low-thrust trajectory design for a cislunar cubesat ... · mission. after an overview of the...

70th International Astronautical Congress, Washington D.C., USA. Copyright © 2019 by the authors. All rights reserved.

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Low-Thrust Trajectory Design for a Cislunar CubeSat LeveragingStructures from the Bicircular Restricted Four-Body Problem

Robert PritchettPurdue University, USA, [email protected]

Kathleen C. HowellPurdue University, USA, [email protected]

David C. FoltaNASA Goddard Space Flight Center, USA, [email protected]

The upcoming Lunar IceCube (LIC) mission will deliver a 6U CubeSat to a low lunar orbit via a ride-shareopportunity during NASAs Artemis 1 mission. This presents a challenging trajectory design scenario, asthe vast change in energy required to transfer from the initial deployment state to the destination orbit iscompounded by the limitations of the LICs low-thrust engine. This investigation addresses these challengesby developing a trajectory design framework that utilizes dynamical structures available in the BicircularRestricted Four-Body Problem (BCR4BP), along with a robust direct collocation algorithm. Maps arecreated that expedite the selection of invariant manifold paths from a periodic staging orbit in the BCR4BP,as these offer favorable connections between the LIC transfer phases. Initial guesses assembled from thesemaps are passed to a direct collocation algorithm that corrects them in the BCR4BP while including thevariable low-thrust acceleration of the spacecraft engine. Results indicate that the ordered motion providedby the BCR4BP and the robustness of direct collocation combine to offer an efficient and adaptable frameworkfor designing a baseline trajectory for the LIC mission.

1. Introduction

Advances in spacecraft technology miniaturizationand an increase in launch opportunities have expo-nentially increased the number of CubeSats launchedin the two decades since the platform was first pro-posed. Early success has motivated concept devel-opment for CubeSat missions beyond the bounds oflow Earth orbit (LEO). In 2018, the two MarCOCubesats, as the first to operate beyond Earth or-bit, were deployed at Mars to monitor the entry, de-scent, and landing of the Insight lander.1 Soon, thir-teen CubeSats will utilize a rideshare opportunity onthe Space Launch System (SLS) during the Artemis1 mission to reach a variety of destinations well be-yond LEO. Several of these spacecraft will reach he-liocentric space, including the Near Earth AsteroidScout (NEA Scout) and the CubeSat for Solar Parti-cles (CuSP), which will fly by a near-Earth asteroidand investigate space weather, respectively. OtherCubeSats on this launch which are destined for theMoon, include Lunar IceCube, Lunar Flashlight, andLunaH-Map. Together, these far-flung CubeSat mis-sions indicate that small spacecraft will play an in-creasingly important role in space exploration.

While the CubeSat revolution has opened excitingopportunities, it brings new challenges. Despite tech-nological advancements, ambitious CubeSat missionsoften require “doing more with less”. For missiondesign, innovative trajectory design approaches arenecessary that fully exploit natural dynamics. TheLunar IceCube (LIC) concept offers an excellent ex-ample of the challenges. This mission will deliver a6U CubeSat to a low lunar orbit (LLO) where LICwill collect data on water transport throughout thelunar surface. A challenging trajectory design sce-nario emerges, as the vast change in energy requiredto transfer from the initial deployment state to LLOis compounded by the limited control authority ofthe LIC low-thrust engine. Furthermore, as a sec-ondary payload, LIC is subject to changes in launchdate and conditions required by the primary mission.These challenges necessitate a trajectory design strat-egy that is flexible and incorporates natural forces toassist with the required energy change.

This investigation proposes a trajectory designframework that addresses the challenges of the Lu-nar IceCube mission by utilizing dynamical struc-tures available in the Bicircular Restricted Four-BodyProblem (BCR4BP) along with a robust direct col-

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location algorithm. Designing in the BCR4BP en-ables the gravitational force of the Sun to be lever-aged to achieve part of the required energy change,while avoiding the additional perturbations of a fullephemeris model. A key feature of the proposed de-sign approach is the use of a staging orbit near theMoon to split the trajectory into two phases: thefirst from spacecraft deployment to the staging or-bit, and the second from the staging orbit to the sci-ence orbit. This choice allows the two halves of theLIC trajectory to be designed mostly independently,thus, simplifying the redesign process if deploymentconditions change. Moreover, a periodic orbit in theBCR4BP is employed as the staging orbit, to lever-age its invariant manifolds for the design of efficientpaths to and from the staging orbit. Another crucialcomponent of the proposed framework is a direct col-location algorithm to correct initial guesses developedin the BCR4BP into optimal low-thrust trajectories.The robust convergence properties of direct colloca-tion facilitate a wider variety of initial guesses despitelarge discontinuities in states or time. Together, thesekey design choices produce a design process that di-rectly addresses the challenges of the Lunar IceCubemission. After an overview of the necessary back-ground, the proposed trajectory design framework isdescribed. Sample Lunar IceCube trajectories areconstructed with the proposed strategies. While thetrajectory design procedure is applied to the LunarIceCube mission, it is sufficiently general for a widevariety of low-thrust missions especially those withlimited control authority.

2. Background

2.1 Previous Work

First proposed in 2015, the Lunar IceCube (LIC)mission is a collaborative effort led by MoreheadState University and supported by Goddard SpaceFlight Center (GSFC), Busek, and Catholic Univer-sity of America2. During this mission the presenceand movement of water in all its forms across a broadswath of the lunar surface is to be investigated. Toenable the science collection, the 6U CubeSat willconduct science operations in a low lunar orbit (LLO)that covers a range of longitudes on the sunlit side ofthe Moon with a perilune altitude of 100 km. Thefull set of Keplerian orbital elements that define thescience orbit are provided in Table 1 and are drivenby the science requirements as well as the desire tomaximize spacecraft lifetime while minimizing thestation-keeping cost.3 Station-keeping and the trans-fer trajectory to the science orbit are achieved via

Table 1: Lunar IceCube science orbit Keplerian or-bital elements defined in a Moon-centered iner-tial frame. Inclination is measured relative to theMoon’s equator and the right ascension of the as-cending node (RAAN) is defined with respect tothe vernal equinox vector.

Orbital Element ValueSemi-Major Axis, a 4271.4 kmEccenctricity, e 0.5697Inclination, i 89.35

RAAN, Ω 65

Argument of Periapsis, ω 355

a BIT-3 Busek ion thruster, which is capable of amaximum thrust of 1.24 mN, a specific impulse (Isp)of 2640 seconds, and storing up to 1.5 kg of propel-lant.4 Given the total 14 kg mass of LIC, these en-gine characteristics equip it with a maximum accel-eration of 8.857×10−5 m/s2. This value is comparedto the maximum acceleration values of several otherlow-thrust spacecraft in Table 2; the low-thrust ca-pability of LIC is clearly of the same order of magni-tude as other recent or proposed low-thrust missions.The mission design challenge for LIC is due less to

Table 2: Representative low-thrust spacecraft accel-eration levels.

