Abstractincreaseinertiaanddecreaseoperatingspeed)；active-jointerrors,comingfromthefiniteresolutionoftheencoders,sensorerrors,andcontrolerrors.*Correspondingauthor.E-mailaddress:ilian.bonevetsmtl.ca(I.A.Bonev).AMechanismandMachineTheory43(2008)445458/locate/mechmtMechanismandMachineTheory0094-114X/$-seefrontmatterC2112007ElsevierLtd.Allrightsreserved.Parallelrobotsareincreasinglybeingusedforprecisionpositioning,andanumberofthemareusedasthree-degree-of-freedom(3-DOF)planaralignmentstages.Clearly,insuchindustrialapplications,accuracyisoftheutmostimportance.Therefore,simpleandfastmethodsforcomputingtheaccuracyofagivenrobotdesignareneededinordertousethemindesignoptimizationproceduresthatlookformaximumaccuracy.Errorsinthepositionandorientationofaparallelrobotareduetoseveralfactors:manufacturingerrors,whichcanhoweverbetakenintoaccountthroughcalibration；backlash,whichcanbeeliminatedthroughproperchoiceofmechanicalcomponents；compliance,whichcanalsobeeliminatedthroughtheuseofmorerigidstructures(thoughthiswouldThree-degree-of-freedomplanarparallelrobotsareincreasinglybeingusedinapplicationswhereprecisionisoftheutmostimportance.Clearly,methodsforevaluatingtheaccuracyoftheserobotsarethereforeneeded.Theaccuracyofwelldesigned,manufactured,andcalibratedparallelrobotsdependsmostlyontheinputerrors(sensorandcontrolerrors).Dexterityandothersimilarperformanceindiceshaveoftenbeenusedtoevaluateindirectlytheinfluenceofinputerrors.However,industryneedsapreciseknowledgeofthemaximumorientationandpositionoutputerrorsatagivennominalconfiguration.Anintervalanalysismethodthatcanbeadaptedforthispurposehasbeenproposedintheliterature,butgivesnokinematicinsightintotheproblemofoptimaldesign.Inthispaper,asimplermethodisproposedbasedonadetailederroranalysisof3-DOFplanarparallelrobotsthatbringsvaluableunderstandingoftheproblemoferroramplification.C2112007ElsevierLtd.Allrightsreserved.Keywords:Parallelmechanisms；Accuracy；Dexterity；Performanceevaluation；Erroranalysis1.IntroductionAccuracyanalysisof3-DOFplanarparallelrobotsSebastienBriota,IlianA.Bonevb,*aDepartmentofMechanicalEngineeringandControlSystems,NationalInstituteofAppliedSciences(INSA),Rennes,FrancebDepartmentofAutomatedManufacturingEngineering,Ecoledetechnologiesuperieure(ETS),Montreal,CanadaReceived21September2006；receivedinrevisedform19March2007；accepted6April2007Availableonline4June2007doi:10.1016/j.mechmachtheory.2007.04.002mostonepassiveprismaticjointinaleg.Themethodisillustratedontwopracticaldesigns:1.A3-RPRplanarparallelrobot.ThisrobotistheplanarprojectionofthePAMINSArobot6andthe446S.Briot,I.A.Bonev/MechanismandMachineTheory43(2008)445458designparametersarethoseoftheprototypemanufacturedatINSAofRennes,France.2.Aplanar3-PRRrobot7.AprecisionparallelrobotbasedonthisdesignhasbeendevelopedintheTech-nicalUniversityofBraunscheig,inGermany8.Theremainderofthispaperisorganizedasfollows.Section2brieflyoutlinesthemathematicaltheoremsusedinthispaper.Section3presentsthemethodusedfortheanalysisoftheorientationandpositionerrors.Finally,Section4coversseveralnumericalexamples,andconclusionsaregiveninthelastsection.Therefore,aspointedoutbyMerlet1,active-jointerrors(inputerrors)arethemostsignificantsourceoferrorsinaproperlydesigned,manufactured,andcalibratedparallelrobot.Inthispaper,weaddresstheprob-lemofcomputingtheaccuracyofaparallelrobotinthepresenceofactive-jointerrorsonly.Inthebalanceofthepaper,thetermaccuracywillthereforerefertothepositionandorientationerrorsofaparallelrobotthatissubjectedtoactive-jointerrorsonly.Theclassicalapproachconsistsinconsideringthefirstorderapproximationthatmapstheinputerrortotheoutputerror:dpJdq；1wheredqrepresentsthevectoroftheactive-joint(input)errors,dpthevectorofoutputerrorsandJistheJacobianmatrixoftherobot.However,thismethodwillgiveonlyanapproximationoftheoutputmaximumerror.Indeed,aswewillproveinthispaper,givenanominalconfigurationandsomeuncertaintyrangesfortheactive-jointvariables,alocalmaximumpositionerrorandalocalmaximumorientationerrornotonlyoccuratdierentsetsofactive-jointvariablesingeneral,buttheseactive-jointvariablesarenotnecessarilyallatthelimitsoftheiruncertaintyranges.Severalperformanceindiceshavebeendevelopedandusedtoroughlyevaluatetheaccuracyofserialandparallelrobots.Arecentstudy2reviewedmostoftheseperformanceindicesanddiscussedtheirinconsisten-cieswhenappliedtoparallelrobotswithtranslationalandrotationaldegreesoffreedom.Themostcommonperformanceindicesusedtoindirectlyoptimizetheaccuracyofparallelrobotsarethedexterityindex3,theconditionnumbers,andtheglobalconditioningindex4.However,inarecentstudyoftheaccuracyofaclassof3-DOFplanarparallelrobots5,itwasdemonstratedthatdexterityhaslittletodowithrobotaccuracy,aswedefineit.Obviously,thebestaccuracymeasureforanindustrialparallelrobotwouldbethemaximumpositionandmaximumorientationerrorsoveragivenportionoftheworkspace1,5oratagivennominalconfiguration,givenactuatorinaccuracies.Ageneralmethodbasedonintervalanalysisforcalculatingcloseapproximationsofthemaximumoutputerroroveragivenportionoftheworkspacewasproposedrecentlyin1.Obviously,themaximumoutputerroroveragivenportionoftheworkspaceisthemostimportantinformationforadesigner.However,thismethodisrelativelydiculttoimplement,givesnoinformationontheevolutionoftheaccuracyofthemanipulatorwithinitsworkspaceandgivesnokinematicinsightintotheproblemofoptimaldesign.Incontrast,averysimplegeometricmethodforcomputingtheexactvalueoftheaccuracyof3-DOF3-PRPplanarparallelrobotswasdescribedin5(inthispaper,PandRstandforpassiveprismaticandrevolutejoins,respectively,whilePandRstandforactuatedprismaticandrevolutejoins,respectively).Thismethodproposestoreplacetheexistingdexteritymapsbymaximumpositionerrormapsandmaximumorientationerrormaps.Whilethismethodcoversthreeofthemostpromisingdesignsforprecisionparallelrobots(oneofwhichiscommercializedandtheothertwobuiltintolaboratoryprototypes),itdoesnotalwaysworkforother3-DOFplanarparallelrobots.Thispapergeneralizesthemethodproposedin5byfollowingadetailedmathematicalproofwhichgivesusimportantinsightintotheaccuracyofplanarparallelrobots.Thepresentstudyconsidersonly3-DOFthree-leggedplanarparallelrobotswithprismaticand/orrevolutejoints,oneactuatedjointperleg,andat2.MathematicalbackgroundAnalyzingthe(local)maximumpositionerrorandthe(local)maximumorientationerrorofaparallelrobot,inducedbyboundederrorsintheactive-jointvariables,isbasicallystudying,onasetofclosedinter-vals