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Traversing

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TraversingTraver

    Traversing

JournalofZhejiangUniversitySCIENCEA

    ;ISSN1673565X(Print);ISSN18621775(Online)

    ;wwwzju.educn~zus;www.springerlinkcom

    ;E-mail:jZUS@zju.edu.ca

    ;Lfefa1./dZhejiangUnivSciA20089(11):1539-1551

    ;

    ;1539

    ;Traversingthesingularityhypersurfacebyapplyingtheinput

    ;disturbancesto6-SPSparallelmanipulator

    ;YutongLI

    ;,

    ;YuxinWANG,ShuangxiaPAN1

    ;,RuiqinGUO

    ;(CollegeofMechanicalEngineeringandEnergy,Zh~iangffHangzhou31002ZChina)

    ;(2DepartmentofMechanicalEngineering,Tong/iUniversity,Shanghai200092,China) ;E-mail:creativetj@263.com

    ;ReceivedOct.20,2007;revisionacceptedApr.16,2008

    ;Abstract:Thesingularpointsofa6.SPSStewartplatformaredistributedonthemultidim

    ensionalsingularityhypersurfacein

    ;thetask.space.whichdividestheworkspaceofthemanipulatorintoseveralsingularity.freeregions.Becauseofthemotionun.

    ;certaintyatsingularpoints.whilethemanipulatortraversesthiskindofhypersurfacefromonesingularityfreeregiontoanother,

    ;itsmotioncannotbepredetermined.InthisPaDer’adetailedapproachforthemanipulatortotraversethesingularityhypersurface

    ;withitsnonpersistentconfigurationispresented.First.thesingularpointtransfordisturbanceandtheposedisturbance.which

    ;maketheperturbedsingularpointtransferhorizontallyandvertically,respectively,areconstructed.Throughapplyingthesedis-

    ;turbancesintotheinputparameterswithinthemaximumlosscontroldomain.theperturbedpersistentconfigurationistransformed

    ;intoitscorrespondingnonpersistentone.Undertheactionofthedisturbances.themanipulatorcantraversethesingularityhy.

    ;persurfacefromonesingularity.freeregiontoanotherwithadesiredconfiguration. ;Keywords:Parallelmanipulator,Singularityhypersurface,Singularity-freemovingregion

    ;doi:10.1631/jzus.A0720034Documentcode:ACLCnumber:TH11

    ;INTRODUCT10N

    ;Itiswellknownthatthereexistmanysingulari

;tiesinthewholeworkspaceforthe6SPS

    ;GoughStewartplatformsf66SPMs)whichleadto

    ;lossofrigidityincertaindirection(s),andunbounded ;loadsatoneormorepassiveioints.Thereforeidenti

    ;ficationandavoidanceofsingularitiesforthiskindof ;manipulatorareofpracticaimportance.andthey

    ;haveattractedasignificantvolumeofresearch. ;Identificationofthesingularitymanifoldofthe ;66SPMsisanactiveareaofresearch.Inrecentyears. ;researchershaveusedvariouscomputationalalgebra ;toolstoarriveattheanalyticalform.StOngeand

    ;Gosselinf1996)usedthesingularityconditionpro

    ;posedin(GosselinandAngeles.1990).andderiveda ;polynomialexpressionforthesingularitymanifoldof ;thegeneralSPM.KimandChungrl999)usedan

    ;Project(Yos.50375111and50675188)supposedbytheNational ;NaturalScienceFoundationofChina

     ;alternateformulationofthe1inearvelocityrelation

    ;shipstoarriveatasimilarexpressionwithaless ;numberofterms.StOngeandGosselin(2000)refined

