ContentslistsavailableatScienceDirect
Sensors
and
Actuators
A:
Physical
jo u r n al hom e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / s n a
Short
communication
Perfect
collocation
using
self-sensing
electromagnetic
actuator:
Application
to
vibration
control
of
flexible
structures
Mohit
Verma
a,∗,
Vicente
Lafarga
b,
Christophe
Collette
b,caCSIR-StructuralEngineeringResearchCentre,CSIRCampus,Taramani,Chennai-600113,India
bUniversitéLibredeBruxelles,PrecisionMechatronicsLaboratory,BEAMSDepartment,50,F.D.RooseveltAv.,1050Brussels,Belgium cUniversityofLiège,DepartmentofAerospaceandMechanicalEngineering,AlléedelaDécouverte9,4000Liège,Belgium
a
r
t
i
c
l
e
i
n
f
o
Articlehistory:Received11May2020
Receivedinrevisedform6July2020 Accepted15July2020
Availableonline17July2020
Keywords: Vibrationisolation Self-sensing Collocatedcontrol Activedamping Electromagneticactuator
a
b
s
t
r
a
c
t
Collocatedcontrolhastheadvantagesofrobustnessandguaranteedstability.Incollocatedcontrol,the
sensorandactuatorareplacedclosetoeachother.Truecollocationcanbeachievedusingself-sensingin
whichtheactuatorcanalsobeusedasasensor.Inthisarticle,wepresentaself-sensingelectromagnetic
actuatorforvibrationcontrolofflexiblestructures.Thebackelectromotiveforce(emf)generatedinthe
coilismeasuredtoevaluatethevelocityofthestructure(andhence,thedisplacement).Theposition
measurementsobtainedfromself-sensingarefoundtohavegoodcorrelationwiththoseobtainedusing
aneddycurrentsensor.Theefficacyoftheproposedtechniquehasbeenverifiedforthevibrationcontrol
ofaflexiblecantileverbeam.Self-sensingofferseconomical,simpleandrobustcontrolarchitecture.It
canbeeffectivelyusedforapplicationswheresensorsandactuatorscannotbecollocatedduetospace
andsizelimitations.
©2020ElsevierB.V.Allrightsreserved.
1. Introduction
External disturbances like ground motion or environmental disturbancescangreatlyaffecttheperformance ofthesensitive equipment.Activesystemscanbeusedtoisolatetheequipment fromtheexternaldisturbances[1].Sensorsandactuatorsinmany activesystemsareoftencollocatedinordertobuildrobustnessinto thesystem.Theopenlooptransferfunctionofcollocatedsystems hasalternatingpolesandzerosontheimaginaryaxis.Insuch sys-tems,thestabilityisguaranteedasthephaselagduetoeachpoleis compensatedbythephaseleadduetothecorrespondingzero[2]. Thecollocationofsensorsandactuatorsishowevernotpossibleat alltimes.Theremightbespacerestrictionsincertainapplications. Onemightencounterasituationwherethesizeofthesensorisnot negligibleincomparisontothemainstructure.Insuchcases,the mechanicalimpedanceofthestructureisgreatlymodifiedwhen thesensorsareinstalledonthestructure.
Self-sensingprovidesanalternativetoovercomethese prob-lems. Perfect collocation can be achieved using self-sensing in whichasingletransducerisusedasanactuatorandsensor con-currently.Itoffersasimple,robustandcost-effectivesolutionfor
∗ Correspondingauthor.
