PATHOS:

AMATLAB‐basedWeakStabilityBoundaryOrbitalTrajectory

SimulatorforUseinInterplanetaryMissionDesign

AThesis

SubmittedtotheFaculty

of

DrexelUniversity

By

EricTran

inpartialfulfillmentofthe

requirementsforthedegree

of

MasterofScienceinMechanicalEngineering

September2012

©Copyright2012

EricTran.AllRightsReserved.

ii

TableofContents

ListofFigures.........................................................................................................................................................iii 

ABSTRACT..................................................................................................................................................................v 

1.  Introduction...................................................................................................................................................1 

2.  Background.....................................................................................................................................................2 

(i)  Two‐BodyProblem..............................................................................................................................3 

(ii)  n‐BodyProblem....................................................................................................................................5 

(iii)  4‐bodyProblemDerivation..............................................................................................................6 

(iv)  TheWeak‐StabilityBoundary.......................................................................................................12 

3.  Motivation.....................................................................................................................................................18 

4.  StatementofWork....................................................................................................................................19 

5.  MethodofSolution....................................................................................................................................19 

(i)  Full‐forceModelTrajectories(PR3BP).....................................................................................19 

(ii)  Full‐forceModelTrajectories(PCR3BP)..................................................................................26 

(iii)  AnomaliesintheFull‐ForceModel.............................................................................................29 

6.  Results&Discussion................................................................................................................................34 

7.  ComparisonofPATHOStoSimilarPropagatorsofitsClass..............................................38 

8.  Conclusion.....................................................................................................................................................41 

9.  FutureWork.................................................................................................................................................42 

10.  ListofReferences.................................................................................................................................44 

iii

LISTOFFIGURES

Figure1.Typical2‐BodySystemConfigurations.......................................................................................4 

Figure2.Visualconceptofconicsections.Eachcross‐sectioncorrespondstoadifferentorbitshape..................................................................................................................................................................5 

Figure3.Spacecraftfree‐bodyforcediagraminthepresenceoflargemasses.............................6 

Figure4.Referencesystemforthe4‐BodyProblemformulation.......................................................7 

Figure5.LocationofequilibriumpointsforthePCR3BPmodel.......................................................11 

Figure6.LocationofLagrangepointsfortheEarth‐SunSystem.(Source:NASAJPL).............13 

Figure7.HeteroclinicConnectionbetweenJupiter'sL1andL2points.(Source:Koonetal1999)...........................................................................................................................................................................15 

Figure8.3‐dimensionalviewoftheGenesisMissiontrajectory.(Source:NASAJPL)..............16 

Figure9.2‐dimensionalviewoftheGenesisMissiontrajectory.(Source:NASAJPL)..............16 

Figure10.TheGenesisMissionTrajectoryoverlaidontoaheteroclinicconnection.(Source:Koonetal1999).....................................................................................................................................................17 

Figure11.IllustrationoftheDeadReckoningmethodappliedtotrajectorycomputation....20 

Figure12.IllustrationoftheDeadReckoningmethodappliedtodiscretenumericaltrajectorycomputation.......................................................................................................................................21 

Figure13.Initialsimulationoutputplot......................................................................................................23 

Figure14.PlotofaLEOcircularorbitfordt=0.01sec.........................................................................24 

Figure15.Divergenceoftheexpectedtrajectoryfordt=0.1sec.....................................................24 

Figure16.Morerefinedtrajectoryplotwithdt=0.001sec..................................................................25 

Figure17.Semi‐majorAxiserrorplotasafunctionoftimestep......................................................26 

Figure18.3‐dimensionaltesttrajectory......................................................................................................27 

Figure19.ParametersforatypicalMolniyaOrbit...................................................................................28 

Figure20.SimulationrecreatedthesameMolniyaorbitshownpreviously................................28 

Figure21.ForcecontourplotofsimulationenvironmentinthevicinityofEarth....................33 

ivFigure22.Enhancedviewsoftheforcecontourplotofthesimulationenvironment.Thedipsandpeaksshownaboveindicatetheequilibriumlocations(i.e.‐Lagrangepoints)..................34 

Figure23.PATHOSprogramoutputplotofaWeak‐StabilitytrajectoryfromEarth‐SunL1vicinitybacktoEarthretrieval.........................................................................................................................35 

Figure24.APATHOS‐generatedheteroclinicconnectionbetweentheEarth‐SunL1andL2points..........................................................................................................................................................................37 

Figure25.IsometricviewofthePATHOSheteroclinicconnectionbetweenL1andL2..........37 

Figure26.Sampleorbitgeneratedby"Sat_Orbit",ofPolitecnicodiMilano.................................39 

Figure27.SampletransfergeneratedbyASTROTIKsimulator.........................................................40 

Figure28.Sampletrajectorygeneratedbythe"InterplanetaryMissionPlanner"....................40 

v

ABSTRACTPATHOS:AMATLAB‐basedWeakStabilityBoundaryOrbitalTrajectory

SimulatorforUseinInterplanetaryMissionDesignEricTran

JinS.Kang,Ph.D.

Trajectorydesignistraditionallyperformedunderverystrictconstraintsorsimplifications.

Thisisbecausefull‐forcemodelshavethusfareludedanalyticalsolution.Onecommonsimplification

isthetwo‐bodyassumption,wheretheonlybodiesconsideredarethespacecraftandacentralmass.

This simplification yields fairly accurate results for a small number of specific cases (binary stars,

low‐Earthorbit).However,oncetheorbitalregimeenterstheinterplanetaryrange,wheremultiple

gravitationalbodies are relevant, simple two‐body calculationsprove inadequate. In response, the

patched‐conic approach was used, where multiple two‐body trajectories would be “patched”

togethertoformanapproximatepathforthespacecraft.Thisapproach,however,stillemployedthe

two‐body simplification and so the hidden constraints of the two‐body problem are carried over.

Consequently, while this method produces useful trajectories, it does not yield the most efficient

ones.

Whilethen‐bodyproblemhadnotbeenexplicitlysolved,numericalmethodswithmodern

computationalsoftwareprogramscanbeusedto identifyextremelyefficient trajectoriesby taking

into account a greater number of bodies. It was recently discovered that gravitational pathways

linkingtheSolarSystem’sLagrangepointscanprovideextremelycheapinterplanetarytravel.While

onlyahandfulofmissionshaveflowntheseso‐called“WeakStabilityBoundary”(WSB)trajectories

inthepast,theyhavethepotentialtogainwidespreadusefortheirextremelylowfuelcosts.

ThisdocumentwilldiscusstheconstructionoftheseWSBtrajectoriesthroughtheuseofa

Dead‐Reckoning numerical simulation tool, called PATHOS, which accounts for at least 3‐body

gravitational effects. The simulation will be used to generate a group of sample trajectories as

validation,aswellascomparedagainstsimilarsoftwaretools.

(Thispageisintentionallyblank.)

1

1. INTRODUCTION

Traditionally, orbital transfers and maneuvers were performed under very strict

constraints or simplifications. One of the main simplifications was the two‐body

assumption, where the only bodies considered in the trajectory calculation were the

spacecraftandacentralmass.Thissimplificationyieldedmostlyaccurateresults for low‐

altitudeorbitsorveryspecificscenarios,likebinarystarsystems.However,oncetheorbital

regimeenterstheinterplanetaryrange,simpletwo‐bodycalculationsproveinadequate,as

the gravitational effects of othermassive bodieswas no longer negligible. In response to

this,thepatched‐conicapproachwasgenerallytaken,wheremultipletwo‐bodytrajectories

wouldbe“patched”togethertoformanapproximatepathforthespacecraft.Becausethis

approach still employed the two‐body simplification, a hidden constraint is carried over:

that all the bodies (as well as the trajectory itself) are in the same plane, among other

assumptions.Whilethismethoddidproduceusefultrajectories,theydonotyieldthemost

efficientones.

