Factorization of rational curves in the study quadric

•We introduce an algorithm for synthesis of linkages generating a prescribed motion.•Any generic rational curve can be obtained as coupler curve of a closed linkage.•The algorithm produces rotational joints and, in special cases, prismatic joints.•The algorithm uses factorization of polynomials over the dual quaternions.•New families of closed 6R linkages are given as special cases of the method.

For every generic rational curve C in the group of Euclidean displacements we construct a linkage such that the constrained motion of one of the links is exactly C. Our construction is based on the factorization of polynomials over dual quaternions.