MODELLING AND CONTROL STRATEGY OF ROBOTIC YO-YO

ABSTRACT- In the paper we address a problem of controlling an oscillating motion with a robot. As the object we have selected a yo-yo. Yo-yo is a toy made of two discs connected with a short thin axle, which can be moved up and down by moving a string tied

ProceedingsofRAAD’03,12thInternationalWorkshoponRoboticsinAlpe-Adria-DanubeRegion

Cassino,May7-10,2003

MODELLINGANDCONTROLSTRATEGYOFROBOTICYO-YO

ˇLeonZlajpah

JoˇzefStefanInstitute

Jamova39,1000Ljubljana,Sloveniaemail:leon.zlajpah@ijs.si

ABSTRACT-Inthepaperweaddressaproblemofcontrollinganoscillatingmotionwitharobot.Astheobjectwehaveselectedayo-yo.Yo-yoisatoymadeoftwodiscsconnectedwithashortthinaxle,whichcanbemovedupanddownbymovingastringtiedtotheaxle.Althoughitisveryeasytoplaywithyo-yoforahuman,designingacontrolsystemforarobotisquiteachallenge.Wepresentasimpli edmodelofayo-yowhichisintendedforcontrolanalysisanddesign.Ourmodelhasonedegree-of-freedomandthebehaviourattheendofthestringismodelledasanimpact.Nextwediscussthecontrolstrategyandpresentacontrolmethodwhichallowsplayingtheyo-yoatselectedtopheight.

Keywords:Robotjuggling,Oscilatorysystems,Yo-yomodelling,VisualfeedbackINTRODUCTION

Inthelastyearstherehasbeenagrowinginterestinrobotsystemsthatarecapableofperformingthecyclictasks.Oneoftheexcitingtasksisjuggling[5,1,6]orplayingwithdifferenttoys[6,7,3],amongwhichisalsoayo-yo[2,4].Commontoallofthemisthatplayingwiththemisusuallymoreorlessaneasytaskforahuman,ly,dexterityofthesystemandsynchronizationwiththetoyarerequired.Ahumancanusehissensestolearnhowtooperateatoy.However,developingaroboticsystemthatcanperformthesamejobrequirescomplexsensorysystemsandadvancedcontrolstrategies.Yo-yoisatoymadeoftwodiscsconnectedwithathinshortaxle.Astringistiedtotheaxleandtheoperatorcontrolsthemotionoftheyo-yobymovingitupanddown(seeFig.1).Theobjectiveistoattainaperiodicmotionoftheyo-yo.Foranef cientroboticyo-yoacorrespondingmodelisneeded.Therearesomemod-elsofyo-yoavailableinliterature.Agoodinsightintothebehaviouroftheyo-yoisgivenin[4].Themotionoftheyo-yoisdividedintofourphases,andeachofthemisanalysed.Asthederivedmodelisverycom-plex,theauthorsproposeasimpli edmodel.Howev-er,someoftheirassumptionsaretoorestrictive,espe-ciallyneglectingthediameterofthestring.In[2],asimpli edmodelisgiven,buttheauthorsassumethattheenergylossisduetothefrictionandtheyneglecttheimpact.Additionally,theyproposecontrolmethodforroboticyo-yo.

Inthepaperwedealwithmodellingandcontrolstrate-giesfortheyo-yo.Inthe rstsectionweanalysethe

behaviouroftheyo-yo.Forthatwehavemademea-surementsofthemotionandforces.Nextweproposeasimpli edoneDOFmodel,whichcapturesallimpor-tantfeaturesoftheyo-yo.Inthethirdsectionwedis-cussthecontrolstrategiesandinthelastsectionweillustratetheproposedcontrolbyasimulation.ANALYSISOFYO-YOMOTION

Beforemodellingwehavemeasuredthemotionofayo-yowithandwithouthumaninteraction.Wehavemeasuredthepositiontrajectoriesandstringforces.Theexperimentalsetupforthemotionanalysiscon-sistsofanopticalsystemElitewhichcanmeasurea3Dpositionusingpassivemarkersandinfraredcam-eras.Forourmeasurementswehaveusedtwocam-eras.Fig.1showsacameraandayo-yowithamarkerattachedtothecenterofadisc.Thesystemoperatesat100Hzandtheaccuracyisaround1mm.Theparame-tersoftheyo-yoaregivenlater.

