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EARTH,

(c) ATTITUDE LIES AT POINT A

Fig. 14-1. Nadir Angle/Nadir Angle Correlation for Data Passes of Different Lengths. Subscripts 1,2, and 3 denote the positions of the Earth and nadir cone at times (,, »j, and l3. See text for explanation.

estimator it is usually assumed that the bias is completely unknown and can take on any value.* It is in this sense that a single measurement with a bias of unknown magnitude provides no attitude information.

Fig. 14- 1(b) illustrates two more widely spaced points corresponding to the ends of a data pass of intermediate length, or to a data pass for which the central data is unavailable. Although we do not know the value of the nadir angle bias, We assume that it is a constant bias for both measurements. Therefore, the attitude must lie along the dotted line. Thus, the horizontal component of the attitude is well determined, but both the vertical attitude component and the nadir angle bias are not determined.

Finally, Fig. 14-l(c) illustrates the information available from a full data pass with a large correlation angle between the measurement at the beginning and at the end. If we assume that the attitude is fixed, then all of the measurements must give the same result if the nadir angle bias has been correctly determined. Because the three solid curves do not intersect in one point, there must be a nadir angle bias. Because the dashed lines do intersect in a point, the attitude must be at that intersection and the nadir angle bias must be equal to Atj. In this case we have used a single measurement type to determine both the attitude and the magnitude of the type II bias associated with that measurement type.

As an example of the above analysis, consider a system similar to the examples of Chapter 11 (such as CTS, SMS/GOES, or SIRIO) consisting of a spinning spacecraft with Sun angle, Sun-to-nadir rotation angle, and nadir angle measurements and possible biases in all three measurements. If data is obtained over a period of less than a day, then the inertial position of the Sun remains essentially fixed. Therefore, if there is a possible Sun angle bias, the Sun angle measurement indicates that the attitude lies on a cone of unknown radius centered on the Sun; that is, there is no information in the Sun angle measurement. Adding Sun angle data to a state estimator and including a Sun angle bias in the state vector solved for will affect neither the attitude results nor the values of any of the other state vector elements. Of course, if the attitude is determined from other data, then the Sun angle measurement provides a measure of the Sun angle bias.

To determine the content of the Sun-to-nadir rotation angle measurement, it is convenient to find a general procedure for determining the correlation angle between a measurement at the beginning of a data pass and that same measurement at some other time during the data pass. As shown in Fig. 14-2, the attitude locus, Lp, for a given Sun angle measurement, )8, remains nearly fixed on the celestial sphere as the spacecraft moves in its orbit. Therefore, the correlation angle, ®m ,mj, between one measurement, m, at any two positions in the orbit is just the difference between the fi/m correlation angles at these two positions. (See Section 10.4 for a discussion of correlation angles.) For example, for m = 4», where \$ is the Sun-Earth rotation angle:

0*/4>(time 1 to time 2) = 0^i/i<tj = 9^/<t(time 2)-©^(time =

•Assuming that the bias is completely unknown is equivalent to setting the state weight matrix, Sa in Section 13.4 to zero. It S0 is nonzero, then a penalty is assigned to deviations of the bias from its nominal value, and a single measurement with a possible bias does constrain the attitude solution. angle, /}, and a given measurement value, m. So long as the time interval t2-l, is short enough so that the Sun remains essentially fixed, then for any measurement m.

The rotation angle correlation angle, Q^o, can be determined from Fig. 14-3, which shows the 0/j/® correlation angle curves at 2-deg intervals over the entire sky for fixed positions of the Sun and attitude and variable positions of the Earth. For example, if the center of the disk of the Earth is at B, then @n/i> (evaluated at the Fig. 14-3. Sun Angle/Rotation Angle Correlation Angle Curves at 2-Deg Intervals for a Sun Angle of Approximately 65 Deg. See text for explanation. The orbit and Earth envelope illustrated are for the CTS transfer orbit At 10-deg intervals, the correlation angle curves are solid lines. The lines at ±2 deg have been omitted to identify the 0 curve. Because the only independent parameter m generating these curves is the Sun angle, p, they may be used for any spacecraft for which /3s»65 deg as shown.

