Rational Motions of Minimal Quaternionic Degree with Prescribed Plane Trajectories

TL;DR

This paper develops a theoretical framework for constructing minimal quaternionic degree rational motions that generate prescribed plane trajectories, with applications in robotics and CAD.

Contribution

It introduces algebraic conditions and methods to compute minimal degree rational motions for given plane trajectories using dual quaternions.

Findings

A rational torse's realizability depends on its Gauss map being rational.

Minimal motion degree relates to the degree drop of the Gauss map and the structure of the plane polynomial.

The framework enables systematic construction of low-complexity rational motions.

Abstract

This paper investigates the construction of rational motions of a minimal quaternionic degree that generate a prescribed plane trajectory (a ``rational torse''). Using the algebraic framework of dual quaternions, we formulate the problem as a system of polynomial equations. We derive necessary and sufficient conditions for the existence of such motions, establish a method to compute solutions and characterize solutions of minimal degree. Our findings reveal that a rational torse is realizable as a trajectory of a rational motion if and only if its Gauss map is rational. Furthermore, we demonstrate that the minimal degree of a motion polynomial is geometrically related to a drop of degree of the Gauss and algebraically determined by the structure of the torse's associated plane polynomial and the real greatest common divisor of its vector part. The developed theoretical framework has potential applications in robotics, computer-aided design, and computational kinematic, offering a systematic approach to constructing rational motions of small algebraic complexity.