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InternationalJournalofEngineeringandTechnologyVolume2No.10,October,2012ISSN20493444©2012–IJETPublicationsUK.Allrightsreserved.1717AnInverseKinematicAnalysisofaRoboticSealerAkinolaA.Adeniyi1,AbubakarMohammed2,AladeniyiKehinde31DepartmentofMechanicalEngineering,UniversityofIlorin,Ilorin,Nigeria2DepartmentofMechanicalEngineering,FederalUniversityofTechnology,Minna,Nigeria3DepartmentofScienceLaboratoryTechnology,RufusGiwaPolytechnic,Owo,NigeriaABSTRACTAplanarroboticsealingorbrandstampingmachineispresentedforanautomatedfactoryline.Theappropriatetimetosealortostampanobjectisbasicallydeterminedbyamotorcontrollerwhichreliescriticallyonwhetherornottheobjectisinthebestposition.Theextentofprotractionandretractionofthepistonheadislargelydictatedbyaninfraredsensor.Giventheextenttoprotractorretractthepistonhead,theangulardisplacementsofthelinkrequiredaredeterminedusingtheInverseKinematicIKtechniques.Theinertiaandgravityeffectsofthelinkshavebeenignoredtoreducethecomplexityoftheequationsandtodemonstratethetechnique.KeywordsForwardKinematics,InverseKinematics,Robotics,Sealer.1.INTRODUCTIONAnautomatedfactoryusesanumberofmechanicallinkselectronicallycontrolledtoachievetasks.Thebenefitsoffactoryautomationaremanyandofstrategicimportancetomanagement1.Standardmechanicallinksareusuallypoweredwithelectricalmotors,pneumaticsystemsorsolenoids.Inamanuallyoperatedmachine,thehumanperformsvisualchecksandotherstandardchecksthataretobereplicatedbyautomation.Theinterestofthisworkiscenteredonahypotheticalsealingmachinewhichisusedforstampingsomesignaturesandlogosasdoneinabrandingfactoryline.Inversekinematicanalysisisappliedtoenableusdetermineangulardisplacementsofthelink.Kinematicsinvolvesthestudyofmotionwithoutconsiderationfortheactuatingforces.InverseKinematicsIKisamethodfordeterminingthejointanglesanddesiredpositionoftheendeffectorsgivenadesiredgoaltoreachbytheendeffectors1.AfeasibilityofusingaPIDcontrollerwasstudiedbyNagchaudhuri2foraslidercrankmechanismbutwithoutanoffset.Tolanietal3reviewedandgroupedthetechniquesofsolvinginversekinematicsproblemsintoseven.ThetechniquesaretheNewtonRaphsonsmethodanditsothervariants.TherearetheJacobianandthevariantswithpseudoinverseotherwiseknownastheMoorePenroseinverseforsquareornonsquareJacobian.Othermethodsarethecontroltheorybasedandtheoptimisationtechniques.Anumberofauthors1,47haveproposedalgorithmsforsolvingIKproblemswhichincludebutnotlimitedtoNeuralNetworkalgorithm,CyclicCoordinateDescentclosureandInexactstrategy,butlikeeveryothertechniquesforagivenproblemthechoiceofmethoddependsonthespecificsoftheproblem.Buss8discussedtheJacobiantranspose,theMoorePenroseandtheDampedLeastSquarestechniques.Intermsofcomputationalcost,theJacobiantransposemethodisthecheapbutcanperformpoorlybasedontherobotconfigurations.InthisworktheJacobiantransposetechniqueillperformedbuttheJacobianInversetechniqueissuitableandmoresoitisasimple2Dplanarrepresentationoftheproblemwithonly4degreesoffreedom.2.OPERATIONSOFTHEROBOTICLINKFig.1showstheschematicdiagramoftheroboticsealingsystem.Thecappingorstampingisachievedwiththepistonorramhead,P.Cistheconveyorline.Thecapsorthebrandingheadsareplacedinpositionandsensedbyaninfraredsensor,S.Theinstructiontosealorbrandisdependentonfeedbackfromthesensor.Iftheitemtobebranded,cappedorstampedisoutofplaceattheinstancewhentheramheadwasgoingtotouch,thesensorfeedbackwillbetoretractthehead.Itcanalsobetonotgotoofar.Therecanbearangeoffeedbacktothemotorcontroller,M.Thiskindofcontrolsystemissimilartowhatahumanoperatorwoulddoifitweremanuallyoperated.Theuseofsensorsandfastrespondingmotorcontrollerwillmakethishypotheticalmachineaveryusefultoolinafactoryperformingthiskindofmundanetask.ThisfactorysublineisasimpleslidercrankmechanismwithactuatorarmA.Inclearerterms