Design of trajectory optimization approach for space maneuver vehicle skip entry problems runqi chai

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Runqi Chai

Al Savvaris

Antonios Tsourdos

Senchun Chai

Design of Trajectory Optimization Approach for Space Maneuver

Vehicle

Skip Entry Problems

SpringerAerospaceTechnology

The SpringerAerospaceTechnology seriesisdevotedtothetechnologyofaircraft andspacecraftincludingdesign,construction,controlandthescience.Thebooks presentthefundamentalsandapplicationsinall fieldsrelatedtoaerospace engineering.Thetopicsincludeaircraft,missiles,spacevehicles,aircraftengines, propulsionunitsandrelatedsubjects.

Moreinformationaboutthisseriesat http://www.springer.com/series/8613

RunqiChai • AlSavvaris • AntoniosTsourdos • SenchunChai

DesignofTrajectory

SkipEntryProblems

RunqiChai

CranfieldUniversity

Cranfield,Bedford,UK

AntoniosTsourdos

CranfieldUniversity

Cranfield,Bedford,UK

AlSavvaris

CranfieldUniversity

Cranfield,Bedford,UK

SenchunChai SchoolofAutomation

BeijingInstituteofTechnology Beijing,China

ISSN1869-1730ISSN1869-1749(electronic)

SpringerAerospaceTechnology

ISBN978-981-13-9844-5ISBN978-981-13-9845-2(eBook)

https://doi.org/10.1007/978-981-13-9845-2

© SpringerNatureSingaporePteLtd.2020

Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart ofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped.

Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthis publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.

Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthis bookarebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernorthe authorsortheeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontained hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.

ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSingaporePteLtd. Theregisteredcompanyaddressis:152BeachRoad,#21-01/04GatewayEast,Singapore189721, Singapore

Preface

Spacevehicletrajectoryplanninghasbecomeincreasinglyimportantduetoits extensiveapplicationsinindustryandmilitary fields.Awell-designedtrajectoryis usuallyakeyforstable flightandforimprovedguidanceandcontrolofthevehicle. Hence,theaimofthisresearchisusuallytoworkontrajectoryoptimization,and thenimproveononeoftheexistingtrajectoryoptimizationmethodsinorderto circumventthelimitationsbroughtbytheclassictechniques.

Thisbookpresentsthedesignofoptimaltrajectoryforspacemaneuvervehicles usingoptimalcontrol-basedtechniques.Itstartsfromabroadintroductionand overviewtothreemainapproachestotrajectoryoptimization.Itthenfocusesonthe designofanovelhybridoptimizationstrategy,whichincorporatesaninitialguess generatorwithanimprovedgradient-basedinneroptimizer.Further,ithighlightsthe developmentofmulti-objectivespacecrafttrajectoryoptimizationproblems,witha particularfocusonmulti-objectivetranscriptionmethods,andmulti-objectiveevolutionaryalgorithms.Finally,thespacecraft flightscenariowithnoise-perturbed dynamicsandprobabilisticconstraintsisstudied.Newchance-constrainedoptimal controlframeworksaredesignedandvalidated.Thecomprehensiveandsystematic treatmentofpracticalissuesinspacecrafttrajectoryoptimizationisoneofthemajor featuresofthebook,whichisparticularlysuitedforreaderswhoareinterestedto learnpracticalsolutionsinspacecrafttrajectoryoptimization.Thebookcanalso benefitresearchers,engineers,andgraduatestudentsin fieldsofGNCsystems, engineeringoptimization,appliedoptimalcontroltheory,etc.

Theauthorshavecarefullyreviewedthecontentofthisbookbeforetheprinting stage.However,itdoesnotmeanthatthisbookiscompletelyfreefromany possibleerrors.Consequently,theauthorswouldbegratefultoreaderswhowillcall outattentiononmistakesastheymightdiscover.

May2019

Acknowledgements

TheauthorswouldliketoexpresstheirsincereappreciationstoProf.YuanqingXia, Prof.GuopingLiu,andProf.PengShifortheirconstructivecommentsandhelpful suggestionswithregardstothetheoreticalpartofthisbook.

The fi rstauthorwouldliketothankothercolleaguesfromcenterof cyber-physicssystems,CranfieldUniversity,forprovidingvaluablecomments. Withouttheirsupport,thewritingofthebookwouldnothavebeenasuccess.

Also,wewouldliketothankallthestaffintheautonomoussystemsresearch group,CranfieldUniversity,foreverythingtheyhavedonetomakethingseasierfor usthroughoutthepreparationofthiswork.

Finally,theauthorswouldliketothankCranfieldUniversity,Schoolof Aerospace,Transport,andManufacturing,andBeijingInstituteofTechnology, SchoolofAutomation,forgivingusthesupporttomaketheworkareality.

3.2SMVTrajectoryOptimizationFormulation

3.2.1DynamicModel

3.2.2SMVInitialandTerminalConstraints

3.2.4TrajectoryEventSequence

3.2.5Interior-PointConstraints

3.2.6SkipEntryPathConstraints

3.2.7ObjectiveFunctionsfortheSkipEntryProblem

3.2.8OverallFormulation

3.3DiscretizationoftheSMVSkipEntryProblem

3.4.1ParametersSpeci

3.4.2OptimalSkipHoppingResults

3.4.3AnalysisofDifferentSkipHoppingScenarios

3.4.4SensitivitywithRespecttoPathConstraint

3.4.5FindingSolutionfor

4.3.3DifferentialEvolution

4.3.4ParticleSwarmOptimization

4.4.1ProblemModi

4.5.1CharacteristicArcsoftheTrajectory

5HybridOptimizationMethodswithEnhancedConvergence Ability

5.1InitialGuessGenerator ................................

5.1.1ViolationLearningDifferentialEvolutionAlgorithm

5.2InnerOptimizationSolver ..............................

5.2.1AnImprovedGradient-BasedOptimizationStrategy

5.2.2MeshRe fi

5.2.3OverallStructure

5.3SolutionOptimalityVeri

5.3.1First-OrderNecessaryConditions

5.3.2TerminalTransversalityConditions

5.3.3HamiltonianFunctionCondition

5.3.4PropertiesoftheControlVariable

5.3.5Bellman’sPrinciple

5.4SimulationResultsforaTime-OptimalEntryScenario

5.4.1OptimalSolutions

5.4.2Veri

5.4.3ComparisonwithExistingEvolutionarySolvers

5.4.4DispersionModel

5.4.5ComparisonAgainstOtherOptimalControlSolvers

6.1.2Multi-objectiveOptimalControlProblems ............

