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Impedance-type kinesthetic haptic displays aim to render arbitrary desired dynamics to a human operator using force feedback. To 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 the rendered dynamics in terms of effective impedances, a conceptual tool that decomposes the displays closed-loop impedance into components with physical analogs. Our results establish bandwidth limits for rendering effective stiffness and damping. The effective stiffness bandwidth is limited by the virtual stiffness and device mass, and the effective damping bandwidth is limited by the cut-off frequency of the low-pass filter which filters the device velocity estimate. We show that a general system impedance can be characterized by a mass, damper, and spring optimally by the solution to a convex optimization problem, and we present a quantitative metric, the Average Distortion Error (ADE), to describe the fidelity of this model. Time delay has no significant effect on characterized stiffness, and reduces characterized damping by the product of virtual stiffness and total time delay. Reducing the low-pass filter cut-off frequency reduces the characterized damping. Experimental data gathered with a Phantom Premium 1.5 validates the theoretical analysis. We also conducted human user experiments to investigate the effects of time delay and low-pass filtering on perceived stiffness and damping. Similar to the characterized dynamics results, we observed no significant effect of time delay on perceived stiffness, and increasing time delay resulted in reduced perceived damping. Lower filter cut-off frequencies resulted in lower perceived damping. This work informs haptic display design by presenting how closed-loop behavior changes with key parameters.

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