ydr bo

Cont ro l of a n Hydraulic Su bsea R obot

by

©Vifci Zhang,l3.C'g.

AthesissubmittedtotheSchoolof Gradu ateStu d ies in partial ful611mentoftherequirements forthedegreeof

Master ofEngineering

Facu lty ofEngineeri ngamiAppliedScience Memorial University ofNewfoundland

Aug ust,1994

St..lohn's Newfoundland Canada

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Nalional library 01Canada

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Abstract

Remotelyoperatedvehicles (ROVs) andunderwaterrohoI8 IuW(.·I~nKf('a lly involved in scientific relatedresearchandcomme rcialutilizatjonoflludt'n\'IlIt'r resources forthepast few decades.Theyare operatedatscenes where('on.li tillllll arebeyondthephysical limita tionof huma ndivers ornrc too haza rdousfor111I'1lI.

To gainexperiencewiththe design,fabrication. controlnnd oporruion^II[

subsearobot,anhydraulicsubsea robotis beingdeveloped at Faculty ofEngl- Deering and AppliedScienceof MemorialUniversityofNewfoundland.ThislIlll'S

Itplunger/waterjet deviceto controldept h.When operating withthephl1lw~r, variationof displaced volume genera tesItpositiveoranegative buoya ncynndllLlll'l

controls therobot'sver ticalmov ementanddepth.Whenoperatingwithwater- jet,movement isccatrclled bythecomb inationofthehUOylUlCYandthethrust generat edfrom the..-aterje tsqueeze.

Thisthesisexplains indetailthe development of the controlsystemofthe robot,fromsimu lationtotest ina water tank.and reviewsthesystemsstahili ty, time res ponseand frequencyresponseperformance.

Acknowledgments

Iwouldliketoex press my thanksto my supervisor, Dr.:\Iicliael Hinchey.

forallowingmeto getinvolved inthisproject,helpingme at everystage of the develo pme nt,andfor his constructivecomments andvaluab lesuggestions.

Special thanksto Mr. Austin Burseyand othertechnicalstaff fortheir concret e help andcontr ibution inmanyof mylaborator y experiments. Ialso wishto expressmy thanks10 my friendandcolleague,Mr.ZhongqunLu.forhis encouragementandcontinuous support.

Finally,I want tothankDr.Micha elHinchey and theFacultyofEngineering andApplied Sciencefor providingthefinan cial assistanceneededto completethis project.

Cont ent s

1 Introduct ion 1.1 ProblemDescription

1.2 Literat ure Review.. ... . .. ... .

1.3 Scopeof theWork . .

1.4 Outline of This Thesis.• •. . . ••..

2 Syst emSim ula tion and StabilityAnalysi s 2.1 RobotDescription

2.2 Computer Simulation...

2.2.1 Proport ionalnllOyancyControlScheme . 2.2.2 PID BuoyancyCont rol Scheme • • . . .. 2.3 Buoyan cy Contro lSysterr. StabilityAnalysis..

2.3.1 NyquistStabilityCriterion . 2.3.2 Nonl inea rDescribing Function . 2.4 WalerjetStabili ty Analysis..

3 Syst emTests

3.1 ComponentsDescription .

3.1.1 Transducers•.• . . • . •. ...•..

iii

\I

13 13 18 211

31 31 38

3.1.2 DataAcquisitionSystem.

3.1.3 TheHydr au lic System ,,. .• ...• •. 56 3.1,4 Buoyancy ControlSystemDesign .. , .. .. ...• .•.. 57 3.1.5 Power

3.3.1 Pre paratio ns .

3.2.2 Simulation .

3.2 BuoyancyControlSystemTest

60 60 61 64 69 74 74 RodPosit ioningTest ... .•. . .... . . ... . .... 3.2.1

3.2.3 SystemPerformanceTest 3.3 TestintheWaterTank• . ..

3.3.2 Buoya.ncy ControlTest. . ... .. . ... .. 76 3.3.3 Waterjet Tesl ... ... .•.... . . • ..•.. . . 81

4 Discussions 83

!) Con clusion 87

Reference 89

A SimulatlcnPrograms 91

B ControlPrograms se

iv

Li st of Figures

2.1 Configurationof the robot .... .... (.I

2.2 Duct and the plunger . 17

2.3 Block diagram of a proportional controller 20

2.4 Block diagramofthesystem with proportionalcontrolscheme, 21 2.5 Variation of robotvelocitywhenpropor tionalgainis1. 23 2.6 Trajectory ofrobot whenproportional gainis1 23 2.7 Variationof robot velocity when proport ionalgainis20. 2·1 2.8 Trajectoryof robot when proportional gainis 20.. ... . 2·1 2.9 Block diagramofpmcontroller.... ... 26 2.10Blockdiagramof thesys temwith.PID control scheme.• , 27 2.11Variation of robot velocitywhenK,

=

10, [(,

=

1,f(,j

=

10. 27 2.12 Traject oryofrobot when[("

=

10, K;

=

^I,I(d=to . 28 2.13Variation of robotvelocity when[(,='laO,[(.=I,[(d=10 29 2.14.Trajectory ofrohot whenK"

=

200,J(j

=

1,/(J

=

10. .. 20 2.15Variationofrobot velocity whenJ(,

=

200,](j

=

1,/(J

=

1 . 30 2.16Trajectoryof robot when[(p=200,Ki=I,/(J=1 ... . 30

2.17Closed-loopsystem.• 32

2.18Blockdiagramofthebuoyancy systemforstabilityanalysis . 33

2.19 Reduced blockdiagram of the system. 33 2.20G(jw)H(jw)loci in the GHplanewhen G{S)H(8)=1.1/8(488³+

242s^2+1892s+W7) 35

2.21 G(jw)H(jw)loci in the GHplane when G(8)H(8)=IlOO/s(488³⁺

242s2+18928+ 197) , . . ,. . ... . ... .. . .. ... . 36 2.22G(jw)H(jw )loci in the GHplanewhen G(8)H(8)=22201'8(4883+

2428²⁺18928+197) . .• ... .. 37

2.23 Input-outputcharacteristiccurveforthe saturation nonlinearity 39 2.24Input andoutputwaveforms for the saturationnonlinearity.. . .. 40

2.25 Systemwith nonlinearity. 42

2.26Plot of-l INand G(jw)H(jw) for stabilityanalysisof buoyancy system•.. . , . . . ... ... .. ... 44 2.27muckdiagram of jet systemfor stability analysis .. 45 2.28 G(jw)H(jw )loci in the GHplanewhen G(s)H(s)=1.6.5/2378³

+

12108²+ 1258 46

2.29G(jw)H(jw)loci intheGHplanewhen G(8)H(8)

=

1650/2378³

+

1210s²+1258 47

2.30Plot of-liN and G(jw)H(jw) for stability analysiso f waterjetsystem48

3.1 Potentiometer in the watertanktcst .•. . .. .. .... ... 53

3.2 Potentiometer andpulleys

.. .. . ..

54

3.3 Direction of signalsin the controlsystem 59

3.4 Blockdiagramof the controlsystem ... 61

3.5 Feedback vcltege vs. piston position

..

63

3.6 Inputvoltage vs.amplifieroutputvoltage 66

3.7Rod posit ioningcontrolsystem

.... . ..

^,

... . . ...

66 3.8 l3lockdiagram ofthe model

. .. .. .. . . . .

^,

.. . .. ...

67 3.9 Transienttimeresponse of thesystem.. .. 68

3.10Bodediagramorthesystem..• •.• . . •..•.. • •.... .. 6S 3.11Excitationwave(rom LVDTwhentheamplitudewas1volt •. . . 70 3.12Errorsignal whenamplitude was 1volt. ... ....• . •... 70 3.13Amplifiederrorsignalof1volt. ..• . .•.•. • • ... ••• • .• 71 3.14Excitatio n wavewhentheamplitudewas2 volts... ... .. 72 3.15Errorsignalwhenamplitudewas2volts .•..•..• ... . . .. 73 3.16AmplifiederrorsignalwhentheamplitudeWIIS2volts. .•.•. .. 73 3.17Water ta nk inthetest ... .... . .... 77 3.18Robottrajectoryinthe buoyancytest ... ..•. 79 3.19Block diagramofthe systemwith PIDcontrolscheme.... 80 3.20 Robottrajectoryinthewaterjettest ..., , , ,,....,., ., 82

Chapter 1

Introduction

1.1 Problem Description

The ocean covers two-thirds ofthe planet and containsa vastamountofmineral resources,oil and gasreserves.Unlike outerspace,the ocean still remainsa poorly exploredrealm.One reasonillthe environmentishostileandcomplexwhen cern- pared to space.Themediumishighly corrosiveand thepressures can exceed many thousan ds of pounds per squareinch.Agents operat ingunderwater must contend withcurrents, thermoclines, unknownobstacles,andchanging bott om top ography.

