A Lunar L2 Navigation, Communication, and Gravity Mission · 2017. 8. 27. · A Lunar L2...

A Lunar L2 Navigation, Communication, and Gravity

Mission

Keric Hill,∗ Jeffrey Parker,∗ George H. Born,† and Nicole Demandante‡

University of Colorado, Boulder, Colorado 80309

Most high-priority landing sites on the lunar surface do not have direct communica-

tion with the Earth. A mission was designed which would involve two spacecraft, a halo

orbiter at Earth-Moon L2 and a microsatellite in a low lunar orbit. The two spacecraft

would travel together on a ballistic lunar transfer and arrive on the Earth-Moon L2 halo

orbit. No insertion maneuver would be required, which would allow the spacecraft to be

about 25% to 33% more massive than if a conventional transfer were used. After that,

the microsatellite would only require a small maneuver to depart from the halo orbit and

descend to where it can be inserted into a low lunar orbit. The spacecraft would relay

communications from the lunar far side and south pole to Earth. They could also be used

to track other lunar missions, or broadcast navigation signals. To lower the operations

cost, a new method of autonomous orbit determination called “Liaison Navigation” would

be used. To perform Liaison Navigation, spacecraft in libration point orbits use scalar

satellite-to-satellite tracking data, such as crosslink range, to perform orbit determination

without Earth-based tracking. Due to the characteristics of libration orbits, relative track-

ing between two spacecraft can be used to estimate the relative and absolute positions and

velocities of both spacecraft simultaneously. The tracking data from the two spacecraft

could also be used to estimate the lunar far side gravity field.

Nomenclature

BLT Ballistic Lunar Transfercomm/nav Communications and NavigationCRTBP Circular-restricted Three-body ProblemDSN Deep Space NetworkEKF Extended Kalman FilterESAS Exploration Systems Architecture StudyJPL NASA Jet Propulsion LaboratoryLiAISON Linked, Autonomous, Interplanetary Satellite Orbit NavigationLL2 Second Earth-Moon Lagrange Point (L2)OD Orbit DeterminationSKM Stationkeeping maneuverSRP Solar Radiation PressureSSST Scalar Satellite-to-Satellite Trackingδ Phase angle for Snoopy-Woodstock orbits (τ − 2Ω)τ Nongeometric angle describing position along a halo orbit∆V Change in velocityΩ Longitude of the ascending node relative to lunar prime meridian

∗Graduate Research Assistant, Colorado Center for Astrodynamics Research, Member AIAA.†Director, Colorado Center for Astrodynamics Research, Campus Box 431, Fellow AIAA.‡Undergraduate Research Assistant, Colorado Center for Astrodynamics Research.

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American Institute of Aeronautics and Astronautics

AIAA/AAS Astrodynamics Specialist Conference and Exhibit21 - 24 August 2006, Keystone, Colorado

AIAA 2006-6662

Copyright © 2006 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

I. Introduction

To fulfill the mandate to return to the Moon outlined in the Vision for Space Exploration, NASA con-ducted the Exploration Systems Architecture Study.1 The ESAS final report issued in November 2005 liststen sample landing sites for manned missions to the Moon. Six out of these ten sites are at the poles, limb,or far side of the Moon where direct communication links to Earth are impossible. Because of this, theESAS study concluded that an orbital communication and navigation (comm/nav) infrastructure would beneeded. According to ESAS, this constellation would be used to help provide support for orbit insertion,landing, surface operations, and Earth transit. It would be used for intervehicle and Moon-to-Earth com-munications as well as to provide tracking data for navigation. It should be designed to evolve seamlesslyas the requirements and numbers of user spacecraft and missions increase. The constellation should also be“capable of uninterrupted, multi-year activity” in addition to being reliable and redundant.

Several concepts for a lunar comm/nav constellation have already been proposed. Farquhar proposedusing halo orbits at Earth-Moon L2 (LL2) to provide communications for the far side of the Moon duringthe Apollo Program.2 Although the idea was not used for Apollo, a communications relay satellite at LL2

could provide communications for the far side and south pole and enable better determination of the far sidegravity field.

Spacecraft placed in halo orbits have several identified advantages and disadvantages when compared tospacecraft placed in low lunar orbits. Carpenter et al. showed that a few halo orbiters could provide thesame coverage as a larger number of lunar orbiters.3 Halo orbiters have longer communication contacts withthe surface, meaning that there are fewer times when it is necessary to break contact with one spacecraftto acquire the signal of another. Tracking antenna pointing is easier because of the slower apparent motionof a halo orbiter when viewed from the lunar surface. Halo orbiters are illuminated by the Sun more thantypical low lunar orbiters, and this paper shows that halo orbits are more fuel-efficient than low lunar orbits:they require less energy to reach from Earth and they can have lower stationkeeping costs. A disadvantage isthat the link distance from the Moon to a halo orbiter would be larger. Disadvantages for all the previously-proposed lunar comm/nav constellations were that they required several launches to build the constellationsand a lot of expensive Deep Space Network (DSN) time to track each spacecraft.

This paper presents a new method for the construction and operation of a comm/nav infrastructure atthe Moon that takes advantage of all of the benefits of halo orbits and reduces the costs of establishing andmaintaining that infrastructure. The operational costs of the comm/nav constellation may be substantiallyreduced using a new method of autonomous orbit determination for halo orbiters called “Liaison Navigation.”The costs of establishing the constellation may be substantially reduced using “Ballistic Lunar Transfers”that are low-energy trajectories spacecraft may use to transfer from the Earth to the Moon for considerablyless ∆V than conventional transfers. The details of these techniques and how they reduce the cost of thecomm/nav infrastructure at the Moon are described in later sections of this paper.

II. Mission Concept

A mission to the Moon is described here which would involve two spacecraft: one called “Snoopy” in ahalo orbit at LL2 and another called “Woodstock” in a low lunar, polar orbit. Snoopy would be able toprovide communications coverage for the south pole of the Moon for extended periods of time. Woodstockwould provide intermittent coverage for the lunar poles as well as other areas of the Moon’s surface. Bothcould be used as tracking stations for spacecraft orbiting, landing, or roving at the Moon. The lunar far sidegravity field has not been very well determined because no direct observations have been made of spacecraftin orbit on the far side.4 This mission would also provide the opportunity to obtain tracking data forspacecraft on the far side, to enable the estimation of the far side gravity field of the Moon.

During the validation phase of the mission, Snoopy would be used to track Woodstock, alone and in con-junction with the DSN. This tracking data could be processed on the ground to determine lunar gravity fieldparameters. During this time, orbit determination and maneuvers for both spacecraft would be computedboth on the ground, and on Snoopy using Liaison Navigation. If Liaison Navigation provides adequate re-sults in operation and proves to be a reliable method of performing autonomous orbit determination, Snoopycould be switched into autonomous navigation mode for the rest of the mission.

During the operational phase of the mission, Snoopy could be used to relay communications from thelunar far side and south pole to Earth. Woodstock could also be used to receive faint signals from rovers

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or landers on the Moon and relay them to Earth, either directly or through Snoopy. During periods whenSnoopy is below the horizon at the south pole, Woodstock would provide periodic coverage since it passesover the poles once an orbit. Snoopy and Woodstock could be used to track other lunar spacecraft or tobroadcast navigation signals. In the future, another halo orbiter could join Snoopy to provide continuouscoverage to the lunar south pole using the method outlined by Grebow, et al.5 With even more spacecraft,continuous coverage for the entire lunar far side and polar regions is possible using a constellation similar tothat proposed by Carpenter, et al.3

A. Ballistic Lunar Transfers

Figure 1. An example ballistic lunar transferviewed in the Sun-Earth rotating frame fromabove the ecliptic.

The lunar comm/nav spacecraft would be placed into theirmission orbits using a single launch vehicle. A low-energyBallistic Lunar Transfer (BLT)6, 7 would be used to trans-fer the two spacecraft together to a halo orbit about LL2.

The launch vehicle would propel the two spacecraft ona 1 − 1.5 million-kilometer journey away from the Earth.They remain at that distance for a period of time underthe gravitational influence of the Earth, Moon, and theSun. As the spacecraft fall back to the Moon, they ballis-tically arrive at the lunar halo orbit, requiring no insertionmaneuver at all. The entire transfer requires much less en-ergy than a conventional direct transfer, allowing payloadsto be 25% to 33% larger in mass, depending on the launchvehicle. Figure 1 shows an example ballistic lunar trans-fer viewed in the Sun-Earth rotating frame from above theecliptic.

