EVALUATING ORBIT DETERMINATION POST-PROCESSING METHODS FOR OPERATIONAL ARTEMIS DATA

Transcription

1 AAS EVALUATING ORBIT DETERMINATION POST-PROCESSING METHODS FOR OPERATIONAL ARTEMIS DATA Bradley W. Cheetham * and George H. Born INTRODUCTION Operating in the highly dynamic Earth-Moon libration point orbit (LPO) region, which is predominantly perturbed by the Earth, the Moon, and the Sun, is a challenge. The Artemis mission operated by the NASA Goddard Space Flight Center and the University of California at Berkeley recently became the first to ever maintain orbits in this regime. The resulting operational data provides significant opportunity for analysis to better understand these orbits and their operational constraints. Future efforts to quantify orbit determination results, recover un-modeled accelerations, realistic uncertainty propagation, and ultimately LPO utilization will grow out of an ability to post-process this operational data for further understanding of the dynamics involved. To prepare for post-processing of this data, this paper quantifies the effects of various contributors to the dynamic models, experimentally models errors in a simulated environment, and outlines areas of future focus. Realistic spacecraft ephemeris and attribute information will be used to the maximum extent possible. Simulations of orbit determination efficacy are performed using the Analytical Graphics Inc. Orbit Determination Tool Kit (ODTK) with appropriate tracking and spacecraft characteristics and known error sources. This work is motivated by the desire to better understand the dynamic environment and operational constraints of spacecraft in the Earth-Moon three-body regions. Specifically of interest are libration point orbits (LPOs) about the Earth-Moon co-linear L 1 and L 2 points as labeled in Figure 1. These orbits exist in a regime perturbed predominantly by the Earth, the Moon, and the Sun, and are chaotic in nature. Orbits in this region are unstable and require stationkeeping maneuvers to maintain their orbits. These stationkeeping maneuvers have previously been found to vary from approximately 1-4 m/s/yr assuming orbit determination accuracy of better than 1 meters 1 to more than 5 m/s/yr including modeling and maneuver errors. 2 The Artemis mission budgeted approximately 15 m/s for the planned ~6 month L 1 /L 2 orbit maintenance portion of the mission. The magnitude of these station keeping maneuvers is very highly dependent on the accuracy of the orbit determination solutions recovered, dynamic models employed, and maneuver execution errors observed. As a result of the driving nature of these error sources on operational implementation of LPO missions, they will become the focus of this and future efforts. In this * Graduate Research Assistant, University of Colorado Boulder Aerospace Engineering Sciences, Colorado Center for Astrodynamics Research, 431 UCB, 839. Director, Colorado Center for Astrodynamics Research, University of Colorado, Boulder Aerospace Engineering Sciences, 431 UCB,

2 chaotic environment, errors in initial conditions and dynamic models used for propagation can have a significant effect on the resulting orbit solution. Thus any improvement in orbit determination accuracy and dynamic modeling directly translates to mission capabilities in the form of reduced stationkeeping maneuver and operations requirements. The orbital period of a LPO in the Earth-Moon L 1 -L 2 region is approximately two-weeks and thus errors in operational assumptions, orbit determination solutions, or maneuvers can quickly change the orbit of the spacecraft. Unlike dynamically similar Sun-Earth LPOs which have an orbital period of about six-months, there is an increased need to understand the driving perturbations and operational constraints for Earth-Moon LPOs due to their condensed time scale. While these Earth-Moon LPOs pose multiple operational challenges, they also provide significant opportunity for scientific exploration, cis-lunar space observations, and future space development. The same dynamics that make LPOs about L 1, L 2, and L 3 unstable also naturally clear these orbits of any generated orbital debris. Additionally, the dynamics involved provide the opportunity to easily transfer between orbits and asymptotically approach and depart the region. Their geometric position in cis-lunar space provides satellites in LPOs with the operational high ground for near-earth operations. These LPOs provide an opportunity to relay communications from the Moon to the Earth, to observe the lunar far side, and to serve as observation posts for Earth orbits to name a few uses. In the case of the Artemis mission, these orbits provide unique positioning to observe the interactions between the Moon and the Earth s magnetic field as it is excited by solar plasma and as intermediary orbits for a transfer to Lunar orbit. The two spacecraft of the Artemis mission were the first to successfully navigated and perform stationkeeping operations in these Earth-Moon LPOs. The observations and navigation solutions of these vehicles are thus of great potential value to better understand the orbital regime. In preparation for eventual post-processing of raw observations of these spacecraft, the dynamic influence of perturbations on the spacecraft has been performed to provide insight into possible modeling errors. Furthermore, simulated data has been generated using the Orbit Determination Took Kit (ODTK). This simulated data is intended to provide orbit determination insight for these specific orbital regimes and experience in synthesizing orbit solutions given a program developed by others. Specifically, the effect of modeling errors on filter performance is evaluated to improve future solution fidelity. Figure 1. Earth-Moon three-body system. 2

