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ACOMPARISONBETWEENIMPLICITANDEXPLICITSPA__CRAFTGYROCALIBRATIONITZHACKY.BAR-ITZHACKFacultyofAerospa__EngineeringTechnion-IsraelInstituteofTechnologyHaifa32000ISRAELRICHARDR.HAR__NFlightDynamics____ysisBranchCode595NASA-GoddardSpaceFlightCenterGreenbeltMD20771USAAbstract:-Thispaperpresentsacomparisonbetweentwoapproachestosensorcalibration.Accordingtooneapproachcalledexplicitanesti__torcomparesthesensorreadingstoreferen__readingsandusesthedifferen__betweenthetwotoesti__tethecalibrationparameters.Accordingtotheotherapproachcalledimplicitthesensorerrorisintegratedtoformadifferententitywhichisthencomparedwithareferen__quantityofthisentityandthecalibrationparametersareinferredfromthedifferen__.Inparticularthispaperpresentsthecomparisonbetweentheseapproacheswhenappliedtoin-flightspa__craftgyrocalibration.Referen__spa__craftrateisneededforgyrocalibrationwhenusingtheexplicitapproach;howeversuchreferen__ratesarenotreadily__ailableforin-flightcalibration.Thereforethecalibration-parametersesti__torisexpandedtoincludetheesti__tionofthatreferen__ratewhichisbasedonattitudemeasurementsintheformofattitude-quaternion.Acomparisonbetweenthetwoapproachesis__deusingsimulateddata.Itisconcludedthatthetwoapproachesyieldcomparableresultsbuttheimplicitsoftwareimplementationislesscomplexthantheexplicitimplementation.Key-Words:-CalibrationGyroscopesSpa__craftKal__n-Filter1IntroductionIn-flightsensorcalibrationisacrucialelementinspa__craftpreparationforasuc__ssfulmission.Duringthefirststageofthecalibrationpro__ssthesensorerrorssour__swhicharecalledcalibrationparametersareesti__ted.Duringthesecondstageerrorscausedbythesecalibrationparametersareremovedfromthesensorreadings.Therearetwoapproachestosensorcalibration;namelyexplicitandimplicit.Intheexplicitapproachanesti__torcomparesthesensorreadingstoreferen__readingsanditusesthedifferen__betweenthetwotoesti__tethecalibrationparameters.Accordingtotheimplicitapproachthesensorerrorisintegratedtoformadifferententitywhichisthencomparedwithareferen__quantityofthisentityandthecalibrationparametersareinferredfromthedifferen__.Ifforexamplethesensorsaregyrosthenusingtheexplicitapproachesti__torcomparesthegyroreadingstoareferen__rate.Iftheimplicitapproachisappliedthenthegyroreadingsareproperlyintegratedtogenerateattitudeandthelatteriscomparedwiththereferen__attitude.Inbothapproachesthedifferen__isusedbytheesti__torwhichesti__testhecalibrationparameters.Obviouslyreferen__rateisrequiredintheexplicitapproachtogyrocalibrationandreferen__attitudeisneededwhentheimplicitapproachisused.Anexamplefortheuseoftheexplicitapproachcanbefoundin
[1]andanimprovementofitispresentedin
[2]wherethecalibrationoftheAQUAsa____itegyroscopeswastreated.Sin__thecalibrationwasdonein-flightareferen__angularratewasnotreadily__ailable.Thereforemeasuredattitudewasusedtoesti__tethereferen__angularratevector.Subsequentlyanesti__torusedthedifferen__betweentheesti__tedreferen__rateandthegyro-measuredratetoesti__tethecalibrationparameters.Actuallythetwoesti__torswerecombinedintooneKal__nfilter.Becausetherateesti__tionwasbasedonEuler’sequationthatdescribesthespa__craftSCangulardynamicstherateesti__tionwasnonlinearwhichg__erisetotheuseofthePseudo-LinearKa__nFilterPSELIKA
[3].Theimplicitapproachisnor__llyusedforcalibratingtheinertialmeasuringunitininertialn__igationsystems
[4].Thepurposeofthepresentworkisthecomparisonbetweentheexplicitandtheimplicitapproachesusingsimulatedgyrodata.2TheGyroErrorModelThegyroerrorsthatareconsideredinthisworkare:misalig__entscalefactorerrorandbiasconstantdriftrate.Thegyroerrormodelisalinearmodelwhichassociates__allerrorsour__stothegyrooutputs.Duetothelinearityofthemodelwecancomputethecontributionofeacherrorsour__independentlyandthensumupallthecontributionsintoonelinearmodel.Westartthedescriptionoftheerrormodelbyderivingtheexpressionforthegyromisalig__ents.
