ROBOT ARMS HYDRODYNAMICLOADSONUNDERWATER

HYDRODYNAMIC LOADS ON UNDERWATER ROBOT ARM S

By

e CarlosRivera,B. Eng.

Athesis submitted tothe School of Grad uateStud ies in parti alfulfillmen t ofthe requ irementsforthedegree of

Master ofEnginee ring

Facuny ofEngineeringand AppliedSciences Memori alUniversi tyofNewfoundland

Ma rch1995

$1.John's Newfoundland Canada

1+1

Naliol"lal Ubrary

oICan"'" :'~f\alionale

AcQUsilionsand Directiondesac:quisilionsel Bibliographic

sevces

Branch desseMces bibliographiques

m~ ~~ ~~ ~

~Or"'f<io ~~l

The author has granted an irrevoca ble non-exclus ive licence allowing the National Ubraryof

Canada to reproduce, loan ,

distribute or sell caples of his/h erthesis byanymeansand In any form or format,making thisthesis available to Interested persons.

The authorretains ownershipof the copyri ghtIn his/herthesis.

Neitherthe thesis nor substantial extracts fromitmaybe printedor otherwise reproduced without his/herpermission.

ISBN 0-612-01912-8

Canada

L'aute uraaccordeunelicence Irrevocable at non exclusive permettant • la Blbllotheque nation ale du Canada de reprodu lre,prfHer,distribuer ou vend re des copiesdesa

these

de quelque manlere at SOllS quelque forme que ce soft pour mettredes exemplalres decette these

a

la disposition des personn es fnteressees.

l'aut eurconservela proprleMdu

droit d'auteur qul prot ege sa these.

Ni la

thesenl des extralts subs tantiels de

celle-cl

ne dolvent etre Impri mes ou autrement reprodults sans son autorisation.

Abstract

ComputedLoadand SlidingModecontrolstrategiesforrobotsmakeuse ofthe overallequationsofmotion.Unfommately. forroholsworkinginamarine environment.

theseequations do^notprovideanaccurate control becauseofa lackofgood hydrodynamics data.

Theobjectiveofthethesisis10incorporatethehydrodynamic loads,actingon robotannsworkingunderwater,intothe equations ofmotion.The thesisdescribesIItwo- link robot annmodeland experimentalsetup designed forthepurposeofgathering hydrodynamic data.

Thethesisalsodescribesthetechniques usedtoanalyzethe obtaineddata.TIICsc includedimensionalanalysisandaneuralnetwork ident ifICationprogram.

Acknowledgements

I would like 10express my grulitudeandthanks to Dr.Michael Hincheyfor his constant sIIPI'()rl "lid valuable glliJunce during the period of my studies in the Master's program.

and in particularduring theresearch part that led tathecompletion af this work.

I also extend my appreciation(0the School of Graduate Studiesand the Faculty of EngincerinKand Applied Sciences[or thefinancialsupport I received.

Iwouldlike10thank Andrew Kuczorafor his help beyond his dillies in the data acquisition und processing,a crucial part ofthis work, and HumphreyDyer and the rest IIItheIJI!ol'l1!in TechnicalServices who helped me construct the experimentalmodel.

I thunkIllywife !)ebbil!jar her III/conditional support.

March /995

Carlos Rivera

iii

Contents

Abstract Acknowledgeme nts List of Figuresand Tables list of Symbols 1. Introduction

1.1Background 1.2 Scope andobjectives 2. LiteratureReview

iii

vii

2.1 Background 4

2.2 AUVIROVStateofthe art 6

2.3 Hydrodynamics loadson cylinders 11

2.3 Hydrodynamicsloadsonrobot arms 24

2.4 Neural networks 28

3. Mathematicalmodel ofhydrodynamics loads onarobot ann 35 3.1 Analysisof atwo-link planar revolutejoint arm 36

3.2 Mechanical losses 52

3.3 The dynamic equations ofmotion 55

4. Wavelankexperiment 62 4. 1 Designof thetwo-linkplanarrevolutejointann model 62

4.2 Testing 66

4.3 Dataacquisitionarxlprocessing 67

4.4 Neural networksystemidentiftcation 71

.5. Experimentalandtheoretical results 76

6. Conclusions 104

7. Rccomme odations 106

References 107

AppendixA.Flow chansofthe neuralnetworkidentification system 113

AppendixB.Programs Listing 127

Appendix C.TrainingData 140

List of Figures

2-1a Backlash compensating mechanism 2-1b Pressure compensating system

2-2 TheJasonROV

2-3 Manipulatorof the JasonROV It,

2-4 Cylind ersin tandemandside-by-side arran gement

'I>

2-5 Drug coefficients for cylinderswithtwofreeends

"

2-6 Torque venusreducedvelocityforacceleratedmouons :n 2-7 Stationarytwo link planar revolutejointarm 25 2-8 Twolinkarmmodelmovinginthree dime nsions ::!r, 2-' ArmIbodylestconfigul'ln;-u

"

2- 10 Process ingcle mentofa ncurolnetwork

"

2-11 AonehiddenlayerMLPinmatrixnotation ^]3

3- ' One-link planar armmodel 17

3-2 Two-linkrevoluteplanararm model 41

3-3 Force profile on link (caseI) 44

3-4 Forceprof ileon link(caseII) 45

4-' Experimentalsetupof atwo-link underwater armmodel'.3

4-2 Driving section ofthesetup h4

4-3 Two-link underwater armmodel 65

4-4 NeuralNetworkwithcommonmiddlelayers 72

4-5 NeuralNetworkwithparallel middlelayers 74

5-1through 5-24 Datagraphs 80-103

A-IlhroughA-lO Flowchartsfor theNeuralNetwoork Identification 116·12(,

A,.HI C/I'

c.;c,

e:".

C~I D

[,of.

If I

L.L, IvII ,N, m, Q, Rt. Rm. Ro.RI

wt.Wo x.y

z . i e

o o

List of Symbols

Coefficientsinthe velocityexpression Drag,Inertia andaddedaaasscoefficients

Centroidcoordinateforthe distributionof the drag/incrtiaforces LinkdiameterIRelatingto hydrodynamicdragassuperscript Error

Drag/inertia force alongthe links Manipulatorinertia tensor matrix

Inertia tensorIRelating to hydrodynamicinertiaassuperscript Indexfor thelink number

Jacobianmatrices for angular and linearvelocitiesrespectively Coefficient

Length oflinkILocationof thecenterof mass Cocfficicrusintheacccicrationexpression Massof links

Generalized load

Inputs.outputs fromthemiddle layer,outputfromthe netand target outputs respectively

Totalkinetic energy/Experimentallymeasured torque Linear velocityofthe linkIVelocityof the carrier Input andoutputweights

Denotesthexorydirection Outputofthenet and desired output Angularposition

Angularvelocity Angularacceleration Kinematicviscosity Density Torquesat thejoints

vii

Chapter 1

Introdu ction

1.1Background

Although theoceanscovermorethan two thirdsof theEarth ' s surface antiarc a vitalpart oftheexistence of Mankind.humans have not yet been able to fullyexplore their finitedepths.Actuall y wenow know more aboutouterspacethanweknow ahuut theoceans.

Modern robotics offers humanity a widearray of economicallyantisociallysound benefits. Autono mousrobo ts canpotentially handletasks inhostile orinaccessible environments,such as.inspace, innuclearreactors andunderwater.Industrialrobotshave alreadyassumedmanyhazardou s,unpleasant orboring tasks.while simultaneously improving theproductivityoffactori esin theindustrialize dworld.Robotarmssuchas Canado.rm havebeen successf ully deployedinspace.Theoceanfloor contains valuable mineralsand fuelsbut the actualeconomic exploitationof these resou rceshashecn restrictedbytechnologicalcapabilitie s nnd theapplicatio nof robotsto underwatertasks hasbeenvery limited.Recentad vances in underwatervehicle technology arc helping to

changethis situation.

Most ofunderwate rvehicles arcmanuallycontrolled.However.recently therehas beena shintowards autonomousordigitallydrivencontro l.This ismainlybecause manual controlhasbeenfound unsat isfactory for certain tasks.Most armsmounted on underwater vehiclesoperate at slowspeeds,wherethe hydrodyna mic forcesare relatively smellbut notnegligible[FarbrothcrlindStacey,1993).However,the armssometimes have to operatein the wakeof amoving vehicle orbeimmersedina current. Inthese cases thehydrodynamicloadsonthem can beconsiderable andunpredictable.The sudden changeinthe dragcoeffic ientsfor a relativelysmallchangeinvelocitymakes the hydrodynamic loadsvery difficult to ascertain. These effectscanbring instabilityto an uutomnticullycontrolled vehicleTherefore.arms operatingin theseconditions willrequire thatcontrolsystemsbe robustand able\0maintainacceptable performance levelsin responsetouncertainties(Farbrother andStacey.1993).