Spacecraft Max Acceleration (m/s2)Deep Space 15 1.892× 10−4

Lunar IceCube 8.857× 10−5

Dawn6 7.473× 10−5

Gateway7 3× 10−5

the small thrust magnitude and more to the fact thatthis engine must deliver a massive change in energy totransfer from the high-energy deployment state nearthe Earth to the low-energy LLO. While the initialvelocity of LIC relative to the Earth is less than es-cape velocity, the subsequent ballistic path includesa flyby of the Moon that causes LIC to escape theEarth-Moon system. Low-thrust maneuvers must beinitiated shortly after deployment to prevent escapefrom occurring. In contrast to this high-energy de-ployment, the LIC science orbit possesses an energylow enough that the spacecraft is securely capturedat the Moon. Despite the challenge of achieving thislarge change in energy, engineers at GSFC have devel-oped a complete baseline trajectory, plotted in Fig-ure 1, that utilizes the current launch date of June25th 2020. However, this baseline trajectory must be

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(a) Earth-Centered J2000 Inertial Frame

(b) Earth-Moon Rotating Frame

Fig. 1: Current baseline for Lunar IceCube trajectorydeveloped assuming a June 25th, 2020 launch date.

redesigned when an updated launch date is releasedin the near future. Previous experience has demon-strated that varying the launch date can significantlyimpact the geometry of the LIC trajectory.

Faced with a challenging trajectory design scenarioand uncertain launch conditions, engineers at GSFChave been informed by the results of several investiga-tions on LIC trajectory design. A strategy that uti-lizes the high-fidelity General Mission Analysis Tool(GMAT) to design an LIC trajectory with a captureorbit at the Moon is offered by Mathur8. An inno-vative design approach for LIC is also presented byBosanac, Folta, Cox, and Howell which subdividesthe Lunar IceCube trajectory into three phases: de-ployment, phasing and energy adjustment, and lu-nar capture. A strategy for linking these phases that

incorporates periapse maps and phasing arcs gen-erated in the Sun-Earth Circular Restricted Three-Body Problem (CR3BP) or the BCR4BP is devel-oped by Bosanac et al.9–11. Particular focus on thedynamics of the lunar capture phase is delivered sep-arately by Folta et al.12. The strategy presented byBosanac et al. is effective, and the current investiga-tion expands upon their work by approaching the de-sign problem with a framework that utilizes BCR4BPdynamical structures and direct collocation. This ap-plication may yield a design procedure that requiresless computational time and is more robust with poorinitial guesses.

In addition to work focused on LIC, this investiga-tion is influenced by a greater body of literature onleveraging the influence of the Sun to design trans-fers from the Earth to the Moon. Belbruno and Millerdemonstrate new types of Earth to Moon trajectoriesby simultaneously incorporating the gravitational in-fluence of the Sun in addition to the Earth and Moon.Strategies for utilizing this acceleration to developlow-energy trajectories from the Earth to the Moonare developed by many authors, including Koon etal.13, Gomez et al.14, as well as Parker and Martin15.Low-energy trajectory design techniques are also ap-plied to design low-thrust trajectories to the Moon byMingotti et al.16 and Zanottera et al.17. The presentwork uses direct collocation to compute low-thrusttransfers building on the work of authors such as En-right and Conway18 as well as Grebow, Ozimek, andHowell19 who also employ this algorithm to gener-ate low-thrust Earth to Moon transfers. Perez-Palauand Epenoy explore similar transfers using an indirectoptimal control approach in a four-body dynamicalmodel20. Some authors exploring low-energy trajec-tory design also demonstrate transfers from Earth-Moon halo orbits to LLO, a strategy used in thisinvestigation. Parker and Anderson21 offer an im-pulsive transfer, while Mingotti et al. demonstratea low-thrust result22. Recently, Cheng et al.23 andCao et al.24 have more closely examined impulsivetransfers from halo orbits to LLO in the CR3BP.

2.2 Bicircular Restricted Four-Body Problem

The BCR4BP builds on the assumptions of theCR3BP. The CR3BP models the path of a third body,P3, under the influence of two more massive primarybodies, P1 and P2. These bodies are assumed tofollow circular Keplerian orbits about their mutualbarycenter, B1. Additionally, the third body is as-sumed to possess negligible mass in comparison tothe primary bodies, and this assumption is reason-

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able when the mass of the third body is quite small,e.g., a spacecraft. Finally, the mass ratio of P1 and P2

is denoted, µ = m2

m1+m2, and is used to characterize

the CR3BP system.

The BCR4BP assumes the addition of a fourthbody, P4; both P4 and the P1-P2 barycenter, B1,move in circular orbits about their mutual barycen-ter, B2. In this investigation, P4 is always the Sun,thus the mass of P4 equals the mass of the Sun, mS .The circular orbits of P1 and P2 are not affected bythe gravitational force of the Sun. As a result of thisassumption, the BCR4BP is not coherent because themotion of P1 and P2 are not influenced by the Sun,i.e., the indirect effects of the Sun are not incorpo-rated. Additionally, as in the CR3BP, the mass of P3

is assumed to be negligible relative to the other threebodies, i.e., m3 m2 < m1 < mS . In general, theBCR4BP does not require that the Sun-B1 orbit becoplanar with the P1-P2 orbit; however, in this inves-tigation a coplanar model is employed. This modelas well as a non-coplanar formulation are presentedby Boudad25.

It is insightful to examine motion in the BCR4BPfrom the perspective of two different rotating refer-ence frames. The first is a reference frame rotatingwith P1 and P2, whose axes are defined by three or-thogonal unit vectors. By convention, the x unit vec-tor of this frame points from P1 to P2, while the z unitvector is parallel to the angular momentum vector ofP2 about P1. Finally, the y unit vector is definedto complete the orthonormal set. A similar secondrotating reference frame is defined for the Sun andB1, but the x′ unit vector is instead in the direc-tion of B1 from the Sun. Quantities expressed in theSun-B1 rotating frame are generally denoted with anapostrophe, e.g., x′ is the x position coordinate ofP3 in the Sun-B1 rotating frame. When the coplanarassumption is made, the orbit of the Sun as viewedfrom the P1-P2 rotating frame is modeled as illus-trated in Figure 2. The position of the Sun in theP1-P2 rotating frame is determined by the Sun angle,θS , and the distance from B1 to the Sun is defined bythe constant value aS . Viewed from this frame, theSun rotates clockwise about B1, thus the value of θSdecreases with time.

