Robot Hydraulic Control

7/29/2019 Robot Hydraulic Control

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AHYDRAULICFLEXIBLEJOINTROBOTSIMULATOR

AThesisSubmittedtothe

CollegeofGraduateStudiesandResearch

inPartialFulfillmentoftheRequirements

fortheDegreeofMasterofScience

intheDepartmentofMechanicalEngineering

UniversityofSaskatchewan

Saskatoon,Canada

By

ShahramDezfulian

CopyrightShahramDezfulianJune2007.Allrightsreserved.


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i

PERMISSIONTOUSE

Inpresentingthisthesisinpartialfulfillmentoftherequirementsforapostgraduatedegreefrom

theUniversityofSaskatchewan,IagreethattheLibrariesofthisUniversitymaymakeitfreely

availableforinspection.Ifurtheragreethatpermissionforcopyingofthisthesisinanymanner,

inwholeorinpart,forscholarlypurposesmaybegrantedbytheprofessororprofessorswho

supervisedmythesisworkor,intheirabsence,bytheHeadoftheDepartmentortheDeanof

theCollegeinwhichmythesisworkwasdone.Itisunderstoodthatanycopyingorpublication

oruseofthisthesisorpartsthereofforfinancialgainshallnotbeallowedwithoutmywritten

permission.ItisalsounderstoodthatduerecognitionshallbegiventomeandtotheUniversity

ofSaskatchewaninanyscholarlyusewhichmaybemadeofanymaterialinmythesis.

Requestsforpermissiontocopyortomakeotheruseofmaterialinthisthesisinwholeorpart

shouldbeaddressedto:

Head,DepartmentofMechanicalEngineering

CollegeofEngineering

UniversityofSaskatchewan

57CampusDrive

Saskatoon,Saskatchewan

Canada,S7N5A9


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ABSTRACT

Theobjectiveofthisprojectwastodesignandimplementanexperimentalhydraulic

systemthatsimulatesjointflexibilityofasinglerigidlinkflexiblejointrobotmanipulator,withtheabilityofchangingthejointflexibilitysparameters.Sucha

systemcouldfacilitatefuturecontrolstudiesofrobotmanipulatorsbyreducing

investigationtimeandimplementationcostofresearch.Itcouldalsobeusedtotestthe

performanceofdifferentstrategiestocontrolthemovementofflexiblejoint

manipulators.

Ahydraulicrotaryservomotorwasusedtosimulatetheactionofaflexiblejointrobot

manipulator.Itwasachallengingtask,sincethecontrolofangularaccelerationwas

required.

Asingle-rigid-link,elastic-jointrobotmanipulatorwasmathematicallymodeledand

implementedusingMatlab.Jointflexibilityparameterssuchasstiffnessanddamping,

couldbeeasilychanged.Thissimulationwasconsideredasafunctiongeneratorto

drivethehydraulicflexiblejointrobot.Inthisstudythedesiredangularaccelerationof

themanipulatorwasusedastheinputtothehydraulicrotarymotorandtheobjective

wastomakethehydraulicsystemfollowthedesiredaccelerationinthefrequencyrange

specified.Thehydraulicsystemconsistedofaservovalveandrotarymotor.

Ahydraulicactuatorrobotwasbuiltandtested.Theresultsindicatedthatiftheinput

signalhadafrequencyintherangeof5to15 Hzanddampingratioof0.1,the

experimentalsetupwasabletoreproducetheinputsignalwithacceptableaccuracy.

Becauseoftheinherentnoiseassociatedwiththemeasurementofaccelerationandsome

severenon-linearitiesintherotarymotor,controloftheexperimentaltestsystemusing

classicalmethodswasnotassuccessfulashadbeenanticipated.Thiswasafirststagein

aseriesofstudiesandtheresultsprovideinsightforthefutureapplicationofmore

sophisticatedcontrolschemes.


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ACKNOWLEDGEMENTS

IwouldliketoexpressmydeepestappreciationtoProfessorsR.T.Burton

andR.Fotouhifortheiradvices,encouragementandpatience.Myspecial

thanksgotoMr.D.V.Bitnerforhistechnicalsupportandnever-ending

helpandarrangements.

Iwouldalsoliketoacknowledgethelong-lastingsupportandloveoffered

bymyparents,FraidoonDezfulianandAfaghDelfani,mywife,KathyAbrishamiand

mydaughter,SheryDezfulian.


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CONTENTS

PermissiontoUse...................................................... i

Abstract..................................................... ........ii

Acknowledgements.................................................... iii

Contents............................................................. iv

ListofTables........................................................ vi

ListofFigures....................................................... vii

Nomenclature........................................................ xii

1Introduction........................................................ 11.1Background.....................................................1

1.2LiteratureReview.................................................2

1.3Objectives......................................................51.4ThesisOutline...................................................6

2FunctionGenerator................................................. 7

2.1Introduction.....................................................7

2.2GoverningEquations.............................................7

3HydraulicSimulator................................................ 193.1Introduction....................................................19

3.2HydraulicSystemAnalysis........................................21

3.3ControllerDesign(Theoretical).....................................303.3.1Closed-LoopController.....................................30

3.3.2Open-LoopInverseCompensationController....................37

4ExperimentalSystemSetup.......................................... 39

4.1Introduction....................................................39

4.2PowerSupply...................................................40

4.3PressureControlServovalve.......................................404.4HydraulicRotaryMotor..........................................44

4.5RobotManipulator..............................................454.6PersonalComputer..............................................48

4.7SignalAnalyzer.................................................49

4.8Accelerometer..................................................504.9AccelerometerCompensator.......................................50


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5ExperimentalResults............................................... 555.1Introduction....................................................55

5.2PressureControlServovalve.......................................55

5.3Accelerometer..................................................59

5.4HydraulicSimulator.............................................63

5.4.1Open-LoopFrequencyResponses.............................635.4.2Closed-LoopFrequencyResponses............................67

5.4.3Open-LoopCompensation...................................695.4.4TransientAccelerationResponses.............................74

5.4.4.1Frequency5HzandDampingRatio0.1.................75

5.4.4.2Frequency10HzandDampingRatio0.1................785.4.4.3Frequency15HzandDampingRatio0.1................80

5.4.5TransientResponseErrors...................................83

6ConclusionsandRecommendations...................................89

6.1Conclusions....................................................89

6.2FutureWork...................................................91

References........................................................... 92


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LISTOFTABLES

2.1Simulationparametersadoptedfrom[9]................................12

3.1Parameterssummaryofopen-loopandclosed-looptransferfunctions.........32

4.1SpecificationofMoogservovalvemodel15-010.........................43

4.2SpecificationsofMicromaticrotarymotorMPJ-22-1V....................44

4.3Importantmanipulatormeasurements..................................48

4.4Matlab-Simulinksetupforexperimentalsystemcontrol....................49

5.1Meansquarederrorvaluesoftransientresponseerrorsindifferentcases......87


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LISTOFFIGURES

2.1Schematicofasingle-linkmanipulatorwithflexiblejoint...................8

2.2Theopen-loopblockdiagramofsinglelinkmanipulatorwithflexiblejoint....112.3Simulationresultsforimpulseinputstothesinglelinkmanipulator.........14

2.4Plotofequation2.29...............................................17

3.1Schematicofthehydraulicsimulator..................................20

3.2Blockdiagramofequation3.11......................................25

3.3Reducedblockdiagramofequation3.11...............................25

3.4TypicalmagnitudeBodeplotofequation3.11when nv

1

1.............26

3.5TypicalmagnitudeBodeplotofequation3.11when nv

1

1.............26

3.6Open-loopmagnitudefrequencyresponseofthehydraulic

simulator(points)withsuperimposedasymptoticlines(experimental:solidlinebasedon40dBperdecadeand

theoretical:dashedlinebasedon20dBperdecade).......................27

3.7Simplifiedopen-loopblockdiagramofthesystem.......................28

3.8Blockdiagramofthesimplifiedtransferfunctionwithinfluenceof

theaccelerometersensitivity.........................................29

3.9Blockdiagramofclosed-loopcontrolsystemwithproportionalgain.........31

3.10Magnitudefrequencyresponse)(

)(

sV

s

e

v oftheopen-loopand

closed-loopsystemswithproportionalcontroller........................32

3.11Transientresponseof v intheopen-loopandclosed-loopmodels

with 2=PK tothefunctiongenerator d of10Hzsignal..................33

3.12Blockdiagramofclosed-loopcontrolsystemwithproportional

andintegralgains.................................................34


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3.13Magnitudefrequencyresponse)(

)(

sV

s

e

v ofopen-loopand

closed-loopsystemswithPIcontroller................................35

3.14Transientresponseofv

intheopen-loopandclosed-loopmodels

with 2=PK and 10=IK tothefunctiongenerator d of10Hzsignal.......36

3.15Blockdiagramofopen-loopcompensatorcontroller.....................37

3.16Transientresponseof v intheopen-loopcompensatedmodeltothe

functiongenerator d of10Hzsignal.................................38

4.1Schematicofthehydraulicsimulator.................................39

4.2Schematicofthepressurecontrolservovalve............................41

4.3Schematicofthetwo-linkrigidmanipulatorinwhichtheshoulderlinkwasfixed............................................46

4.4Photoofthemanipulatorsetupusedintheexperiment....................47

4.5SchematicofShakerTabletest.....................................51

4.6MagnitudeandphaseBodeplotsoftheshakertabletest(pointsaredatafromtheanalyzerandthesolidlineisthestraightline

approximationbasedona20dBperdecadeslope).......................52

4.7Blockdiagramofthesystemwiththeaccelerometercompensator...........54

5.1aMagnitudefrequencyresponseofthepressurecontrolservovalvewithablockedload(pointsaredatafromtheanalyzerandthesolid

lineisthestraightlineapproximation)................................56

5.1bPhasefrequencyresponseofthepressurecontrolservovalve

withablockedload................................................57

5.2aMagnitudefrequencyresponseofpressurecontrolservovalvewith

actualhydraulicsimulatorload(pointsaredatafromtheanalyzer

andthesolidlineisthestraightlineapproximation).....................58

5.2bPhasefrequencyresponseofpressurecontrolservovalvewith

actualhydraulicsimulatorload.......................................58


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5.3aMagnituderatio

ja

ja

po

ac

(

)( versusfrequencyoftheaccelerometerwith

proposedaccelerometercompensator(pointsaredatafromthe

analyzerandthesolidlineisthestraightlineapproximation)...............61

5.3bPhase)()(

jaja

po

ac versusfrequencyoftheaccelerometerwith

proposedaccelerometercompensator.................................61

5.4Open-loopmagnitudefrequencyresponseofthehydraulicsimulator

beforeapplyingtheaccelerometercompensator(pointsaredatafrom

theanalyzerandthesolidlineisthebestfitstraightlineapproximation;40dBperdecadeslope)............................................62

