Am or Us Op a Per 1

8/2/2019 Am or Us Op a Per 1

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/ ) B - I R - o 3 - " ?

EULERANGLESAND QUATERNIONSINSIXDEGREEOFFREEDOMSIMULATIONS

OFPROJECTILES

jJO^-**

byMichaelJ .Amoruso

March1996

Approvedforpublicrelease;distributionunlimited.

REVIEWEDBYAerospace

APPROVEDBY:Branch,MATD,AED

Dr.William Ebihara^--Chief,Materials&AeroballisticsTechnologyDivision,AED

AEROBALLISTICS BRANCHMATERIALS&AEROBALLISTICSTECHNOLOGYDIVISIONARMAMENTENGINEERINGDIRECTORATEARMAMENTRESEARCH, DEVELOPMENTANDENGINEERING CENTERU.S.ARMYARMAMENT, MUNITIONSANDCHEMICALCOMMANDPICATINNYARSENAL,NEWJERSEY07806-5000

2 0 0 3 0 9 2 90 4 9PIQAJIUUMM;


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CONTENTS

ContentsFiguresTablesPreface

Pagei

iiiiiiv

1.0Introduction2.0Review of Matrix AlgebraforOrthogonalTransformation3.0Euler Angles

3.1RotatingCoordinateFrames3.2Plane-Fixed Coordinates 4

4.0Quaternions 85.0Equationsof Motion 76.0IntegrationofEquationsofMotion 3

6.1Plane-FixedEquations 36.2Body-FixedEquations 46.3Aeroballistics(Zero-P)Equations 7

AppendixAlgorithmsforImplementationoftheEquationsofMotioninSixDegreeofFreedom ComputerSimulations 58

DistributionList 0

A(5^^3w^'^f2|


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FIGURESPage

1oordinateSystem2omponentEuler AngleRotations3maginary NumberiInterpretedasaRotationOperator 9

XI


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TABLESPage

1btainingtheQuaternionsfromtheRotationMatrixT 22valuationofARCTAN (A,B)overAllFourQuadrants 363ody-FixedEquations 74imeDevelopmentoftheBody-FixedTransformationMatrixParameters 85lane-FixedEquations 96imeDevelopmentofthePlane-Fixed TransformationMatrixParameters 07eroballisticEquations 18imeDevelopmentofthe AeroballisticTransformationMatrixParameters 2

iix


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PREFACETheuthoraseennvolvednheimulationfuidedrojectilesormanyyears.Differentnvestigatorsdoptifferentoordinaterames,uchsody-fixed,lane-fixednderoballisticzero).omeseheulernglerepresentationoealwithotationsndomeseuaternions.ource'swhichexplainheignificancenddvantagesndisadvantagesfheseariousapproachesreoteadilyvailable,tsdifficultoinderivationsndhereis ackfdvicenncorporatinghesemethodsntoomputerimulations.Theuthorecamespeciallyrustratedhenettemptedoollectheequationsoonvertnxistingixegreeofreedom6DOE)imulationromtheEulerngleohequaternionepresentation.everalourcesforheeededequationswereoundutowogreedxactly.inceittlenhewayfderivationswereprovided,twasotrivialoerifyhequationsoreconciletheiscrepancies.hisocumentesultedromheuthor'sttemptomakesomeenseofthisconfusion.Theirsthapterontainsnverviewofheproblem.nheecondhapter,briefeviewfheareminimumfmatrixlgebrasprovidedoemindhereaderofomeofhemportantpropertiesoforthogonalransformations.hethirdhapterevelopsheEulerngleormalismwithnntroductionohedifferenceetweenody-fixed,lane-fixednderoballisticoordinates.hequaternionlgebrasevelopednhapter.Thissnxtensiveubject.OnlyenoughoftheormalismwasdevelopedoprovideunderstandingofquaternionsandTntroduceheoolseededorhisocument.nhapter heigidodyequationsofmotionredevelopedorhehreeoordinateramesdiscussed,n bothheulernglenduaternionepresentations.heiscussionfhedistinctionetweenody-fixed,lane-fixednderoballisticoordinatessdistributedhroughouthapters o.nhapter,hentegrationfheequationsfmotionsiscussed.iscussionsfhereatmentfCoriolisndcentripetalorrectionsinaflatarthmodel,gravityforanon-flatarth,ndimevaryingmassndmomentfnertiaaveeenncludednhiseport.hesetopicswillereatedn utureeport.heppendixrovides ummaryfthealgorithmsneededforimplementingtheseresultsin DOEimulation.TheuthorwishesohankDr.RichardHaddadfPolytechnicUniversityfNewYork,MessrsRomelCampbellndohnGrauofARDEC,Dover,NJ,ndMr.homasHarkinsfARL,Aberdeen,D,_foraluableiscussionsnd.suggestions.ewishesohankMrSungChungorheckinghemathematicalderivations.

IV


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1 INTRODUCTIONWhenevelopingimulationsfircraft,missilesrun-launchedrojectiles,investigatorsequireoordinateramenhichoollowheotion.Newton'sawsequirennertialunaccelerated)rame.hearthsconvenienteferencerameutsotnertia!incehearthotates.hearthmayonthelesseused,withCoriolisndentripetalccelerationsncludedo accountfortheearth'srotation.However,heprojectilesothranslatingndotating.Thustsonvenientoexpresshequationsfotionfherojectile,issilerircraftncoordinateshatmovelongwithtnomeway.heobvioushoicesody-fixedoordinates.Theseoordinatesrettachedoheprojectileorircraftndroll,itchndyawwitht.heeaderamiliarwithimbalsoryroscopeswillrecognizehatheseEulernglesofoll,pitchndyawrequivalentoimbalangles.nheasefuidedrojectile,heeeker,ateensor,accelerometers,ndontrolmechanismswhetheraerodynamicorreactionontrolalloperateinndareeasiesttodescribenody-fixedcoordinates.Sometimesnonollingcoordinatesredesirable.tisdifficulttonterpretesultsof imulationwhenheointfiewsolling,sheyrewithody-fixedcoordinates.nddition,pin-stabilizeduniredprojectilesotatethundredsofevolutionsperecond.Computerunimesoruchrojectilesusingody-fixedoordinatesecomentolerablyong.hisifficultyrisesecauseheintegrationimetepmustecomextremelymallnordertoeephengleofrollmallduringtheintegrationimetep.fthissnotdone,gravitysmearedoverhengularmotionhatoccursuringhentegrationimetepecauseofthehighollate,givingincorrectresults.Someypefon-rollingoordinateystemssedoealwithhisroblem.Oneolutionsoethe omponentofheoordinateramengularelocitytoero.AnothersoettheEulerollngleoero.Thesewopproachesrenotdentical,swehalleenubsequenthapters.ehalleehathedifferencerisesromheacthatheomponentsofhengularvelocityormanrthogonaletwhereashehreeulernglesootaveutuallyorthogonaletofrotationxes.ChoosingheollEulerngleoeeroliminateshehorizontalomponentof-gravitynlatarthoderentirelyinceehalleehathe-axissconstrainedomoven horizontalplane.hismakesheumericalntegrationinsensitiveoheollate.owever,tstillensitiveoheitchndaw rates.hispproachsypicallyelectedwhenmodelingnunguidedtagefspintabilizedprojectile.Thisypeofframeiscalledplane-fixed.Choosinghe omponentfheoordinateramengularelocityoeeroyieldseroballisticoordinates.Thishoiceoesotompletelyliminatehecomponentofgravityutensitivityoheffectsofollsreatlyeduced.tschief value is the simplification of the equationsfmotion. Coupling terms


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involvinghe omponentfheramengularelocityisappearromheequationsofmotion.furtherimplificationsremadeasednymmetryndlinearityoftheerodynamics,tspossibleoobtainlosedormolutionstoheequationsofmotion^.Withnyoftheseframes,tsnecessaryoegeneratetheramesheprojectilemoves.Thusherametselfasquationsfmotion.Wehalleehatherotationmatrixhatransformshevectorsromhemovingrameohenertia!(earth)rameanexpressednermsfitherhreeEulernglesrourquaternions.hequationsfotionorheulernglesndorhequaternionsreerivedohatheymayentegratedobtainheewrameandpdateheprojectilequationsofmotion.nlywonglesreequiredodescribeheotationfigidodyootllheulernglesrhequaternionsreinearlyndependent.onstraintsuchsormalizationconditionshereforexistandwillederived.ThedvantageofEulernglesoverquaternionsistheirintuitiveness.oll,pitchandawre aturalwayor pilotodescribeorvisualizetheangularmotionofnircraft.TheEulernglesreheaturalariableorescribing eekerorpinningyroscopeimbal.oweverheEulernglelgebrasomewhatmessyndnsymmetrical,orrorsreotlwaysvident.urthermore,hesinendosineofthethreeEuleranglesmustberepeatedomputed,providingcomputationalurdenhatoesotxistithuaternions.hus,lthoughquaternionsreotntuitivenheensehatEulernglesre,heirimplicityandymmetricormmakeerivationsmuchimpler,reessroneomaskerrorsndreomputationallymorefficient.origonometricunctionsrtranscendentalunctionseedoevaluated.Themostomplicateduaternionarithmeticequiresthequareofaquaternionortheproductoftwoquaternions.Forhiseasonuaternionlgebrasesirablenigitalutopilotsoruidedprojectilesecausetlleviatesheomputationalurden.urthermore,Euleranglesreusceptibletoingularitieshatanevoidedyusinghequaternionforrnalism.ThedetailsofheEulernglendquaternionormalismequiredoevelophe6DOFquationsfmotionor igidodynhehreeoordinateramesdiscussedbovewilledevelopednubsequentchapters.

