heading compensation Planar motion controller design for a modular mechatronic device with Mechatronics

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Mechatronics

journalhomepage:www.elsevier.com/locate/mechatronics

Planar motion controller design for a modular mechatronic device with heading compensation

Stefan Ristevski, Melih Cakmakci

Department of Mechanical Engineering Bilkent University, Ankara 06800, Turkey

a r t i c le i n f o

Keywords:

Motion control Friction compensation Mobile robots

a b s t r a ct

MechaCellsaredesignedasclosed,scalableandmodularsemi-autonomousdevicesthatcanbeusedaloneor partofapack.Inthispaper,wediscussalocomotionsystemthatusesthereactionforceproducedbyarotating unbalancethatmovesinasphericaldomainwithasteeringmechanism.Inordertoproducetheprecisemotion capability,amulti-loopcontrollerisdeveloped.Thiscontrollerusesafrictioncompensationalgorithmbasedon themathematicalmodelofthelocomotionsystem.Toimprovetheaccuracyoftracking,conventionalLuGre frictionestimationmodelisextendedforrapiddirectionalchangesoftheMechaCellduringplanarmotion.The linearandrotationalaccelerationofthedeviceisalsoincludedincontrollercalculationssinceitaffectstheloco- motionforcegeneratedbytheunbalancedmass.Theresultingcontrolsystemisvalidatedbothwithsimulations andexperimentsandtheeffectivenessoftheextendedmodelandthecontrollerisverified.Ourresultsshow significantimprovementwhenadetailedfrictioncompensationobserverisusedinthecontrollerthatincludes theeffectofsuddensteeringchangesforprecisepathfollowing.

1. Introduction

Inrecentyears,therehasbeenincreaseduseofsensorsandactua- torsanddeviceswithon-boarddiagnosticsthatcreateusefulcontroller designopportunitiesasreportedin [1–4].Duetotheirmulti-domain nature,designandmanufacturingofthesesmartmechatronicsystems canbeachallengingandacomplicatedtask.Onewaytomanagethe complexitiesistousemodularityduringthedesignandmanufacturing phasesasdiscussedin[5,6].Forexamplebyincreasingthecomponent- sharingmodularity,complicatedsystemscanbebuiltbyusingsimple commonmodulesthatareidenticalinengineeringproperties.

Theapplication of component-sharingmodularityin mechatronic systems,especiallyin thegroundroboticsfield,isstudiedbyvarious researchers. In[7], anoverview of the reconfigurable modularsys- temsisgiven.Buildingscalable,self-repairing,self-sustainingandself- configuringsystemsarepresentedasgrandchallengesinthisfield.A designmethodandexperimentsforwhole-bodylocomotionbyamodu- larrobotispresentedin[8].Inthiswork,thelocomotionofthewhole structureisstudiedratherthanthemodulemotion.In[9],amodular roboticdevicewasdesignedsuchthatmodulescanself-assemblethem- selvestoformaroboticstructure,andcanmoveindependently.How- ever,thelocomotionsystemusedinthismoduleisnotenclosed,which maybeproblematicforprecisionmotionandforun-idealsurfacecon- ditions.Generally,thefinalexternalstructureofthesystemisstudied

ThispaperwasrecommendedforpublicationbyAssociateEditorProf.PaoloRocco.

Correspondingauthor.

E-mailaddress:melihc@bilkent.edu.tr(M.Cakmakci).

ratherthanhowtounifythecapabilitiesandstrengthendevicecollab- oration.In[10],amodularselfconfigurablerobotthatchangesshape horizontallytoconveymicropartsisdeveloped,whichisalsocloseto theapplicationusedinthispaper.

Ourmotivationistoapplytheprincipleofcomponent-sharingmod- ularityandutilizeitspotentialinmechatronicsareabydevelopinga scalablemechatronicmodulethatcanbeusedasthebuildingblockfor morecomplicated systems.Componentsharingmodularityinmecha- tronicsystemsimplythat,complexsystemscanbebuildfromasingle module.Thismoduleshouldbeminiaturizable,havetheabilityofboth sensingandactuationandshouldbestrong/sturdyenoughtoaddress theexternaldisturbancesuchassurfacefriction.Similarexamplesin recentliteratureistheresearchwheretheswarmdevicessuchasKilo- bot[11],Alicerobots[12]weredeveloped. However,oncontraryto swarmdevices,mechatronicmodulesdiscussedherearedesignedtoac- complishtasksthatinvolvephysicalinteractionaspartofalargersys- temwithcontrolledactuationandsensing.Thesedeviceswillbeshortly referredasMechaCells.

InFig.1,aprecisionpositioningtaskperformedbyMechaCellsis illustratedasanexample.AgroupofMechacellscanpositionawork- pieceonaplanesothatalasermachiningoperationisperformed.This typeof handlingcan beimportantespeciallyforthecases whenthe workpieceistoofragiletobeattachedinsideafixtureonapositioning system.Moreover,thisoperationmayneedtobedoneatahardtoreach

https://doi.org/10.1016/j.mechatronics.2019.102257

Received13July2018;Receivedinrevisedform11May2019;Accepted3August2019 Availableonline14August2019

0957-4158/© 2019ElsevierLtd.Allrightsreserved.

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Fig.1. AsmallscalelasermanufacturingsystemillustratingtheuseofMechaCelldevicestoguideaworkpieceonaplanarsurface.