Attempting to generalize the orbit design problem, the n‐body problem was

formulated. As the name suggests, the n‐bodyproblem is formulatedwith “n” number of

gravitationalbodies.Whilethisproblemhasnotexplicitlybeensolved,numericalmethods

withmoderncomputationalsoftwareprogramscanbeusedtoidentifyextremelyefficient

trajectoriesbytakingintoaccountagreaternumberofmasses.Itwasrecentlydiscovered

that gravitational pathways linking the Solar System’s Lagrange points can provide very

cheap interplanetary travel.While only a handful ofmissions have flown these so‐called

“WeakStabilityBoundary” (WSB) trajectories in thepast, they have thepotential to gain

widespreadusefortheirextremelylowfuelcosts.(Belbruno2000)

2

Thisdocumentwilldiscusstheconstructionof theseWSBtrajectoriesthroughthe

useofanumericalsimulationthataccountsforn‐bodygravitationaleffects.Theaccuracyof

thissimulationwillbevalidatedwiththereconstructionofahandfulofsampletrajectories.

Noteabout the languageof thedocument: In this text, the terms “WeakStabilityBoundary

Trajectory” and “low‐energy transfer” are used interchangeably. Similarly, the terms

“Lagrangepoints”and“librationpoints”areusedinterchangeably.

2. BACKGROUND

Inorder toanalyzeorbital trajectories,a classicalNewtonianmechanicsapproach

wasinitiallytaken.Thegoverningequationfortheforceexertedbyamassivebodyonan

objectisNewton'sLawofUniversalGravitation,Eq.1,below.

(1)

where

, theUniversalGravitationalConstant(6.67300 10 / ∙ )

, , themassesofthetwomassivebodies

, thedistancebetweenthetwomassivebodies

Taking a very basic approach, the trajectory of nearly any object under the influence of

gravitycanberoughlyestimatedwithonly theequationabove.For trajectories relatively

close to a massive central body, a very simplified problem can be formulated. This

formulationiscalledthe"Two‐bodyProblem".

3(i) Two‐BodyProblem

TheTwo‐bodyproblemisasimplifiedversionofthe"n‐bodyProblem",whichwill

bediscussedlater.Itdefinesthemotionoftwomassivebodiesundertheeffectsoftheother

body's gravitational field. However, this simplification comes with a set of underlying

assumptions:

I. Thereareonlytwobodiesintheregionandbodiesoutsideoftheregionare

gravitationallyinsignificant

II. Theonlyforceactingonthetwobodiesisgravity(thebodiesarefarenough

apartthattheiratmospheresdonotinteractandtheydonotcollide)

III. Thetwobodiesorbiteachotherinthesameplane

IV. Thetwobodiescanbetreatedaspoint‐particles(i.e.‐theirmassesareeither

uniformlydistributedorconcentratedattheirgeometriccenters)

Thefigurebelowshowsafewofmanypossibletwo‐bodyconfigurations.Themost

familiaroftheseisthethirdone,whichisthecaseforallman‐madesatellites,theMoon,and

theEarth'smotionaroundtheSun.Thatis,thatthecentralmassissignificantlylargerthan

the orbitingmass.While in reality, the twomasses orbit the system’s barycenter, in the

third configuration of Figure 1, the barycenter of the systemcanbe approximated as the

geometric center of the largermass. As a result, only themotion of the smallermass is

examined,asthelargermassisassumedtobestationary.

4

Figure1.Typical2‐BodySystemConfigurations.

While some of these assumptions are nearly true in most practical cases, some

others greatly limit the possibly of trajectory design outside of the "simple orbits", also

knownasKeplerianorbits,regime.Theaboveformulationallowsforsimplifiedcalculations

ofthese"simpleorbits".Underthissetofassumptions,traditionalKeplerianorbitscanbe

definedinverybasicmathematicalterms.Asitturnsout,Keplerianorbitscanbedefinedby

so‐called"conic‐sections",showninFigure2below.

5

Figure2.Visualconceptofconicsections.Eachcross‐sectioncorrespondstoadifferentorbitshape.

Asthenameandfiguresuggests,aconicsectionistheresultingshapethatacross‐

section of a cone produces. Interestingly, the shapes of Keplerian orbits MUST either be

circular,elliptical,parabolic,orhyperbolic(andtheoretically,linear).Alloftheseshapesare

verywell andsimplydefinedmathematically.Thus, trajectoriesandorbits thatariseas a

resultofthetwo‐bodyformulationarealsowell‐definedmathematically.

Because of the great benefit of a simplified mathematical model and the lack of

demandforamoreaccuratemethod,thetwo‐bodyformulationwassufficientformuchof

ourprogressinthefieldofspacetechnologyandresearch.Thisismainlybecausehumanity

has not routinely extended its presence past Near‐Earth space. Since the two‐body

assumptionsstillholdinthisorbitalregime,therewasnoreasontofurtherstudyadvanced

orbitalpropagationtechniques,suchasthen‐bodyproblem.

(ii) n‐BodyProblem

The n‐Body Problem is simply the Two‐body problem extended to include more

than just one object orbiting another. The n‐Body Problem takes into account the

gravitationaleffectsofseveralnearbybodies.Thisisamoreaccuraterepresentation,since

6inreality,therearemanymoregravitationaleffects.Thefigurebelowillustratestheconcept

ofthen‐bodyproblem.Whenanobject(i.e.‐asatellites)isinthevicinityofmorethanjust

onelargemass,itsorbitcanbecomechaotic,ratherthanasimpleconic‐section.(Belbruno

2004)

Figure3.Spacecraftfree‐bodyforcediagraminthepresenceoflargemasses.

In order to accuratelydetermine anobject's trajectory in3‐dimensional space, all

gravitational effectsmust be considered. Limiting the calculation to two bodies at a time

greatly limitsthepotentialapplicationsof thetrajectoriesthataredesigned.Forexample,

trajectories designedwith only a two‐body approach are limited to linear, planar orbits.

However,Weak‐StabilityBoundarytrajectories,whichwillbediscussedinmoredetaillater,

forexample,are3‐dimensional,complex,andnonlineartrajectories.(Koon2001)

(iii) 4‐bodyProblemDerivation

Withmanyinterplanetarymissions,thereare,infact,morethan2‐3gravitationally

relevantbodies.Forexample,an interplanetarymission toMarsoraNear‐EarthAsteroid

(NEA) would have the Earth, Mars (or the NEA) and the Sun, as the entire system is

heliocentric.Afterincludingthespacecraft,thetotalnumberofbodiesisfour.Fornow,we

will consider the most basic interplanetary scenario, one that involves the Moon, our

nearest celestial neighbor. Formulating the scenario such that there are only three

7gravitationally relevant bodies, we can derive the governing equations for a 4‐body

problem.ForamissionbeginningattheEarth,therearethreesuchobjects:theEarth,the

Moon,andtheSun.Thefourthbodyinthisscenarioisthespacecraftitself.Thefirststepis

toestablishasetofreferenceaxes.Forthis,aheliocentriccoordinatesystemwillbeused.

