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Design of One Degree of Freedom Closed Loop Spatial Chains Using Non-Circular Gears

Permanent Link: http://ufdc.ufl.edu/UFE0024363/00001

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Title: Design of One Degree of Freedom Closed Loop Spatial Chains Using Non-Circular Gears
Physical Description: 1 online resource (32 p.)
Language: english
Creator: Harshe, Mandar
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Real world robot manipulators often use open-loop geometry where the motions governing positioning are independent from those controlling the orientation of the objects. These types of robot manipulators need expensive hardware and complicated controllers to manipulate the multiple (generally six) actuators. This research presents the design of one degree-of-freedom spatial mechanisms that use non-circular gears to constrain the motion. The geometry of one degree-of-freedom mechanisms and the design of the non-circular gears that link certain joint axes has already been dealt with. This research is concerned with the design of mechanism parameters like link lengths, joint offsets, and twist angles when the gear profiles are known. In a spatial body-guidance problem, representing the motion by systems of polynomial equations restricts the number of end-effector positions and orientations (end-effector poses) that can be used as inputs for mechanism design. An approach has been developed that takes any number of desired poses as guide points and develops a mechanism that approximately attains the desired poses over the course of its motion. A problem with implementing this design strategy is the inherent difficulty in accounting for orientation and position errors. The approach described here addresses this problem by defining a new error functional, calculated in the joint space domain. As the mechanisms being dealt with are single degree-of-freedom closed chains, the starting position is a crucial decision in the design process. The method outlines the choice of the starting position and details how this error term can be used along with optimization techniques on either the mechanism parameters or the non-circular gears. Numerical examples are presented.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Mandar Harshe.
Thesis: Thesis (M.S.)--University of Florida, 2009.
Local: Adviser: Crane, Carl D.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024363:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024363/00001

Material Information

Title: Design of One Degree of Freedom Closed Loop Spatial Chains Using Non-Circular Gears
Physical Description: 1 online resource (32 p.)
Language: english
Creator: Harshe, Mandar
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Real world robot manipulators often use open-loop geometry where the motions governing positioning are independent from those controlling the orientation of the objects. These types of robot manipulators need expensive hardware and complicated controllers to manipulate the multiple (generally six) actuators. This research presents the design of one degree-of-freedom spatial mechanisms that use non-circular gears to constrain the motion. The geometry of one degree-of-freedom mechanisms and the design of the non-circular gears that link certain joint axes has already been dealt with. This research is concerned with the design of mechanism parameters like link lengths, joint offsets, and twist angles when the gear profiles are known. In a spatial body-guidance problem, representing the motion by systems of polynomial equations restricts the number of end-effector positions and orientations (end-effector poses) that can be used as inputs for mechanism design. An approach has been developed that takes any number of desired poses as guide points and develops a mechanism that approximately attains the desired poses over the course of its motion. A problem with implementing this design strategy is the inherent difficulty in accounting for orientation and position errors. The approach described here addresses this problem by defining a new error functional, calculated in the joint space domain. As the mechanisms being dealt with are single degree-of-freedom closed chains, the starting position is a crucial decision in the design process. The method outlines the choice of the starting position and details how this error term can be used along with optimization techniques on either the mechanism parameters or the non-circular gears. Numerical examples are presented.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Mandar Harshe.
Thesis: Thesis (M.S.)--University of Florida, 2009.
Local: Adviser: Crane, Carl D.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024363:00001


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-RichardFeynman,Physicist(1918-1988) 3