3 BACKGROUND Artemis Mission The Artemis mission, which is used here as a the baseline for simulated data, is a phase two mission which re-purposed two satellites of the original five satellites involved with the THEMIS mission. The ultimate objective of the THEMIS mission is to study the interrelationship between the Sun and the Earth s magnetic field. Specifically of interest is the phenomenon associated with geomagnetic sub-storms. The outer two spacecraft of this constellation are what became P1 and P2 of the Artemis mission. To raise the outer two satellites, P1 and P2, to the Moon they first underwent many phasing loops about the Earth. After a complex transfer including Lunar, Earth, and Solar assists, the vehicles then entered lissajous orbits about the Lunar L 1 and L 2 points. This is where the mission data of relevance to this and future work in this area was gathered. 3,4,5 The region of specific interest for this work is shown in Figure 2 during the Lissajous phase where both satellites P1 and P2 navigated and maintained LPOs about both L 1 and L 2. These orbits simultaneously satisfy both scientific and astrodynamic interests. From a science perspective this phase allows the spacecraft to gather data about how the Moon interacts with the geomagnetic field, specifically as the Moon passes through the wake of the geomagnetic field. From a mission design perspective, these orbits are used to reduce the inclination of the orbits and are ultimately designed to position the spacecraft for their subsequent entry into lunar orbit. They also provide the first-of-its-kind data, which this report simulates, of operations in these dynamic regions. Specifically for the simulations in this paper, the P2 spacecraft was considered during an orbit section about the L 1 point. Figure 2. Artemis LPO Phase 4 3

4 Three-Body Orbits Satellites in the region of consideration are perturbed predominantly by both the Earth and Moon which are both under the influence of gravitational forces from the Sun. These orbits traditionally have been simulated and studied using specific assumptions to facilitate evaluation including the circular restricted three body problem (CRTBP). 6 Using this simplified model, equations of motion can be developed using dimensionless parameters and a better understanding of the theoretical performance of spacecraft is possible. This work has led to many proposed utilizations of these orbits some of which were alluded to earlier in this paper. One of the initial champions of such orbits in the 196s was Robert Farquhar who coined the term Halo orbits for certain LPOs. 7 While these simplified models make study and simulation easier, when implemented for a mission, the full ephemeris must be used as well as forces such as solar radiation pressure. This is possible to simulate, given certain assumptions, in software packages and has been done frequently when considering operational constraints. These simulations, however, have never before had true operational data to validate the modeling and operational constraints involved with spacecraft navigation, orbit determination, or stationkeeping. It is this shortcoming which the Artemis mission now presents the opportunity to address. To begin this process, the work in this paper simulates the operational environment to evaluate the orbit determination constraints and subtleties. Perturbation Evaluation The performance of orbit determination and propagation is dependent on the accuracy and fidelity of the dynamic model used both for filtering observations and predicting satellite states during times without observations. Generally for Earth orbiting satellites, improved accuracy is derived from improving the gravity modeling of the Earth. In the LPO region, however, spacecraft are perturbed by many celestial bodies and forces and thus the focus on dynamical modeling is driven by accurately accounting for relevant gravitational accelerations from celestial bodies and modeling solar radiation pressure effects. In many evaluations of station keeping performance or orbit evolution, simplifying assumptions are made to reduce the complexity of the dynamic models required. To understand the appropriateness of such assumptions and the relevancy of various perturbing forces, Figure 3 was developed. 4

