A Primer on the Differential Calculus of 3D Orientations

TL;DR

This paper introduces a representation-independent approach to 3D orientation calculus, simplifying optimization problems by using abstract notions and exponential maps, addressing common issues with multiple conventions.

Contribution

It proposes a novel, minimal differential framework for 3D orientations based on coordinate mappings and exponential maps, enhancing optimization methods.

Findings

Provides a unified approach to 3D orientation calculus

Simplifies optimization problems involving orientations

Addresses issues with multiple orientation conventions

Abstract

The proper handling of 3D orientations is a central element in many optimization problems in engineering. Unfortunately many researchers and engineers struggle with the formulation of such problems and often fall back to suboptimal solutions. The existence of many different conventions further complicates this issue, especially when interfacing multiple differing implementations. This document discusses an alternative approach which makes use of a more abstract notion of 3D orientations. The relative orientation between two coordinate systems is primarily identified by the coordinate mapping it induces. This is combined with the standard exponential map in order to introduce representation-independent and minimal differentials, which are very convenient in optimization based methods.