Factorization of dual quaternion polynomials without study’s condition. (English) Zbl 1457.16023

Siegele, Johannes; Pfurner, Martin; Schröcker, Hans-Peter

Adv. Appl. Clifford Algebr. 31, No. 2, Paper No. 22, 17 p. (2021).

Summary: 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.

MSC:

16S36	Ordinary and skew polynomial rings and semigroup rings
16H05	Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
70B15	Kinematics of mechanisms and robots

Keywords:

factorization; dual quaternions; dual quaternion polynomials; rational motion; skew polynomial ring; construction of mechanisms

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