IntJAdvManufTechnol(2003)21:604611OwnershipandCopyright2003Springer-VerlagLondonLimitedBacklashEstimationofaSeekerGimbalwithTwo-StageGearReducersJ.H.Baek,Y.K.KwakandS.H.KimDepartmentofMechanicalEngineering,KoreaAdvancedInstituteofScienceandTechnology,373-1Gusung-dongYusung-guDaejon,KoreaAnoveltechniqueforestimatingthemagnitudeorcontributionratioofeachstagebacklashinasystemwithatwo-stagegearreducerisproposed.Theconceptisbasedonthechangeoffrequencyresponsecharacteristic,inparticular,thechangeofanti-resonantfrequencyandresonantfrequency,duetothechangeofthemagnitudeofthebacklashofeachstage,eventhoughthetotalmagnitudeofthebacklashofasystemwithatwo-stagegearreducerisconstant.Thevalidityofthetechniqueisverifiedinaseekergimbalandsatisfactoryresultsareobtained.Itisthoughtthatthediagnosisandmaintenanceofmanufacturingmachinesandsystemswithtwo-stagegearreducerswillbecomemoreefficientandeconomicalbyvirtueofproposedtechnique.Keywords:Anti-resonantfrequency；Backlashestimation；Contributionratio；Frequencyresponsecharacteristic；Resonantfrequency；Seekergimbal1.IntroductionTheautomationofmanufacturingmachinesandthefrequentuseofrobotsandservosystemshavegreatlyincreasedthedemandforservosystemswithservomotors.Withtheadvanceofmotormanufacturingtechniques,servosystemshavebeendevelopedwithdirectdrivetypemotorsthatdonotrequiregearreducers.However,thusfar,servosystemswithgearreducershavebeenusedextensivelyinmanufacturingmachinesinmanyfields,becausetheservosystemvolumeandweightislargerthanthatofthegearreducer,whileitstorqueisrelativelysmallincomparison.Servosystemswithgearreducershavehadproblemsrelatedtogearbacklashsincetheirinception.Accordingly,manystudieshavebeenperformedinordertodealwiththeproblems.Correspondenceandoffprintrequeststo:J.H.Baek,ResearchandDevelopment7Group,LGInnotekCo.,Ltd.,1481Mabuk-riGusung-eupYongin-cityKyonggi-do,449910,Korea.E-mail:Received5February2002Accepted29March2002Inordertodiagnoseandmaintaintheperformanceoftherobotsandservosystems,amethodofmonitoringanddetectingthemagnitudeandchangeofbacklashhasbeendeveloped.DagalakisandMyersusedacoherencefunctionandthemagni-tudeofresonantpeakinthefrequencyresponsebetweenthemotorvoltageandtheaccelerationofarobotlinkasmeasures1.SteinandWangdevelopedatechniquebasedonmomen-tumtransferanalysisinordertodetectandestimatethebacklashofaservosystemwithagearreducer.Theyfoundthatthespeedchangeoftheprimarygearduetoimpactwiththesecondarygearisrelatedtothemagnitudeofthebacklash2.Sakeretal.developedatechniquetocomplementtheworkofSteinandWangusingtheimpulsivetorqueduetoimpact,insteadofthespeedchangeoftheprimarygear3.Panetal.developedatechniquefordetectingandclassifyingthebacklashofarobotbyusingWignerVilledistributionscombinedwithatwo-dimensionalcorrelationoftherelationshipbetweenthesinusoidaljointmotionandtheaccelerationoftherobotlink4.However,thereisnotechniqueforestimatingthemagni-tudeorcontributionratioofeachstageofthebacklashinaservosystemwithamultistagegearreducer,whichisoftenusedinmanufacturingmachinesandrobots.Itisveryimportanttoknowthemagnitudeofeachstagebacklashofsysteminordertoobtainthedesiredmagnitudeofbacklashandtomaintainthatmagnitudeinacorrectrange.Thepurposeofthispaperis,therefore,topresentatechniqueforestimatingthemagnitudeorcontributionratioofeachstagebacklashofaservosystemwithatwo-stagegearreducer.Thecontributionratioisdefinedastheratioofthemagnitudeofthefirststagebacklashtothatofthetotalbacklash.Theconceptforestimat-ingthemagnitudeofeachstagebacklashisbasedonthechangeofanti-resonantfrequency(ARF)andresonantfre-quency(RF)inthefrequencyresponsecharacteristicofaservosystem,accordingtothechangeofthemagnitudeofeachstageofbacklash,eventhoughthetotalbacklashoftheservosystemisconstant.Inordertoverifythevalidityoftheproposedtechnique,twodrivingservosystemsofaseekergimbal,whichareusedinordertostabilisetheorientationofanobject,areconsidered.Oneisanazimuthdrivingservosystem(ADSS)；theotherisanelevationdrivingservosystem(EDSS).Bothservosystemshavetwo-stagegearreducers.BacklashEstimationofaSeekerGimbal6052.ModelofSeekerGimbal2.1ModeloftheADSSintheSeekerGimbalAphotographoftheseekergimbalwithtwo-stagegearreducerswhichisconsideredinthispaperispresentedinFig.1(a).TheADSSandEDSScorrespondtothetwodrivingpartsoftheseekergimbal.InthecaseoftheADSS,thehatchedcomponents,pinion2,shaft1,gear1,pinion1,motorandbearingsrotatewithrespecttotheAAH11032axisexceptthatgear2isattachedonafixedshaftasshowninFig.1(b).Itisassumedthatbearingssupporteachshaftwithoutanyclearance,duetothepreload.Also,theinfluenceofthedampingcharac-teristicisneglected.ThemodeloftheADSSobtainedundertheseassumptionsispresentedinFig.1(c).ThemomentofFig.1.(a)Seekergimbal；(b)structureofADSS；(c)modelofADSS；(d)structureofEDSS；(e)modelofEDSS.inertiaofpinion1isincludedwiththatofthemotor.Thetorsionspringrepresentedattherightsideofgear1indicatesthetorsionstiffnessduetotoothstiffnessbetweenpinion1andgear1.Inthecaseofshaft1,themomentofinertiaislumpedatthecentreofthedistancebetweengear1andpinion2andtorsionsprings,withtwicethevalueoftorsionstiffnessofshaft1,areconnectedwithgear1andpinion2.Becausetheyarefixed,gear2andthefixedshaftaremodelledsothattheyhaveonlytorsionspringswithoutmomentofinertia.Eachbacklashisrepresentedastheanglesofrotationofthegearswhenthepinionsarefixed.Componentsenclosedbyaphantom(doubledot)lineinFig.1(c)indicatetheloadoftheADSS.TheADSSconsideredconsistsofatachometerfilter,amotoramplifierandtheaforementionedstructure.Themotoramplifierisusedtoamplifytheinputvoltageofmotor.Apermanent606J.H.Baeketal.magneticfieldtyped.c.motorwithatachometerisusedasanactuator.Inordertofiltertheoutputvoltageofthetach-ometer,asecond-orderlow-passfilterisused.ThegoverningelectricEq.ofthesecomponentsareasfollows5:Vm=kaVi(1)Ladiadt+Rmia+kbH9258m=Vm(2a)Tm=ktia(2b)Vt=ktsH9258m(3)Vo(s)=Gf(s)Vt(s)(4)TheEq.ofmotionforthemotorisasfollows:JmH9258m+BmH9258m=TmH11002Tg1N1H11002Tf,msign(H9258m)(5)Thetorquetransmittedtogear1isrepresentedasanonlinearEq.,presentedinEq.(6),duetothebacklashbetweenpinionFig.2.Thebodediagram(Vo/Vi)ofADSSaccordingtocontributionratio:(a)case1；(b)case2；(c)case3；(d)case4；(e)case5.(Sim:simulation；Exp:experiment.).1andgear1.Themodelofthedeadzoneisusedasthemodelofthebacklash6.Tg1=H20902kg1(H9258d1H11002H92541),H9258d1H11022H925410,H20841H9258d1H20841H11021H92541kg1(H9258d1+H92541),H9258d1H11021H11002H92541H20903(6)whereH9258d1=H9258m/N1H11002H9258g1(7)TheEq.ofmotionforgear1isasfollows:Jg1H9258g1=Tg1H110022ks1(H9258g1H11002H9258s1)(8)TheEq.ofmotionforshaft1isasfollows:Js1H9258s1=2ks1(H9258g1+H9258p2)H110024ks1H9258s1(9)Besides,theequationofmotionforpinion2isasfollows:Jp2H9258p2=2ks1(H9258s1H11002H9258p2)H110021NrTL(10)BacklashEstimationofaSeekerGimbal607ThetorqueoftheloadisrepresentedinEq.