ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 14 (2019), 341--354
This work is licensed under a Creative Commons Attribution 4.0 International License.
VARIATIONS ON THE THEME EULER ANGLES
Clementina D. Mladenova and Ivaïlo M. Mladenov
Abstract. We discuss different parameterizations of the Lie group SO(3). The well-known Rodrigues formula describes the three dimensional orthogonal matrices in terms of their axes and angles of rotation. In particular, an arbitrary SO(3) element can be described by two real parameters and one angle. In [5] an alternative to Rodrigues representation in which an arbitrary rotation is expressed in terms of two angles and one real parameter is derived. This is done via the Cayley map applied to the canonical form of the so(3) matrices. The relationships between the novel parameterization, the classical Rodrigues representation and the extended SO(3) vector parameterization are established. The composition law in the new coordinates is derived for the composition of two regular rotations. In this paper we cover all possible scenarios for the composition law, including the cases when at least one of the composed matrices is a half-turn. To do this the extended vector-parameter composition law in SO(3) is used.
2010 Mathematics Subject Classification: 70B10; 70B15; 22E60; 22E70
Keywords: Lie algebras; Lie groups; rigid body kinematics; parameterizations; rotations; vector-parameter.
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Clementina D. Mladenova
Institute of Mechanics
Bulgarian Academy of Sciences
Acad. G. Bonchev Str., Block 4
Sofia 1113, Bulgaria
e-mail: clem@imbm.bas.bg
Ivaïlo M. Mladenov
Inst. Biophys. & Biomed. Eng.
Bulgarian Academy of Sciences
Acad. G. Bonchev Str., Block 21
Sofia 1113, Bulgaria
e-mail: mladenov@bio21.bas.bg