ORBIT DETERMINATION OF SATELLITES IN LUNAR ORBIT USING AN OPTIMAL SEQUENTIAL FILTER

AAS 08-103 ORBIT DETERMINATION OF SATELLITES IN LUNAR ORBIT USING AN OPTIMAL SEQUENTIAL FILTER James Woodburn * and John H. Seago INTRODUCTION Determination of orbits about the Moon poses a number of extra challenges compared to those of Earth orbiting satellites. Most problematic for sequential estimation is the uncertainty in the dynamical environment. We describe deterministic and stochastic upgrades made to the Orbit Determination Tool Kit in pursuit of this goal. We present results from the processing of tracking data for Lunar Prospector over a one month period and STEREO during the lunar fly-by portion of the mission. Orbit determination is required for all operational satellite programs. The determination of orbits about bodies other than the Earth poses a number of extra challenges compared to those of Earth orbiting satellites. Measurement modeling must be performed at a higher fidelity, the accuracy and consistent use of planetary ephemerides becomes more important, and the relationships between multiple reference frames must be properly managed. However, these issues are deterministic and can be overcome by careful attention to detail. More problematic is the uncertainty of the dynamical environment. For the case of lunarorbiting satellites, the primary issue is the uncertainty in the lunar gravitational potential on the far side of the Moon. This type of dynamical uncertainty is typically dealt with in batch weighted least squares estimation by adjusting the fit span of the orbit determination run and solving for empirical accelerations with frequencies natural to the problem. Our endeavor is to apply optimal sequential estimation methodologies to the problem of Moon-orbiting spacecraft. A successful application of this goal would result in a sequential estimator that could be continuously employed over the lifespan of the spacecraft. The dynamical uncertainty must be considered in a way that is properly connected to the physics of the problem. This is to satisfy the optimality criteria for the estimate and eliminate the need for operational tuning of the filter for each phase of a mission. In this paper, we present a description of the deterministic and stochastic modeling upgrades that were recently made to the Orbit Determination Tool Kit (ODTK) software product developed by Analytical Graphics. 1 We also present sample results based on the processing of tracking data from the Lunar Prospector mission over a one month period, and results based on processing tracking data for the STEREO mission during the lunar fly-by portion of the mission. Finally, we * Chief Orbital Scientist, Analytical Graphics, Inc., 220 Valley Creek Blvd., Exton, PA, 19341-2380. Astrodynamics Engineer, Analytical Graphics, Inc., 220 Valley Creek Blvd., Exton, PA, 19341-2380. 1

discuss the expected improvement from new lunar gravity fields developed from the far-side tracking of such missions as SELENE. ODTK ENHANCEMENTS A number of enhancements were required to process observations for lunar-orbiting spacecraft in ODTK. Many of the required changes were architectural or related to the user interface, and mainly designed to provide an abstraction of the primary gravitational body. Two major estimation enhancements were also required: providing a means to deal with the uncertainty in the lunar gravity field, and observation modeling for measurements from the NASA/JPL Deep Space Network (DSN). The Lunar Gravity Field The largest challenge to the operational use of a sequential filter for lunar-orbiting spacecraft is the uncertainty in the gravitational potential of the Moon. The Moon holds no appreciable atmosphere, and thus allows for spacecraft to orbit at very low altitudes. This is advantageous for gravity field determination; however, prior to the SELENE mission there had been no tracking data for satellites on the far side of the Moon. The gravity field models developed from tracking of modern lunar missions, such as Lunar Prospector, have the interesting characteristics of being of high degree and order - needed to capture the high frequency information available on the front side of the Moon - while also having large uncertainties on the far side (Figure 1). Figure 1 Geoid uncertainty for the Moon (m), LP150Q 2 When using traditional batch weighted-least-squares estimators in the presence of significant force modeling uncertainties, fit spans are adjusted to find an optimal trade between incorporating more observations, via longer fit spans, and matching the trajectory to the dynamical model which is more readily accomplished with shorter fit spans. The effectiveness of the adjustment is usually gauged by overlap comparisons and residual analysis. 2