After the two spacecraft reach the halo orbit, Woodstock would separate and follow a low-energy transferdown to the Moon like the one shown in Figure 2. Departing from the halo orbit only requires a few metersper second ∆V. However, there is a required injection maneuver to capture into the low lunar orbit. Anylunar orbit is attainable, including polar and equatorial orbits for essentially the same ∆V budget.

In the future, if the constellation is to be reconfigured, Snoopy may follow similar low-energy transfersto new station-locations such as that shown in Figure 3.8 Spacecraft may even be sent from halo orbits backto the Earth along a transfer that mirrors the BLT shown in Figure 1.

Figure 2. An example low-energy transfer froma halo orbit to a low, polar orbit about theMoon.

Figure 3. Example low-energy transfers between orbitsabout the lunar L1 and L2 points, including an optionaldistant retrograde orbit about the Moon.

Dynamically, halo orbits have characteristic stable and unstable directions. If a spacecraft on a haloorbit is perturbed in the unstable direction, the perturbation becomes magnified and the spacecraft fallsaway from the halo exponentially. On the other hand, if the spacecraft is perturbed in the stable direction,

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it will asymptotically return to the nominal orbit. The unstable manifold is the surface defined by all of thetrajectories that leave the halo orbit in the unstable direction and the stable manifold is the surface definedby all of the trajectories that approach the halo orbit from the stable direction. By propagating trajectoriesand creating these manifolds, it is possible to find the low-energy transfers to and from the halo orbits usedin this mission.

B. Autonomous Navigation

In response to the high cost of Earth-based tracking and maneuver design, the DSN is currently looking forways to reduce ground operations costs by making deep space missions more autonomous and by providingmore in situ tracking resources.9 A new method of autonomous orbit determination could be used for justsuch purposes. Spacecraft in libration orbits can use scalar satellite-to-satellite tracking (SSST) data, suchas crosslink range, to perform autonomous orbit determination. This technique has been called LiAISON or“Liaison Navigation.” Due to the characteristics of libration orbits, relative tracking between two spacecraftcan be used to estimate the relative and absolute positions and velocities of both spacecraft simultaneously.10

SSST between two spacecraft only provides information on the size, shape, and relative orientation oftwo orbits. A pair of Keplerian orbits of a certain size, shape, and relative orientation can have any absoluteorientation about the center of mass of the primary body. This means that SSST alone is not sufficientto autonomously determine the absolute orientation of the conic orbits of spacecraft in near Earth orbits.However, in the three-body problem the gravitational influence of the third body can indirectly provideinformation about the direction to that third body and with it, the absolute orientation of the orbits. Inother words, a halo orbit near the Moon is influenced very strongly by both the Earth and the Moon. Becauseof the strong asymmetry of the three-body force field, a halo orbit with a given size and shape can onlyhave a single orientation with respect to the Earth and Moon. This means that a spacecraft in a halo orbitcan track a second spacecraft using crosslink range measurements and determine the absolute positions andvelocities of both spacecraft simultaneously without any Earth-based tracking or mathematical constraints.While one spacecraft must be in a halo orbit, the second spacecraft may be in any orbit either about theMoon, about a libration point, or in transit somewhere in the Earth-Moon system.

Simulations have shown that this technique works well in lunar halo orbits both in the CRTBP11 andin an inertial model with solar gravity and SRP.12 A lunar constellation could use Liaison Navigation todetermine the positions and velocities of all the spacecraft autonomously. This knowledge could be used tocompute stationkeeping maneuvers on board the spacecraft. With automated maneuver planning, the burnscould be performed more often to decrease the amount of fuel needed. Ground stations on Earth wouldonly be needed to uplink commands and receive telemetry and relayed data. Ground stations would notneed to obtain any tracking data, except for small amounts used to verify that the autonomous navigationsoftware is working properly. Two-spacecraft constellations can get the best orbit determination (OD) whenone spacecraft is in a halo orbit and the other is in a lunar orbit. Thus, Snoopy could track Woodstock fromLL2 and use Liaison Navigation to estimate the orbits of both spacecraft and become an asset to, instead ofa burden on, the DSN.

III. LL2 Halo Orbit Selection (Snoopy)

There is a full continuum of orbits in the family of halo orbits about LL2, and the optimum halo wasselected for Snoopy using seven requirements. In addition to satisfying the needs of the current mission, thehalo orbit was chosen so that in the future, a second spacecraft could be placed at LL2 to provide continuouscoverage of the south pole and other missions orbiting the Moon. The requirements are:

1. The LL2 halo orbit should be above 10 elevation over the entire lunar south pole region (80 to 90S)for the estimated length of the first lunar sorties, or seven continuous days of each halo orbit period.

2. With all other requirements satisfied, the LL2 halo orbit stationkeeping costs should be minimized.

3. With all other requirements satisfied, the LL2 halo orbit determination error should be minimized.

4. With all other requirements satisfied, the percentage of the time that Woodstock is not visible fromeither the Earth or Snoopy should be minimized, and tracking data should be obtainable over theentire lunar far side.

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5. With a future second spacecraft in the halo orbit, at least one of the two halo orbiters should be above10 elevation over the entire lunar south pole region at all times.

6. Either the two halo orbiters mentioned above, or Earth-based tracking stations should be able to trackspacecraft in any lunar orbit at all times.

7. It should be possible for Woodstock to separate from Snoopy and, using only a very small perturbationto the trajectory, travel to a point where an insertion into the proper lunar orbit can be performed.

To verify compliance with the requirements, quantifiable metrics were computed for each of them.

A. Requirement 1

For requirement 1, several locations in the lunar south pole regions were selected. For each of those surfacelocations, the azimuth and elevation of a halo orbiter were computed over an entire halo orbit period for awide spectrum of LL2 halo orbits generated in the Circular-restricted Three-body Problem (CRTBP). Theamount of time that a halo orbiter was above 10 was computed as a metric to verify that requirement 1was met. Figure 4 shows the results at 80S 0E, which was the worst case. The Jacobi constant13 for thehalo orbits, C, is computed using the following equation:

C = −V 2 + x2 + y2 + 2

(

1 − µ

r1

+µ

r2

)

.

V is the velocity in the rotating frame. Requirement 1 was satisfied for all halo orbits with Jacobi constantsless than about 3.085.

3.013.023.033.043.053.063.073.083.093.13.113.123.133.140

1

2

3

4

5

6

7

8

9

Jacobi Constant

Days

of c

over

age

Three−body halo7 day requirement

Figure 4. Number of days a halo orbiter is visible above 10 elevation at 80S 0E

B. Requirements 2 and 3

For requirements 2 and 3, a simulation was performed. Nominal trajectories were computed for a spectrumof halo orbits. The altitude of the lunar orbit had not yet been selected, so a nominal polar, lunar orbit at 100km altitude was computed. The orbit propagation included SRP and the gravitational forces of the Moon,Sun, Venus, Earth, Mars, Jupiter, and Saturn, computed using the JPL DE405 ephemeris. The nominalhalo orbit was generated with a Multiple Shooting method14, 15 and spanned four revolutions. A “true” orbitfor each spacecraft was generated by randomly perturbing the initial conditions. Spacecraft-to-spacecraftrange observations were generated and corrupted using white noise with a standard deviation of 1 m. Thestates for both the halo and lunar orbiter were estimated with a Batch Processor and Liaison Navigation.The results from Liaison Navigation were used to compute the stationkeeping maneuvers (SKM) required

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to keep Snoopy near the nominal halo orbit. For each halo orbit generated with a different Jacobi constant,ten simulations were performed. The OD and SKM results were averaged and displayed in Figure 5. TheOD error shown is an average for the two spacecraft.

3.013.023.033.043.053.063.073.083.093.13.113.123.133.140

1

2

3

4

5

6

7

Jacobi Constant

OD error (m)SKM budget (m/s/yr)

Figure 5. Orbit Determination and Stationkeeping simulation results.

The orbit determination results show some noise because the data points are really an average of only10 runs with random initial conditions. But the general trend is for the OD error to trend upwards below aJacobi constant of about 3.08. The stationkeeping results show that there is a large spike in the stationkeepingbudget at a Jacobi constant of about 3.065. This spike is centered on the Jacobi constant for halo orbitswith a period in 2:1 resonance with the Moon’s orbital period about the Earth. If this spike is avoided, therest of the halo orbits have a yearly stationkeeping budget of about 1 m/s.

C. Requirement 4

Compliance with requirement 4 was verified by simulating Snoopy’s halo orbit and Woodstock’s orbit in theCRTBP. A circular, polar, lunar orbit with an altitude of 100 km was used for Woodstock. About every sixminutes for 4 months, if either Snoopy or the Earth was able to see and track Woodstock, then Woodstock’slatitude and longitude was computed and tallied in bins that were 10 of latitude by 10 of longitude. Thedensity of tracking coverage for bin i, j, Di,j , was computed using the number of tracking points tallied inthe bin, Ti,j , the total number of points Ttotal, and the area of that bin on the surface of the Moon, Ai,j :

Di,j =Ti,j/Ttotal

Ai,j

.