Vaughn, Harold R., "Aetailed Development ofhe Tricyclicheory," Sandia Laboratories, SC-M-67-2933, Albuquerque,NM ,968.


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2 REVIEWOFMATRIXALGEBRAFORORTHOGONALTRANSFORMATIONSThishapterontainsriefeviewfhematrixlgebraequirednhisdocument^ ' ^ ^An ymmatrixAsnrrayofelementsa..,= o, = om,f owsnd olumns,whichbeysheollowingawsfdditionndmultiplication.

C= + ..=.. ..2.1)tj ij ijC= .=Ya.,^.2.2)

Forheseoperationsoemeaningful,ertainmatchingestrictionsxistnhenumberofowsndolumns.orddition.A, nd mustllhaveheamenumberfowsndheameumberfolumns.orultiplication,henumberfolumnsf mustmatchheumberofowsof .TheproductC hasheamenumberofowssAndheamenumberofcolumnssB.uch matricesreaidoeonformableIfheumberfowsndolumnsreequal,hematrixsquare.ectoraneepresentedyn y olumnmatrix.Thesuallgebraicawsoldxcepthatultiplicationsotenerallycommutativendhemultiplicativenverseoesotlwaysxistseeelow).Wheneeferohenversef quarematrix,eenerallyeanhemultiplicativenverse.henverseofaquarematrixAsdenotedyA ^ndsdefinedy

AA~^=A" A1 2.3)where sheunitmatrixi.e., longheiagonalnderolsewhere).hiscanlsoewrittennermsofthelementsofthematrix

\a..a~ =ya.7 a., .^ 2.4)

Wylie, .R.,r,AdvancedngineeringMathematics,"McGraw-Hil l ,NewYork,956.2Margenau,HenryandGeorgeMoseleyMurphy,TheMathematicsofPhysicsndChemistry,"VanNostrand,New

York,956.


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where nd ., nd8.,sheKroneckereltaunctionwhichsunitywhenheubscriptsrequal' ^ n derotherwise.neneral,henversematrixoesnotXaysxist.enerally, necessaryndufficientonditionrTmatrixohaven^verseshattmustequarendhedeterminantofheamatrix^f^^^"', matrixsalledoningular. EvenwhenmftrixTnningularfindingLnversesenerallyelious.hiswillotS)nc"rnshereincewewilleealingonlywithorthogonalmatriceswhichreconcerns^^^J*"'^t,inpuiorehalllsoeehathenverseaneound:'u",71ri?Wy 'Fromhs'p.ZXedop.heummationonventionrhichtates'ha. umsimpliedoveranyndexhatisepeatedwee.Thus

jWedefinenorthogonalmatrixsonethatpreservesthelengthofavectoruponwhichperates.hematrixperatorshanotategid-bodyectorsmustpreserveen'Jhinceotationoesottretchrompress igidody-JI JRotationalransformationmuste ubsetoforthogonalransformations.Whatpropertiesa \"weIrJ^eromhisdefinition?onsiderhematrix peratmgonhecolumnmatrixorvectorv.

v' =Thismaylsobewrittenntermsofthelements

V' ...jIftheenothofvspreservedbythistransformationA,hen

v'.v' =.vwherethedotproductisdefinedby

22V.V = V^ + V^ + V3Thus,fAsanorthogonal,ength-preservingransformation

^'i 'i i "iic\ j k^Thisspossibleonlyifa..a.,=8ij ikk

(2.6)

(2.7)

(2.8)

(2.9)

(2.10)

(2.11)ButthisisthedefinitionofthenverseofA.

-1 _ 2.12)Thus,ornorthogonalmatrix,al' ^..Thenverseofa northogonalmatrix


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isbtainedynterchangingheubscriptsrheowsndolumnsfhematrix.hissquivalentoeflectingheatrixboutheiagonal.hisoperationscalledhetransposendwillbedenotedy uperscriptT. ThusA"=A 2.13)definesa northogonalmatrixWeefineheracerpurof matrix.Theracesimplyheumfllheelementsalongthediagonal.Thus

TrA=TrA*^ =X^ii 2-14)Therace isbviouslynvariantoheransposeoperation sinceheiagonalelementsareunchangedunderransposition.Finally,wehowhatthetransposeofaproductoftwomatricesistheproductofthetransposesofthendividaulmatrices,butneverseorder.

[A f ""A 2.15)Theproofisa sfollows:

BA, =. " "A"; =B""A""].2.16)qi kqq qk"^tca nehownthatth eetenninantsnvariantoheransposeoperation.Thereforerom2.3)nd2.13 )wean

concludeha thequarefheetenninantfnorthogonalmatrixs.hu sheeterminantmuste1.Thenegativeeterminantsssociated itheflections,hichbviouslylsoreservesength. Theositiveeter-minantsassociatedwithrotations.


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3ULERANGLES3.1 ROTATINGCOORDINATEFRAMES

Weusearighthandcoordinate system withxpositive forward, ypositiveto the rightandzpositive down,asindicatedin Figure1. Each Euleranglealsoobeys arighthand rulewith respect toit s axis. TheroU>ccursaboutthexaxis,thepitch occurs aboutthey axisndtheya w \|/ occursaboutthezaxis.hus therollis positive clockwise lookingforwanifromtherear,pitch is positiveupward(eventhoughzis positivedown),andtheya w ispositivelooking forward.y convention, therotational transformationftommer-t ialtorotatingaxes consists ofaya w throughangle\|;,followedby apitchthrough angle8ndfinally aroU through angle < p .Theorderoftheserotationsis importantsincero-tiionsonotcommute.hecomponent rotations a re shown in Figure1.Theonginalinertialaxesrendicatedy x,, and.herimes mdicateintermediate axes ofsubsequent rotations.he finalrotating axes a re indicated by".y"Vndz'".Thesecomponentrotationsmay beexpressed a s

cos(i|>) sin(i|f)

0

siii(4f)cos()cos(4>)

(3.3)

*Biakelock,John H.,-AutomaticContrrfofAircraftandMkates,"JohnWileyandSons,NewYork,1965.2 Etkin,Bernard,"Dynamics< r f Flight-StabilityandQmtioL" JohnWileyandSons,New York,1965.^ Etkin,Bernard,"DynamicsofAtmosphericFli^t,"JohnWil^and Sons,NewYork,1912.


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X3O -3OS2

co'- > u-H 0 0-H o a, o X)u (Dea > . aA ux:> wo

o *i(1 > r-tW < aio c'H -1u-I cn o o HM lU hO >!J= o e4J H E"Ho , Hs> . > ^ A pa . OOo ueE


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TT TT 0


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O=ft,+t +i^ (3.6)=j< f ey'4-4>f"

Wewantoesolvehisnmoving-fixedoordinates,wherewecanwriteft=t,x" t,,y'" t, f"3.7)

Theectoruantitiesnheightf3.6)reotnrthogonaletwhereasthosen3.7)rerthogonal.henitectors",y'"nd"reorthogonal.hesemayeesolvedsollows.heectortesolvednheorthogonaloordinatesnhemovingramesbtainedypplyinghepartialEulerotation:ft =

ft4 e

0

' I'

- < j


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0 +eos('"

K) = 0 (3.12)2'"

Weandd vectorially3.8) to3.10) tobtain an explicit representation of(3.6)esolvednhemovingrame. Comparinghiso3.7)ivesheesultweseek,iz.ft,., = -ii2/2(6) i > (3.13)ft,,, = ^sin{ )os{%) osi ) (3.14)ft^,,, = ^cos{ )os{%)- in{

These can be inverted for the derivatives of the Euler angles by the usualalgebraictechniquesogetdroppingheprimes)[ft sin{^) t^cos{ )\

4 , (3.16)cos{Q)

1 1


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4 )ft^ [ft ?fn( \>

(3.19)

Likewise,3.16)hrough3.18)mayewrittenromnspections< t > 1in(d))tan(8)os() -sin(4>)0in(d))/cos(e)os((j))/cos(e) ftftft (3.20)

Expressions3.19)nd3.20)reotymmetricalrlegant.otehathematricesn3.19)nd3.20)ustenversesfnenother.hisaneverifiedyakingheroductfhesewomatricesnderifyinghathenitidentitymatrixesults.1 2


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Thechoiceforthecoordinateframeangularvelocityomponentst^,ftandt willependnhepplication.ody-fixedoordinatesreppropriateorsimulatinguidedrojectile,ocketrircraft.nuidedrojectileheseeker,ensorsndontrolmechanismsreixedoheody.Forhiseasontisasiestndimplesttodescribetheseubsystemsn coordinateframefixedotheprojectileody.heoordinateramengularelocityomponentsrehenequalohenalogousomponentsfheodyngularelocity,whichreconventionallydenotedyP, ndR.Thusnquations3.13)hrough3.20),wemaketheubstitutions

ft = PXft = Q 3.21)yft = R z

Thisshesualhoicemadeor DOFimulationsfuidedrojectilesndmissiles.Forunguidedprojectiles, non-rollingcoordinateframesoftenpreferred.uch arameitchesndawswithheprojectileutoesotollwitht.on-rollingramemighteefinedyettinghe omponentofheramengularveloci'ty,t anishryettingheimederivativeofheollEulerngle,,vanish.rom3.13)weeehathesewopproachesreotdentical.Thecoordinatesobtainedyheirstapproachrecallederoballisticcoordinatesandtheatterlane-fixedoordinates.hedvantagefheormersesimplificationfhequationsfmotioninceheouplingermsnvolvingtdropout,swehalleenChapter.Thispproachsoftenakenwithnalyticorlosed-formolutionsofhequationsofmotion.quation3.21)wouldemodifiedyettingt =.Theplane-fixedpproachsoftensedor DOFomputerimulationsofpinstabilizedrojectiles.pintabilizedrojectilesaveypicalpinatesfhundredsfevolutionserecond.sing ody-fixedepresentationncomputerimulationfuch projectileequiresnxtremelymallntegrationtimetepnd,onsequently,nordinatelylongcomputerrunimes.Theimetepmustemallohatherojectileollsotppreciableuringheimetep.Otherwiseheffectofgravitysmearedcrosshisngle.Whileeroballisticcoordinateswillhelp, moreusefulolutionstoequired /dt { ) forhecoordinaterame.ehalleenChapter hathe omponentofgravitysrigorouslyliminatednheixedlanepproach.Thisliminatesensitivityofthentegrationoherojectileollate,hough imilarensitivitystillpresentorhemuchlowerpitchndawates.hispproachanpeedpsimulationunimebyordersofmagnitude.AllhreepproacheswilleiscussedurtherwhenevelopinghequationsofmotionnChapter. Thelane-fixedoordinatesreerivednheollowing

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sectionromhephysicaliewpointofconstrainingoneofthexesomovensingleplane.