Fig.2. PartsoftheMechaCellandassemblysteps.

areasuchasinthecasesofon-boardrepairs.Theexampledepictedin Fig.1suggeststhatthemechatronicsystemswithmoreprecisetasks willrequiremodulesthathaveadvancedcapabilitiessuchascontrolled motionandforce.

InFig.2,themechatronicmodule,whichwasdesignedanddevel- opedtotesttheideaof applyingcomponent-sharingmodularity(i.e.

buildingsystemsfromcommonmodules)forgenericsystems,isshown.

Thenumbersinthisfigureshow theassemblyorder.Themechanical designof thisdevice andits componentswerediscussedindetail in

[13,14].In[15],thecooperativeoperationstrategiesforMechaCellsis discussedfordextereousmanipulation.Thisdevicecontainsasteering servo-motor(S),aprimaryvibrationmotor(V),athreeaxisaccelerom- eterandasurfacepressuresensorandfinallyanembeddedcomputer withwirelesscommunicationcapabilitiesforinteractingwithotherde- vices.Themodulescanalsobemechanicallyconnectedeachotherwith magnet pairsand/ormechanicalhingesplaced ontheirexternalsur- face.Thetestprototypesdonotincludeapower-packandarepowered externallyusingthinisolatedwires.Inthecurrentdesign,thedevice

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Fig.3. ThelocomotionmechanismoftheMechaCellusinganunbalancedmass.

alsohaveasecondaryvibrationmotorattachedatthetoptocontrolits orientation.

Theintegrationandoperationofsensing,actuationandcomputing hardwareintheMechaCellsystemisexplainedin[14].Themechanism thatprovidesthelocomotionofthemoduleisanovelapplicationofcon- trolledunbalancemotionasshowninFig.3.Rotationalmotionofthe unbalancemassgeneratesaforcethatcausesthetranslationalmotion ofthebasecylinder. Themechanismconsistsofarotatingunbalance massattachedtoasteeringcylinder.Thesteeringcylinderisattached tothebasecylinder.Forrepresentingthemotionofthebodyandthe components,twocoordinatesystemsaredefined,onecoordinatesystem (FCS)fixedattheorigin,Oandamovingandrotatingcoordinatesys- tem(MCS)attachedtothebasecylindersuchthattheZ′axisisparallel totheZaxisasshowninthefigure.Steering-cylinderisattachedtoa basecylinderandcanrotateabouttheZ′axis.TheSvectorpointsto thecurrentvelocityoftheMechaCellandmakesanangleof𝛽 radians withtheX′axis.TheunbalancedmassrotatedbyaDC-motorwithan angularvelocityof𝝎whichisbothperpendiculartotheZ′axisandSD,a vectorpointstodirectionofunbalancedmasslinkintheX’Y’plane.The steeringcylinderisrotatedwithaservomotorhousedinthebaseand itsorientationrelativewiththeX′axisisgivenas𝜙.AlthoughaxesZ andZ′arealwaysparallelpertheproblemsetup,theXYandXY′pairs mayhavedifferent orientationsduetotherotation oftheMechaCell duringmotion.AnimportantparametershownasasidenoteinFig.3is thebasecylinderrotation,𝛾,whichisdefinedastheanglebetweenthe fixedaxisXandmovingandrotatingaxisX′.

Astheun-balancedmass,m,rotatesaplanarreactionforce,𝐅𝑢,is appliedtothebasefromthelinkatconnectorA.TheMechaCellframe movesdepending on thecurrent rotationalspeed of theunbalanced mass,theamountoftheunbalancemassovercomingtheforcesduethe friction,Ff,betweenthebodyandthesurfaceandtheinertia.Theap- plicationdirectionofthereactionforcecanbechangedbyrotatingthe steeringcylinder(i.e.changing𝜙)withrespecttothebaseasshownin Fig.3usingtheattachedservomotor.

Inthis paper, theprecisionplanarmotioncontrolalgorithm ofa MechaCellispresentedwithbothanalysis(i.e.mathematicalmodeland simulations)andexperimentalwork.TheproposedMechaCellsystem, whichis designedasthebuildingblockelementin thesetypesof sys- tems,haveactuationcapabilitywithsensorandwirelessconnectivityin aclosedpackage(i.e.nowheelsand/ormovingextensions).Withthe steadyprogressofelectronicsandsensortechnologies,ourprimarymo- tivationandfocusistodesignareliablemotioncontrolsystemwhich ispreciseandsuitableforusingcollaborativemechatronicapplications suchaspositioningunderstrictconstraints.Theprimarycontributions oftheresearchpresentedinthispapercanbegivenasthemathemati- calmodelforamechatronicdeviceequippedwithanovellocomotion systemthatincludesfrictionandthreedimensionalcentrifugaleffects, afeedbacklinearizingcontrolsystemmethodfeaturinganextendedLu- Grefrictionmodelforrapiddirectionalchangesvalidatedwithtestre- sults.Thecontrolledco-operationofthesedevicesusingthedesignand algorithmpresentedhereisthetopicofanupcomingpublication.

Theoutlineoftheremainderofthepapercanbegivenasfollows.

InSection2,amathematicalmodelthatexplainstheplanarmotionof thedeviceisdeveloped.Then,inSection3,themotioncontrollerde- velopedfortheMechaCelldeviceispresented.Theexperimentalsetup tovalidatethecontrollerperformanceisgiveninSection4.Finally,in Section5,initialconclusionsandfutureworkwillbediscussed.

