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A Tensegrity-Based Compliant Mechanism

Permanent Link: http://ufdc.ufl.edu/UFE0043573/00001

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Title: A Tensegrity-Based Compliant Mechanism Analysis and Application
Physical Description: 1 online resource (137 p.)
Language: english
Creator: Moon, Young Jin
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: compliance -- control -- mechanism -- robot -- tensegrity
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This research presents an analysis of a planar and spatial tensegrity-based mechanisms. In a planar case of this study, a moving body is joined to ground by four compliant leg connectors. Each leg connector is comprised of a spring in series with an adjustable length piston. Two problems are solved in this paper. In the first, the values of the four spring constants and free lengths are given and the lengths of the four pistons are determined such that: (1) the top body is positioned and oriented at a desired pose; (2) the top body is at equilibrium while a specified external wrench is applied; and (3) the total potential energy stored in the four springs equals some desired value. In the second problem, the values for the four spring free lengths are given and the values for the four spring constants and the lengths of the four pistons are determined such that conditions (1) and (2) from above are met and also the instantaneous stiffness matrix of the top body equals a specified set of matrix values. The paper formulates the solutions of these two problems. Numerical examples are presented. The above process is also done for a spatial tensegrity-based compliant mechanism with the same manner. The velocity and acceleration analyses are conducted and the dynamic equation is derived for the spatial case. Finally, it is described that this type of compliant mechanism can be used to control the pose and contact force in connection of the modular robots.
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 Young Jin Moon.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Crane, Carl D.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043573:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043573/00001

Material Information

Title: A Tensegrity-Based Compliant Mechanism Analysis and Application
Physical Description: 1 online resource (137 p.)
Language: english
Creator: Moon, Young Jin
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: compliance -- control -- mechanism -- robot -- tensegrity
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This research presents an analysis of a planar and spatial tensegrity-based mechanisms. In a planar case of this study, a moving body is joined to ground by four compliant leg connectors. Each leg connector is comprised of a spring in series with an adjustable length piston. Two problems are solved in this paper. In the first, the values of the four spring constants and free lengths are given and the lengths of the four pistons are determined such that: (1) the top body is positioned and oriented at a desired pose; (2) the top body is at equilibrium while a specified external wrench is applied; and (3) the total potential energy stored in the four springs equals some desired value. In the second problem, the values for the four spring free lengths are given and the values for the four spring constants and the lengths of the four pistons are determined such that conditions (1) and (2) from above are met and also the instantaneous stiffness matrix of the top body equals a specified set of matrix values. The paper formulates the solutions of these two problems. Numerical examples are presented. The above process is also done for a spatial tensegrity-based compliant mechanism with the same manner. The velocity and acceleration analyses are conducted and the dynamic equation is derived for the spatial case. Finally, it is described that this type of compliant mechanism can be used to control the pose and contact force in connection of the modular robots.
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 Young Jin Moon.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Crane, Carl D.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043573:00001


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ATENSEGRITY-BASEDCOMPLIANTMECHANISM:ANALYSISANDAPPLICATIONByYOUNGJINMOONADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011YoungjinMoon 2

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Tomywife,son,anddaughter 3

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ACKNOWLEDGMENTS Iwouldliketoexpressmygratitudetomyadviser,Dr.CarlD.CraneIIIforprovidingmewithhisguidanceandadvice,andtheopportunitytocompletemyPh.D.study.Ialsowishtothankallmycommitteemembers,Dr.A.AntonioArroyo,Dr.WarrenDixon,Dr.RodneyRoberts,andDr.GloriaJ.Wiens,andmyex-advisor,Dr.ScottBanksfortheireffort.IalsowouldliketothankallmycolleaguesintheCenterforIntelligentMachinesandRobotics,especiallyMosesAnubi,fortheirhelpandsupport.Finally,Iwouldliketothankmywife,parents,andparents-in-lawfortheirpatienceandencouragement. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................... 4 LISTOFTABLES ....................................... 7 LISTOFFIGURES ....................................... 8 ABSTRACT ........................................... 10 1INTRODUCTION .................................... 11 1.1Background ..................................... 11 1.2Motivation ...................................... 12 1.3ProblemStatement ................................. 13 2TENSEGRITYANDTENSEGRITY-BASEDMECHANISM ............. 14 2.1Tensegrity ...................................... 14 2.2TensegrityMechanismandTensegrity-BasedMechanism ............. 14 2.3Tensegrity-BasedCompliantMechanism ...................... 16 3ANALYSISOFAPLANARMECHANISM ....................... 19 3.1MechanismDescription ............................... 20 3.2StiffnessMapping .................................. 21 3.3KinestaticAnalysis ................................. 25 3.4StiffnessSynthesis ................................. 30 3.5NumericalExamples ................................ 33 3.5.1Example1 .................................. 33 3.5.2Example2 .................................. 36 3.5.3Example3 .................................. 38 3.5.4Example4 .................................. 42 4ANALYSISOFASPATIALMECHANISM ....................... 44 4.1MechanismDescription ............................... 44 4.2StiffnessMapping .................................. 45 4.3KinestaticAnalysis ................................. 51 4.4StiffnessSynthesis ................................. 56 4.5NumericalExamples ................................ 59 4.5.1Example1 .................................. 59 4.5.2Example2 .................................. 64 4.5.3Example3 .................................. 66 5DYNAMICSOFATENSEGRITY-BASEDCOMPLIANTMECHANISM ...... 70 5.1VelocityAnalysis .................................. 71 5

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5.2AccelerationAnalysis ................................ 73 5.2.1ProblemStatement ............................. 73 5.2.2AccelerationoftheJoint .......................... 74 5.2.3AccelerationofthePiston ......................... 77 5.3DynamicEquation ................................. 78 5.3.1LagrangeMethod .............................. 79 5.3.2LinearVelocities .............................. 81 5.3.3AngularVelocities ............................. 84 5.3.4KineticEnergy ............................... 87 5.3.5PotentialEnergy .............................. 89 6MODULARROBOTAPPLICATION .......................... 90 6.1Introduction ..................................... 90 6.2ProblemStatement ................................. 93 6.3ConnectorDesign .................................. 94 6.3.1WorkspaceandSingularity ......................... 95 6.3.2SystemDescription ............................. 100 6.3.3DynamicSimulation ............................ 100 6.4DynamicEquation ................................. 103 6.5PoseControl .................................... 111 7CONCLUSION ...................................... 121 APPENDIXDERIVATIONOFTHEDYNAMICEQUATION ................ 123 REFERENCES ......................................... 130 BIOGRAPHICALSKETCH .................................. 137 6

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LISTOFTABLES Table page 4-1Resultsforexample2 ................................... 67 6-1Optimization ....................................... 97 6-2Optimizationresult .................................... 97 6-3Simulationparametersandinitialvalues ......................... 116 7

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LISTOFFIGURES Figure page 2-1ExamplesoftensegritymechanismsbyArsenaultandGosselin ............. 15 2-2Examplesoftensegrity-basedcompliantmechanisms .................. 17 3-1Planarmechanismschematic ............................... 20 3-2Compliantlegconnector ................................. 21 3-3Example1:forcemagnitudesforchangeinmagnitudeoftheexternalwrench ..... 36 3-4Example2:poseofthemechanism ............................ 37 3-5Example3:pathtrajectory ................................ 39 3-6Example3:pistonlengths(solutionA) .......................... 40 3-7Example3:pistonlength(solutionB) .......................... 40 3-8Example3:posechange(solutionA) ........................... 41 3-9Example3:posechange(solutionB) ........................... 41 4-1Spatialmechanismschematic ............................... 45 4-2Coordinateofalegconnector ............................... 46 4-3Example1:mechanismconguration .......................... 62 4-4Example1:forcemagnitudesandpistonlengthsforchangeoftheexternalwrenchintensity .......................................... 64 4-5Example1:forcemagnitudesandpistonlengthsforchangeofthepotentialenergyinallsprings ......................................... 65 4-6Example3:desiredpathoftheoriginandorientationofthetopcoordinatesystem ... 68 4-7Example3:pistonlengths ................................ 69 4-8Example3:poseofthemechanism ............................ 69 5-1HPSlegconnector .................................... 72 5-2Pistonpart ......................................... 77 6-1Examplesofapplicationsusingcompliantparallelmechanisms ............. 91 6-2Changeofthemoduleconnectorsinadesiredpath ................... 94 6-3Resultofoptimizationusingthegeneticalgorithm .................... 97 8

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6-4Workspacefortheselectedorientationangles ...................... 98 6-5Singularareafortheselectedorientationangles ..................... 99 6-6Designedmechanism ................................... 100 6-7Twomodularrobots .................................... 101 6-8SimMechanicsdiagram .................................. 101 6-9SimMechanicsdiagram:pistonpart ........................... 102 6-10SimMechanicsdiagram:springpart ........................... 102 6-11End-effectordisplacementforspiralmotion ....................... 103 6-12Pistondisplacementforspiralmotion .......................... 103 6-13X-Yplotforspiralmotion ................................ 104 6-14Topbodyfreebodydiagram ............................... 104 6-15Legfreebodydiagram .................................. 110 6-16Controlschematic ..................................... 112 6-17Positionandorientation .................................. 117 6-18Velocity .......................................... 117 6-19Pistondisplacement .................................... 118 6-20Pistondisplacementerror ................................. 118 6-21Parameterestimates .................................... 119 6-22Intermediatecontrolinputatthejoint .......................... 119 6-23Controlinputatthepiston ................................ 120 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyATENSEGRITY-BASEDCOMPLIANTMECHANISM:ANALYSISANDAPPLICATIONByYoungjinMoonDecember2011Chair:CarlD.CraneIIIMajor:MechanicalEngineering Thisresearchpresentsatensegrity-basedcompliantmechanism.Planarandspatialtensegrity-basedcompliantmechanismsweredesignedusingthetensegrityconcept.Problemsforthekinestaticanalysisandstiffnesssynthesiswerethenperformedforbothmechanisms.Fortheformer,itwasshownthatamaximumoftworealsolutionscouldexist.Forthelater,auniquesolutionisobtainedunlesscertaingeometricconditionsoccurwhichcausethematriceswhichmodeltheproblemtobecomesingular.VelocityandaccelerationanalyseswerealsoperformedandthedynamicequationwasderivedusingtheLagrangeformulabasedonthegeneralizedcoordinatesoftask-spacevariables.Asanapplication,aplanartensegrity-basedmechanismwasconsideredasaconnectorofagroundvehicletypemodularrobotwhichtracksadesiredpath.Inordertodesignaspecicmechanism,itsworkspaceandsingularitywereinvestigated.Thejointpositionsinthemechanismwerechosensothattheworkspacewasmaximizedandsingularpointswereminimizedbyusingageneticalgorithm.Athreedimensionalmodelwasmadeandsimulatedincommercialprograms.Thesimplieddynamicmodelforcontrolwasderivedandaposecontrollerwasdesignedandsimulated. 10

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CHAPTER1INTRODUCTION Roboticmechanisms,whichcanbedatedtothemechanicalbirddevisedbyaGreekmathematician,Archytas,in425BCE[ 87 ],havebecomenearuniversal.Amongvarioustypesofroboticmechanisms,parallelmechanismshavebeenwidelyusingintheeldsofmanufacturing,motionsimulators,andvibrationcontrollers.Thisdissertationpresentsanddiscussestheresearchactivitiesrelatedtoaspecialparallelroboticmechanismcalledatensegrity-basedcompliantmechanism. 1.1Background Kinematicorroboticmechanismscanbegenerallyclassiedasparallelorserialtypesaccordingtotheirkinematicconguration:closedoropenkinematicchain,respectively.Theparallelmechanismshavebodiesgenerallyatopmovingbodyandbasebodyjoinedbymultipleconnectors,whileinserialmechanisms,thebodiesareseriallyconnectedbyjoints.Therealsoexisthybridmechanismsthatincludebothtypes.Comparedtoserialmechanisms,parallelmechanismsarestructurallystifferandtheycanachievemoreaccuratepositioncontrol.Also,theinversekinematicsoftheparallelmechanismscaneasilybesolved.Ontheotherhand,theforwardkinematicsofparallelmechanismsismorecomplicatedthanthatforserialmechanismsandtheirworkspaceissmaller.Forthesereasons,theparallelmechanismshavebeenusedinenvironmentsthatrequireshighstiffnessandspeedorhighaccuracyandbandwidthinasmall-sizedtaskspace. ThemostcommonparallelmechanismistheGough-Stewartplatform,whosemovingplatformisconnectedtogroundbysixlegswhoselengthsareadjustable.TheGoughmechanismwasconstructedformeasuringtirewearandtheStewartplatformwasproposedasaightsimulator.Themechanismsuseverysimilararchitectures,thussuchtypesofmechanismsarecalledGough-Stewartplatforms.BasedonthisGough-Stewartplatform,varioustypesofparallelmechanismsormanipulatorshavebeenproposed,developed,andstudiedforspatialdevices, 11

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vibrationcontrol,medicalapplications,simulators,industrialmachinetools,andpositioningdevices. Parallelmechanismsthathavecompliantlegconnectorsratherthanadjustableonesareonevariationofparallelmechanisms,whicharedistinguishedfromexuremechanisms.Compliantmechanismsareusefulwhenthesituationrequirescontactwiththeenvironmentbecausetheircompliantcomponentsactlikesprings.Thecompliantcomponentplaysaroleinmeasuringthecontactforcebydisplacementwhenthemechanismisincontactwithanyobjectorenvironment.Duetosuchusage,mostcompliantparallelmechanismsareclassiedaspassivemechanisms,thatistosay,thesemechanismsshouldbecombinedwithotheractiveelementswhenappliedtoacontrolsystem. OnerepresentativeexampleforapplicationusingcompliantparallelmechanismsisthekinestaticcontrolsystemproposedbyGrifs[ 32 ].Thesystemconsistsofapassivecompliantparallelmechanismandaserialrobotmanipulatorasanactivemechanism.Thecontrolinputgovernsthemotionoftheserialrobotmanipulatorandagripperonthecompliantparallelmechanismachievesthedesiredmotionandcontactforcebyinteractingwithanobjectorenvironmentusingthekinestaticcontrolalgorithm.Theroleofthecompliantparallelmechanismintheapplicationisthesameasaremotecenterofcompliance(RCC)device. 1.2Motivation Thisresearchaimstoproposeanewcompliantparallelmechanismandtheoreticallyanalyzingitskinematics,statics,anddynamics,andapplyingacontrolsystem. WhenaclassofGough-Stewartplatformsisappliedtoaforcecontrolproblem,theplatform'sprismaticjointsinthelegconnectorsshouldbeactivecomponentstogenerateappropriateforce.Thepointthatshouldbeconsideredinforcegenerationisthatvibrationmighthappeninrigidcontactbetweentheend-effectoroftheroboticsystemandanobjectorenvironmentoritmightneedtosimulateelasticcontact.Usingthemechanismswithonlytheactuatorinsuchcontactsituationsmightnotbeefcientwithregardstotheuseofpowerbecausetheactuatormayhavetoworkhardtogenerateacompliantforcebasedondisplacementandit 12

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sometimescausestheactuatortooverheatorburnout.Thesesituationsrequiretheuseofalegconnectorwithbothaprismaticactuatorandpassivecompliantdevice. Oneofbigproblemsinrobotmechanisms,eventhoughitisnotlimitedtotheparalleltype,issingularity.Inthesingularpointsorregion,robotmechanismscannotmoveorthemotorsinthejointsmightexertexcessivetorquesothatoperationfails.Suchsingularpointsorregionscanbeavoidedbypre-calculatingthesingularcongurationsthatthemechanismcanformusingkinematics.Thisapproachhasapotentialriskbecausenecontrolisrequired.Thebetterway,whichisusedforaproposedmechanisminthisresearch,isaddingaredundantleg.Thisintrinsicallydecreasesthedimensionofsingularitymanifold,whichisameasurefortheextentofsingularity. Atensegrity-basedcompliantparallelmechanismisproposedwiththepurposeofalleviatingthelimitationsmentionedabove.Thismechanismcanbeusedasapassivecompliantmechanismwhenthedisplacementofthepistonisxedormanuallyadjusted,oracompliantrobotmanipulatorwhenthepistonplaysaroleofanactuatorincontrolapplications. 1.3ProblemStatement Asmentionedintheprevioussection,anewparallelmechanismisproposedtoperformasahybriddeviceanddesignedusingtheconceptoftensegrity.Inthisresearch,thefollowingtopicsareinvestigated.First,kinematicsforthemechanismisanalyzed.Themechanismhaslegconnectors,eachofwhichconsistsofanadjustablepistonwithaspring.Duetotheexistenceofsprings,itisnotpossibletondgeometricalrelationsusingonlykinematics.Thus,themechanismshouldbesimultaneouslyanalyzedusingbothkinematicsandstatics.Thisanalysisisdoneforaplanarandspatialcaseofthemechanism.Second,adynamicanalysisisalsodoneandthedynamicequationoftheproposedmechanismisderived.Finally,applicationsofthemechanismarediscussedandthemechanismisappliedtooneofthemsuchthattheaforementionedkinematicanddynamicanalysiscanbeused. 13

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CHAPTER2TENSEGRITYANDTENSEGRITY-BASEDMECHANISM 2.1Tensegrity Tensegrity,awordcoinedbyFuller,isacombinationoftensionandintegrity[ 25 ].Tensegritystructuresarebasicallycomposedofdiscontinuouscompressivecomponents,referredtoasstruts,andcontinuoustensilecomponents,referredtoasties[ 19 ].Thestrutsandtiesaretypicallystiff.Whenthetensegritystructureisassembled,thetiesareplacedintension,whichpre-stressestheentiremechanismsuchthatthetiesareintensionandthestrutsareincompression.Duetosuchstructuralcharacteristics,tensegritystructureshavesomebenetssuchastensionalstabilityorrigidity,lightweight,anddeployablity.Theseaspectshaveimpelledresearcherstostudytensegrityintheeldsofbiology[ 12 39 71 ],civilengineering[ 1 57 66 ],andmathematics[ 14 34 ]. 2.2TensegrityMechanismandTensegrity-BasedMechanism Especiallyintheareasofmechanismsandrobotics,tensegritymechanismsusingtheconceptoftensegrityortensegrity-basedmechanismsthathaveprestressedtensionalandcompressiblemembershavebeenproposedeveniftheydonotmatchtheexactdenitionoftensegrity. ArsenaultandGosselinintroducedseveralplanarandspatialtensegritymechanisms[ 4 9 ].Theirsingledegreesoffreedom(DOF)planartensegritymechanismismadefromaplanarunactuatedsystemwiththreeDOFaddingadjustablepistons(Figure 2-1 A).Theysolvedthedirectandinversestaticproblemsusingpotentialenergyofthemechanismanddiscussedsingularcongurations,workingcurve,stiffness,anddynamicmodel.TheirtwoDOFplanarmodulartensegritymechanismisaseriesassemblyofmechanismmodulesmadebysubstitutingspringsonbothsidewithcablesassumedtohaveinnitestiffnessrelativetothespringsinthesingleDOFplanartensegritymechanism(Figure 2-1 B).TheyalsodevelopedathreeDOFspatialmodulartensegritymechanism(Figure 2-1 C)fromSnelson'sX-moduletetrahedrontensegrity[ 78 ].Itsbasicconstructionconsistsoftwocompressivebars,fourtensilesprings,then 14

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AAsingleDOFplanarmechanism BAtwoDOFplanarmodularmechanism CAmoduleofthreeDOFspatialmodularmechanism DAthreeDOFspatialmechanism Figure2-1. ExamplesoftensegritymechanismsbyArsenaultandGosselin amechanismmodulecomprisedofthreeX-moduleconstructions.Theybuiltthespatialmodularmechanismwithaserialassemblyofasmallnumberofthemechanismmodulesandsolvedthedirectandinversestaticproblems(Figure 2-1 D).Anotherspatialtensegritymechanismthattheydevelopedisa3-PUP Sspatialtensegritymechanism.ThismechanismisbasedonatriangulartensegrityprismknowntobeinventedbyKarlLogansonandhasthreeprismaticactuatorsascompressivecomponentsandsixspringsandthreecablesastensilecomponents.Theyobtainedanalyticsolutionsforthedirectandinversestaticproblemsundertheconditionofabsenceofexternalandgravitationalloadsandinvestigatedstiffness,singularity,andworkspaceastheirotherresearch.Schenketal.designedandconstructedstaticallybalancedtensegritymechanismbasedonthetensegrityprism[ 74 ].VasquezandCorreaproposedatensegrityrobotmanipulatorwithtwolinearactuatorsandfourtensileties[ 88 ],derivingadynamicequationforthemotion 15

