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

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Mechatronics

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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.

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

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′axisandS_D,a vectorpointstodirectionofunbalancedmasslinkintheX’Y’plane.The steeringcylinderisrotatedwithaservomotorhousedinthebaseand itsorientationrelativewiththeX′axisisgivenas𝜙.AlthoughaxesZ andZ′arealwaysparallelpertheproblemsetup,theXYandX′Y′pairs mayhavedifferent orientationsduetotherotation oftheMechaCell duringmotion.AnimportantparametershownasasidenoteinFig.3is thebasecylinderrotation,𝛾,whichisdefinedastheanglebetweenthe fixedaxisXandmovingandrotatingaxisX′.

Astheun-balancedmass,m,rotatesaplanarreactionforce,𝐅^′_𝑢,is appliedtothebasefromthelinkatconnectorA.TheMechaCellframe movesdepending on thecurrent rotationalspeed of theunbalanced mass,theamountoftheunbalancemassovercomingtheforcesduethe friction,F_f,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

Fig.4.FreebodydiagramofthesysteminFig.3.

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

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

𝐚_𝐦=𝑎_𝑚,𝑥𝐢+𝑎_𝑚,𝑦𝐣+𝑎_𝑚,𝑧𝐤 (1)

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

𝐅_𝐮=𝑚𝐚_𝐦+𝑚𝑔𝐤 (2)

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

Fig.5(a)thecomponentsoftheaccelerationoftheunbalancedmassis shownforasimplelinearmotioncase.Itcanbeseenclearlythatthe acceleration,a_M,ofthebasecylinderisalsoacontributortotheinertial accelerationoftheunbalancedmass.Itisalsoimportanttonotethatad- ditionaltermsmaybecontributorssincetherotationofthebasecylinder andthesteeringmechanismalsocancauserotationtotheunbalanced massasshowninFig.3.Thetotalacceleration,a_m,oftheunbalanced massofthesystemcanbecalculatedbyusingtherelativeacceleration equationgivenin(3)byassumingXYZasthefixedaxis(FCS)andX′Y′Z′ asthemovingaxis(MCS)coordinatesystems.

𝐚_𝐦=𝐚_𝐌+̈𝛄 × 𝐫_𝐮+𝐚_𝐫𝐞𝐥+ ̇𝛄 × (̇𝛄 × 𝐫_𝐮)+2(̇𝛄 × 𝐯_𝐫𝐞𝐥) (3) In(3),a_M,̇𝛄 and̈𝛄 aretheacceleration,angularvelocityandangular accelerationoftheMechaCellbase(i.e.themovingframe)respectively.

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

AsshowninFig.4,forcesactingontheMechaCellsystemarethe weightandnormalforcesintheZ′direction,frictionforce,F_f,inthe oppositeofthevelocitydirection,S,andreactionforce,F_utherotating intheZ′S_Dplane.ApplyingNewton’sSecondLawtotheMechaCellwith mass,M,wecanobtainthedynamicmotionequationsasgivenin(4)– (6).

−𝑀𝑔+𝐹_𝑢sin(𝜃)+𝑁=𝑀𝑎_𝑀,𝑧=0 (4)

𝐹_𝑢cos(𝜃)𝑐𝑜𝑠(𝜙 +𝛾)−𝐹_𝑓cos(𝛽 +𝛾)=𝑀𝑎_𝑀,𝑥 (5) 𝐹_𝑢cos(𝜃)𝑠𝑖𝑛(𝜙 +𝛾)−𝐹_𝑓cos(𝛽 +𝛾)=𝑀𝑎_𝑀,𝑦 (6)

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

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 ]

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

𝐼̈𝛾 =𝑇_𝑠+𝑇_𝑢−𝑇_𝑓 (7)

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

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

𝐅_𝐜=−𝜇_𝑘𝑁𝐞_𝐯=−𝜇_𝑘(𝑀𝑔−𝐹_𝑢sin(𝜃))𝐞_𝐯 (8) where𝜃 isthecurrentorientationoftheunbalancedmasslink,e_visa unitvectorinthedirectionofthevelocityoftheMechaCellanditsinthe directionofthetranslationvectorS.𝜇kisthefrictioncoefficientwhen thedeviceisinmotion.Thecalculationofthefrictionforce withthe Coulombapproachresultsinasimpleandquickestimation.However, thefrictionforceisknowntochangewithvariousfactorsasnotedby manyresearchersandmoredetailedmodelsexist[18–21]includingthe flexibilityofthecontactinteractionsanddirection.Oneofthesemod-

elsisknownastheLuGremodelandcanberepresentedasshownin (9)–(12).

