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2. Matrix of transformation

Every set of attitude coefficients (corresponding to a line in attitude data file) is valid for its own time interval and gives the possibility to get a matrix N of tranformation from the GSE coordinate system to the S/C coordinate system as well as a matrix M of tranformation from the S/C coordinate system to the GSE coordinate system.

Every set consists of 20 parameters. There are:

1.  Line number
2.  Year
3.  Month
4.  Day
5.  Beginning of the time interval (in thousands of sec)
6.  Interval duration (in thousands of sec)

Parameters 7-20 are attitude coefficients:

7-11.  A₁ - A₅
12-16.  B₁ - B₅
17.  ω₁
18.  ω₂
19.  c₁
20.  c₂

Using the coefficients we can get angles α, β, γ:

α is an angle between X_S axis and the Sun projection onto X_SY_S plane,
β is an angle between X_S axis and the Sun projection onto X_SZ_S plane.

The algorithm is based on the assumption that the α,β angles can be approximated by the following functions:

α = A₁ + A₂sin ω₁ t + A₃cos ω₁ t + A₄sin ω₂ t + A₅cos ω₂ t,
β = A₁ + B₂sin ω₁ t + B₃cos ω₁ t + B₄sin ω₂ t + B₅cos ω₂ t.

γ is an angle between Y_S axis and projection of the North Pole of Ecliptic onto X_SY_S plane.
As this direction is close to the one which is orthogonal to the spin axis the linear approximation of the γ angle is valid:

γ = c₁ + c₂ t

ω₁ is the spin rate of the Sun direction around the angular momentum vector;
ω₂ is the spin rate of the the angular momentum vector around the main axis of inertia X_S.

Using α, β angles the Sun direction (which coincides with X_G axis) cosines in the S/C coordinate system can be calculated:

The direction e of the North Pole of Ecliptic plane (which coincides with Z_G axis cosines in the S/C coordinate system can be calculated by formulas:

a_x = s₂ cos γ + s₃ sin γ

So we have the direction cosines of X_G and Z_G axis in the S/C coordinate system.

The third direction p is

p = e x s

and the matrix M of transformation from the S/C to the GSE systems (the direction cosines matrix) is:

M =	s1 s2 s3 p1 p2 p3 e1 e2 e3

and the matrix N of transformation from the GSE system to the S/C system is the transpose of matrix M:

N =	s1 p1 e1 s2 p2 e2 s2 p2 e2

Any vector V_S/C in the S/C coordinate system can be transformed to the GSE coordinate system using M matrix:

V_GSE = M * V_S/C

as well as

V_S/C = N * V_GSE

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