Fig. 14-3. Sun Angle/Rotation Angle Correlation Angle Curves at 2-Deg Intervals for a Sun Angle of Approximately 65 Deg. See text for explanation. The orbit and Earth envelope illustrated are for the CTS transfer orbit At 10-deg intervals, the correlation angle curves are solid lines. The lines at ±2 deg have been omitted to identify the 0 curve. Because the only independent parameter m generating these curves is the Sun angle, p, they may be used for any spacecraft for which /3s»65 deg as shown.

attitude) equals zero. Similarly, if the Earth is at C or D, then 0^ = 268°; if the Earth is at E or F, then ®p/tt> = 272". Figure 14-3 also shows the approximate geometry of the CTS transfer orbit to synchronous altitude. As in Figs. 11-25 and 11-26, the line with vertical tick marks denotes the Earth's orbit about the spacecraft (as seen by the spacecraft), asterisks mark the envelope of the Earth's disk, AP marks the location of apogee, and 1 and 1' mark the interval over which horizon sensor 1 senses the Earth. Thus, horizon sensor 1 picks up the Earth at 1 where 0^ = 268°; 0^/<t, then increases to a maximum of approximately 271° just before apogee and drops to about 266° as sensor 1 loses the Earth at I'\ Therefore, the maximum variation in &p/<t is about 5 deg. From Eq. (14-2), this implies that ©.¡./.j, has a maximum value of about 5 deg. Thus, the rotation angle correlation angle for the CTS geometry is small and there is very little information content in the rotation angle measurement if the possibility of an unknown bias in the measurement is considered.

The above conclusion about minimal information in the CTS rotation angle measurement is generally applicable under certain common conditions. Note that in the vicinity of the spin plane in Fig. 14-3 (between the lines at nadir angles, t), of 85 deg and 95 deg), is approximately 270 deg and is insensitive to the rotation angle, <!>.* Physically this means that if the attitude is near orbit normal, then as the spacecraft moves through an entire orbit, the rotation angle, 4>, goes from 0 deg to 360 deg, but the loci of possible attitudes remains nearly the same for the various positions of the spacecraft in its orbit. Although the measurement is changing through its full range, the information content as to the possible locations of the attitude is nearly the same for all of these measurements. Therefore, whenever the nadir angle remains near 90 deg for an entire pass (i.e., if either the attitude is at orbit normal or the Earth is small and the sensor is mounted near the spin plane as is the case for CTS) and there is the possibility of a rotation angle bias, then there is very little information in the rotation angle measurement.

For the nadir angle measurement, the situation is the opposite of the rotation angle measurement. For an attitude near orbit normal, the measured value of the nadir angle remains approximately fixed, but the corresponding attitude loci rotate through 360 deg as the spacecraft goes around a full orbit. Therefore, the nadir angle measurement contains sufficient information to determine both the attitude and a constant nadir angle bias as illustrated previously in Fig. 14-1. (The nadir angle bias may be a composite of biases in the sensor mounting angle, the angular radius of the Earth, or other parameters.)

In summary, we may determine the information content of any type of measurement in which there may be a constant bias by examining the changing orientation of attitude loci for that measurement. It is the attitude loci, not the reference vector or measurement values, that is important. If there is no rotation of the attitude loci (e.g., the Sun angle measurement) and if a bias in the measurement is solved for, then there is no information about the attitude in that measurement. Conversely, if there is a large rotation of the attitude loci (e.g., nadir angle or Earth-width measurements over a full orbit with the attitude near orbit normal),

*Thc same conclusion can be obtained from Eq. (11-32) or Fig. 11-18.

then that information may be used to solve for both the attitude and the magnitude of the constant type II bias in that measurement These conclusions on the information content of the /?, and ij measurements have been verified on both real and simulated data for the GOES-1 and CTS missions [Tandon, et al., 1976].

4. Geometry of Individual Biases. The procedure described in the preceding paragraphs is only applicable to type II biases. The general procedure for type III biases is to find a region in which the data are very sensitive to the bias in question. Particularly good regions to test in this regard are those where the effect of the bias on the data changes sign or reaches an extremum or where the measurement density is low.

