BacklashDetectionin GearedMechanisms:Modeling, Simulation, and Experimentation

NilanjanSarkar���

RandyE. Ellis���

ThomasN. Moore�� Departmentof MechanicalEngineering�

Departmentof ComputingandInformationScienceQueen’s University, Kingston,CANADA K7L3N6

ABSTRACT

Backlashis acommonfault thatoccursin gearedmechanisms,andcanproduceinaccuracy or uncontrollabil-ity of themechanism.It is shown that,by modelingbacklashasmicroscopicimpact,its presencecanbedetectedandpossiblymeasuredusingonly simplesensors.The first stageof modelingestablishesthe needfor the useof nonlinearelasticanddampingforcesin establishingthegear-teethreactionforce. Thesecondstageof mod-eling useda detaileddigital multibodysimulationto developandtesttheeffectivenessof theproposeddetectionapproach.Experimentsverify theefficacy of themethodologyin practice.Finally, theresultscanbequantifiedfor a testbedmechanism,indicatingthat this methodologymaybeusefulfor conditionmonitoringof industrialmechanisms.

1. Intr oduction

Theuseof gearsin powertransmission,instrumentation,andasanintegralpartof positiondeviceshasbecomealmostubiquitous.Consequently, any faultsin geartransmissionseverelyaffect theperformanceof suchsystems.Thesefaultscanbediverseand,dependingonthesituation,canaffecttheperformanceof thesystemconsiderably.Commonexamplesof suchfaultsarebrokengearteeth,backlashbetweentheteeth,interferencebetweentheteeth,andeccentricmountingof matinggears.But regardlessof thesourceof degradedperformanceandthechancesof preventingthe occurrenceof suchfaultsaltogether, early detectionof the cause(faults) canprevent severedamagingconsequencesin many cases.

Thiswork concentratesonthedetectionof backlash betweenmatinggearteeth,for two reasons.First,it is themostcommonfault foundin any gearedsystems;althoughsomebacklashis essentialfor any geartransmission,lessthan the appropriateamountresultsin interferencebetweenthe teethwhereasexcessbacklashintroducesloosenessinto the system. In either case,the result is poor performanceand possibledamageto the system.Second,theeffect of backlashis mostconspicuouswhenthesystemis subjectedto non-continuousmotionwithfrequentreversalof thedirectionof rotation.Mechanismsthatfall into thiscategory includeroboticmanipulatorswhich have becomeanintegral partof manufacturingautomation.As theuseof roboticmanipulatorsincreases,thedetectionof faultswhichcanseverelyreducetheirperformancebecomesincreasinglyimportant.

Theprimaryobjective of this paperis to proposea measurethat canbeusedto detectbacklashin a gearedmechanism,a goal that is approachedby successively elaboratinga contactmodel. The first stepis the devel-opmentof a mathematicalmodelfor gearmechanismswith backlash.Second,a measureis developedthathas

IN: MechanicalSystemsandSignalProcessing11(3):391–408

thepotentialboth to detectsmallbacklashin themechanismandto describetheprogressionof backlashasthegear-teethwear. Third, a detaileddigital multibody simulatoris developed,to simulatebacklashandto detectits presenceappropriatelyusingtheproposedmeasure.Finally, physicalexperimentsareconductedto verify thepracticalityof theproposedmeasure.

This paperis organizedasfollows. Relevantliteratureis surveyedin Section2, andtheelementarydynamicsof spurgearsis reviewed in Section3. Section4 is devotedto the modelingof contactbetweengearteeth. Adiagnosticmeasurefor backlashis presentedin Section5,andamodelof onesuchmechanismis developedinto adigital dynamicssimulatorin Section6. Theresultsobtainedfrom thesimulationsarepresentedin Section7,withanexperimentaltestbedandexperimentalresultsdescribedin Section8. Quantificationof theresultsis presentedin Section9. Finally thereis a summaryof thecontributionsaswell asdrawbacksof thisapproach.

2. Literatur eSurvey

Whenthereis backlashbetweenthematinggearstheinitial contactcanmodeledasanimpactphenomenon.Gearimpactis generallyapproximatedby a linearmodel[14, 16, 15], althoughthelimitationsof thisapproxima-tion in impactmodelinghave beenfirmly established[13]. Thestudyof the transientbehavior of thegearpair,however, requiresan understandingof the exact form of the impactsbetweenthe gearteethandconsequentlythe researchinto this areaoften bearsmoreresemblanceto impactstudies[6, 7, 12, 13]. The dynamicmodelof matinggearsis actuallya rotary versionof the classic“impact pair”, first conceived by Kobrinskii [17] forthestudyof vibrationsin systemswith clearances.YangandSun[27] proposeda rotaryversionof this impactpair model. Theeffect of gearbacklashhasa destabilizingeffect on the controlleroncethe gearsbecomepartof a closed-loopcontrol system;transienteffectsof gearmotiondueto backlashandothernonlinearitiesactasdestabilizingforces,whichTustin[25] showedcanresultin limit-cycle instabilities.

