Using Rotations to Build Aerospace Coordinate Systems - Defence ...

DSTO–TN–0640Using Rotations to Build Aerospace Coordinate SystemsEXECUTIVE SUMMARYThis paper presents the main techniques necessary to understand three-dimensional rotations.The subject is not new, but can be very difficult to sort out and to explore intextbooks. Mathematics and physics texts that discuss the subject generally do so only inpassing, and probably never cover any standard software protocols. The main literatureon the subject seems to be that generated by the graphics computing community, sinceprogrammers need to use rotations when writing three-dimensional visualisation code.Nevertheless, code ease-of-use and ability to be passed on means that although there aremany internet sites that talk about rotations, their information content is often biasedtowards whatever system or code their authors have become familiar with.We begin by describing the three standard coordinate systems that are used for simulationof aerospace scenarios and the Global Positioning Satellite system, and how theirdefining axes can be built from rotations alone. We then describe the mathematics ofrotations using both matrices and quaternions, defining Euler angles, and concentratingon the important matrix (or equivalently, quaternion) that allows any rotation about anyaxis to be made.As examples of the techniques, we give the necessary calculations for dealing withstandard scenarios involving the Global Positioning Satellite system, for finding line-ofsightdirections on Earth, for visualising the world from a cockpit, and for converting to andfrom the standard software protocol for distributed interactive simulation environments.Appendices then discuss combining rotations, conversions with a particular type ofEuler angle convention, the dangers of confusing Euler angles with incremental rotationsfor software writers, and finally there is a short dicussion of interpolation of rotations incomputing.iii