Based Tool

CROBOTS

A CAD Based Robot Simulation Tool

by

CJohnJoseph O'Leary,B.Eng.

A THESIS SUBMI'ITED TOTHESCHOOLOF GRADUATE STUDIESIN PARTIAL FULFILMENT OF THE

REQUIREMENTS FOR THE DEGREE OF MASTER OF ENGINEERlNG

Faculty of EngineeringandAppliedScience MemorialUnivers ity of Newfoundland

St.John's,Newfoundland

August 1998

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NationaiUbrary

"'Canada

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This thesisisdedicated to my wife Charlotte whose persistence and patience

helped me achieve this goal.

Abstract

This thesis proposes a new CADbasedtoolfor robot simulation (CROBOTS)that canbeusedinthe design. application.andprogramming ofeducational and industrial robots.Thesoftwareiswrittenusingthe AutoLispprogram.minglanguageand runs8.8athirdpartyapplication inside AutoCAD R14.CROBOTScombinestheimpressivegraphics capabilityofAutoCADwithcustom developedtools (orrevolute robot model creation,robot operationcycle planning.and robot contro ller simulation.Customfunctionsenable the automatic creationof a 3Dsolid model repre sentationof robotgeometrygivendimensions.theteachin g of pointpositionsdefining adesiredrobottrajectory, andthe 3D graphi cal simulatio noftbemanipulator'sresponsetocompute dtorque plus

pm

genera ted drive torques.

CROBOTS should prove an effective toolinthe delivery of robotics related curriculum atbothsecondaryand post-secondaryeducational institutions.The time required to become proficientin theuseof CROBOTS sho uldbelesethanthatrequired to learn stand alone commercial80ftware packages forrobot simulation.AutoCAD'ainquiry tools,forexample.maybeusedto facilitatethe teaching of robot design conce pts.Furthermore.CROBOTScan888istwiththe developme ntof robot programs oflline and shouldallow formore efficientuse of limited robothardwareresources ateduca tionalinstitutions.

Future developmen ts ofCROBOTSshould allowmodelingof additional moot mechanical configurations.therecordin g ofsignificant productiondata.and a task levelprogram.m.ing interface.CROBOTS shouldelsebetestedinbeth industrialand educational settings to determineitsrobustneea andtoidentifyneeds foradditional functionality.

iii

Acknowledgement

[extend my sincere appreciation toDr.Michael Hinchey,my supervisor,for -knowingeverythingaboutrobotics andcontrol"and.in particular.forhisunderstandingofhowchallenging itistoachievethis goalwhen employed full-time.

E G H

R Y

p

8W

Nomenclature

Toolcenterpoint positionwithrespect tothe robcebase.

Robotbase rotationangle [rad).

Robotshoul de r rotationangle[r ad].

Robotelbowrotationangle[ra d].

p,+EIra dl.

Robotbaseheightem}. Length oflink 1 [mi.

Length nClink 2 [mi.

Wrist positionwithrespect to therobot beee.

Endeffecto rpitchangle[rad].

Endeffecto rrollangle(rad].

Endeffecto r yaw angle [tad). End effectorapproach vector.

Endeffector orientationvector.

End effector norm alvector End effector rate [radial Jointrate[ra d/e].

Virtualwork [J}.

Vutual displacement[mI.

Joint load (N.m).

q, Joint displacemen t(rad]

T Totalkineticene rgy(J].

V Totalpote ntialene rgy (J).

L Lagrangian,T-V[J].

M Mas.[kg]

~ Base torque [N.m]

~ Shoulder torque[N.m]

Elbowtorque[N.m]

M(q) Inertiamatrix

Vlq.<i! Velocity torquematrix

G(q) Gravity torquematrix

F(q) Frictiontorquematrix

"

^{Distwbancetorque[N.m]}

e(t} Jointtrackin gerror(ra d]

K, Proportionalgain[N.lDv.dj

K. Derivativegain[N.m.s'\rad

K. Integralgain[N.ID.Vad..]

iv ill Tabl eofContents

Abstract Acknowledgement Nomenclature 1.0 Introduction

1.1 IndustrialRobot ProgrammingTechniques 1.2 Commercial Offline Programming Software 1.3 Related Previous Work

2.0 Development of the Robot Model 2.1 RobotClassifications

2.2 MathematicalModeling 13

2.3 Kinematics 14

2.3.1 DirectKinematics 14

2.3.2 Inverse Kinematics 2.4 Singularity

18 22

3.0 Dynamical Equationsof Motion 25

3.1 Lagrangian Equationsof Motion 26

3.2 Equa tionsof Motionfor the RevoluteRobot 28

4.0 Simulation 32

4.1 ComputedTorqueControl 4.2

nn

Control

34

38 vii

'.3 Zero OrderHold 40

. ..

^{NumericalSimulation 42}

5.0 CROBOTS:A CADBaaedTool(orRobotSimulation 44

5.1 Comm ercial Applications 44

5.2 Educa tionalApplications 45

5.3 CROBOTSOverview 47 5.4 Overvie wof AutoCAD Customization 55

5.4.1AutoCADR14 55

5.4.2 TheTempla te Drawing File 57

5.4.3 The ModifiedMenu 58

5.4 .4 The Custo m AutoLisp Functions 60

6.0 Concl usions andObserva n ooa 63

7.0 Recomme ndati ons 66

Refere nce s 68

Appendix AModifiedAutoCADMenuFile 71 AppendixB.ListingofCustom AutoLispFunctions 73

List

or

illustrations

Figure1. CartesianCoD.fiauration

Fi.eure2. Cylia drica1Conficun,tion 10

Figure3. Sp bericalConfiguration 11

Figure 4. RevoluteCon.figun. tion 12

Figure5. SCARAConfiguratio n 13

Figure6. Co-<lrdinateFrames!'oraTwoLinkRevoluteRobot

I.

Figure 7. RobotEndEfIectorOrientationVecton 18 Figure 8. Multip leArmSolutions!'ora RevoluteRobot 22 Figure9. GeoeralSimulatio nBlockDiagram 34 Fig-.ue10. ComputedTorq ueControl BlockDiapam 31

Figure 11. DigitalController 40

Figure12. ZeroOrd erHold 41

Figure 13. Customize dPull-downMenu forCROBOTS 41

Figure14. CROBOTSMainMenu 48

Figure 15. ImageMenufor-Cre ate NewRobot Moder 4'

Figure16. Create NewRobotModel 50

Figure 17. Home PositionCortheRobot 51

Figure18. Teaching theRobot 52

Figure19. Tl'ajeetory Approxima tion Using-Teach- 53

Figure 20. SimulateContro ller 54

Figure21. LayerDialogueBoxforRevsetup. dwt 51 Figure 22. Pf'OIl'aIDFlowchartforSim ula teContro ller 62

Chapter 1

1.0 Intro d uction

Inan effort to remaincompetiti veina dynamicglobal marketplace, today'smanufacturers arechallenged to reduce thelead-timeandcost required to bringaproduct from the conceptstagetothecons um er. Flexibl e manufacturing systems(FMS)provide a numbe rof advantages over traditionallabor intensi ve techniqu eswhich ma y help meetthisneed..FMS can resultinincre asedproductivity, sho rterproductio n time for new prod ucts,reduction of inventory partsintheplant,savingsinlaborcost,and improved. productqu ali ty (Bieke rt,R.etal.1991),

Industrialrobotsrepresenta maiercomponentinFMS duetotheir capa bility tobeusedforanumberof differentapplicatio ns.The main applicationsof robots includeparts handling,asaembly,machineloading and unloading,spraypainting,welding,and inspection(Ryan ,D.•1994),The world' sindustrial-robot population was estimatedat 570,000unitsat theend of1992.Japan accounted for about 60%ofthe world'sstockoperationalin 1992.The worldwide stock of industrial robotsin1992 increased onlyby 8%.

comparedwith16%in1991. Robot orders,however,jumped. by 40%inthe record-settingfirsthalfof 1993,lead bytheauto industryinspot-welding,

coating and material handling. About 48,000robots are at work in American factories, the second largest robot user behind Japan. But Japaninstalls more robots each yearthanthetotalthat the U.S.hasinstalledin the past 32 years. Itisestimated that the numberofindustrial robots operating in Japan willreach 880,000units bythe year 2000 (National Robot Society, 1994).