2. Mathematicalmodel

Inthissection,acontrolorientedtranslationalmathematicalmodel fortheMechaCellunitisdeveloped.Thismodelisusedforinitialsim- ulationsbeforeexperimentalvalidation.Partofthismodelisalsoused astheestimationmodelinthefrictioncompensationalgorithmofthe MechaCell’smotioncontrollerasexplainedinthenextchapter.

ThedynamicmodelforthemotionoftheMechaCelldevicecanbe developedusingplanarmotionprinciplesofarigidbodyunderthein- fluenceofathree-dimensionalexternalforceandafrictionforcedueto

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Fig.4.FreebodydiagramofthesysteminFig.3.

sliding.Oneofthechallengesassociatedwiththesystemsetupisthat themotionforceisgeneratedbytheaccelerationofacomponent(un- balancedmass)attachedtoabaseframethatdoesplanartranslationand rotationmotion.Therefore,arelativevelocityandaccelerationanaly- sisusingthefixed(FCS)andmoving(MCS)coordinatesystemsismore appropriatetofindthecorrectlocomotioneffectoftheunbalancemass.

InFig.4,forcesaffectingthemotionoftheMechaCellisshownusing aprojectedviewfromZSDplanegiveninFig.3.Thebasecantranslate inXandYaxisdirections.Inaddition,theunbalancedmasscanrotate abouttheZ′axisandabouttheaxisbothperpendiculartoSDandZ′.Be- causeofitsmotionitcanbeassumedthattheunbalancedmass,m,has displacement,velocityandaccelerationinthreedimensions.Theaccel- erationoftheunbalancedmass,am,canberepresentedincomponents byusingtheorthogonalunitvectorsi,jandkinX,YandZdirections respectivelyasshownin(1).

𝐚𝐦=𝑎𝑚,𝑥𝐢+𝑎𝑚,𝑦𝐣+𝑎𝑚,𝑧𝐤 (1)

FromFig.4,theforce,Fu,exertedbytherotatingunbalancedmass ontheMechaCell,canbecalculatedusing(2).

𝐅𝐮=𝑚𝐚𝐦+𝑚𝑔𝐤 (2)

As (2)implies, calculation of unbalanced force, Fu, requires the calculationoftheinertialaccelerationoftheunbalancedmass,m.In

Fig.5(a)thecomponentsoftheaccelerationoftheunbalancedmassis shownforasimplelinearmotioncase.Itcanbeseenclearlythatthe acceleration,aM,ofthebasecylinderisalsoacontributortotheinertial accelerationoftheunbalancedmass.Itisalsoimportanttonotethatad- ditionaltermsmaybecontributorssincetherotationofthebasecylinder andthesteeringmechanismalsocancauserotationtotheunbalanced massasshowninFig.3.Thetotalacceleration,am,oftheunbalanced massofthesystemcanbecalculatedbyusingtherelativeacceleration equationgivenin(3)byassumingXYZasthefixedaxis(FCS)andXYZ′ asthemovingaxis(MCS)coordinatesystems.

𝐚𝐦=𝐚𝐌+̈𝛄 × 𝐫𝐮+𝐚𝐫𝐞𝐥+ ̇𝛄 × (̇𝛄 × 𝐫𝐮)+2(̇𝛄 × 𝐯𝐫𝐞𝐥) (3) In(3),aM,̇𝛄 and̈𝛄 aretheacceleration,angularvelocityandangular accelerationoftheMechaCellbase(i.e.themovingframe)respectively.

ru,vrelandarelaretheposition,velocityandaccelerationoftheunbal- ancedmasswithrespecttocenterofthebasecomponent.InFig.5(b) theeffectofomissionofthebaseaccelerationandrotationalcompo- nents andusing onlyarelisshown.Sincetheomissionoverestimates thelocomotionforce,a0.3cmsteadystateerrorisobservedforasingle pulsemotorcommand(givenat100PWM(40%dutycycle)levelfor4s) simulation.Usingarelonlyforunbalancedforceestimationiscommon practice.However,thisdifferenceisimportanttonoteforapplications whereprecisemovementisarequirement.Thereforethemovingframe accelerationcalculationgivenin(3)willbeusedinthecontrollerdesign algorithmsandsystemsimulations.

AsshowninFig.4,forcesactingontheMechaCellsystemarethe weightandnormalforcesintheZ′direction,frictionforce,Ff,inthe oppositeofthevelocitydirection,S,andreactionforce,Futherotating intheZSDplane.ApplyingNewton’sSecondLawtotheMechaCellwith mass,M,wecanobtainthedynamicmotionequationsasgivenin(4)– (6).

𝑀𝑔+𝐹𝑢sin(𝜃)+𝑁=𝑀𝑎𝑀,𝑧=0 (4)

𝐹𝑢cos(𝜃)𝑐𝑜𝑠(𝜙 +𝛾)𝐹𝑓cos(𝛽 +𝛾)=𝑀𝑎𝑀,𝑥 (5) 𝐹𝑢cos(𝜃)𝑠𝑖𝑛(𝜙 +𝛾)𝐹𝑓cos(𝛽 +𝛾)=𝑀𝑎𝑀,𝑦 (6)

Fig.5.AccelerationoftheUnbalancedMass(a)BaseandRelativeAcceleration(b)TheEffectofbaseaccelerationinFucalculations.

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Table1

Parametersusedinthemathematicalmodel.

