DesignofGuidanceand ControlSystemsforTactical Missiles
QiZaikang and LinDefu
SchoolofAerospaceEngineering
BeijingInstituteofTechnology
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—LinDefu
Contents Preface xi Authors xiii 1TheBasicsofMissileGuidanceControl 1 1.1Overview....................................... 1 1.2MissileControlMethods............................... 1 2MissileTrajectoryModels,AerodynamicDerivatives,DynamicCoefficientsand MissileTransferFunctions 7 2.1SymbolsandDefinitions............................... 7
9 2.3ConfigurationoftheControlSurfaces........................ 14 2.4AerodynamicDerivativesandtheMissileControlDynamicCoefficient...... 15 2.5TheTransferFunctionofaMissileastheObjectBeingControlled......... 20 3BasicMissileControlComponentMathematicalModels 28 3.1Seeker........................................ 28 3.2Actuator....................................... 28 3.3AngularRateGyro.................................. 29 3.4Accelerometer.................................... 30 3.5InertialNavigationComponentsandIntegratedInertialNavigationModule.... 30 4AutopilotDesign 31 4.1AccelerationAutopilot................................ 31 4.1.1Two-LoopAccelerationAutopilot...................... 31 4.1.2Two-LoopAutopilotwithPICompensation................. 35
37 4.1.4ClassicThree-LoopAutopilot........................ 44 4.1.5DiscussionofVariableAccelerationAutopilotStructures.......... 48 4.1.6HingeMomentAutopilot........................... 50 4.1.7SeveralQuestionsConcerningAccelerationAutopilotDesign....... 53 4.2Pitch/YawAttitudeAutopilot............................ 58 4.3FlightPathAngleAutopilot............................. 60 4.4RollAttitudeAutopilot................................ 61 4.5BTTAutopilot.................................... 68 4.6ThrustVectorControlandThrusterControl..................... 78 5GuidanceRadar 85 5.1Introduction...................................... 85 5.2MotionCharacteristicoftheTargetLine-of-Sight.................. 85 5.3LoopoftheGuidanceRadarControl......................... 89 5.4EffectoftheReceiverThermalNoiseonthePerformanceofGuidanceRadar... 97 vii
2.2EulerEquationsoftheMissileRigidBodyMotion.................
4.1.3Three-LoopAutopilotwithPseudoAngleofAttackFeedback.......
5.5EffectofTargetGlintonthePerformanceofGuidanceRadar............
5.6EffectofOtherDisturbancesonthePerformanceofGuidanceRadar........
5.6.1EffectofDisturbanceMomentonthePerformanceofTrackingRadar...
5.6.2EffectofTargetManeuvers..........................
6LineofSightGuidance
6.1LOSGuidanceSystem................................
6.2AnalysisoftheRequiredAccelerationfortheMissilewithLOSGuidance.....
7.2ElectromechanicalStructureofCommonlyUsedSeekers..............
7.2.1DynamicGyroSeeker............................
7.2.2StabilizedPlatform-BasedSeeker......................
7.2.3DetectorStrapdownStabilizedOpticSeeker.................
7.2.4Semi-StrapdownPlatformSeeker......................
7.3MechanismAnalysisoftheAnti-DisturbanceMomentoftheSeeker’sStabilization LoopandTrackingLoop...............................
7.4TransferFunctionofBodyMotionCouplingandtheParasiticLoop........
7.4.1TransferFunctionofBodyMotionCoupling................
7.4.2Seeker-MissileCouplingIntroducedGuidanceParasiticLoop.......
7.5.2TestingMethodsforModelingtheRealSeeker...............
7.6OtherParasiticLoopModels.............................
7.6.1ParasiticLoopModelforaPhaseArrayStrapdownSeeker.........
7.6.2ParasiticLoopDuetoRadomeSlopeError.................
7.6.3BeamControlGainError ∆KB ofthePhasedArraySeekerandtheRadome SlopeError Rdom EffectontheSeekerOutput................
7.7StabilizationLoopandTrackingLoopDesignofthePlatform-BasedSeeker....
7.7.1StabilizationLoopDesign..........................
7.7.2TrackingLoopDesign............................
8.1ProportionalNavigationGuidanceLaw.......................
8.1.1ProportionalNavigationGuidanceLaw...................
8.1.2AnalysisofProportionalNavigationGuidanceLawwithNoGuidance SystemLag..................................
8.1.3TheProportionalNavigationGuidanceCharacteristicswiththeMissile GuidanceDynamicsIncluded........................
8.2ExtendedProportionalNavigationGuidanceLaws(Optimal ProportionalNavigation,OPN)...........................
8.2.1OptimalProportionalNavigationGuidanceLaw(OPN1)withthe ConsiderationoftheMissileGuidanceDynamics..............