The expected growth in industrial demand for rcbctica technologywill invariably lead to a need for more efficient engineering design practices, increased productivityinrobot programming, and improved resources in support of research, development, education and training.Thisthesis presentsanew(C)ADbased tool for (Robot) (S)im.ulation or CROBOTS that has been designed to help meet these needs.

Abrief review of robotics fundamentalsisprovided beginningwitha discussion on industrial robot configurations.Secondly,conventional programming techniques are reviewed and the potential advantages of offline programming are discussed.Adetailed overview of the requirementa for mathematical modeling of robotic systems follows.Thissection includes development of the direct kinematics,inverse kinematics,and dynamics models for a revolute manipulator.The application oftheae modelsinthe development ofnumerical8imulations for robotsisthen provided.The capabilities of the CROBOTS software are then highlighted and a diecuaaion on the developmentofCROBOTSispresented. This is followed by an overview oftheeffectivenessofusingthis softwareinrobot design and motion planning. Finally, recommendations for the future enhancement of the CROBOTS software are presented.

1.1 Industrial Robot Programming Techniques

Thereduced timeforprod uct changeovers affordedbyFMSisdue primarilyto the capabilityto pre-program robots to perform.a varietyof differenttasks.Therearethreeprimaryformsofrobotprogrammingnow beingused.inindustry;(1)lead-throughprogramming.(2)teachpendant programming.and(3)ofiline programming (Greenwood,F.,1989 ).InIe ad- throughprogrammingtheoperatorsetstherobot controllerto programming mode.graspsthe manipulator.andleads itthroughthevariouspoint move s whi ch makeupthe operationcycle. Afterthe operator pushes abutton.the controlle rrecords the pceidcnetowhichthe end effectorismoved.During play backthecontroller is ableto duplica te thedefin ed robot moves . Alte rn atively,a band-he ldteach pendantwhich enablesrealtime pn>gramming oftherobotisused,Usingthe pendant, theoperato risableto jogea ch robot axes to positionthe end effectorwheredesired..After the operatorpushes a button onthe pendant,thepointsare recordedbythe contro lle r.Theteach pendantalec enablestheope ra to r to developthe overall robot programwhi ch.inadditio n to pointmoves,may include instructions to controlgripper actuation.conditional logic,instructiODBforcommunication withexternal sensorsandothermachinesthroughinput/outp ut(110) modules,and mostothercommands included with therobotcontrol pn>grammingsoftware.Finally.oft'lineprogramming enablesdevelopmentof thepreliminaryrobot control programonaPCor workstation without the requirem entto taketherobot outofservice.During en-lineprogramming usinglead throughorteach pendant methods.the robotisnot availablefor prod ucti onandmanufacturers mustincurthecost ofthiedown time.

Ind us trialrobots generally provide the operatorwiththe capa bilityto develop programsoftline using anASCntext editorandthendownloadthe program to the robotcontroller.Inaddition.proprietarysoftwareisavailable

from commercialrobot suppliers which enablesdebuggingof theASCnform oCthe program before download. Theseprograms,however,typica:1lydo not provide a 3D graphic:al simulationofthe operation cycle and the operatoris, therefore, still required to test the robot interaction with itaenvironment online.Fortunately,there are a number ofthirdpartycommercial8Oftware packages available whichallowofflineprogramming and 3Dgraphical simulation of the robot operationcycle.The capital costofthese aoftware progra.ms andsupporting hardware mayinsomeC88e8exceedthe 008toC robot hardwareandis.therefore.dif6.cu1t to economically or financially jus tify forindustrial use.Educatio naland R&D institutions have even more limited budgetfunds andpurchase of such commercialsoftw areisgenerally not feasible.

1.2 Commercial Omine Programming Software

There are a wide variety of 3D CADbasedcommercial oftline programming software packages currently available in the marketplace.

Workspace(versio n 3.2),an indwrtrial robot simulationsoftware package.i.e a graphic simulation system and a meansofoff-lineprogramming a robot cell. The software packagewillcreate and simulate robot programsinthe native language of the robot. For example.users

or

Fanuc robots may write robot programsinKarel,and ABBrobotueersmay write robot programsin ARLA.Thereistherefore no need forpostprcceeecre to translate from a simulationlanguagetothe robotlanguage:thefullpowerof the robot languageisavailabletotheuser throughoff-lineprogr amming.It isalso poeaible to transfer exiBting robot program s fromtherobot contro ller back intoWorkspace foroptimization.80cff-Iine progr ammingisatwo-way process .The£u11structure of the robotlanguagesisimplemen ted.including typedvari ables,teachpointa, subro utines.looping,branchingoncondition.

signals,andconditionhandlerinterrupts.All themain ind wrt:rialand

educational robot languages are implemented. and a libraryofover 140 robot modelsisavailable to the user (though itisalsopossible for users to create their own robots).

Robotica is a collectionofrobotics problem solving functions for the Mathematica software package. It has the capabilityofreading external simulation (e.g.,SIMNON)output files and displaying themotionofthe robot when subjected to the sequence of joint variables. It requires Mathematica andX-windows.

Deneb Inc.offers a number of software packages for oft1ine robotics programming and simulation including IDtraArc•The Simulation and Programming Tool for Robotic Arc Welding,IDtraFinish . The Simulation and AnalysisTool for Robot Deburring, Grinding, Polishing and Buffing, UltraPaint-The SimulationandAnalysisTool for Robotic Painting, and tntraSpot.The Simulation and Program..ming Tool for Spot Welding.

Working Model 3Disa PC based software package which allows kinematic and dynamic simulation of a varietyofmechanical systems .The program.isnot specifically designedfor robotics applications but it does provide the utilities necessary to build and simulate manipulator operation.

Ineachofthese cases,the cost of the software. required hardware, and rela ted. training may be difficult to justify for industry and educational customers who have already dedicated. extensive in-bouseresources to CAD software such as AutoCAD.

1.3 RelatedPrevi ousWork

Developmentof theCROBOTS software required considerationofa wide arrayoftopics including robotics design fundamentala. direct kinematics, inverse lrinemat:ic:a,dynamics modeling. inverse dynamics, numerical controller simulation,offline robot progra.m.m.ing,androboti cs related CAD applications.Severalprevious research efforts related to these individual topic areas wereidentified and sourcedinthe developmen tofthi.8 thesis.

Development ofthedirectandinverse kinematicsmodel has been well documentedbya numbe rofsources.The basi cpar ameter susedtodescribe a manipulator were first presentedby DenavitandHartenberg(1955) and furtherdetailedbyPa ul (1981).Pieper(1968) presentedaclosed form solutionfor the inverse kinematics forsimplemanipulators which Paul (1981)later expanded and applied to industrial robots such as the Unimate PUMA

Development of control schemes have been numerous as welL Whitney (1969) proposed the useof resolvedmotion rate control(RMRC)for robcee.

Given a desiredpa thofthetoolendpoint in cartesianeo-ordinates,RRMC can be designed by relating the jointrates to thetoolendpointrate through the manipulator Jacobian.RRMC.however,givesrisetoan accumulationin trackin g errors in position since positionisestablished indirectly by taking an integralover8specifiedtimeinterval. The Computed Torqu emodel was originally pro pose dby Bejczy (1974) andagain byVukobrato vic(1982).

Unfortunately.the Computed Torquevariesfrom the actualtorq ue due to inaccuraciesinthe drnamicemodel resul tingfrom unknownfricti on.pa yload and inertiaparame te rs.Controlschemesbasedon linearization ofLagran ge's equationsby non-linearfeedback werepropoeed. byFre un d (1982).This

achemeenco unteredpro ble msindiscrepenciesbetwee nrequiredand calculatedtorquessimilar tothe Computed Torq uemeth od.Tak egaki and Arimoto(198 1) showedthatProportionalplus Derivative (PO) c:ontrollera can providestablemani pula tor controlifthe gravity term canbecarefully compensated forusingcounterweights. Alternatively,an integralcontrol term resulting in aPIDscheme may beusedtocom pe nsateforthe gravity term.Thisachemestillassumes knowledgeofmanipulatordynamics.which invariablyare inaccurately modeled.