    ;theirearlierworktoreportthealgebraicstructureof ;thesingularitymanifoldoftheSPMforvariousar. ;chitecturalclasses.DiGregorio(2002)presenteda ;newexpressionofthesingularityconditionofthe ;mostgeneralmechanismbasedonthemixedproducts ;ofvectors,andtransformedthesingularitycondition ;intoaninth.degreepolynomialequationwhosesin. ;gularitypolynomialequationiscubicintheplatform ;orientationparameters.HuangandCao(2005)studied ;thesingularitylocianddistributioncharacteristicsof ;thesimplifiedsymmetrictrianglemanipulatorarchi. ;tecture.whereallsingularitiesareclassifiedintothree ;differentIinearcomplexsingularities.WolfandSho. ;ham(2003)usedthelinegeometryandthescrew ;theorytodeterminethesingularpointsofparallel ;manipulatorsandtheirbehaviorsatthesepoints.Liet ;a1.(2006)presentedananalyticfoITflofthe6D ;154OLfeal/JZhejiangUnivSciA20089(11):1539?1551 ;singularitylocusofthegeneralGough-?Stewartplat__ ;form.Whentheorientationofthemanipulatorisgiven, ;thetypeIIsingularities(St-OngeandGosselin,2000) ;aredistributedonthe3DsurfacesintheDescartes ;coordinatesystem.BandyopadhyayandGhosal(2004) ;presentedacompactclosedformsingularitymanifold

;expressionofthe6SPSSPMs.Thesingularity

    ;manifoldisobtainedasthehypersurfaceinthe ;task-space,SE(3),onwhichthewrenchtransforrna? ;tionmatrixforthetopplatformdegenerates. ;Becauseoftheexistenceofthesingularityhy

    ;persurface,thewholeworkspaceofthemanipulatoris ;dividedintoseveralindependentsingularityfree

    ;regions.However,theformofthehypersurfacede

    ;pendsontheposeparametersofthemanipulator. ;Whiletheposeparametersarechanged,theformof ;thehypersurfacewil1bealtered.Whenthemanipu

    ;latormovesfromonesingularityfreeregiontoan-

    ;other,themanipulatorshouldtraversethehypersur

    ;facewithadesiredconfiguration.Duetothemotion ;uncertaintyatthesingularpointswhichdistributeon ;thehypersurface,afterpassingthroughthesingular ;point.themotionofthemanipulatorisuncertain. ;Withtheapplicationofparallelmanipulatorsinsome ;importantaspects,suchasintheaviationandspace- ;flight,themotioncertaintyofparallelmanipulatorsat ;thevicinityofthesingularpointhasbeenpaidmuch ;moreattention.

    ;Bandyopadhyayeta1.(2004)developeda

    ;schemeforavoidingsingularitiesofaStewartplat- ;formbyrestructuringapreplannedpathinthevicinity ;ofthesingularposition.DasguptaandMruthyunjaya ;f20001presentedanalgorithmtoconstructcontinuous ;pathswithintheworkspaceoftheStewartplatform ;f0ravoidingsingularities.Giventwoendposesofthe

    ;manipulator,thealgorithmcanfindoutsafeviapoints ;andplanacontinuouspathfromtheinitialposetothe ;finalone.Seneta1.(20031usedavariationalapproach ;forplanningsingularity-freepathsforparallel ;mechanismsbasedonaLagrangianincorporating ;bothakineticenergytermandapotentialenergyterm ;toensurethattheobtainedpathisshortandsingular

    ;ityfree.Dasheta1.(20031presentedanumerical ;approachforpathplanninginsidetheworkspaceof ;parallelmechanismstoavoidsingularities.Thepath ;ismodifiedtoavoidthesingularconfigurationsbya ;localroutingmethodbasedonGrassmann’sline

    ;geometry.

    ;Thesingularityfreepathplanningmethodcan

    ;beutilizedtoavoidsingularitiesofmanipulatorswith

;aPTPcontrolrequirementwhentwoendposesofthe

    ;manipulatorbelongtothesamesingularity.-freere?- ;gion.However,whenthestartingpointandtheending ;pointbelongtodifferentsingularityfreeregions,no

    ;singularityfreepath,alongwhichthemanipulator ;cantraversethesingularityhypersurfacewithade

    ;siredconfiguration,canbefiguredout(Dasguptaand ;MruthyunJaya,2000).

