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Research Papers

General Identities for Parameterizations of SO(3) With Applications

[+] Author and Article Information
Anton H. J. de Ruiter

Department of Aerospace Engineering,
Ryerson University,
Toronto, ON M5B 2K3, Canada
e-mail: aderuiter@ryerson.ca

James Richard Forbes

Department of Aerospace Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: forbesrj@umich.edu

Manuscript received May 13, 2013; final manuscript received March 6, 2014; accepted manuscript posted March 10, 2014; published online April 3, 2014. Assoc. Editor: Alexander F. Vakakis.

J. Appl. Mech 81(7), 071007 (Apr 03, 2014) (16 pages) Paper No: JAM-13-1196; doi: 10.1115/1.4027144 History: Received May 13, 2013; Revised March 06, 2014; Accepted March 10, 2014

Rotation matrices, which are three-by-three orthonormal matrices with determinant equal to plus one, constitute the special orthogonal group of rigid-body rotations, denoted SO(3). Owing to the three-by-three nature of rotation matrices plus their orthonormality constraint, parameterizations are often used in favor of rotation matrices for computations and derivations. For example, Euler angles and Rodrigues parameters are common three-parameter unconstrained parameterizations, while unit-length quaternions are a popular four-parameter constrained parameterization. In this paper various identities associated with the parameterization of SO(3) are considered. In particular, we present six identities, three related to unconstrained parameterizations and three related to constrained parameterizations. We also discuss rotation matrix perturbations. The utility of these identities is highlighted when deriving the motion equations of a rigid body using Lagrange's equation. We also use them to examine some issues associated with spacecraft attitude determination.

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Free-body diagram of a rigid body

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