Finding mechanisms of exceptional mobility using numerical algebraic geometry

Mechanisms of exceptional mobility, including both overconstrained mechanisms and robots with self-motion, move with more degrees of freedom than predicted by the Grübler–Kutzbach formulas. Although a number of such cases are known, it is difficult to find new examples. This article explains a geometric formulation, called a fiber product, that facilitates finding exceptional mechanisms using tools from numerical algebraic geometry. The purpose of this article is to specialize the mathematical theory developed in A.J. Sommese and C.W. Wampler (2008) to the realm of kinematics and to present simple planar, spherical, and spatial examples that illustrate basic concepts. Although the formulation is general, its application to more complicated mechanisms will require the development of more refined solution techniques that exploit the symmetry inherent in fiber products.