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NAVALPOSTGRADUATE

SCHOOL

MONTEREY,CALIFORNIA

THESISREQUIREMENTSANDLIMITATIONSOFBOOSTPHASE

BALLISTICMISSILEINTERCEPTSYSTEMSby

KubilayUzunSeptember2004

ThesisAdvisor: PhillipE.PaceCoAdvisor: MuraliTummalaApprovedforpublicrelease;distributionisunlimited

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THISPAGEINTENTIONALLYLEFTBLANK

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REPORTDOCUMENTATIONPAGE FormApprovedOMBNo.07040188

Publicreportingburdenforthiscollectionofinformationisestimatedtoaverage1hourperresponse,includingthetimeforreviewinginstruction,searchingexistingdatasources,gatheringandmaintainingthedataneeded,andcompletingandreviewingthecollectionofinformation.Sendcommentsregardingthisburdenestimateoranyotheraspectofthiscollectionofinformation,includingsuggestionsforreducingthisburden,toWashingtonheadquartersServices,DirectorateforInformationOperationsandReports,1215JeffersonDavisHighway,Suite1204,Arlington,VA222024302,andtotheOfficeofManagementandBudget,PaperworkReductionProject(07040188)WashingtonDC20503.1.AGENCYUSEONLY 2.REPORTDATE

September2004 3.REPORTTYPEANDDATESCOVEREDMastersThesis4.TITLEANDSUBTITLE:RequirementsandLimitationsOfBoostPhaseBallisticMissileInterceptSystems6.AUTHOR(S)KubilayUzun7.PERFORMINGORGANIZATIONNAME(S)ANDADDRESS(ES)

CenterforJointServicesElectronicWarfareNavalPostgraduateSchoolMonterey,CA939435000

9.SPONSORING/MONITORINGAGENCYNAME(S)ANDADDRESS(ES)MissileDefenseAgency

5.FUNDINGNUMBERS

8.PERFORMINGORGANIZATIONREPORTNUMBER10.SPONSORING/MONITORINGAGENCYREPORTNUMBER

11.SUPPLEMENTARYNOTESTheviewsexpressedinthisthesisarethoseoftheauthoranddonotreflecttheofficialpolicyorpositionoftheDepartmentofDefenseortheU.S.Government.12a.DISTRIBUTION/AVAILABILITYSTATEMENT 12b.DISTRIBUTIONCODEApprovedforpublicrelease;distributionisunlimited13.ABSTRACT(maximum200words)

Theobjectiveofthisthesisistoinvestigatetherequirementsandlimitationsofboostphaseballisticmissileinterceptsystemsthatcontainaninterceptoranditsguidancesensors(bothradarandinfrared).Athreedimensionalcomputermodelisdevelopedforamultistagetargetwithaboostphaseaccelerationprofilethatdependsontotalmass,propellantmassandthespecificimpulseinthegravityfield.Theradarcrosssectionandinfraredradiationofthetargetstructureisestimatedasafunctionoftheflightprofile.Theinterceptorisamultistagemissilethatusesfusedtargetlocationdataprovidedbytwogroundbasedradarsensorsandtwolowearthorbitinfraredsensors.Interceptorrequirementsandlimitationsarederivedasafunctionofitsinitialpositionfromthetargetlaunchpointandthelaunchdelay.Sensorrequirementsarealsoexaminedasafunctionofthesignaltonoiseratioduringthetargetflight.Electronicattackconsiderationswithintheboostphasearealsoaddressedincludingtheuseofdecoysandnoisejammingtechniques.Thesignificanceofthisinvestigationisthatthesystemcomponentswithinacomplexboostphaseinterceptscenariocanbequantifiedandrequirementsforthesensorscanbenumericallyderived.14.SUBJECTTERMSBoost-PhaseBallisticMissileIntercept,Modeling,Simulation,MissileRequirements,SensorRequirements,ElectronicAttackEffects,ProportionalNavigation,RadarCrossSection,IREnergyRadiationEstimation,RFSensors,IRSensors,DataFusion,Decoys,NoiseJamming

15.NUMBEROFPAGES

16316.PRICECODE

17.SECURITYCLASSIFICATIONOFREPORT

Unclassified18.SECURITYCLASSIFICATIONOFTHISPAGE

Unclassified19.SECURITYCLASSIFICATIONOFABSTRACT

Unclassified20.LIMITATIONOFABSTRACT

ULNSN7540-01-280-5500 StandardForm298(Rev.2-89)PrescribedbyANSIStd.239-18

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Approvedforpublicrelease;distributionisunlimitedREQUIREMENTSANDLIMITATIONSOFBOOSTPHASEBALLISTIC

MISSILEINTERCEPTSYSTEMS

KubilayUzunCaptain,TurkishAirForce

B.S.,TurkishAirForceAcademy,1993Submittedinpartialfulfillmentofthe

requirementsforthedegreesofMASTEROFSCIENCEINELECTRICALENGINEERING

andMASTEROFSCIENCEINSYSTEMSENGINEERING

fromtheNAVALPOSTGRADUATESCHOOL

September2004

Author: KubilayUzun

Approvedby: PhillipE.PaceThesisAdvisor

MuraliTummalaCoAdvisor

JohnP.PowersChairman,DepartmentofElectricalandComputerEngineering

DanC.BogerChairman,DepartmentofInformationSciences

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ABSTRACT

Theobjectiveofthisthesisistoinvestigatetherequirementsandlimitationsofboostphaseballisticmissileinterceptsystemsthatcontainaninterceptoranditsguidancesensors(bothradarandinfrared).Athreedimensionalcomputermodelisdevelopedforamultistagetargetwithaboostphaseaccelerationprofilethatdependsontotalmass,propellantmassandthespecificimpulseinthegravityfield.Theradarcrosssectionandinfraredradiationofthetargetstructureareestimatedasafunctionoftheflightprofile.Theinterceptorisamultistagemissilethatusesfusedtargetlocationdataprovidedbytwogroundbasedradarsensorsandtwolowearthorbitinfraredsensors.Interceptorrequirements

and

limitations

are

derived

as

afunction

of

its

initial

position

from

the

target

launchpointandthelaunchdelay.Sensorrequirementsarealsoexaminedasafunctionofthesignaltonoiseratioduringthetargetflight.Electronicattackconsiderations withintheboostphasearealsoaddressedincludingtheuseofdecoysandnoisejammingtechniques.Thesignificanceofthisinvestigationisthatthesystemcomponentswithinacomplexboostphaseinterceptscenariocanbequantifiedandrequirementsforthesensorscanbenumericallyderived.

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TABLEOFCONTENTS

I. INTRODUCTION.......................................................................................................1A. BALLISTICMISSILEDEFENSE................................................................1B. PRINCIPALCONTRIBUTIONS.................................................................5C. THESISOUTLINE.........................................................................................7

II. TARGETMODELING..............................................................................................9A. BASICDEFINITIONSANDASSUMPTIONS...........................................9B. COORDINATESYSTEMS.........................................................................10C. THEGRAVITYFIELD...............................................................................11D. TARGETVELOCITYREQUIREMENTS...............................................14E. BOOSTINGTARGETMODELING..........................................................16

1. SiloExitVelocity...............................................................................172.

The

Rocket

Equation

and

Consequences........................................17

F. BOOSTINGTARGETINTHEGRAVITYFIELD.................................20G. SUMMARY...................................................................................................23

III. INTERCEPTORMISSILEMODELING..............................................................25A. BASICDEFINITIONSANDASSUMPTIONS.........................................25B. BOOSTINGMISSILEMODELING..........................................................25C. MISSILEGUIDANCE.................................................................................26

1. GuidanceSystemAgainstConstantSpeedTarget........................302. GuidanceSystemAgainstICBMModel.........................................33

D. FLIGHTCONTROLSYSTEM..................................................................36E. MISSILEREQUIREMENTS......................................................................39F. SUMMARY...................................................................................................45

IV. RADARCROSSSECTIONANDIRENERGYRADIATIONPREDICTION..47 A. TARGETSTRUCTURE..............................................................................48B. POFACETSMODELING...........................................................................49C. RCSPREDICTION......................................................................................50D. ESTIMATIONOFPLUMEIRRADIATION...........................................54E. SUMMARY...................................................................................................56

V. SENSORMODELING.............................................................................................59A. TRANSMISSIONDELAY...........................................................................62B. TRACKINGINACCURACIES..................................................................63

1. RFSensorInaccuracies....................................................................632. IRSensorInaccuracies .....................................................................733. DataFusion........................................................................................784. MissilePerformance.........................................................................79

C. SUMMARY...................................................................................................81VI. ELECTRONICATTACKEFFECTS....................................................................83

A. EFFECTOFDECOYS.................................................................................83vii

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1. DecoyTrajectory...............................................................................832. IRDecoys(Flare)..............................................................................863. RFDecoys(Chaff).............................................................................884. TrackTransfertoDecoy..................................................................90

B. EFFECTOFNOISEJAMMING................................................................97C.

SUMMARY

.................................................................................................102

VII. CONCLUSIONS.....................................................................................................103

A. SUMMARYOFTHEWORK...................................................................103B. SIGNIFICANTRESULTS.........................................................................103C. SUGGESTIONSFORFUTUREWORK.................................................106

APPENDIXA CODEFLOWCHART...................................................................109APPENDIXB THEMATLABCODE...................................................................117LISTOFREFERENCES................................................................................................... 139INITIALDISTRIBUTIONLIST...................................................................................... 141

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LISTOFFIGURES

Figure21. TheBasicReferencefortheSimulation,CartesianCoordinateSystemandtheEarthsLocation........................................................................................ 11Figure22. DefinitionsfortheTrajectoryEquation,theCentralAngleandtheRanger.......................................................................................................... 12

Figure23. TargetTrajectorywithaLaunchSpeedof0.91km/s(3,000feet/s)andLaunchAngleof45....................................................................................... 13



Figure26. TargetVelocityRequirementstoHitaGivenDistance.................................15Figure27. 3DOverviewofanICBMAttackfromKilju-kunMissileBase,North

KoreatoSanFrancisco,California................................................................. 20Figure28. GroundDistanceversusHeightfortheSanFranciscoAttack.......................21 Figure29. VelocityversusFlightTimefortheSanFranciscoAttack.............................21 Figure210.VelocityversusFlightTimefortheSanFranciscoAttack(BoostPhase

Only)............................................................................................................... 22Figure211.TotalMassversusFlightTimefortheSanFranciscoAttack(BoostPhase

Only)............................................................................................................... 23Figure31. MissileBlockDiagram...................................................................................27 Figure32. 3DOverviewofaTypicalInterceptfortheConstantSpeedScenario...........30Figure33 TheTargetandtheMissileFlightCharacteristicsfortheConstantSpeed

Scenario:(a)GroundDistanceversusHeight,(b)VelocityversusFlightTime.

...............................................................................................................

31

Figure34. TheTargetandtheMissileClosureCharacteristicsfortheConstantSpeed

Scenario:(a)RangeversusFlightTime,(b)ClosingVelocityversusFlightTime................................................................................................................ 32

Figure35. MissileGuidanceCharacteristics fortheConstantSpeedScenario:(a)MissileLateralAcceleration,(b)MissileLateralDivert................................ 32

Figure36. 3DOverviewoftheInterceptfortheAcceleratingTarget.............................33 Figure37. TargetManeuverduringtheIntercept............................................................34 Figure38 TheTargetandtheMissileFlightCharacteristicsfortheAccelerating

Target:(a)GroundDistanceversusHeight,(b)VelocityversusFlightTime................................................................................................................ 34

Figure39.