The^UBeofCAD to aidinthedesi gn andprogrammingofrobotshas beenthesubjectof several researchefforts.Wu (1984)proposed a mathe mati calCorm ulationthat could beusedtobuilda CADtool which would improve the accuracy of kinematicmodels and thedevelopme nt of offlineprogram s.In thiscase,calibrationof the actual robot hardwareis still necessaryonline since thegeometryofthe manufactured componentsdiffers from theCADmodel. Therobotics facilitiesofthe CAD/CAMCAT lAsy8tem wereproposedbyBcrrel, P.et al (1982).Thisprogram was developed to run oncomp uterworkstations.Hornickand Ravani(1986)proposedaCADtool calledSTAR(Simula tio n ToolforAutomationand Robotics)for offlinerobot motionplanningand programming.Thisprogramwas developed to runon compute r wor kstationsand relied on separate programs for the geometric model andthedyn.amieemodel.Dutt (1991) utilized AutoCADtogenerate geometrical informationrequired forofilinerobotprogra.mmi.ng.Inthiscase.

BASICprogramming wasusedto extract geometrical information characterizing themanipulatorpositionfromthe Drawing Interchange Format(DXF)file outp utofAutoC AD. This couldonlybedone oneposition at a time.Ryan (1994)proposed.the use of AutoCADfor simula ti onofroboti c and auto matedsystems.AutoCADisnot customized inthiscase and essentiallythisproposalisanovervi ew of nativeAutoC AD commandsthat may be used for a static analysis of positiongeomeay.

Chapter 2

2.0 De velopme n t of theRo botModel

TheRobot In.etituteoCAmericadefines arobotas:

•A robot;"areprogrammable. multifunctional manipulatordesignedtomoue moseriols,parts.tools.orspecialized devicesthrough variable programmed motionsforthe performanceof avarietyoftasks."

Asthiadefinitionimplies,robots are availableina widevari etyoCmechanical confi gurati ons but,ingeneral,may beclassifiedinone offivedifferent claeeifications.These includecartesian,cylindrical spherical, revoluteand SCARA(Cri tchlow,A..1985).

2.1 RobotClassifications

Carte s ianConfiguration

Thisconfigur ati on characterizes robotsthat have linear orprismatic motion capabilitythat can be measuredinthefamiliar XYZ cartesianco- ordinates as showninFigure 1.Themanipulator can be moved.linearlyupor downthe verticalZaxisand positionedinthehorizontalplanethroughlinear

motion alongboththe X and Y.ues.Theserobotsareeasiesttoprogram because of the complete independence of their joints and thereisno coupling betweenallaxes for arigidandfrictionleaa structure.Thismotion definesa rectangularworkspacethatissuitable for relativelysimpleapplications such as machine loadinwunloadingandinspection operations.

Figure1: CartesianConfi guration

Cyli ndrical Co nfi guratio n

Thisconfigurationappliestorobots that combine vertical andradial prismatic motion capability with the ability to rotate about the verticalaxis 8Bshown in figure 2.These combined motionsdefine a cylindrical workspace volume.The cylindrical configuration is more versatilethan the rectangular configurationanditisusedfora varietyofproductionapplications. Sin ce thereIetelescopic radial motion,theseunitsrequire relativelysmallspace and reduced dynamics sinceinitialrotationcanbeexecuted withthe mani pula tor retracted.Thisconfigurationiswellsuitedtothe majority of pick andpla ce operations.One diaadvantageof the cylindrical configuration

isthat the manipulator cannot reach below thebedofthe structure.

Redundant degrees of freedom at thewristmaybeusedto overcome this.

Figure 2: Cylindrical Configuration

SphericalConfiguration

The spherical configuration retains the telescopic prismatic radial motion and the capability for rotation about the verticalaxisof the cylindrical configuration.Inaddition it employs the ability to rotate about a horizontal axis through the base as showninfigure 3. The resulting motion capability defines a hemispherical workspace bounded by an inner and outer hemisphere.This configuration requires more sophisticated control systems but itisableto service a larger workspace.

10

Figure 3: SphericalConfi gu r ation

Rev olute Confi gur atio n

The revolutemanipulatoremploys rotarymoti on capabilityaboutthe ba se,the sho ulde randthe elbow88showninfigure4.Threeadditional rotationaldegreesoffreedomare normally addedatthewriBtresultingina 6-axismanipulatorformostindustrial units.Thisconfiguration provides the mostdexteritybutelsedemandsthemoat sophisticatedcontrolle rs.Their smallsize relati veto their wor ksp acecapa bility , ease ofinstallation,high relia bility,andabilityto easily work in enclosedspacesmakethem well suitedfora varietyofindustrial applications.A disadvantageofthis confi gurationisthe degeneratingbehavior of themanipulatornear the workspace boundaries.Inaddition., becausetheshoul derandelbowaxes are parallel andorthogonal to thewaistaxis,these unitshave relati ve ly low stiffness.This mak esthem unsuitedformany highprecision applica tions.

"

Figure4: Rev olute Con fi llUr a t io n

SCARACon fi gu r ation

The SCARA (Selecti ve Compliance Assembly RobotAnn),whichis specifically designed for high precision assemblyanddrillingoperations, overcomes the low st:if£neseproblemofthe revoluteconfigurati on butoffers limited mobility.Inthiaconfiguration shownin figure 5, theshoulderand elbow axes are parallel to thewaist axis.The SCARAhasfourdegreesof freedom:limitedrotationsabout the shoulder andelbow to position thetool rotationabout thebasetolocate the shoulderand elbow wcekepaces,and vertical prismatic:motionof the endeffector.Tbi.smoti oncapabilitymakes it wellsuited forthemajority of eeeemblycperanoneinindustry.

12

Figure5: SCARA Con figu r a t ion

2.2 MathematicalModeling

lndustrial robots are basicallypositioning andhandlingdevices.To effectively completetypical operationssuch8.8weldin g,therobot mustbe able tocontrolita motion. at aminimum,and.inmany cases,theforces it applies toitsenvironmenL Contro l of the endeffector demandsanaccurate analysis of the characteristics orthe mechanical structure,actuators and sensors.Mathematicalmodelingor a robot manipulator is.therefore,a necessarypre-requisiteto developing a successfulcontro ller(Canudas de Wit, 1996) .Modeling or robotmanipulators requires considerationofboth kinematics anddynamics.

2.3 Kinemati cs

Kinematic modelingconcernsthedescription ofthemanipulator motion withrespecttoareferenceframe withoutconsiderationofthe forces andtorques that eauee themotionof the strncture.This formulationofthe kinematic relationship allows study ofbothdirect kinematicsand inverse kinematics.Directkinematicsenablesthe descriptionoCthe end effector motionas a functionof jointmoti on. Inverse kinematics consistsof transformingthe desired end effector motio ninthe workspace into the correspo ndingjoint motion.

2.3.1 Dire ct Kinematic s

Aseriallinkmanipulatorconsists ofa seque nceoflinks connected togetherbyactuated joints.Foran nde greeoffreed o m manipulator.the re will be nlinks andnjoints.The relationahip betweenlink.eof a twolink articulatedrobotcan be developed afterassigningco-ordinate frames to each linkas showninfigure 6.Thetoolcenterpoint(TCP) can be measured locally withrespect to:(1)movingco-ordinate frame uvw whichhasita origin at the wrist,(2)moving c:o-ordinate framenmhwhichhasita originat theelbow.(3) movingco-ordinate frame pqrwhichbaaits origin at the shoulder.(4)moving co-ordinate frame xyz whichhasitel originat thebase. or(6)fixedworldco- ordinate frameXYZwhichalso hasita origin atthe base.Thedirect kinematics(DK)solution Corthiacase allows the determination ofthe Cartesian positionandorientationof the endeffector whengiven the jointco- or dinates.

Figure6: Co-oedlae ee Frames foraT ....o LiDk Re v oluteRobot

For alockedwristcase,the end effectortoolcente r point(TCP)tobase transform ati onis(Hinchey,M.J.,1994):

-Sa

0 0 I CP 0

^Sp

0 I c,

Ca

a

0 0 I 0 0 0

o

I 0

-sp

0

cs

G

-s«

0 0 1 0 0 0 1 0

o

^I

s.