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Iamextremelygratefultomysupervisorycommitteechair,Dr.CarlD.CraneIII,forgivingmeanopportunitytoworkatCIMAR.Hisguidanceandsupportthroughoutthedurationofmystudymadethisworkpossibleandgavedirectiontomywork.IalsothankDr.DavidDooner,ofUniversityofPuertoRico-Mayaguez,withwhomwecollaboratedonthisproject.Hisguidanceandsupporthelpedmeimmenselyinthiswork.IwouldliketothankDr.JohnSchuellerandDr.A.AntonioArroyoforagreeingtobeonmycommittee.IwouldliketoacknowledgethesupportprovidedbytheDepartmentofEnergyviatheUniversityResearchPrograminRobotics(URPR)throughUniversityofFloridagrantnumberDE-FG04-86NE37967.Ireceivedgreatsupportfrommyfriendsandcolleagues,includingVisheshVikas,SankethBhatandPriyankBagrecha.Iamextremelygratefulfortheirwordsofencouragementandsupport.IalsothankfellowstudentsatCIMAR.Finally,Iamindebtedtomyparentsandmysisterfortheirloveandsupport;Iowemysuccessandprogresstothem. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 6 LISTOFFIGURES .................................... 7 ABSTRACT ........................................ 8 CHAPTER 1INTRODUCTION .................................. 10 1.1Motivation .................................... 10 1.2Relatedwork .................................. 11 2PROBLEMSTATEMENTANDAPPROACH ................... 14 3DESIGNMETHOD ................................. 17 3.1ActualPathGeneration ............................ 17 3.2ErrorMeasure .................................. 19 3.3WeightingFactor ................................ 22 3.4HandlingMultipleSolutions .......................... 23 4NUMERICALEXAMPLE .............................. 25 5CONCLUSIONS ................................... 29 REFERENCES ....................................... 30 BIOGRAPHICALSKETCH ................................ 32 5

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Table page 3-1MechanismparametersforAside .......................... 19 3-2MechanismparametersforBside .......................... 19 3-3Systemparameters .................................. 19 4-1MechanismparametersforAside .......................... 25 4-2MechanismparametersforBside .......................... 25 4-3Desiredposesandorientations ............................ 26 4-4Numberofsolutionsbyreverseanalysis ....................... 26 4-5Solutionsselectedformin.errorcriteria ...................... 26 6

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Figure page 1-1A12-link,1-degreeoffreedommanipulator ..................... 12 2-1Acandidatemechanism ............................... 14 3-1CaseStudy:-A-sideanglesvsinputangle ..................... 20 3-2CaseStudy:-B-sideanglesvsinputangle ..................... 20 3-3CaseStudy:-Gearratios(Aside)vsstepnumber ................. 21 3-4Startandendcongurations ............................. 21 4-1Multiplepathsarisingfromdierentstartingcongurations ........... 27 4-2Variationofweightingfactors ............................ 28 7

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Realworldrobotmanipulatorsoftenuseopen-loopgeometrywherethemotionsgoverningpositioningareindependentfromthosecontrollingtheorientationoftheobjects.Thesetypesofrobotmanipulatorsneedexpensivehardwareandcomplicatedcontrollerstomanipulatethemultiple(generallysix)actuators.Thisresearchpresentsthedesignofonedegree-of-freedomspatialmechanismsthatusenon-circulargearstoconstrainthemotion.Thegeometryofonedegree-of-freedommechanismsandthedesignofthenon-circulargearsthatlinkcertainjointaxeshasalreadybeendealtwith.Thisresearchisconcernedwiththedesignofmechanismparameterslikelinklengths,jointosets,andtwistangleswhenthegearprolesareknown. Inaspatialbody-guidanceproblem,representingthemotionbysystemsofpolynomialequationsrestrictsthenumberofend-eectorpositionsandorientations(end-eectorposes)thatcanbeusedasinputsformechanismdesign.Anapproachhasbeendevelopedthattakesanynumberofdesiredposesasguidepointsanddevelopsamechanismthatapproximatelyattainsthedesiredposesoverthecourseofitsmotion.Aproblemwithimplementingthisdesignstrategyistheinherentdicultyinaccountingfororientationandpositionerrors.Theapproachdescribedhereaddressesthisproblembydeninganewerrorfunctional,calculatedinthejointspacedomain.Asthemechanismsbeingdealtwitharesingledegree-of-freedomclosedchains,thestartingpositionisacrucialdecisioninthedesignprocess.Themethodoutlinesthechoiceofthestartingpositionanddetailshow 8