Motion in the BCR4BP is described by a set ofdifferential equations similar to those of the CR3BP,but modified to accomodate the perturbing accelera-tion of the Sun. The P1-P2 rotating frame togetherwith the Sun-B1 rotating frame are commonly em-ployed for analysis in the BCR4BP, and the equationsof motion for P3 may be expressed in either of these

x

y

B1

P1 P2

θS

aS

SunB2

Fig. 2: Definition of the Sun angle in the BicircularRestricted Four-Body Problem.

frames. The equations of motion for P3, expressedin the P1-P2 frame and including a low-thrust force,are,

x = 2y +∂Ψ

∂x+Txm

(1)

y = −2x+∂Ψ

∂y+Tym

(2)

z =∂Ψ

∂z+Tzm

(3)

m =T

ve(4)

where the low-thrust force is represented by the threecomponents of the thrust vector T = Tx Ty Tz,and ||T || = T . The magnitude of the thrust vectorappears in Equation 4, along with the exhaust veloc-ity, ve. Together these parameters define the massflow rate of the spacecraft, m. Additionally, Ψ is thesystem pseudo-potential written in terms of P1-P2 ro-tating frames coordinates. This pseudo-potential andthe pseudo-potential as expressed in Sun-B1 coordi-nates, Ψ′, are,

Ψ =1− µr13

+µ

r23+

1

2(x2 + y2) +

mS

rS3−

mS

a3S(xSx+ ySy + zSz)

(5)

Ψ′ =1

2(x′2 + y′2) +

1− µSB1

r′S3

+

µSB1(1− µP1P2)

r′13+µSB1

µP1P2

r′23

(6)

where the distance from the Sun to P3 is representedby rS3, and µSB1 is the mass ratio of the Sun-B1

system. Together these equations govern motion inthe BCR4BP.

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To facilitate numerical computation, the depen-dent variables in this dynamical model are nondimen-sionalized via a set of characteristic quantities. Thevalues of the characteristic quantities are determinedby the frame that the states are expressed in. Whenstates are expressed in the P1-P2 rotating frame thecharacteristic length, l∗, is defined as the distancebetween P1 and P2; the characteristic mass, m∗, isequal to the combined mass of P1 and P2; and thecharacteristic time is determined such that the nondi-mensional angular velocity of P1 and P2 is equal toone. The characteristic quantities are defined sim-ilarly when states are expressed in the Sun-B1 ro-tating frame, except that the parameters of the Sunand B1 replace those of P1 and P2 in the previouscase. In this investigation, numerical propagation isperformed using the equations of motion expressed inthe P1-P2 rotating frame.

In contrast to the CR3BP, the BCR4BP is not anautonomous system, i.e., motion in this model is timedependent. As a consequence, this system possessesno integral of the motion. However, the Hamiltonian,serves as a useful metric for analyzing the motion ofP3 in the BCR4BP. The Hamiltonian defined in thisinvestigation does not include the low-thrust force,thus it represents only the ballistic energy of the sys-tem. The Hamiltonian may be computed using co-ordinates expressed in either the P1-P2 or Sun-B1

rotating frames.

H = 2Ψ− (x2 + y2 + z2)− σ (7)

H ′ = 2Ψ′ − (x′2 + y′2 + z′2) (8)

The value of H is scaled by a constant parameter σthat is incorporated to offset the high value terms in-troduced by the Sun and ensure that H is of a similarmagnitude to the Jacobi constant value of the Earth-Moon CR3BP. Throughout this analysis, σ = 1690nondimensional units.

The same types of dynamical structures that areavailable in the CR3BP also emerge in the BCR4BP,namely, periodic and quasi-periodic orbits as well astheir invariant manifolds. Because the BCR4BP isnon-autonomous, these structures are not only de-fined by position and velocity states, but also by spe-cific epochs, i.e., Sun angles. A periodic orbit in theBCR4BP requires a repetition of the same positionand velocity states at the same Sun angle. This an-gle requirement implies that all periodic orbits in theSun-Earth-Moon BCR4BP possess a resonance withthe synodic period of the Sun, approximately 29.5days. The Sun angle, θS , completes a full revolu-tion once every synodic period. For example, a pe-

riodic halo orbit in the Sun-Earth-Moon BCR4BP,that is computed about the Earth-Moon L2 librationpoint, is displayed in Figure 3. This orbit possesses a

Fig. 3: 2:1 synodic resonance halo orbit in the Sun-Earth-Moon BCR4BP that is computed about theEarth-Moon L2 libration point. This orbit is usedas a staging orbit in the proposed design frame-work. Projected in the xy-plane of the Earth-Moon rotating frame.

2:1 synodic resonance, i.e., two revolutions along thehalo orbit are completed for every one synodic pe-riod. Similarly, the individual trajectories along theinvariant manifold associated with a periodic orbitin the BCR4BP are associated with unique Sun an-gles. Every path along a stable or unstable manifoldarrives at or departs from the periodic orbit at a spe-cific Sun angle. Structures available in the BCR4BPcan appear similar to those in the CR3BP, but thedependency on Sun angle is critical.

2.3 Direct Collocation

Low-thrust trajectory design is frequently posedas a continuous optimal control problem; at each in-stant in time a thrust vector is selected that mini-mizes a cost, typically propellant consumption, timeof flight, or some combination. A multitude of strate-gies for solving continuous optimal control problemsare available, and the best strategy is dependent onthe characteristics of the problem. In this investi-gation, a direct optimization technique is employedbecause these methods are generally more robust topoor initial guesses than indirect optimization for-mulations and require less computational time thanglobal optimization approaches. This balance of ro-bustness and efficiency is critical for the proposed de-

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sign framework because it aims to rapidly explore alarge search space of potential LIC trajectory solu-tions.

The process of discretizing a continuous optimalcontrol problem to allow a numerical solution is de-noted transcription, and collocation offers one ap-proach to this procedure. A collocation scheme usespolynomials to approximate a solution to the set ofdifferential equations that govern a dynamical model,for example, Equations (1)-(3). A collocation prob-lem is discretized into n segments where the dynamicsalong each segment are approximated by a polyno-mial. Many collocation schemes are differentiated bythe type and degree of polynomial used; this investi-gation employs 7th degree Legendre Gauss polynomi-als. Collocation is the transcription method of choicein this investigation due to its wide basin of conver-gence and amenability to the addition of constraints.Collocation often exhibits a wider basin of conver-gence than other approaches for solving a system ofdifferential equations, i.e., it will converge upon a so-lution despite a poor initial guess, even when othermethods fail. Since this investigation utilizes directoptimization, the overall optimization approach em-ployed here is termed direct collocation.

