1 EVALUATING ORBIT DETERMINATION POST-PROCESSING METHODS FOR OPERATIONAL ARTEMIS DATA Bradley W. Cheetham * and George H. Born † 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 chal- lenge. The Artemis mission operated by the NASA Goddard Space Flight Cen- ter and the University of California at Berkeley recently became the first to ever maintain orbits in this regime. The resulting operational data provides signifi- cant opportunity for analysis to better understand these orbits and their opera- tional 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 dy- namic models, experimentally models errors in a simulated environment, and outlines areas of future focus. Realistic spacecraft ephemeris and attribute in- formation will be used to the maximum extent possible. Simulations of orbit de- termination efficacy are performed using the Analytical Graphics Inc. Orbit De- termination Tool Kit (ODTK) with appropriate tracking and spacecraft charac- teristics and known error sources. INTRODUCTION This work is motivated by the desire to better understand the dynamic environment and opera- tional 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 Fig- ure 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 ma- neuvers 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 10 me- ters 1 to more than 50 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 ac- curacy 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, 80309. † Director, Colorado Center for Astrodynamics Research, University of Colorado, Boulder Aerospace Engineering Sciences, 431 UCB, 80309. AAS 11-513
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1
EVALUATING ORBIT DETERMINATION POST-PROCESSING METHODS FOR OPERATIONAL ARTEMIS DATA
Bradley W. Cheetham* and George H. Born†
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 chal-
lenge. The Artemis mission operated by the NASA Goddard Space Flight Cen-
ter and the University of California at Berkeley recently became the first to ever
maintain orbits in this regime. The resulting operational data provides signifi-
cant opportunity for analysis to better understand these orbits and their opera-
tional 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 dy-
namic models, experimentally models errors in a simulated environment, and
outlines areas of future focus. Realistic spacecraft ephemeris and attribute in-
formation will be used to the maximum extent possible. Simulations of orbit de-
termination efficacy are performed using the Analytical Graphics Inc. Orbit De-
termination Tool Kit (ODTK) with appropriate tracking and spacecraft charac-
teristics and known error sources.
INTRODUCTION
This work is motivated by the desire to better understand the dynamic environment and opera-
tional constraints of spacecraft in the Earth-Moon three-body regions. Specifically of interest are
libration point orbits (LPOs) about the Earth-Moon co-linear L1 and L2 points as labeled in Fig-
ure 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 ma-
neuvers 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 10 me-
ters1 to more than 50 m/s/yr including modeling and maneuver errors.
2 The Artemis mission
budgeted approximately 15 m/s for the planned ~6 month L1/L2 orbit maintenance portion of the
mission. The magnitude of these station keeping maneuvers is very highly dependent on the ac-
curacy 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, 80309. † Director, Colorado Center for Astrodynamics Research, University of Colorado, Boulder Aerospace Engineering
Sciences, 431 UCB, 80309.
AAS 11-513
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 determi-
nation 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 L1-L2 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 perturba-
tions 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 op-
portunity for scientific exploration, cis-lunar space observations, and future space development.
The same dynamics that make LPOs about L1, L2, and L3 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 geomet-
ric 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 per-
form stationkeeping operations in these Earth-Moon LPOs. The observations and navigation so-
lutions 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 the-
se specific orbital regimes and experience in synthesizing orbit solutions given a program devel-
oped 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.
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 L1 and L2 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 L1 and L2. These or-
bits 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 geomagnet-
ic field, specifically as the Moon passes through the wake of the geomagnetic field. From a mis-
sion design perspective, these orbits are used to reduce the inclination of the orbits and are ulti-
mately 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 re-
gions. Specifically for the simulations in this paper, the P2 spacecraft was considered during an
orbit section about the L1 point.
Figure 2. Artemis LPO Phase4
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 tradi-
tionally have been simulated and studied using specific assumptions to facilitate evaluation in-
cluding the circular restricted three body problem (CRTBP).6 Using this simplified model, equa-
tions 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 utiliza-
tions of these orbits some of which were alluded to earlier in this paper. One of the initial cham-
pions of such orbits in the 1960s 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 fre-
quently 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 fi-
delity 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, space-
craft 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 perfor-
mance or orbit evolution, simplifying assumptions are made to reduce the complexity of the dy-
namic models required. To understand the appropriateness of such assumptions and the relevan-
cy of various perturbing forces, Figure 3 was developed.
5
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 4th, 2011. 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.
1.00E-14
1.00E-13
1.00E-12
1.00E-11
1.00E-10
1.00E-09
1.00E-08
1.00E-07
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
0 1 2 3 4 5
Acc
eler
ati
on
(m
/s^
2)
Days from Epoch
Solar Gravity (m/s^2)
Earth Gravity (m/s^2)
Lunar Gravity (m/s^2)
Jupiter Gravitational (m/s^2)
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)
6
Table 1. Parameter Values for Acceleration Calculations
Parameter Value
GM Earth 3.986004418E+014 ⁄
GM Moon 4.902798882E+12 ⁄
GM Sun 1.3271240E+020 ⁄
J2 Earth 0.0010826269
J2 Moon9 0.000203428
(Time Averaged
Solar Luminocity) 3.839E+26 W
c (speed of light) 299792458 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 0.99% 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 85.243 kg were used from mission derived values.
(1)
Where f is the radiative power per unit area. This can be solved for using the folowing
equation,
| | . (2)
Using equations 1 and 2 an approximation for the acceleration on the spacecraft caused by
Sun’s radiation energy can be found.
(3)
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 10.
(
) (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 0.01 cm/s uncertainty and the ‘low’ value is
associated with a 0.001 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 ac-
celerations will be a function of the arc length evaluated and the noise of the measurements.
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 require-
ments 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 ge-
ometry 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 Cali-
fornia 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 Cen-
ter using the Goddard Trajectory Determination System (GTDS). The information provided in-
cludes range and Doppler tracking data. The USN and Berkeley stations provide this in the Uni-
versal Tracking Data Format (UTDF) while the DSN provides range and range-rate information
in the TRK-2-34 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 select-
ed 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 vali-
date 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 condi-
tions, 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 de-
termination solutions provided by the mission team and are displayed in the True of Date refer-
ence frame.
Table 2. Spacecraft Initial Conditions.
Central Body: Earth Initial Epoch: 4 Jan 2011 16:43:30.000 UTCG
True of Date
Initial Position
(x, y, z) 41446.5477729 km -323743.687141 km -138118.185082 km