  We introduce minimal passive physical equivalents 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 help 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 to properly designed plant dynamics. Finally, we provide experimental
validations of our theoretical results and characterizations of the haptic
rendering performance of S(D)EA under VSIC.
