Let p be the angular radius of the Earth and y be the sensor mounting angle; then, from Fig. 11-14, cosp"cosrcosi) + sinrsini)cosy (11-39)
By differentiating Eq. (11-39) and substituting it into Eq. (11-38), we get i5gI^-cot,cot(0/2) (1.-40)
Thus, can change rapidly during a data pass. Specifically, do = 0 when cot? = cot7jcosQ/2 or LAEH,= LAEHa=90°. This means that when Earth-width measurements are used for attitude determination, a singularity occurs when the sensor scans near the middle of the disk of the Earth.
The geometry for the rotation angle density, shown in Fig. 11-15, is considerably more complex than for arc-length densities. In the figure, the attitude changes from A to A' along the direction perpendicular to due to an infinitesimal change in rotation angle from 4> to <t>+A4>. To obtain the arc length s', let B be the intersection of Lq+m with the extension of EA. Then AB is Atj along the constant A direction due to the change A4>; that is, y4i=A»j|A. By definition, the angle BAA'°> Therefore,
The rotation angle density. is
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