The results were plotted and had a marked correlation between the position of Snoopy on its halo orbit, τ ,and the Moon-fixed longitude of the ascending node of Woodstock’s orbit, Ω. τ is a non-geometric anglesimilar to mean anomaly for halo orbits. For LL2 halo orbits, τ is zero when the spacecraft is at the pointclosest to the Moon and goes through 360 in one halo orbit period. The value of Ω goes through 360

every time the Moon rotates in inertial space (≈ 28 days). Since the periods of the LL2 halo orbits areapproximately half of the rotation period of the Moon, there is a near 2:1 resonance between τ and Ω.The relationship between the two parameters can be quantified with the angle δ which is computed in thefollowing way:

δ = τ − 2Ω.

The value of δ for halo orbits only changes slowly, the rate of change depending on how close the halo orbitis to a 2:1 resonance with the Moon’s orbit. By picking the right δ, the total Snoopy-Woodstock trackinggaps on the far side can be minimized. The total tracking gaps are computed as a percentage:

Total tracking gaps =

∑

i

∑

j Ti,j

Ttotal

× 100%.

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3.013.023.033.043.053.063.073.083.093.13.113.123.133.140

1

2

3

4

5

6

7

Jacobi Constant

Gap

s in

trac

king

(%)

best initial δworst initial δ

Figure 6. Gaps in tracking as a percentage of the time when Woodstock is not visible from Earth or Snoopy.

Figure 6 shows the best and worst that could be achieved for each halo orbit.If the best case δ can be achieved, the gaps in tracking data are relatively low until the Jacobi constant

goes below about 3.04.

D. Requirement 5

For requirement 5, the elevation was checked again at various south pole locations using two halo orbiterson the same halo orbit, phased 180 apart. The orbit propagation included SRP and the gravitational forceof the Moon, the Sun, Venus, Earth, Mars, Jupiter, and Saturn computed using the JPL DE405 ephemeris.Each halo orbit was generated with a Multiple Shooting method14, 15 and spanned five halo orbit periods.Gaps in tracking at a surface location occurred when both of the halo orbiters were below 10 elevation.In order for requirement 5 to be met, the tracking gaps had to occur 0% of the time. Figure 7 shows thetracking gaps for each halo orbit at the worst case surface location. The results are also shown when theminimum elevation requirement is increased to 15. This coverage requirement was satisfied for all haloorbits with Jacobi Constants less than about 3.085.

3.013.023.033.043.053.063.073.083.093.13.113.123.133.140

2

4

6

8

10

Jacobi Constant

Trac

king

Gap

(%)

10° min elev15° min elev

Figure 7. Tracking gaps as a percentage of the time neither of two halo orbiters is visible at the worst casesurface locations.

E. Requirement 6

Requirement 6 was checked in the same way as requirement 4, except that a second halo orbiter was simulated180 away from the first. The percentage of the time when Woodstock could be tracked by neither the two

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halo orbiters nor the Earth is plotted in Figure 8. Complete coverage of Woodstock is possible with Earthtracking and two LL2 halo orbiters except for orbits with Jacobi constants less than about 3.04.

3.013.023.033.043.053.063.073.083.093.13.113.123.133.140

1

2

3

4

5

6

7

Jacobi Constant

Gap

s in

trac

king

(%)

worst initial δ

Figure 8. Gaps in tracking as a percentage of the time when Woodstock is not visible from Earth or two haloorbiters.

F. Requirement 7

Requirement 7 was met by first designing Woodstock’s descent to its low lunar orbit in the CRTBP andthen by simulating the descent including the JPL ephemeris model of the solar system. The halo orbitsin the Earth-Moon CRTBP may be used as a starting approximation to quasi-halo orbits in the real solarsystem. The CRTBP halo orbits are perfectly periodic; hence, they are more predictable and structured.But a single CRTBP solution may deviate into many similar solutions in the real solar system. Thus, theCRTBP is useful to narrow down the design-space before simulating Woodstock’s true descent down to itslow lunar orbit.

Figure 9 shows the relationship of the orbital period of lunar halo orbits about LL2 as a function of theirx-axis crossing position and as a function of their Jacobi constant value. Most lunar halo orbits have anorbital period between 13 and 15 days, bracketing an orbit in a 2 : 1 resonance with the Moon’s motionabout the Earth. For reference, the Moon is located in the CRTBP on the x-axis at approximately 379, 729km and LL2 is located on the x-axis at approximately 444, 244 km.

Figure 9. The relationship between the orbital period and (left) x-axis crossing and (right) Jacobi constant ofthe family of halo orbits that exist about LL2 in the CRTBP.

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In order to design Woodstock’s descent, the invariant manifolds of the halo orbits were studied. Theunstable manifold of a halo orbit characterizes where Woodstock could descend for free. The first step inthe design is therefore to identify the range of halo orbits whose unstable manifolds include trajectories thatwould lead Woodstock down to a desirable low lunar orbit. For the purposes of this study, a 50-km polarlunar orbit was chosen.

Figure 10. The unstable manifold of an example haloorbit about LL2. The trajectories in blue and green arein the region of trajectories that encounter the Moon ataltitudes below 50 kilometers. The trajectories are shownin the Earth-Moon synodic reference frame.

Figure 11. The altitude of closest approachof each trajectory in the halo orbit shown inFigure 10 after 30 days of propagation. Thered line indicates the target altitude of 50kilometers.

Figure 12. The inclinations that are obtainable following those trajectories in the unstable manifolds of eachhalo orbit that reach a lunar altitude of 50 kilometers. The trajectories are each propagated for 30 days.

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Figure 10 shows the unstable manifold of an example halo orbit about LL2. The trajectories shown inblue and green are in the region of trajectories that encounter the Moon at altitudes below 50 kilometers.Figure 11 shows a plot of the altitude of closest approach of each trajectory in the manifold shown in Figure10. The red line indicates the target altitude of 50 km, and green dots indicate altitudes less than 50 km.Each trajectory is propagated for approximately 30 days; given more time, the trajectories could approachcloser to the Moon.

Each trajectory that falls within 50 kilometers of the Moon encounters the Moon at a different inclination.The two trajectories shown in blue in Figure 10, i.e., the two trajectories with a periapsis altitude of precisely50 kilometers, encounter the Moon at inclinations of about 83.44 and 93.04. Hence, this example haloorbit nearly meets the initial goal. It is reasonable to suspect that a nearby halo orbit would meet the goalprecisely. Figure 12 shows which inclination values may be obtained by following those trajectories in theunstable manifolds of each halo orbit that reach a lunar altitude of 50 kilometers. Notice that trajectoriesoriginating from halo orbits with Jacobi constants less than about 3.048 do not reach within 50 kilometerswithin 30 days of propagation. For reference, the halo orbit and its unstable manifold shown in Figures 10and 11 have a Jacobi constant of approximately 3.0597.

Since the trajectories in the unstable manifold of an orbit will change when converting a CRTBP haloorbit into the real solar system, it is only immediately useful to identify the range of Jacobi constant valuesthat correspond to CRTBP halo orbits with unstable manifolds that reach near 90. Looking at Figure 12,one expects to find quasi-halo orbits in the real solar system that meet Requirement 7 with Jacobi constantvalues in the ranges of 3.05− 3.06, 3.11− 3.13, and 3.14− 3.15.

G. Selected Halo Orbit

Figure 13. Plot used to select the halo orbit for Snoopy. The numbers in the legend correspond to the orbitselection requirement number. The legend also shows the vertical axis units for each metric. The chosen orbithas a Jacobi constant of about 3.06.

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An orbit selection plot was created and shown in Figure 13. Regions with high and low Jacobi constantwere eliminated because they didn’t provide sufficient coverage of the South Pole or Woodstock, respectively.The orbit selected had a Jacobi constant of 3.0597 because it provided excellent coverage of the south polewith two halo orbiters. At least one of the two halo orbiters would be above 15 elevation at all times.With Jacobi constants higher than this, the stationkeeping costs were high. Orbits with Jacobi constantslower than 3.06 had increasing error in the orbit determination and increasing tracking gaps in the Snoopy-Woodstock coverage. The Jacobi constant for this orbit also falls in the acceptable range mentioned forrequirement 7.