𝐹_𝑓=𝜎0𝑧+𝜎1̇𝑧+𝑓(𝑣) (9)

̇𝑧=𝑣−𝜎0|𝑣|∕𝑔(𝑣)𝑧 (10)

𝑔(𝑣)=𝐹_𝑐+(𝐹_𝑠−𝐹_𝑐)𝑒^{−|𝑣∕𝑣𝑠|𝛼} (11)

𝑓(𝑣)=𝜎2𝑣 (12)

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

IntheoriginalformoftheLuGremodel,frictionforcegenerateddue tothetranslationmotioninthedirectionofmotionisconsidered.How- ever,inoursystemthereisalsorapiddirectionalchangeofmotionthat affectthefrictionforceexperiencedbytheMechaCellduetorapidro- tationofthesteeringbase(i.e.rapidchangeof𝜙).Therefore,asecond stablestate,𝜉 wasaddedthatbecomesactiveasthedevicechangesF_u directionontheXYplaneandquicklyconvergestozeroifthenewdirec- tionisconservedasshownin(13).Incalculations,toobtainareasonable delay,𝜎3isusedas0.04,hfunctionwassetto𝜋| ̇𝜙| and𝜎4=0.05N/rad wasused.Thecalculatedeffectoftherotationalangleextensiontothe LuGrefrictionmodelispresentedinFig.6(b).Thedatapointsshowthe calculatedvalueoffriction,F_f,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−𝐞_𝐯direction.

𝐹_𝑓=𝜎0𝑧+𝜎1̇𝑧+𝑓(𝑣)+𝜎4𝜉 (14)

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

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

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

Fig.7. Comparisonofthedynamicmodelswithdifferentfrictioncalculations.

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.

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

InFig.10,desiredtranslationanddirectiongenerationbasedonthe desiredCartesiancoordinatesoftheMechaCelldeviceispresented.At atypicalinstantMechaCellshouldtravelfromitscurrentposition,R_M, followingthevectorstartingatitscurrentpositionandendingatthe desiredposition,R_D. ThisvectorisshownasS_D.Hence, tomoveto- wardsthedesiredpoint,theservocontrollershouldpositionthesteer- ingcylindersuch that∠𝐒_𝐃=𝜙 +𝛾 where𝛾 is theorientationof the MechaCellwithrespecttothefixedframeXYand𝜙 istheangleofthe unbalancedmasslink withrespecttothemovingandrotatingframe X′Y′.

Thedesiredtranslationreferenceusedbythes-controller,andthe servocontrollerisshownin(15).

𝑠_𝑑𝑒𝑠=|𝐑_𝐃−𝐑_𝐌| (15)

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

𝜙_𝑑𝑒𝑠=∠𝐒_𝐃−𝛾 (16)

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

Duringthesimulationsandexperimentsperformedforthiswork,the desiredorientationoftheMechaCell,𝛾des,iscommandedas0^∘sothat 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.

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,q_s,servocommand,𝜙des,devicetranslationalac- celerationcomponents,{̈𝑥,̈𝑦},rotationalvelocityandacceleration(i.e.

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

Whenthefrictionestimationalgorithmisimplementedwithprop- erly calibrated parameters, theestimationerror, 𝑒_{𝐹 ,𝑓}=𝐹_{𝑓,𝑒𝑠𝑡}−𝐹_𝑓 is small,thecontributionfromthefrictionforcecanbeassumedasadis- turbanceandthecontrollerscanbedesignedaccordinglyasasecond ordersystemwithinertiadynamics.Infact,withe_f≈0thetransferfunc- tionsrepresentingthetranslationdynamics (i.e.𝐺_𝑠(𝑠)=𝑆(𝑠)∕𝑆_𝑑𝑒𝑠(𝑠)) andtherotationdynamics(i.e.𝐺_𝛾(𝑠)=𝛾(𝑠)∕𝛾_𝑑𝑒𝑠(𝑠))in Fig.11can be givenasshownin(17)and(18)respectively.

