IAC-11-C1.2.3

EVOLUTION OF THE OUT-OF-PLANE AMPLITUDE FORQUASI-PERIODIC TRAJECTORIES IN THE EARTH-MOON SYSTEM

Thomas A. Pavlak∗

Purdue University, United States of Americatpavlak@purdue.edu

Kathleen C. Howell†

Purdue University, United States of Americahowell@purdue.edu

The out-of-plane amplitude along quasi-periodic trajectories in the Earth-Moon system is highlysensitive to perturbations in position and/or velocity as underscored recently by the ARTEMISspacecraft. Controlling the evolution of the out-of-plane amplitude is non-trivial, but can becritical to satisfying mission requirements. The sensitivity of the out-of-plane amplitude evolutionto perturbations due to lunar eccentricity, solar gravity, and solar radiation pressure is exploredand a strategy for designing low-cost deterministic maneuvers to control the amplitude historyis also examined. The method is sufficiently general and applied to the L1 Lissajous orbit thatserves as a baseline for the ARTEMIS P2 trajectory.

I. INTRODUCTION

For decades, the collinear libration points have beenviewed as potentially useful locations to support bothcommunications and scientific observations. Since thelate 1970s, a number of missions have successfully op-erated in the vicinity of Sun-Earth collinear librationpoints including ISEE-3,1 WIND,2 SOHO,3 ACE,4

MAP,5 and Genesis.6 Despite the successes of pastSun-Earth libration point missions, no spacecraft hadever flown in the vicinity of an Earth-Moon librationpoint until August 2010 when the ARTEMIS P1 space-craft entered an orbit near the Earth-Moon L2 point.The ARTEMIS P2 probe became the second Earth-Moon libration point orbiter with an arrival near theL2 point in October 2010. Due to the continuing suc-cess of the mission, it seems likely that interest in theEarth-Moon libration points will continue to increase.

The ARTEMIS libration point orbit design featuresLissajous orbits with z-amplitudes, i.e., excursionsnormal to the lunar orbit plane, that vary greatly overthe course of the mission. The evolution of the z-amplitude is highly sensitive to small perturbationsin position or velocity. Deterministic correction ma-neuvers are designed and can deliver specific pointsolutions to ensure that the spacecraft trajectory re-tains the required lunar arrival conditions includinginclination, several months into the future. However,these successfully designed corrections do not yield any

∗Ph.D. Student, School of Aeronautics and Astronautics†Hsu Lo Professor of Aeronautical and Astronautical Engi-

neering

useful generalizations about the trajectory behavior atlater epochs along the path or for future mission ap-plications. Increasing the design intuition concerningthe Lissajous trajectory evolution and the potentialtrade-off relationships in this dynamical environmentis essential for effective future spacecraft operations inthe Earth-Moon regime. It is desirable to understandthe underlying dynamical structure that produces thez-amplitude evolution and to explore the sensitivityof this out-of-plane component to lunar eccentricityand solar perturbations. This analysis is accompaniedby a comparison of results from numerical simulationsin both the restricted three-body model and higher-fidelity Earth-Moon-Sun ephemeris models for the L1

Lissajous trajectory that represents the path of theARTEMIS P2 spacecraft. As will be demonstrated,initial studies indicate that lunar eccentricity, solargravity, and solar radiation pressure create a small butnon-negligible effect on z-amplitude evolution.

Numerical differential corrections algorithms arealso required to both predict and control the evolutionof these unstable trajectories via the determination oflocations along the trajectory that are best-suited fordeterministic maneuvers. This capability is exploredby using some type of initial guess to compute a refer-ence solution that likely does not satisfy specified con-straints. Then, the path is decomposed into a series ofsegments and impulsive maneuvers are applied at vari-ous discrete locations along the trajectory. Ultimately,the process translates into a general search procedureto ensure that the spacecraft satisfies desired end-of-mission criteria at lunar orbit arrival despite the highly

IAC-11-C1.2.3 Page 1 of 11

sensitive Earth-Moon environment.