E-mailaddress:mohitverma@serc.res.in(M.Verma).
collocatedcontrol.Self-sensingreducesthecomplexityby reduc-ingthenumberofactuatorsandsensorsinthesystem.Italsoleads toreductioninthepowerrequirementsandthewiringusedin theinstrumentation.Sensorsaremorepronetofailurecompared totheactuatorsinthecontrolsystem[3].Thus,self-sensingalso leadstoarobustsystemasasingletransducerservesthedual pur-poseofsensingandactuation.Manyself-sensingtransducershave beendevelopedinthepastusingdifferenttypesofactive materi-als.Self-sensingusingpiezoelectricmaterialshavebeenusedfor measuringstress,forceandposition[4–7].Adielectricelastomer actuatorhasbeenusedforpositionsensingandforcecontrol[8,9]. Insomerecentapplications,shapememoryalloys[10]and mag-netorheologicalfluid[11]basedactuatorshavealsobeenusedfor self-sensing.Electromagneticactuatorshavebeenusedforvelocity, positionandforceestimations[12–14].
Inthisarticle,wepresentaself-sensingelectromagnetic actua-torforvibrationcontrolofflexiblestructures.Thepresentscheme doesnotrequireanymodificationstothedesignofthe electro-magneticactuator.Ittakesadvantageofthefactthatthevoltage dropacrossthecoilcanbemeasuredandrelatedtoitsmotion.The electromagneticactuatorisoperatedincurrentmode.Itdoesnot requireanadditionalbridgestructureformeasuringthevoltage dropacrossthecoil(whichisusuallyrequirediftheactuatoris operatedinvoltagemode).Theorganizationofthisarticleisas fol-lows:Section2presentscomparisonofcollocatedcontrolwiththe https://doi.org/10.1016/j.sna.2020.112210
Fig.1. Locationofactuatorandsensorforcollocatedandnon-collocatedcontrol.
non-collocatedcontrol.Theconceptofself-sensingusing electro-magneticactuatorispresentedinSection3.Theproof-of-concept experimentscarriedouttovalidateself-sensingareexplainedin Section4.Theenvisionedapplicationforself-sensingispresented inSection5,whichisfollowedbyconcludingremarksinSection6.
2. Collocatedvsnon-collocatedcontrol
Inordertohighlighttheadvantagesofthecollocatedcontrol overnon-collocatedcontrol,we considera caseofacontinuous beampinnedatboththeends.UsingEuler–Bernoullibeamtheory, thedynamicsofthebeamcanbewrittenasthefollowingpartial differentialequation
∂
2∂
x2 EI∂
2 w∂
x2 +m∂
2 w∂
t2 =p (1)wherewisthedisplacement,EIrepresentsthebendingstiffness, misthemassperunitlengthandpistheexternalloadperunit length.Foracaseoffreevibrationandharmonicsolutionoftheform w(x,t)=(x)ejωt,theabovepartialdifferentialequationisreduced
tothefollowingordinarydifferentialequation d4 dx4 − m EIω 2 =0 (2)
The above equation is an eigenvalue problem whose solutions areωiandirepresentingnaturalfrequenciesandcorresponding
modeshapes,respectively.Forabeampinnedatboththeends,the naturalfrequenciesandthemodeshapesare
ω2 i =(i) 4 EI ml4 i(x) =sinixl (3)
Fig.2. Frequencyresponsesoftheopenlooptransferfunctionfromactuatorforcetothedisplacementmeasuredbythesensorincaseof—(a)collocatedcontroland(b) non-collocatedcontrol.
Fig.4. Modelofelectromagneticactuator.
wherelisthelengthofthebeam.Thefrequencyresponsebetween theactuatorforce,u,atpointxaandsensordisplacement,y,atpoint
xbcanbewrittenas[15] G(s)= y u = ∞
i=1 i(xa)i(xb) i(s2+2iωis+ω2i) (4)whereiisthegeneralizedmass(equaltoml/2forabeampinned
atboththeends)and i isthedampingassociatedwiththeith
mode.Theseriesexpansioncanbelimitedtoafinitenumberof modesincaseofadiscretesystem.Thesensorandactuator loca-tionforcollocatedandnon-collocatedcontrolareshowninFig.1. Incaseofcollocatedcontrolitisassumedthatboththeactuator andsensorarelocatedatxa=xs=0.5l.Fornon-collocatedcontrol,
itisassumed thattheactuatorislocatedatmidspan(x1=0.5l)
whilethesensorisplacedatxs=0.9l.Thebendingstiffness,EI,of
thebeamistakenas100Nm2,massperunitlength,m,istaken
as1kg/mand length,l,istakenas1m. Thefirsttenmodesare consideredforevaluationoftheopenlooptransferfunction.The openlooptransferfunctionsfromactuatorforcetothesensor dis-placementforcollocatedandnon-collocatedcontrolsareshownin Fig.2aandb,respectively.Itisobservedthatthecollocated sys-temhasalternatingpolesandzeros.Thephasedropduetoeach
Table1
Goodnessoffitbetweenthedisplacementsignalsobtainedfromeddycurrentsensor andself-sensing.