Currently,remotely operated vehicles (ROVs) and remotelyoper atedun- derwaterrobots ate beingusedin ocean exploration. However,it is expected thatautonomou sunderwat er vehicles(AUVs) and untethered unmanned vehicles (UUVs)willsooncompete forthis role. Their communi ca tion withthemothe r

shipsdOClinot relyon 11IlumbilicalC'~hle.Tlms, the y haveovercomesom.~ofIl..~

Iimilal ions ofROVsand under waterrohOl~.Their workingIlIl.lrt~^ill(~xl':Jncll~1^ami theyare netrest rictedbythelengthofcabl....c~lII,C('tingllwllIwith their lIIotlU"r ships.Theyrequirelillie orno lIup portfromsurfaceYl'>l.'iCllI,an.'hay"Utihill.·r tobecomeentangledor to hreak(OIidberg,1991).;rhc autonomyof 0pl,.aliullhalt been grea tlyincreased.

Withrespecttothelevelofautonomy,mostexislingHOVsand underwater robotsarcclassifiedastethe redsupervisedvI)hid~!l(Yuh,1!IS!l),whid .illcl;.,,,••.,.

lhatthey arcconnectedtoamot hershtpbyItcablethroughwl.ifh1\1l111t~rouuuu- nication,dat aand powerarctransmitted,Thesevc!lidl'lIcann~I'];Il:I~lunuau.livl·l'li in manycasesin shnllowanddeepwaterwherecllvi rnUlllt~II\.!lare toohil1.imlllllll forthem.Theyperforman enormousvarietyortaskllslIdlalilid.:nti licrt.·tl(·ardl, inspec tionandrepairoroffshor eplatforms,pipelineburial, millec1ispu!l;II,fimlanll identify wrecksand ite mslostorplac ed all theSCII.floor,l,uryil1lt,insl'l,.;tioll1111.1 recovery oftelecommunica tion cables.

SeveralROVs or underwa ter robo tsweredeveloped in recent years.Htlv,..~

anunmanned , untcthcrc dvehicledevelopedhyUniversityorNcwIIIl1l1p~ldr,~,i~

11test bedfor tiledevelopmentofautonomouscontrolCOIICI~I)l~.lJO/.IJIJ1N3K,;1 deeptethered

nov,

developedby.l a panr-.lIIrillcSciencenmlTc~c1ll1ol(Jgy r":1I1,~r (JAMSTEC)isusedfor ocean bottom surveys.lIYlJ/tO1,50,ale tllcr.:clwhidt:

develo pedbyHydroProductslnc.,andSUIl/}RC'CA'I',aldh'~r,~luuderwuter vehi cle,developed byUSAI.Inc.arcbothusedfor inspection(Yuh,1!.I8!').

Due tothoreasonthaltheoceanis anon-unlfonn,unpredict able'~lIviTlm·

ment with greatunccrt ainly andcomplex disturbancesfromtheocean'sUlllltidi·

rec tional currentsand tethers, we cannot adapt the high technologies developed foron-land vehicles orrobotsdirectly to the ROVs andunderwater robots.Many ROVs andunderwaterrobotshave unstreamlinedfeatures,theirhydrodyn amic coefficients are difficultto ascertain, and thedynamic behaviorofthevehiclesis highlynonlinear.Thosemakeprecisemotionandpositioning moredifficult.To improve performance,it istherequirement that ROVs and underwater robots have areliable propulsion and arobustcontrol system.

MostROVsand underwaterrobotsusepropellers or thrusters forpropul- sio n. They arc ductedto improve performance whose characteristics aregenerally definedby their open-water efficiency.They operate withhighefficiency onlyif they are welldesignedand operateunder favorableconditions. Any significant departure from this operating point leadsto a correspondingdrop in efficiency.

Usually a free-swimming ROV has 4to8 propellerein orderto gain six degreesof freedommotion.

Ductcdpropellersorthrusters arccurrentlybeingutili zedby most underwa- terrobotsand ROVs, butpeople usually underestima tetheirdefects.

•They are expensiveand physicallycomplex,so theirnumbersare alwayskept to minimum(Farbrother andStacey,1993).

•Thrusters are subjectto the thruste r/vehicleinteraction,meaningtheblock- age of thethrusters regionof influence by other structu ralcomponents(Far.

brotherand Stacey,1993).

•Extramechanicalcomplexitycan introducethepotential for a greeterdegree of nonlinearityill the thrusterresponse(Farbrotherand Stacey, 1993).

•Significantvelocitydifferentialwhenfluid entersorleavestheth rusterduct- ing willinduce a ch an gein momentumthatisfelt asan additional drag force (Farbrother and Stacey, 1993).

•Theyare subjecttoserious degradat ion due toaxial andcross-flow effects that makethemnotthe ideal force-producingactuators (Ycerger,et al 1990).

•Thethruster dynami csare dependentonpropellerspeed nnd theeffect of

thrusterdynamicsto the vehiclemotionbecomes significant at lowvelo city (Yoerger,et aI1990) .

•Whenhovering,nonlinearthruster response causesthevehiclest~oscillat e aboutthe desired positionwitha very repeatable frequency and magnitude (Yoerger.19!}l)).

•Theil'performancehas beenlimitedbycavit at ion.

Besidesthepropulsion system, thecontrol system isalsoone of themost critical subsystems of ROVsand underwate rrobots. ROVandrobotdynamicsare nonlinear due to inertia,buoyancyand hydrodynamic effects. TItuS, thecontrolof suchvehiclesis difficult.Withthe increas ed utilization of thevehicles in subsea applications, the developmentof autonomous underwater veh iclesbecomeshighly desirabletoenhance oper atorefficiency.ROVs and rohots are always unstrea m- lined andaddingmore sensorsandothercomponents can causesignifica.ntdynamic change. A goodcontrol systemcangreatlyreducethe humanpilot'sworkload, adapt the vehicle intheenvironment when parameterschange andenhancetheve- hicle'sautonomy.Thecurrentcon trolsyste ms of ROVs and underwaterrobotsare

usual lynotrobustandreliableenough whichmakeunderwateroperationsmore difficult tohandle.

1.2 L itera t u re Revi ew

Vehiclesmust maintainacceptableperformanceintheface ofparametric uncer- taintyand unmodellcd disturbance. Toachievetherequired robustnessofthe cont rolsystem,adaptiveelements have toheincorporated into the control system, so thecont rolleriscapableof learningaboutaprocesson-lineand modifyitsown behaviors withthe goalofenha ncing system performance (Farbrother andStacey, 1993).Sliding mode,fuzzylogicandneuralnetwork control have the characterls- tics ofadaptivity,andtheirusageinunderwatervehiclescontrolisexpanding.

Alotof work has been conductedinrecentyearsand manyliteraturespub- lished haveaddressed the usage o(more advanced control!ltrate~estogether with the classical controlstrategies inROVsand underwaterrobots toenhance propul- sion systemperformanceand vehicleautonomy (Hinchey,1994 and Muggeridge, 1994).

Yoerg~,elal(1990) testedtwo conventional linear controllersand adaptive sliding switching controller ina computersimulat ionofthrusterdynamics. Results showedthattheadaptiveslidingcontrollerwas effectiveover the entire operat - iugrangeand cancompensate (oruncertainties anddegradationofthethruster.

Yoerger(1990) solvedthenonlinear thrusterresponse ofavehicleusingsmall thrusters for fine maneuvering. Yuh and Gonugunta(1992) designeda neural

net controller for an underwaterrobotic veh icle(URV) using a parallel recursi ve adaptationalgorithm.Computer simulationshowed thatthe controlsystem can provide high performance in the presence ofthe effect of thruster dynamics and saturation.

Yoerger and Slotine (1985) proposed three single-inputandsingle-output(SISO) controllersusing sliding mode control methodology.The influence ofparametric uncertain ty and the lise of simplifiedmodelswere demonstrated in computerslm u- la tion using a non linearmodelof the Universityof New Hampshire'sExperimental Autonomous Vehicle{EAVE).Yoerger,et al (1986) demonstrated two important elementsofthe supervisorycontrolsystem:the close-loop control of vehicle trans - la tion in pooltes ts with a prototypevehicle,andtheinteractive tra jectoryin simulation trials.Tilecontroltechn iquethey used was lliidingmode control,ami resultsshowedvehicleoperationswere made more efficientand simple r.Yoergcr and Newman (1989)tested both basic automaticcontrol andinterac tive supe r' visory control techniquesfor hovering andtracking of the vehicleJASON

nov.

