Control Engineering Ashish Tewari Optimal Space Flight Navigation An Analytical Approach Control Engineering SeriesEditor WilliamS.Levine DepartmentofElectricalandComputerEngineering UniversityofMaryland CollegePark,MD USA EditorialAdvisoryBoard RichardBraatz MarkSpong MassachusettsInstituteofTechnology UniversityofTexasatDallas Cambridge,MA Dallas,TX USA USA GrahamGoodwin MaartenSteinbuch UniversityofNewcastle TechnischeUniversiteitEindhoven Australia Eindhoven,TheNetherlands DavorHrovat MathukumalliVidyasagar FordMotorCompany UniversityofTexasatDallas Dearborn,MI Dallas,TX USA USA ZongliLin YutakaYamamoto UniversityofVirginia KyotoUniversity Charlottesville,VA Kyoto,Japan USA Moreinformationaboutthisseriesathttp://www.springer.com/series/4988 Ashish Tewari Optimal Space Flight Navigation An Analytical Approach AshishTewari DepartmentofAerospaceEngineering IndianInstituteofTechnology,Kanpur IIT-Kanpur,India ISSN2373-7719 ISSN2373-7727 (electronic) ControlEngineering ISBN978-3-030-03788-8 ISBN978-3-030-03789-5 (eBook) https://doi.org/10.1007/978-3-030-03789-5 LibraryofCongressControlNumber:2018961705 MathematicsSubjectClassification:70F05,70F07,70F10,49N05,49N25,49J15,49K15,49J30,37N05, 34B10 ©SpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered companySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface This book is primarily written to consolidate optimal space flight navigation theory as a discipline. There are several excellent textbooks on optimal control theory and equally good books on orbital space mechanics, but the application of optimal control theory to orbital mechanics is mainly confined to research articles. Over the last several decades, the research literature on optimal space flight navigation has become voluminous and has fragmented into many esoteric subdisciplines.However,amonographwhichorganizesthiscontentandvitalizesit forcontemporaryapplicationishithertolacking. Optimal space flight navigation is a problem of practical interest and has been so ever since humans imagined journeying into space. While visionaries such as TsiolkovskyandGoddardhelpedintakingthefirststepsbydevelopingrocketry,it requiredenthusiaststodevisefuel-optimaltrajectoriesforpracticalspacemissions. The vision of Walter Hohmann, a civil engineer, revealed in his 1925 book Die Erreichbarkeit der Himmelskörper (The Accessibility of Celestial Bodies) is one of the most useful and commonly applied space flight maneuvers—the Hohmann transfer. As recently as in 2014, the Hohmann transfer was successfully applied to efficiently send India’s first mission to Mars. While arising as an elegant idea in Hohmann’s mind, the optimality of the Hohmann transfer can now be rigorously proved in various ways, including by optimal control theory. Similarly, the straightforward and insightful application by D.F. Lawden of the erstwhile nascentoptimalcontroltheorytothespacenavigationproblemproducedin1963in themonographentitledOptimalTrajectoriesforSpaceNavigationhasguidedmany researchers since then. The first footsteps on the Moon would not have appeared if NASA’s engineers (including Richard Battin who passed away in 2014) had not devised simple but efficient techniques to guide the lunar module on a flat thrusting trajectory to a safe landing. Battin in his treatise An Introduction to the Mathematics and Methods of Astrodynamics reminisces about how he came up with the velocity-to-be-gained and cross-product steering as simple but practical guidancestrategiesforspaceflightintheeraofprimitivecomputertechnology.He writes of the enormous buildings required to house the computer mainframes of the day with only very modest computing power. Perhaps the total computational v vi Preface resourcespresentonboardApollo11couldbesurpassedbythoseoftoday’spocket calculator. The feat performed by those pioneering guidance engineers can only be appreciated by the complexity of their task—to send people to the Moon and to bring them successfully back to the Earth. The design and analysis of practical lunarandinterplanetarytrajectoriescouldbeconductedintheanalyticalframework established by Victor Szebehely in his 1967 masterpiece, Theory of Orbits: The RestrictedProblemofThreeBodies. Ratherthanhighlightingthecomputational(andoftenunrealizable)solutionsof mathematically complex problems, this