The Integration of Angular Velocity

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 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.

Control of helical navigation by three-dimensional flagellar beating

“…These reduce to a set of 10 equations for unknowns [31]. The method of quaternions was used to resolve singularities at θ 2 = ±π/2 [32].…”

Compact binary waveform center-of-mass corrections

“…The numbers above each column represent the median of each variable over all simulations, with superscripts and subscripts giving the offset (relative to the median) of the 84th and 16th percentiles, respectively. condition that the waveform vary slowly; it is integrated in time to obtain one such frame [57], but the result is only unique up to an overall rotation. We choose that overall rotation so that the z axis of the final corotating frame is aligned as nearly as possible throughout the inspiral portion of the waveform with the dominant eigenvector [58,59] of the matrix…”

Model-Based Real-Time Motion Tracking Using Dynamical Inverse Kinematics

“…Base position I p B and joints configuration s lie in vector space over R for which most of the numerical integrations methods proposed in literature can be used [39]. The integration of the base angular velocity I ω B (t k ) is not trivial [40], and numerical integration errors can lead to the violation of the orthonormality condition [41] for the base orientation I R B . There are different schemes that successfully solve discrete angular velocity integration making use of quaternion representation [40].…”

“…The integration of the base angular velocity I ω B (t k ) is not trivial [40], and numerical integration errors can lead to the violation of the orthonormality condition [41] for the base orientation I R B . There are different schemes that successfully solve discrete angular velocity integration making use of quaternion representation [40]. Concerning rotation matrix representation, the orthonormality condition can be directly enforced using the Baumgarte stabilization [41], and avoiding change of representation.…”