# Variable Admittance Control in Sliding Mode for Robust Physical Human-Robot Interaction
###### Abstract
The proposed power envelope regulation (PER) approach by [PERSON] and [PERSON] [16] can regulate extremely changed admittance parameters induced by human intention such that the drastic change in the admittance dynamics system is suppressed. Though the power envelope regulation adds a protection layer to system stability, it cannot guarantee system stability analytically. Therefore, a stable controller needs to be applied in addition. The sliding mode controller becomes a natural candidate due to its two merits: (i) dynamic behavior of the system can be tailored by choice of the sliding surface; this flexibility allows the realization of desired admittance and (ii) it is robust against modeling uncertainties and external disturbance [33; 34]. SMC is a nonlinear controller that utilizes high-frequency switching feedback control. The first application of SMC in impedance control was the study conducted by [PERSON] and [PERSON] in which the integral of the desired impedance was selected as the sliding surface [35]. A new velocity signal was introduced for tracking, intended to couple impedance and robot dynamics. However, the theoretical analysis was somewhat superficial. Additionally, terms such as reference velocity, actual velocity, command velocity, and this newly introduced velocity were not clearly distinguished, leading to potential confusion. For admittance control in pHRI, SMC has been mainly applied in the rehabilitation exoskeleton [36; 37], where the sliding surface was adopted as the conventional one discussed in [33]. The admittance equation is solely solved to obtain the response sent to the robot as commands. Nonetheless, simulation results presented in this study show diminished tracking accuracy using the classical method in admittance control in the absence of a prescribed trajectory (or no stiffness term). In addition, in [35; 36; 37], the Coriolis and centrifugal matrix can be uniquely determined due to the system's low degree of freedom (DoF), specifically 2. This unique matrix is then used to derive the reachability of SMC via Lyapunov methods, where the explicit expression of the Coriolis and centrifugal matrix is required. Nonetheless, this approach may not be directly applicable to systems with a higher DoF, where the Coriolis and centrifugal matrix have non-unique solutions. A comparative analysis in Section 3 will better deliver the ideas mentioned above. Chattering is known as the main limitation of implementing SMC in real applications caused by high-frequency oscillation. Two types of techniques, i.e., quasi-sliding mode and higher order sliding mode (HOSM), have been proposed to tackle chattering. In quasi-sliding mode, a boundary layer is added to the sliding surface by replacing the signum function with saturation, relay, or hyperbolic tangent function [38; 39]. Notwithstanding eliminating chattering, the quasi-sliding mode sacrifices robustness and accuracy. Among many HOSM methods, Super-Twisting second-order sliding mode control (STSMC) gained extensive attention in the literature due to its merit of retaining accuracy and robustness while eliminating chattering [39]. Nonetheless, proof of stability of STSMC in the presence of time-varying admittance parameters is challenging and will not be considered in the paper. Acceleration feedback control (AFC) has been shown to improve control performance by increasing system bandwidth in previous studies [40; 41; 42; 43]. [PERSON] et al. conducted a frequency-based analysis for two types of AFC on AT telescope via the Bode plot technique [41]. Analyzing power spectral density shows that acceleration feedback adds another sensitivity transfer function to the servo system, improving performance against perturbation. [PERSON] and [PERSON] experimentally showed the improved performance of the inner position controller on a 1 DoF manipulator by applying AFC along with the friction compensator [42]. Theoretically, the extended bandwidth contributed by acceleration feedback is due to the phase-lead effect.