The

Target

and

the

Missile

Closure

Characteristics

for

the

Accelerating

Target:(a)MissileTargetDistanceversusFlightTime,(b)MissileTargetClosureVelocityversusFlightTime................................................... 35

Figure310.MissileGuidanceCharacteristicsfortheAcceleratingTarget:(a)MissileLateralAcceleration,(b)MissileLateralDivert............................................. 36

Figure311.(a)ControlSystemLag,(b)Detail................................................................. 37Figure312.MissDistanceversusTimeConstantfortheConstantSpeedScenario.........38

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Figure313.LimitationtotheMissileLaunchSiteDistancefromtheTargetLaunchSite:MissileDirectlyatAttackDirection,NoLaunchDelay........................ 40

Figure314.PotentialAttackDirectionsandtheMissileLocation.................................... 41Figure315.LimitationtotheMissileLaunchSiteDistancefromtheTargetLaunch

Site:70AngularError,NoLaunchDelay..................................................... 42Figure316.LimitationtotheTolerableLaunchDelayforGM1LocatedatAttackDirection......................................................................................................... 43Figure317.LimitationtotheMissileLaunchSiteDistancefromtheTargetLaunch

Site:40AngularError,GM3,SanFranciscoAttack.................................. 43Figure318.LimitationtotheTolerableLaunchDelay:40AngularError,GM3,San

FranciscoAttack............................................................................................. 44Figure41. SimpleGeometricalShapesUsedtoConstructtheModel.............................48 Figure42. FacetStructureUsedtoConstructtheModel:(a)Detail,TopViewand

NoseCone,(b)Detail,Nozzle........................................................................ 49Figure43. FullScaleModelsofStages:(a)Stage1,(b)Stage2,(c)Stage3,(d)

Payload............................................................................................................ 50Figure44. TheMonostaticAngle.................................................................................51 Figure45. RCSComparisonofStages:(a)LBand(f=1.5GHz),(b)SBand(f=

3GHz),(c)CBand(f=6GHz),(d)XBand(f=10GHz)............................. 52Figure46. RCSComparisonofFrequencies:(a)Stage1,(b)Stage2,(c)Stage3,

(d)Payload...................................................................................................... 53Figure47. SpectralRadiantExitance,Blackbodyat1400K............................................55Figure48. RadiationIntensityversusTime.....................................................................56 Figure51. TheGeographicScenariofortheBoostphaseBallisticMissileIntercept

includingLocationsoftheSensors,theMissile,andtheTarget....................60Figure52. TheSchematicScenariofortheBoostphaseBallisticMissileIntercept

includingLocationsoftheSensors,theMissile,andtheTarget....................61Figure53. RCSSamplingLocations................................................................................61 Figure54. AverageRCSSeenbyRFSensorasaFunctionofBearingandRange

fromtheTargetLaunchSite........................................................................... 62Figure55. RCSSeenbyRF1andRF2duringtheIntercept:(a)LBand,(b)S

Band,(c)CBand,(d)XBand....................................................................... 64Figure56. RFSensortoTargetRange.............................................................................65 Figure57. EffectofPeakPowertoTrackingAccuracy:(a)Angle,(b)Range...............68Figure58. EffectofHalfpowerBeamwidthtoTrackingAccuracy:(a)Angle,(b)

Range.............................................................................................................. 69Figure59. EffectofPulsewidthtoTrackingAccuracy:(a)Angle,(b)Range.................70Figure510.EffectofPulseIntegrationtoTrackingAccuracy:(a)Angle,(b)Range....... 70Figure

511.

Single

Pulse

SNR

versus

Flight

Time.............................................................

71

Figure512.MagnitudeofPositionErrorversusFlightTime,(a)RF1,(b)RF2...........73Figure513.AtmosphericTransmittanceversusTargetHeight......................................... 74Figure514.SignaltoClutterRatioofIRSensors............................................................ 76Figure515.TheIRsensorTargetTriangle....................................................................... 77Figure516.MagnitudeofPositionErrorversusFlightTime(IR).................................... 78Figure517.MagnitudeofPositionErrorversusFlightTime(Fused)............................... 79

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Figure518.ClosureandGuidanceCharacteristicsfortheMissileGuidedbySensed

TargetPositionData(a)ClosingVelocityversusFlightTime,(b)LateralAccelerationversusFlightTime..................................................................... 80

Figure519.TargetPositionErrorversusMissDistance.................................................... 81Figure61. DecoyTrajectory(Releasedatt=90s)...........................................................84Figure

62.

Decoy

Trajectory

(Ground

Distance

versus

Height)

for

Target,

Missile

and

Decoy.............................................................................................................. 85

Figure63. DecoySeparation,TargetDecoyDistanceversusFlightTime.....................85 Figure64. DecoyVelocityversusFlightTime................................................................86 Figure65 SpectralRadiantExitanceofPlumeandFlareversusWavelength................88Figure66. ProbabilityofAverageRCSofChaffCloudExceedstheTargetRCS

versusNumberofChaffDipolesDispensed................................................... 90Figure67. 3DOverviewoftheIntercept:BothRFSensorTracksCapturedbyDecoy...91Figure68 IncreaseintheLateralAccelerationRequirementswhenbothRFSensor

TracksCapturedbytheDecoy........................................................................ 92Figure69. 3DOverviewoftheIntercept:EitherRF1orRF2TrackCapturedby

Decoy..............................................................................................................

92

Figure610.LateralAccelerationversusFlightTime,EitherRF1orRF2TrackCapturedbyDecoy......................................................................................... 93Figure611.ScenarioforTrackTransfertoDecoyandConsecutiveReacquisitionof

theTarget........................................................................................................ 94Figure612.MissDistanceasaFunctionofDecoyReleaseandReacquisitionTime.......95Figure613 MissDistanceasaFunctionof(a)DecoyReleaseTime,(b)Reacquisition

Time................................................................................................................ 97Figure614.S/JRatioduringtheInterceptfor1kWJammer........................................... 100Figure615.EffectofJammerPower(4GHzBandwidth):RMSErrorversusFlight

Timein,(a)Range,(b)Angle....................................................................... 100Figure616.EffectofJammerBandwidth(1kWPower):RMSErrorversusFlight

Timein,(a)Range,(b)Angle....................................................................... 101Figure617.EffectofJammingonMissDistance............................................................ 102FigureA1. CodeFlowchart(1of7)................................................................................109 FigureA2. CodeFlowchart(2of7)................................................................................110FigureA3. CodeFlowchart(3of7)................................................................................111FigureA4. CodeFlowchart(4of7)................................................................................112FigureA5. CodeFlowchart(5of7)................................................................................113FigureA6. CodeFlowchart(6of7)................................................................................114FigureA7. CodeFlowchart(7of7)................................................................................115

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LISTOFTABLES

Table21. TargetDataMatrix..........................................................................................16 Table22. TheoreticalVelocityCapabilityofTargetModel...........................................19Table31. MissileDataMatrix........................................................................................26 Table32. SummaryofGenericMissileSpecifications...................................................39Table41. ICBMPlumeParameters................................................................................54 Table51. GainandAntennaDiameterversusRequiredHalfPowerBeamwidth..........66Table52. RFSensorParameterstobeExamined...........................................................67 Table53. GenericRadarParameters...............................................................................71Table54. IRSensorParameters......................................................................................76Table55. MissileTestParameters..................................................................................79 Table61. RCSasSeenbytheRFSensorsduringIntercept...........................................89Table62. PossibleBandwidthstobeConsideredbytheJammer..................................99

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ACKNOWLEDGMENTS

IwouldliketothankmywifeOzumforherpatienceandsupport.IwouldliketothanktomythesisadvisorsProfessorPhillipE.PaceandProfessor

MuraliTummalafortheirhelptoconductthisresearch.AlsoIwouldliketothanktoProfessorBretMichaelandtheballisticmissiledefenseteamfortheirvaluableideas.

ThisworkwassupportedbytheMissileDefenseAgency.

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EXECUTIVESUMMARY

Thisresearchinvestigatedthebasicrequirementsandlimitationsofboostphaseballisticmissileinterceptsystems.Inordertoaccomplishthis,acomputercodewasdevelopedtomodelavarietyofsystemcharacteristicsincludingmotioninthreedimensionalspace.Afterdefiningtheballisticmissile(referredtoastargetinthetext)andtheinterceptor(referredtoasmissileinthetext)indetail,theradiofrequency(RF)andinfrared(IR)sensorcharacteristicsandtheirabilitytoguidethemissiletothetargetwereexplored.Thenormaloperationoftheboostphaseballisticmissileinterceptsystemwastestedforseveralscenarios.Finally,theeffectsofthepossibleuseofelectronicattack

on

the

defense

system,

which

aballistic

missile

target

may

employ

during

the

boost

phase,wereinvestigated.

Developingacomputercodetosimulatetheboostphasescenariowasthemethodologyusedforthisresearch.Equationswereusedregardingmissiletrajectories,propulsion,andsensorcalculationstoconstructatheoreticalbasisfortheresearch.Thecomputersimulationresultsforeachstepwereverifiedbyusingsimplecases.Then,allverifiedpartsofthesystemwerebroughttogethertorunthecomplexcases.

Theboostphaseinterceptschemewasconstructedaroundthefollowingscenario.Anintercontinentalballisticmissileislaunchedfromagivenlaunchsite.ThetargetistrackedbytwogroundbasedRFsensorsandtwospacebasedIRsensors.Thetargetpositiondataistransmittedtoafusionprocessortocalculateanaccuratetargetposition.Thefusedtargetpositiondataisusedtoguideamissile.Themissileislaunchedafteracertaindelayfollowingthetargetlaunchandestablishesacollisiongeometrywiththetarget.Atasuitabledistance,akillvehicleislaunchedfromthemissiletoaccomplishtheintercept.Thekillvehiclehitstokillthetarget,andtheinterceptisaccomplished.

Thefirststepinthedevelopmentofthesimulationwasmodelingthemechanicsofthetarget.ThetargetwasmodeledbyevaluatingthesumofallactingforcevectorsinthethreedimensionalCartesiancoordinatesystem.ThechangeinmassduetofuelconsumptionandchangeingravityduetothedistancefromtheEarthscenterwerealsoconsidered.Propulsionwasmodeledbyusingtheconsumptionrateandthespecificimpulse

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ofthefuelused.Thetrajectoryequationandtherocketequationwereusedasatheoreticalbasisfortheflightofthetarget.Alltestrunsshowedthatthecomputermodelreflectedtheresultsoftheequationssatisfactorily.Afterverification,anexamplecaseincludinganintercontinentalballistictargetattacktargetingSanFrancisco,Californiawasconducted,andthemeasureddatayieldedvaluablefindingsregardingallphysicalparametersofthemissileduringitsflight,suchasdistances,heights,andvelocities.

Thesecondstepwastomodelthemissile.ProportionalnavigationinathreedimensionalCartesiancoordinatesystemwasimplemented.Toverifytheresults,asimplecasewithaconstantspeedtargetwassimulated.Testsshowedthattheimplementationofproportionalnavigationworkedsatisfactorilyandthetargetwashit.Themissilemodelwasrunagainstthetargetmodeldevelopedpreviouslyanddatawerecollectedandpresented.Finally,thezerolagcontrolsystemdevelopedsofarwasextendedtoanonzerolagcontrolsystembymodelingthemissiledynamicswithasingletimeconstant,thirdordertransferfunction.Themissdistanceresultsduetomissiledynamicswerepresented.Next,therequirementsregardinglocationofthemissilewereinvestigated.Testrunsyieldedgoodinsightintermsofmissilecapabilityandlocation.