3.2PLANE-FIXEDCOORDINATESPlane-fixedoordinatesitchndawwithheodyutootollwithhebody.enceweefine|) ndd ldt0.orerecisely,nexissconstrainedtoalwaysemainnoneplane,houghtca notateinhatplane.Forexample,he-axisouldeonstrainedoheerticallaneoriginal-zplane).Thisanechievednnnertialomovingi.e.,lane-fixed)rametransformationonsistingin pitchboutheoriginaly-axiswhicheepshe-axisnheriginalitchlane,whichsertical)ollowedy awbouthenew-axis,whicheaveshe-axisunchangedndhereforetillnheerticalplane.Alternatively,hey-axisaneonstrainedohehorizontalplaneoriginal-yplane).Thissoney awbouthe-axisollowedy itchbouthe-axis.eanonstructheotationmatrixsefore,sing3.1)nd3.2).However,unlikeefore,heboveecipeinvolvesnverseransformationsromthenertialohemovingrameatherhanrommovingrameonertial,swasheasenhederivationf3.5)orheody-fixedrame.Thushebovetransformationsomprisedfhenversesfhematrixperatorsreviouslyused.Thus,ecalling2.15)ndhathenversefnrthogonalmatrixststranspose.or

T ^=RT R -TTR =R R = R. =

cos8osijisinil cos8in4osi|i-sinesin6osi]/sin8inij;COS0 (3.22)14


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Thisstheplane-fixednalogof(3.5). Itisoturprisinghathissquivalenttomakinganishn3.5).Likewise3.6)ecomesn n=\ ii^(3.23)Equation3.8)ecomes

n, =0

0

-i|/in(Q)6

+4 ;os{Q)Likewise,3.9)ecomes

^e= > 0

e 0

Equations(3.13)hrough3.15)ecome

(3.24)

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Q,,, = i|;os(Q)1 (3.26)i (3.27)fi^,, = -in(e) = -a ,,,anO) (3.28)

Weaveubstituted3.26)nto3.28).TherimesnheubscriptsfHn(3.26)hrough3.28),swellsn3.13)hrough3.15)anedroppedincethexeseferredorerthogonal.nverting,withomelgebraieldsheanalogsof(3.16)hrough3.18),viz..a a

cos(e) sin(e) (3.29)

< f ) = 0 (3.30)= a (3.31)

Thenalogsof(3.19)nd3.20)rennft 0

0sin(e)0 cos(e)

(3.32)

and0 1 0 tan(e)

0 10 0 i/cos(e) ftftft (3.33)16


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Inummary,heepresentationftheotationmatrixTsuniqueomatterhowitserived,lthoughomeethodsayotlwaysorkecausefsingularities.heotationmatrixTcaneviewedsanoperatorwhichotatesvectorin ixedoordinateystemor,onversely,saotationofthecoordinatesystemwhileheectoremainsixed.nheormerointfiew,heectorhasheameengthutitscomponentsrechangedecauseitsdirectionnpacehashangedueohevector'sotation.nheatterpointfiew,heectorhasheameengthndirectionnpaceuttsomponentsreifferentbecauseoftherotationofthecoordinateframe.Ifweakeheatterpointofview,wea neehatheotationmatrixsusthematrixofthedirectionosines.eti' denotethehreemutuallyorthogonalunitbasisectorsofheprimedrotated)oordinateystemndenotehehreebasisectorsofheunprimedoordinaterame.willehelementsofhetransformationromheunprimedotheprimedrame.Then

i'I 3.34)P9 9Takingheotproductndmakinguseofhemutualorthogonalityofheasisvectors,weanwritei''I,II, 3.35)p /q q I = h =pq qliThushelementsfheotationmatrixouldebtainedyakingheotproductsfhenitasisectorsfhenprimedoordinateramewithhebasisvectorsoftheprimedrame.

17


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4 QUATERNIONSWewillevelopustnoughfhelgebrafuaternionssseededorunderstandingndritingixegree-of-freedom6OF)imulations.Quaternionsre uadrupletofumbersstrictlypeakingperators)hatanbeonsideredoeeneralizationfomplexumbers.ecallhathequantity

i = V~l 4.1)mayeesthoughtfsotationperator.huss0egreecounterclockwiserotationfromhe"real"othe"imaginary"xis.eeFigure.Thesquare"f swouccessive0degreeotations.hissquivalento180egreeotation.hisakesusoheegativeealxis.heubeof s270egreeounterclockwiseotationromheositiveealoheegativeimaginaryxis.tsnhisensehat=1nd=i - .he4thpowerof sjusta360degreeotationwhichgetssackoherealxis.Forquaternionswedefinethreeuchquantities,orrespondingtootationsboutthe,,nd xesespectively.Asnheasefheonventionalmaginarynumber,heperators,,nd -canenterpreteds0egreeotationsabouthe,,nd xes;ndhequaresorrespondo 80egreeotationaboutheppropriateirection,ndoorth.heseperatorsreometimescallhyperimaginaryumbers.ustsheonventionalomplexumbersaneusedoprovide'machineryorreatingotationsn plane,wemightxpecthatthreeimaginary"perators,,nd mightesedoreatotationsboutthreexes,.e.,nhreeimensionalpace.nothersefulropertyfquaternionssthattpermitsusomultiplynddividevectors.hiswilleeentoprovide muchimplermathematicalreatmentthanmatrices.Recallthatotationsrenotnumbersbutoperatorsndonotcommute.husjdoesotquali,ndoorth. ittleitofhoughtndomexperimentationwithotationsil lonvinceheeaderhatheollowinglementaryrelationshipshold.

22/ = = =1 4.2)i = i c = -ji 4.3)j = i = -k 4.4)k = ; = -i 4.5)

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+ ix

REALAXIS+x

-IX

Figure. ImaginaryNumber InterpretedsaRotationOperator

19


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Wetake quaternionoe quadrupleQ= Q Q, Q- Q.ometediouslgebrail lerifyhatultiplyingutwouaternions,akingllpossibleproductsndimplifyingusing4.2)hrough4.5),ivesheollowingresult.W =W^+iW^+ jW^+kW^ Q^+iQ^+jQ^+kQ^]4.6)

=W ^Q^-W^Q.-W^Q.-W.Q.;\+^[iQ^ +jQ,+kQ^]+^[iW ^ +jW ,+ kW,]+i{W^Q.-W^Q,+j[W3G1-WjG3] +k[W^Q.-^Q^]

Uponnspectionf4.6),weeehattheirstinefterheastqualitycontainsexpressionshatesemble otproductnd rossproduct.hisuggestsnalternativeormulation.eanreat, nd otsotationperatorsrhyperimaginaryumbersutsnorthogonaletofunitvectorsndonsiderquaternionormallyoonsistf calarpartnd ectorart.hisetsswritethequaternionproductxpressedn4.6)morecompactly.Thequaternionproductisequivalentlydefineds

yfQ=W^Q -W.QW Q Q^WWxQY4.7)where

WWjjW and Q=GoG 4.8)Furthermore,weanbtaintillnotherlternateormince4.6)a nlsoerearrangednto

+[WjQ^ W^Q^-W^Q^+W^Qj 4.9)+[w,G-M,w,e, W2,]

Thismayeorganizedntohematrixforms

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WQ="1A.

+W(j -Wj -w., -w.+Wj + Wg -w. +w.+W-, +W, +WQ -W +w, -w, +w, +w.

Go Gi

G3

(4.10)

+ GoQiQ-G3+ ^160G3G2+ G;G3GoGi+ 63G2GiGo W o

Compare4.7)nd4.10)o4.6).Theyrequivalent.4.7)nd4.10)remoreompacthan4.6)utrerbitraryfuseds definitionoruaternionmultiplication,someauthorsdo.nheotherhand,4.6)lowsogicallyndautomaticallyromheroperties4.1)o4.5),ndivesnsightntoheconnectionwithotations.ewillusehepproachhatshemostonvenientinachase.Someusefulxpressionsollow.Definetheonjugate

Q*-[Q,-Q] (4.11)Definehenormorabsolutevaluequared

IQ -QQ* Go -G0 *Q Letusdefinennvesendverifytworks.

Q'-^-Go,-G]/(Go +Q'Q)=* Q\

(4.12)

(4.13).

