Euler angle transformations

TLDR

The paper discusses the definition of Euler angle sets and introduces a useful tool for handling their associated mathematics. The authors propose a systematic technique for error analysis of sequences of rotations. They partition the twelve equivalent Euler angle sets into two subsets for right‑handed systems with positive rotations and illustrate how to determine orthogonal infinitesimal rotations equivalent to nonorthogonal increments. The method yields expressions for the angular velocities of the final coordinate system relative to the reference in terms of Euler angle derivatives, and shows that infinitesimal increments between equivalent Euler sets can be computed symbolically with ease.

Abstract

The definition of sets of Euler angles is discussed and a useful tool for treating the mathematics associated with Euler angles is illustrated. Restricting attention to right-handed coordinate systems and positive rotations, twelve distinct but equivalent sets of Euler angles are partitioned into two subsets. The method of determining a set of orthogonal infinitesimal rotations equivalent to nonorthogonal infinitesimal increments on a set of Euler angles is illustrated. It is shown that the same solution yields expressions for the angular velocities of the final coordinate system relative to the reference coordinate system in terms of derivatives of the Euler angles. The ease with which the infinitesimal increments of one Euler set in terms of the increments of another equivalent Euler set can be determined by the symbolic technique is illustrated. This technique offers a systematic approach to error analysis of sequences of rotations.