Ho-HoonLeeDepartmentofMechanicalEngineering,UniversityofSuwon,SuwcnP.0.Box77,Seoul440-600,KoreaModelingandControlofaThree-DimensionalOverheadCraneInthispaper,anewdynamicmodelofathree-dimensionaloverheadcraneisderivedbasedonanewlydefinedtwo-degree-of-freedomswingangle.Thedynamicmodeldescribesthesimultaneoustraveling,traversing,andhoistingmotionsofthecraneandtheresultingloadswing.Foranti-swingcontrol,thispaperproposesadecoupledcontrolschemebasedonthedynamicmodellinearizedaroundthestableequilibrium.Thedecoupledschemeguaranteesnotonlyrapiddampingofloadswingbutalsoaccuratecontrolofcranepositionandloadhoistingforthepracticalcaseofsimultaneoustraveling,traversing,andslowhoistingmotions,whichisalsoprovenbyexperiments.1IntroductionOverheadcranesarewidelyusedinindustryfortransportationofheavyloads.However,thecraneacceleration,requiredformotion,alwaysinducesundesirableloadswing.Largeraccelerationusuallyinduceslargerloadswing.Loadhoistingalsotendstoaggravatetheloadswing.Thisunavoidableloadswingfrequentlycausesefficiencydrop,loaddamages,andevenaccidents.Forsafety,overheadcranesareusuallysetinmotionwiththeirloadshoistedhigherthananypossibleobstaclesandthelengthofhoistingropesiskeptconstantorslowlyvaryingwhilethecranesareinmotion.Variousattemptshavebeenmadetocontroltheloadswing.Thenumberofthecontrolinputsforacranesystemisbynaturelessthanthatofthesystemoutputs,whichmakestherelatedcontrolproblemscomplicated.Thecranecontrolconsistsofcranemotioncontrol,loadhoistingcontrol,andloadswingsuppression.MitaandKanai(1979)solvedaminimum-timecontrolproblemforswing-freevelocityprofilesofacraneundertheconstraintofzeroloadswingatthestartandendofacceleration.Ohnishietal.(1981)proposedananti-swingcontrolbasedontheswingdynamicsoftheload.Starr(1985)proposedanopen-loopcontrolalgorithm,whichrequirestheconstraintofzeroinitialloadswing.Ridout(1987)designedafeedbackcontrollawusingtherootlocusmethod.Yuetal.(1995)proposedanonlinearcontrolbasedonasingularperturbationmethod,whichisvalidonlyiftheloadmassismuchlargerthanthecranemass.MoustafaandAbou-El-Yazid(1996)discussedthestabilityofacranecontrolsystemforhoistingmotionsoftheload.Leeetal.(1997)proposedananti-swingcontrollawthatguaranteedbothrapiddampingofloadswingandaccuratecontrolofcraneposition.Alltheaboveresearchershavefocusedonthecontroloftwo-dimensionaloverheadcranesthatallowonlythetravelingandhoistingmotions；however,inmostfactoriesandwarehousesthree-dimensionaloverheadcranesarenormallyused.MoustafaandEbeid(1988)derivedadynamicmodelofathree-dimensionaloverheadcranebasedonthesphericalcoordinates(Meir-ovitch,1970andGreenwood,1988),andthendesignedatrajectory-dependentcontrolbasedonthedynamicmodellinearizedalongadesiredtrajectory.Theirlinearizedmodeliscoupledanditsparametersaredependentoncranetrajectories；thiscom-pUcatestherelatedcontroldesignsandapplications.Theircon-ContributedbytheDynamicSystemsandControlDivisionforpublicationintheJOURNALOFDYNAMICSYSTEMS,MEASUREMENT,ANDCONTROL.ManuscriptreceivedbytheDynamicSystemsandControlDivisionDecember2,1996.AssociateTechnicalEditor:R.Redfield.troladdressesonlythesuppressionofloadswing；consequently,theircontrolresultsinconsiderablepositionerrorsincranemotionandloadhoisting.Thispaperpresentsapracticalsolutiontothemodelingandcontrolofthree-dimensionaloverheadcranes,wheretheloadswing,cranemotion,andloadhoistingareconsideredalltogetherinthemodelingandcontrol.First,anewtwo-degree-of-freedomswingangleisdefinedassociatedwiththetravelandtraverseaxesofathree-dimensionaloverheadcrane.Thenanewnonlineardynamicmodelforthecraneisderivedbasedonthenewswingangledefinition.Thenewdynamicmodelisequivalenttothatofathree-linkflexiblerobothavingthefirstflexiblemode.Next,thenewdynamicmodelislinearizedaroundtheverticalstableequilibrium.Thentheresultingdynamicmodelisdecoupledandsymmetricwithrespecttothetravelingandtraversingmotionsofthecraneandthemodelparametersareindependentofcranetrajectory,whichsignificantlysimplifiesthecontrolproblems.Withthisresult,thispaperproposesanewdecoupledanti-swingcontrolschemethatguaranteesaccuratecontrolofcranepositionandloadhoistingaswellasrapiddampingofloadswingforthepracticalcaseofsimultaneoustraveling,traversing,andslowhoistingmotions.Theremainderofthispaperisorganizedasfollows.InSection2,anonlineardynamicmodelisderivedbasedonanewtwo-degree-of-freedomswingangle,forathree-dimensionaloverheadcrane.InSection3,thenonlineardynamicmodelislinearized,andthenanewdecoupledanti-swingcontrolschemeisdesignedusingtheloopshaping,rootlocus,andgainschedulingmethods.InSection4,thedecoupledcontrolschemeisappliedtoathree-dimensionalprototypeoverheadcraneforperformanceevaluation.InSection5,theconclusionsaredrawnforthisstudy.2ModelingofaThree-DimensionalOverheadCrane2.1DefinitionofGeneralizedCoordinates.Figure1showsthecoordinatesystemsofathree-dimensionaloverheadcraneanditsload.XYZisthefixedcoordinatesystemandXjYjZristhetrolleycoordinatesystemwhichmoveswiththetrolley.Theoriginofthetrolleycoordinatesystemis(x,y,0)inthefixedcoordinatesystem.Eachaxisofthetrolleycoordinatesystemisparalleltothecounterpartofthefixedcoordinatesystem.Yrisdefinedalongthegirderwhichisnotshowninthefigure.ThetrolleymovesonthegirderintheYj(traverse)directionandthegirderandYjaxismoveintheXT(travel)direction,distheswingangleoftheloadinanarbitrarydirectioninspaceandhastwocomponents:9andOy,where9isJournalofDynamicSystems,Measurement,andControlCopyright1998byASMEDECEMBER1998,Vol.120/471Downloaded18Mar2009to87.RedistributionsubjecttoASMElicenseorcopyright；see/terms/Terms_Use.cfmLoadFig.1Coordinatesystemsofathree-dimensionaloverheadcranetheswingangleprojectedontheXrZrplaneand9yistheswinganglemeasuredfromtheXTZTplane.Thepositionoftheload(Xm,ymZm)inthefixedcoordinatesystemisgivenbyx=X+Isin6cos6y,y,=y+lsinOy,Zm=-ICOS0cos9y,(1)(2)(3)where/denotestheropelength.Thepurposeofthisstudyistocontrolthemotionofboththecraneanditsload.Hencex,y,I,9,and9yaredefinedasthegeneralizedcoordinatestodescribethemotion.2.2DynamicModelofaThree-DimensionalOverheadCrane.Inthissection,theequationsofmotionofacranesystemarederivedusingLagrangesequation(Meirovitch,1970).Inthisstudy,theloadisconsideredasapointmass.Themassandstiffnessoftheropearealsoneglected.K,thekineticenergyofthecraneanditsload,andP,thepotentialenergyoftheload,aregivenas1/?K=-(M,x+Myf+M,l)+-vl,(4)P=mglcos9xcos6y),(5)whereM,My,andMiarethex(traveling),y(traversing),and/(hoistingdown)componentsofthecranemassandtheequivalentmassesoftherotatingpartssuchasmotorsandtheirdrivetrains,respectively；m,g,andudenotetheloadmass,thegravitationalacceleration,andtheloadspeed,respectively；Im=xn+yl,+Zm)ISobtainedasvl=x+f+t+/cos9y9l+/ej+2(sin9,cos9yi+/cos9cos9ydIsin9sin9y9y)x+2ism9yi+lcos9y9y)y.(6)LagrangianLandRayleighsdissipationfunctionFaredefinedas1*ttiL=-M,x+Myf+M,P)+vi+mgl(cos9cosy-1)(7)F=(D,x+Dyf+D,h,(8)where),D,andD,denotetheviscousdampingcoefficientsassociatedwiththex,y,and/motions,respectively.TheequationsofmotionofthecranesystemareobtainedbyinsertingLandFintoLagrangesequationsassociatedwiththegeneralizedcoordinatesx,9,y,6y,and/,respectively:(M,-I-m)x+mlcos9cos9y9:,mlsin9sin9y9y+msin9cos9yl+Dx+2mcos9cos9yi92msin9sin9yWy-mlsin9cos9ydl-2mlcos9Jsin9y9jy-mlsincos9y9y=/,(9)mPcos9y9x+mlcos9cos9yX+2mlcos9yi6x2mlsin9ycos9y9；t9y+mglsin9cos9y=0,(10)My+m)y+mlcos9y9y+msin9yi+Dyy+2OTcos9yi9y-mlsin9y9y=fy,(11)m/y+mlcosyymlsin0:sin5j,jfc+2mll9y+mlcos9ysin+mg/cos6sin5=0,(12)Ml+m)l+msin9cosA+msiny+A-mlcosimlbmgcos9cos9y=fi,(13)where,/j,andarethedrivingforcesforthex,y,and/motions,respectively.2.3RemarksontheDynamicModel.Thedynamicmodelforathree-dimensionaloverheadcranehasthefollowingfeaturesthankstothecharacteristicsoftheproposedswingangledescription.Wheny=y=9y=9y=9y=0,thedynamicmodelofathree-dimensionaloverheadcraneisreducedtothatofatwo-dimensionaloverheadcrane(Leeetal.,1997)movingalongtheXaxis.ThesameistruefortheYaxiswhenx=x=9=9.=h=0.Thedynamicmodelisequivalenttothatofathree-linkflexiblerobothavingthefirstflexiblemode(LucaandSiciliano,1991).Thatis,thedynamicmodel(9)-(13)canberepresentedbythefollowingmatrix-vectorform:M(q)q-I-Z)q-FC(0and9y7r/2；the5X5CoriolisandcentrifugalforcematrixC(q,q)thatsatisfiesM(q)-2C(q,q)=-(M(q)-2C(,q)canbefoundfromtheqandqterms.3DesignofaControlLawInthissection,anewanti-swingcontrolschemewillbeproposed.First,thenonlineardynamicmodelwillbelinearized.Second,anewdecoupledanti-swingcontrollawwillbedesignedforthecaseofconstantropelength.Third,anindependentropelengthcontrollerwillbedesignedandagainschedulingmethodwillbeadopted,forslowlyvaryingropelength.Finally,thestabilityoftheresultingcontrolsystemwillbeanalyzedforslowlyvaryingropelength.3.1LinearizationoftheDynamicModel.Inpractice,themaximumaccelerationofoverheadcranesismuchsmallerthanthegravitationalacceleration,andtheropelengthiskeptconstantorslowlyvaryingwhilethecranesareinmotion.Thisstudyconsidersthesepracticalcases:|ji；ig,yg,l472/Vol.120,DECEMBER1998TransactionsoftheASMEDownloaded18Mar2009to87.RedistributionsubjecttoASMElicenseorcopyright；see/terms/Terms_Use.cfmg,and|/|/|,whichimply|/6Jg,ie,4g,andhencesmallswing(|(9J1and|6,|?1).Then|(9,|1anddy1arealsovalid.Forsmallswing,sin99,sin6y=By,cos9,i,andcos=1.Inthiscase,withthetrigonometricfunctionsapproximated,thehighordertermsinthenonlinearmodelcanbeneglected.Thenthenonlinearmodel(9)-(13)issimplifiedtothefollowinglinearizedmodel:(M,.+m)x+D,x+ml9,-19,+x+gO,=0,My+m)y+Dyy+mWyWy+y+gOy=0,=/；,=/v.(15)(16)(17)(18)Ml+m)l+D,l-ing=/,.(19)Thislinearizeddynamicmodelconsistsofthetraveldynamics(15)and(16),thetraversedynamics(17)and(18),andtheindependentloadhoistingdynamics(19).Thetravelandtraversedynamicsaredecoupledandsymmetric,whichmeansthatthecontrolofathree-dimensionaloverheadcraneistransformedintothatoftwoindependenttwo-dimensionaloverheadcraneshavingthesameloadhoistingdynamics.Inthisstudy,ananti-swingcontrollawwillbedesignedbasedonthetraveldynamicsandwillbeusedforthecontrolofbothtravelingandtraversingmotions,andaropelengthcontrollawwillbedesignedbasedontheloadhoistingdynamics(19).3.2Anti-SwingControlforConstantRopeLength.Inthissection,anewcontrollerdesignmethodfortwo-dimensionaloverheadcranesisproposedbasedonthelinearizedmodelusingtheloopshapingandrootlocusmethods.Theproposedmethodisfreefromtheusualconstraintsontheloadmass(Ridout,1987andYuetal.,1995).3.2.1DesignofaVelocityServoSystem.Inpractice,thedrivingforcef,foracraneisusuallygeneratedbyelectricmotorscontrolledbytorqueservocontrollers,whosedynamicscanbeneglectedsincetheyareusuallyahundredtimesfasterthanthetrolleyandgirderdynamics.Henceinpracticalcases,fj,isproportionaltou,theinputtothetorqueservocontroller:L=K,u(20)whereK,isthecrane-dependentconstant.Thenthedynamicmodel(15)and(16)canberewrittenas(21)(22)M；,x+DjtX-mgO,=K,iit,IL+X+g9,=0.First,mgdi,thecouplingterminthecranedynamics(21),isprecompensated.Thatis,u,isdesignedas=umgBJKs,(23)whereuisthenewinputtobedeterminedbelow.Thenthecranedynamics(21)canberewrittenasMxX+DxX=KgU.(24)LaplacetransformationisappliedtoEq.(24)toobtainthefollowingtransferfunction:G,s)-K.Vis)_Us)M,s+D,(25)where.vistheindependentcomplexvariable；Vs)andUs)aretheLaplacetransformsofv=x)andu,respectively.Second,avelocityservocontrollerKss)isdesignedbasedon.G,s)usingtheloopshapingmethod；first,theopen-looptransferfunctionG(.v)=K,s)G,s)sshapedasGs)=Klsaccordingtotheloopshapingcriteria(Doyleetal.,1992),velocityservocontrollertrolleydynamicsVris)*9-r-A4s+DKsSKvs(s)u(s)KsMrS+DGt(s)V(s)Fig.2SchematicdiagramofvelocityservosystemandthentheresultingK,s)isobtainedfromK,s)=Gs)/GM:KM)=gM,s+D,K,s(26)whereKisthecontrolgain