Rootmeansquared error(RMSE)
Sumofsquarederror (SSE)
Coefficientof determination(R2)
0.0072 1.939 0.9927
poleiscompensatedbythephasegainduetozero.Thephaseof theopenlooptransferfunctionisboundedbetween0and.This interlacingpropertyofthecollocatedsystemprovidesasymptotic stabilityevenwhenthesystemparametersaresubjectedtolarge variations.Collocatedsystemprovidesrobuststabilitytothe sys-tem.However,thispole-zerointerlacingpropertydoesnotexist inthenon-collocatedsystem.Thephaseisnotboundedbetween 0andandthesystemisnotrobustlystable.Thisfactisfurther illustratedbystudyingtherootlocusofthefeedbacksystem.The systemisassumedtohave afollowingleadcompensatorinthe feedbackloop
K(s)= 10s+1
0.005s+1 (5)
The root loci of thecollocated and non-collocatedsystems are showninFig.3aandb,respectively.Itcanbeseenthatfor col-locatedsystemtherootlocusliesentirelyinthelefthalfplane. Thismeansthatthesystemisstableforanyvalueofthegain.The rootlocusofthenon-collocatedsystemdoesnotlieentirelyinthe lefthalfplane.Thisimpliesthatforcertainvaluesofthegainthe closedlooppolesofthesystemwillmovetotherighthalfplane resultinginanunstablesystem.Duetothepole-zerointerlacing propertyandrobustlystablenature,collocatedsystemsareusually preferredwhereverpossible.
3. Self-sensingelectromagneticactuator
ThemodeloftheelectromagneticactuatorisshowninFig.4. Theactuatorcanbedrivenbyeitheravoltageorcurrentsource.The
Fig.6. Comparisonofthefrequencyresponsesobtainedfromtheexperimentand themodel.Agoodmatchbetweentheexperimentsandthemodelisobservedup to100Hz.
Fig.7.Comparisonofthedisplacementresponseofthefreeendofcantileverbeam obtainedfromself-sensingwiththatobtainedfromeddycurrentsensor.
Fig.8. Openloopgainoftheplantwiththedesignedleadcontroller.Thedesigned controllerhasacrossoverfrequencyat36.5Hzandhasaphasemarginof60◦.
modelingoftheactuatordependsonthemodeinwhichitis oper-ated.Theconstitutiveequationsoftheelectromagnetictransducer canbewrittenas
Vm =Zei+Tem
v
F =Tmei+Zm
v
(6) whereVmisthevoltagedropacrosstheterminalsoftheactuator
coil,Zeistheelectricalimpedanceofthecoil,iisthecurrentflowing
inthecoil,Temisacouplingcoefficientrepresentingthebackemf
constantofthecoil,
v
isthevelocityofthemechanicalstructure,Fis theforceactingonthemechanicalstructureattachedtothe actua-tor,TmeisthecouplingcoefficientrepresentingtheforcetocurrentratiooftheactuatorandZmisthemechanicalimpedanceofthe
attachedstructure.Theoperationoftheelectromagneticactuator withacurrentsourceensuresthatthedesiredcurrentflowsinthe coilirrespectiveofthebackemfgeneratedbythemotionofthe mechanicalstructure.Thelossesduetobackemfarecompensated bytheamplifier.UsingKirchoff’slaw,thevoltagedropacrossthe terminalsofthecoilcanbewrittenas
Vm=Ri+Ldi
dt+Tem
v
(7)whereRandLaretheresistanceandinductanceofthecoil, respec-tively.GiventheparametersR,LandTemoftheactuator,theabove
Fig.10.Comparisonofthefrequencyresponsesofthecontrolledanduncontrolled structureusingfeedbackfrom—(a)eddycurrentsensorand(b)self-sensing.