II. THE ARTEMIS MISSION

As the first libration point orbiter in the Earth-Moon system, the Acceleration Reconnection and Tur-bulence and Electrodynamics of the Moons Interac-tion with the Sun (ARTEMIS) mission represents asignificant step in multi-body mission design and op-erations.7 ARTEMIS is an extension of the TimeHistory of Events and Macroscale Interactions dur-ing Substorms (THEMIS) mission that was launchedin 2007.8 The THEMIS mission originally consistedof five spacecraft in elliptical orbits about the Earthcollecting measurements of the magnetosphere. A ren-dering of a THEMIS/ARTEMIS spacecraft appears inFig. 1.

Fig. 1: THEMIS/ARTEMIS Rendering9

The ARTEMIS mission originated in July 2009,when two of the five THEMIS spacecraft, termed P1and P2, initiated a series of orbit-raising maneuversand lunar fly-bys to eventually depart the vicinity ofthe Earth. Closely following Sun-Earth and Earth-Moon manifolds, the P1 and P2 spacecraft each in-serted into an orbit near the Earth-Moon L2 librationpoint in August and October of 2011, respectively.The libration point orbiting phase of the ARTEMISmission incorporates both L1 and L2 quasi-periodicLissajous trajectories. Like most libration point or-bits, the L1 and L2 Lissajous orbits, designed for theARTEMIS spacecraft, are inherently unstable and sen-sitive to perturbations. Since both P1 and P2 space-craft are operating only on propellant remaining fromthe THEMIS mission, efficient stationkeeping and or-bit maintenance strategies are critical.10,11 While thisanalysis focuses on the ARTEMIS P2 orbit, the base-line trajectories for both spacecraft are presented forreference. Note that the information summarized inthis investigation reflects the ARTEMIS baseline de-sign as of August 2010.

II.I The ARTEMIS P1 Trajectory

The Earth-Moon libration point phase of theARTEMIS P1 trajectory begins with insertion into anEarth-Moon L2 Lissajous orbit on August 23, 2010.The P1 spacecraft orbits near the L2 point for approxi-mately 131 days before transferring to an L1 Lissajous

orbit where it remains for an additional 85 days. Afterdeparting the L1 vicinity, P1 is delivered into a retro-grade lunar orbit such that periapsis occurs at an alti-tude of 1500 km on April 9, 2011. Three-dimensionaland planar projections of the ARTEMIS P1 baselinetrajectory appear in a Moon-centered rotating refer-ence frames in Fig. 2(a) and 2(b), respectively. Thefinal 1500 km × 18,000 km (altitude) retrograde lunarorbit is included in red in Fig. 2(b).

−50

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z(×

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) L1

InitialState

L2

Moon

a)

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−6

−4

−2

0

2

4

6

x (× 104 km)

y(×

104

km

)

L1 L2

Moon

b)

Fig. 2: ARTEMIS P1 Baseline Trajectory

II.II The ARTEMIS P2 Trajectory

The ARTEMIS P2 spacecraft arrives in the vicin-ity of the Earth-Moon L2 libration point on October3, 2010 and, rather than entering orbit near L2, theP2 spacecraft immediately shifts to a path that blendsinto an L1 Lissajous orbit where it remains for approx-imately 154 days. The departure from the L1 orbitdelivers P2 to a prograde lunar arrival at a periapsis al-titude of 1500 km. An isometric view of the ARTEMISP2 baseline trajectory is plotted in Fig. 3(a) and aplanar projection with the final 1500 km × 18,000 km(altitude) orbit appears in Fig. 3(b).

III. SYSTEM MODELS

In this analysis, several dynamical models are em-ployed to explore the effects of various perturbations

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StateL1

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Fig. 3: ARTEMIS P2 Baseline Trajectory

on the z-amplitude evolution in a quasi-periodic Lis-sajous orbit and to demonstrate a strategy for placingdeterministic maneuvers that correct an unfavorablez-amplitude evolution. The circular restricted three-body problem (CR3BP) serves as the framework forpreliminary trajectory design activities and as thebaseline dynamical model for the perturbation study.Employing the CR3BP is advantageous in prelimi-nary analysis since the gravity of both the Earth andthe Moon are incorporated simultaneously, but thetime-invariant nature renders a formulation that offerssignificant advantages over higher-fidelity ephemerismodels for initial analysis. An additional benefit ofthe CR3BP is that solutions obtained in this modelcan generally be transitioned to higher-fidelity mod-els using straightforward differential corrections tech-niques.12,13

