外文资料--封口机器人.pdf

InternationalJournalofEngineeringandTechnologyVolume2No.10,October,2012ISSN:2049-34442012IJETPublicationsUK.Allrightsreserved.1717AnInverseKinematicAnalysisofaRoboticSealerAkinolaA.Adeniyi1,AbubakarMohammed2,AladeniyiKehinde31DepartmentofMechanicalEngineering,UniversityofIlorin,Ilorin,Nigeria2DepartmentofMechanicalEngineering,FederalUniversityofTechnology,Minna,Nigeria3DepartmentofScienceLaboratoryTechnology,RufusGiwaPolytechnic,Owo,NigeriaABSTRACTAplanarroboticsealingorbrandstampingmachineispresentedforanautomatedfactoryline.Theappropriatetimetosealortostampanobjectisbasicallydeterminedbyamotorcontrollerwhichreliescriticallyonwhetherornottheobjectisinthebestposition.Theextentofprotractionandretractionofthepistonheadislargelydictatedbyaninfraredsensor.Giventheextenttoprotractorretractthepistonhead,theangulardisplacementsofthelinkrequiredaredeterminedusingtheInverseKinematic(IK)techniques.Theinertiaandgravityeffectsofthelinkshavebeenignoredtoreducethecomplexityoftheequationsandtodemonstratethetechnique.Keywords:ForwardKinematics,InverseKinematics,Robotics,Sealer.1.INTRODUCTIONAnautomatedfactoryusesanumberofmechanicallinkselectronicallycontrolledtoachievetasks.Thebenefitsoffactoryautomationaremanyandofstrategicimportancetomanagement1.Standardmechanicallinksareusuallypoweredwithelectricalmotors,pneumaticsystemsorsolenoids.Inamanuallyoperatedmachine,thehumanperformsvisualchecksandotherstandardchecksthataretobereplicatedbyautomation.Theinterestofthisworkiscenteredonahypotheticalsealingmachinewhichisusedforstampingsomesignaturesandlogosasdoneinabrandingfactoryline.Inversekinematicanalysisisappliedtoenableusdetermineangulardisplacementsofthelink.Kinematicsinvolvesthestudyofmotionwithoutconsiderationfortheactuatingforces.InverseKinematics(IK)isamethodfordeterminingthejointanglesanddesiredpositionoftheend-effectorsgivenadesiredgoaltoreachbytheendeffectors1.AfeasibilityofusingaPIDcontrollerwasstudiedbyNagchaudhuri2foraslidercrankmechanismbutwithoutanoffset.Tolanietal3reviewedandgroupedthetechniquesofsolvinginversekinematicsproblemsintoseven.ThetechniquesaretheNewton-Raphsonsmethodanditsothervariants.TherearetheJacobianandthevariantswithpseudo-inverse(otherwiseknownastheMoore-Penroseinverse)forsquareornon-squareJacobian.Othermethodsarethecontrol-theorybasedandtheoptimisationtechniques.Anumberofauthors1,4-7haveproposedalgorithmsforsolvingIKproblemswhichincludebutnotlimitedtoNeuralNetworkalgorithm,CyclicCoordinateDescentclosureandInexactstrategy,butlikeeveryothertechniquesforagivenproblemthechoiceofmethoddependsonthespecificsoftheproblem.Buss8discussedtheJacobiantranspose,theMoore-PenroseandtheDampedLeastSquarestechniques.Intermsofcomputationalcost,theJacobiantransposemethodisthecheapbutcanperformpoorlybasedontherobotconfigurations.InthisworktheJacobiantransposetechniqueill-performedbuttheJacobianInversetechniqueissuitableandmoresoitisasimple2Dplanarrepresentationoftheproblemwithonly4degreesoffreedom.2.OPERATIONSOFTHEROBOTICLINKFig.1showstheschematicdiagramoftheroboticsealingsystem.Thecappingorstampingisachievedwiththepistonorramhead,P.Cistheconveyorline.Thecapsorthebrandingheadsareplacedinpositionandsensedbyaninfraredsensor,S.Theinstructiontosealorbrandisdependentonfeedbackfromthesensor.Iftheitemtobebranded,cappedorstampedisoutofplaceattheinstancewhentheramheadwasgoingtotouch,thesensorfeedbackwillbetoretractthehead.Itcanalsobetonotgotoofar.Therecanbearangeoffeedbacktothemotorcontroller,M.Thiskindofcontrolsystemissimilartowhatahumanoperatorwoulddoifitweremanuallyoperated.Theuseofsensorsandfastrespondingmotorcontrollerwillmakethishypotheticalmachineaveryusefultoolinafactoryperformingthiskindofmundanetask.Thisfactorysub-lineisasimpleslider-crankmechanismwithactuatorarmA.Inclearerterms,theinstructionswouldbetopressthepistonramtosealifthecapandthecontainerareinline；toreversethepistonincaseofajam；tonotpressthepistonramifeitherthecontainerorthecapisabsent；toInternationalJournalofEngineeringandTechnology(IJET)Volume2No.10,October,2012ISSN:2049-34442012IJETPublicationsUK.Allrightsreserved.1718pressfurtheriftheseallengthisshorterthanexpectedasmaybecausedbywearandtear.Thisclearlyshowsthatthepistondeterminestheangleofthelinkorthedirectionoractionofthemotor.Thisisaninversekinematicsproblem.Thesensorfeedbackpartismuchofacontrolengineeringproblem,notconsideredinthispaper.Fig.1:Theroboticsealingrigschematic3.ANALYSISFig.2isarepresentationoftheslider-crankmechanism.Thereisanoffset,f,ofthepistonaxisfromthemotoraxis,O1.O2istheaxisofthepistonwithmovingcoordinates(x,y).ThemotorrotatesclockwiseorcounterclockwiseaboutO1.Ifthecrankmakesdisplacementsonthepistonplane,itisequivalenttoamotionofexandey.Thismotioniscausedbythecrankmakinganangularmotionclockwiseorcounter-clockwise,.Theanglebetweentheconnectingrodandcrankmakesanangulardisplacementof,.Thisalsomeanstheangularshiftofismadebetweentheconnectingrodandthepistonorramplane.Fig.2:Theoffsetslidercrank(Cartesiancoordinateworld)Inacomputergameapplicationforthese,theangleswouldbeexplicitlyrequiredsothatthelinksdonot“physicallydisjoint”；foraphysicallyconnectedlink,themotorcontrolleronlywouldneedtheinstructiontomoveonlythecrank.3.1TheWorldCartesiancoordinatesystemisadopted.Clockwiseispositiveandmotiontorightandupwardsarepositive.TheTopDeadCentre(TDC)isattainedwhenthecrank,radiusr,andtheconnectingrod,lengthl,areinline.Thisisattainedwhen.fmisthemaximumvariableoffsetbasedonthegeometry.TheBottomDeadCentre(BDC)isreachedwhen.TheTDCandBDCwiththevariableoffsetareshowninFig.3.Fig.3:TheTopandBottomDeadcentreThepistonhasbeenconstrainedtomoveonlyinplanardirection,onthevectorof.Inthiswork,thedirectionvectoris,makingtheplaneat45otothehorizontal.3.2TheForwardKinematicsThedisplacementcausedbythemotormovingclockwisefromthepositioninFig.2isrepresentedinequation(1).Wheresubscripts(i,f)arerespectivelymeaninitialandfinalvalues.Thepositionatfisreachedinrealitysmoothlyforarotatingcrank,butthesmoothnesscanbereachedinfineincrementalsteps,inthenumericalapproach.Attheendofthesteppedincrements,thefinaldisplacementtothegoalisseenasafunctionofangularparametersgivenas:(1)Thelineardependenceoftheangles,inthisproblem,canhelptoreducethenumberofdegreesoffreedomtocomputeinequation(1).Itcanbeshownthat,therebymaking.Usingtrigonometry,theinstantaneousinitial,arbitrary,positionofthepistoninFig.2isgivenbyEquation(2).