Thecontrolstrategies used inrobots can be divided intwomain groups:

conventional(class ical) con trol techniquesand supervisory controltechniques. Someof thetechniquesin both groupscan exhibit adaptivecapabilities.Byadaptive capabilities ismeant theability of the controller to accuratelyrespondwhen faced withchangesinthe environment.which hadnot beenanticipatedor programme d (Westerman,1991).The conventional contro lstrateg ies include,amongothers,the computedloadtechniq ue and the slid ing mode.Theyareboth based on theequationsofmotionand couldbe inaccurateforunderwater applicationsbecause of unce rtainties intheequationsof motion. Good experimentaldata couldbeusedtoreducesuchuncertainty.

Neural networks and fuzzylogic arcconsidered supervisor)' controlstrutcgics, They Iry tomimicahumanoperatorbypicking set po intslindgainsforclassical strategies.

1.2Scopeand Objectives

The determination of hydrodynamics dataisvital 10 pcrform un effic ient CUlIlrtll ofthe mot ion of robotarms inun underwater envuonmcnr. Untor turuuclyverylhtlcdaiu canbefoundinopenliterature.Forthisreasontorque mcasurcrnca tshuvcbeen conducted usingthewave/lowingtankfacilitiesonsingle-linkbody con figurationnnd1111iltwo-link plan ar armconfiguration. Themeasurementswere conducted using steady trunshulonsand rotat ions.A neural network wasusedtoprovideaconvenientlit tothctorqueduta.In addition. a mathematicalmodel basedon the Morison'sequationwas developedforthe hyd rodynamics loads.Morecomplex motionsandgeometrywere avoided because even simpler casesarenot fully understood.

Chapter 2

Literature Review

2.1Background

A rohot ann workingunde r water is constantlyunderhydrodyna micsloads, such liS drag. hydrostatic restoringforce, force due 10 themoorings and the force due to currentsandwaves (Muggcridgeand Hinchey,1991 ).The addedmass effectswouldalso playasubstantialrole inthepower requirementsof theann (Baker andSayer.1990). In additiontounderwater currents, the turbulence createdby thrustersof thehostvehicle itself orotherpassing vehicles, may affect greatly thedynamics ofthe arm(Riveraand Hinchey,1992).Inthis case controlstrategiesare bound10fail sincethe equationsof motionlack good hydrodynamics data.andfor complex configurations they are computationallyexpensive.Inorderto make controlstrategieswork effectively underwater.a modelofarobot armwouldhave tobefabricatedandtested underwater.

Thenwc wouldbeabletolind thedatalit whichiscomputationallyleastexpensiveand mostrobust.

The hydrodynamicsof underwatervehiclesor robotarmsundergoingarhitrury.

unsteady motionsis very difficultto deal withanalytically(Muggcridgcand Hinchey.

1991).The major problemis thecomplex wakesset-up behindthebody.Even steady motionsarcverydifficult to trcat analyticallyand thetreatment relics heavily nil experimental data.Moststudies assumethe 110wtobequasi-steady.in otherwords.nows setup quicklyandresemble steady flows.Thequasi-steadyapproximationis perhaps acceptablefor uniform or slowlyvaryingmotions of the arm hut questionableforgeneral arbitrarymotions.

For steadyflow wakes behind a body.muchdependson the Reynolds number (Re=Il.D/i') (Slaoutiand Stansby,1992). Section3of thehydrodynamicstextby SarpkayaandIsaacson(1981) gives a good overview of this type ofmotion.Al lowRe theflow is viscosity dominated.AthighRc,vortices(symmetric/asymmetric)shed downstreamdueto boundary layer separation.the flow cannot remain attachedttlthe body asit moves into theadversepressuregradientgenerallyround there (Prundtland Tietjens,1934).Turbulenceis a featureof fluid flow,not a property of the fluids themselves.and exhibitirregularityand randomnessin timeand space,diffusivity and rapidmixing, three-dimensional vorticityfluctuations,and dissipationof the kineticenergy of theturbulenceby viscous shearstresses. Ithas also beenobservedthai fluid particles ina turbulentflow travelas randomlymovingfluid masses.whichcausesatany point in theflow,a rapid and irregularpulsation ofvelocityabout a well-definedmean value.

Turbulent boundarylayers donot separateas easilyas laminarboundarylayers.TheIluid in turbulentboundarylayer is rnorc energeticandcan move farther intoanadverse

pressuregrad ient. Bodies withturbulentboundarylayershave smallerwakesandthere fore less wakedrag.

Most realflow problemscanbesolved,atbest,onlyapproximately by analytical ornumerical methods.Thus,experimentsareessentialinveri fying solutions,insuggesting which models arc validorinprovidingresultsthat couldnotbeobtainedby theoretical analysisornumericalsimulation (Szirtes,1992).

All the abovementioned factors have tobe consideredinorder to gaina better understandingofthe hydrodynamicsof underwate rrobot arms.

2.2 AUV!ROVStateof the Art.¹¹

Papersin this area were reviewed withthepurpose of gaining an understand ing ofthecomplexityof thetaskof deep ocean exploration,andtheproblemsfaced in the design and operationof suchcrafts; the hydrodynamicloads onarmshap penstobe one oftilemany.

AutonomousUnderwaterVehicles,AUVs, are stillinthe earlydevelopmen tal stage {Zorpettc,1994) andthe most adv ancedvehiclesknowntoday are the resultof research recentlyunclassifiedby themilitaryinseveralcountries.8lidbcrget al. (I991) give agood overview ofthepresent stateofthe researchin thisarea, whichincl udes capab ilities suchasopera tional depth, mobility,autonomy,etc,along witha chronological develo pment ofthesecraft.Thesamepaper outlinessomeof the non-m ilitaryneedsfor thedevelopment of automatedunderwatertechnology,that are helping to promote a

'AUV: Auton o mous Underwat e rVehicle lROV:Remot ely Operate dVehicle

widespreaduse ofthese vehicles, andtherefore the accesstoamore var iedsourceuf researchresources.

Yoergeretal. (1991) describein detailthe designof thearm used on the JASON ROY.

which has beensuccessfullyused in deep subseaexploration.It shows the ingenillus designsolutions applied to cope withthehigh pressure(approximntcly9,000 psiforII

depth of6.000meters) andextrcmelycorrosiveandelectricallyconductiveenvironment.

whileat thesametime exhibiting no backlashandlowfriction.Themechanismusedttl preventbacklash atthe shoulderof thearm,a reducerwith a system ofpulleys. isshown in figure2·1a.To ensuretheneeded stiffness,four parallel cablecircuitsarc usedonthe outputstage, two circuitsonthe intermediatestage andonlyoneonthe driver stage.

Tensioning ofthe cablecircuitsis adjustedat thedrivepinion. Thepressurecompensating system is shown infigure 2-1b. Themanipulator is pressurecompensated. ullthcparisrun in anoil-filledencasementslightlyabove ambient pressure (9000psi).The overpressure ismaintained by spring-loadedbellows.Mineraloilwas chosenwhichinadditionto providingoverpressure,lubricationandcoolingallowed the researcherstovisually examine the partsduringtesting.The JASON ROY and its armarcshown in ligures 2-2 and 2·3respect ively,

Farbrother andStacey (1993)reviewsome oftheaspectsindesigning ROVs with controlin mind.Thepaper centersits attentionon theproblemofcontrollingthe movement andpositioningof theROYitself. bUIdoes not makeanyreference tothe controlofthe arm.

Figure2·1a Backlash Prevention Reducer,Yocrger et al.,l991

Figure 2·1bManipulator'spressure CompensatingSystem,Yoergeretal.,1991.

10

11 2.3Hydrodynamic sLoadsonCylinders

In thepresent work. wehaveusedcylinders10modelthelinks ofan underwater robot arm, mainlybecausethe case ofhydrodynamicloadsactingon cylindersImshcCII extensivelystudied by manyresearchers.Howeverthevastmajorityofthesestudies concern steady and oscillatoryflowsaroundcylinders of infinitelength.althoughmost practicalapplicationshaveat leastonefreeend,and vcrylewhave devotedefforttoIlows aroundcylinderswith finite span.The caseofarbitrarycomplcxmotionsoffinitespa n cylinders withtwo freeends like thearmin the presentworkhasnotbeenserio usly researched .Thereforemostof thedata concerning hydrodynamic for ces acting nil cylindersare not applicablein this case.