Althoughmuchemphasishasbeenplacedon the dynamicmodelingof gearbacklash,the researchtowardsthedetectionof sucha fault hasbeenminimal. Signalprocessingandnoiseanalysisarewell developedsubjects[21,1], but they mostlydealwith signalsfrom continuouslyrotatingmachines.DagalakisandMyers[5, 4,3] werethefirst to proposea techniqueto detectbacklashin roboticsystems.Their techniquewasbasedon theanalysisof a coherencefunctionbetweenthemotorvoltageandtheaccelerationof the robotarm,given joint excitationby bandlimitedrandomsignals. Their method,althoughit may work in somesettings,hasthreeproblems:1)a coherencefunction is sensitive to noiseandmay not be usefulfor noisysignals,2) mountingaccelerometerson the drivenshaftsmay poseinstrumentationproblemsin somecases,and3) sincetheir methodis empirical,generalizationof the techniqueis not readily apparent.To overcometheseproblems,SteinandWang[22, 23]proposeda techniquebasedon the analysisof momentumtransferto detectbacklashin mechanicalsystems.They have derivedananalyticalexpressionrelatingthemagnitudeof thebacklashto thechangein thespeedoftheprimarygeardueto impactwith thesecondarygear. Both simulationsandexperimentshave beenconductedto verify theiralgorithm.Theiralgorithmis robustandeasyto implement,sinceit requiresonly avelocitysensorattachedto theservomotor. Therealsohavebeensomeeffortstowardsmechanicalfaultdetectionusingtechniquesderivedfrom artificial intelligence[2] aswell astechniquessuchasbuilding aLuenbergerstateobserverandusedit to detectincipientfaults[26].

Our work is, in a sense,complementaryto the work of Stein and Wang [22, 23]. Stein and Wang havedevelopedanalgorithmto sensethechangein velocity betweenthegearteeth,which canarisefrom tooth-toothimpactdueto backlash.Their methodactuallyis anintegrationof theforceof impactto obtainmomentum.We,ontheotherhand,concentrateontheforcedomaindirectly to obtainanimpulsesignalgeneratedat thegearteeth

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Figure1: Schematicdiagramof aspurgearpair.

interfaceandanalyseit to detectthe presenceof backlash.It is hypothesizedherethat backlashis a dynamicphenomenonthatcanbedetectedby physicalmanifestationsof thesmall impactsof thegearteeth. This paperdevelopsa techniquefor detectionof incipientbacklashthatrelieson impactsignals.

3. Gear Dynamics

Gearedmechanismscanvaryconsiderablydependingon thetypeof gearsusedin thesystems.A pairof spurgearswaschosenfor study, becauseit is the mostcommonlyfound gearpair andis amenableto considerablemodeling.Beforepresentationof thebacklashdetectionmodelandrelatedtechniques,it is necessaryto discusssomebackgroundconceptsrelatedto spurgearmeshing.

3.1 Spur-Gear Dynamic Equations

Whentwo involutegearteetharemeshed,thecontactpointwill alwaystravel alongthecommonnormalline,which is tangentialto the two basecircles. Thedetailsof spurgeardynamicequationshave beendevelopedbyYang[27] andGerdes[10], but thefinal equationsarepresentedherefor thesakeof completeness.

Whenthereis nocontact ���� �� �� � (1)����� �� �� � (2)

andwhenthereis contact �����������������! #"$� �� � (3)�����%���&�'�����! #"$� �� � (4)

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where �)( is the appliedtorque, ���*( is the basecircle radius, �( is the momentof inertia and� ( is the angular

displacementof gear+ . Thecontactforceis composedof two components:� — elasticrestoringforceand —dampingforce.Thesignof Equations(3) and(4) will changeif thedirectionof rotationchanges.

To testwhetherthereis contactor not, thereis acontactconstraintequationrelatedto backlash:, ����� � �-�.���&� � � , /10 (5)

where0

is thebacklashwhichmustbemeasuredalongthecommonnormal.

3.2 Number of Contacting Teeth

In orderfor gearsto operatesmoothly, a pair of teethmustcomeinto contactbeforethepreviouspair ceasesto touch. This requirementcreatesa periodof overlapwheretwo pairsof teetharein contactandthe effectivestiffnessof thegearpair increases.Theinteractionsbetweena pair of meshinggearsis a periodicprocessanditrepeatsaftereverybasepitch, 2 � , which is definedas:

2 ���$3'4 ���5 (6)

where5

is the numberof teeth. Thebasepitch is the samefor eachgear. Anotherimportantparameteris thecontactratio 687 , definedas 687 �:9<; 72 � (7)

where9<; 7 is thelengthof contactmeasuredalongtheline of action.

Thedetailedderivationcanbefoundin Gerdes[10]. Suffice it to saythatanalyticalexpressionscanbefoundto determinewhenthereis onepair of teethcontactandwhentherearetwo pairsof teethcontact.It is importantto notethat theseequationsinvolve =?> , theoperatingpressureangle.Since =?> is a functionof theactualcenterdistancebetweenthegearsgivenby

=?> �A@�B�C�D �FE �����G�!���&�H I (8)

thecontactregionsaredependenton theactualcenter-to-centerdistance(H

) andthereforeon backlash.WhenH

increases,thecontactratio decreasesandeventuallyfalls below unity, signifying that therewill beperiodswhentherewill benocontactatall betweenthegearsresultingin increasedvibration.

Tooth thinning is anothersourceof backlash,which mustbe accountedfor in an accuratedeterminationofthenumberof gearteethin contact.An analyticalexpressionfor backlashat any centerdistancecanbefoundinGerdes[10], andfrom it theexpressionfor linearbacklashcanalsobedetermined.