book emphasizes the analytical approach tooptimalspacenavigation.Theneedforsimplicityandpracticabilityofguidance methods is to be contrasted with the unfortunate emphasis which has been placed these days on increasing the complexity of control schemes. Multilevel iterative algorithmsmustbesolvedinreal-timecomputationsandthereforehaveunresolved convergenceissueswhichpreventapplicationstoactualmissions.Itshouldthusbe asked: Is more sophistication always better, and does it lead to any advantage in the practical sense? I think a simpler guidance solution must be sought, wherever possible,duetoitsreliableimplementationinarealisticsituation,whichhasbeen amplydemonstratedbytheanalyticalguidancemethods.Thatiswhyeventhelatest spaceflightmissionsarebaseduponanalyticaltechniques. The complex interplanetary and asteroid (or comet) flyby and rendezvous missions are analytically derived from the application of optimal control theory to multi-body dynamics, such as the optimal transfers between halo and quasi- halo orbits and Lissajous trajectories using the stable and unstable manifolds of the restricted three-body problem. Studying such optimal paths often imparts new insightsintotheproblemofmulti-bodytransfers,whichwouldbelackinginapurely numericalsearchforsuboptimalsolutions.Inasimilarvein,theproblemoforbiting andlandingonanirregularlyshapedbodywithanuncertaingravityfield(suchasan asteroid or a comet) can be solved out by time-optimal and fuel-optimal methods, ashasbeendemonstratedinmanypracticalmissions. ThethoughtofwritingthisbookarosefromthecourseAE-777(OptimalSpace FlightControl),whichIhavetaughtforthepastseveralyearsandforwhichasingle textbookcontainingalltherelevantmaterialwasunavailable.Therefore,thebookis designedtobeusedasatextbookinacourseonoptimalspaceflightatthegraduate and senior undergraduate levels. The first three chapters give a basic introduction tothetopic,namely,optimalcontroltheory(Chaps.1and2)andorbitalmechanics withimpulsivetransfers(Chap.3).Chapter4coversoptimaltransfersinaspherical gravity field with continuous, unconstrained acceleration inputs, whereas Chap.5 extends the treatment to trajectory optimization with bounded acceleration inputs. Finally, Chap.6 introduces the reader to the advanced topics of optimal transfer in time-varying gravity fields, including multi-body transfers. Unfortunately, due to the vast scope of these topics, it is not possible to do them full justice in an introductory text. Hence, references to research articles are provided for the advanced topics. The end-of-chapter exercises test the understanding of the basic concepts, whereas several references are provided for undertaking research and for advanced applications. A basic knowledge of control systems and dynamics is Preface vii assumed of the reader, which can be supplemented by textbooks on these topics, such as those previously published by me (e.g., Modern Control Design, and Atmospheric and Space Flight Dynamics, Automatic Control of Atmospheric and SpaceFlightVehicles). ItisapleasuretoofferthismanuscriptforpublicationwithBirkhäuser.Iwould like to thank the editorial and production staff at Birkhäuser for their invaluable assistance, especially Bill Levine, Benjamin Levitt, Samuel DiBella, and Christo- pherTominich.IwouldfinallyliketothankmywifePrachianddaughterManyafor theirpatienceduringthepreparationofthismanuscript. Kanpur,India AshishTewari September2018 Contents 1 Introduction .................................................................. 1 1.1 OptimalControl ....................................................... 1 1.2 SpaceVehicleGuidance............................................... 3 2 AnalyticalOptimalControl................................................. 7 2.1 Introduction............................................................ 7 2.2 OptimizationofStaticSystems ....................................... 8 2.2.1 StaticEqualityConstraints .................................. 11 2.2.2 InequalityConstraints ....................................... 18 2.3 DynamicEqualityConstraintsandUnboundedInputs .............. 20 2.4 SpecialBoundaryConditions......................................... 26 2.4.1 FixedTerminalTime......................................... 26 2.4.2 FreeTerminalTime.......................................... 