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Todesignthisnewclassofmechanisms,wecaneitheremployexactsynthesisoraccuratesynthesis.Inexactsynthesis,wedesignthemechanismtoguidetherigidbodyorend-eectorexactlythroughtheprecisionpointswhileinaccuratesynthesisthemechanismdesignedissuchthatitoptimizesacertainobjectivefunctionthatdependsupontheprecisionpoints.Thedesignspecicationsarethesetofendeectorpositionsandorientationstobeattainedinagivenorder.Thedesignparametersarethemechanismparameters,suchaslinklengths,jointosetsandtwistangles,andthelocationofthebaseswithrespecttothedesiredposes.Alistingofsuchmethodsthatdealwithgeometricdesignhasbeenpresentedby( 1 ). Methodsinvolvingexactsynthesisrequirethatthenumberofindependentdesignequationsmustbelessthanorequaltothenumberofdesignparameters.Thisputsalimitonthenumberofprecisionpointsthatcanbedenedonapath( 2 { 4 ).Thecomplexityoftheproblemincreasesasthenumberofprecisionpointsincreaseandmethodslikeintervalanalysis( 5 )orpolynomialcontinuation( 6 { 9 )areusedtosolvetheequationsgenerated.Approximatesynthesisallowsustousemoreprecisionpoints,with 10

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Inthecaseofplanarmechanismdesign,methodslikeapproximatesynthesisandaccuratesynthesisareused.Thesemethodsusethepositionofapointonthecouplerlinkastheoutputfunctionanddenetheobjectivefunctionasthedierencebetweenthedesiredandactualpositions.( 10 )showsamethodfordesigningthreedimensionallinksthatalsousesonlythepositionofthecouplerpointtodesignthefourbarlinkage.Others,( 11 ; 12 )usetheconditionnumberasanobjectivefunctiontodesignspatialmanipulators. Inthisresearchthefocusisondeninganerrorfunctionalthatcanbeusedasanobjectivefunctionforoptimizationtechniques.Thiserrorfunctionalshouldincludetheerrorsbetweenthedesiredandactualpositionandorientationsoftheend-eector.Themostcommonwayofdoingthisistorstcalculatethepositionerror.Tocalculatetheorientationerror,theangulardierencesinthecorrespondingx,y,zaxesofthedesiredandactualcoordinatesystemsattachedtotheend-eectorsareconvertedtolengthscales.Thechoiceofthismultiplicationfactorissubjectiveandessentiallydecideswhichofthepositionororientationerrormustbegivenpriority.Thisproblemisaddressedbydeningtheerrorinthejointangledomainspace. 1-1 .Theworkdonein( 13 )focusedonthesamegeometryanddemonstratedasolutiontotheinversepositionanalysisofeachofthetwosubchains.Forthegivengeometry,itwasprovedthatupto16realsolutionsexistforeachsubchainforagivenpose.Furtherworkwasdoneand( 14 )dealtwithdesigningnon-circulargearsforthegivenmechanism,thatwouldallowtheend-eectortoaccuratelyreacheachpose.Howeveritisdiculttoensurethatthegearproleremainsfeasible.Inalmostallcases,thecalculated 11

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A12-link,1-degreeoffreedommanipulator gearratiosvariedfrompositivevaluestonegativevalues.Thisimpliedthatforcertainsectionsofthepath,internalgearswerenecessarywhileforothersectionsexternalgearswouldbeneeded.Implementingsuchsolutionsispracticallynotfeasible. Anotherwayoflookingattheproblemistotreatthegearratiosasxed,anddevelopaschemethatdesignsthemechanismparameterssothatrigidbodyguidanceispossible.Thistypeofdesignisdealtin( 15 ),whereadesignstrategyusingoptimizationforopenloopchainsisexplained.Asimilarmethodisemployedin( 16 )whereFourierAnalysisisusedtoaidthedesignsynthesis.However,inboththeseapproaches,thefocusisonforceorenergyrelateddesigncriteria.Theoptimizationapproachwasusedtodevelopanopenloopchainsuchthattheend-eectorbestapproximatesthedesiredsetofposes.Forthepurposeofoptimization,thepaperdescribesamethodthatusestheorientationtoparameterizethecurvesanddeneacorrespondencepointthatcanbeusedtocalculatetheerror.TheFourierAnalysisapproachaimstoperformnumberanddimensionalsynthesis. Inthisresearch,thegeometryofthemechanismisalreadyknown.Thedimensionalsynthesisistheaimofthedesign.Also,apurelykinematicapproachistakenandanyenergyorforceconsiderationsareignoredinthedesign.Thepathisparameterizedintermsoftheinputanglebutacompletelydierenterrorfunctionalisdeveloped.The 12