A different strategy is used with a sequential filter over the life of a mission. The desire is to always restart the filter with the final state estimate from the previous filter solution. If this goal is achieved, then all observations are effectively processed as one long, recurrent orbit estimate. This lofty goal becomes complicated, however, by the need to adequately model the uncertainties in the dynamical model. Otherwise, if the filter becomes overly optimistic, all tracking data will be rejected and the filter will diverge. Dynamical uncertainties are considered by using a dynamical process-noise model during the filter time update. In simple terms, the state error covariance (which approximates the uncertainty in the estimated state) becomes inflated between observations times to compensate for imperfect knowledge of the forces on the satellite. The dynamical process model for uncertainty in the 3, 4, 5 gravity field used during this study is described in the biblography. The gravity process-noise model used in ODTK averages gravity errors from all latitudes and longitudes over a shell of constant altitude. For the case of the Moon, this averaging causes the process-noise model to be infrequently optimistic (when the satellite is over the back side of the Moon) and pessimistic much more of the time (when the satellite is not over the back side of the Moon). The use of such a process-noise model for lunar-orbiting satellites therefore results in a generally pessimistic covariance. Luckily, a pessimistic covariance keeps the filter stable - we demonstrate this in the sequel by processing one month of low lunar-orbit tracking data in a single arc. DSN measurement models Measurements from the DSN are processed using observation models for two-way sequential range, two-way and three-way Doppler, and two-way and three-way total-count-phase measurements. In the context of DSN measurement modeling, a two-way measurement involves a signal which is transmitted and received by the same ground station. A three-way measurement involves a signal which is transmitted by a different ground station than the one receiving it. The transmitted signals may be ramped or constant in frequency. The mathematical modeling of these measurement types follows Moyer, except as noted in the sequel. 6 The measurement modeling is consistent with DSN operations using Block V receivers following the completion of the Network Simplification Program. The DSN sequential range measurement is an ambiguous range. The length of the unambiguous part of the observable is supplied in the tracking-data file in conjunction with the observation value. Normal DSN procedures involve an internal calibration of antennae prior to each tracking pass. The pre-pass calibration serves to remove most of the bias from the ranging data. The remaining bias is small, on the order of 14 range units or less, and independent from one pass to the next. 7 While ODTK does not estimate a state-parameter called "pass-dependent measurement bias," the time between passes for a particular DSN antenna is long compared to the time between observations; this allows the exponentially correlated Gauss-Markov bias state estimate available in ODTK to absorb the remaining measurement bias. The DSN total-count-phase measurement is a measure of total received cycles of the carrier from a specified epoch. The epoch for the start of the count is supplied in the tracking-data file in conjunction with the observed value. Sequential total-count-phase measurements are differenced during the reading of the tracking-data file so that the resulting measurement processed in ODTK is the received cycles over the current count interval. This differencing operation, which simplifies the modeling of the measurement in a sequential filter, makes the processing of totalcount-phase observations equivalent to processing Doppler observations. No bias estimate is 3

typically used when processing total-count-phase measurements, but a bias may appear if the spacecraft is spin stabilized. The DSN Doppler measurement is a measure of total received cycles of the carrier over the specified Doppler count interval. The count interval is supplied in the tracking-data file in conjunction with the observed value. The Doppler measurement differs slightly from the totalcount-phase measurement in that the time tag on the Doppler measurement is placed in the middle of the count interval while the time tag on the total-count-phase measurement is placed at the end of the count interval. Although the model for this measurement type has been implemented in ODTK, it does not appear that Doppler measurements are routinely reported any longer because the total-count-phase measurements provide the same information content. To process DSN measurements, ODTK was enhanced to read observations from the TRK 2-34 tracking data format. 8 The TRK 2-34 format contains a wealth of information about the configuration of the antenna while observations were taken. Of primary importance to the processing of the total-count-phase measurements is the frequency ramp information contained in the TRK 2-34 file. Ramps are provided for each tracking station and consist of a starting time and frequency and a frequency rate. Modeling of the total-count-phase observations requires a very precise computation of the transmission interval and proper indexing into the frequency ramp information to correctly determine the number of cycles transmitted during that interval. LUNAR PROSPECTOR Approximately one month of Lunar Prospector (LP) tracking data were processed for a time interval starting on 1 March 1999 and ending on 30 March 1999. The approximate orbit parameters for LP during this time frame are given in Table 1. During the tracking-data interval, the Moon completed slightly more than one complete revolution about the Earth. This is significant because all possible tracking geometries were experienced over the period of the tracking data. Under normal operations, the tracking data would have been processed in chunks at a pre-determined frequency (i.e., once per day) to provide orbit updates for mission planning purposes. For our analysis, however, we processed the entire 30 days of data in a single filter run. Since an operational filter would be periodically restarted from data records containing the final filter state from the prior filter run, the results are the same as what would have been achieved by processing the data over many shorter intervals. Table 1 Orbit Elements for Lunar Prospector Epoch: 28 Feb 1999 00:00:0 UTC Semi-major axis 1767.6 km Periapsis Height 20.1 km Eccentricity 53 Apoapsis Height 39.0 km Inclination (J2000) 66.0 deg Inclination (TOD) 89.9 Ascending Node (J2000) 193.4 deg Ascending Node (TOD) 194.3 Argument of Periapsis (J2000) 93.9 deg Argument of Periapsis (TOD) 101.0 The force modeling options selected for LP are shown in Table 2, and the LP100K gravitational field was augmented with matched gravity process-noise inputs generated using the algorithm described in Wright et al. (2008). 3 4

