A5l

From the spherical triangle AES, we have cot^sin/S=cos^cos2 + cot«I>sin2 (11-30)

By differentiating Eq. (11-50) and expressing ^ and 2 in terms of fi, ij and it can be shown that [Wertz and Chen, 1976]

This can be reformulated in terms of A and ij as

Again, by symmetry between the Sun and tin Earth, the correlation angle between ij and 4 measurements can be written in the same form:

11.4 Geometrical Limitations on Single-Axis Attitude Accuracy

Lily C. Chen James R. Wertz

In Section 11.3 we described how to determine the attitude accuracy for given geometrical conditions. However, for most aspects of mission planning—such as hardware configuration studies, maneuver and attitude planning, contingency analysis, or launch window analysis—the inverse problem is more relevant Instead of determining the attitude uncertainty for given conditions, we wish to select the geometrical conditions such that the required attitude accuracy can be achieved. Thus, we would like to understand the effect of any change in the .mission conditions on the attitude uncertainties.

In this section, we present a graphical method to study the geometrical limitations on attitude accuracy by applying the equations derived in Section 11.3. With this method, we obtain an overview of the attitude determination geometry and an insight into the effect of changes in mission parameters. Specifically, the equations of Section 11.3 will be used to identify "poor" geometry regions on the celestial sphere for either the attitude or one of the two reference vectors [Chen and Wertz, 1977],

Two cases are considered. In Section 11.4.1, the two reference vectors are assumed fixed and the attitude direction is treated as a variable. The poor geometry regions on the celestial sphere are defined such that whenever the attitude is inside one of these regions, one or more of the attitude determination methods of Section 11.1 will not provide the required attitude accuracy. In Section 11.4.2, the attitude and one of the two reference vectors are assumed fixed and the other reference vector is treated as the variable. In this case, the poor geometry regions on the celestial sphere are defined such that whenever the variable reference vector falls inside one of these regions, the attitude uncertainty evaluated at the attitude will be high for one or more of the attitude determination methods. Examples of the application of this geometrical study to mission support activities are given in Section 11.4.3. Again, throughout this section, we use for convenience the Sun and the Earth as the two reference vectors. However, the discussion and conclusions can be applied to any pair of known reference vectors. The notation defined in Figs. 2-1 or 11-18 is used throughout.

11.4.1 Limitations on the Attitude Direction Due to Attitude Accuracy Requirements

We wish to determine the regions of single-axis attitude directions on the celestial sphere which give poor attitude accuracy for fixed positions of the Sun and the Earth. As introduced in Sections 10.1 and 11.3, two geometrical factors limit the attitude accuracy: the correlation angle, ©m/„, and the measurement densities, dm and d„. From Eq. (11-36), the attitude uncertainty becomes infinite whenever @m/„ = 0 or 180 deg, or either dm or dn equals zero, That is, poor geometry regions occur when either the correlation between the two measurements is high or the measurement density is low.

Regions of High Correlation. Regions of poor geometry due to high correlations can be defined for each of the three attitude determination methods: /?/tj, /?/<!>, and tj/4>. Although specific attitude accuracy limits are mission dependent, we define a region of "poor geometry" as any region in which the attitude Fig. 11-17. Poor Geometry Regions for the Location of Attitude

| LOW lROTATION r ANQLE I DENSITY

Fig. 11-17. Poor Geometry Regions for the Location of Attitude uncertainty given in Eq. (11-36) is more than five times greater than the measurement uncertainty, assuming Um= U„, Cm = C„ = 0, and dm = d„= 1. From Eq. (1136), this corresponds to a region in which is in the range 0C±11.5C or 180° ±11.5°.

For the /?/tj method, the analysis is simple because 00/,, = <I>, as given in Table 11-2. Thus, the constant correlation angle curves for the p/t\ method are the same as the constant Sun-to-Earth rotation angle curves given in Section 10.3. From Fig. 10*}, it is obvious that the singularity occurs when the attitude lies along the Sun/Earth great circle where 9^ = 0 or 180 deg and the poor geometry regions due to high >3/tj correlation must be regions around this great circle bounded by constant rotation angle curves. This poor geometry region for a Sun/Earth separation of 30 deg is shown as the shaded region labeled "¿8/l correlation" in Fig. 11-17 (preceding page).