5 Acceleration (m/s^2) 1.E-1 1.E 1.E-3 1.E Solar Gravity (m/s^2) Earth Gravity (m/s^2) Lunar Gravity (m/s^2) Jupiter Gravitational (m/s^2) 1.E-5 1.E-6 1.E-7 1.E-8 1.E-9 1.E-1 1.E-11 1.E-12 1.E-13 Solar Radiation Pressure (m/s^2) High In-plane Linear Approx. of Accel. Unc. (m/s^2) Venus Gravity (m/s^2) Saturn Gravity (m/s^2) Earth J2 (m/s^2) Mercury Gravity (m/s^2) Uranus Gravity (m/s^2) Low In-Plane Linear Appox. of Accel. Unc. (m/s^2) Mars Gravity (m/s^2) Neptune Gravity (m/s^2) Lunar J2 (m/s^2) Earth Relativistic Effect (m/s^2) 1.E-14 Days from Epoch Figure 3. LPO Acceleration Contributions The evaluation of the accelerations in Figure 3 is based on the position and velocity of the Artemis P2 spacecraft over a 5-day arc beginning January 4 th, 211. The ephemeris data was generated using operational mission data. Spacecraft parameters were derived from mission information and are detailed in subsequent tables. Gravitational accelerations from the Earth, Moon, and Sun are evaluated at the spacecraft s position. Magnitude of the 2-body gravitational effects are considered individually. Other planetary accelerations are similarly calculated based on their instantaneous position vector relative to the Moon in a heliocentric frame. Gravitational parameters not listed in Table 1 can be found in Reference 8. 5

6 Table 1. Parameter Values for Acceleration Calculations Parameter Value GM Earth E+14 GM Moon E+12 GM Sun E+2 J2 Earth J2 Moon (Time Averaged Solar Luminocity) c (speed of light) 3.839E+26 W m/s To calculate the acceleration due to Solar Radiation Pressure (SRP) a model using the absolute distance of the spacecraft from the Sun ( ) was used. It was observed over a 14-day evaluation of the SRP acceleration, that it varied.99% over what is approximately equivalent to 1 revolution of a LPO or appoximately half of an orbit about the Earth. To evaluate the acceleration due to this SRP force, the refectivity constant ( ) of 1.12, perpendicular area ( ) of.95 meters, and spacecraft mass (m) of kg were used from mission derived values. Where f is the radiative power per unit area. This can be solved for using the folowing equation, (1). (2) Using equations 1 and 2 an approximation for the acceleration on the spacecraft caused by Sun s radiation energy can be found. A simplified acceleration model from Reference 8 was used to evaluate the acceleration caused by J2 forces of both the Earth and Moon. For comparison a first order evealuation of the relativistic effects of the Earth were included in the plot and were derived using the following equation found in Reference 1. ( ) (4) To provide a rough estimate of the ability of various accelerations to be recovered in the OD process, two double-dashed lines in Figure 3 show the first order linear approximations of the accelerations recoverable for uncertainties related to the in-plane velocity component of the spacecraft. The high value is associated with a.1 cm/s uncertainty and the low value is associated with a.1 cm/s uncertainty. These values correspond to initial results associated with the orbit solutions for the mission. Ultimately the ability of the orbit solution to resolve accelerations will be a function of the arc length evaluated and the noise of the measurements. (3) 6