Thethirdstepwasthepredictionoftargetparametersfromthesensorspointofview.Thisincludedtheradarcrosssection(RCS)forRFsensorsandIRradiationestimationforIRsensors.ThemonostaticRCSwaspredictedbymodelingthetargetstructurewithtriangularfacets.Themodeledstructurewasevaluatedbyanothersoftwareprogram(POFACETS).CalculatingtheplumeIRradiationintensityusingPlanckslawandintegratingtheemittedenergyinthebandofinterestsummarizedthesimplisticIRenergyradiationestimation.

Thefourthstepwasmodelingofthesensors.OptimallocationfortheRFsensorswasdetermined.Thetransmissiondelaybetweenthetargettrackdatacollectionandmissileguidancewasmodeled,andtheireffectwasinvestigated.AsetofRFsensorparameterswasproposed,andtheeffectofdifferentradardesignparameterswasinvestigatedindetail.Theresultingradarspecificationswereutilizedtoquantifythesignaltonoiseratioduringtheintercept.TheRMSerrorinangleandrangewerequantifiedusingthecomputermodel.Theprobabilisticnatureoftargetpositionalerrorwaspresented.IRsen

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sordesignparameterswerediscussed.TwoIRsensorswerelocatedatalowEarthorbit,andaprobabilistictargetpositionalerrorintroducedbytheIRsensorswasquantified.Datafusionwasimplementedbyaveragingtargettrackinputs.Thefusedtrackdatawerethenusedtoguidethemissile,andresultswerepresented.Theeffectoftrackingqualityonmissdistancewasinvestigated.

Thefinalstepwastheinvestigationofelectronicattackeffectsintheboostphase.Acommonassumptionisthataballisticmissilehasenoughtimeandopportunitytoattackadefensesystemelectronicallybyusingmanymeasures,suchasusingmultiplewarheads,decoys,and/ormetallizedballoons,ordisguisingitsIRsignaturebycoolingorshroudingthewarheadaftertheboostphase,butthemissiledoesnotprioritizetheelectronicattackwhileitisstillintactandaccelerating.However,thereisnoreasonfortheballisticmissilenottoperformthistypeofattack,althoughitistechnicallymorecomplicated.Toexploretheelectronicattackeffects,thedecoytrajectorywasmodeledinthesimulation,andseparationofthedecoyduetoaccelerationofthemissilewasshown.TheeffectofIRandRFdecoyswasinvestigated.Theamountofchaffdipolesrequiredtoscreenthetargetwascalculated.Theeffectofreacquisitionfollowingatracktransfertodecoywasalsoexaminedaswellastheeffectofnoisejamming.

Milestonesusedtoconstructthemodelledtomanyresultsandcontributions.Mechanicalmodelsofthetargetandthemissileunearthedmanyrequirementsandlimitationsalongwiththeabilitytochoosethecapabilityandlocationofthedefensesystemelements.Theworkalsoshedlightontheeffectivenessofthecommonelectronicattacks,suchasIRandRFdecoysaswellasnoisejamming.

Resultsreportedhereweresignificantsincetheboostphaseinterceptscenarioshavenotbeeninvestigatedpreviouslyasmuchastheotherphases.ThecomputercodeusesathreedimensionalEarthcenteredsystemthatotherresearcherscaneasilyusetoimplementdifferentscenarios.Deductionofmissileparametersintermsofcapabilityandpositionmaycontributetothefuturedecisionsonongoingnationalmissiledefenseplans.ExaminationofRFandIRsensorparametersandlocationsarealsosignificant.Finally,investigatingtheelectronicattackduringtheboostphaseanswersmanyquestionswhileraisingmorequestionsforfutureinvestigations.

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I. INTRODUCTION

A. BALLISTICMISSILEDEFENSEDefendingtheUnitedStatesagainstalongrangeballisticmissileattackhasbeen

anissueformanyyears.Theballisticmissiledefenseplansarosemanytimesandinmanyformsduringthepast50years.AccordingtoFowler,itappearsonceevery17yearsindifferentforms[Ref.1].TheStrategicDefenseInitiative(SDI),alsoknownastheStarWarsProject,wasproposedduringtheReaganadministrationin1985.Effortsusinglaserbaseddefenseprogramsintensifiedinthelate1980sandhavecontinuedwiththeairbornelaser(ABL)mostrecently[Ref.2].

AsaresultoftheobservationsontheWorldsnewmembersofthemissileclub,theUnitedStateshasenactedtheCongressionalNationalMissileDefenseActin1989statingthataprogramtodefendtheUntiedStatesagainstthemissileattacksisacceptedaspolicy[Ref.1].

Duringthelastdecade,sincethecollapseoftheUSSRin1991,thequestionsregardingtheabilityofasmallcountryhavingtheintentandcapabilitytohittheUnitedStateswithlongrangeballisticmissileshavebeenaskedincreasingly.DonaldRumsfeldpresentedareporttoCongressin1998discussingthepresenceofthiskindofthreat[Ref.3].ThereportpointedoutthegrowingmarketintheareafedbyuncontrolledknowhowandpersonnelunleashedbythecollapseoftheUSSRandmotivatedbymoney[Ref.3].

Beforethereport,itwasgenerallybelievedthatindividualmissiletechnologiesdevelopedbysmallcountrieswouldtendtobeoriginal.However,manyanalystsnowbelievethattheproliferationofmissilesismuchmorelikelybyusingsimplermethods,suchasbraindrain,transferringtechnologyintothecountryorsimplybuyingthem[Ref.4].

ItisverywellknownthattheRumsfeldreportstatesthatIranorNorthKoreacanachievesuchacapabilitywithinfiveyearsofdecidingso.Whetherornotthisisanoverestimation,thereportraisescrucialquestions.Canasmallstateobtainsuchacapabilityinthenearfuture?Whatisthemotivationforsuchastatefordoingso?AccordingtoOberg,passionforthirdworldcountriestoseizeonthecapabilityoflaunchingrockets

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servesthreemainobjectives:theabilitytocarrywarheadsontotheterritoryoftheiropponents,contributingtospaceapplications,andbecomingaworldpower[Ref.4].AccordingtotablesprovidedbyZakheim,lesserpowershavingballisticmissileswitharangeofmorethan1,000kmareIndia,Iran,Israel,NorthKorea,PakistanandUkraine[Ref.3].

NorthKoreaisoneofthecountriesattractingspecialconsideration.DoesNorthKoreahavesuchanintentionorcapability?NorthKoreasICBMdevelopmentprogramisevidentintermsofintentaswellascapability.InterrogationofadefectorhasunearthedthatthefinalobjectiveofNorthKoreaistobuildmissilescapableofhittingtheUnitedStates.TaepodongIImissilesarebelievedtohavebeenimprovedforcarryinglargepayloadsto4,0006,000kmwhiletheymayhavethecapabilityofcarryinglighterpayloadsupto10,000km[Ref.4].NorthKoreasNodongtestlaunchin1993focusedtheattentionofscientistsworldwideontheincreasingcapabilityofthiscountry.AlthoughitwasclaimedbyNorthKoreathatthiswasalaunchintendedtocarryasatelliteintoorbit,asofyet,noonehasfoundanyevidenceofthissatellite.Whetherornotitwasafailedsatellitelaunchoratricktohideanearintercontinentalballisticmissiletest,thelaunchshowedthatthethreatisimminent[Ref.4].

AlthoughthethreatisevidentandconfidenceaboutdefendingtheUnitedStatesfromanyconventional,nuclear,chemical,orbiologicalattackisabsolute,manyscientistsdisagreewiththeexistingroadmapoftheNationalMissileDefense(NMD)efforts.AdebatecontinuesontheNMDprogram.Someoftheissues,whichFowlerhaspointedout,areasfollows.Firstly,realizationofsuchacapabilitytoneutralizeexistingintercontinentalhitcapabilityofRussiaandChinamaydrivethemtobuildmorecapabilities.Secondly,launchinganintercontinentalballisticmissilemaynotbethefirstprioritymethodtoplaceaconventional,nuclear,chemical,orbiologicalthreatintheUnitedStates.

Finally,

it

is

far

from

convincing

that

the

existing

ballistic

missile

defense

tech

nologyiscapableofdoingtheintendedjob[Ref.1].

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Regardingthethreatpriority,althoughtheSeptember11thdisasterhaschanged

thethreatperceptionradically,someobserversfeelthatrecentannouncementsindicatethattheBushadministrationisfarfromstoppingtheeffortsregardinganintercontinentalattack,whichmaybeexecutedbyballisticmissiles[Ref.5].

Capabilityisanotherissue.SomefeelthattheexistingNMDplansareconsideredincompleteandnothingmorethanadeterrencetool[Ref.1].Whyareongoingplansconsideredunsatisfactory? Mostoftheproblemsassociatedwiththeexistingplanarerelatedtothemidcourseinterception,andthemidcourseballisticmissileinterceptstilloccupiesamajorityofresourcesusedbytheBallisticMissileDefenseOrganization[Ref.5].

AcommonapproachistodetectthetargetbyusingspacebasedIRsensors,whichcansensethehugeamountofenergyemittedfromtherocketplume.Afterdetection,atargettrackisestablishedbygroundbasedRFsensors(radars)incoordinationwithIRsensorstogenerateusefulandaccuratetargetpositiondatatoguideaninterceptor.Sinceanintercontinentalballisticmissileshouldtravelroughly10,000kmandfly3040minutes,itmakesperfectsensetolookforthecapabilitytointercepttheICBMforthemidcourseorterminalphasebeforeithitsthetarget.Asdetailedbelow,however,severalresearchershaveshownthatthisisnotasstraightforwardasitseems.

Althoughmidcourseinterceptofballisticmissileshasseveraladvantages,suchastheabilitytolocateassetsathomeandadequatetimefordetection,decisionandinterception,ithasmoredrawbacks.Thedisadvantagescanbesummarizedasbeingmoresusceptibletoelectronicattack,probabilityofdebrislandinginfriendlyterritoryifthewarheadisnotcompletelydestroyedandpossibilityofthedefensesystembeingoverwhelmedbyutilizationofsubmunitionsinsteadofasinglewarhead[Ref.5].

Assumingthatthethreatwillcompleteitsflightintactwouldbedangerouslyoversimplifyingtheproblem.Additionally,therearemanytypesofelectronicattack,whichcanbeusedbytheballisticmissileduringitsflight.LewisandPostollisttheseasmultiplesubmunitions,decoys,radarandinfraredstealthbyshrouding,andmaneuvers[Ref.6].

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TheMITCountermeasuresReportalsoemphasizessimilarissuesthatcanbe

summarizedasfollows.Firstly,thereisnoreasontobelievethatacountrycapableofbuildingandlaunchingaballisticmissilecanalsoexploitanelectronicattack.Secondly,anelectronicattackmightverylikelyaffect,overwhelmorfailtheplannedNMDsystem.Finally,anelectronicattackmayincludesubmunitions,falsetargetsincludingreplicadecoys,decoysusingsignaturediversity,anddecoysusingantisimulation(metallizedballoons,shrouds,chaff,electronicdecoys),radarsignaturereduction,infraredstealth,hidingthewarhead,andmaneuver[Ref.7].