Fig. 11-18. Relations Among Attitude (/I). Sun (5), Earth (£), Null (N), and Correlation Angles

For the /?/«& and tj/4» methods, the interpretation of the expressions for and ®v/® from Table 11-2 is more difficult. However, this interpretation may fee simplified by using the Null, N, or Sun-Earth cross product, as shown in Fig. 11-18. Applying Napier's rule to the spherical triangles EAN and SAN, and comparing the results with Èq. (11-52) and (11-54), the rotation angle NAE equals the correlation angle 0^/« and the rotation angle NAS equals the correlation angle 0,/®. From Fig. 11-18, it is clear that the constant 0^/4> and curves are the constant rotation angle curves between the Earth and the nu'.l and between the Sun and the null, respectively. Because the Earth/null and the Sun/null separations are always 90 deg, the rotation angle curves given in Fig. 10-10 can be used to obtain the regions of high /?/4> or 7)/<t> correlation. The poor geometry regions for which ®p/4> and 0,/4> lie within 11.5 deg of 0 deg or 180 deg are shown as the shaded regions labeled "¿8/4» correlation" and "tj/<& correlation," respectively, in Fig. 11-17. Note that the centers of these regions are the Earth/null and the Sun/null great circles, respectively. Fig. 11-18. Relations Among Attitude (/I). Sun (5), Earth (£), Null (N), and Correlation Angles

Regions of Low Measurement Density. Poor geometry regions due to a low measurement density (d) occur only for the rotation angle measurements. As discussed in Sections 10.3 a fid 11.3, the rotation angle density goes to zero, i.e., the attitude uncertainty goes to infinity, when the attitude approaches the null or the antinull. Therefore, poor geometry regions for the attitude due to low rotation angle densities are regions around the null or the antinull bounded by the constant rotation angle density curves. These curves can be obtained by using Eq. (11-46) to obtain a quadratic equation in (sin2|) in terms of Z, i/>, and d: '

[ d2 cos2Z cos2( \$ - Z) ]sin4£ - {d2 [ cos2Z + cos2(^ - Z) ] + sinfy} sin2|+ d2=0

Note that Z is defined in Figs. 11-11 and 11-18. The result in Eq. (11-55) for i/> = 30 deg and d=0.2 is shown as the unshaded region about the null in Fig. 11-17. That is, whenever the attitude lies inside this region, the attitude component Fig. 11-19. Evolution of the Shape of the Low Rotation Angle Density Region for Varying Separation Between the Reference Vectors. Each subfigure is centered on the null.

Fig. 11-19. Evolution of the Shape of the Low Rotation Angle Density Region for Varying Separation Between the Reference Vectors. Each subfigure is centered on the null.

determined from the rotation angle measurements will have an uncertainty at least five times greater than the rotation angle measurement uncertainty. The evolution of the shape of this region for varying Sun/Earth separations is shown in Fig. 11-19 (preceding page).

Combination of High Correlation and Low Density. In Fig. 11-17, only the P/tj subfigure gives the poor geometry region directly because the measurement densities are unity. The /i/<J> and tj/4» regions must be obtained by combining the high-correlation effect with the low-density effect. This can be done numerically using Eq. (11-36) and results are shown in Fig. 11-20. This figure shows the poor geometry regions for the attitude such that within the shaded regions, the attitude uncertainty will be five times greater than the measurement uncertainties for the /?/<!> or ij/4» method, assuming equal uncertainties in the two measurements. Fig. 11-20. Poor Geometry Regions for the Attitude for the p/<S> and ij/O Methods. When the attitude lies inside the shaded region, the attitude uncertainty will be more than five times the measurement uncertainty, based on equal uncertainty in the two measurements. (Compare with Fig. 11-17.)

11.4.2 Limitations on Reference Vector Direction

For many mission support activities, the attitude direction is predetermined while one of the two reference vector directions either remains to be determined or is moving, as in the case of a satellite with an inertially fixed attitude moving around the Earth which is being used as one of the reference vectors. In this case, we wish to obtain the poor geometry regions for the variable reference vectors such that whenever this vector is located inside these regions, one or more of the attitude determination methods will result in poor attitude accuracy.