III.I Restricted Three-Body Model

The circular restricted three-body problem is uti-lized to describe the motion of a body – in this case,a spacecraft – that is influenced by two gravity fieldssimultaneously, that is, the Earth and the Moon, un-der a set of simplifying assumptions. The spacecraft is

considered “massless” and the orbits of the Earth andMoon are modeled to be coplanar and circular relativeto their barycenter. It is convenient to describe theCR3BP in terms of a rotating reference frame centeredat the Earth-Moon barycenter in which the positivex-axis is oriented along the Earth-Moon line, the posi-tive z-axis is parallel to the orbital angular momentumvector that reflects the orbits of the primaries, and they-axis completes the right-handed triad. The massesof the two primaries are related through the nondi-mensional mass parameter, µ, defined as

µ =mM

mE +mM[1]

where mE is the mass of the Earth and mM representsthe mass of the Moon. The six-dimensional space-

craft state vector is denoted x =[x, y, z, x, y, z

]T, with

bold symbols signifying vector quantities and positiondefined relative to the system barycenter. Then, thegoverning equations of motion in the CR3BP are writ-ten as nondimensional second-order scalar differentialequations, i.e.,

x− 2y − x = − (1 − µ) (x+ µ)

d31− µ (x− 1 + µ)

d32[2]

y + 2x− y = − (1 − µ) y

d31− µy

d32[3]

z = − (1 − µ) z

d31− µz

d32[4]

with the scalar relative distances,

d1 =

√(x+ µ)

2+ y2 + z2 [5]

d2 =

√(x− 1 + µ)

2+ y2 + z2 [6]

such that d1 and d2 are measured from the Earth tothe spacecraft and the Moon to the spacecraft, respec-tively. To compute trajectories in the CR3BP as wellas higher-fidelity models, the vector equation of mo-tion is expressed in first-order form as

x = f (t,x) [7]

where position and velocity information are includedin the six-dimensional state vector, x.

III.II Higher-Fidelity Modeling

To explore the sensitivity of z-amplitude evolutionin quasi-periodic orbits to various dynamical pertur-bations and to more accurately compute the magni-tude and placement of deterministic ∆V maneuvers,higher-fidelity dynamical modeling is ultimately re-quired. While the fidelity of the model is adjusted toassess the sensitivity of z-amplitude evolution to spe-cific dynamical influences, the highest-fidelity modelused in this analysis is a Moon-Earth-Sun point massmodel incorporating the JPL DE405 ephemerides and

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solar radiation pressure. Trajectories are computed ina Moon-centered, Earth J2000 reference frame usingthe equations of motion as expressed in second-orderform, i.e.,

rqi = fg + fs [8]

where the vector rqi is the position of the spacecraftwith respect to the central body, “q”, and the vectorsfg and fs represent the acceleration of the spacecraftdue to gravitational forces and solar radiation pres-sure, respectively. Both fg and fs are expressed herein terms of dimensional quantities, but are, in practice,nondimensionalized for use in numerical integrationand differential corrections algorithms.

The terms on the right side of Eqn. [8] representthe components of the vehicle acceleration that aremodeled to be consistent with the dynamics in theproblem. The gravitational acceleration is governedby the familiar N -body relative equations,

fg = − G (mi +mq)

r3qirqi+G

n∑j=1j 6=i,q

mj

(rijr3ij

− rqjr3qj

)[9]

where the vector rqj represents the position of eachperturbing body with respect to the central body. Therelative locations of the celestial bodies are delivereddirectly from DE405 ephemeris data. The vector rijthen corresponds to the position of each perturbingbody relative to the spacecraft, “i”. The terms G andmk are the gravitational constant and the mass of body“k”, respectively, expressed in terms of dimensionalunits. The acceleration due to solar radiation pressureis modeled as

fs =kAS0r

20

cm

rSi

r3Si

[10]

where the vector rSi represents the position of thespacecraft relative to the Sun. The constant k is a ma-terial property based on the reflectivity/absorptivityof the spacecraft, A is the cross-sectional area ofthe spacecraft, and S0 is the solar flux associatedwith the nominal Sun-Earth distance, r0 = 1 AU.The constants c and m correspond to the speed oflight and the mass of the spacecraft, respectively.14–16

These parameters are spacecraft-specific and, in thisinvestigation, are selected to be consistent with theARTEMIS P2 spacecraft to improve the fidelity of thez-amplitude evolution analysis.