0.0 1.1 2.3 3.4 4.5 5.7 6.8 7.9 9.1 10.2 11.3 12.5 13.60

15

30

45

60

75

90

time (days)

elev

atio

n (d

egre

es)

Lunar Halo Coverage at Lunar 90S 0E

Elev1Elev2EarthMin Elevation

Figure 14. Elevation of two halo orbiters (Elev1, Elev2) and the Earth at the lunar south pole. The spacecraftare in the orbit selected for Snoopy, 180 apart. Orbits were propagated using the full ephemeris and SRP.

0

45

90

135

180

225

270

315

60

30

Halo1Halo2Earth

Figure 15. Skyplot of two halo orbiters at the lunar south pole. The spacecraft are in the orbit selected forSnoopy, 180 apart. Orbits were propagated using the full ephemeris and SRP.

For the selected orbit, Figure 14 shows the elevation at the south pole of two halo orbiters on Snoopy’snominal orbit over one halo period. Earth rises above 0 briefly at the beginning of the period. Figure 15shows how the paths of the two halo orbiters across the sky would look to an observer at the south pole.Figure 16 shows that for one halo orbiter, the Earth/Snoopy-Woodstock tracking gaps can be minimized bytargeting a certain value of δ when inserting Woodstock into its lunar orbit. Figures 17 and 18 show mapsof the density of Woodstock’s tracking coverage over the entire lunar surface, centered on the far side.

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0 50 100 150 200 250 300 3500

1

2

3

4

5

δ (deg)Tr

ackin

g G

aps

(%)

Tracking gaps for Lunar halo Snoopy

Figure 16. Earth/Snoopy-Woodstock tracking gaps (% of points).

11

111

1

1 1 1

2222

2 2 2 2

0.6

0.6

0.6

0.60.6 0.6

1

0.6

Tracking coverage density (%points/%area) for Lunar halo Snoopy, δ = 220, 0.4% lost

longitude (deg)

latit

ude

(deg

)

30 60 90 120 150 180 −150 −120 −90 −60 −30−90

−60

−30

0

30

60

90

Figure 17. Earth/Snoopy-Woodstock tracking coverage density (% points/% area) at the optimum value ofδ = 220.

1 11 1

11

1

1

2222

2 2 2 2

0.6

0.6

0.6

0.6 0.4

0.4

0.40.2

0.2

0.6

0.10.6

0.4

0.6

0.2

0.6

0.4

0.1

Tracking coverage density (%points/%area) for Lunar halo Snoopy, δ = 60, 4.5% lost

longitude (deg)

latit

ude

(deg

)

30 60 90 120 150 180 −150 −120 −90 −60 −30−90

−60

−30

0

30

60

90

Figure 18. Earth/Snoopy-Woodstock tracking coverage density (% points/% area) at the worst case value ofδ = 60.

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The density is expressed as the percentage of tracking data points in a latitude-longitude bin divided bythe percentage of the lunar surface area contained in the bin. So, if the tracking coverage were uniform overthe entire lunar surface, the density would be 1. For areas with less than the average coverage, the trackingdensity is less than 1 and greater than 1 for areas with more than the average tracking coverage. BecauseWoodstock is in a polar orbit, it passes over the poles once every orbit and the density is the highest in thehigh latitudes and the density thins toward the equator. Figure 17 shows the tracking coverage density foran optimum value of δ and represents about the most uniform coverage possible for this polar orbit. Figure18 shows how the orbit geometry at the worst value of δ can lead to holes in coverage over the lunar far sidesurface. The regions with low tracking coverage density, represented by the dark shading, are areas whereneither the Earth nor Snoopy can see Woodstock in its orbit. The orbits were propagated using the fullephemeris and the LP100K lunar gravity field up to degree and order 30.

IV. Low Lunar Orbit Selection (Woodstock)

The requirements established for selecting the low lunar orbit for Woodstock are:

1. The low lunar orbit should pass over the lunar South Pole.

2. Woodstock’s orbit should be as low as possible for good estimation of the lunar far side gravity field.

3. The percentage of the time that Woodstock is not visible from either the Earth or Snoopy should beminimized and tracking data should be obtainable over the entire lunar far side.

4. The stationkeeping costs of the lunar orbit should be minimized.

Requirement 1 can be satisfied if Woodstock is in a polar orbit, or an orbit with an inclination closeto 90. Requirement 3 is satisfied with the proper selection of δ as described in the Snoopy orbit selectionprocess. To find a lunar orbit that satisfies requirements 2 and 4, a lunar orbit study was performed. Lunarorbits were propagated in STK v.6.2 using the “Lunar RKV8th9th” integrator in Astrogator. The lunargravity field GLGM-2 was used to model the effects of the Moon’s asymmetric gravity.16

To reduce computation time, the first task in the study was to find the smallest truncation of the gravityfield that still produced accurate orbits. A lunar point mass model as well as truncations to degree andorder 2, 10, 20, 50 and 70 were used to propagate a circular, polar lunar orbit with a radius of 1787 km, ora periapsis altitude of 50 km. The Earth and Sun were modeled as point masses, and the right ascension ofthe ascending node, argument of periapsis and true anomaly were initially set to zero.

0 20 40 60 801730

1740

1750

1760

1770

1780

1790

Radi

us o

f Per

iaps

e (k

m)

Elapsed Time (days)

Point Mass2x210x1020x2050x5070x70

Figure 19. The effect of different gravity fields on the radius of periapse of an initially circular 50-km polarlunar orbit.

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In low lunar orbits, a significant effect of the gravity field is that the altitude of periapsis can decreaselow enough that the spacecraft will impact the lunar surface. Figure 19 shows the effects of different gravityfield models on the periapsis radius over a period of three months. It can be seen that a 2x2 gravity fieldlooks similar to a lunar point mass and is not accurate. The 10x10 gravity field displays the same shapeas gravity fields with higher degree and order. However, as time passes, the difference between the 10x10gravity field and the 70x70 gravity field becomes larger. The differences between the 20x20, 50x50 and 70x70gravity fields are very minimal. Although the results do not change greatly from the 20x20 gravity field tothe 70x70 gravity field, the computation time does. Thus the 20x20 gravity field was chosen due to itssuperior combination of accuracy and computation time.

According to requirement 2, in order to collect the necessary information about the lunar gravity field,the low lunar satellite must orbit close to the Moons surface. However, lower orbits are perturbed moreby the gravity field and are less stable. Circular lunar orbits with radii of 1767 km, 1787 km and 1807 kmwere propagated with the 20x20 gravity field. The radius of the Moon is 1737 km leaving the correspondingperiapsis altitudes as 30 km, 50 km and 70 km. In all cases the right ascension of the ascending node andthe argument of periapsis were initialized to zero.

0 10 20 30 40 50 60 70 80 901720

1730

1740

1750

1760

1770

1780

1790

1800

1810

Radi

us o

f Per

iaps

e (k

m)

Elapsed Time (days)

rp = 1767rp = 1787rp = 1807

Figure 20. The variation of radius of periapsis using different initial periapsis altitudes.

Figure 20 shows the perturbations to the three different low lunar orbits over a time span of three months.A periapsis altitude of 30 km is ruled out immediately because the satellite collides with the Moon afterseven days. In both the 50 km and 70 km orbits, the radius of periapsis and apoapsis shift by approximately30 km and the variations in inclination are similar. Thus the 70 km orbit is not significantly more stablethen the 50 km orbit. Therefore 50 km was chosen as the periapsis altitude due to enhanced data collectioncapabilities.

It is important to note that the radius of apoapsis increases over time while the radius of periapsisdecreases over time. A study of low, eccentric lunar orbits was conducted to see if they would be more stablein the long term.

A. Eccentric orbits

In addition to the circular, 50 km orbit, three different eccentric orbits were propagated with varying apoapsisaltitudes while the periapsis altitude was fixed to 50 km. The apoapsis radii were 1802 km, 1817 km and1832 km, which corresponds to altitudes of 65 km, 80 km and 95 km.

Figure 21 shows the perturbations in the low lunar orbits over a time span of three months. The longperiod oscillations in the periapsis altitudes vary less for more eccentric orbits. In the case of the circularorbit, the periapsis altitude decreases by 37.2 km and the eccentricity increases by 0.02 in the span of threemonths. For the orbit with an apoapsis altitude of 65 km, the periapsis altitude decreases by 23.7 km andthe eccentricity increases by 0.013. For the orbit with an apoapsis altitude of 80 km, the periapsis altitude

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0 10 20 30 40 50 60 70 80 901740

1750

1760

1770

1780

1790

1800

1810

Radi

us o

f Per

iaps

e (k

m)

Elapsed Time (days)

ra = 1787ra = 1802ra = 1817ra = 1832

Figure 21. The variation of radius of periapsis using different initial apoapsis altitudes.

decreases by 11 km and the eccentricity increases by 0.006. For the orbit with an apoapsis altitude of 95 km,the periapsis altitude decreases by only 0.67 km and the eccentricity increases by just 0.0002. All four orbitsappear to have the same amount of short period oscillations. It appears that the further the apoapsis isfrom the Moons surface, the smaller the difference between the initial and final orbit states. Thus, the moreelliptical orbits are more stable. The orbit chosen for Woodstock has an apoapsis altitude of 95 km and aperiapsis altitude of 50 km. This orbit is the most stable of all the orbits examined, and thus stationkeepingcosts will be reduced. The low, 50 km periapsis altitude allows the spacecraft to pass close enough to theMoon to gather the necessary gravity data while periapsis is at low latitudes on the lunar far side.

