We study reduced-order models of series elastic actuation under velocity-sourced impedance control, where the inner motion controller is assumed to render the system into an ideal motion source within a control bandwidth and replaced by a low-pass filter. We present necessary and sufficient conditions for the passivity of this system and prove that the passivity results obtained through the reduced-order model may violate the passivity of the full-order model. To enable safe use of the reduced-order model, we derive conditions under which the passivity bounds of the reduced-order model guarantee the passivity of the full-order system. Moreover, we synthesize passive physical equivalents of closed-loop systems while rendering Kelvin-Voigt, linear spring, and null impedance models to provide rigorous comparisons of the passivity bounds and rendering performance among the full- and reduced-order models. We verify our results through a comprehensive set of simulations and experiments.

We study a realistic model of series elastic actuation (SEA) under velocity-sourced impedance control (VSIC), where the inherent damping of the series elastic element is considered during the analysis, even when only the elasticity of the series damped elastic element is used to estimate the interaction forces. We establish a fundamental rendering limitation when the viscous damping of the physical filter is considered in the plant model and prove that passive rendering of stiffness levels that are higher than the stiffness of the physical filter, as well as passive rendering of Voigt models whose damping levels exceed the physical damping of the plant, are possible. We introduce passive physical equivalents of the closed-loop SEA systems with inherent series damping while rendering Kelvin-Voigt, spring, and null impedance models to provide an intuitive understanding of the passivity bounds and to enable rigorous comparisons of rendering performance among various closed-loop systems with different plant models (including or omitting the series damping) and/or controllers (utilizing different interaction force estimates). We present a comprehensive set of experiments to verify our results and demonstrate the effect of including/omitting the damping of the physical filter in the model of SEA.

We introduce minimal passive physical realizations of series (damped) elastic actuation (S(D)EA) under closed-loop control to determine the effect of different plant parameters and controller gains on the closed-loop performance of the system and to establish an intuitive understanding of the passivity bounds. Furthermore, we explicitly derive the feasibility conditions for these passive physical equivalents and compare them to the necessary and sufficient conditions for the passivity of S(D)EA under velocity-sourced impedance control (VSIC) to establish their relationship. Through the passive physical equivalents, we rigorously compare the effect of different plant dynamics (e.g., SEA and SDEA) on the system performance. We demonstrate that passive physical equivalents make the effect of controller gains explicit and establish a natural means for effective impedance analysis. We also show that passive physical equivalents promote co-design thinking by enforcing simultaneous and unbiased consideration of (possibly negative) controller gains and plant parameters. We demonstrate the usefulness of negative controller gains when coupled with properly designed plant dynamics. Finally, we provide experimental validations of our theoretical passivity results and comprehensive characterizations of the haptic rendering performance of S(D)EA under VSIC.