IV. NUMERICAL SCHEMES

To investigate the sensitivity of the z-amplitudeevolution to various dynamical perturbations and toexamine the placement of deterministic maneuvers tocompensate for unfavorable z-amplitude histories, areference solution is required. Unfortunately, the nu-merical computation of a continuous, end-to-end ref-erence trajectory incorporating multiple revolutions ofa Lissajous trajectory is nontrivial given that most

collinear libration point orbits are unstable and anysimulation departs the orbit after 1-2 revolutions ifnot maintained. Numerical determination of a contin-uous, multi-revolution reference solution is possible,however, by employing a multiple shooting differentialcorrections algorithm. This reference is then incor-porated into both the z-amplitude sensitivity analysisand in the deterministic correction maneuver place-ment strategy.

IV.I Multiple Shooting

The general multiple shooting approach is repre-sented in the schematic in Fig. 4. An initial guessis discretized into n “patch points” connected by n−1trajectory arcs as illustrated in Fig. 4(a). Each patchpoint is represented by a six-dimensional vector, xi,and the integration time associated with each arc is Ti.Differential corrections in time-dependent ephemerismodels also require the epoch at each patch point, de-noted τi. The shortened notation xt

i, identifies the ter-

…

x1, τ1

x2, τ2

xn-1, τn-1

xn, τnx2

txn-1t xn

tTn-1

T1

a) Initial Guess

tTn-1

…x1, τ1

x2, τ2xn-1, τn-1

xn, τnx2

t xn-1t xn

T1

b) Converged

Fig. 4: Multiple Shooting Schematic

minal integrated state along each trajectory segment,xti (xi−1, Ti−1, τi−1). In this application, Newton’s

method is incorporated in the differential correctionsprocess and converges to a solution that is continuousin position and velocity at all internal patch points andsatisfies any additional problem-dependent end con-straints.12

IV.II Designing a Reference Solution

Computing a reference solution is a critical first stepto explore the sensitivity of z-amplitude evolution inquasi-periodic orbits and, ultimately, to determine themost effective placement of deterministic correctionsmaneuvers for meeting a set of end-of-mission require-ments. Note that, in this investigation, “referencesolution” is distinguished from “baseline.” Here, abaseline trajectory is a nominal path computed duringthe mission design phase. However, the term referencesolution is used more broadly to denote a continuous,“end-to-end“ trajectory that serves as an initial guessfor a numerical corrections algorithm. During nomi-nal mission operations, the baseline may be employed

IAC-11-C1.2.3 Page 4 of 11

as a reference solution but if the spacecraft deviatessignificantly from the original design, continually up-dating the reference is more effective. While referencesolution design is largely application-dependent, it issummarized in three general steps:

1) Initial guess generation

2) Convergence in lower-fidelity model

3) Convergence in higher-fidelity model

In the case of libration point trajectories such as thosefor ARTEMIS, it is useful to complete steps 1) and 2)within the context of the circular restricted three-bodyproblem and exploit well-established analysis tools forperiodic orbits and invariant manifolds. For example,initial guesses for ARTEMIS-like trajectories are gen-erated using stacked revolutions of planar Lyapunovorbits and arc segments along stable and unstable in-variant manifolds as discussed by Pavlak and Howell.17

Multiple shooting is used to converge to a referencesolution in the circular restricted three-body problemthat can, in turn, be transitioned to any desired higher-fidelity dynamical models.

IV.III Out-of-Plane Amplitude Correction Strategy

Quasi-periodic orbits in the Earth-Moon systemare highly sensitive and even small errors in posi-tion and/or velocity strongly influence the z-amplitudeevolution downstream along the trajectory. For space-craft such as ARTEMIS, with a relatively strict setof end-of-mission objectives, significant changes in z-amplitude can result in lunar arrival inclinations, forexample, that are unacceptably large and do not sat-isfy mission requirements. There are a variety ofpotential approaches for correcting an undesirable z-amplitude evolution, but, for this initial investigation,a straightforward, systematic strategy is implemented.

