Using fractional order elements for haptic rendering

Abstract

Fractional order calculus—a generalization of the traditional calculus to arbitrary order differointegration—is an effective mathematical tool that broadens the modeling boundaries of the familiar integer order calculus. Fractional order models enable faithful representation of viscoelastic materials that exhibit frequency dependent stiffness and damping characteristics within a single mechanical element. We propose the use of fractional order models/controllers in haptic systems to significantly extend the type of impedances that can be rendered using the integer order models. We study the effect of fractional order elements on the coupled stability of the overall sampled-data system. We show that fractional calculus generalization provides an additional degree of freedom for adjusting the dissipation behavior of the closed-loop system and generalize the well-known passivity condition to include fractional order impedances. Our results demonstrate the effect of the order of differointegration on the passivity boundary. We also characterize the effective impedance of the fractional order elements as a function of frequency and differointegration order.