Effects of position quantization and sampling rate on virtual-wall passivity

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 active (nonpassive) behavior, destroying the illusion of reality. In this paper, we find an explicit upper bound on virtual wall stiffness that is a necessary and sufficient condition for virtual wall passivity. We consider a haptic display that can be modeled as a mass with Coulomb-plus-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 model 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 necessary and sufficient condition for passivity that relies on the Coulomb friction in the haptic device, as well as the encoder resolution. We experimentally verify our results with a one-degree-of-freedom haptic display, and find that the system can display nonpassive behavior in two decoupled modes that are predicted by the necessary and sufficient condition. One mode represents instability, while the other mode results in active tactile sensations.