And Yet It Moves: Paradoxically Moving Linkages in Kinematics

Abstract

The configuration space of a mechanical linkage, consisting of rigid bodies moving in space constrained by joints, is defined by algebraic conditions. If these equations do not define a complete intersection, then the dimension of the configuration space is higher than expected. These linkages violate the Chebychev-Gr\"ubler-Kutzbach formula for the degree of freedom of mobility. Mathematicians developed a variety of methods to describe and understand this phenomenon. This paper explains some of them.