Wehaveanalysedtwosituations:themotionwithoutmovingthehand(stringisrigidlyrestrained)andwhenahumanplayswiththeyo-yo.Figure2showsthemotionoftheyo-yowithouthumaninteraction.Wecannoticethattheamplitudedecreaseswitheachperi-od.Furthermore,somesmalldisturbingoscillationsinxandydirectioncanbeseen.InFig.3theverticalmotionofthehandandtheyo-yoisshownwhenahumanisplayingwiththeyo-yo.Wecanseethatthehumancancontroltheyo-yowithupanddownmotionofthestringsothattheamplitudeoftheyo-yomotionisapproximatelyconstant.

Nextwehavemeasuredtheforcesinthestring

ABSTRACT- In the paper we address a problem of controlling an oscillating motion with a robot. As the object we have selected a yo-yo. Yo-yo is a toy made of two discs connected with a short thin axle, which can be moved up and down by moving a string tied

Figuresystem1:Elite

MeasurementofYo-yomotionwithopticalFigurestring2:3Dpositionsoftheyo-yoselfmotion–becauseisoftherigidlyvisionrestrainedsystemcould(thetrajectoriesnotarebrokenthetheverticalmarkersaxis)

duetotherotationoftracetheyo-yothepositionsaroundFigurewhenhuman3:Verticalisplayingpositionstheyo-yo.

oftheyo-yoandthehandattachedmeasuredtotheaxleoftheandtwoabya6-dimensionalyo-yo.force/torqueTheforcessensorhavebeenJR3aretheshownstringsPCcomputerinwhichFigs.differ(see.5andinFig.4).Wehavecompared6.theirWeelasticity.

canseeTheresultsratheryo-yoever,whensmallreachesthe(proportionalthebottompositionthethatforcesbeforeareyo-yoreachestothetheyo-yobottomweight).positionHow-an

Force sensor

Figure4:Experimentalsetupforforcemeasurement

Figurethestring5:Tensionisnotelastic.

inastringattachedtoyo-yoaxle—Figurethestring6:Tensionismoreelastic.

inastringattachedtoyo-yoaxle—impacttheoccurs.Aftertheimpactashortperiodwhenfree)string guresfollowedisnotbyunderasmallertensionimpact.happens(yo-yois yingforcetheiswegreatercanseewhenthattheamplitudeComparingoftheimpactbothtic.free yingperiodisthelongerstringwhenisnottheelasticandthatonlyAsterbythethemotionoftheyo-yocanstringbecontrollediselas-thethatthestringthemotionshouldstringofthetopendofthestring,itisbet-notisalwaysbeextensible.undertension.Therefore,MODELLINGOFYO-YO

Inbygeneralfreedomastringaattachedyo-yoistoait.freeActually, yingitobjectconstrainedofmotionayo-yo(DOF)isveryandcomplicated.acompletemathematicalhas6degrees-of-modelbalance.ofkineticenergyWhenyo-yoitisessentialHowever,toobservetothemodelenergytheisyo-yoconvertedisbouncingtopotentialupandanddown,vicever-

the

ABSTRACT- In the paper we address a problem of controlling an oscillating motion with a robot. As the object we have selected a yo-yo. Yo-yo is a toy made of two discs connected with a short thin axle, which can be moved up and down by moving a string tied

sa.impactAdditionally,thesaryyo-yo.anddueitdissipatestheenergyatthebottomToobtaintothefrictionoscillatorybetweenmotionthestringandviatosupplyenergytothesystem.Thiscanitisbeneces-forthestringbymovingthehandupanddown.doneAsgywebalance,thecontroltheuseanalysis.