6.2AnImprovedMulti-objectiveEvolutionaryAlgorithms

6.2.1ExtendedNSGA-IIAlgorithm

6.2.2SuperiorityofFeasibleSolutionMethod

6.2.3PenaltyFunctionBasedMethod

6.2.4Multi-objectiveConstraint-HandlingTechnique

6.2.5ComputationalComplexityAnalysis

6.3Multi-objectiveTranscriptionMethods

6.3.1FuzzyPhysicalProgramming

6.3.2InteractiveFuzzyPhysicalProgramming

6.3.3FuzzyGoalProgrammingMethod

6.4SimulationResults ...................................

6.4.1Multi-objectiveSMVTrajectoryPlanning

6.4.2ParetoFrontResultsObtainedUsingMOEAs

6.4.3AnalysisofRelationshipsBetweenDifferent Objectives

6.4.4PerformanceoftheIFPPMethod

6.4.5PerformanceoftheFGPMethod

6.4.6ComparisonwithMOEAs

6.5PotentialApplicationsofDifferentMulti-objectiveSolutions .....

6.5.1DesignofSpacecraft/SatelliteFormationControl Schemes .....................................

6.5.2DesignofIntegratedSpacecraftGuidanceandControl Systems .....................................

6.5.3Database-BasedOnlineGuidanceStrategy

7Real-TimeOptimalGuidanceandControlStrategiesforSpace ManeuverVehicles

7.1RelatedDevelopmentofGuidanceStrategies

7.2MPC-BasedOptimalGuidanceMethods

7.2.1OverallGuidanceFramework

7.2.2NonlinearModelPredictiveControl

7.2.3LinearModelPredictiveControl

7.3SimulationStudyfortheMPC-BasedGuidanceSchemes

7.3.1ReferenceTrajectoryGeneration

7.3.2OptimalTrackingSolutions

7.3.3ComparativeAnalysis ...........................

