by constructing a rational motion such that one point has the given rational curve as trajectory. In order to keep small the number of links and joints, it is advantageous to require a minimal motion degree in the dual quaternion model of rigid body displacements. In [@li15a], we proved that these properties determine a unique rational motion which may be computed in rather straightforward manner by .

This rational motion is parameterized by a certain polynomial with dual quaternion coefficients which we call a “motion polynomial”. General motion polynomials can be written, in several ways, as products of linear motion polynomials [@hegedus13]. These factorizations correspond to the decomposition of the rational motion into products of rotations. The axes are determined by the linear motion polynomials and the rotation angles are linked by a common parameter. On these open chains of revolute joints we base our linkage construction. Unfortunately, there are motion polynomials that do not allow factorizations. Even worse, the motion polynomials of minimal degree with prescribed trajectory typically fall into this category. Therefore, we have to artificially raise their degree in such a way that factorizations exist. This can be done by multiplying them with real polynomials as in 