Table 2 Force modeling for Lunar Prospector Lunar Gravity 3 rd Body Gravity Solar Pressure Field LP100K 100 x 100 Solid Tides On k 2 = 0.03 Sun Earth Satellite model Spherical Cp = 1.2 Shadow model Dual Cone The LP tracking data contained observations from DSN stations 16, 27, 46, and 66. The locations of these tracking stations are illustrated in Figure 2. Stations 16 and 27 are located at Goldstone. Station 66 is in Madrid and station 46 is located at Canberra and station 66 is located at Madrid. All stations provided Doppler, with DSS27 also providing a small amount of ranging data. GTDS outputs were also made available for the time period covered by the tracking data. These output files sometimes show the processing of observations from additional stations that were not included in the provided data sets, and sometimes do not show the processing of data from some stations that were included. Figure 1 Location of DSN tracking stations used for Lunar Prospector The LP tracking data were supplied in the Universal Tracking Data Format (UTDF). The Doppler data in UTDF are not represented as a cycle count on the carrier frequency as reported in TRK-2-34 format. Since the tracking data came in UTDF format, ODTK processed the data using its nominal two-way range and two-way Doppler observation models. The main differences in the 5

nominal two-way range and the DSN sequential range models are: the absence of a model for the gravitational bending of the signal path, and the solution of the light-time equation in the geocentric frame, instead of the solar system barycentric frame. For a lunar orbiter, the differences in the observation models are small. A summary of the tracking data processed for LP is given in Table 3. The indicated numbers of measurements include only measurements that were accepted by the filter. All tracking was performed in S band. Table 3 Tracking data summary for Lunar Prospector Station Measurement Type # Observations DSS16 Doppler 18430 DSS27 Range 11570 Doppler 18961 DSS46 Doppler 32562 DSS66 Doppler 37212 Precise tracking-station locations were provided at the start of this study, but the measurement white noise and bias uncertainties for the tracking stations and the transponder delay were not known. These values were iteratively determined by repeatedly processing the tracking data and tweaking input values until the resulting residual patterns seemed reasonable. The initial value of the transponder delay was determined by plotting range residuals from orbit-determination runs fit only to Doppler data. The results of this calibration process are shown in Table 4. An observed bias was added to the two-way Doppler measurements to account for the fact that Lunar Prospector was spin stabilized. 9 The same calibration values were used for all stations. Additional calibration could be done on a per-station basis, but little accuracy gain would probably have been achieved due to the significant process noise associated with lunar orbits. Table 4 Measurement statistics Measurement Type Constant Bias Bias Est. Bias Sigma White noise Range 0.0 m Yes 10 m 1.5 m Doppler -0.112 cm/s No N/A 0.1 cm/s Transponder delay 405 m A number of quality indicators is used to scrutinize the orbit determination results obtained using ODTK. Two of the most important are the apparent randomness of the residuals, and satisfaction of the McReynolds filter-smoother consistency test. Histograms of normalized residuals - residuals divided by the square root of the measurement error variance - are presented in Figures 4a-4b. The theoretical normal curve is superposed on the histograms. Residual scatter 6

plots for typical range and Doppler passes are shown in Figures 5a-5b. There is no apparent indication from these figures of significant measurement modeling errors. It is, however, interesting to note the expansion of the 3-sigma measurement uncertainty envelope for the Doppler passes. This expansion is largely due to the pessimistic gravity error process-noise model. 3 This expansion is also the main reason for the severely peaked Doppler histograms in Figure 4b. The McReynolds filter-smoother consistency statistic is computed by dividing the difference between the filter and smoother estimates by the difference between the filter and smoother root variances for specific state elements. 10 This test statistic should be normally distributed, so that under ideal conditions, the test statistic should remain within ± 3 for 99% of the time. If it is highly abnormal, then this is interpreted as a defect in the dynamical and/or measurement modeling. 13.00 Process: Satellite: LP 12.00 Meas.Type: Range Tracker: DSNetwork.DSS27 Sample Size: 11570 11.00 1 9.00 Histogram 0.42 0.40 0.38 0.36 0.34 0.32 0.30 Percentage Found 8.00 7.00 6.00 5.00 4.00 3.00 2.00 1.00 0.28 0.26 0.24 0.22 0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 Normal Density -3.000-2.850-2.550-2.250-1.950-1.650-1.350-1.050-0.750-0.450-0.150 0.150 0.450 0.750 1.050 1.350 1.650 1.950 2.250 2.550 2.850 3.000 Number of Sigmas Percentage Found Normal Density Percentage Predicted Figure 4a Range residual histogram Percentage Found 42.00 4 38.00 36.00 34.00 32.00 3 28.00 26.00 24.00 22.00 2 18.00 16.00 14.00 12.00 1 8.00 6.00 4.00 2.00 Process: Satellite: LP Meas.Type: Doppler Tracker: DSNetwork.DSS16, DSNetwork.DSS27, DSNetwork.DSS46, DSNetwork.DSS66 Sample Size: 18430, 18960, 32563, 37114 Histogram -0.450 2.550-1.050 1.950-1.650 1.350-2.250 0.750 3.000 Number of Sigmas Percentage Found Normal Density Percentage Predicted 1.40 1.30 1.20 1.10 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 Normal Density Figure 4b Doppler residual histograms 7

