IAC-15-C1.2.10RAPID TRAJECTORY DESIGN IN THE EARTH-MOON EPHEMERISSYSTEM VIA AN INTERACTIVE CATALOG OF PERIODIC AND

QUASI-PERIODIC ORBITS

Davide Guzzetti, Natasha Bosanac, Amanda Haapala and Kathleen C. HowellSchool of Aeronautics and Astronautics, Purdue University, West Lafayette, IN, USA

{dguzzett,nbosanac,ahaapala,howell}@purdue.edu

David C. FoltaNational Aeronautics and Space Administration/ Goddard Space Flight Center, Greenbelt, MD, USA

david.c.folta@nasa.gov

Abstract

Upcoming missions and prospective design concepts in the Earth-Moon system are extensively leveragingmulti-body dynamics that may facilitate access to strategic locations or reduce propellant usage. Toincorporate these dynamical structures into the mission design process, Purdue University and the GoddardNASA Flight Space Center have initiated the construction of a trajectory design framework to rapidlyaccess and compare solutions from the circular restricted three-body problem. This framework, basedupon a ‘dynamic’ catalog of periodic and quasi-periodic orbits within the Earth-Moon system, can guidean end-to-end ephemeris design. In particular, the inclusion of quasi-periodic orbits further expands thedesign space, potentially enabling the detection of additional orbit options. To demonstrate the concept ofa ‘dynamic’ catalog, a prototype graphical interface is developed. Strategies to characterize and representperiodic and quasi-periodic information for interactive trajectory comparison and selection are discussed.A sample application is explored to demonstrate the efficacy of a ‘dynamic’ catalog for rapid trajectorydesign and validity in higher-fidelity models.

INTRODUCTION

With the increasing complexity of space missions,there is significant interest in trajectory design ap-proaches that require fewer resources and deliver re-sults sustainable over long term scenarios. Such goalsmay be achieved by leveraging the natural dynami-cal structures in the Earth-Moon system to guide theselection of a baseline trajectory. A well-informedtrajectory design process may be particularly benefi-cial for several upcoming mission concepts includingexoplanet observatories, in-situ exploration of aster-oids as well as redirect concepts, and lunar cubesatmissions.[1, 2, 3, 4, 5] The design of a baseline trajec-tory is nontrivial in a dynamically sensitive environ-ment. In fact, in a higher-fidelity multi-body regime,the comparison of a large set of candidate solutionsdemands significant, and often prohibitive, time andcomputational resources. However, the well-studiedCircular Restricted Three-Body Problem (CR3BP)can provide a reasonable approximation to the actualdynamical environment. The dynamical structuresavailable in this model have been successfully lever-aged by several missions in the Sun-Earth system aswell as in early demonstrations in the Earth-Moonsystem.[6, 7, 8]

For rapid trajectory design in a multi-body regime,knowledge of the dynamical structures in a simplifiedmodel may facilitate a better understanding of thedesign space than a set of point solutions in the com-plete ephemeris model. Many software packages, forexample, Systems Tool Kit (STK) and NASA’s Gen-eral Mission Analysis Tool (GMAT), offer a graphicalenvironment for trajectory design incorporating grav-itational fields at various levels of fidelity.[9, 10] How-ever, the focus is generally directed towards the deliv-ery of trajectory point designs and other operationalmission support capabilities. Thus, they may be notspecifically structured to offer guidance and insightinto the available dynamical structures throughoutthe region. To supply a framework for incorporat-ing knowledge of the dynamical accessibility in theEarth-Moon system, Purdue University and NASAGoddard Space Flight Center have been developingan interactive adaptive design process exploiting areference catalog of solutions from the CR3BP to en-hance efficient trajectory design in such complex en-vironments. In this simplified model, periodic andquasi-periodic orbits govern the underlying dynam-ics and are approximately retained in higher-fidelitymodels. The current effort is devoted to creating di-rect links between the problem understanding and

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its practical application by exploring the Earth-Moondesign space. There exists a wide array of known or-bits with significant potential for parking, staging andtransfers within the Earth-Moon system. Previouslydeveloped software tools, e.g., AUTO, can also supplya selection of these solutions as well as some insightinto the local dynamics and the evolution of a setof orbits along any family.[11] In particular, AUTOenables the computation of periodic orbits and theirnumerical continuation into orbit families, as well asthe detection and analysis of bifurcations. Neverthe-less, such tools do not provide a basic “blueprint” tosupport rapid, efficient and well-informed decisionsregarding the use of fundamental solutions in multi-body dynamical environment for any mission prior toan end-to-end trajectory design.

To overcome the challenges associated with identi-fying candidate trajectories in a chaotic multi-bodyregime, the available dynamical structures may beexplored interactively. Previous studies on the ap-plication of interactive visual analytics to trajectorydesign has been conducted by Schlei for various ap-plications in multi-body regimes [12]. In addition, aprototype software to assemble trajectories via point-and-click arc selection in multi-body scenarios is in-troduced by Haapala et al.[13] This design suite,Adaptive Trajectory Design (ATD), offers an inter-active interface to facilitate exploration of missiondesign options. First, the dynamics in the Earth-Moon and Sun-Earth systems are approximated us-ing the CR3BP. Dynamical structures in the formof periodic and quasi-periodic orbits and manifoldsmay be computed on-demand to construct an initialguess for an end-to-end trajectory, along with maneu-vers. Ultimately, the constructed initial guess can becorrected both in the CR3BP and in an ephemerismodel, and even exported to NASA’s GMAT. [10] Asa supplement to ATD, a ‘dynamic’ catalog has beenconstructed to identify and characterize periodic or-bit that may aid in trajectory design and selectionwithin the Earth-Moon system.[14] This informationand a preliminary classification of orbits are com-piled into a graphical environment, allowing the userto directly interact with data that can not be ade-quately represented by a static set of tabular data.As a result, a ‘dynamic’ and interactive catalog mayovercome some of the challenges associated with con-structing a predefined trade space to analyze a largeset of solutions for a general mission concept. [15]

In this investigation, the Earth-Moon catalag ofperiodic solutions is expanded to include nearlybounded motion. Quasi-periodic motion, which in-herits the behavior of a nearby periodic orbit, fur-ther expands the set of design options, thereby al-

lowing identification of trajectories that may satisfythe mission requirements when transitioned to anephemeris model. Families of quasi-periodic solu-tions are precomputed numerically and sampled toconstruct a representative set, which the user canimmediately access in the catalog. [16, 17] Whilea quasi-periodic orbit may partially retain the char-acteristics of a nearby periodic solution, it may alsopossess unique and independent features that may beexploited in the mission design process. Accordingly,quasi-periodic motions are characterized in terms ofquantities that may be used for a preliminary eval-uation of the mission constraints. The utilization ofthis information within a user graphical interface isdiscussed and a prototype is demonstrated via appli-cation to a sample mission concept.

