Convergence Analysis of an Inverse Flexible Manipulator Model Algorithm

Abstract

In this paper we present a convergence analysis of an iterative method to obtain the inverse model in the virtual joint space of a class of flexible manipulators. Virtual joints are defined so as to be related kinematically to the workspace. Thus, when the desired trajectory can be transformed from the workspace to the virtual joint space, the inverse model in the virtual joint space can be used to obtain a trajectory tip-tracking controller. Because the robot model in the virtual joint space is a non-minimum phase system, its inverse is unstable. The inverse method is then based on a causal-anticausal integration approach used in an iterative manner. The convergence analysis of this inverse algorithm is based on the contraction-mapping theorem and provides a very simple way to expand the desired trajectory to ensure its convergence. Finally, the inverse algorithm and its convergence analysis are validated by simulating a controller, based on the inverse model, that drives a two-link manipulator made up of one rigid and one flexible link. The simulation results demonstrate the good performance of the inverse algorithm and its convergence analysis presented here.