Illustration of Numerical Algebraic Methods for Workspace Estimation of Cooperating Robots after Joint Failure

Abstract

We consider the estimation of the post-failure workspace of two two-link serial robots where the free-swinging failure of one robot's last joint is handled by having the functional robot grasp the final link of the broken robot. We present an algorithm for finding the optimal placement of such synergistic robot arms and the optimal grasp point on the final link of the broken robot. To determine whether a point in the pre-failure workspace of the broken robot remains in the post-failure workspace, we solve an inverse kinematics problem in the form of a polynomial system. Homotopy continuation provides an efficient means for solving such polynomial systems, so we use the software package Bertini. Finally, Monte-Carlo methods are used to estimate the post-failure workspace.

Paramotopy: Parameter Homotopies in Parallel

“…Kinematics problems can often be cast in polynomial systems language; see e.g. [19,8] and a slew of others. Robotic arms, whether prismatic or revolute in nature, can be described in terms of now-canonical parameters [10].…”

“…Parameter homotopies are not new and have been used in several areas of application [8,17,25,26] and implemented in at least two software packages for solving polynomial systems: Bertini [5] and PHCpack [29]. These implementations allow the user to run a single parameter homotopy from one parameter value p 0 with known solutions to the desired parameter value, p 1 , with the solutions at p 0 provided by the user.…”