Our model consistsof two cylind ersoffinit e span linkedbya revolutejoint.The three dimensionalityofthe cylinderswith twofree ends.plus theirinfl uencetineach other, causes thehydrodynamicforcestodifferfromthose found in the caseofinfinite cylindersorquasitwo dimensionalcylinde rs (endplates). Also,the specificsIll'the kinematics of the robotarms can causedifferentIlowregimestotake place.even ulnng the same link.

When a body movesthrougha fluid a force is exertedonthe body by thefluid . The componentof thisforceparallel to thefreestreamrelativevelocity iscalleddrag, whereasthecomponentperpendicular to the freestream velocityiscalledlili. Both pressure,p,and viscousshear,T..,contributeto the resultant forces. Theoretically,thedrag andliftforces ona movingbody couldbecalculatedif we weretoknowtheprcs.~ureand viscous stressesdistributionover the entire surface of thebody,equation2·1and2-2.

Drag"

J

(psinO"''T"cosO)dA

Uft"f(pcoss-T..sin6JdA

12

2-1

2-2

Here0isthe polar anglefrom thecentreofthe cylinderto thepoint onthe surface.

Unfortunately itis extremelydifficultto compute exactly thecompleteflowand pressure pattern uverthe entire relevant surface of most bodies.Equations2-)and2<!, although formal.arcof little practicalvalue.In practice. for themostpart,drag andlift arc calculated using empirica lorsemi-empirical methods.

Morison' sequation.equation2-3, wasdeve loped to describethehori zontal wave forceper unitlength,!,acting on averticalpile extendingfrom the seabed 10 theocean surface(Chakrabart i,1987).

2-3

He re(~,is theinertia coefficient.

elJ

is thedragcoefficient.pisthe density of thefluid, D the diameterofthe cylinderand/Ivthe componentof the velocitynormal to the cylinder'snxis.Usually(~Iandellare functions ofthe Reynoldsnumber,defined earlier.

andtheKculegun-Carpenterdimensionlessparameter.KC=lI_uT/D,whereTis the period ofthe wave.The equationconsistsof twoterms,inertia anddrag.Aparticle of water passing arounda circularcylinderfirst acceleratesand thendecelerates, and thisapplies aforce onthe cylinder,This force is described bythefirst term inequation 2·3.The

1J principal cause fordrag is theexistence of alowpressureregionon thedownstream side ofthe cylinderandahigherpressurezoneon the upstream side(PrnndtlandTicticns, 1934).

Morison'sequationis semi-empiricaland itsapplication\0acomplex lime dependent separated flowis questionable. However,attempts to deriveotherfonn ulutious tobetterdescribethephenomenahavebeenunsatisfactory.Someproposalshave been madetoimprovethe existingequation(Sarpkayaand Isaacson.1981 )hyadding hiyhcr harmonicterms. Other more complexsolutionshavebeen attempted(Naill,19(0).But overall the original Morison' sequation hasbeenreliablein predict ingIluid forces011 smallmembers.The equationcanbe extendedtothe computationof fluid forces on inclined cylinders by writing it intermsofthe normal velocityand accelerationvectors (Chakrabarti,1987,Binand Yu-Cheng,1991 andOtsukaand Ikeda,1991).

TheMorison' s equationwasdevelopedtomatch thepeak forces due10asuparuted flow.However, forattachedflowsthe forcesare notwell describedbyMorison's equation(Anarluketal., 1992).The essenceofsome newmodels (Anartuketul.,IlJ92 andSarpkaya,1991) isthat the mean rateoftheworkdone bythedrag ordamping force mustbeequal to therate of energy dissipation. Eachof thesemodelshowever ussumcs that theflowremainsattached at all times which makestherange of itsuscruthernarrow, Apotentialflowmodel(Chewetal.,1992)hasactuallyvalidatedthe much criticized Morison'sequationfor unsteady flow. Themethod isbased on a two dimensional potentialflowmodel that simulates the effectof flow separation in unsteady flowbyplacing surfacesourceswith time dependent strengthandangular positions on the

14

rear wetted surfaceof thebodyand downstrea msinks in the flow to form aclosed wake model. Then theunstead y Bernoulliequationis used to obtain the timedependent pressure. The in-lineforte equationobtained bythis modelis comprised of uncoupled drag andine rtiaterms.Withsome simplifications the equationreduces tothe classical Morison's equation.This worksupports the applicability ofthe Morison'sequation to unsteadyseparated flow s. The cases consideredshowedsmall variat ionsofthe dragand inert ia coefficie ntsovera cycle.ThisiswhyMorison'sequationwithmean coefficients predictsthe in-line force rather precisely. This method, however. did not take the surface TOughness inconsideration,afa ctor that affects greatlythe drag force (Sarpkaya,1990).

SinceMorison's equatio n was proposedin 1950, ithas beena challenge to find a correlation forthe dragandinertia coefficients. In fact, satisfactorycorrelationsof withthe relevantdimensionlessparameters havenot been achieved.Datacorrelationsto date usually plot each dynamic coetlicicntversusoneof the dimensionlessparameters ignoringthe others(Makhortikh andShcheglove,1990 and ShcheglovaandMakhortikh, 1988).This hasled to scatter correlations(Hortonet al.,1992,andRao et el., 1992).

Generally flows around circular cylindersarcturb ulent and wakes have a three dimens ionalunsteadycharacter(Slaouti and Stansby,1992). Thereis no generalturbulent modelto incorporateintoanumerical simulationof unsteady separatedflows andthe limitationsof computi ngpowerwillnot allowthe direct three dimensiona lnumerical solutionofthe Navier-S tokesequationsin thenearfuture (Slaoutiand Stansby,1992 and Snrpkaya,199 1). Atlow Reynolds numbers,theflow andforceson a cylinder may be reasonably predictedbytwodimensionallaminar computations(Kawamura etal.,1986),

15 although even here.theflows areknown10contain some threedimensionalchamctcristics (Tatsuno, 1989.and Siaout i andStansby. 19( 2).

Therehavebeena fewnumerical studiesof two cylinders inside-by-sideami tandem arrangements usingtherandom vortex method (Sin. 19K7).This method is believed10beversatileandefficient,notexhibitingsomeof thenumeric aldiffusionlind dissipationproblemsassociatedwith fixedmesh schemes.Ithas beenobserved thaifur theside-by-sidearrange ment(Sin.1987).forlarge spacings(TID>3.5),whereT is the distance between the two cylindersandD theirequivalent diameter (figure2-4).the interactionbetweenthe two cylindersis negligible.and the twocylindersform their own vortexstree ts inde pende ntly . Spacingsintherange of 2-cTID<3.5causesthe cylinders 10 producevortexshedd ingin anti-p hase.Spacings of 1.2-cTID<;2 results inslrong interaction, producing onenarrowncar-wake finda wide ncar-wa ke.Thetwonca r-Wilkes occasiona llyswitchoveranda largescale vorte x streetis formed further downstream.For spacingsof 1-cTID<1.2 the flow velocity in the gap is sma llandalargeformation regionoccurs, resultingin the formationofalarge scalevortexstreet. Sources (SllIouti andStansby.1992 andZdravkovic het 0.1.,1989) show drag coefficie ntslindStrouhal numbers foreachcylinder fordifferent spacingratios.Therepulsivefo rcebetwee nfho twocylinders,as expected,decreasesgraduallywithincreasingspacing ratio1'/1),and disappears whenTID>3 .

Inthe tande marrange ment,source (Slaoutl andSta nsby, 19(2 )found that acritlcnl spacingexists whichdivides theflowregimeinto twocatego ries.Forspacing ratiosof TID>3.8 eachcyli nderproduces a vortex wake andforTID<3.4 vortex shedding

DirectionoftheFlow

16

T

'j

Direction oftheRow

b)

Figure2-4CylindersinSidc-by-Side , a), and TandemArrangements.b).

17

doesnotocc ur between the two cylinders.At spacingratiosof 1<TIJJ«1.1 the^t\V\1 bodiesbehave likeasingleslender body.tbcscparan..-dshear layersoftlJ.:upsu...OIbody encloselhedownstreambodyanda vortexslrc&:tformsdown streamofthercerht"'] }'.

ell

showsasudden increase asTIDexceeds the criticalvalue.

