Stability of Nonlinear Time-Delay Systems Describing Human–Robot Interaction

Abstract

In this paper, we present sufficient conditions for the stability analysis of a stationary point for a special type of nonlinear time-delay systems. These conditions are suitable for analyzing systems describing physical humanrobot interaction (pHRI). For this stability analysis a new human model consisting of passive and active elements is introduced and validated. The stability conditions describe parameterization bounds for the human model and an impedance controller. The results of this paper are compared to stability conditions based on passivity, approximated time-delays and to numerical approaches. As a result of the comparison, it is shown that our conditions are more general than the passivity condition of Colgate and Schenkel. This includes the consideration of negative stiffness and nonlinear virtual environments. As an example, a pHRI including a nonlinear virtual environment with a polynomial structure is introduced and also successfully analyzed. These theoretical results could be used in the design of robust controllers and stability observers in pHRI. Index Terms-Impedance control, Lyapunov-Krasovskii functional, nonlinear time-delay systems, physical humanrobot interaction (pHRI). I. INTRODUCTIONI N PHYSICAL human-robot interaction (pHRI), a robot and a human share a common workspace and the human guides the robot via direct physical contact. This technology is based on sensing the forces and torques of the human's arm and the implementation of an impedance controller. The pHRI is used in industrial tasks for teaching procedures and lifting heavy objects. The stability analysis for a pHRI system is a complex problem. The reason for this lies in the difficulty of modeling

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“…The varying parameters include the damping of the impedance control, the compliance of the human and the compliance of the environment. The stability of such systems has been considered in several work, including Müller et al (2019), but here several parameters are considered to be constant. The presented stability criterion relaxes this assumption.…”

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