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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|month = July|year=1964|volume=2|issue=7|pages=1350-1351|doi=10.2514/3.2555}}</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|year=1990|month=July-Aug|volume=13|issue=4|pages=577-585|doi=10.2514/3.25373}}</ref> . As evident from the literature, TRIAD represents the state of practice in spacecraft attitude determination, well before the advent of the Wahba <ref>{{cite journal|last=Wahba|first=Grace|title=A Least Squares Estimate of Satellite Attitude, Problem 65.1|journal=SIAM Review|year=1966|month=July|pages=385-386}}</ref> problem 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.
===Summary===
We consider the linearly independent reference vectors <math>\vec{R}_{1}</math> and <math>\vec{R}_2 </math>. Let <math> \vec{r}_1, \vec{r}_2</math> be the corresponding measured directions of the reference unit vectors as resolved in a body fixed frame of reference. Then they are related by the equations,
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|{{EquationRef|9}}}}
Note that computational efficiency has been achieved in this procedure by replacing the matrix inverse with a transpose.
===Handedness of Vectors: Effect on the Direction Cosine Matrix Estimation===
It is of consequence to note that the Triad method always produces a proper orthogonal matrix irrespective of the handedness of the reference and body vectors employed in the estimation process. This can be shown as follows.
Let us re-write Eq. ({{EquationNote|8}}) in a matrix form given by
{{NumBlk|:|
<math>
\Gamma = A \Delta
</math>
|{{EquationRef|10}}}}
where <math>
\Gamma := \left[ \hat{S} ~ \vdots ~ \hat{M} ~\vdots~ \hat{S} \times \hat{M} \right]
</math>
and
<math>
\Delta = \left[ \hat{s} ~\vdots~ \hat{m} ~\vdots~ \hat{s} \times \hat{m} \right].
</math>
Note that if the columns of <math> \Gamma </math> form a left handed triad, then the columns of <math> \Delta </math> are also left handed because of the one-one correspondence between the vectors. This is because of the simple fact that, in Euclidean geometry, the angle between any two vectors remains invariant to coordinate transformations. Therefore, the determinant <math> det \left( \Gamma \right) </math> is <math> 1 </math> or <math> -1 </math> depending on whether its columns are right or left - handed respectively (similarly, <math> \Delta = \pm 1</math> ). Taking determinant on both sides of the relation in Eq. ({{EquationNote|10}}), we conclude that
{{NumBlk|:|
<math>
det \left( A \right) = 1.
</math>
|{{EquationRef|11}}}}
This is quite useful in practical applications since the analyst is always guaranteed a proper orthogonal matrix irrespective of the nature of the reference and measured vector quantities.
==Applications==
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