Passive wave variable control of haptic interaction with an unknown virtual environment

The wave variable transformation cannot be exploited for the control of sampled-data and discrete-time systems without precaution. This paper shows that connecting a haptic interface to a discrete time virtual environment through a wave variable controller can inject energy into the haptic feedback loop and thus jeopardize the stability of the haptic interaction. The connection involves a one step computational delay when the virtual environment is not known prior to starting the interaction. Using the Jury-Marden stability criterion, the paper investigates the effect of this computational delay on the stability of wave variable control of haptic interaction with a virtual wall. It also develops a time domain passivity analysis to compute the energy injected in the wave variable transformation by the computational delay. Then, it proposes an algorithm for dissipating the extra energy and restoring the passivity of the wave variable transformation. The paper concludes with the experimental validation of the energy dissipating algorithm.