    ;Exceptthesingularity-freepathplanningmethod ;foravoidingthesingularities,thedeterminationof ;singularityfreezonesintheworkspaceofparallel ;mechanisms(Lieta1.,2007),andthesingularity ;avoidancemethodbyaddingredundantdegreesof ;freedomintothenonredundantmanipulator

    ;(ChoudhuryandGhosal,2000;WangandGosselin, ;2004;Kimeta1..20051havebeenstudied.

    ;ForlettingthemanipulatorWOrkinthewhole ;workspacewithadesiredconfiguration,westudied ;thepossibilityfora3.DOFplanarmanipulatortopass ;throughsingularpositionswithadesiredconfigura

    ;tionthroughaddingadisturbingcontrolfunction ;(Wangeta1.,2003)intothesystem.Takingaplane ;fivebarlinkageasanexample.thedisturbancefunc

    ;tionforthemechanismtopassthroughthesingular ;pointwasconstructed(WangandLiu,2004).Under ;theactionofthedisturbancefunction,theconfigura

    ;tioncurves.whichintersectinthesingularpointwhile ;thedisturbanceisnotintroduced.areseparated.In ;thisway,themechanismcanpassthroughthesingu

    ;larpointwithadesiredconfiguration.Withtheaidof ;theuniversa1unfoldingapproach.itisfoundthatall ;configurationbranchesconvergedinthesamesin

    ;gularpointintheunperturbedsystemforthe ;semi..regularhexagons6..SPSGough..Stewartma.. ;nipulator(SRHGSMP)(St-OngeandGosselin,2000) ;willbeseparatedinthedisturbedsystemfWangand ;Li,2008).Inthispaper,basedonourpreviousre

    ;searches(WangandWang,2005;WangandLi,2008), ;thedetailedapproachforthemanipulatortotraverse ;thesingularityhypersurfacewithadesiredconfigu

    ;rationispresented.

    ;TYPE.IlSINGULARITIES

    ;AsshowninFig.1,thefixeddimensionsofthe ;SRHGSMPare:R1isthedistributionradiusofsix

;Lfefal/JZhejiangUnivSciA20089(11):15391551

    ;sphericaliointsAonthemovableplatform;R2isthe ;distributionradiusofsixsphericaljoints ;Bi(xE,YzB,onthebase.Therelativeanglebe’

    ;tweentwoequilateraltrianglesformedbyjointsA1, ;3,A5andjointsA2,A4,A6isG1,andtherelativeangle ;betweentwoequilateraltrianglesformedbyjointsBI, ;B3,B5andjointsB2,B4,B6is.1i(1,2,…,6)are

    ;thelengthsofthesixextendablelegs,whichareutil

    ;izedasindependentvariables.

    ;

    ;Fig.1Semi-regularhexagons6-SPSGough-Stewartma. ;nipulator

    ;ThefixedcoordinatesystemOxyzofthe

    ;SRHGSMPonthebaseframeissetupas:theorigin ;ofthefixedcoordinatesystemisinthecenterofthe ;baseframe.1ineOB1asxaxis,andthenormal1ineof

    ;thefixedflameasz-axis.Themovablecoordinate ;systemO1XlYlZIonthemovableplatformissetupas: ;lineO1A1asx】一axis,andthenormallineofthemov

    ;ableplatforrnasz1axis.JointAinthefixedcoordi

    ;natesystemisAg(xA,Y,z),andinthemovable ;coordinatesystemisAit(xf,Y,zi).UtilizeP,Q]

    ;toexpresstheposeofthemanipulator,here,P(x,Y,z) ;isthecentercoordinateparametersofthemovable ;platform,istheyawangleofthemovableplatform ;withrespecttothex-axis8isthepitchangleofthe ;movableplatformwithrespecttotheYaxis,andis

    ;therollangleofthemovableplatformwithrespectto ;thezaxis.