4. Contact Models

Thenatureof contactforcebetweenthegearteethis of utmostimportancefor thedevelopmentof a backlashdetectionmodel. Contactmodelscanbe distinguishedby differentmodelsof elasticforce,dampingforce andcoefficientof restitution.A brief discussionof someof themoreimportantmodelsfollows.

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4.1 Natureof Elastic Force

Theelasticforcedevelopedin gearteethcontactcanbeapproximatedby applyingHertziancontacttheorytotheteeth,assumingthattheinvoluteprofilesof theteetharecylindersat thepointof contact.Theexactanalyticalexpressionfor thehalf-widthof thesurfaceof contactcanbeobtainedfrom Timoshenko [24] or Goldsmith[11].YangandSun [27] have usedthis expressionto derive an expressionfor penetrationwhich givesa nonlinearrelationshipbetweentheappliedforceandtheelasticinterpenetration.They havealsolinearizedtherelationshipwithoutappreciablelossof accuracy. In suchacase,it ispossibletoobtainalinearforce-displacementrelationshipin theform �J�LKNM (9)

wherethestiffnessis KO� P�QSRTVU � DXWZY\[ , ] is thethicknessof thegears, is theYoung’smodulus,and _ is thePoisson’sratio.

Whentwo pairsof teetharein contact,thestiffnessis assumedto bedoublethis value. It shouldberemem-beredthat two main assumptionsaremadein deriving this expression:the radii of curvaturearemuchgreaterthanthe half-width, andthe axesof the cylinder areperfectlyparallelsuchthat ideal line contactoccurs. Al-thoughthefirst of theseassumptionsseemssafewith respectto thegearteeth,thevalidity of thesecondonemaybe questionable.To addressthis issue,we make useof Hertziancontacttheoryin its generalizedform whereforce-displacementrelationshipobeysapower-law �`�LKNM�a (10)

where b �dc for line contactand b �fe� for point contact.Hunt andCrossley [13] have suggestedthat the lawgivenin Equation(10)beusedwith apower chg b g e� .4.2 Natureof Damping Force

Differentmethodshave beenproposedto simulatethedampingforce,includinga linearmodel[6], a power-law relationshipof interpenetrationand approachvelocity [13], and a direct solution of the surfaceequation[12, 18].

Althougha linearmodelof thedampingforce is simplest,sucha modelexhibits two characteristicsthatareinconsistentwith the physicalsystems.The Kelvin-Voigt model,which is a linear modelof both stiffnessanddampingof theform � 7 �iKNMj�%kmlM predictsphysicallyimpossibleforcephenomena:adiscontinuityin theforceuponimpact,anda tensilecomponentin thehysteresisloop. (For a detailedstudyseeHunt [13]). A hysteresisloop thatdoesnot exhibit the inconsistenciesof a linearmodelcanbeobtained[13] from a contactforceof theform

`�:n3po KNM a lM (11)

whereo is foundfrom astraightline approximationof thecoefficientof restitutionasa functionof initial impactvelocity lMq( r �scj� o lMq( (12)

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A further refinementof this model, suggestedby HerbertandMcWhannell[12], incorporatesa nonlinearcoefficientof restitutionof theform

r �tcj� o lM�u( .A final approachfor obtainingthecontactforceis to assumea coefficient of restitutionandsolve thesurfaceequationdirectly undertheboundaryconditionssuchthatthedampingforcegoesto zerowhen Mv�tw andwhenlMx�Lw [12, 18]. YangandSunhave takenthisapproach[27] andtheirexpressionfor thecontactforceis:

� 7 �iKNM a � y{z cj� r�|� z 3 r ��c | � � n " K lM)( M a lM (13)

Thehysteresislooppredictedby Equation(11)andEquation(13)differsby c�}�~ percent[12]. Equation(11) isusedto simulatethedampingforcefor ourbacklashdetectionmodelbecauseit exhibitsgreaternumericalstabilityat low contactvelocities.

4.3 Coefficientof Restitution

Approximationof thecoefficient of restitutionof an impactasa linear functionof impactvelocity causesaseverereductionin theaccuracy of thecontactmodel[13], whichis especiallysignificantwhentherearerepetitiveimpacts.Herbert[12] showedthataccuracy is improvedconsiderablyby modelingthecoefficientof restitutionasr �Jc�� o lM u( . Thevaluesof o and� canbeobtainedfromexperimentalimpactdata,e.g.fromGoldsmith[11]. Onepointthatneedsto beraisedwith respectto theuseof thisdatais thedependenceof thecoefficientof restitutiononthegeometryof the impactingbodies.Sincethevibrationestablishedin anobjectby animpactremoveskineticenergy from the system,the coefficient of restitutiondecreasesas the vibrational energy increases.Spheres,cylinders,andgearteetharevery differentobjects,soanattemptto useexperimentaldatafrom onesuchobjectto find coefficient of restitutionfor anotheris not strictly valid. However, sincethereis no reliableexperimentaldatafor spur-geartoothimpact,thiswork usesthebestpublisheddata[11].

5. DiagnosticApproach

Thissectiondescribestheprinciple,thehypothesis,andaproposedqualitativemeasurefor detectionof back-lashin gearedmechanisms.

