53.20.000 56.MO.000 00.00.000 03.20.000

SS.00.000 SB. 20.000 01.HO.000 05.00.000

TIME IHH.MM.SS)

Fig. 9-4. Curve Fitting for GEOS-3 Attitude Data pitch, X^roll, y=yaw) Using a Third-Degree Chebyshev Polynomial geometry. The sensor data was validated but not otherwise preprocessed. Figure 9-5 illustrates a difficulty with preprocessed sensor data. In this case, Sun sensor and magnetometer data were preprocessed and the smoothed value used for deterministic attitude solutions. Note that the observed structure in the attitude data is artificial and the apparent high accuracy of the attitude solution is misleading.

A major problem with batch process curve fitting is the asymmetric treatment of the data and the difficulty of obtaining a satisfactory compromise between a polynomial of degree high enough to avoid truncation error and low enough to reduce random noise.

Filtering. With data filters, some of the problems of curve fitting can be avoided by first fixing the degree and then selecting a data interval about each measurement. This approach is often called a moving arc filter because it is sequentially centered oh the measurement to be smoothed and is therefore symmetric.

A useful filter is the least-squares quadratic filter, which has the following expressions for the smoothed function and its derivative [Budurka, 1967]:

53.20.000 56.MO.000 00.00.000 03.20.000

SS.00.000 SB. 20.000 01.HO.000 05.00.000

TIME IHH.MM.SS)

Fig. 9-4. Curve Fitting for GEOS-3 Attitude Data pitch, X^roll, y=yaw) Using a Third-Degree Chebyshev Polynomial

Fig. 9-5. GEOS-3 Attitude Solutions Using Smoothed Sun Sensor and Magnetometer Data. Solutions appear unrealistically accurate as discussed in the tract The resolution of the GEOS-3 attitude sensors is approximately 0.5 to 1 deg, and the only frequencies which should be observed in the data are related to the orbital period. The observed data span covers only 8% of an orbital period.

Fig. 9-5. GEOS-3 Attitude Solutions Using Smoothed Sun Sensor and Magnetometer Data. Solutions appear unrealistically accurate as discussed in the tract The resolution of the GEOS-3 attitude sensors is approximately 0.5 to 1 deg, and the only frequencies which should be observed in the data are related to the orbital period. The observed data span covers only 8% of an orbital period.

where the filter coefficients are

Ck = {P-Qyk+Ryl)/D |
(9-2 la) |

Dk = {-Q+Tyk-Syl)/D |
(9-2 lb) |

'-ljilji-(lti)' |
(9-2 lc) |

Q=Yyk*Zyt-YylYyl |
(9-2 Id) |

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