Results showedth a t the vehicle had betterperformanceto interactwith theenvl- ronmentwith the supervisory control techniq uesthanwith theconventionalcon- troltechn ique.Yuh(1990) proposeda mu lt ilayer neuralnetwork controller for a URV, and error-backpropagat ionmethod wasused as the training algorithm, Simulatio nresults showed thatthe controlsystem could respond to changes in tlte vehicle andits environment,and the networkweightscouldbe adjustedto provide th e proper controlsignal.Yoerger,et01(199 1)appliedadaptiveslidingcontrol to theautonomo usbenthic explorer(AilE). Thetests showedthatthecontrolsystem could detectchanges in control gain on multi pledegreesof freedom, automatically re fine the vehicle model until performanceachieved a levelcorresponding to a

prescribedlevel ofmodel uncertainty,thenautomatical ly revert to a nonada ptive configuration.Zheng(1992)implemented layeredcontrol,ontheARCS vehicle and THESEUSvehicle. Test and validation of thecontrolsystem showed that layered controlis a simpleand powerful concept. The quickness withwhich it can he implementedmakesita goodcandida teforpract icalAUV applicatio n.Yuh andLakshm i(1993) developed amultilaye red neuralnetwork controllerusing a criticequat ionforanROV.Threelearning algorit hms:errorbackpropagational- gorithm(EI3P) ,parallelrecursivepredictionerroralgorit hm(PRPE)and modified parallel recursivepredictionerror algorith m(M PR PE)weretested.Effects of the nu mberof layers inthenetwork,thenumberofneuronsin thehidden layer,initial weig htsfor the networkand the critic coefficient were investigate dby computer sim ulation. Results showedthattheneuralnet controllercanbe effectiveinthe presenceof parametricuncertai ntyandrandom noise.Yuhand Choi(1993) have developedtwocontro llersforan omni-directionalunderwater robotic vehicle:a nonlinear controller and a discrete-ti meadaptive controller.Sincethehydrody- namlccoefficientswere poorlyknown, the Parameter AdaptationAlgorithm( PAA) was o.pplied inthe adaptivecontroller.

Allofthecontro lstrategiesoutlined in thissectionhave beenshownto pos- sese considerablerobustnessto parametricuncertainty and arecapable of dealing withncnlinearities.With reg ard to underwate rvehicles,theyhave been tested on lyin compute rsimulations.The muchharderphysicalimplementation has yet to becarriedout.

1.3 Scope of the Work

At Memorial UniversityofNewfoundland(MUN), an hydraulic subsearobot is being developed, The projectwasinitiatedbyDr. M. Hinchey and one of his students K. Muggeridgethreeyears ago.K. Muggerldge designed the mcchnnicnl setup, as shownonpage 14 in next chapter.Due to the lack of completesoftwnre andhardwar e in the control system, this rig could not be testedin n watertank at that time. AfterI got involved in this project,I designed thecontrol system, and tested thissystem togetherwiththe rig in a small watertank.

Rightnow,the robot does not havea manipulatorIUl l1can notdo any kind of work in the water.In the present workwefocused only ondepthcontrol. The robotcan be used later onto gatherdata (pressure,temperature,conductivity, salinity)in shallowwaterif proper sensors areimplemented.The purpose ofthe projectwas to gain experience with thedesign,fabrication,control and operation of a subsearobot.

In designingits propulsion system,Muggeridge(1994) deviatedfrom the ordinary motordriven propellersor thrustersand used an hydraulic system,which includes an accumulator,a storage chamber.an hydraulicactuator and a servovalve together with a plunger/duct system,The plungercontacts the ducttightlyby threeO'ringaas shown on page17, The plungeris screwed onlite actuatorpiston rod andmoved insidethis duct whose diameteris6 inches, Fordepthcontrolthia movementcan change the buoyancyofthe robotor create astrong watnrjet .The long term goalof the projectis to comparethe buoyancy and waterjetcontrol, and the goal of my workwas to check the buoyancyandhave a preliminarylookIII

waterjet.

Compa redwith propellers,ourduetor water jet propulsionsystem has the followingadvan tages and disad vantages.

Advantages

•Itisphysicallymuch sim pler and easiertoattac h on ROVs androbots than propellers.

•Its powcrsupplyishydraulic oil. There is no need for DC motor.The size of thetether can be reduced,together with the nonlinear effectscausedby the tether .

•When opera ting,it has much smal lervibration than propellers.

• Degradat ionof performance causedbythe cavitationisgreatly reduced.

•Cleaning oceanappendages ismuch easierforthe ductthanthe propellers.

•WheI'operating , itissaferthanpropellersinshallow water.

Disad vantage

•Therobotat th is moment can only move vertically inthe water,thus its maneuverabilityislimited.

•Itsefficiency isusuallylower thanpropellers.Thethrustgener atedby pro- pellers isalways higherthanother marinepropulsion systems.

•Inour project,the amountofoilin the accumula tor det ermines thetask duration .Presentlythis is muchshorterthan propeller operatedsystems.

10

• Due tothe high pressure indeepwater,thevelocityofwat erjet canbe reduced.Thus the devicein deepwaterwillnot performas efficientl yasin shallow water.

As for the controlsystem of the robot, weused conventionalcontrolstrate- gies. These strategies give goodperformanceandhave beenwidely usedinindus.

triesfordecades. Thestabilityofthecontrolsystem was checkedusing Nyquist stability criterion and nonlinear describingfuncti on.Thetransient response and frequency response performancewas investigatedinsimu lationandtest.

1.4 Outline of This T h esis

Thisthesis gives a detaileddescriptionofthe cont rolsyste mforourrobot Including its computersimulation,stabilityanalysisand realtimecontrol andtestin a water tank.Itcontainsfive chaptersincluding theintr o duction.

Chaptertwointroduces the dynamicsequations of therobotanddemon- st ratesclassical controlstrategiessuchaspropo rtional.integral-derivativc (PID) and proportional controlof buoyancyincompute rsimulation.Thestabilityis im- alyzedusing thenonlineardescribingfunction with the Nyquist stabilitycriterion for buoyancy andwat erjet devices.

Chapte r thr ee details the componentsofth erobot andtheequipmentneeded to testit ina wa tertank.Proceduresfortesting the hydrau lic system usingpropor- tionalcontrol acti on and simulationarepresente dnext.Stabilityis examinedhy Bode diagram ,andthebuoyancy and wa tcrjet testsinthewater tankare described

11

respectively.

Chapterfour discussesthe advantagesand disadva ntages ofthesystembased onthe resultswe gotfrom thecomputer simulation and theexperimentin the wntertank.Potenti almodifications tothe setu p to lessenthe disadvantages are elucidated .

Chapte r fivegivestheconclusion from the project , andillustratestheexpe- ricncewehave gainedthroughthis project, aswell asourfuture works.

12

Chapter 2

System Simulation a n d Stability Analy sis

2.1 Robot Description

Theconfiguration oftherobotis a cuboidshape withmainlyn supportingIrnme, twocylinders, an acluator, ll servovalveandac1Ucl/wntcrjcl deviceasc1t~pid L..1in Figure 2.1. The robotcanbe expandedby instal lingalillitiollll!components.Tim supportingframeisweldedtogether,andismadeof aluminum.TileIlinll:ns inll(If the rigis O.6m inlength,O.45min widthand107m inheight,nnl~the wdJlUis approximate ly160 kg.

13

nitrogen chargingvalve

Figure 2.1:Configura tion

o r

^therobot

The two cylinders,the actuatorandtheservovalvemake upthehydraulic system.The shorteroneofthetwocylindersisanMTSaccumulato r.Inside,when fully charged,ithas 3,000 psihigh pressureoilundernit rogen gas. Thenitrogen ischarged througha smallpneumaticvalveontop of theaccumulator. Theoil is usedtoactivatethe pistonrod ofthe actua to r.Thevelocityand positionof thepistonrod are controll ed by theservovalve. After theoil goes throughthe actuatorand servovulve,it goes to the longer cylinder. This cylinderisjust nn emptychamberwhose purpose isto store thelow pressureoilwhichhas passed through the actuatorand servovelve.We callthis cylin derthe storage chamber.

ThereisalOmm diamete r hole withthread.ontop of thestoragechamber.Itis usuallykeptclosed by aboltexcept whenthe oil is drained from the cha mber,in which case the bolt istakenoffsothat theair pressureinside thischamberis the sameas the air pressureouts ideandthe oilcanbe easily releas ed.As moreoil passes through the servovalve, the pressureofthe oil left insidethe accumulator goesdown. When the minimumof lOOOpsioperatingpressure isreached, the system stops to work. The oilinside thestoragechamber needstobedrained,and accumulatorneeds tobe recharged.