7.4AnIntegratedGuidanceandControlAlgorithm

7.56-DOFSMVEntryTrajectoryOptimization

7.5.1RotationalEquationsofMotion

7.5.2State/Control-RelatedConstraints

7.5.3ObjectiveandOptimizationModel

7.6Bi-levelTrajectoryandAttitudeControlMethod

7.6.1Offl ineTrajectoryEnsembleGeneration

7.6.2DNN-DrivenControlScheme

7.6.3OverallAlgorithmFramework

7.7NumericalResults

7.7.1Mission/Vehicle-DependentParameterSetup

7.7.2TrajectoryEnsembleGeneration

7.7.3DNN-BasedControlResults

7.7.4ComparativeCaseStudy

8StochasticTrajectoryOptimizationProblemswithChance Constraints

8.1.1Chance-ConstrainedSpacecraftTrajectory Optimization

8.1.2Chance-ConstrainedSpacecraftTrajectory Optimization:StochasticDynamics

8.2Chance-ConstrainedStochasticTrajectoryOptimization Methods

8.2.1InitialTransformationofChanceConstraints

8.2.2DiscretizedCCSOCPFormulation

8.2.3Chance-Constraint-HandlingStrategy

8.2.4DeterministicCCSOCPModel

8.3Chance-ConstrainedStochasticSpacecraftEntryTrajectory Planning:SystemModeling

8.3.1StochasticDynamicsandObjectiveFunction

8.3.2HardConstraintsandChanceConstraints

8.4SimulationStudiesandAnalysis

8.4.1ParameterSpeci

8.4.2PerformanceoftheChance-Constraint-Handling

8.4.3SensitivitywithRespecttoControlParameter c

8.4.4SensitivitywithRespecttoSampleSize

8.4.5OptimalTrajectoriesfortheStochasticEntryProblem

Acronyms

Acceptableprobabilityofoccurrence

a Angleofattack

r Bankangle

b Boundaryfunction

u Controlvariable

q0 Densityoftheatmosphereatsealevel

CD Dragcoefficient

D Dragforce

c Flightpathangle

l Gravitationalparameter

w Headingangle

Isp Impulse

/ Latitude

CL Liftcoefficient

h Longitude

m Mass

U Mayercost

E Numberofequalityconstraints

I Numberofinequalityconstraints

M Numberofobjectivefunctions

J Objectivefunction

g Pathfunction

L Processcost

r Radialdistance

Re RadiusoftheEarth

X Self-rotationrateoftheEarth

x Statevariable

T Thrustforce

t Time

n Uncertainparameter

V Velocity

w Weightedparameter

ABCArti ficialbeecolony

ACAntcolony

ADEMGTAdaptivedifferentialevolutionandmodi fiedgametheory

ASMAdaptivesurrogatemodel

CCChanceconstraint

CCOChance-constrainedoptimization

COVCalculusofvariation

DDPDifferentialdynamicprogramming

DEDifferentialevolution

DPDynamicprogramming

EMOEvolutionarymulti-objectiveoptimization

FGPFuzzygoalprogramming

FONCFirst-ordernecessarycondition

FPPFuzzyphysicalprogramming

FSGPFuzzysatisfactorygoalprogramming

GAGeneticalgorithm

GPGoalprogramming

gPCGeneralizedpolynomialchaos

HBVPHamiltonianboundary-valueproblem

IFPPInteractivefuzzyphysicalprogramming

IPInteriorpoint

IPSQPInterior-pointsequentialquadraticprogramming

LEOLowearthorbit

LPLinearprogramming

MCMCMarkovchainMontoCarlo

MOABCMulti-objectiveartificialbeecolony

MOEA/DMulti-objectiveevolutionaryalgorithmbasedondecomposition

MOPSOMulti-objectiveparticleswarmoptimization

MOTMulti-objectivetranscription

MOTOMulti-objectivetrajectoryoptimization

NLPNonlinearprogramming

NPGANichedparetogeneticalgorithm

NSGA-IINondominatedsortinggeneticalgorithm-II

PDFProbabilitydensityfunction

PPPhysicalprogramming

PPPIOPredator–preypigeon-inspiredoptimization

PSOParticleswarmoptimization

RORobustoptimization

SASimulateannealing

SDDPStochasticdifferentialdynamicprogramming

SDEStochasticdifferentialequation

SDPSemidefiniteprogramming

SMVSpacemaneuvervehicle

SOCPSecond-orderconeprogramming

SOPSingle-objectiveproblem

SPPSOStrengthparetoparticleswarmoptimization

SQFStochasticquadratureformula

SQPSequentialquadraticprogramming

TSTabusearch

VLDEViolationlearningdifferentialevolution

WSWeightedsum

Chapter1 Introduction

Abstract Formostatmosphericorexo-atmosphericspacecraftflightscenarios,a well-designedtrajectoryisusuallyakeyforstableflightandforimprovedguidance andcontrolofthespacevehicle.Amongtheseflightmissions,trajectoryoptimization forreentryvehiclesisusuallyrecognizedasachallengingproblemandithasbeen widelyresearchedforacoupleofdecades.Oneofthecurrentobjectivesisthe developmentofspacemaneuvervehiclesforadynamicmissionprofile.Thischapter brieflyoutlinesthebackgroundandcurrentdevelopmentofthereentrytrajectory designproblem.Typicalspacecraftskipentrymissionscenariosarealsointroduced. Followingthat,theoverallaimsandobjectivesofthebookwillbesummarized. Finally,thestructureoftheentirebook,togetherwithsomehighlightsofeachchapter, willbegiven.

1.1Background

Inthelastcoupleofdecades,numerousachievementsandmassiveeffortshavebeen witnessedinordertomovehumanbeingsintospace.Nowadays,aerospacescience andtechnologyhasbroughtvariouschangesinnotonlythemilitaryfieldbutalso scientificandengineeringapplications.Amongthem,thedevelopmentofspacecrafttechnologyhasattractedsignificantattention[1, 2].Sofar,severalgenerations ofspacecrafthavebeendesigned,manufactured,launched,andsuccessfullyimplementedindifferentmissionprofilessuchascommunications[3],interplanetarytravel [4],regionalreconnaissance[5],environmentalmonitoring[6],andsoon.Among them,thespacemaneuvervehicles(SMVs)willplayanincreasinglyimportantrole inthefutureexplorationofspace,sincetheiron-orbitmaneuverabilitycangreatly increasetheoperationalflexibilityandaremoredifficultasatargettobetrackedand intercepted.However,becauseofthelongdevelopmentcycle,highoperatingcost, andlimitedresources,itisusuallydesiredbyaerospaceengineersthatthespace vehiclecanfulfillthemissionwithsomeperformancemetricstobeoptimized,orin otherwords,inanoptimalornear-optimalway.Toachievethisgoal,apropertreatmentoftheflighttrajectoryforthespacevehicleisoftenrequired,andthisstimulates thedevelopmentoftrajectoryoptimizationtechniques.

©SpringerNatureSingaporePteLtd.2020

R.Chaietal., DesignofTrajectoryOptimizationApproachforSpace ManeuverVehicleSkipEntryProblems,SpringerAerospaceTechnology, https://doi.org/10.1007/978-981-13-9845-2_1

Trajectorydesignforspacevehicleshasbeeninvestigatedwidelybysome researchers.Thistypeofproblemisusuallytreatedasanoptimalcontrolproblem. Thecoreaimofthiskindofproblemistodetermineafeasiblepathortrajectory,fora givenvehicle,toachieveaprespecifiedtargetandoptimizeapredefinedperformance index.Duringthetrajectoryplanningphase,anumberofconstraintsshouldbetaken intoaccountinordertoachievethemission-dependentrequirementsandprotectthe structuralintegrity.

Duetothehighnonlinearcharacteristicsandstrictpathconstraintsoftheproblem, itisdifficulttocalculatetheoptimalsolutionusinganalyticalmethods.Therefore, numericalmethodsarewidelyappliedtocalculatetheoptimaltrajectories.Although extensiveresearchworkhasbeencarriedoutonthedesignofspacecrafttrajectories andmanynumericaloptimizationtechniquesweredeveloped,itisstillchallenging tofindageneralapproachwhichcanbeappliedtoproduceoptimalsolutionsfor differentmissionprofilesandfulfilldifferentmissionrequirements.

1.2MissionScenarios

Inthissection,abriefdescriptionofsomeongoingprojectsinthefieldofaerospace engineering,especiallyinthefieldofspacecraftreentry,isprovided.

1.2.1SkipReentryMission

Themissionscenarioinvestigatedintheskipreentryproblemfocusesontheatmosphericflight,targetingtheentryintotheatmospheredowntoapredetermined positionsetatthestartofthemission.Studiescanbefoundintheliteratureregarding theskipreentryofdeep-spacespacecraftwithhighspeedoverfirstcosmicvelocity. AgraphicalmissionillustrationcanbefoundinFig. 1.1.

Generalskipreentrycanbedividedintofivephases:initialroll,downcontrol, upcontrol,Kepler,andfinalentry.Themaindifficultyofanatmosphericentryisto dealwiththerapidchangeintheaerodynamicenvironment.Thealmostimmediate switchfromspaceflightdynamicstoatmosphericflightcontrolisachallengeto performandanalyze.

1.2.2RegionalReconnaissance

Themissionscenariooftheregionalreconnaissanceprojectinvestigatedinissimilartothegeneralskipentrymission.Itfocusesontheatmosphericskiphopping, targetingtheentryintotheatmospheredowntodifferentpredeterminedtargetpoints forobservationandgatheringofinformationofinaccessibleareas.Oncethesetarget

Fig.1.1 SMVreentrymissionprofile

pointsarereached,thespacecraftstartstheascentphase,exitingtheatmosphere andreturningbacktolowEarthorbit(LEO).Duringthemission,theSMVcan flyineithertheunpoweredexo-atmosphericflight,poweredexo-atmosphericflight, unpoweredatmosphericflight,orpoweredatmosphericflight.Theoverallmission profileisdesignedandillustratedinFig. 1.2

ItisworthnotingthatasshowninFig. 1.2,thedashedlinephasesmayrepeat severaltimes(e.g., n 1times).Thisisbecause,inthisproject,itisexpectedforthe vehicletohaveamultiple-hoptrajectoryinordertooverflydifferenttargetregions andcompletethereconnaissancemission.Therefore,basedonthemissiondefinition statedabove,thetypicalskipentryproblemcanalsobetreatedasasubproblemof theregionalreconnaissancemission.