V. Design of the Descent to Lunar Orbit

Although trajectories from the LL2 halo orbit to a lunar orbit insertion point were found in the CRTBP,moving into the real solar system ephemeris model involves an additional degree of freedom, namely, thereference Julian Date, since the bodies in the real solar system do not move in coplanar, circular orbits.The result is that a single trajectory in the CRTBP translates into a range of possible trajectories in thereal solar system. Figure 22 shows the difference between a halo orbit about LL2 produced in the CRTBPcompared with the same halo orbit differentially-corrected into the JPL Ephemeris model of the real solarsystem using a multiple-shooting method. One can see that the real halo orbit is quasi-periodic, tracing outthe same vicinity of space on each orbit, but never truly retracing itself.

The trajectories on the unstable manifold of the quasi-halo orbit in the real solar system descend downto the Moon in a similar manner as those originating from a true halo orbit in the CRTBP. However, thosetrajectories in the real solar system that approach the near-vicinity of the Moon will diverge quickly fromtheir counterparts in the CRTBP. The result is that the trajectories whose perilune is at 50 kilometers willencounter the Moon at different orbital inclinations.

Figures 23 - 26 show the results of propagating each trajectory in the unstable manifold of a quasi-haloorbit down to its closest approach to the Moon after 30 days. The quasi-halo orbit was constructed bydifferentially-correcting twelve halo orbits from the CRTBP into the JPL Ephemeris model of the solarsystem. As Figure 22 shows, the orbit is not periodic, so the trajectories propagated to produce the plotsshown in Figures 23 - 26 originate from one period of the quasi-periodic orbit; namely, from one positive x-axiscrossing to the next. As such, the plots resemble the results from the CRTBP model, but one “quasi-period”does not necessarily match a single true halo period.

Figure 23 shows the radius of closest approach that each trajectory makes with the Moon as it is propa-gated for 30 days from the quasi-halo orbit toward the Moon. The points shaded in green approach within0 − 500 kilometers from the Moon. Figures 24 and 25 show what inclinations and δ-values are reached at

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Figure 22. A comparison between a halo orbit produced in the CRTBP and a quasi-halo orbit produced inthe JPL Ephemeris model of the real solar system. The orbits are shown in the Earth-Moon synodic referenceframe.

Figure 23. The radius of closest approach that eachtrajectory makes with the Moon as it is propagatedfor 30 days from the quasi-halo orbit toward theMoon.

Figure 24. The inclination that is reached at per-ilune by the trajectories in the manifold shown inFigure 23.

Figure 25. The δ-value that is reached at periluneby the trajectories in the manifold shown in Figure23.

Figure 26. A parametric plot of inclination ver-sus δ, using the data plotted in Figures 24 and 25,respectively.

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Figure 27. Parametric plots that show the lunar inclination and δ combinations that are obtainable at periluneof each trajectory of a manifold produced from a quasi-halo orbit with a given reference Julian Date. The

nominal Julian Date is 2455891, namely, November 25th, 2011 at noon; each quasi-halo orbit produced uses areference Julian Date equal to the nominal Julian Date plus the given ∆JD value. The points shown in greenhave a perilune altitude between 0 − 500 km.

perilune by the trajectories in the manifold shown in Figure 23. In each case, the curves are plotted in greenwhen they encounter the Moon between an altitude of 0 − 500 kilometers, and the target inclination and δvalues are indicated by red lines. Finally, Figure 26 shows the parametric plot of inclination versus δ.

Figure 27 shows the same parametric plot as in Figure 26, except that it spans many reference JulianDates. The nominal reference Julian Date is 2455891, namely November 25th, 2011 at noon; each curve inFigure 27 represents the quasi-halo orbit constructed using a reference Julian Date equal to the nominal JDplus some ∆JD, indicated by the legend in each plot. It is interesting to notice how the curves graduallychange as the reference Julian Date changes. Furthermore, the features in the plots repeat after approxi-mately one month: the bottom-left parametric plot resembles the top-left plot and the bottom-right plotresembles the top-right plot. In this way, one can identify which reference Julian Date produces a trajectorythat meets a set of specific constraints.

To meet the requirements of this comm/nav constellation, it is desirable to target a trajectory with theproper δ, as close to the target inclination as possible, with a perilune altitude in the range of 0 − 500kilometers (those points shown in green in Figure 27). The curves that best meet these requirements areindicated by ∆JD values of about 19, 47, and values that are multiples of one month.

After refining the search, it was found that a quasi-halo orbit produced in the JPL Ephemeris model ofthe solar system with a reference Julian Date of approximately 2455910.45. Its perilune altitude fell in therange of 0− 500 kilometers, with an inclination very close to 90 and a δ-value in the vicinity of 80. Figure28 shows the first 28 revolutions of this quasi-halo trajectory about LL2, spanning approximately 381.5 days.

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Figure 28. The first 28 revolutions of the final quasi-halo trajectory about LL2, spanning approximately 381.5days. The trajectory is shown in the Earth-Moon synodic reference frame.

Figure 29. The trajectory that Woodstock would follow to depart Snoopy and descend down to its low-lunarorbit. This trajectory encounters a 189-km altitude orbit with an inclination of approximately 90 after about28 days. A 397.4-m/s maneuver is performed to inject into a 189×2000 km polar orbit. The trajectory is shownin the Earth-Moon synodic reference frame.

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The trajectory demonstrates that Snoopy’s orbit is well-approximated by a CRTBP halo orbit, despite theperturbations of the solar system ephemeris model. Additional orbits may be further added on to the endof this trajectory for very little cost.

Woodstock’s departure from Snoopy and subsequent descent down to its low-lunar polar orbit may takeplace for very low amounts of energy once per month. Figure 29 shows the 28-day descent that Woodstockwould take to arrive at its low-lunar orbit injection point approximately 189 km above the lunar surface atan inclination of approximately 90. The departure can be performed with a minimal maneuver, althoughit does require a large maneuver to become captured in its low-lunar orbit. Since 50 km is very close to thelunar surface, Woodstock was instead placed into a higher parking orbit before descending down to its finalorbit. The parking orbit was polar with an altitude of 189 × 2000 km, requiring an insertion maneuver of397.4 m/s. Additional maneuvers would be performed later to reduce Woodstock’s orbit to a 50 × 95 kmaltitude.

VI. Launch and BLT Design

Given knowledge of Snoopy’s quasi-halo orbit, dynamical systems theory can be used to construct alow-energy ballistic transfer from the Earth to this orbit. The ballistic transfer may be constructed bypropagating the quasi-halo orbit’s stable manifold backwards in time. A search is then performed for aspecific trajectory on the stable manifold that intersects a desirable low-Earth orbit, i.e., an orbit with aninclination of about 28.5.

Figure 30 shows a plot of a portion of the stable manifold of one of the quasi-halo orbit revolutions inthe Sun-Earth synodic frame. The trajectories that intersect the Earth’s vicinity are shown in blue.

Figure 30. A portion of the stable manifold of one of the quasi-halo orbit revolutions about LL2, shown in theSun-Earth synodic frame. The trajectories shown in blue intersect low-Earth orbits.

Figure 31 shows a plot of the perigee altitude of each trajectory in the quasi-halo orbit’s stable manifoldthat would arrive at the quasi-halo orbit with enough time to allow Woodstock to descend to the Moon.Figure 32 shows the inclination value of the perigee points shown in Figure 31. Finally, Figure 33 shows theparametric plot of the perigee altitude versus the perigee inclination. In each plot, the target altitude rangeand target 28.5 inclination are indicated with red lines.

The insert shown in Figure 33 indicates that a trajectory exists in the stable manifold of the lunar haloorbit with a perigee altitude of about 500 km at an inclination of 28.5. This means that if the spacecraftwas launched into a 500-km parking orbit, the spacecraft could then perform a single injection maneuver to

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Figure 31. The perigee altitude of each trajectoryin the quasi-halo orbit’s stable manifold that wouldarrive at the quasi-halo orbit with enough time toallow Woodstock to descend to the Moon. Thetarget perigee altitude range is indicated with redlines.