Thatis,theoriginofthesystemisplacedatthecenteroftheSun.Figure4belowillustrates

thereferencesystemforthefollowingproblemformulation.

Figure4.Referencesystemforthe4‐BodyProblemformulation.

Thepositionsof theEarth,Moon,andsatellitearedefined inreferenceto theSun,

with the entire system revolving around the Sun,which is fixated at the origin. In other

words,we'reonlyconcernedwiththerelativemotionofourspacecraftwithintheconfines

of the Solar System. It should be noted that the position vectors in the figure above are

three‐dimensional vectors and that the z‐axisof the reference systempointsout towards

thereaderfromtheplaneofthepage.

Withthisreferencecoordinatesystem,thevector‐equationofmotioncanbederived

bytakinganEuler‐Lagrangeapproach.

Y

X

rMoon

rEarth

rSatellite

8 (2)

WhereListheLagrangian,TisthekineticenergyandVisthepotentialenergyofthesystem.

The mathematical representations for kinetic and potential energy are taken from the

classicalNewtonianmechanicalequations:

12

→

∴ → (3)

OncetheLagrangiantermiscalculated,wecanapplytheLagrangeEquation(ofthesecond

kind). This is also known as the Euler‐Lagrange Equation. This equation is applicable

because our system is a conservative system, meaning that all forces (only gravity) are

functionsofpositiononlyandnotvelocity.TheEuler‐Lagrangeequationis:

(3)

There is only one kinetic energy term, because the spacecraft only has one velocity.

However, since there are 3 gravitational forces acting on the spacecraft (from the three

largebodies),thepotentialenergyoftheLagrangianwillcontain3terms:

12

∴ 12

Rewritingthis,wehave:

12

9SubstitutingthisintotheEuler‐Lagrange,Eq.3,above,weget:

→

0

0

0 (4)

This is the vector equation of motion of our spacecraft under the gravitational

effectsofthreenearbybodies.Initscurrentform,thisdifferentialequationhasnoanalytical

solution.Thatis,wecannotsolvethisdifferentialequationtoobtainthespacecraft'sexact

positionasafunctionoftime.Ifplottednumerically,thespacecraft'spositionwouldbehave

rather chaotically, varyinggreatlywitheven slightperturbations in initial condition.As a

result,simplificationshavebeenusedtodesignorbitsinregionsofspacewherethereare

morethanonegravitationallyrelevantbody.

The Euler‐Lagrange formulation above would be useful for a full‐force model

approach.Thatis,aninertialframewouldbeadoptedforthesetofequationstoapply.Since

we’re interested inmissions that originate or terminate at the Earth, a few adjustments

mustbemade to view the system froma rotating frame, following theorbit of theEarth

aroundtheSun.

If we apply a few simplifying assumptions,we canmore clearly see the system’s

equationsofmotionwithrespect toanEarth‐rotating frame.Onecommonapproach is to

usethePlanarCircularRestrictedThree‐BodyProblemmodel(PCR3BP).Theequationsof

10motion for this system with the PCR3BP model are well studied and are shown below.

(Howell1984)

2 ; 2 ; (5)

where

(6)

1

is themassratioof thesmallergravitationalbodyto thesumof themassof the

system.Thatis:

wherem1 andm2 are themasses of the central and smaller gravitational bodies,

respectively.

Inaddition,theeffectivepotentialfunctionisgivenas:

, (7)

This potential function, Eq. 7, will be used in generating a potential field plot in

ordertolookattheenergyenvironmentofoursimulationlaterinthedocument.

While the above set of equations has eluded an analytical solution since its

derivation, it is still beneficial as a starting point. Studying these equations can provide

usefulinsightsintothebehaviorofobjectsinthevicinityofthesetofequilibriumpointsin

thePCR3BP,calledtheLagrangepoints,shownbelowinFigure5.

11

Figure5.LocationofequilibriumpointsforthePCR3BPmodel.

It is in the vicinity of these 5 equilibrium points, labeled L1 – L5, that complex

dynamical interactions occur that produce the extremely nonlinear trajectories that this

workaimstogenerate.

AlthoughananalyticalsolutionhasnotbeenfoundforthePCR3BP,therearemany

approachesthataimtosimplifytheproblemtoproduceusefulorpracticalresults.Onesuch

approach is the"patched‐conic"method,mentionedpreviously.Asthenamesuggests, the

patched‐conicmethod"stitches"differentsegmentsof two‐bodyorbits together to forma

full flight path. This method is useful for designing interplanetary trajectories. It has

allowedforsimplecalculationsoftrajectoriesbetween3‐4planetarybodieswithouthaving

to solve the full 3‐body equation of motion. This method also produces fairly practical

trajectories and is sufficient for trans‐lunar injections, resulting in lunar orbits. The

drawbackscomefromtheunderlyingassumptionsassociatedwiththetwo‐bodyapproach.

The main drawback is the fact that the conic sections must be two‐dimensional, by

definition.While this was not initially seen as an issue, the success of the patched‐conic

12approach has hindered, and possibly even precluded, the research and development of a

moreaccuratemodelinorbitalpropagation.

The 3‐body system is an example of a system that is governed by the dynamical

systems theory.This is anareaofmathematicsused todescribe thebehavior of complex

dynamicalsystems,usuallybyemployingdifferentialequationsordifferenceequations.A

dynamical system, like our Sun‐Earth‐Moon or Sun‐Earth‐Spacecraft system, can be

describedbyasetofgoverningequationsthatdetermineitsevolutionasafunctionoftime.

In this case, the governing equations are those ofNewton's andEinstein's describing the

forceofgravity.Morespecifically,ifweareinterestedinthetrajectoriesofandaroundthe

Weak‐Stability Boundary, the system is more appropriately described as a deterministic

chaoticsystem.Thismeansthatalthoughthesystemisconsideredtobedeterministic,the

time evolution of a spacecraft's position in the system is highly sensitive to its initial

conditions.Itisintheseseeminglychaoticoutcomesthatuseful,Weak‐StabilityBoundary,

trajectoriesarise.

(iv) TheWeak‐StabilityBoundary

TheWeak‐StabilityBoundary(WSB)refers toaregionofspace(primarily located

aroundandthroughtheareasoccupiedbyplanetarylibrationpoints)wherethetransition

between gravitational capture and escape becomes fuzzy or unclear. (Koon 2000) The

mathematics of the gravitational field around these libration regions becomemuchmore

complex than what is generally needed for simple two‐body trajectory calculations.

However, the benefit of examining these regions is that they can provide extremely low

fuel‐costpathwaystofartherregionsofspace.

Librationpoints,alsoknownasLagrangepoints,aretheresultoftherotatingthree‐

bodyproblemandarepresentinanyrotatingthree‐bodysystem.Intuitively,thelibration

pointscanbeseenasgravitationalequilibriumpoints,insomesense.Figure6belowshows

13theLagrangepointsoftheEarth‐Sunsystem.Inanythree‐bodysystem,therearefivesuch

points,labeledL1throughL5,shownbelow.

Figure6.LocationofLagrangepointsfortheEarth‐SunSystem.(Source:NASAJPL)

L1isthelibrationpointlocateddirectlybetweentheEarthandSunandisthemost

intuitively understood of the points, mostly because it is the only libration point that is

presentinanon‐rotatingsystem.Theother4pointsariseasaresultoftherotationofthe

system.WSBtrajectoriesprimarilyutilizethefirsttwolibrationpointsofeachsystem,and

thuswillbethefocusofourattentionlaterinthisdocument.