1.3BookAimsandObjectives

Theprimaryaimofthisbookistopresentthelatestprogressthathasbeenachieved inthedevelopmentofspacecrafttrajectoryoptimizationtechniques.Specifically,the mainfocuswillbeontherecentlyproposedoptimizationmethodsthathavebeen utilizedinconstrainedtrajectoryoptimizationproblems,multi-objectivetrajectory

optimizationproblems,andstochastictrajectoryoptimizationproblems.Oneindividualobjectiveofthisbookistosummarizethemainadvantagesanddisadvantagesof applyingdifferentoptimizationmethodsinspacecrafttrajectoryoptimizationproblemsbasedontheresultsreportedinthenewlypublishedworks.Apartfromthat, wealsoputeffortsontheimprovementofthesenumericalmethodsinordertocircumventthelimitationsbroughtbytheclassictechniques.

Consequently,alltheobjectivesofthisbookcanbesummarizedasfollows:

1.Systematicallyintroducedifferentoptimizationapproachestospacecrafttrajectoryoptimizationproblems.

2.Proposeanenhancedtrajectoryoptimizationmethodinordertocircumventthe limitationsbroughtbytheclassictechniques.

3.Improvetherobustnessandstabilityofthedesignedtrajectoryoptimization approach.

4.Reducethecomputationalcomplexityoftheproposedalgorithm.

5.Presentnewmulti-objectiveapproachestosearchtheoptimaltrade-offsolution withpreferencerequirements.

Fig.1.2 SMVorbitalhoppingmissionprofile

6.Providein-depthanalysisofstochastictrajectoryoptimizationproblemswiththe considerationofchanceconstraints.

1.4ChapterLayout

Therestofthisbookisorganizedasfollows.Chapter 2 reviewsthestate-of-theartdevelopmentinspacecrafttrajectoryoptimizationproblemsandoptimalcontrol methods.Aparticularfocuswillbegiventothedesignofnumericaltrajectoryoptimizationalgorithmsandtheirapplications.Seniorundergraduatestudentsandpostgraduatestudentswhoaredoingresearchorareinterestedintrajectoryoptimization methodswillgetabetterunderstandingofthelatestdevelopmentonthistopic.

FollowingthatinChap. 3,themodelingoftheSMVtrajectoryoptimizationproblemwillbepresented.AnonlinearconstrainedoptimalcontrolformulationisconstructedandusedtosearchtheoptimaltrajectoryoftheSMV.Twosetsofflight dynamicsareestablishedinordertorepresentthemovementofthespacecraftduringtheexo-atmosphericandatmosphericflightphases.Inaddition,acoupleof interior-pointconstraintsareintroducedtoconnectthetrajectorybetweendifferent flightphases.

Chapter 4 analysestheperformanceofdifferentoptimizationstrategiesforcalculatingtheoptimaltrajectories.Twotypesofoptimizationstrategies,namelythe gradient-basedmethodandthederivative-freemethod,areappliedtosolvetheSMV trajectorydesignproblem.Theadvantagesanddisadvantagesofusingthesewelldevelopedalgorithmsarediscussedandconcludedindetail.

Chapter 5 introducesanewhybridoptimalcontrolsolvertosolvetheconstrained SMVtrajectoryoptimizationproblem.Aderivative-freealgorithm-basedinitial guessgeneratorisdesignedandappliedtodecreasethesensitivityproblem.Inaddition,animprovedgradientsolverisproposedastheinneroptimizer.Thistwo-nested structurecanoffertheusermoreflexibilitytocontroltheoptimizationprocess.

InChap. 6,theSMVtrajectoryoptimizationproblemisreformulatedandextended toamulti-objectivecontinuous-timeoptimalcontrolmodel.Multi-objectiveoptimizationevolutionarytechniquesaredesignedandappliedtocalculatetheParetooptimalsolution.Furthermore,inordertotakeintoaccountthedesigner’spreference requirements,differentmulti-objectivetransformationtechniquesarealsoproposed toproducecompromisedsolutions.

TheworkpresentedinChap. 7 focusesonthedevelopmentofreal-timeoptimal guidancestrategiesforthespacemaneuvervehicles.Twotypesofoptimalguidance strategies,namelytherecedinghorizoncontrol-basedmethodsandthedeepneural network-drivenalgorithm,areproposedtoproducetheoptimalcontrolcommandin realtime.Detailedsimulationstudieswerecarriedouttoverifytheeffectivenessand real-timeapplicabilityoftheproposedstrategies.

Chapter 8 investigatesacomputationalframeworkbasedonoptimalcontrolfor addressingtheproblemofstochastictrajectoryoptimizationwiththeconsideration ofprobabilisticconstraints.Adiscretizationtechniqueisemployedtoparametrize

theuncertainvariableandcreatethetrajectoryensemble.Besides,asmoothand differentiablechanceconstrainthandlingmethodisproposedtoapproximatethe probabilisticconstraint.Simulationresultsareobtainedtopresenttheoptimalflight trajectories.Basedonthenumericalsimulation,somekeyfeaturesoftheobtained resultsarealsoanalyzed.