Inmost practicalapplications.likeinourmodel,there isatlm\1onerrecend.^;I case whichhas attracted little attention(Zdrevkovichctcl.,19119).Fur cylinderswith length\0diameterratioUD>6 . ithas been observed(Zdravkovich0:1nl.,]I)IN)thai vortex shedding occursalongmostofihe spanexcept ncar thefreeend.How visualization(Zdravkovich ctal.,1989)has demo nstrated thattheintcructinnuf'tbcvortex sheddingand the free endwasextremelycomplex.Also"eyes"were visualizedncar the surface nearthefree end.Each"eye"isthought tobeassociated withtowpn..'S.'iUrcwhich isresponsibleforanincreaseinthe local dl<lgcoefficient.ForJ)J)=5inphase('f\-'S...urc fluctuations have been reported(YeungetaI.•1993) onlh~two sidesuflhecylinder.Arc shapedvorticeshavebeen observedbeingformed behindshortcylinders.The cbegc-ove- fromanalternatetoIIsymmetric vortexstreetoccurred ncarIJD=2.5.A shuteylindl.'f withIWI)free endshastwointeracting.sideFlows which inhibit the vortex shl..'t.Iding (Eckerle andAwad,1991 ).A shortcylinderwithtwohemisphericalendsarrnlllimOltC!i theshapeofan ROY (BakerandSaye r. 1990).Reduction ofUDdecreases the drug coefficient(figure 2-5).Zdravkovich et al.(1989) arguethutthcexplanationshouldhe soughtintheventing of ancarwakewhichoccursbehind^itbluffbody.i.e.allinl1uwtlf thefluidintothencar wakespaceonitsway around thecylinder' sends.lieneethe pressure overthebackside of thebodywould rise resultinginasmallerdrag.Itshould

I~

s

A

. ⁰ _~

a

I ]

. 0 ~ "

"

0 ~

:::J

^"1

'" " . "

,.., ₀

0 N

~

.0 N

0 4 a

" .

~

~

^0N

g

-e

'" ^.li _~

. _'"

⁰

1

f)tI>r>:cl(l(

'"

r>cD

J(

'"

..

. 0 a

" ³ .~

"

:~

^aM

a . g

a ::!

^.~

J>

: . D D

" s

^~

" " ]

o a

"

⁰

o 00

'"

;~~ 8

0

o. a

. &

0 0 0

" ~

⁰

0"

" ;;:

e

~ ~

<.'J

::! _~ '"

^{0 e-,Q}

^{~ d}

'"

be pointedourthat ittakes some timefortheinflow\0turn antireach thencarwnkc.l'hc result is a low pressureregion ncarthe free endscausing ahigh local drag coefficient .

Zdravkovichct at. (1989) observedthattheseparatedshearlayersfrom shnrp ellgc\!

circular endsformtwocounter-rotatingswirlingvortices. which separatefromthe Free endswherethenowaroundtherestof thecylinderdocs andnrccarried downstream as strcamwisevortexpair.Theeffect of theinflowon thencar wake along the span docsnul merelyincrea sethebasepressure butalsodisplaces thevortexfonmnion region fu rther

downstream andwidens theseparatedshearlayersbeforeroll-up into vortices [Nakumuru etal. •1992).Thiscontributesto areduction

o r

thedrag coefficientandcauses1Idecrease infrequency vortexshedding. In additionthe Stroubnl numbermeasured hchind tinite cylindersisalwayssmallerthanbehindinfinite cylinders. The ruu-til'full tlfthednlj;

coefficientbecomesnegligibleforUD<6.Thisindicatesthatwithfurthe r reductionIll' UD thedragbecomes governed bythe shape ofthetwo freeends. Whenusing hemisphericalendsthedragcan bereduceduntil fjlJ=I andthe cylinder becomes :111 sphere (Zdravkovichctal.,1989).

The effectof theReynolds number issimilar to thatfound^(IncylindersIll'infinite length.However. theseparation linesarcdisplaced graduallytowards thestagnationpoint lineas

un

decreases.andeventuallybecomes bow-shapedfor/JIJ'"2.Thecouccpr ofa universal Strouhal number (Zdravkovichctel..1989 nodNakamura ct at,19lJ2l whichisbased on datafrom infinitecylindersisnotapplicable inrcgillos

u r

three dimensionalflows suchasneartheends ofafinitecylinder (Koctal.,1(87).Thethree dimensionaleffects or.the drag coefficientof a finitelength circularcylinderis quite

20 sim ilar tothai in steadynow(Nakamura ctal.•1992). Thevalues oftheadded mass coefficient,L'~,for a finitespan cylinderarc aboutequal to thosefor quasi two dimensionalcircular cylinders for Ke ulegan -Carp enter numbersKC~8 and larger than thoselor KC>8 (SarpkayaandIsaacson. 1981). Thevalues of theaddedmass coefficientforinfinitetwo dimensionalcylindersdecrease in theinterval 8<KC <25.

hutfinitecylindersdo not exhibitthis behaviour.The

1m

coefficientsarc alsosmaller for finilcspancy lindcrs(NakllmuractaJ..1992),

Unlikeinfini te twodirnensioual cylinders,theproblem of finitespan cylinders placedin a uniformcross llow has not been studiedextensive ly.Ourunder sta nd ingof the vortexsheddingphenomenonandtheassociatedflowinduced unsteady forcesis rather poor.Thisispartiallydueto thecomplexityofthe separated flow.its threedimensional natureandits interactionwiththe wakeflowand partly becauseof alack ofreliable techniquesto resolvetheunsteady forcesinduced onfinite span cylinders(Bahanet al., 1989).

In spiteoftheinterestinunstea dyflowpast a bluffbodyand many experimental mea surements ofthe unsteady now inducedforces,thereare very fewnumericalworks in this area duetoits complexity.Analyticalsolutionsfor separated lime dependentflows arcnotyet possible even for relativelyidealizedsituations{Kiyaand Tamu ra,1989).

Whenu twodimensional cylinderisplacedin a crossflow,alternate vortex sheddi ngfromboth sides of the cylinderinducefluctuat ing

lin

and dragforces (Bahan ct al.,1989.Blevins.1990.Sarpkayaetal.,198 1andSarpkaya 1991).The frequencyof vortexshedding is giveninterms ofthe Strouhalnumbe r:S=nD/l/~•whereIfisthe

"

frequencyofvortexsheddin\; from oneside ofthecylinder.J)isthediamete r ofthe cylinderand".isthefreesueamvelocity.TheStrou hal numberdependsin tumon the Reynoldsnumber.IftheReynolds number isinthesubcnucalrangethefll,."IUI.."IlC)'\lrlhe dragisabouttwicethaiaftnevertexshedding (Bahan ctOIl..11)8¹

» .

II has beenU~T\'I..'I.I that thefluctuatingdrag signal doesnOI correlatewellwith thevortexslll..-ddinl:\(l\."qUl.'lu:y.

Ithas beenshownthat the complex.thr ee dimensionalno wwhichexistsbehind11K' cylinderinhibitsthe periodic shedding of vortices.Thiseffectillvariable01111111:\ the cylinder'sspan(Sin.1987.lindNiemannandHoelscher.ICJq{)}.

Alarge volumeof literaturedealswith fluctuatingforcesandassociatedvortex sheddingfrombluff bodiessubjected to steady ambientflow .NumerousClllllpululillnal studies havebeenperformed onnow aboutcircularcylinde rsattemptingto predictthe Strouha1numberinsteadyambientnowandtheshapeand growth ofthe wakeregionin impulsivelyslarted flow (Gopalkrishnan et al.,1992:mdwang end Dalton.II)tJl).The complete Navier•Stokesequations invorticity/stream function formhavebeensillvcxl (Borde,1990) using acombinationof second and fourthordercompact finitediffcrclM;c schemes. Shorttimesymmetric-wakesolutionsfor Reynold s numberstlf3(M).550and 1000 were obtained showing good agree mentwithvisualizatio n results for hlllh vortex size and centerposition(WestandApelt.1990).

The evolution ofthedrag coefficientdocsnotdependsignificantlyun thereduced velocity.u"IID.whereu"is thelinearvelocityorfree streamvelocity. Iisthetime und Dthe diameter of thecylinder beyond theperiod ofconstant accelera tion.figure2·CI.1\

drag overshoot occurs betweenu..IID=4 andu.'ID""S(at(I;relativc distancecovered in

22 approximately half the vo rtexsheddingperiod).Thedragovershootoccursbefore the vortices exhibit noticeableasymmetry.The reasons forhighlocaldrag couldbethe following (Sarpkaya. ]991):Duringthe initia landsymmetricdevelopmentof vortices (for1I~II/)<4),the vorticityrapidlyaccumulatesin the wake. The symmetricvortices grow10sizesconsiderablylargerthanthosefoundinlater stagesof themotion wherethe vorticesshed altcrnalingly.But,this is not enoughtoexplainthe dragovershoot. Also.

during the early stages of motionthevorticity fluxis expected 10 be considerablylarger thanthe cross wake transfer ofoppositelysignedvorticity. Thevortices grow andexhibit three dimensionality.The combination

o r

tileabovementioned factorscontributes to the rapid riseand fall of thedrag coefficientat or nearlIol l D:::::4.5(Yeung et al..1993).

