In this paper, we consider the existence of a factorization of a monic, bounded motion polynomial. We prove existence of factorizations, possibly after multiplication with a real polynomial and provide algorithms for computing polynomial factor and factorizations. The first algorithm is conceptually simpler but may require a high degree of the polynomial factor. The second algorithm gives an optimal degree. Let H(t) be the ring of univariate polynomials with quaternion coefficients, with the variable t commuting with the coefficients. The existence of factorizations ofquaternion polynomials into linear factors is a classical result (2). In (3), motion polynomials are defined as elements of DH(t) - the ring of univariate polynomials with dual quaternions - with real norms; these can be used to parametrize rational motions in Euclidean 3- space. The main result there is that a factorization into linear factors allows to construct a mechanical linkage that generates the desired motion. An adaption of the algorithm by (2) to the dual quaternion case indeed allows to factorize "generic" polynomials in DH(t), namely those whose primal part has no strictly real factors. For fixed degree, the set of generic motion polynomials is open and dense in the set of all motion polynomials. Since 2012, we have been wondering which non-generic motion polynomials do allow factorization into linear factors. One reason for our curiousity is a paradoxical fact: ratio- nal motions that are parametrized by generic motion polynomials have special properties, namely that their orbit curves have full cyclicity. The question is still not completely solved, but in this paper we give an affirmative answer for "bounded"motion polyno- mials. They always admit factorizations into products of linear rotation polynomials, possibly after multiplication with a real polynomial. This changes the motion polynomial but not the motion it parameterizes. Bounded motion polynomials are defined by the condition that the norm polynomials has no real roots. The kinematic meaning of this condition is that the orbits are bounded curves. It is also quite obvious that motions that can be generated by linkages with revolute joints (in particular, no translational joints)
['Zijia Li', 'Josef Schicho', 'Hans-Peter Schröcker']
Factorization of Motion Polynomials