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A successful match occurs when the observation m vector is sufficiently close to the m vector of distances between four catalog stars. The disadvantages of the polygon match are that it requires more data than a pairwise distance match and that the computation time is longer. In the most efficient models, the computation time will increase approximately as TV2.

The angular separation match technique is difficult to analyze statistically; this makes the choice of e and /i more difficult than for the direct match method, e must be large enough to allow for the error in the initial attitude estimate plus the error caused by inaccuracies in the motion model. For SAS-3, e = 5 deg gave satisfactory results, ju need only be large enough to allow for the distortion caused by the motion model inaccuracy. It should be set to the maximum anticipated error in attitude at the end of the interval of analysis, assuming that the initial attitude was perfect. If ju is too large, ambiguous identifications and misidentifications will arise; if it is too small, no identification will be possible. The analyst must choose p. on the basis of the data accuracy and previous experience with the particular algorithm.

7.73 Phase Match Technique

The phase match technique computes a phase angle about a known spin axis by matching observation longitudes and catalog star azimuths about that spin axis. To use this technique, the frame of the sensor observations must be nearly undistorted; i.e.,

where «I» is the phase or azimuth difference and 6, is an error tolerance to allow for distortion. (For the HÉAO-1 attitude acquisition algorithm, 6, = 1 deg.) For the phase match technique to work, the spin axis of the spacecraft must be known to an accuracy substantially better than 5,. The phase about the spin axis need not be known, however.

To implement a phase match, compute the phases of all observations in an estimatedCC frame with an arbitrary zero phase. Next, extract from a star catalog all stars, S, meeting the requirement cos(0 + f + fi2) < Z • S < cos(0 -f-S2) (7-127)

where Z is the spin axis unit vector; 0 is the angle between the spin axis and the sensor optical axis; f is the radius of the sensor field of view; and S2 is the maximum anticipated error in the spin axis position. Compute the longitude of each catalog star as discussed in Section S.6. Divide the entire azimuth circle (0-to 360 deg) into bins of equal width, 8, such that

where the second term allows for errors in catalog star longitudes caused by errors in the spin axis position. The score, R, is given by

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