Triad method: Difference between revisions

Revision as of 08:45, 11 December 2011

Revision as of 10:07, 18 December 2011

Line 1:

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=1350-13511350–1351|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=577-585577–585|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=385-386385–386}}</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,

Line 69:

|{{EquationRef|9}}}}

Note that computational efficiency has been achieved in this procedure by replacing the matrix inverse with a transpose.

===Triad Attitude Matrix and Handed-ness of Measurements===

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

Line 94 ⟶ 95:

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===

Triad was used as an attitude determination technique on-board the Transit system (used by the U.S. Navy for navigation). It is interesting to note that the principles of the Transit system gave rise to what we now know as the GPS 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 Triad 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.