TherearedistinctadvantagesanddisadvantagestoWeak‐stabilityBoundary(WSB)

maneuvers.Themostnotableadvantageistheirextremefuelefficiency.Thesetrajectories

arisefromthecomplexitiesofthegravitationaldynamicsaroundlibrationpointsandthey

14areprimarilytheresultofgravity.Therefore,themajorityoftheworkdonerequireslittleto

no fuel. All that is required of the spacecraft is a "nudge" or small impulse in the right

direction at the right time. Themagnitude of this impulse ismuch less thanwhatwould

typically be needed to produce such a long distance gain. The drawback to taking these

pathways is that, because they rely on gravity, they can be relatively slow. For example,

takingacloserlookattheGenesisMissiontrajectoryabove,thedirectflightfromEarthto

Earth‐L1 took less than three months, whereas the return flight took over 5 months.

However,thereturnflightusedalmostnofuelandwaspoweredbyaverycalculatedand

precise"fall"backtowardstheEarth.

Interestingly,aspacecraftcanalso"fall"awayfromtheEarth,ifittakesadvantageof

thedynamicsbetweentheEarth‐MoonLibrationpointsandtheEarth‐Sunlibrationpoints.

Asitturnsout,theenergypotentialdifferencebetweenLunar‐L1andEarth‐L1isonlyabout

50 m/s ΔV. In other words, if a spacecraft is in orbit around Lunar‐L1, it can reach the

energypotentialofanorbitatEarth‐L1,providedthat itcangenerateat least50m/sΔV.

Thisisnearlynegligible,consideringthatittakesontheorderofkilometerspersecondΔV

toreachLunar‐L1inthefirstplace.Thisfortunatecoincidenceofnatureisthereasonthat

Low‐energytransfersoutofEarth'sneighborhoodareaviableoption.(Belbruno2004)

One of the primary WSB maneuvers commonly utilized is what is called a

“heteroclinic connection”.This is a low‐energy transferbetween theL1andL2pointsof a

system. It typically links two period orbits about the two Lagrange points. (Koon 1999)

Figure7belowdepictsaheteroclinicconnectionbetweenJupiter’sLagrangepoints.

15

Figure7.HeteroclinicConnectionbetweenJupiter'sL1andL2points.(Source:Koonetal1999).

Althoughtheseexotictrajectorieshaveonlyrecentlybeeninvestigated,ahandfulof

missions have already applied thismethodwith great success. Perhaps one of themore

notablemissions thatusedaWSBtrajectorywas theNASAGenesismission.Thismission

wasanunmannedroboticsamplecollectionandreturnmission.Itsobjectivewastocollect

solarwind samples from a halo orbit aroundEarth‐L1. The experiment also required the

spacecraft to return the samples back to Earth, where it was to be intercepted by

helicopters.However,duetoamiscalculation,thespacecraftmadeacrashlandinginUtah,

ratherthanasoftlandingbyhelicopter.Fortunately,thesampleswerenotdestroyedupon

impactandthemissionwasconsideredasuccess.(Koonetal1999)

However, the trajectory of the Genesis mission is what caught the attention of

astrophysicists.Figure8andFigure9,below,showtheunconventionalandnonlinearpath

thatwastakentoreachthemission’sfinaldestinationatEarth‐L1.Itshouldbenotedthat

the shape of the Genesis Mission trajectory closely resembles that of the heteroclinic

connectionshownabove.ThetwotrajectoriesareoverlaidinFigure10,below.

16

Figure8.3‐dimensionalviewoftheGenesisMissiontrajectory.(Source:NASAJPL)

Figure9.2‐dimensionalviewoftheGenesisMissiontrajectory.(Source:NASAJPL)

Thismission performed its experiment at Earth‐L1 and used aWSBmaneuver to

return to the Earth for very little fuel cost. Figure 9 is an illustration of the Genesis

Spacecraft'spaththroughspaceasseenintheX‐Yplane.

17

Figure10.TheGenesisMissionTrajectoryoverlaidontoaheteroclinicconnection.(Source:Koonetal

1999)

LikemanyWSB trajectories, the Genesis Spacecraft passed through one or more

libration points. These points are the "gateways", so‐to‐speak, that link the low‐energy

pathwaysthroughspace.Thatis,inordertoaccessthesepathways,aspacecraftmustpass

throughornearoneofthelibrationpoints.(Ross2006)Thisalsoaddstothecomplexityof

computingaWSBtrajectorybecauselibrationpointsare3‐dimensionalinnature.Whatthis

means is that the dynamics around a libration point cannot realistically or practically be

simplified for a planar‐restricted flight path, as many other trajectories can. Likewise, a

WSB simulatormust be able to process the increased computation load associated with

increasing the degree of freedom of a 3‐dimensional model over a more typical 2‐

dimensionalmodel.

Aslongasamissioncanaccommodatealongertimeofflight,itcantakeadvantage

of the significantly reduced fuel costs of a WSB trajectory. This is ideal for unmanned

roboticexplorationmissions,suchasGenesis.Onecommonlyproposedmissionforwhich

18thistransitmethodologywouldbeidealistheexplorationoftheJovianplanetarysystem.As

mentioned, WSB trajectories link libration points throughout the Solar System. This

includesthelunarlibrationpointsofJupiterandSaturn.Withthepropersimulationtool,a

flight path linking the libration points of the Jovian moon system could potentially be

realized.

TheproblemwithutilizingaWSBapproach is thatcomputingthesetrajectories is

verydifficult todoanalytically.Withmoresophisticatedcomputingdevicesateven lower

costs, numericalmethods havemade tremendous improvements in their algorithms and

capabilities.Methods,suchastheWSBapproach,thatpreviouslywereunsolvablecannow

beconsideredusingnumericalcomputation.Orbitalpropagationofmultiplebodysystems

isonesuchproblemthatcanbetackledwithcontemporarynumericalcomputing.Personal

computerprogramssuchasMATLABcan,andhavebeen,usedtowriteorbitalpropagators

formanydifferentapplications.

3. MOTIVATION

For hundreds of years, orbit determination and techniques for trajectory design

have been refined and analyzed by mathematicians and physicists. While the study of

analytic solutions to the more complex orbit problems require a deep mathematical

understanding of the physics of multiple‐body systems, the computation of these

trajectoriescanbedonewithaverybasicgraspofthephysicsofgravity.Greatprogresshas

beenmadeonbothfronts.Byutilizingnumericalmethodstocomputecomplextrajectories,

insightscanbemadeaboutthenatureofthesetypesoftrajectorieswithouthavingtodelve

intothecomplicatedmathematics.

Many orbital propagators use a patched‐conic approachwith planar‐restricted 3‐

bodyconstraintstofindinterplanetarytrajectories.Whilethismethodallowsforthedesign

19ofasimpletrajectory,itgreatlylimitsthetypesofpathscreated.Inordertoallowformore

fuelefficientoptions,anorbitalpropagatorwiththecapabilitytogenerateWSBtransfersis

needed.Currently,therearealimitednumberofpropagatorswiththiscapability.