^{0 0}

0 1

^{Uv ]}

o

C& H E+ w

o

^{0 I I}

(I) where {Jh.Pr.pJisthecartesianposition of theTCPwithrespectto

thebeseframeX.Y,Zin[ml_

(u,v,w)isthecartesianpositionoftheTCPwithres pect tothe wrist frame uvwin[m].

Eisthe baseheigh tin[m].

G isthele ngth oflink1in[m].

Histhele ngthoflink2in[m].

Caandsimilar notationsare usedtoabbrevia te coainelsine of respective jointanglesin[rad].

Atthewrist. u

=

v

=

w=0 andthewristtobase transformationistherefore:

-Sa_Ca 00 0

0r cp

0 0I

ss

0

or

0 ^SEE0 ^]

o

I 0

- sp

0

cp

G H +CEE

0 0 1 0 0 0 1 I

(2) where <.p-p,...Pro)represe ntsthe positionofthewristororigin ofthe

endeffector framewithrespect to thebase referenceframe.

Matrix multiplicationyields:

-Sa

Ca

o o r

CPSEE+SP(H +Ct£) ]

o

0 0

I 0G-SpsEE+CP(H+Ct£)

o

^{I I}

(3)

From whichit follows:

p,.

=

ea(Cp&E+Sfl(H+C<E) p,.

=

Sa(Cp&E+Sfl(H+C<E) Pm

=

G-Sp&E+CP<H+C<E)

(4)

(5)

(6)

LettingK lJ--e,these equationsreduce to:

'6

p,.

=

&(S<E+S~H) Pzo= G-+CPH + CKE

Itshouldbenotedthat inspection of geometry yieldsthe sameequation s.

Foratwolinkrevolu te robotwithaRoll Pitch Yaw(RPY)wristforend effector orientation, thetoolcente rpoint(TCP)tobase transformation is:

(7)

(8)

(9)

[

fh] r ca -Sa

⁰

° I C ^P

⁰ sp

°I ce

⁰

Se 0 1'

Pi' Sa Ca 0 0 0 I 0 0 0 I 0 0 0

p: ::: 0 0 1 0

-ss

0

cs

G

-se

0Ce H 0

1 0 0 0 1 0 0 0 1 0 0 0 1 0

[

crcr - CPSY SP 0 IU]

CRSY+SRSPCY CRCY-SRSPSY -SRCP 0 v SRSY-CRSPCY SRCY+CRSPSY

cscr

0 w

o

0 0 I I

where P isthepitchanglein{ra d].

Y is theya w anglein[rad] . Ristherollanglein[rad] .

Multiplication gives anequa tion of theform:

o 0 ]

o

0 I E

o

I

(10)

(11)

TheTCP ofthe end effecto r may.there fore . be expressed in terms of base co-ord:inatesthrougha rotatio nmatrix R=[n0a]represe ntingthe orientationof the endeffector anda position vectorPorepresentingthe positionofthewrist.These values are denoted collectively88thearm T mat:riLAsshown in figure7.the approach vectorarepresentsthereach direction.Theorientation vector0represe ntsthe directionspecifyingthe orienta tionoftheendeffectorfrom fingertip to fingertip.ThenormalvectorD ischosentocompletethe defini tio nof a right-handedco-ordinatesystem.

Figure7: Robot EndEffe ctor Orientation Vectors

2.3.2InverseKinematics

TheOKsolutionallows thedetermination oftheTCP positio nwith respect tothebaseframegiventhewristpositio nandthe en d effector orienta tion. Whilethissolutio nisanimportantpart of mathematical modelin g ofrobots,theinversekinema tics(lK)sol utionoffersmore practi cal informa tion that can be used inthe development ofcontrolle rstrategies.The 1Ksolutionforthe revolute robotisaimed at determiningthe joint angles

18

neceeeary to produce a given positionandorie ntationof theend effector.The IK solutionfor the wristoftherevoluterobot. previouslyshown in Figure1, allowsthedeterminationof thejointangles0..P.andIf.corresponding toa give n(p-.p,..p.)location.

Manipulationofequations(4)and(5)yields:

p.,Sa

=

SaCa(So<E•S~II)

(12) p,.Ca

=

CaSa(So<E •S~II)

(13)

Subtractingequa tion(13)from(12) gives:

p,.Ca-p.,Sa=0

(14)

SolvingCora.implies;

(151 whe reatan2isacomputer representationofatanwhich accountsforthefact thattwopossiblequa dranta providethesame solutio nfore,

Additionalmanipulation of equati ons(12)and(13)yields:

p..Ca=CaCa(So<E •S~II)

(16) p,.Sa=SaSa(So<E •S~II)

(17)

, .

Summation gives:

p..Ca +

p,.sa =

S<E+S~H

Manipulation ofthisequationyields:

SK= (p..Ca +p,.8a.S~H)IE= P +RS~

Manipulation ofequation(9) gives:

CK=(p..-G·C~H)IE=Q+RC~

Squaringboth sides of equations (19) and (20) yields:

SKSK+CKCK

=

(P+ RSfJ,p+(Q+ RCfJ)2

=

1

Expandingthisgives anequation of the form:

IC~+JS~=K

General analytical inverse kinematicsformula canbeused to showthat:

p=otan2(J fI )+otJm2(±"/11 +J1_K1IK

(18)

(19)

(20)

(21)

(22)

(23)

20

The P.Q,Requations areoftheform:

SK=V and CK=W

wherewithaandf3known,Vand Warealsoknown.VandWcanthen be used to determine:

K

=

atan2(V1W)

(24)

(25)

Exanrination of equations(15), (23),(24)and(25)reveals thattheIKsolution isnotunique.There are two possiblesolutionsforCLThere are inturn,for each a, two possiblesolutions (orf3andE.Thus,there may be fourad.missible solutionsobtained.for thewristposition according tothevaluesofa.P.andE.

These include two elbow-up positions and two elbow-down positionsas illustrated.infigure8.

Figure 8: MultipleArmSolutionsCora Rev oluteRobot

2.4 Singularity

For certain configurationsofa manipulator, finite end effectorrates require infinite joint rates.Theseconfigurations are saidtobesingular.Join t loads gener ate d by a controlle rare usually excessive near singular configuratio ns.To determinesingularconfigurationsforthe wrist of the revoluterobot.consider the wristpositionequa tions previouslydeveloped:

pm= Ca(SKE+S~H)

(7) p,.

=

Sa(SKE+S~H)

(8) Pzo= G+CIJH+CKE

(9)

22

Differentiationwithrespecttotime gives:

p-=-Sa(SIl£+SfI/}a+Ca( CK£ +CfH )jJ+ CaCK£i

(26) p". ""Ca( SKE+SfI/}a+Sa( CK£+CfI/) P+SaC K£i

(2 1) p-.-{S/H+ SK£lP -SK£&

(28)

Solvingequation(28) (ori:

&=[-P.-(S/iH+SK£)PIISK£

(29)

Substitution into equations(26) and (27) yields:

p-=-Sa(SKE+SfI/}a +Ca(CK£+CfI/) p .CaCKfSq - p.. -(S/lH+SJC£)iJ1 (30)

p".z::-Ca (SKE+SfJH)it.+Sa(C K£ +C{JH)P.SaCKfSq -p.. - (S/iH+SK£),8J (3 1)

Manipulationgives:

SKi>-+C<£:,."..-SaSK(SK£+S/H l<i+ [CaSK{CK£+C/H)-C<£:K{S/iH+ S.£)IP (32)

SKi>-+S<£:,.".=-CaSK{SK£ +S/iHl<i+[SaSK{CK£+C/iH )-S<£:K{S/iH+SK£lIP (33)

Multiplying equation (32)byCoand equation(33)bySaandsummingthe resultsyields:

Ca{S.p.. +caC'¢-) +Sa{S.p,. + SaC'¢-l

=

CajcaS«C<£+CfR/)-caC«SfR/ +S<£) !P +Sal saS«C<£ +CfR/)- saC«SfR/+S<£J!P

(34) Equation(34)isof the formA=Bp.lnfinite pocc:urs when B=0:

CalCa5«C<£+C/lH )-caC«SfR/ +Sdi)]

+SalSa5«C<£+C/lH)-saC«S/lH+S<£1I-0

(35) Expansion andsimplification gives:

(CaCa+SoSa)SK(CJC£+CfJH)-(CaCa+SaSa)CK(SfJH+SIC£)

=

0 (36)

SKC~·C.s~=0 S"'CK=Sfl/C~

tan(o<)

=

^tan(~)

(37)

Equation(37)hastwosolutions:It

=

P whenE:

=

0 andIt

=

P+11"whenE:

='II"Theseangles correspondtothe outerand innerlimits of the workspace.