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ThemechanismtobedesignedhasthegeometryasshowninFig. 2-1 .Thepairsofaxesthatareparalleltoeachotherare($1;$2),($3;$4),($5;$6),($7;$8),($9;$10)and($11;$12),where$ireferstothejointbetweenlinksiandi+1.Forthistypeofgeometry,themechanismparametersaregivenbythecondition: (2{1) Figure2-1. Acandidatemechanism Themechanismcanbetreatedastwo6linkopenchainswithcoincidentend-eectors.Thesidecontaining3gearpairsiscalledtheAsideandthesidecontaining2gearpairsiscalledtheBside.CoordinatesystemsareattachedtothebaseoftheAandBsideandalsoattheend-eector.The`A xed'coordinatesystemhastheoriginon$12).The`B xed'coordinatesystemhastheoriginxedon$1withthezaxisalong$1/.The`tool'coordinatesystemisxedtotheend-eectorandrepresentsthelocationandorientationoftheend-eector.Theparametersneededfordesigningthemechanismare: fixedA fixedT: xed'coordinatesystemwithrespectto`B xed'coordinatesystem 14

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fixedtoolT: xed'coordinatesystem Acandidatedesignisevaluatedbysweepingtheinputangleofacandidatemechanismtoreachthegivendesiredposes.Piisthe44matrixthatdenesthedesiredpositionandorientationoftheendeectorattheithpose.Areverseanalysisasoutlinedby( 13 )isperformedwhichresultsinupto16realsolutionsforeachpose.Wenotethateachoftherealsolutionsisvalidandcouldbeachieved. Sincethisisarigidbodyguidanceproblem,thepathisdenedbyasetofmatricesPiwhichmustbereachedinaparticularorder.Inordertoeectivelyuseapproximatesynthesiswithoptimization,thepathmustbedescribedusingaparametricrepresentation.Thisparameterwillalsoallowustodenecorrespondingpointsonsimilarpaths.Thegearscouplingtheparalleljointaxesresultinthemechanismbeingasingledegree-of-freedomserialchainandhence,thepathtracedoutbytheend-eectorcanberepresentedintermsofoneparameter.Theinputangle(therstangleontheBside)isusedasthisparameter.Thefocushereistodescribethemotionintermsofjointanglesasopposedtointermsofthepathcoordinates. Ifthestartingcongurationofthecandidatemechanismisknown,theinputanglecanbeeectivelyusedtodescribethemotionoftheend-eector,forthatstartingconguration.Thiscanbeachievedbyusingascrewtheoryapproach( 17 ; 18 )anditerativelyusingthevelocityequationstondthepositionoftheend-eectorcorrespondingtoeachvalueoftheinputangle.ItisthusnotedthateachsolutioninthesetofjointanglesUj1willgiverisetoadierentbehavioroftheend-eector,andalsoadierentpath. Thus,weendupwiththedesiredpathandsetsofactualpathsallrepresentedintermsofthestartingcongurationandparameterizedintermsoftheinputjointangle.Pointsondierentpathsthatarerepresentedbythesamevalueofinputanglearedenedtobecorrespondencepoints.Theerroristhendenedintermsofthedierenceinthe 15