7 Tracking Stations and Data Types Tracking both P1 and P2 in the region near the Moon and during the trans-lunar phase, which saw them travel approximately 1.5 million km from Earth, significantly increased the requirements on the data and tracking systems as compared to the systems used previously for the THEMIS baseline mission. To accommodate these increased demands on both distance and geometry for tracking, the Artemis mission employs tracking assets from the Deep Space Network (DSN), the Universal Space Network (USN), and an 11-meter antenna at the University of California at Berkeley. Primarily the DSN assets used are the 34-meter antennas at Goldstone, CA, Canberra, Australia, and Madrid, Spain. The USN assets are the 13-meter antennas in Australia and Hawaii. The specific locations of the stations used for simulation of the data in this report and their associated parameters will be discussed in the following section. Data formats from these stations are currently processed by NASA Goddard Space Flight Center using the Goddard Trajectory Determination System (GTDS). The information provided includes range and Doppler tracking data. The USN and Berkeley stations provide this in the Universal Tracking Data Format (UTDF) while the DSN provides range and range-rate information in the TRK-34 format. 11,12 This DSN format is complex and non-trivial to convert. Embedded functionality within GTDS, however, is able to convert this data source to the UTDF. Solutions are then found using the embedded GTDS batch least squares algorithm. TEST DESCRIPTION As mentioned previously, the objective of this work is to simulate observational data for the Artemis mission in the Earth-Moon LPO regime. For this initial study one spacecraft was selected for simulation and evaluation. The focus of analysis was on understanding the effects of errors on filter performance given a known truth orbit as well as intricacies and nuances of the ODTK system. Such understanding will benefit future studies in the efforts to perform analysis and validate independent analyses performed with custom filters and scripts. Specifically explored in the following sections is the effect of known modeling errors on the orbit determination filter performance for a data arc given operationally informed initial conditions, tracking schedule, observation noise estimates, and spacecraft parameters. The remainder of this section will outline the parameters used as the baseline for this study with variations and their effects discussed later. The initial spacecraft position and velocity information was derived from operational orbit determination solutions provided by the mission team and are displayed in the True of Date reference frame. Table 2. Spacecraft Initial Conditions. Central Body: Earth Initial Epoch: 4 Jan :43:3. UTCG True of Date Initial Position (x, y, z) Inertial Velocity (vx, vy, vz) km km km km/s km/s km/s 7

8 Having defined the initial conditions for position, velocity, and epoch from operational mission solutions in Table 2, the spacecraft parameters were derived from mission information. The spacecraft mass is specific for spacecraft P2 and represents the mission team estimate of the mass of the vehicle at the evaluation epoch. The solar pressure area value was derived from operational mission information on what the spacecraft orbit determination solution has consistently converged on. While there may be room for improvement on all of these parameters they are considered sufficient given the current status of analysis and errors in their modeling will be evaluated and discussed in the following section. Using the nominal spacecraft values as defined will thus allow for the evaluation of filter susceptibility to errors or variation in such parameters. The spacecraft is a spinning vehicle, however for simplicity in this simulation it was assumed to be aligned/constrained with the body alignment vector in the negative Z direction and the body constraint vector in the X direction. Additionally the center of mass was assumed to be at the center of the spacecraft for simplicity. This simplification is most relevant to the perpendicular cross sectional area of the spacecraft for evaluation of SRP forces. It is acknowledged that this will insert modeling errors which will likely need to be resolved in the future. Mass (kg) Table 3. Spacecraft Parameters Solar Pressure: Area Solar Pressure: CPNominal Attitude m^ Aligned/Constrained * Utilizing the spacecraft parameters from Table 3, the next information required to simulate the spacecraft behavior is the force model with which to propagate the initial conditions and to which the filter will attempt to fit the observational data. This baseline force model is shown in Table 4 and incorporates gravitational perturbations from all planetary bodies in the Solar system in addition to a high fidelity gravity model of the Earth. Table 4. Spacecraft Force Model (baseline) Degree and Order 4 Tides Solid: OFF Ocean: OFF Variational Equations Third Bodies Degree: 2 Order: 2 Sun, Moon, Mars, Mercury, Venus, Jupiter, Saturn, Uranus, Pluto Perturbations SRP: ON Drag: OFF * Note that the Artemis spacecraft are spinners with a spin rate of approximately 17 RPM. This was omitted for simplicity in this simulation but should be included for future studies. 8