Cylindersmay heclassifiedas semi-streamlined bodies wherethe positionof separation is varyingdependingon thefree streamvelocity andflow profile. turbulence ofthcFree stream.geometryandthe roughnessofthe body'ssurface (Niemannand lloclschcr ,1990).Theflow aroundacircularcylinderdepends on Reynolds numbe r.since thc stateofthc boundary layer may be eitherlaminar or partlyturbulent (Bedret aJ..

1989).In turbulent conditio ns.additionalenergyisfed to the boundary layerbyturbulent momentum exchange and henceit gainsaddit ionalcapabilityofovercomingtheadverse pressuregradient behind the pressureminimum.producinga smallerwidthofthewake.

recoveryofthe wakepressureand smaller drag. Most researchers agree about these features hut the steps by which the transition from laminar to turbulentoccursis not completely understood (NiemannandHoelscher, 1990 andSarpkaya, 1991).In the case ofacylinderoscillating in the presence ofasteadycurrent.the flow maybe characteri zed

g

~ ~

l

w'NJanbroj,

2J

0 11 e 0

~

j

~

1 E

i·

~ ~ ] e

:§,E

~

E

~ .= ~

~'

~

24 by three non-dimensional parameters,thereduced velocity,theReynoldsnumber and the amplitudeparameter(Badrctal.,1989).

2.4HydrodynamicsLoads on Robot Arms

Verylittle has been fo und intheliterature regardingthis specific topic.the central partof this thesis.Most of thepapers revieweddealwith hydrodynamicsandcontrol of ROYs,or with hydrodynamic forceson offshorestructures.Onlythreerelevantreferences have beenfo und.Fukuda and I-lara (1989),Liceaga-Cestrc etal.(1991)and Lee(1993).

Liccaga -Castroct al (1991) proposed a mathematica lmodel based onMorison's equation.which uses drag andinertia(added mass) coefficientsdetermined for infinite circularcylinders.Inthe caseofthe added mass coefficients,a constantvalueofCa==/.8 isused.Themanipulator modeldescribed is atwo-link planararmwith two degreeof freedom.figure2·7:thetwojoints rotate inoneplane andthereis no translation.The effec tsthat onelinkhas on thc other were not considered.The hydrodynamicloads on thcmodel wereincorporated intothederivations ofthc equationsofmotionusing the Newton-Eulermethod. In thisthesisit wasdecidedto use theLagrangeformulation.

Fukudaand Hera (I989)(in Japanese) use a simplifiedmathematicalmodelof the hydrodynamic forces incorporated intoan adaptivecontroller for anunderwater robotic manipulato r.Themodelstatesthat the equivalentdrag force acts atthemidspan of the link.The physical model testedis atwo-link arm withjoints rotatinginpcrpendicular planes. fig ure 2· 8.the rotation of the first jointoccurs along theaxis of thefirst link.A translationofn1epaper wasunavailablebutmuchcouldbeinferredfrom theequations.

Thethird source (Lee. 1993) isavideo demonstrationora one-linkrevolute,An adaptive

26

27 controllerusedwas testedfor angularpositioningof thelink.The demonstration lusted barelyfewseconds andonly mentions thai researchis underway.

Rivera and l-linchey (1992)conductedexperimentson allurrn/bodyconfiguration (figure 2·9) whichconsisted of a sphe ricalbody.0.25m in diameter.with11box cross- sectionsinglelinkhorizontalarmfixed tothebody nnd0.25IIIlong and 0.05111wide.

During tests thearmwas position I mbelowthewater surface intheMUN wave/towing tankfacility: thetank itself beingapproximately2mdeep. The motion ofthe carriage was used to generate steady translation anda DCmotorWllSused to generatesteady rotation.Torque datawere obtainedfor various combinationsof steadytmnshuion und rotation.Tests showed thatthe hydrodynamicsforcesacting on thc armbehave randomly when the arm is positioneddownstreamfromthebody:forthecombinationshownin figure 2·9(steadytranslationand no rotation)the torq ue mcasun..'1iread sometimes counterclockwiseandothers clockwise.

2.4 Neural Networks System Identification

The environmentin which ROVs andAUVs work emphasize the rcqutrcmenrthat controlsystemsheextremely robustandcapableof maintainingacceptable performance levelsin theface ofparametricuncertaintiesandunmodelleddisturbances[Farbmthcr and Stacey,1993).Recentadvancesincontrolstrategies for RO Vs andAUVs includenotonly neuralnets, butthere is anadaptive element in almosteverynewproposal {i.e.Pup et ul., 1991. Zomayaand Tarek,1993 andWesterman, 1<;91). But maintaining anon-line adaptivecapability(i.e. after the neural networkhasbeentrained)is computationally expensive for present practicaluse(Farbrother andStacey. 1993).Westerman(1991) has

28

- ( ^o

beenable 10maintainan on-lineadaptivecontrollerfor onlytwo degreesoffreedomuf five degreeoffreedom arm:hardwarelimitationshave been thesource\11'this shortcoming. Zomaya and Tarek(1993)have used decentralized ncuro-cdaptivccontrollers (acontroller perjoint) for the control of some industrialrobots.Pap ct,II.(1991) describe a very similarprocedurefor underwaterroboticoperations.

Artificialneuralnetworkshave biologically inspired architectures.hiltitwouldbe a misconceptiontothink that neuralnets mimicaccuratelybrain activity.Most

or

the workingapplicationsofneuralnets todayhavemore incommonwithstatisticsanddmu manipulationthanwiththe simulationof grey matter (Westerma n.199I). Busieully ncurnt networksarc a meansof correlatingdata using multiplehighlyinterconnectedprocessing elements.The elementsform a transfer functionthat maps input datatooutputdmn (Caudill, 1988). Neural netsdo not needspecificmathematicalmodelscorrelatingthe input datato output data.Theyare able 10comeup with arelationship bctW\.'C IIthe input variablesandtheoutput. Anetwork can be trainedbypresentingitwithknown and reliableinput/output samples,adaptingthemselvestomap one10the other.Allcr a sufficientnumber of trainingcycles.thenetworkis able to generalizeand producesIIbest estimateoutput,witha correspondingmeasureof certainty,in respon:-ic to some newdata, Neural networkswillexhibitdifferentpropertiesdependingon howtheylearn and uuin.

A wayoftraining a networkisto presentinputsamples with correspondingoutputs (Zomaya andNabhan,1993). For eachsample set,thenetwork readstheinput and estimatesthe output. Itthen comparesits estimatewith thecorrectresponse,usingthe error to adjustthe weightsintheprocessing clements.Thisis calledsupervised learning.

30

Ittries10leachthe networktherightanswerfor any corresponding input duringthe training phase. Fromthe above wecanconcludethaiitisimportanttotrain thenetwith

a·good~sampleset.Determiningwhat a good samplesetof inputsandoutputsis.fora

given neuralnetwork.turns our tobea mostdifficultissue.Anetmaypay attentionto rcteuonsinthedata samplethaimay seemunimportant10people.Ifa networkis undcrtralncd,itwill nolyield reliableresponses.If.onthe other hand.itisovertrained.

it justmemorize s thetaughtcases anddoes not acquire theabilityto generalize if posed withnew data.

A netwo rk uhlc to generalize can inte rpola te betweencasesfo undinthetraining SCIS.In this casewhattheneuralnetwork reallydoesistoclassifydata:itis a wayof m:Hching signalstolearned patterns.Neuralnetscanalso generatecomplex sign,:'1with relative case,which findsroominapplicationssuchas speech reconnitlcnandsynthesis.

and roboticcontrol.(Craig. 1988, Kawatc etal.,1988,Miyamoto et al.,1988and Pap etal.,1993).Anotheradvantageofneuralnetsis the abilitytolindanswersinspiteof incompleteand imprecise available data.Neuralnetworks havealsobeen used todevelop adaptivecontrol systems and we intendto use it to analyatheobtained data.