References

1.Betts,J.T.:Surveyofnumericalmethodsfortrajectoryoptimization.J.Guid.Control.Dyn. 21(2),193–207(1998). https://doi.org/10.2514/2.4231

2.Conway,B.A.:Asurveyofmethodsavailableforthenumericaloptimizationofcontinuous dynamicsystems.J.Optim.TheoryAppl. 152(2),271–306(2012). https://doi.org/10.1007/ s10957-011-9918-z

3.Lavaei,J.,Momeni,A.,Aghdam,A.G.:Amodelpredictivedecentralizedcontrolschemewith reducedcommunicationrequirementforspacecraftformation.IEEETrans.ControlSyst.Technol. 16(2),268–278(2008). https://doi.org/10.1109/TCST.2007.903389

4.AlonsoZotes,F.,SantosPenas,M.:Particleswarmoptimisationofinterplanetarytrajectories fromearthtojupiterandsaturn.Eng.Appl.Artif.Intell. 25(1),189–199(2012). https://doi.org/ 10.1016/j.engappai.2011.09.005

5.Chai,R.,Savvaris,A.,Tsourdos,A.,Chai,S.,Xia,Y.:Optimalfuelconsumptionfinite-thrust orbitalhoppingofaeroassistedspacecraft.Aerosp.Sci.Technol. 75,172–182(2018). https:// doi.org/10.1016/j.ast.2017.12.026

6.Bogorad,A.,Bowman,C.,Dennis,A.,Beck,J.,Lang,D.,Herschitz,R.,Buehler,M.,Blaes,B., Martin,D.:Integratedenvironmentalmonitoringsystemforspacecraft.IEEETrans.Nucl.Sci. 42(6),2051–2057(1995). https://doi.org/10.1109/23.489252

Chapter2

OverviewofTrajectoryOptimization Techniques

Abstract Thischapteraimstobroadlyreviewthestate-of-the-artdevelopmentin spacecrafttrajectoryoptimizationproblemsandoptimalcontrolmethods.Specifically,themainfocuswillbeontherecentlyproposedoptimizationmethodsthathave beenutilizedinconstrainedtrajectoryoptimizationproblemsandmulti-objectivetrajectoryoptimizationproblems.Anoverviewregardingthedevelopmentofoptimal controlmethodsisfirstintroduced.Followingthat,variousoptimizationmethodsthat canbeeffectiveforsolvingspacecrafttrajectoryplanningproblemsarereviewed, includingthegradient-basedmethods,theconvexification-basedmethods,theevolutionary/metaheuristicmethods,andthedynamicprogramming-basedmethods.In addition,aspecialfocuswillbegivenontherecentapplicationsoftheoptimized trajectory.Finally,themulti-objectivespacecrafttrajectoryoptimizationproblem, togetherwithdifferentclassesofmulti-objectiveoptimizationalgorithms,isbriefly outlinedattheendofthechapter.

2.1SpacecraftTrajectoryOptimizationProblems andOptimalControlMethods

Overthepastcoupleofdecades,trajectoryoptimizationproblemsintermsofreentry vehiclehaveattractedsignificantattention[1–3].Ithasbeenshowninmanypublished worksthatthetrajectorydesigncomponentplaysakeyrolewithregardtostableflight andimprovedcontrolofthespacevehicle[4, 5].Acomprehensiveoverviewofthe motivationfortheuseoftrajectoryoptimizationindifferentspacemissions,together withvariousrelatedtrajectoryoptimizationapproaches,wasmadebyConwayin 2011[6].Inthisreviewarticleandthereferencestherein,severalimportantpractical exampleswerehighlightedsuchastheorbitaltransferproblems[7, 8],thespacecraft rendezvousanddocking[9, 10],andtheplanetaryentry[11–13].Theseproblems weresummarizedinageneralformandtreatedasoptimalcontrolproblems[14].It isworthnotingthataccordingtoBetts[15],aninterchangeddesignationbetweenthe term“optimalcontrolproblems”and“trajectoryoptimizationproblems”canalways befoundintheliterature.Acompletedescriptionandanalysisofthedifferences betweenthesetwostatementscanbereferredto[16, 17].

©SpringerNatureSingaporePteLtd.2020

R.Chaietal., DesignofTrajectoryOptimizationApproachforSpace ManeuverVehicleSkipEntryProblems,SpringerAerospaceTechnology, https://doi.org/10.1007/978-981-13-9845-2_2

Amongtheseapplications,oneofthecurrentobjectivesisthedevelopmentof spacemaneuvervehicles(SMV)foradynamicmissionprofile[8, 18, 19].The Machnumberandtheflightaltitudeoftheentryvehiclevarylargelyduringthe wholeflightphase,theaerodynamicfeatureofthevehiclehaslargeuncertaintiesand nonlinearities[12, 20].Duetothesereasons,numericalalgorithmsarecommonly usedtoapproximatetheoptimalsolution[21–23].

Fromthecurrentdevelopmentofoptimalcontroltheory,onthewhole,thedevelopment/applicationofnumericaltrajectoryoptimizationmethodsforatmosphericor exo-atmosphericspacecraftflightscenariosleadstotwodifferenttrends.Thefirstone isthatsystemdiscretizationtendstobecomemorereliableandadaptivesuchthatit canmaximallycapturethecharacteristicsofthedynamicalsystem[14, 24].Theother isthatoptimizationbecomesmoreaccurateandcomputationallyfriendlysothatthe solutionoptimality,togetherwiththereal-timecapability,canbeimproved.Dependingontheorderofdiscretizationandoptimization,numericaltrajectoryoptimization methodscanbeclassifiedintotwomaincategories.Thatis,theso-calledindirect methods(“optimizationthendiscretization”)andthedirectmethods(“discretization thenoptimization”)[24].Theformertypeofmethodaimstoapplythecalculusof variations(COV)andsolvethefirst-ordernecessaryconditionsforoptimalitywith respecttothespacecrafttrajectoryoptimizationproblems.Successfulexampleshave beenreportedintheliteratureforaddressingproblemswithoutconsideringinequalityconstraints[25, 26].Intheseworks,thefirst-ordernecessaryconditionswere formulatedastwo-pointboundaryvaluedifferential-algebraicequations.However, intermsofproblemsinthepresenceofinequalityconstraints,thistypeofapproach mightnotbeeffective.Thisisbecauseitisdifficulttodeterminetheswitchpoints wheretheinequalityconstraintsbecomeactive,thuslimitingthepracticalapplicationofthistypeofmethod.Moreover,theHamiltonianboundaryvalueproblem (HBVP)shouldalsobeconstructedandthisprocessusuallybecomescostlydueto thecomplexityofthedynamicmodelandpathconstraints.

Asforthedirectmethod,thefirststepistodiscretizethecontrolorthestate andcontrolvariablessoastotransformtheoriginalformulationtoastaticnonlinear programmingproblem(NLP).Followingthat,differentwell-developedoptimization techniquesareavailabletoaddresstheoptimalsolutionoftheresultingstaticproblem. Comparedwiththeindirectstrategy,itismucheasiertoapplythedirectmethodto handlethespacecrafttrajectorydesignproblem.Moreover,thewayofformulating constraintstendstobemorestraightforward.Therefore,applyingthe“discretization thenoptimization”modehasattractedmoreattentioninengineeringpractice.

