The “virtual wall” is the most common building block used in constructing haptic virtual environments. A virtual wall is typically based on a simple spring model, with unilateral constraints that allow the user to make and break contact with a surface. There are a number of factors (sample-and-hold, device dynamics, sensor quantization, etc.) that cause virtual walls to demonstrate energetic behavior, destroying the illusion of reality. Passive objects are incapable of generating energy, so in this paper, we find an explicit upper bound on virtual wall stiffness that is a sufficient condition for virtual wall passivity. We consider a haptic display that can be modeled as a mass with Coulomb and viscous friction, being acted upon by two external forces — an actuator and a human user. The system is equipped with only one sensor — an optical encoder measuring the position of the mass. We explicitly consider the effects of position resolution, which has not been done in previous work. We make no assumptions about the human user, and we consider arbitrary constant sampling rates. The main result of our analysis is a sufficient condition for passivity that relies on the Coulomb friction in the haptic device.

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