DYNAMICALBACKGROUND

The rapid and intuitive exploration of the dynamicalstructures in the Earth-Moon system is first based onthe CR3BP. This dynamical model, which serves as areasonable approximation to the actual gravitationalfield, reflects the motion of a massless spacecraft un-der the influence of the point-mass gravitational at-tractions of the Earth and Moon. These two primarybodies are assumed to move in circular orbits abouttheir mutual barycenter. The motion of the vehicleis described relative to a coordinate frame, xyz, thatrotates with the motion of the Earth and Moon. Inthis frame, the spacecraft is located by the nondimen-sional coordinates (x, y, z). By convention, quantitiesin the CR3BP are nondimensionalized such that theEarth-Moon distance, as well as the mean motion ofthe primaries, are both equal to a constant value ofunity. In addition, the Earth and Moon have nondi-mensional masses equal to 1− µ and µ, respectively,where µ equals the ratio of the mass of the Moon tothe total mass of the system. In the rotating frame,the equations of motion for the spacecraft are writtenas:

x− 2y =∂U

∂x, y + 2x =

∂U

∂y, z =

∂U

∂z(1)

where the pseudo-potential function,

U =1

2(x2 + y2) +

1− µd

+µ

r,

while

d =√

(x+ µ)2 + y2 + z2

and

r =√

(x− 1 + µ)2 + y2 + z2 .

2

This gravitational field admits five equilibriumpoints: the collinear points L1, L2, and L3 are lo-cated along the Earth-Moon line; and two equilateralpoints, L4 and L5, form equilateral triangles with thetwo primaries. Since the CR3BP is autonomous, aconstant energy integral exists in the rotating frameand is equal to the Jacobi constant, JC:

JC = 2U − x2 − y2 − z2 (2)

At any specific value of the Jacobi constant, thereare infinite possible trajectories exhibiting a wide ar-ray of behaviors. However, any trajectory may begenerally classified as one of four types of solutions:equilibrium point, periodic orbit, quasi-periodic or-bit, and chaotic motion. Each of these solutions canbe identified using numerical techniques and subse-quently characterized using concepts and quantitiesfrom dynamical systems theory.

CATALOG OF PERIODICORBITS

To capture the dynamical structures available in theCR3BP, families of periodic orbits are exploited. Thecharacteristics of a periodic orbit generally reflectqualities of the nearby dynamics, potentially indi-cating the presence of additional structures, such asnearby manifolds or bounded motions. The charac-terization and classification of families of periodic so-lutions is, therefore, valuable in creating an efficientframework for mission design and preliminary mis-sion trade-offs.[15]

The catalog adopts the classification system forCR3BP periodic orbits, the one most currently ac-cepted in the astrodynamics community. Families ofperiodic orbits in the CR3BP are gathered into fourclasses: Libration Point Orbits (LPO), Resonant Or-bits (RES), Moon-Centered Orbits (P2), and Earth-Centered Orbits (P1). Classes are designated accord-ing to the dynamical origin of the families. Orbitsidentified as LPO emanate from the vicinity of theequilibrium points, such as the sample of axial, halo,Lyapunov, and vertical families in Figure 1(a). Reso-nant orbits, i.e., RES families, are each derived froman integer ratio between the orbital period and theperiod of the Moon’s motion around the Earth; theresonance is denoted p : q, where p is the number ofMoon revolutions about the Earth by the time thatthe vehicle accomplishes q orbits along the reference.Some representative orbits from the 3:1, 3:2, 2:1 fam-ilies are depicted in Figure 1(b). Orbits classified asP2 originate from the Moon’s dynamical neighbor-hood; this class includes distant prograde orbits, low

prograde orbits and distant retrograde orbits suchthose displayed in Figure 1(c). Similarly, P1 fami-lies originate from the Earth’s dynamical neighbor-hood. The families currently included in the cata-log are listed in Table 1 along with the abbreviationsadopted to identify each set of orbits. Presently, conicarcs and P1 families are not included in the catalog,but may be considered for future expansion of thedata set.

Table 1: List of families of periodic orbits currentlyavailable in the catalog.

Class Family Tag

LPO Li Lyapunov LyiLi Halo HiLi Axial AiLi Vertical ViLi Short Period SPiLi Long Period LPiLi Horseshoe HS

RES Planar Resonant n:m rpnmSpatial Resonant n:m rsnm

Moon- Planar Distant Retrograde DROCentered Spatial Distant Retrograde DRO3D

Distant Prograde DPOLow Prograde LoPO

For each family of periodic orbits, characteristicparameters can be analyzed by the user to identifyorbits that exhibit a desired behavior. Such quanti-ties enable the creation of user-defined design spacesfor the rapid performance of simple trades. The ap-plication of Keplerian parameters to a preliminarydesign framework may be ineffectual due to the timevariation along a generic arc and the dependence onthe selected central body. A characterization of pe-riodic orbits in the CR3BP for mission design is dis-cussed in [14]. The focus is a set of quantities thatincludes the geometrical amplitudes of the orbit, itsorbital period and Jacobi constant. As a first approx-imation, the geometrical amplitudes may be usefulin evaluating preliminary size and constraints, whilethe period of the orbit provides an approximate timescale for use in maneuver and communications plan-ning. In addition, the Jacobi constant, as defined inEq. (2), indicates the energy level of the trajectoryand can be linked to the minimum transfer cost be-tween two orbits. These quantities are complementedby estimates for the operational costs. Accordingly,simple periodic orbit insertion and station keepingcosts for a large variety of families are included, en-

3

23

45x 10

5 −1

0

1

x 105

−1

−0.5

0

0.5

1

x 105

y [km]x [km]

z [k

m]

Lyapunov

VerticalAxialHalo

Moon (x3 scale)

(a) Libration Point Orbits.

−4 −2 0 2 4

x 105

−5

−4

−3

−2

−1

0

1

2

3

4x 10

5

x [km]

y [k

m]

rp31

rp21rp32

(b) Resonant Orbits.

2 3 4 5 6

x 105

−2

−1.5

−1

−0.5

0

0.5

1

1.5

x 105

x [km]

y [k

m]

DRODPOLoPO

L1

L2

Moon (x3 scale)

(c) Moon-Centered Orbits.