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ofthemanipulatorusingtheLagrangemethodandsimulatedpath-trackingofthemanipulatorwithcomputedtorquecontrolbasedonfeedbacklinearization.SchmalzandAgrawalconductedastudyontheworkspacefortwodifferenttypesoftensegritymechanisms;onewithaxedrodandafreerodandtheotherwithapivotconstraintandafrictionlessslidingconstraint[ 75 ]. Fromtheconceptthatthemusculoskeletalsystemisanalogoustoatensegritystructure,somepeoplenotedapplicationtolocomotionusingthedynamiccharacteristicsoftensegrity.Pauletal.proposedtwotensegrityrobotsbasedonthree-andfour-struttensegrityprisms[ 63 ].AldrichandSkeltonpresentedarobotictensegritysnakewithtwomotorizedpulleysandsimulateditsmaneuveringinaconstrictedenvironment[ 2 ].Rieffeletal.exploredmorphologicalcomputationusingatesegrityrobotconsistingoffourstrutsandsixteenstrings[ 67 ].RoviraandTurdesignedasimulatortostudythemovementofatensegrityrobotbasedonatriangularprism[ 30 ]. 2.3Tensegrity-BasedCompliantMechanism Someotherstudiesattemptedtograftthetensegrityconceptontoparallelcompliantmechanisms.Tranetal.introducedatensegrity-basedparallelmechanismwithelasticties[ 80 81 ](Figure 2-2 A).Inthemechanism,threetieshavebothcompliantandnon-compliantelementsandtheothertiesonthetopandbottomarenon-compliant.Theformers'lengthscanvaryaslegconnectorsinparallelcompliantmechanismsandtrianglescomprisedofthelatterplaythesameroleasthetopandbaseplatforminparallelmechanisms.Thus,thismechanismiscalledatensegrity-basedcompliantmechanism.Marshallproposedanothertensegrity-basedcompliantmechanismclosertoaparallelmechanismsthanTran's[ 49 ](Figure 2-2 B).Themechanismhasthetopandbaseplatformandtwodifferentkindsoflegconnectors;oneisastrutandtheotherisacablewithaspring.Craneetal.presentedaplanartensegrity-basedmechanismanalogoustothecross-sectionofahumankneejoint[ 15 89 ](Figure 2-2 C).Fumagallietal.designedatensegrity-liketendon-drivenneckforahumanoidrobot[ 26 ](Figure 2-2 D)andAzadietal.introducedavariablestiffnessspringbasedonatensegrityprism[ 10 ]. 16

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ATran'smechanism BMarshall'smechanism CKneejointmechanism DTendondrivenneck Figure2-2. Examplesoftensegrity-basedcompliantmechanisms Inspiredbythemechanismsin[ 49 81 ],anewplanartensegrity-basedmechanismisintroducedwherethetiesthatconnectthetopsofthestrutsandthetiesthatconnectthebottomsofthestrutsarereplacedbyrigidbodies.Thestrutsand`sideties'arereplacedbyanadjustablelengthpistoninserieswithaspringelement.Thepointthatalegconnectorconsistsofbothacompliantelementandaadjustablepistonallowsthemechanismtobehaveasnotonlyatensegritybutalsoacompliantparallelmechanismwhiletheformermechanismsin[ 49 81 ]playaroleofavariationoftensegrity.Whenthisnewtensegrity-basedmechanismisineffectofanexternalwrenchandthelengthsofthepistonsarexed,themechanismisthesameastwoformermechanisms.Whenthetopbodyofthemechanismisincontactwithanobjectandthelengthsofthepistonsarexed,themechanismcanbeusedasapassivecompliantdevicelikearemotecenterofcompliancedevice.Whenthelengthsofthepistonsareadjustedbyacontrolalgorithm, 17

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themechanismcanbeusedasanactiveparallelmanipulatorforposition,force,orsimultaneouspositionandforcecontrolproblem. 18

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CHAPTER3ANALYSISOFAPLANARMECHANISM Kinematicanalysisinparallelmechanismsexaminestherelationbetweenthejointcoordinateofthelegconnectorandtheposeofthetopmovingplatform.Theanalysisincludestwotypes:forwardandreverseanalyses.Forwardanalysisisdenedasndingtheposeinformationforagivenjointcoordinateandthereverseanalysisisitsreverse.Itisgenerallyknownthatitisdifculttosolvetheforwardkinematicsforparallelmechanismswhileitiseasytosolvethereversekinematics,andthisfactistheoppositetothatofserialmanipulators. Inatensegrity-basedcompliantmechanism,thereverseanalysisdoesnotmeanonlythekinematicproblembecausethismechanismincludesacompliantcomponent,simplyaspring,inthelegconnectorandtheexistenceofthespringadditionallyrequiresstaticsonthekinematicanalysis.Thatistosay,thereversekinematicanalysisforthismechanismmeanssimultaneouslysolvingbothkinematicsandstatics.Thisanalysisiscalledakinestaticprobleminthisdissertationfromtheterminology'kinestaticcontrol',whichGrifsrstcoined.Inthischapter,thekinestaticproblem,stiffnessmapping,andstiffnesssynthesisproblemfortheplanarcaseandthespatialcaseinthefollowingchapterarecovered. Astherstpart,aplanartensegrity-basedmechanismisintroducedwherethetiesthatconnectthetopsofthestrutsandthetiesthatconnectthebottomsofthestrutsarereplacedbyrigidbodies.Thestrutsand`sideties'arereplacedbyanadjustablelengthpistoninserieswithaspringelement.Followingaformaldescriptionofthemechanismtobeanalyzed,thesectionrstderivestheplanarstiffnessmatrixusingamethodsimilartothatusedin[ 31 ].Threeproblemsarethenaddressed:tworeverseproblemsandoneforwardproblem.Intherst,thelengthsofthepistonsaredeterminedthatwillpositionandorientthetopbodyasspeciedsuchthatthemechanismisinequilibriumwhenagivenexternalwrenchisappliedtothetopbody.Multiplesolutionsarepossibleanditisshownhowasinglesolutioncanbeobtainediftheuseralsospeciesthedesiredtotalpotentialenergythatisstoredinthesprings.Inthesecondproblem,itisnecessarytodeterminethepistonlengthsandspringconstantssuchthat(1)thetopbodycanbe 19

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Figure3-1. Planarmechanismschematic positionedandorientedasdesired,(2)thetopbodyisinequilibriumasagivenexternalwrenchisapplied,and(3)thestiffnessmatrixforthetopbodyequalsadesiredsetofmatrixvalues.Inthenalproblem,thepositionandorientationofthetopbodyandelongatedlengthsofthespringsaredeterminedwhenthemechanismisinequilibriumforagivenexternalwrenchappliedtothetopbodywithinputpistonlengths.Numericalexampleswillbepresented. 3.1MechanismDescription Theplanartensegrity-basedcompliantmechanismtobeconsideredhereisshowninFigure 3-1 .Thetopbodyisrigidandthebaseisxedtotheground.Thetwobodiesareconnectedatarbitrarypositions,pi,i=1;:::;8byfourlegconnectors.Eachlegconnectorisanadjustablelengthpistoninserieswithaspringelement.Thespringconstantsarenamedki,i=1;:::;4accordingtotheindexofthejointsinthebasebody.Twocoordinatesystemsaredened.CoordinatesystemfTgisxedtothetopbody.Itsoriginisatp5andp6liesalongtheXaxis.CoordinatesystemfBgisattachedtothebasebody.Itsoriginislocatedatp1andp2alsoliesalongtheXaxis.TheanglesofthelinesalongthelegconnectorsasmeasuredwithrespecttotheXaxisofthefBgcoordinatesystemarelabeledi,i=1;:::;4.Thetotalinstantaneouslengthofthelegconnectors,Lti,arewrittenas Lti=Li+li=Li+loi+i;i=1;:::;4 (3) 20

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Figure3-2. Compliantlegconnector whereLiisthepistonlength,l0iisthefreelengthofthespring,andiisthespringelongation. 3.2StiffnessMapping Threeparametersarerequiredinordertodescribethepositionandorientationoftheplanarmechanism.Thesearetwodistancecomponentsfortwoperpendicularaxesintheplane,generallyxandy,andarotationangleabouttheaxisperpendiculartotheplane, .Thestiffnessmatrixwhichmapsaninnitesimalchangeineachoftheseparametersintoachangeintheinnitesimalappliedwrenchisa33matrix.Toderivetheplanarstiffnessmatrix,theequationsofforceandmomentgeneratedbythespringtensionorcompressioninthelegsareobtainedrstas fx=4Xi=1ki(li)]TJ /F7 11.955 Tf 11.96 0 Td[(loi)cify=4Xi=1ki(li)]TJ /F7 11.955 Tf 11.96 0 Td[(loi)simo=4Xi=1riki(li)]TJ /F7 11.955 Tf 11.96 0 Td[(loi) (3) wherei=1;:::;4,andci,siandrirepresentcos(i),sin(i),andtheperpendiculardistancefromreferencepointointheplanetopointsinthelineofthelegs,respectively.Therefore,the 21

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wrenchequationcanbewritteninmatrixformas w=fxfymoT=266664c1c2c3c4s1s2s3s4r1r2r3r4377775266666664k1(l1)]TJ /F7 11.955 Tf 11.96 0 Td[(lo1)k2(l2)]TJ /F7 11.955 Tf 11.96 0 Td[(lo2)k3(l3)]TJ /F7 11.955 Tf 11.96 0 Td[(lo3)k4(l4)]TJ /F7 11.955 Tf 11.96 0 Td[(lo4)377777775: (3) Aninnitesimalchangeinthewrenchcanbedenedbymultiplicationofthestiffnessmatrixandaninnitesimalchangeinmotionas ^w=266664fxfymo377775=K^D=266664k11k12k13k21k22k23k31k32k33377775266664xy 377775:(3) Inequation( 3 ),thereareeightparameters,iandli,i=1;:::;4.Inordertoobtaintheformofequation( 3 ),partialdifferentiationcanbetakenfortheeightparameters.Thefollowingequationsareobtainedas fx=4Xi=1kicili)]TJ /F9 7.97 Tf 18.47 14.95 Td[(4Xi=1ki(li)]TJ /F7 11.955 Tf 11.96 0 Td[(loi)siify=4Xi=1kisili+4Xi=1ki(li)]TJ /F7 11.955 Tf 11.96 0 Td[(loi)ciimo=4Xi=1kirili+4Xi=1ki(li)]TJ /F7 11.955 Tf 11.95 0 Td[(loi)ri: (3) 22

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Equation( 3 )maybewritteninmatrixformas ^w=266664c1c2c3c4s1s2s3s4r1r2r3r4377775266666664k1l1k2l2k3l3k4l4377777775+266664)]TJ /F7 11.955 Tf 9.29 0 Td[(s1)]TJ /F7 11.955 Tf 9.3 0 Td[(s2)]TJ /F7 11.955 Tf 9.3 0 Td[(s3)]TJ /F7 11.955 Tf 9.3 0 Td[(s4c1c2c3c4r1 1r2 2r3 3r4 4377775266666664k1(1)]TJ /F7 11.955 Tf 11.96 0 Td[(1)Lt11k2(1)]TJ /F7 11.955 Tf 11.96 0 Td[(2)Lt22k3(1)]TJ /F7 11.955 Tf 11.96 0 Td[(3)Lt33k4(1)]TJ /F7 11.955 Tf 11.96 0 Td[(4)Lt44377777775 (3) where1,2,3and4aretheratiosofthefreespringlengthandthelengthofthepistontothetotallengthofthelegconnector,i=(loi+Li)=Lti. Inordertodescribetherightsidevectorsofeachtermintherightsideofequation( 3 )intermsof[xy ]T,tworelationsusedin[ 31 ],li=$Ti^DandLtii=($i+Lti)T^Dcanbeappliedwhere$iistheunitizedscrewofthelineofthei-thlegconnector1,$iisinnitesimalof$iandLti=[0;0;Lti]T.Therefore,thefollowingequationisobtained. ^w=jkijT^D+jki(1)]TJ /F7 11.955 Tf 11.95 0 Td[(i)jT^D+jki(1)]TJ /F7 11.955 Tf 11.96 0 Td[(i)LtiT^D (3) 1Theplanarunitizedscrew,$isgenerallydenedas31vectorofST;rSTinscrewtheorywhereSisadirectionandrisadistancevectorfromtheoriginofthereferencetoanypointontheline.Therefore,$=[cos;sin;rxsin)]TJ /F7 11.955 Tf 11.96 0 Td[(rycos]Twhererxandryarexandydirectioncomponentsofrforeachlegconnector. 23

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where j=266664c1c2c3c4s1s2s3s4r1r2r3r4377775;ki=266666664k10000k20000k30000k4377777775;j=266664)]TJ /F7 11.955 Tf 9.3 0 Td[(s1)]TJ /F7 11.955 Tf 9.3 0 Td[(s2)]TJ /F7 11.955 Tf 9.3 0 Td[(s3)]TJ /F7 11.955 Tf 9.3 0 Td[(s4c1c2c3c4r1 1r2 2r3 3r4 4377775;ki(1)]TJ /F7 11.955 Tf 11.96 0 Td[(i)=266666664k1(1)]TJ /F7 11.955 Tf 11.96 0 Td[(1)0000k2(1)]TJ /F7 11.955 Tf 11.95 0 Td[(2)0000k3(1)]TJ /F7 11.955 Tf 11.95 0 Td[(3)0000k4(1)]TJ /F7 11.955 Tf 11.96 0 Td[(4)377777775;Lti=26666400000000Lt1Lt2Lt3Lt4377775: (3) Finally,thestiffnessmatrixofthemechanismisobtainedas K=jkijT+jki(1)]TJ /F7 11.955 Tf 11.95 0 Td[(i)jT+jki(1)]TJ /F7 11.955 Tf 11.96 0 Td[(i)LtiT: (3) Thematrixisnotsymmetric2becauseofthelasttermintheright-handsideinequation( 3 ).Therearetwoconditionsthatthisplanarstiffnessmatrixbecomessymmetric;oneisthatthematrix[j][ki(1)]TJ /F7 11.955 Tf 11.96 0 Td[(i)][Lti]Tisequaltozeroandtheotheristhatthevector[j][ki(1)]TJ /F7 11.955 Tf 11.96 0 Td[(i)][Lti]T[0;0;1]Tisthesameas[00m33]Twherem33isanarbitrarynumberthusthethirdtermofequation( 3 )isadiagonalmatrix.Therstconditionmeansthateachindividualspringisattheunloaded 2ThisplanarstiffnessmatrixispartiallysymmetricduetothelasttermofEq.( 3 ).Thatistosay,theupper-left22partialmatrixofthematrixissymmetric. 24

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position,andthesecondconditionmeansthatthewrenchesoftheloadedspringscancelouteachothersothattheoverallnetwrenchwouldbezero.Thesesatisescorollary2in[ 13 ]. 3.3KinestaticAnalysis Thefollowinginformationisgivenforthekinestaticanalysisproblemofaplanartensegrity-basedcompliantmechanism: Thedesiredpositionandorientationofthetopbodymeasuredwithrespecttothebaseasrepresentedbyfxo;yo;g,wherexo;yoarethecoordinatesoftheoriginofthetopcoordinatesystemmeasuredwithrespecttothebottomcoordinatesystemandistheanglemeasuredfromtheXaxisofthebottomcoordinatesystemtotheXaxisofthetopcoordinatesystemmeasuredinaright-handsenseabouttheZaxis. Thecoordinatesofpoints1through4inthebasecoordinatesystemandthecoordinatesofpoints5through8inthetopcoordinatesystem. Themechanismistobeatstaticequilibriumwhileaknownexternalwrench^wextisappliedtothetopbody. Thespringconstantsandfreelengthsforthefoursprings,kiandloi,i=1:::4. Thedesiredtotalpotentialenergytobestoredinthefoursprings,U. Theobjectiveistodeterminethelengthsofthefourpistons,Li,i=1:::4,thatwillsatisfytheserequirements. Fortherotationangleandthedistancevector[xoyo]T,the33transformationmatrixthatrelatesthebasecoordinatesystem,fBg,andthetopcoordinatesystem,fTg,intheplaneisgivenas BTT=266664cos())]TJ /F8 11.955 Tf 11.29 0 Td[(sin()xosin()cos()yo001377775:(3) 25

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Thecoordinatesofthelegconnectorendpointsmaybewrittenas Bp1=[x1y1];Bp2=[x2y2]Bp3=[x3y3];Bp4=[x4y4]Tp5=[x5y5];Tp6=[x6y6]Tp7=[x7y7];Tp8=[x8y8]: (3) Forconvenienceincalculation,allvectorsandmatriceswillbeexpressedintermsofthebasecoordinatesystem.Thusthepositionsofthepointsinthetopbodymeasuredwithrespecttothebasecoordinatesystemmaybewrittenas Bp5=BTRTp5+BpTorigBp6=BTRTp6+BpTorigBp7=BTRTp7+BpTorigBp8=BTRTp8+BpTorig: (3) whereBTRistherotationmatrixofthetopcoordinatesystemwithrespecttothebasecoordinatesystem,whichisthesameastheupperleft22partialmatrixinthematrixinequation( 3 )andBpTorigisthepositionvectoroftheoriginofthetopcoordinatesystemwithrespecttothebasecoordinatesystem,whichisthesameasthevectorincludingthetoptwoelementsofthelastcolumninthematrixinequation( 3 ).Thereforethepositionsofthejointsinthetopplatformcanbewrittenas Bp5=x5c)]TJ /F7 11.955 Tf 11.96 0 Td[(y5s+xox5s+y5c+yoT;Bp6=x6c)]TJ /F7 11.955 Tf 11.96 0 Td[(y6s+xox6s+y6c+yoT;Bp7=x7c)]TJ /F7 11.955 Tf 11.96 0 Td[(y7s+xox7s+y7c+yoT;Bp8=x8c)]TJ /F7 11.955 Tf 11.96 0 Td[(y8s+xox8s+y8c+yoT (3) 26

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wherecandsdepictcos()andsin(),respectively. Fromthepositionsofthejoints,thecoordinatesofthelinesofthelegconnectorscanbecalculated.ItcanberepresentedintermsoffL;M;Rg,thePluckercoordinatesintwo-dimensionalspace.Thersttwocomponentsaredimensionlessandtheunitofthelastcomponentismeters,m. $L1=266664x5c)]TJ /F7 11.955 Tf 11.95 0 Td[(y5s+xo)]TJ /F7 11.955 Tf 11.96 0 Td[(x1x5s+y5c+yo)]TJ /F7 11.955 Tf 11.96 0 Td[(y1x1(x5s+y5c+yo))]TJ /F7 11.955 Tf 11.96 0 Td[(y1(x5c)]TJ /F7 11.955 Tf 11.96 0 Td[(y5s+xo)377775;$L2=266664x7c)]TJ /F7 11.955 Tf 11.95 0 Td[(y7s+xo)]TJ /F7 11.955 Tf 11.95 0 Td[(x2x7s+y7c+yo)]TJ /F7 11.955 Tf 11.96 0 Td[(y2x2(x7s+y7c+yo))]TJ /F7 11.955 Tf 11.96 0 Td[(y2(x7c)]TJ /F7 11.955 Tf 11.96 0 Td[(y7s+xo)377775;$L3=266664x6c)]TJ /F7 11.955 Tf 11.95 0 Td[(y6s+xo)]TJ /F7 11.955 Tf 11.95 0 Td[(x3x6s+y6c+yo)]TJ /F7 11.955 Tf 11.96 0 Td[(y3x3(x6s+y6c+yo))]TJ /F7 11.955 Tf 11.96 0 Td[(y3(x6c)]TJ /F7 11.955 Tf 11.96 0 Td[(y6s+xo)377775;$L4=266664x8c)]TJ /F7 11.955 Tf 11.95 0 Td[(y8s+xo)]TJ /F7 11.955 Tf 11.95 0 Td[(x4x8s+y8c+yo)]TJ /F7 11.955 Tf 11.96 0 Td[(y4x4(x8s+y8c+yo))]TJ /F7 11.955 Tf 11.96 0 Td[(y4(x8c)]TJ /F7 11.955 Tf 11.96 0 Td[(y8s+xo)377775 (3) Themagnitudeofthetotallengthsofeachlegconnectorcanbecalculatedbythedifferencebetweentworelevantpointvectors. Lt1=kBp5)]TJ /F6 7.97 Tf 11.96 4.94 Td[(Bp1k;Lt2=kBp7)]TJ /F6 7.97 Tf 11.96 4.94 Td[(Bp2k;Lt3=kBp6)]TJ /F6 7.97 Tf 11.96 4.93 Td[(Bp3k;Lt4=kBp8)]TJ /F6 7.97 Tf 11.96 4.94 Td[(Bp4k: (3) 27

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wherekkisEuclideannormonR2.Then,theunitizedscrewsofthelinesofthelegconnectorsarecalculatedas $i=1 Lti$Li;i=1;:::;4:(3) Thelengthofeachlegconnectorisrelatedtotheelongationofthespringandtheforceexertedonthespring.Bytheequilibriumcondition,theforcesandtheexternalwrench,fext$ext,appliedtothetopbodysatisfythefollowingequation. f1$1+f2$2+f3$3+f4$4+fext$ext=0(3) Therearefourunknowns,theforcemagnitudesoff1,f2,f3andf4,forthethreeequationsinequation( 3 ).Inordertosolvetheequationsforforcemagnitudes,onemoreequationisrequired.Inthisproblem,thetotalenergyinthesprings,U,isgivenasanadditionalcondition.TranandMarshallsolvedthisproblembydeningnewparameters,whichareratiosofforcestooneforcemagnitude[ 49 80 ].However,thosearethesolutionsofsevenparameters,sixforcesofmagnitude,andoneexternalwrenchmagnitudeforsixequationsonspatialmechanismsandthereisnoconstantterm,whiletheequationinthissectionhasaknownexternalwrenchmagnitude,whichisaconstantterm.Thereforeitisnotpossibletousethesamemethodtosolvethisequationbecausetheratiooftheconstanttermtooneforceincreasesthenumberofparameters.Asawaytosolvetheequationundertheexistenceoftheconstantterm,itcanbemodiedtoformtheforcesintolinearfunctionsofoneforcemagnitude.Theequationbelowisanexampleoff1beingusedasaparameterofthefunctionstoformotherforceswhentherstmatrixintheright-handsideintheequationisnotsingular. 266664f2f3f4377775=)]TJ /F10 11.955 Tf 11.29 16.85 Td[($2$3$4)]TJ /F9 7.97 Tf 6.58 0 Td[(1(f1$1+fext$ext)=)]TJ /F3 11.955 Tf 9.3 0 Td[(J0)]TJ /F9 7.97 Tf 10.8 0 Td[(1(f1$1+fext$ext) (3) 28

