Representing Attitudes as Sets of Frames

A general framework for representing continuous sets of frames with the unit quaternion representation is presented. The determination and control of the attitude of a rigid body is important in a wide range of applications and has been given much attention in the control community. Not always, however, must the desired attitude be restricted to one given orientation, but can be given as a discrete or continuous set of orientations subject to some restriction. An attitude can be represented by the four-parameter unit quaternion without the presence of singularities. It is shown how continuous sets of frames can be described by the unit quaternion representation. It is also shown how this set can be reorientated into an arbitrary coordinate system by the quaternion product. Some work is done on finding the attitude that is closest to the desired orientation when the desired orientation is out of reach due to some restriction on the allowed orientations or rotations.