8.2.2OptimalProportionalNavigationGuidanceLaw(OPN2)Consideringthe ConstantTargetManeuver.......................... 187
viii Contents
101
102
102
104
105
105
107 6.3AnalysisoftheLOSGuidanceLoop......................... 111 6.4LeadAngleMethod................................. 119 7Seekers 122
122
7.1Overview.......................................
123
123
129
132
133
133
136
7.2.5StrapdownSeeker..............................
7.2.6Roll-PitchSeeker...............................
137
140
140
144
147
147
7.5ARealSeekerModel.................................
7.5.1ARealSeekerModel.............................
150
152
152
153
154
157
157
158 8ProportionalNavigationandExtendedProportionalNavigationGuidanceLaws 161
161
161
165
170
182
182
8.2.3ExtendedProportionalNavigation(OPN3)ConsideringBothConstant TargetManeuversandMissileGuidanceDynamics.............
8.2.4EstimationofTargetManeuverAcceleration................
8.2.5OntheEstimationof
8.2.6ProportionalNavigationGuidanceLawwithImpactAngleConstraint...
8.3OtherTypesofProportionalNavigationLaws....................
8.3.1GravityOver-CompensatedProportionalNavigationLaw..........
Contents ix
190
194
tgo 194
195
197
197
201 8.4TargetManeuverAccelerationEstimation...................... 203 8.5OptimumTrajectoryControlDesign......................... 216 Appendices 223 Bibliography 232 Index 233
8.3.2LeadAngleProportionalNavigationGuidanceLaw.............
Preface
Overtheyears,theauthor,QiZaikang,hasbeenengagedinteachingandscientificresearch inthefieldofmissileguidanceandcontrolsystemdesignbothathomeandabroad.Withmany textbooksalreadypublished,themotivationstowriteanewreferencebookwerethefollowing:
1.Mostoftheexistingtextbooksprimarilyaddressedtheoperatingprinciplesofmissile guidanceandcontrolsystemsandtheircontrolcomponents,leavingoutdetailsonsystem designmethods.
2.Thewidespreadapplicationofinertialnavigationandintegratedinertialnavigationonboardthemissilehasmademanynewguidanceandcontrolsolutionsfeasible.Yet,due tothelackofnecessaryhardware,thesenewsolutionscouldnotpreviouslybeapplied. Inaddition,theever-increasingdemandsforhighermissileguidanceaccuracyandfaster interceptionresponsemakeitnolongeracceptabletosimplifytheguidanceandcontrol modeltotheextentthatwasallowedinthepast.Therefore,thecurrentdesignmodel,as wellasguidanceandcontrolstrategy,hasseensignificantchangesincomparisonwith earliermissiledesign.
Againstthisbackground,theauthorshopethatthisbookcanintroduceasmanyup-to-dateguidanceandcontrolsolutionsandimproveddesignmodelsaspossible,whileatthesametimepresentingmorepracticalandvaluabledesignmethods.
DuetoQiZaikang’smanytechnicalexchangeswithasignificantnumberofWesternandRussianexpertsduringhisworkabroad,andhisdomesticengagementintherelatedscientificresearch, thisbookwillcoverthelatestpracticaltechnologiesinthefield,bothathomeandabroad,bygiving considerationtodesignconceptsofboththeEastandWest.
Thisbookcanbeusedasatextbookforundergraduateandgraduatestudentsinthisfield,as wellasahelpfulreferencebookforpracticalengineeringdesigners.
Theinnovativecontentofthisbookiscloselyrelatedtotheresearchworkoftheauthors’many graduatestudents.Theauthorswishtoextendtheirsincerethankstothemall.
Finally,theauthorswouldalsoliketoexpresstheirappreciationtoMissXuJiaoforhercontributiontothisbook.Overthepastyears,shehasworkedfull-timeonthisbook’smathematical simulations,drawings,typingandreview.Itshouldbesaidthatwithoutherdedication,thisbook couldnothavebeenfinished.
QiZaikang SchoolofAerospaceEngineering,BeijingInstituteofTechnology Beijing,China,December,2018
LinDefu SchoolofAerospaceEngineering,BeijingInstituteofTechnology Beijing,China,December,2018
xi
Authors
QiZaikang isaChiefTechnicalExpertinWeaponsGuidanceandControlTechnology.Heserves aschairandprofessorintheAircraftGuidanceandControlDesigndepartmentattheBeijingInstituteofTechnology.HeisalsotheDirectoroftheInstituteofUAVAutonomousControl.Hehas beenengagedinteachingandresearchinthefieldofaircraftsystemsdesignfor60years.Notonly hasheadvisedmanyexcellentyoungscholarsandengineers,hehasalsoservedaschiefengineer forseveraladvancedresearchandmodelscientificresearchprojects.HeisamemberoftheState CouncilAcademicDegreesCommitteeDisciplinaryAppraisalTeamofthePeople’sRepublicof Chinaandaninternationaleditorof ComputersinMechanicalEngineering
Dr.LinDefu receivedhisPhDdegreefromtheBeijingInstituteofTechnology.HeisDirectorof theInstituteofUAVAutonomousControl.Hehasmorethan20years’experienceintheoverall designandguidanceandcontrolofflightvehicles.Hehasworkedasprincipleinvestigatorfor severalkeynationalprojects.Duetohisoutstandingresearchwork,hehasbeenawardedsecond prizeintheNationalScientificInventionandNationalDefenseScienceandTechnologyAward.He hasauthoredorcoauthoredmorethan80journalpublicationsandservesasmemberofmultiple academiccommittees.