Mass of the Mechacell M 0.049 [ kg ] Mass of the rotating unb. m 0.006 [ kg ] Inertia of the Mechacell I 0.0000123 [ kgm 2 ] Radius of the Mechacell r M 0.025 [ m ] Radius of the rot. unb. r 0.004 [ m ] Angular speed of the unbalance 𝜔 Swept [ rad / s ] Coefficient of static friction 𝜇s 0.2 Coefficient of kinetic friction 𝜇k 0.15 Force prod. by the unbalance F u Calculated [ N ] Torque prod. by the sec. motor T s Calculated [ Nm ]

Friction force F f Calculated [ N ]

Friction torque T f Calculated [ Nm ]

Normal force N Calculated [ N ]

Gravitational constant g 9.81 [ m / s 2 ]

whereNisthenormalforceappliedonthebasefromthesurfaceandFfis themagnitudeofthefrictionforceexperiencedbythebaseinthedirec- tionoppositetothemotion.TherotationaldynamicsoftheMechaCell structurebyusingthederivativeofthetotalangularmomentumabout itscenterofmassasshownin(7).

𝐼̈𝛾 =𝑇𝑠+𝑇𝑢𝑇𝑓 (7)

In(7),Tsisthetorqueprovidedbythesecondarymotor,Tuisthere- actiontorquegeneratedbytheunbalancedmassmotionontheMecha- Cellstructure.Sinceitinvolveshigherordersinusoidalsmultipliedwith small momentarm values,the contributionof Tu is very small and currentlydisregarded.Asanexample,thetorquegeneratedbytheun- balancedmasscanbecalculatedbyusingthephysicalparameters.At themaximumcondition when𝜃 =0, thetorque can beestimated as 𝑇𝑢=𝑟𝑚𝑟(̈𝛾 + ̈𝜙),consideringtherotationalmotionofunbalancedmass,m asviewedfromtheXY′inFig.3.Thesteeringcylinderhasamaximum rotationalaccelerationof ̈𝜙 =500 rad/s2 (device specifications)and thebasecylinderrotationrateislessthanapproximately1000rad/s2 basedonexperimental data,thisgivesusaunbalancedtorquevalue of𝑇𝑢≈ 8× 10−5Nmasanestimatedmaximumvalueusingthevalues giveninTable1.In(7),Iand𝛾 aretherotationalinertiaandrotationof theMechaCellaboutZ′axisrespectively.Tfisthefrictiontorquegen- eratedbythefrictionforcebetweentheplanesurfaceandtheMecha- Cellfloor andcan beapproximated by𝑇𝑓=𝑟𝑀𝐹𝑓 (frictionforceap- pliedattherim)whererMistheradiusoftheMechaCelldeviceusing thesinglefrictionclutchplatecalculationsgiveninreferencessuchas [16]. Likewise,we canestimate thestaticfrictiontorque,Tf,sby us- ingtheformula𝑇𝑓,𝑠=𝑟𝑀(𝑀+𝑚)𝑔𝜇𝑠≈ 1× 10−2Nm.Tsvariesoncom- mand,however, comparisonof thestatic frictiontorque andthees- timatedunbalance torquevaluessupportsthedecisiontoneglectTu, in(7).

ForprecisecontroloftheMechaCellmotionthecalculationofthe frictionforceiscrucial.Thefrictionforcebetweenthesurfaceandthe lowersurfaceoftheMechaCellbasecanbeestimatedusingvariousmod- els.Forexample,usingtheCoulombFrictionModel,thefrictionforce canbecalculatedbyusingthenormalforce,N,actingontheMechaCell padsandafrictioncoefficient[17].ACoulombfrictionforceestimation, Fccanbecalculatedasshownin(8).

𝐅𝐜=−𝜇𝑘𝑁𝐞𝐯=−𝜇𝑘(𝑀𝑔𝐹𝑢sin(𝜃))𝐞𝐯 (8) where𝜃 isthecurrentorientationoftheunbalancedmasslink,evisa unitvectorinthedirectionofthevelocityoftheMechaCellanditsinthe directionofthetranslationvectorS.𝜇kisthefrictioncoefficientwhen thedeviceisinmotion.Thecalculationofthefrictionforce withthe Coulombapproachresultsinasimpleandquickestimation.However, thefrictionforceisknowntochangewithvariousfactorsasnotedby manyresearchersandmoredetailedmodelsexist[18–21]includingthe flexibilityofthecontactinteractionsanddirection.Oneofthesemod-

elsisknownastheLuGremodelandcanberepresentedasshownin (9)–(12).

𝐹𝑓=𝜎0𝑧+𝜎1̇𝑧+𝑓(𝑣) (9)

̇𝑧=𝑣𝜎0|𝑣|∕𝑔(𝑣)𝑧 (10)

𝑔(𝑣)=𝐹𝑐+(𝐹𝑠𝐹𝑐)𝑒|𝑣𝑣𝑠|𝛼 (11)

𝑓(𝑣)=𝜎2𝑣 (12)

Thefrictionmodelgivenbetween(9)and(14)havemanyapplica- tionspecificparameters.Fs,representthestictionforceandiscalculated using𝐹𝑠=𝜇𝑠𝑁.f(v)istheviscousfrictionforcetherefore𝜎2isthevis- cousfrictioncoefficientwhichdeterminedtobe around0.3Ns/m.𝛼 infunctiong(v)istakenas1basedonthediscussiongivenin[22].z isaninternalstateof thefrictionmodelandspringanddampercon- stants,𝜎0and𝜎1aretakenas13N/mmand6Ns/mmrespectively.vs istheexponentialdecayconstantinfunctiong(v)andavalueof8mm/s isused.Althoughphysicalinterpretationsexists,inordertodetermine thesevalues,theestimatedvalueofthefrictionforce,Ff,wasdetermined fromtheexperimentaldatapointsusing𝐹𝑓=𝐹𝑢,𝑚𝑎𝑥−(𝑀+𝑚)𝑎𝑀 and recordedwiththecorrespondingvelocity.Sinceateachdatapointmo- torcommandisknown,buttheangularpositionoftheunbalancedmass, 𝜃,isnotmeasured,amaximumunbalancetorqueFu,maxwascalculated usingthemotorcommanddata.Then,usingthecalculatedFfvs.vpairs basedontheexperimentaldatapointsgiveninFig.6(a),fittingLuGre modelparameterswerecalculated.Themodelfitisgiveninthesame figureasasolidline.