Figure 1: Sample members of well-known orbit families in the Earth-Moon system, plotted in the rotatingframe.

abling rapid estimation of the deterministic DV. Suchinsight is valuable in preliminary identification of re-gions in the Earth-Moon system that are accessiblefor a given mission scenario. In the catalog, a se-lected set of periodic orbit insertion costs are based onsimple straightforward transfers from LEO, includingdirect transfers and powered flyby transfers. The sta-tion keeping maneuvers implemented to predict theorbit maintenance costs are based on the long-termstrategy discussed in [15].

The trade spaces in the ‘dynamic’ catalog use sim-ple statistical indexes, such as mean, range or stan-dard deviation to provide a compact global represen-tation of a large set of orbits. For example, con-sider Figure 2. Each box in the figure represents afamily of orbits (identified by the associated label);the boxes are portrayed in a two dimensional designspace, where the horizontal axis corresponds to anestimate of the annual station-keeping cost and thevertical axis represents the direct insertion transferDV. Within this space, each box is centered at themean value of the associated quantities along the fam-ily. The size of each box is proportional to the stan-dard deviation of the quantities computed along thefamily. Using this simplified representation, the usercan simultaneously estimate and compare the station-keeping and transfer costs for multiple families of or-bits. Since this representation is implemented in aninteractive graphical interface, the user can modifythe visible information, such as the quantities on eachaxis or the families included in the plot. For a com-plete representation of the characteristic quantitiesalong the family, the actual characteristic curve canbe displayed on-demand. Allowing the user to displayinformation only when desired, reduces the complex-ity in visualizing multiple characteristic curves thatare overlaid on a single two-dimensional plot. A moredetailed description of this representation of periodic

orbits and its implementation appear in [15].

0 10 20 30 40 50 60 70 80300

400

500

600

700

800

900

1000

1100

1200

Annual ∆VSK [m/s]

∆V PO

I [m/s

]

DRO

A1

H2

H1

Ly1

Ly2

A2

SP5

Figure 2: Simplified representation of transfer andstation-keeping cost of each family using boxes.

EXISTENCE OFQUASI-PERIODIC ORBITS

In the CR3BP, the existence of important dynami-cal structures, such as manifolds and quasi-periodicorbits, is associated with the linear stability of a pe-riodic orbit. Upon linearization of the equations ofmotion for the CR3BP, in Eq. (1), the general so-lution to the linear variational equations is writtenas

δx(t) = eA(t−t0)δx(t0) , (3)

where δx(t) is the variation relative to some referencesolution, A = A(t) is the Jacobian matrix of first par-tial derivatives of Eq. (1) evaluated along the refer-ence (A is generally not constant). The state transi-tion matrix (STM) is defined as Φ(t, t0) = eA(t−t0),essentially indicating (in linear approximation) the

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(a) Libration Point Orbits. (b) Moon-Centered Orbits. (c) Resonant Orbits.

Figure 3: Number of nontrivial modes associated to a center subspace across sample families in the CR3BPdenoted by color (Blue = 0, Green = 1, Red = 2).

sensitivity of a given state along the reference pathat time t to any variations in the initial state at t0.For a periodic orbit, the STM integrated for exactlyone orbital period, known as monodromy matrix, isleveraged to predict the orbital stability.

The eigenvalues λi of the monodromy matrix,which occur in reciprocal pairs, reveal the stability ofthe reference periodic orbit and, therefore, the behav-ior of the nearby flow. For each eigenvalue, there isa corresponding eigenvector, vi; together, all 6 eigen-vectors span R6. In the CR3BP, the monodromy ma-trix for a periodic orbit possesses two trivial eigenval-ues equal to unity, due to periodicity of the solutionand the existence of a family. The eigenvectors as-sociated with |λi| > 1 define an unstable invariantsubspace, where the nearby trajectories depart thevicinity of the orbit. The eigenvectors correspondingto |λi| < 1, however, identify a stable invariant sub-space, with trajectories that asymptotically approachthe orbit. Each of these subspaces include manifoldsthat may connect the orbit to other regions of theEarth-Moon system. Families of quasi-periodic or-bits are incorporated in an invariant center subspace,EC, which is predicted by the existence of eigenval-ues that lie along the unit circle, i.e., |λi| = 1. Eachquasi-periodic motion in the center subspace tracesout a closed surface or torus. Although these tra-jectories do not exactly repeat over time, they arebounded solutions. If the monodromy matrix pos-sesses a pair of nontrivial eigenvalues that lie on theunit circle, one family of quasi-periodic orbits exists.Two pairs of unitary nontrivial eigenvalues, however,indicate the existence of two quasi-periodic families.In this case, nearby motions at the same energy levelneither depart nor approach the reference, but, ratherremain in its vicinity indefinitely. Similar to the evo-lution of periodic orbits along a family, quasi-periodicorbits also evolve along a family. In designing trajec-

tories that satisfy a given set of mission constraints,quasi-periodic orbits supply structures that may offera better alternative than the corresponding periodicorbit. Furthermore, the boundedness of a torus maybe retained in an ephemeris model when combinedwith small maintenance maneuvers.

As the stability of a periodic orbit evolves alonga family, so too does the number of associated fam-ilies of quasi-periodic orbits. In Figure 3, the num-ber nm of pairs of unitary nontrivial eigenvalues issummarized, and, equivalently, the number of quasi-periodic families. In this figure, each plot correspondsto sample members of a class of periodic orbits. InFigure 3, each periodic orbit is represented by a sin-gle marker in a two-dimensional space: the verticallocation corresponds to the Jacobi constant of theperiodic orbit, while the horizontal axis indicates theassociated period. Marker colors indicate the value ofnm and, therefore, the number of quasi-periodic fam-ilies corresponding to a periodic orbit: none (blue),1 (green), and 2 (red). Examination of Figure 3 pro-vides a simple overview of the qualitative stability ofsample of periodic orbits in the CR3BP for the Earth-Moon system. First, consider the resonant 1:3 planarfamily (rp13). The absence of red markers indicatesthat each member of the resonant 1:3 planar familypossesses stable/unstable manifolds, potentially en-abling relative low ∆V transfers to the orbits. Fur-thermore, green markers identify members of the res-onant 1:3 planar family with nearby quasi-periodicmotion, that may be exploited during mission design.Conversely, the L4/L5 short period family (SP45) iscomprised solely of members with two quasi-periodicfamily in their vicinity. The L1 axial orbits, how-ever, possess no quasi-periodic motion along the en-tire family. Accordingly, L1 axial orbits (A1) do notprovide feasible options for missions that leverage theboundedness of motion along a torus, e.g., a forma-

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tion of spacecraft. Considering these examples, thedescription in Figure 3 enables preliminary identifi-cation of reference orbits that produce quasi-periodicmotions, that may be incorporated into the missiondesign process.