Whilethepatched‐conicmethodworkstodesignatrajectory,itoftenprecludesthe

possibilityof findingamoreefficient path that takingaWSBapproachmayproduce.The

motivationforcreatinganorbitaldesigntoolthatallowsforaWSBmethodisto findand

utilizethesemoreefficientroutesthroughspace.Thegoalofthisdocumentistodetailthe

concepts,approach,implementation,andoutcomesofdesigningthe“Planetary&Asteroidal

Trajectories&HeteroclinicOrbitsSimulator”(PATHOS)program,anorbitalpropagatorthat

canproduceWSBtrajectoriesforinterplanetarymissions.

4. STATEMENTOFWORK

The objective of this work is to design and implement the PATHOS program, an

orbital propagator that uses a Planar Circular‐Restricted ThreeBody model to compute

trajectories that are not restricted to Keplerian trajectories. PATHOSwill allow formore

flexibility inmissiondesign andplanning. To validate thework, PATHOSwill construct a

handful of well‐known orbits as well as a heteroclinic connection between L1 and L2. In

addition,acomparisonofPATHOStootherorbitalsimulatorsofitsclasswillbediscussed.

5. METHODOFSOLUTION

(i) Full‐forceModelTrajectories(PR3BP)

Orbitaltrajectoriesintheregimeofcis‐lunarspaceandbeyondarepredominantly

governedbythegravitationofmultiplemassivebodies.TheeffectoftheEarth’s(andother

planets’)atmosphereisnegligibleonceaspacecrafttravelsintocis‐lunarandinterplanetary

space.(Koonetal2001)

20

As previously discussed, traditional orbital calculations attempt to make

simplificationsbyconstrainingtheorbitingbodiestoatwo‐dimensionalplane.Thisiscalled

thePlanar‐RestrictedThreeBodyProblem(PR3BP). In thisproblem, the trajectoryof the

negligiblemass(i.e.‐thespacecraftorsatellite)canonlyliewithintheplanethatholdsthe

two largerbodies.However,WSB trajectories takeadvantageof gravitational equilibrium

points, calledLagrangepoints, and generally containportionsof halo orbits or Lyapunov

orbits,whichhavethreedimensionalcomponents.(McCaine2004)Therefore,thePR3BRP

will not be sufficient. To get an accurate trajectory estimate, the most basic governing

equations will be used, namely Eq. 1, above. By basing the trajectory plot solely on the

fundamental force equations,we can let the physics propagate and observe the resulting

trajectories.Thefigurebelowillustratestheconceptbehindthismethod.

Figure11.IllustrationoftheDeadReckoningmethodappliedtotrajectorycomputation.

21

Givenasetofinitialpositionsandvelocities,theideaistouseonlytheequationfor

gravitational force and numerical integration to solve for the next set of positions and

velocities.Thisisillustratedinthefigurebelow.

Figure12.IllustrationoftheDeadReckoningmethodappliedtodiscretenumericaltrajectory

computation.

This method of computing the trajectory is an example of "Dead Reckoning"

navigation. Itcanalsobe thoughtofas treating thesystemasadiscretedynamicalsystem.

Dead‐reckoningcomputationsarethosethatrelysolelyonpreviouslyknownorcalculated

positions to calculate the next position. Since the projected positions are based on

previouslycalculatedvalues,errorsinthepositionswilltendtopropagatethroughtherest

ofthecalculations.Themagnitudeofthesedeviationswilldependonthesizeofthetime‐

stepselectedforthecomputations.Thebasicstructureoftheprogramisshownbelow.

22%% I - Setup constants and initialize variables x = []; %Position vector, [km] v = []; %Velocity vector, [km/s] dt = 1; %Timestep, [s] %% II - Define initial conditions x(1,:) = [x y z]; v(1,:) = [vx vy vz]; %% III - Computation loop for (i = 1:maxiterations) Fmag(i) = -(G*Mea*Msa)/(norm(x(i,:)))^2; %Gravitational force a(i,:) = F(i)./Msa; %Gravitational acceleration dv(i) = a(i,:)*dt; %Change in velocity vector dx(i) = v(i,:)*dt; %Change in position v(i+1,:) = v(i,:) + dv(i); %New velocity x(i+1,:) = x(i,:) + dx(i); %New position end

This code functions to generate basic orbits and is based on the parameters provided in

SectionIofthescript.TheseparametersincludethemassesoftheEarthandspacecraft,as

wellas theuniversal gravitational constant,G.Setting the following initial conditionsand

runningthescript,weget:

% Initial conditions v(1,:) = [0 10 0]; %[km/s] x(1,:) = [6728 0 0]; %[km]

23

Figure13.Initialsimulationoutputplot.

Calculatingtheexpectedsemi‐majoraxisandcomparingitagainsttheplottedsemi‐

majoraxis,thepercentdeviationbetweenthetwoislessthan0.1%.Ofcourse,thisisover

thecourseofonlyafeworbits.BecausetheerrorsinDead‐reckoningalgorithmspropagate

through the remaining data points, it is especially important to minimize any errors

wheneverpossible.

Inadditiontoconcernswitherrorsintheoutputtrajectory,thePATHOScode,asit

isshownabove, isverycomputationallyslowandinefficient.Sincetheprimaryconcernis

theoveralltrajectoryofaspacecraft,thecodedoesnotnecessarilyneedtocomputeevery

singlepoint,especiallywithatimestepofafractionofasecond.Inanefforttoreducethe

amount of raw data passed into the final trajectory plot, a nested‐loop structure was

adoptedforthecode,wheretheinnerloopofthecodewouldcomputethefinerresolution

datapointsandtheouterloopwouldplotthefirstandlastpointsfromtheinnerloop.The

resultswere trajectories thatwere "dotted lines".This reduces theamountofpoints that

wereplotted,butretainedthedesiredinformation.

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1

x 104

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

4

24

Figure14.PlotofaLEOcircularorbitfordt=0.01sec.

The above plot is a verification of the functionality of the nested‐loop structured

code.Itusesatimestepofdt=0.01seconds.ItcanbeseenthatthePATHOSalgorithmisvery

reliantonthesizeofthetimestepused.Forexample,ifweincreasethetimesteptodt=0.1

seconds,wegetthefollowingplot.

Figure15.Divergenceoftheexpectedtrajectoryfordt=0.1sec.

25

Because this algorithm is employs a “dead reckoning” approach, in other words,

each data point relies solely on the previous data point, any error in the trajectory

calculation will propagate throughout the following data points. However, if we further

refine the size of the step, to dt = 0.001 seconds, for example, we get a much cleaner

trajectoryplot,shownbelow.

Figure16.Morerefinedtrajectoryplotwithdt=0.001sec.

Whilethisfigureisamuchclearertrajectoryandmuchcloserthanwhatisexpected

(astrictlycircularorbit),therearestillminuteerrorsthatwillpropagateasthetrajectory

extendsintospace.Wecanapproximatelydeterminethemagnitudeoferrorsexpectedby

runningthecodeatseveraltimestepsandcalculatingthedeviationfromtheexpectedsemi‐

major axis. Figure 17, below, shows a graph of the relationship between the error

percentages of the algorithm’s outputted semi‐major axis versus what is theoretically

expectedandthetimestepusedinthealgorithm.

26

Figure17.Semi‐majorAxiserrorplotasafunctionoftimestep.

