— sin 4>—Sip sin A cos <j> Sip sin sin A cos 4> cosacos A cos<í>—¿tysinÁsin</> — Sipcos 4>—sin A sin ^ sinocos A cos A cos A sin A
Note that in the example shown, slit A is nominally parallel to the spacecraft X-Y plane and the spacecraft Z-axis is in the sensor Y-Z plane.
One-Axis Digital Sensor. The geometry of a ray incident on a block of material with index of refraction n is illustrated in. Fig. 7-6. Snell's law relating the angle of incidence, 9, and the angle of refraction, 9', is nsin0' = sinfl
where the index of refraction of space is unity. The detectors beneath the reticle pattern of the sensor yield a signed, digitized output, N, proportional to the deflection, x, such that x = kN (7-11)
where k is the reticle step size. From Eq. (7-10) and simple trigonometry, we have nkN
Expanding in a trigonometric series in t = x/h = kN/h«.l and retaining terms through t5, we obtain
0«/i£ - n(3 - n2y3/6 + n(l5 - 10/i2 + 3/i4)ts/40 (7-13)
where 6 is in radians. A design goal of digital sensors is a linear relation between the sensor output, N, and the measured angle. For a material with nzsfi, the term dependent on c3 becomes negligible, yielding the approximate result
A further useful simplification results if the reticle geometry is chosen such that 180m/c/(7tA)= 1. In this case, BazN where B is now expressed in degrees.
When the Sun angle measurement, B, is made, the Sun vector in sensor reference coordinates is (-sin0,O,cos0)T. In spacecraft coordinates, the Sun vector is
from which the azimuth and elevation of the Sun in spacecraft coordinates may be computed.
Two-Axis Digital Sensors. The derivation of the data reduction equations for two-axis Adcole sensors is analogous to that for the single-axis sensors [Adcole, 1975]. The geometry is shown in Fig. 7-7. Note that OZs is the optical null (or boresight) of both the A and B sensors. The refracted ray (OP') is deflected by the slab with index of refraction n and strikes the Gray-coded rear reticle at P' with coordinates (b,a). Application of Snell's law yields sin0 = /isin0'
By analogy with Eq. (7-11), the output of the A and B components denoted by NA and NB, respectively, is converted to a displacement by a = km(NA-2m~l + 0.5)
NA and NB are unsigned decimal equivalents of the m-bit Gray-coded sensor output and km is a sensor constant. (See Table 7-2 for representative values of the sensor constants.) The form of Eq. (7-17), particularly the addition of 0.5 to NA and NB, is a consequence of the Adcole alignment and calibration procedure.
Right triangles OO'P' and O'Q'P' yield the relations
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