𝐺𝑠(𝑠)= 𝐶_𝑝𝐾_𝑣𝑏

𝑚𝑠²+𝐶_𝑣𝐾_𝑣𝑏𝑠+𝐶_𝑝𝐾_𝑣𝑏 (17)

𝐺_𝛾(𝑠)= 𝐶_𝑟𝑝𝐾_𝑣𝑏𝑇

𝐼𝑠²+𝐶_𝑟𝑣𝐾_𝑣𝑏𝑇𝑠+𝐶_𝑟𝑝𝐾_𝑣𝑏𝑇 (18)

whereC_p,C_vvelocityandpositioncontrollerparametersforthetransla- tionloopandC_rp,C_rvaretherotationalpositionandvelocitycontroller parametersfortherotationlooprespectively. ConstantsK_vbandK_vbT aresignal-to-effortservomotorconstantsasdiscussedbefore.Basedon theinertiapropertiesoftheMechaCelldevicestheseunitygaintransfer

functionsaretunedtohavecriticallydampednaturesothattheresponse obtainedwillhavenoovershootbutfastresponse.Thevalueselected forC_p alsoaffectsthenoiserejectionproperty ofthesystem.TheC_p andC_vvaluesselectedforthes-controlloopare10and29respectively.

Forthe𝛾-controlloop,0.1and180areusedforC_rvandC_rpparameters 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,x_err,and,y_err, 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

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^∘ and180^∘andchangesrotationdirectionforanglesbetween180^∘ and 360^∘asexplainedin[14].Therefore,itnegotiatesthissharpturnintwo commandsasshowninthesecondsubplotwheresteeringcommand𝜙 isgiven.Indashedbox3,theMechaCelldevicereachestotheendof thepath,howeverduetothecyclicfeedingofourreferencealgorithm thedevicetriesthenextB-shapebeforetheexperimentisstopped.This extramovementcanalsobeseeninFig.14.Deviationfrompatherror parameter’svalueisinsub–centimeterscale,whichisasuccessfulpath trackingforoutrequirements.

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.

References

[1] Cakmakci M, Ulsoy AG. Improving component-swapping modularity using bidirec- tional communication in networked control systems. IEEE/ASME Trans Mechatron 2009;14(3):307–16. doi: 10.1109/TMECH.2008.2011898 .

[2] Cakmakci M, Ulsoy AG. Swappable distributed MIMO controller for a VCT engine. IEEE Trans Control Syst Technol 2011;19(5):1168–77. doi: 10.1109/

TCST.2010.2080275 .

[3] Grosu V, Rodriguez-Guerrero C, Grosu S, Vanderborght B, Lefeber D, et al. Design of smart modular variable stiffness actuators for robotic assistive devices. IEEE/ASME Trans Mechatron 2017. doi: 10.1109/TMECH.2017.2704665 . 1–1

[4] Suryadevara NK, Mukhopadhyay SC, Kelly SDT, Gill SPS, et al. WSN-based Smart sensors and actuator for power management in intelligent buildings. IEEE/ASME Trans Mechatron 2015;20(2):564–71. doi: 10.1109/TMECH.2014.2301716 .

[5] Ulrich K , Tung K . Fundamentals of product modularity. New York: ASME; 1991.

p. 73–9 .

[6] AlGeddawy T. A DSM cladistics model for product family architecture design. Pro- cedia CIRP 2014;21:87–92. doi: 10.1016/j.procir.2014.03.122 .

[7] Yim M, Shen W-m, Salemi B, Rus D, Moll M, Lipson H, Klavins E, Chirikjian G, et al. Modular self-reconfigurable robot systems [grand challenges of robotics]. IEEE Robot Autom Mag 2007;14(1):43–52. doi: 10.1109/MRA.2007.339623 .

[8] Kamimura A, Kurokawa H, Yoshida E, Murata S, Tomita K, Kokaji S, et al. Automatic locomotion design and experiments for a modular robotic system. IEEE/ASME Trans Mechatron 2005;10(3):314–25. doi: 10.1109/TMECH.2005.848299 .

