Fig. 7-14. Bolometer Offset Model Geometry the rotation angle about the spin axis from the Earth-in to the nadir vector, and L0 is the rotation angle about the spin axis from the nadir vector to the Earth-out.
B + (Q/2-L,)+tsl(ww-us) + Qb= 360° or 720° (7-55)
where is the wheel rate, ws is the body spin rate, and <PS is the phase of the bolometer offset. For a fixed bolometer position, the four unknowns in these equations are t), L„ L0, and B. Equations (7-54) and (7-55) determine B and L0 in terms of L,, and these are substituted into Eq. (7-53), whereupon tj and L, are solved for, usually in an iterative fashion. Alternatively, for a fixed spacecraft attitude, the observed WHS data may be used to compute the bolometer offset parameters o and <&fl (Liu and Wertz ).
73 Sun Sensor/Horizon Sensor Rotation Angle Models
In this section, we describe observation models for the following Sun sensor/horizon sensor rotation angle measurements: Sun-to-Earth-in, Sun-to-Earth-out, and Sun-to-Earth-midscan. Related azimuth biases are discussed for body-mounted horizon sensors and panoramic scanners (Section 6.2). For additional modeling procedures, see Joseph, et a/., . In every case, the observable quantity is a time difference, A/. For the Sun-to-Earth-in model, A/= — ts, where t, is the horizon-in crossing time and ts is the Sun sighting time. (Note that these times are measured by different sensors at different orientations in the spacecraft.) For the Sun-to-Earth-out and the Sun-to-Earth-midscan models, t, is replaced by the horizon-out crossing time, rQ, and the midscan crossing time, tm = 1 /2(t, +10), respectively.
The relevant geometry for the Sun-to-Earth-in model is shown in Fig. 7-15. We assume that the Earth is spherical; that the spin rate, to, is constant; and that there is no nutation. Therefore, the total rotation angle change between ts and t, is w('/~ = u-A/, and the observation model is
Here <!>, is the rotation angle from the Sun, S, to the horizon in-crossing, H,; <bH is the azimuthal mounting angle between the Sun sensor and the horizon sensor onboard the spacecraft; and n=? ± 1, or 0. Qr can be calculated from
where A is the spin axis attitude, S is the Sun unit vector, and H, is a unit vector along the horizon sensor line of sight at the time / = t,. Here A is assumed known, S
is provided by an ephemeris (Section 5.5) evaluated at / = ts, and H, is calculated below.
Equation (7-57a) is derived as follows: Let Sp and Hp lie the normalized components of S and H, in the spin plane, Le, the plane whose norma) isA. Then
Performing the dot product of Hp and S,, we obtain the following expression for cosO,:
cos*,=.H,S,= [ S - H, - (S - A)(H;- A) ] /(Ds£>„ ) (7-57b)
where Ds and D„ are the denominators in the expressions for Sp and H,, respectively. Using a different manipulation of Hp, Sp, and A, we obtain the following expression for sin®,:
sin » A X Sp- Hp » A- Sp X Hp ° A • (S X ft,)/(£>$£>« ) (7-57c)
Equation (7-57a) is dien obtained by dividing Eq. (7-J7c) by Eq. (7-57b).
Jhe unit vector H, is calculated as follows: Let M be a unit vector perpendicular to both A and E. Then
where.ij is the. angle between A and E (the nadir angle). Let N be a unit vector perpendicular to feot]i E and M. Then E, M, and N form an orthononnjl triad such that N°ExM. Because E-H,=cosp, where p is the angular radius of the Earth, H# can be written as
where A is a phase angle which can be determined from the dot product between A and H,. This is done as follows: If y is the horizon sensor mounting angle, then
which simplifies to cosf^cospcosij + sinpsinijcosA (7-S8a)
cosApCOST-COSPCOS, sin p sin i]
Because Eq. (7-58a) is the law of cosines applied to the spherical triangle A EH, in Fig. 7-15» the phase angle A must be the rotation angle about E between A and H,. Due to our choice of the unit vectors M and N, the negative sign in Eq. (7-58c) is associated with Hf and the positive sign with Hc. The nadir vector E is determined from the spacecraft ephemeris.
For the Sun-to-Earth-out and Sun-to-Earth-midscan models, the procedure is identical with that for the Sun to Earth-in model, except that the quantities t,, and H/ are replaced everywhere by tQ, and H0 or by <Pm, lm, and Hm. Here, Hm is a unit vector in the direction of H7 + H0. can also be calculated directly, using the following relation (see Fig. 7-15):
An expression for 4>n is obtained by applying the law of cosines to triangle SAE in Fig. 7-5, yielding cosxp = cosr)cos/3 + sinr)sin/?coswhich becomes, upon solving for
An expression for the Earth width, S2, is obtained analogously from triangle H,AE:
The quantities r>, ft, and are computed from ephemerides evaluated at the proper times.
Figure 7-16 shows the relevant geometry when biases in the orientation of both sensors are included and the horizon sensor is assumed to have a fixed mounting angle, y. In Fig. 7-16, /JM is the measured Sun angle, ft is this true Sun angle, and ts and A/J are the inclination and elevation biases which cause the difference between ftM and ft. A9S is the resulting rotation angle bias. Similarly, y^ is the nominal mounting angle of the horizon sensor line of sight, relative to the spin axis; Ay is the difference, y~yN, between yN and the true mounting angle, y; A3>w is a constant bias on <J>W, the nominal azimuthal mounting angle difference between the sensors; pc is the computed angular radius of the Earth: and Ap is a fixed angular
bias on pc, resulting primarily from a constant bias on the triggering threshold of the horizon sensor (see Sections 6.2 and 7.2).
In terms of the above quantities, the observation model becomes
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