Itollowsthat

2 1


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QQ' 0] 4.14)Ifheormanishes,hequaternionsaidoeingularndhenversedoesnotxist.tsasyohowhathenormof productqualsheproductofthenorms.henverseof productsheproductofhenversesneverseorder.Theonjugatef roductsheroductfheonjugatesneverserder.Thus

IQ1Q2I = IQ1IIQ2I 4.15)

[QjQj]* = Q2* Qi* 4.17)Ineneral,uaternionrithmeticwilleamiliarxceptoron-commutativityofmultiplication.Commutationreaksownormultiplicationecausefhecrossroductermn4.7).therwisellhethersualawsrebeyed.Quaternionrithmeticsistributivendssociative,utommutativenlyoraddition.dentitylementsxistorothdditionndmultiplication,iz.0,0)and1,0).nverseslsoxistordditionorllquaternions.multiplicativeinversexistsornyon-zerouaternion.ee4.13).heulesordifferentiationreheamiliarnes,xceptareusteakenecausequaternionsootommute.snxample,onsiderheerivativefheproductoftwoquaternions.

[QjQ2'=Q{Q2 QiQ2' i'22'1^^^)Sinceheormf roductqualsheroductfheorms,twouldeemplausiblehat uaternionfunitormouldeusefuloreatotationsincerotationsustreserveheengthfectors.ehalleeaterhathisconjecturesssentiallyorrect.nnticipationf uaternionormalismorrotation,omeelationsforunitquaternionswillnoweprovided.Considerheunitquaternionsometimesalled versororEulerquaternion)e=e^e] e^ ^^e^] 4.19)

wheree2 = ee4.12).Thus2222e + +. +, = \ 4.20)22


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From4.15),he"length"orormof quaternionspreservedwhenmultipliedby unitquaternion.

|eQ=Q 4.21)Sincehenormofaunitquaternionsunity,4.13)ellsushathenverseofunitquaternionsequaltotsconjugate,.e.

e ^ [e,-I]=[e,,-e-e-e] =*4.22)Also

i \i e e e ] 4.23)I 0 1 2 3 JBydifferentiating = 1,weobtain

. -1-1e = eApplying4.16)o4.24)ives

r.if'e = e J (4.24)

-1r 11[e J 4.25)Sincehenverseof nituaternionsqualotsonjugate,henversesn (4.25)aneeplacedyconjugation.

r.* e = e J * r= [e J 4.26)

Vectorsca nereatedsquaternionswith erocalarpart.Notehat

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V = = ? V 4.27)Someuthorseferouaternionhataserocalarartsurequaternion.Generally,hequaternionproductofwopurequaternionsvectors)isot purequaternionvector)incehecalarpartfheroductsusuallynotero.ometimesheroductfwouaternionswithon-zerocalarartyields purequaternion.Forxample,from4.26)

r.e =- J (4.28)Thissnlyossiblefheboveroductsyperimaginary,.e.,urequaternionorvector.Weowryoormulate otationperatornermsfquaternionsperatingon vectori.e., quaternionwith erocalarpart).heimplesthingtoryismultiplicationf ectorromheightrefty nituaternion. nitquaternionshoseninceotationspreservetheengthorormofavector,ndweaneerom4.21)hatmultiplicationy unituaternionwillothangelength.irsteillryultiplicationromheeft.ehalleehatmultiplicationromheeft'only(orromherightonly)sunsatisfactory.Wechoosefortheotationoperatortheunitquaternion

X [x^ - [cos( p) inO) J 4.29)wherethe"hat"denotes unitvectorand

X '+X=.,'+x'+x'+3 4.30)Forthevectorwechoose quaternionepresentationwitherocalarpart

q= [0, ] 4-31)Thenwetryepresenting otationy

q'= q= [XQ [0q] = 4.32)

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-\.q XQ9 +X^ j Forquaternionq'oe vector",hecalarpartmustvanish.Butmultiplyingaureectory uaternionroducesnotheruaternionwith_non-vectorcomponent.Thus,unlesswemakenrthogonalityssumptionhat.q othecalarpartanishes,heuaternionmultiplicationoesoomuch.utuc hanssumptionsooestrictive.Alternatively,eouldrymultiplyingyfromheright.Thisgives

q '= qX .= [0,q\ [\ ^ J= [-^-^> ^ o ^ ^ ^ J (4.33)Notethat,rom4.22)weca nwrite

q '=qX '^= [+?.X V+ " ^ ^ J (4.34)Thecalarpartsof(4.32)nd4.34)haveoppositeign.Thisuggestswemighttryombining4.32)nd4.33)nto imilarityransformationnhehopehatwemightbebleogetthecalarpartoropout.histrategyturnsouttoeaoodne.ehalleehathispproachoesotequirenyestrictiveassumptionsuchsherthogonalityfheuaternionndheector.naddition,tillurnutoequivalentoheatrixotationperatordescribedn3.5).

q'Expanding

q'= [XQX] 6, ]Xp -X](4.35)

(4.36)

[-X,jX. +Xjj^.X XX .X+XJ'^ +\kxq+.qk-^^X X-XX^xXjUsinghedentities - 4 xBJ.I= 0ndAxBxC [A.C\-C[A.B^,ithhenormalizationonditionortheunitquaternion ,viz.,4.29),hisbecomes

q'= ( ^ -V r-0, (2X^^-1)9 +2{\.q)k 2XJXX ;I (4.37)25


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Thusq'efinedy4.35)still ectororureuaternionndtsengthspreserved,sequiredor otation.eeedomanipulatehesetermsohatqppearsnheightwithomeoperatorxpressionotseft.hisvectorpartcanewrittennmatrixomponentformyusingtheollowingdentities

k - 1 XX j ....X.q,ijk j k (4.38)0 -X.

0 -X,0

^1

^2^3

Also,- ^

\(--^(--rij{[\.q)k\ =^^^.X. =KA^g^ = LUX )q\^S S^I

XjX^ XjXjX^Xj X^X^ X^Xj

x,x,3 2 ss^1^2^3

(4.39)

where the superscriptenotes the transpose (interchange of rows andcolumns). Thusweanwrite._q'.= y[2\ -ljl XX . +X^Xjg (4.40)

= 2K, , H L

1 zoA.,A -\ XjXj0^3XQ j % X.X. QX, X,X- + XyX,A,A~,-.A.2 j1\(j j

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=X j X j XQX J

XjX --QX ,%[XQ-XJ+X,-X3]A


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X= [cosO) inO)nj 4.29)where

X jj=osO)1/2

sinO) [-j,^II. II 1.1.1.1.1.n

sinO)Wewillstablishheconnectionbetween4.35),4.40)nd3.5)ymeansofanexampleatherhanomprehensivendigorousroof,yealingwithrotationbout ingleoneofthecoordinatexes.Ofcourse, completegeneralrotationaneuiltupyeveraluchomponentotations.akehepecialcasewhere ieslonghey-axis.Then^=osO),^=inO)nd^ ^ =.Theeaderaneadilyerifyyubstitutionnto4.40)hat3.3)illegeneratedxceptorheuriousacthathenglej >squalo^!hisprovidesnsightntohenterpretationfheotationssociatedithquaternion.nEulerquaternionenerates otationboutnxisdeterminedbythevectorpart.Thehalfangleofrotationsdeterminedyhercangentoftheatiooftheengthofthevectorparttothecalarpart.Thematrix sheotationmatrixrommovingxesonertialxes.tsnumericallydenticaloheuchessymmetricndomputationallyorecomplexxpressionn3.5).utsimpleshisuaternionormfherotationmatrixppearsoe,ecalltsusthematrixndectorartsf(4.37)ndermatrixlgebra.ut4.37)sdenticalo4.35),hichsheexpressionf otationf ectorepresenteds quaternionwitherocalarpart.tsothingorehannituaternion ultipliedyheectormultipliedyhenversefhenituaternion,wheremultiplicationmeansquaternionultiplication.xamine3.5)ndoteowumbersome,ndcomputationallywkwardts.Thenxamine4.40)ndinally4.35).owdeceptivelyimple,legantndomputationallyfficient4.40)nd4.35)re!Byowheeaderhouldeginoavenppreciationorheowerfhequaternionormalism.heresrawack,owever.neanasilyintuitivelyraspheEulerngles.Theouruaternionomponentsreotoeasilyubjecttontuition.Because of this, it is more convenient to define the initial conditions of a

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simulationnermsofEulernglesatherhanhequaternionshemselves.ecanuseheEulernglesogenerate otationmatrixT.thenemainsoindthequaternionshatwouldgenerateheameotationmatrix.WealreadynowhowogenerateTromhequaternions ..Howoweohenversion,btainthe .romT?Thexpression4.35)anenvertedoiveheuaternionsnermsfherotationmatrix n4.40).hisnversionaneccomplishedsollows.Fromheiagonallementsf4.40)ndhenituaternionormalizationcondition4.29),weobtainhefollowingourimultaneousquations,wherewewillesingherace,Tr ,whichsefinedsheumfheiagonalelementsofthematrixT,viz.,Tr =n"*"^22' ' ' ' ^33"

22^22=\-1 +2 ~3" (4.42)^3 3\ -S -2 +S2

1 = >-o +1 +2 * "3Theseca neolvedimultaneouslyogivethefollowing.