.LargerKyleadstobettercommandtracking,butKmustnotbetoolargeforrobuststabilityandsensornoiseattenuation.TheschematicdiagramofthevelocityservosystemisshowninFig.2.ThenthetransferfunctionofthevelocityservosystemG,(.?)isobtainedasG,(.v)-VKV,s+K,(27)whereV,.denotesthereferenceinputtothevelocityservosystem.Thecranemotorsaresometimescontrolledbyvelocityservocontrollersinsteadofthetorqueservocontrollers.ThenthedesignmethodproposedbyLeeetal.(1997)canbereadilyapplied.3.2.2DesignofaPositionServoSystem.Figure3showstheschematicdiagramofthepositionservosystem,whereK,ss)isthepositionservocontroller,G,(i)isthevelocityservosystem,Ds)isthevelocitydisturbances,andl/.sistheintegratorforconversionofcranevelocitytocraneposition.TheslipofcranewheelsisanexampleofD(.y).Asabove,A.(.v)isdesignedbasedonGy,s)usingtheloopshapingmethod；theopen-looptransferfunctionG,o(s)=Ks)Gyss)/s)isshapedasG.,s)Ks+K,/KpKSSS+Ky(28)whereKpandK,aretheconstantssatisfyingKilKpK,K0；slKpSH-K,)isadoptedonpurposetocancelKpS+Ki)lsofKsis).ThenKgs)becomesalagcompensatorwhentheoutputofKDS)isdirectlyinputtedtothevelocityservosystemGis).Inthisway,theswingangleandcranemotioncanbeseparatelycontrolled.Figure5showstherootlocusoftheoverallcontrolsystemfor/=1m,K,=24.0,Kp=1.6,K,=0.08,K=1.5,andK=0.6.TheoptimumvalueofKacanbedeterminedfromtherootlocus.TheoverallcontrolsystemisshowntobestableregardlessofthevalueofK,.However,thesystemmaybecomeunstableforalargevalueofK,sincethenonlineardynamicswereneglectedinthelinearizationprocess.3.2.4RemarksontheControlPerformance.TheperformanceoftheoverallcontrolsystemcanbeanalyzedusingthetransferfunctionsfromeachinputtoeachoutputinFig.4.ThetransferfunctionsforZ=1mandKa=3.55withthecontrolgainsobtainedaboveareasfollows:X38.4(s+0.05)(s+0.6)(j+9.8)-38.4(j+0.05)(s+0.6)Gcis)s(s+0.6)(s+24)(s+9.8)0XrAGAs)_-ss+0.6)(5+24)(33)(34)(35)(36)whereGds)isdefinedasG,(.s)=(s+0.05)(s+O.S2)(s+3.1)X(s+n.26)(s+1.69)+1.47.(37)Asexpected,theclosed-looppolesshowninGs)areallstablewithsufficientdamping；accordingly,soarethetransferfunctions(33)-(36).X/XshowsexcellentcommandtrackingrealaxisFig.5Rootlocusoftheoverallcontrolsysteminthelowfrequencyregion.0/X,showsthatthesteadystateswingangleiszeroforramppositioncommands.AccordingtoX/Dand&/D,thesteady-statecranepositionisnotinfluencedbystepdisturbancesandthesteadystateswingangleisnotaffectedbyparaboladisturbances.3.3Anti-SwingControlforSlowlyVaryingRopeLength.Anewdecoupledanti-swingcontrollawhasbeendesignedaboveforthecaseofconstantropelength.Inpractice,however,theropelengthsometimesneedstobesetslowlyvaryingforloadhoistingwhilecranesareinmotion.Accordingly,thispracticalcasewillbeconsideredhere.3.3.DesignofaRopeLengthServoController.Aswiththecranedynamics(21),theloadhoistingdynamics(19)canbewrittenas(Ml+m)l+D/l-mg=K,iU,i,(38)whereu,iistheinputtothetorqueservocontrollerofthehoistingmotorandKiisthecrane-dependentconstant.As