    ;Accordingtothelengthconstraintequationsfor ;sixextensiblelegs,

    ;(/4-B)T-B!,--0,i=1,2,…,6,(1)

    ;1541

    ;theconfigurationequationsoftheSRHGSMPare ;writtenas

    ;=++z++_l2+2[cospcosy

    ;+yA(singsin,8cos)’cosasin)’)xBIx

    ;+2cos,ssin),+(singsinflsin),+cosacos)’)

    ;

    ;yBlv2(x~sinflyAsingcos,8)zx~y,cos,8sinr

    ;+2[x’~xB,

    ;cos,8cos)’xB

    ;,

;(singsin,8cos),

    ;

    ;COSGSin),)(COSGCOS),+singsin,8sin=

    ;i=1,2,…,6.(2)

    ;Letting[,1,12,…,/6]T,theintegratedformof

    ;Eq.(2)is

    ;(,)=(,,,,,O6)=o,

    ;whereistheinputparametervectorasindependent ;variablestoanalyzetheconfigurationbifurcation ;behaviors;Xisaposevectorofthemovableplatform. ;Thetype-IIsingularitiescorrespondingtoEq.(3) ;aredeterminedasfollows:

    ;f(X0,)=0,

    ;IdetlO~(Xo,o)/axl=o

    ;Usually,allsixactuatorsofthemanipulator ;shouldbedriventogiveouttheoutputs,sothatthe ;movableplatfornlcanmovealongaspecificpathin ;theworkspacewithgiventime-varyingorstaticori ;entationparameters.Becausethedistributionhyper

    ;surfaceofthetypeIIsingularitieshasacloserela

    ;tionshipwiththegivenorientationparameters,when ;theorientationparametersarechanged,theformof ;thehypersurfacewillbealerted.Therefore,inthis ;paper,theinputparametersaretakenastheinde- ;pendentvariablestoinvestigatetheconfiguration ;bifurcationcharacteristicsofthemanipulator. ;Forthestructuralsemisymmetryofthemanipu

    ;lator,withoutlOSSofuniversality,selecttheinput ;parameter1asanindependentvariable,andkeepthe ;otherfiveinputparametersconstant.Whenthedi- ;mensionsofthemanipulatorareR1=O.2m,R2=0.4m, ;G1=10.,a2=22.,li=O.4m,i=2,3,…,6,withthe

    ;homotopymethod(WangandWang,2005),from ;Eq.(3),theconfigurationbifurcationcurvesexpressed ;

    ;Lfea1./JZhejiangUnivSciA20089(11):1539-1551 ;Table1TypeIIsingularpointsoftheSRHGSPM(z=0) ;1543

    ;configurationcurvesforthemanipulatorswithir

    ;regularlydistributedsphericaljointsbothonthebase ;andonthemovableplatforrnhavebeenanalyzed.The ;resultsshowthattheyaresimilarwiththecurvesas ;showninFig.2.Therefore.thesingularityavoidance ;methodsetupbasedontheconfigurationcurvesas

    ;showninFig.1canbeappliedtothesingularity ;avoidanceofthemanipulatorunderactionofthe ;multipleinputparameters.

    ;APPROACHT0TRAVERSETHESrNGULARITY

    ;HYPERSURFlACE

    ;Traversingthesingularity

    spaceofthe ;Thetask

    ;beendividedintoseveral

    ;hypersurface

    ;parallelmanipulatorhas

    ;differentsingularityfree

    ;regionsbythesingularitydistributionhypersurface ;correspondingtoagroupotgwenposeparameters,as ;showninFig.5in(BandyopadhyayandAshitava, ;2006).Itsprojectioninoneplanecanbeexpressed ;withFig.3.Inthisfigure,theconfigurationalong ;whichthemanipulatormovesclosetothesingular ;pointiscalledthepersistentconfiguration,andthe ;correspondingconfigurationcurveiscalledtheper

    ;sistentconfigurationcurveorbranch.Theothercon

    ;figurationbranch,whichintersectswiththepersistent ;configurationbranchatthesingularpoint.iscalled ;thenonpersistentconfigurationbranch.Thesingu. ;1arityhypersurfacedenotedwithheavylinehasdi. ;videdtheworkspaceintotwosingularity.freeregions: ;thesingularityfreeregion1andthesingularityfree

    ;region2.