As an illustration of sensitive regions for particular biases, consider the case of a negative bias on the angular radius of the Earth, as illustrated in Fig. 14-4. The solid line is the nominal Earth disk and the dotted line is the sensed or biased Earth disk. If there is a bias, then as the sensor scan moves downward across the disk of the Earth, a measured Earth width corresponding to a scan at A will imply that the scan was crossing at A' where the Earth width for a nominal Earth disk would be the same size as for thé biased disk at A. Thus, the computed nadir angle would be significantly larger than the real nadir angle. Similarly, a real scan at B will imply that the scan crossed the nominal Earth at B' and the computed nadir angle will be significantly smaller than the real nadir angle. Thus, as the sensor scans across the diameter of the Earth going from A to B, there will be a large discontinuity in the computed nadir angles if there is an unresolved bias on the angular radius of the Earth. Making the computed attitudes agree (even if the value of the attitude is not particularly well known) as a horizon sensor sweeps across the diameter of the Earth provides a very sensitive measure of the bias on the angular radius of the Earth. This procedure was used on the CTS mission to determine the Earth radius bias to about 0.02 deg on a very short span of data taken as the horizon sensor scan crossed the diameter of the Earth. Fig. 14-4. Sensitivity to Bias on the Angular Radius of the Earth. Earth width is the same on the nominal Earth disk at A' and B' as it is on the sensed Earth disk at A and B.

Although an analysis of this type is necessary for type III biases, such as a bias on the angular radius of the Earth or an orbital in-track error, it may be used for other biases as well. For example, a bias in the mounting angle of an Earth horizon sensor causes a shift in opposite directions on opposite sides of the orbit. Thus, two data passes on opposite sides of the orbit with the spacecraft at a constant attitude were used to successfully determine the sensor mounting angle bias for sensors on the CTS and GOES-I spacecraft [Tandon and Smith, 1976]. A similar procedure was used for the Panoramic Attitude Scanner on RAE-2 [Werking,e/ a/., 1974] and magnetometer data on the SAS-1 mission [Meyers, et al., 1971].

References

1. Branchflower, G. B., et al., Solar Maximum Mission (SMM) Systems Definition Study Report, NASA GSFC, Nov. 1974.

2. Camahan, Brice, H. A. Luther, and James O. Wilkes, Applied Numerical Methods. New York: John Wiley & Sons, Inc., 1969.

3. Chen, L. C. and J. R. Wertz, Analysis of SMS-2 Attitude Sensor Behavior Including OABIAS Results, Comp. Sc. Corp., CSC/TM-75/6003, April 1975.

4. Freund, John E., Mathematical Statistics, >Englewood Cliffs, NJ: Prentice-Hall, Inc., 1962.

5. Hotovy, S. G., M. G. Grell, and G. M. Lerner, Evaluation of the Small Astronomy Satellite-3 (SAS-3) Scanwheel Attitude Determination Performance, Comp. Sc. Corp., CSC/TR-76/6012, July 1976.

6. Levitas, M., M. K. Baker, R. Collier, and Y. S. Hoh, MAPS/MAGSAT Attitude System Functional Specifications and Requirements, Comp. Sc. Corp., CSC/SD/78-6077, June 1978.

7. Meyers, G. F., M. E. Plett, and D. E. Riggs, SAS-2 Attitude Data Analysis, NASA X-542-71-363, GSFC Aug. 1971.

8. Tandon, G. K., M. Joseph, J. Oehlert, G. Page, M. Shear, P. M. Smith, and J. R. Wertz, Communications Technology Satellite (CTS) Attitude Analysis and Support Plan, Comp. Sc. Corp., CSC/TM-76/6001, Feb. 1976.

9. Tandon, G. K. and P. M. Smith, Communications Technology Satellite (CTS) Post Launch Report, Comp. Sc. Corp., CSC/TM-76/6104, May 1976.

10. Werking, R. D., R. Berg, K. Brokke, T. Hattox, G. Lerner, D. Stewart, and R. Williams, Radio Astronomy Explorer-B Postlaunch Attitude Operations Analysis, NASA X-581-74-227, GSFC, July 1974.

11. Wertz, J. R., C. E. Gartrell, K. S. Liu, and M. E. Plett, Horizon Sensor Behavior of the Atmospheric Explorer-C Spacecraft, Comp. Sc. Corp., CSC/TM-75/6004, May 1975.

12. Werte, James R. and Lily C. Chen, "Geometrical Limitations on Attitude Determination for Spinning Spacecraft," J. Spacecraft, Vol. 13, p. 564-571, 1976.