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.
Highlights
Factorization of dual quaternion polynomials is a powerful tool for the systematic construction of mechanisms which can perform a given rational motion [3,4]
In [5] it has been shown that rational motions can be parametrized by motion polynomials, which are defined as polynomials over the ring of dual quaternions with real norm polynomial (Study’s condition)
We study the factorization theory of general dual quaternion polynomials which is interesting in its own right and provides us with additional possibilities for the construction of mechanisms
Summary
Factorization of dual quaternion polynomials is a powerful tool for the systematic construction of mechanisms which can perform a given rational motion [3,4]. In [5] it has been shown that rational motions can be parametrized by motion polynomials, which are defined as polynomials over the ring of dual quaternions with real norm polynomial (Study’s condition). Factorization algorithms for motion polynomials are presented in [3,7]. They decompose a given polynomial into products of linear motion polynomials. This allows for the construction of mechanisms with multiple “legs”, each corresponding to one factorization. One leg consists of revolute or prismatic joints, obtained from linear factors in one factorization
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