5.5Open-loopmagnitudefrequencyresponseofthehydraulicsimulatorafterapplyingaccelerometercompensator(pointsaredatafromthe

analyzerandthesolidlineistheexpectedstraightlineapproximationoftheplant;20dBperdecadeslope)..................................63

5.6aOpen-loopmagnitudefrequencyresponseofthehydraulicsimulator

withinputamplitudeof0.5vor 2/5.11 srad (solidlineistheideal

theoreticalfrequencyresponse)....................................65

5.6bOpen-loopphasefrequencyresponseofthehydraulicsimulator

withinputamplitudeof0.5v( 2/5.11 srad )...........................65

5.7aOpen-loopmagnitudefrequencyresponseofthehydraulicsimulator

withinputamplitudeof0.1v( 2/3.2 srad )............................66

5.7bOpen-loopphasefrequencyresponseofthehydraulicsimulator

withinputamplitudeof0.1v( 2/3.2 srad )...........................66

5.8Closed-loopmagnitudefrequencyresponseofthehydraulicsimulator

withinputamplitude1.0vor 2/0.23 srad (pointsaredatafromthe

analyzerandthesolidlineisthestraightlineapproximation,

20dBperdecadeslope)............................................67

5.9Closed-loopmagnitudefrequencyresponseofthehydraulic

simulator(inputamplitude0.5vor 2/5.11 srad )........................68

5.10Closed-loopmagnitudefrequencyresponseofthehydraulic

simulator(inputamplitude0.1vor 2/3.2 srad ).........................68


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5.11Open-looptransientresponseofthehydraulicsimulatortoa

unitstepinputwithamplitudeof1v( 2/0.23 srad ).......................69

5.12Transientresponseofthemodel(71.15

4.0

+s

s)tothestepinput

signal(redlineistheinputandbluelineistheoutput)....................70

5.13aOpen-looptransientresponseofthehydraulicsimulatorto5Hz

sinewavewithamplitudeof1v(compensations

s

8.0

71.15+).................72

5.13bOpen-looptransientresponseofthehydraulicsimulatorto5 Hz


s

8.0

71.15+).................72

5.14aOpen-looptransientresponseofthehydraulicsimulatorto15 Hz


s

8.0

71.15+).................73

5.14bOpen-looptransientresponseofthehydraulicsimulatorto15 Hz


s

8.0

71.15+).................74

5.15Open-looptransientresponseofthehydraulicsimulatortothefunctiongenerator5Hzand0.1dampingratiosignal(notcompensated).............76

5.16Open-looptransientresponseofthehydraulicsimulatortothefunction

generator5Hzand0.1dampingratiosignal(compensatedbys

s8.0

4+ )........76



s

8.0

8+)........77



s

8.0

71.15+).....77


generator10Hzand0.1dampingratiosignal(notcompensated)............78



s

8.0

4+).......79


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generator10Hzand0.1dampingsignal(compensatedbys

s

8.0

8+)............79



s8.0

71.15+ )....80

5.23Open-looptransientresponseofthehydraulicsimulatortothefunctiongenerator15Hzand0.1dampingratiosignal(notcompensated)............81



s

8.0

4+)............81



s

8.0

8+)............82



s

8.0

71.15+)....82

5.27TrackingerrorofFigure5.15(frequency5 Hzanddampingratio0.1,

notcompensated).................................................83


compensatedbys

s

8.0

71.15+),..........................................84


notcompensated),................................................84


compensatedbys

s

8.0

71.15+).........................................85

5.31TrackingerrorofFigure5.23(frequency15 Hzanddampingratio0.1,notcompensated)................................................85


compensatedbys

s

8.0

71.15+)..........................................86


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NOMENCLATURE

linkangularacceleration )/( 2srad

d desiredangularacceleration(functiongeneratoroutput)(

2

/srad )v linkangularaccelerationinvolts(v)

vc linkangularaccelerationinvoltwithcompensator(v)

motorviscousdamping )/( radsmN

12P nozzlepressuredifference(Pa)

ABP loadpressuredifference(Pa)

jointdampingratio

v valvedampingratio

linkangularposition(rad)

&

linkangularvelocity(rad/s)&& linkangularacceleration( 2/srad ) Laplacetransformof(rad) transferfunctiontimeconstant(s)

1 loadtimeconstant(s)

a servo-amplifiertimeconstant(s)

1CL closed-looptimeconstantwithPcontroller(s)

o open-looptimeconstant(s)

motorangularposition(rad)

& motorangularvelocity(rad/s)

&& motorangularacceleration( 2/srad )

Laplacetransformof(rad)

linkangularvelocity(rad/s)

n jointnaturalfrequency(rad/sorHz)

nv valveundampednaturalfrequency( rad/sorHz)

1a partial-fractionexpansionparameter



4

a partial-fractionexpansionparameter

aca measuredaccelerometersacceleration(v)

poa calculatedaccelerationbydoubledifferentiationofposition(v)

AA spoolringarea(2m )

SA spoolendarea( 2m )

Djointviscousdamping( radsmN / )

mD motorvolumetricdisplacement )/(3 radm


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ie accelerationerror(v)

1F firsttermofequation2.19

2F secondtermofequation2.19

ac

G accelerometertransferfunction

1G idealaccelerometercompensator'

ei servo-amplifieroutputcurrent(a)

Ilinkmomentofinertia( radsmN /2 )

eI inputcurrent(ma)

'

eI Laplacetransformof'

ei (a)

LI linkmomentofinertia )/(2 radsmN

Jmotormomentofinertia( radsmN /2

)

1k transferfunctiongain(4

/ smNrad )

Kjointstiffness( radmN / )

1K loadgain( sParad / )

aK servo-amplifiergain(a/v)

1CLK closed-loopgainwithPcontroller(s)

conK conversiongainfactor( 2/srad

v)

eK servo-amplifierconversiongain(ma/a)

IK integralgain(1/s)

oK open-loopgain( svrad / )

OLK open-loopoverallgain(s)

PK proportionalgain

vK valvegain(Pa/ma)

lrotationarm(m)

Lservo-amplifierinductance(H)

MSEmeansquarederrorvalue(2v )

LP loaddifferentialpressure(Pa)

SP supplypressure(Pa)

Rservo-amplifierresistance()

sLaplaceoperator(1/s)

lS accelerometerlinearcalibratedrating( 2/sm

v)

ttime(s)

LT motortorque( mN )

umotorinputtorque( mN )

ULaplacetransformofu( mN )

ev servo-amplifierinputvoltage(v)


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eV Laplacetransformof ev (v)

pox measuredposition(v)


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CHAPTER1

INTRODUCTION

1.1Background

Arobotmanipulatorconsistsoflinks,joints,anddrivecomponents.Inordertoachieve

accuraterobotpositioning,thelinksandjointsareusuallymadeasrigidaspossible.

Rigidrobotmanipulatorsarebasedontheassumptionthatthetransmissionsarestiff

andthatthelinksarerigid.Suchmanipulatorsareheavy(e.g.,load-carryingcapacityis

typicallyonly5%to10%oftheirownweight),consumeconsiderablepower,andare

generallyimpracticalforhighspeedmaneuvers.Ontheotherhand,flexible

manipulatorshaveseveraladvantagesoverrigidmanipulators,suchassmallersizeand

mass,rapidresponse,lowerpeakpowerrequirements,andenergyuse.Lightweight

manipulatorshavelinksthatdeflectsignificantlyinhighspeedoperationssotheir

flexibilitycannotbeignored.Flexibility,however,canalsobepresentinthejoints.

Jointflexibilityexistswhenthereisadifferencebetweentheangularpositionofthe

drivingactuatorandthatofthedrivenlink.Itisknownthatthejointflexibilitycan

causeoscillationsinrobotmanipulators.Therefore,whenanaccuratetrajectorytracking

oftheend-effecterisneeded,jointflexibilityisconsideredasaproblem[1].Inother

words,jointelasticityisthemainsourceofcomplianceinmostcurrentmanipulator

design.Jointelasticityoriginatesfromseveraltransmittingcomponents,suchas

elasticityinthegears,belts,hydrauliclines,etc.[2].


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Thecontroltasks,incaseoftheelasticjointmanipulators,aremorecomplicatedthan

theequivalentrigidjointmanipulators.Thisistrueduetothefactthatintheelasticjoint

manipulators,thenumberofcontrolinputtorquesappliedbytheactuators,islessthan

thenumberofdegreesoffreedom(DOF),alsoknownasunder-actuatedsystems.This

happensbecauseforeachjointthereisoneactuator,andtwodegreesoffreedom.This

conditionimpliesthattheimplementationofafullstatefeedbackcontroltaskneeds

additionalsensorsformeasuringthestatevariablesoftheactuatoraswellasthelinks.

Thepurposeofcontrolistoappropriatelydealwiththeoscillationcreatedbythejoint

elasticityinordertogetfastpositioningandaccuratetrajectorytrackingofthe

manipulatorend-effecter.Therefore,propercontrolofelasticjointrobotmanipulators,

withdifferentcharacteristicexpectations,andwithareasonablenumberofsensors,is

stillachallengingproblemthatneedstobeaddressedandindeed,isoneofthe

motivatingfactorsforthisresearchproject.

1.2LiteratureReview

Inthissectionseveralpapersassociatedwiththreedifferenttopicsareinvestigated.First

dynamicmodelingofflexiblejointrobotmanipulatorsisbroughtintofocus.Then

differentstrategiesforcontrolofflexiblejointrobotmanipulatorsarebrieflystudied.

Finallyafewpapersregardingtheanalysisofhydraulicactuatedrobotsarereviewed.It

shouldbenotedthatthereareotherpaperspublishedonthetopicofcontrolofflexible

andstiffmanipulatorsbuttheseareonlymarginallyrelatedtothetopicofthisthesis.

Thefollowingrepresentsasampleofthepublicationsinthisarea.

MarioandSpongstatedthatrecentexperimentalworksshowedthatjointelasticitywas

themainsourceofcomplianceinmostcurrentmanipulatordesigns[2].Thisjoint

elasticityoriginatedfromseveralmotiontransmittingcomponents,suchaselasticityin

thegears,belts,hydrauliclines,etc.Inthispaper,asinglerigidlinkmanipulatorwithan

elasticjointwasmodeled.Twocontrolsolutionswerederivedandcompared:onewas


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basedonthefeedbacklinearizationtechnique,andtheotheronebasedonthenonlinear

compositetechnique.Theresultsindicatedthatthefeedbacklinearizationcontrol

neededafullstatefeedback(thepositionsandvelocitiesofthelinkandmotor);

however,thenonlinearcompositecontrolrequiredonlythepositionandvelocityofthe

linkasthestatefeedback.