Fordirectmethods,onetraditionaltechniquewhichhasbeenusedinpractical problemsisthedirectmultipleshootingapproach[27–29].Inashootingmethod,only thecontrolvariablesareparametrized[30–32].Thenexplicitnumericalintegration (e.g.,theRunge–Kuttamethod)isusedtosatisfythedifferentialconstraints[33–35]. Anotherwell-developeddirecttranscriptiontechniqueisthecollocationmethods. Generally,therearetwomainkindsofcollocationschemes:localcollocationmethods andglobalcollocationmethods.Relativeworksondevelopingthelocalcollocation methodscanbefoundinliteratures.Forexample,Yakimenkoetal.[36]appliedan inversedynamicsinthevirtualdomaincollocationmethodtogenerateanear-optimal

aircrafttrajectory.TheworkofDuanandLi[37]presentsadirectcollocationscheme forgeneratingtheoptimalcontrolsequenceofahypersonicvehicle.

Inrecentyears,globalcollocationtechniqueshaveattractedextensiveattentions andalargeamountofworkisbeingcarriedoutinthisfield[38].Forexample,Fahroo andRoss[39]developedaChebyshevpseudospectralapproachforsolvingthegeneralBolzatrajectoryoptimizationproblemswithcontrolandstateconstraints.In theirfollow-upwork[40],apseudospectralknottingalgorithmwasdesignedsoasto solvenonsmoothoptimalcontrolproblems.Inaddition,Bensonetal.[41]developed aGausspseudospectral(orthogonalcollocation)methodfortranscribinggeneraloptimalcontrolproblems.Inapseudospectralmethod,thecollocationpointsarebased onquadraturerulesandthebasisfunctionareLagrangeorChebyshevpolynomials. Incontrasttothedirectcollocationmethod,pseudospectralmethodusuallydivides thewholetimehistoryintoasinglemeshintervalwhereasitscounterpart,direct collocation,dividestimeintervalintoseveralequalstepsubintervalsandtheconvergenceisachievedbyaddingthedegreeofthepolynomial.Toimproveaccuracyand computationalefficiencyusingpseudospectralmethod,ahp-strategyisproposedand analyzedin[42–44].Byaddingcollocationpointsinacertainmeshintervalordividingthecurrentmeshintosubintervalssimultaneously,theaccuracyofinterpolation canbeimprovedsignificantly.

Itisworthnotingthattheprimarygoalofthisbookistopresentthelatestprogress thathasbeenachievedinthedevelopmentofspacecrafttrajectoryoptimizationtechniques.Specifically,themainfocuswillbeontherecentlyproposedoptimization methodsthathavebeenutilizedinconstrainedtrajectoryoptimizationproblems, multi-objectivetrajectoryoptimizationproblems,andstochastictrajectoryoptimizationproblems.Therefore,comparedwiththeoptimizationprocess,thediscretization processisrelativelyless-importantandwillonlybebrieflypresentedinthefollowingchapters.Adetailedandseriousattempttoclassifydiscretizationtechniquesfor spacecrafttrajectorydesigncanbefoundin[8, 17].

2.2OptimizationTechniquesandApplications

Nevertheless,allthedirectmethodsaimtotranscribethecontinuous-time-optimal controlproblemstoanonlinearprogrammingproblem(NLP)[45–47].Theresulting NLPcanbesolvednumericallybywell-developedoptimizationalgorithms.

Themainobjectiveofthissectionistoreviewthestate-of-the-artoptimization strategiesreportedintheliteratureforcalculatingtheoptimalspacecraftflighttrajectories.Basedonthereportedresults,onemaybeabletogainabetterunderstanding intermsoftheperformanceandbehaviorsofdifferentalgorithmsforaddressing variousspacevehicleflightmissions.Moreover,itispossibletoguidethereaderto improveoneofthesetechniquesinordertocircumventthelimitationsbroughtby theclassicmethods.

Intheliterature,fourtypesofoptimizationstrategiesareusuallyappliedtosolve thespacecrafttrajectoryoptimizationproblems.Specifically,thegradient-based,

Table2.1 Populardeterministicoptimizationalgorithmsavailablefortrajectoryoptimizationproblems

Deterministicoptimizationalgorithms

Sequentialquadraticprogramming(SQP)[7]

Interior-pointmethod(IP)[48]

Interior-pointsequentialquadraticprogramming(IPSQP)[49]

Linearprogramming(LP)[50]

Second-orderconeprogramming(SOCP)[47]

Semidefiniteprogramming(SDP)[51]

Dynamicprogramming(DP)[52]

Differentialdynamicprogramming(DDP)[53]

Stochasticdifferentialdynamicprogramming(SDDP)[54]

Table2.2 Popularstochasticoptimizationalgorithmsavailablefortrajectoryoptimizationproblems

StochasticOptimizationAlgorithms

Geneticalgorithm(GA)[1]

Differentialevolution(DE)[55]

Violationlearningdifferentialevolution(VLDE)[3]

Particleswarmoptimization(PSO)[10]

Predator–preypigeon-inspiredoptimization(PPPIO)[56]

Antcolony(AC)[57]

Artificialbeecolony(ABC)[37]

Simulateannealing(SA)[58]

Tabusearch(TS)[59]

convexification-based,dynamicprogramming-based,andderivative-free(heuristicbased)optimizationtechniquesareusedtocalculatetheoptimaltimehistorywith respecttothespacecraftstateandcontrolvariables.Thesealgorithmscanbefurther groupedintothedeterministicandthestochasticapproaches.

Themostpopularoptimizationmethodsamongthesetwogroupsaresummarized andtabulatedinTables 2.1 and 2.2.Itshouldbenotedthatnotalltheoptimization algorithmsundereachcategoryarelistedinthistable.Alternatively,onlysome importantexamplesarereviewedandthesetechniquesarediscussedindetailin thefollowingsubsections.Alargenumberofnumericalsimulationswerecarried outinrelatedworks.Theresultsindicatedthatthesenewlyproposedoptimization strategiesareeffectiveandcanprovidefeasiblesolutionsforsolvingtheconstrained spacevehicletrajectorydesignproblems.