2.1Misalig__entModelItcanbeshown
[1]thattherateerrorduetomisalig__entofabodymountedgyrotriadis:1whereistheprojectionofthei-gyrosensitive-axisonthebody-axis.Sin__thesensitiveaxisisdescribedbyaunitvectorandduetotheproximityofthegyrosensitiveaxistotherespectivebody-axisisamisalig__entangleassumedtobe__all.Theelementsaretheangularvelocitycomponents.Let2and3wheredenotesthetranspose.ThenEq.1canberewrittenas
42.2ScaleFactorErrorModelAsmentionedothererrorsour__sthatcausesthedifferen__betweenthecorrectvalueoftheactualratesandtheirmeasurementsarethescalefactorerrors.Theerrormodelforthescalefactorerrorsissimply5whereisthescalefactorerrorofgyro.Eq.5canbewrittenasfollows6Define
7.aand
7.bthenEq.6canbewrittenas
7.c
2.3BiasModelThebiaserrormodelisquite______andisgivenby8whereisthethreedimensionalidentity__trixand9andxyzarethecorrespondinggyroaxes.
2.4TheAugmentedGyroErrorModelThetotalgyroerroristhesumofalltheerrordiscussedbefore;namelymisalig__entscalefactorandbiaserrors;thatis
10.aUsingEqs.
47.cand
810.bThelastequationcanbewritteninthefollowingform
10.cDefineasfollows
10.dalsolet
10.ethenEq.
10.ccanbewrittenas
10.f3TheExplicitGyroCalibrationAlgorithmTheinfor__tionthatweh__ein-flightisattitudeandratesmeasuredbythegyrosbutunlikegroundcalibrationthereferen__ratesneededforexplicitcalibrationarenotreadily__ailable.Thereforeasmentionedearlierinthiscaseweh__etoesti__tetheangularratevectorwhileesti__tingthecalibrationparameters.Weusethentheattitudeinfor__tiontoesti__tetheangularrate.Theattitudeinfor__tioncanbesuppliedinvariousways;namelyitcanbeintheformofrawvectormeasurementsoritcanbegiveninanalreadypro__ssedformasattitudequaternionforexample.Theesti__tionoftheangularratevectorhingesonEuler’sequationthatdescribestheSCangulardynamicsasspecifiedbelow:11whereIistheSCinertiatensoristhecrossproduct__trixofthevectoristheangularmomentumofthemomentumwheelsandistheexternaltorqueoperatingontheSC.Asmentionedbeforethereferen__rateisesti__tedfromthemeasuredattitude.Thisne__ssitatestheinclusionoftheattitudedynamicsintheesti__tordynamicsequation.SupposethattheattitudeismeasuredbyautonomousstartrackersAST
[5]thatyieldthemeasuredquaternion.Itsdynamicsequationisexpressedby
[6]
12.awhere
12.bEquation
12.acanbealsowrittenas
12.cwhere
12.dBecauseEq
10.eisaconstantvectoritobeysthefollowingdifferentialequation13WecancombineEqs.