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Thematrix,J0,issingular3ifthecoordinatesofthelinesalongthesecond,third,andfourthlegconnectorsarelinearlydependent.Theproblemcanbereformulated,however,ifasetofthreeofthefourlinecoordinatescanbefoundthatarelinearlyindependent.Anequationsimilartoequation( 3 )canthenbeobtainedwherethecolumnsofthematrixtobeinvertedarecomprisedofthecoordinatesofthesethreelinearlyindependentlines. Goingbacktothesolutionprocess,f2,f3andf4canbewrittenasafunctionoff1,thatis,fj=ajf1+bjwherej=2;3;4andajandbjareconstants.Theequationofthetotalenergyinthespringsisafunctionofspringconstantsandspringelongations,andcanberewrittenintermsoftheforcesas U=4Xi=1kii2 2=4Xi=1fi2 2ki=f12 2k1+4Xj=2(ajf1+bj)2 2kj:(3) Thusequation( 3 )canbewrittenasaquadraticfunctionoff1, Af21+Bf1+C=0:(3) Aquadraticequationhastworoots.Typicallythesetofrootsshouldbeoneofthreecases:(1)tworealroots,(2)onerepetitiveroot,and(3)twocomplexroots.Tworealrootsoftheequationmeantwopossiblesetsofthelengthsofeachpiston,onerepetitiverootmeansasinglepossibleset,andthethirdmeansanimpracticalsituation.Rootdeterminationcanbeobtainedfromequation( 3 ).Thesignofthefollowingexpression,discr,identiesthesolutioninthecasewhereanexternalwrenchandadesiredgeometricposeofthemechanismcanbecheckedas discr= 4Xj=2ajbj kj!2)]TJ /F8 11.955 Tf 11.96 0 Td[(4 1 2k1+4Xj=2a2j 2kj! 4Xj=2b2j 2kj)]TJ /F7 11.955 Tf 11.96 0 Td[(U!: (3) 3Generally,allpossiblecombinationsofJ0arenonsingularwhenthematrix,[$1;$2;$3;$4]hasafullrank.Whenitfails,apolynomialsystembuiltfromgeometricconstraints,staticequilibrium,andthepotentialenergyinthespringsisunderdeterminedsystem,thennumericalmethodsarerequiredtosolvethesystem.Thefailurecaseshappenwhenallfourlinespassingthroughthelegconnectorsareparallelorconcurrent.Thosecasesarerareandtheycanbeavoidedbyaspecicjointconguration. 29

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Oncef1isdetermined,theremainingforces,f2,f3andf4,canbecalculatedfromequation( 3 ),andthenthespringelongationscanbealsoobtainedas i=fi ki:(3) Finally,fromthetotallengthsoflegconnectors,thefreespringlengthsandtheelongations,thelengthsofthepistonscanbecomputedas Li=Lti)]TJ /F8 11.955 Tf 11.96 0 Td[((loi+i);i=1;:::;4:(3) 3.4StiffnessSynthesis Theinformationtosolvethestiffnesssynthesisproblemforaplanartensegrity-basedcompliantmechanismisgivenas: Thetopbodyispositionedandorientedatadesiredpose. Thetopbodyisatequilibriumwhileaspeciedexternalwrenchisapplied. Theinstantaneousstiffnessmatrixofthetopbodyequalsaspeciedsetofmatrixvalues. Theobjectiveistodeterminethelengthsofthefourpistons,Li,i=1;:::;4,andthespringconstantsofallsprings,ki,i=1;:::;4,thatwillsatisfytheserequirements. Inthisproblem,theplanarstiffnessmatrixisknown,whichmeans9equationsareobtainedfor9elements,asseeninequation( 3 ).However,asdescribedinthesectiononstiffnessmapping,twoelementsinthematrixarethesame,thusthereareactually8equationsinequation( 3 ).Fromtheproblemstatement,thereare8unknownparameters,Liandkiwherei=1;:::;4.Thelengthratios,iwherei=1;:::;4,canbeconsideredasparameterstobesolvedintheequationbecausetheparametersLihavealinearrelationwithi.Substitutingri=xisi)]TJ /F7 11.955 Tf 13.14 0 Td[(yiciandri=i=xici+yisiintoequation( 3 ),theequationsin 30

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equation( 3 )arewrittenas k11=4Xi=1ki(1)]TJ /F7 11.955 Tf 11.95 0 Td[(is2i);k12=k12=4Xi=1kiicisi;k13=4Xi=1kificisixi)]TJ /F8 11.955 Tf 11.95 0 Td[((1)]TJ /F7 11.955 Tf 11.96 0 Td[(is2i)yi)]TJ /F8 11.955 Tf 11.96 0 Td[((1)]TJ /F7 11.955 Tf 11.96 0 Td[(i)Ltisig;k21=k12;k22=4Xi=1ki(1)]TJ /F7 11.955 Tf 11.95 0 Td[(ic2i);k23=4Xi=1kif(1)]TJ /F7 11.955 Tf 11.96 0 Td[(ic2i)xi)]TJ /F7 11.955 Tf 11.96 0 Td[(icisiyi+(1)]TJ /F7 11.955 Tf 11.96 0 Td[(i)Lticig;k31=4Xi=1kifixicisi)]TJ /F8 11.955 Tf 11.95 0 Td[((1)]TJ /F7 11.955 Tf 11.96 0 Td[(is2i)yig;k32=4Xi=1kif(1)]TJ /F7 11.955 Tf 11.96 0 Td[(ic2i)xi)]TJ /F7 11.955 Tf 11.96 0 Td[(icisiyig;k33=4Xi=1kif(1)]TJ /F7 11.955 Tf 11.96 0 Td[(ic2i)x2i+(1)]TJ /F7 11.955 Tf 11.95 0 Td[(is2i)y2i;)]TJ /F8 11.955 Tf 9.29 0 Td[(2icisixiyi+(1)]TJ /F7 11.955 Tf 11.96 0 Td[(i)Lti(cixi+siyi)g: (3) Equations( 3 )arenotalinearequationsetfortheparameterskiandi,buttheycontainthesamecombinationastheparameters,thatis,kii.Usingthisfact,equation( 3 )canbewritteninmatrixformas AKXK=bK(3) where AK=AK11AK12AK13AK14...AK21AK22AK23AK24XK=k1k2k3k4k11k22k33k44TbK=k11k21k31k22k32k13k23k33T: (3) 31

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ThecolumnvectorsinthematrixAK,AK11toAK14,havethesameexpression,excepttheindexi=1;:::;4andAK21toAK24alsohavethesameform,whichdependsontheindexi=1;:::;4.ThereforetheymaybewrittenastheformofAK1iandAK2ias AK1i=266666666666666666666410)]TJ /F7 11.955 Tf 9.3 0 Td[(yi1xi)]TJ /F7 11.955 Tf 9.3 0 Td[(yi)]TJ /F7 11.955 Tf 11.95 0 Td[(Ltisixi+Lticix2i+y2i+Lti(cixi+siyi)3777777777777777777775; (3) AK2i=2666666666666666666664)]TJ /F7 11.955 Tf 9.3 -.01 Td[(s2icisixicisi+s2iyi)]TJ /F7 11.955 Tf 9.29 0 Td[(c2i)]TJ /F7 11.955 Tf 9.3 0 Td[(c2ixi)]TJ /F7 11.955 Tf 11.96 0 Td[(cisiyicisixi+s2iyi+Ltisi)]TJ /F7 11.955 Tf 9.3 0 Td[(c2ixi)]TJ /F7 11.955 Tf 11.96 0 Td[(cisiyi)]TJ /F7 11.955 Tf 11.95 0 Td[(Ltici)]TJ /F8 11.955 Tf 9.3 0 Td[((cixi+siyi)2)]TJ /F7 11.955 Tf 11.95 0 Td[(Lti(cixi+siyi)3777777777777777777775: (3) IfAKisnotsingular,thecolumnvector,XKcanbeobtained.ThespringconstantscanbedirectlyobtainedfromtherstfourelementsinXK,andthelengthsofthepistonscanbeobtainedfromequation( 3 )andthedenitionofi. XK=A)]TJ /F9 7.97 Tf 6.58 0 Td[(1KbK(3) ItisclearthatthesingularmatrixAKmeansitsdeterminantiszero.However,itisverydifculttoanalyzewhatgeometricalmeaningisrelatedtotheconditionthatjAKj=0becausethematrix 32

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is88anditsdeterminantequationisextremelylongandevenamathematicaltoolprogramcannotsolveit,causingamemoryproblem.Asanalternativeway,itmaybeabletochecklineardependencyofanytwoormorerowsandcolumnsofthematrix,respectively.Whenthiswayistaken,494cases,whosenumbercanbecountedfrom28i=28Ci,shouldbeconsidered.Forexample,theconditionthattherstandsecondcolumnsarelinearlyindependentisthefollowing. x1=ax2y1=ay2x1+Lt1c1=a(x2+Lt2c2y1+Lt1s1=a(y2+Lt2s2x21+y21+Lt1(c1x1+s1y1)=afx22+y22+Lt2(c2x2+s2y2)g whereaisnonzeroconstant,x1,x2,y1,andy2arexandycoordinatevalueofthejoint1and2,respectively.Asshownintheexample,thistaskisalsoverydifcultandtime-consumingproblem.Therefore,itisnotcoveredinthisdissertation. 3.5NumericalExamples Fortheproblemsdiscussedintheprevioussections,somenumericalexamplesareshown. 3.5.1Example1 Thisexampleistosolvetheforceequationwiththetotalenergyinthespringswhenanexternalwrenchisappliedonthetopbody.Thiscanbeusedtoestimateforceappliedtothespringinthelegconnectorwithoutusingasensorinapracticalproblem.Thelinecoordinatesofthelegconnectorsatthedesiredpose,thespringconstants,thespringfreelengths,thetotal 33

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energy,andtheexternalwrencharegivenasfollows; $1$2$3$4=266664)]TJ /F8 11.955 Tf 9.3 0 Td[(0:311)]TJ /F8 11.955 Tf 9.3 0 Td[(0:6710:4040:151)]TJ /F8 11.955 Tf 9.3 0 Td[(0:950)]TJ /F8 11.955 Tf 9.3 0 Td[(0:741)]TJ /F8 11.955 Tf 9.3 0 Td[(0:915)]TJ /F8 11.955 Tf 9.3 0 Td[(0:9890:000)]TJ /F8 11.955 Tf 9.3 0 Td[(1:112)]TJ /F8 11.955 Tf 9.3 0 Td[(3:089)]TJ /F8 11.955 Tf 9.3 0 Td[(4:704377775;k=20202020TN/cm;lo=1:01:01:01:0Tcm;U=10Ncm;fext=2:75N;$ext=2666640:1850:983)]TJ /F8 11.955 Tf 9.3 0 Td[(2:384377775 (3) Notethatthetermsinthetoptworowsof[$1$2$3$4]and$extaredimensionlessandthetermsinthethirdrowhaveunitsofcm.Usingtheprocedureinsection 3.3 ,theexternalwrenchcanbeexpressedintermsoftheforcesinthefourlegconnectorsas $1$2$3$4f=266664)]TJ /F8 11.955 Tf 9.3 0 Td[(0:311)]TJ /F8 11.955 Tf 9.3 0 Td[(0:6710:4040:151)]TJ /F8 11.955 Tf 9.3 0 Td[(0:950)]TJ /F8 11.955 Tf 9.3 0 Td[(0:741)]TJ /F8 11.955 Tf 9.3 0 Td[(0:915)]TJ /F8 11.955 Tf 9.3 0 Td[(0:9890:000)]TJ /F8 11.955 Tf 9.3 0 Td[(1:112)]TJ /F8 11.955 Tf 9.3 0 Td[(3:089)]TJ /F8 11.955 Tf 9.3 0 Td[(4:704377775266666664f1f2f3f4377777775=)]TJ /F7 11.955 Tf 9.3 0 Td[(fext$ext=)]TJ /F8 11.955 Tf 9.3 0 Td[(2:752666640:1850:983)]TJ /F8 11.955 Tf 9.3 0 Td[(2:384377775: (3) Thematrix,[$1$2$3$4]hasrank3anditsconditionnumberis10.936,thusitisnotsingularnorill-conditioned.Asinequation( 3 ),f1ischosenasaparameterandequation( 3 )is 34

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rearrangedas 266664)]TJ /F8 11.955 Tf 9.3 0 Td[(0:6710:4040:151)]TJ /F8 11.955 Tf 9.3 0 Td[(0:741)]TJ /F8 11.955 Tf 9.29 0 Td[(0:915)]TJ /F8 11.955 Tf 9.3 0 Td[(0:989)]TJ /F8 11.955 Tf 9.3 0 Td[(1:112)]TJ /F8 11.955 Tf 9.29 0 Td[(3:089)]TJ /F8 11.955 Tf 9.3 0 Td[(4:704377775266664f2f3f4377775=)]TJ /F7 11.955 Tf 9.3 0 Td[(f1266664)]TJ /F8 11.955 Tf 9.3 0 Td[(0:311)]TJ /F8 11.955 Tf 9.3 0 Td[(0:9500:000377775)]TJ /F8 11.955 Tf 11.96 0 Td[(2:752666640:1850:983)]TJ /F8 11.955 Tf 9.3 0 Td[(2:384377775: (3) Thenthefollowingforcemagnitudescanbeobtainedintermsoff1as f2=)]TJ /F8 11.955 Tf 9.3 0 Td[(1:087f1+3:867;f3=)]TJ /F8 11.955 Tf 9.3 0 Td[(1:500f1+7:985;f4=1:242f1)]TJ /F8 11.955 Tf 11.96 0 Td[(7:551: (3) Usingtheaboveequations,thepotentialenergyequationcanbechangedtoaquadraticfunctionofonlyaparameter,f1as 0:149f21)]TJ /F8 11.955 Tf 11.96 0 Td[(1:278f1)]TJ /F8 11.955 Tf 11.96 0 Td[(6:607=0: (3) Tworootsoftheabovequadraticequationhaverealvalues,whichmakethefollowingtwosetsofforces. 266666664f1f2f3f4377777775a=26666666412:183)]TJ /F8 11.955 Tf 9.3 0 Td[(9:382)]TJ /F8 11.955 Tf 9.3 0 Td[(10:2987:585377777775N;266666664f1f2f3f4377777775b=266666664)]TJ /F8 11.955 Tf 9.3 0 Td[(3:6337:81513:429)]TJ /F8 11.955 Tf 9.29 0 Td[(12:058377777775N (3) Thesolutionwasrepeatedfordifferentvaluesofthefext.Themagnitudeoftheexternalwrench,fextwasvariedfrom-12Nto12Ninincrementsof0.012Nm.Thesolutionfi;i=1:::4andplottedforeachcaseinFigure 3-3 .Smallcirclesat11.72Noffextindicaterepeatedrootsandcomplexrootsarenotshown. 35

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Figure3-3. Example1:forcemagnitudesforchangeinmagnitudeoftheexternalwrench 3.5.2Example2 Thisisanexamplefortherstreverseproblem.Thepositionsofthejointsinthetopandbasebodiesaregivensothatthemechanismformssimilarlytocongurationshowninsection 3 .Theplanartransformationmatrixthatrelatesthetopandbaseplatformcoordinatesystemsaswellastheinformationgivenintheformerexamplesaregivenas BTT=2666640:8660:5001:080)]TJ /F8 11.955 Tf 9.3 0 Td[(0:5000:8663:3000:0000:0001:000377775;Bp1=0:0000:000T;Bp2=1:5000:000T;Bp3=3:2000:400T;Bp4=4:800)]TJ /F8 11.955 Tf 9.3 0 Td[(0:270T;Tp5=0:0000:000T;Tp6=1:3000:000T;Tp7=2:8001:000T;Tp8=3:5000:650T; 36

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Figure3-4. Example2:poseofthemechanism lO=2666666641:0001:0001:0001:000377777775cm;k=26666666420:020:020:020:0377777775N/cm;U=10Ncm;fext$ext=6:7752666640:1850:983)]TJ /F8 11.955 Tf 9.3 0 Td[(2:384377775: (3) Throughtheproceduredescribedinsection 3.3 ,thefollowingresultscanbecalculated. 266666664f1f2f3f4377777775a=26666666416:852)]TJ /F8 11.955 Tf 9.3 0 Td[(8:830)]TJ /F8 11.955 Tf 9.3 0 Td[(5:6832:395377777775N;266666664f1f2f3f4377777775b=2666666644:1484:98313:375)]TJ /F8 11.955 Tf 9.29 0 Td[(13:382377777775N; (3) 37

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2666666641234377777775a=2666666640:843)]TJ /F8 11.955 Tf 9.3 0 Td[(0:442)]TJ /F8 11.955 Tf 9.3 0 Td[(0:2840:120377777775cm;2666666641234377777775b=2666666640:2070:2490:669)]TJ /F8 11.955 Tf 9.29 0 Td[(0:669377777775cm;La=2666666641:6303:1731:7441:291377777775cm;Lb=2666666642:2652:4830:7912:080377777775cm: TheposeofthemechanismisshowninFigure 3-4 .Thesquaresandtrianglesdepicttheconnectionpointsbetweenthespringsandthepistonsinthelegconnectorsfortheabovetwosolutions. 3.5.3Example3 Thisexampleisanalysisdescribedastherstproblem,speciallywhenthetopbodyismovingalongaspeciedpath.Thiscanbesaidtobequasi-staticproblemthatcansimulatethemotionofthetopbodywhenthemovingpartssuchasthelegconnectorsandthetopbodyarelightenoughtoneglect.Thepositionsofallpoints(5,6,7and8)inthetopbodycoordinatesystemareafunctionoftimetotheglobalcoordinatesystem,fGg.Theplanartransformationmatrixbetweenthetopplatformandglobalcoordinatesystemsandtheinitialtransformationmatrixbetweenthetopandbaseplatformsaregivenas GBT=2666641:0000:00022:3000:0001:00012:5000:0000:0001:000377775BTT=2666640:8660:5001:080)]TJ /F8 11.955 Tf 9.3 0 Td[(0:5000:8663:3000:0000:0001:000377775: 38

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Figure3-5. Example3:pathtrajectory Bp1=0:0000:000T;Bp2=1:5000:000T;Bp3=3:2000:400T;Bp4=4:800)]TJ /F8 11.955 Tf 9.3 0 Td[(0:270T;Tp5=0:0000:000T;Tp6=1:3000:000T;Tp7=2:8001:000T;Tp8=3:5000:650T;lO=2666666641:0001:0001:0001:000377777775cm;k=26666666420:020:020:020:0377777775N/cmU=10Ncm;fext$ext=3:5722666640:1850:983)]TJ /F8 11.955 Tf 9.3 0 Td[(2:384377775: Fortherotationangle andthedistancevector[xoyo0]T,thetransformationmatrixthatrelatestotheglobalandbasecoordinatesystemsintheplanecanbedened.Thetrajectoryoftheoriginofthetopbody,point5inthisexampleisshowninFigure 3-5 .Thethinlinerepresents 39

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Figure3-6. Example3:pistonlengths(solutionA) Figure3-7. Example3:pistonlength(solutionB) orientationofthex-axisattachedtothetopbodycoordinatesystem,thatisafunctionoftheangle, 40

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Figure3-8. Example3:posechange(solutionA) Figure3-9. Example3:posechange(solutionB) x5(t)=0:1t+22:05cm;y5(t)=0:24sin(0:7t)]TJ /F8 11.955 Tf 11.96 0 Td[(0:35)+15:8cm; (t)= 8sin(0:2t)radian: 41

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ThelengthofthepistonsasafunctionoftimeiscalculatedandisshowninFigure 3-6 and 3-7 .Figure 3-8 and 3-9 showtheposechangeofthemechanismin0.5secondsteps.Thecirclesaretheconnectionpointsbetweenthespringpartandthepistonsinthelegconnectors. 3.5.4Example4 Thisexampleisthesecondanalysiswiththeplanarstiffnessmatrix.Asdescribedinsection 3.4 ,itistodeterminevaluesforthefourspringconstantsaswellasthelengthsofthepistonsforthesameinformationexceptthataplanarstiffnessmatrixisgiveninsteadofthetotalpotentialenergyinthesprings.Thisexamplecanbeappliedtothecompliancecontrolproblem.Ifthemechanismincludesavariablestiffnesselementinthelegconnector,insteadofaspringwithaxedspringconstant,theneachvariablestiffnessdevicecanbeadjustedsothattheirspringconstantsfollowtheresultofthisexampleforthegivenstiffnessmatrix.Valuesofthefollowingstiffnessmatrixarerandomlyselectedas K=26666410:8062:874)]TJ /F8 11.955 Tf 9.3 0 Td[(22:9012:87432:33182:838)]TJ /F8 11.955 Tf 9.3 0 Td[(9:25276:572306:526377775 wheretheunitsofthematrixareN/mfor22upper-leftpartition,Nforupper-rightandlower-leftpartitions,andNmforlower-rightpartition. ThematricesAK1andAK2canbecalculatedfromequation( 3 ),andthevectorbKcomesfromthegivenstiffnessmatrix.ParametervectorXKcanbesimplycalculatedbyinverseofAKbecauseitsrankis8andconditionnumberisnotverylarge,whichmeansthematrixisnotsingular.Then,fromXK,valuesforfourspringconstantsandtheratioofthefreespringlengthstocurrentspringlengthsisobtained.Thelengthsofthepistonscanbecalculatedbyequation( 3 ). 42

