Triad method: Difference between revisions

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{{short description|Solution to the spacecraft attitude determination problem}}

TriadThe is'''TRIAD onemethod''' ofis the earliest andpublished simplestalgorithm solutionsfor to[[Spacecraft theattitude determination|determining spacecraft attitude]], determinationwhich problem,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|month date= July|year= 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|yeardate=July–August 1990|month=July-Aug|volume=13|issue=4|pages=577–585|doi=10.2514/3.25373|bibcode = 1990JGCD...13..577B }}</ref><ref>{{Cite duejournal |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 Haroldboth Blackframes. 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. As evident from the literature, TRIAD representsrepresented 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|yeardate=July 1966|monthpages=July385–386|pagesdoi=385–38610.1137/1008080|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|yeardate=April–June 1993|month=April-June|volume=41|issue=2|pages=261-280261–280|url=http://www.malcolmdshuster.com/FC_Markley_1993_J_FOAM_JAS_MDSscan.pdf|accessdate=April 18, 2012}}</ref> .

==Summary==

WeFirstly, one considerconsiders 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. ThenFollowing that, they are then related by the equations,

{{NumBlk|:|

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|{{EquationRef|1}}}}

for <math> i = 1,2 </math>, where <math> A </math> is a rotation matrix (sometimes also known as a proper [[orthogonal matrix]], i.e., <math>A^{T}A = I, det(A) = +1 </math>). <math> A </math> transforms vectors in the body fixed frame into the frame of the reference vectors. Among other properties, rotational matrices preserve the length of the vector they operate on. Note that the direction cosine matrix <math> A </math> also transforms the cross product vector, written as,

{{NumBlk|:|

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|{{EquationRef|2}}}}

TriadTRIAD 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. TriadTRIAD incorporates the following elegant procedure to redress this problem. To this end, weone definedefines unit vectors,

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|{{EquationRef|7}}}}

to be used in place of the first two columns of equation ({{EquationNote|3}}). Their cross product is used as the third column in the linear system of equations obtaining a proper orthogonal matrix for the spacecraft attitude given by the following:

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|{{EquationRef|8}}}}

While the normalizations of Equationsequations ({{EquationNote|4}}) - ({{EquationNote|7}}) are not necessary, they have been carried out to achieve a computational advantage in solving the linear system of equations in ({{EquationNote|8}}). Thus an estimate of the spacecraft attitude is given by the proper orthogonal matrix as

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<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}}}}

Note that computational efficiency has been achieved in this procedure by replacing the matrix inverse with a transpose. Equation({{EquationNote|8}})This showsis thatpossible because the matrices usedinvolved forin computing attitude are each composed of ana orthogonal triadTRIAD of [[Orthonormality|orthonormal]] basis vectors. "TRIAD" derives its name from this observation.

==TriadTRIAD Attitude Matrix and Handed-nessHandedness of Measurements==

It is of consequence to note that the TriadTRIAD 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.: In a matrix form given

Let us re-write Eq. ({{EquationNote|8}}) in a matrix form given by

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\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 triadTRIAD, 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}}), weone concludeconcludes that

{{NumBlk|:|

<math>

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==Applications==

TriadTRIAD was used as an attitude determination technique to process the telemetry data from the Transit satellite system (used by the U.S. Navy for navigation). It is interesting to note that theThe principles of the Transit system gave rise to whatthe weglobal nowpositioning know as the GPSsystem satellite constellation. In an application problem, the reference vectors are usually known directions (e.g., stars, Earth magnetic field, gravity vector, etc.). Body fixed vectors are the measured directions as observed by an on-board sensor (e.g., star tracker, magnetometer, etc.). With advances in micro-electronics, attitude determination algorithms such as TriadTRIAD have found their place in a variety of devices (e.g., smart phones, cars, tablets, UAVs, etc.) with a broad impact on the modern society.

==See also==