Motion Polynomials Admitting a Factorization with Linear Factors

TL;DR

This paper establishes a precise condition under which bounded motion polynomials can be factored into linear components, enabling the design of mechanisms through motion decomposition.

Contribution

It provides a necessary and sufficient criterion for factorization of bounded motion polynomials and introduces an improved algorithm for computing such factorizations.

Findings

Bounded motion polynomials always admit a linear factorization after suitable multiplication.

The paper offers an efficient algorithm for computing factorizations.

Factorization enables mechanism construction via simple rotations or translations.

Abstract

Motion polynomials (polynomials over the dual quaternions with nonzero real norm) describe rational motions. We present a necessary and sufficient condition for reduced bounded motion polynomials to admit factorizations into monic linear factors, and we give an algorithm to compute them. We can use those linear factors to construct mechanisms because the factorization corresponds to the decomposition of the rational motion into simple rotations or translations. Bounded motion polynomials always admit a factorization into linear factors after multiplying with a suitable real or quaternion polynomial. Our criterion for factorizability allows us to improve on earlier algorithms to compute a suitable real or quaternion polynomial co-factor.