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AK1=26666666666666666666641:0001:0001:0001:0000:0000:0000:0000:0000:0000:2700:000)]TJ /F8 11.955 Tf 9.3 0 Td[(0:4001:0001:0001:0001:0000:0004:8001:5003:200)]TJ /F8 11.955 Tf 9.3 0 Td[(3:300)]TJ /F8 11.955 Tf 9.3 0 Td[(2:113)]TJ /F8 11.955 Tf 9.3 0 Td[(2:766)]TJ /F8 11.955 Tf 9.3 0 Td[(2:6501:0804:4364:0052:2060:00020:7236:0078:1193777777777777777777775;AK2=2666666666666666666664)]TJ /F8 11.955 Tf 9.3 0 Td[(0:903)]TJ /F8 11.955 Tf 9.3 0 Td[(0:977)]TJ /F8 11.955 Tf 9.3 0 Td[(0:549)]TJ /F8 11.955 Tf 9.3 0 Td[(0:8370:296)]TJ /F8 11.955 Tf 9.3 0 Td[(0:1490:498)]TJ /F8 11.955 Tf 9.3 0 Td[(0:3700:000)]TJ /F8 11.955 Tf 9.3 0 Td[(0:9800:746)]TJ /F8 11.955 Tf 9.3 0 Td[(0:848)]TJ /F8 11.955 Tf 9.3 0 Td[(0:097)]TJ /F8 11.955 Tf 9.3 0 Td[(0:023)]TJ /F8 11.955 Tf 9.3 0 Td[(0:451)]TJ /F8 11.955 Tf 9.3 0 Td[(0:1630:000)]TJ /F8 11.955 Tf 9.3 0 Td[(0:150)]TJ /F8 11.955 Tf 9.3 0 Td[(0:676)]TJ /F8 11.955 Tf 9.3 0 Td[(0:3753:3001:4033:5121:407)]TJ /F8 11.955 Tf 9.3 0 Td[(1:0800:214)]TJ /F8 11.955 Tf 9.3 0 Td[(3:1810:6190:0001:407)]TJ /F8 11.955 Tf 9.3 0 Td[(4:7711:4213777777777777777777775;bK=10:806;2:874;)]TJ /F8 11.955 Tf 9.3 0 Td[(9:252;32:331;76:572;)]TJ /F8 11.955 Tf 9.3 0 Td[(22:901;82:838;306:526T;XK=10:522;9:625;10:825;7:643;7:969;9:946;9:200;6:972T;L=1:630;1:491;2:173;1:244T(cm);k=10:522;9:625;10:825;7:643T(N=cm): 43

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CHAPTER4ANALYSISOFASPATIALMECHANISM Asthesecondstep,aspatialtensegrity-basedmechanismthatoriginatesfromitsplanarversionisintroducedwherethetiesthatconnectthetopsofthestrutsandthetiesthatconnectthebottomsofthestrutsarereplacedbyrigidbodies.Thestrutsand`sideties'arereplacedbyanadjustablelengthpistoninserieswithaspringelement.Followingaformaldescriptionofthemechanismtobeanalyzed,thespatialstiffnessmatrixwillrstbederivedusingthemethodusedin[ 41 ].Threeproblemsarethenaddressed:tworeverseproblemsandaforwardproblem,asintheprevioussection.Intherstproblem,thelengthsofthepistonsaredeterminedthatwillpositionandorientthetopbodyasspeciedsuchthatthemechanismisinequilibriumwhenagivenexternalwrenchisappliedtothetopbody.Multiplesolutionsarepossibleanditisshownhowasinglesolutioncanbeobtainediftheuseralsospeciesthedesiredtotalpotentialenergythatisstoredinthesprings.Inthesecondproblem,thecompliancecontrolproblemisdescribedfromtheresultofthestiffnessmatrix.Inthisproblem,itisnecessarytodeterminethepistonlengthsandspringconstantssuchthat(1)thetopbodycanbepositionedandorientedasdesired,(2)thetopbodyisinequilibriumasagivenexternalwrenchisapplied,and(3)thestiffnessmatrixforthetopbodyequalsadesiredsetofmatrixvalues.Inthethirdproblem,theforwardproblem,positionandorientationofthetopbodyandelongatedlengthsofspringsaredeterminedwhenthemechanismisinequilibriumforagivenexternalwrenchappliedtothetopbodywithinputpistonlengths.Numericalexamplesforeachproblemwillbepresented. 4.1MechanismDescription Thespatialtensegrity-basedcompliantmechanismisaspatialversionoftheplanaroneintheprevioussectionandisshowninFigure 4-1 .Itconsistsofrigidtopandbasebodiesandsevenlegconnectors.Thelegconnectorsjointhebodiesat14points;sevenonthebasebody,p1throughp7andsevenonthetopbody,p8throughp14.Forgenerality,theshapesoftwobodiesareassumedtobearbitrarywiththeassumptionbeingthatnotwolegconnectorpointscoincide. 44

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Figure4-1. Spatialmechanismschematic Twocoordinatesystemsaredenedinthetwobodies:fBgonthebasebodywhoseoriginisp1andfTgonthetopbodywhoseoriginisp8. Thesevenlegconnectorsrepresentaredundancyfortypicalparallelplatformswhoselegconnectorsarestiff.Thisisthesameconceptastheplanarversionthathasfourlegconnectors[ 55 ].Eachlegconnectoriscomprisedofspringandpistonpartsandassumedtobelinearlyrigid.Therelationoflengthofthespringsandpistonsinthelegconnectorsisgivenby Lti=Li+l0i+ifori=1;:::;7 (4) whereLtiistotallengthofthelegconnector,Liisthepistonlength,l0iisthefreelengthofthespring,andiisthespringelongation. 4.2StiffnessMapping Thestiffnessmatrixmapsaninnitesimalchangeofpositionandorientationofthetopbodyintoaninnitesimalchangeinthewrenchappliedtothetopbody,i.e.^w=[K]^Dwhere^wisinnitesimalchangeinthewrenchappliedtothetopbody,[K]isstiffnessmatrix,and^Disinnitesimalchangeofpositionandorientationofthetopbody.Accordingto[ 41 ],thestiffnessmatrixofthemechanismcanbederivedas 45

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Figure4-2. Coordinateofalegconnector [K]=7Xi=1ki$i$Ti+7Xi=1 ki(1)]TJ /F7 11.955 Tf 11.96 0 Td[(i) @$0i @i@$00i @iT+@$i @i@$00i @iT!! (4) where i=Li)]TJ /F7 11.955 Tf 11.95 0 Td[(l0i Lti;$i=264SipiSi375@$0i @i=264@S0i @ipi@S0i @i375;@$00i @i=264@S0i @ipj@S0i @i375@$i @i=264@Si @ipi@Si @i375;@$00i @i=264@Si @ipj@Si @i375;Si=[sinicosi;sinisini;cosi]T@S0i @i=[)]TJ /F8 11.955 Tf 11.29 0 Td[(sini;cosi;0]T andandaretheanglesdenedinFigure 4-2 46

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Fromequation( 4 ),atotalof33equationscanbeobtainedbecause3arerepeatedequationsduetothesymmetryofthe33upper-leftpartitionwhicharethe(1,2),(1,3)and(2,3)elementsofthetotal66matrix.Thus, k11=7Xi=1kis2ic2i+ki(1)]TJ /F7 11.955 Tf 11.95 0 Td[(i)(s2i+c2ic2i) (4) k12=7Xi=1kis2icisi+ki(1)]TJ /F7 11.955 Tf 11.95 0 Td[(i)()]TJ /F7 11.955 Tf 9.3 0 Td[(sici+c2icisi) (4) k13=7Xi=1[kisicici)]TJ /F7 11.955 Tf 11.95 0 Td[(ki(1)]TJ /F7 11.955 Tf 11.96 0 Td[(i)sicici] (4) k14=7Xi=1[kisici(yici)]TJ /F7 11.955 Tf 11.95 0 Td[(zisisi)+ki(1)]TJ /F7 11.955 Tf 11.95 0 Td[(i)fzjsici)]TJ /F7 11.955 Tf 11.96 0 Td[(cici(yjsi+zjcisi)g] (4) k15=7Xi=1[kisici(zisici)]TJ /F7 11.955 Tf 11.95 0 Td[(xici)+ki(1)]TJ /F7 11.955 Tf 11.96 0 Td[(i)fzjs2i+cici(zjcici+xjsi)g (4) k16=7Xi=1[kisici(xisisi)]TJ /F7 11.955 Tf 11.96 0 Td[(yisici)+ki(1)]TJ /F7 11.955 Tf 11.95 0 Td[(i)f)]TJ /F7 11.955 Tf 15.28 0 Td[(si(xjci+yjsi)+cici(xjcisi)]TJ /F7 11.955 Tf 11.95 0 Td[(yjcici)g] (4) k21=k12 (4) k22=7Xi=1kis2is2i+ki(1)]TJ /F7 11.955 Tf 11.95 0 Td[(i)(c2i+c2is2i) (4) k23=7Xi=1[kisisici)]TJ /F7 11.955 Tf 11.96 0 Td[(ki(1)]TJ /F7 11.955 Tf 11.95 0 Td[(i)sisici] (4) k24=7Xi=1[kisisi(yici)]TJ /F7 11.955 Tf 11.95 0 Td[(zisisi))]TJ /F7 11.955 Tf 11.96 0 Td[(ki(1)]TJ /F7 11.955 Tf 11.95 0 Td[(i)fzjc2i+cisi(yjsi+zjcisi)g (4) 47

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k25=7Xi=1[kisisi(zisici)]TJ /F7 11.955 Tf 11.95 0 Td[(xici)+ki(1)]TJ /F7 11.955 Tf 11.95 0 Td[(i)f)]TJ /F7 11.955 Tf 15.28 0 Td[(zjsici+cisi(zjcici+xjsi)g] (4) k26=7Xi=1[kisisi(xisisi)]TJ /F7 11.955 Tf 11.95 0 Td[(yisici)+ki(1)]TJ /F7 11.955 Tf 11.96 0 Td[(i)fci(pj;xci+yjsi)+cisi(xjcisi)]TJ /F7 11.955 Tf 11.96 0 Td[(yjcici)g] (4) k31=k13 (4) k32=k23 (4) k33=7Xi=1kic2i+ki(1)]TJ /F7 11.955 Tf 11.95 0 Td[(i)s2i (4) k34=7Xi=1[kici(yici)]TJ /F7 11.955 Tf 11.95 0 Td[(zisisi)+ki(1)]TJ /F7 11.955 Tf 11.95 0 Td[(i)si(yjsi+zjcisi)] (4) k35=7Xi=1[kici(zisici)]TJ /F7 11.955 Tf 11.95 0 Td[(xici))]TJ /F7 11.955 Tf 9.3 0 Td[(ki(1)]TJ /F7 11.955 Tf 11.96 0 Td[(i)si(zjcici+xjsi)] (4) k36=7Xi=1[kici(xisisi)]TJ /F7 11.955 Tf 11.95 0 Td[(yisici))]TJ /F7 11.955 Tf 9.3 0 Td[(ki(1)]TJ /F7 11.955 Tf 11.96 0 Td[(i)si(xjcisi)]TJ /F7 11.955 Tf 11.96 0 Td[(yjcici)] (4) k41=7Xi=1[kisici(yici)]TJ /F7 11.955 Tf 11.95 0 Td[(zisisi)+ki(1)]TJ /F7 11.955 Tf 11.95 0 Td[(i)fzicisi+()]TJ /F7 11.955 Tf 9.3 0 Td[(yisi)]TJ /F7 11.955 Tf 11.95 0 Td[(zicisi)cicig] (4) k42=7Xi=1[kisisi(yici)]TJ /F7 11.955 Tf 11.96 0 Td[(zisisi))]TJ /F7 11.955 Tf 9.3 0 Td[(ki(1)]TJ /F7 11.955 Tf 11.96 0 Td[(i)fzic2i+(yisi+zicisi)cisig (4) k43=7Xi=1[kici(yici)]TJ /F7 11.955 Tf 11.96 0 Td[(zisisi)+ki(1)]TJ /F7 11.955 Tf 11.95 0 Td[(i)(yisi+zicisi)si] (4) 48

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k44=7Xi=1ki(yici)]TJ /F7 11.955 Tf 11.95 0 Td[(zisisi)2+ki(1)]TJ /F7 11.955 Tf 11.96 0 Td[(i)fzizjc2i+(yisi+zicisi)(yjsi+zjcisi)g] (4) k45=7Xi=1[ki(yici)]TJ /F7 11.955 Tf 11.95 0 Td[(zisisi)(zisici)]TJ /F7 11.955 Tf 11.96 0 Td[(xici)+ki(1)]TJ /F7 11.955 Tf 11.96 0 Td[(i)fzizjcisi)]TJ /F8 11.955 Tf 11.95 0 Td[((yisi+zicisi)(zjcici+xjsi)g] (4) k46=7Xi=1[ki(yici)]TJ /F7 11.955 Tf 11.95 0 Td[(zisisi)(xisisi)]TJ /F7 11.955 Tf 11.96 0 Td[(yisici))]TJ /F7 11.955 Tf 9.29 0 Td[(ki(1)]TJ /F7 11.955 Tf 11.95 0 Td[(i)fzici(xjci+yjsi)+(yisi+zicisi)(xjcisi)]TJ /F7 11.955 Tf 11.95 0 Td[(yjcici)g] (4) k51=7Xi=1[kisici(zisici)]TJ /F7 11.955 Tf 11.95 0 Td[(xici)+ki(1)]TJ /F7 11.955 Tf 11.96 0 Td[(i)fzis2i+cici(zicici+xisi)g (4) k52=7Xi=1[kisisi(zisici)]TJ /F7 11.955 Tf 11.96 0 Td[(xici)+ki(1)]TJ /F7 11.955 Tf 11.96 0 Td[(i)f)]TJ /F7 11.955 Tf 15.28 0 Td[(zicisi+(zicici+xisi)cisig] (4) k53=7Xi=1[kici(zisici)]TJ /F7 11.955 Tf 11.96 0 Td[(xici))]TJ /F7 11.955 Tf 9.29 0 Td[(ki(1)]TJ /F7 11.955 Tf 11.95 0 Td[(i)(zicici+xisi)si] (4) k54=7Xi=1[ki(yici)]TJ /F7 11.955 Tf 11.95 0 Td[(zisisi)(zisici)]TJ /F7 11.955 Tf 11.95 0 Td[(xici)+ki(1)]TJ /F7 11.955 Tf 11.96 0 Td[(i)fzizjcisi)]TJ /F8 11.955 Tf 9.29 0 Td[((zicici+xisi)(yjsi+zjcisi)g] (4) k55=7Xi=1ki(zisici)]TJ /F7 11.955 Tf 11.96 0 Td[(xici)2+ki(1)]TJ /F7 11.955 Tf 11.96 0 Td[(i)fzizjs2i+(zicici+xisi)(zjcici+xjsi)g] (4) 49

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k56=7Xi=1[ki(zisici)]TJ /F7 11.955 Tf 11.96 0 Td[(xici)(xisisi)]TJ /F7 11.955 Tf 11.96 0 Td[(yisici)+ki(1)]TJ /F7 11.955 Tf 11.96 0 Td[(i)f)]TJ /F7 11.955 Tf 15.28 0 Td[(zisi(xjci+yjsi)+(zicici+xisi)(xjcisi)]TJ /F7 11.955 Tf 11.95 0 Td[(yjcici)g] (4) k61=7Xi=1[kisici(xisisi)]TJ /F7 11.955 Tf 11.95 0 Td[(yisici)+ki(1)]TJ /F7 11.955 Tf 11.96 0 Td[(i)f)]TJ /F8 11.955 Tf 15.28 0 Td[((xici+yisi)si+(xicisi)]TJ /F7 11.955 Tf 11.96 0 Td[(yicici)cicig] (4) k62=7Xi=1[kisisi(xisisi)]TJ /F7 11.955 Tf 11.96 0 Td[(yisici)+ki(1)]TJ /F7 11.955 Tf 11.96 0 Td[(i)f(xici+yisi)ci+(xicisi)]TJ /F7 11.955 Tf 11.96 0 Td[(yicici)cisig] (4) k63=7Xi=1[kici(xisisi)]TJ /F7 11.955 Tf 11.96 0 Td[(yisici))]TJ /F7 11.955 Tf 9.3 0 Td[(ki(1)]TJ /F7 11.955 Tf 11.95 0 Td[(i)(xicisi)]TJ /F7 11.955 Tf 11.95 0 Td[(yicici)si] (4) k64=7Xi=1[ki(yici)]TJ /F7 11.955 Tf 11.95 0 Td[(zisisi)(xisisi)]TJ /F7 11.955 Tf 11.96 0 Td[(yisici))]TJ /F7 11.955 Tf 9.3 0 Td[(ki(1)]TJ /F7 11.955 Tf 11.95 0 Td[(i)f(xici+yisi)zjci+(xicisi)]TJ /F7 11.955 Tf 11.96 0 Td[(yicici)(yjsi+zjcisi)g] (4) k65=7Xi=1[ki(zisici)]TJ /F7 11.955 Tf 11.96 0 Td[(xici)(xisisi)]TJ /F7 11.955 Tf 11.95 0 Td[(yisici)+ki(1)]TJ /F7 11.955 Tf 11.96 0 Td[(i)f)]TJ /F8 11.955 Tf 15.28 0 Td[((xici+yisi)zjsi+(xicisi)]TJ /F7 11.955 Tf 11.96 0 Td[(yicici)(zjcici+xjsi)g] (4) k66=7Xi=1ki(xisisi)]TJ /F7 11.955 Tf 11.96 0 Td[(yisici)2+ki(1)]TJ /F7 11.955 Tf 11.96 0 Td[(i)f(xici+yisi)(xjci+yjsi)+(xicisi)]TJ /F7 11.955 Tf 11.96 0 Td[(yicici)(xjcisi)]TJ /F7 11.955 Tf 11.95 0 Td[(yjcici)g] (4) wheresisin;sisin;cicos;cicos;s2isin2i;s2isin2i;c2icos2i,c2icos2i,andpj=[xj;yj;zj]Tandpi=[xi;yi;zi]Tarethepointsinthetopandbasebodies,respectively. 50

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4.3KinestaticAnalysis Thereversekinestaticproblemaimstosolvefortheinstantaneousdisplacementofthelegconnectorsinordertopositionandorientthetopplatformasdesired.Thefollowinginformationisgivenforthereversedisplacementproblem: Thedesiredpositionandorientationofthetopbodymeasuredwithrespecttothebaseasrepresentedbyfxo;yo;zo;;; g,wherexo,yo,andzoarethecoordinatesoftheoriginofthetopcoordinatesystemmeasuredwithrespecttothebasecoordinatesystemand,,and areEulerangles4thatrepresenttheorientationofthetopcoordinatesystemwithrespecttothebasecoordinatesystem. Thecoordinatesofpoints1through7areintermsofthebasecoordinatesystem,p1,:::,p7,andthecoordinatesofpoint8through14areintermsofthetopcoordinatesystem,p8,:::,p14. Themechanismistobeatstaticequilibriumwhileaknownexternalwrench^wextisappliedtothetopbody. Thespringconstantsandfreelengthsforthesevensprings,kiandl0i,i=1;:::;7. Thedesiredtotalpotentialenergytobestoredinthesevensprings,U. Theobjectiveistodeterminetheforcemagnitudesappliedtothesevensprings,fiandthelengthsofthesevenpistons,Li,i=1;:::;7,thatwillsatisfytheserequirements. Thetransformationmatrixthatrelatesthetwocoordinatesystemsofthetopandbasebodiescanbegivenas BTT=264BTRBPTorg0131375 (4) 4Theseangleparametersarere-denedhere.Therefore,theyareirrelevanttotheparametersusedintheanalysisfortheplanarcase. 51