The tool that implements direct collocation forthis investigation is labelled COLT (Collocation withOptimization for Low-Thrust) and was developed incollaboration with Daniel Grebow at the Jet Propul-sion Laboratory. The direct collocation framework inCOLT generally follows the scheme developed by Gre-bow and Pavlak and implemented in their MColl soft-ware26. This scheme is demonstrated to successfullysolve many types of of low-thrust trajectory designproblems.27–29 Solution of this broad range of trajec-tory design scenarios is aided by the availability of avariety of problem constraints in COLT. While someconstraints must be enforced to obtain a solution tothe low-thrust trajectory optimization problem, oth-ers are only included for specific scenarios. For exam-ple, it is frequently beneficial to include minimum ra-dius constraints with respect to gravitational bodies.This constraint enforces a “keep-out” zone aroundthese bodies such that the trajectory remains beyondthe zone. Other useful constraints include constrain-ing the states, energy, and/or orbital elements for theinitial and final boundary points along a trajectoryarc. Specific low-thrust trajectory design scenariosmay require other constraints, and these are straight-forward to incorporate in a collocation framework.

Several additional features of COLT are availablein the current version of the algorithm. The design

variable and constraint framework for implementingcollocation is paired with the optimizer IPOPT30 tocompute mass optimal low-thrust trajectories. Fora more tractable optimization problem, the designvariables are frequently bounded within a desiredrange. Some design variables possess obvious upperand lower bounds; for example, necessary constraintson the control and mass variables are enforced sim-ply by applying bounds. Bounds on other variables,such as the position and velocity states, are definedrelative to their initial values and are defined as userinputs. Convergence to an optimal solution is alsoaided by the ability to scale the design variables andconstraints. Finally, collocation is typically pairedwith a mesh refinement scheme to generate a suffi-ciently accurate solution. Mesh refinement schemesadjust the spacing of boundary points along a solu-tion to evenly distribute and ultimately reduce error.The mesh refinement strategies leveraged in this in-vestigation, developed by Grebow and Pavlak, are la-belled Control Explicit Propagation (CEP) and Hy-brid CEP.26 Bounding the design variables, scalingthe problem, and conducting mesh refinement ensurethat COLT is a robust and efficient direct collocationalgorithm.

2.4 Nearest Neighbor Search

The proposed trajectory design framework em-ploys maps to aid the construction of initial guessesthat are passed to the direct collocation algorithm.Maps capture the returns of trajectory segments toa particular hyperplane, Σ. Frequently, maps areused to facilitate the identification of close connec-tions between two sections of a spacecraft trajectory,e.g., one propagated forward in time and the otherbackward. Points along these trajectories that in-tersect the selected hyperplane are displayed on themap. Example hyperplanes include a plane in con-figuration space, e.g., the xy-plane, or the occurrenceof a specific epoch. In this investigation the Sun an-gle, θS , is used to define hyperplanes for two differ-ent maps. Parameters such as position, velocity, orenergy at the hyperplane intersections may be dis-played on the map. The maps in this analysis in-clude points, i.e., hyperplane crossings, from manytrajectories, and each trajectory can possess multiplereturns to the hyperplane. Due to the large num-ber of points and multiple dimensions of each pointplotted on the map it can be challenging to visuallyidentify the best connections between trajectory seg-ments. Therefore, a nearest neighbor search algo-rithm aids the identification of points on maps that

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share similar characteristics.Nearest neighbor (NN) algorithms are employed

in many computer science fields under a variety ofnames.31 Fundamentally, the nearest neighbor prob-lem involves locating the point p in a set of pointsP with the shortest distance to a given point q, as-suming all points occupy a space of dimension d.32

In the present application, events along the forwardpropagated group of trajectories provide one set ofpoints, while events on the backward propagated tra-jectories comprise the other set. Thus, an NN searchis ideally suited for identifying close connections be-tween these two sets. The tool Poincare, developedat JPL, employs NN algorithms for this purpose.33

In this investigation, Matlab’s knnsearch algorithmis employed for the NN search. Furthermore, a stan-dardized Euclidean distance metric is used to com-pute the distances between points. As NN searchsupplements visual identification of close connectionson a map, only parameters visible on the map areincluded in the search. In this case, these parame-ters are x and y position in the Earth-Moon rotatingframe, the value of the Hamiltonian, H, and the an-gle of the xy-plane projection of the velocity vectorwith respect to x. Because these parameters can pos-sess different magnitudes, the distance metric appliesscaling to prevent one set of parameters from biasingthe search. The standard deviations of each of theNN search parameters are used as scaling factors. Ifdesired, the scaling factors may be weighted to em-phasize close matches in specific parameters. Thisweighting is achieved by decreasing the value of thescaling factors for the parameters that are to be em-phasized. The NN search algorithm is a useful toolthat complements visual inspection of maps to iden-tify close connections between trajectory segments.

3. Trajectory Design Framework

The proposed trajectory design framework is dis-tinguished by three key features: modeling in theBCR4BP, employing a staging orbit, and comput-ing low-thrust transfers with direct collocation. To-gether, these design choices deliver a flexible and ro-bust procedure for constructing the LIC trajectory. Astaging orbit near the Moon is used to divide the mis-sion design challenge into two phases, as illustratedin Figure 4. Phase 1 occurs from deployment to thestaging orbit and Phase 2 passes from the stagingorbit to the low lunar altitude science orbit. Thesetwo phases are solved nearly independently, where thespacecraft mass is the only parameter, aside from theselected staging orbit, carried over between phases.

Fig. 4: Schematic of trajectory design framework.

In the BCR4BP, any discrepancies in epoch betweenthe two phases are overcome simply by waiting inthe staging orbit until the desired departure epoch isreached. Initial guesses for both phases of the trajec-tory design framework are assembled with the aid oftwo different maps. These maps display intersectionswith the hyperplanes Σ1 and Σ2 which are definedby the Sun angles θS1

and θS2, respectively. The first

map captures intersections of Σ1 by forward propa-gated deployment trajectory arcs, D, and backwardpropagated paths on the stable manifold of the stag-ing orbit, WS . Similarly, the second map capturesintersections of Σ2 by forward propagated trajecto-ries on the unstable manifold of the staging orbit,WU , and backward propagated capture trajectoryarcs, C. Close matches between hyperplane intersec-tions of forward and backward propagated trajecto-ries are identified and their corresponding trajectorysegments are assembled into an initial guess for thedirect collocation tool COLT.

