Extension of colgate´s passivity condition for variable-rate haptics

Colgate´s passivity conditions for the virtual coupling and the virtual wall have been one of the most important results in haptics. However, due to its development being rooted in the frequency-domain, its theoretical justification has been limited to only uniform-rate haptics. In this paper, relying on analysis/derivation in the time-domain, we extend this Colgate´s passivity condition to variable-rate haptics. We also present a similar passivity condition for the haptic rendering based on the recently proposed passive set-position modulation (PSPM).