Triad method

Triad is one of the earliest and simplest solutions to the spacecraft attitude determination problem, due to Harold Black. 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 represents the state of practice in spacecraft attitude determination, well before the advent of the Wahba's 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. Covariance analysis for Black's classical solution was subsequently provided by Markley.    (1)    (2)    (3)    (4)    (5)    (6)    (7)    (8)    (9)    (10)    (11) Triad is one of the earliest and simplest solutions to the spacecraft attitude determination problem, due to Harold Black. 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 represents the state of practice in spacecraft attitude determination, well before the advent of the Wahba's 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. Covariance analysis for Black's classical solution was subsequently provided by Markley. We consider 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. Then they are 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. Note that the direction cosine matrix A {displaystyle A} also transforms the cross product vector, written as, Triad proposes an estimate of the direction cosine matrix A {displaystyle A} as a solution to the linear system equations given by where ⋮ {displaystyle vdots } have been used to separate different column vectors. The solution presented above works well in the noise-free case. However, in practice, r → 1 , r → 2 {displaystyle {vec {r}}_{1},{vec {r}}_{2}} 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, we define unit vectors