Historicallessonslearnedshowthattheattackerdoesnotneedtopossessthesophisticatedtechnologyasthedefendertodefeatthedefensesystem.Duringthe1991GulfWar,probablyunintentionalbreakupandtumblingofalHusaynmissilesresultedinthealmosttotalfailureofPatriotdefenses[Ref.6].Allsolutionsassociatedwiththepostboostphasedefensemustconsideracommonfactthatwhentheaccelerationofthemissileends,thepossibilityisgreatforthedeploymentofdifferentelectronicattacks.Forlongrangeballisticmissiles,eachdeployedparticlefromthemainpayloadfollowsthesametrajectoryregardlessofitsmass.Thus,asmallchaffdipoleweighingontheorderofgramsisnotdifferentfromaheavywarheadinouterspacewhereatmosphericeffectscanbeneglected.

Thissituationnaturallydirectsmindstoanotheroption,whichistheboostphaseintercept.Boostphaseisdefinedastheinitialstageoftheballisticmissileflightlastingfrommissilelaunchtotheburnoutofrocketengines[Ref.5].Duringthepoweredflightwherethemissileisstillintact,technicalconsiderationsdiffer[Ref.6].Theboostphaseballisticmissileintercept,althoughnotaffectedbythefactorsassociatedwiththeotherphases,hasotherproblemstomanage.Theseareusuallydetectionanddecisionrequirements,whichinturn,resultinaneedtolocatetheinterceptorsveryclosetothetargetlaunch

site.

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Anothercrucialquestionarisesimmediately:Whatiftheelectronicattackisexe

cutedduringtheboostphase?TheresultingpointconcerningtheNMDeffortsistheassumptionthatbyusingaboostphaseinterceptplan,mostoftheproblemsassociatedwiththemidcourseinterceptcouldberesolved.Howvalidisthisassumption?Whatarethestrongandweakpointsofthedefensesystemagainstthiskindofattack?B. PRINCIPALCONTRIBUTIONS

Thisresearchinvestigatedthebasicrequirementsandlimitationsofboostphaseballisticmissileinterceptsystems,whichwasaccomplishedbydevelopingacomputercodetomodelavarietyofsystemcharacteristicsincludingmotioninthreedimensionalspace.Afterdefiningtheballisticmissile(referredtotargetinthetext)andtheinterceptor(referredtoasmissileinthetext)indetail,theradiofrequency(RF)andinfrared(IR)sensorcharacteristicsandtheirabilitytoguidethemissiletothetargetwereexplored.Thenormaloperationoftheboostphaseballisticmissileinterceptsystemwastestedforseveralscenarios.Finally,theeffectsofthepossibleuseofelectronicattackonthedefensesystem,whichaballisticmissiletargetmayemployduringtheboostphase,wereinvestigated.

Themethodologyusedforthisresearchconsistedofdevelopingacomputercodetosimulatetheboostphasescenario.Thecomplexityoftheinteractingdynamicsinthescenariousuallylimitstheopportunitytodefineallelementswithsimpleequations.Therefore,astepbystepapproachwasfollowedtoverifytheaccuracyoftheresults.Equationswereusedregardingmissiletrajectories,propulsion,andsensorcalculationstoconstructatheoreticalbasisfortheresearch.Thecomputersimulationresultsforeachstepwereverifiedbyusingsimplecases.Then,allverifiedpartsofthesystemwerecombinedtorunthecomplexcases.

Theboostphaseinterceptschemewasconstructedaroundthefollowingscenario.An

intercontinental

ballistic

missile

is

launched

from

agiven

launch

site.

The

target

is

trackedbytwogroundbasedRFsensorsandtwospacebasedIRsensors.Thetargetpositiondataistransmittedtoafusionprocessortocalculateanaccuratetargetposition.Thefusedtargetpositiondataisusedtoguideaninterceptmissile.Themissileis

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launchedafteracertaindelayfollowingthetargetlaunchandestablishesacollisiongeometrywiththetarget.Atasuitabledistance,akillvehicleislaunchedfromthemissiletoaccomplishtheintercept.Thekillvehiclehitstokillthetarget,andtheinterceptisaccomplished.

Twoimportantelementsofthescenarioarebeyondthescopeofthisresearch.Thefirstisthedatafusion.Thetrackdataarefusedherebyusingasimpleaveragingmethod.Thesecondiskillvehicleflight.Themissileisallowedtoflyuntilithitsthetargetinsteadoflaunchingakillvehicle.

Theworkreportedhereconsistsofmanyresultsandcontributions.Athreedimensionalcomputermodelwasdevelopedforamultistagetargetwithaboostphaseaccelerationprofilethatdependsontotalmass,propellantmassandthespecificimpulseinthegravityfield.Theradarcrosssectionandinfraredradiationofthetargetstructurewasestimatedasafunctionoftheflightprofile.Theinterceptorisamultistagemissileusingfusedtargetlocationdataprovidedbytwogroundbasedradarsensorsandtwolowearthorbit(LEO)infraredsensors.Interceptorrequirementsandlimitationswerederivedasafunctionofitsinitialpositionfromthetargetlaunchpointandthelaunchdelay.Sensorrequirementswereexaminedasfunctionofthesignaltonoiseratio(SNR)duringthetargetflight.Electronicattackconsiderationswithintheboostphasearealsoaddressed,includingtheuseofdecoysandnoisejammingtechniques.Mechanicalmodelsofthetargetandthemissileunearthedmanyrequirementsandlimitationswiththeabilitytochoosethecapabilityandlocationofthedefensesystemelements.Theworkalsoshedlightontheeffectivenessofthecommonelectronicattacks,suchasIRandRFdecoysaswellasnoisejamming.

Thesignificanceofthisinvestigationisthatthesystemcomponentswithinacomplexboostphaseinterceptscenariocanbequantifiedandrequirementsforthesensorsnumericallyderived.ThecomputercodeusesathreedimensionalEarthcenteredsystemthatotherresearcherscaneasilyusetoimplementdifferentscenarios.Deductionofmissileparametersintermsofcapabilityandpositionmaycontributetofuturedecisionsforongoingnationalmissiledefenseplans.ExaminationofRFandIRsensorpa-

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rametersandlocationsarealsosignificant.Finally,investigatingtheelectronicattackduringtheboostphaseanswersmanyquestionswhileraisingmorequestionsforfutureinvestigation.C. THESISOUTLINE

ChapterIIisdedicatedtothedevelopmentofthecodeandmodelingthemechanicsofthetarget.ThescenariowhereanintercontinentalballisticmissileattacksSanFrancisco,Californiaisconducted,andtheresultspresentedintermsofthetargetparameters,suchasdistances,heights,andvelocities.

ChapterIIImodelsthemissile.TheproportionalnavigationisimplementedinthreedimensionalCartesiancoordinatesystem.Themissilemodelisrunagainstthetargetmodeldevelopedpreviously,anddataarecollectedandpresented.Afterfinishingmodelingmissilemechanics,therequirementsregardinglocatingthemissileareinvestigated.Resultsoftestrunsarepresented.

ChapterIVpredictstargetparametersfromthesensorspointofview.ThemonostaticRCSispredictedbymodelingthetargetstructurewithtriangularfacets.Themodeledstructurewasevaluatedbyanothersoftwareprogram(POFACETS).CalculatingtheplumeradiationintensitybyusingPlanckslawandintegratingtheemittedenergyinthebandofinterestsummarizethesimplisticIRenergyradiationestimation.

ChapterVmodelsthesensors.OptimallocationsfortheRFsensorsarefound.Thischaptermodelsthetransmissiondelaybetweenthetargettrackdatacollectionandmissileguidance.AsetofRFsensorparametersisproposed,andtheeffectsofdifferentradardesignparametersontrackingqualityareinvestigatedindetail.ThemodellocatestwoIRsensorsinaLEOandquantifiestheprobabilistictargetpositionalerrorintroducedbythesensors.Thefusedtrackdataareusedtoguidethemissile,andtheresultsarepresented.

ChapterVIinvestigatestheeffectofelectronicattackintheboostphase.Toexploretheelectronicattackeffects,thesimulationmodelsthedecoytrajectory,andalsotheseparationofthedecoyduetoaccelerationofthemissile.TheeffectofIRandRFde-

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coysisinvestigated.Theamountofchaffdipolesrequiredtoscreenthetargetiscalculated.Theeffectofreacquisitionfollowingatracktransfertodecoyisalsoexaminedaswellastheeffectofnoisejamming.

ChapterVII

provides

the

concluding

remarks.

AppendixAshowsadetailedchartforthecodeflow.AppendixBprovidesa

completelistingoftheMATLABcodedevelopedforthisresearch.

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body,m isthetotalmass(inkg),andaisthenetaccelerationvector(inm/s2).Thereare

Thethrustvectorisassumedtobeinthedirectionofthevelocityvectorv.Inthe

dt g.

II. TARGETMODELINGThischapterpresentsathreedimensionaltargetmodelthatoperatesinthe

Earthsgravityfield.Thesimulationmodelsamultistage,boostingtargetcapableofreachingthevelocityof6.5km/sthatenablesittoreachintercontinentaldistances.Thebasicdefinitionsandassumptionsforthemodel,thecoordinatesystemsused,thegravityfieldeffects,andthetargetvelocityrequirementsarediscussedasfollows.A. BASICDEFINITIONSANDASSUMPTIONS

ThetargetbodyobeysNewtonsSecondLawthatcanbedefinedinvectorformasEquationChapter2Section1

Fma (2-1)

whereFistheforcevector(inN)actingonthecenterofgravity(CG)ofthetarget

onlytwotypesofmajorforcevectorsactingontheCGofthetargetbody.Thesearethe

thrustTandtheweightW.ThenetforcevectorFnetcanbewrittenas

FnetTW. (2-2)

model,thedirectionofthethrustvectorisnotmodified(i.e.,thrustisnotvectored)meaningthatthetargetmakesagravityturn[Ref.8:p.255].Todevelopthethrustvectormagnitude T,thestagespecificimpulseIsp(ins)isfirstexpressedas[Ref.8:p.255].

Isp TW (2-3)

wherechangeinweightovertimeWcanbedefinedasafunctionofchangeinmassovertimeorinstagefuelconsumptiondmdt(inkg/s)andgravitationalaccelerationatthecurrentdistancefromthecenteroftheEarthg(inm/s2)as

W dm (2-4)Substituting(2-4)into(2-3)andsolvingforTgives

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dt gIsp.T dm (2-5)Thestagespecificimpulseisassumedtobeconstant,andfuelisassumedtode

creaselinearlyduringthestage.TheweightvectorisinthedirectionofthecenteroftheEarth.Themagnitudeof

theweightvectorWcanbewrittenasWmg. (2-6)

Sincemostoftheinterceptionoccursintheexoatmosphericregion,dragisneglected.Also,sincethescopeofthisstudyisonlyontheboostphase,whichoccursatrelativelysmalldistancesfromtheEarthandshorttimeswithrespecttotheoveralltargetflight,theEarthisassumedtobeaperfectnonrotatingspherewitharadiusof6,370km.Theabovedefinitionsandassumptionswereusedtobuildthemodel.B. COORDINATESYSTEMS

Threecoordinatesystemsareusedwithinthemodel.ThefirstistheEarthcentered,Earthfixed(ECEF)Cartesiancoordinatesystem.