Theactuatorisplacedbetween theaccum ulatorand the storagechamber, withthe servovalveat tachedonit.Theoilfrom the accumulato rto the actuator and from theactuato rto thestoragechamberpossesthroughtwo rubbe r pipes andeachpipeis switchedbyaball valve.Thesetwo ballvalvesare only opened whentheexperiment is in process, so theoilcanbecircu lated from theaccumu- latorto thestoragechamber via the servovalve.Otherw ise,the twovalvesare usual lyclosed.Thereareanot her twoball valvesconnectedto'hebottoms of the accumulatorand thestoragechamber respectively. When the ballvalveconnected

IS

to the accumulato ris open, oilcan bepressedinto the accumulator froma pump and after the pressureof oilreaches3,000psi, ithas tobekept closed allthe time.

The ballvalve connectedto the storagechamberiskeptclosed unlessoilinthe chamberneedstobe drained.Allthe fourball valvesare designed tohold3,000 psi oil pressure.

A plasticductis assembled onaflatplate by severalsmallbolts andstuck to itbytheeealent Silicon. The flatplateis made of aluminumand is Hxedby four connecting bolts atthe bottomofthe supporting frame. Its inner diameteris 6 inches andthe lengthis 15 inches. Theplungeris 2incheshighwith a diameter slightlyless than 6 inches. It isalsomade of plastic.The plunger is tightly contacted withtheduct by three O'rings, andit is drivento moveinsidethe duct bythepistonrod.The plungerseparates the plast icductinto two parts.Above the plungeristheupperpart whichopenstothe water.Belowthe plunger is the lowerpar t and onlycontains air. Forbuoyancycontrol tests , tygontubingattached toasm allventhole is usedto ccnnectthisair to atmosphere.When thepiston rodmoves, the amountof air and waterinside the ductchanges,and so does the buoyancy.Withthe variationof thebuoyancy, the robotcan move verticallyin the water.

Fo r waterjettests, a 1.5"diameter holewas drilledinto the bottomplate.

When tileplunger moves inside the duct,therobotcanbe drivenby thethrustof thewa terjetthroughthis hole. Theductwith theplunger is shownin Figure2.2.

16

plunger

---

I .'"'

Figure 2.2:Ductandthe plunger eJring

d....

2.2 Computer Simulation

Testing an ROVoranunderwater robot in the oceanoreveninawatertank is difficultandexpensive. Simulation isaversatilemethodthaiprovides for diagnosis and correctionof system faults.Inorderto ensure completerelia bility,thecontrol strategiesand software needto be tested in computer simula tio nbefore therobot is deployed in water.Thecontents ofthissection demonstrate the use ofsimulat ionas an approach fordesignin g,developing , evaluat ing and comparing differentcontrol strate gies andsoftware.

During deploymentofthe robot inawater tan k orseawater. the robotshould be in a neutrally buoyant state,beforeany action takesplace.For thebuoyancy control teststhis wasmade to be so whenthe pistonrod of the act uator was atits middleposition(halfwayin and halfwayout ). Whenthe piston rodstretches out fromitsmiddleposition,theplungermovesdownwardssothevolumeof the upper partincreaseswhilethatof the lower part decreases. Partoftheairinsidethe duct ispressedoutto the surface bythe plung..sand morewater comes into the upperpart.So the buoyancyforcebecomesnegativeandtherobot sinks.When the pistonrod movesin opposite way,volume of thelower part expands andmore air is drawn fromthe surfaceandsome waterin the upperpartis expelled out.So, buoyancyforce becomes positive,and therobotfloatstowards thesurface. The huoyant force is changeddue to the movement of the pistonrod .To hoverat ncertaindepth,the pistonrod must stayatthe middlepositionagain.For the watcrjettests, atrial and errorprocedurewas usedto getneutralbuoyancy.

Newton'sSecond Lawwas assumedto governthevertical movement of the

18

robot.

~x

^{(Fa+FJ -FD)! ,\-[}

f,X =

V

~V

^{(Fa+FJ- FD)//Il}

whereXisthe depthofthe rob ot ,Vis the velocity,FRis the buoyancy force,FJ

is the jetforceandFDis the drag force,Mis the muss. In the setu p,FB and FJ

werenot active atsame time.Fs ,^FJandFDcanbe expressedby:

FJ M·U

FD

~P '

CD

, Aro~l

^'V²'sign(V)

A"'udis the ductinnercross area,herewe assumeis thesameas theplungerarea, LrorJ.is the pistonrod position,whichbeing meas ured relative to the neutralpiston rodposition,Ar<>hOIis theareaof therobot facingthewa ter inthe movingdirection, pisthe waterdensity,9 isthegravit a tional acceleration, andA-fandUnrc the mass flow rate and velocity of thejetrespectively.

Thesimp lestnumericalintegrat ionprocedu rewas used10 marchthesimu- lationforward intime(Muggeridge and Hinchey1992).For atime step 6t:

X(t) +t>,·V{' ) 19

V(I

+" ,) ~

^V,')

+ ,, <.

-dd VU)

,I [ . I I ]

diV,')~ FH,')

+

FAI) -" .CD'A.•••,. V·IVI1M (2.1)

Inthe followingpar~ofthischapter, weare going touse severalconventional cont rol strategiesandcompa retheireffects. From these, we canchooseonewh ich givestile best performance.

2.2.1 ProportionalBuoyancy Co nt r olScheme

We Ilrst lise the proportionalcontro lschemeto evaluatethe robol performa nce in the waLer. For a controllerwith proport ion al contr olaction,therelationsh ip betweentheoutpu t ofthecontrollermel)andthe actuating erroreft)is

m(I)~J(.'(I)

or,inLaplece-t raneformed quantities,

whereICpister medthe proporlionalsensitivityorthe gain.Theblock diagram of aty pical proportionalcontr olleris show n in Figure2.3.

Figure 2.3: Block diagramof a proportional controller

In ourCI\8C,thecontroller outputisthe servcvelvesignal,andthe error signalis the difference betwee n the commandpositionwhere the robot is supposed

20

tohover andthe robot 's actual depth . The blockdiagram ofthe whole system withproportionalcontrolscheme is shown in Figure2.4.

Figure 2.4: Block diagram of the system with proportionalcontrol scheme

In Figure 2.4, the innercontrol loopistheblockdiagram ofthe hydraulic system which positionsthepistonrod .This rod positioningcont rolschemewill be discussedin nextchapter. Here,we can simply consideritasagain.Kisthe overallgain of thewholesystem,J(Fis thefeedbackgain.

Before the simulation,we assumetheviscousdragcoefficientCDis O.Sinour program. Alltheother parameters such as massM,plungerdiameter,piston rod strokeand cross area^ArQbolcan be measuredfromthe robot.We assumethe robot startsat a certai n depth and moves towardsanotherdepth,andstopsthere. We call thesetwodepthsthestartdepthand theend depth (in ourprogram theslart depth is 2m andenddepthis5m).Weput all thesedataina file,togetherwiththe proport ionalgain which we need to changeto evaluatethe performance. Whenthe programexecutes,itfirst reads alltheinput data from thisfile,aftercomputa tion, it writesits output (depth, velocityandtime) to another file. From this file, we

21

canuhsurvc tile trajectoryofI,OWthe robotreaches 1I1c enddepth,and t.he timeit

1I1~~lstoreadlthis finalspot. Wilen itreachesthis spot, the robo t should hefully

lIlolill"l e:;.~.ThisHIllYre sult endless com pute rcomputationandmakesthe output filefairly large. 111ourprogram ,whentiledifference of theroho t's posit ion and final spotisless thanO.0 5m and rohot's velocityisless tha nO.02m l s,we assume tIll! robo treachesllll~final spotandismotionless~The simulatio nprogram with proporuional contro lscheme is inAppen d ixA.

FollowingMI)UUlplot.s of velocity versus depth andtiledepthversustime.

Theproportionalgain weuSC(1was 1,0.Inprogram,we set lime stept:.lto 0.0015, wllidlill H1it!tillll: tf:St, is 1101.exactly thisvalue.

cquntions. Themethod doesnothave good accuracy and stahility.Toha ve better nccurncy,WI)dlOose step lIii(l)6/ a verysmall value,0.00 15.Alt houghtheaccuracy wouldIu:lost overulo ng timeperiod,itwillnet affecttilesimulatio nresults, dilrl)rellt cont rolschemes couldst.illLccompared.

