Interior Parameters, Exterior Parameters, and a Cayley-Like Transform

Abstract

A RGUABLY, the most fundamental representation for describing the relative orientation of a rigid body is the orientation matrix, also called the direction cosine matrix. Few believe, however, that this matrix provides the most convenient parameterization of orientation. For that, many instead turn to one of a collection of three-parameter sets, even though it is known that every three-parameter orientation set will encounter some kind of singularity, a fact shown by Stuelpnagel [1]. Others turn to the Euler parameters, which are a nonsingular four-parameter set. The direct relationship between each of the various parameter sets and the orientationmatrix is considered to be of basic importance. So, too, is the time derivative of such relationships, although the angular velocity vector or matrix is usually substituted for the time derivative of the orientationmatrix. The purpose of this note is to investigate the direct relationship between a particular three-parameter orientation set, viz., the modified Rodrigues parameters (MRP), and the orientation matrix. The modified Rodrigues parameters were originally developed by Wiener [2], rediscovered by Marandi and Modi [3], and investigated quite fully by Tsiotras and Longuski [4] and Schaub and Junkins [5]. Nevertheless, there is more to say.