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where BTR=266664cc )]TJ /F7 11.955 Tf 9.3 0 Td[(cs sssc +cs )]TJ /F7 11.955 Tf 9.29 0 Td[(sss +cc )]TJ /F7 11.955 Tf 9.3 0 Td[(sc)]TJ /F7 11.955 Tf 9.3 0 Td[(csc +ss css +sc cc377775 (4) and BPTorg=[xo;yo;zo]T: (4) Thepositionvectorsinthetopandbasebodies,p1top14,withrespecttoeachcoordinatesystemcanbewrittenas Bp1=[0;0;0]T;Bpi=[pix;piy;piz]Tfori=2;:::;7 (4) and Tp8=[0;0;0]T;Tpj=[pjx;pjy;pjz]Tforj=9;:::;14: (4) Thepositionvectorsinthetopbodyinequation( 4 )withrespecttothebasebodycoordinatesystemcanbeobtainedbythefollowingequation. Bpj=BTRTpj+BPTorgforj=9;:::;14 (4) Fromequations( 4 )and( 4 ),thelengthsandtheunitizedlinecoordinatesofthelegconnectorscanbecalculatedas Lti=kBpj)]TJ /F6 7.97 Tf 11.95 4.93 Td[(Bpik2 (4) 52

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and $i=1 Lti264Bpj)]TJ /F6 7.97 Tf 11.95 4.34 Td[(BpiBpi(Bpj)]TJ /F6 7.97 Tf 11.95 4.34 Td[(Bpi)375 (4) wheretheindicesarei=1;:::;7;j=8;:::;14andkk2isEuclideannorm.Equation( 4 )denesPluckerlinecoordinatesintermsoffL;M;N;P;Q;Rg.Theunitsoftherstthreetermsaredimensionlessandtheunitsofthelastthreetermsarelength,m. Thedisplacementanalysisforcompliantmechanismsrequiresnotonlykinematics,butalsostaticsbecauseelongationofcompliantcomponents,springelongationinthispaper,canbeaffectedbytheforce.Therefore,thefollowingwrenchequationcanbeconsideredasaconstraint. 7Xi=1fi$i=fext$ext(4) wherefextand$extarethemagnitudeandthescrew(linewithpitch)ofthegivenexternalwrench,^wext.Equation( 4 )canbewrittenas Hf=fext$ext (4) whereHisa67knownmatrixandfisa71unknownvectorwhichconsistsofforcemagnitudes,fi;i=1;:::;7.Thisequationisunder-determinedsystemwith7unknownsin6equations,thusonemoreequationshouldberequiredforanalyticalsolutionoftheunknowns.Asanadditionalequation,thetotalpotentialenergyinthesprings,Ucanbeused. Astherststep,aterm(column)inHcanbeselectedandmovedtotheright-handsideinequation( 4 ).Forexample,therstcolumn,f1$canbechosenandequation( 4 )canbethenwrittenas H0f0=fext$ext)]TJ /F7 11.955 Tf 11.96 0 Td[(f1$1 (4) 53

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whereH0=[$2$3$4$5$6$7]andf0=[f2f3f4f5f6f7]T.Ifthe66squarematrixH0isnotsingular,thefollowingequationcanbewritten. f0=H0)]TJ /F9 7.97 Tf 6.59 0 Td[(1(fext$ext)]TJ /F7 11.955 Tf 11.96 0 Td[(f1$1) (4) Thereforetheforcemagnitudesinf0canbeexpressedas fi=aif1+bifori=2;:::;7 (4) whereaiandbiareconstants.Equation( 4 )canthenbesubstitutedintothetotalpotentialenergyequationas U=7Xi=1kii2 2=7Xi=1fi2 2ki=f12 2k1+7Xj=2(ajf1+bj)2 2kj: (4) Equation( 4 )canbewrittenasaquadraticfunctionoff1as Af12+Bf1+C=0 (4) where A=1 2k1+7Xj=2aj2 2kjB=7Xj=2ajbj kjC=7Xj=2bj2 2kj)]TJ /F7 11.955 Tf 11.95 0 Td[(U: (4) Typicallythesetofrootsinaquadraticfunctionshouldbeoneofthreecases:(1)twodistinctrealroots,(2)onerepeatedroot,and(3)twocomplexroots.Thersttwocasesresultintheactualforceintherelevantspring,butthelastcasereectsanimpracticalsituation.Thefollowingequationidentiesthesolutioncasewhereanexternalwrenchandadesiredgeometricposeof 54

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themechanismcanbeachieved. B2)]TJ /F8 11.955 Tf 11.96 0 Td[(4AC0 (4) Meanwhile,inequation( 4 ),itisimportantwhatcolumninHischosenbecausesomechoicemightcauseH0tobesingular.Inotherwords,thesingularityhappenswhenrank(H0)<6orrank(H)<6.Thelateriscoincidentwiththeconditionofforcesingularityforstaticallyredundantparallelmanipulators[ 17 ].Thisrankdeciencycanbeinterpretedasgeometry.Ifallsevenlinesthatpassthroughthelegconnectorsofthemechanismareparallelorintersectatonepoint,HhastherankdeciencyandH0issingularnomatterwhatcombinationofcolumnsinHisachievedforH0,andanalyticsolutionfortheproblemdiscussedinthissectioncannotbeobtained.Ifsixlinesareparallelorintersectatonepoint,HhasafullrankandH0isnon-singular,andtheproblemcanbeanalyticallysolved.Comparedtothespatialmechanismwithsixlegconnector,itcanbesaidthatthedegreeofredundancyofthismechanismreducespossibilityofsingularityinsolvingthereversekinestaticproblem. Oncetheforcemagnitudeinequation( 4 ),f1intheexample,isdetermined,theremainingforces,f2throughf7,canbecalculatedfromequation( 4 ).Thenthespringelongationscanbealsoobtainedas i=fi ki:(4) Finally,fromthespringelongationsandnaturallengths,thelengthsofthelegsexcludingspringlengthscanbecomputed. Li=Lti)]TJ /F8 11.955 Tf 11.95 0 Td[((l0i+i)fori=1;:::;7(4) Inthemeantime,itmaybedesirabletosolvethisproblemwithanotherconstraint,minimumpotentialenergy,insteadofthegivenvalueofpotentialenergy.Therefore,theproblemstatementisthesameasmentionedabove,exceptforthepotentialenergyinthesprings.Fromequation 55

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( 4 ),anotherequation,differentfromequation( 4 ),canbewrittenas U(f1)=Af21+Bf1+C0 (4) where C0=7Xj=2bj2 2kj: (4) Nowthepotentialenergyinthespringsisafunctionoff1.Inordertominimizetheenergy,thederivativecanbetakenwithrespecttof1as @U @f1=2Af1+B: (4) Thepotentialenergy,U,hasaminimumvaluewhenf1is)]TJ /F7 11.955 Tf 9.3 0 Td[(B=(2A)becauseAisalwayspositive,soUisanupward-openparabola.Inthiscase,theforcemagnitudescanbeobtainedas f1=)]TJ /F7 11.955 Tf 13.06 8.08 Td[(B 2Afj=)]TJ /F7 11.955 Tf 9.3 0 Td[(ajB 2A+bjforj=2;:::;7: (4) Thespringelongationsandpistonlengthsforthiscasecanthenbeobtainedusingtheequationsdescribedintheprevioussection,equations( 4 )and( 4 ). 4.4StiffnessSynthesis Thisproblemistosolvefortheinstantaneousdisplacementofthelegconnectorsandthespringconstantsofallspringssuchthatthestiffnessmatrixmatchesthegivendesiredstiffnessmatrixvalue.Thefollowinginformationisgivenforthecompliancecontrolproblem. Thesameinformationasdescribedinsection 4.3 ,exceptforthetotalpotentialenergyinallsevenspringsandthespringconstantsofallsprings. Desiredstiffnessmatrixvalue. Theobjectiveistodeterminethelengthsofthesevenpistons,Li,i=1;:::;7,andthespringconstantsofallsprings,ki,i=1;:::;7,thatwillsatisfytheserequirements. 56

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Someparametersofthestiffnessmatrixinequation( 4 )areexpressedintermsofanglesbetweenthelegconnectorsandthexyplaneofthebasecoordinatesystem,suchas1;:::;7;and1;:::;7,asshowninFigure 4-2 .ThisisrequiredtoconvertthegivenvaluesdenedintheCartesiancoordinatesystemtotheanglesdenedinthepolarcoordinatesystem.Therelevanttrigonometricfunctionscanbewrittenas ci=xj)]TJ /F7 11.955 Tf 11.95 0 Td[(xi p (xj)]TJ /F7 11.955 Tf 11.95 0 Td[(xi)2+(yj)]TJ /F7 11.955 Tf 11.95 0 Td[(yi)2 (4) si=yj)]TJ /F7 11.955 Tf 11.95 0 Td[(yi p (xj)]TJ /F7 11.955 Tf 11.95 0 Td[(xi)2+(yj)]TJ /F7 11.955 Tf 11.95 0 Td[(yi)2 (4) ci=zj)]TJ /F7 11.955 Tf 11.96 0 Td[(zi kpj)]TJ /F3 11.955 Tf 11.96 0 Td[(pik (4) si=p (xj)]TJ /F7 11.955 Tf 11.95 0 Td[(xi)2+(yj)]TJ /F7 11.955 Tf 11.95 0 Td[(yi)2 kpj)]TJ /F3 11.955 Tf 11.95 0 Td[(pik (4) Theanglesifromequations( 4 )and( 4 ),andifromequations( 4 )and( 4 )canbedetermined. Fromequation( 4 )through( 4 ),itcanbeseenthatallequationscontaintheindependenttermskiandki(1)]TJ /F7 11.955 Tf 12.54 0 Td[(i).RecallingthatiincludesLi,asdenedinequation( 4 ),thetermscorrespondtotheparameters,thespringconstantsandthepistonlengths,intheproblemstatement.Becausethenumberofunknownsis14andthesizeoftheunknownvectoris141,only14equationsofthetotalof33areenoughtosetupthematrixequation. A...BX=CX=d (4) 57

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where A=266664A1;1;:::;A1;7.........A14;1;:::;A14;7377775B=266664B1;1;:::;B1;7.........B14;1;:::;B14;7377775X=[k1;:::;k7;k1(1)]TJ /F7 11.955 Tf 11.96 0 Td[(1);:::;k7(1)]TJ /F7 11.955 Tf 11.95 0 Td[(7)]Td=[d1;:::;d14]T (4) andA1;itoA14;iandB1;itoB14;irepresenttherespectivetermscorrespondingtokiandki(1)]TJ /F7 11.955 Tf 10.71 0 Td[(i)fori=1;:::;7inthe14equationsselectedfromthe33inthestiffnessmatrix,andcjforj=1;:::;14representsthegivenvaluesofthestiffnessmatrixcorrespondingtotheselectedequations. IfCisnotsingular,theunknownvectorcanbecomputedas X=C)]TJ /F9 7.97 Tf 6.58 0 Td[(1d: (4) Thespringconstantscanbedirectlyobtainedfromtheresultofequation( 4 )andthepistonlengthsaredeterminedfromthefollowingcalculation. Li=l0i+Lti1)]TJ /F7 11.955 Tf 13.15 8.08 Td[(Xi+7 kifori=1;:::;7 (4) whereXi+7isi-thvalueofthelastsevenelementsoftheunknownvector,X. However,accordingtosomeresearchesbyCiblakandLipkin[ 13 ]andGrifs[ 32 ],ageneral66stiffnessmatrixhas26independentelements.Thedifferencebetweentwonumbers,33and26,iscausedbythefactthatastiffnessmatrixcanbedecomposedintoasymmetricandaskew-symmetricpartsandtheskew-symmetricpartisafunctionofanexternalwrenchappliedtothemechanism.Therefore,thematrixCintheabovesolutionprocessmightbesingularforsome 58

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setswhich14equationsareselected.Inordertoavoidsuchasingularcase,14equationsshouldcomefromthesymmetricmatrixafterdecomposition. Thedecompositioncanbesimplydoneas [K]=[K]+[K]T 2+[K])]TJ /F8 11.955 Tf 11.96 0 Td[([K]T 2=[K]sym+[K]skew: (4) With14equationsselectedfrom[K]sym,thesameprocessinequations( 4 )through( 4 )canbeusedtosolvetheproblem.5 4.5NumericalExamples 4.5.1Example1 Thisexampleisofthereversekinestaticprobleminsection 4.3 .Thefollowingdataisgivenandthepistonlengthsareparametersaretobeobtained. Bp1=[0:00;0:00;0:00]TBp2=[1:22;)]TJ /F8 11.955 Tf 9.3 0 Td[(1:31;0:35]TBp3=[4:42;0:41;0:12]TBp4=[4:87;3:92;0:29]TBp5=[1:52;5:75;0:55]TBp6=[)]TJ /F8 11.955 Tf 9.3 0 Td[(1:14;4:09;0:35]TBp7=[2:37;3:12;0:31]T (4) 5All33equationscanbeselectedtosolvethisproblemusingtheleastsquaremethod,X=(C0TC0))]TJ /F9 7.97 Tf 6.59 0 Td[(1C0Td0whereC0is3314matrixandd0istheknownvector.However,onlyanalyticsolutionsareintroducedinthisresearch. 59

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Tp8=[0:00;0:00;0:00]TTp9=[1:23;1:97;0:38]TTp10=[1:55;3:96;0:27]TTp11=[)]TJ /F8 11.955 Tf 9.3 0 Td[(1:25;2:57;0:13]TTp12=[)]TJ /F8 11.955 Tf 9.3 0 Td[(0:92;2:22;0:34]TTp13=[)]TJ /F8 11.955 Tf 9.3 0 Td[(2:05;1:26;0:26]TTp14=[0:54;0:86;0:21]T (4) (Allunitsofabovepositionvectorsarem.) =7:3;=23:5; =19:1BpTorg=[1:75;0:96;3:84]Tmk=[11:213;9:827;11:921;11:723;10:812;11:792;12:591]TN/ml0=[1:234;1:592;1:032;1:922;1:572;1:105;1:276]TmWext=[1:956;2:240;1:815;13:745;14:964;11:998]TU=10Nm wheretheunitsoftherstthreecomponentsofWextareNandtheunitsofthelastthreecomponentsareNm.ThetransformationmatrixwhichrelatesthetopandbaseplatformscanbecomputedusingtheEulerangleswiththethreeangles,;and andthedistancevector,BpTorgas BTT=2666666640:866)]TJ /F8 11.955 Tf 9.3 0 Td[(0:2770:4151:7500:3000:9540:0090:960)]TJ /F8 11.955 Tf 9.3 0 Td[(0:3990:1170:9103:8400001377777775: (4) 60

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Withthetransformationmatrix,thepositionofthejointsonthetopplatformwithrespecttothebasecoordinatesystemcanbecalculatedas Bp8=[1:750;0:960;3:840]TBp9=[2:429;3:212;3:925]TBp10=[2:110;5:205;3:929]TBp11=[0:010;3:038;4:756]TBp12=[0:480;2:805;4:775]TBp13=[)]TJ /F8 11.955 Tf 9.29 0 Td[(0:267;1:549;5:041]TBp14=[2:067;1:944;3:915]T (4) whereallunitsarem.Thetotallengthsofthelegconnectorsfortwocorrespondingjointscanbeobtainedas Lt=[4:328;5:889;6:545;6:659;5:254;5:406;3:805]Tm: (4) Theunitizedscrewsfortherespectivelegconnectorscanbecalculatedas $1=[:404;:222;:887;0:000;0:000;0:000]T$2=[:205;:768;:607;)]TJ /F8 11.955 Tf 9.3 0 Td[(1:064;)]TJ /F8 11.955 Tf 9.29 0 Td[(0:669;1:206]T$3=[)]TJ /F7 11.955 Tf 9.3 0 Td[(:353;:733;:582;:151;)]TJ /F8 11.955 Tf 9.3 0 Td[(2:615;3:383]T$4=[)]TJ /F7 11.955 Tf 9.3 0 Td[(:730;)]TJ /F7 11.955 Tf 9.3 0 Td[(:133;:671;2:667;)]TJ /F8 11.955 Tf 9.3 0 Td[(3:478;2:216]T$5=[)]TJ /F7 11.955 Tf 9.3 0 Td[(:198;)]TJ /F7 11.955 Tf 9.3 0 Td[(:561;:804;4:932;)]TJ /F8 11.955 Tf 9.3 0 Td[(1:331;:286]T$6=[:161;)]TJ /F7 11.955 Tf 9.3 0 Td[(:470;:868;3:714;1:046;)]TJ /F7 11.955 Tf 9.3 0 Td[(:125]T$7=[)]TJ /F7 11.955 Tf 9.29 0 Td[(:080;)]TJ /F7 11.955 Tf 9.3 0 Td[(:309;:948;3:053;)]TJ /F8 11.955 Tf 9.3 0 Td[(2:271;)]TJ /F7 11.955 Tf 9.3 0 Td[(:484]T (4) Therstthreeelementsaredimensionlessandthelastthreeunitsofm.Whenf1istakenastheparametertosolvetheforcemagnitudes,the66matrixH0inequation( 4 )isnon-singularfromequation( 4 ).Therefore,theforcemagnitudescanbeexpressedintermsoff1asin 61

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Figure4-3. Example1:mechanismconguration equation( 4 )as f2=1:455f1+7:663f3=)]TJ /F8 11.955 Tf 9.3 0 Td[(1:808f1)]TJ /F8 11.955 Tf 11.96 0 Td[(0:289f4=1:177f1)]TJ /F8 11.955 Tf 11.96 0 Td[(2:786f5=2:076f1+13:696f6=)]TJ /F8 11.955 Tf 9.3 0 Td[(1:444f1)]TJ /F8 11.955 Tf 11.96 0 Td[(0:137f7=)]TJ /F8 11.955 Tf 9.3 0 Td[(2:030f1)]TJ /F8 11.955 Tf 11.96 0 Td[(12:338: (4) Usingequation( 4 ),thefollowingquadraticequationcanbeobtained. 0:800f12+5:535f1+8:049=0: (4) Fromequation( 4 ),twosetsofforcemagnitudes,springelongationsandpistonlengthscanbecalculated.TheposeofthemechanismincludingtheconnectionpointsbetweenthespringandpistonfromtheresultsisshowninFigure 4-3 .Thetwosolutionfortheforcesinthelegs, 62

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displacementofthespringsandlengthofthepistonsaredeterminedas fa=[)]TJ /F8 11.955 Tf 9.3 0 Td[(2:076;4:643;3:464;)]TJ /F8 11.955 Tf 9.3 0 Td[(5:228;9:386;2:859;)]TJ /F8 11.955 Tf 9.3 0 Td[(9:125]TNfb=[)]TJ /F8 11.955 Tf 9.3 0 Td[(4:844;0:615;8:471;)]TJ /F8 11.955 Tf 9.3 0 Td[(8:485;3:638;6:856;)]TJ /F8 11.955 Tf 9.3 0 Td[(2:505]TN (4) a=[)]TJ /F8 11.955 Tf 9.3 0 Td[(0:185;0:472;0:291;)]TJ /F8 11.955 Tf 9.3 0 Td[(0:446;0:868;0:242;)]TJ /F8 11.955 Tf 9.3 0 Td[(0:645]Tmb=[)]TJ /F8 11.955 Tf 9.3 0 Td[(0:432;0:063;0:711;)]TJ /F8 11.955 Tf 9.3 0 Td[(0:724;0:336;0:581;)]TJ /F8 11.955 Tf 9.3 0 Td[(0:199]Tm (4) La=[3:279;3:825;5:222;5:183;2:814;4:058;3:174]TmLb=[3:526;4:235;4:802;5:461;3:346;3:719;2:728]Tm: (4) Inordertondtherangeoftheforcemagnitudeoftheexternalwrenchwhichresultsinrealsolutions,thecoefcientforforcemagnitudeoftheexternalwrenchisdenedascW2[)]TJ /F8 11.955 Tf 9.3 0 Td[(2;2]andforcesintherespectivelegconnectorsandcorrespondingpistonlengthsareinvestigated.Thus,themagnitudeoftheexternalwrenchforthisexampleisgivenas fext;new=cWfext: (4) Figure 4-4 Aand 4-4 Barethecurvesforforcemagnitudesandpistonlengths,respectively.WhencWvalueis1.088,therepeatedsolutioncanbeobtained.ThatmeanstworealsolutionscanbeobtainedatcW2()]TJ /F8 11.955 Tf 9.3 0 Td[(1:088;1:088).Thecomplexsolutionsareplottedasstraightlinesnormalizedinthegures. Asanotherinvestigation,theforcemagnitudeandthepistonlengthscanbeobtainedforsomerangeofthepotentialenergyinthesprings.ThecoefcientforthetotalpotentialenergyisdenedascU2(0;2]. Unew=cUU (4) Figure 4-5 Aand 4-5 BshowthatthesystemhastworealsolutionswhencU>0:84.Therefore,thepotentialenergyintheallspringshasminimumvalueatcU=0:84.Thelinesplottedfor 63

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AForcemagnitudes BPistonlengths Figure4-4. Example1:forcemagnitudesandpistonlengthsforchangeoftheexternalwrenchintensity cU<0:84aremeaninglessvalues,andjustindicatetheforcemagnitudesandlengthshavethecomplexsolutions. 4.5.2Example2 Thisexampleisforthestiffnesssynthesisproblem.Thesameinformationasexample1exceptthespringconstantsandtotalpotentialenergyinthespringsisgiven.Thedesiredstiffnessmatrixisalsogivenas 64