IntheECEFCartesiancoordinatesystem,allcomputationsoccurinthreeorthogonalaxes.TheEarthislocatedattheorigin.Inarighthandedsystem,thepositivexaxispassesthrough0N,0E,thepositiveyaxispassesthrough0N,90E,andthepositivezaxispassesthrough90N.AllelementsdefinedindifferentcoordinatesystemsaretranslatedintotheECEFCartesiancoordinatesystem.Inthefollowingpages,onlythenameCartesiancoordinatesystemreferstotheECEFcoordinates.Figure21illustratestheCartesiancoordinatesystemandtheEarthslocation.

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eration(inms )ismodeledas[Ref.10:p.326]

Figure21. TheBasicReferencefortheSimulation,CartesianCoordinateSystemandtheEarthsLocation.Thesecondisthegeodeticcoordinatesystem.Alllocationsincludingthetarget,

themissile,andthesensorsaredefinedinthegeodeticcoordinatesysteminN/Sddmm.mmmE/Wddmm.mmmformat.ThetargetlaunchsitecontainedinthemodelislocatedatN4100.000E12900.000andrepresentstheKiljukunmissilebase,NorthKorea.

Thethirdisthetopocentrichorizoncoordinatesystem[Ref.9:p.53].Launchanglesaredefinedinthetopocentrichorizoncoordinatesystemwherethefirstelement,azimuth,ismeasuredfromtruenorthindegrees,andthesecondelement,elevation,ismeasuredfromthelocalhorizonindegrees.Thetopocentrichorizonnotationisusefulindefiningtheinitialdirectionoftargetvelocityvectorandisindependentofthetargetlocation.Aswithallothervectors,targetlaunchanglesarealsotranslatedtotheCartesiancoordinatesystembeforecomputations.C. THEGRAVITYFIELD

Whentheconcernisintercontinentalranges,theflatEarthapproximationwithaconstantgravitationalaccelerationisnolongervalid.ThedirectionoftheweightvectoristowardstheEarthscenter(roundEarthmodel),andthechangeinthegravitationalaccel

2

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(2-7)

of6.671011 m/(kgs),M istheEarthsmass[Ref.10:p.A4],whichhastheap

r0V

GMr2

whereGisthegravitationalconstant[Ref.10:p.323],whichhastheapproximatevalue3 2

proximatevalueof5.981024kg,andristhedistancefromthecenteroftheEarth(inm)assumingthattheEarthisauniformdensity,nonrotating,perfectsphere.

Giventhelaunchangleandtheinitialdistancer0fromthecenteroftheEarth,thetargetdistancer(inm)asafunctionofthecentralanglecanbecalculatedbyusingthetrajectoryequationas[Ref.8:p.235]

r r0cos21coscoscos()

(2-8)

wheretheparameterdependsontheinitialranger0,launchvelocityV,gravitationalconstantG,andtheEarthsmassMasgivenby[Ref.8:p.234]

. (2-9)GM

ThecentralangleisdefinedastheanglebetweentheinitiallaunchpositionandthepositionofthetargetinflightmeasuredatthecenteroftheEarthasshowninFig.22.

Figure22. DefinitionsfortheTrajectoryEquation,theCentralAngleandtheRanger.12

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Giventhetheoreticalvaluesofdistanceversusheightbythetrajectoryequation,it

ispossibletotestthegravityfieldbehaviorofthemodel.Notethat,withaspecifiedlaunchvelocityandlaunchangle,thetargettrajectoryisindependentofthemass.Whensettingthethrustofthemodelto0,theinitialvelocitytoVandthelaunchelevationangleto,thetheoreticalandsimulatedtrajectoriesmustmatch.ForaninitialvelocityofV0.91km/s(3,000feet/s)andalaunchangleof45,theoreticalandsimulationresultsareillustratedinFig.23,whichshowsthattheymatchexactly.

Figure23. TargetTrajectorywithaLaunchSpeedof0.91km/s(3,000feet/s)andLaunchAngleof45.

Figures24and25showthetargettrajectorieswithinitialvelocitiesofV1.83km/s(6,000feet/s)andV7.32km/s(24,000feet/s),respectively,whilekeepingthelaunchangle45.Insummary,Figures23,24,and25showthatthetargettrajectoryinthemodelsgravityfieldyieldsaccurateresultsindicatingthatthesimulationcurvesareexactlythesameasthecurvesplottedusing(2-8).Velocitiesof3,000,6,000and24,000feet/sarechosentocomparegravityfieldmodelingwiththefindingsgivenin[Ref.8:pp.225233].

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Figure24.

Target

Trajectory

with

aLaunch

Speed

of

1.83

km/s

(6,000

feet/s)

and

LaunchAngleof45.

Figure25. TargetTrajectorywithaLaunchSpeedof7.32km/s(24,000feet/s)andLaunchAngleof45.

D. TARGETVELOCITYREQUIREMENTSTherequiredvelocity(inms)tohitatargetataspecifieddistancealongthe

Earthssurfaceisgivenby[Ref.8:p.242]V GM(1cos)

r0cos[(r0cos /re)cos()]14

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(2-11)

wherereistheradiusoftheEarth(inm)andthetotalcentralangletraveled(inrad)canbecalculatedas[Ref.8:p.241]

dre

wheredisthespecifieddistance(inm)alongtheEarthssurfacetoaccomplishthehit.ItisassumedthattheICBMistobelaunchedataninitialvelocityofVfromsea

level.GiventhedistanceoftheICBMstargetd,itispossibletocalculatetheinitialvelocityoftheICBMusing(2-10).Figure26illustratesthevelocityrequirementsforvarioustargetdistances.Astherequireddistancetobehitincreases,therequiredvelocityofthetargetincreases.ThevelocityrequirementsfortwomajorcitieschosenfromtheEastandWestCoastsoftheUnitedStatesareillustrated.AnICBMlaunchedfromKiljumissilebase,NorthKoreahastotravel8,668kmattrueheading050tohitSanFrancisco,California,and10,771kmattrueheading020tohitWashington,D.C.Toreachthisrange,theICBMshouldbelaunchedatavelocityof6.95km/s,and7.32km/sforSanFrancisco,CaliforniaandWashington,D.C.,respectively.

Figure26. TargetVelocityRequirementstoHitaGivenDistance.

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Notethat,whenmodelingaboostingtarget,velocityrequirementswillbeslightly

differentsincethetargethasalreadytraveledsomegrounddistanceandaltitudeatburnout.However,theinitialvelocitylaunchmodelprovidesgoodinsightintothevelocityrequirementsofthetargettobemodeled.E. BOOSTINGTARGETMODELING

Whentheconcernisthemidcourseorreentryphaseofanintercontinentalballisticmissile(ICBM),theinitialvelocity/launchanglemodelmayprovideanadequatebasisforsimulations.However,forboostphaseinterceptmodels,thisapproachisnolongeruseful.ICBMsthatcanthreatentheUnitedStatesburnoutinabout34minutesreachingavelocityof67km/s.Speedsrequiredforhittingtargetsatcertaindistancescanbecomputedbyusing(2-10).However,ICBMdesignisbeyondthescopeofthisresearch.Theaimistoachievearealisticboostphasetrajectoryandspeedprofileforatargetcapableofhittingthecitiesdiscussedabove.

Inthesimulation,theboostingcapabilityismodeledbyaminimalsetofparametersincludingtotalandpropellantmassesaswellasthespecificimpulsesandstageburntimes.Generictargetmodelsareused;however,allparametersareextractedfromactualmissilespecificationsusingtheU.S.Peacekeepermissile[Ref.11].Table21illustrateshowthetargetismodeled.Thistableiscalledthedatamatrix,andthereisoneoftheseforeachtargetandmissileinthesimulation.ThetargetwiththedatamatrixshowninTable21iscapableofboostingupto6.5km/satburnoutand,iflaunchedfromKiljuMissileBase,NorthKorea,willhittheWestCoastoftheUnitedStates(specifically,SanFrancisco).ThisisathreestagetargetandthetotalmassanddimensionofeachstageisthesameastheU.S.Peacekeepermissile[Ref.11],with85%ofthetotalmassofeachstagebeingthepropellantmass.Eachstageisassumedtobeusingafuelwithaspecificimpulseof300sandburnouttimeof60s.Thetotalboostphasetakesthreeminutes.Thistargetisassumedtobecarryingapayloadof5,000lbs.

Stage1 Stage2 Stage3 PayloadTotalMass(lb) 108,000 61,000 17,000 5,000PropellantMass(lb) 91,800 51,850 14,450 0SpecificImpulse(s) 300 300 300 0InstageBurnTime(s) 60 60 60 0

Table21. TargetDataMatrix.16

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Fnet T W0 T m0g0m0

0

1. SiloExitVelocityWhenlaunched,thetargettravelsinsidethesilo(thelengthofthesiloisthe

lengthofthemissile).Assumingthatthetargettravelsverticallywithaconstantaccelerationduringthisphase,thetargetspeedatsiloexitv(inm/s)canbewrittenas

evadtae

0 (2-12)

whereaistheconstanttargetacceleration,andeisthetimewhenthetargetexitsitssilo(ins).Inthiscase,thedistancetraveledcanbewrittenas

e elvdtaedtae2

0 0 (2-13)

wherel(inm)isthetargetlength.Theaccelerationa(inms2)canalsobewrittenasa 0 0

m0 m0 (2-14)

whereFnetisthenetforceactingonthetarget,m0isthetargetmass(sumofthetotalmassofeachstage), T0isthethrust,W0istheweight,andg0isthegravitationalaccelerationatlaunch.Bysubstituting(2-14)into(2-13),thesiloexittimee(ins)canbefound

as

e lm0

T0m0g0 (2-15)

Bysubstituting(2-15)into(2-12),thesiloexitvelocityvcanbefoundasvT0m0g lm0

m0 T0m0g0 (2-16)Thesimulationmodelstartswiththesiloexitvelocityastheinitialvelocityofthetarget.Forthetargetdefinition,thisvalueisapproximately18m/s.

2. TheRocketEquationandConsequencesTheincreaseinvelocityV(inm/s)providedbyasinglestagerocketisgivenby

therocketequation[Ref.8:p.247]as17

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1mf

m

VIspgln

wherethemassfractionmfisdefinedas[Ref.8:p.248](2-17)

mf WpWpWs (2-18)

whereWp(inN)isthepropellantweight,andWs(inN)isthestructuralweightincludingthenonpropellantpartofthetargetandpayload.

Therocketequationconveysthattheachievedvelocitybyacertainrocketcanbeenhancedbyincreasingthespecificimpulse(orexhaustvelocity)and/orincreasingthemass

fraction.

The

mass

fraction

can

be

increased

by

reducing

the

structural

parts

of

a

rocketotherthanthepropellantand/orreducingthepayload.

Themostimportantconsequenceoftherocketequationisthefactthatalargermissiledoesnotnecessarilymeanafastermissile.Inordertomakeamissilefaster,weightefficiencyshouldbeincreasedoralesseramountofpayloadshouldbeused.Inthistargetmodel,85%ofmassofanystageisassumedtobethepropellant.Inreality,theactualmassfractionisalargerpercentagethanthatusedinthismodel.

Toincreaseweightefficiency,stagingisused[Ref.8:p.249].Ithasbeendemonstratedthat,asthenumberofstagesapproachesinfinity,thetotalweightrequiredtoobtainthedesiredvelocityisminimized[Ref.8:p.251].Ithasalsobeenconcludedthatuseofthreestagesyieldsresultsveryclosetotheoptimal[Ref.8:p.251].Sincenoreasonexiststobelievethattheevolutionofpotentialtargetswillbelessthanoptimal,athreestagemodelisused.

