# Variable Admittance Control in Sliding Mode for Robust Physical Human-Robot Interaction
## 3 Reachability in VAC
### Conventional Sliding Surface  
SMC allows the system's dynamic behavior to be tailored by choice of the sliding surface. In terms of tracking, the general sliding surface is defined as a function of tracking errors in the state space \(\mathbb{R}^{2}\):  
\[\sigma(\mathbf{x},t)=\bigg{(}\frac{d}{dt}+\mathbf{\lambda}\bigg{)}^{z-1}\widetilde{ \mathbf{x}} \tag{7}\]where the tracking error is defined as \(\widetilde{\mathbf{x}}=\mathbf{x}-\mathbf{x}_{r}\); \(\lambda\) is a positive definite diagonal matrix. For a second-order system, Equation (7) is simply a weighted sum of velocity and position errors:  
\[\mathbf{\sigma}=\dot{\widetilde{\mathbf{x}}}+\lambda\widetilde{\mathbf{x}} \tag{8}\]  
In direct tracking, the sliding surface and a corresponding proper control yield good performance, as shown in Figure 1a,b in both joint and Cartesian spaces, where a simple 2-DoF manipulator is simulated. It is important to note that to avoid confusion between the x-dimension and the pose of the end-effector in Cartesian space (both typically denoted by 'x'), this paper uses the numerals 1 and 2 to represent the translational dimensions of x and y, respectively. This notation convention will be consistently used in all subsequent figures.  
In admittance control, the desired response is computed from Equation (1). SMC is then applied to impose a robust inner tracking control. If the classical approach (i.e., Equation (8)) is applied, then the sliding surface is expressed as  
\[\mathbf{\sigma}=\left(\dot{\mathbf{x}}-\dot{\mathbf{x}}_{c}\right)+\lambda(\mathbf{x}-\mathbf{x}_{ c}) \tag{9}\]  
Theoretically, as long as the system states stay on this surface, the desired admittance can be realized via accurate tracking. The sliding surface in Equation (9) was widely adopted in applications of impedance/admittance control [35; 36; 37]. The prescribed trajectory and stiffness term were included in admittance in these studies. However, we found that applying Equation (9) gives less accurate results in the human-leading scenario where the targeted admittance is expressed as  
\[\mathbf{M}_{d}(t)\ddot{\mathbf{x}}_{c}+\mathbf{D}_{d}(t)\dot{\mathbf{x}}_{c}=\mathbf{f}_{\mathbf{h}} \tag{10}\]  
The comparison is presented in Figure 1e,f for the robot being a passive follower and a compliant leader, respectively, with the same force profile. In the human-leading case, no reference trajectory is defined, and the tracking is less accurate. In contrast,  
Figure 1: Conventional SMC is implemented on a 2-DoF planar manipulator that appeared in (**d**); (**a**,**b**) display direct tracking in joint and Cartesian space, respectively. The reaching phase is observed in (**a**) when the initial position differs from the reference; (**c**) displays the interaction force. (**e**,**f**) present tracking under admittance control for human-leading and human-following, respectively. In (**e**), tracking is between actual states and desired response of admittance. In contrast, two tracking goals are required in (**f**): (_i_), responses of admittance system track referenced trajectory; (_ii_), actual trajectory tracks responses of admittance system. Subscription ’1’ represents the x-axis dimension._  
accurate tracking for both referenced trajectory and the robot's desired response in the human-following case is observed.  
In fact, the inner controller in human-leading is a velocity controller since Equation (10) is a first-order system. Hence, the sliding surface should be updated as a function of velocity error only according to the definition in Equation (7):  
\[\mathbf{\sigma}=\dot{\mathbf{x}}-\dot{\mathbf{x}}_{\mathbf{c}} \tag{11}\]  
However, tracking precision is further declined, as shown in Figure 2.