From Figure2.5 fwdFigure2.6,we noticethattherobot oscillat esat the 511I 1111pt.lI,withitreducing aillplit lllieuuulilslo psutthe511I.Thedee pes t spot itreachesill5.75711, 0.75mdeepertlmnthefinaldepth,wename thisvaluethe m'l~r~hoot.Tilerolwt llf!cllsaboutI[..l~tcrcuchther'nmmaur]spot,wename this timethe setup lime,FromFigure 2.5,weIindthemaxi mum speeditreadi es is OAm /~. Hyillcrcil~i ngtheproporlionalgaintodifferentvalue,thelim efor the rohotto reach theendI]cllthbecomes sho rte runtiltheproportional gainequals 1,0:!O,thenumberof oscillationand the setup limeis minimum ,9.5s ,as shownill Vigun : 2.7 and Figure 2.8.

22

Figure 2.5: Variation of robot velocity when proportionalgainis I

Figure2.6:Traj ectory of robot whenproport ionalgainis1

23

Figure 2.7:Variation or robot velocity when proportionalgain is 20

Figure 2.8:Trajectory of robot when proportionalgain is 20

24

Wc noticcfrom theaboveplots,thatno mutter whattheproportlonnlg" in is,therob ot alwaysreaches5.7fJmandthenCOllll~Sbark.Sothe propo rt hmn lgain docsnotaffcctthe overshoot.Inorder tof(~ducetheoscillation~Ilt.11t~1'1111IIt'pt.h,or to reducetheovershoo t Lominimum,wearcgoillgLotry illloLlwrl"ulIt.rollltr.ll ,,!,!;y, the PIO con trolscheme,inthe followingsection.

2.2.2 PIDBuoyancyControlScheme

Tileana logPII)controlsystemhasllC'ellIIst'll successfullyintuauyindu sl,rial controlsystems (oroverhaira cent ury. ILhit.1wc~olUhillll lionflfpro[lo rtiuII1l 1 contr olacti on,deriva tive contro lectlonaudiUlc'g rillClmlrnli1t:t ioll.'I'het:olllhillt:tl nctlou hastheativall lilges ofeachofUwUll"l~t)iudividuu l contl"ullId-ious.'I'he equationofitcon troller withthis combin edactionisgivenhy

m(l}

= u,

[e(l)

+ 1 ~ L

^c(t)dl

+

^'1;1

d~l~t)]

or thetransferfunct ion is

where e(i)is theinputto thecontroller (theilclu~ltingerrorsignal),mel )i.~llrt~

rmt pu t ofthecontr oller[themaulpulatiug signal), /(,, isth{!prollort iollillgaill,T;

is the integralLime (orresettime), and'/;,islilt:d(~ r i va ti vt:tirrlll(or ratl:tillw).

The blockdiagramofarIDcont roller is showninFijJ;u l"l: 2,9.

We lisevelocit yformPIDcontrolschemeill0111"SilllUlillitlllprogralll(Og'Llil 1(87), Toder ivethevelocity formPIDcontrolequa tion,considerlIm1",. :kwlll"11

25

M(a)

Figure 2.9: Dlock diagramof PID controller difference inm{kT),thatis,the differencebetweenm(kT)andm((k - I)T):

where

6mlkT) m(kT)-

m ilk

- liT )

K{' lkT )-'I(k -liT)

+ k

^l«kT)

⁺

^{'Ilk -l)T )1}

+¥

['(kT) - ',«k- I)T)

+

^{,((k - ')TJI)}

J(p [,(kTI-'I(k- I)T)1+J(,'(kT) +J(D[, l kT) - 2,((k- I)T)

+

'1(.\-2)T)]

J(p==K -

¥

^==J(-

~

⁼proportional gain K1==

[~~

⁼integral gain

[<D

= J <;<l =

derivativegain

Thesimulation program of PIDcontrolscheme is in Appendix A. We add nnothcr two parametersin theinput data file, the integralgain K; and derivative gainKd.First,we randomlychoose three valuesfor threegains,10for prope r- tiounl gain, I forintegral gainand 10 for derivativegain, nnd observethe robot's performanceinthewater,Thistime,itstill moves from 2m to Sm. Thevelocity

VB.depth is inFigure2.11 and depthvs.timeplotisinFigure2.12.

'6

Figure 2.10:Block diagramof thesystemwithPIOcontrolscheme

Figure2.11:Variationofrobot velocitywhenJ(,:::10,J(o:::1,!(d:::10

27

Figure2.12: Trajectoryofrobot when [(,

=

10,J(,

=

1,[( d

=

10 FromFigure 2.11and 2.12, we findthat the setup time is 12.58and the numberof oscillation has not been reduced. Thus we need to adjust thethree gains to see if a better performance can be achieved.

Wefix the integralgainJ(,andderivative gainK~,and increase!(pstep by stepandwe watch the performance. Whenit reaches200, we findthat as Figure 2.13and 2.14show,the overshootis zero and setup timetheshortest,1.32s.So the robot will not need to oscillate around the end depthandthe time neededto reach the finalspotis muchreduced.

We fix the proportionalgainK,and thederivativegain[(d,changethe integralgain [(" what we gotwas eitherlonger setup timeor largerovershoot.We thenfix the proportionalgainJ(,andintegralgain[(" and changethe derivative gain [(4,to 1,wefound thattileK~does nothave mucheffect in the robot performance.The plotswhen [(,

=

200,[(j

=

1andJ(,j

=

1areinFigure2.15

28

Figure2.13:Variationof robot velocitywhenK,=200,[(I=I,1\"<1=10

• ·f ··· ···· ···

!"f ···.' ·/

O.B

_ .,

0,8

Figure2.14 :Trajectoryofrobo twhen 11.·,

=

200,11.";

=

I,1(1

=

10

29

and 2.16.

i _{f O ,2}

,

0.15

35 •

d,plh(ml

\

Figure2.15:Variation of robotvelocitywhenJ\p

=

200,A";

=

1,A"d

=

1

Figure2.16: TrajectoryofrobotwhenA"p

=

200.K;=1,KJ=1

From theuboveobservationof performan ce.wecanconcludethn ttherID

controlstrategyoutperforms the proporti onalcoutrol s1ratl'ltr.TheIWrli.lr1u1\1lt'l·

in PIDcont roldependsontheproper choiceofIhi'thrt,l'~aim",1\,..1\',and1\'", Followingsection willdiscussthe systel1l'sstnbilltv

2.3 Buoyancy Control Syst em St ability Ana ly- sis

The simulationilltheprevioussectionisjus tlikedillcrcut kinduftl'St ,~,FnuuIhl' resu ltsofthesimulation , weknowthe controlsystemis,~I nbl<',SilW"1\'"\\"11111III ana lyzeitsstabilityfromitsfrequ encyresponseperfor manc e,\\"1' USt'.Ill':'>!Yllllist stabilitycriterion,

2.3.1 Nyquist Stability Criter ion

Forsta bil ityanalysisofthesystem,we usc Nyquist »rnbilitycnturioust:IIl't!below. This crit erion isusefulincontrolengineeringhl'caHSIllheubsnlntest ab ili1yof IIll' close-loop syst em canbedetermined graphica lly from"IWn-l u lJ]lFrequencyn~p(lllSt' curvesandthereisnoneedfornctunllvrlctennininuthednS1~-I(J(J]l l'lIl,'s.

Nyq uiststahili tycrit erionstates that intill'systemSII/lWIlillFi~lll'l' 2.17, iftheopen-loop transferfunctionG(..;)H (..;)!IBSkpoll's intheri~lll- h:df.\plms-, then forstability theGtj-.;)H(j;,;)IOI:llS,as ;.;variesInun-x.tu ox"must_{"l ldr,.],~}

the-1

+

jOpointktimesintheeountereloekwisellin 'I'l itllJ.

Tounalyzethestability ufthe sysl'·IJl.\\"1' firM plotIlwsysn-mJ,]Il(:kdi;r~',:alll

:H

C(.,)

Figure2.17 :Closed-loopsystem

whichisthesame asFigure2.4 on page21. Dut here weconsider the robot as a dampedsecondorder system.Its dynamicequationis

whereIIIis robot mass,Ris the actualdepthhere,bis acycled averageddamping rnellicient andFis force.Very roughca lcula tio ns giveb=25

~~

^{s' Afterthe}

Lapl ace transformation.it becomes (m52+bs)R=F

Theblock diagra m ofthewholesystemis shown in Figure 2.18.The inner loopisthepistonrod positioningsystem.

IIIFigure 2.18,KG is theproportionalgain,Kc,istheLVDT gain(t hese two~(ainsarc givenindepth.h filein Appendix B),1(4,is the amplifier gainand K ..illthegainofthe ecrvovalve,Tis thesetup time. The prod uct ofI\'G,K,A and h'scan he expr essed as

ic.

