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This paper analyzes inherent conflicts between model-matching goals of haptic rendering and passivity requirements for coupled stability. We apply results from complex analysis to prove that certain linear passive virtual environments cannot be rendered passively with a desired level of accuracy and over a given finite bandwidth. One practical consequence is that, under appropriate hypotheses, passivity will be violated when accurately rendering inertia in a virtual environment that is less than the inertia of the uncompensated hardware dynamics. In a related result, we show that there exists a waterbed-type tradeoff between performance and phase lag in the rendered dynamics. Both design constraints arise from feedback-bandwidth limitations and not sampled-data effects, quantization, or nonlinearities. The key to our analysis is an interpretation of a Bode gain-phase integral relationship that relates magnitude at low frequencies to phase at high frequencies. The performance limitation and the waterbed tradeoff are illustrated through an experimental study.
Date of Publication: Feb. 2011