where a and e are the azimuth and elevation, respectively, in the inertial system. Using Eq. (16-102) and the third row of Eq. (16-103) becomes
•Equation (16-103) may be used to show that if the spin rate is measured by observing the times when an inertial vector, S, crosses a body-fixed plane containing the z axis (taken as the x-z plane without loss of generality), then, for small 9, crossings occur whenever, sin(ij>+^)-0. Thus, for S sufficiently far from L (Le., 9SZ/Sx<t:\), the average measured spin rate is or
and because S,=sini«^), where is the elevation of the observation in the body frame, we have sin(t.)—sint
—--= [ 9 cos ] sin($f - a) + [ 9 sin 1 costyt -a) (16-1%)
For small c the left-hand side approximately equals the elevation residual, e^ — c, and is referred to as the reduced elevation residual. Because $ is known from Eq. (16-101), this can be solved for a given set of observations with simple linear regression by recognizing that it is of the form y-C,JIT,+ <yra (16-107)
with appropriate sign checks to determine oa the range 0 to 360 deg.
Finally, if g is the observed phase angle of the body x-axis, we may approximate
Integrating, then, or
and we have thus determined 9 and the initial values of <fr and ^
This technique was used successfully to determine the dynamic motion of the SAS-2 spacecraft from telemetered star sensor data. Figure 16-6 is a plot of the left-hand side of Eq, (16-105) (the reduced elevation residual) versus time (modulo the period of <>) for a selected orbit of SAS-2 star sensor data, where I J If"* l-06^ and the average measured spin rate was «*» 1,061 deg/sec. Fitting this data to Eq. (16-106) using the technique of Eq. (16-107) resulted in 16.73 deg, d«0.58 deg, ^=343.27 deg, whefe from Eq. (16-101), ¿«4,75 deg/sec, -3.69 deg/sec. For convenience the star sensor may be considered as mounted along the body x axis. Because the sensor field of view is small, the left-hand side of Eq. (16-105) is approximately the difference between the observed elevation in the nutating body, frame and the elevation in the (nonnutating) inertial frame. The amplitude of the plot is then approximately The approximate phase, may be obtained from the
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