COMPUTATION OFQUASI-PERIODIC ORBITS

Given the complexity involved in numerically defin-ing a torus in a chaotic dynamical system, the com-putation of quasi-periodic orbits in the CR3BP alsopresents numerous challenges. One of the earliestproposed methods for locating quasi-periodic orbitsnear libration points leverages analytical approxima-tions (from first- to fourth-order).[18, 19, 20] How-ever, these approximations are only accurate in asmall neighborhood of the libration point, and arenot applicable in other regions of configuration space.Several numerical algorithms have been proposed byprevious researchers to compute quasi-periodic orbitsin any region of the CR3BP.[21, 20, 22] However, therange of validity of existing numerical methods forcomputing families of quasi-periodic orbits is limitedand depends on both the family and the complex-ity of the nearby flow. Nevertheless, two recentlydeveloped strategies are employed in combination tocompute quasi-periodic orbits for this investigation.The first method, introduced by Pavlak, involves onlyenforcing continuity constraints to compute boundedmotion that likely lies near a quasi-periodic orbit.[16]Although the computed solutions are not preciselyquasi-periodic, this methodology yields relatively ac-curate predictions of the existence and shape of toriwithin a family of quasi-periodic orbits. A secondapproach employed by Olikara and Scheeres, recov-ers the exact tori that emanate from a periodic orbitwith little computational difficulty. However, the nu-merical detection of precisely quasi-periodic solutionsis limited to tori that can be described by two nonres-onant frequencies. Thus, such limitations may yieldfew quasi-periodic motions in a region with nearbyhigher-order periodic orbits. These two approachescan be used in combination to produce a vast selec-tion of quasi-periodic orbits for use in a dynamicalcatalog of ordered motions in the Earth-Moon sys-tem.

The differential corrections algorithm described byPavlak supplies a relatively robust and computa-tionally inexpensive method for computing approx-imately quasi-periodic motions in various regions ofconfiguration space.[16] In particular, this strategyis based on a similar scheme used to compute peri-

odic orbits, and can be straightforwardly extended tohigher-fidelity models. This methodology begins witha periodic orbit, possessing a nontrivial center mani-fold, that is decomposed into several arcs. These arcsare repeated several times and result in multiple revo-lutions of the periodic orbit as the initial guess. Next,a perturbation is applied to the initial state along thefirst arc in either the y- or z-direction of the rotat-ing frame. An approximately quasi-periodic solutioncan be generated from this slightly perturbed solutionby using a multiple shooting algorithm that enforcescontinuity between neighboring arcs and the value ofthe Jacobi constant. For subsequent solutions alongthe family, a natural parameter continuation schemeis employed, and the initial state along the first arc isincreasingly displaced from the original periodic or-bit. Since the converged solutions are not preciselyquasi-periodic, the last several revolutions along thesolution are discarded. Nevertheless, the resultingtrajectory generally reflects the structure and behav-ior of quasi-periodic orbits along a given family.

An alternative method for locating a quasi-periodicorbit in the CR3BP relies on numerical computa-tion of the corresponding two-dimensional invarianttorus using a stroboscopic mapping, as presented byOlikara and Scheeres [23]. In particular, each torus isdescribed by two frequencies. One frequency, ω0, cor-responds to motion along the associated periodic or-bit and possesses a value that is near the inverse of theorbital period. The second frequency, ω1, indicatesrotation in the transverse direction and is related tothe complex eigenvalue of the limiting periodic or-bit. This numerical method employs the concept ofan invariant circle, formed by the intersection of thetorus and a higher dimensional plane. States that liealong this invariant circle, when propagated forwardfor a time equal to T0 = 2π/ω0, return to the circleand also experience a rotation related to the trans-verse frequency, ω1. To numerically compute the two-dimensional torus using this stroboscopic mappingconstraint, an odd number of states along the invari-ant circle are differentially corrected, along with thecorresponding frequencies. Although this procedureproduces an exact torus, it experiences numerical dif-ficulties when the frequencies pass through low-orderresonances. In such cases, alternate numerical strate-gies may be employed. Nevertheless, to construct aninitial guess, a periodic orbit that possesses a nontriv-ial center subspace, along with the eigenvalues andeigenvectors of its monodromy matrix, is employed.In particular, at a selected location along the refer-ence periodic orbit, an eigenvector that correspondsto the center subspace is scaled by a small numberand rotated by 360 degrees, forming an initial guess

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for the invariant circle. Upon computing a small torusnear the reference periodic orbit, the differential cor-rections process is continued to produce additionaltori along the family of quasi-periodic orbits at agiven energy level. Both procedures by Pavlak andOlikara are implemented to compute quasi-periodicorbits at various energy levels for different families inthe catalog, including halo, Lyapunov, vertical, dis-tant retrograde, 2:1 and 3:1 resonant trajectories.

CHARACTERIZATION OFQUASI-PERIODIC ORBITS

In addition to families of periodic orbits within theEarth-Moon system, the existence of nearby quasi-periodic motions significantly expands the designspace. In particular, these tori possess additionaldegrees of freedom, while still retaining the bound-edness that may be valuable in the design of long-term space applications. Quasi-periodic orbits alonga family may exhibit a larger range of in-plane andout-of-plane amplitudes than their periodic counter-parts, enabling significant flexibility in the trajectorydesign process. In fact, while a periodic orbit mayfail to meet a desired set of mission requirements,arcs along a torus may provide viable alternatives.Furthermore, the boundedness of quasi-periodic mo-tion can be approximately retained when the selectedarcs are transitioned to a higher-fidelity model, suchas ephemeris. For instance, NASA’s ARTEMIS mis-sion successfully exploited segments of quasi-periodictrajectories near L1 and L2 in the Earth-Moon sys-tem for the science orbits of its two spacecraft.

Similar to the inclusion of periodic orbits in an ef-ficient trajectory design framework, the characteriza-tion of quasi-periodic motions is warranted to facili-tate their inclusions in simple trade spaces. First con-sider a geometric description of a quasi-periodic orbit,essentially describing the shape and size of the asso-ciated torus. Such geometric quantities may includeorbit box sizes, cone angles, periapsis and apoapsisdistances. This information may be useful in the de-sign of trajectories for missions that are constrainedby requirements on altitude or even directionality forcommunications, line-of-sight or shadow avoidance.Each of these quantities that may characterize quasi-periodic orbits can be equally applied to the exactlyperiodic motion.

