Stability and Transparency in Series Elastic Actuation: A Two-Port Analysis

Series Elastic Actuation (SEA) is a widely-used approach for interaction control, as it enables high fidelity and robust force control, improving the safety of physical human-robot interaction (pHRI). In the design of pHRI systems, safety is an imperative design criterion that limits interaction performance, since there exists a fundamental trade-off between the stability robustness and rendering performance. The safety of interaction necessitates coupled stability to ensure the closed-loop stability of a pHRI system when coupled to a wide range of unknown operators and environments. The frequency-domain passivity framework provides powerful analysis tools to study the coupled stability of linear time-invariant systems. In the literature, coupled stability of one-port models of SEA has been studied for various controllers while rendering certain basic environments, and the necessary and sufficient conditions for such passive terminations have been derived. In this study, we extend the one-port passivity analyzes provided in the literature and provide the necessary and sufficient condition for two-port passivity of SEA under velocity-sourced impedance control. Based on the newly established conditions, we derive non-conservative passivity bounds for a virtual coupler. We also prove the need for a physical damping term in parallel to the series elastic element to ensure two-port passivity, even when a virtual coupler is present. Finally, we validate our theoretical results through numerical simulations and by reproducing one-port passivity results as special cases of our results that correspond to appropriate one-port terminations.