Stable haptic display of slowly updated virtual environment with multirate wave transform

This paper proposes a control method that overcomes the instability of a haptic interface arising from both the slow update rates of the virtual environment (VE) and the time delay in a communication line. The instability arising from the time delay in a communication line can be overcome using the wave variables since the time delay on the wave variables becomes passive. The so-called multirate control scheme is applied to cope with the problems presented by the slow update rates of the VE. Since digital sample and hold (rate transitions) can be represented as a series of time delay, the passive nature of time delay on the wave variables is applied. By computing the norm of the scattering matrix, it was verified that rate transitions on the wave variables become passive, so a stable haptic interface was derived with the multirate wave transform. Various experiments have shown that a stable force display can be achieved with the multirate wave transform.