robot hydraulic control
TRANSCRIPT
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AHYDRAULICFLEXIBLEJOINTROBOTSIMULATOR
AThesisSubmittedtothe
CollegeofGraduateStudiesandResearch
inPartialFulfillmentoftheRequirements
fortheDegreeofMasterofScience
intheDepartmentofMechanicalEngineering
UniversityofSaskatchewan
Saskatoon,Canada
By
ShahramDezfulian
CopyrightShahramDezfulianJune2007.Allrightsreserved.
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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
sinewavewithamplitudeof2v(compensations
s
8.0
71.15+).................72
5.14aOpen-looptransientresponseofthehydraulicsimulatorto15 Hz
sinewavewithamplitudeof1v(compensations
s
8.0
71.15+).................73
5.14bOpen-looptransientresponseofthehydraulicsimulatorto15 Hz
sinewavewithamplitudeof2v(compensations
s
8.0
71.15+).................74
5.15Open-looptransientresponseofthehydraulicsimulatortothefunctiongenerator5Hzand0.1dampingratiosignal(notcompensated).............76
5.16Open-looptransientresponseofthehydraulicsimulatortothefunction
generator5Hzand0.1dampingratiosignal(compensatedbys
s8.0
4+ )........76
5.17Open-looptransientresponseofthehydraulicsimulatortothefunction
generator5Hzand0.1dampingratiosignal(compensatedbys
s
8.0
8+)........77
5.18Open-looptransientresponseofthehydraulicsimulatortothefunction
generator5Hzand0.1dampingratiosignal(compensatedbys
s
8.0
71.15+).....77
5.19Open-looptransientresponseofthehydraulicsimulatortothefunction
generator10Hzand0.1dampingratiosignal(notcompensated)............78
5.20Open-looptransientresponseofthehydraulicsimulatortothefunction
generator10Hzand0.1dampingratiosignal(compensatedbys
s
8.0
4+).......79
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5.21Open-looptransientresponseofthehydraulicsimulatortothefunction
generator10Hzand0.1dampingsignal(compensatedbys
s
8.0
8+)............79
5.22Open-looptransientresponseofthehydraulicsimulatortothefunction
generator10Hzand0.1dampingratiosignal(compensatedbys
s8.0
71.15+ )....80
5.23Open-looptransientresponseofthehydraulicsimulatortothefunctiongenerator15Hzand0.1dampingratiosignal(notcompensated)............81
5.24Open-looptransientresponseofthehydraulicsimulatortothefunction
generator15Hzand0.1dampingsignal(compensatedbys
s
8.0
4+)............81
5.25Open-looptransientresponseofthehydraulicsimulatortothefunction
generator15Hzand0.1dampingsignal(compensatedbys
s
8.0
8+)............82
5.26Open-looptransientresponseofthehydraulicsimulatortothefunction
generator15Hzand0.1dampingratiosignal(compensatedbys
s
8.0
71.15+)....82
5.27TrackingerrorofFigure5.15(frequency5 Hzanddampingratio0.1,
notcompensated).................................................83
5.28TrackingerrorofFigure5.18(frequency5 Hzanddampingratio0.1,
compensatedbys
s
8.0
71.15+),..........................................84
5.29TrackingerrorofFigure5.19(frequency10 Hzanddampingratio0.1,
notcompensated),................................................84
5.30TrackingerrorofFigure5.22(frequency10 Hzanddampingratio0.1,
compensatedbys
s
8.0
71.15+).........................................85
5.31TrackingerrorofFigure5.23(frequency15 Hzanddampingratio0.1,notcompensated)................................................85
5.32TrackingerrorofFigure5.26(frequency15 Hzanddampingratio0.1,
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
2a partial-fractionexpansionparameter
3a 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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1
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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7
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
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8
Rotational
SpringRotational
DamperD
K
MotorJ
Link
I
u
(a)Linearequivalentschematicofthesingle-linkmanipulator
(b)Dynamicequivalentschematicofthesingle-linkmanipulator
Figure2.1-Schematicofasingle-linkmanipulatorwithflexiblejoint
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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:
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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:
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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
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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
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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
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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
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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)
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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:
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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.
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Thetwosolutionsforaccelerationresponseusingthecomputersolverandtheclosed
solutionformyieldthesametheresultsaswouldbeexpected.Thefunctiongenerator
canberealizedeitherfromthesimulationusingMatlaborbyimplementingtheclosed
solutionformofequation2.29.Inthiswork,thesimulationformwasusedbecauseof
theeaseofchangingtheparametervalues.
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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.
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-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.
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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)
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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 / )
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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;
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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.
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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
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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
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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
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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
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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
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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.
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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
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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
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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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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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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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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
Figure3.14-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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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
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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)
Accelerationamplitude(v)
FunctionGenerator
Compensatedopen-loop
Figure3.16-Transientresponseof v intheopen-loopcompensatedmodeltothe
functiongenerator d of10Hzsignal
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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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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
Torque
( mN )
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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Figure4.4-Photoofthemanipulatorsetupusedintheexperiment
Thepressurecontrolservovalvewasinstalledatthetopofthefixedshoulderlinkwhich
wastheshortestdistancetotheoperatingactuator(elbowactuator)tominimizethe
compressibilityeffectofthehydraulicfluid.
Table4.3listssomeparticularsofthemanipulator[13].AsindicatedintheTable4.3,
theconstructionoftherobotmanipulatorwasstrongenoughtobeconsideredasarigid
manipulator.
PressureServovalve
Rotar Motor
Accelerometer
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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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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)
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-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
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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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Figure4.7-Blockdiagramofthesystemwiththeaccelerometercompensator
Itshouldbenotedthatthiscompensatorwasutilizedforallexperimentaltests.
Accelerometer
Compensator
eV
00.5
85.7
+
+
s
s
v vc Plant
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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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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
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-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
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-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
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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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-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