  In this paper we investigate factorizations of polynomials over the ring of
dual quaternions into linear factors. While earlier results assume that the
norm polynomial is real ("motion polynomials"), we only require the absence of
real polynomial factors in the primal part and factorizability of the norm
polynomial over the dual numbers into monic quadratic factors. This obviously
necessary condition is also sufficient for existence of factorizations. We
present an algorithm to compute factorizations of these polynomials and use it
for new constructions of mechanisms which cannot be obtained by existing
factorization algorithms for motion polynomials. While they produce mechanisms
with rotational or translational joints, our approach yields mechanisms
consisting of "vertical Darboux joints". They exhibit mechanical deficiencies
so that we explore ways to replace them by cylindrical joints while keeping the
overall mechanism sufficiently constrained.
