Human-in-the-Loop Stability Analysis of Haptic Rendering With Time Delay by Tracking the Roots of the Characteristic Quasi-Polynomial: The Effect of arm Impedance

Abstract

The uncoupled stability of haptic rendering has been analyzed extensively, predominantly using passivity considerations. Yet, the role of the operator in improving the stability of haptic systems has received less attention. Here, towards a human-in-the-loop stability analysis of haptic rendering with delay, we study the effect of the operator impedance on the stability boundaries. We employ a method that tracks the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$j\omega$</tex-math></inline-formula> crossings of the roots of a characteristic second order equation of a coupled system that includes the impedance of the operator, the haptic device, the rendered virtual stiffness and damping, and the time delay. We found that the consideration of a realistic operator impedance leads to significantly less conservative stable time delay margins. Moreover, for certain arm impedance parameters, the coupled system may allow for: (1) assuring delay-independent stability, and (2) being stabilized by increasing the delay. These results indicate that a human-centered approach can push the boundaries of safe haptic feedback, especially in applications where the delays are substantial and unavoidable, such as in teleoperation.