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13 ; 19 ; 20 ).Consideringtheend-eectorasapartoftheA(orB)sideofthemechanism,thevelocityoftheend-eectorisgivenbyEqn.( 3{1 )( 21 ). Here,i1!irepresentstheangularvelocityoflinki+1withrespecttolinki.i1$irepresentsthescrew(orthePluckercoordinates)aboutwhichlinki+1moves,withrespecttolinki.Sincealljointsareassumedrevolutehere,i1$irepresentthejointaxes.$7representsthescrewaboutwhichtheend-eectormoveswithrespecttogroundand!7isthecorrespondingangularvelocity. However,duetothepresenceofgears,theequationssimplifygreatly.FortheAside,thethreegearpairsreducetheequationtooneinthethreeunknownangularvelocities,asgiveninEqn.( 3{2 ).TheBsidereducestoEqn.( 3{3 )in4angularvelocities,ofwhichoneistheknowninputangularvelocity(0!1)oftherstjoint. 17

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3{2 andEqn. 3{3 givesEqn. 3{4 whichrepresent6scalarequationsin6unknownangularvelocities.TheseequationsarethensolvedandtheBsidejointanglesareupdatedforasmallchangeoftheinputangle,byusingEqns. 3{5 .AforwardanalysisoftheBsideisdonetogettheupdatedendeectorposeandthisisthenusedtogettheexactjointanglesfortheAsidebyinverseanalysis.Asinverseanalysisgiverisetomultiplesolutions,thenewanglesthatareclosesttotheprevioussetofanglesarechosen. (0'1B)i+1=(0'1B)i+'(12B)i+1=(12B)i+(1!2 0!1)'(23B)i+1=(23B)i+(2!3 0!1)'(34B)i+1=(34B)i+(3!4 0!1)'(45B)i+1=(45B)i+(4!5 0!1)'(56B)i+1=(56B)i+(5!6 0!1)' Acasestudyforthismethodshowsusthatthismethodindeedgivesthecorrectmotionfortheclosedloopchainastheinputangleisswept.ThiswasdonebyperformingthecalculationsasgivenbyEquations 3{5 togetnewangles.Thenusingthesenewangles,thedisplacementanalysesoftheAandBsideareperformedtogetallthescrewaxescoordinates.ThisiterativemethodusingEquations 3{4 and 3{5 ,givesusthejointanglesfortheentirerangeofmotion.ThemechanismparameterschosenforthiscasestudyaregiveninTable 3-1 andTable 3-2 withtheotherparametersgiveninTable 3-3 Figure 3-1 andFig. 3-2 showthevariationoftheanglesastheinputanglesweepsfromitsinitialvalueby0.5radians.Thegearswereallchosentohave1:1ratios.Figure 3-3 showsthatallcalculatedgearratios(byforwarddierencemethod)closelyapproximatetheactualratiosof1.Asmallervalueofstepsize'givesvaluescloserto 18

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MechanismparametersforAside Osetdistance(in.)Linklengths(in.)Twistangles(deg) S2=6a12=312=0S3=0a23=1023=90S4=6a34=334=0S5=0a45=1045=90S6=6a56=356=0 Table3-2. MechanismparametersforBside Osetdistance(in.)Linklengths(in.)Twistangles(deg) S2=6a12=412=0S3=0a23=1423=90S4=6a34=434=0S5=0a45=1245=90S6=6a56=456=0 1.InFig. 3-3 ,thevaluesA12,A34andA56denotethethreegearratiosforthegearpairsontheAside.A12isthegearratiobetweenjoints1and2,A34betweenjoints3and4,andA56betweenthejoints5and6.Asseenfromthegure,thecalculatedvaluesofgearratiosarewithin99.5%oftheactualvalues.Thisisaverygoodestimateandthusensuresthatthemethodgivesamotionthatisaccuratetoagreatextent. Table3-3. Systemparameters fixedA fixedT0BB@1001800150101200011CCA6BtoolT0BB@10040102001300011CCA6B6AT0BB@10080012010400011CCA

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CaseStudy:-A-sideanglesvsinputangle Figure3-2. CaseStudy:-B-sideanglesvsinputangle 20

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CaseStudy:-Gearratios(Aside)vsstepnumber BEnd:Inputanglesweep=0.5radians Startandendcongurations 21

