Triad method

The TRIAD method is the earliest published algorithm for determining spacecraft attitude, which was first introduced by Harold Black in 1964. 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 and its several optimal solutions. Covariance analysis for Black's solution was subsequently provided by Markley.

Summary

Firstly, one considers the linearly independent reference vectors

R

→

1

{\displaystyle {\vec {R}}_{1}}

and

R

→

2

{\displaystyle {\vec {R}}_{2}}

. Let

r

→

1

,

r

→

2

{\displaystyle {\vec {r}}_{1},{\vec {r}}_{2}}

be the corresponding measured directions of the reference unit vectors as resolved in a body fixed frame of reference. Following that, they are then related by the equations,

for

i

=

1

,

2

{\displaystyle i=1,2}

, where

A

{\displaystyle A}

is a rotation matrix (sometimes also known as a proper orthogonal matrix, i.e.,

A

T

A

=

I

,

d

e

t

(

A

)

=

+

1

{\displaystyle A^{T}A=I,det(A)=+1}

).

A

{\displaystyle A}

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.