For convenience, the Earth will be used as the varying reference vector. However, due to the symmetry between the Sun and the Earth, the results can be equally applied if the Sun position is treated as the variable instead, as will be shown below for the launch window analysis. I

Regions of High Correlation. As in Section 11.4.1, poor geometry regions can be defined for each of the three attitude determination methods. For the /}/tj method, Eq. (11-48) can be used directly. Because 6^,, = \$, the poor geometry region lies between two great circles which intersect at the attitude at an angle on either side of the Sun-attitude great circle, as shown in Fig. 11-21 for &p/v = -11.5 deg.

For the /9/\$ correlation, the poor geometry regions can be obtained from Eqs. (11-51) and (11-52). From Eq. (11-52), A must be a right angle when ©^=0 or 180 deg. That is, a singularity occurs when the Earth lies .on the 90-deg or 270-deg constant rotation angle curve between the Sun and the attitude. This is equivalent to the attitude lying on the Earth/null great circle, as shown in Fig. 11-17. The boundaries of this poor geometry region may be obtained by reformulating Eq. (11-51) into an expression for 77 in terms of P and as shown in Fig. 11-21 for ©£/<■>= ±11.5 deg about 0 deg or 180 deg. The evolution of the shape of this region as a function of ft is shown in Fig. 11-22. As seen most clearly in Fig. 11-22(c) and (d), this region is not symmetric under an interchange of the Sun and the attitude. Except for a Sun angle of 90 deg, the /3/\$ correlation region consists of two unconnected areas, one near the attitude and the other near the antiattitude.

For the r\/<b correlation, Eq. (11-54) may be used directly. When 0^/4>=O or 180 deg, 2=90 deg or 270 deg and the Earth lies on the great circle through the Sun perpendicular to the Sun-attitude great circle. The poor geometry region Fig. 11-21. Poor Geometry Regions for the (Earth) Reference Vector Due to High Correlations or Low Rotation Angle Densities. (Contrast with Fig. 11-17 showing poor geometry regions for the attitude with fixed Earth/Sun positions.)

around this great circle is bounded by two great circles intersecting at the Sun and the antisolar point and making a constant angle with the 2=90 deg or 270 deg great circle. The shaded area in Fig. 11-21 labeled "i}/\$" shows this region for ©,/(»=±11.5 deg.

Regions of Law Measurement Density. Finally, in addition to the poor geometry regions due to measurement correlations, Eqs. (11-44) and (11-46) can be rx ? -v \ -I ! UL" s or jons and deg i for poor in be (c) 0 - 80° Id) (3-80°

Fig. 11-22. Evolution of the Shape of Poor Geometry Regions for the Earth Due to /3/<6 Correlation used to obtain the poor geometry regions due to low rotation angle density. Equation (11-46) shows that the rotation angle density approaches zero whenever if/ or £ is near 0 or 180 deg. £=0 deg or 180 deg implies that the attitude is at the null or the antinull and both /3 and -q are equal to 90 deg. For a given /? other than 90 deg, the poor geometry region due to low rotation angle density depends strongly on \p> the rotation angle density becomes low when the Earth is close to the Sun or the antisolar point.

We may determine two regions around the Sun and the antisolar point, such that if the Earth lies inside either region, the rotation angle density, evaluated at the attitude, will be less than a specified value. The boundary of this region can be obtained from Eq. (11-44) by substituting \$ in terms of 2, \f>, and /? and reformulating the equation to yield

cosj8cosZ±

This region around the Sun for /? = 30 deg and d= 0.2 is shown in Fig. 11-21 and the evolution of the shape of this region for varying ft is shown in Fig. 11-23. Note that when /?=90 deg, die low rotation angle density regions become a single continuous band bounded by small circles of fixed nadir angle such that sinS,= _!— (11-57) (a) 0 = 30P (b) 0 = 60° Fig. 11-23. Evolution of the Shape of the Low Rotation Angle Density Region for Positions of the Earth. (The Sun is at the center of each plot.) When the Earth lies inside the darkly shaded region, the rotation angle density, d, at the attitude is less than 0.1. The lightly shaded and unshaded regions are for 4=0.2 and 0.5, respectively.

Fig. 11-23. Evolution of the Shape of the Low Rotation Angle Density Region for Positions of the Earth. (The Sun is at the center of each plot.) When the Earth lies inside the darkly shaded region, the rotation angle density, d, at the attitude is less than 0.1. The lightly shaded and unshaded regions are for 4=0.2 and 0.5, respectively.