1/ 2 \Q= % [ r(T)]1/ 2 X , % [ T-Tr{T)] (4.43)

X^= % [ 7^2-'(T)]1/2X 3= % [ 73 3-7r(T)]

Theres ignmbiguityoesolve.Thehiefonstraintonhelgebraicignsofhe sthattheotationmatrix romwhichthe weregeneratedhoulderegeneratedwhenheseXreubstitutedn4.40).Notethatoegativematrixelementsanrisefllhe haveheameign,whetherpositiveoregative.Thushehoicefignsotrivial.heiagonallementsfToseoconstraintincehe ccurnlysquaresnheiagonal.Theff-diagonalelementsappearscrossermsndrethekeyoourask.From4.40),wean takellpossiblecombinationsofT : ii^j.29


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(4.44)V2= f l3- 31 SS= ^f^23 ^32^S= ^f^32-^23^

Ineneratinghesexpressionsonvestgateheignmbiguity,weeehat(4.44)rovidesnlternateormoretermininghe ,rovidinghatneofthe .snown.Forxample,upposeweobtain rom4.43).Thenand anebtainedrom4.44)yividingheppropriatexpressionsn(4.44)yX Thusweobtain

k^=%[1 Tr[T]\1

A. =1

1 A =24

1 A .

4

[ 7 - 3 2--23] 4.45)

[^13 ^3lJ

IT TI 2 1 ^12jThishybridolutionombining4.43)nd4.44)asnnexpecteddvantage.Theignmbiguityemainsnlyor hisig nmbiguitysotignificantsincehangingheig nf^illhangeheig nfllhe..uthequaternionslwaysppearsproductsofpairsofquaternionsorheSquareofa quaternion.husheotationmatrix-isnvariantnder ignhangeorSee4.40).lthoughhe.signonventionoesn'tmatter,tsonventionallychosenoepositive.Thebovenversionanaveumericalroblemsnomputation,owever.The or=1,2,3,ecomell-definedwhenheXnhedenominatorvanishes.Thiswilloccurfheraceofheotationmatrixquals1 ,whichwouldhappen

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whenji= ndj )=-iradiansor80egrees.Furthermore,umericaloundoffrrorsoccurnhemmediateneighborhoodofthisegionwhensing4.45)foromputation.Itsnotdifficultovoidhisprobleminceweidothaveotartwith n (4.43).notherhoiceouldaveeenmadeeforenvoking4.44)orheother ...hisouldeoneyvaluatingllour .n4.43),ickinghelargestndhensinghisdominant.oindheother'xn4.44).ewouldagainobtainesultsthatwouldensensitivetoignmbiguitybutwouldoweinsensitivetoomputationaloundoffrrorsswellyputtingsfarawayfromanyingularity.ThislgorithmsescribednTablewithwomodifications.oronsistency,theignsrexaminedtthendofthealgorithm.fX ^ jsnegative,heignsofall.reeversedoeep ositive,ccordingouronvention.Weavealrea'dyeenhathangingheverallignfllourquaternionshasoffectonheotationmatrix.econdly,valuatingllfourquaternionsusing4.43)scomputationallywasteful.tsufficientoompareheracendhehreediagonallementsfoelectheominantuaternion.heompletealgorithmsshownnTable.Inorderousehequaternionswemusthave chemeordeterminingheimerateofhangeofhequaternionsohatheseatescanentegratedoprovideupdatedquaternionssheystemvolvesnime.hesewouldexpressionsthatlay oleompletelynalogouso3.16)hrough3.18)orheEulerangles.hiserivationouldaveeenccomplishedytandardatrixalgebraictechniques.owever,hederivationwouldhavebeenxceedinglyongandedious.Usingquaternions,tisratherimplendtraightforward.Since ndq'reectorsrureuaternions,heybeyheransformationla wgivenn4.35),

q=XqX =XqXq' XqX =XqX

(4.46)

Nowwewishoaketheimederivativeof inhemovingcoordinaterame.Asiswell-known,ermusteddedueohengularelocity,fherotatingcoordinateframe. Takinghederivative

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Table.btainingheQuaternionsromheRotationMatrixTDefineTjjp=Tr[Tl.CompareT..here =0,1,2,3.Findthedominantquaternionmostpositiveorleastnegative.)TheindexofthedominantT eterminesthedominantX tobeosedn4.43).

Determinetheotherthreek.,j=in(4.44).Thefoorcases,withthedominantXirstare

X =%[l--rr(T)]\=

K'4X.

4\.

\j=% [1+2rjj-Tr(J)]1

4X.

4X .

S=

Xj=% [1+2Tj,-Tr(T)f1

=

\=

4X.

4X ,

X j=% [1+aTjj+Tr(J)]1

\= 4X .X^=.. 4X .

^= 4X . [ ^ : 3 * ^ ^ 3 . 1ExaminethealgebraicsignofXIfnegative,hangethesignofallfour ..

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dq_^_= ~~[X. q'] axq]dtt= X~'X x'^q'X "^q'X 0 x^"]4.47)

Allhesetermsarepurequaternionsorvectors. SinceX"%' ~^ ~^ = 4.48)

weconcludethatX q'X x'^q'X -[ ixq= +[0 qxft

or 4.49)X XqX~\+x"\qX~ X= +[0 qxO]

.-11-= X q X X Differentiating X=1,0]ives X - X .Thusfrom4.7)

[0qxH = "^ -"^X = qx[X" X ] 4.50)Recallfrom4.28)hatX Xsapurequaternionorvector.Thus

^1-ft= X 4.51)Weusehe,, otationescribedn4.1)hrough4.6),incethisormalismmakesitasiertogrouptermsndind matrixoperatorquivalent.Weobtain

2X~^X=t=[ t =0 ft^+ft^ ftj = 4.52)2[k -iX -jX -kk ]k^ k^ k^ k^]

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2[Vo+Vl+SS+SS] / [ -X^XQ+XXJ +X3X2-X2X3J^^-^^-S^^^Ws]+ ^-s^+VrMs+Vs]

Thismayexpressedintothematrixormrecallquation4.10)0ftXftyft

=+ X ^+s +S +S-\ +\ +S -^2 -^2 -S +^ +S

X3 +X^ s +\

(4.53)

Theoplementnheolumnmatrixorhengularelocityaneeenovanishyakingheimeerivativefheormalizationonditionn,iz.,(4.29).imilarly,heirstinef4.52)aneuaternion-multipliedromherightby oyieldX .=%Xft =%[XQ+x^+\ +K^MQ+ft^ ft^ :ftj (4.54)

%XjjO-Xjft^-x^ft^-Xjftj+y.i[\ Q+\ ti -\ a + \ a +%j[X20+X3ft +XQft^-XjftJ M[X30-X2ft +Xjft^,+XQft^

Thismayeorganizedntohematrixormusing4.10)

=X, -X3X 3 +X^

+^2 +S +^0 -\+X 3 -x +x ^ +x

0ftXft> ft

(4.55)

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Seequation4.10)Notehatoth4.53)nd4.55)rereefingularities,unlikeheirnalogs3.19)nd3.20).Bothmatrices(4.53)nd4.55)requalto unitmatriximes ddedonntisymmetricmatrix.hesematricesreinversesfnenotherswellsransposesfnenother.Henceheyreorthogonalndpreserveheengthfheectorsheyperaten.hushevectors ndOH,ireelatedy otationn-spacewithheatternavingonly hree-vec^tor^part.omparetheselegantpropertiesandheimplicityndowomputationalburdenssociatedwithheuseoftheseequationsversus3.19)nd3.20).fthebovederivationsnotimplenoughforyou,hematrixn4.55)mayeobtainedyealizingitsthenverseofthematrixn4.53),ndaneobtainedyubstitutingintohematrixn4.53)heinverseoftheunitquaternion .Butfrom4.22) eealizehatthenverseofa unituaternionsbtainedmerelyyhangingheignfheasthree"vector-like")omponentsofheuaternion.Thusllhateedsoeoneoinverthismatrixsohangeheig nfheff-diagonallements.heressomethinghereorveryone:hemathematician,hephysicist,hengineerndtheprogrammer.Sincequaternionsarenota sintuitivea sEulerngles,tisometimesdesirabletomoveackndorthetweenheEulernglenduaternionepresentations.GoingromEulerngleoquaternionsepresentationanechievedyusingtheEulernglesovaluatetheransformationmatrixusingquation3.5).Thetransformationmatrix anheneutnto4.41)obtainheuaternions.Notehatfter4.41),twasotedhatheignor^sotmportant.Whythissobecomesapparenthortly.romhefirstowndastcolumnof3.5),and4.40)

sine-7^3^=Hk^k^-X^k^]Tr/2


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Table2. EvaluationofARCTAN(A,B)verAllFourQuadrants

IFB>0 tan'^A/B)IFB=0,A>0 IT/2IFB=0,A


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5 EQUATIONSOFMOTIONInhisection,heigidodyquationsofmotionwilleevelopedorhreetypesfoordinateystems:ody-fixed,eroballisticndlane-fixed.ody-fixedoordinatesotateroll,itch,aw)withheodyf rojectile.encethengularvelocityftofthecoordinateaxessequalothengularvelocityooftheprojectilebody.ee(3.13)hrough3.15).Forbody-fixedcoordinates

i =o = ijrin j > 5.1)XXft =0) =Q |isin4>os8-1-6os |) yft =i ) = osd )os inf ) z

RecallhatnerodynamicsheomponentsfhengularelocityfheprojectileodyareonventionallyenotedyP, ndR.Theseoordinatesarehenaturalhoiceorguidedprojectilesinceheeekerndensoroutputs,actuatorarameters,nd,soorthremostimplyndaturallyxpressedn bodyoordinates.owever,orpintabilizedrojectiles,erymallintegrationimetepsequired.therwiseheoordinateystemillollthroughtoogreatnngleduringthetimetepndmearhedirectionoforcessuchsgravity.oealwithhis,lane-fixedoordinatesreutilized.hesecoordinatesitchndawithherojectileutootollitht.nparticular,onexissconstrainedoemainingn ingleplane.nourcase,hey-axiswilleonstrainedoheorizontallane.eequations3.22)ndfollowing.hismpliesthatheollEulerangleoftheplane-fixedfi^ameatisfiestherelationsdanishesorheixedlanexes. However,hecomponentofhengularvelocityoftherame,iz.t doesotanishndtof the frame does not equal P (the x component of the projectile angularvelocity). Theseelationsmaybefoundinquations(3.26)hrough3.28).ft =i} /ine RaneXO^2-= -e 5.3)yi = os8 Z