A schematic for a deterministic ∆V placement strat-egy to rectify an unfavorable out-of-plane amplitudeevolution appears in Fig. 5. The process begins with acontinuous reference solution as in Fig. 5(a). Possiblemaneuver locations, i.e., patch points, appear in redand the green “X” denotes the desired end-of-missionrequirement. The search for a low-cost maneuver toproduce a favorable z-amplitude evolution commencesby introducing a ∆V at each patch point – employ-ing multiple shooting with any remaining segmentsalong the reference solution as an initial guess. Theprocess results in an entire set of trajectories, eachwith a ∆V in a different location; each arc satisfiesthe end-point constraints. The potential determinis-tic maneuvers along a sample reference path appearin Fig. 5(b). Each of these maneuvers represents one∆V in a corrected trajectory that satisfies the desiredset of end constraints though the individual resultingpaths are not pictured. Note that, in general, a cor-rections scheme may not be successful in delivering

one ∆V from an arbitrary point on the path that sat-isfies the end-point constraints. Thus, a single ∆Vthat yields the desired end conditions may not ex-ist for every patch point along a reference trajectory.The final step in the out-of-plane amplitude correc-tion strategy is to select a single ∆V – typically themaneuver with the lowest magnitude – from the setof all possible maneuvers. In the schematic, ∆V2 inFig. 5(b) has the smallest magnitude and is appliedin Fig. 5(c) as a deterministic maneuver along thereference trajectory that is plotted in black. The con-tinuous, post-maneuver trajectory that satisfies the setof end-of-mission requirements is depicted in blue. It

x1, τ1x2, τ2

x3, τ3 x4, τ4

x5, τ5

a) Reference Trajectory

x1, τ1x2, τ2

x3, τ3 x4, τ4

x5, τ5ΔV2

ΔV1

ΔV3

ΔV4

b) Potential ∆V Maneuvers

x1, τ1x2, τ2

x3, τ3

x4, τ4

x5, τ5

ΔV

c) Corrected Trajectory with Minimum ∆V

Fig. 5: Deterministic ∆V Placement Schematic

is important to note that future iterations of this al-gorithm could incorporate a strategy to optimize boththe location and magnitude of the maneuvers. Thecurrent procedure, however, demonstrates the ben-efits of employing a reference solution and suppliesa reliable and straightforward approach for correct-ing unfavorable out-of-plane amplitude evolution in aquasi-periodic libration point orbit using a single de-terministic ∆V maneuver.

V. ARTEMIS P2 APPLICATION

An important scientific goal for the ARTEMIS mis-sion is the collection of two-point measurements of theEarth’s magnetotail and the solar wind over a widerange of locations in the Sun-Earth and Earth-Moonregimes.8 In the final phase of the mission, the twoARTEMIS spacecraft enter lunar orbit to record alarge number of “lunar wake crossings” in which theMoon is directly between the spacecraft and the Sun.To produce the desired sequence of wake crossings,it is critical that both probes achieve a near-planar

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lunar orbit insertion at a specific Julian date as dic-tated by scientific requirements. In the highly sensi-tive Earth-Moon system, however, small perturbationsare capable of altering the z-amplitude evolution ofquasi-periodic orbits such as those employed by theARTEMIS spacecraft in such a way that lunar arrivalrequirements are not satisfied. In fact, during missionoperations for the P2 spacecraft, small errors in theL2 injection state did result in an unfavorable out-of-plane evolution for the subsequent L1 Lissajous orbit.A deterministic ∆V maneuver to correct the evolv-ing z-amplitude had to be incorporated. Thus, theP2 trajectory is employed in this analysis to examinethe effect of perturbations on z-amplitude evolution inquasi-periodic orbits and to demonstrate the applica-tion of a deterministic maneuver placement strategy.