A neural networkis medeup of simple.highlylnrerconnecredprocessingelements.

whichrespondillparallel to inputspresentedina dynamic,self-organizingway.Certain knowledge iscontained withinthenetworkin the waythe processingelementsarc connectedand the weightingassigned toeach neuroninput.Byitself,asingleprocessing clement.figure2·\O,;srelativelyuseless:itis theinterconnectionofmanythatworks.

Theprocessing clementsarc organizedinlayers andconnectedindifferent waysto

o o o o

31

32 providefor the differentlearningdynamicsfor solvingdifferentproblems.Theconnection weights lITCabout the onlything thatchanges in a neural netasne w datu becomes availableand thus provides forits adaptability.Theweights varyin ordertores pondto dutu currclmion.They arc like the coefficientsof a multi-variablesystem of equations, evolvingtolitthe datano matter howco mplex .Itis possible thatthe transferfunction be made variable inorder 10 increase thelearning speed. but thisis still a matter of research.In the scopeof thisthesiswewillusc a typeof architecture known as fcedforwardarchitecture.The activityflows onlyin the forward direction.origi natingin theinputlayer.In conjunction with this sche me we willuse11back propagation learning ulgnrithm,In this typeof supervisedlearning,the desiredoutputmust be provided along with theinput.

Sorensen (1993)describesamulti-la yer perceptron system able to perform multidimensionalcurve-fitti ngin a multi-variable dynamic process.The neuralnetwork in question. figure 2-11, contains aninputla yer.a hidden layerwith asufficient number ofnon-linearneuronsquashingfunctionsincluding offsets.and anoutputlayer withlinear neuron functions.In thiscaseWIis a matrixrepresentingthe weights between the input layerand thehidde n layer.the "I" input provides the necessary offse t. ThematrixIV!

representsthe weights between the hiddenla yer andthe output layerand Frepresentsthe neural function, a tanhfunction lor all clements.The errore,which representsthe mismatchbetweenZ.the desired output, and

i ,

the output from thenet,isused inthe Back Propagation Error Algorithm,widely described inthe literature on the subject Sorensen(1993 ) extractsa so called Gain Matrixfroma trainedmulti-layerperceptron,

>"1

e. ~

_~"

~

=~

0...,

~

_~

Jl

:;, " g

.~

!

_~ ^.~

¹¹

⁰

'0 ~

:s!

::1 ^::;;

}

0

,,- _~

-e 1]

! ,,'

^N

_.~

"5

.5

u,

::..

33

J4 whichisthe derivative ofthe outputvector,

i .

to theinputvector of the net.V.This allows atrained multi-layerpcrccprronto workas an identificationparameterroutine.

lS

Chapter 3

Ma the ma tica l Model of Hydrod ynamic Loads on a Robot Arm

In thischapter a mathematicalmodelofnplanar two-linkrevolutejointroboturm isdeveloped ,The hydrodynamictorquesweredevelopedbase d on Morison 'sequatiun anddimensionalanalysis.The hydrodynamicmodel and the dissipatingforces werethen incorporated into the equationsofmotionofthe two-linkarm,Themodel isnut expected to performaccurately,dueto alack of hydrodynamicdatafor cylinders with two1'r~'C ends moving randomlyin theplane and theeffects of one cylinderonthe other. The model is useful howeverin identifyingthe dimensionalanddimensionlessparamete rsthat affect thejoint torques. and in what measuretheydo so.These parametersconstitu teune of the input setstriedin the neural networkdescribedinchapter5,

36 3.1Ana lysisofa Two-Link Plan arRevoluteJoint Arm

Applying dimensionalanalysis(Sharpetal.1992) tothe one-link con figuration undergoingsteady translationsandrotations shownin figure3-1,weobtaina functionlor theto rq ue dependingontheindependentparame ters thatinfluence itsdetermination:

I'0¢

(u"

b,6,L,D,p,,)

3-1-1 where:

I.-Lengthofthe link j)-Diameterofthelink

",_•Normalvelocity (1/,,"'II..(:0.1'0 )

II"-Translational velocity ofthe base

.,I' -Torque at thejoint p•Densityof the fluid II -Kinem aticviscosityofthe fluid

o-

Ang lebetwee n thelinkandII~

(j-Rotational velocity

[m) [rn]

[m/sec]

[m/sec } [N.m]

[Kg/m']

[m!/sec] [radians) [rad/sec]

As aflrst casewe analyze thedragforceon the singlelinkwhen thetranslationalvelocity

II"isconstantand the rotationalvelocityiJequa lszero.Inthis case the torqueatthe joint,

dueto thedragfo rceonthelink,may beexpressed throughthe followingfunc tional equation:

3-1-2

Y.

37

x"

' i@nill]J ^".

- . _ . _

^{. .}

__

^.

o

<-llj 9< L

Figure3·1One-Link Planar ArmModel(VelocityProfile) iI,_

)8

Usingdimensionalanalysisthe above equationcan be reducedtoafunction of dimensionlessparameters.

Compoundingthe pi termsgives:

)-1-)

ApplyingMorison'sequation10this case asim ilarexpresion isobtainedfor the torque ut(hejoint:

)-1-4 In thiscasethe dragcoefficientell •the term in the right handsideofequation 3-1-3.isafunction ofthe Reynoldsnumberand theUDparameterandisconstant along litelink.exceptncarthe freeends.

Ina secondcase.wherethetranslationalvelocityis zero and the rotationalvelocity isconstant.dimensionalanalysis yields the followingfunctionalequation after

]9 compounding:

," [0 L' L )

pDL4/i

" q, ----;-'0

3-1-5

Heredimensionalanalysis yields the socalled rotmional Reynolds number.the tirsl tenn insidethe parenthesesin equation 3-1-5(Oldshuc.I(83 ):theconventionalReynolds number wouldbedifferentalong the lengthof the link. Once againupplyiugMurison's equationand integratingalong the linkgives:

]-1-6

Inthis case the drag coefficientdependsonthedtmcnstonlcsspununcrcrsinside the parentheses .

Athird more generalcaseinvolvessimultaneous rotationandtranslatio n.'1'11\:

significant functional equationsobta inedfrom dimensionalanalysisarc:

,', (0 L' OL I.)

pDtPL4 =

q, ---;-'

u"cosl1'

0

^3-1-8

Following isanattemptto generalizethe case of arevolute multiplelink ma nipulatormoving in theplanebyapplyingMorison's equation(figure3·2 ).thele ll hand sideterminequation3·1-7 or 3-1·8 co nstitute the dragcoefficient.

40 The verticalandhorizontalvelocitydistribut ion along a linkicanbewrittenas:

3-1-9

3·1-10 Inthe above equationsx,andy,denotethe local coordinatesystem,x,equalszero at the jointi andL,at the joint;+1.Equation (3·1·10)isofthetypeI(x,) ;::A/x,+8"

whereA,is equal 10 the sumof allangula rveloc ities precedinglinkt,includingthe angular velocity oflinki,andH;isthe sum of the secondand thirdtermsin equation (3-1- 10).J,pandqare counters.The coefficientsA,and 8,canbefoundfrom:

A,'

, EO ., ,

3-1-11

3-1-12

Thejoint displacements9,(I)and thetransla tionalvelocityof the base arc functions of time r.In the scope of this thesisonlysteady translations and rotationswere tried.butevenalconstemtrenslauonalvelocity(U,,(l)=constant) and constant rotational velocity

(8,(/)

=constant),accelerationsexistsincethe velocitiesII_J,/(x,.I)and uAt) alsodepend on.theconfiguration orthe arm. The velocitiesandaccelerationsareusedto calculatethedrag and inertiaterms inMorison'sequation.

Thefollowingexpressionscomputetheaccelerations in they,andx/directions:

41

42

3-1-14

In order 10 find the dragcomponentoftheforceacting along thelinkl,wemustintegrate thefirst termof Morison'sequat ionfrom 0toLioHoweverthe solutionof the integral of the expressiondocsnot have a commonclosed form for everyconfigurationof the arm.

In thecase where-B/ A,<=0 and-R,fA;>'"L/.it is sufficienttohave oneinterval for theintegrationfrom 0toL, .Howeverif 0<-B, IA,<L, then two integration intervals areneeded.Theinteg ration intervalsare from 0to-O,lA,andfrom-B/A,toL,.

If.n/A,<=0or-B,IAI>=L1(caseI,figure3-4) then:

I .,

f::; '"

^1PD;CV),/sgll (Uy/(Lj))!(A/ X;+B/ )2dx / o

e..

•.~.

1.

