{2}\sum_{k=1}^{n}\left(\frac{\partial H _{ik}}{\partial x_{j}}-\frac{\partial H_{jk}}{\partial x_{i}}\right)\dot{x}_{k} \tag{28}\]  
In order to prove the stability of a controller, the analytical model of robot dynamics is required. This analytical model is usually obtained via the Euler-Lagrange method and yields  
\[\mathbf{H_{w}}(\mathbf{x})\ddot{\mathbf{x}}+\mathbf{S_{w}}\big{(}\mathbf{x},\dot{\mathbf{x}}\big{)}+ \mathbf{G_{w}}(\mathbf{x})=\mathbf{f_{c}}+\mathbf{f_{h}} \tag{29}\]  
where \(\mathbf{S_{w}}\big{(}\mathbf{x},\dot{\mathbf{x}}\big{)}\) is Coriolis and centrifugal force. And factorization of \(\mathbf{C_{w}}\) by \(\mathbf{S_{w}}\) is not unique. A problem then naturally arises: How does the proof using a specific solution of \(\mathbf{C_{w}}\), i.e., \(\mathbf{C_{w}^{*}}\), guarantee overall reachability for all solutions of \(\mathbf{C_{w}}\)? To address this issue, the unique \(\mathbf{S_{w}}\big{(}\mathbf{x},\dot{\mathbf{x}}\big{)}\) is adopted, and the term \(\mathbf{C_{w}^{*}}\) is modeled as a source of uncertainty (refer to the following proof). Furthermore, time-varying admittance parameters are taken into account. **Control law for SMC in VAC. ** The system is stable in the sense of Lyapunov with the control law considering uncertainties and disturbances:  
\[\mathbf{u}=\hat{\mathbf{H}_{w}}(\mathbf{x})\ddot{\mathbf{x}_{s}}+\hat{\mathbf{S}_{w}}\big{(}\mathbf{x},\dot{\mathbf{x}}\big{)}+\hat{\mathbf{G}_{w}}(\mathbf{x})-\mathbf{\Delta}-\mathbf{f_{h}}-K_{smc} sign(\mathbf{\sigma}) \tag{30}\]  
where  
\[K_{smc}\geq\ \delta_{H}\parallel\ddot{\mathbf{x}_{s}}\parallel+\delta_{S}+\delta _{G}+\delta+\eta \tag{31}\]  
and,  
\[\parallel\ddot{\mathbf{x}_{s}}\parallel\leq\frac{\lambda(\mathbf{D_{max}})}{\lambda( \mathbf{M_{min}})}\parallel\dot{\mathbf{x}}-\dot{\mathbf{x}_{r}}\parallel+\frac{\lambda( \mathbf{K_{max}})}{\lambda(\mathbf{M_{min}})}\parallel\mathbf{x}-\mathbf{x_{r}}\parallel \tag{32}\]  
where \(\delta_{H}\), \(\delta_{S}\), and \(\delta_{G}\) denote the bounds of uncertainties of dynamic modeling, and \(\delta\) represents the approximate bounds of uncertainties from other resources. In Equation (30), \(\hat{\mathbf{H}_{w}}\), \(\hat{\mathbf{S}_{w}}\), and \(\hat{\mathbf{G}_{w}}\) represent the approximation of dynamic modeling; \(K_{smc}\) denotes the sliding mode gain; other uncertainties, for example, unmodeled friction force, computational errors, and \(\mathbf{C_{w}}\) term (will appear in Equation (39)) are assigned to \(\mathbf{\Delta}\). **Proof.