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Thus,forposePi,thereexistsjointanglesij;desiredandij;actual,wherejreferstothejointnumber.Foreachpose,theerrorcanbedenedas: 2(3{6) Here,wiistheweightingfactorthatisdescribedinthenextsection.Theerrorfortheentirepath,canbedenedas: Letr0bethelocationoftheend-eectorinthegivenconguration.Tocalculatetheweightforjointi,theangleiisvariedbyasmallamount,.Letribethenewpositionoftheend-eectorifthejointiisvariedby.Then,theweightwiisdenedas: 22

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LetibethepathgeneratedbythesolutionUi1.Toevaluatethescoreforthesecondposeonthispathwithrespecttothedesiredsecondpose,thesetofsolutions^U2mustbechosenfromallUj2,suchthattheposecorrespondingto^U2liesini. Uj2f1gischeckedforallj.IfUj2f1g=ikf1gforsomejandk,thetwojointanglesetsaresaidtocorrespondtothesamepose.TheikisthejointanglesetfortheactualposewhiletheUj2representsthedesiredsecondpose.Theerrorej2isthencalculatedforthispose.Thesamecalculationsareperformedforallj,ktogetallej2.Theerrorforthesecondposeisthendenedas Theerrorforeachposeiscalculatedusingthesamemethod,usingEqn.( 3{6 )togiveejl.Theerrorfortheposeisthendenedas Thus,usingEqn.( 3{7 ),theerrorforthepathiisgivenbyEi.Sinceeachpathiwillgiverisetoadierenterror,thebestpathmustbechosentodenethemechanismerror.ThisisdonebydeningtheerrorEas 23

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Table4-1. MechanismparametersforAside Osetdistance(in.)Linklengths(in.)Twistangles(deg) S2=6a12=312=0S3=0a23=1023=90S4=6a34=334=0S5=0a45=1045=90S6=6a56=356=0 Table4-2. MechanismparametersforBside Osetdistance(in.)Linklengths(in.)Twistangles(deg) S2=3.4947a12=14.23812=0S3=0a23=0.741123=59.2992S4=1.3465a34=12.900934=0S5=0a45=6.134945=76.8924S6=6a56=10.378256=0 Theresultsofusingthismethodtogeneratetheerrormeasureofamechanismonanexamplesetofposesisgivenbelow.Table 4-1 andTable 4-2 listthemechanismparametersfortheAandtheBsiderespectively.ThesixdesiredposesPiarelistedinTable 4-3 andthesolutionstoreverseanalysisforeachposefortheAsideandBsidearecalculated.Forthegivencombinationofposesandmechanismparameters,thenumberofsolutionsobtainedforeachsideislistedinTable 4-4 .Forthisparticularexample,theallgearpairsareassumedtobecircular,withgearratio1. Sincethereare2solutionsforAsideand14fortheBsideforthestartingpose,thereare28possiblestartingcongurationsforthemechanism.Thepathstakenbytheend-eectorineachofthe28casesareshowninFig. 4-1 .Attherstpose,theerroriszero.Eachofthesecongurationsgivesusapaththatthemechanismend-eectorfollows.ThemethoddescribedinthispaperchoosesthesolutionsgiveninTable 4-5 forthestartingconguration.Theinputjointanglesfortheother5posesarealsolisted.Sincethemechanismisaonedegree-of-freedomserialchain,specifyingonlytheinputanglesforthesuccessiveposesdenesthemechanismconguration. 25