Figure 32. The inclination value of the perigeepoints shown in Figure 31. The target inclinationvalue is indicated with a red line.

Figure 33. A parametric plot of the perigee altitude versus the perigee inclination using the data displayed inFigures 31 and 32. The target altitude range and inclination are indicated with red lines.

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get onto this trajectory - the spacecraft would then follow a purely ballistic trajectory and eventually arriveonto the lunar halo orbit for free. In reality, there would be trajectory correction maneuvers en-route, butthose maneuvers would be small once the errors in the launch are corrected. The 500-km altitude is fairlyhigh for a parking orbit about the Earth; if the spacecraft began in a lower orbit the injection maneuverwould be slightly larger and an additional small maneuver might be required later in the mission to arriveonto the lunar halo orbit. Nonetheless, this trajectory demonstrates the concept.

Figures 34 - 37 show the ballistic lunar transfer from three perspectives. Figure 34 shows the transfer inan Earth-centered inertial frame of reference. In this frame the transfer may be described using two-bodyarguments. The spacecraft first departs the Earth on a very eccentric orbit. As the spacecraft approachesand traverses the orbit’s apogee, the Sun’s gravity substantially perturbs the spacecraft’s motion. The Sun’sgravity ultimately raises the orbit’s perigee until the orbit’s perigee intersects the Moon. As the spacecraftfalls back toward the new perigee point, the spacecraft encounters the Moon. Figure 35 shows the entiretransfer in the Sun-Earth synodic reference frame. From this perspective, one can see that the spacecraftdeparts the Earth toward the vicinity of the Sun-Earth L1 point. The spacecraft doesn’t approach theLagrange point directly, though, but instead remains on the near-Earth side of the point so that it falls backtoward the Earth. Its out-bound motion actually shadows the stable manifold of a Sun-Earth three-bodyorbit, and its in-bound motion shadows the unstable manifold of the same orbit. Figure 37 shows the entiretransfer in the Earth-Moon synodic reference frame. From this perspective, one can see that as the spacecraftapproaches the Moon, the spacecraft does indeed asymptotically approach and eventually arrive on the lunarhalo orbit. Figure 36 shows the final lunar halo approach from a three-dimensional perspective. This BLTincludes a lunar flyby as the spacecraft departs the Earth. This lunar flyby benefits the mission, furtherreducing the energy required to transfer the payload to the Moon.

Figure 34. The ballistic lunar transfer shown in the Earth-centered inertial reference frame.

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Figure 35. The ballistic lunar transfer shown in the Sun-Earth synodic reference frame.

Figure 36. The final lunar halo approach and free insertion shown in the Earth-Moon synodic reference frame.

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Figure 37. The ballistic lunar transfer shown in the Earth-Moon synodic reference frame.

After differentially-correcting the entire mission, we arrived at a solution with the following performancemetrics:

• Launch from a LEO parking orbit at an altitude of 534 ± 10 km at an inclination of 28.5± 0.02 on

September 20, 2011 at 2:00 am using approximately 3.11 km/s of ∆V.

• The transfer to the lunar halo orbit requires approximately 114 days and no deterministic maneuvers.There is a lunar flyby with a closest approach of about 22,474 km (altitude of 20,736 km) on September22, 2011 at 8:05 am. The launch cost may be further reduced by approaching closer to the Moon.

• The lunar halo insertion requires no deterministic maneuver and occurs on approximately January 12,2012.

• The sum total of the patchpoint ∆V’s, which theoretically may be removed, is equal to about 4.5mm/s, including the entire transfer and eight halo revolutions.

• On February 2, 2012 at 6:39 am, Woodstock performs a 1.18 m/s maneuver to depart Snoopy.

• On February 5, 2012 at 12:53 am, Woodstock performs a 0.97 m/s maneuver to adjust its descenttrajectory.

• On March 1, 2012 at 6:18 am, Woodstock arrives at perilune at an altitude of 189 km and an inclinationof 90, after following its descent for about 28 days.

• Woodstock performs a 397.4 m/s maneuver to enter its temporary 189 × 2000 km orbit, followed byadditional maneuvers totalling 287.9 m/s to lower the orbit to a final 50 × 95 km polar orbit withperiapsis near the lunar equator.

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VII. Spacecraft Design

The link distance between Snoopy and Woodstock is large, approximately 75,000 km, which means thateach spacecraft requires high-gain communication antennas. These highly directional antennas would needgood pointing accuracy and the ability to point in many different directions. Because of these pointingrequirements, both Snoopy and Woodstock would need to be three-axis stabilized spacecraft. The spacecraftwould also need precise clocks to perform autonomous orbit determination or generate stable navigation bea-cons. Snoopy would be assigned to perform the on-board orbit determination for both spacecraft, althoughWoodstock could compute its own estimates as a cross-check. Both spacecraft would need photovoltaics togenerate electricity. Snoopy’s halo orbit is ideal for solar power since no eclipses occur in the trajectorygenerated, although that does not rule out the possibility that an eclipse could occur in a different timeperiod.

A simple link budget analysis was done using Ka-band radio frequencies. Woodstock would have onehigh-gain antenna (41 dB with 1.3 beamwidth) for communicating with Earth and Snoopy. It would alsohave omni-directional antennas for communicating with assets on the lunar surface. Snoopy would have twohigh-gain antennas. One of these would have a narrow beam (1 beamwidth) to send and receive data toEarth at a high rate. The other would have a somewhat wider beam (3 beamwidth) so that the beamcould cover the entire lunar surface while pointing at the center. Each spacecraft could recieve data from theMoon at a rate of 18 Mbps using the narrow-beam antennas. This would require tracking the lunar surfacetransmitter with the antennas. A data rate of 2 Mbps would be possible with the wide beam antennas,which would not need to track the transmitter on the surface. These link budgets were made using a 1 mdish on the lunar surface with 10 W of transmit power.

Woodstock would need a large propulsion system for the lunar orbit insertion maneuver and the lunarSKM’s. Snoopy’s propulsion system could be much smaller due to the low cost of SKM’s in the halo orbit.Table 1 shows estimated ∆V budgets for both spacecraft. Snoopy’s SKM budget as computed from theLiaison Navigation simulation was only 16.2 cm/s per year, which is amazingly small. The longest nominalhalo trajectory generated using the DE405 ephemeris was about a year. Since it is difficult to patch togetherdifferent nominal trajectories without a ∆V, there would be at least one planned ∆V of about 1 m/s in ayear. That would probably mean that the SKM budget for Snoopy would be at least 1 m/s per year. TheWoodstock Trajectory Correction Maneuver (TCM) budget was based on the Lunar Reconnaissance Orbiter(LRO) estimates.17 The SKM budget for Woodstock (56 m/s per year) was much less than that estimatedfor LRO (140 m/s per year) due to Woodstock’s eccentric orbit and higher apoapsis.

Table 1. STATIONKEEPING BUDGETS FOR BOTH SPACECRAFT IN TERMS OF ∆V

Spacecraft Maneuver Type ∆V

Snoopy SKM 16.2 cm/s/yr

Woodstock TCM 75 m/s

Woodstock Descent from Halo 1.9 m/s

Woodstock Orbit Insertion 397.4 m/s

Woodstock Orbit Lowering 287.9 m/s

Woodstock SKM 56.3 m/s/yr

VIII. Mission Performance

An analysis of the mission using the final selected orbits was performed. This analysis included a realisticsimulation of autonomous orbit determination performance and a simulation of the use of Snoopy andWoodstock as tracking stations.

A. Liaison Navigation performance

To simulate the on board performance of Liaison Navigation, there were three types of orbits generated.The first was a nominal halo orbit generated at LL2 using the multiple shooting methods mentioned before.

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The nominal orbit is divided into four segments per halo orbit period. The state at the beginning of eachsegment is called a patch point. At the time of each patch point, Snoopy is never exactly on this nominalorbit, so a SKM must be performed to get Snoopy closer to the nominal orbit. A differential corrector isused to target the next patch point using the best estimate of the current position. The current velocity iscorrected until the trajectory hits the next patch point. The SKM is performed to get Snoopy’s velocity tomatch the corrected velocity. In this simulation, maneuver execution errors were modeled in each componentdirection as normally distributed with a standard deviation equal to 5% of the maneuver magnitude in thatcomponent.