T1Lissubsys tem ofthe rodpositio ning sys temca n be reducedtoits transferfunc tio n.

_G_= ~/(l+ i-""KL )= Ii· _ 1

+

GH .'1(Ts

+

1)

«r, +

1) .5(T s+1)+l\"KL TIl\' subblockBUOYinFigure 2.18 converts actuatordisplacement tobuoy"

32

Figure2.18:Blockdiagra mofthe buoyan cy systemfor stabili ty nnnlvsis nncy force.Itcanbe considerednsItgain which is the product ofthe watord"lIsity p,plungerareaApI~..^gorand gravitati on al acceleration9.We canobtniuils valllt' from the following equation.

BUO Y

=

^p.Api...,.. '9

=

^{lOOOkg/m3x O.0l824m2x}9.8IN/kg

=

179N/m

Figure 2.19:Reduced blockdiagram ofthesystem

The block diagram of the systemcanhe reducedlISshownisF'i~ur( ':UU.

Its open-looptransferfunctio nG(s)H(s)is shown111Equatio n 2.2.

!7!.lxKt\"fIF

s('f'msJ

+

(m

+

-;rb )s'+

(-" m-"/i'-'-f{~,,-+~b)-'-+~bl'(~f{~d

By rc:plild ngs withjw , we gel the following cqu a tiou

(2.2)

G(j w )lI(j w )= . 179xgRIIF •

jw([(mKJ(L

+

b) - Tmw'!jw

+ I b /l

^{/{L -}(111+'I'b)co2]) Wlu:1Ifd( lit,-(IU+'J'b)w'

=

0, G(jw)lI(jw) locushasrealaxiscross-over,amiit occurswhen

SoUUw)fI(jw)hC:nlll ll:s

G(jw)lI(jw)

=

179 XI\ [.; fl'F ."

_w

1 [(llIk

^J{/,

+

I,) -'I'm x

;~I/~/~/:b]

Since

illpositl vc,1I0thecross-overoccu rstonega l ive reidaxis.

WenowcvnluetclI11J vahrcaofthoparamet ersinEqnalion 2.2. \Vcfirst

;ISSll Ill Ctileoverall gainIIis0.1, andbyexperience,setuptime Tis0.2s andbis

25.l\isthoproduct of/{G,I(Aand/(5,anditisgiven avalue0.2onpage 66next l:lIu]ller.lIf'is thegail.' ofthesystemfeedback,whichcallheobta inedfromthe cxpe rimcnLnlsetup.Tilevalueisa.3.1m/Jl m.misthomassclthcrobot. Because ofiunrtiaofsurroundin g water, robot appearsheavier tha n it actuallyiswhich we callit.ad dc:dlllilS lI.The added mMShereiscstim.\tedtobecuehalf oltheactual

I\li\ Sll15Skg,so III is 237kg.Ill,is theLVD T gain ,which is1/0.0254

=

39.4 V/m.

From theabovevalue,thoopen-lo optransferfun ctionG(,s)lI(,s)bec omes 3·1

Thelrllw.;fl ~rInnctioulmslourlHllt's, U1WisHIlilt'(,rig;ll,111l'ollu-rtll\'I'I':11'0';11 -0 .1055,-2.'168+5.7275jand-2.'168-5.7275j('l's prd i\'d .v.SinceIlu-renrcIIIl poles 011theright-halfoftheG( s)ll(.~)plunu,theG(jw)ll(jw) lor-us shulIllllint en circle the-1

+

jOpoint ifthesystemisstnble.

Dyexaminingthe G(jw)H(jw) locus,we lisethe ControlToolboxillI\lallall, a munerienlcomputat ion]liH'];:I P;t~ ,to1'1111theNytillist1:111'\'('11.Givillg",lill'"l't'1l1 rungeflffn:qllcllI:iesw,we/;cLIoru-G{jw )lf (j w l lul ~iill Fi/;Hrl'~.:!n

Figure2.2ll:G(jw)F1(jw)lodillt1H~Oll)l11111l~when(,'(H}If( JI)

=

1.I/~{48.~:l

+

242s~

+

18D29

+

197)

FromFigure2.20,weIiudthattheG(jw)ll{jw ) locusdOl 'S1I0 t(!w:in:l.!till!

-1

+

^{jO point Thesystemis}stnble for tIL()0PC/1. loOllfuuctinnofG( Ii)/l(S)

=

1.1/ 11(4883+242s²

+

18029+1(7).Siller.the oVt'riIll gain1\iii('Slil lll1l t;( I,Llw n.':>llll maynot be nccurnto.Now we cheek tilestuhllity1mgivingal1i1f.!rt:lllvalve ofg.

35

Thistun«weincrease theWdll from 0.11.0 InO,and~clnllolllcrG(jw)H {j w)locus inFiglll't: 2.21.TbeG(jw)fl(j w)10(~llsdocsnotencircletho-1

+

^{jO point},the

':r ' E '" , ' =[ 8 ] "

Z : .

II '; ;

I ' . r -• .... f

^e

r- .

·05

+ .

.soo.f- , ;

:/00 ^'50110011""10° 50 "tryOO '50Ro.1lA'i'0 50

Fil-\llW2.21:G(jw)f1(j w)lodiuthc GHpluuc whenG(s)lI(s)

=

!1ll0/ s(IISs³

+

2·12.~2

+

18!)28

+

Hl7)

NolV WI!dlllllgC!/11eovcmllguiu[0200,alld theG(j w)H{jw)loc usisshown ill Figure 2.22 .Fromthisplot, we findthelocus encirclesthe-I

+

jO point twicc doc:kwii':iC', lmtill:systmubecomesuustublcat large gain.

III 01"11(,1'tomakethesyste mtohestnblc,the gain Inctcrill theopen-loop transferInuctionG(s)H(s )must maintain atlow value.

WeIIS()Itouth's stn bili ty criteriontochecktheresult. The characteri s ti c eqnnfion is48.~"

+

2"2.~3

+

I892s²

+

197s

+

^II=O.Allthe coefficientsarcpositive.

Theurruyofeoolllclcntsis

[EJ , " ", ill'

J

^{T I '}

I, r r,a r

:10; .100_ -,-,0 IDO ~ .,00_ _ 0 '00

' B "Ml

I~ '- ' C LlJ

"~I ^·1,'~IAJtI '05 ⁰ -O'!a ·' AuI ...O , Figure2.22:G(j w}H(jw)loclintheGHplanewhenG(s)H(.~)=2220/.,(·18.,3+ 2425'

+

lS92.!'

+

197)

J4 48 1892 11

,,3

^{242 197}

$2 1853 11

,,' 240

,,' 11

Routh's stabilitycriterion state"!thatthe numberof roots ofthe chafi\("If'ristj("

equa ti onwithpositive realpartsisequaltothenumbe rofchangesinsiJtIl"fIll"

coefficientsofthefirstcolumn of thearray.Sincethereis110slgnc11mLJ;(~illtlll~

first columnin ourcoefficient arrey,thusthere isnoc1osc·lclfIll]loll'intill '1!!ft·llIllf splane.Soth e system isstable,

37

2.3.2 Non linear Describing Function

WeUlIIll\ll.vuselineardifferential equationsto simplifythesyetcme.Actually,all

physical systemsare nonlineararid havetime-va rying parametersin some degree.

Tlil:rd ore ,we havetousc nonlinearcontrolstability criterion together with the Nyquist criterion.

The importantdifferencebetween nonlinearcont rolsystemsandlinear con- t.rolsystemis that inlinearcont rolsystem, the shape of the time responseis il1(lependentofthe size of theinput or initialcondition, and stabili ty,orthe lack (If it , isI~propertyof timsystem,In a nOlllinl!1Irsystem,on theotherhand,the natureoftim limeresponse,and in factsta bility,is usuallydependent on theinput.

or initialcondition.Newfrequencies,harmonicsand subharmc nics, of the input frequencies andlimit cycleswhicharcperiodicoscillation s of fixed frequencyand amplitude, aregeneratedbynonlinearcomponents (Ogata1970).

One way to analyzea particulargroup of nonlinearcont rolsystems,in which the degreeof nonlinearity is small, isto use equivalent linearizationtechniques and to solvethe rcsulting linenriscdprohlern.The describingfunction methodisone ortheeqnivalcntHncariaatlonmethods.