Inathreestagerocket,themassfractionofseparatestagescanbecalculatedbyusing

mf,n 3in

mp,nt,impay

18

(2-19)

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V Vi

wherenisthestagenumber,mp,n(inkg)isthestagepropellantmass,mt,i(inkg)isthestagetotalmass,andmpay(inkg)isthepayloadmass.Inagivenstage,allweightsexceptthepropellantweightofthatstageisthestructuralweight.EachofthethreemassfractionsyieldsaseparateVwherethetotalvelocitycapabilityoftheoverallsystemisgivenby

3i1 (2-20)

whereViisobtainedbysubstituting(2-19)into(2-17)andtheoverallincreaseinthevelocityisobtainedbysummingindividualincreasesforeachstage.

Toprovethattheboostingtargetsimulationworkssatisfactorily,thetheoreticalspeedscomputedusingtherocketequationarecomparedtothoseofthesimulation.Toaccomplishthis,gravityfieldeffectsareremovedfromthesystemtemporarily.Table22liststhetheoreticalvaluesfrom(2-17).Computationsshowthatthetargetreachesatheoreticalspeedof1.925km/sattheendofStage1,4.811km/sattheendofStage2,and7.96km/satburnout.Thesecomputationsassumethatnogravityfieldispresentandaconstantexhaustvelocityof2943m/s(whichisaproductofthesealevelgravitationalaccelerationandthespecificimpulse)isused.

Stage1 Stage2 Stage3StageMassFraction 0.480 0.625 0.657StageV(km/s) 1.925 2.886 3.149

Table22. TheoreticalVelocityCapabilityofTargetModel.Thesimulationmodelcomputestheaccelerationbyevaluating(2-5).Itcontinu

ouslyintegratestheaccelerationtocomputethevelocityandcontinuouslyintegratesvelocitytocomputethepositionofthetarget.Testrunsofthesimulationyieldedspeedsof1.928km/sattheendofStage1,4.811attheendofStage2,and7.959km/sattheendofStage3.SmallerrorsinV(maximumof3m/s)betweenthetheoreticalandthesimulationvaluesareduetothesmallinitialvelocityofthetarget(otherthanzero)andtheintegrationerror.Comparisonoftheoreticalandsimulationspeedsshowsthattheaccelerationcapabilityofthistargetmodelreflectsthetheorysatisfactorily.

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F. BOOSTINGTARGETINTHEGRAVITYFIELD

Sofar,thegravityfieldperformanceofthemodelwherethetargethaszerothrustandtheboostingperformanceofthemodelinthelackofgravityfieldhavebeenvalidated.Thus,twomajorforceswereinvestigatedactingonthebodyindependently.Inthelightoftestsandfindings,itispossibletoconcludethattheboostingtargetinthegravityfieldworkssatisfactorily.

Inpractice,theoreticalspeedscannotbereachedsincetheworkisdoneagainstgravity.Byrunningthesimulationunderrealconditions,majoraspectsofthetargettrajectorywereexamined.Figures27through211areresultsofasimulationofanICBMattackfromKiljuMissileBase,NorthKoreaagainstSanFrancisco,Californiawithaninitiallaunchangleof84.Figure27isathreedimensionalillustrationoftheattackonEarthssurface.

Figure27. 3DOverviewofanICBMAttackfromKilju-kunMissileBase,NorthKoreatoSanFrancisco,California.

Figure28showsthetraveledheightversusgrounddistance.Thetargethitsthegroundatapproximately8,640km,reachinganapogeeofapproximately1,560km.Thisplotshowstherealisticperformanceofthetargetbyincludingthegravityfieldandrealisticthrustparameters.

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Figure28. GroundDistanceversusHeightfortheSanFranciscoAttack.Figure29showsthevelocityprofilefortheentireflight.Thetargethasreached

avelocityofapproximately6.5km/sattheendoftheboostphase.Afterburnout,itdeceleratesduetogravityuntiltheapogeeisreached.Followedbythat,thetargetbeginstoaccelerateduetogravity.

Figure29. VelocityversusFlightTimefortheSanFranciscoAttack.

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AcloserlookattheboostphasepartofthevelocityversusflighttimeplotinFig.

29revealstheaccelerationprofileduetostagingasshowninFig.210.Thesimulationresultsforthisspecificrunyielded1.43km/sspeedand26kmaltitudeattheendofStage1,3.86km/sspeedand107.5kmaltitudeattheendofStage2and6.53km/sspeedand250.5kmaltitudeatburnout.Thevelocitiesreachedinsimulationarelowerthanthetheoretical(nongravity)velocities.Thetargethasauniqueaccelerationprofilecomingfromitsindividualstagethrustsandfuelconsumption.Sincefuelconsumptionandthrustareconstantduringastage,theinstageaccelerationincreasesasthefuelisconsumedandtheweightisdecreased.

Figure210.VelocityversusFlightTimefortheSanFranciscoAttack(BoostPhaseOnly).

Figure211illustratesthechangeinmassduringtheboostphase.Themassdecreaseslinearlyduringeachstageduetoconstantfuelconsumption.However,stagetransitionshavediscontinuities.Discontinuitiesarearesultofcanisterjettisoningattheendofthestage.Aftertheboostphase,thetargetcontinueswiththepayloadonly.

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Figure211.

Total

Mass

versus

Flight

Time

for

the

San

Francisco

Attack

(Boost

Phase

Only).

G. SUMMARYThischapterdevelopedathreedimensionalboostingtargetmodel.Equationsre

gardinggravityfieldandthrustwereusedtoconstructatheoreticalbasisforthismodel.Later,thesimulationwasrunmanytimesfordifferentcasesinordertocomparethesimulationresultswiththetheoreticalvalues.Alltestsshowedthat,undergivenassumptions,the3Dtargetmodelworkssatisfactorily.Thisisanimportantstepfordevelopingtheboostphaseinterceptsimulation.Thesimulationwasrununderrealisticconditionsinanexampleofanintercontinentalattack,anddatawerecollected.Theresultinggraphsprovidedanunderstandingoftheboostandtheotherphasesoftheattack.Thenextstepistoexaminethemissilecharacteristics.

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where T(inN)isthethrustcomponentalongthedirectionofvelocityvectorv (inm/s)vand T(inN)isthethrustcomponentperpendiculartothevelocityvectorv.Theoverallp

III. INTERCEPTORMISSILEMODELINGInthischapter,athreedimensionalmultistageinterceptormissilemodelthat

operatesinEarthsgravityfieldisdeveloped.Theboostinginterceptoriscapableofinterceptingamultistageboostingtargetwithintheboostphasewithaminimumlateralaccelerationandsmallmissdistance.Below,thebasicdefinitionsandassumptionsaregivenalongwithadescriptionofthemissileguidanceanddynamics.A. BASICDEFINITIONSANDASSUMPTIONS

Thebasicrulesusedtodevelopthetargetmodelalsoapplytotheinterceptormis-

silemodel.Themissileoperatesundertwomajorforcevectors,thethrustT(inN),and

theweightW(inN).TheweightvectorWalwaysactsinthedirectiontowardsthecen-

teroftheEarthsimilartothetargetmodel.ThethrustvectorTisalsoalignedwiththevelocityvectorwiththeexceptionthatitsdirectionismodifiedtoobtaintheguidingforce

(lateralacceleration).Inthiscase,thenetforcevectorFnetactingonthebodycanbewrittenasEquationChapter3Section1

FnetTvTpW (3-1)

magnitudeofthetwothrustcomponentsparallelandperpendiculartothevelocityvector

isalwaysequaltothetotalthrustTprovidedbytherocketengine.Otherdefinitionsandassumptionsusedforthetargetmodelalsoapplytothemissilemodel.B. BOOSTINGMISSILEMODELING

FortherealistictargetmodeldevelopedinChapterII,atargetdesigncapableofobtainingapproximately6.5km/satburnoutwaspresented.ItwasshownthatthistargetdesigncanhittheWestCoastoftheUnitedStates.Amissilecapableofinterceptingthistargethasbeendesigned.Velocitiesrequiredforthistypeofinterceptaregreaterthantheballistictargetvelocityasdetailedbelow.

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Themissiledesignstartswiththesamesetofparametersaswereusedtodefine

thetargetboostingcapability;however,thedesignismoreefficientandhasasmallerpayload.Table31summarizesthemissileparametersusedhere.ThisisathreestagemissilehavingtotalmassesanddimensionsthesameastheU.S.Peacekeepermissile[Ref.11];however,95%ofthemassofeachstageisassumedtobethepropellantmass.Eachstageusesafuelwithaspecificimpulseof300sandburntimeof60s.Thetotalboostphasetakesthreeminutes.Thismissilecarriesapayloadof1500lbs.Notethatthepayloadoftheinterceptormissileisthekillvehicle.Theobjectiveofthemissileistocarrythekillvehicletoanoptimalpositioninspacetoallowittocompletetheintercept.

Stage1 Stage2 Stage3 PayloadTotalMass(lb) 108,000 61,000 17,000 1,500PropellantMass(lb) 102,600 57,950 16,150 0SpecificImpulse(s) 300 300 300 0InstageBurnTime(s) 60 60 60 0

Table31. MissileDataMatrix.Thesameprinciplesusedinthedesignofthetargetareused.Toachieveamore

efficientperformance,massfractionsareimprovedandthepayloadisreduced.Thesamelaunchelevationangleof84isusedforbettercomparisonwiththetargetperformance.

Testrunsofthesimulationyielded1.845km/sattheendofStage1,5.17km/sattheendofStage2,and10.31km/sattheendofstage3.Thesevelocitiesareobtainedwithouttheguidanceforceappliedandreflectthefreeflightperformanceofthemissile.Whenguided,theenergyusedtoguidethemissiletrajectoryeffectivelyreducestheobtainablevelocities.C. MISSILEGUIDANCE

Proportionalnavigationisusedformissileguidance.Theproportionalnavigationisoptimalforconstantvelocitytargets[Ref.12].Itisemphasizedherethattheclassicalproportionalnavigationguidancelawissuboptimalfortheboostphaseintercepttypeofapplication.Againstacceleratingtargets,ithasbeenshownthatsaturationisalwaysreachednearinterception[Ref.13];however,theterminalphaseoftheinterceptisout

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LOSratevector

tionvectornc.TheflightcontrolsystemusesthecommandedlateralaccelerationtochangetheattitudeofthemissileresultingintheachievedlateralaccelerationvectornL.T eac eve atera acce erat onvectornL s ntegrate a ongw t t eot eracce era-t onsact ngont esystemresu t ng nanewm ss epos t onrm.

t

sidethescopeofthisresearch.Themissilesobjectiveistocarrythekillvehicletoasuitablepositiontoterminatetheintercept.Althoughtheterminalphase(killvehicle)wasnotinvestigated,themissilefliesuntilitpassesthetargetinordertomeasurethemissdistanceandassesstheeffectivenessoftheguidancealgorithm.

Figure31showsablockdiagramofmissileprocessing.Themissiletakesthe

positionvectorofthetargetrandcomputesthelineofsight(LOS)vectorbysub

tractingitsownpositionvectorrm.TheLOSvector isdifferentiatedtocalculatethe

andclosingvelocityVc.Thecalculatedparameters andVcaremultipliedbythenavigationcoefficientN tocalculatethecommandedlateralaccelera

Figure31. MissileBlockDiagram.Forproportionalnavigation,thecommandedaccelerationisappliedperpendicular

totheLOSandgiveninscalarformas[Ref.8:p.12]ncNVc.