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AForcemagnitudes BPistonlengths Figure4-5. Example1:forcemagnitudesandpistonlengthsforchangeofthepotentialenergyinallsprings Kd=266666666666666461:7820:826)]TJ /F8 11.955 Tf 9.3 0 Td[(1:060)]TJ /F8 11.955 Tf 9.3 0 Td[(10:006266:260)]TJ /F8 11.955 Tf 9.3 0 Td[(167:5060:82663:638)]TJ /F8 11.955 Tf 9.3 0 Td[(1:310)]TJ /F8 11.955 Tf 9.3 0 Td[(274:5710:78777:512)]TJ /F8 11.955 Tf 9.3 0 Td[(1:060)]TJ /F8 11.955 Tf 9.3 0 Td[(1:31072:248197:693)]TJ /F8 11.955 Tf 9.3 0 Td[(93:3279:219)]TJ /F8 11.955 Tf 9.3 0 Td[(10:006)]TJ /F8 11.955 Tf 9.3 0 Td[(36:470172:778574:056)]TJ /F8 11.955 Tf 9.3 0 Td[(122:196)]TJ /F8 11.955 Tf 9.3 0 Td[(0:38428:1590:787)]TJ /F8 11.955 Tf 9.3 0 Td[(136:592)]TJ /F8 11.955 Tf 9.3 0 Td[(508:155322:448)]TJ /F8 11.955 Tf 9.3 0 Td[(85:241)]TJ /F8 11.955 Tf 9.3 0 Td[(142:591120:7779:219)]TJ /F8 11.955 Tf 9.3 0 Td[(465:864)]TJ /F8 11.955 Tf 9.3 0 Td[(680:226530:8883777777777777775: (4) 65

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wheretheunitsofthematrixareN/mfor33upper-leftpartition,Nforupper-rightandlower-leftpartitions,andNmforlower-rightpartition.Forcomparison,twowaysdescribedinsection 4.4 aretried;oneistoselect14equationsoutofthe33equationsandtheotheristoselect14equationsoutofthe21equationsfromthesymmetricmatrixafterdecomposingthegivenstiffnessmatrix.Fortheformer,thepossiblecasesare8e+8from33C14andtheyare116280from21C14forthelater.Inthisexample,threecases(1,2,and3)fortheformerandonecase(4)forthelaterareconsideredasshowninTable 4-1 .Theresultforcases2,3,and4canbeacceptedbecausetheirconditionnumbersofCarerelativelysmallwhilethecase1showsill-conditionedC.Inthetable,theequationscorrespondingtothekijvaluesareexpressedasfijorfsij. 4.5.3Example3 Thisexampleisrelatedtothepathtrackingproblem.Thesameinformationasexample1isgivenexceptfor Wext=[1:1736;1:344;1:089;13:745;14:964;11:998]TU=12Nm: TheunitsofWextaresameasthoseinexample1.Thedesiredpathfortheoriginofthetopbodycoordinatesystemisalsogivenas px(t)=px0+0:1t)]TJ /F8 11.955 Tf 11.96 0 Td[(0:5py(t)=py0+0:02sin(0:1t)]TJ /F8 11.955 Tf 11.96 0 Td[(0:5)pz(t)=pz0+0:1cos(0:7t)]TJ /F8 11.955 Tf 11.95 0 Td[(0:15) (4) wherepx0,py0andpz0arethreeelementsofBpTorginexample1.Thedesiredorientationisgivenastheunitdirectionvectorforeachaxisofthetopbodycoordinatesystem,insteadofEulerangle 66

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Table4-1. Resultsforexample2 CaseEquationsusedtoformCcond(C)Result(1strow:ki,2ndrow:Li) 1ff11;f22;f33;f44;f55;f66;f12;7.813e+16[50.121,3.422,23.914,-56.063,46.836,17.269,15.547]Tf13;f14;f15;f16;f23;f24;f25g[1.596,0.247,1.069,8.630,2.495,1.725,7.281]T2ff11;f22;f33;f44;f55;f66;f12;1.329e+3[11.213,9.827,11.921,11.723,10.812,11.792,12.591]Tf13;f14;f15;f16;f23;f24;f25g[2.468,3.184,2.064,3.844,3.144,2.210,2.552]T3ff11;f22;f33;f44;f55;f66;f12;1.385e+4[11.213,9.827,11.921,11.723,10.812,11.792,12.591]Tf13;f14;f15;f16;f23;f24;f25g[2.468,3.184,2.064,3.844,3.144,2.210,2.552]T4ffs11;fs22;fs33;fs44;fs55;fs66;fs12;1.566e+3[11.213,9.827,11.921,11.723,10.812,11.792,12.591]Tfs13;fs14;fs15;fs16;fs23;fs46;fs56g[2.468,3.184,2.064,3.844,3.144,2.210,2.552]T 67

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Figure4-6. Example3:desiredpathoftheoriginandorientationofthetopcoordinatesystem parameters,asfollows; xd(t)=1 kdpto(t) dtk2dpto(t) dt (4) zd(t)=)]TJ /F9 7.97 Tf 10.5 4.71 Td[(100 7dpz(t) dt010 7T k)]TJ /F9 7.97 Tf 10.49 4.7 Td[(100 7dpz(t) dt010 7Tk2 (4) yd(t)=zd(t)xd(t) (4) wherepto(t)=[px(t);py(t);pz(t)]T.TheX-axisofthedesiredorientationisdenedasatangentvectorofthedesiredpathinequation( 4 ).TheZ-axisisdenedasavectorperpendiculartotheX-axisinequation( 4 )andtheY-axisisperpendiculartothosetwoaxes.TherotationmatrixiscomprisedofthesethreevectorsandEulerangleparameterscanbefoundusingequation( 4 ).ThedesiredpathandeachaxisofthedesiredorientationareshowninFigure 4-6 .TwosolutionsforthepistonlengthsareinFigure 4-7 andtheposechangeforoneofthesolutionsisshowninFigure 4-8 68

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Figure4-7. Example3:pistonlengths Figure4-8. Example3:poseofthemechanism 69

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CHAPTER5DYNAMICSOFATENSEGRITY-BASEDCOMPLIANTMECHANISM Thedynamicanalysisofparallelmechanismscanbeclassiedasforwardandreversedynamics,likeforthekinematicanalysis.Forwarddynamicsmeansdeterminingthedisplacement,velocity,andaccelerationofthemovingplatformortheend-effectorforthegivenforce/torqueinactuators.Reversedynamicsmeansdeterminingtheforce/torqueinactuatorsforthegivendisplacement,velocity,andaccelerationofthemovingplatform. Theefforttosetthedynamicmodelsofparallelmechanismshasbeenmadethroughvariousapproaches,suchastheNewton-Eulerequation,Lagrangianformulation,virtualwork,Kane'smethod,andothers.Eachapproachgenerallyhasbothadvantagesanddisadvantagesinaspectsofcomplexity,computationcost,andtypedependency.Forexample,theLagrangianformulationisbasedonthecalculationofkineticandpotentialenergy,butitneedsgreatersymboliccomputationtocalculatepartialderivativesofeachenergyterm.IntheNewton-Eulerapproach,suchcomputationisnotrequired.SomevirtualworkapproachesaremoreefcientincomputationcostthantheNewton-Eulerapproach. ForageneralGough-Stewartplatform,DoandYangsolvedtheinversedynamicproblem[ 18 ],andDasguptaandMruthyunjayaobtainedtheclosed-formdynamicequationusingtheNewton-Eulerequation[ 16 ].FattahandKasaeiderivedtheirdynamicequationofmotionforkinematicallyredundant3DOFparallelmechanismusingtheNewton-Eulerapproach[ 23 ].LeeandShah[ 45 ],Gengetal.[ 28 ],andYiuandLi[ 98 ]usedtheLagrangeformulationtoderiveadynamicequationfor3DOFparallelmechanism,generalGough-Stewartplatform,andplanar2DOFredundantparallelmechanism,respectively.ZhangandSonginvestigatedcomputationalefciencyinaninversedynamicproblemusingtherecursiveNewton-Eulermethodandthevirtualworkprincipleapproach[ 100 ].TsaisolvedtheinversedynamicsofthegeneralGough-Stewartplatformusingthevirtualworkprinciple[ 83 ].Fora2DOFparallelmechanismandgeneralGough-Stewartplatform,Gallardoetal.conducteddynamicanalysisbasedonthevirtualworkprincipleandthescrewtheory,whichisapowerfulmathematical 70

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tooltorepresentthemotionofrigidbodies.[ 27 ].Wangetal.devisedadynamicequationofaspatialparallelmechanismusingboththevirtualworkprincipleandtheLagrangeformulation[ 90 ].Baiges-ValentinderivedadynamicequationforthespatialparallelmechanismwhoselegconnectorshavebothcouplinganddecouplingstagesusingtheKane'smethod[ 11 ].OtherapproachesincludingtheHamilton'sprinciple[ 54 ]havebeenpublished[ 42 97 ]. Inthefollowingsections,velocityandaccelerationanalysesintermsofscrewsarecarriedout.Velocityandaccelerationanalysesidentifyparameterscalledvelocitystateandaccelerationstate,whichdenethevelocityandaccelerationofanypointonarigidbodywithrespecttoareference.Firstofall,thevelocityandaccelerationofapointonthetopbody,whichisconnectedtothelegconnectors,vpandapshouldbefoundintermsofangularandlinearvelocitiesandaccelerationsandlinecoordinates.Magnitudesofangularandlinearvelocitiesandaccelerationcanthenbecomputedthroughreversevelocityandaccelerationanalyses.Finally,thedynamicequationisderived. 5.1VelocityAnalysis Thereversevelocityanalysisforaspatialtensegrity-basedcompliantmechanismisperformed.Thefollowinginformationisgivenfortheproblem: Thedesiredangularandlinearvelocitiesofthetopbodymeasuredwithrespecttothebaseasrepresentedbyf!o;vogundertheassumptionthateachlegconnectorisjointothebasebodywithahookjointandtothetopbodywithasphericaljoint. Theobjectiveistodeterminelinearvelocitiesineachprismaticjoint,vLi,andangularvelocitiesineachhookjointinthelegconnector,!i1and!i2forindexofeachlegconnector,i=1:::7,thatwillsatisfytheserequirements. Inordertoanalyzevelocitycharacteristics,thejointinthelegconnectorofthemechanismisconsideredasHPS(Hook-Prismatic-Spherical)typewhoselegconsistsofthehookjointonthebasebody,theprismaticjointontheleg,andthesphericaljointonthetopbodyasshowninFigure 5-1 .Threevectorsofs1,s2ands3forthehookandprismaticjointscanbedenedsothats1isparalleltothex-axisofthereferencecoordinatesystem(orthebasebodycoordinatesystemifthebasebodyisxed),s3isthelongitudinaldirectionvectorfortheprismaticjoint,and 71

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Figure5-1. HPSlegconnector s2=s1s3.Theotherthreevectorsforthesphericaljoint,s4,s5ands6arenotdened.Thelinearcomponentofthevelocitystate,vocanbewrittenas vo=!1s0L1+!2s0L2+vLsL3+!4s0L4+!5s0L5+!6s0L6 (5) wheres0L1tos0L6arethelowerthreemembersintheunitizedscrewforeachlegconnectorandtheindexiforrecognizingeachlegisremovedforthesakeofsimplicity.ThelinearvelocityofthepointPcanbewrittenas vp=vo+!opop; (5) andthiscanbewrittenas vp=!1Lts2+!2Lts23+vLs3 (5) 72

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usingtherelationsof s0L1=pops1)]TJ /F3 11.955 Tf 11.95 0 Td[(pqps1=pops1)]TJ /F7 11.955 Tf 11.96 0 Td[(Lts3s1=pops1+Lts2s0L2=pops2)]TJ /F3 11.955 Tf 11.95 0 Td[(pqps2=pops2+Lts2s3=pops2+Lts23s0L4=pops4s0L5=pops5s0L6=pops6: (5) Threevectorsinequation( 5 )areperpendiculartoeachother.Therefore,twoangularandonelinearvelocitiescanbeobtainedbytakingdotproductwiths2,s23,ands3,respectively. !1=1 Ltvps2!2=1 Ltvps23vL=vps3 (5) Meanwhile,thevelocityoftheprismaticjoint,vL,canbeobtainedbyanotherwayasfollowing. vL=vLs3=vos3+!opops3=264!ovo375$3 (5) whereisreciprocaloperatorand$3istheunitizedscrewfortheprismaticjoint.Thelastequationsinequation( 5 )andequation( 5 )aresamebecause$3hasonlylowerthreecomponentsandupperthreearezeros. 5.2AccelerationAnalysis 5.2.1ProblemStatement Thereverseaccelerationanalysisforaspatialtensegrity-basedcompliantmechanismisperformed.Thefollowinginformationisgivenfortheproblem: 73

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Thedesiredaccelerationofthetopbodymeasuredwithrespecttothebaseasrepresentedbyfo;ao)]TJ /F18 11.955 Tf 11.96 0 Td[(!ovogunderthesameassumptionasinthevelocityanalysis. Theobjectiveistodeterminelinearaccelerationineachprismaticjoint,aLandtwoangularaccelerationsineachhookjoint,01and12,thatwillsatisfytheserequirements. 5.2.2AccelerationoftheJoint ThevelocityatthepointConthetopbodywithrespecttothebasecoordinatesystemcanbeexpressedas bvtC=bvtO+b!ttpOC (5) wherethesuperscriptsbandtindicatethecoordinatesystemsorbodies6andtpOCisthedistancevectorfromthepointOtothepointC. Differentiatingtheaboveequation,accelerationatthepointCcanbewrittenas bd dtbvtC=bd dtbvtO+bd dt(b!ttpOC) (5) batC=batO+bd dt(b!t)tpOCb!tbd dt(tpOC)=batO+bttpOC+b!t(b!ttpOC): (5) TheaccelerationatthepointPonthetopbodycanbealsoobtainedbythesamepatternasequation( 5 )andbere-writtenbysubstitutingequation( 5 )as batP=batO+bttpOP+b!t(b!ttpOP)=batC)]TJ /F6 7.97 Tf 11.95 4.94 Td[(bttpOC)]TJ /F6 7.97 Tf 11.96 4.94 Td[(b!t(b!ttpOC)+bttpOC+b!t(b!ttpOC)=batC+bttpCP+b!ttpPC: (5) wheretpCP=)]TJ /F6 7.97 Tf 9.29 4.34 Td[(tpPC. 6ThesesuperscriptsshouldbeBandTforconsistency,butsmalllettersarereplacedtoavoidconfusionwithToftranspose. 74

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Thelinearaccelerationpartoftheaccelerationstatecanbecalculatedas bato=01So1+12So2+2a3So3+34So4+45So5+56So6+(0!1)2S1So1+(1!2)2S2So2+(3!4)2S4So4+(4!5)2S5So5+(5!6)2S6So6+20!1S1(1!2So2+2v3So3+3!4So4+4!5So5+5!6So6)+21!2S2(2v3So3+3!4So4+4!5So5+5!6So6)+23!4S4(4!5So5+5!6So6)+24!5S55!6So6: (5) Substitutingequation( 5 )intoequation( 5 ),theaccelerationatthepointPcanbewrittenas batp=01So1+12So2+2a3So3+34So4+45So5+56So6+(0!1)2S1So1+(1!2)2S2So2+(3!4)2S4So4+(4!5)2S5So5+(5!6)2S6So6+20!1S1(1!2So2+2v3So3+3!4So4+4!5So5+5!6So6)+21!2S2(2v3So3+3!4So4+4!5So5+5!6So6)+23!4S4(4!5So5+5!6So6)+24!5S55!6So6+bttpop+b!t(b!ttpop) (5) 75

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Thevectors,Soifori=1;2;4;5;and6canbewrittenas So1=popS1+LtS2So2=popS2+LtS23So4=popS4So5=popS5So6=popS6 (5) whereS23=S2S3. Substitutingequation( 5 )intoequation( 5 ),theaccelerationatthepointPcanbewrittenas batp=01popS1+01LtS2+12popS2+12LtS23+2a3S3+34popS4+45popS5+56popS6+(0!1)2S1(popS1)+Lt(0!1)2S1S2+(1!2)2S2(popS2)+Lt(1!2)2S2S23+(3!4)2S4(popS4)+(4!5)2S5(popS5)+(5!6)2S6(popS6)+20!1S1(1!2popS2+2v3S3+3!4popS4+4!5popS5+5!6popS6)+2Lt0!1S23+21!2S2(2v3S3+3!4popS4+4!5popS5+5!6popS6)+23!4S4(4!5popS5+5!6popS6)+24!5S55!6(popS6)+bttpop+b!t(b!ttpop): (5) 76

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Figure5-2. Pistonpart Aftercancelingoutallcorrespondingterms,thiscanbere-writtenas batp=01LtS2+12LtS23+2a3S3+2Lt0!1S1S23+2Lt0!12v3S1S3+21!22v3S2S3=01LtS2+12LtS23+2a3S3+2Lt0!1(S1S3)S2+2Lt0!12v3S2+21!22v3S23: (5) TheangularandlinearaccelerationscanbeobtainedasfollowsbytakingdotproductwithS2,S23,andS3,respectively. 01=1 Lt(batpS2)]TJ /F8 11.955 Tf 11.95 0 Td[(20!12v3)]TJ /F8 11.955 Tf 11.96 0 Td[(2Lt0!1) (5) 12=1 Lt(batpS23)]TJ /F8 11.955 Tf 11.96 0 Td[(21!22v3) (5) 2a3=batpS3 (5) 5.2.3AccelerationofthePiston Intheprismaticjointofthelegconnector,themotionofthelowerandupperpartsofthepistoninFigure 5-2 canbedenedasthevelocityandaccelerationateachmasscenter.The 77

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velocityatthemasscenterofthelowerpistononthebody2withrespecttothebasecoordinatesystemcanbewrittenas bv2lC=bv2Q+0!22pQlC (5) wherelCandQmeanthemasscenterpointofthelowerpistonandthejointonthebasebody,respectively.Differentiatingthis,theaccelerationatthemasscenterofthelowerpistoncanbeobtainedas ba2lC=022pQlC+0!2(0!22pQlC)=llC02S3+llC0!2(0!22S3) (5) wherellCisthelengthbetweenthepointQandthemasscenterofthelowerpistonlCanditisaconstant. Bythesamemanner,thevelocityatthemasscenteroftheupperpistononthebody2withrespecttothebasecoordinatesystemcanbewrittenas bv2uC=bv3Q+0!23pQuC+3vuC; (5) whereuCmeansthemasscenterpointoftheupperpiston.Differentiatingthis,theaccelerationatthemasscenteroftheupperpistoncanbealsoobtainedas ba3uC=luC02S3+luC0!2(0!22S3)+luC LLS3+2luC L_L0!2S3 (5) whereluCisthelengthbetweenthepointQandthemasscenterofthelowerpistonuC,whichisavariabledenedasafunctionoftimeandLisdisplacementofthepiston. 5.3DynamicEquation Thedynamicequationofmotionisrequiredtoperformtheinversedynamicanalysisthatndsthewrenchneededoneachjointfromthegivendisplacement,velocityandacceleration.Theequationgenerallytakesoneofthefollowingforms. 78

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=M(q)q+Vm(q;_q)+G(q)=M(q)q+Adyn(q;_q)=Bdyn(q;_q;q) whereMismassmatrix,VmisacombinationofthecentrifugalandCoriolisterms,Gisthegravitationalterm,Adynisthesumofthecentrifugal,Coriolis,andgravitationalterms,andBdynisalltermsontheright-handsideintherstequation. Theseformsaredependentonthespecicapproachforthedynamicanalysis,suchastheNewton-Eulermethod,theLagrangeformulation,thevirtualworkprinciple,andothers.Eachapproachhasbothbenetsanddrawbacks.TheNewton-Eulermethodrequirescomputationofvelocitiesandaccelerationsformanypoints,mainlythecenterofmassandtheendoflinksonasystem,butitdoesnotneedacomplexmathematicalprocesslikeaderivative.TheLagrangeformulationusestermsforkineticenergyandpotentialenergyandneedseachderivative.Suchcomputationisoccasionallysimple,butnotsometimes.Forexample,thisapproachisgenerallyeasyandsimplewhenusedforaserialmanipulator,butitinvolvesamuchmorecomplicatedderivationprocessesforaparallelone.Therefore,theapproachfordynamicmodelingshouldbeselectedbasedonthetypeofobjectivesystemandhowcomplexthecomputationprocessis.Inthisstudy,theLagrangeformulationisused. 5.3.1LagrangeMethod Totalkineticenergyinthetopbodyandlegconnectorscanbewrittenas EK=mt 2bvCTbvC+b!tTIt 2b!t+7i=1fmpu 2bvucTbvuc+mpl 2bvlcTbvlc+0!2TIpu+Ipl 20!2gi (5) wheremandIaremassandmomentofinertiamatrix,whoseindices,t,pu,andpl,meanthetopbody,theupperpartofthepistonincludingthespring,andthelowerpartsofthepiston, 79

