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

DSTO–TN–06405 Concluding RemarksAll of the commonly-needed techniques in three-dimensional rotations rely on our abilityto do two basic procedures:1. Rotate a vector about any axis whatsoever, and2. calculate components using dot products.These calculations need to be done in just one frame. The usual one is the ECEF, whichis good for all of the usual motions we associate with moving around Earth. (If we needto deal with satellites, a nonrotating frame is needed, such as one fixed to the stars.)Rotating about arbitrary axes and taking dot products within the ECEF is all that hasbeen done for the examples in this report.Rotations themselves can be carried out using matrices or quaternions. Both of theseare simply ways of writing down the three numbers needed to encode a rotation (anangle, and two numbers for the axis). The extra redundancy that matrices have overquaternions allows for simpler algebra when using matrices to analyse rotation on paper,but quaternions, being essentially pared-down versions of the rotation matrix, can be easierto prevent from degrading numerically inside a computer routine (but see the comment atthe end of Sect. 3.4). Performance questions like this are critical in that three-dimensionalmodels can have 10,000+ points to rotate at e.g. 30 frames per second.When solving a problem involving three-dimensional rotations, a good approach is tobreak the scenario down into its building block rotations and to handle each one separately,all within the ECEF. This step-by-step approach lends itself well to being modularised insoftware, and is what we have used repeatedly in this report.ReferencesMathematics essential to three-dimensional rotations is described in:Lyons, L., (1998) All You Wanted to Know About Mathematics But Were Afraid toAsk, Cambridge University PressVarious useful calculations involving GPS coordinates are contained in:Farrell, J., Barth, M. (1998), The Global Positioning System and Inertial Navigation,McGraw-Hill20