Nicosiaetalexaminedsomeproblemsofdynamicalcontrolofmanipulatorswithhigh

speedcontinuousdisplacement[3].Firsttherigidmodelwasdevelopedandthenelastic

anddissipativejointswereincluded.Acontrolstrategybasedonanonlinearfeedbackof

amodelofthelocaljointvariableswasapplied,andthentheperformanceofthe

controlledmanipulatorwasevaluatedbymeansofsimulation.Theresultsshowedthat

theuseofelasticjointsinducedvibrations.Theresultswerenotverifiedexperimentally.

Spongderivedasimplemodeltorepresentthedynamicsofelasticjointmanipulators

[4].Twobasicassumptionsaboutthedynamiccouplingbetweenthemotorandthelink

wereusedtoderivetheequations.First,itwasassumedthatthekineticenergyofthe

rotorwasduetoitsownrotation;thesecondassumptionwasthattherotor/gearinertia

wassymmetricabouttherotoraxisofrotation.Twodifferentcontrollawswerethen

examined.Thefirstonewasthefeedbacklinearizationcontrol,andthesecondonewas

correctivecontrol,basedontheintegralmanifoldlaw.Theresultsindicatedthatthe

correctivecontrollawdemonstratedbettertrajectorytracking.

Yuetalpointedoutthattheirexperimentalinvestigationsindicatedthatthejoint

elasticityhadtobeconsideredinthedynamicsofmanipulators,especiallywhensome

components,suchasharmonicdriveswereincludedinthejoint[5].Ontheotherhand,

considerationofjointelasticitycomplicatedthedynamicequationsinsuchamanner

thatthecurrentstrategiesforrigidjointrobotscontrolcouldnotbeuseddirectly.Inthis

paper,themodeldevelopedbySpong[4]wasadopted.Anadaptiveobservercontrol

wasthenappliedforthetrajectorytrackingoftherobotmanipulatorswithflexiblejoint.

Thesimulationresultsshowedthattheestimatedandactuallinkpositionswerewell

matchedwiththedesiredlinkposition.


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Albu-SchafferandHirzingerincludedthedampingeffectsofthetransmissiondevicesin

thedynamicmodelproposedbySpong[6].Differentcontroltechniqueswerebriefly

discussed.Itwaspointedoutthatsometechniques,suchasfeedbacklinearization,were

theoreticallycomplete,buttheirimplementationwereextremelydifficultduetothe

measuringand/orcomputingofallthestatevariables.Theproposedcontrolstrategy

startedusingthestatefeedbackcontroller;thecontrollergraduallywasextendedby

addingmoredetailedrobotdynamics.Theresultsprovedthatthevibrationscausedby

jointflexibilitywereeffectivelydampedbyusingtheproposedcontroltechnique.

Thummeletalshowedthatvibrationinducedbyelasticjointscouldbesignificantly

reducedbyusingfeedforwardcontrol,basedoninversedynamicmodels[7].The

modelconsistedoftheinertiaofmotorandlinkconnectingbythemeansofaspringand

adamperthatdescribedtheelasticityanddampingofthegearbox,respectively.Also

velocitydependent,frictiontorquesinthebearingsforbothmotorsideandlinkside

wereincluded.Introducingamorerealisticmodel,theelasticityinthejointwas

modeledasanonlinearspring.Somecontrolalgorithmswerethenreviewed.By

extendingthefeedforwardpartofthecontrollerandusingthemorecompleterobot

dynamics,smalleramplitudeofvibrationwasexperienced.

Habibietalmodeledalargehydraulicrobottoderiveageneralmathematicalfunction

forhydraulicactuationmanipulators[8].Thepaperfocusedonthedynamicsof

servovalveandactuatorbywritingtheflowequationsrelatedtothesecomponents.A

comparisonwasmadeamongthedifferentflowterms.Itwasconcludedthatthe

compressionflowsweresignificantandsubsequently,theywereincludedinthederived

model.

Insummary,itcanbeseenthatsomeresearchworkhasbeendevotedtothecontrolof

theflexiblejointmanipulators,butmoreworkneedstobedonetofullyunderstandthe

performanceoftheflexiblejointmanipulators.Itisanticipatedthatsignificant

advancementsinthisareamightoccurinnextdecades,inadvancingtheunderstanding


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ofthefundamentalsofflexiblejointsandindevelopingpracticalandcosteffective

controllawsforsuchmanipulators.

Oneoftheconstraintsthatresearchershaveindevelopingcontrolalgorithmsforflexible

jointsisverifyingtheapproachesexperimentally.Atpresent,commercialflexiblejoints

haveeithernoadjustmentcapabilityoriftheydo,requiremechanicaldeviceswhich

mustbereplacedexternally.Itishighlydesirabletohaveaflexiblejointinwhichthe

operatingcharacteristicsofthejointsuchasdampingratioandnaturalfrequencycanbe

adjustedonthefly.Thiswouldallowcontrolalgorithmswhichhavebeendesignedto

reducetheeffectsofflexibilityinjoints,tobetestedunderawiderangeofconditions

andtoconditionswhichchangeintimecontinuously.Designingaroboticjointwhich

wouldallowadjustableflexibilitythusbecamethefocusofthisresearch.

1.3Objectives

Theobjectiveofthisresearchprojectwastodesignandimplementanexperimental

hydraulicactuatedrobotmanipulatorthatsimulatesrobotjointflexibilityinthespecific

rangeoffrequenciesanddampingratiowiththeabilityofchangingthejointflexibilitys

parameters.Thedampingratioforsuchmanipulatorsisnormallylessthan0.1andthe

naturalfrequencybetween5to15Hz[9].Theapproachthatwastakeninthisresearch

isasfollows.Amodelbasedtransferfunctionwhichrepresentsthedynamicsofflexible

jointmanipulatorswasmodeled.Applyinginputimpulsesignalstothetransferfunction,

anoscillatoryangularaccelerationoutputsignalwasderivedbyusingcomputerbased

simulation.Forthisresearchproject,thismodelbasedarrangementwasdefinedasa

functiongenerator.Theoutputofthefunctiongeneratorwasemployedtodrivean

experimentalhydraulicsystemwhichconsistedofaservovalveandarotaryhydraulic

actuatorconnectedtoasingle-rigid-linkmanipulator.Thisexperimentalarrangement

wasdefinedasahydraulicsimulator.Theaimofthecontrollerwhichdrivesthe

systemwastomakethehydraulicsimulatorreproducethefunctiongeneratorsignalin

thespecificrangeoffrequenciesanddampingratioofthefunctiongeneratorsignal.


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1.4ThesisOutline

Thissectionincludesanoutlineofthisthesis.Chapter2introducesthedynamicmodel

andsimulationofaflexiblejointrobot.Chapter3analyzestheoreticallyanactual

hydraulicactuatedmanipulator.InChapter4theexperimentalsetupisincluded.

Chapter5showsanddiscussestheexperimentalresultsandfinallyChapter6concludes

theresultsandrecommendspossiblefutureworks.


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CHAPTER2

FUNCTIONGENERATOR

2.1Introduction

Criticaltothedevelopmentoftheflexiblejointsimulatorwasthefunctiongenerator

whichinputtheappropriatesignaltothehydraulicjoint.Thegoverningequationswhich

describethebehavioroftheflexiblejointareconsideredinthisChapter.Theobjective

ofthischapter,then,istopresentdynamicmodelingofasingle-rigid-linkrobot

manipulatorwithflexiblejoint.Themodelisthensimulatedusingtypicalparameters

foraflexiblejointinordertodemonstratethebehaviorofthelinkangularposition,

velocity,andaccelerationforaspecifiedtorqueinput.

2.2GoverningEquations

Asingle-rigid-linkmanipulatorwitharevoluteflexiblejointisshowninFigure2.1.In

thisfigure,theflexiblejointismodeledasatorsionalspringandatorsionaldamperin

paralleltoeachother(forsimplicity,linearequivalentsymbolsofspringsanddampers

areadopted).IandJarethelinkandthemotormomentsofinertia(withrespecttothe

centerofrotation),respectively,uisthemotorinputtorque,Kisthejointstiffness,and

Disthejointviscousdamping.Theangles andaretheangularpositionsofthelink


23/108

8

Rotational

SpringRotational

DamperD

K

MotorJ

Link

I

u

(a)Linearequivalentschematicofthesingle-linkmanipulator

(b)Dynamicequivalentschematicofthesingle-linkmanipulator

Figure2.1-Schematicofasingle-linkmanipulatorwithflexiblejoint


24/108

9

andthemotor,respectively.Thelinkandmotorangularpositionsarechosenasthe

generalizedcoordinates.Thelinkmotionisassumedtobeinthehorizontalplane.

Jointstiffnessanddampingareincludedinthedynamicsofthemodelpresentedhere.

Forsimplicity,gravityeffectsarenotconsideredinthemodel;thisisconsistentwiththe

experimentalsetup(hydraulicsimulator)usedinthisstudywhichisahorizontalplanar

robotmanipulator.

ForthelinkshowninFigure2.1thedynamicequationsareasfollow:

0)()( =++ KDI &&&& ,(2.1)

uKDJ =++ )()( &&&& ,(2.2)

or:

=

+

+

uKK

KK

DD

DD

J

I 0

0

0

&

&

&&

&&

.(2.3)

Theinputtothesinglelinkmanipulatoristorque uwhichisassumedtobeanimpulse

signal.TakingtheLaplacetransformofequations(2.1)and(2.2),andassumingzero

initialconditions,yields:

0)]()([)]()([)(2 =++ ssKssssDsIs ,(2.4)

)()]()([)]()([)(2 sUssKssssDsJs =++ .(2.5)

Rearrangingequations(2.4)and(2.5)gives:


25/108

10

0)()()()( 2 =+++ sKDssKDsIs ,(2.6)

)()()()()( 2 sUsKDssKDsJs =+++ .(2.7)

Solvingfor )(s inequation(2.6):

)()(

)(2

sKDs

KDsIss

+

++= .(2.8)

Substitutingequation(2.8)intoequation(2.7),yields:

)()()()())((

22

sUsKDssKDs

KDsIsKDsJs=+

+

++++,(2.9)

)()()(])())([( 222 sUKDssKDsKDsJsKDsIs +=+++++ ,(2.10)

)()()(])()([ 234 sUKDssKsJIDsJIIJs +=++++ .(2.11)

Finally,thetransferfunctionwhichrelateslinksrotation toinputtorqueuisas

follows:

])()([)(

)(22 KJIDsJIIJss

KDs

sU

s

++++

+=

,(2.12)

or:


26/108

11

22

1*

)()()(

)(

s

J

K

I

Ks

J

D

I

Ds

IJ

Ks

IJ

D

sU

s

++++

+

=

.(2.13)

Thisindicatesthattheundampednaturalfrequenciesofthesystemarezeroand

)(J

K

I

K+ .Zeronaturalfrequencyrepresentstherigidbodymotionofthebaseandthe

link.