Combination of High Correlation and Low Density. Similar to the discussion in Section 11.4.1, among the four regions in Fig. 11-21, only the /tj region provides the poor geometry area directly. For the /?/<& and tj/<& methods, the correlation regions must be combined with the low rotation angle density region to obtain the regions corresponding to a factor of five between the attitude uncertainty and the measurement uncertainties. This can be done by substituting Eqs. (11-45) and (11-52) or Eqs. (11-44) and (11-54) into Eq. (11-36) and expressing tj in terms of /?, and / (the ratio between attitude uncertainty and the measurement uncertainty). The results for 0 = 30 deg and f=5 are shown in Fig. 11-24. Fig. 11-24. Poor Geometry Regions for the Earth for and tj/O Methods. When the Earth is in the shaded region, the attitude uncertainty will be more than five times the measurement uncertainty, based on equal uncertainty in the two measurements. (Compare with Fig. 11-21.)

11.4.3 Applications

The geometrical study of the limitations on attitude accuracy described in this section has been applied in both prelaunch and postlaunch analysis for SMS-2; GOES-1, -2, and -3; AE-4, and -5; CTS; and SIRIO and in prelaunch analysis for ISEE-C and IUE [Wertz and Chen, 1975, 1976; Chen and Wertz, 1975; Tandon, et al., 1976; Chen, et ai, 1976, 1977; Chen, 1976; Lerner and Wertz, 1976; Rowe, et ai, 1978]. To illustrate the procedure, we will discuss the attitude determination accuracy for SMS-2 and the attitude launch window constraints for SIRIO. The profile for both missions is similar to that of CTS, as described in Section I.I. An alternative formulation is given by Fang ,

SMS-2 Attitude Determination. The Synchronous Meteorological Satellite, SMS-2, was launched into an elliptical transfer orbit on February 6, 1975. Shortly after launch, the attitude wets maneuvered to that appropriate for Apogee Motor Firing (AMF). On the second apogee, the AMF put the spacecraft into a circular near-synchronous drift orbit over the equator. Over the next 3 days, the attitude was maneuvered to orbit normal with two intermediate attitudes. The data collected in both the transfer and drift orbits allowed the measurement of 20 attitude bias parameters on five Earth horizon sensors and one Sun sensor. The geometrical methods described here were used extensively in the analysis of SMS-2 attitude and bias determination and contributed substantially to the result obtained [Chen and Wertz, 1975; Wertz and Chen, 1975, 1976.]

As the spacecraft moves in its orbit, the attitude determination geometry changes due to the motion of the position of the Earth (as seen by the spacecraft) relative to the Sun and the attitude. A convenient vehicle for examining this changing geometry is a plot of the celestial sphere as seen by the spacecraft, with the directions of the Sun and the attitude fixed. Figures 11-25 and 11-26 show examples of such plots* for the nominal transfer orbit and apogee motor firing attitude near apogee. The region around perigee was of less interest because the spacecraft was then out of contact with the Earth.

As usual, the spacecraft is at the center of the sphere. The heavy solid line is the orbit of the Earth around the spacecraft as seen from the spacecraft. The Earth is moving toward increasing right ascension, i.e., from left to right on the plots. Tic marks denote the time from apogee in 10-minute intervals. The dotted line surrounding the orbit denotes the envelope of the disk of the Earth as it moves across the sky. AP marks the location of the Earth when the spacecraft is at the apogee. S "A, and A ~1 mark the location of the antisolar point, attitude, and negative attitude axis, respectively.

The small solid circles labeled ES1 and ES 4 and centered on the A/ A~1 axis are the fields of view (FOV) for two of the five SMS-2 Earth horizon sensors as the spacecraft spins about the AIA~l axis. Arrowheads on the FOV lines indicate the direction in which the sensors scan the sky. Acquisition of signal (AOS) and loss of signal (LOS) of the Earth by each sensor are marked by arrowheads along the orbit with primed numbers for LOS and unprimed numbers for AOS.

The three dashed curves in Fig. 11-25 are the central lines of the poor geometry regions for the position of the Earth due to strong correlations, and the fM Fig. 11-25. SMS-2 Attitude Determination Geometry for the Transfer Orbit and Apogee Motor Firing Attitude. See text for explanation.