xiii
TheBasicsofMissileGuidanceControl
CONTENTS
1.1Overview
Thepurposeofmissilecontrolistomakethemissilehitthetargetattheendofitsflight.In ordertoachievethisgoal,itisessentialforthemissiletoconstantlyacquirethemotioninformation ofthetargetandofthemissileitselfinthecourseoftheflightandadoptatactic(thatisaguidance law)todecidehowtochangethemissile’svelocitydirectionbasedonthecurrentmissileandtarget relativemotion,allowingthemissiletofinallyhitthetarget.Therelationshipbetweentheangular velocity θ ofthemissilevelocityvectoranditsnormalacceleration a ofthemissile,isasfollows:
Therefore,thecommandofaguidancelawthatisgeneratedtochangethemissilevelocityvector directionisusuallythenormalacceleration ac ofthemissile.Thismissileandtargetinterception controlloopisquitedifferentfromtheconventionaltrackingcontrolloop;theformerisatimevaryingcontrolsystem,anditsanalysismethodiscompletelydifferentfromthegenerallineartimeinvarianttime-domainandfrequency-domainanalysismethod.Soaspecialterm(guidanceloop) hashistoricallybeengiventothisparticularmissilecontrolouterloop.
Withthehelpofautopilots,themissileoutputacceleration a willfollowtheaboveguidance accelerationcommand ac .Undertheassumptionsofsmallperturbation,linearizationandconstant systemparametervalue,thisautopilotloopisalineartime-invariantsystemandsodifferenttraditionalcontroltheorydesignmethodscanallbeapplied.Therefore,theautopilotloopthatactsas theguidanceinnerloophistoricallyisstillreferredtoasthecontrolloop.
Themissilepositionandvelocityinformationneededintheguidanceprocessareobtainedby aninertialnavigationorintegratedinertialnavigationsystem.Theprocessofobtainingthemissile positionandorientationinformationiscallednavigation.Itisnoteworthythatthetermnavigation heredoesnotrefertothehistoricaldefinitionofdirectingthecourseofashiporanaircraft. Fig. 1.1 showstherelationshipsbetweenthetermsnavigation,guidanceandcontrolinmissilecontrol loops.
1.2MissileControlMethods
Ithasbeenstatedbeforethatthetaskofamissilecontrolsystemistousemissilenormalaccelerationtochangethemissile’svelocitydirectionaccordingtotheguidancelawcommand.For
1
1.1Overview 1 1.2MissileControlMethods .............................................. 1
θ = a V (V isthemissilevelocity) (1.1)
1
Fig.1.1:Blockdiagramofthemissileguidanceandcontrolloops
tacticalmissilesflyingintheatmosphere,thisnormalaccelerationisgeneratedbynormalaerodynamicforces.Asweknow,whenthemissilehasanangleofattackwithrespecttoitsvelocity vector,thecorrespondingliftwillproduceanormalacceleration.However,forthemissiletomaintainasteadyangleofattack,thisangleofattackinducedaerodynamicmomentmustbebalanced bythecontrolsurfacedeflectioninducedcontrolmoment.
Whenthecenterofgravityofthemissileislocatedinfrontofthecenterofpressure,theangle ofattackgeneratedaerodynamicmomentwilldecreasetheexistingangleofattackandmeanwhile, the x-axisofthemissilebodywilltrytocoincidewiththemissilevelocityaxis.Thistypeofaerodynamiclayoutisknownasastaticallystableaerodynamicconfiguration(Fig.1.2).However,when thecenterofpressureofthemissileisinfrontofitscenterofgravity,theexistingangleofattack willcontinuouslyincreaseundertheactionofitscorrespondingdestabilizingaerodynamicmoment. Therefore,themissileisinadivergentstate.Thisaerodynamiclayoutiscalledastaticallyunstable aerodynamicconfiguration(Fig.1.3).