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rtrm

previous previous

Vc . (3-6)

ProportionalnavigationreliesontheLOSbeingconstantortheLOSratebeingzero.Inotherwords,themissileandthetargetareonacollisiontriangle.TheseekersusedintacticalmissilesusuallyprovidetheLOSrate.Themissiledesigninvestigatedcomputesitsownguidancecommandsusingthepositiondatasuppliedbyoffboardsensorsviaadatalink.

TheinstantaneousLOSvectoriscomputedfirstas

(3-3)TheinstantaneousLOSvectorisnormalizedtoobtaintheLOSunitvector.In

thenextsampletime,thenewLOSiscomputedbyusing(3-3)andalsoconvertedtotheunitvector.VectorsubtractionofthesetwounitvectorsisthedirectioninwhichtheaccelerationcommandisappliedandisalwaysperpendiculartotheinstantaneousLOS.Afternormalizingtheaccelerationcommand,theunitvectoris

nc (3-4)wherencistheunitaccelerationcommandvectorperpendiculartotheLOS,isinstantaneousunitLOSvector,andpreviousisthepreviousunitLOSvector.Notethatthisisonlythedirectionoftheaccelerationcommandtobeappliedforguidance.ThemagnitudeoftheLOSratecanbeobtainedby

(3-5)

twheretisthesimulationsteptime.

TheclosingvelocityVcisalsorequiredandcomputedasarangerate.Therange

betweenthemissileandtargetisthemagnitudeoftheLOSvector.Thismagnitudeiscalculatedforeachsteptimeofthesimulationanddifferentiated.Dividingthedifferenceintherangebythesimulationsteptimeyieldsclosingvelocityas

t

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nc c cnNV

wherev istheunitvelocityvector.Next,theaccelerationvectorcomponentparallelto

nc|| nccos

nc||vnc||

ncncnc||. (3-11)

Themagnitudeofthecommandedaccelerationiscomputedbymultiplyingthenavigationratio(unitlessconstant),theclosingvelocity(scalar),andmagnitudeoftheLOSrate.Multiplyingthemagnitudeoftheaccelerationcommandwiththeaccelerationcommandunitvectoryieldsthecommandedaccelerationcommandvector.Thiscanbewrittenas

(3-7)

Forazerolagsystem,theachievedaccelerationnLisalwaysequaltothecom-mandedaccelerationncand,forthemoment,itisassumedthatthemissiledynamicsarefreeoflags.

ThecomputedaccelerationcommandisperpendiculartotheLOS;however,missileaccelerationcommandscanonlybeappliedperpendiculartothemissileattitudeorthevelocityvector.Thus,onlythecommandedaccelerationcomponentperpendiculartothevelocityvectorcontributestothemissileguidance.

Toignoretheparallelcomponentandcalculatetheperpendicularcomponent,thefollowingprocedureisused.First,theanglebetweenthecommandedaccelerationvectorandthevelocityvectoriscalculatedas

cos1(ncv) (3-8)

thevelocityvectorisobtainedas

(3-9)Theaccelerationvectorparalleltothevelocityvectorcanbecalculatedbymulti

plyingthevelocityunitvectorandthemagnitudeofthecommandedaccelerationvectorcomponentparalleltothevelocityvectoras

(3-10)Theaccelerationvectorcomponentperpendiculartothevelocityvectorisob

tainedbysubtractingtheparallelcomponentfromtheoriginalaccelerationvectoras

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Tp

nc

mm

Thecommandedaccelerationvectornccanbeappliedbyvectoringthethrust(movementofthenozzle),controlsurfaces,orlateralthrustersattheCGofthemissile.Therequiredthrustcomponentperpendiculartothevelocityvectoristhen

(3-12)

wheremmisthemissilemassatthecurrentsampletime.

From(3-1),themagnitudeofthethrustcomponentalongthevelocityvectoris 2 2

Tv TTp. (3-13)1. GuidanceSystemAgainstConstantSpeedTargetToensurethattheproportionalnavigationguidancesystemisworkingproperly,

constantspeedmissileandtargettestscenariosareusedwiththegravityfieldandthrustdeactivated.Thetargetvelocitywassetto6.5km/sandthemissilevelocityto10km/s.Thetargetandthemissilewerelaunchedinageometrythatintroducesaheadingerrorinordertoexaminetheaccelerationcommandsgenerated.RepresentativetargetandmissileflightsduringthetestrunareshowninFig.32.

Figure32. 3DOverviewofaTypicalInterceptfortheConstantSpeedScenario.Figure33(a)showstheheightoftheinterceptormissileandthetargetasafunc

tionofthegrounddistance.Thetargetandthemissilereachanapproximatealtitudeof30

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145kmatthetimeoftheintercept.Themissiletravelsanapproximategrounddistanceof420kmwhilethetargettravels250kmsincethemissileisfaster.Figure33(b)showsthevelocityofthetargetandmissileasafunctionoftheflighttime.Figure33(b)revealsthatthemissilespeeddoesnotchange,illustratingthatthevelocityvectorandaccelerationcommandsareorthogonal.

(a) (b)Figure33 TheTargetandtheMissileFlightCharacteristicsfortheConstantSpeed

Scenario:(a)GroundDistanceversusHeight,(b)VelocityversusFlightTime.Figure34(a)showstheLOSmagnitudebetweenthemissileandthetarget.Fig

ure34(b)

shows

the

closing

velocity

as

afunction

of

time.

From

Fig.

34(a),

the

range

betweenthemissileandthetargetdecreaseslinearlysincetheyareconstantspeedbodies.Figure34(b)confirmsthatasthecollisioncourseisestablished,closurevelocitystabilizesaswellastheLOS.

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(a) (b)Figure34. TheTargetandtheMissileClosureCharacteristicsfortheConstantSpeed

Scenario:(a)RangeversusFlightTime,(b)ClosingVelocityversusFlightTime.Figures35(a)andFig.35(b)showthemissilelateralaccelerationandthemis

silelateraldivertresults,whicharetypical[Ref.8:pp.19-23].Theheadingerrorintroducedatthebeginningofthesimulationcausestheinitialaccelerationcommandandcorrespondinglateraldivert.Asthecollisioncourseisestablished,themissileaccelerationdecreasestozeroandthemissilehitsthetarget.Thecollecteddataafterthesimulationfinishedindicatesthatthetargetandmissiletraveledgroundrangesof248kmand419km,respectively.Theintercepttimewas0.755minutes.Themissdistancewasunder1meter,andthefinallateraldivertwas1023m/s.

(a) (b)Figure35. MissileGuidanceCharacteristicsfortheConstantSpeedScenario:(a)

MissileLateralAcceleration,(b)MissileLateralDivert.32

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Insummary,theconstantspeedtargettestsshowedthattheproportionalnaviga

tionimplementationinthismissiledesignworkssatisfactorily.2. GuidanceSystemAgainstICBMModelWithgravityandthrustactivated,themissilemodeldevelopedhereandthetarget

modeldevelopedinChapterIIaresimulatedtogethertoillustrateaninterception.Themajordifferenceinthistypeofinterceptisthelargeaccelerationsprovidedbyboththemissileandthetargetandfluctuationsinaccelerationduetostaging.

Figure36illustratesa3Doverviewoftheinterceptfortheacceleratingtarget.AsseeninFig.36,thetrajectoriesofboththemissileandthetargetarenolongerstraightlineswhencomparedtoFig.32.

Figure36. 3DOverviewoftheInterceptfortheAcceleratingTarget.Figure37showstheaccelerationprofileofthetarget;onlyaccelerationperpen

diculartotheLOSisrelevantandplotted.TargetaccelerationperpendiculartoLOSisalsoknownastargetmaneuver.Althoughthetargetisnotmaneuveringdeliberately,accelerationduetorocketenginesandinterceptgeometrycausesthemissiletoencounteratargetmaneuverupto5g.Abiggerproblemisthetargetmaneuverdiscontinuitiesduringstagechanges.Allthesefactorscauseunexpectedguidancecommandsasshowninthefollowingsections.

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Stage1 Stage2 Stage3

Figure37. TargetManeuverduringtheIntercept.Figure38showsplotsofgrounddistanceversusheightandflighttimeversus

velocityforthemissileandthetarget.Thetargetandthemissilereachanapproximatealtitudeof120kmatthetimeoftheintercept.Themissileandthetargettravelanapproximategrounddistanceof400kmand300km,respectively.InFig.38(b),sincethemissileissuperiortothetargetincapability,itreacheshighervelocities.Also,Fig.38(b)illustratesthevelocityprofileduetostaging.Sincethemissileandthetargetarelaunchedsynchronously,velocitydiscontinuitiesoccuratthesametime.

StageChange

(a) (b)Figure38 TheTargetandtheMissileFlightCharacteristicsfortheAcceleratingTar

get:(a)GroundDistanceversusHeight,(b)VelocityversusFlightTime.34

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Figure39illustratesdistanceandclosurevelocityduringtheintercept.Asshown

inFig.39(a),thechangeinthedistanceisnolongerlinearsincebothbodiesareaccelerating.Fig.39(b)showstheuniqueclosingvelocityprofileduetoaccelerationandpositionofthemissileandthetarget.Notethattheclosingvelocityisapproximately10km/satthetimeofhit.Thissituationcannotbeseeninconventionalinterceptcasesandisachallengingaspectoftheboostphaseballisticmissileinterceptproblem.

StageChange

(a) (b)Figure39. TheTargetandtheMissileClosureCharacteristicsfortheAccelerating

Target:(a)MissileTargetDistanceversusFlightTime,(b)MissileTargetClosureVelocityversusFlightTime.

Figure310illustratesthelateralaccelerationandlateraldivertversusflighttime.Figure310(a)showsthatmissilelateralaccelerationcommandsareusuallyunder0.4gandincreaseupto1.4gattheterminalphase.Figure310(b)showsthatthelateraldivertofthemissileincreasesupto250m/s.Bothresultsarehighlydependentontheinitialheadingerrorbetweenthemissileandthetargetatthetimeoflaunch.Itcanbeconcludedthatbothresultsarereasonableandcanbeachievedbythemissileflightcontrolsystem.

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(a) (b)Figure310. MissileGuidanceCharacteristicsfortheAcceleratingTarget:(a)Missile

LateralAcceleration,(b)MissileLateralDivert.Insummary,inthissimulation,thetargetandmissiletraveledagrounddistance

of294kmand407km,respectively.Intercepttimewas2.4766minutes.Missdistanceandtotallateraldivertwere1.3mand248.3m/s,respectively.D. FLIGHTCONTROLSYSTEM

Sofar,themissileguidancesystemisperfect.Inotherwords,theachievedaccelerationnLisalwaysequaltothecommandedaccelerationnc.Thistypeofmodelisknownasazerolagguidancesystem.Sincethecontrolsystemcanrespondtoaccelerationinputsimmediately,eventhoughtheaccelerationlevelsorlateraldivertsdiffer,themissdistancewillalwaysbeequaltozero[Ref.8:p.32].

Guidancesystemshavelags(ordelays)intheirresponse.Inthissection,themodelisexpandedtosupportapracticalflightsystemcontrolresponse.Thesystemresponseismodeledasannthordertransferfunction(Laplaceform).Althoughitisverycommonpracticetomodelmissiledynamicsasa3rdordertransferfunction[Ref.8:p.98],themodeldescribedhereisabletosupportanyorder.Itshouldbeemphasizedthatthemechanicalmodelingofthecontrolsystemdynamicsarebeyondthescopeofthisresearch.Theobjectivewastomodelthesystemresponsegiventhenthordertransferfunctiontimeconstants.Forthisreason,arbitrarytimeconstantsareused,whichcaneasilybereplacedbyrealisticonestomodelaspecificmissile.

