Nonlinear Stability of Hybrid Control

A theoretical and experimental investigation on the stability proper ties of the hybrid control scheme was performed using Lyapunov's theory for both the original scheme, which uses the Jacobian in verse for mapping Cartesian errors to joint errors, and a scheme using the Jacobian pseudoinverse. Both schemes result in position and force controllers that are statically coupled in the task space. Stability analysis shows that the pseudoinverse scheme is asymp totically stable, whereas the inverse scheme may become unstable depending on the manipulator attitude and the environmental stiff ness. In the manipulator workspace, where kinematic instabilities have been reported to exist even away from kinematic singularities, the Jacobian inverse affects negatively the Lyapunov function's posi tive definiteness and the negative sign of its derivative; this effect may become dominant when the environmental stiffness is zero or very low. Experimental results for a 2- and 3-degrees-of-freedom planar manipulator using a PUMA 560 were performed both in free space where stiffness is zero and in contact with a stiff surface. Experi mental results in free space have confirmed the stability properties of the two schemes as predicted by the theoretical analysis and are in agreement with previously reported simulation and experimental results. Experimental results in contact with a stiff wall gave stable results for both schemes.

Multivariable Stability of Force-Reflecting Teleoperation: Structures of Finite and Infinite Zeros

This paper presents a stability analysis of force-position teleoperation under general end-effector contact. The analysis is based on the finite and infinite zero structure of the multivariable root-locus resulting from modulation of the environment stiffness. The starting point is an analysis of the stability of robot force control, motivated by the observation that the human-operator in a force reflection loop acts as a “force servo,” generating position commands in response to reflected force. Asymptotic root loci properties are used to establish passivity conditions on force feedback to give root locus interpretations of the well-known results that (1) feedback via the inverse joint Jacobian can lead to (kinematic) instability and that (2) passivity is preserved by kinematically proper force feedback through the transpose of the joint angle Jacobian. It is demonstrated that a fully constrained force-position teleoperation loop has an identical infinite zero structure to that of a slave manipulator under kinematically proper force control and that the dominant vibration modes of a force-position loop are fully described by a multivariable analogue of the single-input single-output pseudo-system investigated in a study by Daniel and McAree. Extension of the analysis to cover partial end-effector constraint provides a design tool for teleoperation control and serves to aid selection of teleoperation slave-arms. The paper concludes by giving a passivity condition for multiple-input multiple-output force-position teleoperation for stable contact against all environments.