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Desiredposesandorientations PoseTransformationmatrix(inrad.&in.) Pose10BB@0:47710:47520:739310:10430:59940:43930:66917:91450:64280:76240:075240:652400011CCAPose20BB@0:44980:46690:76149:57690:59490:47920:64538:11360:66620:74320:062270:219900011CCAPose30BB@0:41120:47730:77669:04670:57210:52810:62768:37700:70970:70230:05590:017000011CCAPose40BB@0:35750:50620:78488:52760:52440:58650:61728:80860:77280:63220:05580:137300011CCAPose50BB@0:28030:55140:78578:0230:44080:65310:61579:41160:85270:51900:06000:122200011CCAPose60BB@0:16830:60340:77957:51960:30400:72050:623310:17750:93770:34190:06210:015900011CCA Numberofsolutionsbyreverseanalysis PoseAsideBside Pose1214Pose2214Pose3212Pose4210Pose528Pose626 Table4-5. Solutionsselectedformin.errorcriteria Startingjointangles,Aside0:58850:58850:70941:49482:87491:3041 Pose3Inputangle2.2890 Pose4Inputangle2.3890 Pose5Inputangle2.4890 Pose6Inputangle2.5890 26

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Thedierentpathstracedbyend-eectorwhendierentstartingcongurationsarechosen.Allpathsstartfromthesamestartingpose,highlightedinred. 27

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Variationoftheweightingfactorsasthemechanismsweepsovertheselected"best"path ThevariationoftheweightingfactorsfortheselectedpathareshowninFig. 4-2 .Whilethechangesintheweightingfactorsarenotlarge,theydemonstratethefactthatthecontributionoftheanglesdependsonthecurrentorientation.Ascanbeseenfromthegure,the2and3haveagreatereectthan1ontheend-eectorerror.Forthemechanismpresented,theerrorwascalculatedtobe42.49mm. 28

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Amethodforthedesignofonedegreeoffreedom,closedloopspatialchainsthatutilizeplanarnon-circulargearswaspresentedinthisthesis.Themethodaddressedanimportantproblemofaccountingfororientationerrorsalongwithpositionerrorsintherigidbodyguidanceproblem.Byutilizingthejointspacedomain,theerrortermdenedhascombinedthetwoerrorsbyavoidingtheissueofmixeddimensionsornon-homogeneousterms.Theworkpresentedsetsupaschemethatcouldbeusedtoapplyoptimizationtechniquesinthedesignofotherclosedloopmechanisms. Thismethodiscomputationallyintensiveduetotheiterativenatureofthealgorithm.ThemethodwasimplementedusingMatlab7.0softwareonaIntelCore2T5300(1.73Ghz)processor,witha2Gbram.Withparallelprocessingdisabled,asinglerunforthecodetookabout996.32CPUsecondstocalculatetheerrorfortheexamplelistedpreviously.However,sinceeachpathisindependentoftheother,thismethodissuitedforparallelarchitectureandparallelprocessingwillgreatlyimprovethespeed. Thegeometryofhavingconsecutiveaxesparallelgreatlyincreasesthesimplicityofthegearsthatareusedtocouplethejointaxes.Theproblemcanbebroadenedwherethemechanismgeometryischangedtosimplifytheinverseanalysisproblemandnon-planarnon-circulargearsareusedtocoupleconsecutiveaxes.Theerrorfunctionalinsuchacasewouldbethesameastheonepresentedinthispaper.Theproblemofextendingthismethodtomechanismswithadierentgeometryisbeinginvestigated. 29