Fig.1.2:Missileinastaticallystableaerodynamicconfiguration
Ingeneral,therearethreetypesofaerodynamicconfigurationsforthegenerationofamissile controlmoment:
(1)Normalaerodynamicconfiguration
Inthisaerodynamicconfiguration,themissileactuatorisarrangedatthetailofthemissile(Fig. 1.4).Thebenefitofthisconfigurationisthatwhenthecontrolmomentisbalancedbytheangleof attackproducedmoment,thecontrolsurfaceincidentangleisthedifferencebetweenthecontrol
2 DesignofGuidanceandControlSystemsforTacticalMissiles
Fig.1.3:Missileinastaticallyunstableaerodynamicconfiguration
surfacedeflectionangleandtheangleofattack,whichisthemostefficientwayofusingthecontrol deflectionangle,thusallowingtheuseofalargercontrolsurfacedeflectionandlargerangleofattack formaneuvering.Butthedrawbackisthatthepositionoftheactuatorinthisconfigurationcoincides withtherearendmotor,whichplacescertainrestrictionsonthesizeoftheactuator.Inaddition, whenthemissileistomaneuver,thecontrolsurfaceforceisintheoppositedirectiontotheangle ofattackproducednormalforce,whichwillcausesometotalnormalforceloss.However,taking theseadvantagesanddisadvantagesintoaccount,thisconfigurationisstillthemostcommonlyused aerodynamicconfigurationfortacticalmissiles.
Fig.1.4:Normalaerodynamicconfiguration
(2)Canardaerodynamicconfiguration
Inthisaerodynamicconfiguration,theactuatorispositionedattheheadofthemissile(Fig.1.5). Thebenefitofthisarrangementisthatthemissilemotorcanbearrangedindependently,avoiding theneedtocontendforspacewithothersubsystems.Inaddition,whenthemissileistomaneuver, thecontrolsurfaceforceisinthesamedirectionastheangleofattackproducednormalforce, thusachievinghighermaneuveringforceutilizationefficiency.However,inthisconfiguration,the actuatorincidentangleisthesumoftheactuatordeflectionangleandthemissileangleofattack.As themaximumallowedcontrolsurfaceincidentangleislimited,alargeangleofattackmaneuvering cannotbeachieved.Therefore,nowadaysthisaerodynamicconfigurationislesscommonlyseenin missileapplications.
Fig.1.5:Canardaerodynamicconfiguration
(3)Moving-wingscheme
Withthisaerodynamicconfiguration,themissilewingcanbeturnedasacontrolsurface(Fig. 1.6),andthefullcenterofpressureispositionedinfrontofthecenterofgravity,similartothe
TheBasicsofMissileGuidanceControl 3
Fig.1.6:Moving-wingaerodynamicconfiguration
canardaerodynamicconfigurationbutwithashortcontrolarm.However,therequiredliftformissile maneuveringisessentiallyprovidedbythewingdeflection,thisisbecausethewinghasaverylarge liftingsurface.Forthisreason,theangleofattackrequiredformissilemaneuveringcouldbesmall. Therefore,itisparticularlysuitabletobeusedwhenthemissileturbineengineforcruisingflight isnotallowedtoworkatalargeangleofattack.However,duetothehigherpowerrequirement forthewingactuator,itsoperatingfrequencybandwidthislimited,andsoistheresponsespeed oftherelatedautopilot.Forthisreason,thisaerodynamicconfigurationisrarelyusednowadaysin engineeringpractice.
Fig.1.7 and Fig.1.8 showthesituationsinwhichthecontrolmomentandtheaerodynamic momentareinanequilibriumstatewhenthereisasteadystateangleofattackforstaticallystable andstaticallyunstablemissiles.
Fig.1.7:Momentequilibriumofastatically stablemissile
Fig.1.8:Momentequilibriumofa staticallyunstablemissile
Itisnoteworthythatforastaticallystablemissile,thecontrolmomentgeneratedbytheactuator deflectionangle δ willmakethemissilerotateintherequireddirectiontoproduceanangleofattack. Whentheaerodynamicstabilizingmomentthatincreaseswiththeangleofattackincreasestothe samelevelasthecontrolmoment,thecorrespondingangleofattackwillbeatanequilibriumstate. Therefore,missileswithsufficientstaticstabilitycanalsobedesignedwithoutautopilot.However, thistypeofaerodynamicfeedbacksolutionhaslessprecisemissilenormalaccelerationcontrol comparedwithanaccelerationautopilotsolution.Butforstaticallyunstablemissiles,asteadystate angleofattackcanonlybegeneratedthroughautopilotclosed-loopcontroltomaintainarequired equilibriumbetweenthecontrolmomentandaerodynamicmomentgeneratedbytheangleofattack.
Asmentionedabove,asteadystateangleofattack α isachievedwhenthecontrolmomentand theaerodynamicmomentareinequilibrium,thatis:
4 DesignofGuidanceandControlSystemsforTacticalMissiles
Mδ z · δ = Mα z · α. (Controlmoment)(Aerodynamicmoment)
Thetransferfunctionwiththeactuatordeflectionangle δ astheinputandtheangleofattack α astheoutput,shownbelow,canberegardedastheobjectbeingcontrolledfortheautopilot(Fig. 1.9).