equationshowsthatitispossibletoevaluatethevelocityofthe
mechanicalstructurebymeasuringthevoltagedropacrossthe ter-minalsofthecoil.Thevelocityofthemechanicalstructureisrelated toVmandibythefollowingequation
v
=Vm−Ri−Ldi dt
Tem (8)
TakingtheLaplacetransformoftheaboveequation,weget
v
=Vm−(R+Ls)iTem (9)
Fig.11.Comparisonofthetimedomaindisplacementresponseofthebeamwith andwithoutcontrolobtainedusingself-sensing.
wheresistheLaplacetransformvariable.Thedisplacementofthe mechanicalstructureisgivenby
x= 1 sTemVm−
(R+Ls)
sTem i (10)
In-housemanufacturedcurrent amplifieris usedin order to operatetheelectromagneticactuatorincurrentmode.Thecircuit diagramofthecurrentamplifierisshowninFig.A.13ofAppendix A.
4. Proofofconceptexperiment
Thevalidationoftheself-sensingelectromagneticactuatoris carriedoutonacantileverbeamclampedatoneendandattached tothevoicecoilactuatorontheotherend.Thetestsetupusedis showninFig.5.Aneddycurrentsensorisusedtotrackthemotion ofthefreeendofcantileverbeam.Self-sensingincurrentmodeis usedfortheproofofconceptexperimentduetoitssimplicity,ease ofimplementation,andavailabilityofthecurrentsource.
4.1. Identificationoftheplant
Theplant isidentified experimentally byevaluatingthe fre-quencyresponsefromtheinputcurrenttothedisplacementofthe freeendofthecantileverbeam.Awhitenoisesignalisinjectedin theactuatorandthedisplacementisrecordedusingtheeddy cur-rentsensor.Atransferfunctionoftheplant(G(s))isthenobtained usingfitfrdfunctioninMatlab[16].Theorderofthetransfer functionischosensuchthatitisabletocaptureallthemodesof theplantupto200Hz.Thetransferfunctionoftheplantis G(s)= 76.386(s2+25.82s+1494)(s2+1.803s+1.23e05)(s2+144.2s+1.268e06)
(s2+25.87s+1491)(s2+6.88s+2932)(s2+3.461s+1.528e05)(s2+144s+1.451e06) (11)
Thefrequencyresponsesobtainedfromtheexperimentand the modelarecomparedinFig.6.Thefirstbendingmodeofthebeam isat8.6Hz. Itisobservedthatthefrequencyresponseobtained fromtheexperimentsmatcheswellwiththatobtainedfromthe modelupto200Hz.
4.2. Displacementfromself-sensing
Thedisplacementofthefreeendofthebeamisevaluatedusing Eq.(10).TheparametersoftheelectromagneticactuatorR,Land Temarefoundtobe6.2,1.23mHand3.9Vs/m,respectively.The
Fig.12. Configurationoftheproposedisolationsystemfordronecamerastabilization.
displacementobtainedusingEq.(10)ispassedthrough a high-passfilterinordertoavoidthedriftduetointegration.Thecutoff frequencyofthefilterissetto0.2Hz.Thedisplacementresponse obtainedfromself-sensingiscomparedwiththatobtainedfroman eddycurrentsensorinFig.7.Thedisplacementresponseobtained fromself-sensingandtheeddycurrentsensorarefoundtomatch wellwitheachother.Thegoodnessoffit,definedintermsofroot meansquarederror(RMSE),sumofsquarederror(SSE)and coef-ficientofdetermination(R2),betweenthedisplacementresponses
ofthebeamobtainedfromtheeddycurrentsensorandself-sensing aregiveninTable1.