3.1 Phase 1: Deployment to Staging Orbit

Design of Phase 1 of the LIC trajectory is facil-itated by the creation of maps that display inter-sections of Σ1 along trajectories propagated forwardfrom deployment and backward on the stable invari-ant manifold of the staging orbit. To expand theoptions available on these maps, a range of D andWS trajectory arcs are generated. Different WS arcsare obtained by changing the state and epoch of thedeparture point from the periodic orbit. In contrast,because the deployment state and epoch cannot bechanged, a span of D trajectory segments is gener-ated by varying thrust direction prior to the first Lu-nar flyby. The deployment state and epoch used inthis investigation are the same as those used to gen-erate the baseline trajectory displayed in Figure 1.

Trajectories propagated forward in time from thedeployment condition, D, are divided into threeparts, an initial coast period, a thrust segment, and a

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second coast period. An example of this subdivisionis displayed in Figure 5. The duration of the first

Fig. 5: Sample range for deployment trajectories inthe creation of a Phase 1 map, plotted in theEarth-Moon rotating frame. The first coast pe-riod is 0.8 days, the thrust segment is 3 days, andthe second coast section is 4 days. Thrust vec-tors span a range of α angle values from 0 to 270

in the VNB frame. Trajectories that impact theMoon are omitted.

coast segment is set to 0.8 days and is dictated bythe time required to perform initial spacecraft sys-tems checkouts and obtain tracking data. Followingthis coasting period, a multi-day thrust segment is in-troduced. A thrust segment of three days is used inthis investigation; however, this value can be alteredto change the post-flyby behavior of the deploymenttrajectory. The direction of the thrust vector alongthis segment is varied over a user-defined span of an-gles to generate a range of deployment trajectories.The angle, α, used to determine the thrust vector di-rection is defined relative to the v unit vector in thevelocity-normal-binormal (VNB) frame. This frameis defined such that the v unit vector is in the direc-tion of the velocity vector of the spacecraft expressedin the rotating frame. Additionally, the n unit vectoris in the direction of the spacecraft’s angular momen-tum vector relative to B1, and the b unit vector isdefined to complete the orthonormal set. The an-gle α determines the direction of the thrust vectorin the vb-plane, and no out-of-plane, i.e., n compo-nent of the thrust vector, is introduced. By varyingα from αmin to αmax many different post-flyby tra-jectories are generated, as seen in Figure 5. In thisinvestigation, αmin = 0 and αmax = 270. Following

the thrust segment, a second coast segment is propa-gated for a user-defined number of days. Intersectionsof these D trajectory arcs with the hyperplane Σ1 arerecorded and used to generate the map.

The forward propagated D trajectory arcs arelinked to trajectories propagated backward in timealong the stable manifold, WS , of the staging orbit.The stable invariant manifolds of periodic orbits offerefficient paths onto the orbits. Thus, using these tra-jectories to guide LIC to the staging orbit should leadto a solution that requires less propellant than otherpotential insertion paths. Figure 6 displays trajecto-ries along the stable manifold. The initial trend of

Fig. 6: Stable (blue) and unstable (magenta) mani-fold trajectories plotted in the Earth-Moon rotat-ing frame and originating from the 2:1 resonant L2

halo orbit displayed in Figure 3.

these manifold paths is either in the positive or neg-ative x direction. In the former case, trajectories onthe manifold tend to escape the Earth-Moon system,and these paths offer more useful connection pointswith the D trajectory arcs. Stepping off the periodicorbit and onto the stable manifold at different statesand epochs around the orbit generates a variety ofmanifold paths.

A map is created to join the two halves of thePhase 1 LIC trajectory by recording intersections ofthe D and WS trajectory arcs with Σ1. Trajectoriesare propagated until either a maximum time limit ora maximum distance from the Earth is reached. Inthis case, a maximum time of 100 days and a maxi-mum Earth distance of 3× 106 km are used for boththe D and WS propagation. An example map isdisplayed in Figure 7 where the Sun angle at whichevents are recorded is θS1

= 135. The events that oc-

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Fig. 7: Map of Σ1 intersections of the forward propa-gated D and backward propagated WS trajectoryarcs in the BCR4BP. Intersections are projectedin the xy-plane of the Earth-Moon rotating frame,and the Sun angle selected for Σ1 is θS1

= 135.

cur along the D trajectories are plotted as diamonds,while the events on the WS arcs are marked as as-terisks. All events are plotted in the Earth-Moonrotating frame, and each marker is colored accordingto the Hamiltonian, H, of the spacecraft at the timeof the event. Additionally, the spacecraft’s xy-planevelocity direction is plotted as an arrow centered atthe marker. Adding this extra information to eachplot aids the identification of close matches betweenD and WS trajectories. By selecting intersectionsthat match closely in position space along with en-ergy and velocity direction, useful initial guesses forthe direct collocation algorithm can be obtained. Aclose match, identified by the nearest neighbor algo-rithm, is highlighted in Figure 7. The D and WS

trajectories are propagated to the selected intersec-tion times and used as an initial guess for the directcollocation algorithm in the Sample Trajectory De-sign section.

3.2 Phase 2: Staging Orbit to Science Orbit

Phase 2 of the LIC trajectory consists of the trans-fer from the staging orbit to the science orbit. An ini-tial guess for this phase is assembled in a similar man-ner to Phase 1; intersections with Σ2, along a rangeof forward and backward propagated trajectories, areplotted on a map used to select the initial guess. Inthis case, paths along the unstable manifold, WU ,of the staging orbit make up the forward propagatedtrajectory segments. The backward propagated seg-

ments, C, consist of trajectories propagated with low-thrust, in reverse time, from different true anomalyvalues on the science orbit. The map used to link theforward and backward propagated trajectories con-sists of intersections with Σ2, defined by the Sun an-gle θS2 , along these trajectory segments.

Trajectories along the unstable manifold, WU ofthe staging orbit offer energy efficient paths for de-parting the orbit and beginning the spiral down tothe science orbit. Apart from their inverse direction,these trajectories behave similarly to those on the sta-ble manifold and are displayed in Figure 6. Paths onWU offer a variety of locations and epochs at whichto depart the staging orbit, and intersections with Σ2

along these trajectories populate the Phase 2 map.

