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Then, the solutions for A are given by

Equation (ll-3e) gives the two possible ambiguous attitude solutions. If the radicand in Eq. (11-3c) is negative, then no real solution exists; i.e., the cones do not intersect. Utility subroutine CONES8, described in Section 20.3, may also be used to solve for the intersection of two cones.

Figure 11-2 shows case (b), in which an arc-length measurement and a rotation angle measurement are combined to solve for the attitude. The arc-length measurement, say /3, constrains the attitude to lie on a small circle of radius /3 centered at S. This small circle is called the Sun cone, where the reference vector is the Sun vector. In addition, the rotation angle measurement, requires that the attitude, A. lie at the intersection of the great circles defined by (A,S) and (A,E), where S and E are known reference vectors. To solve for the attitude, we first solve for 17. Using the law of cosines for the sides of spherical triangle AES (Appendix A), it follows that cosxp = cosPcosi] + sin/3 sin 17 cos $ (11-4)

Shapes Sphere
Fig. 11-2. Single-Axis Attitude Solution Using an Arc-Length and a Rotation Angle Measurement, Case (b)

where ip is the arc length between E and S. Solving for i\ gives two possible solutions'.

cos ß cos \p±s\nß cos <&\sin2ß cos24>+cos2ß - cos\p

sin2/Jcos2<I>-l-cos2/i

Each arc length, i), defines a small circle about E. When E represents the nadir vector, this small circle is called the nadir cone. The evaluation of A now reduces to case (a) as the attitude is constrained to lie at the intersections of the small circles with radii equal to arc-length measurements P and ij. Because of the twofold ambiguity in i), a maximum of four possible solutions can be obtained for A. However, of the two possible attitudes computed for each value of ij, only one member of each pair will be consistent with the original rotation angle, O; hence, the fourfold ambiguity is reduced to a twofold ambiguity as in case (a).

Adjustments to the data are required for certain sensor types, before deterministic solutions can be computed. For visible-light Earth sensors, terminator crossings must be differentiated from horizon crossings and removed from further processing. The problem of identification of terminator crossings is described in Section 9.3. Attitude determination methods which use the angular radius of the Earth must obtain a value based on an oblate Earth. Methods for modeling the Earth's oblateness are described in Section 4.3. Both Earth oblateness and spacecraft orbital motion cause distortion in the observed Earth width. This may be corrected by constructing a fictitious Earth width—i.e., that which would have been observed if the spacecraft were stationary—as shown in Fig. 11-3. The two disks represent the position of the Earth at horizon-in and -out crossing times, l, and ta. Each disk has a radius (j>,,p0) equal to the angular Earth radius for the appropriate horizon crossing. Ey and E0 are the nadir vectors evaluated at t, and tQ. The open dots represent the horizon crossing events for an Earth fixed at its position at time t,, and the solid dots represent the horizon crossing events for an Earth fixed at its position at tQ. The observed Earth width, ft, is corrected to the

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