Coupled stability of multiport systems — Theory and experiments

This paper presents both theoretical and experimental studies of the stability of dynamic interaction between a feedback controlled manipulator and a passive environment. Necessary and sufficient conditions for “coupled stability” —the stability of a linear, time-invariantn-port (e.g., a robot, linearized about an operating point) coupled to a passive, but otherwise arbitrary, environment— are presented. The problem of assessing coupled stability for a physical system (continuous time) with a discrete time controller is then addressed. It is demonstrated that sucha system may exhibit the coupled stability property; however, analytical, oreven inexpensive numerical conditions are difficult to obtain. Therefore, anapproximate condition, based on easily computed multivariable Nyquist plots, is developed. This condition is used to analyze two controllers implemented on a two-link, direct drive robot. An impedance controller demonstrates thata feedback controlled manipulator may satisfy the coupled stability property. A LQG/LTR controller illustrates specific consequences of failure to meet the coupled stability criterion; it also illustrates how coupled instabilitymay arise in the absence of force feedback. Two experimental procedures — measurement of endpoint admittance and interaction with springs and masses — are introduced and used to evaluate the above controllers. Theoretical and experimental results are compared.