4.3. Activevibrationcontrol
Adisplacementfeedbackcontrollerisdesignedtocontrolthe vibrationsofthe freeend of cantileverbeam.A lead controller is designed to have sufficient gain margin near the crossover frequency.Thedesignedcontrollerhasacrossoverfrequencyat 36.5Hzandhasaphasemarginof60◦.Thetransferfunctionofthe designedcontroller,H(s),is
H(s)=6094(s+95.54)
(s+1404) (12)
The open loop gain (G(s)H(s)) of the plant is shown in Fig. 8. Theblockdiagramforexperimentalverificationofself-sensingis showninFig.9.Abeamisexcitedbyinjectingawhitenoisesignal intheactuator.Thiswhitenoiseactsasanexternaldisturbance. Thedisplacementfeedbackfromtheeddycurrentsensorand self-sensingareusedtoreducetheeffectoftheexternaldisturbanceon themotionofthefreeendofthebeam.Thecontrolforceobtained usingthedisplacementfeedbackisthensubtractedfromthewhite noise.First,thefeedbackfromtheeddycurrentsensorisusedto seetheeffectivenessofthedesignedcontrolleronvibration reduc-tion.Thefrequencyresponsesoftheuncontrolledandcontrolled structurearecomparedinFig.10a.Itcanbeseenthatthedesigned controlleriseffectiveinreducingthevibrationofthecantilever beam.
Afterverificationofthecontroller,thesignalobtainedfrom self-sensingisusedforthefeedbackcontrol.Thenoiseintheself-sensed signaldrivestheactuatorwhich in turnforcesthestructure to followthenoise. Hence,thesignalobtainedfromtheeddy cur-rentsensor providesbetterindication of the performance. The uncontrolledandcontrolledfrequencyresponsesofthebeamwhen self-sensing is usedas feedback are compared in Fig. 10b. The feedback fromself-sensing is found toexhibit thesimilar per-formanceasthatobtainedbyusingfeedbackfromeddycurrent sensorforfrequenciesabove0.2Hz.Thecomparisonofthe con-trolledanduncontrolleddisplacementtimehistoriesofthebeam
obtainedusingself-sensingisshowninFig.11.Thisdemonstrates thatself-sensingcanbeeffectivelyusedforsensingandactuation simultaneously.Animportantadvantageofself-sensingisthatit resultsinperfectlycollocatedcontrol.
5. Envisionedapplication:Dronecamerastabilization
Dronecameraissubjectedtovibrationsfromthedronewhich hampersthequalityofthecapturedimages.Thecurrentsystem usespassiverubbermounts.Thesemounts,althougheffectivein reducing theresponsenearresonance, degradetheresponse at highfrequencies (adding dampingto thesystem). Many active mountshavebeendevelopedbuttheyworkonlyinonedirection [17,18].Thismotivatedthedevelopmentofasixdegreesof free-domisolationsystemfordronecamerawherehighqualityimages arerequired.Theisolationsystemisbasedonthecubic configu-rationoftheStewartplatform.Theproposedisolationsystemis showninFig.12.Theuniquefeaturesoftheisolationsystemare —useofnon-contactelectromagneticactuators,noflexiblejoints (clearancebetweenthecoilandmagnetisusedtoallowfor rota-tions),3Dprintedlightweightpartsandhighfrequencyroll-offas thereisnostiffnessinthelegs.Duetothecompactarchitecture andweightlimitations,itisnotpossibletohaveanarrayof sen-sorswhichcanbecollocatedwiththeactuators.Thismotivates theuseofself-sensing.Itdoesnotrequireanymodificationsinthe electromagneticactuators.Thevoltagedropacrosseachcoilcanbe measured.Themeasuredvoltagedropcanthenbeusedtoevaluate themotionineachofthelegsoftheisolator.Thisisthenusedas afeedbacktoactivelydamptheresonancesoftheisolationsystem withoutamplifyingthehighfrequencyresponse.Theproperties ofthesystemlikestiffnessmaychangewithtimeduetofatigueor otherenvironmentalfactors.Self-sensingprovidesarobustcontrol strategywhichcanbeusefulinsuchsituations.Currently,workis underwaytodevelopaprototypeofsuchisolationsystemusing self-sensingforcollocatedcontrol.
