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外文翻译---运动学分析和优化设计3-PPR平面平行机械手 英文版.pdf

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外文翻译---运动学分析和优化设计3-PPR平面平行机械手 英文版.pdf

528KSMEInternationalJournal,Vol.17No.4,pp.528537,2003KinematicAnalysisandOptimalDesignof3PPRPlanarParallelManipulatorKeeBongChoiRobotControlGroup,IntelligencePrecisionMachineDept.,KoreaInstituteofMachineryandMaterials171,JangDong,YuseongGu,Daejeon,305343,KoreaThispaperproposesa3PPRplanarparallelmanipulator,whichconsistsofthreeactiveprismaticjoints,threepassiveprismaticjoints,andthreepassiverotationaljoints.Theanalysisofthekinematicsandtheoptimaldesignofthemanipulatorarealsodiscussed.Theproposedmanipulatorhastheadvantagesoftheclosedtypeofdirectkinematicsandavoidfreeworkspacewithaconvextypeofborderline.Forthekinematicanalysisoftheproposedmanipulator,thedirectkinematics,theinversekinematics,andtheinverseJacobianofthemanipulatorarederived.Altertherotationallimitsandtheworkspacesofthemanipulatorareinvestigated,theworkspaceofthemanipulatorissimulated.Inaddition,fortheoptimaldesignofthemanipulator,theperformanceindicesofthemanipulatorareinvestigated,andthenanoptimaldesignprocedureiscarriedoutusingMinMaxtheory.Finally,oneexampleusingtheoptimaldesignispresented.KeyWordsPlanarParallelManipulator,Kinematics,Jacobian,Workspace,OptimalDesign,MinMaxI.IntroductionParallelmanipulatorsconsistingofclosedloopmechanismshavemanyadvantagescomparedtoserialmanipulatorsintermsofpayload,accuracy,andstiffness.Itiswellknownthatparallelmanipulatorshaveahigherpayloadtoweightratio,higheraccuracy,andhigherstructuralrigiditythanserialmanipulatorsBenHorinetal.,1998.RecentlysomemachinetoolsKimelal.,2001Wangetal.,2001havebeendevelopedutilizingtheseadvantages.AmanipulatortbrfinemotionRyuetal.,1997alsoadoptedtheparallelmechanismratherthantheserialone,sincetheparallelmechanismcanbemanufacturedmonolithically.Emailkbchoikimmre.krTEL82428687132FAX82428687135RobotControlGroup,IntelligencePrecisionMachineDept.,KoreaInstituteofMachineryandMaterials171.JangDong,YuseongGu,Daejeon.305343.Korea.ManuscriptReceivedMay22,2002RevisedDecember13,2002CopyrightC2003NuriMediaCo.,Ltd.Amongtheparallelmanipulators,theplanarparallelmanipulatorisamanipulatorforplanemotion.PlanarparallelmanipulatorshavetwodegreeoffreedomDOFmotionthatistwotranslations,or3DOFmotion,consistingoftwotranslationsandonerotation.Itiswellknownthat231variationsof3DOFplanarparallelmanipulatorsexist,whichareRRR,RRP,RPR,RPP,PRR,PRP,andPPR,dependingonthecombinationsofprismaticjointsandrotationaljoints,excludingaPPPcombination,wheretheprismaticandrotationaljointsarerepresentedbyPandRMerlet,1996and2000.ThesolutionsofthedirectkinematicslbrpossiblearchitecturesoftheplanarparallelmanipulatorswerealsoalreadyproposedMerlet,1996,butmoreconcretesolutionsandkinematicanalysesofthearchitecturesarestilrequired.Most3DOFplanarparallelmanipulatorshavedisadvantagesthatthemanipulatorshavepolynomialtypesofcomplexdirectkinematicsandsmallworkspaceswithuselessvoidsaswellasconcavetypesofborderlines.AstheorderofKinematicAnalysisandOptimalDesignof3PPRPlanarParallelManipulator529thepolynomialsofthedirectkinematicsincrease,solvingequationsaswellaschoosingapropersolutionbecomesagreatburden.Moreovertheconcavetypesofborderlinesinducenonstraightmotionsfromaneighboroftheborderlinetotheothers.ThereforeitisimportantthataparallelmanipulatorhasaclosedtypedirectkinematicsandavoidtYeeworkspacewithaconvextypeborderline.Inthispaper,a3PPRplanarparallelmanipulator,inwhichPisanactiveprismaticjoint,isproposedtoovercometheaforementioneddisadvantages,i.e.,theproposedmanipulatorhasaclosedtypedirectkinematicsandavoidfreeworkspacewithaconvextypeofborderline.Forthekinematicanalysisofthismanipulator,firstthedirectkinematics,inversekinematics,andinverseJacobianoftheproposedmanipulatorarederived.Second,rotationallimitsandworkspacesareinvestigated.Also,fortheoptimaldesignofthismanipulator,performanceindicesofthemanipulatorareinvestigatedandthenanoptimaldesignprocedureiscarriedoutusingMinMaxtheory.Finally,oneexampleusingtheoptimaldesignispresented.2.Descriptionof3PPRPlanarParallelManipulatorFigureIshowstheschematicconfigurationofa3PPRplanarparallelmanipulatorthatconsistsofthreeactiveprismaticjoints,threepassiveprismaticjoints,threepassiverotationaljoints,amovingplate,andlinks.Theactivejointscanbeactuatedbyelectricrotationalmotorsandballscrewsformotiontransformation.Thethreelinksformotionoftheactivejointsarefixedtoabaseframewithtwoendsofeachlink.ThedegreeoffreedomDOFoftheplanarmanipulator,m,isrepresentedbyMerlet,2000zm3ln1di1i1wherelisthenumberofrigidbodies,nisthenumberofjoints,anddlisDOFofjointi.SincethismanipulatorhaseightrigidbodiesCopyrightC2003NuriMediaCo.,Ltd.Fig.1AtlveprisrticjoirtSchematicconfigurationof3PPRplanarmanipulatorincludingthebase,andninejointswithatotalofnineDOF,itsDOFisthreei.e.,twotranslationsandonerotationontheplane.Inaddition,whentheactivejointsofthemanipulatorarelocked,theDOFofthemanipulatorbecomeszerobecausetheninejointshaveonlysixDOF.ThereforethismanipulatorhasthreeDOFwhentheactivejointsareactivated,whereasitbecomesastaticstructurewhentheactivejointsarelocked.3.DirectKinematicsThecoordinatesandthegeometricparametersofthismanipulatorareshowninFig.2.Themovingplateisacircle,whichcontainsanequilateraltriangle,witharadiusr.Thecentersoftherotationaljointsareontheverticesofthetriangle.TheactiveprismaticjointscantravelonthesidesoftheouterequilateraltrianglewhichcontainsacirclewithradiusR.WhenthecoordinateofeachactivejointAiisxi,yi,wherei1,2,and3,themovingplatehastheposeoftranslationx,yandrotationfromthereferencepointO.TheneachlengthofthepassivelinkbecomesLi.Becausetheinnerandoutertrianglesareequilateral,theangleofthetriangles,00,is00zc/32530KeeBongChoiA,x/7771olAjrI,Fig.2CoordinatesystemfordirectkinematicsThesidelengthoftheinnertriangle,e,isef3r3Theinclineoftheinnertriangle,9,isexpressedbythetermofrotationofthemovingplate,¢,9¢34TherelativedisplacementsoftheactivejointsareLxcosaecos9L2cosa0ox2x5LsinaesinpL2sinaOoyzy,6L1cosaecos009L3cosaOoxaxl7Ltsinaesin009L3sina00y3y8FromEqs.5and6,thelengthsLaandL2areLIL2cosasin0oY2ylesin99sinax2Xlecos9sin001xzxecos9cosa00COSasinOoyzylesin910COSa.,tama0ox2xlecos9Also,fromEqs.7,9,and10,thelengthL3isCopyrightC2003NuriMediaCo.,ltd.1Lscosa0o{x3XleCos19o9}cosacosa,.,y2ylesin9jllsina0ocosa4cosa0osin0oX2xlecos9SubstitutionofEqs.911intoEq.8derivesthefollowingequationCC2cos9C3sin9012whereClcosa00y3ysina00x3xcosa00y2ylsina00x2xxC2e{sinasina_Oo}13C3e{cosacosa00}Equation12canbesolvedintroducingaparameterTasfollows1Tzcos9IT2Tsin9lT2SubstitutingEq.14intoEq.14orderpolynomialinTisobtainedasC1C2Tz2C3TCIC2015Equation15offerstheclosedformsolutionsofTC3C22C2Cz16TC1C2FromEqs.4and14,therotationofthemovingplateis2Tx17¢tan1ITz3Thus,thetranslationofthemovingplateisxxlLIcosarsin¢18yRLxsinarcos¢ItisremarkablethatthismanipulatorhasatmosttwosolutionsfordirectkinematicsaccordingtoEq.16,andmoreovertheclosedformsolutionsofEqs.17and18.12,asecondKinematicAnalysisandOptimalDesignof3P_PRPlanarParallelManipulator5314.InverseKinematicsandInverseJacobianFigure3showsacoordinatesystemfordescribingtheinversekinematicsofa3PPRplanarparallelmanipulator.WhenthecenterofthemovingplatemovesfromoriginOtoOwithtranslationx,yandrotationb,thevertexoftheplateBisexpressedasxBixrsinb19ysiyrCOS0gbwhere2i10,r203andi1,2,3.TheoriginoftheactiveprismaticjointOiisdepartedbyRfromtheoriginO.Providedthatuiandviaretheunitvectorsoftheaxese,and,whicharetheaxesoftheactiveprismaticjoint,thecoordinateofthevertexB,expressedbythetermsof,andi,iseBiOBiui21giROBiviThusthepositionoftheactiveprismaticjoint,a,isYB,,L0Rvo,Fig.3CoordinatesystemforinversekinematicsBi,cota22DirectdifferentiationofEq.22withrespecttotheposex,y,bderivestheinverseJacobianjiasfollowsI1c0tart,cos4cotasm4j4.513c0ta23c0talrc0sc0tasin23Il,,c0tag3c0tarc0sc0tasin}TheelementsoftheinverseJacobianofEq.23donothavethesamedimension.Firsttwocolumnscorrespondingtotranslationaredimensionless,whereasthelastcolumncorrespondingtorotationhasthedimensionoflength.Bymakingthethirdcolumndimensionless,ahomogeneousinverseJacobianwithnondimensionalelementsisobtainedbyByun,1997l°a1JJ0124ooI/R5.RotationalLimitandWorkspaceTherotationofthemovingplateisrestrictedbytheinterferencebetweenthelinksandtherotationaljoints.Fig.4showstheconfigurationofthismanipulatorwithrotationallimitsoftheclockwisecaseaandcounterclockwisecaseb.Assumingthewidthofthelinksisnegligible,therotationbisboundedby_4zccz3a253Providedthatb0istheinitialrotationdefinedby0te26Therotationrange25ismodifiedaswhere57c56zr627qSb028Itisconcludedthatthemovingplateofthe3PPRplanarparallelmanipulatorisboundedby57c/6fromtheinitialrotation.CopyrightC2003NuriMediaCo.,Ltd.

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