Closed-loop stiffness and damping accuracy of impedance-type haptic displays

Impedance-type kinesthetic haptic displays aim to render arbitrary desired dynamics to a human operator using force feedback. To effectively render realistic virtual environments, the difference between desired and rendered dynamics must be small. In this paper, we analyze the closed-loop dynamics of haptic displays for three common virtual environments: a spring, a damper, and a spring-damper, including the effects of time delay and low-pass filtering. Using a linear model, we identify important parameters for accuracy in terms of “effective impedances,” a conceptual tool that decomposes the display's closed-loop impedance to components with physical analogs. Our results establish bandwidth limits for rendering effective stiffness and damping. The stiffness bandwidth is limited by the virtual stiffness and device mass, and the damping bandwidth is limited by the cut-off frequency of the low-pass filter. Time delay reduces the effective damping of spring and spring-damper displays, reduces the effective mass for damper displays, and can introduce effective jerk feedback; otherwise delay has negligible effect on accuracy (when the system is stable). Experimental data gathered with a Phantom Premium 1.5 validates the theoretical analysis. This work informs haptic display design by presenting how closed-loop behavior changes with key parameters.