Triad method: Difference between revisions

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Triad is one of the earliest and simplest solutions to the spacecraft attitude determination problem,<ref>{{cite journal|last=Black|first=Harold|title=A Passive System for Determining the Attitude of a Satellite|journal=AIAA Journal|date=July 1964|volume=2|issue=7|pages=1350–1351|doi=10.2514/3.2555|bibcode = 1964AIAAJ...2.1350. }}</ref><ref>{{cite journal|last=Black|first=Harold|title=Early Developments of Transit, the Navy Navigation Satellite System|journal=Journal of Guidance, Control and Dynamics|date=July–Aug 1990|volume=13|issue=4|pages=577–585|doi=10.2514/3.25373|bibcode = 1990JGCD...13..577B }}</ref> due to Harold Black. Black played a key role in the development of the guidance, navigation and control of the U.S. Navy's Transit satellite system at Johns Hopkins Applied Physics Laboratories. As evident from the literature, TRIAD represents the state of practice in spacecraft attitude determination, well before the advent of the [[Wahba's problem]] <ref>{{cite journal|last=Wahba|first=Grace|title=A Least Squares Estimate of Satellite Attitude, Problem 65.1|journal=SIAM Review|date=July 1966|pages=385–386|doi=10.1137/1008080 |volume=8}}</ref> and its several optimal solutions. Given the knowledge of two vectors in the reference and body coordinates of a satellite, the TRIAD algorithm obtains the direction cosine matrix relating both frames. Covariance analysis for Black's classical solution was subsequently provided by Markley.<ref>{{cite journal|last=Markley|first=Landis|title=Attitude Determination Using Vector Observations: A Fast Optimal Matrix Algorithm|journal=The Journal of Astronautical Sciences|date=April–June 1993|volume=41|issue=2|pages=261–280|url=http://www.malcolmdshuster.com/FC_Markley_1993_J_FOAM_JAS_MDSscan.pdf|accessdate=April 18, 2012}}</ref>

==Summary==

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Triad proposes an estimate of the direction cosine matrix <math> A </math> as a solution to the linear system equations given by

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|{{EquationRef|3}}}}

where <math> \vdots </math> have been used to separate different column vectors.

The solution presented above works well in the noise-free case. However, in practice, <math> \vec{r}_1, \vec{r}_2 </math> are noisy and the orthogonality condition of the attitude matrix (or the direction cosine matrix) is not preserved by the above procedure. Triad incorporates the following elegant procedure to redress this problem. To this end, we define unit vectors