Attitude Parameterizations as Higher-Dimensional Map Projections

Abstract

A class of vectorial attitude parameterizations that are formulated as a product of the unit rotation vector and various functions of the rotation angle is examined. When related to a four-dimensional unit quaternion, these vectorial parameterizations are shown to be analogous to higher-dimensional azimuthal projections from a threedimensional unit hypersphere. Several types of these projections are examined. Singularities are identified and numerical accuracy is evaluated based on the singular value decomposition of the attitude kinematics. It is shown how shadow parameterizations can be constructed in order to alleviate the kinematical singularities. It is also shown that the kinematical passivity and optimality of the Rodrigues and modified Rodrigues parameters are special cases of themore general result that holds for awider range of vectorial parameterizations. This result is used to formulate and compare passivity-based control laws using various parameterizations.