28 2.5 SufficientConditionforOptimality................................... 31 2.6 Pontryagin’sMinimumPrinciple ..................................... 31 2.7 Hamilton–Jacobi–BellmanFormulation.............................. 32 2.8 Time-InvariantSystems ............................................... 34 2.9 LinearSystemswithQuadraticCostFunctions...................... 35 2.10 IllustrativeExamples .................................................. 43 2.11 SingularOptimalControl ............................................. 55 2.11.1 GeneralizedLegendre–ClebschNecessaryCondition ..... 58 2.11.2 Jacobson’sNecessaryCondition ............................ 67 2.12 NumericalSolutionProcedures....................................... 69 2.12.1 ShootingMethod............................................. 70 2.12.2 CollocationMethod.......................................... 71 Exercises....................................................................... 71 3 OrbitalMechanicsandImpulsiveTransfer .............................. 77 3.1 Introduction............................................................ 77 3.2 KeplerianMotion...................................................... 78 3.2.1 ReferenceFramesofKeplerianMotion..................... 80 ix x Contents 3.2.2 TimeEquation................................................ 83 3.2.3 Lagrange’sCoefficients...................................... 86 3.3 ImpulsiveOrbitalTransfer ............................................ 88 3.3.1 MinimumEnergyTransfer .................................. 92 3.4 Lambert’sTransfer..................................................... 95 3.4.1 StumpffFunctionMethod ................................... 96 3.4.2 HypergeometricFunctionMethod .......................... 99 3.5 OptimalImpulsiveTransfer........................................... 101 3.5.1 CoastingArc ................................................. 102 3.5.2 HohmannTransfer........................................... 105 3.5.3 OuterBi-ellipticalTransfer.................................. 110 Exercises....................................................................... 112 4 Two-BodyManeuverswithUnboundedContinuousInputs ........... 115 4.1 Introduction............................................................ 115 4.2 AMotivatingExample ................................................ 115 4.3 EquationsofMotion................................................... 117 4.4 OptimalLow-ThrustOrbitalTransfer ................................ 120 4.4.1 CoplanarOrbitalTransfer ................................... 122 4.4.2 PlaneChangeManeuver..................................... 122 4.4.3 GeneralOrbitalTransfer..................................... 123 4.5 VariationalModel...................................................... 124 4.6 OptimalRegulationofCircularOrbits ............................... 127 4.6.1 CoplanarRegulationwithRadialThrust.................... 128 4.6.2 CoplanarRegulationwithTangentialThrust ............... 131 4.7 GeneralOrbitalTracking.............................................. 132 4.8 BasicGuidancewithContinuousInputs.............................. 138 4.9 Line-of-SightGuidance ............................................... 139 4.10 Cross-ProductSteering................................................ 145 4.11 Energy-OptimalGuidance ............................................ 148 4.12 Hill–Clohessy–WiltshireModel ...................................... 151 Exercises....................................................................... 155 5 OptimalManeuverswithBoundedInputs ............................... 159 5.1 Introduction............................................................ 159 5.2 OptimalThrustDirection.............................................. 159 5.2.1 ConstantAccelerationBound ............................... 166 5.2.2 BoundedExhaustRate....................................... 166 5.3 Time-InvariantGravityField.......................................... 168 5.4 Null-ThrustArcinCentralGravityField............................. 169 5.4.1 Inverse-SquareGravityField................................ 172 5.5 Intermediate-ThrustArc............................................... 173 5.6 Lawden’sSpiral........................................................ 174 5.7 PoweredArcs .......................................................... 180