This paper presents a passivity condition of a haptic interface with the multirate wave transform (MWT). Although rate transitions at the wave communication line are achieved passively, the sample-and-hold for the continuous system still generates energy. Since the update rates of the sample-and-hold are high (or assumed to be high) in most cases, its effects are often ignored. However, as the update rate of the sample-and-hold decreases, more energy is generated and the whole haptic interface consequently becomes active. A passivity condition is derived for a system consisting of the sample-and-hold, MWT, and virtual environment (VE). An example study has shown that the minimum update rate of the sample-and-hold can be obtained by investigating the passivity condition. Experiments also have shown that stable force displays can be achieved with the proposed update rates.
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