To generate a variety of C trajectories that insertonto the final orbit, backward propagation is initi-ated from true anomaly values on the orbit that spanthe full 360 range. The backward propagation as-sumes a maximum thrust anti-velocity control law -that is, the thrust vector is always oriented along the−v direction of the VNB frame and has a magnitudeequal to the maximum thrust of LIC. Recall that theVNB frame is defined relative to the rotating framevelocity vector. Application of this control law pro-duces a trajectory that, in forward time, graduallyspirals down to the final science orbit, as shown inFigure 8(a). The C trajectory arcs must assume anepoch for insertion onto the science orbit as well asa spacecraft mass at insertion. Reasonable inferencesfor these values made during initial guess formulationare later adjusted by the direct collocation algorithmto ensure a continuous final result. Previous analy-ses have indicated that the total duration of the LICtransfer is approximately one year, thus a date ofJune 1st, 2021 is used to obtain a Sun angle at sci-ence orbit insertion (SOI) of 105.7. Similarly, earlierinvestigations indicate that, at most, LIC consumeshalf of the available propellant mass to execute thetransfer from deployment to science orbit, thereforea final mass at SOI of 13.25 kg is assumed. Evenif an intuition for the epoch and mass at SOI is notavailable, the robustness of the direct collocation al-gorithm increases the likelihood that poor estimatesfor these values can still produce useful initial guesses.While these two quantities are constant for all back-ward propagated trajectories, using a range of trueanomaly values to initialize propagation ensures thateach trajectory evolves differently, as seen in Figure8(b). Trajectories are propagated until either a max-imum time limit or a maximum distance from theEarth is reached. In this case, WU arcs are propa-

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(a) Single backward propagated trajectory in the Moon-centered J2000 inertial frame.

(b) Multiple backward propagated trajectories in the Earth-Moon rotating frame generated by initiating propagation atdifferent true anomaly values on the science orbit.

Fig. 8: Backward propagation from the science or-bit with a constant maximum thrust anti-velocitythrust vector.

gated for a maximum time of 100 days while C trajec-tory segments are propagated up to 200 days. Bothtypes of trajectories are propagated to a maximumdistance from the Earth of 6 × 105 km. Intersec-tions with Σ2 along each propagated trajectory arerecorded and added to the Phase 2 map.

Intersections with Σ2 along the WU and C trajec-tory segments generated for Phase 2 are projected onthe xy-plane and colored according to their H value.These three parameters, along with the in-plane di-rection of the velocity vector, are used to identifyclose matches between trajectory segments. An ex-ample map is displayed in Figure 9(a) where the Sun

angle that defines Σ2 is θS2 = 75. The intersections

(a) Earth-Moon Rotating Frame

(b) Phase 2 Map

Fig. 9: Map of Σ2 intersections of the forward propa-gated WU and backward propagated C trajectoryarcs in the BCR4BP. Intersections are projectedin the xy-plane of the Earth-Moon rotating frame,and the Sun angle selected for Σ2 is θS2

= 75.

that occur along the C arcs are plotted as diamonds,while those on the WU paths are displayed as aster-isks. A close match, identified by the nearest neigh-bor algorithm, is highlighted in Figure 9(b). The de-ployment and manifold trajectories propagated to theselected intersection times are used as an initial guessfor the direct collocation algorithm in Section 4.

Modifications are made to COLT’s nominal collo-cation scheme to enable design of a continuous low-thrust transfer from the staging orbit to the final sci-ence orbit for LIC. The low-thrust spiral required totransfer between these two orbits is long and includes

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many revolutions. This type of trajectory is challeng-ing to optimize using the collocation framework im-plemented in COLT, which employs Cartesian coor-dinates. Other collocation schemes that utilize modi-fied equinoctial elements (MEE) have successfully op-timized low-thrust spiral trajectories.34,35 However,rather than implement a complex multi-phase collo-cation scheme that uses Cartesian and MEE coordi-nates, a simplified approach is used. This strategydivides Phase 2 into two halves: one is solved withdirect collocation and the other is explicitly propa-gated backward in time from science orbit insertion(SOI).

The backward propagated section of the LIC tra-jectory is updated in the direct optimization processby the addition of three design variables and a con-straint. The three design variables govern: the back-ward propagation time from SOI, the true anomalyvalue on the science orbit at insertion, and the space-craft mass at SOI. By including these variables inthe corrections process, the evolution of the spiralingLIC trajectory is allowed to change and can be joinedwith the section of the transfer that departs fromthe staging orbit. A constraint is added to ensurestate and mass continuity between these two halvesof the LIC trajectory. With the addition of the threedesign variables and constraint, a single direct col-location problem is solved to generate a continuouslow-thrust transfer from the staging to the scienceorbit. Because a sub-optimal control law is used forthe spiraling portion of the trajectory, the result ofthe direct collocation algorithm is not a fully opti-mized low-thrust transfer. However, optimization isused to minimize the propellant consumed before theexplicitly propagated spiraling phase begins. Addi-tionally, as the selected control law ensures the max-imum rate of change of the spacecraft’s energy, andthis reduces the time required to achieve Lunar cap-ture, propellant consumption is reduced. This ap-proach for computing a trajectory from the stagingto the science orbit generates a continuous low-thrusttransfer without the complexity of a multi-phase col-location scheme.

4. Sample Trajectory Design

The proposed trajectory design framework offersa procedure for developing an initial guess for thefull Lunar IceCube trajectory, from deployment toinsertion on the science orbit. After maps are gener-ated for Phases 1 and 2, close matches between for-ward and backward propagated trajectory segmentsare identified and used to construct an initial guess

for the direct collocation algorithm. This robust algo-rithm is frequently able to converge even when givenvery discontinuous initial guesses.

4.1 Phase 1: Deployment to Staging Orbit

An initial guess for Phase 1 of the Lunar IceCubetrajectory is assembled by identifying a close matchbetween Σ1 intersections on D and WS trajectories ina Phase 1 map. An example of such a match is high-lighted by black markers in Figure 7, where the blacksquare indicates the deployment event and the blackfive-pointed star denotes the manifold event. Thismatch is identified using the NN search algorithmand the trajectories that correspond to the selectedevents are shown in Figure 10. While a discontinu-


(b) Sun-B1 Rotating Frame

Fig. 10: Initial guess for Phase 1 trajectory. Corre-sponds to the match identified in the Phase 1 mapcreated with θS1

= 135 displayed in Figure 7.