Theblockdiagramoftheopen-loopsystembasedonequation(2.13)isshowninFigure

2.2.Inthisfigure:

IJ

Kk =1 ,

K

D= , )(

J

K

I

Kn += and )

11(

1

2 JIK

D+=

Figure2.2-Theopen-loopblockdiagramofsinglelinkmanipulatorwithflexiblejoint

Itisevidentthatthesystemhastwointegratorswhichwillsignificantlyfilteroutany

higherfrequencyperturbationswhichmightbeintroducedbythejoint.

TheequationswereimplementedusingMATLAB.Inthissimulation,the

manipulatorsphysicalparametershavebeentakenfrom[9]andarelistedinTable2.1:

22

1

2

)1(

nnss

sk

++

+

s

1

s

1


27/108

12

Table2.1-Simulationparametersadoptedfrom[9]

Symbol Description Quantity Unit

I Linkmomentofinertia 5.000 radsmN /2

J Motormomentofinertia 5.000 radsmN /2

K Jointstiffness 1.000E4 radmN /

D Jointviscousdamping 31.63 radsmN /

Applyingtheseparameterstoequation2.13yields:

22

1*

400065.12

)100316.0(400

)(

)(

sss

s

sU

s

++

+=

.(2.14)

Comparingwiththestandardtransferfunction

222

1 1*2

)1(

)(

)(

sss

sk

sU

s

nn

++

+=

,(2.15)

thenaturalfrequencyanddampingratioareasfollows:

sradn /25.634000 == =10.00Hzand 1.0= 0

Itshouldbeemphasizedthattherangeofvaluesusedinthisprojectarequitecommon

forflexiblejointrobotmanipulators.

Theinputtothesimulationwasanapproximateimpulse(representingatorque

impulse).SomeofthesimulationresultsareshowninFigure2.3.Inthisparticular

simulationtwotimedelayedpulseinputtorqueswithzeroand1.5 sphasedelay,period


28/108

13

of3s,pulsewidthof0.1%ofperiod,andamplitudeofu=+3000 mN andu=-3000

mN havebeenused.Theoutputresultsshowninthefigurearethelinkangular

position,velocity,andacceleration,respectively.Itshouldbenotedthatacontinuous-

timeintegrationsolverofRunge-Kuttamethodwithzeroinitialconditionandafixed-

stepsizeof0.001insecondshasbeenapplied.

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time(s)

AngularPosition(rad)

(a)Angularposition


29/108

14

-1

-0.5

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time(s)

AngularVelocity(rad/s)

(b).Angularvelocity

-60

-40

-20

0

20

40

60

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time(s)

AngularAcceleration(rad/s2)

(c)Angularacceleration

Figure2.3-Simulationresultsforimpulseinputstothesingle-linkmanipulator


30/108

15

Theresultsshowthatbyapplyinganimpulsetorqueinput,adampedoscillationis

observableinangularaccelerationandvelocity,butthemagnitudeishighlyfilteredin

angularpositionofthelink.Thesevibrationsmaybefelt,butnotobservedvisually.

Sincetorqueonthejointisdirectlyrelatedtoangularacceleration,thelowdampingcan

havesignificantimplicationswhencontrolisimplemented.

Aclosedformsolutionofequations2.14or2.15doesexistandcanbesolvedby

applyingPartial-FractionExpansion.Considerequation2.14:

400065.12

)100316.0(400

)(

)(2

2

++

+=

ss

s

sU

ss,(2.16)

or

)(400065.12

400265.1)(

2

2sU

ss

sss

++

+= .(2.17)

Inordertocreatethesameformoftheinputsignalasthesimulation(anapproximate

impulse),twoseparatestepinputsareused:thefirstastepof3000N.matzerosteptime

andasecondstepat0.003s(0.1%ofthe3speriodtime)steptimeandfinalvalueof

-3000N.m.Inthiscase, s

e

ssU

s003.030003000)(

= andequation2.17becomes:

++

+=

s

e

sss

sss

s003.0

2

2 30003000

400065.12

400265.1)( ,(2.18)

or:

)()()400065.12(

)12000003795(

)400065.12(

12000003795)( 212

003.0

2

2 sFsFsss

se

sss

sss

s

=

++

+

++

+=

.(2.19)

Knowingthat;

)9285.62325.6)(9285.62325.6(400065.122 jsjsss +++=++ ,(2.20)


31/108

16

andthePartial-FractionExpansionoffirsttermofequation2.19:

400065.12)400065.12(

120000037952

321

2++

++=

++

+

ss

asa

s

a

sss

s,(2.21)

multiplyingbothsidesbythedenominatorofitslefthandsideofequation2.20yields:

sasassas 32

2

2

1 )400065.12(12000003795 ++++=+ ,(2.22)

or

131

2

21 4000)65.12()(12000003795 asaasaas ++++=+ .(2.23)

Comparingbothsidesofequation2.23,itcanbeshownthat:

021 =+ aa ,(2.24)

379565.12 31 =+ aa ,(2.25)

12000004000 1 =a ,(2.26)

.

Solving,theparametersarefoundtobe:

3001 =a , 3002 =a and 03 =a .

Therefore:

400065.12

300300

)400065.12(

12000003795

)( 221 ++=++

+

= ss

s

ssss

s

sF ,(2.27)

or:


32/108

17

22219285.62)325.6(

)9285.62(1.0)325.6(300)

1(300

)400065.12(

12000003795)(

++

+=

++

+=

s

s

ssss

ssF .(2.28)

Considering )()( 1003.

2 sFesFs

= ,usingequation2.28andapplyingtheinverseLaplace

transformtoequation2.19yields:

( ) )003.0()]003.0(9285.621.0)003.0(9285.62[300300)()9285.621.09285.62(300300)(

)003.0(325.6

325.6

=

tutSintCose

tutSintCoset

t

t&&

(2.29)

Figure2.4showstheplotoftheanalyticalsolutionofequation2.29.

0 0.5 1 1.5-40

-30

-20

-10

0

10

20

30

40

50

Time(s)

AngularAccele

ration(rad/s2)

Figure2.4-Plotofequation2.29.


33/108

18

Thetwosolutionsforaccelerationresponseusingthecomputersolverandtheclosed

solutionformyieldthesametheresultsaswouldbeexpected.Thefunctiongenerator

canberealizedeitherfromthesimulationusingMatlaborbyimplementingtheclosed

solutionformofequation2.29.Inthiswork,thesimulationformwasusedbecauseof

theeaseofchangingtheparametervalues.


34/108

19

CHAPTER3

HYDRAULICSIMULATOR

3.1Introduction

Thehydraulicsimulatorisessentiallytheexperimentalhydraulicsystemwhichreceives

theinputsignalfromthefunctiongeneratorandoutputsanaccelerationwhichis

controlledtofollowthedesiredinputsignal.Themainreasonforchoosingahydraulic

actuationsystemwasbecauseofthehighervalueofitsnaturalfrequencyincomparison

totheusualrobotmanipulatorsnaturalfrequency.Hydraulicsystemsarealsoreadily

foundinmanyroboticapplicationswhichmakecompatibilityissuesmarginalized.It

wasrecognizedthatthedesignofasystemtoreproduceaccuratelysuchwaveforms

wouldbeachallengingprocess,giventhataccelerationtendstobenoisyanddifficultto

control.Aswillbeillustratedlater,thepresenceofaseverenonlinearity(suspectedto

bedead-band)intherotarymotorcompoundedthecontrolproblem.Figure3.1shows

theoverallarrangementofthesystem.


35/108

20

-60

-40

-20

0

20

40

60

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time(s)

AngularAcceleration(

rad/s.sq)

Figure3.1-Schematicofthehydraulicsimulator

ThesystemshowninFigure3.1consistedofapressurecompensatedhydraulicpumpdrivenbyanelectricmotor,apressurecontrolservovalve(MOOGmodel15-010),

andarotarymotor(MICRO-PRECISIONROTACmodelMPJ-22-1V),whichdrovea

single-linkmanipulator.Therotarymotorcouldrotatethelink270inthehorizontal

plane.

Themeasuringinstrumentsconsistedofanaccelerationsensor(Bruel&Kjaermodel

4370)andamplifier(Bruel&Kjaerchargeamplifiertype2635).Theexperimental

systemwascontrolledbyaPCcomputerusingMatlab-Simulinksoftware.

Aflowcontrolservovalvewasinitiallyconsideredbutbecausetheflowrateswerevery

small,becausethecut-offfrequencyofpressureservovalvesisknowntobeveryhigh

(relativetoaflowcontrolservovalve),andbecausepressure(force)isdirectlyrelatedto

acceleration,apressurecontrolservovalvewaschosenasthehydrauliccontrolling

device.AllcomponentswereofftheshelffromtheFluidPowerLaboratoryatthe

UniversityofSaskatchewanandhence,thisbecameaconstraintforthehydraulic

systemdesign.


36/108

21

3.2HydraulicSystemAnalysis

Asstatedabove,thefunctionofthehydraulicsimulatorwastobeabletoreproducethe

inputsignalwhichinthisparticularcasewastheangularaccelerationsignalcreatedby

thefunctiongenerator.Tofacilitatethedesignofacontroller,ananalysisofthe

hydraulicsystemwasperformed.

Acommonformofatransferfunctionapproximationoftherelationshipbetweenthe

loaddifferentialpressureandtheinputcurrenttotheservovalvehasbeenderivedtobe

[10]:

12)(

)(2

+

+

=

ss

K

sI

sP

nv

v

nv

v

e

L

,(3.1)

inwhich:

LP loaddifferentialpressure(Pa) eI inputcurrent(ma)

vK valvegain(Pa/ma)

nv valveundampednaturalfrequency(Hz)

v valvedampingratio

sLaplaceoperator(1/s).

Thetorqueonthehydraulicrotarymotorisgivenby;

+== LmLL IDPT ,(3.2)


37/108

22

inwhich:

LT motortorque( mN )

mD

motorvolumetricdisplacement )/(

3 radm

LI linkmomentofinertia )/(2 radsmN

motorviscousdamping )/( radsmN

linkangularvelocity(rad/s)

linkangularacceleration )/( 2srad ,

TakingtheLaplacetransformofequation3.2yields;

)1()(

)(

+

=

sI

D

sP

s

L

m

L

,(3.3)

or:

)1()(

)(

+

=

sI

sD

sP

s

L

m

L

.(3.4)

let;

mDK =1 and

L

I=1

therefore;

1)(

)(

1

1

+

=

s

sK

sP

s

L

,(3.5)

inwhich;

1K loadgain( sParad / )


38/108

23

1 loadtimeconstant(s).

