# Variable Admittance Control in Sliding Mode for Robust Physical Human-Robot Interaction
## 3 Reachability in VAC
### Sliding Surface in Admittance Control  
Knowing the limitations of interpreting SMC as direct tracking in realizing desired admittance, a new sliding surface is derived. It is then extended to be a general sliding surface for admittance control. A rigid manipulator with \(n\)-DoF is considered, and its dynamics, exclusive of friction and uncertainties, are expressed in joint space as  
\[\mathbf{H}(\mathbf{q})\ddot{\mathbf{q}}+\mathbf{C}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}\dot{\mathbf{q}} +\mathbf{G}(\mathbf{q})=\tau_{\mathbf{c}}+\tau_{\mathbf{h}} \tag{12}\]  
where \(\mathbf{q}\) denotes the robot's joint angles, and \(\dot{\mathbf{q}}\) and \(\ddot{\mathbf{q}}\) represent accelerations and velocities. \(\mathbf{H}(\mathbf{q})\) represents a symmetric positive definite inertia matrix; \(\mathbf{C}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}\) is Coriolis and centrifugal matrix; \(\mathbf{G}(\mathbf{q})\) is a torque vector generated by gravity; \(\tau_{\mathbf{c}}\) and \(\tau_{\mathbf{h}}\) represents applied torque and external torque, respectively. To adapt to the admittance implemented in the workspace, Equation (12) is written in the Cartesian space as  
\[\mathbf{H}_{\mathbf{w}}(\mathbf{x})\ddot{\mathbf{x}}+\mathbf{C}_{\mathbf{w}}\big{(}\mathbf{x},\dot{\mathbf{x}} \big{)}\dot{\mathbf{x}}+\mathbf{G}_{\mathbf{w}}(\mathbf{x})=f_{\mathbf{c}}+f_{\mathbf{h}} \tag{13}\]  
where subscript '\(\mathbf{w}\)' represents a fixed Cartesian coordinate at the base of the manipulator in world space, and the mapping from joint to Cartesian space of the end-effector is simply the Jacobian-based operation:  
\[ \begin{gathered}\mathbf{H}_{\mathbf{w}}=-{J^{-1}}^{\mathbf{T}}\mathbf{H}(\mathbf{q}){ J^{-1}},\\ \mathbf{C}_{\mathbf{w}}=-{J^{-1}}^{\mathbf{T}}\mathbf{H}(\mathbf{q}){J^{-1}}{J^{-1}}+{J^ {-1}}^{\mathbf{T}}\mathbf{C}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}{J^{-1}},\\ \mathbf{G}_{\mathbf{w}}={J^{-1}}^{\mathbf{T}}\mathbf{G}(\mathbf{q}),\\ f_{\mathbf{c}}={J^{-1}}^{\mathbf{T}}\mathbf{\tau}_{\mathbf{c}},\\ f_{\mathbf{h}}={J^{-1}}^{\mathbf{T}}\tau_{\mathbf{h}}\end{gathered}  \tag{14}\]  
In [35], the sliding surface was defined as the desired impedance in the integral form. The impedance equation includes system states that are not rigorous unless the perfect tracking (e.g., \(\mathbf{x}=\mathbf{x}_{\mathbf{c}}\)) is assumed. No proof was given in the paper of how to arrive at the suggested control law. In addition, replacing \(\dot{\mathbf{x}}\) and \(\ddot{\mathbf{x}}\) with the new states (i.e., \(\dot{\mathbf{x}}_{\mathbf{s}}\) and  
Figure 2: SMC for human-leading, where Equation (11) is selected as the sliding surface. \(\ddot{x}_{s}\)), while leaving \(\mathbf{H}\), \(\mathbf{C}\), and \(\mathbf{G}\) as functions of the original system states, is incorrect and creates confusion. A new sliding surface is proposed with theoretical proof to address the above limitations. Unlike conventional SMC imposing direct tracking, tracking desired response in admittance control can be interpreted as indirect tracking: _(i) realizing desired admittance; (ii) tracking desired admittance response_. A sliding surface shall be selected to tailor the dynamic behavior in line with Equation (10), i.e., desired admittance regarding human leading. A straightforward option could be:  
\[\sigma=\big{(}M_{\mathbf{d}  
where \(\mathbf{v}\) is known as the new input. Substituting Equation (23) into (18) arrives at  
\[\dot{\mathbf{\sigma}}=-\mathbf{M}_{\mathbf{d}}(t)\ddot{\mathbf{x}}_{\mathbf{c}}+-\mathbf{D}_{\mathbf{d}}(t) \dot{\mathbf{x}}_{\mathbf{c}}+\mathbf{f}_{\mathbf{h}} \tag{24}\]  
The right side of Equation (24) is exactly the targeted admittance and is always equal to zero. Equation (24) is exactly the targeted admittance and is always equal to zero. It also suggests that Equation (18) is equal to zero with the unique solution, \(\dot{\mathbf{x}}=\dot{\mathbf{x}}_{\mathbf{c}}\). Thus, it also allows satisfaction of the second goal: tracking desired admittance response.