IntheoriginalformoftheLuGremodel,frictionforcegenerateddue tothetranslationmotioninthedirectionofmotionisconsidered.How- ever,inoursystemthereisalsorapiddirectionalchangeofmotionthat affectthefrictionforceexperiencedbytheMechaCellduetorapidro- tationofthesteeringbase(i.e.rapidchangeof𝜙).Therefore,asecond stablestate,𝜉 wasaddedthatbecomesactiveasthedevicechangesFu directionontheXYplaneandquicklyconvergestozeroifthenewdirec- tionisconservedasshownin(13).Incalculations,toobtainareasonable delay,𝜎3isusedas0.04,hfunctionwassetto𝜋| ̇𝜙| and𝜎4=0.05N/rad wasused.Thecalculatedeffectoftherotationalangleextensiontothe LuGrefrictionmodelispresentedinFig.6(b).Thedatapointsshowthe calculatedvalueoffriction,Ff,whenthedevicehaveavelocitybetween 0.005and0.05m/srangewithchangingsteeringcylindervelocity,𝜙.

Theeffectoffrictionisseeminglyhighinthelow ̇𝜙 valuessincethisis alsothehighaccelerationstartarea.

Theparametesin theLuGreextensiongivenin(13) and(14) are chosenaccordinglyaspresentedinFig.6(b).

̇𝜉 = ̇𝜙 −𝜎3| ̇𝜙|∕ℎ(̇𝜙)𝜉 (13)

Thenthetotalfrictioncalculationgivenin(9)canbemodifiedasshown in(14).Thedirectionofthefrictionforceistheoppositeofmotion,i.e.

inthe−𝐞𝐯direction.

𝐹𝑓=𝜎0𝑧+𝜎1̇𝑧+𝑓(𝑣)+𝜎4𝜉 (14)

Theforceappliedbytheunbalancedmasstothesystemisafunction ofthemotorrotationalspeed𝜔sincetheunbalancedmassandthelink lengthdonotchange.Themotorrotationalspeedisafunctionofthe commandsignalwhichisgivenasaPulseWidthModulation(PWM) signal.PWMsignalisastandardinputsignalusedinpowerelectronics andits typicaldesignator isthewidthmodulation[23]givenin the unitsofdutycyclepercentage%DC.TheconstantthatrelatesthePWM signaltothemaximumunbalanceforce,Fuiscalculatedexperimentally as𝐾𝑣𝑏=13𝑥10−6where𝐹𝑢=𝐾𝑣𝑏× %𝐷𝐶mN.InTable1,allthemajor parameterswithnumericalvalues(ifapplicable)usedinthesimulation ofthemathematicalmodelaresummarized.

In Fig. 7, comparison between the mathematical model and ex- perimental datafromthereal systemisshown forasingledirection

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Fig.6. (a)IdentifyingLuGreModelParameters(b)Theeffectofbasespeedonfrictioncalculations.

Fig.7. Comparisonofthedynamicmodelswithdifferentfrictioncalculations.

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Fig.8.Comparisonofthemathematicalmodelsinplanarmotionwiththeexperimentaldata.

translation(rectilinearmotion)case.Thespeedcommandvalueisin- creasedfrom0to100in10s,thenthepolarityisreversed(i.e.direc- tionofrotationischanged)andthecommandisdecreasedfrom100to 0withconstantslope.Thecalibrationformotorcommandandangular speedisgivenas14.65rad/s/%DC.Itcanbeseenthatthemodelusing LuGrefrictionmodel,whichincludesthevelocityandflexibleinterac- tions(effective intheencircledregions)andtheeffectofbaseaccel- erationofthecontactingsurfacesispredictingtherealworldresponse betterthanconventionalCouloumbfrictionmodelwhichonlyincludes thenormalforce.Inthiscase,sincethesteeringwheelisnotmoving, thereisnochangeinthedirectionofMechaCellandthestate,𝜉,isal- ways0andthereforenotcontributingtothefrictioncalculations.The frictionidentificationexperimentsalsoshowadelayedstartthatpoints totheexistenceofmotordrivercircuitandcommunicationrelatedde- lays.Thisdelayisnotstudiedinthiswork.InFig.8,insteadofsingle directionmotion,aplanartranslationprofilewasusedwherethesteer- ingwheel rotatesduringthemotionaswell.Itcan beseenfromthe resultsthatmodificationontheLuGremodel,whichcompensatesfor thedirectionchangesoftheslidingsurface,predictsthefrictionforce betterfortheplanarmotion.Theencircledregionsinthefigureshow thiscompensationinaccordtotheexperimentdatawhenthesteering systemoftheMechaCelldeviceisused.Bettermatchwithexperimental dataispossible,however,onlythecorrectdirectionofcompensationis targettedtoavoidover-calibrating.