Attheselimits,radialmotionisimpossible and a degree of freedomislost.

Itshouldbenotedthatingeneralendeffectorrates,p,and jointrates,q, areconnectedby aJacobianmatrix,J.of derivati vesthrough the equation p=Jq.Manipulationgivesq⁼J-tp.Itturnsout that thesingular configurations occur wherethedeterminantofJiszero because the determinantappearsintheden ominator of eachcomponent ofJ·I.

2'

Chapter 3

3.0 Dynamical Equations of Motion

Simulationsbasedonlyon thekinem atic modelforarobot assumeit willfaithfullyfollow itscomm an ded trajectory.Inpractice,industrial rcbcea areusuallyrequired to move at highspeedsto achievehigherprod ucti vity and dynamiceffect8 may begin to dominate.Acontroll erdesignbaaedsolely on thekinematicmodelmay resultindrive saturation. Nonlin ear dynamic coupling and inertialeffecte canalec make therobotdevia te significantly fromthe designtrajecto ry,cause excessiveovenhoot, and destabilize the sys te m. Finally,thedi&c:rete-timeand time-delaycharacteristicsofthe robot contro lcompute rsome times interactwiththe robotdynamics to further degradeand deetabilizethesystem'sperformance.

These problema must beconsi deredinthe robot design.The drive motorsandcircuitsmustbe sizedtoproduce thedesired performance.The links mustbeconstructedtominimizethe dynamiceffects,andthecontrol system mustbedesigned.to ensure dynamicperformance overthe entire workprofileof the robot.Inordertoaccom plishthese goals,goodanalytical dynamicmodelsorsimula tio nsofthe robotare required (Andeen,G. etel, 1988 ).

Thefirststepindevelopingthedynamicsimulationfora rigidlink robotistowritetheequationsofmotionforthesystem. Two formulationsare mainlyused to derivethedynamicmodel.:theLagrangianformulation and the Newton-Balerformulation.TheLagrangianformulationissimpler and more systematic whiletheNewtcn-Bulerformulationisconsidered. more efficient from a computational point ofview(CanudasdeWit, S.,1996).

3.1 Lagrangian Equationsof Motion

Oneapproachtothe dynamicsproblemistoconsider the instantaneousequili bri umofeverypartof the system Forthis.a freebody diagramisconstructedseparatingeachbodyfromotherparts ofthesystem, andtheirinfluence is substitutedby the initially unknownreaction forces.

Whena syste mcons istscf numeroua bodies,however.thistechnique becomes very cumbersomeandrequiresdealingwith largesystems of equations.In the dynamics analyBie methodsbased.onthe principleofvirtualwork.the coIl8training effectson a bodyofotherbodiesinthe systemare not consi dered bythe introductionofunknown reactions.Ineeead,imaginary infinitesimal displacements which could be given tovariouspart8oftbe systemwithout violating theconst:raint conditionsareconsi dered. These displa cements are calledvirtualdisplacements.

Virtual workis definedas an increment ofwor k which aforce . F.

actingonaparticle mightperformon avirtualdisplacementofa particle and iswritten as(Rivin,E., 1988 ):

&W=F8qcosa.

(38)

,.

where 8Wisthevirtualwork Sqisthevirtualdisplacement

a.istheanglebetween directions offurceanddisplace me nt

Itcan beshown that:

L8W=O

(39)

where 2:&Wisthe sum.ofvirtualworkson anyvirtualdisplacementof thesystem.

Thisoonstitutestheprinci plecfvirtual wor k.

TheLagrangianformulationisbasedon theprincipleofvirtualwork and D'A1embert'sprinciple whi ch states that reactionloads due toinertiacan be treatedas staticloadsinthevirtualworksta te ment.Lagrangeaequations of motion Corageneral multi degreeof freedomsystem aregiven by (Arimoto, S.•I996) ,

d(3UiJiW<b- iJLlaq,••

(40) where L=T-Visknown88thesystemLagrangian

T isthe totalkineticenergyof the syste m Visthetotal potentialenergyofthe syste m qlrepresentsthegeneralized displa cements.

rlrepresentsthegeneralized loads .

Frictionloa ds can be added to theLagrangianform ula tionusingthe RayleighDiasipation function although itisofteneasiertoconside rthem partof thegeneralizedloa d. By definition.thegeneralizedload.T,fora particular generalized displacement,q,isequal to the virtual work.8W,done by external loadsduring avirtualdisplacement,8q,dividedbythevirtual displacement:

(41)

(42) where jisthe joint load and a andbare constantsassociatedwithwet

anddryfrictionrespectively.

Jointllink complianceboth addtoVandthis isoften modeledusing finite elementdiscretization <Hinchey.M.J.,1994).

3.2 EquationsoCMotionCorthe RevoluteRobot

Cons idera rigid revoluterobotstructurewitha concentratedpayload mass,M,atthe wrist.The kinetic energy ofMis:

(43) Similarly, the potential energy of Mis:

V::Mg[G+HCP+EC~]

(44)

2.

For a spherical payloadwith rotary inertiaI,wewouldadd toT:

(45)

To simplify the presentation, thepay load massisassumed dominant and the IcontributionstoT areignored.Inthiscase.the Lagrangian is:

L=T-V

L =MI2([E2+H2+2EHC elp2+E 2&2+2[£2+EHCeJPt +[HSp +ESK)2

a

²⁾

-Mg[G +HCp+£CK]

(46 ) The equations of motionare:

d(iJLl iJti)/dt- iJLliJa

=,

(47) d(aLlap )ldt-a Liap''1'

(48) d(iJUiJt )/dt-iJLliJe^=OJ

(49)

whereq"rp,and w are thejointloads.Manipulation of equation (47) gives:

iJLliJa = O

d(iJL/iJti )/Jt

=

M(HSfJ+ ESIC)la+2M(HSfJ+ ESICXHCfJ+ £CIC)tZiJ + 2M(HSfJ+ESc}ECKtiE

Equation (47) becomes:

;=M(HSfJ+£SIC)lii +2M(HSfJ+£SICXHCfJ+ECIC)ciP+2M(HSfJ +£SIC)ECKtiE (SO)

where 4'isthebeeetorque

Notethe gyroscopic terms

ap

and

as

inequation(50).

Manipulationofequatio n(48) gives:

iJLliJfJ=Mg(HSfJ+ESK)+ M(HSfJ+£SICXHCfJ+ £CK)ci'J

d(iJLliJP}/dt

=

M( EJ+HJ +2£HCe}p+M(E J+£ HC&)i- MEHSdJ-2MEHSEPt

Equation(48)becomes:

rp=M(£J+Hl+2£HCe)p+ M(El+£HCe}£-MEHSet l- 2MEHS&iJi -Mg( HSP +£SK)- M( HSp +£SKX HCP +£CK)ti J

(5 1)

where If)istheshoul der torque

Notethe centrifugaltermst^landa^land the CoriolistermPt.

:lO

Manipulationof equation (49) gives:

Equation(49) becomes:

where(lJisthe elbow torque.

Note thecentrifugal terms

iJ1

and liz .

(52)

Chapter 4

4.0 Simulati on

Robotsimulationisthe solutionof the equationsofmotionof the manipulator that yields thepositions,velocities,andaccele r a ti onsofthe systemele ments88functionsoftime.Insome eases,these simulationsalec yieldthe drive forcesandtorq ues required to producemotion and the resulting internal forcesinthe systems mechanicaleleme nts.Thedriveloads are importantinthedesign of the system'sactuatorsand control circuits.The internalforcesarerequired for the designofthe mechanica.l compone nt8of themanipulator.CADbaaed simulations enablegenerationofdisplaysof the manipulator moti anatoaidinevaluationand interpretationofsystem performance.