To characterize the size of a quasi-periodic or-bit in configuration space, a simple geometric box isbe employed. In particular, consider the smallest boxthat encloses the entire quasi-periodic orbit with eachside aligned and parallel to the x, y, z unit vectors in

Figure 4: Box sizes schematics.

the CR3BP, as depicted in Figure 4. The length ofeach box dimension is defined as Bx, By and Bz, rep-resenting the maximum end-to-end excursion of thetrajectory in each spatial direction as defined in therotating frame. Note that, by this definition, eachbox dimension does not directly correspond to theorbit amplitudes Ay and Az, which are defined asthe maximum magnitudes of the y and z variablesalong the trajectory. For example, consider an L1

northern halo orbit, which is not symmetric acrossthe xy-plane. In this case, the value of Az is equalto the maximum excursion of the trajectory in thepositive z direction. This quantity is greater thanthe maximum extension of the trajectory in the neg-ative z direction. In contrast, the value of Bz is equalto the difference between the most positive and mostnegative values of z along the trajectory. Accord-ingly the value of Bz is not equivalent to twice theamplitude Az, i.e., Bz 6= 2Az. Despite the simplic-ity of this geometric representation, Bx, By, and Bz,supply a straightforward characterization of the ‘size’of the quasi-periodic solution. Consider, for exam-ple, quasi-periodic orbits within the center subspaceof an L1 vertical orbit. As the family evolves awayfrom the periodic orbit, the trajectories expand in they direction. This variation in the size of the quasi-periodic orbit may be straightforwardly visualized byan increasing value of By. Similarly, a planar peri-odic orbit may possess an associated family of quasi-periodic orbits that extend out of the Earth-Moonplane. The parameter Bz offers an estimate of themaximum variation of z along each trajectory. Suchinformation may be valuable in preliminary evalua-tion of geometrical mission constraints.

In addition to size, the maximum angular devia-tion from a given direction is included in the geomet-rical description of a quasi-periodic orbit. Such in-formation can be represented using cone angles, withan axis of symmetry aligned along the x axis of the

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Figure 5: Cone angles schematics.

rotating frame. The base point for each cone is lo-cated at one of the primary bodies: either the Earthor the Moon. Using this reference point, two conesare defined: one external cone and one internal cone.The external cone, as depicted in Figure 5 is con-structed as the smallest conical surface that encom-passes the entire trajectory. The corresponding coneangle is equivalent to the largest angular deviation ofthe orbit with respect to the x-axis of the rotatingframe. Similarly, the internal cone is constructed asthe largest cone about the x-axis that lies inside theentire trajectory path. As portrayed in Figure 5, theassociated cone angle denotes the minimum angulardeviation of the orbit with respect to the x-axis. Fororbits that pass directly through the x-axis, an inter-nal cone cannot be constructed. Additionally, if thebase point of the cone lies within the orbital path,neither the internal nor external cones can be com-puted. Despite the simplicity of this geometric rep-resentation, cone angles supply valuable informationfor preliminary assessment of directional constraintssuch as visibility requirements, possibly due to sci-ence, communication or power requirements. For ex-ample, consider an L2-based orbit. A constraint oncontinuous communication can be translated to a re-quirement on lunar occultation avoidance. To main-tain this direct link to the Earth the spacecraft must,therefore, remain outside a small 0.25 degree internalcone. By characterizing quasi-periodic motions viacone angles, such constraints can be rapidly evalu-ated for a large set of trajectories, thereby allowingthe user to explore the design space.

Characterization of quasi-periodic motions via sim-ple geometrical quantities facilitates the comparisonof a large set of periodic and quasi-periodic orbits. Tovisualize that comparison, plots such as Figure 6 areemployed. Displayed on this plot, dots correspondto periodic orbits in a given family, colored by the

corresponding Jacobi constant. The value of Bz foreach periodic orbit, i.e., a dot in the figure, is plottedas a function of an integer index that simply repre-sents the order the trajectories appear in the dataset,without any sorting. For the case of the DROs, eachmember is planar, resulting in a series of periodicorbits that aligns with the horizontal axis, Bz = 0.Using Figure 6 as a reference, each family of quasi-periodic solutions is represented as a single verticalline originating at the periodic orbit and encompassesthe range of values of Bz along the computed portionof the quasi-periodic family. The color of each ver-tical bar reflects the number of nontrivial complexmodes nm within the center subspace. By employingsimple representations such as the plot in Figure 6,the potential for quasi-periodic motions to satisfy agiven set of mission constraints may be rapidly evalu-ated. For instance, three-dimensional periodic DROsonly exist for a limited range of y-amplitudes, Jacobiconstants and periods. However, three-dimensionalquasi-periodic DROs can significantly expand the setof solutions that both exhibit the motion typical ofthe DRO family and extend out of the xy-plane.

Figure 6: Comparison of the out-of-plane excursionin terms of Bz for periodic and quasi-periodic DROorbits.

Arcs along the complete quasi-periodic trajectoryare often more valuable than the entire torus. Asdemonstrated in previous applications, history pro-files for selected variables are an effective way to iden-tify portions of a quasi-periodic orbit that may sat-isfy certain mission requirements.[24] These time his-tory plots depict the temporal evolution of a param-eter of interest; a high number of revolutions is war-ranted to sufficiently capture the global behavior ofthe quasi-periodic motion. Mission requirements canoften be translated into thresholds on variables of in-terest. Such variables may vary significantly along

8

a quasi-periodic orbit, especially in comparison tothe reference periodic solution. For example, Fig-ure 7 depicts the time history for the y-componentof the state along an L2 vertical quasi-periodic orbit.The y-component oscillates within a [-40000,40000]km range for the quasi-periodic solution, which is re-markably larger than the [-13000,13000] km rangefor the periodic motion. For some time intervalsalong the quasi-periodic orbit, the variable may re-main within some of the given thresholds. Such in-tervals are quickly identified on a time history plot.An approximate envelope curve may also be numeri-cally computed and overlaid on the time history, as inFigure 7. To calculate the envelope the extreme val-ues over 1-3 revolutions are detected. Then, the en-velope is estimated using a spline interpolating thoseextreme values. The spline approximation becomesinaccurate at the boundaries of the history profile,due to extrapolation from the last available interpo-lating point. In general, the selected time intervalscorrespond to arcs of the quasi-periodic orbit thatmeet some of the mission specifications and can aidin the construction of a feasible trajectory.