Comparingthisxpressionorftith5.1),weeethatj)sheollateoftheprojectileithespectofane-fjxedoordinaterame.5.1)a lTereconstructedrom5.3)yddingj )o5.3)ndotatingwith3.3).hisexpressionnsefulfheulerngleepresentationssed.ithhequaternionepresentationorlane-fixedoordinates,emakesef3.5),(4.40)nd{ > oobtain37


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Thus

- 3: I' 0 2Jtanw= =r p 22"^33 \-\-2 2X, 3 2/?[x^X 3-^ X ^ .

ii = * 22222[x,X,-XX^1

(5.4)

(5.5)

Thedenominatorsn5.4)nd5.5)anvanish,orrespondingo ='ir/2.n definingplane-fixedoordinates,weadomposeEulerngleypelgebran theuaternionsndaveorruptedhemwithEulerngleypeingularities.Thus,uaternionsnlane-fixedoordinateshouldotesednerticaltrajectories.seody-fixedoordinatesnstead.hissoturdensomecomputationallyinceheirectionfgravityslonghexisofoll.inally,weouldhoosetn.5 .1)oreroballisticoordinates.hishoicewillsimplifyquationsofmotionInheollowingevelopment,esultswilleerivedsingheerm Thiswillllowllthreeoordinateformalismstoedevelopedimultaneously.Attheend,heesultsanepecializedooneorheotheryettingtqual ortheody-fixedaserqualeroorheeroballistic"zeroP")aserqual Ran8ortsquaternionquivalent ee5.3)nd5.5)).i nd2Rorllhreeases.rom3.13)weeehatheefinitionsoreroballisticandody-fixedramesreotquiva:lentutecomeheamenheimitfsmall ormalld /dt.hedvantageofchoosingplane-fixedcoordinatesfora non-rollingystemshatthaso componentofgravityn latarthmodel,thusliminatinghepossibilityofgravitymearingduetoollduringntegrationofthequationsofmotion.Thiswillbehownnquation5.10)elow.)Inummary,heody-fixedoordinatesoll,itchndawwithherojectileandctsfhysicallyttachedoherojectile.lane-fixedoordinatesitchandyawwithheprojectileutootollwitht.hey-axissonstrainedomovenhehorizontalplane.eeheiscussionorquations3.22)o3.33).TheEulerngleotationmatrixorhelane-fixedaseanebtainedromtheody-fixedmatrix3.5)yetting|) .Equivalently,3.3)seplacedytheunitidentitymatrix.^ Vaughn,HaroldR., "Aetailed Development of the Tricyclicheory," Sandia Laboratories, SC-M-67-2933,

Albuquerque,NM,968.

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TheormfNewton'saw mv/dtsalidnlynnertiali.e.,on-accelerating)oordinaterames.fheoordinateystemsotating,Newton'slawwillotevalidecause rotationsancceleration.owever,heawan bemendedorotationalrames.sswell-known,Newton'saworinearaccelerationsndorcesn otatingrameakesheormtheuperscriptM denotesthemovingrame,ody-fixed,plane-fixedoreroballistic)''^'*F + mg = mV+ ExmV mV + mflxV5.6)

dtFontainsppliedorcesuchshrustnderodynamicorces.incehecoordinateystemson-inertial,tlsoontainsfictitious"ermsuchscentrifugalndCoriolisforces".erivativesofnertialpropertiesuchsmasswilleomittednhisdevelopment^efining^ ,V,nd^ W,thecomponentsoftheabovevectorquationsnow

MF+mg =U+ m[QW-RV]F +mg^ V+ m[RU-lW] 5.7)y*

MF+mg =W+m[aV-QU]zRearranging F^

U g"'-QW+RVXm^ JohnH. lakelock,AutomaticControlofAircraftandMissiles,"JohnWileyan dSons,New York,1965.^ KeithK.Symon,Mechanics ,"Addison-Wesley,Reading,Mass ,960.* Goldstein,Herbert,ClassicalMechanics" ,p136 ,AddisonWesley,Reading,Mass ,959."* Landau,L.D.,ndE.M.Lifshitz,ClassicarMechanics",p128,AddisonWesley,Mass ,960." "Th ethrustofareactionnginethat smeasurednateststandlreadyontainstheeffectsoftherateof changeof th emass .incehisnformationsusuallyvailableornputnto imulationratherthannozzlepressures,derivativesfthemassootppearnhequationsfmotion.owever,fhehrustsoe-reconstructedromressuremeasurementstheozzle,masserivativeermsndheelocityfhexhaustaseswouldaveoeakenntoaccountnth eequationsofmotion. Thiswillbediscussedin detailnafuturereport.

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FV+ g -RU+ a 5.8)

mF M w=+ g ^-a v+QU m

Theomponentsor aneobtainedymultiplyingheravityectormheearthrameyheppropriate matrix.Forhepecialaseor latarth,gonlyhas verticalor componentpointingownward.Coriolisndentripetalaccelerationorrectionshouldemadeince latarthsoteallynertialsincehearthotates.fistanceslownndimeflightrehort,hesecorrectionsarenegligible.)Weeedhenertialoodyransformation romhenverseof4.40) ndtherepresentationofthegravityvectorin flatarths(

g^=[X^X^-K^K^]gg""=nk k +\\ ]g orflatarth (5.9)

Ifheserwishes,heomponentsor nheeroballistic.lane-fixedrbody-fixedramesor latarthouldeobtainednermsoftheEulerngles-ITinsteadofquaternionsyusinghexpressionorTn3.5)nsteadof(4.40).

f'^g^ =gineg^ ^sinose

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Notehatheresolane-fixed-axisomponentfravityinceinj )orplane-fixedoordinates,dvantagefheefinitionfplane-fixedoordinatesadoptednhisevelopment.twouldoteruef^ washosen.ngeneral,f as r omponentsnnertialoordinates,heullotationmatrixTwouldhavetoeused.See3.5)nd4.40).RecallinghatOshengularelocityofheoordinateramewithespectothenertialrame,oshengularelocityofheodytselfwithespecttoheinertialrame,ndhemomentfnertiaensorsymmetric,.e.,.. .;Newton'sawsforangularvelocitiesandmomentsareoftheform

M lxL7(0-1-jc[/(o] (5.11)M M M

+1 -Iyx -I

-I y+J yy

-I-I

-I1ZV2 1 ci^ 11: > 1 '1 A

TimederivativeoftheangularvelocityofthebodyAngularvelocityofcoordinateystem

Va /

X -/II-/4 - nn yjcA

xy +1

-I-I

crossproduct

yyI1zyz (O(0C O

AAngularvelocityofthebody

4 1


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48/75

+ 1 PI QI R XXyz-I P1 QI R yxy*"z-I PI Q1 R zxyzQ-1 P-I Q1 R- -I P1 Q-I RzxyzxyzR-\-I P-I Q-I R-a [-I P-I Q1 RXXyzxyza [-1 P1 Q1 R]- +1 P-I Q-I RXxyzxyz

ThismayewrittenL P XX ( 5 .15 )

IZ y\ 1/ -/ \QR < -Vanisheswithaxialsymmetryr 2i+/ \R -Q J+/ \-QP-R\-l \RP-Q\


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IfnseplacedyP,quations5.14)r5.15)reheeneralquationsofmotionorhengularelocityomponentsfigidodynody-fixedcoordinates.oronveniencenrogramming,ermshatanishhenheproductsfnertiaanishaveeenroupedogether.ody-fixedoordmatesareppropriateornsymmetricodiesndoruidedrojectiles,incehesensorsndontrolystemrenaturallydescribednody-fixedcoordinatesthatrollwithheody.Notehatheomponentsofhemomentofnertiaensorwillnotchangedueotheotationofthebody-fixedoordinates.Forxially-symmetricpin-stabilizedrojectiles,lane-fixedoordinatesremoreppropriate.fbody-fixedoordinatesreusedwithapidlypinningpinstabilizedrojectiles,hentegrationimetepn egree-of-freedomsimulationsrivenoeverymall,ncreasingheimulationunime.Thissnecessaryoeepheollngleuringhentegrationimetepmall.hisavoidsmearingravitynhexpressionsorx, nd.ee5.7)r5.8).Withplane-fixedoordinatesndxialymmetry,heproductsofnertiaand anish,nd 2^seplacedsing5.13)n5.14)r3.15).Notetfiatthecomponeiitsof"hemomentofnertiaensor wouldgenerallyvarywithimeftheprojectilewereotatingbutheramewerenot.otonlywouldthise omplicationuttwouldriveownhentegrationimetepndincreasentegrationime.imilarlyorheerodynamics.incehemotivationforlane-fixedoordinatessreateromputationalpeed,tsointlessoeliminategravitationalmearndubstitutenertialorerodynamicmear.Thusaxialymmetrynmasspropertiesnderodynamicsisgenerallyassumed.Afteronemoreesultsobtained,heseormulaewilleollectednTables o8.rom4.55),usingheappropriatexpressionori^,wecanwriteownhetimederivativesofthequaternionsfortheseframes.