[9] Wei H, Cai Y, Li H, Li D, Wang T, et al. Sambot: a self-assembly modular robot for swarm robot. In: 2010 IEEE international conference on robotics and automation.

IEEE; 2010. p. 66–71. doi: 10.1109/ROBOT.2010.5509214 .

[10] Piranda B, Laurent GJ, Bourgeois J, Clévy C, Möbes S, Fort-Piat NL, et al. A new con- cept of planar self-reconfigurable modular robot for conveying microparts. Mecha- tronics 2013;23(7):906–15. doi: 10.1016/j.mechatronics.2013.08.009 .

[11] Rubenstein M, Ahler C, Nagpal R. Kilobot: A low cost scalable robot system for collec- tive behaviors. In: 2012 IEEE international conference on robotics and automation.

IEEE; 2012. p. 3293–8. doi: 10.1109/ICRA.2012.6224638 .

[12] Caprari G., Balmer P., Piguet R., Siegwart R., et al. The autonomous micro robot

“Alice ”: a platform for scientific and commercial applications. In: MHA’98. pro- ceedings of the 1998 international symposium on micromechatronics and hu- man science. – creation of new industry - (Cat. No.98TH8388). IEEE. 231–235.

doi: 10.1109/MHS.1998.745787 .

[13] Cakmakci M., Ristevski S. A motion system. 2018. https://patents.google.

com/patent/US20170066134A1/en .

[14] Ristevski S, Cakmakci M. Mechanical design and position control of a modular mechatronic device (MechaCell). In: 2015 IEEE international confer- ence on advanced intelligent mechatronics (AIM). IEEE; 2015. p. 725–30.

doi: 10.1109/AIM.2015.7222623 . vol. 2015-Augus

[15] Ristevski S., Cakmakci M. Mathematical model for coordinated motion of modular mechatronic devices (mechacells). In: Volume 2: diagnostics and detection; drilling;

dynamics and control of wind energy systems; energy harvesting; estimation and identification; flexible and smart structure control; fuels cells/energy storage; hu- man robot interaction; HVAC building energy M. ASME. 2015, p. V002T34A010.

10.1115/DSCC2015-9896 .

[16] Dolcini PJ, Canudas de Wit C, Béchart H. Dry clutch control for automotive appli- cations. London: Springer; 2010. doi: 10.1007/978-1-84996-068-7 .

[17] Armstrong-Helouvry B. Control of machines with friction, 128. Springer;

1991 . http://www.springer.com/gp/book/9780792391333#otherversion = 978146 1367741 .

[18] Astrom KJ, Canudas-de Wit C. Revisiting the lugre friction model. IEEE Control Syst 2008;28(6):101–14. doi: 10.1109/MCS.2008.929425 .

[19] Lee TH, Tan KK, Huang S. Adaptive friction compensation with a dynamical friction model. IEEE/ASME Trans Mechatron 2011;16(1):133–40. doi: 10.1109/

TMECH.2009.2036994 .

[20] Ruderman M. Mechatronics presliding hysteresis damping of LuGre and Maxwell-slip friction models. Mechatronics 2015;30:225–30. doi: 10.1016/

j.mechatronics.2015.07.007 .

[21] Keck A, Zimmermann J, Sawodny O. Friction parameter identification and com- pensation using the elastoplastic friction model. Mechatronics 2017;47:168–82.

doi: 10.1016/j.mechatronics.2017.02.009 .

[22] Tustin A. The effects of backlash and of speed-dependent friction on the stabil- ity of closed-cycle control systems. J Inst Electr Eng– Part IIA 1947;94(1):143–51.

doi: 10.1049/ji-2a.1947.0021 .

[23] Holmes DG, Lipo TA. Pulse width modulation for power converters : principles and practice. John Wiley; 2003 . http://cds.cern.ch/record/1480869 .

[24] Freidovich L, Robertsson A, Shiriaev A, Johansson R, et al. LuGre-model-based friction compensation. IEEE Trans Control Syst Technol 2010;18(1):194–200.

doi: 10.1109/TCST.2008.2010501 .

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.