0 200 400 600 800 1000

−4

−2

0

2

4x 10

4

Elapsed time [days]

y [k

m]

PeriodicRange

Quasi−PeriodicRange

History approximate envelope

Figure 7: Representative history plot for a L2 verticalquasi-periodic orbit.

IMPLEMENTATION ININTERACTIVE GRAPHICAL

INTERFACE

Using a simple set of characteristic quantities, quasi-periodic solutions are incorporated into an efficientframework for orbit comparison and design. This ad-ditional framework extends a pre-existing interactivecatalog of periodic orbits for the quick selection of tra-jectories. The interactive catalog serves as a resource

to obtain preliminary arcs prior to the use of an end-to-end design tool for complete path construction andtransition to ephemeris. For demonstration, the cata-log is implemented as a prototype Graphical User In-terface (GUI).[15] The prototype is assembled in theMATLAB R© environment. A supplemental and inter-active module is built to visualize and compare quasi-periodic solutions from the CR3BP, as displayed inFigure 8. The analysis originates in the upper-leftpanel, where selected characteristics for a family ofperiodic orbits and their corresponding quasi-periodicmotions are displayed in the form of a representationsimilar to Figure 6. Relevant characteristics are dis-played on the vertical axis and can be selected by theuser. The subplot within this panel displays the num-ber of quasi-periodic families associated with eachperiodic orbit. A selector, such as a sliding bar onthe graph allows the user to select the quasi-periodicfamily of interest. A quasi-periodic trajectory that isrepresentative of the selected family of tori appears inthe upper-right panel and is plotted in configurationspace. The lower panel reports the time history fora selected quantity of interest. Using the functionsassociated with this panel, the user can set thresh-olds on a given variable and enable automatic detec-tion of the time intervals along a quasi-periodic orbitsatisfying these pre-defined limits. Time intervals ofinterest can also be manually selected. Ultimately,the time history enables the selection of segments orarcs along a quasi-periodic solution that satisfy themission requirements. Following arc selection, theuser can export any candidate solutions for use in anadvanced trajectory design suite for further exami-nation. Given the iterative nature of the trajectorydesign process, such a catalog may be used interac-tively to explore additional candidate orbits.

Figure 8: Screenshot of the GUI prototype for thequick comparison and selection of quasi-periodic arcs.

9

SAMPLE APPLICATIONS

Since the dynamical environment in the Earth-Moonsystem is reasonably approximated by the CR3BP,an interactive catalog of periodic and quasi-periodicsolutions can be used for preliminary trajectory de-sign in various mission scenarios. Given a missionconcept and a set of constraints on the path of thespacecraft, the interactive catalog can be employedto reveal a variety of orbital options, thereby allevi-ating the complexity of searching the design space ina higher-fidelity ephemeris model. Based on the in-tended region in the Earth-Moon space for the space-craft, the user can first reduce the set of candidateorbits by identifying feasible regions of configurationspace, e.g., in the vicinity of a given libration pointor near the Moon. The user can then simultaneouslyexplore two-dimensional trade-spaces involving eachfamily to examine whether any of the members sat-isfy the given mission constraints. For periodic orbitfamilies that possess a nontrivial center subspace, theuser may also explore one family at a time and ana-lyze the characteristics of any nearby quasi-periodicmotion. Through this selection strategy, the user canidentify and export orbit options for analysis in ahigher-fidelity model. This process is demonstratedfor two mission design applications.

L2 Gateway Operations

Consider a long-term Multi-Purpose Lunar Fraction-ated System (MPLFS) consisting of a formation ofspacecraft stationed near the Earth-Moon L2 gate-way. The spacecraft comprising the MPLFS couldprovide various services for operations in cislunarand/or interplanetary space. Locating the MPLFSnear L2 could enable the exploitation of naturalmulti-body dynamics either for a spacecraft that mayrendezvous with the formation or upon deployment ofany of the individual modules to their final destina-tion. For instance, an MPLFS that may facilitatelunar operations could consist of several spacecraftincluding a fueling depot, multiple on-orbit storagevehicles, as well as a communications relay. Sincethe MPLFS must be located near the L2, the setof candidate orbits for this mission can be signif-icantly reduced. Depending upon the mission sce-nario, it may be crucial to maintain continuous con-tact with the Earth. This requirement can be trans-lated into a geometrical constraint on the path, suchthat the MPLFS should maintain a constant line-of-sight to the Earth. In addition, the L1 gateway maybe used as a low-cost transfer mechanism betweenthe Earth and lunar vicinities. Accordingly, a vehicle

that originates near the Earth may pass through theL1 gateway in order to rendezvous with the MPLFS.To ensure that any such transfers for a crewed ve-hicle, for example, are possible and not prohibitivein terms of fuel consumption, a loose constraint canbe placed on the energy level of the candidate orbitfor the MPLFS. Specifically, the Jacobi constant ofthe MPLFS orbit should be comparable to the Ja-cobi constant value of L1. Operationally, it is alsoconvenient for all the vehicles to remain in a closeformation. Orbit options that satisfy each of theseconstraints can be rapidly identified using the inter-active catalog.

The interactive catalog enables rapid identificationof a preliminary baseline solution for the MPLFS con-cept. First, consider the ranges of Jacobi constantvalues for the L1 and L2 families of periodic orbitsavailable in the catalog. The simple bar plots in thecatalog reveal that the JC ranges along these familiesgenerally overlap.[15] This correspondence in Jacobiconstant is indicative of potentially low-cost transfersbetween the L1 and L2 regions. Thus, each of thefamilies of L2 orbits possesses members that satisfythe loose constraint on the Jacobi constant. Next,the type of baseline motion for the MPLFS - periodicor quasi-periodic - is selected. For multiple space-craft that lie along the same periodic orbit, there arelimited orbital geometries that enable a configura-tion where the first and last spacecraft remain closeover time. Alternatively, placing several spacecraftalong the surface of a quasi-periodic orbit may en-sure that each member of the spacecraft formationremains in sufficiently close proximity to the othermodules. The feasibility of leveraging quasi-periodicmotion in formation flight has been demonstratedby previous researchers.[25] Accordingly, the quasi-periodic selection tool within the catalog is employed.Recall from Figure 3, that quasi-periodic solutionsare only available along the L2 Lyapunov, vertical,and halo orbits. Since the L2 axial family does notpossess any members with a center subspace, theseorbits are discarded from the set of candidate solu-tions. Instead, each of the L2 Lyapunov, vertical,and halo orbit families are analyzed individually toidentify any solutions that satisfy the continual line-of-sight constraint.