1112.cand13intoonedynamicsequationaddwhitenoisetotheangulardynamicsequationtoaccountformodelun__rtaintyadda__allquantityofwhitenoisetothecalibrationparameterdynamicstobettermodelthemincasetheyarenotreallyconstantbutrathervaryslowlyandaddwhitenoisetothequaterniondynamicsequationtoaccountformodelinginaccuracies.Asaresultweobtainthefollowingdynamicsequation…14whereisthenoisevectoraddedtotheangulardynamicsisthenoisevectoraddedtothecalibrationparametersandisthewhitenoiseaddedtothequaterniondynamics.Becausethedynamics__trixisafunctionofaswellasthedynamicsequationpresentedinEq.14isnonlinearsubsequentlyanonlinearesti__torisneededforesti__tingthestatevector.Themostappropriateesti__torforthenonlinearitystructureofthisdynamicsmodelisthePSELIKAalgorithm
[2].Asexplainedbeforeweh__etwomeasurementstoconsider;namelythegyrorate-measurementsandthequaternionmeasurement.Theformerconsistsofthetrueratetheerrorduetothecalibrationparametersandsomehighfrequencynoisewhichismodeledasazero-meanwhitenoise.Consequentlyweh__e
15.aUsingEq.
10.fthelastequationcanbewritteninthefollowing__trixform
15.bThequaternionmeasurementismodeledbyacombinationofthetruequaternionandazero-meanwhitemeasurementnoisevector.
15.cAsexplainedintheintroductionwecombinetheesti__torsthatesti__tethecalibrationparametersandthatwhichesti__testheratevectorintooneesti__tor;thereforeweh__etoproperlycombinethetwomeasurementequations.ThecombinationofEqs.
15.band
15.cyields
15.dEquations14and
15.dconstituterespectivelythedynamicsandthemeasurementequationsfortheaugmentedesti__torusedintheexplicitapproachtogyrocalibration.4TheImplicitGyroCalibrationAlgorithmAsinthepreviouscaseourgoalnowistoesti__teandforthatweneedtoknowhowinfluen__stheattitudeesti__tion.Thetruequaternionobeysthedifferentialequation
12.awhereisafunctionofthetrueangularratevectorwhichwedonotknow.Weknowthoughandthereforeweratherusethemeasuredratevector.AccordingtoEq.
15.atherefore
16.aandsimilarly
16.bwhere
16.candwhereandarefunctionsofandrespectivelyandareinthefor__tofEq.
16.c.UsingEq.
16.bEq.
12.acanbewrittenas:
17.aThelattercanbewrittenas
17.bwhereQisasinEq.
12.d.UsingEq.
10.finEq.
17.byields
17.cAugmentingthelastequationwithEq.13towhichasbeforeweaddyields
17.fwhere
17.gEq.
17.fisthedynamicsmodelusedbytheesti__torwhenweapplytheimplicitapproachtogyrocalibration.FromtheexplanationofthewaythisapproachworksitisclearthatthemeasurementintheKFsenseisjusttheattitudemeasurementandnotthegyromeasurementswhichinourcaseissimply.FromEq.
15.citisobviousthat18Eqs.
17.fandEq.18constituterespectivelythedynamicsandthemeasurementequationsfortheesti__torusedintheimplicitapproachtogyrocalibration.5CompensationTocompletethecalibrationpro__ssweneedtoperformthecompensationstageusingtheesti__tedcalibrationparameters.FromEq.
10.cweobtainthefollowingesti__tesofthegyroerrors
19.aSin__
19.bthenacalibratedgyromeasurementofiscomputedasfollows
19.c6ImplementationConsiderationsThedynamics__trixinEq.14isafunctionofandwhicharenot__ailable.Initiallyhoweverwecanevaluatethe__trixusingratherthanbutwhentheattitudeconvergeswecanusetheesti__tedquaternionwhichiscloserto.Similarlyinitiallywecanuseratherthantoevaluatethisdynamics__trixthemeasurement__trixofEq.