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respectively,andindicesforbvucandbvuc,ucandlcmeanthepositionsofthemasscentersoftheupperpistonpartincludingthespringandlowerpistons,respectively. Thetotalpotentialenergyinthetopbodyandlegconnectorscanbewrittenas EP=f(mpuLuc+mplLlc)g(s3T^k+k2 2gi (5) wheres3isunitdirectionvectorthatpassesalongthelegconnectorand^kistheunitvectorperpendiculartotheground. Thegeneralizecoordinatesforthedynamicequationcanbechosenas q=[xo;yo;zo;x;y;z]T (5) whereallelementsrepresentspatialmotionofthetopbody.ThesegeneralizedcoordinatesaredifferentfromthegeneralGough-Stewartplatformswhosegeneralizedcoordinatesaregenerallydenedasprismaticjointvariablesinthelegconnectors.Inthetensegrity-basedcompliantmechanism,itisimpossibletoderivetheLagrangeformulausingprismaticjointvariablesbecausethepresenceofthespringinthelegconnectormakesitdifculttondananalyticsolutionoftheforwardkinematicproblemandthusthepartialderivativeintheformulaisnotpossible.Thatistosay,jointspacevariablescanbeexpressedasfunctionoftaskspacevariables,butvice-versaisimpossible. Thus,thedynamicequationcanbewrittenas d dt@EKP @_q)]TJ /F7 11.955 Tf 13.15 8.08 Td[(@EKP @q= (5) whereEKP=EK)]TJ /F7 11.955 Tf 11.95 0 Td[(EP. 80

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5.3.2LinearVelocities ThepositionvectorfromtheorigintothepointCwhichismasscenterofthetopbodycanbewrittenas tpC=[xcyczc]=btRtpC+bptorg=266664cyczxc)]TJ /F7 11.955 Tf 11.96 0 Td[(cyszyc+syzc+xoxc(sxsycz+cxsz)+yc()]TJ /F7 11.955 Tf 9.3 0 Td[(sxsysz+cxcz)+zc()]TJ /F7 11.955 Tf 9.3 0 Td[(sxcy)+yoxc()]TJ /F7 11.955 Tf 9.3 0 Td[(cxsycz+sxsz)+yc(cxsysz+sxcz)+zc(cxcy)+zo377775 (5) where bptorg=[xoyozo]btR=266664cycz)]TJ /F7 11.955 Tf 9.29 0 Td[(cyszsysxsycz+cxsz)]TJ /F7 11.955 Tf 9.3 0 Td[(sxsysz+cxcz)]TJ /F7 11.955 Tf 9.3 0 Td[(sxcy)]TJ /F7 11.955 Tf 9.3 0 Td[(cxsycz+sxszcxsysz+sxczcxcy377775: (5) Differentiatingequation( 5 ),thevelocityatthepointCcanbederivedas bvC=2666666666666666666664xc()]TJ /F7 11.955 Tf 9.3 0 Td[(sycz_y)]TJ /F7 11.955 Tf 11.96 0 Td[(cysz_z)+yc(sysz_y)]TJ /F7 11.955 Tf 11.96 0 Td[(cycz_z)+zcsy_y+_xoxc(cxsycz_x+sxcycz_y)]TJ /F7 11.955 Tf 11.96 0 Td[(sxsysz_z)]TJ /F7 11.955 Tf 11.95 0 Td[(sxsz_x+cxcz_z)+yc()]TJ /F7 11.955 Tf 9.3 0 Td[(cxsysz_x)]TJ /F7 11.955 Tf 11.95 0 Td[(sxcysz_y)]TJ /F7 11.955 Tf 11.95 0 Td[(sxsycz_z)]TJ /F7 11.955 Tf 11.95 0 Td[(sxcz_x)]TJ /F7 11.955 Tf 11.96 0 Td[(cxsz_z)+zc()]TJ /F7 11.955 Tf 9.3 0 Td[(cxcy_x+sxsy_y)+_yoxc(sxsycz_x)]TJ /F7 11.955 Tf 11.95 0 Td[(cxcycz_y+cxsysz_z+cxsz_x+sxcz_z)+yc()]TJ /F7 11.955 Tf 9.3 0 Td[(sxsysz_x+cxcysz_y+cxsycz_z+cxcz_x)]TJ /F7 11.955 Tf 11.95 0 Td[(sxsz_z)+zc()]TJ /F7 11.955 Tf 9.29 0 Td[(sxcy_x)]TJ /F7 11.955 Tf 11.95 0 Td[(cxsy_y)+_zo3777777777777777777775: (5) 81

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Thepositionvectorfromtheorigintothemasscenterfortheupperpartofthelegconnectorcanbewrittenas bpuC=bpQ+bpQuC=bpQ+luCbpP)]TJ /F6 7.97 Tf 11.96 4.34 Td[(bpQ kbpP)]TJ /F6 7.97 Tf 11.96 3.45 Td[(bpQk=bpQ+luCbttpP+bptorg)]TJ /F6 7.97 Tf 11.96 4.34 Td[(bpQ kbttpP+bptorg)]TJ /F6 7.97 Tf 11.96 3.45 Td[(bpQk=bpQ+luC Lt(bttpP+bptorg)]TJ /F6 7.97 Tf 11.95 4.94 Td[(bpQ)=266666666664qx+luC Lt(cyczpx)]TJ /F7 11.955 Tf 11.96 0 Td[(cyszpy+sypz+xo)]TJ /F7 11.955 Tf 11.96 0 Td[(qx)qy+luC Ltf(sxsycz+cxsz)px+()]TJ /F7 11.955 Tf 9.3 0 Td[(sxsysz+cxcz)py+()]TJ /F7 11.955 Tf 9.3 0 Td[(sxcy)pz+yo)]TJ /F7 11.955 Tf 11.96 0 Td[(qy)gqz+luC Ltf()]TJ /F7 11.955 Tf 9.3 0 Td[(cxsycz+sxsz)px+(cxsysz+sxcz)py+(cxcy)pz+zo)]TJ /F7 11.955 Tf 11.95 0 Td[(qz)g377777777775=266664qx+luC LtUCxqy+luC LtUCyqz+luC LtUCz377775 (5) wherebpQ=[qxqyqz]TandbpP=[pxpypz]T. Differentiatingequation( 5 ),thevelocityatthemasscenterforupperpartofthelegconnectorcanbederivedas bvuc=d dtbpuc=266664_luC LtUCx)]TJ /F6 7.97 Tf 13.15 5.05 Td[(luC L2t(UCx)_Lt+luC Ltd dtUCx_luC LtUCy)]TJ /F6 7.97 Tf 13.15 5.04 Td[(luC L2t(UCy)_Lt+luC Ltd dtUCy_luC LtUCz)]TJ /F6 7.97 Tf 13.15 5.04 Td[(luC L2t(UCz)_Lt+luC Ltd dtUCz377775: (5) 82

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Inthesamemannerastheabove,thepositionvectorfromtheorigintothemasscenterforthelowerpartofthelegconnectorcanbewrittenas bplc=bpQ+bpQlc=bpQ+llCbpP)]TJ /F6 7.97 Tf 11.95 4.34 Td[(bpQ kbpP)]TJ /F6 7.97 Tf 11.95 3.45 Td[(bpQk=bpQ+llCbttpP+bptorg)]TJ /F6 7.97 Tf 11.96 4.34 Td[(bpQ kbttpP+bptorg)]TJ /F6 7.97 Tf 11.96 3.45 Td[(bpQk=bpQ+llC Lt(bttpP+bptorg)]TJ /F6 7.97 Tf 11.96 4.94 Td[(bpQ)=266666666664qx+llC Lt(cyczpx)]TJ /F7 11.955 Tf 11.96 0 Td[(cyszpy+sypz+xo)]TJ /F7 11.955 Tf 11.96 0 Td[(qx)qy+llC Ltf(sxsycz+cxsz)px+()]TJ /F7 11.955 Tf 9.3 0 Td[(sxsysz+cxcz)py+()]TJ /F7 11.955 Tf 9.3 0 Td[(sxcy)pz+yo)]TJ /F7 11.955 Tf 11.95 0 Td[(qy)gqz+llC Ltf()]TJ /F7 11.955 Tf 9.3 0 Td[(cxsycz+sxsz)px+(cxsysz+sxcz)py+(cxcy)pz+zo)]TJ /F7 11.955 Tf 11.96 0 Td[(qz)g377777777775=266664qx+llC LtLCxqy+llC LtLCyqz+llC LtLCz377775: (5) Differentiatingequation( 5 ),thevelocityatthemasscenterforlowerpartofthelegconnectorcanbederivedas bvlc=d dtbplc=266664)]TJ /F6 7.97 Tf 10.49 5.04 Td[(luC L2t(LCx)_Lt+luC Ltd dtLCx)]TJ /F6 7.97 Tf 10.5 5.04 Td[(luC L2t(LCy)_Lt+luC Ltd dtLCy)]TJ /F6 7.97 Tf 10.49 5.05 Td[(luC L2t(LCz)_Lt+luC Ltd dtLCz377775: (5) 83

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5.3.3AngularVelocities Theangularvelocityofthetopbodywithrespecttothebasebodycanbefoundfromthefollowingequation; d dt(btR)=bt!btR=bt!btR (5) wherebt!isthespatialcrossproductoperatorforbt!=[wxwywz]. Fromtheelementsof(1,3)6forthebothsidesinequation( 5 ), cy_y=wzsxcy+wycxcy; (5) thederivativeoftheEulerangleforthey-directioncanbederivedas _y=wzsx+wycxforcy6=0: (5) Fromtheelementsof(2,3)and(3,3)forthebothsidesinequation( 5 ), )]TJ /F7 11.955 Tf 11.95 0 Td[(cxcy_x+sxsy_y=)]TJ /F7 11.955 Tf 9.3 0 Td[(wxcxcy+wzsy; (5) )]TJ /F7 11.955 Tf 9.3 0 Td[(sxcy_x)]TJ /F7 11.955 Tf 11.96 0 Td[(cxsy_y=)]TJ /F7 11.955 Tf 9.3 0 Td[(wxsxcy)]TJ /F7 11.955 Tf 11.95 0 Td[(wysy; (5) thefollowingmatrixequationcanbewritten. 264)]TJ /F7 11.955 Tf 9.29 0 Td[(cxcysxsy)]TJ /F7 11.955 Tf 9.3 .01 Td[(sxcy)]TJ /F7 11.955 Tf 9.3 .01 Td[(cxsy375264_x_y375=264)]TJ /F7 11.955 Tf 9.3 0 Td[(wxcxcy+wzsy)]TJ /F7 11.955 Tf 9.3 0 Td[(wxsxcy)]TJ /F7 11.955 Tf 11.95 0 Td[(wysy375 (5) 6Thenumbersinparenthesismeanrowandcolumnofthematrix. 84

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Thedeterminantofequation( 5 )iscysyandthederivativesoftheEuleranglesforthexandydirectionscanbewrittenas 264_x_y375=1 cysy264)]TJ /F7 11.955 Tf 9.3 0 Td[(cxsy)]TJ /F7 11.955 Tf 9.29 0 Td[(sxsysxcy)]TJ /F7 11.955 Tf 9.3 0 Td[(cxcy375264)]TJ /F7 11.955 Tf 9.3 0 Td[(wxcxcy+wzsy)]TJ /F7 11.955 Tf 9.3 0 Td[(wxsxcy)]TJ /F7 11.955 Tf 11.95 0 Td[(wysy375=264wx+wysxsy cy)]TJ /F7 11.955 Tf 11.95 0 Td[(wzcxsy cywzsx+wycx375 (5) forcy6=0orsy6=0.Thus, _x=wx+wysxsy cy)]TJ /F7 11.955 Tf 11.96 0 Td[(wzcxsy cy: (5) Fromtheelementsof(1,1)or(1,2)forthebothsidesinequation( 5 ), )]TJ /F7 11.955 Tf 11.96 0 Td[(cysz_z=)]TJ /F7 11.955 Tf 9.3 0 Td[(cxszwz+cxszwy; (5) thederivativeofEulerangleinz-directioncanbederivedas )]TJ /F7 11.955 Tf 11.96 0 Td[(cysz_z=)]TJ /F7 11.955 Tf 9.3 0 Td[(cxszwz+cxszwy;_z=cx cywz)]TJ /F7 11.955 Tf 13.15 8.09 Td[(sx cywyforcy6=0;sz6=0: (5) Fromequations( 5 ),( 5 ),and( 5 ),thefollowingmatrixequationcanbewritten. 266664_x_y_z377775=2666641sxsy cy)]TJ /F6 7.97 Tf 10.5 5.87 Td[(cxsy cy0cxsx0)]TJ /F6 7.97 Tf 10.5 4.71 Td[(sx cycx cy377775266664wxwywz377775 (5) Takingtheinverseofequation( 5 ),theangularvelocityofthetopbodywithrespecttothebasebodycanbeobtainedas b!t=266664_x+sy_zcx_y)]TJ /F7 11.955 Tf 11.96 0 Td[(cysz_zsx_y+cxcy_z377775: (5) 85

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Withtheassumptionthatthejointonthebasebodyisthehooktypeasinthevelocityandaccelerationanalyses,threeunitdirectionvectorscanbedened;s1=[s1xs1ys1z]Tforaxedaxis,s3forthelinepassingalongthelegconnector,ands2perpendiculartotheformertwovectors.Thevectors3canbewrittenas s3=1 Lt(bpP)]TJ /F6 7.97 Tf 11.95 4.93 Td[(bpQ)=1 Lt266664UCxUCyUCz377775 (5) Thevector,s2,canbealsowrittenas s2=s1s3 (5) =[s1]s3=2666640)]TJ /F7 11.955 Tf 9.3 0 Td[(s1zs1ys1z0)]TJ /F7 11.955 Tf 9.3 0 Td[(s1x)]TJ /F7 11.955 Tf 9.29 0 Td[(s1ys1x03777751 Lt266664UCxUCyUCz377775=266664s1yUCz)]TJ /F7 11.955 Tf 11.95 0 Td[(s1zUCys1zUCx)]TJ /F7 11.955 Tf 11.95 0 Td[(s1xUCzs1xUCy)]TJ /F7 11.955 Tf 11.95 0 Td[(s1yUCx377775: (5) Thevectors23whichisrequiredtoobtainarelevantangularvelocitycanbewrittenas s23=s2s3 (5) =[s2]s3=1 L2t266664s1yUCxUCy)]TJ /F7 11.955 Tf 11.96 0 Td[(s1xUc2y+s1zUCxUCz)]TJ /F7 11.955 Tf 11.96 0 Td[(s1xUC2zs1xUCxUCy)]TJ /F7 11.955 Tf 11.95 0 Td[(s1yUC2x+s1zUCyUCz)]TJ /F7 11.955 Tf 11.96 0 Td[(s1yUC2zs1xUCxUCz)]TJ /F7 11.955 Tf 11.95 0 Td[(s1zUC2x+s1yUCyUCz)]TJ /F7 11.955 Tf 11.96 0 Td[(s1zUC2y377775: (5) 86

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Using0!1and1!2obtainedbyequation( 5 ),theangularvelocity,0!2,canbewrittenas 0!2=0!1s1+1!2s2=2666640!1s1x+1!2 Lt(s1yUCz)]TJ /F7 11.955 Tf 11.96 0 Td[(s1zUCy)0!1s1y+1!2 Lt(s1zUCx)]TJ /F7 11.955 Tf 11.96 0 Td[(s1xUCz)0!1s1z+1!2 Lt(s1xUCy)]TJ /F7 11.955 Tf 11.96 0 Td[(s1yUCx)377775: (5) 5.3.4KineticEnergy Thelinearvelocityofthetopbodyderivedinequation( 5 )canbere-writtenas bvC=266664_q1+v15_q5+v16_q6_q2+v24_q4+v25_q5+v26_q6_q3+v34_q4+v35_q5+v36_q6377775 (5) wherevariables,v15,v16,v24,v25,v26,v34,v35,andv36,canbefoundbyrearrangingequation( 5 )intermsof_q. Thekineticenergybylinearmotionofthetopbodycanbewrittenas EK1=mt 2bvCTbvC=mt 2(_q1+v15_q5+v16_q6)2+mt 2(_q2+v24_q4+v25_q5+v26_q6)2+mt 2(_q3+v34_q4+v35_q5+v36_q6)2: (5) ThepartialderivativeofEK1withrespectto_qanditstimederivativeisderivedintheAppendix. Thekineticenergybylinearmotionoftheupperpartofthelegconnectorcanbewrittenas EK2=mpu 2 _luC LtUCx)]TJ /F7 11.955 Tf 13.15 8.09 Td[(luC L2tUCx_Lt+luC Ltd dtUCx!2+mpu 2 _luC LtUCy)]TJ /F7 11.955 Tf 13.15 8.09 Td[(luC L2tUCy_Lt+luC Ltd dtUCy!2+mpu 2 _luC LtUCz)]TJ /F7 11.955 Tf 13.15 8.09 Td[(luC L2tUCz_Lt+luC Ltd dtUCz!2: (5) ThedetailsforthepartialderivativeandtimedifferentiationofEK2arewrittenintheAppendix. 87

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Thekineticenergybylinearmotionofthelowerpartofthelegconnectorcanbewrittenas EK3=mpl 2 )]TJ /F7 11.955 Tf 13.15 8.09 Td[(llC L2tLCx_Lt+llC Ltd dtLCx!2+mpl 2 )]TJ /F7 11.955 Tf 13.15 8.09 Td[(llC L2tLCy_Lt+llC Ltd dtLCy!2+mpl 2 )]TJ /F7 11.955 Tf 13.15 8.09 Td[(llC L2tLCz_Lt+llC Ltd dtLCz!2: (5) ThedetailsforthepartialderivativeandtimedifferentiationofEK3arewrittenintheAppendix. Thekineticenergybyangularmotionofthetopbodycanbewrittenas EK4=1 2(_x+sy_z)2Itxx+1 2(cx_y)]TJ /F7 11.955 Tf 11.95 0 Td[(cysz_z)2Ityy+1 2(sx_y+cxcy_z)2Itzz+(_x+sy_z)(cx_y)]TJ /F7 11.955 Tf 11.95 0 Td[(cysz_z)Itxy+(_x+sy_z)(sx_y+cxcy_z)Itxz+(cx_y)]TJ /F7 11.955 Tf 11.95 0 Td[(cysz_z)(sx_y+cxcy_z)Ityz (5) where It=266664ItxxItxyItxzItxyItyyItyzItxzItyzItzz377775: (5) ThedetailsforthepartialderivativeandtimedifferentiationofEK4arewrittenintheAppendix. 88

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Thekineticenergybyangularmotionofthelegconnectorcanbewrittenas EK5=1 2f0!1s1x+1!2 Lt(s1yUCz)]TJ /F7 11.955 Tf 11.95 0 Td[(s1zUCy)g2(Iuxx+Ilxx)+1 2f0!1s1y+1!2 Lt(s1zUCx)]TJ /F7 11.955 Tf 11.96 0 Td[(s1xUCz)g2(Iuyy+Ilyy)+1 2f0!1s1z+1!2 Lt(s1xUCy)]TJ /F7 11.955 Tf 11.96 0 Td[(s1yUCx)g2(Iuzz+Ilzz)+f0!1s1x+1!2 Lt(s1yUCz)]TJ /F7 11.955 Tf 11.95 0 Td[(s1zUCy)gf0!1s1y+1!2 Lt(s1zUCx)]TJ /F7 11.955 Tf 11.96 0 Td[(s1xUCz)g(Iuxy+Ilxy)+f0!1s1x+1!2 Lt(s1yUCz)]TJ /F7 11.955 Tf 11.95 0 Td[(s1zUCy)gf0!1s1z+1!2 Lt(s1xUCy)]TJ /F7 11.955 Tf 11.96 0 Td[(s1yUCx)g(Iuxz+Ilxz)+f0!1s1y+1!2 Lt(s1zUCx)]TJ /F7 11.955 Tf 11.95 0 Td[(s1xUCz)gf0!1s1z+1!2 Lt(s1xUCy)]TJ /F7 11.955 Tf 11.95 0 Td[(s1yUCx)g(Iuyz+Ilyz) (5) where Ipu=266664IuxxIuxyIuxzIuxyIuyyIuyzIuxzIuyzIuzz377775andIpl=266664IlxxIlxyIlxzIlxyIlyyIlyzIlxzIlyzIlzz377775: (5) ThedetailsforthepartialderivativeandtimedifferentiationofEK5arewrittenintheAppendix. 5.3.5PotentialEnergy Fromequations( 5 )and( 5 ),thepotentialenergy,EP,canbewrittenas EP=f(mpuluC+mplllC)g Lt(s1xUCy)]TJ /F7 11.955 Tf 11.95 0 Td[(s1yUCx)+k2 2gi (5) wherei=1:::7. ThedetailsforthepartialderivativeandtimedifferentiationofEParewrittenintheAppendix. 89