,theinstructionswouldbetopressthepistonramtosealifthecapandthecontainerareinlinetoreversethepistonincaseofajamtonotpressthepistonramifeitherthecontainerorthecapisabsenttoInternationalJournalofEngineeringandTechnologyIJET–Volume2No.10,October,2012ISSN20493444©2012–IJETPublicationsUK.Allrightsreserved.1718pressfurtheriftheseallengthisshorterthanexpectedasmaybecausedbywearandtear.Thisclearlyshowsthatthepistondeterminestheangleofthelinkorthedirectionoractionofthemotor.Thisisaninversekinematicsproblem.Thesensorfeedbackpartismuchofacontrolengineeringproblem,notconsideredinthispaper.Fig.1Theroboticsealingrigschematic3.ANALYSISFig.2isarepresentationoftheslidercrankmechanism.Thereisanoffset,f,ofthepistonaxisfromthemotoraxis,O1.O2istheaxisofthepistonwithmovingcoordinatesx,y.ThemotorrotatesclockwiseorcounterclockwiseaboutO1.IfthecrankmakesdisplacementΔsonthepistonplane,itisequivalenttoamotionofΔexandΔey.Thismotioniscausedbythecrankmakinganangularmotionclockwiseorcounterclockwise,Δ.Theanglebetweentheconnectingrodandcrankmakesanangulardisplacementof,Δ.ThisalsomeanstheangularshiftofΔismadebetweentheconnectingrodandthepistonorramplane.Fig.2TheoffsetslidercrankCartesiancoordinateworldInacomputergameapplicationforthese,theangleswouldbeexplicitlyrequiredsothatthelinksdonotphysicallydisjointforaphysicallyconnectedlink,themotorcontrolleronlywouldneedtheinstructiontomoveonlythecrank.3.1TheWorldCartesiancoordinatesystemisadopted.Clockwiseispositiveandmotiontorightandupwardsarepositive.TheTopDeadCentreTDCisattainedwhenthecrank,radiusr,andtheconnectingrod,lengthl,areinline.Thisisattainedwhen.fmisthemaximumvariableoffsetbasedonthegeometry.TheBottomDeadCentreBDCisreachedwhen.TheTDCandBDCwiththevariableoffsetareshowninFig.3.Fig.3TheTopandBottomDeadcentreThepistonhasbeenconstrainedtomoveonlyinplanardirection,onthevectorof.Inthiswork,thedirectionvectoris,makingtheplaneat45otothehorizontal.3.2TheForwardKinematicsThedisplacementcausedbythemotormovingclockwisefromthepositioninFig.2isrepresentedinequation1.Wheresubscriptsi,farerespectivelymeaninitialandfinalvalues.Thepositionatfisreachedinrealitysmoothlyforarotatingcrank,butthesmoothnesscanbereachedinfineincrementalsteps,inthenumericalapproach.Attheendofthesteppedincrements,thefinaldisplacementtothegoalisseenasafunctionofangularparametersgivenas1Thelineardependenceoftheangles,inthisproblem,canhelptoreducethenumberofdegreesoffreedomtocomputeinequation1.Itcanbeshownthat,therebymaking.Usingtrigonometry,theinstantaneousinitial,arbitrary,positionofthepistoninFig.2isgivenbyEquation2.23TheJacobianmatrixforisgiveninequation4andsimplifiedtoequation5.InternationalJournalofEngineeringandTechnologyIJET–Volume2No.10,October,2012ISSN20493444©2012–IJETPublicationsUK.Allrightsreserved.1719J4J5Computingthenewpistonpositioninvolvessolvingequation1.ThenewcoordinateofthepistonbythefirsttermofexpansionoftheTaylorseriescanbeshowntobegiveninequation6.isthevectoroftherobotangulardisplacementsfortherelatedlinks.Mathematically,.Here,wehave.Thereforethecurrentpositionofthepistonorthepressingheadisapproximatelygiveninequation6.Itshouldbenotedthatcanbemeasuredfromthehorizontaltofurtherreducetheequationsets,thisisreferredtoaselsewhereinthispaper.J63.3InverseKinematicsTheproblemisnotthatofsolvingforXfgivenXiandbutitisthatofsolvingforgivenXi,andthedesiredXf.Thisisiterativelyimplementedsuchthatthetargetdisplacementofthepistonisgivenas.Thisisavectorofthepistondisplacementandcanberepresentedas.Sincethisisaplanarproblemwithnodisplacementsintheotherdirections,itreducestoa.Tosmoothenthepossiblejerkorjumpyeffect,thiscanbesteppedusingafactorofwhichcanbeselectedintuitivelybasedontheratioofrtoLbutandJistheinverseofJacobianmatrix.Thealgorithmchecksifthetargethasbeenreachedornot.Iterationisstoppedwhenthesolutioniswithinapredeterminedleveloferrororamaximumnumberofiterations.Thechoiceoftheselimitingvaluesshoulddependontheresponsetimeacceptable.Thiscanbecriticalforarealtimeapplication.J74.RESULTANDDISCUSSIONSConsideracurrentorientationoftheroboticarmatanyarbitrarypositionwiththepistonheadatapositionP1.SupposethesensorsystemrequiresthepistontomovetoatargetnewpositionP2.Thesimulationisdoneforseveralarbitrarystartingpositionsofthecrankandresultsaresimilarforreachabletargets.Supposingthecrankangleisatacurrentorientationwithcrankangleof5o,andthereisaninstructionfromthesensortoretractthepistonramheadby0.1timesthecrankarmlength.Thesimulationinstructsthecrankproceedstocounterclockwiseby15.58o,thiscorrespondstoanincreaseofto19.26oandcorrespondingly,reducesto86.32o.Fig.4showsthesimulationprogressofthepistonheadfromacurrentpositionP1tothenewtargetP2andthenumberofiterationsdone.Fig.4CrankPositionandIterationwiththeJacobianInverseMatrixThetechniqueusedistheJacobianinversetechnique.TheJacobiantransposetechniqueisnotpredictableforthesameproblemandinthiscase,thesolutionsettlestoalocalminimumforonlyoneoftheanglesbuttheconvergencerateisfaster,seeFig.5.Fig.5CrankPositionsusingtheInverseandTransposeoftheJacobianMatrixIfthereisarequesttoaphysicallyunreachabletarget,suchastoamorethantheTDCorBDClocations,P3,thesimulationrunsandstopsafterthemaximumnumberofiterationsoriftheJacobianMatrixbecomesuninvertible,Fig.6.010203040506070809010020100CrankAnglePercenttoTargetCrankPositions0500010000NumberofIterationsInternationalJournalofEngineeringandTechnologyIJET–Volume2No.10,October,2012ISSN20493444©2012–IJETPublicationsUK.Allrightsreserved.1720Fig.6UnreachableTargetsituation5.CONCLUSIONThispaperisfocusedontheapplicationoftheInverseKinematicstechniquetotheanalysisofaroboticlink,suchasobtainedinasealerofanautomatedfactory,withoutconsiderationfortheeffectsofinertiaeffects.TheJacobianinversetechnique,asmentionedinliteratures,ismorereliableinthisapplication.TheJacobiantransposeapproachisnotreliable.Thispaperhasdemonstratedtheapplicationoftheinversekinematicstoasimpleroboticsealerthepistonisinstructedtoretractby0.1unitsasatestcase.ThenewcrankanglewasfoundmoreaccuratelywiththeJacobianInversetechniquebetterthattheJacobianTransposetechnique.Theproblemcanbeextendedtoincludethedynamicsforpossibleselectionoftheoptimaldrivingtorqueorelectricmotorselectionforthedrivingparts.REFERENCES1S.TejomurtulaandS.Kak,InverseKinematicsinroboticsusingneuralnetworks,InformationSciences,vol.116,pp.147164,1999.2A.Nagchaudhuri,MechantronicRedesignofSliderCrankMechanism,inASMEInternationalMechnicalEngineeringCongressExpositionIMECE2002,NewOrleans,Louisiana,2002.3D.Tolani,A.Goswami,andN.I.Badler,RealTimeInverseKinematicsTechniquesforAnthromorphicLimbs,GraphicalModels,vol.62,pp.353388,2000.4S.K.SahaandW.O.Schiehlen,RecursiveKinematicsandDynamicsforParallelStructuredClosedLoopmultibodySystems,MechanicsofStructuresandMachines,vol.29,pp.143175,2007.5X.Wang,Abehaviorbasedinversekinematicsalgorithmtopredictarmprehensionposturesforcomputeraidedergonomicevaluation,JournalofBiomechanics,vol.32,pp.453460,1999.6A.C.Nearchou,Solvingtheinversekinematicsproblemofredundantrobotsoperatingincomplexenvironmentsviaamodifiedgeneticalgorithm,MechanismandMachineTheory,vol.33,pp.273292,1998.7M.J.D.Powell,SomeGlobalConvergencePropertiesofavariablemetricAlgorithmforMinimizationwithoutExactlinesearches,inSymposiuminAppliedMathematicsoftheAmericanMathematicalSocietyandtheSocietyforIndustrialandAppliedMathematics,NewYorkCity,1976.8S.R.Buss,IntroductiontoInverseKinematicswithJocobianTranspose,PseudoinverseandDampedLeastSquaremethods,UniversityofCalifornia,SanDiego2009.010020030040050060070080090010001000100CrankAnglePercenttoTargetCrankPositions05001000NumberofIterations
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