inmodellingratherdesignthenitisimportanttoconsiderener-sometheassumptionshighfrequencywhichoscillations,simplifyAssumptionmoving1Thecenterofmassoftheyo-yois

yoaxisisasrotatingonlyinalwaysonlytheperpendicularalongverticalthedirectionaxle.andtheyo-totheTheverticalrotationalaxis.Assumptionandthemass2Theofstringthestringis exiblecanbebutneglected.notextensible

Assumptionviscous3Alldissipativeforcesareduetotionalvelocity.frictionwhichisproportionaltotherota-the

Assumptionat4bythebottomTheandrotationthedirectiontimeneededdoesnotchange

aroundπatthetheaxle)bottomcan(whenbeneglected.nostringforiswrappedrotationAssumptionrestitution5Thecoef cientstringisisalwayszero.stretchedandthe

Assumptionhandvelocities6Motionarecontinuous.ofthehandissmooth,i.e.

Thetwo rstassumptionenablesustomodelundesiredDOFsystem.theyo-yoasaremotionAlthough,whenplayingtheyo-yobasicpresentdisturbliketheswinging,motionandyawingcanevenandpitchingmodelup-downisprimarybouncing,intendedweforneglectthecontrolthemdesign.becausebreakthetheTheandsecondassumptionislessrestrictiveastheeddiameter[4].Testsofdifferentyo-yoshaveshowninthat[2]motion.

becauseitofin uencestheropethecanperiodnotbetimealwaysofbouncingneglect-Assumptionmodel.5allowsfurthersimpli cationsofthemotionyo-yoitcanandIftheberotationstringmodelledareisstretched,thentheverticalasdependentoneDOF(constrained)system.Inand[4]impactisexplainedimpactwhenthatfreemotioncanoccuronlyafterantomusuallyyo-yooccursissetwhenoffwithextrafreestring.Therestitutionpositionandthewholethestringyo-yoisunwinded.reachesthebot-theingimpactcoef cienttheisgreaterthenzero,thenIfafterthearenotoftheconstrained.stringverticaldueHowever,tovelocitytherotationisgreaterthanwind-theenergyandbothlossmotionsafter

the

Figure7:Schematicpictureofayo-yo

bottomcientimpactisindependentoftheafterifthetransitionphase(seriesofrestitutionminorimpactscoef -itAnotherisreasonablethebottomtoimpact)setrestitutioniscompleted.Therefore,motion.reasonforimpactscanbedrasticcoef cientupwardtozero.handticjusti ed.

motionAscantheupwardbeprevented.motionisSo,underthiscontrolassumptionadras-isFig.thatthewhen7showstheastringdetailedispictureofyo-yo.Wecanseeanglevertical is

positionofstretchedtheyo-yotheyandrelationtherotationalbetweeny=h l+r( )| |(1)wherethelisthetotallengthofthestring,h=h(t)is

andheighttypesristheofinnerthetopradiusend1.ofWethefoundstringout(handthatposition)ly,motionstomal.ofbene tthatthegapbetweenthediscsismini-itthechangestringConsequently,iswrappingthearoundinnerradiusincreaseswhenwinds,ofinnerradiusisnotthelineraxle.whenAlthoughthefollowingthisrelation

dependencycanbeapproximatedthebystringther=ro+kr| |

(2)

whereinkristheeffectiveradiusofthestring.NotethatthatthekthecasediameterthattheofthegapstringbetweenthenthethediscseffectiveisgreaterradiusvelocitiesrissmallertiatingEq.andthan(1)accelerationstheactualdiameterandconsideringcanofthestring.TheEq.bederived(2)

bydifferen-y˙=h

˙+rsign( ) ˙+kr ˙(3)y¨=

h

¨+(rsign( )+kr ) ¨+2kr ˙2(4)

1Actually,

wehadtouse| | π/2butbyapplyingtheAssump-tion4| |canbeused.

ABSTRACT- In the paper we address a problem of controlling an oscillating motion with a robot. As the object we have selected a yo-yo. Yo-yo is a toy made of two discs connected with a short thin axle, which can be moved up and down by moving a string tied

Notethat

sign( )=0for =0.