Multiplyingequations3.1and3.5,gives;

11

2)(

)(

1

1

2+

+

+

=

s

sK

ss

K

sI

s

nv

v

nv

v

e

.(3.6)

Thefollowingequationisvalidfortheservo-amplifierwhichshouldprovideDCcurrent

intotheservovalve:

Lt

di

Riv

e

ee

''+=

,(3.7)

inwhich:

ev servo-amplifierinputvoltage(v)

'

ei servo-amplifieroutputcurrent( a)

Rservo-amplifierresistance()

Lservo-amplifierinductance(H).

TakingtheLaplacetransformoftheequation3.7yields;

1

1

)(

)('

+

=

sR

LR

sV

sI

e

e ,(3.8)

LetR

Ka1

= andR

La = ,therefore;

1)(

)('

+=

s

K

sV

sI

a

a

e

e

,(3.9)

inwhich;


39/108

24

aK servo-amplifiergain(a/v)

a servo-amplifiertimeconstant(s)

Itshouldbenotedthataccordingto[11]fortheservo-amplifier, R=2000,L=9.7H.

Therefore vaKa /0005.0= and sa 00485.0= .

Sincetheunitofservo-amplifieroutputcurrent( )('sIe )isampereandthatofservovalve

inputcurrent( )(sIe )ismilliampere,aconversiongainof eK isintroducedtoconvert

theunits.Therefore, amaKe /1000= .Applyingthegaintoequation3.9gives;

1)(

)(

+=

s

KK

sV

sI

a

ea

e

e

,(3.10)

inwhich;

eK servo-amplifierconversiongain(ma/a).

Multiplyingequations3.6and3.10yields;

11

21)(

)(

1

1

2+

+

+

+=

s

sK

ss

K

s

KK

sV

s

nv

v

nv

v

a

ea

e

(3.11)

Theblockdiagramofequation3.11isshowninFigure3.2.


40/108

25

Figure3.2-Blockdiagramofequation3.11.

Itshouldbenotedthat )(s hasunitsofrad/s2.

Since a 1 ,theservo-amplifiertransferfunctioncanbereducedto eaKK .Thisreduced

formisillustratedinFigure3.3:

Figure3.3-Reducedblockdiagramofequation3.11

Nowtwopossiblescenariosareexpectedwithrespecttothetransferfunctionsofthe

servovalveandtheload.Thefirstscenariooccurswhen nv

1

1,where nv isnowin

Hztoensureunitconsistency.Atypicalmagnitudeasymptoticapproximation

(magnitudeBodeplot)forthisfirstcaseisshowninFigure3.4.

LoadServovalveServoamplifier

)(sVe

)(sPL )(sIe

1

22

+

+

s

s

K

nv

v

nv

v

11

1

+s

sK

)(s

1+s

KK

a

ea

LoadServovalveServoamplifier

)(sVe )(sPL )(sIe

12

2

+

+

s

s

K

nv

v

nv

v

11

1

+s

sK

)(s

eaKK


41/108

26

Figure3.4-TypicalmagnitudeBodeplotofequation3.11whennv

1

1

Thesecondscenariohappenswhen nv

1

1;atypicalBodeplotforthiscaseis

demonstratedinFigure3.5.

Figure3.5-TypicalmagnitudeBodeplotofequation3.11when nv

1

1

1

1

nv

Slope-20

(dB/decade)

Slope+20

(dB/decade)

)(Hz

)(dB

Ve

Slope-40

(dB/decade)

)(Hz

)(dB

Ve

Slope-40

(dB/decade)

Slope+20

(dB/decade)

1

1

nv


42/108

27

TheactualfrequencyresponseofthehydraulicsimulatorisshowninFigure3.6,where

theinputisthesignalfromtheanalyzerunitandtheoutputisthemeasuredsignalfrom

theaccelerometer.DetailsonobtainingthisplotaregiveninChapter5.

-40

-30

-20

-10

0

10

20

0.1 1 10 100

Frequency(Hz)

Figure3.6-Open-loopmagnitudefrequencyresponseofthehydraulicsimulator

(points)withsuperimposedasymptoticlines(experimental:solidlinebasedon40dB

perdecadeandtheoretical:dashedlinebasedon20dBperdecade)

ItshouldbenotedthatthetwooddpointsatlowerfrequencieswereignoredinFigure

3.6.

Acomparisonbetweentheexperimentalresultandthetheoreticalscenariosshowsthat

thefirstpossiblescenario( nv

1

1)betterdescribestheactualexperimentalsetupfor

frequencieslessthan20Hz.

Itshouldbenotedthattheslopeofasymptoticline(solidline)inFigure3.6is+40dB

perdecadewhichisdifferentfromthatofthetheoreticalmodel(dashedline)ofFigure

3.6.Measurementoftheoutputangularaccelerationwasaccomplishedusingan

accelerometer(tobediscussedingreaterdetailbelowandinChapter4).Itwasbelieved

)(

)(

)(

dB

jV

j

e


43/108

28

thatthepoorperformanceoftheaccelerometeratlowfrequenciescausedthe

discrepancy.Inthenexttwochaptersitwillbeshownthatanaccelerationcompensator

couldbeusedtocorrecttheperformanceoftheaccelerometeratlowfrequencies.

Theseresultsimplythat,1

1

nv ,andthereforethemodelcouldbeeasilyreducedto

thefirstordertermofthetransferfunctionofequation3.11Hence,

1)(

)(

+=

s

sK

sV

s

o

o

OLe

,(OLinthiscasemeansopen-loop.)(3.12)

inwhich;

oK open-loopgain( svrad / )

o open-looptimeconstant(s).

Itshouldbenotedthat 1KKKKK veao = and 1 =o .Thereforetheopen-loopblock

diagramissimplifiedtothatshowninFigure3.7.

Figure3.7-Simplifiedopen-loopblockdiagramofthesystem

Intheaforementionedequations, hasaunitofrad/s2.Sincealinearaccelerometer

wasusedtoindirectlymeasuretheangularaccelerationofthejoint,alinearsensitivity

andcalibrationfactorwhichconvertsrad/s2tovoltagewasintroduced.Thecalibrated

outputratingofthelinearaccelerometerwasdeterminedtobe )/

(1.02sm

vS l = .The

linearaccelerometerwascapableofmeasuringtangentialacceleration.However,itwas

theangularaccelerationthatwasdesired.Sincethedistancebetweenthecenterofthe

installedlinearaccelerometerandtheaxisoftherotation(rotationarm)wasknownvery

accurately(l=0.435m),itwaspossibletoconvertthelineartangentialaccelerationto

)(sVe

1+s

sK

o

o

)(s


44/108

29

angularacceleration.Bymultiplyingthelinearsensitivity lS byl,theangularsensitivity

(theconversiongain)oftheaccelerometerwasobtainedas:

)/

(0435.0)(435.0)./

(1.0.22 srad

vmsmvlSK lcon === (3.13)

Theeffectofthe conK isshownintheblockdiagramofFigure3.8;

Figure3.8-Blockdiagramofthesimplifiedtransferfunctionwithinfluenceofthe

accelerometersensitivity

IntheblockdiagramofFigure3.8;

conK conversiongainfactor( 2/srad

v

)

d desiredangularacceleration(functiongeneratoroutput)(2/srad )

linkangularacceleration( 2/srad )

v linkangularaccelerationinvolts(v),

therefore;

1)(

)(

+=

s

sK

sV

s

o

OL

OLe

v

,(3.14)

inwhich;

OLK open-loopoverallgain(s).

Itshouldbenotedthat conoOL KKK = .

)(sv )(s

)(sVe )(sd

conK

1+s

sK

o

o

conK

Function

Generator


45/108

30

3.3ControllerDesign(Theoretical)

Todesignthecontrollerwhichwouldprovidetherequiredclosed-loopresponse,the

firststepwastoconsiderthehydraulicsimulatorinamathematicalformandtodesign

thecontrollerforthissystem.Thesecondstepwastoapplythiscontrollertothe

experimentalhydraulicsimulatorandtotunethecontrollertooptimizethe

performance(seeChapter5).

Inthissection,aclosed-loopcontrolsystembasedonatheoreticalmodelisstudied.The

closed-looptransferfunctionisderivedanditsparametersarecomparedwiththeopen-

looptransferfunction.Anapproachofcascadeinversecompensationforimprovingthe

open-loopsystemratherthanusingaclosed-loopcontrollerisintroduced.

3.3.1Closed-LoopController

Theobjectiveofanycontrolleristoforcetheoutputtofollowtheinputsuchthat

CLe

v

sV

s

)(

)(=1(CLinthiscasemeansclosed-loop).Thismeansthehydraulicsimulator

wouldfollowexactlythedesiredsignalfromthefunctiongenerator.Thissection

considerstheimplementationofseveralclassicalcontrollersinanattempttomeetthe

aboveobjective.Forallanalysis,thesimplifiedopen-looptransferfunctionofEquation

3.14isassumed.

Asimpleproportionalcontrollerwasfirstexamined.Theclosed-loopblockdiagramis

illustratedinFigure3.9.


46/108

31

Figure3.9-Blockdiagramofclosed-loopcontrolsystemwithproportionalgain

InFigure3.9;

PK proportionalgain

Theclosed-looptransferfunctionisderivedwithrespecttotheblockdiagramshownin

Figure3.9asfollows;

11

1

)(

)(

1

++

+=

s

sKK

s

sKK

sV

s

o

OLp

o

OLp

CLe

v

,(3.15)

whichsimplifiesto:

1)(

)(

1

1

1

+=

s

sK

sV

s

CL

CL

CLe

v

,(3.16)

inwhich; OLpCL KKK =1 ,and OLpoCL KK+= 1 .

Itisclearthattheobjectiveofhaving 1)(

)(

1

=

CLe

v

sV

sisnotsatisfiedatallfrequencies

withthiscontroller.Table3.1providesacomparisonoftheopen-loopandclosed-loop

transferfunctionparameters.

)(sv

+

_

)(sVe

1+s

sK

o

OL

pK


47/108

32

Table3.1-Parameterssummaryofopen-loopandclosed-looptransferfunctions

Open-loop Closed-loop

Gain conoOL KKK=

OLPCL KKK=

1

Timeconstanto OLPoCL KK+= 1

TheoreticallythisTableillustratesthattheclosed-loopcontrolgain(if 1PK )andtime

constantareincreasedincomparisontotheopen-loopsystem.Usingtypicalparameter

values,acomparisonoftheopen-loopandclosed-loopfrequencyresponses(magnitude)

fortwocontrollergainsisshowninFigure3.10.