(2)(3)TheJacobianmatrixforisgiveninequation(4)andsimplifiedtoequation(5).InternationalJournalofEngineeringandTechnology(IJET)Volume2No.10,October,2012ISSN:2049-34442012IJETPublicationsUK.Allrightsreserved.1719J(4)J(5)Computingthenewpistonpositioninvolvessolvingequation(1).ThenewcoordinateofthepistonbythefirsttermofexpansionoftheTaylorseriescanbeshowntobegiveninequation(6).isthevectoroftherobotangulardisplacementsfortherelatedlinks.Mathematically,.Here,wehave.Thereforethecurrentpositionofthepistonorthepressingheadisapproximatelygiveninequation(6).Itshouldbenotedthatcanbemeasuredfromthehorizontaltofurtherreducetheequationsets,thisisreferredtoaselsewhereinthispaper.J(6)3.3InverseKinematicsTheproblemisnotthatofsolvingforXfgivenXiandbutitisthatofsolvingforgivenXi,andthedesiredXf.Thisisiterativelyimplementedsuchthatthetargetdisplacementofthepistonisgivenas.Thisisavectorofthepistondisplacementandcanberepresentedas.Sincethisisaplanarproblemwithnodisplacementsintheotherdirections,itreducestoa.Tosmoothenthepossiblejerkorjumpyeffect,thiscanbesteppedusingafactorofwhichcanbeselectedintuitivelybasedontheratioofrtoLbutandJistheinverseofJacobianmatrix.Thealgorithmchecksifthetargethasbeenreachedornot.Iterationisstoppedwhenthesolutioniswithinapre-determinedleveloferrororamaximumnumberofiterations.Thechoiceoftheselimitingvaluesshoulddependontheresponsetimeacceptable.Thiscanbecriticalforarealtimeapplication.J(7)4.RESULTANDDISCUSSIONSConsideracurrentorientationoftheroboticarmatanyarbitrarypositionwiththepistonheadatapositionP1.SupposethesensorsystemrequiresthepistontomovetoatargetnewpositionP2.Thesimulationisdoneforseveralarbitrarystartingpositionsofthecrankandresultsaresimilarforreachabletargets.Supposingthecrankangleisatacurrentorientationwithcrankangleof-5o,andthereisaninstructionfromthesensortoretractthepistonramheadby0.1timesthecrankarmlength.Thesimulationinstructsthecrankproceedstocounterclockwiseby15.58o,thiscorrespondstoanincreaseofto19.26oandcorrespondingly,reducesto86.32o.Fig.4showsthesimulationprogressofthepistonheadfromacurrentpositionP1tothenewtargetP2andthenumberofiterationsdone.Fig.4:CrankPositionandIterationwiththeJacobianInverseMatrixThetechniqueusedistheJacobianinversetechnique.TheJacobiantransposetechniqueisnotpredictableforthesameproblemandinthiscase,thesolutionsettlestoalocalminimumforonlyoneoftheanglesbuttheconvergencerateisfaster,seeFig.5.Fig.5:CrankPositionsusingtheInverseandTransposeoftheJacobianMatrixIfthereisarequesttoaphysicallyunreachabletarget,suchastoamorethantheTDCorBDClocations,P3,thesimulationrunsandstopsafterthemaximumnumberofiterationsoriftheJacobianMatrixbecomesun-invertible,Fig.6.0102030405060708090100-20-100CrankAnglePercenttoTargetCrankPositions0500010000NumberofIterationsInternationalJournalofEngineeringandTechnology(IJET)Volume2No.10,October,2012ISSN:2049-34442012IJETPublicationsUK.Allrightsreserved.1720Fig.6:UnreachableTargetsituation5.CONCLUSIONThispaperisfocusedontheapplicationoftheInverseKinematicstechniquetotheanalysisofaroboticlink,suchasobtainedinasealerofanautomatedfactory,withoutconsiderationfortheeffectsofinertiaeffects.TheJacobianinversetechnique,asmentionedinliteratures,ismorereliableinthisapplication.TheJacobiantransposeapproachisnotreliable.Thispaperhasdemonstratedtheapplicationoftheinversekinematicstoasimpleroboticsealer；thepistonisinstructedtoretractby0.1unitsasatestcase.ThenewcrankanglewasfoundmoreaccuratelywiththeJacobianInversetechniquebetterthattheJacobianTransposetechnique.Theproblemcanbeextendedtoincludethedynamicsforpossibleselectionoftheoptimaldrivingtorqueorelectricmotorselectionforthedrivingparts.REFERENCES1S.TejomurtulaandS.Kak,InverseKinematicsinroboticsusingneuralnetworks,InformationSciences,vol.116,pp.147-164,1999.2A.Nagchaudhuri,MechantronicRedesignofSliderCrankMechanism,inASMEInternationalMechnicalEngineeringCongress&Exposition:IMECE2002,NewOrl