[@gallet15; @li15b]. This does not change the underlying motion and may be advantageous in certain applied situations. However, the bounds in are only obtained by a refinement of this procedure. We right-multiplying the motion polynomial with a certain quaternion polynomial. This changes the motion but not the trajectory in question. A curious side-effect of this approach is that it may turn a planar or spherical motion into a spatial motion.

Having constructed a factorizable motion polynomial to the prescribed trajectory, we have to construct a linkage. In general, it is possible to combine the open chains obtained from different factorizations to form a linkage with one degree of freedom [@hegedus13; @li13a; @li14ck; @li14ark]. We do, however, not pursue this approach because it seems difficult to prove that the resulting linkage has *always* (not just in general) only a single degree of freedom. Moreover, spurious motion components do exist [@li13a; @li14ck]. Instead, we adapt the scissor-linkage construction of [@gallet15] to the spatial case, replacing the anti-parallelograms of planar linkages by spatial Bennett linkages. This automatically guarantees that the configuration space is of dimension one and has no spurious components.

It should also be mentioned that all algorithms presented here require exact 