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ity between the forward and backward propagatedtrajectories is evident, the criteria used to identifymatches between Σ1 crossings on the map yield apromising initial guess. This initial guess is passed toCOLT, which eliminates the discontinuity by insert-ing additional thrust segments. The optimized tra-jectory resulting from this initial guess is displayedin Figure 11, and consumes 0.23 kg of propellant toreach the staging orbit in 140 days. The collocation



Fig. 11: Direct collocation result for Phase 1 transferusing the initial guess in Figure 10. This transferrequires 140 days and 0.23 kg of propellant.

algorithm computes this solution with relatively fewiterations; moreover, the geometry of the initial guessis generally preserved in the direct collocation result.These two factors indicate that the initial guess iden-tified with the Phase 1 map was useful in guiding thealgorithm towards a solution.

The strong influence of the initial guess on the fi-nal result is more evident when alternate solutionsare examined. A different Phase 1 map, one gener-ated with θS1

= 180, is used to construct the initialguess shown in Figure 12. This initial guess includes



Fig. 12: Initial guess for Phase 1 trajectory that in-cludes an Earth flyby. Corresponds to a closematch identified in a Phase 1 map at θS1 = 180.

an Earth flyby, and this flyby is maintained in the op-timized COLT solution displayed in Figure 13. Thistransfer reaches the staging orbit in approximately115 days and requires 0.27 kg of propellant mass.By experimenting with maps generated using differ-ent values of θS1

and selecting various crossing pairson those maps, a variety of initial guess geometriescan be obtained that lead to an array of optimizedsolutions. The flexibility of this approach and thediversity of solutions it offers makes it adaptable to

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Fig. 13: Direct collocation result for Phase 1 transferusing the initial guess in Figure 12. This transferrequires 115 days and 0.27 kg of propellant.

different mission constraints and deployment condi-tions.

4.2 Phase 2: Staging Orbit to Science Orbit

A solution for Phase 2 of the LIC trajectory is com-puted in a similar manner to Phase 1, and becausea staging orbit is used, little information on the con-verged results of Phase 1 is required to develop aninitial guess for Phase 2. A close match between Σ2

crossings on a sample Phase 2 map is identified inFigure 9, and the trajectories corresponding to theselected intersections are displayed in Figure 14(a).Because the staging orbit and the science orbit aresignificantly out-of-plane, the xy-plane view of thePhase 2 map shown in Figure 9 can be deceptive.

(a) Initial Guess

(b) Converged Result

Fig. 14: Initial guess and direct collocation resultfor Phase 2 transfer in the Earth-Moon rotatingframe that remains in the Lunar vicinity. Theinitial guess corresponds to the close match iden-tified in the Phase 2 map at θS2

= 75 displayedin Figure 9. The resulting transfer requires 228days and 0.52 kg of propellant.

Events that appear to overlap in this map may differsignificantly when the z position and velocity compo-nents are considered. As a result, the NN search isespecially useful for identifying matches on the Phase2 map, because the algorithm can be weighted to em-phasize close matches in energy, a parameter that in-cludes information on the out-of-plane components ofeach event. The close connection point between theunstable manifold and low-thrust spiral trajectoriesdisplayed in Figure 14(a) actually occurs under theMoon, i.e., z < 0, making it is difficult to see, even

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in the zoomed in view of the Phase 2 map shown inFigure 9(b). Utilizing the NN algorithm ensures thatthis close match is not overlooked.

The initial guess identified in Figure 9 is passedto the direct collocation algorithm, COLT, for con-vergence. In general, it is more difficult to achieveconvergence for this phase of the LIC trajectory be-cause the initial guesses generated by the map pos-sess larger state discontinuities and the majority ofthe trajectory is near the Moon, which increases thesensitivity of the optimizer. The likelihood of con-vergence is increased by bounding the design vari-ables, scaling, and including a minimum radius con-straint with respect to the Moon. Additionally, con-vergence is improved by providing COLT more trajec-tory, and therefore more time, with which to achievethe desired transfer. This is done by “stacking” ad-ditional revolutions of the staging orbit prior to de-parture on the unstable manifold path.28 Four ad-ditional revolutions on the staging orbit are addedto the initial guess displayed in Figure 14(a). Thus,the initial guess passed to COLT consists of thesefour revolutions, the unstable manifold trajectory,and the backward propagation parameters, i.e., thethree additional design variables, that produce thelow-thrust spiral displayed in Figure 14(a). With thisinitial guess, COLT computes the low-thrust transferin Figure 14(b), which requires approximately 228days and 0.52 kg of propellant to achieve. The solu-tion displayed in Figure 14(b) shows that the stackedstaging orbit revolutions included in the initial guessare distributed by the direct collocation algorithminto a quasi-periodic-like structure. This new mo-tion is even more evident in Figure 15(a), a three-dimensional view of the transfer. After moving alongthis structure, the spacecraft departs the vicinity ofthe staging orbit and conducts a powered lunar flybyto insert on the explicitly propagated low-thrust spi-ral trajectory. This geometry is influenced by the ini-tial guess, which also includes an unstable manifoldtrajectory with a close lunar flyby.

Several important differences exist between theconvergence process for Phase 1 and 2 transfers.First, the initial mass used to converge a Phase 2transfer must be the final mass of a converged Phase1 transfer. In this case, the final mass of the transfershown in Figure 11 is used as the initial mass for thetransfer in Figure 14(b). The spacecraft mass is theonly parameter from Phase 1 that must be carriedover to compute Phase 2. Furthermore, the space-craft mass and the maximum thrust, Tmax, determinethe maximum acceleration that the spacecraft can

(a) Near Moon Transfer

(b) Around Earth Transfer

Fig. 15: Out-of-plane views in the Earth-Moon rotat-ing frame of two Phase 2 transfer options.

produce. When convergence difficulty is encountered,it is frequently advantageous to temporarily raise themaximum available acceleration by first computinga transfer that uses a Tmax value marginally higherthan that available to LIC. Then, a natural param-eter continuation process can be used to lower Tmax

to the correct value. Continuation is employed toobtain both of the staging to science orbit transferspresented in this investigation. Finally, as stated pre-viously, because the low-thrust spiral phase of thestaging to science orbit transfer is explicitly propa-gated with a sub-optimal control law, the resultingtransfer is not fully optimal. Instead, the objectiveof the direct collocation problem in this phase is tomaximize the mass at the connection point betweenthe trajectory computed with collocation and the ex-

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plicitly propagated low-thrust spiral.Phase 2 transfers generally fall into one of two

categories: transfers that remain near the Moon, asseen already, and those that include at least one looparound the Earth. The latter of these two optionsis examined by generating a new Phase 2 map. Aclose connection is identified on a Phase 2 map gen-erated with Σ2 defined by a Sun angle of θS2

= 315,and the resulting initial guess is displayed in Figure16(a). In this case, the difference in position between

