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.528537,2003KinematicAnalysisandOptimalDesignof3-PPRPlanarParallelManipulatorKee-BongChoi*Robot&ControlGroup,Intelligence&PrecisionMachineDept.,KoreaInstituteofMachineryandMaterials171,Jang-Dong,Yuseong-Gu,Daejeon,305-343,KoreaThispaperproposesa3-PPRplanarparallelmanipulator,whichconsistsofthreeactiveprismaticjoints,threepassiveprismaticjoints,andthreepassiverotationaljoints.Theanalysisofthekinematicsandtheoptimaldesignofthemanipulatorarealsodiscussed.Theproposedmanipulatorhastheadvantagesoftheclosedtypeofdirectkinematicsandavoid-freeworkspacewithaconvextypeofborderline.Forthekinematicanalysisoftheproposedmanipulator,thedirectkinematics,theinversekinematics,andtheinverseJacobianofthemanipulatorarederived.Altertherotationallimitsandtheworkspacesofthemanipulatorareinvestigated,theworkspaceofthemanipulatorissimulated.Inaddition,fortheoptimaldesignofthemanipulator,theperformanceindicesofthemanipulatorareinvestigated,andthenanoptimaldesignprocedureiscarriedoutusingMin-Maxtheory.Finally,oneexampleusingtheoptimaldesignispresented.KeyWords:PlanarParallelManipulator,Kinematics,Jacobian,Workspace,OptimalDesign,MinMaxI.IntroductionParallelmanipulatorsconsistingofclosed-loopmechanismshavemanyadvantagescomparedtoserialmanipulatorsintermsofpayload,accuracy,andstiffness.Itiswellknownthatparallelmani-pulatorshaveahigherpayloadto-weightratio,higheraccuracy,andhigherstructuralrigiditythanserialmanipulators(Ben-Horinetal.,1998).Recentlysomemachine-tools(Kimelal.,2001:Wangetal.,2001)havebeendevelopedutilizingtheseadvantages.Amanipulatortbrfinemotion(Ryuetal.,1997)alsoadoptedtheparallelmec-hanismratherthantheserialone,sincethepar-allelmechanismcanbemanufacturedmonolithic-ally.*E-mail:kbchoikimmre.krTEL:+82428687132:FAX:+82-42868-7135Robot&ControlGroup,Intelligence&PrecisionMa-chineDept.,KoreaInstituteofMachineryandMaterials171.JangDong,Yuseong-Gu,Daejeon.305343.Ko-rea.(ManuscriptReceivedMay22,2002；RevisedDe-cember13,2002)Copyright(C)2003NuriMediaCo.,Ltd.Amongtheparallelmanipulators,theplanarparallelmanipulatorisamanipulatorforplanemotion.Planarparallelmanipulatorshavetwodegree-of-freedom(DOF)motion；thatistwotranslations,or3-DOFmotion,consistingoftwotranslationsandonerotation.Itiswellknownthat(23-1)variationsof3-DOFplanarparallelmanipulatorsexist,whichareRRR,RRP,RPR,RPP,PRR,PRP,andPPR,dependingonthecombinationsofprismaticjointsandrotationaljoints,excludingaPPPcombination,wheretheprismaticandrotationaljointsarerepresentedbyPandR(Merlet,1996and2000).Thesolutionsofthedirectkinematicslbrpossiblearchitecturesoftheplanarparallelmanipulatorswerealsoalreadyproposed(Merlet,1996),butmorecon-cretesolutionsandkinematicanalysesofthearchitecturesarestil!required.Most3-DOFplanarparallelmanipulatorshavedisadvantagesthatthemanipulatorshavepolynomialtypesofcomplexdirectkinematicsandsmallworkspaceswithuselessvoidsaswellasconcavetypesofborderlines.AstheorderofKinematicAnalysisandOptimalDesignof3-PPRPlanarParallelManipulator529thepolynomialsofthedirectkinematicsincrease,solvingequationsaswellaschoosingapropersolutionbecomesagreatburden.Moreovertheconcavetypesofborderlinesinducenon-straightmotionsfromaneighboroftheborderlinetotheothers.Thereforeitisimportantthataparallelmanipulatorhasaclosedtypedirectkinematicsandavoid-tYeeworkspacewithaconvextypeborderline.Inthispaper,a3-PPRplanarparallelmani-pulator,inwhichPisanactiveprismaticjoint,isproposedtoovercometheaforementioneddis-advantages,i.e.,theproposedmanipulatorhasaclosedtypedirectkinematicsandavoidfreeworkspacewithaconvextypeofborderline.Forthekinematicanalysisofthismanipulator,firstthedirectkinematics,inversekinematics,andinverseJacobianoftheproposedmanipulatorarederived.Second,rotationallimitsandworkspacesareinvestigated.Also,fortheoptimaldesignofthismanipulator,performanceindicesofthemani-pulatorareinvestigatedandthenanoptimaldesignprocedureiscarriedoutusingMin-Maxtheory.Finally,oneexampleusingtheoptimaldesignispresented.2.Descriptionof3-PPRPlanarParallelManipulatorFigureIshowstheschematicconfigurationofa3-PPRplanarparallelmanipulatorthatconsistsofthreeactiveprismaticjoints,threepassiveprismaticjoints,threepassiverotationaljoints,amovingplate,andlinks.Theactivejointscanbeactuatedbyelectricrotationalmotorsandballscrewsformotiontransformation.Thethreelinksformotionoftheactivejointsarefixedtoabaseframewithtwoendsofeachlink.Thedegreeoffreedom(DOF)oftheplanarmanipulator,m,isrepresentedby(Merlet,2000)?zm=3(l-n-1)+di(1)i=1wherelisthenumberofrigidbodies,nisthenumberofjoints,anddlisDOFofjointi.SincethismanipulatorhaseightrigidbodiesCopyright(C)2003NuriMediaCo.,Ltd.Fig.1A:tlveprisrticjoirtSchematicconfigurationof3PPRplanarmanipulatorincludingthebase,andninejointswithatotalofnineDOF,itsDOFisthree；i.e.,twotranslationsandonerotationontheplane.Inaddition,whentheactivejointsofthemanipulatorarelocked,theDOFofthemanipulatorbecomeszerobe-causetheninejointshaveonlysixDOF.There-forethismanipulatorhasthreeDOFwhentheactivejointsareactivated,whereasitbecomesastaticstructurewhentheactivejointsarelocked.3.DirectKinematicsThecoordinatesandthegeometricparametersofthismanipulatorareshowninFig.2.Themovingplateisacircle,whichcontainsanequi-lateraltriangle,witharadiusr.Thecentersoftherotationaljointsareontheverticesofthetriangle.TheactiveprismaticjointscantravelonthesidesoftheouterequilateraltrianglewhichcontainsacirclewithradiusR.WhenthecoordinateofeachactivejointAiis(xi,yi),wherei=1,2,and3,themovingplatehastheposeoftranslation(x,y)androtationfromthereferencepointO.TheneachlengthofthepassivelinkbecomesLi.Becausetheinnerandoutertrianglesareequi-lateral,theangleofthetriangles,00,is00=zc/3(2)530KeeBongChoiA:,(x/7771olAj(rI,)Fig.2CoordinatesystemfordirectkinematicsThesidelengthoftheinnertriangle,e,ise=f3r(3)Theinclineoftheinnertriangle,9,isexpressedbythetermofrotationofthemovingplate,¢,9=¢+3(4)TherelativedisplacementsoftheactivejointsareLxcosa+ecos9+L2cos(a-0o)=x2-x(5)Lsina+esinp+L2sin(a-Oo)=yz-y,(6)L1cosa+ecos(00+9)+L3cos(a+Oo)=xa-xl(7)Ltsina+esin(00+9)+L3sin(a+00)=y3-y(8)FromEqs.