Workspace Multiplicity and Fault Tolerance of Cooperating Robots

Cooperating robotic systems, especially in the context of fault-tolerance of complex robotic mechanisms, is an important question for theoretical and applied studies. In this paper, we focus on one measure of fault tolerance in robots, namely, the multiplicity of the configurations for reaching a particular point in the workspace, which is difficult to measure using traditional methods. As a particular example, we consider the case of a free-swinging failure of a robotic arm that is handled by having a cooperating functional robot grasp the link adjacent to the failed joint. We present an efficient method to compute the multiplicity measure of the workspace, based on the tools from numerical algebraic geometry, applied to the inverse kinematics problem re-cast in the form of a polynomial system. To emphasize the difference between our methods and more traditional approaches, we compute the measure of workspace based on the multiplicity of configurations, and optimize placement of synergistic robot arms and the optimal grasp point on each link of the broken robot based on this measure. keywords: Workspace mapping, joint failure, homotopy continuation, Monte Carlo methods