(a) Initial Guess

(b) Converged Result

Fig. 16: Initial guess and direct collocation resultfor Phase 2 transfer in the Earth-Moon rotatingframe that includes a loop around the Earth. Theinitial guess corresponds to a close match iden-tified in a Phase 2 map at θS2

= 315. The re-sulting transfer requires 212 days and 0.52 kg ofpropellant.

the end of the unstable manifold trajectory and the

beginning of the low-thrust spiral is especially large.However, these two points are relatively close in en-ergy and in velocity direction. Thus, the NN searchalgorithm identifies this initial guess that may havebeen disregarded by a visual inspection. Despite theseemingly large discontinuity, COLT uses this initialguess to compute the low-thrust transfer shown inFigure 16(b). This transfer delivers the spacecraftfrom the staging orbit to the science orbit in 212 daysand requires roughly the same propellant mass as theprevious case, 0.52 kg. Figures 15(b) and 16(b) il-lustrate that rather than using repeated revolutionsnear the staging orbit, much of the plane change re-quired to begin the spiral down to the science orbit iscompleted during the transit around the Earth. Thisbehavior, and the lack of a low-altitude Lunar flyby,make this transfer and others like it easier to convergebecause the larger distance from the Moon decreasesthe sensitivity of the optimization problem.

Combining the results computed for Phases 1 and2 of the LIC trajectory yields complete deploymentto science orbit trajectories. The sample LIC trajec-tories computed in the BCR4BP using the proposedframework are summarized in Table 3. The Phase1 transfer without an Earth flyby, shown in Figure11, is labeled A while the transfer that includes theEarth flyby, displayed in Figure 13, is B. Similarly,the Phase 2 transfer that remains near the Moon, de-picted in Figure 14(b), is labeled C and the transferthat loops around the Earth, shown in Figure 16(b),is D. Note that for all transfer combinations, up toa month of additional transfer time is added to ac-count for the phasing time that must be included inthe staging orbit following Phase 1 of the transfer.This time allows the spacecraft to reach the state andSun angle required to begin Phase 2. In this case, themaximum phasing time required is 25 days and theminimum is 0.5 days. Over a third of the total time offlight (TOF) is spent in the transfer from the stagingorbit to the beginning of the low-thrust spiral, anda quarter of the TOF is occupied by the subsequentspiral. In total, all four transfers reach the scienceorbit about one year after deployment.

The final mass of all four transfer combinations inTable 3 is nearly the same. This similarity is likely aresult of the initial guess used for the mass at SOI,mSOI , in the Phase 2 map generation process. In thiscase, mSOI = 13.25 kg, so while mSOI is a variablein the direct collocation algorithm, the value of thisquantity is only altered enough to achieve continuity.One approach for overcoming the bias of the initialguess is to set the maximization of mSOI as the ob-

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jective of the direct collocation problem. However,the sensitivity of the problem to this variable makesachieving convergence difficult when mSOI is used asthe objective. Instead a higher initial mass at SOImay be achieved by iterating the value of the initialguess for mSOI employed in the Phase 2 map gen-eration process. To do this, a higher value of mSOI

is used to generate new Phase 2 maps then the pro-cess of assembling and converging an initial guess isrepeated. Changes in mSOI affect the rate of changeof LIC’s energy, however the behavior of the low-thrust spiral is not significantly different. Thus, sim-ilar initial guesses can be found as mSOI is iteratedby changing the value of θS2

at which Phase 2 mapsare generated. A process of iterating the estimatedvalue of mSOI for the Phase 2 transfer can be usedto increase the mass delivered to the science orbit.Ultimately, this procedure reaches a limit where notransfers with higher mSOI values can be computed.

The transfer times and propellant consumptionvalues provided in Table 3 are comparable to the cur-rent baseline trajectory shown in Figure 1. Whilethe transfers computed with the proposed frameworkall require more time and propellant than the base-line, several steps are available that may reduce thetime and change in mass of these transfers. First,an iterative process of estimating successively highervalues for mSOI can be undertaken, as described pre-viously. Second, while the additional staging orbitrevolutions added to the initial guess for Phase 2are useful to achieve convergence, it may be possi-ble to remove some of them after a transfer is con-verged. For example, some of the revolutions nearthe staging orbit that appear in Figure 15(a) couldlikely be removed because they consist largely of longcoast segments. If possible, removing excess trajec-tory will reduce transfer time. Finally, using a multi-stage collocation algorithm to solve Phase 2 will offerthe greatest improvement in delivered mass becausethe majority of propellant consumption occurs in thisphase. A multi-stage approach would permit the low-thrust spiral to the science orbit to be fully optimized,thereby increasing mSOI . These three steps can leadto reductions in both time of flight and propellantconsumption for the LIC transfer.

5. Concluding Remarks

The proposed framework for constructing a LICbaseline trajectory addresses the need for a designmethodology that is both robust and flexible. Em-ploying a dynamical model that includes the Sun en-ables the perturbing acceleration of this body to be

Table 3: Summary of sample transfers. All resultsare from the given deployment state to the sci-ence orbit. Given characteristics are time of flight(TOF), mass at science orbit insertion (mSOI),and total change in mass (∆m).

Transfer TOF [days] mSOI [kg] ∆m [kg]

A → C 369.25 13.25 0.75

B → C 369.63 13.25 0.75

A → D 353.02 13.24 0.76

B → D 323.50 13.24 0.76

Baseline 318 13.31 0.69

leveraged to achieve the desired transfer despite LIC’slimited control authority. Moreover, utilizing directcollocation to converge initial guesses developed inthe BCR4BP allows a wider range of guesses to beused due to the robustness of this algorithm. Flexi-bility is further enhanced by using a staging orbit toseparate the LIC trajectory into two halves that canbe designed independently. Together, these designchoices offer a framework to systematically generatea variety of transfer configurations with time of flightand propellant consumption values comparable to thecurrent baseline. Further improvements to this pro-cedure could increase the efficiency of the process,enhance the quality of the solutions it generates, andbroaden its applicability to missions beyond LIC.

Acknowledgements

The authors thank the Purdue University Schoolof Aeronautics and Astronautics for facilities andsupport, including access to the Rune and Bar-bara Eliasen Visualization Laboratory. Additionally,many thanks to the members of the Purdue Multi-Body Dynamics Research Group for interesting dis-cussions and ideas. Special thank you to NicholasLaFarge for the idea of employing a nearest neigh-bor search algorithm and for guidance on its im-plementation. This research is supported by NASASpace Technology Research Fellowship, NASA GrantNNX16AM42H. Some analysis was conducted at theNASA Goddard Space Flight Center.

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