On Passive Non-Iterative Variable-Step Numerical Integration of Mechanical Systems for Haptic Rendering

Abstract

This paper consists of three parts. First, with a slightly-different, yet, more physically-plausible, discrete supply-rate (i.e. power), we propose a non-iterative (i.e. fast) and variable-step numerical integration algorithm for (scalar) discrete-passive mechanical systems, consisting of constant mass and damper, and a certain class of nonlinear spring. In the second part, we propose a fast passive collision handling algorithm with a spring-damper type virtual wall, which, to detect exact time of contacts, requires at most three intermediate non-iterative computations within each integration-step. We then propose a way of how to passively connect this discrete-passive, non-iterative, and variable-step mechanical integrators (with passive collision handling) to a continuous haptic device.

Passive set-position modulation approach for haptics with slow, variable, and asynchronous update

“…coordination during the contact (or in free-space, resp.). See [4], where even smaller virtual mass 0.001kg is used. This separation between the device servoloop and the virtual world, which was originally envisioned in [3], can be achieved here, because our framework, relying on the PSPM and the discrete-passive integrators, enforces two-port hybrid passivity of the closed-loop haptic device, essentially individually for the device side and virtual environment (only to the extent of energy shuffling).…”

“…The virtual mass is 0.02kg, and makes contact with a virtual wall with K w = 30kN/m and B w = 100Ns/m. For this, we use the (non-iterative) passive virtual-wall rendering algorithm of [4], so that, with passive humans, the two-port hybrid passivity of Th. 1 can imply stable interaction of the closed-loop haptic system.…”

“…In fact, in [4], it is shown that there indeed exists a class of discrete-passive non-iterative linear mechanical integrators L dp , which, for our purpose here, can be written as:…”

“…Then, from the well-known passivity theorem, the stable interaction of the closedloop haptic system with any passive (continuous) human users and (discrete) virtual environments (e.g. passive virtual wall of [4]) can be guaranteed. Here, the only premise the PSPM requires for the incoming data sequence (e.g.…”

“…Yet, the closed-loop haptic system is a two-port system (one-port between the haptic device and the human user, and the other between the virtual mass and its surrounding virtual environment), thus, it is desirable, if not necessary, to enforce its two-port passivity. For this, in this paper, we will utilize our recently proposed non-iterative (thus, can be simulated fast) discrete-time passive integrator [4] to simulate the virtual world, whose discrete-passivity property, then, allows us to extend the PSPM to the discrete virtual world, so that we can again employ (discrete) spring-damper like connection between the virtual mass's position x k and the sequence of the set-position signal y k received from the haptic device (i.e. sampled device's position), while enforcing passivity.…”