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IntJAdvManufTechnol200321604–611OwnershipandCopyright2003SpringerVerlagLondonLimitedBacklashEstimationofaSeekerGimbalwithTwoStageGearReducersJ.H.Baek,Y.K.KwakandS.H.KimDepartmentofMechanicalEngineering,KoreaAdvancedInstituteofScienceandTechnology,3731GusungdongYusungguDaejon,KoreaAnoveltechniqueforestimatingthemagnitudeorcontributionratioofeachstagebacklashinasystemwithatwostagegearreducerisproposed.Theconceptisbasedonthechangeoffrequencyresponsecharacteristic,inparticular,thechangeofantiresonantfrequencyandresonantfrequency,duetothechangeofthemagnitudeofthebacklashofeachstage,eventhoughthetotalmagnitudeofthebacklashofasystemwithatwostagegearreducerisconstant.Thevalidityofthetechniqueisverifiedinaseekergimbalandsatisfactoryresultsareobtained.Itisthoughtthatthediagnosisandmaintenanceofmanufacturingmachinesandsystemswithtwostagegearreducerswillbecomemoreefficientandeconomicalbyvirtueofproposedtechnique.KeywordsAntiresonantfrequencyBacklashestimationContributionratioFrequencyresponsecharacteristicResonantfrequencySeekergimbal1.IntroductionTheautomationofmanufacturingmachinesandthefrequentuseofrobotsandservosystemshavegreatlyincreasedthedemandforservosystemswithservomotors.Withtheadvanceofmotormanufacturingtechniques,servosystemshavebeendevelopedwithdirectdrivetypemotorsthatdonotrequiregearreducers.However,thusfar,servosystemswithgearreducershavebeenusedextensivelyinmanufacturingmachinesinmanyfields,becausetheservosystemvolumeandweightislargerthanthatofthegearreducer,whileitstorqueisrelativelysmallincomparison.Servosystemswithgearreducershavehadproblemsrelatedtogearbacklashsincetheirinception.Accordingly,manystudieshavebeenperformedinordertodealwiththeproblems.CorrespondenceandoffprintrequeststoJ.H.Baek,ResearchandDevelopment7Group,LGInnotekCo.,Ltd.,148–1MabukriGusungeupYongincityKyonggido,449–910,Korea.Emailjhbaekblginnotek.comReceived5February2002Accepted29March2002Inordertodiagnoseandmaintaintheperformanceoftherobotsandservosystems,amethodofmonitoringanddetectingthemagnitudeandchangeofbacklashhasbeendeveloped.DagalakisandMyersusedacoherencefunctionandthemagnitudeofresonantpeakinthefrequencyresponsebetweenthemotorvoltageandtheaccelerationofarobotlinkasmeasures1.SteinandWangdevelopedatechniquebasedonmomentumtransferanalysisinordertodetectandestimatethebacklashofaservosystemwithagearreducer.Theyfoundthatthespeedchangeoftheprimarygearduetoimpactwiththesecondarygearisrelatedtothemagnitudeofthebacklash2.Sakeretal.developedatechniquetocomplementtheworkofSteinandWangusingtheimpulsivetorqueduetoimpact,insteadofthespeedchangeoftheprimarygear3.Panetal.developedatechniquefordetectingandclassifyingthebacklashofarobotbyusingWigner–Villedistributionscombinedwithatwodimensionalcorrelationoftherelationshipbetweenthesinusoidaljointmotionandtheaccelerationoftherobotlink4.However,thereisnotechniqueforestimatingthemagnitudeorcontributionratioofeachstageofthebacklashinaservosystemwithamultistagegearreducer,whichisoftenusedinmanufacturingmachinesandrobots.Itisveryimportanttoknowthemagnitudeofeachstagebacklashofsysteminordertoobtainthedesiredmagnitudeofbacklashandtomaintainthatmagnitudeinacorrectrange.Thepurposeofthispaperis,therefore,topresentatechniqueforestimatingthemagnitudeorcontributionratioofeachstagebacklashofaservosystemwithatwostagegearreducer.Thecontributionratioisdefinedastheratioofthemagnitudeofthefirststagebacklashtothatofthetotalbacklash.TheconceptforestimatingthemagnitudeofeachstagebacklashisbasedonthechangeofantiresonantfrequencyARFandresonantfrequencyRFinthefrequencyresponsecharacteristicofaservosystem,accordingtothechangeofthemagnitudeofeachstageofbacklash,eventhoughthetotalbacklashoftheservosystemisconstant.Inordertoverifythevalidityoftheproposedtechnique,twodrivingservosystemsofaseekergimbal,whichareusedinordertostabilisetheorientationofanobject,areconsidered.OneisanazimuthdrivingservosystemADSStheotherisanelevationdrivingservosystemEDSS.Bothservosystemshavetwostagegearreducers.BacklashEstimationofaSeekerGimbal6052.ModelofSeekerGimbal2.1ModeloftheADSSintheSeekerGimbalAphotographoftheseekergimbalwithtwostagegearreducerswhichisconsideredinthispaperispresentedinFig.1a.TheADSSandEDSScorrespondtothetwodrivingpartsoftheseekergimbal.InthecaseoftheADSS,thehatchedcomponents,pinion2,shaft1,gear1,pinion1,motorandbearingsrotatewithrespecttotheAAH11032axisexceptthatgear2isattachedonafixedshaftasshowninFig.1b.Itisassumedthatbearingssupporteachshaftwithoutanyclearance,duetothepreload.Also,theinfluenceofthedampingcharacteristicisneglected.ThemodeloftheADSSobtainedundertheseassumptionsispresentedinFig.1c.ThemomentofFig.1.aSeekergimbalbstructureofADSScmodelofADSSdstructureofEDSSemodelofEDSS.inertiaofpinion1isincludedwiththatofthemotor.Thetorsionspringrepresentedattherightsideofgear1indicatesthetorsionstiffnessduetotoothstiffnessbetweenpinion1andgear1.Inthecaseofshaft1,themomentofinertiaislumpedatthecentreofthedistancebetweengear1andpinion2andtorsionsprings,withtwicethevalueoftorsionstiffnessofshaft1,areconnectedwithgear1andpinion2.Becausetheyarefixed,gear2andthefixedshaftaremodelledsothattheyhaveonlytorsionspringswithoutmomentofinertia.Eachbacklashisrepresentedastheanglesofrotationofthegearswhenthepinionsarefixed.ComponentsenclosedbyaphantomdoubledotlineinFig.1cindicatetheloadoftheADSS.TheADSSconsideredconsistsofatachometerfilter,amotoramplifierandtheaforementionedstructure.Themotoramplifierisusedtoamplifytheinputvoltageofmotor.Apermanent606J.H.Baeketal.magneticfieldtyped.c.motorwithatachometerisusedasanactuator.Inordertofiltertheoutputvoltageofthetachometer,asecondorderlowpassfilterisused.ThegoverningelectricEq.ofthesecomponentsareasfollows5VmkaVi1LadiadtRmiakbH9258˙mVm2aTmktia2bVtktsH9258˙m3VosGfsVts4TheEq.ofmotionforthemotorisasfollowsJmH9258¨mBmH9258˙mTmH11002Tg1N1H11002Tf,msignH9258˙m5Thetorquetransmittedtogear1isrepresentedasanonlinearEq.,presentedinEq.6,duetothebacklashbetweenpinionFig.2.ThebodediagramVo/ViofADSSaccordingtocontributionratioacase1bcase2ccase3dcase4ecase5.SimsimulationExpexperiment..1andgear1.Themodelofthedeadzoneisusedasthemodelofthebacklash6.Tg1H20902kg1H9258d1H11002H92541,H9258d1H11022H925410,H20841H9258d1H20841H11021H92541kg1H9258d1H92541,H9258d1H11021H11002H92541H209036whereH9258d1H9258m/N1H11002H9258g17TheEq.ofmotionforgear1isasfollowsJg1H9258¨g1Tg1H110022ks1H9258g1H11002H9258s18TheEq.ofmotionforshaft1isasfollowsJs1H9258¨s12ks1H9258g1H9258p2H110024ks1H9258s19Besides,theequationofmotionforpinion2isasfollowsJp2H9258¨p22ks1H9258s1H11002H9258p2H110021NrTL10BacklashEstimationofaSeekerGimbal607ThetorqueoftheloadisrepresentedinEq.11,likeEq.6.TLH20902k2H9258d2H11002H92542,H9258d2H11022H925420,|H9258d2|H11021H92542k2H9258d2H92542,H9258d2H11021H11002H92542H2090311whereH9258d2H9258p2/NrH11002H9258L12Here,theequivalenttorsionstiffnessbetweengear2andshaft2isasfollows7k2kg2ks2kg2ks213Finally,theequationofmotionfortheloadisasfollowsJLH9258¨LTLH11002Tf,LsignH9258˙L14TheresponseoftheoutputvoltageofthetachometerfilterwithrespecttotheinputvoltageofthemotoramplifierisobtainedfromtheseEq..Inaddition,therelationbetweenthetotalbacklashandeachstagebacklashisasfollowsbtb21Nrb115wherebi360°H9254i/H9266i1,2162.2ModeloftheEDSSintheSeekerGimbalInthissubsection,theEDSSmodelsandEq.ofmotionarederived.ThestructureoftheEDSSispresentedinFig.1d.Becausegear2isdirectlyattachedtotheload,themomentofinertiaofgear2isincludedwiththatoftheloadandgear2hasonlyatorsionspringmodel,asshowninFig.1e.TheEq.ofmotionfortheEDSSbetweenthemotoramplifierandtachometerfilterarethesameasthoseoftheADSS,exceptforreplacingEqs10–13andEq.15withEqs17–20asfollowsJp2H9258¨p22ks1H9258s1H11002H9258p2H110021N2TL17TLH20902kg2H9258d2H11002H92542,H9258d2H11022H925420,|H9258d2|H11021H92542kg2H9258d2H92542,H9258d2H11021H11002H92542H2090318whereH9258d2H9258p2/N2H11002H9258L19btb21N2b120FromEqs1–9,Eq.14andEqs17–20,theresponseoftheoutputvoltageofthetachometerfilterwithrespecttotheinputvoltageofthemotoramplifierisobtained.3.SimulationItiswellknownthatanincreaseinthetotalbacklashinasystemcausesthefrequencyresponsecharacteristic,oftheoutputvoltageofthetachometerfilterwithrespecttotheinputvoltageofthemotoramplifier,tochangebecauseitreducestheeffectiveequivalenttorsionalstiffnessofthesystem8.However,ithasnotbeenreportedyetthatalthoughthetotalbacklashmagnitudeisconstant,aservosystemwithadifferentbacklashmagnitudeateachstagehasdifferentfrequencyresponsecharacteristic.Inthiswork,eachstageofbacklashofaservosystemisexaminedbythisphenomenonandhypothesis.Inordertoverifythishypothesis,thefrequencyresponsecharacteristicofADSSisinvestigatedaccordingtothecontributionratio.ThebodediagramsofADSSobtainedfromthesimulationarerepresentedinFig.2.ThespecificationsusedforthesimulationarepresentedinTable1.ThecombinationsofthemagnitudeofbacklashofeachstageobtainedaccordingtothechangeofcontributionratioarelistedinTable2.TheyareobtainedfromEqs15and20.InordertoobtainthesimulationresultsofFig.2,theequationofmotionoutlinedintheprevioussectionareconvertedintoablockdiagram.ThesimulationisthenperformedusingMATLABSimulinkV.6.1software.Thepeakamplitudeofthesinusoidalvoltagesuppliedtothemotoramplifieris2.5Vandthesamplingtimeusedis10H9262sec.BodediagramsofFig.2aremadefromthefrequencyanalysistoextractonlytheexcitedfrequencycomponentfromtheoutputvoltageofthetachometerfilterwithrespecttothesinusoidalvoltagesuppliedtothemotoramplifier.TheARFandRFobtainedaresummarisedinTable2andarerepresentedinFig.3a.ThedifferencebetweentheARFandRFisshowninFig.3b.FromFig.3aandb,itisfoundthatthefrequencyresponsecharacteristicofaservoTable1.SpecificationsforADSSandEDSS.ParameterADSSEDSSGearratio1,N15.946.41Torsionstiffness,kg1mH11554N/rad3.40E44.74E4Momentofinertiaofgear1,Jg1kgH11554m22.34E53.69E5Torsionstiffnessofshaft1,ks1mH11554N/rad22.81.54E2Momentofinertiaofshaft1,Js1kgH11554m28.30E82.04E7Momentofinertiaofpinion2,Jp2kgH11554m22.21E74.84E7Gearratio,Nr,N210.57.75Equivalenttorsionstiffness,k2,kg2mH11554N/rad7.74E42.54E5Momentofinertiaofload,JLkgH11554m22.75E31.44E2Staticfrictiontorqueofload,Tf,LmH11554N7.0E37.1E3Totalbacklash,btdeg.0.0660.276Motorinductance,LaH8.50E4Motorresistance,RmH90244.10BackEMFconst.,kbVH11554s/rad3.44E2Torquesensitivity,ktmH11554N/A3.49E2Momentofinertiaofmotor,JmkgH11554m28.60E6Staticfrictiontorqueofmotor,Tf,mmH11554N1.40E2Gainofmotoramplifier,ka4.11Tachometersensitivity,ktsVH11554s/rad8.60E2Transferfunctionoflowpassfilter,Gfs723439s21710s723439Viscousdampingcoeff.ofmotor,BmmH115541.6E4N/rad/s
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