Attitude Analysis in Flatland: The Plane Truth
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Attitude Analysis in Flatland: The Plane Truth 1 Malcolm D. Shuster/ Abstract Many results in attitude analysis are still meaningful when the attitude is restricted to rotations about a single axis. Such a picture corresponds to attitude analysis in the Euclidean plane. The present report formalizes the representation of attitude in the plane and applies it to some well-known problems in a two-dimensional setting. In particular we carry out a covariance of a two-dimensional analogue of the algorithm OLAE, recently proposed by Mortari, Markley, and Junkins for optimal attitude determination.
Introduction I call our world Flatland, not because we call it so, but to make its nature clearer to you, ... who are privileged to live in Space. -A. Square in Flatland [1] The treatment of attitude, because of the nonlinearity and noncommutivity of the composition rule, is much more difficult to treat than position, for which components may be combined by simple addition. The complexity of the attitude composition rule leads to virtually all attitude problems being intrinsically three-dimensional or, in the case of the quatemion, four-dimensional. There is, however, a class of attitude problems which are much simpler, namely, single-axis problems, and the study of these will in many cases illuminate the more complex problems. The present report attempts to formalize such a treatment.
Attitude in Flatland Having amused myself til a late hour with my favourite recreation of Geometry, I had retired to rest with an unsolved problem in my mind. Let us imagine that the world, Flatland, has only two dimensions and a constant isotropic Euclidean metric. Such a world was imagined by Edwin Abbott Abbott [1], 'Dedicated to John L. Junkins on the occasion of his sixtieth birthday. 2Director of Research, Acme Spacecraft Company, and acting manager, Planecraft Division, 13017 Wisteria Drive, Box 328, Germantown, MD 20874. email: [email protected].
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Shuster
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with the intent of satirizing the social and political foibles of his day as much as of clarifying the concepts related to the dimensionality of space. Our interest here is more limited than Abbott's. We develop the mathematical structure of Flatland somewhat further in order to better understand those aspects of attitude which do not depend on the dimensionality of space. The quotations which appear in this report are from reference [1]. Following Abbott we will refer to our three-dimensional world as Space. In Flatland, vectors are, of course, two-dimensional
v
=
(1)
[::]
The "dot" product takes the usual form u ·y=
UIVI
+
U2V2
= UTy = v'Iu = y · u
(2)
while the "cross product" is now a scalar u
X
v=
UIV2 -
= det[u:y] = yTJu = -y
U2Vl
X
u
(3)
There is, therefore, no vector product, and as alternate names to scalar and vector products we might prefer symmetric and antisymmetric products. The lack of a meaningful vector product in two dimensions was ultimately the cause of many years of grief for Hamilton [2-4]. The attitude matrix in two dimensions i
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