Next,followingthemotionequation

oftheyo-yocanbedescribedbythe

I

¨+B ˙= rsign( )F(5)my¨=F mg

(6)

whereandImandBaretheinertia,themassoftheyo-yo

tensionviscoseonofthefrictionstring,coef cient,andgisgravityrespectively,FistheF>Assumption0.Therefore,5thethestringaboveisequationsalwaysconstant.stretchedBasedcancecom-andbinedandweobtain

I ¨+B ˙= rsign( )m(y¨+g)

(7)

Substitutingcalculations

Eq.(4)intoEq.(7)yieldsaftersome

¨= sign( )mr(h

¨+2kr ˙2+g)+B ˙+mrkr| |

(8)

Inisverypracticesmalltheandfrictioncanbebetweenneglected,thei.e.ropeBand=0.thediscsEq.down(8)thebottommotion.describeswhereThethethecomplicatedmotionoftheimpactoccurs.partyo-yoisduringupanLetthe(·)motion atthedenotethestatestheimpact.immediatelyApplyingAssumptionbeforetheimpact2and(·)+aftermainlyhighlyduringonthecomplexpropertiesdynamicand4,weneglectofthemotionwhichdependsisTherotatedthisodimpactfornegligibleshortperiodrope.timeWewhenassumetheyo-yothatoccursπnochangeaftertheinrotation.theyo-yoDuringvelocitythisoccurs.peri-tionsthe4innerand6radiusimplyis +constant,r=ro,= =0andh

˙and+Assump-=h˙ =h˙,respectively.sign.

Notethataftertheimpact changesitsTheusingvelocitiesaftertheimpactcanbeobtainedbymomentumtheprinciplesbeforeandofimpactafterthedynamics.impactisTheconservedangularI ˙++mrosign( +)y˙+=I ˙ +mrosign( +)y˙

SubstitutingEq.(3)fory˙ andy˙+yields(9)

I

˙++mrosign( +)(h˙+r=I

˙ osign( +) +mrosign( +)(h˙˙+)=+rosign( ) ˙ )Aftersomecalculationsweobtain

(10)

˙+=1o

I ˙++mr2osign( +)sign( ) ˙ Since changesthesignatthebottom(11)

sign( +)sign( )= 1

Eq.(11)canbesimpli ed

˙+

=I mr2o˙ o (12)

ThebiningverticalEqs.(12)velocityand(3)

canbeeasilyobtainedbycom-y˙+=h

˙+rosign( +) ˙+(13)

Assign( +)=sign(

˙+)wegety˙+=h

˙+ro| ˙+|(14)whichupiftheshowsstringthatisstretchedaftertheimpact(Assumptiontheyo-yo5).

ismoving

Summarizing,systemtheyo-yocanbemodelledasciency”consistingoftheyo-yoofEqs.canbe(1),described(8)andby(12).theTheone-DOFfactor

“ef -ζ=I mr2o

o

(15)

FromtheenergyEqs.(12)lossandduring(14)theweimpactcaneasilyisproportionalconcludethat

toζ2during.BecauseharderI≤mrtoimpactincreasing2playtheζalsoroimpliesthattheenergylossyo-yoincreases,withlargeitexplainswhyitiso

itisimpossibletooperateyo-yo.ro.Ultimately,if

CONTROLSTRATEGY

Theamplitudeobjectiveofplayingtheyo-yoistokeepthethatmovingthemotionoftheyo-yoatadesiredlevel.Itisevidentmationthefreeofendtheofyo-yocanbecontrolledonlybyingaboutthestatestheofthestringsystemandisthatsomeinfor-tant.withyo-yoNamely,theyo-yoitrevealswhichinformationneeded.isimpor-Play-towithclosediseyes.practicallyThereasonimpossibleistoplaytheitmoveupwardbeforereachingthebottomthattheandhandhence,hasingisbeimportantdetermined.thenecessaryforceintoTherefore,thepredictstringtheonlybottomtime.Bymeasur-knowingthetimetheheightofimpactiscanbetemobtainedthaniscrucialonly”feeling”theforce.Astheheightmorecanforroboticbyvisualyo-yoinformation,control.