3. Motioncontroller

InFig.9,theoverviewoftheclosed-loopcontrolsystemusedinthe motioncontroloftheMechaCelldeviceispresented.Thedesiredpla- narpositionandorientationiscommunicatedtothedeviceusingwire- lessmessaging.SincetheMechaCellmovesusingtheunbalancedmass

Fig.9. PositionfeedbackcontrolschemeoftheMechaCell.

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reactionforcethatgeneratesmotioninonedirection,thedesiredposi- tionvaluesareconvertedtodesiredtranslationanddirectioninforma- tionusingthe“TargetPathMapping” calculations.Thetestbedusedin thisstudyhavethehighdefinitionimagecapturecapabilityandmea- suredaccelerationissenttoeachdeviceasawirelessmessage.Allof thisinformationisthenusedbythe“MotionController” togeneratethe translationandrotationcontrolsignalsenttotheMechaCellmotors.In theremainderofthissectiontargetpathmappingandmotioncontroller featuresshowninthisfigurewillbediscussed.

InFig.10,desiredtranslationanddirectiongenerationbasedonthe desiredCartesiancoordinatesoftheMechaCelldeviceispresented.At atypicalinstantMechaCellshouldtravelfromitscurrentposition,RM, followingthevectorstartingatitscurrentpositionandendingatthe desiredposition,RD. ThisvectorisshownasSD.Hence, tomoveto- wardsthedesiredpoint,theservocontrollershouldpositionthesteer- ingcylindersuch that∠𝐒𝐃=𝜙 +𝛾 where𝛾 is theorientationof the MechaCellwithrespecttothefixedframeXYand𝜙 istheangleofthe unbalancedmasslink withrespecttothemovingandrotatingframe XY′.

Thedesiredtranslationreferenceusedbythes-controller,andthe servocontrollerisshownin(15).

𝑠𝑑𝑒𝑠=|𝐑𝐃𝐑𝐌| (15)

where,RDandRMarethedesiredandMechaCell’scurrentpositionvec- torsw.r.t.theinertialcoordinatesystem.Thedesireddirectionofthe translationcanbecalculatedbyusingthedirectionofthesdesandcur- rentorientationoftheMechacellontheXYplane,𝜃 whichisasensor inputasshownin(16).

𝜙𝑑𝑒𝑠=∠𝐒𝐃𝛾 (16)

Fig.10. TopviewoftheMechaCell,itspositionvector,positionvectorofthe desiredposition,MechaCell’sorientationandsteering-cylinderorientation.

Duringthesimulationsandexperimentsperformedforthiswork,the desiredorientationoftheMechaCell,𝛾des,iscommandedas0sothat thefixedandmovingframesarekeptaligned.Inothercases,depending ontheapplication,theangularorientationcanalsobecontrolledasa functionoftimeusingthereferenceinput.

InFig.11,interactionofthe“MotionController” and“Plant” blocks given in Fig. 9 are explained in detail. The motion controller has

Fig.11. TheMotionControlalgorithmfortheMechaCellDevice.

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Fig.12. FrictionObserver(fromFig.11)withheadingandinertialcompensa- tion.

threesubsystems:anopenloopsteeringservocommandfunctionthat turns the steering cylinder to the desired angle, a translation con- troller(s-controller)controlsthemotionoftheMechaCellatthecurrent headingbycontrollingmagnitudetotheactuatingforcethroughrota- tionalvelocityof theunbalancedmass,andanorientationcontroller (𝛾-controller)-controllingorientationoftheMechaCelldevicebycom- mandingareactiontorquegeneratedbythesecondvibrationmotorat- tachedatthetopoftheMechaCell.

Thecontrolalgorithmalsofeaturesanobserverthatpredictsthefric- tionforceexperiencedbythedeviceandthecommandedunbalanced massrotationalspeediscompensatedaccordingly.Thefrictionobserver blockshowninFig.11isgivenindetailinFig.12.Thebasisforthefric- tioncompensatoristhefrictionmodelexplainedintheprevioussection basedontheunderlyingequationsgivenin(1)–(14).Thesesetofnon- linearequationsreceivethecurrenttranslationalacceleration,current controllercommand,qs,servocommand,𝜙des,devicetranslationalac- celerationcomponents,{̈𝑥,̈𝑦},rotationalvelocityandacceleration(i.e.

̇𝛾 and̈𝛾)astheinputandcalculatetheestimatedfrictionforceacting onthesystem.Intheimplementationofthefrictionestimator,firstthe accelerationoftheunbalancedmass,amcanbecalculatedusingthesen- sordataandthecontrollersignals,thenamaximumvaluefortheun- balancedforce,Fucanbefoundusingthecontrollercommandbyalso includingthecurrentaccelerationoftheunbalancedmass.Thisvalue andthevelocityinformationcanbe usedinEqs.(8)and(9)toesti- matethecurrentfrictionforceFfexperiencedonthebasesurface.The stabilityoftheLuGrefrictionmodelalgorithmwasstudiedbymanyre- searchers[18,24].Itcanalsobeseenthattheextendedmodelusedin thispaperisalsostablesincetheaddedstate,𝜉,givenin Eq.(12) is stableanddecoupledfromtherestoftheplantdynamics.