Inmost cases,threelevelsofsimulationare usedto study a manipulatordesign.Thefirstle velkinem a ti c simulation,assumesthe motion of themanipula toriiideterminedbythe comman ded joint displace ments.The positionsand velociti es of themani pulatorarecalculated usingstandardkinemati cmodelsand thedynamics ofthemanipulator and itscontrolsystem are ignored..The secondandmore comple xform of eimulationusee arigidlinkdynamicmodeloftheann.Atthislevel of

32

simulation. the dynamic characteristics of the control systems, drive actuators.transmissions.and control computer maybe considered. Inthis case,the equationsofmotion are coupled sets of nonlinear. algebraic, dift'erential and difference equations whose solution requires the use of numerical forward integration techniques. This form of simulationismost usefulindesigning and evaluating the manipulator's control systems.which include its computer and its drive system.Inthethird and most complex form of simulation, the distributed mass and flexibility of the manipulator's mechanical elements are includedinthemodelinaddition to the control and drive properties containedinthe rigidlinkanalysis.Finite element methods typically mustbe used toinorder consider the linksofthe geometric com plexity foundinindustrial manipulators (Andeen, G.,1988).

The basic structure ofallthree levelsofsimulationisshowninfigure 9.Inthis structure,thedata describing the system parameters and commands are input andallcalculations that are not time dependent are performed.These data maybe obtained directly from CAD modelsofthe manipulator's elements. Based on user defined or defaultinitial conditions for the end effector position vector, p(to), the equations of motion are then evaluated.. The vectorp<t)is then integrated. using standard numerical forward-integration algorithmsto obtain the state of the system atto+.1.t. This provides the new initial conditions to repeat the proceee, so the solution to the equations of motion advancesintimeinthe manner ofclassical forward interaction(Kern,G.,1978).

Time Algebraic Integration Time

Invari ant Solution lnvariant

Parameter

- JI.·· -r

^{w+1I Data}

CeJculations j(.)=F(.... ) Evaluatio n

&.1=&

R.-p~T.' N.tan- Utc:t-

Figure9: Gener alSimulation BlockDiagram

4.1 Con trol Torque Con t r ol

A basic problem in rontro llin g robotsisto makethemanipulator follow apreplanned desired.~ectory.Inmore advancedapplica tions, the robot con trolle r must ena bletrajectory planning which involvesfindingthe prescribed path. collisionavoidance. andecceeclofactuatorsaturatio n.For pointtopointpomtiODcon tro l thereare numberofcontroleche meswhich may beusedtoprovi de acceptablepositio nin gaccuracy (An.C.et

at

1988).

There havebeen numerousrobotcontro lschemes proposedwhich can beconsidered.8Sspecialcasesof theclaeeofCompu ted Torque Contro llers.

TohelpdevelopthiafOnDofcontro ller for a revoluterobot,theLagrangian equ a ti ons of motion(50), (51)and (52) maybe generalizedinthe following form<Lewis.F.etel,1993) :

34

M(q)q+Y(q.ti)+G(q)""T

(53)

where M(q)isan nxninertia mat::rix. q andits derivatives are nx1 vectorsofgenenilizedco-ordinates,andY(q.iU,G{q)andTaren x 1 vectors con taining velocity-dependenttorques,gravity torques.and input torquesrespectively.

To accoun t forfriction and distrurbances.,thisgene ralizedrobot dynamical equation becomes:

M(q)ij+V(q.ti)+G(qj+F(q)+u_T

(54)

Alternatively,this equationmaybewrittenas:

M(q)q+N(q.ti)+U=T

(56 )

where the nonlineartermsare represented. byN(q.V,.Y(q.q) + G(q) + F(q).

FOT adesired triQectory,qd(t).an output trackingerro r may be defined to ensuretrajectory trackingby thejointvariable 8S follows:

e(t )=qd(t)-q(t)

(66)

Differentiating twice yields:

Substituting forijinequation(55) gives:

(57) Definingthe control input function as:

(58)

Thisfeedback linearizingtransform ati onmaybeinvertedtoyield.:

(59) Thisisreferredto88theComputed Torque Control Law.

The useofthecontrol input,u,hasconverted8complicated non-linear con tro lsproblem into8simpledesignfor8linearsystem.Theblock diagram for a Computed Torq ueControlschemeshowninfigure 10illustrate s that this formofcontrolutilizes bothan inner oontrolloopand an oute rcentrol loop.Thecompu te d torquedependson the inversion of the robotdynamics andissometimescalledinver se dynamics control.r(t)iscom putedinthe inner loopby substitutingij.-ufor ijinequation (55).Hthedynamic model isexact, the nonlinear dynamic perturbations areexactly canceledand what

36

isleft is a decoupled linear systemthat can be controlled according to standard techniques (.An, C.et al, 1988). Unfortunately.dynamicmodelsare never exact and an outer loopfeedbacksignal istherefore requiredto counteract tnQectorydri£l;or error growth. Independent joint control using proportional plus integral plus derivative (PID) control maybe usedto compute the corrective torqueinthe outer loop.

L J

Figur-e 10:

4.2 PIDContro l

Theouterloopcontrol feedback signalmaybegeneratedusingpm contro linaccordancewiththeequation:

(60)

whereKp,K...andleiare the proportional, derivative and integralgains repsectively.

The proportionalcontro l madeproduces achan geinthe controlleroutput proportionalto the error signal.Withthismodeofcontrol at steadysta te thereisa residualerro r dueto gravity.Thiswoul d implythat the propo rtionalgainsho ul dbeashighaspossiblesinceincreasing thegain should reducetheresidualerrorrequiredtoproduce achangeinthe controlle routput.lncreasingthegain..howe ver.increasesthetendency for oscillatio nofthemanipulatoraboutthepositio n .eetpoint.Toeliminatethe residualerro r, aninteg:ra.lcontrolterm.is added. The integralmodechanges the con tro lleroutp utbyan amountproportionaltotheintegralofthe error signalAs lon g aathereisanerro r,theintegral modewill changetheoutp ut atarateproportional tothesumof the errorover time.Thederivative contr olmodemay beaddedtolimit cecillatione.This modeof control changes theoutput of thecontrollerpropo..'1ionaltotherate of change ofthe error signal.The derivativemodeis an attempttoanticipate an errorby observing how faBtthe errorischanging,andusingtherate ofchangetoprod uce a contro l actiontha twillreducethe expected erro r .Derivative contro l contri bu te s to theoutp u tof thecontrolleronly whenthe errorischanging

38

andis.therefore.alwaysusedin combinationwith theproportional. or proportionalplus integral controlmodes(8ate8OD,R..1996).

ThePIDgainsmwrtbeselectedforeach jointofthe manipulator separatelysincethe Computed Torqu ecootrollerdoes notresultina decoupledcontrolstrategyinthe inner loop.Thus, information onalljoint positionsand velocitiesisneeded to compute thecontro l torquefor anyone joint. The PID gain.amaybeselected.basedonthenatural frequencyofthe mani pula tor andthedesired.dampinginthesyste m. ThePD gainsmaybe selected as (Lewis.F.eeal,199 3):

(61)

K..=2tw..

(62)

where Ciathe desireddamping ratio.

w...isthenatural frequencyfor jointerroriin[ra d/s].

Sinceitisundesirable for the robottoexhibit overs hoot, the POgainsare usually selectedforcritical dampingt

=

1.In thiscase:

(63) Itcanbeshown that forclceed-Iccpstability.theintegralgainis su bjectto thecondi tion:

(64)

4.3 Zero Order Hold

While moocontrollersare designedinoontinuoustime.theyare implemented on robots digitally.In thiscase,thecontrolsigna.laare updated at discreet instantsoftimeusingamiceoproceeece. Toverifythatacontroller willope ra te as expected.itishighlydesirabletosimulateitinitsdigitized form prior toactualimplementation.