The quasi-periodic orbits that lie in the center sub-space of the L2 Lyapunov family are examined usingthe catalog to identify whether any solutions can pro-vide continuous communications to the Earth. A con-tinuous line-of-sight to the Earth from a spacecraft inthe MPLFS is possible when the formation lies out-side of the lunar Earth shadow. This requirementis straightforwardly translated to a simple geomet-

10

ric constraint that the candidate orbit must not passwithin the angular radius of the Moon as observedfrom Earth, i.e., the orbit must remain outside a 0.25degree cone constructed with an axis of symmetryaligned with the x-axis and its base point located atthe Earth. Consider first an L2 Lyapunov periodicorbit. Since this orbit lies within the xy-plane, itperiodically passes through the lunar shadow everyhalf revolution. The out-of-plane oscillations associ-ated with nearby quasi-periodic orbits may enable re-peated passages that are outside of this shadow cone.In fact, Figure 9 depicts the out-of-plane extensionvia the box size, Bz, for the largest available quasi-periodic solutions associated to each L2 Lyapunovorbit with a center subspace. For this family of or-bits, the evolution of the nearby quasi-periodic orbitsexhibits one of two behaviors. For Jacobi constantvalues within the range JC = [3.15, 3.17], the cor-responding families of quasi-periodic tori connect amember of the L2 Lyapunov orbit to a member of theL2 vertical family at the same energy level. Accord-ingly, quasi-periodic orbits along a family initiallyresemble a Lyapunov orbit. As the family evolves,however, the tori gradually transform to resemble avertical orbit, as depicted in Figure 9(a). For an in-terval of Jacobi constant values JC = [2.95, 3.00], theL2 Lyapunov and vertical orbits are no longer linkedvia a family of tori. Rather, the quasi-periodic orbitsevolve towards the geometry depicted in Figure 9(b).For brevity, only quasi-periodic orbits with a Jacobiconstant in the interval JC = [3.15, 3.17] are inves-tigated to identify orbits that satisfy the continuousline-of-sight requirement for the MPLFS mission.

index

(a)

(b)

Figure 9: Out-of-plane extension and geometry ofavailable quasi-periodic solutions associated to dif-ferent L2 Lyapunov orbits (indentified by the catalogindex).

To assess whether L2 Lyapunov quasi-periodic

motion may satisfy the mission constraints for theMPLFS, the angular time histories for the tori thatexist at JC = 3.17 are rapidly examined using thecatalog. In addition to enabling visualization of theconfiguration space, the catalog interface also dis-plays the time history of a user-selected variable ofinterest. In this sample mission scenario, the in-stantaneous angular deviation of the spacecraft fromthe x-axis, as measured from the Earth, is leveragedto determine whether continuous communication toEarth is possible for a given quasi-periodic orbit. Forthe selected torus, the catalog includes a capabilityfor automatic detection of segments of the orbit thatsatisfy a given constraint on one of the included char-acteristic parameters. As an example, this automaticdetection is applied to an L2 quasi-periodic orbit thatexists at a Jacobi constant of JC = 3.17, as dis-played by the gray toroidal surface in Figure 10(b).The time history for the angular deviation of motionalong this torus is displayed in Figure 10(a). A seg-ment of the quasi-periodic orbit that remains outsideof the lunar shadow cone is highlighted in red. Recallthat for the MPLFS mission scenario, the baselineorbital motion must allow a continuous line-of-sightto the Earth. However, the highlighted segment inFigure 10(a) only spans 150 days. For a differentmission scenario, 150 days may be a sufficiently longvisibility window and, therefore, render this quasi-periodic orbit a candidate solution. Nevertheless, forthe MPLFS example, the L2 Lyapunov quasi-periodicorbits do not provide a solution that remains perma-nently outside of the lunar shadow cone.

(a) Angle separation history. (b) y − z view of thetorus.

Figure 10: L2 Lyapunov quasi-periodic arc that sat-isfies the continuous coverage requirement for theMPLFS mission example.

Quasi-periodic motions emanating from the three-dimensional L2 vertical families may also providecandidate orbits for the MPLFS spacecraft forma-tion. Using the catalog interface, several L2 verticalquasi-periodic motions are examined. Similar to theL2 Lyapunov family, quasi-periodic orbits near some

11

members of the L2 vertical family only slightly de-viate from the originating periodic solution. OtherL2 vertical orbits, however, possess nearby tori thatexhibit significantly different geometries across thefamily. Despite this variability in the behavior of theL2 quasi-periodic orbits, each torus still passes be-hind the Moon. Accordingly, the MPLFS formationwould pass within the lunar shadow cone. For in-stance, consider a sample L2 vertical quasi-periodicorbit as displayed in Figure 11. The correspondingtime history for this quasi-periodic motion is depictedin Figure 11(a). Since the angular deviation along thetrajectory passes within the lunar shadow cone angletwice within the plotted time period, only a 100 daysegment along the L2 vertical quasi-periodic orbit sat-isfies the line-of-sight constraint. Accordingly, the L2

vertical family does not provide a viable candidatesolution for the MPLFS mission concept.

(a) Angle separation history. (b) y−z view ofthe torus.

Figure 11: L2 vertical quasi-periodic arc that satisfiesthe continuous coverage requirement for the MPLFSmission example.

Additional candidate quasi-periodic orbits em-anate from the L2 halo family. Since the major-ity of members in this family that possess a non-trivial center subspace also satisfy the continuousline-of-sight constraint for the MLPFS mission con-cept, quasi-periodic solutions near the reference pe-riodic orbit also remain continuously visible fromEarth. As a family of L2 halo quasi-periodic or-bit evolves away from the reference periodic solu-tion, however, some tori may pierce the lunar shadowcone. Accordingly, an interactive catalog of periodicand quasi-periodic solutions provides a straightfor-ward and rapid method for identifying the space offeasible solutions. Consider the large L2 halo quasi-periodic orbit in Figure 12(b), which possesses a Ja-cobi constant of JC = 3.15. For this orbit, the cata-log automatically highlights trajectory segments thatprovide a line-of-sight to the Earth. As portrayed in

