Effects of discretization on the K-width of series elastic actuators

Rigid haptic devices enable humans to physically interact with virtual environments, and the range of impedances that can be safely rendered using these rigid devices is quantified by the Z-Width metric. Series elastic actuators (SEAs) similarly modulate the impedance felt by the human operator when interacting with a robotic device, and, in particular, the robot's perceived stiffness can be controlled by changing the elastic element's equilibrium position. In this paper, we explore the K-Width of SEAs, while specifically focusing on how discretization inherent in the computer-control architecture affects the system's passivity. We first propose a hybrid model for a single degree-of-freedom (DoF) SEA based on prior hybrid models for rigid haptic systems. Next, we derive a closed-form bound on the K-Width of SEAs that is a generalization of known constraints for both rigid haptic systems and continuous time SEA models. This bound is first derived under a continuous time approximation, and is then numerically supported with discrete time analysis. Finally, experimental results validate our finding that large pure masses are the most destabilizing operator in human-SEA interactions, and demonstrate the accuracy of our theoretical K-Width bound.