5.1 Principle

A proposalof thispaperis thatbacklashcanbemodeledasanimpactgeneratedbetweenapair of gearteeth.In the forcedomain,an impactwill producean impulsesignal. Sincesuchan impulseis dueto a small impactcausedby the clearancebetweenthe teeth,it is necessarythat no other(large) signalsovershadow the impulsesignal. It is thereforefurtherproposedthat theoutputlink of a gear-drivenmechanismbefixedsothat it cannotmove in space,which would ensurethat theeffect of gravitationalandinertial torquesdo not interferewith anymeasurement.To maximizethe impulseresponseof the impactdynamics,the geartrain shouldbe driven bysquarewave signalsandtheimpulseresponseof thegearteethtransducedby torquesensorsattachedto thegearshafts.

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5.2 Hypothesis

It is hypothesizedthat,if thegeartrain is drivenby square-waveexcitationandtorquesignalsaregatheredbysensorsmountedon the gearshafts,noticeabledifferenceswill appearbetweenthe signalsbetweena gearpairwith nobacklashversusa pair with a smallamountof backlash.Thedifferencewill bedueto thedetailednatureof thedynamicsof theimpactingbodies.Thepresenceof backlashwill resultin anincreasedimpactvelocityand,consequently, an impactforce thatproducesa larger torquesignal. It is thereforeproposedthatbacklashcanbedetectedby exploiting this impactphenomenon.

5.3 Qualitati veMeasures:Amplitude Probability Functions

Theamplitudeprobabilitydensityis apotentiallyusefulmeansof characterizingcontinuoustime-domaindatafrom a statisticalpoint of view [19, 20]. This functionrepresentstheprobabilitya randomvariablewill take ona specificamplitudevaluewithin a specifiedinterval. Experimentaldatanever comesfrom a continuoussamplespacebecauseof sensorlimitations,sooneis alwaysinterestedin probabilitiesconnectedwith intervalsandnotwith isolatedpoints.If 2 z�� | is theprobabilityof � where� liesbetween� � and � � then

2 z&� �F� � � � � | ����� Y����� z��|�� �

where� z��|

is theprobabilitydensityfunction.

Thevaluesof a randomvariablemustbeacquiredfrom acontinuousspaceby sampling,sothesetof discretedatacanonly approximatetheactualphysicalvariable.Consequently, theamplitudeprobabilitydensitymustalsobeconstructedfrom discretevalues.Consider

5datavaluesof a variable��z&� | . Thediscreteprobabilitydensity

functioncanbeestimated[1] by �� z��

| � 5 �5�� (14)

where�

is anarrow interval centeredat � , and5 � is thenumberof datavaluesthatfall within therange���s� � .

Henceanestimate

�� z��

|is obtaineddigitally by dividing thefull rangeof � into anappropriatenumberof equal-

width classintervals,tabulatingthenumberof datavaluesin eachclassinterval,anddividing thenumberof valuesin a classinterval by theclasswidth

�andthesamplesize

5. It shouldbenotedthat theestimate

�� z��

|is not

unique,sinceit dependson thenumberof classintervalsandtheir width selectedfor theanalysis.We have usedafixednumberof classintervalswith equalwidth, to ableto comparetheresultsobtainedfor differentcases.Thewindow size,in our analysis,hasbeentakento beonecompletewave lengthof thetime waveformof thetorquesignal.

Thediscreteamplitudeprobabilitydensitycanbeconsideredastheamplitude probability distribution (APD),which is ameasurethatdescribestheamplitudedistributionof thesignalbasedoncomputationof theprobabilitydensityfunction. The APD shows the percentageof time for which the signal lies within a certainamplituderange,andis particularlyapplicableto analysisof signalsfor machinemonitoring.

In the context of backlashdetectionby gear-teethimpact,it is expectedthat for a constantsquare-wave ex-citation the dynamicresponseof the gearswill differ with the amountof backlash(that is, with the clearancebetweenthe matingsurfacesof the teeth). This is becausethe clearancepermits“windup” of the driving gear,andan increasein its kinetic energy. As backlashincreases,thedriving gearwill strike the loadinggearwith agreaterinitial velocity andtherewill bemoreenergy to dissipate.Someresearchers[22, 23] have analysedthis

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phenomenonin time domain.We, however, useanAPD techniquewhich we have earliershown to beeffectivein signalanalysis[20]. In theanalysisof gearbacklash,we suggestthatthemotioneffectsresultingfrom impactshouldproducemarkedchangesin theAPD of thetorquesignalfrom eithergearshaft.

6. Modeling and Simulation

In orderto testthe veracityof the proposeddiagnosticapproach,a modelof a single-linkmanipulatorwasdeveloped.This modelwasthenincorporatedinto a detailedmultibody-dynamicssimulatorin orderto digitallytesttheproposedapproach.

6.1 Modeling

To testtheproposedmethodof backlashdetection,wemodelasystemconsistingof:� aDC electricservo-motor;� aflexible couplingto thegear-box;� apiniongear;� a loadgear;� aflexible couplingto theexternalload;and� a rigid-bodylinkage.

In theabovesystem,theservo-motoris usedto excite thegeartrain thatconsistsof thepinionandloadgears.Theflexible coupling,bothat thedriveranddrivenside,representsensorswhosetorsionaldeformationsareusedto detectthe torquesgenerateddueto impact. In reality, straingaugesaremountedon themto measuretorqueaswill beseenin a latersectiondescribingexperimentalset-up.The rigid-body linkagerepresentsload on thegeartrain. Thustheabovesystem,eventhoughsimple,capturestheessentialelementsof a realgeartransmissionmechanism.