V.I Reference Trajectory

A continuous, “end-to-end” reference trajectory rep-resenting the ARTEMIS P2 spacecraft is required forthe z-amplitude sensitivity analysis and also serves asan integral part of the deterministic correction ma-neuver placement strategy. The P2 reference solutionis designed consistent with the mission requirementsdescribed in Section II.II. Recall that this baselinepath only incorporates the Earth-Moon libration pointphase of the ARTEMIS P2 spacecraft and, thus beginsat the L2 orbit insertion. The trajectory is initiallycomputed in the circular restricted three-body prob-lem to exploit the wealth of dynamical systems theoryknowledge in this regime. The entire process is de-tailed by Pavlak and Howell,17 but is summarized here.The initial guess for the reference is comprised of fourdistinct phases:

1) Unstable L2 Lyapunov manifold arc

2) Stable L1 Lyapunov manifold arc

3) 10 “stacked” revolutions of L1 Lyapunov orbit

4) Unstable L1 Lyapunov manifold arc

These individual segments are discretized into patchpoints and the multiple shooting algorithm returns acontinuous reference solution in the CR3BP. Duringthis process, the initial position is constrained to bethe fixed value obtained from orbit determination dataon October 3, 2010. If required, the CR3BP trajectoryis transitioned to be continuous in an ephemeris modelthat might serve as a higher-fidelity reference, e.g., theARTEMIS P2 baseline trajectory as displayed previ-ously in Fig. 3. The trajectories employed throughoutthis analysis are computed using the parameters fromTable 1. Numerical integration and differential cor-rections processes are conducted using quantities thatare nondimensionalized by the characteristic values oflength and time, denoted as l∗ and t∗, respectively.The radius of the Moon is rM , the gravitational pa-rameters of the various bodies are represented by µk,

Parameter Value Unitsl∗ 385,692.50 kmt∗ 377,084.152667038 s

µM 398,600.432896939 km3/s2

µE 4902.80058214776 km3/s2

µS 132,712,440,017.987 km3/s2

rM 1738.2 kmk 1.17A 1 m2

m 85.3527 kg

S0 1358.098 W/m2

r0 149,597,927 kmc 299,792.458 km/s

Table 1: Problem Constants

and the remaining parameters as defined in SectionIII.II.

V.II Out-of-Plane Amplitude Sensitivity

The ARTEMIS P2 spacecraft emerged from theEarth-to-Moon transfer and entered the vicinity of theL2 libration point on a trajectory with an unfavorableout-of-plane amplitude evolution prior to lunar orbit.A principal goal of this investigation is to examineif this phenomenon is the result of specific dynami-cal perturbations – namely, lunar eccentricity, solargravity, and/or solar radiation pressure – or a productof the fundamental Earth-Moon multi-body gravita-tional environment. To explore the z-amplitude, letthe baseline CR3BP ARTEMIS P2 solution in the lu-nar region serve as the “reference trajectory.” Then,a second path, termed the “perturbed reference tra-jectory”, is generated with an initial velocity that isaltered from that of the reference to produce an unfa-vorable z-amplitude evolution. These two trajectoriesand their associated z-amplitude histories appear inFig. 6.

The sensitivity of the out-of-plane evolution to vari-ous perturbing effects is evaluated by using the shoot-ing algorithm to numerically produce continuous solu-tions in dynamical models of increasing fidelity andcomparing the resulting z-amplitude profiles. Thereference and perturbed reference trajectories in theCR3BP are employed as the initial guesses for themultiple shooting scheme. The four dynamical modelsinclude:

1) CR3BP

2) Moon-Earth Point Mass

3) Moon-Earth-Sun Point Mass

4) Moon-Earth-Sun Point Mass with SRP

Employing the circular restricted-three body solutionsas initial guesses, the shooting algorithms use the ref-erence and perturbed trajectories to converge in thethree higher-fidelity ephemeris models. For each or-bit, the initial position is always fixed and the epoch

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b) Perturbed Reference

0 50 100 150 200−1

0

1

Time (days after epoch)z-amplitude(×

104km)

c) Reference

0 50 100 150 200−1

0

1

2

Time (days after epoch)

z-amplitude(×

104km)

d) Perturbed Reference

Fig. 6: The z-Amplitude Evolution for Two CR3BP Trajectories

and periapsis radius are constrained at lunar arrival.Note, however, that no constraint is placed on finalinclination.