9 Having the spacecraft and force models sufficiently defined the next critical aspect which must be defined is the locations and properties associated with ground stations. These ground stations will subsequently be used to simulate range and Doppler observations for filtering with alterations to the filter force models. For this study several stations are included in the modeling and represent a simplification of the operational capabilities employed by the Artemis mission. Table 5 displays the approximations of the the ground station locations as used in this simulation. Station Table 5. Ground Station Locations Geodetic Latitude (deg) Geodetic Longitude (deg) Altitude (m) Berkeley Canberra (DSN) Goldstone (DSN) Madrid (DSN) Australia (USN) Hawaii (USN) These station coordinates thus allow for the simulation of range and Doppler measurements given the defined initial epoch and satellite state, force models, and satellite parameters. Additional information which has a non-trivial effect on the ability of the ODTK filter to converge on an orbit solution is the inclusion of noise and bias factors on the ground stations. The following measurement statistics were implemented for this simulation based on published estimates. 13 Ultimately evaluation of the observational data will be done to evaluate bias and noise values within the observation residuals. The parameters outlined in Table 6 are the Bias, a constant offset between the value of the measurement as predicted by the measurement model and the observed value of the measurement. The White Noise Sigma is defined as the square root of the variance representing the random uncertainty in the measurements and the Weight Sigma is defined as the square root of the variance of a Gauss Markov bias associated with the measurement model. Measurement Type Table 6. Station Measurement Statistics Bias White Noise Sigma Weight Sigma Range 15. m 3 m 2. Doppler. m.1 m/sec.1 With the modeling information provided to this point it was then possible to create simulated observations of the P2 spacecraft as it orbits the Earth-Moon L 1 point. To maintain validity of the simulation, the operational tracking schedule over this time period was used and is outlined in Table 7. 9

10 Table 7. Observation Intervals Station Berkeley Canberra (DSN) Goldstone (DSN) Madrid (DSN) Australia (USN) Hawaii (USN) Tracking Intervals 5 Jan :19:15. UTCG - 5 Jan :9:15. UTCG 6 Jan :54:. UTCG - 6 Jan :44:. UTCG 6 Jan :1:35. UTCG - 6 Jan :39:. UTCG 7 Jan 211 2:44:4. UTCG - 7 Jan 211 3:14:5. UTCG 8 Jan :33:3. UTCG - 8 Jan 211 2:23:3. UTCG 8 Jan 211 8:2:. UTCG - 8 Jan 211 1:2:. UTCG 5 Jan 211 2:1:. UTCG - 5 Jan :25:. UTCG 7 Jan :55:. UTCG - 7 Jan :5:. UTCG 7 Jan 211 :3:. UTCG - 7 Jan 211 1::. UTCG 8 Jan ::. UTCG - 8 Jan :3:. UTCG To validate that the simulation of data and the filter were working properly, observations were generated with no added errors. This would provide quantification of the errors caused by noise and bias on the observation data. Filter performance was evaluated based on the aforementioned modeling conditions with the ODTK sequential filter which resulted in 3D position RMS of.1362 m and 3D velocity RMS of e-8 m/s. To further refine the solution, the smoother function within ODTK was implemented and found to result in 3D position RMS of 7.276e- 4 m and 3D velocity RMS of 6.476e-9 m/s which can be considered the lowest attainable RMS errors given the noise on the data and observation lengths. The lack of process noise added through the filtering process renders this smoother result as merely a mapping of the final state solution backward in time. RESULTS Using the previously described simulation set-up, three error sources were evaluated and the following results outline the effect of known errors in the modeling of planetary ephemerides, solar radiation pressure reflectivity constant, and solar radiation pressure cross-sectional area. These results are meant to quantify the impact on accuracy of the sequential filter and the smoother within ODTK based on known errors. Such information provides insight into the impact and relative importance of spacecraft parameters needed for future work post-processing the orbit solutions for improved accuracy and dynamical knowledge. The first error which was evaluated using this simulation was the absence of planetary perturbations. An initial simulation of this effect was done with third body perturbations in the filter including only the Moon and Sun with the central body in the filter being the Earth. This is a common approximation used to simplify calculations in this regime. Calculated with the mission observations over the arc under consideration, the following results demonstrate the impact of omitting these planets in the force modeling for the filter and subsequently the smoother function. 1