* Since their initial use for the SMS-2 mission in 1975, global plots of the sky as seen by the spacecraft, such as Fig. 11-25, have been used by the authors for each of the missions they have supported. These plots have been very convenient for examining sensor fields of view and optimum sensor placement, Sun and Earth coverage, attitude uncertainties, the relative geometry of reference vectors, and other aspects of mission analysis. See Section 203 for a description of the subroutines used to generate these plots. With practice, they may also be drawn quickly by hand using the blank grids and methods given in Appendix B. Fig. 11-26. Poor Geometry Regions for the Earth for the Geometry in Fig. 11-25. (Relative to Fig.

11-25, the center of the plot has been shifted down to the celestial equator and to the right of the antisolar point) Here the poor geometry regions are bounded by a correlation angle of 23 deg, whereas Fig. 11-21 shows the same regions bounded by correlation angles of 11.5 deg.

Fig. 11-26. Poor Geometry Regions for the Earth for the Geometry in Fig. 11-25. (Relative to Fig.

11-25, the center of the plot has been shifted down to the celestial equator and to the right of the antisolar point) Here the poor geometry regions are bounded by a correlation angle of 23 deg, whereas Fig. 11-21 shows the same regions bounded by correlation angles of 11.5 deg.

four shaded areas shown in Fig. 11-26 are the poor geometry regions for the Earth analogous to those shown in Fig. 11-21. Thus, whenever the Earth moves inside one of these regions, one or more of the attitude determination methods will give poor results. By comparing the Earth coverage regions (from AOS to LOS) for each of the five sensors with the poor geometry regions for the Earth, we can easily choose the preferred attitude determination method for each of the five data passes. For example, ES1 sees the Earth in a region of poor geometry due to high correlation between the Sun angle and the nadir angle measurements. Therefore, the attitude determined by the {if ij method would yield high uncertainties and the other two methods should be used instead. Similar results can be obtained for the data passes from other sensors. None of the data passes falls inside the low rotation angle density region. Therefore, attitude uncertainties due to low rotation angle density were not a problem during the SMS-2 transfer orbit.

S1RIO Launch Window Constraints. SIRIO is an Italian satellite launched in August 1977, which uses the Sun angle data and the IR Earth sensor data to determine the spinning spacecraft attitude, similar to SMS and CTS. We briefly describe analysis of the attitude determination constraints on the SIRIO launch window. (For additional details, see Chen  and Chen, et a!., .) The purpose of this analysis is to obtain the launch window (in terms of right ascension of the orbit's ascending node versus the launch date) which will give the required attitude accuracy.

Figure 11-27(a) shows the nominal geometry for SIRIO attitude determination in the transfer orbit, in a plot analogous to Fig. 11-25. The position of the antisolar point is plotted for a January 15 launch. As the launch date changes, so does the Sun position and attitude determination geometry. Thus, determining the launch window constraints is equivalent to determining the constraints on the position of the Sun to obtain good attitude determination geometry. Thus, instead of considering the attitude and the Sun to be fixed, as in the previous example, we consider the attitude and the Earth as fixed and treat the Sun position as a variable.

The position of the Earth can be determined by the sensor coverage. Attitude determination is most important before AMF. Therefore, we require that the attitude be determined to within the specified attitude accuracy from a data pass (a) NOMINAL GEOMETRY (b) (J/tj METHOD Fig. 11-27. SIRIO Attitude Determination Geometry in the Transfer Orbit Shaded areas give Sun locations for which the attitude determination geometry is poor. See text for explanation. (Compare with Figs. 11-21 and 11-24.)

Fig. 11-27. SIRIO Attitude Determination Geometry in the Transfer Orbit Shaded areas give Sun locations for which the attitude determination geometry is poor. See text for explanation. (Compare with Figs. 11-21 and 11-24.)

covering the time period of 150 minutes before AMF to 30 minutes before AMF; this is the darkened region between / and F along the orbit in Figs. 11-27(b), (c), and (d).

The method described in Section 11.4.2 can be used to obtain the poor geometry regions for the position of the Sun using the three attitude determination methods, having the Earth located at / and F, respectively. Figures 11-27(b), (c), and (d) show these poor geometry regions for P/tf, fi/Q and 17/\$ methods, respectively (compare with Figs. 11-21 and 11-24). In each figure, two regions are plotted, corresponding to Earth positions at I and F. Thus, the overlapping regions in Figs. 11-27(b) to (d) give the positions of the Sun (or the antisolar point) such that for all locations of the Earth between I and F, the attitude determined by that particular method will not give the required accuracy.

