A Primer on the Differential Calculus of 3D Orientations
This addresses a problem for researchers and engineers in fields like robotics or computer vision who struggle with suboptimal solutions due to complex conventions, though it appears incremental as it builds on existing concepts like the exponential map.
The paper tackles the challenge of handling 3D orientations in optimization problems by proposing an alternative approach using an abstract notion of orientations and representation-independent differentials, resulting in a method that is convenient for optimization-based methods.
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.