# Variable Admittance Control in Sliding Mode for Robust Physical Human-Robot Interaction
## Appendix C  
This appendix lists some key assumptions made in this study.  
**Assumption A1**.: \(\mathbf{x}=\mathbf{x_{c}}\)__  
_Admittance control imposes constraints on the reference trajectory and expected trajectory of the robot when interacting with humans. This expected trajectory acts as the control commands directed to the robot. The objective of the SMC is to ensure that the robot's actual response aligns with this expected or desired outcome. Based on this premise, assuming \(\mathbf{x}=\mathbf{x_{c}}\) basically sets a control goal. As a result,a specific sliding surface (i.e., Equation (15)) can be derived. This ensures two main objectives are met: (i) achieving the desired admittance; (ii) ensuring that the robot's response tracks the desired outcome under variable admittance control._  
**Assumption A2**.: \(\mathbf{C_{w}^{\star}}\)_can be modeled as a source of bounded uncertainty_  
_In many robotic systems, particularly higher DOF systems, the Coriolis and centrifugal term is not unique; that is, multiple matrices can satisfy the equations of motion. This is a result of the properties of the equations of motion and the flexibility in how they can be represented. The skew-symmetric property of \(\Big{(}\dot{\mathbf{H_{w}}}-2\mathbf{C_{w}}\Big{)}\) is only valid for a unique \(\mathbf{C_{w}}\). However, the analytical expression of unique \(\mathbf{C_{w}}\) (i.e., \(\mathbf{C_{w}^{\star}}\)) is practically difficult to obtain. The Coriolislis and centrifugal term (\(\mathbf{C}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}\)) implicitly appears in the equivalent control in sliding mode control for manipulator in the Cartesian Space (e.g., \(\mathbf{u_{eq}}=\mathbf{H_{w}}(\mathbf{x})\ddot{\mathbf{x}}+\mathbf{S_{w}}\big{(}\mathbf{x},\dot{\mathbf{x}} \big{)}+\mathbf{G_{w}}(\mathbf{x})-\mathbf{f_{h}}\), \(\mathbf{S_{w}}=-\mathbf{J^{-1}}^{T}\mathbf{H}(\mathbf{q})\mathbf{J^{-1}}\mathbf{J}^{-1}+\mathbf{J^{-1}}^{T} \mathbf{C}(\mathbf{q},\dot{\mathbf{q}})\mathbf{J}^{-1}\), which contributes to deriving the sliding mode gain. In Equation (39), \(\mathbf{C_{w}^{*}}\) 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 \(\mathbf{C_{w}^{*}}\mathbf{\sigma}\) is bounded, such that it can be modeled as bounded uncertainty.  
**Assumption A3**.: \(K_{smc}(\mathbf{x})=K_{smc}(\mathbf{x_{s}})\)__  
_In Equation (25), \(\bar{\mathbf{x_{s}}}\) is derived as the equivalent acceleration signal. Equation (25) represents perfect tracking of desired admittance response if \(\ddot{\mathbf{x}}_{s}=\ddot{\mathbf{x}}\). A corresponding sliding surface is defined as Equation (26). Considering the finite reach time of sliding mode control and subtraction (i.e., \(K_{smc}(\mathbf{x})-K_{smc}(\mathbf{x_{s}})\)) in Equation (49), it can be fairly assumed that \(K_{smc}(\mathbf{x})=K_{smc}(\mathbf{x_{s}})\) during the sliding mode under constraints of the boundary layer. This assumption results in Equations (50) and (51), in which \(K_{smc}\) is a low-pass-filtered version of \(K_{smc}^{0}\). \(K_{smc}^{0}\), calculated from Equation (31), which is a conservative sliding mode gain, guarantees system stability. Hence, the new low-pass-filtered gain does not distort the stability condition. In addition, the boundary layer becomes a function of sliding mode gain, such that it takes system uncertainties into account and is adapted to system dynamic changes._