This paper proposes a control method that overcomes the instability arising from both the low 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 time delay on the wave variables is passive. The so-called multirate control scheme is applied to cope with the problems presented by the low update rates of 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 are passive, thereby deriving a stable haptic interface with the multirate wave transform. Various experiments have shown that a stable force display is achieved with the multirate wave transform.

It is well known that the wave variables are robust to time delay of a system from the viewpoint of passivity. During implementation of the wave transformation on the sampled-data system, a unit time delay arises due to causality of the reflection wave. The unit time delay on the reflection wave occasionally becomes a passive element. From the passivity condition of the sampled-data system, the wave impedance can be designed such that the wave transformation using delayed reflection provides the effect of positive damping and thus the stable haptic interface is achieved on the sampled-data system. Various experiments for a 2-linked passive haptic device show that stable haptic interface can be accomplished for the wall-following task by use of the wave transformation using delayed reflection.

Wave variables have been used to overcome the transmission delay of a system from the viewpoint of passivity. During implementation of the wave transformation for the sampled-data system, a unit delay arises due to causality of the reflection wave. The property of the wave transformation using delayed reflection, which is different from the standard wave transformation, can be verified with the passivity condition of the scattering matrix. The unit delay on the reflection wave occasionally becomes a positive damping. From the passivity condition of the sampled-data system, the wave impedance can be designed such that the wave transformation using delayed reflection wave provides the effect of positive damping. As an example, the wave transformation using delayed reflection wave makes it possible that the stable haptic interface is achieved on the sampled-data system with appropriate wave impedance. The simulation and the experiment results show that the stable haptic interface can be accomplished by use of the wave transformation using delayed reflection.