Figure 12(a), the MPLFS can maintain a 550 day-long Earth-visibility window. While this time inter-val may be feasible for many mission scenarios, theselected quasi-periodic orbit does not provide contin-uous line-of-sight for a multi-year mission. To iden-tify a baseline trajectory with indefinite visibility atthis same energy level, the torus size may be reduced.For instance, consider the smaller torus displayed inFigure 13(b). As evident from the angular deviationtime history in Figure 13(a), motion along the quasi-periodic orbit never passes within the lunar shadowcone. Accordingly, this quasi-periodic orbit suppliesa candidate baseline orbit for the MPLFS missionscenario. To identify an alternate candidate orbitthat satisfies the continuous visibility requirements,tori near other members of the L2 halo family areexplored. Near rectilinear halo orbits, for example,produce viable candidate tori. For instance, motionalong the torus depicted in Figure 14(a) remains en-tirely outside the lunar shadow cone. The quasi-periodic motion along this torus, identified and ex-ported from the catalog, can be transition to a higher-fidelity design environment such as ATD. The result-ing solution, differentially corrected for continuity inan point mass ephemeris model of the Earth, Moonand Sun, is plotted in Figure 14(b) and requires nomaintenance maneuvers. As demonstrated throughthis example, a catalog of periodic and quasi-periodicsolutions in the CR3BP facilitates exploration of or-bit options that satisfy the constraints associatedwith a general mission concept in the Earth-Moonsystem. Furthermore, the resulting analysis reflectsthe dynamical structures that are retained in a highfidelity ephemeris model.

In-Orbit Lunar Facility

Rapid access and visualization of families of quasi-periodic solutions also supports a well-informed ex-amination of natural trajectories that enables mis-sion scenarios for a single spacecraft. Consider, forexample, the preliminary selection of a baseline orbitfor a long-term human habitat in the lunar vicinity.In addition to establishing a long-term lunar pres-ence, exploration of the polar regions of the Moonhas recently garnered interest for its scientific return.Accordingly, a space-based infrastructure that orbitsthe Moon should also support manned excursions tothe lunar poles. This requirement can be translatedinto a constraint that the spacecraft must be able toview the north (or south) polar regions from above (orbelow). Due to their favorable stability properties,members of the DRO family are frequently proposedas reference orbits for lunar infrastructure. Smaller

12

(a) Angle separation history.

(b) y − z view of the torus.

Figure 12: L2 halo quasi-periodic arc that satisfiesthe continuous coverage requirement for the MPLFSmission example.

periodic members of the DRO family lie entirelywithin the Earth-Moon orbital plane, limiting the vis-ibility of the polar regions. In a preliminary analysisof orbit options, the interactive catalog is used toidentify any nearby quasi-periodic DRO orbits thatregularly possess an out-of-plane extension above thelunar radius. Thus, the geometric size of a candi-date quasi-periodic solution in the z-direction, Bz,should be greater than the diameter of the Moon. Infact, some quasi-periodic members of the DRO fam-ily do extend out-of-plane, as depicted by the nonzerovalues of Bz corresponding to the shaded regions inFigure 15. While periodic members of the DRO fam-ily may not support the described lunar infrastruc-ture mission, a catalog of orbits and their character-istics rapidly reveals that quasi-periodic orbits cancertainly meet the mission requirements. Consider,for example, a small DRO with an orbital period of13.2 days, similar to the proposed reference orbit forthe asteroid redirect mission.[2] Since this orbit pos-sesses a two-dimensional center subspace, there arethree-dimensional quasi-periodic orbits that exist inits vicinity. Using the catalog, a large torus from thisfamily of quasi-periodic orbits is selected. The auto-matic detection feature in the quasi-periodic moduleof the catalog is employed to identify arcs along thequasi-periodic trajectory that pass above the lunarradius in the z-direction. These segments are high-lighted in the z-component time history of the three-dimensional quasi-periodic motion, portrayed in Fig-ure 16(a). As evident in this figure, a spacecraft in

0 200 400 600 800 10000

1

2

3

4

5

Elapsed time [days]

θ M [d

eg]

Moon angular radius

(a) Angle separation history.

(b) y − z view of the torus.

Figure 13: L2 halo quasi-periodic torus that satisfiesthe continuous coverage requirement for the MPLFSmission example.

(a) Quasi-periodic torus inCR3BP.

3.84

x 105

−202 x 104

0

2

4

6

8

x 104

y [km]

z [k

m]

x [km]

(b) Solution converged inephemeris.

Figure 14: Comparison of near rectilinear halo tra-jectory for different Earth-Moon system models.Ephemeris solution converged in ATD for Jan 1, 2023,including Earth, Moon, and Sun gravity.

this reference orbits frequently passes above the northpole, potentially supporting crewed or robotic excur-sions to these regions. In fact, these passages provide8 days of continuous coverage every 10 days. A sam-ple segment along the gray torus is highlighted inred in Figure 16(b). Longer or more frequent polarviewing windows may be obtained by selecting an al-ternate torus, or varying the reference periodic DRO.This complex design space is rapidly explored using acatalog of periodic and quasi-periodic solutions. Theresulting analysis can be verified by correcting the se-lected solutions in a higher-fidelity ephemeris model.

13

index

Moon

z

Figure 15: Out-of-plane extension and geometry ofavailable quasi-periodic solutions associated to differ-ent DROs (indentified by the catalog index).

(a) Out-of-plane component history.

(b) x− z view of the torus.

Figure 16: DRO quasi-periodic arc that enables cov-erage of the lunar poles for an in-orbit space habitnear the Moon.

CONCLUDING REMARKS

To support the design of trajectories in the Earth-Moon system, a graphical environment for the com-parison and selection of candidate orbits is explored.This framework is based on an interactive catalog ofsolutions for the CR3BP, a simplified gravitationalmodel that enables preliminary trajectory design theEarth-Moon system. The set of readily available so-lutions incorporated in the catalog include both pe-riodic and quasi-periodic orbits. These bounded mo-tions are straightforwardly characterized via simplegeometrical quantities that reflect both the size andbehavior. Along with interactive visualizations ofthe trade space, these characteristics parameters areuseful in identifying ordered motions that satisfy agiven mission constraint. In fact, the capabilities ofan interactive framework for exploring and conduct-

ing trade-offs during the preliminary mission designphase are apparent in the design of mission conceptsthat support operations within either cislunar or in-terplanetary space. Furthermore, this early analysisof candidate orbits in the CR3BP is validated whentransitioned to an ephemeris model, thereby demon-strating the utility of a catalog framework.

Acknowledgements

This work was completed at Purdue University underNASA Grants NNX13AM17G and NNX13AH02G.The authors wish to thank the Purdue College of En-gineering for the partial financial support provided toattend this conference.

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