Fig.1.9:Theobjectbeingcontrolledfortheautopilot
Themissile’sstaticstabilityisdirectlyproportionaltothedistancebetweenitscenterofgravity anditscenterofpressure,andthisdistanceissmallformissileswithlowstaticstability.Therefore, whencenterofgravityorcenterofpressureofthemissilewithlowstaticstabilitydeviatesfromits designedvalue,thevalueof Mδ z andthegainofthetransferfunction Mα z Mδ z fromtheactuator δ tothe angleofattack α willchangegreatlyfromitsdesignedvalue,whichmeansthattheopen-loopgain oftheautopilotloopwillalsochangegreatly.Thisisunacceptableforanormallydesignedcontrol loop.Therefore,toreducetheautopilotopen-loopgainchange,themissilestaticstabilityisoften takenataround4-8%.Formissilesthatmusthavealowstaticstabilityaerodynamicconfiguration forotherconsiderations,thegainfrom δ to α couldbestabilizedbydesigningapseudoangleof attackfeedbackloop.Foradetaileddiscussionofthisoption,seetheautopilotdesignsection.
Atpresent,askid-to-turn(STT)controlschemeisadoptedinmosttacticalmissiles.Thatis,in theCartesiancoordinatesystem,amissilepitchturnisachievedbythegenerationofangleofattack α,andayawturnisachievedbythegenerationofsideslipangle β,asshownin Fig.1.10
Fig.1.10:Skidtoturn(STT)polardiagram
Suchacontrolschemehasaveryfastresponse,butitisnecessarytoberollstabilized.Clearly, STTismostsuitableforaerodynamicallysymmetricalmissiles.
Anothercontrolschemeisbank-to-turn(BTT).Thisschemeisgenerallyusedforsurfacesymmetricalmissiles,especiallywhenthereisabigdifferencebetweenthepitchandyawliftsurface areasofthemissile.Inthisscheme,themissilemustturnthemainliftsurfacebyanangle φ with thehelpofarollcontrolautopilottohavethemissileangleofattackintherequiredmaneuvering direction(Fig.1.11).
TheBasicsofMissileGuidanceControl 5
Fig.1.11:Bank-to-turn(BTT)polardiagram
ForthemissilewithBTTcontrol,whenthemissilemaneuveringdirectionneedstobechanged, itispossiblethatthemissilehastorollalargerollangletoanewdirection,andclearlythisleads toaslowmissilemaneuveringresponse.Forthisreason,BTTcontrolismoresuitableformissile midcourseguidancephase.
6 DesignofGuidanceandControlSystemsforTacticalMissiles
CONTENTS
2.1SymbolsandDefinitions
2.4AerodynamicDerivativesandtheMissileControlDynamic
2.5TheTransferFunctionofaMissileastheObjectBeingControlled
2.1SymbolsandDefinitions
Theoriginofthemissilebodycoordinatesystem Oxbybzb isdefinedatthecenterofgravity ofthemissile,andeachaxisisdefinedasfollows(supposethatthemissileisanaxisymmetricor plane-symmetricrigidbody,see Fig.2.1):
Rollaxis Oxb:liesinthesymmetryplane.Pointingforwardispositive.
Yawaxis Oyb:locatedinthesymmetryplaneofthemissilebody,withupwardsasthepositive direction.
Pitchaxis Ozb:formstheright-handedrectangularcoordinatesystemtogetherwithaxes Oxb and Oyb.
Table2.1 definesthesymbolsforaerodynamicforces,momentsactingonthemissile,linear velocities,andangularvelocities,aswellasmomentsofinertia(asshownin Fig.2.1).Themoment ofinertiaaroundeachaxisisdefinedas:
Theproductofinertiaaroundeachaxisisdefinedas:
2 MissileTrajectoryModels,AerodynamicDerivatives, DynamicCoefficientsandMissileTransferFunctions
.............................................. 7
9 2.3ConfigurationoftheControlSurfaces 14
2.2EulerEquationsoftheMissileRigidBodyMotion
Coefficient 15
.. 20
Jx = mi y 2 i + z 2 i , (2.1) Jy = mi z 2 i + x 2 i , (2.2) Jz = mi x 2 i + y 2 i . (2.3)
Jyz = miyizi, (2.4) Jzx = mizi xi, (2.5) Jxy = mi xiyi. (2.6) 7
NOTE: O is the center of gravity of the missile
Fig.2.1:Definitionsofaerodynamicforce,moment,etc.,ofthemissile
Theplane Oxbyb isthepitchplaneandtheplane Oxbzb istheyawplane.Therelevantanglesare definedasfollows:
α—angleofattackinthepitchplane;
β—angleofattackintheyawplane(angleofsideslip);
αT —totalangleofattack;
λ—angleofattackplaneangle.
