The integration of angular velocity

TL;DR

This paper compares methods for determining an object's orientation from angular velocity, finding quaternion-based integration to be most efficient and accurate, with broad applicability in physics and engineering.

Contribution

It demonstrates that quaternion-based integration is superior for orientation determination from angular velocity, offering improved efficiency and stability over traditional methods.

Findings

Quaternion integration is most efficient and accurate.

Norm of the quaternion is irrelevant for the solution.

Methods are stable and suitable for larger differential systems.

Abstract

A common problem in physics and engineering is determination of the orientation of an object given its angular velocity. When the direction of the angular velocity changes in time, this is a nontrivial problem involving coupled differential equations. Several possible approaches are examined, along with various improvements over previous efforts. These are then evaluated numerically by comparison to a complicated but analytically known rotation that is motivated by the important astrophysical problem of precessing black-hole binaries. It is shown that a straightforward solution directly using quaternions is most efficient and accurate, and that the norm of the quaternion is irrelevant. Integration of the generator of the rotation can also be made roughly as efficient as integration of the rotation. Both methods will typically be twice as efficient as naive vector- or matrix-based methods. Implementation by means of standard general-purpose numerical integrators is stable and efficient, so that such problems can be readily solved as part of a larger system of differential equations. Possible generalization to integration in other Lie groups is also discussed.