1 Tracker IDs: DSNetw ork.dss27 9.00 Satellite IDs: LP Meas. Types: Range 8.00 Units: m 7.00 A: Accepted RE: Rejected (A, RE) 9 Mar 1999 08:38:20 UTCG 6.00 5.00 4.00 3.00 2.00 1.00-1.00-2.00-3.00-4.00-5.00-6.00-7.00-8.00-9.00-1 3500 3520 3540 3560 3580 3600 3620 3640 3660 3680 3700 3720 3740 3760 3780 3800 3820 3840 3860 3880 3900 3920 3940 3960 3980 4000 Minutes since 9 Mar 1999 00:00:00 UTCG A DSNetwork.DSS27 Meas Residuals DSNetwork.DSS27 3-Sigma Meas Residual RE DSNetwork.DSS27 Meas Residuals DSNetwork.DSS27-3-Sigma Meas Residual Figure 5a Range pass residuals 2.00 1.80 1.60 1.40 1.20 Tracker IDs: DSNetwork.DSS66, DSNetwork.DSS16, DSNetw ork.dss46, DSNetwork.DSS27 Satellite IDs: LP Meas. Types: Doppler Units: cm/s A: Accepted RE: Rejected (A, RE, LE) 1 Mar 1999 00:00:20 UTCG 1.00 0.80 0.60 0.40 0.20-0.20-0.40-0.60-0.80-1.00-1.20-1.40-1.60-1.80-2.00 4320 4330 4340 4350 4360 4370 4380 4390 4400 4410 4420 4430 4440 4450 4460 4470 4480 4490 4500 4510 4520 Minutes since 1 Mar 1999 00:00:00 UTCG A DSNetwork.DSS66 Meas Residuals RE DSNetwork.DSS66 Meas Residuals RE DSNetw ork.dss16 Meas Residuals A DSNetwork.DSS16 Meas Residuals RE DSNetwork.DSS46 Meas Residuals A DSNetwork.DSS46 Meas Residuals RE DSNetwork.DSS27 Meas Residuals A DSNetw ork.dss27 Meas Residuals DSNetwork.DSS66 3-Sigma Meas Residual DSNetwork.DSS16 3-Sigma Meas Residual DSNetwork.DSS46 3-Sigma Meas Residual DSNetwork.DSS27 3-Sigma Meas Residual DSNetwork.DSS66-3-Sigma Meas Residual DSNetwork.DSS16-3-Sigma Meas Residual DSNetwork.DSS46-3-Sigma Meas Residual DSNetwork.DSS27-3-Sigma Meas Residual Figure 5b Doppler pass residuals Earth in LP orbit plane 2.00 1.80 1.60 1.40 1.20 Tracker IDs: DSNetwork.DSS66, DSNetwork.DSS16, DSNetw ork.dss46, DSNetwork.DSS27 Satellite IDs: LP Meas. Types: Doppler Units: cm/s A: Accepted RE: Rejected (A, RE, LE) 1 Mar 1999 00:00:20 UTCG 1.00 0.80 0.60 0.40 0.20-0.20-0.40-0.60-0.80-1.00-1.20-1.40-1.60-1.80-2.00 14660 14670 14680 14690 14700 14710 14720 14730 14740 14750 14760 14770 14780 14790 14800 14810 14820 14830 14840 14850 14860 14870 14880 Minutes since 1 Mar 1999 00:00:00 UTCG DSNetwork.DSS66 Meas Residuals RE DSNetwork.DSS66 Meas Residuals RE DSNetw ork.dss16 Meas Residuals A A DSNetwork.DSS16 Meas Residuals RE DSNetwork.DSS46 Meas Residuals A DSNetwork.DSS46 Meas Residuals RE DSNetwork.DSS27 Meas Residuals A DSNetw ork.dss27 Meas Residuals DSNetwork.DSS66 3-Sigma Meas Residual DSNetwork.DSS16 3-Sigma Meas Residual DSNetwork.DSS46 3-Sigma Meas Residual DSNetwork.DSS27 3-Sigma Meas Residual DSNetwork.DSS66-3-Sigma Meas Residual DSNetwork.DSS16-3-Sigma Meas Residual DSNetwork.DSS46-3-Sigma Meas Residual DSNetwork.DSS27-3-Sigma Meas Residual Figure 5c Doppler pass residuals Earth direction perpendicular to LP orbit plane 8

Figure 6 shows the filter-smoother consistency statistic for the satellite position components expressed in the radial, in-track, and cross-track directions. The consistency metrics are very small over most of the month long tracking data interval. This is evidence that the filter and smoother position variances are too large, and is primarily due pessimistic gravity error process noise. There are short spans of time when the Earth-Moon line lies in the spacecraft orbit plane and the gravity process noise becomes optimistic. This is reflected in the filter-smoother consistency metrics as well. Inconsistency between the filter and smoother state on the 24 th of March is due to a maneuver which is further analyzed in the sequel. There are a small number of additional spikes in the filter-smoother consistency metrics that are not fully understood, but it may not be possible to determine their cause without detailed information on spacecraft activities during the period in question. Finally, we wish to discover the stability and expected accuracy of the orbit resulting from the filter-smoother process. During normal operations, the sequential filter process would be run to process observations and produce a predicted trajectory to be used for mission planning. Each filter run begins with the final state and covariance from the prior run. If the filter is stable, this process can continue for the life of the mission without the need to re-initialize the filter. The smoother would be run, using output from the filter, to produce a high precision, time-lagged ephemeris for mission analyses. 7.00 6.00 5.00 4.00 3.00 2.00 1.00 Satellite(s): LP Target: Filter Reference: Smoother (Target-Reference) Position Consistency Statistics Earth to orbit plane 1 Mar 1999 00:00:00 UTCG Test Statistic -1.00-2.00-3.00-4.00-5.00-6.00-7.00-8.00-9.00 Earth in orbit plane Maneuver -1 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 22000 24000 26000 28000 30000 32000 34000 36000 38000 40000 Minutes since 1 Mar 1999 00:00:00 UTCG LP Radial LP In-Track LP Cross-Track UpperBound LowerBound Figure 6 Filter-smoother consistency statistics for LP position Each state estimate from the filter contains information about all tracking data taken prior to the current time in the filter. Smoother estimates contain information from all tracking taken prior to the current time in the smoother plus all tracking data after the current time in the filter up to the time of the smoother initial state. The smoother runs backwards in time. Since smoother estimates are conditioned by more information at each time that their corresponding filter estimates, the uncertainty in the smoother estimates are smaller than those from the filter. 9