11 C-position Cdot I-position Idot R-position Rdot Inertial Position RMS [x, y, z] (m): , , Inertial Velocity RMS [vx, vy, vz] (m/s): [ e-6, e-6, e-6] 3D Position RMS (m): D Velocity RMS (m/s): e-6 RIC Position Errors (red), 1-Sigma Covariance Envelope (green) Observation # 2 x 1-5 RIC Velocity Errors (red) x x Observation # Figure 4. Dynamic Model Error Evaluation I In order to display the data some plots do not include the 1-sigma covariance envelope into which all solution errors are found. Introducing Jupiter and Saturn into the force model for the filter and the smoothed results are shown below in Figure 5. Inertial Position RMS [x, y, z] (m): , , Inertial Velocity RMS [vx, vy, vz] (m/s): [ e-7, e-7, e-7] 3D Position RMS (m): D Velocity RMS (m/s): e-7 Figure 5. Dynamic Model Error Evaluation II Looking next to the effect of an error in modeling the solar radiation coefficient of reflectivity, multiple cases were run and a representative error of 5% is shown in Figure 6. These results are from running the filter with a solar radiation coefficient of reflectivity equal to 1.64 compared to the nominal truth value of In these results, as observed in other figures, the error in the radial direction is much smaller than the in-track and cross-track error directions. Inertial Position RMS [x, y, z] (m): , , Inertial Velocity RMS [vx, vy, vz] (m/s): [ e-5, e-5, e-5] 3D Position RMS (m): D Velocity RMS (m/s): e-5 11

12 C-position Cdot I-position Idot R-position Rdot C-position Cdot I-position Idot R-position Rdot 2 RIC Position Errors (red), 1-Sigma Covariance Envelope (green) Observation # 2 x 1-4 RIC Velocity Errors (red) x x Observation # Figure 6. Coefficient of Reflectivity Evaluation Looking again at the force of solar radiation pressure and its effect on the orbit solution filter performance, the cross sectional area used to calculate the solar radiation pressure force was altered in the third simulation case. Similar to the previous simulation, this value was modeled in the filter with ~5% error. Instead of the nominal.95 square meter area, it was instead modeled with a cross sectional area of.9 square meter. The resulting errors are displayed in the Figure 7. Inertial Position RMS [x, y, z] (m): , , Inertial Velocity RMS [vx, vy, vz] (m/s): [ e-5, e-5, e-5] 3D Position RMS (m): D Velocity RMS (m/s): e-5 1 RIC Position Errors (red), 1-Sigma Covariance Envelope (green) 2 x 1-4 RIC Velocity Errors (red) x x Observation # Observation # Figure 7. Cross Sectional Area Evaluation 12

13 C-position Cdot I-position Idot R-position Rdot A final simulation was run including all of the previous error sources in a single filter run. These results are shown in Figure 8 and represent a worst-case scenario given the modeled error sources. Inertial Position RMS [x, y, z] (m): , , Inertial Velocity RMS [vx, vy, vz] (m/s): [ e-5, , ] 3D Position RMS (m): D Velocity RMS (m/s): RIC Position Errors (red), 1-Sigma Covariance Envelope (green) 5 x 1-4 RIC Velocity Errors (red) x x Observation # Observation # DISCUSSION Figure 8. Cumulative Error Performance As presented in Figure 3, the relative perturbations on the 5-day arc of the LPO provide initial insight into the impact of force modeling on propagation and filter performance. This figure is meant to exhibit the force impact in the specific case of the P2 spacecraft during a 5-day section of its orbit in January 211. This figure is not meant to apply universally to other LPO regions or time-periods although future efforts to generate longer time-scale and higher fidelity information has the potential to provide valuable insight in the preparation of filtering schemes or the simplification of force modeling. Of the resulting acceleration magnitudes identified, solar radiation pressure presents the most difficulty in modeling or accurately quantifying. Planetary ephemerides can be incorporated into filter force models or into propagation schemes relatively easily as compared to the dynamic effects of varied solar radiation pressure on a spacecraft. This problem was outlined recently by McMahon, who put forth an analytical theory to model solar radiation pressure effects on spacecraft. 14 Additional unmodelled accelerations or unaccounted for perturbations are undoubtedly present in the true operational environment. The first result discussed as a validation step in this simulation demonstrated that using published estimates for the noise on observations and given a nominal tracking schedule for the Artemis mission, filter performance as measured by 3D RMS errors and after being smoothed were found to be 7.276e-4 m in position and 6.476e-9 m/s in velocity. These are admittedly created under ideal simulated circumstances but are demonstrative of achievable fidelity. Future work investigating the observation residuals for the Artemis data has the potential to refine knowledge of the true observation noise present and thus the attainable fidelity with which the spacecraft states can be solved for. A better quantification of this fidelity will open the door to advanced schemes for recovering unmodelled accelerations such as the use 13

