Construction of overconstrained linkages by factorization of rational motions

Abstract

We prove that for any sufficiently generic rational curve C of degree n in the group of Euclidean displacements, there exists an overconstrained spatial linkage with revolute joints whose linkage graph is the 1-skeleton of the n-dimensional hypercube such that the constrained motion of one of the links is exactly C. The synthesizing algorithm is based on the factorization of polynomials over the dual quaternions. The linkage contains n! open nR chains, so that low degree examples include Bennett's mechanisms and are related to overconstrained 5R and 6R chains.