10-2

10-1

100

101

102

-60

-50

-40

-30

-20

-10

0

Magnitude(dB)

Frequency(Hz)

Open-Loop

Closed-Loop

Gain=2.0

Gain=1.5

Figure3.10-Magnitudefrequencyresponse)(

)(

sV

s

e

v oftheopen-loopandclosed-loop

systemswithproportionalcontroller

Figure3.10indicatesthattheclosed-loopcontrolwithproportionalgainincreasedthe

bandwidthofthesystem,butonlymarginallyimprovedthemagnituderatio.Itshould


48/108

33

benotedthatinexperiments,therewasalimitedroomforincreasingtheproportional

gainbecauseofservovalvesaturation.

TypicalvaluesofKOLand o wereextractedfromtheexperimentalfrequencyresponse

oftheactualsystem(Figure3.6): sKOL 025.0= and so 064.0= .

Theabilityofthesystem(open-loopandclosed-loop)tofollowtheinputwas

demonstratedinthetimedomainbyinputtinga10 Hz.jointvibration(theoutputfrom

thefunctiongenerator)intothesystem.Figure3.11comparestheresponsesoftheopen-

loopandclosed-loopmodels.Itisevidentthattheclosed-loopcontrollermarginally

increasestheamplitudeoftheoutputaccelerationwhencomparedtotheopen-loop

system.Thisisconsistentwiththegainincreaseinthefrequencyresponseshownin

Figure3.10at10Hz.Increasingthegaineventuallysaturatestheinputsignaltothe

servovalveandnofurtherimprovementwaspossible.Itisclearthatthisisnotan

acceptableresponse.

1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Time(s)

Accelerationamplitude(v)

FunctionGenerator

Closed-Loop

Open-Loop

Figure3.11-Transientresponseof v intheopen-loopandclosed-loopmodelswith

2=PK tothefunctiongenerator d of10Hzsignal


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34

AclassicalPIcontrollerwasthenimplemented.Theclosed-loopblockdiagramis

showninFigure3.12.

Figure3.12-Blockdiagramofclosed-loopcontrolsystemwithproportionaland

integralgains

InFigure3.12;

PK proportionalgain

IK integralgain(1/s)

Theclosed-looptransferfunctioninthiscaseisderivedwithrespecttotheblock

diagramshowninFigure3.12asfollows;

1

)(1

1

)(

)(

)(

2

+

++

+

+

=

s

KsKK

s

KsKK

sV

s

o

IPOL

o

IPOL

CLe

v

,(3.17)

whichissimplifiedto;

)1()(

)(

)(

)(

2

+++

+=

OLIoOLP

IPOL

CLe

v

KKsKK

KsKK

sV

s

.(3.18)

__

+ )(sv

s

KK IP +

1+s

sK

o

OL

)(sVe


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35

Usingthesamegainandtimeconstantvaluesasbefore,

i.e. sKOL 025.0= and so 064.0= ,acomparisonoftheopen-loopandclosed-loop

frequencyresponses(magnitude)fortwocontrollergainsisshowninFigure3.13.

10-2

10-1

100

101

102

-60

-50

-40

-30

-20

-10

0

Magnitude(dB)

Frequency(Hz)

Open-Loop

Closed-Loop

P=2.0,I=10

P=1.5,I=5

Figure3.13-Magnitudefrequencyresponse)(

)(

sV

s

e

v ofopen-loopandclosed-loop

systemswithPIcontroller.

ItcanbeobservedfromFigure3.13thattheproposedPIcontrollercouldimprovethe

responseatlowerfrequenciesbetterthantheproportionalcontroller,butthemagnitude

ratiowasstillaproblem.

Figure3.14comparesthetimeresponsesoftheopen-loopandclosed-loopmodelsfora

10Hzjointfunctiongeneratorsignal.


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36

1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Time(s)


FunctionGenerator

Closed-Loop

Open-Loop

Figure3.14-Transientresponseof v intheopen-loopandclosed-loopmodelswith

2=PK and 10=IK tothefunctiongenerator d of10Hzsignal

ItisevidentfromFigure3.14thatthePIcontrolleralsomarginallyimprovesthat

amplitudeoftheoutputbutwasnotsufficientforthefrequencyrangeofinterestinthis

study.

InChapter5itwillbeshownthattheP(proportional)orPI(proportional-integrator)

closed-loopcontrollerscouldnotbeimplementedphysicallybecauseofnoiseinthe

accelerometer(sensor),significantnonlinearitiesintherotarymotorandhighsystem

internalgains.Thereforeanalternateapproachwasproposedandisnowconsidered.


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37

3.3.2Open-LoopInverseCompensationController

Toachievethestatedobjectiveatthebeginningofthissection,acascadecompensator

controllerwasproposedwhoseformwassimplytheinverseoftheopen-looptransfer

functionshowninequation3.14.Ontheotherhand,sincetheplanttransferfunction

includedazeroattheorigin(Root-locusapproach),thecontrollerdeemedtobean

exampleofmarginallystablepole-zerocancellationwhichmightmakethesystem

unstableinsomeconditions(e.g.withconstantdisturbanceornoiseintheinputsignal).

Thereforethecontrollerpolewasdesignedtobeontherealaxes,atthelefthandsideof

imaginaryaxesandveryclosetotheorigin.Thispoleusuallyshouldbeabout

o /1.0= awayfromtheoriginsothatthemagnifyingsignaldoesnotexceedmore

than10timesoftheconstantinputnoise.Theblockdiagramofthecontrolleris

illustratedinFigure3.15.

Figure3.15-Blockdiagramofopen-loopcompensatorcontroller

Itshouldbenotedthattheopen-looppureinversecompensatorwhichwasproposed

earliermightsaturatethehydraulicsimulatorinfrequencieslowerthano

1(2.5Hzor

15.71rad/sbasedonactualsystemmagnitudeBodeplotshowninFigure3.6),however

itwasprovedbysimulationthatthepureinversecontrollerworkedproperlyinthe

frequencyrangeofinterestwhichwas5to15 Hz.Therefore wassettozeroandthe

controllerwasadoptedtobeintheformofsK

s

OL

o 1+ .Thetransferfunctionofthesystem

becomes

OLe

v

sV

s

)(

)(=1,whichdoesmeettheoriginalobjective.Thecontrollermustnot

PlantController

)(

1

+

+

sK

s

OL

o

1+s

sK

o

OL

)(sVe )(sv


53/108

38

outputasignalwhichwouldsaturatetheinputtothehydraulicsimulator.Theformof

thecontrollerisquiteidealinthatitinherentlylimitstheinputsignaltotheplant

(assumingofcoursethatthelimitislessthansaturationvalues).Thedisadvantageof

thiscontrolleristhatitisopen-loopandanysmalldriftintheplantinputwouldbe

integratedtwicebytheplantresultinginsubstantialoutputpositiondrift.To

demonstratethetimeresponseofthesystemwiththisinversecontroller,thesamesignal

fromthefunctiongeneratorwasinputtothecompensatedservosystemandasexpected,

theoutputaccelerationfollowsthedesiredinputsignalperfectly.Thisisshownin

Figure3.16.

1.5 1.6 1.7 1.8 1.9 2 2.1 2.2-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.52

2.5

Time(s)


FunctionGenerator

Compensatedopen-loop

Figure3.16-Transientresponseof v intheopen-loopcompensatedmodeltothe

functiongenerator d of10Hzsignal


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39

CHAPTER4

ExperimentalSystemSetup

4.1Introduction

InChapter2,thedevelopmentofthefunctiongeneratorwasconsidered.Inthis

chaptertheexperimentalsystem(hydraulicsimulator)setupisexplainedindetail.

Figure4.1illustratestheschematicoftheexperimentalsystemwhichincludesthe

powersupply,pressureservovalve,hydraulicrotarymotor,andasinglelink

manipulator.

Figure4.1-Schematicofthehydraulicsimulator


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ThesystemwascontrolledbymeansofapersonalcomputerusingMatlab-Simulink

softwarewhichsubmittedcontrolledinputsignalstothepressureservovalve.For

analyzingtheexperimentalsystem,aSignalAnalyzerUnitwasused.Alsotheangular

accelerationofthelinkwasmeasuredbyanaccelerometer.Inthefollowingsections,

eachpartandcomponentisconsideredindetail.

4.2PowerSupply

Thepowersupplyconsistedofahydraulicpumpandapressurereducingvalve.The

hydraulicpumpwasavariabledisplacement,pressurecompensatedSundstrand22

seriespumpwhichdeliveredamaximumflowrateof121.28L/min.(32GPMUS)at

1740RPM.Thedead-headpressureofthepumpwassetat172.5bar(2500psi).A

DenisonmodelRR12535pressurereducingvalvelimitedthedownstreamsystem

pressureto86.25bar(1250psi).

4.3PressureControlServovalve

AsdiscussedinChapter3,apressureservovalvewasusedbecauseofitshighfrequency

response,animportantdesigncriterioninthissystem.TheElectrohydraulic,two-stage,

four-way,nozzle-flappertypepressurecontrolservovalveofMoogmodel15-010was

utilizedtodirecttheflowfromthepowersupplytotherotarymotor.Figure4.2showsa

schematicoftheservovalve.


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Figure4.2-Schematicofpressurecontrolservovalve

Theservovalveconsistedofthreemainparts:torquemotor,hydraulicamplifierand

valvespoolassembly.Thistypeofservovalveisdesignedtocontrolloadpressure

difference( ABP ).Asummaryofitsoperationisasfollows.

Theinputcurrentcreatesmagneticforcesontheendsofthearmatureinthetorque

motorcoils.Thiscausesthearmatureandflapperassemblytorotateaboutaflexuretube

support(notshowninFigure4.2).Theflappermovesbetweenthenozzlesandbuildsup

adifferentialpressure 12P whichisproportionaltotorquecreatedbytheinputcurrent.

Theflapperrestrictsflowpassingthroughoneofthenozzles,forexamplenozzle(1)in

Figure4.2.Becauseofthefixedorificeupstreamtothenozzle,areductioninflow

Nozzle21

SP

T

SP SP

F

l

ap

p

e

r

SP

A B

PermanentMagnet

RotatingArmature

SA

AA


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42

resultsinanincreaseinthepressurejustdownstreamofthefixedorificewhichis

sensedattheendofthespool(lefthandside).Meanwhile,flowontheothersideofthe

nozzleincreases(wideropeningandlessrestriction)whichmeansthatthepressureon

theupstreamsideofthenozzle(downstreamfromthefixedorificeontherighthand

side)decreasesandissensedbytherighthandsideofthespool.Apressuredifferential

acrosstheendsofthespoolnowexistsandhence,thespoolmovestotheright(inthis

case).Fluidfromthesupplypressure( SP )isnowportedtoonecontrolportandfluidin

thesecondoutletisportedtothetank(T).Asflowisportedtotheloadfrom SP ,the

pressurebuildsup(decreasesontheothersideoftheactuator)andisfedbacktothe

righthandsideofthespool(thedownstreamsidepressureisfedbacktothelefthand

sideofthespool).Theloadpressuredifference( ABP )buildsupafeedbackforcewhich

eventuallybecomesequalto SAB AP onthespoolend.When ASAB APAP = 12 the

spoolstopsmovingandtheoutputpressuredifferentialiscontrolled.InTable4.1some

specificationsfortheservovalvearepresented[11].