Whenthefrictionestimationalgorithmisimplementedwithprop- erly calibrated parameters, theestimationerror, 𝑒𝐹 ,𝑓=𝐹𝑓,𝑒𝑠𝑡𝐹𝑓 is small,thecontributionfromthefrictionforcecanbeassumedasadis- turbanceandthecontrollerscanbedesignedaccordinglyasasecond ordersystemwithinertiadynamics.Infact,withef≈0thetransferfunc- tionsrepresentingthetranslationdynamics (i.e.𝐺𝑠(𝑠)=𝑆(𝑠)∕𝑆𝑑𝑒𝑠(𝑠)) andtherotationdynamics(i.e.𝐺𝛾(𝑠)=𝛾(𝑠)∕𝛾𝑑𝑒𝑠(𝑠))in Fig.11can be givenasshownin(17)and(18)respectively.

𝐺𝑠(𝑠)= 𝐶𝑝𝐾𝑣𝑏

𝑚𝑠2+𝐶𝑣𝐾𝑣𝑏𝑠+𝐶𝑝𝐾𝑣𝑏 (17)

𝐺𝛾(𝑠)= 𝐶𝑟𝑝𝐾𝑣𝑏𝑇

𝐼𝑠2+𝐶𝑟𝑣𝐾𝑣𝑏𝑇𝑠+𝐶𝑟𝑝𝐾𝑣𝑏𝑇 (18)

whereCp,Cvvelocityandpositioncontrollerparametersforthetransla- tionloopandCrp,Crvaretherotationalpositionandvelocitycontroller parametersfortherotationlooprespectively. ConstantsKvbandKvbT aresignal-to-effortservomotorconstantsasdiscussedbefore.Basedon theinertiapropertiesoftheMechaCelldevicestheseunitygaintransfer

functionsaretunedtohavecriticallydampednaturesothattheresponse obtainedwillhavenoovershootbutfastresponse.Thevalueselected forCp alsoaffectsthenoiserejectionproperty ofthesystem.TheCp andCvvaluesselectedforthes-controlloopare10and29respectively.

Forthe𝛾-controlloop,0.1and180areusedforCrvandCrpparameters respectively.

4. Simulationandexperiments

Afterthemathematical modelandthemotioncontrollerarepre- paredasexplainedintheprevioussections,anexperimentalsetupwas constructedforvalidation.Theexperimentalsetupinthelaboratorycon- sistsoffivemajorcomponents:aPCwithBluetoothcapability,atable topplatform,anoverheadcamera,apower-source,MechaCellsanda workpieceasshowninFig.13.

Devicetrackingalgorithmisbasedoncolorrecognitionandtheinfor- mationisthensendtotheMechaCellsthroughBluetooth.Experiments areperformedonaplatformoverwhichtheoverheadcameraisplaced formotiontracking.Theplatformis50×50cmanditsfloorismadeof Styrofoam.MechaCellsarepoweredwithanexternalpower-sourceof 10[V].ToeliminatetheinfluenceofatetheronMechaCell’smotiona verythinwire(magneticwire)isused.

Usingtheconstructedexperimentalsetup,singleMechaCellpathfol- lowingexperimentsareperformedtoseetheeffectivenessofthecontrol algorithmdeveloped.Fortherealexperimentssamecontrollercalibra- tionisusedasintheMatlab/Simulinksimulationsasexplainedinthe previoussectionandFigs.9and11.Pathtrackingsimulationandex- perimentresultsarepresentedinFig.14.InTable2asnumericalerror parameters,deviationfromdesiredpositioninxandy,xerr,and,yerr, respectivelyanddeviationfrompath,𝜖,fromtheexperimentandsimu- lationsarelisted.

AsitcanbeseenfromFig.14,simulationresultsaremuchbetter thanreal-lifeexperimentsbecauseofthenoiseanddisturbanceinthe realsetupmostimportantlycomingfromtheoverheadtrackingcam- era.Startingpointsofalldataisnormalizedfor(−9cm,−10cm)point forfaircomparison.However,whentheexperimentsarecomparedto eachother,theextendedLuGrecompensationwhichincludestheeffect

Fig.13. Experimentalsetupinthelaboratory.

Table2

Numericalvaluesoftheerrorparametersfromtheexperiment andthesimulations.

x err [ m ] y err [ m ] 𝜖[ m ]

Simulation: Extended Lugre

0.00030 0.00035 0.00015

Experiment 1: Original LuGre

0.00490 0.00360 0.00140

Experiment 2: Extended LuGre

0.00420 0.00290 0.00100

Experiment from [14] : PID + Coulomb

0.00470 0.00330 0.00120

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Fig.14. ResultsofthepathfollowingtasksofasingleMechaCellwithexperimentsandsimulation.

of rapiddirectionchange of theMechaCelltracksthepathbetter in duringsharpturns.Whenthedirectionchangeisslow,bothalgorithms behavethesimilarasexpected.Thetrackingresultsarealsotabulatedin Table2.TheextendedLuGrecompensatorerrorresultsarebetterasthe systemperformsbetterinsharpturnsofthe“B” shape,andoverall14%

improvementinthecontourerror.TheextendedLuGrealgorithmalso performedbetterthanthehighlycalibratedPIDcontrollerusedin[14]. Ourexperimentsandcalculatederrordatashowthat asuccessful motioncontrollerisdevelopedfortheMechaCellsystemincludingadi- rectionawareLuGrefrictioncompensation algorithm.Thiscontroller givesbettertrackingresultsthanusingaconventionalfrictioncompen- satoronly.InFig.15,themotorcontrolcommandandthefrictioncom- pensationisgivenfortheB-shapeexperimentpresentedinFig.14“Ex- tendedLuGre” version.Inordertoshowtheeffectofthedirectional compensationtheimportantpartsthatcorrespondstodifferentpartsof