For both the reference and perturbed trajectories,the four converged paths lie relatively close together,thus, it is more useful to plot the difference in z-amplitude over time. Using the CR3BP solution asbaselines, the resulting ∆z-amplitude histories for thereference and perturbed trajectories constructed inhigher-fidelity models appear in Fig. 7. The blue linedenotes the difference in out-of-plane amplitude as de-termined between the circular restricted three-bodyproblem and an Earth-Moon point mass ephemerismodel; thus, the blue curve illustrates the impact oflunar eccentricity on z-amplitude. In comparison, thered line represents the variation between the CR3BPand an Earth-Moon-Sun point mass model. The lineassociated with the addition of solar radiation pressureto the model is not visible in this figure due to the factthat it is so close to the red curve, indicating that SRPhas relatively little effect on the z-amplitude evolutionof the ARTEMIS P2 spacecraft trajectory. This pointis further illustrated by comparing the difference inout-of-plane amplitude produced in the Earth-Moon-Sun model and the Earth-Moon-Sun model incorpo-rating SRP, respectively, as apparent in Fig. 8. Thez-amplitude is altered by less than 5 km due to SRPfor both the reference and perturbed trajectories. Theexact cause of the unfavorable out-of-plane amplitudeevolution that is experienced by the ARTEMIS P2trajectory is difficult to determine definitively given

the chaotic nature of the Earth-Moon system. Thus,it is possible that both the reference and perturbedCR3BP trajectories exhibit very similar z-amplitudebehavior in various higher-fidelity models. This resultindicates the likelihood that the z-amplitude evolutionof the ARTEMIS P2 path is due, not to a specific dy-namical perturbation, that is, lunar eccentricity, solargravity, or solar radiation pressure, but to the funda-mental sensitivity associated with unstable librationpoint orbits in the Earth-Moon-spacecraft three-bodyproblem.

V.III Out-of-Plane Amplitude Correction Example

For future spacecraft to operate efficiently in quasi-periodic libration point orbits, the out-of-plane am-plitude evolution must be controlled with minimal∆V . Low-cost solutions are particularly important forspacecraft with limited propellant such as ARTEMIS.Here, an out-of-plane amplitude correction strategy, aspresented in Section IV.III, is applied to the ARTEMISP2 trajectory as an example.

The perturbed ARTEMIS P2 trajectory, i.e., thetrajectory with an unfavorable z-amplitude evolutionin a Moon-Earth-Sun ephemeris model with SRP, iscomputed as in Section V.II. Note that the initial posi-tion is constrained to be fixed at the spacecraft arrivalvalue on October 3, 2010. The trajectory is plotted inFig. 9(a) and potential maneuver locations – indicatedin red – are then identified at various downstream loca-tions along the continuous perturbed reference trajec-tory. Approximately 130 maneuver locations are intro-

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0 50 100 150 200−2000

0

2000

Time (days after epoch)

∆z-amplitude(km)

a) Reference

0 50 100 150 200−5000

0

5000

Time (days after epoch)

∆z-amplitude(km)

b) Perturbed Reference

Fig. 7: Effect of Lunar Eccentricity (Blue) and Lunar Eccentricity + Solar Gravity (Red) on z-Amplitude

0 50 100 150 200−5

0

5

Time (days after epoch)

∆z-amplitude(km)

a) Reference Trajectory

0 50 100 150 200−5

0

5

Time (days after epoch)

∆z-amplitude(km)

b) Perturbed Trajectory

Fig. 8: Effect of SRP on z-Amplitude Evolution

duced in this example. Multiple shooting is employedto target from each potential maneuver location to afavorable, i.e., nearly planar, lunar arrival condition.Thus, ideally, the process yields 130 opportunities tomeet the arrival constraints, with various ∆V levels,one for each ∆V option. However, the differentialcorrections algorithm rarely converges on a solutionfrom all possible maneuver sites. A plot of the deter-ministic ∆V costs as a function of maneuver locationappears in Fig. 9(b) and a zoomed view is displayedin Fig. 9(c). Each point represents a unique trajec-tory that satisfies the set of desired end constraints.Note that, as expected, it is generally less costly froma ∆V perspective to implement the z-amplitude cor-rection maneuver early along the path. Also indicatedin Fig. 9(b) and 9(c) are the two low-cost maneuveroptions computed in this simulation (red). The firstmaneuver – represented by the first red dot – is theleast expensive of any maneuver computed along theperturbed reference trajectory and occurs just 1.8 daysafter L2 insertion at a cost of 13.5 cm/s. The correctedtrajectory appears in Fig. 10(a) and the correspond-ing z-amplitude evolution profile is displayed in Fig.10(b). The red and blue lines represent the pre- andpost-deterministic maneuver sections along the paths,respectively.