Therefore:
Theaxialvelocityofthemissilebody Vxb isalargebutslowlyvaryingvariable,anditsvariation isusuallylessthanafewpercentpersecond.However,theangularvelocity ωx,ωy,ωz andvelocity components Vyb, Vzb ofthepitchandyawaxesareusuallysmall.Theycanbepositiveornegative, andtheycanhavelargeratesofchanges.
8 DesignofGuidanceandControlSystemsforTacticalMissiles ybV y y M y J bX xbV T o bZ zbV yb xb zb Relative flow velocity Yb z z M z J x x M x J
tan α = tan αT · cos λ,
tan β = tan αT sin λ. (2.8)
α = arctan tan αT · cos λ , (2.9) β = arctan tan αT sin λ (2.10)
(2.7)
Thatis,
Table2.1:Definitionofsymbols
Angularvelocity(missilebody coordinatesystem)
Velocitycomponent(missilebody coordinatesystem)
Forcesactingonthemissile (missilebodycoordinatesystem)
Momentsactingonthemissile (missilebodycoordinatesystem)
2.2EulerEquationsoftheMissileRigidBodyMotion
Thesix-degree-of-freedommodelofamissilemotioninspaceconsistsofsixdynamicequations (threecenterofgravitymotiondynamicalequationsandthreerotationaldynamicalequations)and sixkinematicequations(threecenterofgravitymotionkinematicequationsandthreerotational kinematicequations).
Thecoordinatesystemsinvolvedinthestudyofmissileguidanceandcontrolproblemsinclude theearthcoordinatesystem,themissilebodycoordinatesystem,thetrajectorycoordinatesystem andthevelocitycoordinatesystem.The x-axisofthelasttwocoordinatesystemscoincideswith themissilevelocityvector.However,the y-axisofthetrajectorycoordinatesystemisinthevertical plane,andthe y-axisofthevelocitycoordinatesystemisinthelongitudinalsymmetricalplaneof themissilebody.Thetransformationbetweenthefourcoordinatesystemscanbeaccomplishedby aseriesofrotations(Fig.2.2).Detaileddescriptionsofthesecoordinatesystemscanbefoundin generalflightdynamicstextbooks.
Forexample,therotationtransformationfromtheearthcoordinatesystemtothemissilebody coordinatesystemisshownin Fig.2.3.
Inthestudyofcoordinatetransformation,itisnecessarytoknowthreebasiccoordinatesystem transformationmatrixesaboutaxes x, y, z:
MissileTrajectoryModels 9
RollaxisYawaxisPitchaxis xb yb zb
ωx ωy ωz
Vxb Vyb Vzb
Xb Yb Zb
Mx My Mz Momentsofinertia Jx Jy Jz Productofinertia Jyz Jzx Jxy
Fig.2.2:Transformationfromtheearthcoordinatesystemtoothercoordinatesystems
Fig.2.3:Relationshipbetweentheearthcoordinatesystemandthemissilebodycoordinatesystem
Rotationmatrixthatdoesrotationaboutthe x-axisbyangle ϕx:
Rotationmatrixthatdoesrotationaboutthe y-axisbyangle ϕy:
Rotationmatrixthatdoesrotationaboutthe
10 DesignofGuidanceandControlSystemsforTacticalMissiles
Lx(ϕx) = 100 0cos ϕx sin ϕx 0 sin ϕx cos ϕx .
Ly(ϕy) = cos ϕy 0 sin ϕy 010 sin ϕy 0cos ϕy
z-axisbyangle
: Lz(ϕz) = cos ϕz sin ϕz 0 sin ϕz cos ϕz 0 001
ϕz
Definethefollowingvariables:
V—missilebodyvelocity;
ψV ,θ—missileflightpathangle;
γV —missilesymmetricalplanedeflectionangle;
ψ, ϑ, γ—missilesyawangle,pitchangle,rollangle;
α, β—angleofattack,angleofsideslip;
Vx, Vy, Vz—velocitycomponent(earthcoordinatesystem);
Vxb , Vyb , Vzb —velocitycomponent(missilebodycoordinatesystem);
ωx, ωy, ωz—missileangularvelocitycomponent(missilebodycoordinatesystem);
Fxt , Fyt , Fzt —resultantforcecomponentactingonthemissile(trajectorycoordinatesystem);
Fxb , Fyb , Fzb —resultantforcecomponentactingonthemissile(missilebodycoordinatesystem).
