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外文翻译--三自由度并联机器人精度分析 英文版.pdf

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外文翻译--三自由度并联机器人精度分析 英文版.pdf

Abstractincreaseinertiaanddecreaseoperatingspeed–activejointerrors,comingfromthefiniteresolutionoftheencoders,sensorerrors,andcontrolerrors.Correspondingauthor.Emailaddressilian.bonevetsmtl.caI.A.Bonev.Availableonlineatwww.sciencedirect.comMechanismandMachineTheory432008445–458www.elsevier.com/locate/mechmtMechanismandMachineTheory0094114X/seefrontmatterC2112007ElsevierLtd.Allrightsreserved.Parallelrobotsareincreasinglybeingusedforprecisionpositioning,andanumberofthemareusedasthreedegreeoffreedom3DOFplanaralignmentstages.Clearly,insuchindustrialapplications,accuracyisoftheutmostimportance.Therefore,simpleandfastmethodsforcomputingtheaccuracyofagivenrobotdesignareneededinordertousethemindesignoptimizationproceduresthatlookformaximumaccuracy.Errorsinthepositionandorientationofaparallelrobotareduetoseveralfactors–manufacturingerrors,whichcanhoweverbetakenintoaccountthroughcalibration–backlash,whichcanbeeliminatedthroughproperchoiceofmechanicalcomponents–compliance,whichcanalsobeeliminatedthroughtheuseofmorerigidstructuresthoughthiswouldThreedegreeoffreedomplanarparallelrobotsareincreasinglybeingusedinapplicationswhereprecisionisoftheutmostimportance.Clearly,methodsforevaluatingtheaccuracyoftheserobotsarethereforeneeded.Theaccuracyofwelldesigned,manufactured,andcalibratedparallelrobotsdependsmostlyontheinputerrorssensorandcontrolerrors.Dexterityandothersimilarperformanceindiceshaveoftenbeenusedtoevaluateindirectlytheinfluenceofinputerrors.However,industryneedsapreciseknowledgeofthemaximumorientationandpositionoutputerrorsatagivennominalconfiguration.Anintervalanalysismethodthatcanbeadaptedforthispurposehasbeenproposedintheliterature,butgivesnokinematicinsightintotheproblemofoptimaldesign.Inthispaper,asimplermethodisproposedbasedonadetailederroranalysisof3DOFplanarparallelrobotsthatbringsvaluableunderstandingoftheproblemoferroramplification.C2112007ElsevierLtd.Allrightsreserved.KeywordsParallelmechanismsAccuracyDexterityPerformanceevaluationErroranalysis1.IntroductionAccuracyanalysisof3DOFplanarparallelrobotsSe´bastienBriota,IlianA.Bonevb,aDepartmentofMechanicalEngineeringandControlSystems,NationalInstituteofAppliedSciencesINSA,Rennes,FrancebDepartmentofAutomatedManufacturingEngineering,E´coledetechnologiesupe´rieureE´TS,Montreal,CanadaReceived21September2006receivedinrevisedform19March2007accepted6April2007Availableonline4June2007doi10.1016/j.mechmachtheory.2007.04.002mostonepassiveprismaticjointinaleg.Themethodisillustratedontwopracticaldesigns1.A3RPRplanarparallelrobot.ThisrobotistheplanarprojectionofthePAMINSArobot6andthe446S.Briot,I.A.Bonev/MechanismandMachineTheory432008445–458designparametersarethoseoftheprototypemanufacturedatINSAofRennes,France.2.Aplanar3PRRrobot7.AprecisionparallelrobotbasedonthisdesignhasbeendevelopedintheTechnicalUniversityofBraunscheig,inGermany8.Theremainderofthispaperisorganizedasfollows.Section2brieflyoutlinesthemathematicaltheoremsusedinthispaper.Section3presentsthemethodusedfortheanalysisoftheorientationandpositionerrors.Finally,Section4coversseveralnumericalexamples,andconclusionsaregiveninthelastsection.Therefore,aspointedoutbyMerlet1,activejointerrorsinputerrorsarethemostsignificantsourceoferrorsinaproperlydesigned,manufactured,andcalibratedparallelrobot.Inthispaper,weaddresstheproblemofcomputingtheaccuracyofaparallelrobotinthepresenceofactivejointerrorsonly.Inthebalanceofthepaper,thetermaccuracywillthereforerefertothepositionandorientationerrorsofaparallelrobotthatissubjectedtoactivejointerrorsonly.Theclassicalapproachconsistsinconsideringthefirstorderapproximationthatmapstheinputerrortotheoutputerrordp¼Jdqð1Þwheredqrepresentsthevectoroftheactivejointinputerrors,dpthevectorofoutputerrorsandJistheJacobianmatrixoftherobot.However,thismethodwillgiveonlyanapproximationoftheoutputmaximumerror.Indeed,aswewillproveinthispaper,givenanominalconfigurationandsomeuncertaintyrangesfortheactivejointvariables,alocalmaximumpositionerrorandalocalmaximumorientationerrornotonlyoccuratdifferentsetsofactivejointvariablesingeneral,buttheseactivejointvariablesarenotnecessarilyallatthelimitsoftheiruncertaintyranges.Severalperformanceindiceshavebeendevelopedandusedtoroughlyevaluatetheaccuracyofserialandparallelrobots.Arecentstudy2reviewedmostoftheseperformanceindicesanddiscussedtheirinconsistencieswhenappliedtoparallelrobotswithtranslationalandrotationaldegreesoffreedom.Themostcommonperformanceindicesusedtoindirectlyoptimizetheaccuracyofparallelrobotsarethedexterityindex3,theconditionnumbers,andtheglobalconditioningindex4.However,inarecentstudyoftheaccuracyofaclassof3DOFplanarparallelrobots5,itwasdemonstratedthatdexterityhaslittletodowithrobotaccuracy,aswedefineit.Obviously,thebestaccuracymeasureforanindustrialparallelrobotwouldbethemaximumpositionandmaximumorientationerrorsoveragivenportionoftheworkspace1,5oratagivennominalconfiguration,givenactuatorinaccuracies.Ageneralmethodbasedonintervalanalysisforcalculatingcloseapproximationsofthemaximumoutputerroroveragivenportionoftheworkspacewasproposedrecentlyin1.Obviously,themaximumoutputerroroveragivenportionoftheworkspaceisthemostimportantinformationforadesigner.However,thismethodisrelativelydifficulttoimplement,givesnoinformationontheevolutionoftheaccuracyofthemanipulatorwithinitsworkspaceandgivesnokinematicinsightintotheproblemofoptimaldesign.Incontrast,averysimplegeometricmethodforcomputingtheexactvalueoftheaccuracyof3DOF3PRPplanarparallelrobotswasdescribedin5inthispaper,PandRstandforpassiveprismaticandrevolutejoins,respectively,whilePandRstandforactuatedprismaticandrevolutejoins,respectively.Thismethodproposestoreplacetheexistingdexteritymapsbymaximumpositionerrormapsandmaximumorientationerrormaps.Whilethismethodcoversthreeofthemostpromisingdesignsforprecisionparallelrobotsoneofwhichiscommercializedandtheothertwobuiltintolaboratoryprototypes,itdoesnotalwaysworkforother3DOFplanarparallelrobots.Thispapergeneralizesthemethodproposedin5byfollowingadetailedmathematicalproofwhichgivesusimportantinsightintotheaccuracyofplanarparallelrobots.Thepresentstudyconsidersonly3DOFthreeleggedplanarparallelrobotswithprismaticand/orrevolutejoints,oneactuatedjointperleg,andat2.MathematicalbackgroundAnalyzingthelocalmaximumpositionerrorandthelocalmaximumorientationerrorofaparallelrobot,inducedbyboundederrorsintheactivejointvariables,isbasicallystudying,onasetofclosedintervals,themaximaoffunctionsDXandD/,definedasDX¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxC0x0Þ2þðyC0y0Þ2qð2ÞD/¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið/C0/0Þ2qð3Þwherex0,y0and/0aretheCartesiancoordinatescorrespondingtothenominaldesiredplatformposepositionandorientationofthestudiedparallelrobot,andx,yand/aretheactualplatformcoordinates.Inthecaseofa3DOFplanarfullyparallelrobot,DXandD/arefunctionsofthreevariablestheactivejointvariablesoftherobottheinputs,whichwillbedenotedbyqiinthispaper,i1,2,3.Thus,wehavetofindthemaximaofDXandD/onthesetofintervalsqi2½qi0C0eqi0þeC138,whereqi0aretheactivejointvariablescorrespondingtothenominalposex0,y0,/0oftheplatformintheselectedworkingmode,i.e.,theselectedsolutiontotheinversekinematicsandeistheerrorboundontheactivejointvariablesFig.1.Tosimplifyourerroranalysis,wewillmakethepracticalassumptionthatthenominalconfigurationissufS.Briot,I.A.Bonev/MechanismandMachineTheory432008445–458447Fig.1.Inputerrorboundingbox.ficientlyfarfromType1andType2singularities.Type1singularities9areconfigurationswhereaparallelrobotlosesitsdesiredfunctionality–itlosesoneormoredegreesoffreedom.Thesearetheinternalandtheexternalboundariesofworkspace.Forthisreason,theusableworkspaceofanindustrialparallelrobotwillbeawayfromthesesingularities.Similarly,Type2singularities9areanotherkindofconfigurationswhereaparallelrobotlosesitsdesiredfunctionality–thistimeitlosescontrolofthemobileplatform.Furthermore,neartheseconfigurations,theoutputerrorincreasesexponentially.Forthesereasons,industrialparallelrobotsaredesignedtoexcludesuchsingularities.Therefore,wewillobviouslyperformourerroranalysisonlyforconfigurationsthataresufficientlyfarfromsingularities,i.e.,fornominalconfigurationsfromwhichtherobotcannotenterintosingularitywhiletheactivejointvariablesstaywithintheirerrorboundedintervals.Oncewehavemadethispracticalassumption,weaddresstheproblemoffindingtheglobalmaximaofDXandD/.Itiswellknownthatthemaximumofacontinuousmultivariablefunction,f,overagivensetofintervalscanbefoundbyanalysingtheHessianmatrix,Hh1q1fðq1q20þeq30þeÞh7q2fðq10þeq2q30C0eÞIfsuchFinTheseFinandDThesingularity–TheFig.coulddenotedcentre448S.Briot,I.A.Bonev/MechanismandMachineTheory432008445–458withintheintervalstudied.twistofthemobileplatform,whenlegsjandkj,k1,2,3,i5j5karefixed,isapuretranslation.2representsthemobileplatformofarobotlinkedtothreeactuatedlegs,throughrevolutejointsthesebeprismaticjointsaswell.EachlegappliesawrenchRionthemobileplatform,whosecenterisbyP.TheintersectionpointO3ofthewrenchesR1andR2representstheinstantaneousrotationiiThesederivativesareequaltozeroifo//oqi0orif/C0/00.Obviously,however,amaximumcanexistonlyifo//oqi0.Fora3DOFplanarparallelrobot,twodifferentsituationscorrespondtotheconditiono//oqi0–TherobotisataType1singularity.However,wealreadyassumedthattherobotcannotenteraType1partialderivativesofD/2aregivenasoðD/2Þoq¼2o/oqð/C0/0Þð5Þ3.Analysisoftheorientationandpositionerrors3.1.Maximumorientationerrorh6q2fðq10C0eq2q30þeÞh12q3fðq10C0eq20C0eq3Þpointsexist,wewillcallthemmaximaofthethirdkind.ally,theglobalmaximumoffcouldalsobeononeoftheeightcornersoftheinputerrorboundingbox.eightpointswillbereferredtoasextremaofthefourthkind.dingtheglobalmaximaoffunctionsDXandD/isequivalenttofindingthemaximaoffunctionsDX2/2.Inthenextsection,wewillstudytheextremaofthefunctionsDX2andD/2.h2q1fðq1q20þeq30C0eÞh8q2fðq10C0eq2q30C0eÞh3q1fðq1q20C0eq30þeÞh9q3fðq10þeq20þeq3Þh4q1fðq1q20C0eq30C0eÞh10q3fðq10þeq20C0eq3Þh5q2fðq10þeq2q30þeÞh11q3fðq10C0eq20þeq3Þ22310235121230g3ðq1q3Þfðq1q20þeq3Þg6ðq1q2Þfðq1q2q30C0eÞIfsuchpointsexist,wewillcallthemmaximaofthesecondkind.Theglobalmaximumoffcouldalsobeontheedgesoftheinputerrorboundingbox.Thistime,wehavetostudythemaximaoftwelveunivariatefunctionsg1ðq2q3Þfðq10þeq2q3Þg4ðq1q3Þfðq1q20C0eq3ÞgðqqÞfðqC0eqqÞgðqqÞfðqqqþeÞH¼o2foq21o2foq1oq2o2foq1oq3o2foq22o2foq2oq3symo2foq23266664377775ð4ÞUsingthisHessianmatrix,thesetofvariablesq1m,q2m,q3m,whereqim2½qi0C0eqi0þeC138,leadstoamaximumoffifofoqiðq1mq2mq3mÞ¼0andHisnegativedefinite.Ifsuchapointexistsq1m,q2m,q3m,wewillcallitamaximumofthefirstkind.TheglobalmaximumoffcouldalsobeonthefacesoftheinputerrorboundingboxshowninFig.1.Thistime,wehavetostudythemaximaofsixfunctionsoftwovariableseach,definedasofthemobileplatformwhenactuators1and2arefixedandthethirdactuatorismoving.Thus,if

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