We can see that the trend‐line is roughly parabolic. That is, the error percentage

increasesat a rateof thesquareof the timestep.Therefore, tominimizeerrors, the time

stepwillneededtobekepttoaminimum.However,thedrawbacktominimizingthetime

step is that the codewill takemuch longer to execute. For the sake of accuracy, a lower

timestepwasprioritizedoveralowerruntime.Thus,thetimestepchosenfortherestofthe

simulationwas 0.001 .Thismeanthavingaroughly1.393 10 %errorperloop.

(ii) Full‐forceModelTrajectories(PCR3BP)

ThegoalofthePATHOSsimulationisultimatelytosimulateaWSBtransfertoother

planets or NEAs. In order to accomplish this, a full‐force three‐dimensional model

environment was created. Previously, a two‐dimensional environment was used to

illustrate the proof of concept of a simulation that is based solely on gravitational

interactions. Inotherwords, ifweonlysimulate the forcesofgravitationalattraction,can

proper and accurate trajectories arise? The answer turned out to be yes, towithin some

certainty(whichdependedonthetimestepused).

y = 242.62278x2.08036

0

1

2

3

4

5

6

7

8

9

0 0.05 0.1 0.15 0.2

Error Percentage

 [%]

Time Step [s]

27

Inreality,Weak‐StabilityBoundarytransfersarenotplanar‐restricted.Infact,they

aretypicallythree‐dimensional.Assuch,atwo‐dimensionalmodelwillnotsuffice.However,

the proof of concept still stands; by simulating the interactions betweenmassive bodies

resultingfromgravity,feasibletrajectoriescanbegenerated.Usingthesameprocessasthe

two‐dimensionalsimulation,but increasing thedegreeof freedomto three,amoreuseful

simulationenvironmentwascreated.Thus,three‐dimensionaltrajectories,suchastheone

showninFigure18,below,weregenerated.Itshouldbenotedthatwhilethetrajectoryis

nowallowedintothethirddimension,thetwoprimarygravitationalbodiesarestillplanar‐

restricted.

Figure18.3‐dimensionaltesttrajectory.

However,beforeweusethethree‐dimensionalmodel,aquickverificationwasdone

usingawell‐understoodthree‐dimensionalorbit, theMolniyaOrbit.Asetofknowninitial

conditionswasgiventothescriptandthesimulationwasallowedtopropagateonitsown.

If the simulation environment is sufficiently accurate, the resulting orbitwould have the

expected characteristics of aMolniya orbit. Those characteristics are shown in Figure 19

below.

28

Figure19.ParametersforatypicalMolniyaOrbit.

Torecreatethis,asetofinitialconditionswerecalculatedbasedonbasicorbitalmechanics

andtheconceptofconservationofmomentum.Theinitialconditionswere:

% Initial conditions x(1,:) = [-3303.57 0 -6597.1]; %[km] v(1,:) = [0 -9.6457 0]; %[km/s]

TheresultingtrajectoryisshowninFigure20below.

Figure20.SimulationrecreatedthesameMolniyaorbitshownpreviously.

OrbitalPeriod 12hoursSemi‐majorAxis 26,500kmEccentricity 0.72Inclination 63.4°ArgumentofPerigee ‐90°

29

Bylocatingtheapogeeonthefigureandcalculatingtheresultingpropertiesofthe

trajectory,we findadeviationof roughly~100km from theexpectedvalue.Compared to

the semi‐major axis of 26,500km, this is approximately 0.189% in error.The error likely

arisesfromthedeadreckoningnatureofthealgorithm.

(iii) AnomaliesintheFull‐ForceModel

Upuntil this point, themodel used in thePATHOSprogramhadonlybeen tested

usingNear‐Earthtrajectories.NoneofthetrajectoriesextendedeventowithintheMoon’s

vicinity.ThoughtheresultsfororbitsandtrajectoriesneartheEarthhavebeenfavorable,

attempts to plot trajectories to cis‐lunar or interplanetary space had been unsuccessful.

Many trajectories produced with this model did not agree with established models and

expected results, and some even diverged, going off into infinite velocities. By plotting a

forcecontourofthesimulationenvironment,itwasevidentthattherewasacomponentof

the forcecalculation thatwasabsent.Thesimulationenvironment’s librationpointswere

not located in the correct positions. Instead of lying roughly ±1.5million km fromEarth,

whichisanestablishedvalue,theyweremuchcloser(roughly±260,000km),whichwasan

extremely large error that could not possibly be explained by errors arising from Dead

Reckoning.

Bylookingatthisanomalyandseeingthatthelibrationpointsarenearlyanorderof

magnitudeclosertotheEarththantheyshouldbe,wecaninferthatalargecomponentof

the force fromthemodel isunaccounted for. Judging fromthenatureof theanomaly, the

missingforcecanbepostulatedtoexistasaforcethatactsradiallyoutwardfromtheSun.

Fromthis,itisevidentthatthemissingforceisonethatisassociatedwiththefactthatin

reality,theEarth‐Sunsystemisarotatingsystem.Inthesimulationenvironmentthusfar,

theassumptionofastationaryEarthinaninertialframehasnothadalargeimpactonthe

30trajectories computed, and thus it was gone unnoticed. However, when dealing with a

heliocentrictrajectory,suchasonethatventurestothelibrationpoints,theexistenceofa

rotatingframecannolongerbeignored.

Inarotatingframe,basicNewtonianmechanicsarenolongersufficienttodescribe

the dynamics of a spacecraft in transit. However, for the sake of aesthetics, it would be

beneficialtoviewthedynamicsfromafixed‐framepointofview.Toreconcilethis,theforce

calculations must account for the forces that arise from the dynamics of the Earth and

spacecraftaroundtheSun.Theseextracomponentscomeintheformofthe:

i. Centrifugalforce

ii. Coriolisforce

iii. Eulerforce

The centrifugal force is the largest component of the list above. It acts in the direction

oppositetothegravitationalforcefromtheSunandhasthefollowingmathematicalform.

Where istheangularvelocityoftherotatingsystem.

All three fictitious forces, listed above, are consequences of the effects of inertia

arising from the rotation of the system. They are labeled “fictitious” because there is no

physicalexternalcauseoftheforce,likewithtypicalnormalforcesorgravity.Anobjectwill

feeltheforcewhileundertheeffectsofrotation,buttherewillbenophysical“cause”.The

centrifugal force ismost intuitivelyunderstoodof the3 fictitious forces. Itseffectscanbe

seeninnearlyanyrotatingsystem,suchastheshapeofthewaterinaspinningbucketorin

theoblatenessofarotatingsphere.

31

TheCoriolis force is thenext largest contributing factor.However, its effects are

muchlessnoticeablethanthoseofthecentrifugalforce.Itaccountsfortheapparentmotion

ofanobjectintherotatingframeduetotheeffectsofrotation.

Itsmathematicalformisshownbelow.

2

The coriolis force is typically only taken into consideration over very large systems. For

example, their effects become significant in meteorology, as the coriolis force is directly

responsible forcertainwindpatternsandoceancurrentson thesurfaceof theEarth.For

orbital mechanics, it is a small factor, which may be negligible in most cases. For our

purposes,itwillbeincludedbecausethereisverylittlecomputationalcostassociatedwith

incorporatingitintotheforcecalculation.

32ThethirdfictitiousforceistheEulerforce.Itsmathematicalformisshownbelow.

Ascanbeseenfromthemathematicalformulation,theEulerforceonlyariseswhen

there is a non‐uniform rotation present, i.e.‐ 0. In our simulation environment, we

assume a constant, fixed rotation consistentwith the revolution of the Earth around the

Sun.Thus,theEulerforceisneglected.