Thetotalforce F actingonthemissileconsistsofaerodynamicforce R =
(missilebody coordinatesystem),thrust P =
(missilebodycoordinatesystem),andgravity G =
(earth coordinatesystem).Themomentactingonthemissilebodyis M =
(missilebodycoordinate system).Theprojectionsofrelatedcomponentsinothercoordinatesystemsareshownin Table2.2
Thesix-degree-of-freedommissilemodelcanbegiveninthetrajectoryorthemissilebody coordinatesystem.Whenthesix-degree-of-freedommodelisgiveninthetrajectorycoordinate system(Equation(2.11)),thestatevariablesofthethreedynamictranslationalandthree-rotational equationsaretakenasthevelocity V,θ,ψV (trajectorycoordinatesystem)andtheangularvelocity ωx,ωy,ωz (missilebodycoordinatesystem).Thestatevariablesofthesixkinematicequations arerespectivelytakenasthepositioncomponent x, y, z (earthcoordinatesystem)andtheEuler angle ϑ,ψ,γ (missilebodycoordinatesystem).Otherdependentderivedparametersinclude α,β and γV .
˙ x = V cos θ cos ψV , y = V sin θ, z = V cos θ sin ψV , (2.11) ˙ ϑ = ωy sin γ + ωz cos γ, ψ = (ωy cos γ ωz sin γ)/cos ϑ, γ = ωx tan ϑ(ωy cos γ ωz sin γ), sin β = cos θ[cos γ sin(ψ ψV ) + sin
(cos
V
MissileTrajectoryModels 11
Xb Yb Zb
P 0 0
0 G 0
Mx My Mz
mV = Fxt , m ˙ Vθ = Fyt , mV cos θ ˙ ψV = Fzt , Jxωx (Jy Jz)ωyωz Jyz(ω 2 y ω 2 z ) Jzx(˙ωz + ωxωy) Jxy(˙ωy ωxωz) = Mx, Jyωy (Jz Jx)ωzωx Jzx(ω 2 z ω 2 x) Jxy(˙ωx + ωyωz) Jyz(˙ωz ωyωx) = My, Jz ˙ ωz (Jx Jy)ωxωy Jxy(ω 2 x ω 2 y ) Jyz(˙ωy + ωzωx) Jzx
ωx ωzωy
z
γ
ϑ
cos β sin
(˙
) = M
,
ϑ sin γ sin(ψ ψV )] sin θ cos ϑ sin γ, sin α = {cos θ[sin ϑ cos γ cos(ψ ψV ) sin γ sin(ψ ψV )] sin θ cos ϑ cos γ}/ cos β, sin γ
=
α sin β sin ϑ sin α sin β cos
cos
+
γ cos ϑ)/ cos θ.
Aerodynamic force R
Table2.2:Relatedprojectionsindifferentcoordinatesystems
Earthcoordinatesystem Trajectorycoordinate system Missilebody coordinatesystem
Resultant force F (F = R + G + P)
Whenthesix-degree-of-freedommodelofthemissileisgiveninthemissilebodycoordinate system(Equation(2.12)),asidefromthestatevariablesofthethreetranslationaldynamicequations changingtothevelocitycomponents Vxb, Vyb, Vzb (missilebodycoordinatesystem),theremaining statevariablesarethesameasthetrajectorysystem.Thatis,thestatevariablesofthethreedynamic rotationalequationsaretakenastheangularvelocitycomponents ωx,ωy,ωz (missilebodycoordinatesystem).Thestatevariablesofthesixkinematicequationsaretakenasthepositioncomponents x, y, z (earthcoordinatesystem)andtheEulerangle ϑ,ψ,γ (missilebodycoordinatesystem), respectively.Otherusefuldependentderivedparametersare Vx, Vy, Vz, V,θ,ψV ,α,β and γV .
12 DesignofGuidanceandControlSystemsforTacticalMissiles
Lx( γv)Ly( β)Lz( α) Xb Yb Zb Xb Yb Zb Gravity G 0 G 0 Lz(θ)Ly(ψv) 0 G 0 Lx(γ)Lz(ϑ)Ly(ψ) 0 G 0 Thrust P Lx( γv)Ly( β)Lz( α) P 0 0 P 0 0
Fxt Fyt Fzt Fxb Fyb Fzb
moment M Mx My Mz Velocity V Vx Vy Vz Ly ( ψ)Lz ( ϑ)Lx ( γ) Vxb Vyb Vzb Vxb Vyb Vzb
Aerodynamic
areshownin Table2.2), θ = arctan Vy/ V 2 x + V 2 z ,
ψV = arctan ( Vz/Vx) ,
α = arctan( Vyb /Vxb ),
β = arcsin(Vzb /V),
V = arcsin (cos α sin β sin
Typically,aerodynamicforce R andmoment M arefunctionsofMachnumberMa,angleof attack α,angleofsideslip β,three-channelcontrolsurfacedeflectionangles δx, δy, δz,andthree angularvelocities ωx, ωy, ωz.