^{2 P}

D

^{,CDrl.rgn{U,;(Li}

II [f",L,.B

3A

d _ !!II

j

3A

j

3-1-16

3-1-17

If-B/AI>0and-n/AI <Lj(caseII,figuTC3-4)then:

"

f.~, ^• ~ ^{p D,C}

^{D"'8"(-U"(L,II}

I f",x,.B ,)'

,n,

, ., Ib~1

^{"t pD;C/)!isgl/(Il, ;{L; )J}

J

^{fA;xj+Bj)2dx,}

-;4,D,

3-1-)8

3-1-20

3-1·21

Thetorqueon joint jcausedbythey;drag component ofthehydrodynamic force onlinkIby1/,/canbedetermined from the following expression:

3-)-22

y

44

",-

- ~

~ 1;

~

.:I

§

;::

.

'0

~

::l

~

0

" is ^£

&

~

;!;

&;J 5

46 Excludingthe links precedingwecan lind anexpressionforthe totaltorque on joint}causedbythey,drag component of the forces acting on the linksfollowinglink

3-1-23

Takingintoaccountequations J- )-7 and 3-1-8fromdimensionalanalysis.and equations3-1-1Iand3-1- 12.the drag coefficientscan be describedbyeitherofthe lollowingfunctional equations:

3-1-24

For caseI(figure 3-4)

em

and C",canbeobtained from :

3-1-25

47 For caseII(figure3. 5):

"

"j

(Ar ;+BdxicU;

o _ 1Bi

H, --4~

-A;

[ (A,V

B ,)'dx,

c.~.-"-;;----

t-,

J

(A;x/+Bd xi tU';

:;;

e,

f (A,x,. B,)'dx, .!!..:

A,

3-1-26

In a similar waywe can determinethe dragX;componentofthehydrodynamic force acting on linki:

3-1-27 Thetorqueonjoint jcaused bythe dragX;component ofthe hydrodynamicforce onlink Iby11,/can be determined from thefollowing expression:

3-1-28

Excludingthe linkspreceding,wecanfind an expressionfor the total torque on joill!

i

48 caused bythex, dragcomponentof the forces actingonthelinks following linkj:

3·1·29

The total torque on jointjcausedbythe drag force on thefollowinglinks, includinglink}can be determinedfrom the summation:

3-1-30 The inertia component ofMorison's equation,in they,andx,directions,canbe calculatedina way similarto the drag component. This inertiacomponentwillbe zero inthecase ofa single linkundergoing either constant rotation or constanttranslation.

Howeverthesimple combinationofconstantrotation and constanttranslationwillresult inaccclcrctlon.For the generalcase of accelerated motion.8,-.eOandu"-.eO,the acceleration distribution functionalongthelinkiis of thetypej{x)=Mjx,+N;where:

M,'

t ,. , 9 ,

3-1-31

3-1-32

'"

In the casewhere-N,IM, <=0and-N/AI,>=L, •it is sufficient 10 hnvc one interval for the integrationfrom0 toLt.If0<-N,IM; <l.,then therearetwointegration intervals.Theintegrationintervalsarefrom0to.N,IM ,lindfrom.N,IM,\0L~If.N,IM,

<0::0and.N,IAI;>=L;(caseI)then:

If.N,IM, >0and-N,IM; <L;(case 11)then:

N,

'M;

fa~; =

e

My;p*D;2sgn(-u,.dL ;))[(M,.x;+N;)dx,

,., f:' ;

=CMyl

P~D/

^SSII{liy;{L;))

£

⁽Mjx;+N;) dx;

.-!.

M,

3·1·33

3·1·3'

Thetorque on jointjcaused bytheYiinertia componentof the hydrodynamicforce on

50 link;by

u "

can be determined from the followingexpression:

Excludi ng thelinkspreceding}we canfind anexpression forthe total torqueonjoint jcausedbyrhcy,inertiacomponentoftheforces actingon the linksfollowing link}:

3-1-36

Theinertia coefficientscan be describedbyeitherof thefollowing functionalequations:

3-1-37

For caseI('''IandeNcan be obtainedfrom:

3-1-38

Clt~ ^'"0

51 For ease II:

", -M;

[(M;X/+NdX /dti 1N

- ;if, : -3M;

i

f

(M,x, 'N;) dx, c

L,

f

(M,x,' N;) x,dx,

N, -~

t,

f

^{(M,x,'N,j dx,}

N,

"AT,

The inertia componentofthcforce in thex,directionis equalto:

3- )·39

3- )·40

Thetorque onjointj causedby the inertiax,component ofthe hydrodynamicforce on link i due toIi.."can bedetermined fromthe followingexpression:

3-1-41

Thetotaltorqueon jointjcausedbythe inertiaforce onthe followinglinks,

52 excludinglink } canbedetermined from:

3-1-42

The total torque onjointjcausedby theinertia force on the followinglinks.

includinglink} canbedetermined fromthesummation:

3·}-43 Thenthelotal torque causedbythehydrodynamics forceson linkjis equalto:

3-] -44 Inthe case ofourmodelitis important to considerthe effectsof therelative positionof thetwo links,theparalleldistanceand the planar anglebetween the twolinks, onthe drag and inertia coefficients.Thesefactorswillbeincludedin theneuralnetwork identification routinedescribed in chapter5.

3.2Mech anical Losses

Frictionis oftenneglectedbecause it isdifficultto modelandpoorly understood.

Inrobotic manipulators friction can accountforasmuchas onethirdof the applied torque.This is 110t the case ofamanipulator moving underwater,where the hydrodynamic forcesusually dwarfthe frictional anddynamicforces.Neverthelessfriction isnOI negligible becauseunderwaterrobotarmsoftenhave specialsealingsystemsable tocope

53 with highpressures.and theseincreasefriction atthe joints(Y~'ltcr ~Iat.199 1).

Inc

static-kineti c friction transition ncarzerovelocitycausesstick-slipbehavio ur 111.11limits the fidelityof position and control.Inaddition the load andvelocityOcpL, kicI1I.."Ctil' fricliondegradesthetracking abilityofsimplecontrollers.

The purpose ofincludinga briefaccountof mechanicallossesinIhi~wor kisItl

try10isolatethehydrodynamic forces.Ratherthan come up withanew mudd.fhc (rea/men! of frictionherewillbepatternedaficr Dupont.I993 and Dhanarai,]9C)O,

OurmodelWillidesignedbearingin mindtheneedIII isolatethehydrodynamic forces.andalthough thisisnotentirelypossible.stepswere taken in orderttlminimize the effect of frictionandotherdissipating pheno mena.Infacicalibration shows (hut the overall dissipatingtorques.incJudincdynamictorques.arcinsigr;licantbut notnegligibh:

incomparisonwith theoverall torque(lessthan2%of themeasuredtorque ).·1l1e calibration wasconductedon thephysicalmodelbymeasuringtheto rques atthejoints whileunloaded andoutofthewater.

The unloadedfrictionaltorquefora lubricatedball bearingappearstnfollow the Petrofflaw(Dimarogonas.1989):

,

r ! '"

f"d^J(P.II )l 3·2· 1

Intheabove exprcsskj

J:.

is afunction of the type of bcuring foundinmn.~1 manufacturers'catalogues,d is thepitch diameter of thebearing,1Jis thedynamic viscosity ofthelubricant andnisthespeed ofrotationin RPM.A loaded bearing,

54 however.seemstofollowthe Coulomblaw:

3-2-2

where /,andcdcp:;orllluponthetypeofbearing,Wis the appliedequ ivalentradiallcad.

and C:,isthestatic loadcapacityofthebearing.Notice that the unitsin the above expressionsshouldbeconsistent. Roughestimatesofcoefficients offrictioncan befound in manufacturers catalogues.

Thefrictio n ofalipoil sealismuchmore difficultto model.Itisnon-linear and depends onmany para meters (Czcrnikand Horve1993).Agood approximationof this frictional torquecanbeobtained from:

where.I,;is thecoefficie ntoffrictionbetween the materielof theseal andthematerialof the shaft (for asha ftofpolishedsteel!.:::(l.5).R,.istheinterferenceloadbetweenthe shaft and theseal(RI.:::{),08for recom mende dvaluesoftolerances) andR,is theradius oftheshalt

Thus.the total torqueata givenjoin tdueto frictional effectsis equalto:

-r

^l=

-rf,

⁺

-rf

⁺

.,f

3-2-4 Itshouldbe notedlhalthe direction ofthe frictionaltorque isalways opposite10the

55 rotationofthe joint.a signum function addedto theexpressionaccounts forit.Thcabove equations areempirical.andwhileitis possible to identify thedominant sourcesof frictionina givensystem, theactualparametervaluesshouldhe identified hy experiments.sincethe above expressionsdo notprovide exactresults.Because of the complexityoffrictionmodelsin individualcomponents,robotic researchers typically consider an aggregate frictionmodel for each jointof theroboticarm (Dupont.19tH ).