These fictitious forces are correction factors that allow for the use of Newtonian

dynamicsforrotatingsystems,asviewedfromaninertialreferenceframe.Thatis,wecan

seethedynamicswithrespecttotheEarth,whilekeepingtheEarthfixed.Thisispreferred,

astheEarthisoneoftheprimarypointsofinterest.

Incorporating the fictitious forces into the force calculation, a corrected force

contourplotwasgenerated.Thesecontourplotswillaidintheidentificationofthelocation

of the simulation environment’s libration points. Figure 21 below is an updated contour

plot.BecausetheplotwasgeneratedspecificallyfortheangularvelocityoftheEarth,there

isadipintheplotwherethegravitationalattractionoftheSunandthecentrifugalforceof

therotationbalanceout.Thiswilloccuralongtheorbitof theEarth,asshownbythered

regionofthecontour.

33

Figure21.ForcecontourplotofsimulationenvironmentinthevicinityofEarth.

Thecontoursweretruncatedto limittheplottotheregionnearEarth’sorbit.The

above plot is to scale. The distance from the Earth to its libration points is roughly two

ordersofmagnitudelessthanthedistancefromtheEarthtotheSun.Asaresult,theimage

aboveappears tohaveno indicationsof thepresenceofLagrangepoints.However, ifwe

takea closer lookat the region immediately aroundEarth,we can locate evidenceof the

libration points. Since these are force plots, the existence of libration points will be

manifestedasvalleysintheplot.InFigure22,theforceplotwasinvertedtohighlightthe

locationoftheLagrangepoints,manifestingaspeaksinsteadofvalleys.

34

Figure22.Enhancedviewsoftheforcecontourplotofthesimulationenvironment.Thedipsandpeaks

shownaboveindicatetheequilibriumlocations(i.e.‐Lagrangepoints).

TheblackcircleinthecenteroftheplotistheMoon’sorbit,drawnasareference.

TherearetwopeaksoneithersideoftheEarththatsignifytheexistenceofagravitational

“well”. That is, those are the locations where the gravitational forces balance, i.e.‐ the

librationpoints.ComputingthedistancefromthesepointstotheEarth,wecanseethatthey

are indeed located roughly 1.5million km away, which is expected. PATHOS now has a

suitablesimulationenvironment.

6. RESULTS&DISCUSSION

The main goal of this work was to demonstrate and generate Weak‐Stability

Boundary trajectories using basic physical equations and a numerical computation

simulation. Therefore, once the PATHOS simulation environment is deemed suitable, the

main verification would be to create a WBS trajectory. One of the main uses of a WBS

trajectoryiswhatiscalleda“heteroclinicconnection”,discussedpreviously.Thesearelow‐

35energytransfersthatconnectperiodicorbitsinLagrangepoints.Typically,theyarepartof

periodic halo orbits. Thus, initial conditionswere used that placed a spacecraft at a halo

orbit.Givenonlytheinitialvelocity,theresultingdynamicswereallowedtopropagate.Asin

nature, the complex interactions of the Lagrange points are all that is needed to guide a

spacecraft through these transfers. Figure 23 below is a trajectory plot for the following

initialconditions:

% Initial conditions x(1,:) = [1.48050000e8 -250000 0]; %[km] v(1,:) = [-0.115 0 0]; %[km/s]

Figure23.PATHOSprogramoutputplotofaWeak‐StabilitytrajectoryfromEarth‐SunL1vicinitybackto

Earthretrieval.

The significance of this trajectory is that the amount of fuel required to get from

Earth‐SunL1backtoEarthreturnisminimal.Inotherwords,aftertheinitialimpulsethrust

fromtheL1haloorbit,thedynamicsareallowedtopropagateontheirown,withnearlyno

36fuel expenditures from the spacecraft itself (excluding mission‐specific attitude control

maneuvers).Thisdemonstratesthebiggestadvantageofincorporatingthisclassoforbital

maneuvers into a mission profile. The sheer distance traveled, essentially for free, is

definitely a beneficial option to consider. The main disadvantage, however, as stated

previously in the document, is themuch extended time of flight. For example, the above

trajectory,fromL1halotoEarth‐boundreturn,takesapproximately382days,orabout12.5

months. By contrast, a direct flight from Earth‐Sun L1 to Earth return could typically be

completedinabout2.5months.

To reiterate, since the PATHOS program treats the environment as a chaotic

dynamical system, varying the initial conditions even slightly, can wildly change the

trajectoriesthatareproduced.TheplotinFigure23,above,isatransferfromEarth‐SunL1

backtotheEarth,butiftheinitialconditionsarechangedto:

% Initial conditions x(1,:) = [1.47900000e8 0 70000]; %[km] v(1,:) = [0 0.35 0]; %[km/s]

wegetthetrajectoryshownbelowinFigure24andFigure25.Notethatthemagnitudeof

theinitialvelocityisonlydifferingbyabout200m/s,whichcanreadilybeproducedbya

multitudeofexistingpropulsionsystems.

RecallthatthedynamicsoftheregionsaroundLagrangepointsarechaotic.Thatis,

evenminisculevariationsintheinitialconditionscanproducewildlydifferentoutcomes.As

seen fromthedifferencebetweenFigure23andFigure24,roughly200m/s isenoughto

changeanEarth‐boundreturn trajectory into theheteroclinic connectionbetweenL1 and

L2. If the trajectory isallowedtocontinue, itswingsbyEarthonceagain, terminating ina

heliocentrictrajectoryaroundtheSun.Tofullyunderstandwhythishappens,studiesofthe

37invariant manifolds that underlie these complex dynamics have been done by

mathematiciansandphysicists.However,thiswasnotwithinthescopeofthecurrentwork.

Figure24.APATHOS‐generatedheteroclinicconnectionbetweentheEarth‐SunL1andL2points.

Figure25.IsometricviewofthePATHOSheteroclinicconnectionbetweenL1andL2.

38

The notable point drawn from the trajectory plots above is that the PATHOS

simulationiscapableofpropagatingWSBflightpaths.Figure25showsaflightpaththathas

the characteristics of a WSB trajectory. Although further quantitative analysis and

validation will need to be performed, a brief qualitative assessment concludes that the

PATHOStoolcanaccommodateWSBtrajectoriesinitscurrentform.

Onecaveat,however,isthattheabovetrajectorieswerenotproducedwithprecise

initial conditions. Typically, halo orbits are designed using optimal control algorithms to

find the precise positions and velocities needed to maintain a stable orbit around the

Lagrangepoints.However,thatwasnotwithinthescopeofthisproject.Thus,ahandfulof

educated estimates were used to generate these trajectories. Careful refinements to the

initialconditionsusedwouldproducemorerobusttrajectories.

Ithasbeenshownthatthecomplex,nonlinearWeak‐StabilityBoundarytrajectories,

used by the Genesis mission and studied by astrophysicists, can be generated by the

PATHOS simulation. The tool in its current form can be used to plot potential routes to

interplanetarydestinationsasabaselineandprecursortoacompletetrajectoryanalysis.

7. COMPARISONOFPATHOSTOSIMILARPROPAGATORSOFITSCLASS

PATHOSisaMATLAB‐basedorbitalpropagationtooldesignedwithinterplanetary

missions in mind. A search of the MathWorks MATLAB Central File Exchange reveals

severalsimilarprogramsdesignedbyindividualsfromotheruniversitiesandtheaerospace

industryforsimilarpurposes.Severaloftheseprogramswereassessedtogaugethelevelof

functionalitythatiscurrentlyavailableintheareaoforbitalpropagation.

