Motion Polynomials Admitting a Factorization with Linear Factors

TL;DR

The paper resolves when reduced bounded motion polynomials in dual quaternions factor into monic linear motion factors, establishing a necessary-and-sufficient condition expressed via $c g igm| D D^*$ with $g$ the real gcd of $c$, ${Q}^*D$, and $D{Q}^*$ for $M = cQ + oldsymbol{
psilon } D$. It then provides two constructive proofs and several algorithms to compute such factorizations, including a primary-norm decomposition into $M_i$ with $ u(M_i)=N_i^{n_i}$ and a Bennett-flip technique to extend to general bounded polynomials. The methods not only certify factorizability but also enable direct mechanical synthesis by decomposing rational SE(3) motions into simple rotations or translations, improving on earlier co-factor-based approaches. The work lays groundwork for enhanced mechanism design tools (e.g., Kempe linkages) and suggests future extensions to spinor polynomials and numeric real-GCD implementations.

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.

Theorems & Definitions (47)

• Theorem 1
• Definition 1
• Definition 2
• Lemma 1: ABC-lemma
• proof
• Lemma 2: AB-Lemma
• proof
• Theorem 2: li19
• Proposition 1
• proof