1000 900 800 Process: Filter Satellite(s): LP Position Uncertainty (0.95P) Earth to orbit plane 1 Mar 1999 00:00:00 UTCG Maneuver Two Sigmas (m) 700 600 500 400 Earth in orbit plane 300 200 100 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 22000 24000 26000 28000 30000 32000 34000 36000 38000 40000 Minutes since 1 Mar 1999 00:00:00 UTCG LP 2-Sigmas Radial LP 2-Sigmas Intrack LP 2-Sigmas Crosstrack Figure 7a LP position uncertainty from filter 1000 900 800 Process: Filter Satellite(s): LP Position Uncertainty (0.95P) Earth to orbit plane 1 Mar 1999 00:00:00 UTCG Maneuver Two Sigmas (m) 700 600 500 400 Earth in orbit plane 300 200 100 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 22000 24000 26000 28000 30000 32000 34000 36000 38000 40000 Minutes since 1 Mar 1999 00:00:00 UTCG LP 2-Sigmas Radial LP 2-Sigmas Intrack Figure 7b LP radial and in-track position uncertainty from filter 10

Two Sigmas (m) 250 240 230 220 210 200 190 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 Process: Smoother Satellite(s): LP Earth in orbit plane Position Uncertainty (0.95P) Earth to orbit plane 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 22000 24000 26000 28000 30000 32000 34000 36000 38000 40000 Minutes since 1 Mar 1999 00:00:00 UTCG 1 Mar 1999 00:00:00 UTCG Maneuver LP 2-Sigmas Radial LP 2-Sigmas Intrack LP 2-Sigmas Crosstrack Figure 8a LP position uncertainty from smoother Two Sigmas (m) 250 240 230 220 210 200 190 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 22000 24000 26000 28000 30000 32000 34000 36000 38000 40000 LP 2-Sigmas Radial Process: Smoother Satellite(s): LP Earth in orbit plane Position Uncertainty (0.95P) Earth to orbit plane Minutes since 1 Mar 1999 00:00:00 UTCG LP 2-Sigmas Intrack 1 Mar 1999 00:00:00 UTCG Maneuver Figure 8b LP radial and in-track position uncertainty from smoother 11

The position uncertainty from the filter is shown in Figure 7a for the radial, in-track, and cross-track components. To provide better insight into the behavior of the radial and in-track uncertainties, the cross-track uncertainty is removed in Figure 7b. The important feature of these plots is that the uncertainty, while showing the effects of the tracking geometry, is neither growing nor shrinking in a secular fashion. We also did not experience any increase in the rate of tracking-data rejection by the filter over the duration of the run. The filter, therefore, appears to be stable without indicating the need for re-initialization. The position uncertainty from the smoother is shown in Figure 8a and Figure 8b. Note that the effects of the orbit-earth geometry can be clearly seen in the gross behavior of the lines. Lunar prospector trajectories are available from the Planetary Data System, making a comparison of results possible. 11 These trajectories originated from the Goddard Trajectory Determination System (GTDS). The trajectory differences for March 3 are shown in Figure 9. 2.40 2.10 1.80 1.50 1.20 0.90 0.60 0.30-0.30-0.60-0.90-1.20-1.50-1.80-2.10-2.40 3 Mar 1999 00:00:01.000 3 Mar 1999 06:00:01.000 3 Mar 1999 12:00:01.000 3 Mar 1999 18:00:01.000 (UTCG) 3 Mar 1999 00:00:00 to 3 Mar 1999 23:59:00 Radial (km) In-Track (km) Cross-Track (km) Maneuver Figure 9 Trajectory difference between smoother and GTDS solutions No maneuver information was known during the time span of the available tracking data. Initial processing of the data, however, indicated some type of modification to the trajectory during a significant part of one orbit period on 24 March 1999. The start time of the unknown activity, at approximately 17:30 UTC, was detected via the sudden and unexpected rejection of Doppler tracking data (Figure 10). The end time of the unknown activity was determined by adding a zero-magnitude finite maneuver of large uncertainty, and adjusting the end time of the maneuver until data were processed normally by the filter. Once the approximate end time of the activity was determined to be 18:25 UTC, the large uncertainty in the unknown finite maneuver was reduced in an iterative manner to determine the approximate smallest amount of process noise which was required to allow the filter to resume nominal processing at the end of the activity period. It was possible to stop at this point, and let the filter span the maneuver period courtesy of the process noise injected by the unknown maneuver model; however, we continued to investigate the event. 12