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Forbody-fixed.^0 = % \= % s= % s= %

-px-G\,-/?X3+pXj,-X 3 ex +PX 3+Q\^-?x^ -PXj+GXj ?X Q

(5.16)

Forplane-fixed,using(4.55)nd5.5)Xo= -2J?X, 13 0 2,2,[XQ-X -X.+XJ] -ox.-R\,=-%

X.? x,(2+

f 2,1 [Xo-X,-X,+X 3 ] J

\ =& +2R'Kr XjXj XQX.XQ-X -X.+XJ'] -x. X ^ (5.17)

X.? =-% X ,Q

[XQ-X'-XJ+XJ]]

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X , ii+lRX A. A 3nAjL[Xo-X;-X2+X3]

+ GXQ-X^

=% >^oC X jR2 2 ; 2,1X(,-X^-X,+X-]J

X,= -2i?X. ^1^3 ^0^2[Xo-Xj'-x;+x:] +X j ?X j

=% X i?X iG\ 2,1[Xo-Xi-X,+X,]J

Aingularitywillccurfheenominatoranishes.rom3.22)nd4.40),thisenominatorsustTos(e),whichanishesor =17/2.Thisshesameingularityiscussedfter5.5).tsotdvisableoselane-fixedcoordinatesorearerticalrajectories.seody-fixedoordinatesnstead.Forerticalrajectories,gravitymearingdueoollatesnotheproblemtsforothertrajectories.AllhequaternionndEulerangleesultsorheinearndngularquationsofmotionreollectednheollowingablesorody-fixedndorlane-fixedcoordinatesespectively.Forompleteness,weotehatheeroballisticramequationsfmotionanbebtainedyettingianishn5.8)nd5.15),ndettingtanishn(4.55)ranishn3.16).sithlane-fixedoordinates,eurthersimplifyyeglectingheproductsofnertia.hisormwilleoundnTable5.urtherimplificationesultsromssumingxialymmetryndonstantmass.eeTables nd.

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Table3. Body-FixedEquations

U g -QW+RVmF

V= g RU+PW 5.1),(5.8)mF z

W g"-PV+ QUzm-1ThecomponentsofgravitationalccelerationreobtainedusingT Forheflatarthpproximation,use5.9)nd5.10).

L P XXL" wJ QR < -Vanishesforaxialsymmetry 5.15)

\ ^2! , r ^ - 1 , f - 1 +/ \R Q \+I \ -QP-R\+I \RP-Q\


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Table4. TimeDevelopmentoftheBody-FixedTransformationMatrixParameters

X=^

x=X,=

PX,-Q\,-?X ,+PX-Q, R\\+PX 3 XQ-?xj-PX2+2X j JXpj

(5.16)

or

^

t t >

[Q 5i()J C05 (9)

8Go^(4>)- ini


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Table. Plane-FixedEquations(AxialSymmetry, =/ =/,productsfinertiavanish)yy zz I " ^

u g QW+RVXm ym (5.8)

M W \-g -D,+ QU m

ThecomponentsofgravitationalaccelerationrebtainedusingTFortheflatearthpproximation,use(5.9)nd5.10).L P XXM Q + PR-RntX(5.15)N R- QP +a QtX^xwhereft isobtainedromxft =RaneX (5.3)

or

ft = 2 ^ KS-V2]2222 (5.5)49


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Table. TimeDevelopmentofPlane-FixedTransformationMatrixParameters

k,=% \,KQ [21 [XQ-XJ-XJ+X-JJ-% Xj? x,G [2,1 [Xo-Xj-X.+X-Jj (5.17)

X. +V 4 ^oC \ R2 2 2 2- 1Xo-X^-X +Xj] ^

X ,= +% X^ i? \QXo-Xi'-x:+x;]]

or( } >= 04 ) = 0

R ^ = co5(e)eQ (3.16)(3.18)

C T / z eabovexpressions5.17)nd3.16)reingularnearthevertical. Useody-fixedcoordinatesnstead.Seethediscussionnthetext.)

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Table7 . AeroballisticEquationsTheserebtainedyettingI = 0,ndssumingoroductsfnertia.furtherimplificationa nemadebyetting/ = =7 (axialymmetry).

u F ^ g^-QW+RVm

V ym F M

(5.1),5.8)

W g ^QUm

1 ThecomponentsofgravitationalccelerationreobtainedusingTFortheflatearthpproximation,use5.9)nd5.10).L P 5.15)M +1Q+I R PtXN +1R-I PQtX ^

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Table8. TimeDevelopmentoftheAeroballisticTransformationMatrixParameters\= % Q R). \= % ."GX3 + Rk s= ^ +Q Rk^ ^3 = ^ [ ^ Q+ Rk

(5.16B)

or

^

4 >

[Qini4>) os(i^))cos(Q)

[Qin(^) os(


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6. INTEGRATIONOFEQUATIONSOFMOTION6.1 Plane-FixedEquationsRecallortheplane-fixedase,heorcequationsfromTable re

FU g^QW+RV

mF

V= 1-^ -RU+aW 6) mF MW g^IV+ QVm

wherel^sgiveny5.13)nd siveny5.9)or latarthrmoregenerallyy hereheubscript efersonnertialrame.Thesexpressionsarereadilyntegratednumerically.Forheplane-fixedase,hemomentquationsromTable a neputntonuncoupledformorntegration,where/ = = .ty2yL p

IXXe [M-/^^;?7'+/^/?n;J 6.2)

I

R \N+IP-IanIX X \I " tThesereeadilyntegratednumerically.53


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6.2 Body-FixedEquationsForbody-fixedase,t = androm5.8)

U g^QW+RVm

V = ym

M -RU^PW

w =Fzm

B -PV+ QU

whichreeadilyintegratednumerically.Forhebody-fixedcase,quations5.15)mayewritten

P [L+I^^Q+lJ-fXp,Q,R]]

Iyy

whereI

/ // ]QR+I \R^-Q^\-IP+IP ' Iz yy r^ J Xy(6.3)

Q[M+I ^P+I ^R-f^[p,Q.R]] 6.4)

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^2=-^..]^^+^.2^+^l^'-^'j-^.,e/?6 . 5 ) / = f/ -I ]PQ-I PR+I QR+I \Q^-P \3y X X yzxy-^

Aswritten,hesequationsreoupled.fheroductsfnertiaanish,henumericalntegrationstraightorwardecausehequationsecomeuncouplednhederivatives,sn6.2)bove.venftheyootvanish,heproductsfnertia ^nd reenerallyuitemall.hisuggestssimplepproximation.ThequationsouldeolvedysingheerivativesdP/dt,dQ/dtanddR/dtonheightsideofequation(6.2)romheprevioustimestep.inceheproductsofnertiareypicallymall,hispproximationhouldbedequatenpractice.ormoreprecision,heesultsouldeterated.hatis,heesultsorthederivativesdP/dt,dQ/dtanddR/dtnheeftideouldeputackntoightideobtain etterpproximationeforeperformingnintegration.Ifheroductsfnertiareotmallrfneishesovoidhisapproximation,equations(6.4)nd6.5)aneputntoheform

L-f. +/ P-I Q-I R 1XyxzA/-/,=-/ P+I Q-1 R 6.6)yyz "-"/N-f^ -IP-I Q+I R 3zzz

TheyaneolvedimultaneouslyouncoupleheerivativesofP,Qnd yinvertingthematrixofmomentofinertiaomponents. Formallywecanwrite

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pQ R

+///XXyz-I1Ixyyz-II[xzzz N-f, (6.7)Evaluatinghisnversesomewhatediousutheroceduresornvertingmatrixrewellknown . Ifwedenotethematrixoeinvertedy

j" det ^ / / -/yy zz yzI I +1 Ixz yz xy zzr1xyz xz yy

I I +1 Ixz yz xy zzIi'X Xzz xzr1xyxz XXyz

I I +1 Ixy yz xz yyI I +1 Ixyz X Xz

2I I -IXXyy xy(6.8)

wheredet = 2 2 2III--III --III - -II-// - /X Xyyzz xyzz xz xy yz yy xz zz xy XXyz (6.9)

^Gelb,Arthur,tal..AppliedOptimalEstimationTheory",p7,MITPress,Cambridge,Mass,974.

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6.3 AeroballisticEquationsRecallortheeroballisticcase,heforcequationsfromTable are

u = FXm

M -QW+ RV

V =F ym

My RU

w .^ Fz M- + g.QUm

ThemomentquationsromTable reL P J XX

X X^N+ 1 QP\IXt

Thesereeadilyntegratednumerically.

(6.10)

^MIRP\ 6.11)

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APPENDIX

ALGORITHMSFORMPLEMENTATIONOFTHEEQUATIONSOFMOTIONINSIXDEGREEOFFREEDOMCOMPUTERSIMULATIONS

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TRANSITIONSBETWEENEULERANGLESANDQUATERNIONSTheInitializationProblem

SincemostpeopleremorentuitivelyomfortablewithEulerngleshanwithquaternions,heEulerngleouaternionransformationaneusedonputinitialonditionsnulerngleormatorheonveniencefheserndconvertoquaternionsornternalusen imulationfodesired.Conversely,quaternionssednternallyy imulationaneonvertedoEulernglespriortogeneratingoutput,ortheconvenienceoftheuser.QUATERNIONSTOEULERANGLES:TheEulernglesanevaluatedirectlyromheuaternionsrndirectlyfromotationatrixhatadeenevelopedromheuaternions.se(4.56).Notehatheenominatorfhexpressionsor|;ndor n4.56)anvanish.ThealgorithmnTable inhisdocumentca nhandlethiscasecorrectly.EULERANGLESTOQUATERNIONS:1 )valuateheransformationatrix romheEulernglessing3.5).(Withlane-fixedoordinates,heollulerngle|)usteetoero:Ahernatively,3.22)aneused.)2)EvaluatethequaternionsusingTable.