(11),likeEq.(6).TL=H20902k2(H9258d2H11002H92542),H9258d2H11022H925420,|H9258d2|H11021H92542k2(H9258d2+H92542),H9258d2H11021H11002H92542H20903(11)whereH9258d2=H9258p2/NrH11002H9258L(12)Here,theequivalenttorsionstiffnessbetweengear2andshaft2isasfollows7:k2=kg2ks2kg2+ks2(13)Finally,theequationofmotionfortheloadisasfollows:JLH9258L=TLH11002Tf,Lsign(H9258L)(14)TheresponseoftheoutputvoltageofthetachometerfilterwithrespecttotheinputvoltageofthemotoramplifierisobtainedfromtheseEq.Inaddition,therelationbetweenthetotalbacklashandeachstagebacklashisasfollows:bt=b2+1Nrb1(15)wherebi=360H9254i/H9266(i=1,2)(16)2.2ModeloftheEDSSintheSeekerGimbalInthissubsection,theEDSSmodelsandEq.ofmotionarederived.ThestructureoftheEDSSispresentedinFig.1(d).Becausegear2isdirectlyattachedtotheload,themomentofinertiaofgear2isincludedwiththatoftheloadandgear2hasonlyatorsionspringmodel,asshowninFig.1(e).TheEq.ofmotionfortheEDSSbetweenthemotoramplifierandtachometerfilterarethesameasthoseoftheADSS,exceptforreplacingEqs(10)(13)andEq.(15)withEqs(17)(20)asfollows:Jp2H9258p2=2ks1(H9258s1H11002H9258p2)H110021N2TL(17)TL=H20902kg2(H9258d2H11002H92542),H9258d2H11022H925420,|H9258d2|H11021H92542kg2(H9258d2+H92542),H9258d2H11021H11002H92542H20903(18)whereH9258d2=H9258p2/N2H11002H9258L(19)bt=b2+1N2b1(20)FromEqs(1)(9),Eq.(14)andEqs(17)(20),theresponseoftheoutputvoltageofthetachometerfilterwithrespecttotheinputvoltageofthemotoramplifierisobtained.3.SimulationItiswellknownthatanincreaseinthetotalbacklashinasystemcausesthefrequencyresponsecharacteristic,oftheoutputvoltageofthetachometerfilterwithrespecttotheinputvoltageofthemotoramplifier,tochangebecauseitreducestheeffectiveequivalenttorsionalstiffnessofthesystem8.However,ithasnotbeenreportedyetthatalthoughthetotalbacklashmagnitudeisconstant,aservosystemwithadifferentbacklashmagnitudeateachstagehasdifferentfrequencyresponsecharacteristic.Inthiswork,eachstageofbacklashofaservosystemisexaminedbythisphenomenonandhypoth-esis.Inordertoverifythishypothesis,thefrequencyresponsecharacteristicofADSSisinvestigatedaccordingtothecontri-butionratio.ThebodediagramsofADSSobtainedfromthesimulationarerepresentedinFig.2.ThespecificationsusedforthesimulationarepresentedinTable1.ThecombinationsofthemagnitudeofbacklashofeachstageobtainedaccordingtothechangeofcontributionratioarelistedinTable2.TheyareobtainedfromEqs(15)and(20).InordertoobtainthesimulationresultsofFig.2,theequationofmotionoutlinedintheprevioussectionareconvertedintoablockdiagram.ThesimulationisthenperformedusingMATLABSimulinkV.6.1software.Thepeakamplitudeofthesinusoidalvoltagesuppliedtothemotoramplifieris2.5Vandthesamplingtimeusedis10H9262sec.BodediagramsofFig.2aremadefromthefrequencyanalysistoextractonlytheexcitedfrequencycomponentfromtheoutputvoltageofthetachometerfilterwithrespecttothesinusoidalvoltagesuppliedtothemotoramplifier.TheARFandRFobtainedaresummarisedinTable2andarerepresentedinFig.3(a).Thedifferencebet