6. Concludingremarks
In many applicationsperfect collocationis not possible due tospace and size limitations. In suchcases, onemay resortto self-sensingwhereasingletransducerservesthedualpurposeof sensingandactuation.Thisbriefpresentedself-sensingusing elec-tromagneticactuator.Theactuatorisoperatedincurrentmode.The voltagedropacrossthecoilcanbedirectlymeasuredtoevaluatethe motionofthecoil.Noadditionalbridgestructuresarerequired.The measurementsfromself-sensinghavebeenvalidatedwiththose obtainedusingeddycurrentsensor.Themeasurementsfrom
self-sensinghavealsobeenusedtoactivelycontrolthevibrationsof theflexiblecantileverbeam.Theperformanceoftheactivecontrol usingself-sensingarefoundtobeatparwiththeactivecontrol usinganeddycurrentsensor.Self-sensingprovidesarobust, cost-effectiveandsimplecontrolarchitecture.Itresultsinaperfectly collocatedsystemwiththeguaranteedstabilityastheopenloop transferfunctionhasalternatingpolesandzeros.Itisenvisioned tobeusedfortheapplicationswherealargearrayofsensorscould notbeusedduetospaceandsizelimitations.
Authors’contribution
MohitVerma: conceptualization, methodology, experiments, data curation, investigation, software, writing – original draft. VicenteLafarga:experiments,datacuration,investigation, valida-tion,writing–review&editing.ChristopheCollette: conceptualiza-tion,fundingacquisition,supervision,writing–review&editing.
DeclarationofCompetingInterest
Theauthorsreportnodeclarationsofinterest.
Acknowledgements
Theauthors gratefullyacknowledgetheWallonieRegion for funding this research (WALInnov Projet Trusteye, Grant No. 1710040).The authorswouldliketothank MichelOséefor his inputsoncurrentamplifier.ThetechnicaldiscussionswithDimitri PironandAhmadPaknejadarealsoacknowledged.
AppendixA. Circuitdiagramofcurrentamplifier
Inordertousetheelectromagneticactuatorintheself-sensing mode,currentamplifierismanufacturedin-house.Thecircuit dia-gramofthecurrentamplifierisshowninFig.A.13.
AppendixB. Supplementarydata
Supplementarydataassociatedwiththisarticlecanbefound,in theonlineversion,athttps://doi.org/10.1016/j.sna.2020.112210.
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Biographies
MohitVermareceivedhisPh.D.inStructuralEngineeringfromAcSIR,India.After thathejoinedPrecisionMechatronicsLaboratoryatULB,BelgiumasPostdoctoral ResearchAssociate.Currently,heisworkingasSeniorScientistatCSIR-Structural EngineeringResearchCentre,India.Hisareaofinterestisvibrationcontrolof struc-tures.
VicenteLafargaisadoctoralcandidateatthePrecisionMechatronicsLaboratory atULBsince2017.Hisresearchareafocusesondevelopmentofaactiveisolation systemforspaceapplications.
ChristopheCollettereceivedaM.Sc.andPh.D.degreesinMechanical Engineer-ingfromtheUniversitéLibredeBruxellesin2003and2007.ChristopheColletteis ProfessorofMechatronicsintheFacultyofEngineeringattheUniversitéLibrede BruxellesandUniversityofLiége.HeistheDirectorofthePrecisionMechatronics Laboratory.Hismainresearchinterestsincludetheactiveandpassivecontrolof vibration,opticalinertialsensorsandfuturespacetelescopes.