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bs cs

nc T Ts s Ts1

Ifthesystemlagismodeledasa1stordertransferfunction,therelationbetweencommandedandachievedaccelerationcanbewrittenas[Ref.8:p.32]

nLnc 1 (3-14)1sT

wheresisthecomplexfrequencyandTisthesystemtimeconstant.Thegeneralformofannthorderallpoletransferfunctioniswrittenas

nLnc n n1

a...ds2esf (3-15)

wherea,b,c,,fareconstantscharacterizingthesystempoles.The3rdordersingletimeconstantflightcontrolsystemusedwithinthemodelhas

thetransferfunctionnL 1

3 23

27 32 . (3-16)

Figure311showshowthesystemlagaffectsthecommandedversusachievedaccelerationforatimeconstantofT1s.Figure311(a)showsncandnLforthecompleteflight,andFig.311(b)showsacloseupviewofthediscontinuities.Notethatthesystemresponselagsbehindthecontrolinputbecausethemissilemodelisnolongerazerolagmodel.Thisimpliesthatevenifaccuratetargetpositiondataisprovided,themissilewillexperiencesomemissdistance.

(a) (b)Figure311.(a)ControlSystemLag,(b)Detail.

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Toevaluatethemissdistance,the3rdordersingletimeconstantsystemwastested

againstaconstantspeedtarget.Toexcludeothereffects,thegravityandthethrustweredeactivatedinthesimulation.Themissileandtargetarelaunchedsimultaneouslywithvelocitiesof10km/sand6.5km/s,respectively.Inthisscenario,theflighttimetfisapproximately45seconds.TimeconstantTisvariedfrom0to45secondsin0.1secondincrements.Missdistancedatawascollectedforeachrun.Sincethedirectionofthemissisnotofconcern,themissdistancemeasurementsarethemagnitudeofthedistancevectoratthetimeofmiss.TestrunsresultedinthecurveshowninFig.312,whichisnormalizedwithrespecttothemaximumvaluesofeachaxes.

Figure312.MissDistanceversusTimeConstantfortheConstantSpeedScenario.FromFig.312,ifthetimeconstantislessthanonetenthofthetotalflighttime,

themissdistanceisnegligibleagainstaconstantspeedtarget.Asthetimeconstantincreases,themissdistancereachescertainpeakswhilecontinuingtoincrease.Thisresultconformstotheresultsreportedintheliterature[Ref.8:pp.31-50]withsomedifferencesinthenotation.Thisconcludestheeffortsinthedevelopmentofarealistictargetmissilemodel,whichwillbethebasisfortheremainingdiscussioninthisthesis.

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E. MISSILEREQUIREMENTS

Toexaminethemissilerequirements,threedifferentmissilemodelsthathavedifferentcapabilitiesaredefined.Thefirstonehasavelocitycapabilitysimilartothatofthetarget;thesecondandthethirdaresuperiormissiles.Table32summarizestheparametersdefiningthesemissilemodels.

StagePropellantMassFraction Payload(lb) VatBurnout(km/s)

(%ofTotalMass)GenericMissile1 85% 5,000 ~6.5GenericMissile2 90% 3,250 ~8GenericMissile3 95% 1,500 ~10

Table32. SummaryofGenericMissileSpecifications.GiventhemissilecapabilitiesasinTable32,theeffectofthemissilelocationon

theflighttimewasinvestigated.Giventheobjectivethatthetargetshouldbeinterceptedinthefirstthreeminutes,itispossibletousethesimulationtodeterminethemaximumdistancebetweenthemissileandthetargetlaunchsite.Thebestcasescenariohappenswhenthemissileislocatedintheattackdirectionofthetarget,andthelaunchdelayequalszero.Usually,thissituationcannotbefulfilledduetoterritoriallimitations,anddetection/decisionrequirements;however,examinationofflighttimeunderthesecircumstancesshowsthetheoreticallimitationsofpossiblemissilesitelocations.

Thesimulationwasusedtoexamineseveralscenarioswheremissileswithdifferentcapabilitieswerelocatedindifferentdistancesfromthetargetintheattackdirection.Foreachcase,theresultingintercepttimewasrecordedasillustratedinFig.313.Sincetheinterceptionsexceedingthe3minutelimit(totalboostphase)areconsideredfailures,thecorrespondingmissiletargetsitedistancewhereeachcurvecrossesthe3minuteintercepttimelinecanbeinterpretedasthelimitationtothemissilelaunchsitelocation.Figure313demonstratesthat,evenintheattackdirection,GM1,GM2andGM3canneverbelocatedmorethan941,1038and1140kmfromthetargetlaunchsite,respectively.

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Figure313.LimitationtotheMissileLaunchSiteDistancefromtheTargetLaunchSite:MissileDirectlyatAttackDirection,NoLaunchDelay.Themissilemaynotbelocateddirectlyattheattackdirection.Asnotedinthe

targetmodeling,anattacktargetingSanFrancisco,CaliforniaorWashington,D.C.fromthechosentargetlaunchsiteshouldbeatanapproximatetrueheadingof50and20,respectively.ForthescenariowherethetargetislaunchedfromNorthKorea,thesebearingsremaininsidetheterritoryofRussianFederation.Thisscenarioforcesthelocationofthe

missile

at

easterly

bearings

in

the

Sea

of

Japan.

Figure

314

illustrates

the

missile

lo

cationandtheprobableattackdirections.Thefigureshowsthatanyinterceptattemptmayverylikelyencounterangularerrorsof40to70.Thefollowinginvestigationassumestheworstcasescenario.IftheattackisinthedirectionofWashington,D.C.andthemissileislocatedeastofthetargetlaunchsite,themissilelocationisseverelyconstrainedbecauseoftheangulardeviationintroduced.

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Washington,DC

70 SanFrancisco, CA40 Missile

Figure314.PotentialAttackDirectionsandtheMissileLocation.Figure315illustratestheimpactofa70angularerrorbetweenthemissilepositionandtheattackdirection.Byusingthecurvescorrespondingtodifferentmissiles,itcanbeconcludedthatGM1,GM2andGM3canneverbelocatedmorethan325,477and593kmfromthetargetlaunchsite,respectively.Figure315alsohighlightsthefactthatthemoresuperiorthemissile,themoreflexibilitypossiblewhenpositioningthemissile.BycomparingFig.315withFig.313,theintroducedangularerrorapproximatelyhalvedtherequireddistanceforGM3whileitcausedapproximatelyonethirddegradationforGM1.Thisshowsthatthesuperiormissilecantoleratelocationandangulardeviationsbetter.

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Figure315.LimitationtotheMissileLaunchSiteDistancefromtheTargetLaunchSite:70AngularError,NoLaunchDelay.Anotherimportantfactoristhelaunchdelay.Followingthetargetlaunch,thede

tectionandthedecisionprocesstointerceptthemissiletakesplace.Thismissilelaunchdelayalsointroducesadditionallimitations.Figure316illustratestheeffectoflaunchdelayforGM1forthreedifferentmissiletotargetdistanceswhenthemissileislocatedexactlyintheattackdirection.Asthedistancefromthetargetincreases,tolerancetolaunch

delay

decreases.

For

example,

if

GM-1

is

located

at

adistance

of

700

km

from

the

targetlaunchsite,anymissilelaunchattemptwithadelayofmorethanapproximately32swillfail.

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Figure316.LimitationtotheTolerableLaunchDelayforGM1LocatedatAttackDirection.ReturningtotheSanFranciscoattack,itispossibletoinvestigatethelimitations

intermsoflocationandlaunchdelay.ForGM3,theeffectofmissilelocationcanbeillustratedasshowninFig.317.ToaccomplishaboostphaseinterceptwithGM3,anattacktargetingSanFrancisco,Californiarequiresamissilelocationoflessthan992kmtotheeastofthetargetlaunchsite

Figure317.LimitationtotheMissileLaunchSiteDistancefromtheTargetLaunchSite:40AngularError,GM3,SanFranciscoAttack.

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Figure318showsthetolerablelaunchdelayasafunctionofthemissileto

targetdistanceatlaunch.Forthisspecificscenario,itiseasilypossibletocalculatethemaximumtolerablelaunchdelayforagivendistancetothelaunchsite.Forexample,assumeadeployedcruisercarryingthemissiletotheSeaofJapanatalocation600kmeastoftargetlaunchsite.ByusingFig.318,itispossibletocalculatethatthemissilemustbelaunchedwithinapproximately31sfollowingthetargetlaunch.

Figure318.LimitationtotheTolerableLaunchDelay:40AngularError,GM3,SanFranciscoAttack.

Theinvestigationofmissilerequirementsyieldedthefollowingresults.Givenasuitableposition(inangleanddistance)andlaunchangles,allpotentialmissilesdefinedinTable3-2accomplishedtheboostphaseinterceptwithinreasonablemissdistanceandlateraldivertvalues.Thecapabilityofthemissilebecameimportantwhenpositionandlaunchdelaydeviationswereintroduced.Generally,themorecapablethemissile,themoretolerableitistolessthanidealcircumstances.Thepositionaladvantagewasthebestwhenthemissilewasdirectlyintheattackdirectionandwithazerolaunchdelay.Asthedeviationsfromtheidealwereintroduced,locationandlaunchdelaytolerancesdecayedquickly.Itwasshownthat,givenanangulardeviationand/oracceptablelaunchdelay,themaximumdistancethatthemissilecanbelocatedcouldbeestimatedbyusingthesimulation.

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F. SUMMARY

Thischapterdevelopedamultistage,boostingmissilecapableofinterceptingarealistictargetmodeldevelopedinChapterII.Theproportionalnavigationin3Dwasimplemented.Testrunsshowedthattheproportionalnavigationalgorithmworkedsatisfactorilyagainstconstantspeedandrealistictargets.Alsodevelopedwasanonzerolagmodeldefinedbya3rdordertransferfunction.Thesystemlagagainsttheconstantspeedtargetmodelwastestedandconfirmedthetheory.Themissilerequirementswerebrieflyinvestigatedintermsofcapabilityandposition.Theeffectsofdistanceandangulardeviationsaswellasthelaunchdelayweredemonstrated.Thisconcludedthedevelopmentofthetargetmissilemodel,whichisthebasisfortheworkinthefollowingchapters.Sofar,physicalcharacteristicsofthetargetandthemissilemotionwereexamined.Thenextstepistodeterminetargetcharacteristicsthataffectthesensorsdetectionandprocessingcapability.

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IV. RADARCROSSSECTIONANDIRENERGYRADIATION

PREDICTION

Inthefirstpartofthischapter,themonostaticradarcrosssection(RCS)ofathreestagegenericintercontinentalballisticmissile(ICBM)ispredicted.Thisisaccomplishedbymodelingthephysicalshapeofeachstagebyusingfacets.Asoftwareprogramwasthenusedtocalculatetheresultsfordifferentaspectangles.Allseparatestagesofthetargetaremodeledtoquantifythediscontinuitiesbetweenstages.Toinvestigatetheeffectoffrequency,theRCSispredictedforLBand(1.5GHz),SBand(3GHz),CBand(6GHz),andXBand(10GHz).

TheinvestigationofthemonostaticRCSiscrucialsincetheaccuracyoftheRFsensortrackisdirectlyproportionaltothebackscattercharacteristicsofthetarget.Forthetarg

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