14 of second order Gauss-Markov processes. 15 Moreover, the use of filtering schemes such as an Extended Kalman Filter (EKF) to update the reference trajectory or an Uncented Kalman Filter (UKF) for improved non-linear filtering and quantification of realistic uncertainty bounds will be explored. The simulations presented in this work demonstrate that uncertainty in spacecraft parameters has the potential for more significant filter errors than does planetary ephemerides or gravitational models. In addition to the added susceptibility of orbit determination filters to errors in modeling solar radiation pressure in the LPO region, there is also an inherent problem with observation geometry in this regime. Although not seen in all figures due to the emphasis on displaying orbit errors, in Figures 4, 6, 7, & 8, the radial direction in all cases has lower uncertainty as compared to the in-track and cross-track directions. This is a result of having Earth based observations which are geometrically constrained to favor the radial direction. Better observation geometry would provide mission designers with improved performance relative to uncertainty and ultimately improved accuracy potential. Adding inter-spacecraft data to the filter process has the potential to provide this added geometric robustness. Prior work has demonstrated the ability of both relative and absolute orbit determination capabilities for spacecraft in a LPO due to the asymmetry of the gravitational field and the resulting uniqueness of orbit behavior. 16 Alternately, recent work by Tombasco has investigated a similar constraint for satellites in GEO with poor geometric observability. This work has demonstrated that the selection of differing element sets for orbit determination can improve filter performance in the presence of poor geometric observability. 17 CONCLUSIONS Simulating the operational environment for the P2 spacecraft of the Artemis mission has led to the quantification of individual relative acceleration contributions on the satellite. It has also demonstrated the effect on filter/smoother performance within ODTK of observation noise, errors in modeling gravitational contributors and uncertainties associated with solar radiation parameters. Filter performance was shown to be minimally effected by observation noise as modeled. Omitting planetary perturbations resulted in modest RMS errors. Incorrect spacecraft modeling with respect to solar radiation pressure was seen to exhibit non-negligible RMS errors. The ultimate goal of this work is to improve understanding of operations in the Earth-Moon LPO regime. This is significantly driven by the need for orbit determination solutions that enable frequent stationkeeping maneuvers to remain in highly dynamic orbits. Future improvements to obtaining orbit solutions for spacecraft will require improvements in data sources, filter schemes, and operational execution to minimize cost and increase operability. Opportunities for follow-on studies in this area are bountiful and will be based upon this preliminary work to further study operational constraints to efficient orbit determination. Some areas that this future work may include are the study of solar radiation pressure on orbit accuracy, the potential for specific element sets or frames to improve orbit determination filter performance, the benefit of modeling spacecraft maneuvers in the orbit determination filter over longer filter times, the evolution of non-gaussian uncertainty throughout the orbit and given differing observation geometries, and the true measurement noise by evaluating operational measurement residuals. Information on relevant forces, appropriately modeling such influences, and realistically quantifying the uncertainty associated with orbit solutions are significant motivating factors for studying Artemis operational data. These forces, modeling parameters, and uncertainties have a direct impact on the ability of orbit determination filters to produce accurate results. As was demon- 14