2.2.1Gradient-BasedMethods

Oneofthemostcommonlyusedoptimizationalgorithmsforoptimizingthespacecraftflighttrajectoryistheclassicgradient-basedmethod.Amonggradient-based methods,thesequentialquadraticprogramming(SQP)methodandtheinterior-point (IP)methodareusedsuccessfullyforthesolutionoflarge-scaleNLPproblems[60]. In[61],afuel-optimalaeroassistedspacecraftorbitaltransferproblemwasfirsttransformedtoastaticNLPviaapseudospectraldiscretizationmethod.Then,thestatic NLPwassolvedbyapplyingthestandardSQPmethodtogeneratethefuel-optimal flighttrajectory.Similarly,in[7]theSQPmethodwasappliedastheprimaryoptimizertosearchthetime-optimalflighttrajectoryofalow-thrustorbitaltransfer problem.

AlthoughSQPmethodscanbeusedasaneffectivealgorithmtoproducetheoptimalflighttrajectory,mostoftheSQPimplementationsrequiretheexactsolution ofthesubproblem.Thismayincreasethecomputationalburdenofthesolversignificantly[49].Moreover,sincemostSOPmethodsutilizetheactivesetstrategy tohandleinequalityconstraints,thecomputationalburdenmaybeincreasedifthe activesetisinitializedinanimproperway.

ApartfromtheSQPmethod,analternativegradient-basedmethodistheinteriorpoint(IP)methoddevelopedduringthelastdecade.InvestigationsofIPcanbefound inalargeamountofwork.Toapplythismethod,theinequalityconstraintsneedtobe transcribedtoequalityconstraintsbyintroducingsomeslackvariablessuchthatthe problemcanbesolvedinasimplerform.AnapplicationoftheIPmethodinspace vehicletrajectorydesignproblemcanbefoundin[48].Inthiswork,aspaceshuttle atmosphericreentryproblemwasconsideredanddiscretizedviaashootingmethod. TheresultingstaticNLPproblemwasthenaddressedbyapplyingtheIPmethod. Simulationresultsprovidedinthisworkconfirmedtheeffectivenessofapplyingthe IPmethod.However,itisworthnotingthatfortheIPmethod,themainchallenge istodefinethepenaltyfunctionsandinitializethepenaltyfactorintheaugmented meritfunctioninordertomeasurethequalityoftheoptimizationprocess.

In[49],combiningtheadvantagesoftheSQPandIPmethods,theauthorsproposedatwo-nestedgradient-basedmethod,namedinterior-pointsequentialquadratic programming(IPSQP),forsolvingtheaeroassistedspacecrafttrajectorydesignproblem.Oneimportantfeatureofthisapproachisthataninnersolution-findingloopwas embeddedinthealgorithmframework,therebyallowingtheQPsubproblemtobe solvedinexactly.Inthisway,thedesigncanhavemoreflexibilitytocontroltheoptimizationprocessandthealgorithmefficiencycanalsobeimprovedtosomeextent. Simulationresultsandcomparativestudieswerereportedtoshowtheeffectiveness aswellasthereliabilityofthisimprovedgradient-basedmethod.

2.2.2Evolutionary-BasedMethods

Inanoptimizationproblem,ifitishardtogetthegradientinformationoftheobjectivefunctionsorconstraints(i.e.,duetothehighnonlinearityinvolvedinthese functions),theclassicgradient-basedmethodmightnolongerbereliableoravailable.Inthiscase,theevolutionary-basedmethods,alsoknownasglobaloptimization methods,becometheonlywaytoproducetheoptimalsolution,asthereisnoderivativeinformationrequiredinanevolutionaryapproach.Thisindicatesthatitwillnot sufferfromthedifficultyofcalculatingtheJacobianaswellastheHessianmatrix.

Evolutionaryalgorithmsorglobaloptimizationmethodsusetheprincipleof“survivalofthefittest”adoptedtoapopulationofelementsrepresentingcandidatesolutions[6, 11, 62].Comparedwithclassicgradient-basedalgorithms,thereisno initialguessvaluerequiredbythealgorithmsasthepopulationisinitializedrandomly.Thankstothenatureoftheevolutionaryalgorithm,ittendstobemorelikely thanclassicgradientmethodstolocatetheglobalminimum[10].

Therearemanytypesofevolutionaryalgorithmsthatareavailabletoproducethe optimalsolutionofanengineeringoptimizationproblem.Forexample,thegeneric classofevolutionaryalgorithmssuchasthegeneticalgorithm(GA)anddifferentialevolution(DE),theagent-basedclasssuchastheparticleswarmoptimization (PSO)andthepigeon-inspiredoptimization(PIO),andthecolony-basedclassof algorithmssuchastheantcolonyoptimization(ACO)andtheartificialbeecolony (ABC)algorithm.Relativeworksondevelopingorapplyingtheseglobaloptimizationmethodsinspacecrafttrajectorydesignarewidelyresearchedintheliterature. In[63],aconstrainedspacecapsulereentrytrajectorydesignproblemwasaddressed byapplyingamodifiedGA.Similarly,Kameshetal.[64]incorporatedahybridGA andacollocationmethodsoastoaddressanEarth–Marsorbitaltransfertask.The authorsin[62]producedtheoptimalpathforaspaceroboticmanipulatorbyusing astandardPSOmethod.

Conwayetal.[6]combinedglobaloptimizationalgorithmswithastandard gradient-basedmethodinordertoconstructabi-levelstructuraloptimalcontrol method.Intheirlatestwork,PontaniandConway[10]utilizedamodifiedparticleswarmoptimizationalgorithmtogloballyoptimizetheflightpathofacycling spacecraft.

Anenhanceddifferentialevolutionapproachincorporatedwithaviolationdegreebasedconstrainthandlingstrategywasconstructedinourpreviousworktoapproximatetheoptimalflighttrajectory[3]ofaspacemaneuvervehicleentryproblem.In thiswork,asimplex-baseddirectsearchmechanismwasembeddedinthealgorithm frameworkinordertoimprovethediversityofthecurrentpopulation.Besides,a learningstrategywasusedtoavoidtheprematureconvergenceofthealgorithm.

