# Variable Admittance Control in Sliding Mode for Robust Physical Human-Robot Interaction
## 3 Reachability in VAC
### Acceleration Feedback in SMC  
In Figure 7, the overshooting phenomenon is observed. One reason could be that the limited bandwidth of the system results in a delayed reaction. Therefore, this section introduces acceleration feedback to improve control performance.  
Figure 6: Chattering phenomenon is observed when boundary layer thickness is decreased from 0.1 in Figure 5c to 0.03.  
The equivalent control is interpreted as the continuous control that maintains system motion restricted to the sliding mode in the absence of disturbances and uncertainties. It consists of exact system dynamics (best approximation). Therefore, we include acceleration error, \(\dddot{\tilde{\mathbf{x}}}=\ddot{\mathbf{x}}-\ddot{\mathbf{x}}_{\mathbf{s}}\), in the equivalent control such that the bandwidth of system can be influenced (i.e., increased),  
\[\mathbf{u}_{\mathbf{eq}}=\mathbf{\hat{H}}_{\mathbf{w}}\ddot{\mathbf{x}}_{\mathbf{s}}+\mathbf{\hat{S}}_{\mathbf{ w}}+\mathbf{\hat{G}}_{\mathbf{w}}-\mathbf{\Delta}-\mathbf{f}_{\mathbf{h}}-\mathbf{\hat{H}}_{\mathbf{w}}\mathbf{Q} \dddot{\tilde{\mathbf{x}}} \tag{52}\]  
where \(\mathbf{Q}\) is a positive definite dynamic matrix gain. Since the equivalent control is obtained by setting \(\dot{\sigma}=0\), then a new sliding surface shall be derived inversely to adapt to the introduced acceleration feedback. However, it is learned that Equation (26) also satisfies SMC with acceleration feedback. Derivation in Appendix B shows that adding the acceleration feedback does not change the stability of the original system. The extended system bandwidth could explain the improved performance in tracking and robustness against uncertainties [40; 41; 42; 43]. In fact, the way the acceleration error feeds into the control loop in the present paper is similar to what has been studied as a Plug-in structure in [41]. It has been shown that acceleration has no impact on stability conditions in this structure.  
In the end, numerical simulation is conducted by compiling the proposed approaches in Section 3 together, yielding results shown in Figure 8. Comparing Figures 7 and 8b, it concludes that adding acceleration feedback results in smooth and perfect tracking in the presence of significant uncertainties. In addition to the robot acting as a passive follower, our approach performs well in human-following cases. A sinusoidal trajectory, \(\mathbf{x}_{\mathbf{r}}\), is prescribed and tracked by the robot to provide physical support to the human. Assuming the interaction is normal and safe (recall the human intention framework in Section 2), human interaction is recognized as a disturbance. The stiffness is increased proportional to the human's applied force, then regulated by PER. The proposed SMC not only ensures the tracking between \(\mathbf{x}_{\mathbf{c}}\) and \(\mathbf{x}_{\mathbf{r}}\) but also between \(\mathbf{x}\) and \(\mathbf{x}_{\mathbf{c}}\).  
Figure 7: The variable boundary layer approach is applied in implementing SMC in VAC, in which the same VAC models and force profile are employed as in Figure 5c. Tracking is relatively accurate, and the boundary layer thickness does not need to be defined by trial and error. It is also observed that the new sliding mode gain is simply a low-pass filtered signal of Equation (31). The modeling errors are designed the same as in Figure 5.