MODELLING AND CONTROL STRATEGY OF ROBOTIC YO-YO(4)

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