Furthermore,theauthorsin[65]establishedanantcolonyinspiredoptimization algorithmsoastoplanamulti-phasespacevehicleorbitalflighttrajectory.Anautomatedapproachbasedongeneticalgorithmandmonotonicbasinhoppingwas appliedin[66]toaddressalaunchvehicleinterplanetarytrajectoryproblem.

Althoughtheaforementionedworkshaveshownthefeasibilityofusingheuristicbasedmethodsforaddressingspacecrafttrajectorydesignproblems,thevalidationof solutionoptimalitybecomesdifficult.Moreover,thecomputationalcomplexitydue totheheuristicoptimizationprocesstendstobeveryhigh[67].Therefore,itisstill difficulttotreatheuristic-basedmethodsasa“standard”optimizationalgorithmthat canbeappliedtosolvegeneralspacecrafttrajectoryplanningproblems.Mucheffort isexpectedtoimprovethecomputationalperformanceofthiskindofalgorithm.

2.2.3Convexification-BasedMethods

Recently,agrowinginterestcanbefoundinapplyingconvexification-basedmethods forgeneratingtheoptimalspacecraftflighttrajectories[68].Animportantfeatureof applyingthiskindofmethodisthatitcanbeimplementedwiththeoreticalguarantees withregardtothesolutionandcomputationalefficiency.Sincemostofthepractical spacecrafttrajectoryoptimizationproblemsareusuallynonconvex,inordertoapply aconvexoptimizationmethod,variousconvexificationtechniquesaredeveloped totransformtheoriginalproblemformulationtoaconvexversion.Thiscanalsobe understoodasusingaspecificconvexoptimizationmodeltoapproximatetheoriginal nonconvexformulation.Commonly,therearethreetypesofconvexoptimization existingintheliterature:

1.Linearprogrammingmodel(LP),

2.Second-orderconeprogrammingmodel(SOCP), 3.Semidefiniteprogrammingmodel(SDP).

IntermsoftheLPmodel,itshouldbenotedthatiftheconsideredproblemis relativelycomplex(i.e.,thenonlinearityofthesystemdynamics,objectives,orconstraintsishigh),thentheLPmodelmightnotbesufficientandreliabletoapproximate theoriginalproblemformulation.Ontheotherhand,asfortheSDPmodel,although ithasthemostaccurateapproximationabilityamongthethreemodelslistedabove, thetransformedconvexformulationisoftennotwell-scaled,therebyresultingin anincreasewithregardstothecomputationalcomplexity.Onthecontrary,agood balancebetweentheapproximationaccuracyandthecomputationalcomplexitycan beachievedbyapplyingtheSOCPmodel.Thisstrategyapproximatestheproblem constraintsusingthesecond-orderconesuchthatthetransformedproblemcanbe solvedwitharelativelysmallcomputingpower.

Contributionsmadetoimplementconvexification-basedoptimizationmethodsto solvespacevehicletrajectorydesignproblemscanbefoundintheliterature.For example,in[69, 70],theplanetarylandingproblemwasaddressedbyusingthe convexoptimizationmethodundertheconsiderationofnonconvexthrustmagnitude constraints.Also,in[71],theSOCPmethodwasappliedtoproducetheoptimal trajectoryofthespacecraftentryplanningproblem.Inthiswork,nonconvexcollision avoidanceconstraints,aswellasthenavigationuncertainties,werealsotakeninto accountandreformulatedintoconvexconstraintsduringtheoptimizationphase.

2.2.4DynamicProgramming-BasedMethods

Themotivationfortheuseofdynamicprogramming-basedmethodsreliesontheir enhancedabilityinachievingstableperformanceandindealingwithlocaloptimal solution,thatnaturallyexistinnonlinearoptimalcontrolproblems.Inthissubsection, twotypicaldynamicprogramming-basedalgorithmsarereviewedsuchasthestandarddynamicprogramming(DP)method,andthedifferentialdynamicprogramming method(DDP).

MotivatedbytheBellman’sprincipleofoptimality,DPisproposedandapplied tosolveengineeringoptimizationproblems[52].TheprimaryideaoftheBellman’s principleisthattheoptimalsolutionwillnotdivergeifotherpointsontheoriginal optimalsolutionarechosenasthestartingpointtore-triggertheoptimizationprocess. Basedonthisprinciple,DPcalculatestheoptimalsolutionforeverypossibledecision variable.Hence,itishighlylikelytoresultinthecurseofdimensionality[54].

InordertodealwiththemaindeficiencyfacedbythestandardDP,theDDP approachhasbeendesigned[72].Inthismethod,thesolution-findingprocessis performedlocallyinasmallneighborhoodofareferencetrajectory.Subsequently, thismethodcalculatesthelocaloptimalsolutionbyusingabackwardandaforwardsweeprepeatedlyuntilthesolutionconverges.TheDDPmethodhasbeen successfullyappliedtocalculatetheoptimalsolutionofsomespacemissions.For example,in[73, 74],acomprehensivetheoreticaldevelopmentoftheDDPmethod, alongwithsomepracticalimplementationandnumericalevaluationwasprovided. In[72],aDDP-basedoptimizationstrategywasproposedandappliedtocalculate therendezvoustrajectorytonear-Earthobjects.

However,mostoftherecentDDPworkdoesnottakethemodeluncertaintiesand noisesintoaccountintheprocessoffindingthesolution.Consequently,thesolutionfindingprocessmightfailtoproduceanominalsolutionwhichcanguaranteethe feasibilityallalongthetrajectorywhenuncertaintiesormodelerrorsperturbthe currentsolution.

Accordingtoalltherelativeworksreported,itcanbeconcludedthatalthough theresultsgeneratedfrommostexistingoptimizationalgorithmscanbeacceptedas near-optimalsolutions,thereisstillroomforimprovementwithrespecttoapplying theseoptimizationstrategiesinspacecrafttrajectorydesignproblems.

2.3Multi-objectiveTrajectoryOptimizationOverview

Traditionaltrajectorydesignusuallyaimsatonesingleobjective,forexample,minimizingtheaerodynamicheating,maximizingthecrossrange,etc.However,itis worthnotingthatinmanypracticalspacecraftflightoperations,multipleperformanceindicesmustfrequentlybeconsideredduringthetrajectorydesignphase andthisbringsthedevelopmentofmulti-objectivetrajectoryoptimization(MOTO) [75–77].