MODELLING AND CONTROL STRATEGY OF ROBOTIC YO-YO(3)

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.