Nonlincaritlcausuallyrefertosa t ura tion, hysteresis,deadzone,backlash and statk orcoulombIrlction,etc. Inour case, robotbuoya nt forceis proportionalto thepistonrorl posilionwithin thepiston rodst roke,<1inches.The piston rod position is proportionalLo thesignalwesend fromthe computer. Any signal which movesthe piston rodbeyond its stroke will notbeachieved, andwillonly resultillitsstroke. So thecommandsig nal we send from thecomputer is sat ura ted.

38

Thus thesystem is subject to saturationnonlinearity.

A typical input-outputcharacteristiccurve forthesat uration non!incnrity is shown in Figure 2.23.Forsmall inputsignals,the output of nsaturation element is proportional 10 the input. Forlargerinput signals the output will not increase proportionally,and finallyfor very large input signalsthe outputit;

constant,Figure2.24depictsthe input and output waveformsforthe saturati on nonlinearity.

Output

-5

-

^{Slope k}

⁼

^I

Figure 2.23:Input-out putcha racterist ic curveforthesaturationlIoulill,..arily

In Figure 2.24,theinput to the systemis sinusoidal. TILeout put isnut sinusoidal.In the describingIunctiouanalysis,only the fundamentalharmonic componentisconsidered importan t,The reason is that higherhnrmonicsolten IIlIV'!

39

C

Yt(t)

=

Yjsin wt

Figure 2.24:Inp ut and output waveformsforthe saturationnonlinear ity

40

smalleramplitude tha ntha t of the fundamentalharmoniccomponent.Also most cont rolsys tems arelow-pass filters,withtheresulttha tthe higher har monicsare very much attenuat ed.The descri bing function orsinuso ida l ll(·s crihinl!;funct ionof anonlinear eleme ntis defined to hethe co mplex ratio of the fundrnu cntn l harmonic component oftheout putto theinput. That is

where

N

=

describingfunction X

=

amplitudeofinput sinusoid

Y1

=

am plitudeofthe fundamentalharmonic co mpone ntofnutput

~l=phaseshift of thefundamen ta l ha rmo nic componentofout put

Incalcula t ing the describing function for a~~;vennonlinearelement, welWt..'i1 to findthe fundamental harmoniccompon entof the outpu t. Fur thesinus oida l inputx(t )

=

Xsi n wt tothe nonlinearelement , theoutput!I(! )llla.ybuexpressed as aFourierseriesasfollows:

y(t )

Ao +~(An Cos nwt +Dn Sill nwt)

Ao+

~

^Ynsin(nwt+

(~n )

where

tin

= '; t"

y{t)cosTlw t d(Wf) En

= ;' t"

^y(t)s i n nwt d( wt)

r, =

.jA~+D~

41

Ifthe nonlinearityis symmetric,thenAo⁼O. In Figure2.24,the outputis anoddfunction.Foranyodd function,wehaveAll

=

^O(n

=

^0,1,2,"') Hence, tJu~fundamental componentofy(t)is

YI(t)=B.sinwt where

ill

= ~ ["

y(t)sinwtd(wt)

= ';

^[ Y(t )sinwtd(wt)

The describing functionforan elementwithsaturation Callbeobtainedas

forX>SandN=k forX<=5, whereX isthe amplitudeofthe sinusoidal input, k:andScan befound in Figure 2.23.

Thedescribingfunct ion can be usedfor stabilityanalysis ornonlinearcontrol system.The system block diagran:withsaturationnonlineari tyis showninFigure 2.2.5.

~ _.~

Figure2.25:Systemwith nonlinearity

42

whereN denotesthedescribing function ofthesatu ration uo nliuear ity,GHisthe open-loo p transferfunction. Thentheclosed-loop frequency responsebe...-omcs

R(j w) NG(jw) HU w) e(jw)=1

+

NG(jw) H{jw) Thecharacte risticequatio nis

1+ NG(jw) H{jw)=0

G(jw)H(jw)=

- N

In thedescribingfuncti on analysis,the convent ionalfrequency response nnnl- ysis is modifiedso thatthe entire-l INlocus beco mes alocus of critical poi nts.

Thus, therelati ve location ofthe-l INlocus andG(j w)H(j w)locu s willprovide the stability information.

The criterion for sta bility is that if the-lI Nlocus is notenclosed hy the G(jw)H(jw)loc us, then the systemissta ble,or thereisno limit cycle at stead}' state. Ifthe-liNlocus inenclosedhytheG(j w)H(j w)locus,then the system is unsta ble.Ifthe-lINlocusandtheG(j w)H(j w)locus intersect, the system output may exhibita eust .ained oscillation , orlimitcycle.

To analyze thestab ility, we first sketchthe- li Nlocu s from Equation 2.3.

Sincethe valuesofslopekandS are constant,-lI Nisfunction uf onlythe amplitudeof sinusoidal inputX.When X<=S,the-lI Nis1.whenX>S, and we letXcha nge from S to00,wefindthatN in Equat ion2.3i~Immk to 0, so the describin gfunction-li Nis from-11kto-00. AsWI!can makethe slopekin Figure 2.23 to be 1,thus,the-lINisfrom -1 to-oc 011Reaxis ill

G(jw)H(jw)planewhen input X is from0to00.

Weplot-lINlocus as X changesfrom0 to00inFigure 2.2G.Atthe same time, we alsoplottheG(jw)H{jw)loci we got fromFigure 2.21,and usethe freq uency range of-00to00.

Lm

R,

Fig ure2.26: Plotof-lINandG(jw)H{jw )forstabilityanalysis ofbuoyancy system

From Figure2.26. wefind thatthe-lINlocusis tothe leftof theG(j w)H(jw) locus.sotheG{j w)H(j w)locu s docsnot enclose the-l I Nlocus or int ersectwith the-lINlocus. Asfo rXin range0 to S,-l I N

=

^{-1,the Ny qu istplot docsnot}

incl ude-li Nlocus, and thus does not encircle-1

+

^jOpoint.So,iflin ear syste m is stable,sa turationwillnotca useinstabilityorlimit cycle.We cancon cludefrom

Figure2.26thatthe systemisst able,a\HIanyosclllutlonswhichmay occurilltill!

systemoutputas aresultofdisturbanceswilldieantand lin sustnincrloscillation willexistatsteady state.

2.4 W aterjet Stability Analysis

Forthewaterj ctcase, theblockdiagram is as showninFigure 2.27.TIIe.lblock conve rts rodmotiontoforce:

' - - - -- - - -- IKp.I - - -- - -- - -

Figu re2.27:Blockdia gramofjc~systemfor lllnhilityi1l1l\lysis

whereJisItconstan t. Forlinearmot ionaboutItcommandposition,,1

=

0, Pro ceduressuchasNyquistshowthatinthis case, 111l: systemisIllJf(lt~rli lll~ Sl;lblt~.

Using anonlinear contro ller,onecouldgetItdescribingInactionNor(:Irl:cliw:./

whichin nonzero,By roughcomputationandexperience,weest imatelilt:vallie

or

Jtohe 50.The open-looptransfer function0(3)/1(3)ClU Jhe (:xpmssl:dliS:

Jxl\lO(r s{m.~+b)(Ts+1)

l.G5

2378³

+

121Os²+1255 (2.4)

Till:Nyquist plot of thetransferfunction 2.4is shown in Figure 2.28.The pn]l:soft111~oJllm-l(JflptransferfunctionG(.~)ll(s) n~.eattheorigin,-0.1055

+

^jO

alld-fJ+jOpoints,there nrcnopoles intheri~hthnlfofs plane. Sincethe Nyquistplo t,Inesnutincludethe-1

+

jn point,the systemis stebte.

Figure2.28:G(jwlH(jw)lociilltheG1T plan ewhenG(s)H(s) ==1.65/ 237ss+

12I1Js~+12[;s

When tile overallgain11."und.1arechosentoheverybig,theNyquist plot wouldlooklike Figure 2.29.Theplot encircles the -1

+

jO pointtwiceclockwise, sothe system isuustnblc atverylarge gain.

Like the buoyancy system,thewnterjctsystemalsohas saturatio nnonlin- 46

Figure 2.29:GUw)H(jw)lociill theGlI pl1111e whe nG( s)ll ( .~ )

=

IGr;Il/~:I7.~~+

12 1Os²

+

125.~

earitycharac teris ti cs .Thedcscrihiug functionNis slill l1lf'sntur:nil IIII' bIlIlY:IIWY oneon Page <12.

The-lINlocus stillchun gcefrom-1point to -00 usallllllillldl : Xl~h;\lIW'd from0to 00.SincetheNyquist plo t docs1I0 lcncircle1I1c-I

+

jOpIJilll,iLwillnut include-lINlocus .Iftholinear systemis stuble, thensuturutiouwillnot eaUMl any inst abili tyorlimitcycle. The-lINlocuswith1I1c Nyquis t plot un:sbowuill Figure2.30.