Eachcomponentof the systemis modeledas a linear second-orderlumped-parametersystem. The onlysourceof nonlinearitycomesfrom the modelingof contactinteractionbetweenthe gearteeth. This contactinteractioncouplesthedriverandthedrivensideof thegeartrain. Theresultis asetof couplednonlinearsecond-orderdifferentialequationswhich cannotbesolved in closedform, but which canbenumericallysimulated.Aschematicof suchasystemis givenin Figure2.

6.2 Dynamic Equations

Thenomenclaturefor thedynamicmodelis given in Table1. Using this nomenclature,themotor/gear/linksystemcanbetreatedasa pair of multibodysystems,which arecoupledby thenonlineartoothinteractionforceasdescribedin Section4. Theequationsof motionof themotor/input-coupling/pinionsystemare

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τ� M� θ� M�, InputFlexi� bleCoupling

Rigid-bodylinkage

OutputFlexi� bleCoupling

τ�L� θ� L

�,

GeartoothInterac� tionDynamics

θ� G�P�θ�

Figure2: A gear-train,with excitationandloadlink.

Table 1: NomenclatureFor DynamicsModel��� z&� | Time-varyingmotorexcitationtorque� � Motor position � Motor inertia  � Motor dampingcoefficient��¡Relativepositionof inputcoupling ¡ Input-couplinginertia¢ ¡Input-couplingstiffness  ¡Input-couplinginternaldampingcoefficient� � Relativepositionof piniongear� � Relativepositionof loadgear� Tooth-toothelasticrestoringforce Tooth-toothinternaldampingforce��£Relativepositionof outputcoupling £ Output-couplinginertia¢ £Output-couplingstiffness  £Output-couplinginternaldampingcoefficient� R Relativepositionof loadlink R Inertiaof loadlink¤ R Massof linkH R Distanceof centerof massof link fromaxisof rotation¥¦ Gravitationalvector

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M �x§¨© �� ����¡�� �ª�«¬ � §¨©   �Ll� �L�­��� z&�

|  ¡ l��¡ � ¢ ¡Z��¡�®���������¯�% x"ª�«¬

whereM � is theinertiamatrix,computedin theusualmanner. Similarly, theload/output-coupling/linksystemisdescribedas

M ��§¨© �� ����£�� Rª�«¬ � §¨© �x�®�&�������! #"  £ l��£ � ¢ £°��£¤ R H R C²±´³ z � R | ¥¦

ª�«¬where,again,M � is a inertiamatrix.

6.3 Digital Simulator

In orderto verify the usefulnessof this dynamicsmodel– andof the ideathatbacklashcanbedetectedbygear-toothimpact– anextensiveC-languagecomputersimulationwasdeveloped.Thesimulatoris a multibody-dynamicscodethatis intendedfor applicationswherehighaccuracy is desired(asopposedto, for example,super-real-timeperformance).In orderthat futurework couldeasilymodelbacklashin complicatedrobotic linkages,thecodeemploysFeatherstone’s articulated-bodyinertias[9] asthefundamentaldynamicsrepresentation.

As shown in previous work [8], the choiceof numericalintegrationroutinescanhave dramaticeffectsonthesimulatedbehavior of a system.A simpleexampleof suchdifferencesariseswhencomputingthehysteresisloop of gear-toothinteractionusingHunt andCrossley’s model[13]. Figure3 shows a simulationconductedforc�~ msof modeledtime of theinteractionwhena back-differentiationformulais used– theBDF code,developedspecificallyfor simulationof mechanisms.1 Choosingahigh-orderRunge-Kuttaintegration– theDVERK code–producesthecorrectbehavior, asshown in Figure4. NotethattheBDF codeoverestimatestheinertial effectsofthegear, andtheteethreboundandbegin to interpenetratein theoppositedirection.This phenomenonwasalsofoundin earlierwork [8], whereit led to dynamicinstabilities. theDVERK integrationroutinewasthusfor thesimulator.

7. Simulation Results

Extensive computersimulationswereconductedto verify thebacklashdetectionmethodology. A numberofparametervariationswerestudied;potentiallyrelevantparameterswereidentified,andfor variousvaluesthedy-namicsweresimulatedandtheresultingtime-domainsignalswereplottedfor visualcomparison.First,wevariedtheflexibility of both the input andoutputshaftsby meansof thecouplersthat representtorquesensors.It wasobservedthatfor materialstiffnessvaluestypicalof metalssuchasaluminumandsteel,therewasnoappreciablequalitative changein the time waveformof the torquesignal. Hence,we chosevaluesof shaftflexibilities fromstandardengineeringtexts. Similarly, realisticvaluesof viscousfriction thatwereobtainedfrom differentmotordatasheetsdid not causeany qualitative changein the torquesignal. Both the amplitudeandfrequency of the

1All integratorsusedin thesesimulationsareavailableon theInternet,from netlib@ornl.gov, andarerefereedcodes.

