Closed Bounded Rational Framing Motions

TL;DR

This paper introduces a method to construct all bounded rational motions that frame a space curve, ensuring smooth, infinitely differentiable motions linked to Pythagorean Hodograph curves, with practical optimization techniques for implementation.

Contribution

It develops a comprehensive theory for bounded rational framing motions based on Pythagorean Hodograph curves parameterized over the projective line, including geometric conditions and optimization methods.

Findings

All bounded rational framing motions correspond to bounded rational Pythagorean Hodograph curves.

A geometric condition on the spherical part guarantees the existence of a bounded, rational framing motion.

Semidefinite optimization can ensure the positive rational speed distribution in practical applications.

Abstract

We present a method for constructing all bounded rational motions that frame a space curve $\mathbf{r}(t)$. This means that the motion guides an orthogonal frame along the curve such that one frame axis is in direction of the curve tangent. Existence of (bounded) framing motions is equivalent to $\mathbf{r}(t)$ being a (bounded) rational Pythagorean Hodograph curve. In contrast to previous constructions that rely on polynomial curves with smooth self-intersection, our motions and curves are infinitely differentiable. To this end, we develop the theory of Pythagorean hodograph curves parameterized over the projective line. We also provide a simple geometric necessary and sufficient condition on the spherical part of the motion, given by the homogeneous quaternionic preimage of the Pythagorean hodograph curve, that ensures the existence of a corresponding bounded, rational, and even regular framing motion. The translation part comes from the speed distribution, which must be a special positive rational function. This can in practice be ensured by semidefinite optimization methods. We illustrate our findings with a number of examples.