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Table4.1-SpecificationofMoogservovalvemodel15-010

Inputcurrent 10.00ma(rated)

TorqueonArmature/Flapper

0.0100 mN (rated)

HydraulicAmplifierFlowtoDrivethe

Spool

3.770 scm /3 (max.)

ServovalveFlow,No-Load

902.0 scm /3 (rated)

SpoolDisplacement

0.5080mm(rated)

HydraulicAmplifierDifferentialPressure

61.41bar(rated)

LoadDifferentialPressure

207.0bar(rated)

SpoolDrivingArea

26.45 2mm

SpoolFeedbackEndArea

7.8712

mm

Apressureservovalvewaschosenbecauseofitshighcutofffrequencycharacteristics

andthefactthatrelativelysmallflowrateswererequired.Theservovalvewasinstalled

ascloseaspossibletotherotarymotorinordertominimizetheeffectsof

compressibilityofthehydraulicfluidandtomaximizethestiffnessofthesystemby

reducingthevolumeoffluid(loadside).


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4.4HydraulicRotaryMotor

Asinglevanehydraulicrotarymotor(MicromaticmodelMPJ-22-1V)wasusedasthe

actuator.Table4.2illustratessomespecificationofthismotor[12].

Table4.2-SpecificationsofMicromaticrotarymotorMPJ-22-1V

[email protected]

[email protected]

Torque

( mN )

[email protected]

62.61per270

VolumetricDisplacement

( 3cm )

13.36perradian

Max.OperatingPressure

(bar)

69.00

Max.RotatingAngle

270.0

AswillbepresentedinChapter5,thisrotarymotordisplayedsignificantnonlinear

characteristics.Sinceitwastheonlyoneavailableinthelaboratory,ithadtobeused.

Thiscertainlyposedsomeinterestingchallengesforthecontrollerdesign.


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4.5RobotManipulator

Theoriginalmanipulatorusedinthisresearchprojectwasdesigned,constructedand

reportedinathesisentitledPlanarManipulatorwithThreeRevoluteJoints[13].The

schematicfrontviewisshowninFigure4.3andaphotoisshowninFigure4.4.Some

modificationsweremadetotheoriginalmanipulatortoprepareitforthepurposeofthe

project.Thiswasatwo-degree-of-freedomrigidlinkmanipulator,whichincludedtwo

hydraulicrotarymotorsattheshoulderandelbowjoints,andtworigidshoulderand

elbowlinks.Duetothefactthattheshoulderactuator(aChar-Lynnseries2000)was

adiscvalvetyperotarymotorwithsignificantviscousfrictionandanarrowrangeof

motion(180),thedecisionwasmadenottousethisactuator.Instead,theelbow

actuatorwasusedwhichhadlessviscousfrictionandawiderrangeofmotion(270).

Tostabilizethemanipulator,theshoulderlinkwasfixedandsecuredbyutilizinga

numberofcablesandplateswithboltsandnuts.Thewristactuatorwassubsequently

usedasaloadattheendoftheelbowlink.Figure4.3illustratestheschematicofthe

manipulator.


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Figure4.3-Schematicofthetwo-linkrigidmanipulatorinwhichtheshoulderlinkwas

fixed


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47

Figure4.4-Photoofthemanipulatorsetupusedintheexperiment

Thepressurecontrolservovalvewasinstalledatthetopofthefixedshoulderlinkwhich

wastheshortestdistancetotheoperatingactuator(elbowactuator)tominimizethe

compressibilityeffectofthehydraulicfluid.

Table4.3listssomeparticularsofthemanipulator[13].AsindicatedintheTable4.3,

theconstructionoftherobotmanipulatorwasstrongenoughtobeconsideredasarigid

manipulator.

PressureServovalve

Rotar Motor

Accelerometer


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48

Table4.3-Importantmanipulatormeasurements

Shoulderlinklength

0.6100m

Shoulderlinkcrosssection

0.0760msquaresteeltube

0.0060mwallthickness

Elbowlinklength

0.4600m

Elbowlinkcrosssection

0.0750msquaresteeltube

0.0050mwallthickness

Shoulderlinkweight

(withoutservovalve)

15.90kg

Elbowlinkweight

8.600kg

4.6PersonalComputer

InadditiontousingapersonalcomputerwithMatlab-SimulinkRealTimeWindows

Targetforsoftwarealgorithmprogramming,aNationalInstrumentsPCI-6025edata

acquisitioncardwitha12bitA/DandD/Awasusedtointerfacetheinputsignalfrom

thecomputertotheservovalveandtheaccelerometersignalbacktothecomputer.The

functiongeneratorintheSimulinkprogramwasutilizedtoprovidetheinputsignalto

thehydraulicsimulator.Table4.4presentssomethepertinentsettingsthatwereused

intheMatlab-Simulinkalgorithm.


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49

Table4.4-Matlab-Simulinksetupforexperimentalsystemcontrol

SolverType

Fixed-step

Solver

ode4(Runge-Kutta)

Fixed-StepSize(FundamentalSample

Time)

0.001s

Analoginput NationalInstrumentsPCI-6025E

(auto),inputrange10v

Analogoutput NationalInstrumentsPCI-6025E

(auto),outputrange10v

4.7SignalAnalyzer

Asignalanalyzerunit(Bruel&Kjaertype2035)wasusedtoobtainthefrequency

responsedataofthesystem.TheanalyzerprovidedtheactualBodemagnitudeand

phaseplotsoftheexperimentalsystem.

Theanalyzergeneratesarandomsignalofvariousfrequenciesrangingfrom0 Hzto200

Hz(forthistest).Otherfrequencyrangeswerepossiblewiththisunit.Thesignalwas

interfacedtotheservovalveviaonechannelasthesysteminputsignal.Themeasured

acceleration(outputsignal)wasreturnedtotheanalyzerviaanotherchannel.


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4.8Accelerometer

ApiezoelectricchargeaccelerometerofBruel&Kjaertype4370wasattachedtothe

linkage(seeFigures4.3and4.4).Theaccelerometerwasconnectedtoacharge

amplifierofBruel&Kjaertype2635.Thecalibratedoutputratingofthecharge

amplifierwasadjustedto )/

(1.02sm

v.

Theaccelerometerwascapableofmeasuringtangentialacceleration.However,itwas

theangularaccelerationthatwasdesired.Sincethedistancebetweenthecenterofthe

installedaccelerometerandtheverticalaxisoftherotarymotor(rotationarm)was

known,itwaspossibletoconvertthelinear(tangential)accelerationtoangular

acceleration(detailswereprovidedinChapter3).

4.9AccelerometerCompensator

InChapter3itwasmentionedthattheactualfrequencyresponseoftheexperimental

system,showninFigure3.6,wasbestapproximatedbythetheoreticalfrequency

responseillustratedinFigure3.4.Inreality,theslopeoftheactualBodeplot(Figure

3.6)wasgreaterthanthetheoreticalplotatlowfrequencies.Itwasbelievedthatthe

poorperformanceoftheaccelerometerandamplifieratlowfrequencieswasapossible

reasonforthisdiscrepancy.Uponfurthertesting,(asfollows)itwasdeterminedthatthe

accelerometer/amplifierdisplayedthecharacteristicsofahigh-passfilter.

Anaccelerometertestwasperformedonashakertableutilizinganaccuratelinear

positionsensor(SchaevitzDCLVDT)alongwiththetestaccelerometer(B&K4370)

whichwerebothinstalledonthesameplaneofmotion.Therandomsignalgeneratedby

thesignalanalyzerwasinterfacedtotheshakertable.Theoutputsignaloftheposition

transducerwasconnectedtotheanalyzer.Theanalyzerwassettoconverttheposition


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signaltoanaccelerationsignalbydifferentiatingittwice.Theoutputsignalofthetest

accelerometerwasalsointerfacedtotheanalyzer.Theanalyzerwasutilizedtocompare

amplitudeandphaseangleofacceleration(derivedfromthepositionsignal)withthose

ofthetestaccelerometersignal.Figure4.5illustratestheschematicofthetest

arrangement.

Figure4.5-SchematicofShakerTabletest

InFigure4.5;

aca measuredaccelerometersacceleration(v)

pox measuredposition(v)

poa calculatedaccelerationbydoubledifferentiationofmeasuredposition( v)

Figure4.6showsthemagnitudeandphaseBodediagramoftwicedifferentiated

positionsignalandthetestaccelerometersignal.

pox

aca

RandomSignal

Anal zerUnit

poa

ShakerTable

Accelerometer

2s

Position

(DCLVDT)


67/108

52

-10

-5

0

5

10

0.1 1 10 100

Frequency(Hz)

-180

-135

-90

-45

0

45

90

135

180

0.1 1 10 100

Frequency(Hz)

Figure4.6-MagnitudeandphaseBodeplotsoftheshakertabletest(pointsaredata

fromtheanalyzerandthesolidlineisthestraightlineapproximationbasedona20dB

perdecadeslope).

)(

(

)(

dB

ja

ja

po

ac

)(

)(

)(

Degrees

ja

ja

po

ac


68/108

53

Anapproximatedtransferfunctionofthetwoaccelerationsignalscanbeextractedfrom

anasymptoticanalysisofthemagnitudeBodeplotwhichisshowninFigure4.6.Itis

evidentthatatlowfrequencies,theaccelerometerproducedresultswhichwere

attenuated.Thisattenuationcouldbeapproximatedbyatransferfunction.Analysisof

Figure4.6indicatesthatthecornerfrequencywas1.25 Hz(7.85rad/s).Thereforethe

transferfunctionwhichreflectedthislowfrequencyattenuationwasestimatedtobe;

85.7+=s

sGac (4.1)

Tocompensateforthisattenuation,acompensatorwaschosentobe 00.5

85.7

+

+

s

s

inorderto

expandthelowerendofbandwidthoftheaccelerometer.

Itshouldbenotedthatanidealcompensatorshouldbes

s 85.7+butaswillbediscussed

inChapter5,theform00.5

85.7

+

+

s

sprovidedbettercompensation.Moredetailswillbe

discussedonthismatterinthefollowingchapterinwhichexperimentaldatais

considered.

Inimplementingthisaccelerometercompensatorexperimentally,theproposed

compensatorwasinstalledaftertheplantandphysicalaccelerometer.Figure4.7

illustratestheplacementofthecompensatortoatheoreticalapproximationofthe

accelerometerfrequencycharacteristics.Chapter5willconsidertheimplementationof

thecompensatorfromanexperimentalpointofview.