The calibrationof the dissipatingtorques has beenincludedintheidcmificution routine.

The dissipation of energyin the geartrain andinthetimingbelt transmissionwere also consideredfollowing themanufacturer's recomendations.

3.3Thedynamicequations ofmotion

Theequations of motion ofthe manipulatorarc a descriptionof therelationship betweenthe inputjointtorques and the motionofthearm linkage. Theinverse dynamics problemis solved to calculate the inputtorques necessarytoobtainudesired output.The forward dynamics problemsolution, the calculationofthe motion given the input torques, is mainlyused in simulations,The above mentioned processyields theclosed-form dynamic equations ofmotion whichcan be derivedthroughdifferent methods abounding inthe literature,

In the scopeofthis thesis.however,efficient algorithms for the solutionof the inverse dynamicsproblemwill notbediscussed:we willbe almostentirelyrelying on AsadaandSiotine(1986)andSchilling(1990) forthe derivation of the equationsof motion,The choiceofthisthesis isthe Lagrangian formulation,which describes the

56 behaviourofthe dynamic system interms energy.in. read of dealingwith forcesand moments ofindividual membersasthe Newton-Eulerformulationdocs.TheLagrangian approach has the advantagethateachofthe terms inthefinal closed-formequationhas a simple physical interpretation interms of suchthings as manipulator inertia, gravity, friction, hydrodynamic.and Coriolisandcentrifugalforces.The Lagraniancanbewritten

whereTisthe totalkineticenergy andUisthe potentialenergystoredinthe dynamic system. Ihe equationsofmotion arc given by:

3-3-2

HereQjis thegeneralizedload correspondingto thegeneralizedcoordinateq,The generalizedloads ate residualloads actingonthe arm once the loads associatedwith kinetic and potentialenergyareremoved.In ourcase,the frictionandhydrodynamicloads arcgeneralizedloads.

Thekinetic energyof a linktinaplanarrevolute manipulator(figure3-2)hasthe form:

3-3-3 Then thetotal kinetic energy ofthelinkage is:

3-3-4

Inthe above expressio nnilisthe mass ofthelinkl,I,istheinert ia tensoratthecentroid ofthe link, expressedin the base ccordinetes,lidisthe velocity vector ofthe centroid of thelinkj,WIis the angular vectorwithreferenceto the base coordinateand,I,.and'/.1 are the Jacobian matricesfor linearand angularvelocities respectively.

For atwolinkplanarrevolutejointurm the result is:

u •

[0

"

1 3-3-5

3·3·6

Usingtherotation matrixwe determinethe inertiatensorof link1 (ahollow cylinder)andfor link2 (a solidcylinder) in thebase coordinates (figure 5·3):

-sin 6lcos6_J cas²9_J

3-3-7

58

[

sinl(S.+8z> -sineS,+9₂)CO!l ( 9,+92)

m1 1 2

/2'"'48(3D2 +4Li) -sin(8. +9Jcos(9 .+92) cas²(S,+92)

o

0

ThemanipulatoriT'lC'rtiatensormatrixisamatrix based ontheindividualinertiatensors:

HZ2²ml/../\+/,+IIlZ

{ L,

^Z

+ L }'z

+2L.Lc1cos9z)+/z HZ) =m1L.Lct cos92^+mZL}2+/2

H₁₂=mzL,Lc_2cos8₂+1II2L,22^+/z HH~m 2L}2+/2

Nowthe firsttermillthe Lagrangianformulationcanbecalculated:

3-3-9

3-3-10

3·3·11

Excluding thegravityforceswe obtainan expression for the general izedloud atjointi:

where:

"

"

.

Q, "

EH'j4j ' EE"'j,Qj4,

j_1 j-lk.l 3·3-12

3-3-13

Thefirst term of equeuca3-2-12represents the inertiatorques.includinginteraction torques, and the secondaccounts forthe Coriolisandcentrifugaleffec ts.Noticethatthe gravityterm has notbeen included since the two link planarmodelmoves in uplane perpendicular to the direction ofthegravityandbuoyancy forces.Howeverthese IlJTC.:S and reaction momentswill have tobetaken into accountwhile calculating the friction at the joints.The friction,includ ing the efficiencyof thetransmission, and thehydrodynamic forces willbe includedinthegeneralizedloadQi 'However because theyUTedifficult In modelthefrictionandtransm issioneffectswere experimentallycalib rated.

60 ThetorquesatthejointsIand2dueonlytodynamicloadsare:

T ;

^IH:llllu +H22i)r+H2.l1)2j +(j1{j211 223~iJ2 11 2.U [

Z 1111{3D1Z+

d12 +4L IZ

) 2

=m.tfl+ 48 +nlZ(LI+L(2+ 2 L IL'2cos 82l

3·3-15

The manipulatormodeldevelopedforthiswork hasthe secondlink remotely driven.as shown infigure 4·3 .bya timingbeltpowered althe base.Thiseliminates the reactiontorque utjoint2and onlyan insignificantfrictiontorqueremains.

61 Therefore the virtualdynamic to rques exertedatthejointsTI andT: are:

3-J·16

A program forthe computationof the jointtorque s based on lheuunhcnuulc atmodcl is listed in AppendixD.2.Outputfrom the modelwillbecomparedwithexperimental data inChapter5.

62

Chapter 4

Wave Tank Experiment

4.1 DesignoftheTwo-Li nkPlanarRevoluteArm Model

111c experimentalmodeldesigned10measurejoint torqueswasa two-linkplanar, fullyrevoluteann. The modelitselfhastwo degreesoffreedom, movingonlyinthe horizontal plane.IIwasmountedon the lowingcarrier inthe wavelank facility at the MemorialUniversity ofNewfoundland .

Apic tureofthe setupQuIof thewaterispresented in figure4-1.The two links arcindependentlydriven,thesecondlink isdriven from thebase(sec equation3-3·16).

The firstlink, figure4·3. ishollowandisdrivenbya DC motor, ageartrainwitha reductiontateof 25and a verticalhollowshaft. figure4-2. A torque sensorhasbeen

mou ntedbetweenthe DC motorand the geartrain,coupled with compensating couplings as recommendedbythesenso r's manufacturerinorder to eliminate radialand axialloads.

Thegeartra inandtheverticalhollowshaftaremounted on highprecision self-aligning bearings.This compensates formanufacturingerrors and minimizesfriction.Thegears

Figure 4-1 Experimental Setup of a Two-Link Underwater ArmModel 63

64

Compensating Couplings

Figure 4-2 Detail of the Driving Section of the Setup

65

Figure ...·3 Two-Link Underwater ArmModel

66 arcalso highprecisionand withminimalbacklashcharacteristics.Thisenablesus to measurereverseloadsacc uratelyand 10minimizefrictionaswell.

Theseco ndlink isdrivenfromthe basebyasecond DC motor,a gea r trainof the same characteristics asthefirstanda sha ft mountedon self -aligningbearing sinside the hollowsha fiattached\0thefirstlink. Atimingpu lleyismountedatthe end oftheshaft and thro ugh ati m ingbeltdrivesasecondpulleymountedat the en d ofth ehollowlink, figure4-3.Adouble-lipoil seal wasinstalledbetweenthe two links.fortu natelythe frictiondue\0the seal docs not depend onthe radial loa d applied,anddoesnotva ry significa ntlywiththe relativecircular spee dbetwe e n theshanand theseal.

Inorder(0meas u re theposi tionof the links.potent io meters wereinstalled in the shunsof the gear trains.

4.2Testin g

Itisimpossibletotesteverypossiblecomb i nation of configur ationand velocities sincethenumberistoo great.

ThefirstSingeof theexperimentincluded towingthemodelatspe ed sranging from0.3 101.25nvsecwhilethe jointswerefixedat spe ci ficang u lar positions (figure ]·2).Fo r exampleforthe angularpositio n ofjoin t 101=0.the ang ularpos it ionof join t 2wasalsoset to0)"'0,themodelwasthe n towcdatOJ m/sec.This constitutes one data file.For tile sa m eangu lar positions themodelwas towed atdiffe rentspee d sfrom OJ uvsccto t.25m/sec.Thesameprocedurewasrepeatedfordifferentvaluesof 91and 02' Splicedby15degrees.Theuppe r limitof1.25m/secwas set 10avoid high stressesand vibration s on the mechanism,Th elowerlimit of OJm1secisdue to thefactthat the