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A Pseudo-Reversing Theorem for Rotation and its Application to Orientation Theory.

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Abstract

We state and prove a useful theorem on manipulating rotation order which, while not new, is barely present in the literature. This theorem allows the order of a sequence of rotations to be reversed, provided that the sense of the axes of rotation is changed from “body� to “space fixed� or vice versa. We use the theorem to aid calculations in geodesy (constructing a local north–east– down coordinate system) and aerospace theory (relating yaw–pitch–roll rates to vehicle angular velocity). The new notation here sheds light generally on the field of orientation theory, as well as giving insight to standard terms relating to wind direction used for treating ship motion. Although we present our analyses in the style of a tutorial in the general subject of spatial orientation theory, there is new notation here, along with alternative and novel ways of treating problems that are often seen as difficult or obscure by practitioners. This report follows on from the 2005 DSTO report DSTO–TN–0640, but is completely self contained, and DSTO–TN–0640 need not be read beforehand.

Executive Summary

This report is a much-evolved follow-on from the 2005 DSTO publication DSTO–TN–0640, “Using Rotations to Build Aerospace Coordinate Systems�, that explained the construction of coordinate systems used in aerospace calculations. That report followed a step-by-step approach to implement its calculations. In the current report we rephrase those calculations in a more efficient language, while incorporating a very useful theorem that is known but almost absent from the literature. This “Pseudo-Reversing Theorem� allows the order of a sequence of rotations to be reversed, provided that the sense of the axes of rotation is changed from “body� to “space fixed� or vice versa. The current report is self contained, so that familiarity with the content of DSTO–TN–0640 is not necessary.; The current report places the theorem and the reworked examples of DSTO–TN–0640 into the greater context of orientation/rotation theory. We first introduce the theorem, then establish a solid mathematical language necessary for quantifying the orientation of an object. We cover the background of how to rotate a vector, using either a matrix or a quaternion. We then rework the examples in DSTO–TN–0640: constructing a local north–east–down set of axes from a given latitude and longitude, and calculating where a pilot must look to see a distant aircraft. We also make an extended revisit to the subject of conversions within the Distributed Interactive Simulation environment for handling orientation information, since this often causes problems to practitioners who must deal with several coordinate systems at once. We end the main report by showing how the Pseudo-Reversing Theorem can be used to simplify some of the concepts behind dead reckoning an object’s changing orientation.; The report ends with an appendix that applies its notation and general approach to the task of constructing the appropriate course a ship must steer in order for the wind to appear to come from some given direction with some given speed. This is a nontrivial problem that is handled well in a novel way by the orientation-matrix language of this report, although its solution doesn’t require the Pseudo-Reversing Theorem.

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