10

0 10 20 30 40 50 60 70 80 90 100-10

0

10

20

30

40

50

60

70

80Hysteresis loop, BDF integrator

Figure 3: Impacthysteresisloop,back-differentiationsimulation

0 10 20 30 40 50 60 70 80 90 100-10

0

10

20

30

40

50

60

70

80

Interpenetration (microns)

Rea

ctio

n fo

rce

(New

tons

)

Hysteresis loop, DVERK integrator

Figure4: Impacthysteresisloop,DVERK simulation

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squarewaveexcitationwerevaried,andit wasfoundthatthequalitativecharacteristicsof thetimewaveformsre-mainlargelyunchangedprovidedthattheamplitudeandfrequency arenotextremelyhighor low. Theamplitudeof theexcitationsignalis chosensothatit providesenoughimpulseto bedetectedby thetorquesensor, but notsohigh asto saturatethesensor. Thefrequency, on theotherhand,is chosensothatit doesnot excite resonanceorchaosin thesystem.Basically, ourobjective is to extractsignaturesusinganexcitationsignalof smallamplitudeandmoderatefrequency.

We did not studytheeffect of Coulombfriction andassumedit to benegligible. Sensornoise,eventhoughimportant,wasalsonot studied. The rationalewasto matchthe ideal simulationresultswith the experimentalresults:if theexperimentalresultsmatcheswell, thenneglectedvariablesarenot likely to besignificantfactors.

Ourfindingscanbeexplainedbyusingtheresultsof asamplesimulation.Thissimulationemployedparametervaluesderived from specificationsof the componentsof our experimentalsystem,which will be presentedinSection8.2 Thevaluesfor thekey parametersto thissimulationarepresentedin Table2.

Table2: Valuesof thekey parametersusedin thesimulation.No. of teethin inputgear µ wNo. of teethin outputgear µ wGearangle 3 w degreeGearwidth w¶}·w�w y n ~ meterGearpitchdiameter w¶}·w n c�¸¹~ meterGearinertia— input w¶}·w�w�w�w 3 c Kg-m

�Gearinertia— output w¶}·w�w�w�w 3 c Kg-m

�Motor inertia 3 } 3 µ<º c�w DX» Kg-m

�Motor damping w¶}·w�w µ½¼ N-m/rad/sInput-sensorstiffness 3 ¼ }�¸�c N-m/radInput-sensordampingcoefficient w¶}·w½~ n N-m/rad/sExcitationfrequency c�w HzExcitationAmplitude w¶}·w½~ N-m

Although the modelandsimulatorhave the provision for incorporatinganoutputtorquesensor(i.e. torquesensorat thedrivensideof thegeartrain), theoutputsensorwasnotusedin thedetectionmethodology.

The time waveformsof the torquesignalswith andwithout backlashareshown in Figure5. Whenthereisno backlash,thereis no impactbetweentheteethandconsequentlytheamplitudeof thesignalis low. Thesmallripplesaredueto theelasticityof thegeartooth.But, assoonasa very low ( ~¾w microns)backlashis introduced,a muchhigheramplitudeof torqueis predicted.If theAPD of thesetwo wave forms is plotted(Figures6), twoconspicuousfeaturesappear:the peaksaredistinctly lower when thereis backlash,and the amplitudesshowa greaterspreadin the presenceof backlash. Both of theseresultsare the direct manifestationof the impactphenomenon.

2It cansometimesbe instructive to tunethe simulationparameterscloseto the realparametersandseehow the simulationresultsfareagainstexperimentalresults,but theeffort is usuallytime-consumingandin high-ordernonlinearsystemscanbeverydifficult. Thegoalof thispaperis detection,ratherthanexactdynamicsmodeling,soparametertuningwasnotperformed.

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0 0.02 0.04 0.06 0.08 0.1 0.12-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Time (sec.)

Tor

que

(N-m

m)

Figure5: Simulationresults:Timewaveformof thetorquesignalsof theinputtorquesensor;solid line is withoutbacklashanddottedline is with ~¹w micronsbacklash.

-0.3 -0.2 -0.1 0 0.1 0.2 0.30

5

10

15

20

25

30

35

Amplitude

Per

cent

Amplitude Probability Distribution

Figure 6: Simulationresults:APD of the input torquesensorsignalwhenthereis no backlash(solid line) andwhenthereis ~¹w micronsbacklash(dottedline).

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8. Experimental Results

In additionto the detailedsimulationstudies,physicalexperimentationwasconductedto testthe proposeddiagnosticapproach.A singledegree-of-freedommanipulatorwasbuilt, comprisedof thefollowing components:� oneYasakawaDC servo motor(DM- n ¸ n ) with apeaktorqueof w¶}¿¸ N-m,� two samesize À n w n stainlesssteelspur gears(NORDEX LAS-D c - µ w ), eachwith µ w teeth, 3 w degree

pressureangle,and w¶}·w n c�¸¹~ meter( c�} 3 ~ inch)pitchdiameter,� input torquesensorwhich is a hollow aluminumcylinder instrumentedwith four strain gauges(CEA-13-062UV-350),mounteddiametricallyopposite(two gaugestogetheron eachside)to eliminatebendingeffects.This is attachedto themotoroutputshaftandinputgearshaft,� severalbearings,and� anoutputlink

A functiongenerator(WAVETEK model c ¼ ) andapoweramplifier(Bruel& Kjærpoweramplifiertype 3 ¸¹w y )wereusedto excite themanipulatorat c�w Hz.,andWAVEPAK dataacquisitionandanalysissoftwarewereusedtodetectincipientbacklash.Theexperimentaldatapresentedhereis obtainedwith asamplingfrequency of c�w¹w�w Hz.A pictureof thetestbedis givenin Figure7.

