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. 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, one defines unit vectors,
and
to be used in place of the first two columns of equation (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:
While the normalizations of equations (4) - (7) are not necessary, they have been carried out to achieve a computational advantage in solving the linear system of equations in (8). Thus an estimate of the spacecraft attitude is given by the proper orthogonal matrix as
Note that computational efficiency has been achieved in this procedure by replacing the matrix inverse with a transpose. This is possible because the matrices involved in computing attitude are each composed of a TRIAD of orthonormal basis vectors. "TRIAD" derives its name from this observation.
TRIAD Attitude Matrix and Handedness 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: In a matrix form given
where Γ := [ S ^ ⋮ M ^ ⋮ S ^ × M ^ ] {\displaystyle \Gamma :=\left[{\hat {S}}~\vdots ~{\hat {M}}~\vdots ~{\hat {S}}\times {\hat {M}}\right]}
and
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