Factorization of Dual Quaternion Polynomials Without Study's Condition

TL;DR

This paper develops a new method for factorizing dual quaternion polynomials without relying on Study's condition, enabling the design of novel mechanisms with unique joint configurations.

Contribution

It introduces a sufficient condition for polynomial factorization over dual quaternions and presents an algorithm for constructing mechanisms with vertical Darboux joints.

Findings

New factorization algorithm for dual quaternion polynomials

Construction of mechanisms with vertical Darboux joints

Potential to replace Darboux joints with cylindrical joints

Abstract

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.