Content deleted Content added
some things Tags: Reverted references removed Visual edit Newcomer task Newcomer task: copyedit |
incorrect extra period removed |
||
| (4 intermediate revisions by 4 users not shown) | |||
Line 1:
{{short description|Solution to the spacecraft attitude determination problem
The '''TRIAD method''' is the earliest published algorithm for [[Spacecraft attitude determination|determining spacecraft attitude]], which was first introduced by Harold Black in 1964.<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–August 1990|volume=13|issue=4|pages=577–585|doi=10.2514/3.25373|bibcode = 1990JGCD...13..577B }}</ref><ref>{{Cite journal |last=Markley |first=F. Landis |date=1999 |title=Attitude Determination Using Two Vector Measurements |url=https://www.researchgate.net/publication/4706531_Attitude_Determination_Using_Two_Vector_Measurements |journal=1999 Flight Mechanics Symposium |pages=2 |via=ResearchGate}}</ref> Given the knowledge of two vectors in the reference and body coordinates of a satellite, the TRIAD algorithm obtains the direction cosine matrix relating to both frames. Harold 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. TRIAD represented the state of practice in spacecraft attitude determination before the advent of [[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. Covariance analysis for Black's 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
{{NumBlk|:|
Line 28 ⟶ 29:
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,
{{NumBlk|:|
Line 65 ⟶ 66:
{{NumBlk|:|
<math>
\hat{A} = \left[ \hat{S} ~ \vdots ~ \hat{M} ~\vdots~ \hat{S} \times \hat{M} \right] . \left[ \hat{s} ~\vdots~ \hat{m} ~\vdots~ \hat{s} \times \hat{m} \right]^T
</math>
|{{EquationRef|9}}}}
Line 86 ⟶ 87:
\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-handed or left-handed respectively (similarly, <math> \Delta = \pm 1</math> ). Taking determinant on both sides of the relation in Eq. ({{EquationNote|10}}),
{{NumBlk|:|
<math>
| |||