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20085395Delta3DeltaDelta,,,,::TP24:AWorkspaceResolutionandDimensionalSynthesisonDeltaParallelMechanismGaoXiulanLuKaijiangWangJuanping(BaojiUniversityofArtsandSciences,Baoji,721007,China)AbstractAmethodofworkspaceresolutionanddimensionalsynthesisonDeltaparallelmechanismswaspresented.Theresolutionexpressionofthesingle2open2chainsub2spaceboundarysurfaceandthereachableworkspaceoftheDeltaparallelmechanismwereobtainedbasedoncurveenvelopetheory.Thenthemaximalinscribedspheroidcontainedinworkspacewaseduced,andtherelationshipbetweenitspositionandthedimensionswasestablished.TakingthespheroidasthedesignspaceandusingtheuniversalaveragevalueofJacobianmatricesconditionnumberindesignspaceasmechanismperformanceindexfordimensionalsynthesis,theoptimaldimensionsfortheDeltaparallelmechanismwereobtainedlastly.KeywordsParallelmechanism,Workspace,Dimensionalsynthesis:20072012083(:05JK2138),721007,,63,Delta3,[1][24],,[56]Delta,,,,1Delta23,1,11,1DeltaFig.1KinematicdiagramofDeltaparallelmechanism,,3,3Oxyz,,xA1,yA3A2OAiAiBii,r0r,BiCibi=rcosisini0+l1cosicosicosisinisini(i=1,2,3)ci=r0cosisini0+xyz(i=1,2,3)l1xyzPiAiCiBiCi=l2fi=(x-xi)2+(y-yi)2+(z-zi)2-l22=0(i=1,2,3)(1)xi=l1cosicosi+(r-r0)cosiyi=l1sinicosi+(r-r0)sinizi=l1sini(2)2211,(2)i,,i,Ni((r-r0)cosi,(r-r0)sini,0),l1(1)Si(xi,yi,zi),l2,ii,PiSii,,Ni,ni,i,3G(0,0,l21-(r-r0)2),22Fig.2Graphicofsingle2open2chainsub2spaceboundarytoruscenterline3Fig.3Formingtheoryofsingle2open2chainsub2spaceboundarytorusni,3,M1(x1,y1,z1),[x1-(l1+r-r0)cosi]2+[y1-(l1+r-r0)sini]2+z21=l22z1=0(3)M1M1niNiM1Ni,nx(x-x1)+ny(y-y1)+nz(z-z1)=0(x-x0)2+(y-y0)2+(z-z0)2=(x1-x0)2+(y1-y0)2+(z1-z0)2(4)(3)(4)x1y1z1xyz,[x2+y2+z2-2(r-r0)(xcosi+ysini)+l21-l22+(r-r0)2]2=4l21[x2cos2i+y2sin2i+z2+2xycosisini-2(r-r0)(xcosi+ysini)+(r-r0)2](5)7415:Delta212,,,,,,G,Gl2,,:l1=712cm,l2=412cm,r=3cm,r0=2cm,:(1)zmax=l21-(r-r0)2+l2zmin=l21-(r-r0)2-l2(2)Oxyz(3),zj=zmax-(j-1)z(5),zji,4a,min{i|i=1,2,3},zj,4b(4)j,,5(5)zj,,G,64Fig.4Diagramsofworkspacescross2sectioncontour(a)(b)56:(1)Delta,zj,(2)G,5DeltaFig.5WorkspaceboundarycurveofDeltaparallelmechanisms6Fig.6Reachableworkspaceanddesign(a)z=1015cm(b)z=7113cm(c)z=3177cm3,,311(1)t,8412008J=A-1BA=5f15x5f15y5f15z5f25x5f25y5f25z5f35x5f35y5f35zB=-5f1515f2525f3535fi5x=x-l1cici-(r-r0)ci5fi5y=y-l1cisi-(r-r0)si5fi5z=z-l1si5fi5i=5fi5xi5xi5i+5fi5yi5yi5i+5fi5zi5zi5i=l1xsici+l1ysisi-l1(r-r0)si-l1zcisi=sinici=cosici=cosisi=sini312,cond(J)=maxminmaxminJTJ,,F(J)=mwcond(J)dwmwdww,,JxyzRx=x0+Rsincosy=y0+Rsinsinz=z0+Rcos(R[0,l2],[0,],[0,2])x0y0z0F(J)=20d0dl20cond(J)5(x,y,z)5(R,,)dR5(x,y,z)5(R,,)=R2sin:(l2)(Gh),l1rr0,F(l1,r,r0)313:l2=412cm,h=516cm,:l1=519642cm,r1=219911cm,r0=019912cm,411379111060116136,77Fig.7Conditionnumbervariabletendencyinworkspace4Delta,,,1,.