(5)and(6),thelengthsLaandL2areLI=L2-cos(a-&)sin0o(Y2-yl-esin9)(9)sin(a-&)(x2-Xl-ecos9)sin001(xz-x-ecos9)cos(a-00)COSasinOo(yz-yl-esin9)(10)COSa.,tama-0o)(x2-xl-ecos9)Also,fromEqs.(7),(9),and(10),thelengthL3isCopyright(C)2003NuriMediaCo.,ltd.1Ls=cos(a+0o)x3-Xl-eCos(19o-9)cos(a-&)cosa,.,-y2-yl-esin9j(ll)sin(a-0o)cosa4cos(a+0o)sin0o(X2-xl-ecos9)SubstitutionofEqs.(9)-(11)intoEq.(8)deri-vesthefollowingequation:C+C2cos9+C3sin9=0(12)whereCl=cos(a+00)(y3-y)-sin(a+00)(x3-x)+cos(a-00)(y2-yl)-sin(a-00)(x2-xx)C2=esina+sin(a_Oo)(13)C3=-ecosa+cos(a-00)Equation(12)canbesolvedintroducingapara-meterTasfollows:1-Tzcos9=I+T2Tsin9=l+T2SubstitutingEq.(14)intoEq.(14)orderpolynomialinTisobtainedas(C1-C2)Tz+2C3T+(CI+C2)=0(15)Equation(15)offerstheclosedformsolutionsofTC3+C+22-C2-Cz(16)T-C1-C2FromEqs.(4)and(14),therotationofthemov-ingplateis2Tx(17)¢=tan-1(I-Tz)3Thus,thetranslationofthemovingplateisx=xl+LIcosa-rsin¢(18)y=-R+Lxsina+rcos¢ItisremarkablethatthismanipulatorhasatmosttwosolutionsfordirectkinematicsaccordingtoEq.(16),andmoreovertheclosedformsolutionsofEqs.(17)and(18).(12),asecondKinematicAnalysisandOptimalDesignof3-P_PRPlanarParallelManipulator5314.InverseKinematicsandInverseJacobianFigure3showsacoordinatesystemfordes-cribingtheinversekinematicsofa3-PPRplanarparallelmanipulator.WhenthecenterofthemovingplatemovesfromoriginOtoOwithtranslation(x,y)androtationb,thevertexoftheplateBisexpressedasxBi=x+rsin(&+b)(19)ysi=y+rCOS(0g+b)where2(i-1)0,-r(20)3andi=1,2,3.TheoriginoftheactiveprismaticjointOiisdepartedbyRfromtheoriginO.Providedthatuiandviaretheunitvectorsoftheaxese,and,whicharetheaxesoftheactiveprismaticjoint,thecoordinateofthevertexB,expressedbythetermsof,andi,iseBi=OBi"ui(21)gi=R+OBiviThusthepositionoftheactiveprismaticjoint,a,isYB,(,L0Rvo,Fig.3Coordinatesystemforinversekinematics=Bi-,cota(22)DirectdifferentiationofEq.(22)withrespecttothepose(x,y,b)derivestheinverseJacobianj-iasfollows:I1-c0tart,cos4-cotasm4)j4=.5-(-1%3c0ta)2(；3+c0talr(c0s-c0tasin)(23)I-(l+,c0ta)g:-3+c0ta；r(c0s&c0tasin)TheelementsoftheinverseJacobianofEq.(23)donothavethesamedimension.Firsttwoco-lumnscorrespondingtotranslationaredimen-sionless,whereasthelastcolumncorrespondingtorotationhasthedimensionoflength.Bymak-ingthethirdcolumndimensionless,ahomogen-eousinverseJacobianwithnon-dimensionalele-mentsisobtainedby(Byun,1997)l°a1J=J01(24)ooI/R5.RotationalLimitandWorkspaceTherotationofthemovingplateisrestrictedbytheinterferencebetweenthelinksandtherotationaljoints.Fig.4showstheconfigurationofthismanipulatorwithrotationallimitsoftheclockwisecase(a)andcounterclockwisecase(b).Assumingthewidthofthelinksisnegligible,therotationbisboundedby_4zc+cz<<3+a(25)3Providedthatb0istheinitialrotationdefinedby0=te-(26)Therotationrange(25)ismodifiedaswhere57c<<56zr6(27)=qS-b0(28)Itisconcludedthatthemovingplateofthe3-PPRplanarparallelmanipulatorisboundedby+-57c/6fromtheinitialrotation.Copyright(C)2003NuriMediaCo.,Ltd.