where the FORTRAN function ATAN2 is used in Eq. (7-18) to resolve the quadrant ambiguity. If we substitute Eq. (7-16) into (7-19), and rearrange terms, we get

The angles $ and 9 are the azimuth and coelevation, respectively, of the Sun vector in sensor coordinates which have the positive pole along the sensor boresight and the reference meridian along the + XS axis. The Sun vector may be transformed into spacecraft coordinates by using Eq. (7-9). Two-axis digital Sun sensor data are commonly reparameterized in terms of the angles between the projections of the sunline on the Ys-Z, and Xt-Z, planes and the Z5-axis, as illustrated in Figs. 7-8 and 7-9. The angles a and /} are rotations about the -X, and Ys axes, respectively, given by tanj8 = tanflcos«i>=nA//{

where

The specified field of view of the Adcole two-axis sensor is "square" as illustrated in Fig. 7-9 for a 128- by 128-deg sensor. The effective FOV is often considered circular with radius 64 deg because this is the maximum angle of incidence which guarantees valid sensor data (i.e., sufficient intensity) independent of For Sun angles near the "corners" of the FOV, <f> = ± 45 or ±135, valid sensor

data are obtained for 0 up to 71 deg.* In Section.2.3, we proved that five two-angle sensors may be dispersed to provide Air sr coverage with a maximum 9 angle of 63.S deg. We now see that this result is valid for two-axis digital sensors independent of sensor alignment about the boresight. For an 8-bit sensor with n= 1.4553, the coordinates of various grid points within the FOV, expressed as (NA, NB), are shown in Fig. 7-9. For n 1, lines of constant a or fi are not lines of constant NA or NB and, in particular, the grid point corresponding to [a,/?]=[64°, 0°] is (255, 127.5) and [64°, 64°] is (226, 226). The boresight is at the center of the four grid points (127, 127), (127, 128), (128, 127) and (128, 128). Because of the refractive sensor medium, a ray normal to the boresight at 0=90 deg and <>=45 deg will

Fig. 7-9. Two-Axis Digital Sun Sensor Field of View. See text for explanation of coordinates.

♦From Eq. (7-21), we have <(.=45 deg and /}=64 deg; hence, tan0=tan0/cos45° = tan64°/cos45° or 8-70.97 deg.

Fig. 7-9. Two-Axis Digital Sun Sensor Field of View. See text for explanation of coordinates.

♦From Eq. (7-21), we have <(.=45 deg and /}=64 deg; hence, tan0=tan0/cos45° = tan64°/cos45° or 8-70.97 deg.

reach the reticle pattern (with zero intensity) and fall at (236, 236). Sensor data with grid points corresponding to 9 > 90 deg are necessarily anomalous and an application of Eqs. (7-17) and (7-22) to such data would yield R2<0.

To simulate data, we must solve for NA, NB, and the selected sensor in terms of the Sun vector in sensor coordinates,

where vB and $ss are the Sun vector in spacecraft and sensor coordinates, respectively. Using Eqs. (7-21), (7-22) and Fig. 7-8, we obtain the result,

where

7 = [A7(«2-*2-K2)]>0 (7-25) Finally, the sensor output is

where INT(x) is the integral part of x and NA and NB are Gray coded by the reticle pattern. The Sun is visible to a specific sensor (although the intensity may be below the ATA threshold) if both y and Zs are positive. The selected sensor for multisensor configurations is determined by the ATA output, i.e., the sensor with the largest (positive) Zs.

For state estimation, the digital sensor angular outputs may be computed using Eqs. (7-26) but the sensor identification for multisensor configurations cannot be reliably predicted. The actual sensor selected is a function of the precise threshold settings whenever the Sun is near the Earth's horizon or is between the fields of view of adjacent sensors. Sensor identification should be used merely to validate sensor data for state estimation.

Fine Sun Sensors. The operation of the fine Sun sensor described in Section 6.1 is illustrated in Fig. 7-10 (compare with Fig. 6-9). In the figure, the horizontal axis has been expanded to illustrate the effect of the 32-arc-minute angular diameter of the Sun (from near the Earth), which requires the use of an analog sensor rather than a finely gridded digital sensor. Incident sunlight falling on the entrance slits with spacing s produces the photocell current shown schematically in Fig. 7-10(d). The nearly sinusoidal output signal is a consequence of the Sun's finite size. If four reticle patterns are offset by s/4 the photocell current, /, beneath each pattern may be written as a function of x = 2itw/s where w=/tana, / is the distance between the two reticle patterns, and a is the Sun angle. I is given by l2=f(2vw/s + v/2) /3=/(2ww/J + W) = f(2vw/s + 3 w/2) (7-27)

(MCiOBMT SUNLIGHT

(MCiOBMT SUNLIGHT

OUTPUT AT MEDIAN (b)

OUTPUT AT WINIMUW tc>

Fig. 7-10. Schematic Representation of Fine Sun Sensor Photocell Output Current. Rays coming from different directions represent light from opposite sides of the Sun. (The angular spread of these rays is greatly exaggerated.)