avisionsys-Frompledamplitudeoscillatorsthecontrolueofoneandpointoscillatortheofcontrolviewwedealwithtwocou-(yo-yo)taskatisthetostabilizedesiredtheoscillatorbychanginglatorsbyphase(robottheregulation.

hand),patternandandtosynchronizetheamplitudebothofoscil-otherval-Thedemonstrationrhythmicmotionpatterncanfunctions.[3]oritcanbecomposedbelearnedofbysmoothhumanlationsandToimpactspreventduringundesiredupwardhighmotionfrequencytheoscil-hand

ABSTRACT- In the paper we address a problem of controlling an oscillating motion with a robot. As the object we have selected a yo-yo. Yo-yo is a toy made of two discs connected with a short thin axle, which can be moved up and down by moving a string tied

motionreasonsshouldcycleandthethepositionnotbeendshouldofdrasticthebehandandequal,atjerky.theForpracticalh(tbeginningofthestart)=h(tend).TounderstanddesigntheThehowclose-loopthecontrolleritisimportanttotionenergydependsmainpointistheyo-yoenergymotiontransfer.canbeThecontrolled.topposi-thestoredoninthetheenergyyo-yoatofthethetopyo-yo.positionThepotentialenergypotentialbylostduringenergylastinimpacttheprevioustopposition,issumtheofplaysthedownthehandessentialandyo-yoduringlastcycle.Observingandtheenergyhowsuppliedahumanthenup.(Fig.From3)weEq.can(7)seeitthatthehandgoesBasedweonwhenmeasurementsthehandfollowsthatitisandacceleratesusingtheanddecelerates.thehaveupwardenergyfoundfromoutthethathandthetomosttheyo-yoef cientderivedisiftransfermodel,themajoroftomafterimpactaccelerationisperformedjustbeforethebot-theoftoptimingtheimpact.andtheFordownwardillustration,accelerationwehavetakestestedplacehowheightsofany ofaccelerationtwoconsecutivepulsein uencescyclesµ

theratioµ=

y i+1

i

Fig.8aofshowstheheightoftheyo-yorelativethehand(maltanddelayandthatthethetime tbetweenFig.8bgivestheratioandµacceleration

versustheaccelerationofimpacttheendsthandaccelerationpulsep)i, justt=beforeti tp.theItimpact,isopti- bet=impact.

shown0.Similarly,thatitisforoptimaldownwardifitaccelerationstartsjustafteritcouldtheWeisde nedproposeas

acontrolstrategywherethehandmotionh=khhn(ktτ),

0≤τ≤1

(16)

wherenalhnisthenominalmotionpattern,τisthenomi-ofthethepatternhandtime,motion,khandisthegaintoadjusttheamplitudetimerealandnominaltime;ktisτthe=0scalingandτfactorbetweeniswhenthemotionstarts(t=1indicatethestart)andwhenonecycletcompleted(tt<tend),respectively.Betweentwocycles,

andend,i<h

¨=0.Itstartis,obviousi+1,handthatisnottmoving,h=0,h

˙=0thanend,i tstart,imustbetbeforethecycletheimpact.oftheyoyo.AsitTheisimpossiblemomenttbelesstostartpredictmustheightstartdirectly,thehandmotionisstartedatchangingbeforethetheamplitudetheyo-yoreaches(gainkbottomposition.acertainByh)bemotion(thefactorkandthedurationoft)theyo-yopeakheightcanincreasingcontrolled.kThepeakheightcanbeincreasedbyhordecreasingktandviceversa.

a)time

response

b)ratioµoftopversusheightstheofdelaytwoconsecutive tcycles

Figure8:TopheightversustimeofaccelerationSIMULATIONEXAMPLE

Tostrategyillustrateeterswegivethesomecapabilitiessimulationoftheresults.proposedThecontrol00..05kg,ofthe0036m,I=yo-yoand2.k610 used5kgmin2,theB=simulation0,L=0.are:param-7m,rm=o=r=1.610 4nominalm.WehaveselectedtheThishandmotionpatternasshowninFig.9.ments.pattern0.15maboveThehandsatis esthebottommotionallpreviouslymentionedrequire-position.

startedwhenyo-yohasbeenThemotion.desiredpeakheighty dhasaccordingly.Therefore,Wehavegainsusedkbeenthefollowingkchangedduringhandthadtodependencies

bechangedkh=0.5(y d y )+0.05kt=

1

dwhereNoteywhenthat dandky arethedesiredandactualpeakheights.handkstantduringpeakyo-yotheremainingheighttarechangedisreachedonlytime).