Thischapterhas givenabriefint ro d uctionofthe robot(:llllfj~lI rati(1lIalHl lllls dis cussedbuoyanc y dept hcontrol withtheproportion alcont rolSdlelll l:nullPJI)

47

1m

G(jw )H(jw)

--..

R,

Figure2.30:Plot of-lINamiG{jw)H(jw)Ior stabilityanalysis ofwaterjetsystem

cont rolschemein computersimulatio n.Theresults show thill.till!PII> schemeout- perfo rmstheproport ion al controlscheme!iu ourrobotcontrolsys tem.WithPI]) control, therob otreachest.he finalsp ot faster andhas110oscillat ion. The system sta b ilityforboth buoyancy andwaterjet have heen evaluatedwill.1~()IlSi tll!rat, illll ofnonlinearity using describingfunc tio n withthe Nyquiststab ilitycriterlou.III the next chapterweare going tointroduce the proceduresandresults

o r

the test of thehydraulic system with proport iona lsrh"llI~,1111<1till!l,l'st inWill-PI"tank

r ur

hoth buoyan cyami waterjetsystems.Slncc propurt.lonal schemoluIll'()\'I~1la t('!"ill thetestthatit workedcllcctlvclyinthohydraulic sys tem 1.0posltlon tilepiston rod , wedidnotapplyrIOcontrol.

49

Chapter 3

System Tests

3.1 Compon ents Des cription

Thissectionisgoingto describeall the equipment or components neededinthe experimentaltests.

3.1.1 Transducers

Transducersnre importantin feedbackcontrolsystems.They canlargelyenhance thecontrolhardwareand software functionsif well designed andon thecontra ry, theycallalso deterioratethecontrolsystem capabilities.

In ourproject, the robotwasimplemented wit htwo transducers.OneW:lS

usedinthe hydrnuliceystemcontrol,theLVDT(LinearVariableDifferentialTrans- 50

fe rme r].It was mountedinsidethe MTSactuat or.Anotlu-r nneWll~au poteu- tiomder,used to measure the dept hoftl1t~mho\. ill 1,111' wilt,t!rtunktest.,

LVDT

The LVDTprodu cesanelectricaloutp utproportioualtoLhe di:ipliln'Il1"lIl.ofII separa te mova ble core. Motionofthe1l01lctllliad illg1I1;1~1ll'l.i",'un' \'Mil'stIll' mutualindu cta nceof eachsecondaryrelativeto theprimary,whichIlt'l "'"' lliuPH thevoltageinduced from theprima rytocadisecomlnry.LV!)'!'hast.lu:follo wing characlcristics(A!lo/"C;\ tlllciSlll a rt l!lfH):

•Beca usethe re is110physica lcontact hell\,('(mrornand {,nil,Oil!lIwrJ11lIl il";,1 componentofLVD'I'donot wear0111or dctcelorate.

•LVDThasgoodresolu tion findsmall hystnresis.

•Ithas excelle ntlinearitywithinits nominnllinenr range.

• Ithas good vibrationiUlilenvironmentaonaitivity,

•It111\9 good response and dynemiccha rectc riatlcs.

LVDTis used hereto monitor1I1(~displa r:I!lIIlmLoftill:l'i"~tnlll"fIllIIfthr- actuator,and thusthepositionof theplungerilltim dUdTlil~Oulp ll lllrtlll~"YD'!' isdemodul ated byDayll'Ollic2MCtransducer.!xd Ler rlemorlulnto r,whicbalso suppliesreg ulated excitationto theIN O'l'.Thedetnodulu to r prOllllt:r!S;1 rilt.f!rt:d DC out putsignal whichis propnrt.ional Iothomechanic..1 illplll,overthefullplus andminusrange ortileLVOT,andfeedsthis regul a ted sigll;dto till:A/I) t:fIllw:rtr:r

51

inth,~,lataaCljuiliit io/landcont rolsystem[(eitMey570.Tile excitat ionsup plied hytheOflytronic20l Cis:'Jvolts Ilt3kHz;Theout pu tlevelofDaytronietOICis lilllitceillytrall~d'lcercharacteristics andrange ,typica lly10mVper.001- ofcore lleflr.ctio/l,andthe power requirementislOS- 125volt s,50- 400liz,10waus.

Pot entiometer

WeII:<L'<I apotentiometerill!the dept hSCIlROrto mea surethodepthof the robot

juwatert..\IIkt,~~t.Tile reas on wechosethepoten tiomet eras ourdepthsensor insteadorotherpressuretrnnsducersWilSitispbysic nllysimpleanditsoutputis

ncvoltage!whicheaucilllilybereadhymultimeterorf(dl/'/cy570wit hcomputer, therefore,thereillnoncccifor extra hardware,suchasdemod ulato r or amplifier.

Thepotcnucmeteris basicallya conductiveresistance witha wiper. Itis (lc.'ligllc<1 toprovidean e1edriclllsigna l(volta ge)proporfioualtoangulardisplace' ment of its shart.Theout putcanberea d cu tona.digit al voltmeter.InonrCASC, wefedtheoutputto theKeithley570 sys1e01andread by the compute r.

Inthewat er tank test, whatwewereinterested inwastherobo t'sdepth whichisl\lineardisplacement.Totransrormthe angulardis placement tolinear one,we Pllta pulleywhosediameter isI05mm ontheshaftorthepoten ti ometer and 1lI0011lted themtogether onabeamasshow nin Figure3.1.

Thebe amwasusedillthetesttosup port the potentlcm e terandpulleyabove thewaterlevelwhen thetestwasconducted in a waterta nk.Thepulleywaswinded by11.wire.One endofthewirewas connect ed to the robot,andtheotherendwas COIllIL'Ctcdtoa\veighl.Sowhen robotdescendedin thelank,theweightmoved

"

beam

Figure3.1:Potentiometerin theIl..atertanktest

.:'3

llpward~andwhen therobotascended, thcweightmoved downwards. Since we (lid110twaul HII!wdghttomoveill thewa ter ,weputanothersmallerpulley at the endorthohcalll ItSshown in Figure 3.2,so tileweightmoved outside the water tank.

connectto robot

connecttoweight

Figure3.2:Potentiometer and pulleys

Whentherobot moved in thetank,it pulledthe wire so thepulleyrotated the shaftorpote ntio meter,andthesignal was redto the!(titltley570. ADC power supplywasusedto provide power for the potenti ometer . The pot entiomet er shaft couldrotate ten times, so the maximumlineardisplacem ent we couldmeasure was

ten timesthecircumferenceof thepulley,thatis3.3m.Thutmceus themuxlnunu dep th wecallmeasure is 3.3m.Wccalibratedthe output voltage of the potcmtiume- terbyadjusting the power supplyoutput,50that the potc ntknucteroutpu t ra nge wasfro m0to10voltsin itstenround ro tat ion . Anylinearflill]llarc~l1l(,llloftlu~

robotin the watertank correspo ndedavolta ge readingfromtileP,)tcml.inmd ,·r.

3.1.2 DataAcqu isitionSystem

Thedata acquisitionsystem we used wns/i.'cil/dey 570byKuithleyl);-,laAffillilli.

lion&ControlInc.. Il FunctionedMIthe interfa cebetweenCOlllplltl~rllaue!ntllc~r comp o nents which we neededto collect'1igna lsIromorsellflsiglmls to,suchastill:

ecrvovalve, r,VDT and thepotentiometerillOlll'case.

The[(eilhley570requiresthehos tco mputertohaveatleast2Iili!{hytell of RAM,with8088 or 80286CPU,The Keilhley570came withanother illll:rfilf:e card mounte dinone oftheslotsin computernnd connects withtill:l\'ci/Ucy,~711 usinga15fool shieldedinterfacecable.The1\eithley570hnsinbuiltA/D.111I1 D/ A converterswith32 single-ended,orIii diffcreutialuualoginputchnnncls, 2 analogoutput cha nnels, 16digit al input and1Iidigitaluut.putchnuuels. TILe AIDconvert erinput rangeinKeilldey570can beswit chedto+/- IIIvults.

Becau seAIDco nverter has 12bitsresoluti o n, there arc 109" pllllllihll! volt age!

levels, specifiedwithdig it a lvalues0--lonG.Iftheinpnt rangn i!!+/-I[Jvulh , thenvoltage of eachstep is20/4006or0.00-l882,and0r(:preSI~llts-10 volts,;1I1l1 4095represent slOvolts ,anyvolt agebetween -10to]{Jvoltscan be WIII/llltf:l[

fromthis equation,V

=

-JO

+

(Dx 0.001882),whereDisthedigital value from

55