    ;Whenthemanipulatorapproachestothesingular ;point,whichisdistributedonthesingularityhyper

    ;surface.ithasatleasttwokindsofconfigurationsto ;embody:itspersistentconfigurationornon.persistent ;configuration.Therefore,themotiondirectionofthe ;manipulatoratthesingularpointisuncertain.Togive ;outaconcretemotionwithinthewhole

    ;Fig.3Manipulatortraversesthesingularityhypersurfaee ;workspace,themanipulatorshouldhavethecapabil

    ;itytotraversethesingularityhypersurfaceandmove ;fromonesingularity-freeregiontoanotherwitha ;desiredconfiguration.Therefore.investigationofthe ;methodforthemanipulatortotraversethesingularity ;hypersurfacewithadesiredconfigurationisofim- ;portantsignificanceforthemotioncertaintyofthe ;manipulatorinthewholeworkspace.

    ;Sincethepersistentconfigurationandthe

    ;non-persistentconfigurationcorrespondtothesame ;groupoftheinputparameters,ifthemanipulatorcan ;traversethesingularityhypersurfacealongtheper- ;sistentconfigurationcurveandthenonpersistent

    ;configurationcurvebeforeandaftertraversingthe ;hypersurface,themanipulatorcantraversethesin

    ;gularityhypersurfacealongotherconfigurationcurve ;undoubtedly.Thereasonisthatinthelattercase,one ;groupofinputparametershasonlyoneconfiguration ;tocorrespond;However,intheformercase,one ;groupofinputparametershasatlesttwoconfigura

    ;tionstocorrespond.Comparingwiththeformercase, ;thelatterisrelativelysimple.Therefore,inthispaper, ;toinvestigatetheapproachfortheparallelmanipu

    ;latortotraversethesingularityhypersurface,thecase ;thatthemanipulatormovesalongthepersistentand ;nonpersistentconfigurationcurvesbeforeandafter ;traversingthesingularityhypersurfaceistakenasthe ;object.

    ;1544Lfefa1./dZhejiangUnivSciA20089(11):15391551

    ;Maximumlosscontroldomain

    ;Aconceptofthemaximumlosscontroldomain ;(MLCD)(Wangeta1.,2005)iselucidatedbrieflyin ;Fig.4.Inthisfigure.istheoutputprecisionofthe

    ;extendablelegs.whichisguaranteedbythecontrol ;system,Ppiatorcanobtainlocal

    ;freemotionfromPpitoPnpiontheconfiguration ;component.Therefore.themotionofthemanipu

    ;latorinthislocalzoneisuncertain.Inordertoobtain ;theconcretemotionatthevicinityofthesingular ;point,themanipulatorshouldnotarrivetotheborders ;definedbypointspandPnpi,i.e.,min{,pi}

    ;shouldbegreaterthanThentheMLCDforone

    ;componentoftheconfigurationisdefinedas ;MP,pi=k~

    ;

    ;,

    ;i

    ;wherek>1.5isasafetycoefficient

    ;i=1,2,…,6,(6)

    ;E

    ;IU=1,2,…,6

    ;l,

    ;Nonpersi:

;configura

    ;Fig.4Maximumlosscontroldomain

    ;Then,theMLCDis

    ;max{~,1,2,…,6}

    ;Obviously,whenthemanipulatormovesbeyond ;theMLCDatthevicinityofthesingularpoint,the ;motionofthemanipulatorisconcrete.Thecalculation ;methodfortheMLCDhadbeenpresentedin(Wanget ;a1.,2005).

    ;Approachtotraversethesingularityhypersurface ;Ithadbeenfoundin(WangandLi,2008)thatall ;configurationbranchesconvergedinthesamesin’

    ;gularpointintheunperturbedsystemwillbesepa- ;ratedinthedisturbedsystemwhilethesuitabledis

    ;turbancesareintroducedintothesystem.Itgivesusa ;hinttosetupamethodforthemanipulatortotraverse ;thesingularityhypersurface.In?

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