ThefirstisaprogramwrittenbyastudentatthePolitecnicodiMilano,auniversity

inItaly.Figure26,below,showsasampleplotoftheENVISATsatelliteflightpath.Thetool

39iscalledtheSAT_Orbitsimulator.ItusesaSGP4perturbationmodelinitssimulation.While

thisisanaccuratemodelforNear‐Earthorbits,asitaccountsfortheoblatenessoftheEarth

as well as atmospheric effects, its application to interplanetary missions is greatly

diminished.Theprogram,initscurrentform,doesnotaccommodatenonlineartrajectories,

as it uses polynomial curve fitting to extrapolate its orbit, rather than allowing it to

propagateonitsown.However,theprogramisveryaccurateforNearEarthorbits.Italso

displaysamultitudeofextrainformationattherequestofitsuser,includingerrormodels

andtheprogressionoftheorbitasafunctionoftime.

Figure26.Sampleorbitgeneratedby"Sat_Orbit",ofPolitecnicodiMilano

The next orbital simulator is called ASTROTIK.While it providesmany tools and

manyoptions,itusesaRestrictedTwo‐bodymodel.Theorbitalmaneuversitproducesare

generallyKeplerian,as itusesaKepleriancurve‐fittingalgorithmto findorbital transfers.

Figure 27, below, displays a trajectory calculated between a sample orbit A and a highly

inclined second orbit B. This tool, while great for generating Hohmann transfers to

interplanetarydestinations,seemstoexcludeWSBorLow‐Energytransfers.

40

Figure27.SampletransfergeneratedbyASTROTIKsimulator.

Another simulator, called the “Interplanetary Mission Planner” is written by a

studentattheUniversityofTechnologyofSydney,Australia.Itgeneratesaninterplanetary

trajectory through a series of gravitational swingbys. Figure 28, below, shows a planned

trajectory to one of the outer planetswith two gravitational swingbys. Although the tool

robustly creates these trajectories, it uses a Hohmann transfer for each segment of its

trajectory.Indoingso,itprecludesthepossibilityoflow‐energytransfers.Inthesamesense

as the previously discussed orbital propagators, the Interplanetary Mission Planner is a

goodtoolfortraditionaltrajectorydesign,butexcludesnonlinearorbitalflightpaths.

Figure28.Sampletrajectorygeneratedbythe"InterplanetaryMissionPlanner".

41 Given the orbital propagators available for interplanetary mission planning, the

PATHOS program has the distinct advantage of including Weak Stability Boundary

trajectories as possible outcomes. (Recall that WSB trajectories provide extremely fuel

efficientflightpathstointerplanetarydestinations.)Inaddition,PATHOShasdemonstrated

its success in generating comparable Keplerian trajectories to its competitors, yet still

retaining the ability to consider nonlinear flight paths. While still in the early stages of

development,thePATHOSprogramhasgreatpotentialtoovertakeitscompetitorsinmany

aspects.

8. CONCLUSION

The stated goal of this work was to develop a simulation tool to facilitate the

trajectory analysis of interplanetary missions. The “Planetary&AsteroidalTrajectories&

Heteroclinic Orbits Simulator” (PATHOS) program has effectively demonstrated the

generation of all types of orbits, ranging from traditional Keplerian orbits to the more

complex,nonlinearWeak‐StabilityBoundarytrajectories,includinganexampleheteroclinic

connection between Earth‐Sun L1 and L2. A comparison to currently available MATLAB‐

basedorbitalsimulationtoolshasconcludedthattheinclusionofnonlinear,Weak‐Stability

Boundary trajectories has not been a priority or focus for many tools. The distinct

advantageofthePATHOStoolisinitsabilitytoincludeeveryorbitalmaneuverclassasan

optionwhengeneratinginterplanetarymissiontrajectories.

PATHOSusesaPCR3BPmodelwith threerotationalcorrection factors inorder to

maintain the dynamics of a fixed‐view rotating frame. The simulation environment was

validatedbyobservingthelocationofthesimulatedLangrangepoints.Thesepointsarose

fromthemathematicsofthesimulation,ratherthanbymanualplacement.Thus,itservesas

confirmation for the simulation’s equilibrium points to match those observed in nature

42without manual intervention. By recreating a known low‐energy transfer between two

libration points, known as a heteroclinic connection, PATHOS demonstrated its ability to

generatenonlinear trajectoriesakin to thosestudiedbyWeak‐StabilityBoundaryTheory.

The fidelityofnear‐EarthorbitsgeneratedbyPATHOShasbeenvalidatedwith two‐body

calculations;however,thenonlineartrajectoriesgeneratedhaveyettobeanalyzedinthis

sense.

For interplanetary mission design,Weak‐Stability Boundary trajectories have the

tremendous advantage of a much more fuel efficient pathway than traditional orbital

transfers,suchasaHohmannTransfer.Forexample,utilizingaWSBmaneuver,aspacecraft

inahaloorbitaroundLunar‐L1canreachEarth‐L1,locatedroughly1.5millionkmaway,for

~50 m/s ΔV! (Ross 2006) By including this class of orbital trajectories, potential

interplanetarymissionscanbenefitfromthistypeoffuel‐savingmaneuver.

ThePATHOStool,initscurrentform,canbeusedasabaselineforfuturetrajectory

analysis for interplanetarymissiondesign.Byopeningtrajectorydesigntothepossiblyof

WSB flight paths, this tool provides a huge advantage in fuel efficiency over similar

propagatorsofitsclass.

9. FUTUREWORK

WhiletheoriginalscopeofthisworkwasmeanttoincludeatestscenariotoaNear‐

EarthAsteroidorMars, thesimulation in itscurrentstatecouldnotbeadjustedforthose

interplanetarymissionswithoutsignificantalterations.Thus,itisrecommendedaspartof

anyfuturedevelopmentofthePATHOStooltoextenditsfunctionalitytoincludeMarsand

possibleNEAs.Inaddition,asimulationisonlyasgoodasitsusability.Accordingly,arobust

andintuitivegeneraluserinterface(GUI)shouldbeconsideredtoaccompanythistool.

43 Currently, the PATHOS tool uses a Dead‐Reckoning approach to compute its

trajectories. This method, while intuitive and reasonably accurate, is extremely

computationally inefficient. A full trajectory computation typically takes on the order of

severalhourstogenerate. Usingamoreadvancedcurve‐fittingalgorithmmayreducethe

runtime to a more practical level. This should be considered in any future work on the

project.

Inaddition,thesimulation,initscurrentform,doesnotcalculateanydelta‐Vvalues

fortrajectories.Thesevaluescan,however,becalculatedmanuallyfromthedatapointsin

theoutput trajectory.Because thedelta‐Vbudget isavery importantdesign factor in the

analysisofpotentialtrajectories,anoptiontodisplayandaddorsubtractdelta‐Vshouldbe

includedinfutureversionsofPATHOS.

Finally, since the main advantage of PATHOS over other similar MATLAB‐based

simulations is its ability to generateWSB trajectories, the simulation should be verified

using theGenesismission trajectory. The exact data points for theGenesismissionwere

unavailable for the timeframe of this project; however, a reproduction of the Genesis

missionflightpathwouldvalidatetheaccuracyofthistoolforWSBdesign.

44

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