Ageneralized.digitalcontrol scheme mayberepresentedinsimplified.

block:diagram form.88shown infigure11(Lewis,F.•1993).Theplant or system to be controlled.isacontinuo us-time system and K(z)isthe dynamic digitalcontrollerwherezisthe Z·transformvariable .The referenceinputr{t) isthe desired trajectory thaty(t)should follow,andellisthe discrete tracking

~T ~T~

_ K(z) Hold Plant ~

Figure 11: DigitalCon t r o ll e r

'"

The samplerwith sample peri od Tisan analog·to-digital (AID) converter that taies the samples y(k'I')atthe output y(t) that arerequiredby the80ftware controller.Inrobot control y(t) might represent thevector composed of q(t) andq(1).The hold deviceisa digital.to-analog(DIA) converter that convertsthe discrete control samplesua.computed. bythe software controllerK(z)unto the continuous time control u(t) requiredbythe plant.

Zero-orderhold (ZOH)isgenerallyusedforcontrols purposes.For ZOH.the input^Ukandtheoutput u(t)are showninfigure12. Note thatu(t) isheldcontinuousuntil updated at time s kT.

~~

o T 2 T 3 T 4T STST 1T

T-

Figure12: Zero Order Hold

4.4 Numerical Simulation

Theequations olmotion(50}.(51), and (52) forthe revoluterobot can be expreeeedinmat:ri.J: f'arm88:

[::

O O A . I l a BJ

~ ~I~]=[::]

(65) where All

=

M(E'+H'+2EHCe)

AI,=M(£' +£HCe}

A21=M(E'+EHCe) .A..2J=A/E' An=M(HSP+ESr)'

BI='P+M(HSP+ESr XHCp+ ECr)ti 2+Mg(HSP+ESK) + 2MEHSePt+MEHSa2

BJ=OJ+M(HSP+ESK)ECkti '+MgESK-MEHS4J' BJ= ;-2M( HSP +ESKXHCP + £CK)tiP-2M( HSP+ESc )ECl iU

Multipli cati on of bothsidesof thematrix equationbythe inverse of'th e A matrix givesequations (or

a.

if,andi:

(66) if=A:D.BI -AuBl

AIIA:D. -AuAlI

(6 7)

_ AnB2-AlIBI

&=- - - -

A,,~-Aa2Al,

(68)

Lettingti=Kl,p=K2,t=K 3.a=a.b=

iJ.

andc=t .these becomeaset of sixfirstorderor dinary differentialequations:

P=b

Usingasimple Eule rone ste pintegrationschemegives:

(a--t%""')lliI-a- (a- -a- )IAI=KI""

(p..-p,.)/6J.b~

(...-~)/6J=K 2.M (r-. -~)/AJ=Ce4I (e--c..w)/!!J=KJ-t

where subscri pt oldindicatesvalues at thebeginningof a timeste p and subscript new indicate s values at the end ofatime step.

Usin gthisintegra tion scheme.the equations of motioncanbesolved88a functionoftime and themanipulator motion canbesimulatedusingthe Computed. Torque PIDcontrol model

Chapter 5

5.0 CROBOTS:A CADBa sedToolfo rRob ot Simulatio n

CROBOTSisaproposed software simulationtoolthatisdesigned to assistusersinthedesign.,application,and programming of educationaland industrial robots.Thesoftwareiswrittenusingthe AutoLispprogramming languageand runs as athirdpartyapplication inside AutoCAD R14.

CRO BOTS combinestbeimpressive graphicscapa bilityofAutoCADwith custom developedtoolsfor robot modelcreation, opera tio n cycle planning, and simulationofrobot controllers.Whilethesoftw areisprimarily designed asan educational toolforrobotics relatedcurricula.italsohaspotential for commercial applicati ons.

5.1 CommercialAp plicatio ns

Inmost industrialapplications.motion planningisperformed bythe pro gr ammerrather than a com pu te r. The actualrobot hardwareinitswork environm e ntisusedtoplan8strategy forperforming atask.Thisusually involvesusinga teach pendantor ahighlevel programming languageto facilli ta tethe teachingor theprogranuning ofthe robot.Once the

programmingiscomplete dfor a task,the robot hardwareisusedinplayback mode to test the program and the manipulation strategy.Debuggingis accomplishedthrough a teach-playback loop.The robot programmer can use CROBOTSinccniuncticn withnati ve AutoCAD functionality to assess preliminary robot operationcycle designs.Thiscould reduce unprod uctive programming of theactual robot hardware.

The robot'sperformanceinresponse to various control strategies may alsobegraphicallysimulated using CROBOTS.This may aid designersinthe prelim.inary stagesofcontro llerdesign.

5.2 Educational Applications

CROBOTSoffersa number of potential benefitsfor educational inBtitutionswho are faced withthe challenge of maintaining current and adequate resourcesrelated to the field of robotics with only limited budgets.

Firstly,CROBQTS iswrittenas athirdparty application forAutoCAD.

AutoCADhas dominated the PC CAD market andisrecognized8Sthe industry standard. Instruction in the useofAutoCADisa fundamental componentofengineering technology and degree programsinmany Colleges and Universitiesbothnationally and internationally.Thelearning curvefor CROBOTSisinsignificant for st ude ntsand educatorswho alreadyhaveskills in AutoCAD.Robotics instructionutilizing CROBOTS mayther eforebe focused on thelearning of key concepts andnot on how to usethe software.

Secondly,the ease of progranuninginAutoUSP shouldfacilitatethe developmentof additionalCROBOTS routinesby studentsand educato rs.

Studentswill alsobenefit fromthisopportunityto learn AutoUSP programming since thisskillmay be important for future related employmentinindustry.Furtherm ore,the graphical simulation capabilityof CROBOTS allows stude ntsto actually see the effectsof manipulating

dynamic andcontrolparameters onrobot performance.Thisma y help reinforce theoretical concepts deliveredusingtraditionallectures.Finally, thecostof robothardwareisofte nprohibi ti ve Cormoo educational inBtitutiona.CROBOTS provides studentswithanoffline tool which canbe usedto learn basic conce ptsbeforemoving ontotheactualrcbce hardware.

Inaddi ti on, since most engin ee ring educationalinstitutions already use AutoCAD,theyshould nothavetoincurtheadded C08tofagraphicsengine as wouldbe thecasewithcommercial oftlinerobotics progranuning software.

..

5.3 CROBOTS Overview

CROBOTSincl udes customizedmenus and new AutoLispfunctions which providetheuser withspecialized commands that facilitate the use of AutoCAD for robot geometric modeli ng,operationcycle plan ni ng,an d contro lle rsim ula tion. These specia lize d functi ons are mad e ava ila ble thr ou gh custo miza ti onoftheAutoCA Dmen u file as show n in figur e 13.

~_jj_~l;~~~-- ~f ^~1

~.l!~"".D 31...·- ::J~ <~I .r

/

..

0

l

0

..

r 0

"

D

~

b ~

.

u

'"

"

^r

.

_~ ^r

_.-

j

~

^~

lSI

r1~~3_~c~:';f~il/0D~~- ~~'-"C&d~I>'

ac ad ><>!>1

~ ^.. ~

Figure 13: Customized Pull-Down Menu Cor CROBOTS

Themodifiedmenufilepresen tstheuse rwitha CRO BOTS pull-downmenu whichinitiallypr ovid esonly a"LoadCROBOT Sⁿopt ion.Selectio nof this opt ion loads a secondcus to m menu whic h thenmak es allCROBOTS functions availa bletotheuser8Sshow n in figure 14.

~ttJ7~=i ti;~=--- .~-ti -1

!.!1SF"~'··

:: jj--

^3.=.1 =~

.-

^~

~lll -

'Ill

fl:i

.~ , ~

~

~

e

'¥

r

A

,

~ ^I,

~

... 1. . 1 _ ...

---. ---;:- _.!tJ~)1

?-1::::: ~7- ~ - - -

Figure 14: CROBOTS Main Menu

Themain menu feat u res five menu options including "Create New Robot Model","Home", "Simula te IK",Simulate Controller",and"Unl oad Robot Modeler",

Create New Robot Model

This menuopt ionprovidesa customized paramet ric drawin g tool which allows the usertoquickly generate a functionalgeometric model of the robot.Selection of this option displays an image box from which the usermay choose the robot configuration desired.The image box includes options forall five mechanical configurationsofrobots including revolute. cartesian, cylindrical,sphericaland SCARA88shownin figure15.