Further refinement of the unknown maneuver was accomplished by turning on maneuver estimation. The filter was able to lock onto an initial maneuver estimate, but began rejecting tracking data suddenly about three-quarters of the way through the maneuver span. It appeared that the net thrusting direction changed suddenly at this time from mainly radial to mainly crosstrack. To enable the filter to continue processing data, we ended the first modeled maneuver at the apparent time of change of thrust direction, 18:11 UTC, and added a second maneuver of unknown nominal thrust, and having significant cross-track uncertainty, over the remainder of the maneuver time span. The result of our maneuver modeling selections is shown in terms of the Doppler residuals across the thrusting interval in Figure 11. 100 90 80 70 60 50 40 30 20 10-10 -20-30 -40-50 -60-70 -80-90 Tracker IDs: DSNetwork.DSS27, DSNetwork.DSS66 Satellite IDs: LP Meas. Types: Doppler Units: cm/s A: Accepted RE: Rejected (A, RE) -100 720 750 780 810 840 870 900 930 960 990 1020 1050 1080 1110 1140 1170 1200 1230 1260 1290 1320 1350 1380 1410 1440 Minutes since 24 Mar 1999 00:00:00 UTCG 24 Mar 1999 00:00:10 UTCG A DSNetwork.DSS27 Meas Residuals RE DSNetwork.DSS27 Meas Residuals RE DSNetwork.DSS66 Meas Residuals A DSNetwork.DSS66 Meas Residuals DSNetwork.DSS27 3-Sigma Meas Residual DSNetwork.DSS66 3-Sigma Meas Residual DSNetwork.DSS27-3-Sigma Meas Residual DSNetwork.DSS66-3-Sigma Meas Residual Figure 10 Rejection of tracking data indicates unknown satellite activity 5.00 4.00 3.00 Tracker IDs: DSNetwork.DSS27, DSNetwork.DSS66 Satellite IDs: LP Meas. Types: Doppler Units: cm/s A: Accepted RE: Rejected (A, RE) 24 Mar 1999 00:00:00 UTCG 2.00 1.00-1.00-2.00-3.00-4.00 Maneuver 1 start -5.00 720 750 780 810 840 870 900 930 960 990 1020 1050 1080 1110 1140 1170 1200 1230 1260 1290 1320 1350 1380 1410 1440 Minutes since 24 Mar 1999 00:00:00 UTCG Maneuver 2 start A DSNetwork.DSS27 Meas Residuals RE DSNetwork.DSS27 Meas Residuals RE DSNetwork.DSS66 Meas Residuals A DSNetwork.DSS66 Meas Residuals DSNetwork.DSS27 3-Sigma Meas Residual DSNetwork.DSS66 3-Sigma Meas Residual DSNetwork.DSS27-3-Sigma Meas Residual DSNetwork.DSS66-3-Sigma Meas Residual Figure 11 Residuals over satellite activity interval with unknown finite maneuver modeled 13

Table 5a Force modeling for STEREO, selenocentric Lunar Gravity 3 rd Body Gravity Solar Pressure Field LP100K 8 x 8 Solid Tides Off Sun Mercury Venus Earth Mars Jupiter Saturn Neptune Uranus Model Spherical Cp = 1.2 Shadow Dual Cone Earth,Moon Table 5b Force modeling for STEREO, geocentric Earth Gravity 3 rd Body Gravity Solar Pressure Field GGM02C 8 x 8 Solid Tides Off Sun Mercury Venus Moon Mars Jupiter Saturn Neptune Uranus Model Spherical Cp = 1.2 Shadow Dual Cone Earth,Moon Table 5c Force modeling for STEREO, heliocentric Sun Gravity 3 rd Body Gravity Solar Pressure Field Point mass Moon Mercury Venus Earth Mars Jupiter Saturn Neptune Uranus Model Spherical Cp = 1.2 Shadow Dual Cone Earth,Moon 14