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54

Figure4.7-Blockdiagramofthesystemwiththeaccelerometercompensator

Itshouldbenotedthatthiscompensatorwasutilizedforallexperimentaltests.

Accelerometer

Compensator

eV

00.5

85.7

+

+

s

s

v vc Plant


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55

CHAPTER5

EXPERIMENTALRESULTS

5.1Introduction

Theobjectiveofthischapteristopresentanddiscusstheresultsofthetestsperformed

ontheexperimentalsystemwiththesetupexplainedinChapter4.Generally,thetests

madeuseofsweptfrequencies(aprocessinwhichtheinputfrequencywas

systematicallyincreasedoveraselectedfrequencyrange)andfunctiongeneratorinputs

toproducebothfrequencyandtimedomainresults.Formanyofthetests,magnitude

andphaseBodediagramsweredevelopedfromthedatatoillustratethedynamic

performanceofthesystemandcompensationschemes.Todemonstratethedesireduse

ofthehydraulicsimulator,thetemporalresponsestotheoutputfromthefunction

generatorwereconsidered.TheChapterwillthenillustratethetransientresponse

trackingerrorsandmeansquarederrorvalues.

5.2PressureControlServovalve

Thepressurecontrolservovalvewasusedinthehydraulicsimulatorbecauseofits

superiorfrequencybandwidthatlowflowrateswhencomparedtoaflowcontrol

servovalve[14].Thepressurecontrolservovalvedynamicperformance(thefrequency

responsedata)isgenerallypresentedintwoforms;oneinwhichtheloadportsofthe


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56

servovalveareblocked(blockedload)andtheotherwhentheportsareconnectedtoa

load.Theblockedloadtestisnormallyconducedbypluggingtheloadports(for

exampleportsAandBinFigure4.2).Theloadpressuredifference( ABP )andtheinput

voltagearemeasuredforawiderangeofsinusoidalfrequencies(inthefollowingcase

between0.5to400Hz).Intheblockedloadcasetheservovalveshoulddemonstratea

highstiffnessbecauseoftheverysmallfluidvolumeontheloadsideresultingina

highernaturalfrequency.

Figures5.1aandbillustratetherelationshipofblockedloaddifferentialpressureto

inputvoltagetotheservo-amplifier.AscanbeobservedintheBodeplot,thebandwidth

ofthevalvewasapproximatelyzeroto60Hzwhichwassomewhatlessthanthatstated

inthemanufacturersspecifications[11].Athigherfrequencies,aslopeof-

60dB/decadeinthemagnitudeplotindicatedthatthesystemwasapproximatelythird-

order.

-40

-30

-20

-10

0

10

20

0.1 1 10 100 1000

Frequency(Hz)

Slope-60dB/decade

Figure5.1aMagnitudefrequencyresponseofthepressurecontrolservovalvewitha

blockedload(pointsaredatafromtheanalyzerandthesolidlineisthestraightline

approximation).

)(

)(

)(

dB

jV

jP

e

L


72/108

57

-360

-315

-270

-225

-180

-135

-90

-45

0

45

0.1 1 10 100 1000

Frequency(Hz)

Figure5.1bPhasefrequencyresponseofthepressurecontrolservovalvewitha

blockedload.

Whenthepressurecontrolservovalvewasconnectedtotheexperimentalsystem,the

frequencyresponsewasdeterminedandisshowninFigures5.2aandb.

)(

)(

)(

Degrees

jV

jP

e

L


73/108

58

-40

-30

-20

-10

0

10

20

0.1 1 10 100

Frequency(Hz)

Slope-40dB/Decade

Figure5.2a-Magnitudefrequencyresponseofpressurecontrolservovalvewithactual

hydraulicsimulatorload(pointsaredatafromtheanalyzerandthesolidlineisthe

straightlineapproximation).

-180

-135

-90

-45

0

45

0.1 1 10 100

Frequency(Hz)

Figure5.2b-Phasefrequencyresponseofpressurecontrolservovalvewithactual

hydraulicsimulatorload.

)()(

)(

dB

jV

jP

e

L

)(

)(

)(

Degrees

jV

jP

e

L


74/108

59

Itisobservedthatthebandwidthisabout40 Hzandtheorderofthesystemhaschanged

fromthirdtosecond.Theinfluenceoftheloadisthusapparent.Thehighfrequency

affectsoftheservo-amplifierarenotvisiblefromtheseresultsandhencetheassumption

thattheservo-amplifiercouldbeapproximatedbyasimplegaininChapter3was

justified.

Thefrequencyresponseoftheactualloadpressuretoinputvoltage(Figure5.2aandb)

demonstratesthattheselectedpressurecontrolservovalvewascapableofanappropriate

performanceinthefrequencyrangeofinterest(5-15 Hz).

5.3Accelerometer

InChapter4itwasstatedthattheaccelerometerwasdemonstratingpoorperformanceat

thelowerfrequencies.Itwasbelievedthatsomemethodofcalibratingtheacceleratorat

lowfrequencieswasnecessary.AsdiscussedinChapter4,ashakertabletestfacility

wasavailableinthelaboratorywhichcouldbeusedtoprovideacalibratedposition

measurementandhenceareliableaccelerationcalculation.Boththeaccelerometerand

positiontransducerweremountedonthetableandtheshakertablefrequencyrange

sweptfrom1to25Hz.Thecalibratedaccelerationwasdeterminedbyadouble

differentiationofthepositionsignal.Themagnitudeandphaseratioofthe

accelerometer aca outputtothecalculatedacceleration poa fromthepositiontransducer

wereshowninFigure4.6.Ideallythemagnituderatioshouldbeunityandthephase

shiftzero.ObservationofFigure4.6illustratesthattheperformanceofthe

accelerometerdeterioratesatfrequencieslessthan7.85rad/s(1.25 Hz).

AsdiscussedinChapter4,anidealcompensatoroftheforms

sG

85.71

+= should

extendthelowerfrequencyrangeoftheaccelerometer.Whenappliedtotheactual

system,theresultswerenotsatisfactoryinthatthecompensatorovercompensatedthe

accelerometerssignalsatfrequenciesbelow7.85rad/s(1.25 Hz).Theassumptionthat


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theactualdatafollowedthetransferfunction 1G isonlyapproximateinthattheslopes

arecomparablebutinthetransitionregionsbetweenthetwoslopes,theexperiential

datawasfoundtobelargerthanthatpredictedbythetransferfunction 1G .Thusdatain

thatregionwasamplifiedbeyondthatrequired;thusovercompensated.Inaddition,thedataforfrequencieslessthan1Hzwasveryerraticandtheuseof 1G wouldonly

amplifythescatteratlowerfrequencies(seeFigure4.6).Thusacompensatorofthe

formbs

as

+

+whichwouldnotamplifythelowerfrequenciesbutcompensateatthebreak

pointwasused.Thiscompensatorwastunedtobeintheformof00.5

85.7

+

+

s

swhichwas

thenappliedtoallmeasuredaccelerationdata.

Figures5.3aandbindicatetherelationshipbetweentheaccelerometeroutputsignal

anddoubledifferentiatedsignalfromthepositiontransducer,withtheproposed

compensator.AcomparisonofFigure5.3toFigure4.6inChapter4showsthatthe

lowerendofbandwidthoftheaccelerometerhasbeenextendedfrom3 Hztoabout1

Hz.However,thereisstillaslightaffectontheamplituderatiointherangefrom1to10

Hz,whichisnotdesirable.Theoverallimprovementusingthiscompensatorwas

consideredmarginalbutwasstillimplementedinallsubsequentstudies.


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61

-10

-5

0

5

10

0.1 1 10 100

Frequency(Hz)

Figure5.3a-Magnituderatio

ja

ja

po

ac

(

)( versusfrequencyoftheaccelerometerwith

proposedaccelerometercompensator(pointsaredatafromtheanalyzerandthesolid

lineisthestraightlineapproximation).

-180

-135

-90

-45

0

45

90

135

180

0.1 1 10 100

Frequency(Hz)

Figure5.3b-Phase)(

)(

ja

ja

po

ac versusfrequencyoftheaccelerometerwithproposed

accelerometercompensator.

)(

)(

)(

Degrees

ja

ja

po

ac

)(

(

)(

dB

ja

ja

po

ac


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Thecompensatorwasthenappliedtotheaccelerationdatafromthehydraulicsimulator

andthedynamicperformance(magnitudewiseonly)beforeandaftercompensationis

comparedinFigures5.4and5.5.Itisapparentthatinthelowerfrequencyrange(less

than2Hz),theslopeofthefrequencyresponseafterapplyingthecompensatorisnow

20dbperdecadewhichisconsistentwithwhatwaspredictedfromthetheoretical

modelofthesystem(Chapter3).However,consistentwiththeovercompensating

natureoftheaccelerometerinthefrequencyrangeof1to10 Hz,thereappearssome

distortiontothefrequencyresponseintheplantdataaswell.However,becauseinthe

lowerfrequencyrange,theslopewiththecompensatedaccelerometerapproachedthat

whichwaspredictedanalytically,itwasconcludedthattheaccelerometercompensation

wasacceptableforthisstudy.

-40

-30

-20

-10

0

10

20

0.1 1 10 100

Frequency(Hz)

Figure5.4-Open-loopmagnitudefrequencyresponseofthehydraulicsimulatorbefore

applyingtheaccelerometercompensator(pointsaredatafromtheanalyzerandthesolid

lineisthebestfitstraightlineapproximation;40dBperdecadeslope).

)(

)(

)(

dB

jV

j

e

v


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-40

-30

-20

-10

0

10

20

0.1 1 10 100

Frequency(Hz)

Figure5.5-Open-loopmagnitudefrequencyresponseofthehydraulicsimulatorafter

applyingaccelerometercompensator(pointsaredatafromtheanalyzerandthesolid

lineistheexpectedstraightlineapproximationoftheplant,20dBperdecadeslope).

5.4HydraulicSimulator

Theexperimentalresultsofthehydraulicsimulatorsuchasopen-loopandclosed-loop

frequencyresponses,open-loopcompensatedfrequencyresponse,open-looptransient

responsesandtransientresponseerrorsarepresentedanddiscussedinthissection.

5.4.1Open-LoopFrequencyResponses

Theopen-loopbehaviorofthehydraulicsimulatorisshowninFigure5.5.Itisobserved

thatthegainis-8dBandhasabandwidthofapproximately2to30 Hz.Anunknown

dipinthemagnitudeoccursat25Hz;atthispointnoreasonableexplanationcanbe

forwardedforthisbehavior.

)(

)(

)(

dB

jV

j

e

vc


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Thefrequencyresponseoftheopen-loophydraulicsimulatorforsmallerinput

amplitudesof0.5v(asmallaccelerationof 2/5.11 srad )isshowninFigure5.6aandb.

Itisevidentthattheperformancedeterioratesinthatthelo

Robot Hydraulic Control

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