,themaximaoffunctionsDXandD/,definedasDXxC0x02yC0y02q；2D/C0/02q；3wherex0,y0and/0aretheCartesiancoordinatescorrespondingtothenominal(desired)platformpose(posi-tionandorientation)ofthestudiedparallelrobot,andx,yand/aretheactualplatformcoordinates.Inthecaseofa3-DOFplanarfully-parallelrobot,DXandD/arefunctionsofthreevariables:theactive-jointvariablesoftherobot(theinputs),whichwillbedenotedbyqi(inthispaper,i=1,2,3).Thus,wehavetofindthemaximaofDXandD/onthesetofintervalsqi2qi0C0e；qi0eC138,whereqi0aretheactive-jointvari-ablescorrespondingtothenominalpose(x0,y0,/0)oftheplatform(intheselectedworkingmode,i.e.,theselectedsolutiontotheinversekinematics)andeistheerrorboundontheactive-jointvariables(Fig.1).Tosimplifyourerroranalysis,wewillmakethepracticalassumptionthatthenominalconfigurationissuf-S.Briot,I.A.Bonev/MechanismandMachineTheory43(2008)445458447Fig.1.Inputerrorboundingbox.ficientlyfarfrom(Type1andType2)singularities.Type1singularities9areconfigurationswhereaparallelrobotlosesitsdesiredfunctionalityitlosesoneormoredegreesoffreedom.Thesearetheinternalandtheexternalboundariesofworkspace.Forthisreason,theusableworkspaceofanindustrialparallelrobotwillbeawayfromthesesingularities.Similarly,Type2singularities9areanotherkindofconfigurationswhereaparallelrobotlosesitsdesiredfunctionalitythistimeitlosescontrolofthemobileplatform.Furthermore,neartheseconfigurations,theoutputerrorincreasesexponentially.Forthesereasons,industrialparallelrobotsaredesignedtoexcludesuchsingularities.Therefore,wewillobviouslyperformourerroranalysisonlyforconfigurationsthataresucientlyfarfromsingularities,i.e.,fornominalconfigurationsfromwhichtherobotcannotenterintosingularitywhiletheactive-jointvariablesstaywithintheirerror-boundedintervals.Oncewehavemadethispracticalassumption,weaddresstheproblemoffindingtheglobalmaximaofDXandD/.Itiswellknownthatthemaximumofacontinuousmultivariablefunction,f,overagivensetofinter-valscanbefoundbyanalysingtheHessianmatrix,H:h1:q1!fq1；q20e；q30e；h7:q2!fq10e；q2；q30C0e；IfsuchFinTheseFinandDThesingularityTheFig.coulddenotedcentre448S.Briot,I.A.Bonev/MechanismandMachineTheory43(2008)445458withintheintervalstudied.twistofthemobileplatform,whenlegsjandk(j,k=1,2,3,i5j5k)arefixed,isapuretranslation.2representsthemobileplatformofarobotlinkedtothreeactuatedlegs,throughrevolutejoints(thesebeprismaticjointsaswell).EachlegappliesawrenchRionthemobileplatform,whosecenterisbyP.TheintersectionpointO3ofthewrenchesR1andR2representstheinstantaneousrotationiiThesederivativesareequaltozeroifo/oqi=0orif/C0/0=0.Obviously,however,amaximumcanexistonlyifo/oqi=0.Fora3-DOFplanarparallelrobot,twodierentsituationscorrespondtotheconditiono/oqi=0:TherobotisataType1singularity.However,wealreadyassumedthattherobotcannotenteraType1partialderivativesofD/2aregivenasoD/2oq2o/oq/C0/0:53.Analysisoftheorientationandpositionerrors3.1.Maximumorientationerrorh6:q2!fq10C0e；q2；q30e；h12:q3!fq10C0e；q20C0e；q3:pointsexist,wewillcallthemmaximaofthethirdkind.ally,theglobalmaximumoffcouldalsobeononeoftheeightcornersoftheinputerrorboundingbox.eightpointswillbereferredtoasextremaofthefourthkind.dingtheglobalmaximaoffunctionsDXandD/isequivalenttofindingthemaximaoffunctionsDX