Figure7: Experimentaltestbed.

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0 0.02 0.04 0.06 0.08 0.1 0.12-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Time (sec.)

Tor

que

(N-m

m)

Figure 8: Experimentalresults:Time waveform of torquesignalsfrom input torquesensor;whenthereis nobacklash(solid line) andwhenthereis ~¾w micronsbacklash(dottedline).

-0.3 -0.2 -0.1 0 0.1 0.2 0.30

5

10

15

20

25

30

35

Amplitude

Per

cent

Amplitude Probability Distribution

Figure9: Experimentalresults:APD of torquesignalsfrom input torquesensor;whenthereis nobacklash(solidline) andwhenthereis ~¹w micronsbacklash(dottedline).

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The time-domainwaveform andAPD datafrom the experimentareshown in Figures8 and9 respectively.Theseresultsconfirmthevalidity of theproposedbacklashdetectiondiagnostic,asdothesimulationresults.As averysmallbacklash( ~¾w microns)is introducedby changingthecenterdistancebetweenthetwo gearsby insertingpreciseshims,thereis amarkedincreasein amplitudeof thetorquesignalbecauseof theimpact.TheAPD resultsconfirmtheloweringof peaksandspreadingof thedataaway from thepeakwith theintroductionof backlash.

8.1 Discussion

It is evident from the experimentalresultsthat it is possibleto detectbacklashin a physicalsystemby theproposeddiagnosticapproach.Additionally, the resultsobtainedfrom the computersimulationscloselymatchthosefrom thephysicalexperiments.This conformityindicatesthatthemathematicalmodelingembodiedin thedigital simulatorcapturestheessentialfeaturesof thephysicalworld thatis analysed.

The efficacy of the diagnosticapproachis evident. It is a qualitative measure,but it can be mademoremeaningfulby quantificationof the resultsso that the shapeof the APD canbe translatedinto a measurethatgivesa betterperspective to the users. The following sectiondescribesa meansof quantifyingthe changeinshapeof theAPD of thetorquesignalassociatedwith increasingbacklash.

9. Quantification of the Results

Thevariationsin the shapeof the APD curvesarevisually apparent,but it is necessaryto developa meansof quantifyingsuchdifferencesif this informationis to be employed in an automateddiagnosticsystem.Withthis goal in mind, two measureswereexamined.Thefirst waskurtosis which describesthe“peakedness”of thefunctionandis definedasthe µ th moment[19]. This measureis sensitive to changesthat affect the tails of thefunction,andis anexcellentcandidatefor detectingtheappearanceof isolatedpeaksin thesignal;however, sincetheoccurrenceof isolatedpeakswasnotobservedin backlashAPD’sthekurtosismeasurewasnot investigatedindetail.Thesecondmeasurewasskewness, whichcharacterizesa function’sdegreeof asymmetryaroundits meanvalue[19]. Althoughthe time-domaindatahave almostthesamemeanregardlessof backlash,theAPD dataisvastlydifferentandprovidesawell definedtrendif theskewnessmeasureis appliedto theAPD datadirectly.

Skewnessappearsto capturethevariationsof APD in a quantitative manner. Becausethesimulationresultscloselymatchphysicalexperimentation,andit is easierto exactlyvarysimulationparametersthanto varyphysicalparameters,thequantitativeanalysisof backlashwasconductedusingdigital simulations.

Theexpressionusedto computeskewnessis:

skewness � c5 Á (ÄÃp� E � (?��Å�; I ewhere

5is thenumberof datapoints, � ( is theindividualdatavalue, � is themeanof the

5datavaluesand ; is

thestandarddeviationof the5

datavalues.

Changesin theskewnessof theAPD arecorrelatedwith changesin theamountof backlashin thegears.Thisis shown by startingfrom zerobacklashin thegears,computingthedynamicresponseof thegearedmechanismtoasquare-waveexcitation,findingtheAPD of theinputtorquesensorfor afull periodof excitation,andcalculatingtheskewnessof theAPD.Thebacklashis thenintroducedin stepsof 3 ~ micronsupto atotalof c�w�w microns.ThecorrespondingAPD plotsareshown in Figure10; notethecorrespondingskewnessvalues.Sincetheamplitude

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-0.2 0 0.20

10

20

30

Amplitude

Per

cent

Skewness = 4.6346

(a)

-0.2 0 0.20

10

20

30

Amplitude

Per

cent

Skewness = 3.7013

(b)

-0.2 0 0.20

10

20

30

Amplitude

Per

cent

Skewness = 2.9010

(c)

-0.2 0 0.20

10

20

30

Amplitude

Per

cent

Skewness = 2.2009

(d)

-0.2 0 0.20

10

20

30

Amplitude

Per

cent

Skewness = 1.5762

(e)

0 50 1001

2

3

4

5

Backlash (microns)

Ske

wne

ss

Monitoring Curve

(f)

Figure 10: APD of torquesignalsfrom input torquesensorfor differentbacklashvalues— (a) zero, (b) 3 ~microns,(c) ~¹w microns,(d) ¸�~ microns,and(e) c�w�w microns.(f) shows thecorrespondingconditionmonitoringcurve

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of the time waveform of the torquesignal increaseswith backlash,the peaksof the APD aresmaller, andthespansarebroader. Also, becausethereis moretime of freeflight of thepinion gearfor greaterbacklash-inducedclearance,thereis a steadydevelopmentof a peakin thezero-amplituderegion asbacklashincreases(Figure10(e)). All of thesephenomenaarecapturedby theskewnessmeasureand,asa consequence,theskewnessvaluedecreasesmonotonicallywith the increaseof backlash.Basedon this observationonecanconstructa conditionmonitoring curve (Figure 10 (f)) which can play an importantrole in diagnosingthe incipienceof unwantedbacklash.