The poor geometry regions shown in Figs. 11-27(b) through (d) provide the constraints on the position of the Sun relative to the ascending node. As the launch date changes, the Sun position changes, and the ascending node and the attitude are rotated to maintain the relative positions of the Sun and the node. Therefore, the Sun constraints can be transformed into constraints on the right ascension of the ascending node versus launch date, as desired. Figures 11-28(a) through (c) show such results for the three attitude determination methods for a full year, and Fig. 11-28(d) gives the constraints on the launch window where none of the three attitude determination methods would give required attitude accuracy. o 30 60 so 130 160 180 310 240 2)0 (el METHOD

days prom january 1

0 30 60 60 iso 1b0 180 310 240 770 300 (d) FINAL RESULTS WITH ALL METHODS days from january 1

Fig. 11-28. SIRIO Attitude Constraints on- the Launch Window for AMF Attitude Determination Accuracy. (See text for explanation.)

o 30 60 so 130 160 180 310 240 2)0 (el METHOD

days prom january 1

0 30 60 60 iso 1b0 180 310 240 770 300 (d) FINAL RESULTS WITH ALL METHODS days from january 1

Fig. 11-28. SIRIO Attitude Constraints on- the Launch Window for AMF Attitude Determination Accuracy. (See text for explanation.)

Further Applications. Both of the examples discussed above show the applications of the poor geometry regions for the reference vectors as discussed in Section 11.4.2. However, the poor geometry regions for the attitude discussed in Section 11.4.1 may also be used in mission planning activities, especially in maneuver planning. For example, if we plot an attitude maneuver on the geometry plots given in Fig. 11-17 or 11-20, it is clear where along the maneuver the attitude can be best determined using any attitude determination method. Thus, we can stop the maneuver at the appropriate position for attitude and bias determination. Alternatively, we can change the route of a maneuver to provide attitude accuracy for maneuver monitoring, or we can plan an attitude maneuver purely for the purpose of attitude and sensor bias determination. Activities of this type were used successfully on AE-4 and -5 to evaluate attitude sensor biases. Similar analyses have been performed to provide optimal Sun sensor configurations for SEASAT [Lerner and Wertz, 1976] and to examine Earth and Moon coverages as ISEE-C transfers to the Sun-Earth libration point approximately 6 1unar orbit radii from the Earth [Rowe, et al„ 1978].

11.5 Attitude Uncertainty Due to Systematic Errors

Lily C. Chen James R. Wertz

The causes of single-frame attitude uncertainty may be separated into the two categories of random and systematic errors. A random error is an indefiniteness of the result due to the finite precision of the experiment, or a measure of the fluctuation in the result after repeated experimentation. A systematic error is a reproducible inaccuracy introduced by faulty equipment, calibration, or technique. The attitude uncertainty due to random errors can be reduced by repeated measurement. When a measurement is repeated n times, the mean value of that measurement will have an uncertainty Vn times smaller than the uncertainty of each individual measurement. However, this statistical reduction does not apply to systematic errors. Therefore, the attitude uncertainty due to systematic errors is usually much larger than that due to random errors.* Therefore, to reduce the attitude uncertainty, we must identify and measure as many as possible of the systematic errors present in each of the attitude measurements. In this section, we compare the behavior of the single-frame attitude solutions with and without systematic errors, discuss the singularity conditions for various attitude determination methods, and introduce the concept of data filters and state estimation to solve for the systematic errors.

11.5.1 Behavior of Single-Frame Solutions

Although systematic errors cannot be reduced by measurement statistics, they will usually reveal themselves when the same measurements are repeated at different times along the orbit under different geometrical conditions. Thus, a

* If the random errors dominate, normally more measurements will be taken until the uncertainty is again dominated by the systematic error.

study of the behavior of the single-frame attitude solutions as a function of time can help reveal the existence of systematic errors.

For an ideal case in which no systematic error exists in any of the attitude measurements or attitude determination models, the behavior of the single-frame attitude solutions may be summarized as follows:

1. For each attitude determination method, the attitude solution should follow a known functional variation with time except for the fluctuations due to random errors. If the attitude is inertially fixed and nutation and coning are small, the attitude solutions should remain constant in time.

