  Series Elastic Actuation (SEA) has become prevalent in applications involving
physical human-robot interaction as it provides considerable advantages over
traditional stiff actuators in terms of stability robustness and fidelity of
force control. Several impedance control architectures have been proposed for
SEA. Among these alternatives, the cascaded controller with an inner-most
velocity loop, an intermediate torque loop and an outer-most impedance loop is
particularly favoured for its simplicity, robustness, and performance. In this
paper, we derive the \emph{necessary and sufficient conditions} to ensure the
passivity of this cascade-controller architecture for rendering two most common
virtual impedance models. Based on the newly established passivity conditions,
we provide non-conservative design guidelines to haptically display a null
impedance and a pure spring while ensuring the passivity of interaction. We
also demonstrate the importance of including physical damping in the actuator
model during derivation of passivity conditions, when integral controllers are
utilized. In particular, we show the adversary effect of physical damping on
system passivity.