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TRANSITIONSBETWEENBODY-FIXED,PLANE-FIXEDANDAEROBALLISTICCOORDINATESPlane-fixedoordinatesreorefficientorodelingpin-stabilized,unguided,otationallyymmetricprojectiles.heresoomponentfgravityoutsidefhe-zlanenlatarthodel.husheseoordinatesreinsensitiveoravitymearingecausefoll.owever,lane-fixedcoordinateshave ingularityorverticaltrajectoriesandody-fixedoordinatesarereferredoruchrajectories.ody-fixedoordinatesrelsooreappropriateoruidedtagesrthertageshaton'taveheequiredsymmetry.imilarly,eroballisticcoordinatesrelsomorefficientorxially-symmetric,pin-stabilizedprojectileshanody-fixedoordinates.Furthermore,thequationsofmotionreimplerorhishoicehanheotherwoandidatecoordinateframes.Thisdocumentpermitsheevelopmentof degreeofreedomimulationn whichheoordinaterameanehangedromnetageonotherousehecoordinateframehatsmostappropriateorfficientnachparticulartageofa trajectoryimulation.enerally,therhanhanginghequationsfmotion,nothingpecialneedsoeonewhenransitioningetweenneypeoframeandnother.heresnxceptionorransitioningoplane-fixedoordinatesfromotherframes.

Whenransitioningolane-fixedoordinatesheresiscontinuouschangenherientationftheoordinateystem,ince ustanishn plane-fixedcoordinates. Thisrequirestheollowingadjustments:1)TherojectilengularelocityectorP,Q,R)usteemporarilytransformedoon-movingi.e.,nertial)oordinatesusingheastvalueoftherotationmatrix.2)ewotationatrix usteenerated.Ifheuaternionrepresentationseingsed,hequivalentulernglesusteregeneratedirst,shownn4.56).)etoordinaterameEulerollanglej)oero,etainingheegenerateditchndawngles.RecalculateheotationmatrixwithheseewEulernglesusing3.5)r(3.22).3 )Usehismatrixootateherojectilengularelocityectornhenon-movingrameackohemovingplane-fixed)rameobtainhenewP, ndR.4)fusinghequaternionormalism,egeneratehequaternionsromherotationmatrixusingTable.

Resumealculationssingheppropriatequationsfmotionorheypefcoordinate frame being used, for either the Euler angle or quaternion60


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representationsescribedlsewherenhisocument.Theitch ndaw)i Eulernglesorheodyndorplane-fixedxeswilledentical.heplane-fixedrameollEulernglej)s onstantnddenticallyero.heollEulerangleorheeroballisticframesnotonstantutgenerallywillotvarymuchfromtsvaluewhenransitionoccurred.Whensinglane-fixedreroballisticoordinates,heodyngularelocitycomponent usteeconstructedorsewithheerodynamics.hentransitioningackobody-fixedoordinates, ollngleaneestoredyirstconstructinghenversei.e.,ranspose)fheollotationmatrixrom3.3).Theewulloationmatrixsbtainedyakinghematrixroductfheexistinglane-fixedotationmatrixndheollmatrix,nhatrder.hisprocedurewillnotcorrupttheothervariablesnthequationsofmotion.

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BODY-FIXEDCOORDINATESUSINGQUATERNIONSRotationmatrixfrombody-fixedtoinertialcoordinates:Calculate T( ^.^,^, ) from4.40).Thenverseotationmatrix,romnertialoordinatesoody-fixed,sobtainedbyakinghetransposeofthismatrix.

Timederivativesofthequaternions:Calculate \ ,^,^,^ from5.16).

OtherEquationsofMotion(SeeTables and4.)

a )Forcequations:Use(5.8)with l ^ .See5.1).Obtainomponentsfravity.inody-fixedrameromhenertialframebyusingT .Forflatearth,use(5.9).b)Momentquations:Use(6.4)nd6.5)or(6.9)o6.11).

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PLANE-FIXEDCOORDINATESUSINGQUATERNIONSRotationmatrixfromplane-fixedtoinertialcoordinates:Calculate T( , ^, ,^) using4.40)Thenverseotationmatrix,romnertialoordinatesoody-fixed,sobtainedbytakinghetransposeofthismatrix.

Timederivativesofth equaternions:Calculate \ , ^, ^,^ from5.17).with

a 2^[V3-V2]2 2 22from5.5).Thereisaconstraintn4.57).

OtherEquationsofMotionSeeTables and):a )Forcequations:Use.(5.8)witht from5.5).eeabove.See5.1).Obtainomponentsofgravitynody-fixedrameyusingToi rflatarth,se5.9).ecausefheonstraint4.57),hereso omponentofgravity.b)Momentquations:Use6.2).

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AEROBALLISTICCOORDINATESUSINGQUATERNIONSRotationmatrixfrombody-fixedtoinertialcoordinates:Calculate T (^, ,^,^) from4.40).Thenverseotationmatrix,romnertialoordinatesoody-fixed,sobtainedbyakinghetransposeofthismatrix.

Timederivativesofthequaternions:Calculate \ , , ,\, from5.16B),or(5.16)withP=0.

OtherEquationsofMotion(SeeTables7and8.)a )Forcequations:Use(5.8)withi^-.See5.-1).Obtainomponentsfravitynody-fixedrameromhenertialframebyusingT .Forflatearth,use(5.9).b)Momentquations:Use6.11).

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BODY-FIXEDCOORDINATESUSINGEULERANGLESRotationmatrixfromplane-fixedtoinertialcoordinates:Calculate T(} ) ,,); from3.5).Thenverseotationmatrix,romnertialoordinatesoody-fixed,sobtainedbyakinghetransposeofthismatrix.

TimederivativesoftheEulerangles:ft=P=Q=R5 .1)X (iQin(), j ,-^ 3.16)cos(Q)4 >p + (Qin(


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PLANE-FIXEDCOORDINATESUSINGEULERANGLESRotationmatrixfromplane-fixedtoinertia!coordinates:Calculate T(j)=0,6,);) from3.5).Thenverseoiationmatrix,romnertialoordinatesoody-fixed,sobtainedbyakinghetransposeofthismatrix. Notethatj )szeron3.5).

TimederivativesoftheEulerangles:Use j >=t )= instead3.17)ndt = Qndft^.=.

R4 from3.16).eQ from3.18)OtherEquatioBSofMotionSeeTables and6.)a )Forcequations:Use(5.8)witht from5.3).See5.1).Obtainomponentsofgravitynplane-fixedramebyusingT .Forflatarth,use5.10)withj)=0.b)Momentquations:Use6.2).

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AEROBALLISTICCOORDINATESUSINGEULERANGLESRotationmatrixfromplane-fixedtoinertialcoordinates:Calculate T(),,j; from3.5).Thenverseotationmatrix,romnertialoordinatesobyakingtheransposeofthismatrix. body-fixed,sobtained

TimederivativesoftheEulerangles:X y R (5.1)

(Qin() (3.18)

OtherEquationsofMotionSeeTables7and8.)a )Forcequations:Use(5.8)withO^ .See5.1).Obtainomponentsofgravitynody-fixed-frameyusingflatarth,use5.10). .Forb)Momentquations:Use(6.11).

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CONSTRAINTS,LIMITATIONSANDPRACTICALREQUIREMENTSI.Normalizationconstraintnquaternions:

Weequire k^ ^ ^ ^ from4.29).Atachntegrationtimetep(ortleastatfrequentntervals),normalizeydividingach..by

-1/2f 2III.Constraintforplane-fixedcoordinates:

Werequire X^ 3+0 1" ^^^4.57).Checkhisconstraintregularly.fitbeginstoail:

a )egeneratetheEulernglesromhematrix using4.56),b)et j)=,ndc)egeneratequaternionsfrom4.42).

III.Euleranglesingularity:Terminateimulationf sooloseo90egreesecauseofheingularityathatngle.ee3.16)nd3.17).N.B.,heEulerngleotationsfoll,pitchndawayehosenooveheingularityoccurlonghehorizontalatherhanheerticalxes. etterolutionsoseuaternionswithody-fixedoordinates.orlane-fixedoordinates,eearagraphVbelow.IV.uaternionsingularityforplane-fixedcoordinates:ThisingularitysimilaroIIxcepttonlyoccursorplane-fixedoordinatesandotorody-fixedreroballisticoordinates.ody-fixedoordinateschouldeusedorverticaltrajectoriesatherthanplane-fixed.

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V. Axialsymmetryrequirementforplane-fixedandaeroballisticcoordinates:

a ) =b)Productsofinertial^^,I^^,I^^,/^,,/>/,,llvanish.c)Aerodynamiccoefficientsdonotdependnollangle.

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DISTRIBUTIONLISTCommanderArmaments Research,DevelopmentandEngineeringCenterUSArmyTankAutomotiveandArmamentsCommandATTN: AMSTA-AR-AE,AMSTA-AR-AET,AMSTA-AR-AET-A,

B.BusheyW.EbiharaM.Amoruso(15)E .F .BrovmS .ChungA.FarinaA.FiorelliniT.ForteJ.GrauS .HanH.HudginsW.KoenigW.KuhnleC .LivecchiaG.MalejkoJ.MurnaneC.NgD .PedersenJ.SnyderW.ToledoR.TrohanowskyJ.ThomasovichE .VazquezJ.WhyteB .WongL .Yee

AMSTA-AR-FSP-A, T.HaritosR.ColletteK-KendlM.RosenbluthPicatinnyArsenal,NJ07806-5000

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