48

~^~i)_.!!l;l)!J~.:rJ.'!I~J1~~.!J^{'''',,,.o 31-- 3~j}

-

^3~

I

~I:I

+

§?- e l: ^I r _~

r

,~

~ , .,.

. _~

.

'it,)

"

^r-

• r

A ,

::::::'J

---..J

=

--"".J

~

I ~

--

=~!'I_d_^•••~CllOlIOTII

::-" iliIJ

j

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Figure 15: ImageMenufor"Cre a teNe w Robot Model"

Followingselectionof theconfiguration type,the user isonlyrequiredto input the desiredrobot dimensions.In the case of therevoluterobot,for exam ple, theuserisrequiredtoinput thebaseheigh t,thelength of link I, and thelength of link 2.A 3Dsolid model of therobot geo metry isthen automatically generatedbasedon thisinput8Sshowninfigure16.

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Figure16: Cre a t e NewRob otModel

Home

The home function simulates a standard capability of industrial robots toassume a reference or default start-up position.Industrial robots,which incorporate incremental optical encoders forposition measurement,require thiscapabilitytoallow reset of counters before programmi ng.In the CROBOTSprogram,th isfun ct ion similar ly init iali zes relevantAutoLisp variables andautomatically re-positions the robot to its requiredstartup position as showninfigure 17.

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Figure17: Hom ePos it ionfo rtheRobot

Sim u la te IK (Invers eKinema ti c s)

Thismenuoption providesthe user wit htwo subme nuoptions"Teach"

and"Run Cycle".The"Teach" optionallows theusertosimulate the

industrialon -lineprogramming practiceofteachingan drecordinga series of pointswhich defin e the robottrajector y.Thisfunctioncomputesthe in verse kinem at ics solution for themanipula torwhich thenallowsautomaticreo positioningofthe robotTCP.Followingselectionof thisoption,theuser specifiesthedesir ed pointlocat ion by eit her"cur sor picki ng"alocat ion in the draw in gwindow or bykeyboard entr yof thepointco-ordinatesat the Auto CAD comma nd prompt.In eit he rcase,specifica tionofthedesired point repositionsthemanipulator TCP8Sshowninfigure18 . Theuser is also provided withthe optio ntorecordthe poin tco-ordinates.Recordedpoints are

stored in an ASCIItext filewhich may belater used in CROBOTS or for downlo adto industrialrobotcon tro llers.

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Figure18: Teaching the Robot

Thefunction may be usedinconjunct ion with nativeAutoCAD commandsto simulate path trackin g by therobot.For example,if a circularpathis required, AutoCAD can be used todraw the circle andth en subdivide it into a number of segments. The teach fun ct ion can then be usedtoapproximate trackingof the circle with point to point moves from the beginning of a segmenttothe end of asegment . Figure19 illustrates asegmentedelli pt ical path with segment endpoints displayed.AutoCAD's object snap options ena ble the user to usethe "Tea ch" function to individually record each point and reposition themanipulator accordin gly.

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Figure19: TrajectoryApproximationusin g the "Teach"Function

The"Ru n Cycle"function enables the user to"playback" the repositioning of

the manipulator defined through the series of points recorded in the ASCII text file.In effect, this function allows graphical simulation of the P'TP motion of the manipulator along the approximated trajectory.

Simu lateController

This menu option enablesnumerical simulatio nof a computed torque with PIOcontroller and graphical simulation of the corresponding manipulation for PTP motion of the robot.Following selection of this option. the user is prompted for gain values,cycle time,and command angles as shown in figure 20. The function then providesa graphical simulation of the manipulator motionin response to torques generated based on the modified

computedtorque model.The overal loutputtorqueateachjoin tis calculated based on the dynamical equat ionsof moti on , the PIDcont rolsigna l,the relay controlsignal,and the use ofzeroor der hold,ifdesired .

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Theuser is also providedwithautilitywhich allowsautomaticgenerationof a 3D polylin epathwhich graph icallysum marizes themanipulatormotionfor the given con t rol param eter s .This capa bility maybeused to compa rethe effectsof changing controlparameterssuchas the ZOHsamplingrate.

Un load Robot Mode ler

Thismenu optionisincluded to allow the user to un loadCROBOTS andrestorethe standard AutoC ADmenu display.

54

5.4 Overviewof AutoCAD Customizati onforCROBOTS

The developmento( CROBOTS involvedthemodification ofthe AutoCADR14menufile.thecreationoCatemplatedrawingfile.thecreation o(slidefiles,thecreationo( blocks.and thecreationoCnew AutoLisp functions.Two modified menufiles werecreated.toprovide theCRO BOTS pull-downmenu: crobotal.mnuandcrobota2.mnu.Secondly.atemplate drawingfilewiththerequired layers and viewportsettings wasdeve loped.

Finall y.Autolisp functions were created to enablethe menu options"Cre ate New RobotModel", 'Teach","RunCrcle",and"Sim ula te Contro ller" .

5.4.1AutoCAD R14

AutoCADR14isageneral purpose ComputerAidedDesign(CAD ) draftingapplication(orthePC.Thesoftware providesan open architecture that permitstheuser to cuatomizeandextendmanyAutoCADfeaturesto sui t the particular requirementaathand. Thishasledto the developm ent o(

numerousthirdpartysoftwareapplicatio nswhichare specificallydesigned to improvethe productivityofCAD draft.i.nginparticulartechnical diBciplines such asArchitecturalDesign.AutoCAD cuetomizationcapabilityhas improved witheach new release and many pastcustomuser developed.

functions have become partoCthe standard. toolsoffered in(ollowin g releases of AutoCAD(Auto Desk.199 7).

AutoCADcus to mi za ti on capabilityenables use rs to:

Developcustom menus Programtheirown dialogue boxes

Create RCriptatoautomaterepetitive command sequences

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Define yourowntext:fon ts Define yourownline types Define yourownhatchpatterns Createcustomaymbola andpartslibraries Create templa te drawingswithcustomdefa ul tsettings ExportlImport OXFfiles to share drawing geometrywithother appli ca tions

Gene ra te slides or postBcript files

• UtilizetheWlnd ow8OLEcapa bili ti es

UseAutolisp,Diesel, andARX(Au toC ADRuntimeExtension) programming languagestoperformcalculations,automate repetitive tasks. and createnewAuwCADcommands

Gen erate3Dsolid modelsand extractenginee ringdata

ThesecustomizatiODcapabili tiescoupled with thealreadyimpressive graphics crea ti onandediting capa bili ti esofAutoCADmadeit wellsuitedfor thedevelopme nt of CROBOTS.Inaddi tion,sinceAutoCADistheaccepted indus try standardfor CADonthe PC.thetimeandC06tthatstudents, educators, robot designers,and robotprogramme rsincur tobecomeproficient intheuseofCROBOTSandrelevant AutoCADcommandsshould be reduced. Studen tsofrobotics shouldalsobenefitfurther sincetheymay acquire new AutoC ADskilla. suchas customi za tion,thatpote ntialem ployers maydesire.Inaddition,since moststudentshave alrea dy usedAutoCAD for EngineeringGraphicscour ses,itsuseinroboticscourseswillallowthem to focus on the learningofkeyrobotics conceptsand notthelearningofanew graphicssoftware pack age.

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5.4.2 TheTemplateDrawing File

Templates are drawing files with pre-establishedsettingsfor new drawings.These files allow userstopre-definedefaults for numerous drawing parameters including layers,text styles,dimensionstyles,symbollibraries, and a variety ofsyste m variableswhichcontrolthe performance of AutoCAD. TheCROBOTS template,revsetup.dwt,pre -definesthelayer structure and viewport setup as shownin figure21.This layer setup allowsthe automatic selectionof the manipu latorgeometryin a numberof CROBOTSAutoLisp functions. The viewport setu p is pre-definedtoautomaticallyprovidetheuser withatopviewdispl ay,a front viewdisplay,anda 3D viewdispl ay of the manipulator .

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