15.dandthedynamics__trixinEq.
17.f.Whenintheimplicitapproachtheesti__torconvergesitisbettertoswitchtotheesti__tedratedenotedbyandwhenintheexplicitapproachtheesti__teofconvergesitisbettertousethecalibratedrateratherthanbecausethecalibratedgyroreadingsareclosertomorethanis.7TestResultsThealgorithmwastestedusingsimulatedtelemetryflightdata.AutonomousStarTrackerASTandgyrodataweresimulated.TheASTprovidesthemeasuredquaternion.Themodeledcalibrationparametervalueswere:radiansrad/secThebiasvaluesarerespectively
0.1-
0.2and
0.3degreespersecond.Eachsensorprovideddataata1Hzrate.Thecalibration__neuversstartedwithazeroinertialrateperiodof200secondswhichwasusedtoesti__tethegyrobiases.Thisinertialperiodwasfollowedbythreesequential
0.1deg/sec__neuversaboutthexyandz-axesrespectivelylasting200secondseach.Therunlengthwas1400seconds.Thesecalibration__neuversareshowninFig.
1.Fig.1:SCratesduringthegyroalig__ent__neuvers.
7.1ResultsusingtheexplicitapproachThegyrocalibrationparameterswereesti__tedwell.Howeverextensivetuningofthefilterparameterswasrequiredinordertoachievetheseresults.Attheendofthissimulationtheresultinggyrocalibrationper__ntageerrorscomputedaswherewasthecalibrationparameterwere:Evidentlytheworstesti__tionerrorwasbelow8%;thatisevenintheworstcasenolessthan92%ofthecalibrationparameterwasesti__ted.
7.2ResultsusingtheimplicitapproachHeretoothegyrocalibrationparameterswereesti__tedwellwithoutperforminganytuningonthefilterparameters.Attheendofthesimulationtheresultinggyrocalibrationper__ntageerrorswere:Heretheworstesti__tionerrorwaslessthan11%;thatisevenintheworstcasenolessthan__%ofthecalibrationparameterwereesti__ted.
8.ConclusionsInthispaperwecomparedtwofiltersforin-flightesti__tionofthecalibrationparametersofspa__craftgyroscopes.Onefilterwasarealizationoftheexplicitapproachandtheotherwasarealizationoftheimplicitapproachtogyrocalibration.Thefilterwhichwasusedintheexplicitapproachneeded3morestatesthanthefilterusedintheimplicitapproachandthereforewasmorecumbersome.Alsothatfilterrequiredextensivetuning.Ontheotherhanditwasconcludedthattheperfor__n__ofbothfilterswascomparableReferen__s:
[1]ASA-GoddardSpaceFlightCenterMultimissionThree-AxisStabilizedSpa__craftMTASS553-FDD-93/032R0UD01933pp.
3.
3.2-1–
3.
3.2-
10.
[2]Bar-ItzhackI.Y.andHar__nR.R.“In-Spa__CalibrationofaSkewedGyroQuadrupletAIAAJ.ofGuidan__ControlandDynamicsVol.25No.5Sept.-Oct.2002pp.852-
859.
[3]Bar-ItzhackI.Y.andHar__nR.R.PseudolinearandState-DependentRiccatiEquationFiltersforAngularRateEsti__tionAIAAJ.ofGuidan__ControlandDynamicsVol.22No.5Sept.-Oct.1999pp.723-
725.EngineeringNote.
[4]ChatfieldA.B.FundamentalsofHighAccuracyInertialN__igationVol.174ProgressinAstronauticsandAeronauticsAIAA1997pp.93-
106.
[5]BezooijenR.W.H.ASTCapabilitiesLockheed__rtinAdvan__dTechnologyCenterPaloAltoCA95304-
1191.Slidepresentation.
[6]WertzJ.R.Ed.Spa__craftAttitudeDynamicsandControlReidelPublishingCo.DordrechtHolland1978p.
512.PAGE1。