(theirinvaluethemomentiscon-TheseemotionthatsimulationwiththeresultsareshowninFig.10.Wecandesiredheight.

isstableproposedandthatthecontrolpeakstrategy,heighttrackstheyo-yothe

ABSTRACT- In the paper we address a problem of controlling an oscillating motion with a robot. As the object we have selected a yo-yo. Yo-yo is a toy made of two discs connected with a short thin axle, which can be moved up and down by moving a string tied

h

˙¨tstart

t

τ

tend

Figure9:Handmotionpattern

a)desiredyo-yoandtophandheightaccelerations

andactualyo-yoheight

b)yo-yoheightheightandshouldhandbeaccelerationincreased

whentop

Figure10:Simulationofroboticyo-yo

CONCLUSION

Thisyopaperpresentsacontrolstrategyforplayingtaskwithrobot.foraahuman,robot.Although,itisanexactingplayingpieceyo-yoofworkisanaeasyyo-secondly,Firstonlytheofmotionall,notalloftheoftheyo-yostatesforacanmeasurablebecontrolledandstandbythemovingsystemthewefreehaveendanalysedofthestring.theyo-yoTomotion

under-andonethenwehavedevelopedamodel.Theproposedyo-yoDOFicandmodeliscapturesallimportantfeaturesofthenominalyo-yorobothandtwosuf cientthingsforcontroldesign.Forrobot-motionarepatternimportant:andtotoselectsuitablehavemotionwiththeyo-yo.Experimentssynchronizewithyo-yotheingbetheshownthatvisualfeedbackisessentialforplay-hasstartedyo-yo,cantobepredicted.beforebecausebottomthehandmotionupwardshouldWeimpactandhence,thetimetrolbeveri eddesign.controlledTheandproposedprovidediscusshowthetopheightcontrolsomeguidancemethodhasforcon-possiblebytoplaysimulationsyo-yowithandatherobot.resultsshowthatbeenitisREFERENCES

[1]M.PlanningBuehler,Catching–14,1994.Tasks.andD.E.ControlKoditschek,Int.J.ofofandP.J.Kindlmann.

RoboticRoboticResearchJuggling,6(1):3and[2]K.ControlHashimotoandT.Noritsugu.ModelingIntionIEEEof,pagesInt.RoboticYoyowithVisualFeedback.and

2650Conf.–2655,onMinneapolis,RoboticsandMinnesota,Automa-1996.[3]A.J.RhythmicIjspeert,J.Nakanishi,andS.NonlinearMovementsIEEE/RSJInt.Oscilators.byShaal.Learning

Conf.OnInDemonstrationusingIntelligentProc.ofRobotsthe2002andSystems,pages958–963,Lausanne,Suisse,2002.[4]H.-L.ics:tionSequenceJinandofM.CollisionsZackenhouse.Yoyodynam-tems,Effect.Measurement,Trans.ofandASMECapturedControlJ.ofbaaRestitu-,Dynamic124(3):390Sys-–397,2002.[5]A.A.tialRiziandD.E.Koditschek.ProgressinSpa-andRobotAutomationJuggling.,pagesInProc.775Int.–780,ConfNice,onRoboticsFrance,1992.[6]S.ControlSchaalIEEEConf.StrategiesandC.G.RoboticsforAtkeson.andRobotAutomationJuggling.OpenLoopSTable

,pagesInProc.913–918,Atlanta,Georgia,1993.[7]ingWilliamson.RythmicRobotArmConf.OnOscilators.IntelligentInRobotsProc.and1998SystemsIEEE/RSJCOntrol

,pagesInt.77–83,Victoria,Canada,1998.

本文来源：https://www.bwwdw.com/article/m2x1.html