OUTPUT AT MEDIAN (b)

OUTPUT AT WINIMUW tc>

Fig. 7-10. Schematic Representation of Fine Sun Sensor Photocell Output Current. Rays coming from different directions represent light from opposite sides of the Sun. (The angular spread of these rays is greatly exaggerated.)

where angles are measured in radians. The fine Sun sensor electronics forms the quantity arctan^=arctan[(/,- /3)/(/2- /4)], which is related to the Sun angle by arc tany = tan a + small error term

Equation (7-28) may be derived as follows. The function f(x) is periodic with period s and has a maximum zlx*=w/2. Because f(x) is symmetric about x=n/2, it may be expanded in a Fourier cosine series as [Markley, 1977]:

f(x) = a0+ a,cos(jc - w/2) + a2cos(2x - w) + a3cos(3x - 3 w/2) H----

The fine Sun sensor electronics forms the quantities /, — I3,12 — I4,y = (It —I3) /(/2— I J, and arctanjr, which are approximated as follows:

/, —13 = a, [sin* - sin(x + w) ] — a2[cos2x - cos(2x + 2 w) ]

=2a,sinx[l -a3(4cos2x- 1 )/a,]+ • • • (7-30a)

J2 - J4 = 2a,cos.x[ 1 + a3(4cos2x - 3 )/ar] + • • • (7-30b)

/«tanxp -4fl3(2cos2jc— l)/a,j=lanjc[l —4a3cos2jc/a,] (7-30c) Equation (7-30c) can be rewritten in the more convenient form arc tan^ = x — arc tan t (7-30d)

where t is a small error term, by taking the tangent of both sides of the above equation, and using the trigonometric identity tan(o+b)=(tan a ± tan b)/{ 1 + tan a tan b) (7-31 a)

to obtain

(tan* - c)/(I + e tanjc)=tanx- t(l + tanlc)+ 0(e2) (7-31b) Comparing this result with Eq. (7-30c) to obtain c, we have arc tancas* - arc tan(a3sin4x/a,)»x — a3sin4jc/a, (7-32) For small w = t tan a, we obtain arctan^=^tana-%in(^íM«j (7-33)

which is the desired result.

Thus, if the photocell output is adequately represented by the first three terms of a Fourier cosine series, the output of the fine Sun sensor electronics, arctanp, is given by a term proportional to the tangent of the incident angle, a, plus a sinusoidal error term.

In practice, the inverse of Eq. (7-33) is required for sensor data processing. The digital sensor output, NA, is related to the analog output by arc tan/ = k1(NA)+k2 (7-34)

where At, and k2 are sensor constants. Equation (7-33) can be rewritten as i&na=~(J(xNA+k2)+^^sm\~-Xa.na^ (7-35)

Defining the sensor constants

^3 = sa3/2w/a, (7-36c) then successive approximations, aw, to a are given by tan aw> = Ai+A 2NA (7-37a)

tan a<" + ,) = tana(n) + A 3sin( ^ tan aM) (7-37b)

or, to the same order as Eq. (7-33), tañará, +A2NA +A$m(AANA +AS) (7-37c)

For IUE, the sensor output is encoded into 14-bit (0-16,383) words and the ±32-deg field of view is measured with a 14 arc-second least significant bit. The transfer function is as follows [Adcole, 1977]:

o = a0 + arctan[/4, + /42/V/4 + ^3sin(,44M4 + ,4j)-M6sin(,47M4 + /4g)]

/3=/30+arctan[fl, + B2NB + fl3sin(fl4tffl + B5) + fl6sin(fl7tffl + fl8)] (7-38)

where NA and NB denote the digitized sensor output; the parameters Ait Bt, a0, and /90 are obtained by ground calibration; and a and P are defined in Fig. 7-8.

The parameters defining the slab thickness, index of refraction, alignment, and resolution vary depending on the sensor model and specific calibration. Table 7-2 lists values which are representative and convenient for simulation and error analysis. (See also Table 6-1.)

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