STEREO DSN tracking data for the STEREO-B spacecraft were obtained for the lunar fly-by portion of the STEREO-B trajectory. The STEREO-A and STEREO-B spacecraft utilized lunar gravity assists to direct their trajectories out of the Earth-Moon system towards their heliocentric mission orbits ahead (STEREO-A) and behind (STEREO-B) the Earth. "Definitive" ephemerides were also provided to allow for post-fit trajectory comparisons; these ephemerides were in binary SPICE format and originated from the Goddard Trajectory Determination System (GTDS). Because the fly-by - the selenocentric phase of the trajectory - represented a transition from the geocentric phase to the heliocentric phase, we took the opportunity to use each of the three bodies as the central-body origin for spacecraft orbit determination, and then compared the estimation results. Tables 5a-5c provide the nominal force modeling selections used for each run. In ODTK, the specification of the primary central body for a satellite implies that it is to be the origin for trajectory integration. It is also the only celestial body for which the full gravitational field is evaluated during trajectory integration. It is also the only celestial body for which gravity process noise is available. The preliminary estimation runs showed reasonable agreement for most estimates but did show a small difference in the transponder delay. An investigation into the source of the difference revealed two primary causes: the value of the gravitational parameter for the Moon differed depending on if the Moon was the primary body or not, and the number of lunar gravitational harmonics in the selenocentric case differed. To remedy the first cause, the source code was updated to allow the option to specify the gravitational parameter used for third-body perturbations. This option will be included in a future release of ODTK. To remedy the second cause, an input setting was changed and the pass of tracking data closest to the lunar periapsis passage was excluded. The estimation process was repeated and the estimates from all three cases were then nearly identical (Figure 12). The agreement between estimation results using different centers for trajectory integration provides a strong validation of the measurement model implementations across very different frames of reference. While simplifying the gravity modeling for the primary body allowed us to achieve consistency, for best accuracy we still recommend using a more complete gravitational potential for the Moon. Filter-smoother consistency test results are shown in Figure 13. Observations were provided from DSN stations 25, 26, 45 and 54 during this time period with all tracking being done in X-band. Common measurement statistics were used for all stations and are provided in Table 6. One pass of tracking data from station 45 during the actual lunar flyby exhibited larger than expected residual scatter resulting in the rejection of a significant number of total-count-phase observations, see Figure 14. The cause of the difficulty in processing the totalcount-phase measurements during the flyby has not yet been determined. We also note the brief violation of the filter-smoother consistency test during this time period in Figure 13. Table 6 Measurement statistics Measurement Type Constant Bias Bias Est. Bias Sigma White noise Sequential Range 0.0 m Yes 3 m 1.0 m Total Count Phase 0.0 Cycles No N/A 3 Cycles Transponder delay 425 m 15

0.05 0.04 0.03 0.02 0.01-0.01-0.02-0.03 Tracker IDs: DSNetwork.DSS45, DSNetwork.DSS25, DSNetwork.DSS54, DSNetwork.DSS26 Satellite IDs: StereoBseleno Meas. Types: DSN TCP Units: A: Accepted RE: Rejected (A, RE) 18 Jan 2007 23:26:51.000 UTCG -0.04-0.05 1407 2407 3407 4407 5407 6407 7407 8407 9407 10407 11407 Minutes since 18 Jan 2007 00:00:00 UTCG 0.05 0.04 0.03 0.02 0.01-0.01-0.02-0.03 Tracker IDs: DSNetwork.DSS45, DSNetwork.DSS25, DSNetwork.DSS54, DSNetwork.DSS26 Satellite IDs: StereoBgeo Meas. Types: DSN TCP Units: A: Accepted RE: Rejected (A, RE) 18 Jan 2007 23:26:51.000 UTCG -0.04-0.05 1407 2407 3407 4407 5407 6407 7407 8407 9407 10407 11407 Minutes since 18 Jan 2007 00:00:00 UTCG 0.05 0.04 0.03 0.02 0.01-0.01-0.02-0.03 Tracker IDs: DSNetwork.DSS45, DSNetwork.DSS25, DSNetwork.DSS54, DSNetwork.DSS26 Satellite IDs: StereoBhelio Meas. Types: DSN TCP Units: A: Accepted RE: Rejected (A, RE) 18 Jan 2007 23:26:51.000 UTCG -0.04-0.05 1407 2407 3407 4407 5407 6407 7407 8407 9407 10407 11407 Minutes since 18 Jan 2007 00:00:00 UTCG Figure 12 Total phase count residuals using different integration centers 16

7.00 6.00 5.00 4.00 Satellite(s): StereoB Target: Filter Reference: Smoother (Target-Reference) Position Consistency Statistics Lunar Flyby 18 Jan 2007 23:27:01.000 UTCG 3.00 Test Statistic 2.00 1.00-1.00-2.00-3.00-4.00 1407 2407 3407 4407 5407 6407 7407 8407 9407 10407 11407 Minutes since 18 Jan 2007 00:00:00 UTCG StereoB Radial StereoB In-Track StereoB Cross-Track UpperBound LowerBound Figure 13 Filter-smoother consistency test results for STEREO 0.18 0.15 0.12 0.09 0.06 0.03-0.03-0.06-0.09-0.12-0.15-0.18 Tracker IDs: DSNetwork.DSS45, DSNetwork.DSS25, DSNetwork.DSS54, DSNetwork.DSS26 Satellite IDs: StereoB Meas. Types: DSN TCP Units: A: Accepted RE: Rejected (A, RE) 18 Jan 2007 23:27:01.000 UTCG 4700 4710 4720 4730 4740 4750 4760 4770 4780 4790 4800 4810 4820 4830 4840 4850 4860 4870 4880 4890 4900 4910 4920 4930 4940 4950 Minutes since 18 Jan 2007 00:00:00 UTCG Figure 14 Residuals during tracking interval when STEREO was closest to the Moon 100 90 Process: Filter Satellite(s): StereoB Position Uncertainty (0.95P) 18 Jan 2007 23:27:01.000 UTCG 80 Two Sigmas (m) 70 60 50 40 Lunar Flyby 30 20 10 1407 2407 3407 4407 5407 6407 7407 8407 9407 10407 11407 Minutes since 18 Jan 2007 00:00:00 UTCG StereoB 2-Sigmas Radial StereoB 2-Sigmas Intrack StereoB 2-Sigmas Crosstrack Figure 15 STEREO selenocentric position uncertainty from filter 17