R = R(Ma,α,β,δx,δy,δz), M = M(Ma,α,β,δx,δy,δ
Theexactexpressionofthesefunctionsandtheirreasonablesimplificationcanbeobtained throughwindtunneltestsandtestdataanalysis.
Missileguidanceandcontrolisachievedthroughcontrolsurfacedeflection δx, δy, δz commandedbyguidanceandcontrollaws.Modelswithguidanceandcontrolwillcontainmoreequations.Forexample,themathematicalmodeloftheentiresystemwillalsoincludetheseekerdynamic mathematicalmodel,autopilotmathematicalmodel,commandguidanceradarmathematicalmodel, thecontrolsurfaceservomechanismmathematicalmodel,etc.
Theabovedynamicequationscanoftenbesimplifiedinspecificmathematicalsimulations.For example,foraxisymmetricmissiles,theircrossinertiamoment Jxy, Jyz, Jzx canbesafelyomitted. Forthree-channelcontrolmissiles,therelated ωx, ωy, ωz aresosmallthattheirproduct ωxωy, ωyωz, ωzωx, ω2 x, ω2 y , ω2 z canalsobeomitted.Furthermore,sincetheprojectionsofthevelocityvectorson themissilebodycoordinatesystem Vyb, Vzb arealsoofasmallquantity,theirproductwiththe componentof ω canalsobeomitted.Therefore,thedynamicequationsrepresentedinthemissile bodycoordinatesystemcanbesimplifiedas:
MissileTrajectoryModels 13
V
b
Vzb ωy Vyb ωz) = Fxb = Xb G sin ϑ + P, m
˙ Vyb + Vxb ωz Vzb ωx) = Fyb = Yb G cos ϑ cos γ, m(Vzb + Vyb ωx Vxb ωy) = Fzb = Zb + G cos ϑ sin γ, Jxωx (Jy Jz)ωyωz Jyz(ω 2 y ω 2 z ) Jzx(˙ωz + ωxωy) Jxy(˙ωy ωxωz) = Mx, Jyωy (Jz Jx)ωzωx Jzx(ω 2 z ω 2 x) Jxy(˙ωx + ωyωz) Jyz(˙ωz ωyωx) = My, Jzωz (Jx Jy)ωxωy Jxy(ω 2 x ω 2 y ) Jyz(˙ωy + ωzωx) Jzx(˙ωx ωzωy) = Mz, ˙ x = cos ψ cos ϑ Vxb (cos ψ sin ϑ cos γ sin ψ sin γ) Vyb + (cos ψ sin ϑ sin γ + sin ψ cos γ) Vzb , y = sin ϑ Vxb + cos ϑ cos γ Vyb cos ϑ sin γVzb , z = sin ψ cos ϑ Vxb + (sin ψ sin ϑ cos γ + cos ψ sin γ) Vyb (sin ψ sin ϑ sin γ cos ψ cos γ) Vzb , ˙ ϑ = ωy sin γ + ωz cos γ, ψ = (ωy cos γ ωz sin γ)/ cos ϑ, (2.12) γ = ωx tan ϑ(ωy cos γ ωz sin γ), V = V 2 xb + V 2 yb + V 2 zb = V 2 x + V 2 y + V 2 z
Vx, Vy
Vz
m (
x
+
(
(Expressionsof
,
α sin β cos γ cos
γ cos ϑ)/cosθ
γ
ϑ sin
ϑ + cos β sin
.
z,ωx,ωy,ωz)
mVxb = Fxb , (2.13)
Whenpresentedinthetrajectorycoordinatesystem,theabovetranslationaldynamicequations canbedescribedas:
2.3ConfigurationoftheControlSurfaces
Thesequentialnumberingofthecontrolsurfaceisshownin Fig.2.4.Thedeflectionangles δ1, δ2, δ3, δ4 generatedbyturningclockwisealongeachcoordinateaxispositivedirectionsaredefined aspositive.Therespectivedeflectionanglesaredefinedasfollows:
Rollcontroldeflectionangle: δx = 1 4 (δ1 + δ2 + δ3 + δ4)
(Whenonlyapairofactuatorsismoved,thereis δx = (δ1 + δ3)/2or δx = (δ2 + δ4)/2.)
Pitchcontroldeflectionangle: δz = 1 2 (δ1 δ3) .
Yawcontroldeflectionangle: δy = 1 2 (δ4 δ2) .
Fig.2.4:Definitionofcontrolsurfaceangles
14 DesignofGuidanceandControlSystemsforTacticalMissiles m(Vyb + Vxb ωz) = Fyb , (2.14) m( ˙ Vzb Vxb · ωy) = Fzb , (2.15) Jx ωx = Mx, (2.16) Jy · ωy = My, (2.17) Jz ωz = Mz (2.18)
m ˙ V = Fxt , (2.19) mVθ = Fyt , (2.20) mV cos θψV = Fzt (2.21)