[J].,1999,10(10):11031107.WangJinsong,HuangTian.Parallelmachinetoolaforthcomingopportunityandchallengetothemachinetoolindustry[J].ChinaMechanicalEngineering,1999,10(10):11031107.(inChinese)2,.[J].,2005,36(8):103105.YuShunnian,MaLzhong.Positionandworkspaceanalysisofparallelmechanismwithtwotranslationalandonerotationaldegreeoffreedom[J].TransactionsoftheChineseSocietyforAgriculturalMachinery,2005,36(8):103105.(inChi2nese)3,,,.[J].,2003,13(8):843850.YuJingjun,ZhaoTieshi,BiShusheng,etal.32dynamiccomprehensivestudyofparallelinstitutions[J].ProgressinNaturalScience,2003,13(8):843850.(inChinese)(145)9415:Delta[67],,,,1,,,,,,5(1),,,(2)ANSYS,,,,1,,,.[J].,2006,37(11):9699.YangQizhi,GuoZonghe,MaLzhong,etal.Kinematicsofanovel32DOFtranslationalfullycompliantparallelmicromanipu2lator[J].TransactionsoftheChineseSocietyforAgriculturalMachinery,2006,37(11):9699.(inChinese)2,,,.[J].,2005,36(1):104107.YangQizhi,YinXiaoqin,MaLzhong,etal.Numericalanalysisforestablishingstiffnessmatrixofthecompliantjoint[J].TransactionsoftheChineseSocietyforAgriculturalMachinery,2005,36(1):104107.(inChinese)3,,.MEMS[J].,2001,9(1):15.YuJingjun,ZongGuanghua,BiShusheng.FullycompliantmechanismsandMEMS[J].OpticsandPrecisionEngineering,2001,9(1):15.(inChinese)4LarryLHowell.Compliantmechanisms[M].JohnWiley&Sons,2001.5JensenBrianD,LarryLHowell.Themodelingofcross2axisflexuralpivots[J].MechanismandMachineTheory,2002,37(5):461476.6,,,.[J].,2007,38(7):133136.ShenJianying,YangShixi,ZhouQinghua,etal.Finiteelementanalysisofdisplacementperformancesofflexurehingemech2anismaffectedbymachiningerrors[J].TransactionsoftheChineseSocietyforAgriculturalMachinery,2007,38(7):133136.(inChinese)7,,.[J].,2007,38(3):136140.JiHuawei,YangShixi,WuZhaotong.Erroranalysisandmodelingofmonolithicmicromanipulator[J].TransactionsoftheChineseSocietyforAgriculturalMachinery,2007,38(3):136140.(inChinese)8,.[J].,2007,5(1):14.QinYu,FengZhijing.Simulationandexperimentofnano2positioningcompliantmechanismsformotionreduction[J].NanotechnologyandPrecisionEngineering,2007,5(1):14.(inChinese)9RajeshLuharuka,PeterJHesketh.Designoffullycompliant,in2planerotary,bistablemicromechanismsforMEMSapplica2tions[J].SensorsandActuatorsA,2007,134(1):231238.(149)4,,,.3[J].,2005,15(2):212220.LiuXinjun,WangJinsong,WangQiming,etal.A32DOFparallelspacerobotworkspaceandrotati