While it is most cost-effective to implement a ∆Vas early as possible, it may be operationally undesir-able or infeasible to plan and execute a deterministiccorrection maneuver only 1.8 days after L2 insertion.However, a relatively low-cost maneuver to correct theout-of-plane amplitude evolution may still be possi-ble during the L1 Lissajous phase of the ARTEMIS

P2 trajectory. The second sample deterministic ∆Vpossesses a magnitude of 28.9 cm/s and occurs dur-ing the first revolution of the L1 Lissajous orbit, 27.5days after the L2 insertion. The location and result-ing corrected z-amplitude evolution for this secondmaneuver option are indicated in Fig. 11. In thisinitial investigation, while some low-cost options existnear the xy-plane, it is typically less expensive to in-troduce deterministic maneuvers at locations of max-imum y-excursion which also correspond to regions ofmaximum out-of-plane amplitude for the ARTEMISP2 trajectory. However, it is difficult to draw defini-tive conclusions from these unoptimized preliminaryresults. It is also important to note that optimizationtechniques could be used in future iterations of thisout-of-plane amplitude corrections strategy to moreefficiently design the deterministic corrections maneu-vers.

VI. CONCLUSIONS

As the first mission to exploit Earth-Moon libra-tion point orbits, ARTEMIS spacecraft trajectoriesoffer a number of challenges including the control ofthe z-amplitude evolution along a quasi-periodic orbit.Future missions in this regime will likely be requiredto address the same issues. This preliminary analy-sis employs the ARTEMIS P2 trajectory to considerthe sensitivity of the out-of-plane evolution to variousdynamical perturbations in the Earth-Moon system;a proposed orbit maintenance strategy to computedeterministic ∆V maneuvers that correct the unfavor-able z-amplitude profiles is introduced.

The source of an unfavorable z-amplitude evolution

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Fig. 9: ∆V Cost at Various Maneuver Locations

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Fig. 11: Deterministic ∆V 27.5 Days After L2 Insertion

is not attributed to a single dynamical contribution. Itis demonstrated that favorable and unfavorable out-of-plane evolution histories are observable in dynamicalmodels of varying degrees of fidelity. Perturbationssuch as lunar eccentricity, solar gravity, and solar ra-diation pressure are not the sole cause of unfavorableoscillations in z-amplitude such as those observed dur-ing operations of the ARTEMIS P2 spacecraft. Asuccessful set of maneuvers for the P2 spacecraft diddeliver the vehicle to the appropriate lunar orbit.11 Analternative deterministic maneuver design approach isintroduced, however, one that is reliable and efficientas an option for future application.

The out-of-plane amplitude corrections strategy issuccessfully applied to the ARTEMIS P2 spacecrafttrajectory as a global-type search for deterministicmaneuvers along the trajectory. It is typically moreefficient to implement correction maneuvers as early aspossible, but relatively inexpensive solutions are alsoavailable at a variety of locations along the P2 path.In general, maneuvers are less costly when performedat locations of maximum y-excursion which also cor-respond to regions of maximum z-amplitude. Thiscorrection strategy is sufficiently general and also high-lights the utility of a continuous reference solution inhighly sensitive dynamical regimes such as the Earth-Moon system.

ACKNOWLEDGEMENTS

The authors wish to thank David Folta and MarkWoodard for their assistance with research efforts atGoddard Space Flight Center. Portions of this workwere completed at NASA Goddard and at Purdue Uni-versity under NASA/GSFC Grant No. NNX10AJ24Gand were also supported by the NASA Space Technol-ogy Research Fellowship (NSTRF) under Grant No.NNX11AM85H.

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