the factorization of certain polynomials over a non-commutative ring that describe the motion of one of the links. The upper bounds on the number of links and joints for curves of degree $d$ reduce dramatically to $3d+2$ and $\frac{9}{2}d+1$, respectively. In this article, we extend the ideas of [@gallet15] to rational space curves. Our aim is to provide a construction that works for all rational space curves and, at the same time, to reduce the number of links and joints as far as possible. For this purpose, we introduce several new ideas that also improve the planar case.

We use rational motions of minimal degree in the dual quaternion model of rigid body displacements for the link that draws the given space curve. This acknowledges the importance of circularity. If the rational space curve is entirely circular, the motion degree is particularly low [@li15b; @li15a] and factorization of the motion polynomial is straightforward, without the need for prior degree elevation. In the non-circular case, a degree elevation is necessary but in contrast to [@gallet15; @li15b], we only preserve one relevant trajectory, not the complete rational motion. This allows to keep the degree lower and saves links and joints. If the 