15 strated in the simulated cases for this paper, accuracy in orbit solutions is tied to the ability to know spacecraft parameters with a high degree of confidence. In a broader sense this knowledge relates to the fundamental challenge facing LPO utilization: orbit determination. Sufficient performance has been demonstrated using traditional software and tracking capabilities by the Artemis mission team to plan station keeping maneuvers and maintain orbits in the region. 18 As the Artemis spacecraft begin the Lunar orbit phase of their mission untold additional science and engineering data will be generated for evaluation. Future missions to the Earth-Moon LPO region will benefit tremendously from the work done by the Artemis mission team. It is the hope of the authors of this work to build upon this incredible achievement and to improve our fundamental knowledge of the Earth-Moon region with the ultimate goal of enabling robust utilization of libration point orbits in cis-lunar space. ACKNOWLEDGMENTS This work is made possible by the close collaborative assistance of David Folta and Mark Woodard within the Flight Dynamics Branch at NASA Goddard Space Flight Center. This work was supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program. REFERENCES 1 Hill K., J. Parker, G. H. Born, and N. Demandante, A Lunar L2 Navigation, Communication, and Gravity Mission, Paper AIAA , AIAA/AAS Astrodynamics Specialist Conference, Keystone, Colorado, August 214, Folta, D., and F. Vaughn, A Survey of Earth-Moon Libration Orbits: Stationkeeping Strategies and Intra-Orbit Transfers, Paper AIAA , AIAA/AAS Astrodynamics Specialist Conference, Providence, Rhode Island, August August, Woodard, M., David Folta, and Dennis Woodfork, ARTEMIS: The First Mission to the Lunar Libration Orbits Broschart, S.B.,M-K.J. Chung, S.J. Hatch, J.H. Ma, T.H. Sweetser, S.S. Weinstein-Weiss, and V. Angelopoulos, Preliminary Trajectory Design for the ARTEMIS Lunar Mission, AAS Paper 9-382, AAS/AIAA Astrodynamics Specialists Meeting, Pittsburgh, PA, August 1-13, Angelopoulus, V., The ARTEMIS Mission, Space Science Reviews, August 21. doi:1.17/s Szebehely, V., Theory of Orbits: The Restricted Problem of Three Bodies, Academic Press, New York, Farquhar, R. W., Lunar Communications with Libration-Point Satellites, Journal of Spacecraft and Rockets, Volume 4(1), Vallado, D.A. and W.D. McClain, Fundamentals of Astrodynamics and Applications, Space Technology Library, Springer Link Konopliv, A.S., et al. Improved Gravity Field of the Moon from Lunar Prospector, Science 281(1476), doi:1.1126/science LUNAR PAPER 1 Montenbruck, O. and E. Gill, Satellite Orbits Models, Methods, and Applications, Springer-Verlag, Heidelberg, Deep Space Network Services Catalog, DSN No. 82-1, Rev. E, JPL D-192, December 17, TRK-34 DSN Tracking System Data Archival Format, DSN No , Rev I-1, JPL D-16765, February 29,

16 13 Schanzle, A., D. Kelbel, and D. Oza, Error Sources and Nominal 3-sigma Uncertainties for Covariance Analysis Studies using ODEAS (Update No. 2), Task Assignment , NASA GSFC, May 31, McMahon, J.W., An Analytical Theory for the Perturbative Effect of Solar Radiation Pressure on Natural and Artificial Satellites, Ph.D Thesis, University of Colorado Boulder Leonard, J.M., F.G. Nievinski, and G.H. Born, Gravity Error Compensation Using Second-Order Gauss-Markov Processes, AAS 11-52, AAS/AIAA Astrodynamics Specialist Conference, Girdwood, Alaska, July 31 Aug. 4, Hill, K., Autonomous Navigation in Libration Point Orbits, Ph.D Thesis, University of Colorado Boulder, Tombasco, J., Orbit Estimation of Geosynchronous Objects Via Ground-Based and Space-Based Optical Tracking, Ph.D. Thesis, University of Colorado Boulder, Woodard, M., D. Cosgrove, P. Morinelli, J. Marchese, B. Owens, and D. Folta, Orbit Determination of Spacecraft in Earth-Moon L1 and L2 Libration Point Orbits, AAS , AAS/AIAA Astrodynamics Specialist Conference, Girdwood, Alaska, July 31 Aug. 4,