10 9 Process: Smoother Satellite(s): StereoB Position Uncertainty (0.95P) 18 Jan 2007 23:27:00 UTCG 8 7 Two Sigmas (m) 6 5 4 Lunar Flyby 3 2 1 1407 2407 3407 4407 5407 6407 7407 8407 9407 10407 11407 Minutes since 18 Jan 2007 00:00:00 UTCG StereoB 2-Sigmas Radial StereoB 2-Sigmas Intrack StereoB 2-Sigmas Crosstrack Figure 16 STEREO selenocentric position uncertainty from smoother 0.50 0.40 0.30 19 Jan 2007 23:27:00 20 Jan 2007 23:27:00 21 Jan 2007 23:27:00 22 Jan 2007 23:27:00 23 Jan 2007 23:27:00 24 Jan 2007 23:27:00 25 Jan 2007 23:27:00 0.20 0.10-0.10-0.20 19 Jan 2007 19:00:00 21 Jan 2007 14:00:00 23 Jan 2007 09:00:00 25 Jan 2007 04:00:00 (UTCG) 18 Jan 2007 23:27:00 to 25 Jan 2007 23:58:00 Radial (km) In-Track (km) Cross-Track (km) Figure 17 Trajectory difference selenocentric smoother result vs GTDS result STEREO position uncertainty from the filter in a Moon-centered frame is presented in Figure 15. The corresponding smoother result is given in Figure 16. We note increased observability of the orbit during the lunar fly-by. In fact, while the estimation process is fairly insensitive to the transponder delay for the majority of the trajectory, the ability to process data during the fly-by is greatly degraded by errors in the transponder delay. Orbit determination results from GTDS were provided with the STEREO tracking data making a comparison of results possible. Figure 17 shows the ephemeris differences between the selenocentric filtersmoother trajectory and the GTDS result. CONCLUSION We have demonstrated the ability to process approximately one month of tracking data on a low altitude lunar-orbiting satellite with a single run of the sequential filter. During this time period, the filter showed no signs of divergence or degradation. While no ad-hoc process noise 18

was required to achieve this goal, the existing gravity process-noise model is overly conservative most of the time and optimistic during short periods of time. The non-optimal nature of the gravity error covariance function for the lunar gravity field is mainly a result of the large variation in gravity field uncertainty over the lunar globe. Lunar gravitational potentials developed based on SELENE should be both more accurate and have more homogeneous uncertainty. This should result in more accurate orbit determination and more realistic orbit covariance. We have demonstrated the capability to process DSN sequential range and total-count-phase measurements during a lunar flyby. Estimation was performed using Earth, Moon and Sun centered frames for trajectory integration with similar results. ACKNOWLEDGMENT The authors wish to thank Mark Beckman of NASA Goddard for providing tracking data. We also wish to acknowledge the invaluable assistance provided by Tomas Martin-Mur of JPL in helping us understand the DSN tracking data format and measurement models. REFERENCES 1 Hujsak, Richard S., Woodburn, James W., Seago, John H., The Orbit Determination Tool Kit (ODTK) Version 5," AAS 07-125, AAS/AIAA Space Flight Mechanics Meeting, Sedona, AZ, January 2007. 2 Roncoli,Ralph B., Lunar Constants and Models Document, JPL D-32296, Sept 2005. 3 Wright, James R., Woodburn, James, Truong, Son, Chuba, William, Orbit Gravity Error Covariance," AAS 08-157, AAS/AIAA Space Flight Mechanics Meeting, Galveston, TX, January 2008. 4 Wright, James R., Woodburn, James, Truong, Son, Chuba, William, Sample Orbit Covariance Function and Filter- Smoother Consistency Tests," AAS 08-159, AAS/AIAA Space Flight Mechanics Meeting, Galveston, TX, January 2008. 5 Wright, James R., Woodburn, James, Truong, Son, Chuba, William, Orbit Covariance Inner Integrals with Polynomials," AAS 08-161, AAS/AIAA Space Flight Mechanics Meeting, Galveston, TX, January 2008. 6 Moyer, Theodore D., Formulation for Observed and Computed Values of Deep Space Network Data Types for Navigation, Monograph 2, Deep Space Communications and Navigation Series. 7 Martin-Mur, Tomas (2007), Personal Communication. 8 TRK-2-34 DSMS Tracking System Data Archival Format, 820-013, Deep Space Mission System External Interface Specification JPL D-16765, Revision I. 9 Konopliv, A.S., S.W. Asmar, E. Carranza, W.L. Sjogren, and D.N. Yuan (2001), "Recent Gravity Models as a Result of the Lunar Prospector Mission", Icarus, Volume 150, Issue 1, March 2001, Pages 1-18. 10 McReynolds, S.R. (1984), Editing Data Using Sequential Smoothing Techniques for Discrete Systems. Paper AIAA-1984-2053, Proceedings of the AIAA/AAS Astrodynamics Conference, Seattle, WA, Aug 20-22, 1984. 11 http://pds.nasa.gov/ 19