Distortion Bounds of Subdivision Models for SO(3)

TL;DR

This work addresses the distortion incurred when using a cubic subdivision representation of $SO(3)$ within the $SE(3)$ configuration space for robot path planning. It develops a framework that reduces distance distortion to metric distortion via the pullback of Riemannian metrics and derives the exact metric distortion range for the cubic map $\mu_3: \widehat{SO}_{3} \to S^3$ as $MD_{\mu_3}=[1/4,1]$, yielding a distance distortion range $D_{\mu_3}=[1/4,1]$ and a distortion constant $C_0(\mu_3)=4$. The composite map to $SO(3)$ through the quaternion projection has distortion $D_{\overline{\mu}_3}=[1/2,2]$, establishing sharp, implementable bounds for resolution-exact subdivision in 3D rotations. These results enable rigorous subdivision-based planning in SE(3) and suggest broader applications to rotation estimation problems in motion capture and related robotics tasks.

Abstract

In the subdivision approach to robot path planning, we need to subdivide the configuration space of a robot into nice cells to perform various computations. For a rigid spatial robot, this configuration space is $SE(3)=\mathbb{R}^3\times SO(3)$. The subdivision of $\mathbb{R}^3$ is standard but so far, there are no global subdivision schemes for $SO(3)$. We recently introduced a representation for $SO(3)$ suitable for subdivision. This paper investigates the distortion of the natural metric on $SO(3)$ caused by our representation. The proper framework for this study lies in the Riemannian geometry of $SO(3)$, enabling us to obtain sharp distortion bounds.

Figures (1)

• Figure 1: The Cubic model ${\widehat{SO}_{3}}$ for $SO(3)$ (taken from sss2)

Theorems & Definitions (6)

• theorem thmcountertheorem: Distance Distortion Range for $\mu_3$
• lemma 1.1: Composition of Distortion Range
• theorem thmcountertheorem: Parametric Cubic Models
• lemma 1.2
• theorem thmcountertheorem: Metric Distortion
• theorem thmcountertheorem: Metric Distortion Range for $\mu_3$