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[1] M.Bodduluri,J.Ge,M.J.McCarthy,B.Roth,TheSynthesisofSpatialLinkages,ModernKinematics:DevelopmentsintheLastFortyYears,J.WileyandSons,NewYork,1993. [2] B.Roth,Analyticdesignofopenchains,in:O.Faugeras,G.Giralt(Eds.),ThirdInternationalSymposiumofRoboticResearch,MITPress,1986. [3] L.W.Tsai,Designofopenloopchainsforrigidbodyguidance,Ph.D.thesis,StanfordUniversity(1972). [4] C.H.Suh,C.W.Radclie,KinematicsandMechanismDesign,WileyandSons,NewYork,NewYork,1978. [5] E.Lee,C.Mavroidis,J.P.Merlet,Fiveprecisionpointsynthesisofspatialrrrmanipulatorsusingintervalanalysis,JournalofMechanicalDesign126(5)(2004)842{849. [6] A.P.Morgan,C.W.Wampler,Solvingaplanarfour-bardesignproblemusingcontinuation,JournalofMechanicalDesign112(4)(1990)544{550. [7] C.W.Wampler,A.P.Morgan,A.J.Sommese,Numericalcontinuationmethodsforsolvingpolynomialsystemsarisinginkinematics,JournalofMechanicalDesign112(1)(1990)59{68. [8] B.Roth,F.Freudenstein,Synthesisofpathgeneratingmechanisms,JournalofEngineeringforIndustry85B(1963)298{306. [9] C.W.Wampler,A.P.Morgan,A.J.Sommese,Completesolutionofthenine-pointpathsynthesisproblemforfour-barlinkages,JournalofMechanicalDesign114(1)(1992)153{159. [10] V.I.Kulyugin,Designofthreedimensionalfour-linkmechanismsconformingtothetravelofthedrivenlinkandcoecientofincreaseinvelocityofthereversestroke,in:AnalysisandSynthesisofMechanisms,AmerindPublishingCo.Pvt.Ltd.,1975,pp.152{162. [11] J.K.Salisbury,J.J.Craig,Articulatedhands:Forcecontrolandkinematicissues,TheInternationalJournalofRoboticsResearch1(1)(1982)4{17. [12] L.-W.Tsai,S.Joshi,Kinematicsandoptimizationofaspatial3-upuparallelmanipulator,JournalofMechanicalDesign122(4)(2000)439{446. [13] J.R.Mckinley,C.Crane,D.Dooner,Reversekinematicanalysisofthespatialsixaxisroboticmanipulatorwithconsecutivejointaxesparallel,in:InternationalDesignEngineeringTechnicalConferences&ComputersandInformationinEngineeringConference,no.DETC2007-34433,ASME,LasVegas,2007. 30

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J.R.Mckinley,Three-dimensionalrigidbodyguidanceusinggearconnectionsinaroboticmanipulatorwithparallelconsecutiveaxes,Ph.D.thesis,UniversityofFlorida(2007). [15] V.Krovi,G.K.Ananthasuresh,V.Kumar,Kinematicandkinetostaticsynthesisofplanarcoupledserialchainmechanisms,JournalofMechanicalDesign124(2)(2002)301{312. [16] Y.-W.Pang,V.Krovi,Kinematicsynthesisofcoupledserialchainmechanismsforplanarpathfollowingtasksusingfouriermethods,in:ASMEDesignEngineeringTechnicalConferences,no.DETC2000/MECH-14188,ASME,Baltimore,Maryland,2000. [17] R.S.Ball,ATreatiseontheTheoryofScrews,UniversityPress,Cambridge,1900. [18] K.H.Hunt,KinematicGeometryofMechanisms,ClarendonPress;OxfordUniversityPress,Oxford;NewYork,1978. [19] J.Duy,C.Crane,Adisplacementanalysisofthegeneralspatial7-link,7rmechanism,MechanismandMachineTheory15(3)(1980)153{169. [20] H.-Y.Lee,C.-G.Liang,Displacementanalysisofthegeneralspatial7-link7rmechanism,MechanismandMachineTheory23(3)(1988)219{226. [21] J.M.Rico,J.Gallardo,J.Duy,Screwtheoryandhigherorderkinematicanalysisofopenserialandclosedchains,MechanismandMachineTheory34(4)(1999)559{586. 31

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MandarHarshewasbornin1985inMumbai,India.HedidhisschoolinginE.N.N.SSchool,PuneandattendedjuniorcollegeatFergussonCollege,Pune.HethenattendedM.E.SCollegeofEngineering,aliatedwiththeUniversityofPune,andgraduatedwithaBachelorofEngineeringinmechanicalengineeringinJune2007.HedidhisdegreeprojectatNirmitiStampingsPvtLtd,Pune,Indiawhereheworkedondevelopingalowcostroboticarm.InAugust2007,hejoinedtheCenterforIntelligentMachinesandRobotics(CIMAR)attheUniversityofFlorida,andreceivedhisMasterofSciencedegreeinthespringof2009.Heplanstopursuetohisdoctorateintheareaofmechanismtheoryfocusingonkinematicsandcomputationalgeometry. 32