2. The attitude solutions obtained from different attitude determination methods should give consistent results. Most spacecraft provide redundant measurements for attitude determination to avoid problems of sensor inaccuracies or failure.* Therefore, more than one attitude determination method is generally available. If no systematic error exists, the same attitude solution should result from all methods at any one time in the orbit, to within the random noise on the data.

3. Near an attitude solution singularity, the attitude solutions will have large fluctuations about a uniform mean value because these uncertainties are due entirely to random errors. An attitude singularity is any condition for which the uncertainty of the attitude solution approaches infinity.

Figure 11-29 shows the behavior of single-frame solutions for a near-ideal case. In the figure, the spin axis declination from one real SMS-2 data pass obtained when the spacecraft was in near-synchronous orbit is plotted against the frame number. In obtaining the plotted results, the biases obtained from a bias determination subsystem (as described in Section 21.2) have been used to compensate for most of the systematic errors present in the data. Consequently, apart from the beginning of the data pass, the solutions obtained from the four different attitude determination methods show, nearly constant and consistent results throughout. Also, the solutions near singularities fluctuate about the mean value, as most easily seen from solution 2 near frame 160. The inconsistency in results near the beginning of the data pass and the small deviation in the solutions from a constant value indicate the presence of small residual systematic errors.

The ideal situation normally does not exist in a real mission using nominal parameters. In general, systematic errors are difficult to avoid and contribute most of the uncertainty in single-frame attitude solutions. The systematic errors usually encountered in attitude determination fall into three categories: (1) sensor and modeling parameter biases, which include all possible misalignments in the position and orientation of the attitude sensors and erroneous parameter values used in the models; (2) incorrect or imperfect mathematical models, which include all possible erroneous assumptions or errors in the mathematical formulation of the attitude determination models, such as the shape of the Earth, the dynamic motion of the attitude, or unmodeled sensor electronic characteristics; and (3) incorrect reference vector directions, which include all possible errors in the instantaneous orientation

•In some cases, the same sensors may be used to provide attitude solutions based on different targets. For example, the Earth and the Moon provide redundant information for RAE-2 [Werking, el at^ 1974] and 1SEE-1. Fig. 11-29. Behavior of Single-Frame Solutions With Small Systematic Errors for Real SMS-2 Data.

Numbers on plots indicate solution method: I = Sun angle/Earth-in crossing, 2 = Sun angle/Earth-out crossing, 3 = Sun angle/Earth width, 4= Sun angle/Earth midscan.

Fig. 11-29. Behavior of Single-Frame Solutions With Small Systematic Errors for Real SMS-2 Data.

Numbers on plots indicate solution method: I = Sun angle/Earth-in crossing, 2 = Sun angle/Earth-out crossing, 3 = Sun angle/Earth width, 4= Sun angle/Earth midscan.

of reference vectors, such as orbit errors; errors in ephemeris information for the Sun, the Moon, the planets, and the stars; time-tagging errors; and the errors in direction of the magnetic or gravitational field.

.. Because of systematic errors, the real behavior of the single-frame attitude solutions are generally quite different from the ideal situation. Specifically, the following items characterize the behavior of single-frame solutions with significant systematic errors:

1. For each attitude determination method, the attitude solution departs from the known functional variation with time. This behavior is most easily observed for the spin-stabilized spacecraft where, ideally, the attitude should remain constant in time, and in the presence of systematic errors it shows an apparent time variation.

2. The solutions obtained from different attitude determination methods give different attitude results and show relative variations with time.

3. Near attitude determination singularities, attitude solutions tend to diverge drastically from the mean value.

Thus, the analyst can normally identify the existence of systematic errors by examining the time dependence of the attitude solution from each method, the consistency of results from different methods, and the behavior of solutions nearing singularities.

The behavior of single-frame solutions with significant systematic errors is illustrated in Fig. 11-30, which shows the spin axis declination determined from the same data set as that in Fig. 11-29, except that here the systematic errors have not been removed (i.e., nominal parameters for all sensors were used). Note that the vertical scales are different in the two figures and that solutions which are outside the scale are not plotted. Here, solutions vary strongly in time, show substantial inconsistency among different methods, and diverge rapidly near singularities.