Nearly Geodesic Riemannian Cubics in SO(3)

TL;DR

This paper develops elementary function approximations for nearly geodesic Riemannian cubics in SO(3), addressing a gap in understanding their short-term behavior crucial for practical motion planning applications.

Contribution

It introduces new approximations for nearly geodesic Riemannian cubics in SO(3), enhancing practical computation and understanding of their short-term dynamics.

Findings

High-quality approximations depend on specific mathematical properties of Riemannian cubics.

Approximations are expressed in terms of elementary functions.

Addresses the previously missing understanding of short-term behavior.

Abstract

{\em Riemannian cubics} are curves in a manifold $M$ that satisfy a variational condition appropriate for interpolation problems. When $M$ is the rotation group SO(3), Riemannian cubics are track-summands of {\em Riemannian cubic splines}, used for motion planning of rigid bodies. Partial integrability results are known for Riemannian cubics, and the asymptotics of Riemannian cubics in SO(3) are reasonably well understood. The mathematical properties and medium-term behaviour of Riemannian cubics in SO(3) are known to be be extremely rich, but there are numerical methods for calculating Riemannian cubic splines in practice. What is missing is an understanding of the short-term behaviour of Riemannian cubics, and it is this that is important for applications. The present paper fills this gap by deriving approximations to nearly geodesic Riemannian cubics in terms of elementary functions. The high quality of these approximations depends on mathematical results that are specific to Riemannian cubics.