Starting with an initial deviation from the nominal halo orbit, a true orbit was generated for Snoopy.Snoopy was modeled as a 1000 kg spherical spacecraft with a cross-sectional area of 5 m2. The JPL DE405ephemeris was used to find the positions of the Earth, Moon, Sun, Venus, Mars, Jupiter and Saturn. Eachplanet was modeled as a point mass. The Moon’s gravity field (up to degree and order 2) was modeledusing the LP100K gravity model. SRP was modeled using an umbra/penumbra eclipse model with the solarpressure computed based on the distance from the Sun. A true orbit was generated for Woodstock in thesame way, except the lunar gravity field parameters up to degree and order 20 were included as well. Aftergenerating the true orbits, range observations were computed every 60 seconds and normally distributednoise with a standard deviation of 1 m was applied. These observations were used to estimate the states ofthe spacecraft. The initial states for the estimated orbits were computed with randomly generated position,velocity, and reflectance errors. The initial position errors had a 1σ in each component of 100 m and 10 mfor Snoopy and Woodstock, respectively. The velocity errors were about 6 × 10−4 m/s and 6 × 10−5 m/s,respectively.

A real-time state update would provide the spacecraft with accurate orbit determination quickly so thatthe spacecraft could be used as navigation support for other spacecraft. An Extended Kalman Filter (EKF)provides real-time state estimates, so it was ideal for this purpose. A summary of the EKF comes from Tapley,et al.18 and uses their notation. The spacecraft state consists of three position and velocity components foreach spacecraft. For two spacecraft, the state vector would be written:

X = [ x1 y1 z1 x1 y1 z1 x2 y2 z2 x2 y2 z2 ]T ,

where the subscripts denote the spacecraft number. The equations of motion are written:

X = F (X, t), Xi ≡ X(ti).

An observation at time ti is denoted by Yi, and the state deviation and observation residuals are written:

x(t) = X(t) −X∗(t), y(t) = Y(t) −Y∗(t),

where X∗ is the reference solution, or the “best guess” orbit, and Y∗ is an observation computed using thereference solution. The observations can be related to the state with the H matrix, which is

Hi =

[

∂G

∂X

]

∗

i

.

G is a function that computes observations from a state vector in the following way:

Yi = G(Xi, ti) + εi,

where εi is the observation error. If crosslink range, ρ, is the observation type then Hi is of the form:

Hi =

[

∂ρ

∂x1

∂ρ

∂y1

∂ρ

∂z1

∂ρ

∂x1

∂ρ

∂y1

∂ρ

∂z1

∂ρ

∂x2

∂ρ

∂y2

∂ρ

∂z2

∂ρ

∂x2

∂ρ

∂y2

∂ρ

∂z2

]

i

State deviations at time t may be mapped back to an epoch, tk, with the state transition matrix Φ in thefollowing way:

x(t) = Φ(t, tk)xk.

The state transition matrix is integrated along with the state using the following relation:

Φ(t, tk) = A(t)Φ(t, tk) where A(t) =∂F (X∗, t)

∂X(t)and is of the form:

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A =

∂x1

∂x1

∂x1

∂y1

∂x1

∂z1

· · ·∂x1

∂z2∂y1

∂x1

∂y1

∂y1

∂y1

∂z1

∂z1

∂x1

∂z1

∂y1

∂z1

∂z1

...

.... . .

∂z2

∂x1

· · ·∂z2

∂z2

∗

If it is assumed that the observation errors can be modeled as white noise, and the standard deviation inthe observation noise is denoted σρ, a weighting matrix R−1 can be used to weight the observations, where

R = σ2ρ.

The EKF requires an a priori covariance, P0. The EKF works by propagating the state deviation and thecovariance at time k − 1 to the next observation epoch, k, using the “time update” equations:

Pk = Φ(tk, tk−1)Pk−1ΦT (tk, tk−1).

Pk−1 is the covariance matrix at the time of the last observation, and Pk is the same matrix propagatedforward to the time of the next observation. The state and covariance are updated with the observation attime k using the “measurement update” equations:

Kk = PkHTk

[

HkPkHTk + R

]

−1

xk = Kk

[

yk − Hkxk

]

.

Pk =[

I − KkHk

]

Pk

The state is then updated with x in this way

X∗

new = X∗ + x

The newly updated state is then propagated forward to the time of the next observation and the time andmeasurement update is repeated. At each time update, the estimation errors represented in the covariancematrix become smaller and smaller. This can cause the Kalman gain, Kk, to become so small that theobservations are effectively ignored. This is called filter saturation. To prevent this, state noise compensationcan be used to inflate the covariance matrix slightly, which increases the Kalman gain and assures that therecent observations are not ignored. State noise compensation is performed by adding a covariance matrixdue to process noise to the time update of the estimation error covariance matrix. This matrix can bederived by assuming that the spacecraft velocity is constant between measurement times, ∆t.18 A variancefor the process noise, σ2

a,i is assumed for each spacecraft. Here, i = 1, 2 represents the two spacecraft. This6×6 matrix is added to the position and velocity portions of the estimation error covariance matrix for eachspacecraft:

∆t3

3σ2

a,i 0 0∆t2

2σ2

a,i 0 0

0∆t3

3σ2

a,i 0 0∆t2

2σ2

a,i 0

0 0∆t3

3σ2

a,i 0 0∆t2

2σ2

a,i

∆t2

2σ2

a,i 0 0 ∆tσ2a,i 0 0

0∆t2

2σ2

a,i 0 0 ∆tσ2a,i 0

0 0∆t2

2σ2

a,i 0 0 ∆tσ2a,i

The values of σa,i were tuned until they resulted in a realistic covariance matrix and low state errors. Stateerrors were computed by differencing the estimated state, X∗

new , with the true state at each measurementtime.

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The state estimates from the EKF were used to compute the SKM for both spacecraft. Woodstock’sSKM’s were designed to restore the periapsis and apoapsis altitudes to 50 and 95 km, respectively, andalso had 5% errors in execution. The velocity portion of the covariance matrix was increased accordinglyafter each maneuver, so the EKF could be used properly to estimate the true velocity after the maneuver.Snoopy’s SKM’s were performed four times per halo orbit period as described earlier, and Woodstock’sSKM’s were performed once per halo period, halfway between two of the Snoopy SKM’s. A range bias wasadded to the measurements and the bias was successfully estimated to within several meters.

Because Snoopy and Woodstock would need to perform communication relay duties as well as providetracking support to user spacecraft, they would not be available to track each other all the time. Figure 38shows how each antenna on the spacecraft would be used over a day. The times shown are Central StandardTime, to coincide with “Houston Time” that manned missions would be using. This schedule limits thetime when Snoopy and Woodstock can track each other, however, the schedule for each day was shifted inten minute increments (to a maximum of one hour) to synchronize the Snoopy-Woodstock tracking withthe times that Woodstock was not out of sight behind the Moon. During the eight-hour working day of amanned outpost on the Moon, Snoopy provides constant two-way communication relay services to Earth,with the exception of a one hour break at lunch to track Woodstock. The schedule would change whenSnoopy is not visible from the lunar surface, with Woodstock providing the relay intermittently as it passesoverhead (not shown).

Figure 38. Daily schedule of comm/nav support for Snoopy and Woodstock

To make the simulation more realistic, the force model used in the EKF was different than the forcemodel used to generate the true orbits. In the EKF, the two types of unmodeled accelerations came in theform of gravity model errors and radiation model errors.

A clone of the gravity field was created by randomly perturbing the gravity field coefficients using thecovariance matrix for the LP100K model. The vector cgrav includes all of the normalized coefficients in thegravity model and the corresponding covariance matrix is Pgrav . A clone of the gravity model can be createdby factoring Pgrav :

Pgrav = ST S.

S is upper triangular and can be computed using Cholesky Decomposition. If e is a vector of normallydistributed random numbers with zero mean and variance 1, a clone of the gravity model, cclone would be

cclone = cgrav + ST e.

Errors in the clone are described by Pgrav . With the EKF, the reflectance for each spacecraft was estimatedto much less than 1%. However, since an error in radiation modeling was desired, it was simulated by fixingerrors in the reflectance for each spacecraft and not estimating the true reflectance. The reflectance errorswere designed so that they resulted in acceleration errors on the order of 1 × 10−9 m/s2, which is a typicalmagnitude for interplanetary missions.

Figures 39 through 42 show the resulting accuracy of the EKF over two halo periods starting on 5 March2012. The RSS for the position error (computed after day 8) for Snoopy was 78 m and the velocity error was0.4 mm/s. For Woodstock, the RSS for the position error was 6.9 m and 6 mm/s for velocity. It took about3-6 days for Snoopy’s orbit estimate to converge. Woodstock’s orbit converged in about 2 days. Even withinitial position and velocity errors 100 times larger, the estimates still converged in the same time frame.