thetravelarehighlightedinboxeswithnumbers.Indashedboxnum- bered1,thedevicemakesitsfirstsharpturnandthecompensationre- actstothatwithincreasedcommandeventhoughthespeed(slopeof thefirstplot)didnotchangeconsiderably.Indashedbox2,thisisre- peatedforthenextsharpturnintheB-shape.Thereasonforthedual correctionisthefactthatoursteeringmotoronlyrotatesbetween0 and180andchangesrotationdirectionforanglesbetween180 and 360asexplainedin[14].Therefore,itnegotiatesthissharpturnintwo commandsasshowninthesecondsubplotwheresteeringcommand𝜙 isgiven.Indashedbox3,theMechaCelldevicereachestotheendof thepath,howeverduetothecyclicfeedingofourreferencealgorithm thedevicetriesthenextB-shapebeforetheexperimentisstopped.This extramovementcanalsobeseeninFig.14.Deviationfrompatherror parameter’svalueisinsub–centimeterscale,whichisasuccessfulpath trackingforoutrequirements.

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0 5 10 15 20 25 30 35 40 0

0.2 0.4 0.6 0.8

Displacement S [m]

0 5 10 15 20 25 30 35 40

-300 -250 -200 -150 -100 -50 0

Heading Angle [degree]

0 5 10 15 20 25 30 35 40

0 100 200 300 400 500

Motor Command [PWM]

0 5 10 15 20 25 30 35 40

0 1 2 3 4 5 6

Friction Compensation [PWM]

-10 -9 -8 -7 -6 -5 -4 -3 -2

x [cm]

-10 -8 -6 -4 -2 0 2 4 6 8 10

y [cm]

Reference

Fig.15. ResultsofthepathfollowingtasksofasingleMechaCellwithexperimentsandsimulation.

5. Conclusion

Workpresentedinthispaperoutlinesthemotioncontrollerdesign ofamodularmechatronicdevice,theMechaCell.MechaCellissimple modulethatcontainssensor,actuatorandcontrolfeaturesandasapack, MechaCellsmayabletoovercomecomplicatedtaskssuchasworkpiece manipulationonasurface.Ascalableandnovellocomotionsystemis integratedinthedesignontheMechaCellthatusesamechanismthat convertssteerableunbalancedmassmotionintocontrollerandprecise translationalmotion.

Themotioncontrollerfeaturesthreesub-systems:atranslationalcon- troller(s-controller),steeringandorientationcontrollers(𝜙 and𝛾 con- trollers).Thecontrollerusesanimprovedfrictioncompensationalgo- rithmwithcorrectedinertialeffectsthatlinearizestheplantdynamics.

Frictionforceisestimatedusingthemathematicalmodeldevelopedfor theMechaCell.Apopularfrictionestimationapproach,LuGrefriction model,isusedandthismethodisextendedsothatrapidchangesinthe directionofmotionisalsocompensated.Moreover,theestimationalso usestheinertialaccelerationoftheunbalancedmassbyconsideringmo- tionofthepartsofthedevice.Oursimulationandexperimentalresults showthedesignedtranslationsystem(controllerandthemechanism) workswellincludingtheextensiononthefrictionalgorithmwhichpro- videsadditionalimprovementsupto14%forcontourtrackingerroras comparedtotheresultsgivenin[14]thatwasobtainedusingPIDcon- trollers,estimationbasedonCoulombfrictionandrelativeacceleration oftheunbalancedmassonly.Webelievecurrentresultsareagoodstep intherightdirectionforprecisemotionapplicationsalthoughstillinits earlystages.However,theremaysomeapplicationsinthenearfuture, forexample,torepairthesurfaceofhardtoreachsurfacesbyadditive manufacturingorlasersinteringoffragilesurfaceswheretheprecision requirementscanbemoreforgiving.

Futureworkwillinclude,improvingtheorientation(𝛾-)controlus- ingdetailedplantdynamics,miniaturizationoftheMechaCelldevice translation moduleandimplementationof thissystem withmultiple MechaCellsformingapacktoperformaprecisiontaskaspositioningfor lowimpactmanufacturingdevices.Communicationdelayisnotstudied inthisworkandisafactorwhenmorethanonedevicesarecontrolled togetheraspartofapack.

DeclarationofCompetingInterest

Theauthorsdeclarethattheyhavenoknowncompetingfinancial interestsorpersonalrelationshipsthatcouldhaveappearedtoinfluence theworkreportedinthispaper.

Acknowledgments

Research supported by European Commission through Seventh Framework Program for Research and Technological Development underthecontractPIRG07-GA-2010-268420.

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Stefan Ristevski was a graduate student at Bilkent Univer- sity Mechanical Engineering Department where he received his master’s degree. After spending some time in the industry gaining experience in real-time control systems and HMIs, he is currently a Ph.D. Candidate at University of South Florida.

Melih Cakmakci is an Assistant Professor of Mechanical En- gineering at Bilkent University in Ankara, Turkey. He re- ceived his B.S degree in Mechanical Engineering from M.E.T.U Ankara in 1997. He received his M.S and Ph.D. in Mechanical Engineering Degrees from University of Michigan in 1999 and 2009 respectively. His research areas include modeling, anal- ysis and control of dynamic systems, Prior to joining Bilkent University, he was a senior engineer at the Ford Scientific Re- search Center at Dearborn, Michigan, USA. He is a member of ASME, IEEE and SAE.

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