9.1 GeneralDiscussion

The proposedapproachmay be appropriatefor usein monitoringmachinecondition,but aswith mostap-proachessomecaremustbetaken. First, sinceit is a relativemeasure,theskewnessvalueof theAPD shouldberecordedwhenthemechanismis in goodoperatingcondition(e.g. whennew). Thenthis valuecanbeusedasa basisfor comparison.Second,userscanconstructtheir own conditionmonitoringcurve by usinganauthenticsimulatorandcanuseit asa guideto analyzethephysicalsituation.Alternatively, it is possiblefor themanufac-turerof themechanismto providesuchmonitoringcurvesothatatany timetheusercanestimatethemagnitudeofthebacklashin thesystem.Third, basedontheapplicationsandtheexperienceof theuser, suchmonitoringcurvecanbedividedinto regionscorrespondingto differentgradesof “good” regionsto “shut down” regions.Finally,againbasedon experience,this approachcanbeautomatedusinga computerhookedup with a dataacquisitionsystem.Becauseall thecomputermustdo is to checktheskewnessvalueof theAPD, this approachis ideal forautomatedmonitoring.

Theproposedapproachindicatesthat it is possibleto detectbacklashusingforce-sensorsignals.Unlike theapproachof SteinandWang[22, 23], in whichbacklashis detectedusingvelocitysignalsensedby a tachometermountedon the servomotor, our approachusesa torquesignalobtainedfrom a torquesensormountedon theprimary shaft. Our backlashdetectionmethodmay be harderto implementon existing systems,becauseit isusuallyeasierto mounta tachometerona motorthanto sensetorquefrom a motor-outputshaft.It canbearguedhowever, thatsinceimpactis thefundamentalmanifestationof backlash,oneis betterequippedto detectminutebacklashby sensinga force signal thanby sensingor estimatinga velocity signal. Alternatively, the invasiveinstrumentationourmethodrequirescouldbeovercomeby usingaccelerometersinsteadof torquesensors.

10. Conclusions

Wehavedevelopedanovel backlash-detectionapproachthatis capableof detectingextremelysmallbacklashwithout any ambiguity. This approachis especiallyapplicableto roboticmanipulatorswhich play an increasingrole in manufacturingautomation.A detailedmodelandcomputersimulationhave beendevelopedthat incor-poratea complicatednonlinearcontactmodelto simulaterealisticcontactconditions.A qualitativemeasurehasbeenpresentedthatis capableof detectingtheincipienceof backlashin thegears,andsimulationsshowedthatthemeasurecorroboratesintuition. A singledegree-of-freedommanipulatorwasconstructed,andextensive experi-mentationclearlydemonstratedtheefficacy of theproposeddiagnosticapproach.Finally a quantitativeanalysiswasexploredsothatthisapproachcanpotentiallybeautomated.

Thedisadvantageof theapproachexaminedhereis thatit is invasive. However, preliminaryexperimentationwith an alternative instrumentation– accelerometersmountedon the gearshaftsvia lever arms– indicatethatthe invasive natureof the original instrumentationcanbe surmounted.Therearemany issuesto be explored

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whenchangingsensortechnologies,not the leastbeingaccelerometersensitivity to vibrationsfrom physicallyadjacentmechanismsotherthantheoneunderdiagnosis;thus,auserinterestedin detectingincipientbacklashisnot restrictedto onesensorbut facestrade-offs.

It is instructive to notethatalthoughit is maybedifficult to inserta torquesensorinto theexisting robotsitis quiteeasyto make provisionsfor built-in torquesensorsin the input shaftsof variousjoints for futurerobots.In suchcases,with sosimplea methodologyaspresentedhere,backlashcanreadilybedetectedandits effectsnulledby appropriatecontrolschemes.

Finally, theproceduredescribedin this paperrequiresfixing theoutputlink for thediagnostictest.Althoughsucharequirementmayseemartificial andcontrived,mostcalibrationmethodologiesalsorequirethateachrobotundergoesa calibrationtestat varioustimesduringits operation.During suchtests,robotsaremadeto move inanartificial andconstrainedway, in orderto setthereferenceframesfor eachjoint; if backlashinducessimilarlyundesirablebehavior in a mechanismthatmustbepreciseor safe,fixation (or inertial loading)of theoutputlinkmaybea smallpriceto payfor accurateassessmentof theconditionof themechanism.

11. Acknowledgments

This researchwassupportedin partby theManufacturingResearchCorporationof Ontario,theNaturalSci-encesandEngineeringResearchCouncilof Canada,andthefederalgovernmentundertheInstitutefor RoboticsandIntelligentSystems(which is aNationalCentresof Excellenceprogramme).

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