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0 5 10 15 20 25 30−200

0

200

x sa

t 1 (m

)

Day (RMS = 37.94 m) 64.7% < 1σ

EKF sat 1 position error with state noise 2.68e−009 m/s2, RSS = 77.78 m

x error2σ stdev

0 5 10 15 20 25 30−200

0

200

y sa

t 1 (m

)

Day (RMS = 42.57 m) 52.3% < 1σ

y error2σ stdev

0 5 10 15 20 25 30−200

0

200

z sa

t 1 (m

)

Day (RMS = 52.90 m) 66.1% < 1σ

z error2σ stdev

Figure 39. Snoopy position accuracy using the EKF.

0 5 10 15 20 25 30−2

0

2x 10−3

dx s

at 1

(m/s

)

Day (RMS = 2.78e−004 m/s) 40.4% < 1σ

EKF sat 1 velocity error with state noise 2.68e−009 m/s2, RSS = 3.94e−004 m/s

dx error2σ stdev

0 5 10 15 20 25 30−2

0

2x 10−3

dy s

at 1

(m/s

)

Day (RMS = 1.63e−004 m/s) 68.8% < 1σ

dy error2σ stdev

0 5 10 15 20 25 30−2

0

2x 10−3

dz s

at 1

(m)

Day (RMS = 2.27e−004 m/s) 75.0% < 1σ

dz error2σ stdev

Figure 40. Snoopy velocity accuracy using the EKF.

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0 5 10 15 20 25 30−20

0

20

x sa

t 2 (m

)

Day (RMS = 5.70 m) 90.8% < 1σ

EKF sat 2 position error with state noise 1.34e−008 m/s2, RSS = 6.87 m

x error2σ stdev

0 5 10 15 20 25 30−20

0

20

y sa

t 2 (m

)

Day (RMS = 2.40 m) 71.3% < 1σ

y error2σ stdev

0 5 10 15 20 25 30−20

0

20

z sa

t 2 (m

)

Day (RMS = 2.99 m) 91.0% < 1σ

z error2σ stdev

Figure 41. Woodstock position accuracy using the EKF.

0 5 10 15 20 25 30−0.02

0

0.02

dx s

at 2

(m/s

)

Day (RMS = 2.53e−003 m/s) 91.9% < 1σ

EKF sat 2 velocity error with state noise 1.34e−008 m/s2, RSS = 5.791e−003 m/s

dx error2σ stdev

0 5 10 15 20 25 30−0.02

0

0.02

dy s

at 2

(m/s

)

Day (RMS = 2.70e−003 m/s) 70.5% < 1σ

dy error2σ stdev

0 5 10 15 20 25 30−0.02

0

0.02

dz s

at 2

(m)

Day (RMS = 4.46e−003 m/s) 89.2% < 1σ

dz error2σ stdev

Figure 42. Woodstock velocity accuracy using the EKF.

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B. Navigation Support

Snoopy and Woodstock could be used as mobile tracking stations after using Liaison Navigation to determinetheir orbit states. Snoopy and Woodstock could track other spacecraft in transit, in lunar orbits, or in haloorbits and send the observation data to Earth for processing. They could also broadcast navigation signalsthat would provide additional navigation accuracy during landings or surface explorations. The errors in orbitestimates for Snoopy and Woodstock would translate into errors in this tracking data, so some simulationswere performed to see how useful the tracking data would be.

The first user spacecraft simulated was in a circular, low lunar orbit with an altitude of 100 km. Dependingon the orientation of the orbit plane, Woodstock might not be visible very often from the point of view of thisuser spacecraft. Because of this, only Snoopy was used as a tracking station. Observations were generatedusing Snoopy’s true orbit and the true orbit of the user spacecraft. If the user spacecraft was visible, theobservations were produced every 60 seconds during the times that Snoopy was scheduled to track userspacecraft, for a 2.5-day fit span. The initial state of the user spacecraft was perturbed and a Least-SquaresBatch Processor was used to estimate the initial state of the user spacecraft. This was repeated for runsstarting on March 15, 18, 21, and 28 of 2012 using range or range-rate. The same simulations were performedwith a lunar orbiter at 1000 km altitude for 3-day fit spans. This higher lunar orbiter was tracked by bothSnoopy and Woodstock. The average 3D position and velocity errors over the four runs are shown in Table2. It appears that in these simulations, Snoopy did very well tracking the user in the 100 km orbit, but notas well tracking the user in the 1000 km orbit, although the results are still not bad. Woodstock was moreeffective for tracking the 1000 km user, and would be effective tracking other spacecraft in low lunar orbit ifthe orbit geometry were such that Woodstock could see the user a fair amount of the time.

Table 2. ORBIT DETERMINATION ACCURACY USING SNOOPY AND WOODSTOCK AS TRACKINGSTATIONS

User altitude Tracking s/c obs. type pos. error (m) vel. error (m/s)

100 km Snoopy Range 8.76 0.012

100 km Snoopy Range rate 9.50 8.6× 10−3

1000 km Snoopy Range 84.1 0.032

1000 km Snoopy Range rate 25.6 9.9× 10−3

1000 km Woodstock Range 6.34 3.3× 10−3

1000 km Woodstock Range rate 10.0 4.7× 10−3

IX. Conclusion

Although further research could reveal a more optimum low lunar orbit, or a descent from the LL2 halothat results in a better value of δ, it was shown that this mission to LL2 would provide needed communicationand navigation support for future lunar missions as specified in ESAS. Snoopy and Woodstock would providetracking support for Earth transit, orbit insertion and landing on the Moon. They could also be usedfor intervehicle and Moon-to-Earth communications. The constellation could evolve seamlessly by addingspacecraft when new capabilities are needed. The utilization of halo orbits would allow the constellation tobe reconfigured with very little fuel in the event of the loss or addition of spacecraft. This flexibility wouldadd considerably to the reliability and redundancy of the constellation.

In addition to satsifying the needs outlined in ESAS, the mission costs would be reduced by usinglow-energy transfers and Liaison Navigation. Low-energy transfers such as BLT’s would enable more massto be placed in lunar halo orbits than conventional lunar transfers. This can all be done using provenchemical propulsion technology, and several spacecraft could be deployed using a single launch vehicle.Liaison Navigation would enable the constellation to navigate autonomously to reduce the workload for theDSN, and the tracking data from the low lunar orbiter would allow better estimates of the lunar far sidegravity field.

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References

1“NASA’s Exploration Systems Architecture Study,” November 2005, NASA-TM-2005-214062.2Farquhar, R. W., “The Control and Use of Libration-Point Satellites,” 1970, NASA Technical Report TR R-346.3Carpenter, J., Folta, D., Moreau, M., and Quinn, D., “Libration Point Navigation Concepts Supporting the Vision

for Space Exploration,” 2004, AIAA 2004-4747, AIAA/AAS Astrodynamics Specialist Conference, Providence, Rhode Island,August 16-19.

4Konopliv, A., Asmar, S. W., Carranza, E., Sjogren, W. L., and Yuan, D., “Recent Gravity Models as a Result of theLunar Prospector Mission,” Icarus, Vol. 150, 2001, pp. 1–18.

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11Hill, K., Born, G. H., and Lo, M. W., “Linked, Autonomous, Interplanetary Satellite Orbit Navigation (LiAISON) inLunar Halo Orbits,” 2005, AAS 05-400, AAS/AIAA Astrodynamics Specialist Conference, Lake Tahoe, CA, Aug. 7-11.

12Hill, K., Lo, M. W., and Born, G. H., “Liaison Navigation in the Sun-Earth-Moon Four-body Problem,” 2006, AAS06-221, AAS/AIAA Space Flight Mechanics Conference, Tampa, Florida, January 22-26.

13Murray, C. D. and Dermott, S. F., Solar System Dynamics, Cambridge University Press, Cambridge, UK, 1999.14Howell, K. and Pernicka, H., “Numerical Determination of Lissajous Trajectories in the Restricted Three-body Problem,”

Celestial Mechanics, Vol. 41, 1988, pp. 107–124.15Wilson, R., “Derivation of Differential Correctors Used in GENESIS Mission Design,” JPL IOM 312.I-03-002.16Lemoine, F., Smith, D., Zuber, M., Neumann, G., and Rowlands, D., “A 70th degree lunar gravity model (GLGM-2)

from Clementine and other tracking data,” Journal of Geophysical Research, Vol. 102, 1997, pp. 16339–16359.17Folta, D., “LUNAR Reconnaissance Observer ∆V Budget Estimate,” NASA Goddard Space Flight Center, April 2, 2004.18Tapley, B. D., Schutz, B. E., and Born, G. H., Statistical Orbit Determination, Elsevier Academic Press, Burlington,

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