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CHAPTER6MODULARROBOTAPPLICATION Thissectioncovershowthemechanismpresentedinthisresearchcanbeappliedtopracticalsystemsandwhatperformancecanbeexpected.Theexistingcompliantparallelmechanismsarebeingappliedtovariouspositionorforcecontrolsystems.Thetensegrity-basedcompliantparallelmechanismcanbeusedinsuchsystems,butitwouldbebetterifthismechanismdemonstrateagoodperformancewhenitisusedinasysteminwhichitsexclusivecharacteristics,comparedtoothers,canbeshown.Thismechanismisclassiedasaparallelcompliantmechanismbecauseofthespringinthelegconnector,aparallelrobotmanipulatorifthepistoninthelegconnectorisconsideredasanlinearactuator,andaredundantmechanismbecauseitsmobilityhasthreeandsixDOFwithfourandsevenlegconnectors,respectively.Thus,thekeypointforchoosingtheapplicationiswhethertwocharacteristics,hybridofactiveandpassiveoperationandredundancy,arewellreected.Fortherstcharacteristics,thelegconnectorcanbeapassivecomponentoraactivecomponentwithapassiveenergystorageelement(spring).Redundancyinparallelmechanismswouldyieldsomebenetrelatedtoforwardkinematicproblemsorsingularityanalysiseventhoughithasconstraints,likedecreasedworkspace[ 24 44 51 62 ].Inthisresearch,suchcharacteristicsareusedtocontrolposeofamodularrobotsystemwhentwomodulesconnecteachother. 6.1Introduction Compliantparallelmechanismscanbeusedinapplicationsrelatedtoforcecontrolorpositioncontrol.Traditionally,parallelmechanismshavebeenappliedtoightsimulators,vehiclemotionsimulators,manufacturingtools,positioningdevicesfortelescope,activevibrationsuppressors,surgicaltoolsandthelike.Thefollowingarerelativelyrecentapplicationsusingparallelmechanisms,includingcompliantparallelmechanisms. Anexampleshowingimportanceofcomplianceorstiffnessincompliantmechanismsiswalkingrobotsincludingcompliantelement.Alexanderidentiedthreeusesofspringelementsinanimalleglocomotion[ 3 ]andGeyeretal.modeledahumanlegasacompliantlegofasimple 90

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AAnklerehabilitationrobot BSelf-recongurablerobot CCompliantleginawalkingrobot DComplianthipinawalkingrobot Figure6-1. Examplesofapplicationsusingcompliantparallelmechanisms mass-springsystemandshowedtherelationbetweengaitpatternandchangeofsuchparametersasstiffness[ 29 ].Suchcharacteristicsofaspringoracompliantcomponentcanbeadoptedintheeldofwalkingrobotsorhumanoidrobots.Yipresentedacompliantanklemechanismforeasylandingingaitandreducingweightandcostoflegsinabipedwalkingrobot[ 95 ].Farrelletal.investigatedtheeffectofapassivespringasacompliantankleinabipedwalkingrobotandfoundthattheexistenceofspringsisefcientinthedoublesupportphase,butnotinthesinglesupportphase[ 21 ].Yangetal.designedaparallelcompliantkneeforaplanarbipedwalkingrobotandinvestigatedtheenergeticefciencyofwalkingthroughseveralcasestudies[ 94 ].Scheintetal.exploredtheeffectsonenergyandgaitwithacompliantankleinabipedwalkingrobot[ 72 73 ].Someresearchershaveusedcomplianceforrunningorhoppingrobots.Scarfoglieroetal.builtbio-inspiredleggedrobotswithcompliantjoints[ 70 ]andHurstetal.developedanactuatorwithmechanicallyadjustableseriescompliance[ 37 ].Iidaetal.designedahuman-likecompliantleginwhichankle,knee,andhipareconnectedbyafewspringsthatcan 91

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beconsideredasaplanarcompliantmechanism[ 38 ](Figure 6-1 C).Mettinetal.alsodevelopedaplanarbipedwalkingrobotwithacomplianthipsimilartoaplanarcompliantmechanism[ 53 ](Figure 6-1 D).Otherresearchsoncompliantcomponentsofwalkingrobotsusingtheconceptofcompliantparallelmechanismscanbefoundin[ 22 43 61 64 ]. Forcecontrolinrobots,regardlessofwhetheritisparallelorserial,requirescontactwithanobjectorenvironmentwheretherobotisused,auserwhointendstousetherobottofulllanobjective,andanothersystemthatcooperateswiththerobot.Parallelmechanismscanbealsoappliedwhenhumanusersneedtobeincontactwiththerobot.Tsoietal.designedaparallelrobottorehabilitatepatients'ankles[ 84 85 ].Therobotwasdesignedasaspatialparallelmechanismsothatitscontrollercouldregulatetherelationshipbetweenanklemomentandanklemotionandallowtheuserstoexercisetobuildthemusclestrengthattheirankles.Sagliaetal.conductedsimilarresearchwiththesamepurpose[ 68 ](Figure 6-1 A).Tianetal.developedaspatialparallelrobotforarticialcervicaldiscreplacementsurgery[ 79 ].Theyuseditforgrindingthebonesurfacetomatchaprosthesiswiththeboneforexactpositioning.Othersurgicalapplicationsofparallelrobot-basedmedicaltoolsormachinesareusedinsurgeriesrelatedtosofttissue.Itisclearthatforcecontrolorregulationisessentialinsuchapplications. Oneoftheeldsforforcecontrolusingparallelmechanismsisanassemblytaskcalledthepeg-in-holeproblem[ 46 56 ].GrifsandDuffyusedaspatialcompliantparallelmechanismasaremotecentercompliancethatcanhelpavoidjammingandreducecontactforce[ 33 ].Trongetal.attemptedanassemblytaskusingapassivecompliantdevicesimilartoaplanarcompliantparallelmechanism[ 82 ].MascaroandAsadaappliedaplanarcompliantmechanismtoawheelchairandbedsystem[ 50 ].Theydesignedabumperthatreducestheimpactbetweenawheelchairandabedwhenthedocking.Huietal.developedaspacedockingsystemusingaspatialparallelmechanismasthedockingpartandacompliantmechanismasthecounterpart[ 36 ].Leeetal.[ 47 ],Yanetal.[ 93 ],andHaoetal.[ 65 ]builttestsystemsforspacecraftdockingwithspatialparallelmechanisms.Dunetal.designedanauto-dockingsystemforthefuelloadingrobotusedinhazardousenvironmentsusingathreeDOFspatialparallelmechanism 92

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[ 20 ].Wangetal.[ 91 ]andYuetal.[ 99 ]usedthreeDOFparallelmechanismforrigiddockingofeldmodularrobotswithself-recongurablepurpose(Figure 6-1 B). Modularrobotsgenerallyperformvariousmissionsusinggroupformationofmodules.Oneoftheirmissionscanbepathcongurationinaplaneandshapecongurationinaspace.Inthisresearch,aplanartensegrity-basedcompliantmechanismisappliedtothemodularrobotsystemsimilarto[ 91 99 ].Aplanarmechanismisdesignedasaconnectorforformingthedesiredpathcongurationofthemodules.Themoduleinmodularrobotscouldbeablock,amobilerobot,orakinematicmechanism[ 35 40 58 60 69 76 77 86 92 96 ].Amongthem,asmallgroundvehicleischosenasamodule,anditsdynamics,motion,andsensorsembeddedarenottreatedinthisresearch.Forthisapplication,twocharacteristicsofaplanartensegrity-basedcompliantmechanismareshown.Thismechanismconnectorcanplaytheroleofapassivecompliantmechanismortheroleofaactivecompliantmanipulatorsothatismakescompliantcontactwhenthemodulescongureadesiredpose.Forceredundancyofatensegrity-basedcompliantmechanismcanreducesingularpointswithinitsworkspaceduetodecreasedthedimensionofthesingularitymanifold. 6.2ProblemStatement Inordertoapplyaplanartensegrity-basedcompliantmechanismtoaconnectorofamodularrobot,thefollowinginformationandconditionsaregiven: Itisassumedthatallmodulesmoveonevengroundfortheplanarapplication. Congurationanddockingofonlytwomodulesareconsideredinthisproblem. Thedesiredcurvethateachmodularrobotformsandthecorrespondingposeofeachmodulearegiven. Twomoduleslieinlinestangentialtothedesiredpath.Themotionofmodulesisnotconsidered. Atargetmoduleisrestedandkeepsomefriction. Theobjectiveistocontrolpose,[xEEyEEEE]Toftheconnector. 93

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Figure6-2. Changeofthemoduleconnectorsinadesiredpath AnexampleschematicfortheproblemisshowninFigure 6-2 .Thethicklineisadesiredpathforamodularsystem,thedottedrectanglesarethegroundvehiclemodules,andthequadrilateral(blue/red)areconnectorsattachedtothemodules.TherstgurerepresentscongurationofthemodulesandposeoftheconnectorsatanytimeT,thesecondgurerepresentsthoseatthetimeT+t,andthelastoneshowsposechangeoftheconnectorswhentimestepgoesbyandallthemodulesmoveforward.Itischangedfrombluetored.Theapplicationisthattheconnectorformsadesiredposefromthischange.Thedesignofaconnector,dynamicmodel,andcontroldesignarefollowed. 6.3ConnectorDesign Inordertodesignaplanartensegrity-basedcompliantmechanismasaconnectorofasmallgroundvehiclemodularrobot,thefollowingfactorscanbeconsidered. Mobility Workspace 94

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Singularity Dexterityorothers Theconnector'smobilityischosenasthreebecausethisapplicationisavailableinaplaneasdescribedintheproblemstatement.Itcanbealsocalculatedfromthemobilityformulaasfollows; m=(n)]TJ /F7 11.955 Tf 11.96 0 Td[(j)]TJ /F8 11.955 Tf 11.95 0 Td[(1)+ji=1fj=3(10)]TJ /F8 11.955 Tf 11.96 0 Td[(12)]TJ /F8 11.955 Tf 11.96 0 Td[(1)+12=3 (6) wherenisthenumberofbodies,jisthenumberofjoints,istheorderofmotion,andfiisdegreeoffreedomofi-thjoint.Asdescribedinthepreviouschapters,aplanartensegrity-basedmechanismhasfourlegconnectorswhichcanbeusedasactuators.Thismismatchbetweenthemobilityandthenumberofactuator,threedegreeoffreedomandfouractuators,canmakeforceredundancydenedbyDascuptaandMruthyunjaya[ 17 ].Thisisdistinguishedfromkinematicredundancyinserialmanipulators,whichisredundancyofmotion.Redundancyinpureparallelmanipulators7isrelatedtoforcewhichcancarryload.Suchdualitybetweenserialandparallelmechanismscanbeappliedtosingularity.Singularityinparallelmechanismsisdenedasstaticsingularity,whichconstrainload-carryingwhilesingularityinserialmechanismsisdenedaskinematicsingularity,whichhampersmechanisms'motion.Also,suchrestrictioncanberemovedbykinematicorforceredundancyinserialorparallelmechanisms,respectively.Therefore,fordecreaseofrestrictiononload-carryingcapability,singularityshouldbeadesignparameter.Workspaceisalsoanimportantparameterindesignofthemechanism.Dexterityisnotconsideredinthedesignofthisplanarconnector. 6.3.1WorkspaceandSingularity Firstofall,themechanismisdesignedunderthefollowingconditions. 7This`pure'parallelmanipulatormeansaparallelmanipulatorwithoutanyseriallink.Thatistosay,ahybridparallel-serialmanipulatorisnotapplicabletosuchamanipulator. 95

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1. Themechanismhasashapeofaquadrilateralwhosetwooppositesidesarethetopandbaseplatforms. 2. Thelengthsofthespringsandpistonsarebounded;theyhaveminimumandmaximumlengths. 3. Itcanhaveatwo-layerstructuretoavoidinterferencebetweenthelegs. 4. Themostleft-sidejointsinbothplatformsaretheoriginsforbothbodycoordinatesystems,respectively. Undertheseconditions,themainfocusinmechanismdesignishowtochoosepositionsofthejointsonthetopandbaseplatform.Eachjointshouldbelocatedsothatthemechanismhasenoughworkspaceandminimizesthesingularareainworkspace.Thiscanbethoughtasaoptimizationproblem.Thejoints,p1,p4,p5,andp8,canbechosenfromthesizeofthetopandbaseplatforms.Designvariablesarethendenedasp2x,p2y,p3x,p3y,p6x,p6y,p7x,p7yforp2=[p2x;p2y]T,p3=[p3x;p3y]Tonthebaseplatform,p6=[p6x;p6y]T,andp7=[p7x;p7y]Tonthetopplatform.Theplaneoffx2[)]TJ /F8 11.955 Tf 9.3 0 Td[(10;15];y2[)]TJ /F8 11.955 Tf 9.3 0 Td[(10;10]gisgivenastheareatocheckworkspaceandsingularity.Thisareaisdividedinto10001000segmentstocalculatetnessvalueforacostfunction.Thecostfunctionisdenedas fv=wa cws+wbcsg (6) wherewaandwbareweights,cwsistheratioofworkspacetothetotalarea,#ofworkspacesegments #oftotalsegments,andcsgistheratioofsingularpointstothetotalarea,#ofsingularsegments #oftotalsegments.csgisthevaluethatcomesfrombinarymaskoperationtothematrixincludingworkspacesegments.WorkspacecanbecalculatedbykinematicsandsingularareainthetotalareacanbecalculatedbycheckingtheconditionnumberoftheJacobianmatrixH=[$1;$2;$3;$4]T.Inthisstudy,Hisconsideredassingularwhencond(H)>5000. Theoptimizationmethodtominimizefvischosenasthegeneticalgorithmbecausethismethodissuitableforthesituationwhenthesystemisgivenasanon-explicitfunctionwhichrequirespointwisecalculationbyiteration.ThedetailsofconstraintsandotherparametersforoptimizationcanbefoundinTable 6-1 96

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Table6-1. Optimization OptimizationmethodgeneticalgorithmCostfunctionfv=wa cws+wbcsgwa=10,wb=1e+6ExplorespaceR22fX2[)]TJ /F8 11.955 Tf 9.3 0 Td[(10;15];Y2[)]TJ /F8 11.955 Tf 9.3 0 Td[(10;10]gIndividualsx2R8Constraintsx12[0:1;3:0];x22[)]TJ /F8 11.955 Tf 9.3 0 Td[(2:0;1:0]joint2x32[3:1;6:4];x42[)]TJ /F8 11.955 Tf 9.3 0 Td[(1:0;2:0]joint3x52[0:1;2:5];x62[)]TJ /F8 11.955 Tf 9.3 0 Td[(1:0;2:0]joint6x72[2:6;4:4];x82[)]TJ /F8 11.955 Tf 9.3 0 Td[(2:0;1:0]joint7Populationsize10Crossover0.8Elitecount2MutationGaussiandistributionTolerance1e-2 Table6-2. Optimizationresult Optimizedvalues[0:9857;0:5620;5:4942;0:6561;0:6837;1:6453;4:3917;0:5448]Finaltness55.5850Generation51 Figure6-3. Resultofoptimizationusingthegeneticalgorithm ThetnessvalueisconvergedasshowninFigure 6-3 andtheresultisshowninTable 6-2 .Figures 6-4 and 6-5 showworkspaceandsingularpointsforthemechanisminwhichtheoptimizedjointpositionsareused. 97

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AEE=)]TJ /F23 9.963 Tf 7.75 0 Td[(60 BEE=)]TJ /F23 9.963 Tf 7.75 0 Td[(30 CEE=0 DEE=30 EEE=60 Figure6-4. Workspacefortheselectedorientationangles 98

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AEE=)]TJ /F23 9.963 Tf 7.75 0 Td[(60 BEE=)]TJ /F23 9.963 Tf 7.75 0 Td[(30 CEE=00 DEE=30 EEE=60 Figure6-5. Singularareafortheselectedorientationangles 99

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Figure6-6. Designedmechanism 6.3.2SystemDescription Usingtheresultfromtheprevioussection,amechanismmodelismadeinSolidWorksasshowninFigure 6-6 .TwomoduleswiththemechanismconnectorsisshowninFigure 6-7 6.3.3DynamicSimulation The3-dimensionalmechanismmodelcanbeimportedtoMatlabSimMechanics,whichisadynamicsimulationprogramformechanicalsystems.TheMatlabSimulinkdiagramsforreversedynamicsimulationareshowninFigures 6-8 6-9 ,and 6-10 Forthereversedynamicsimulation,topbodymotionisgivenasaspiralinputanditsequationiswrittenas qin=[0:8tcos(2t);)]TJ /F8 11.955 Tf 9.3 0 Td[(0:8tsin(2t)]T: (6) 100

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Figure6-7. Twomodularrobots Figure6-8. SimMechanicsdiagram 101

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Figure6-9. SimMechanicsdiagram:pistonpart Figure6-10. SimMechanicsdiagram:springpart TheinputtrajectoryandXYplotareshowninFigures 6-11 and 6-13 ,andthepistonlengthsinthelegsareshowninFigure 6-12 102

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Figure6-11. End-effectordisplacementforspiralmotion Figure6-12. Pistondisplacementforspiralmotion 6.4DynamicEquation Althoughthedynamicequationofaspatialtensegrity-basedcompliantmechanismwasderivedinthepreviouschapter,itisverylongandcomplicated.Thatequationisnotappropriateforcontrolbecauselesscomputationtimeinaplantmodelisbetterintheaspectofbandwidthoftheclosed-loopcontrolsystem.Thereforeasimplieddynamicmodelisnecessaryforthisparallelmechanismwhosedynamicsbecomesmorecomplicatedbyexistenceofthesprings.Theapproachusedin[ 52 ]couldbehelpfultoderiveasimplieddynamicequation. 103

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Figure6-13. X-Yplotforspiralmotion Figure6-14. Topbodyfreebodydiagram InFigure 6-14 ,CisthemasscenterandEistheend-effectorofthetopplatform.ThecontactforceapplyingoneachpointPiwhichisjointedwiththelegconnectorscanbewrittenas fPi=fPini+fPNifori=1;:::;4 (6) wherefPiisforcemagnitudesactingalongthelineswiththevectorsni2R21whichareunitdirectionvectorsforthelegconnectorsandfPNi2R21aretheforcecomponentsperpendiculartotheforces,fPi2R21.Theresultantforceandmomentabouttheend-effectorposition,Efor 104

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thosecontactforcescanbewrittenas F=4i=1fPini;s+FN (6) ME=4i=1pEPi;sfPini;s+MEN (6) whereni;s=nTi;0T,FN=4i=1fTPNi;0T,pEPi;s=pTEPi;0T=[xEPi;yEPi;0]T,andMEN=4i=1pTEPi;0TfTPNi;0T. LetWN=FTN;MTENT2R61andW=FT;MTET2R61.Then,wrenchequationonthetopbodyis W=2644i=1fPini;s4i=1pEPi;sfPini;s375+WN=264n1;sn2;sn3;sn4;spEP1n1;spEP2n2;spEP3n3;spEP4n4;s375266666664fP1fP2fP3fP4377777775+WN=HsfP+WN (6) whereaspatialcross-productoperator8, pEPi=26666400yEPi00)]TJ /F7 11.955 Tf 9.3 0 Td[(xEPi)]TJ /F7 11.955 Tf 9.3 0 Td[(yEPixEPi0377775fori=1;:::;4: 8Thisisaspatialcross-productoperatorthatdenedasaskew-symmetricmatrixwhichconsistsofcomponentsinthecorrespondingvectorandtherearesomezerosinoff-diagonalbecausepEPiisavectordenedinaplane. 105

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Fortwopoints,EandC,whichisthecenterofmassofthetopbody,momentaboutCandlinearaccelerationatEcanbewrittenas MC=ME+pCE;sF (6) aE=aC+!(!pCE;s)+pCE;s (6) wherepCE;s=pTCE;0T=[xCE;yCE;0]TisspatialformofthepositionvectorfromthepointCtoE,a2R31islinearaccelerationatthepointswhichcorrespondtothesubscripts,!2R31isangularvelocity,and2R31isangularaccelerationofthetopbody.Totalforcesonthetopbodycanbewrittenas F+mtg=mtaC (6) wheremtismassofthetopbodyandg=[0;)]TJ /F7 11.955 Tf 9.3 0 Td[(g;0]T2R31isgravitationalacceleration.Usingequation( 6 ),thiscanbewrittenas F=mtaE)]TJ /F7 11.955 Tf 11.95 0 Td[(mt!(!pCE;s))]TJ /F7 11.955 Tf 11.95 0 Td[(mtpCE;s)]TJ /F7 11.955 Tf 11.96 0 Td[(mtg (6) Meanwhile,momentaboutthepointCcanbealsoexpressedas MC=It (6) whereItismomentofinertiaofthetopbody.UsingEq.( 6 )and( 6 ),momentaboutthepointEcanbeobtainedas ME=(It)]TJ /F7 11.955 Tf 11.95 0 Td[(mtpCE2))]TJ /F7 11.955 Tf 11.96 0 Td[(mtpCEaE+mtpCE(!(!pCE;s)+g) (6) wherepCEisaspatialcross-productoperatorforpCE;sandgivenas pCE=26666400yCE00)]TJ /F7 11.955 Tf 9.3 0 Td[(xCE)]TJ /F7 11.955 Tf 9.29 0 Td[(yCExCE0377775: 106

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Equation( 6 )and( 6 )canbewrittenas 264FME375=264mtI3mtpCE)]TJ /F7 11.955 Tf 9.3 0 Td[(mtpCEIt)]TJ /F7 11.955 Tf 11.96 0 Td[(mtpCE2375264aE375+264)]TJ /F7 11.955 Tf 9.3 0 Td[(mt!(!pCE;s))]TJ /F7 11.955 Tf 11.96 0 Td[(mtgmtpCE(!(!pCE;s)+g)375 (6) whereI3is33identitymatrix.Thedimensionoftheleft-handsidetermoftheaboveequationis61.Thiscanbesimpliedas31equationforaplanarsystemlike 264FplanarME;planar375=264mtI2mt pCE)]TJ /F7 11.955 Tf 9.3 0 Td[(mt pCEIt)]TJ /F7 11.955 Tf 11.96 0 Td[(mtpCETpCE375264aE;planar375+264mt!2pCE)]TJ /F7 11.955 Tf 11.95 0 Td[(mtgp)]TJ /F7 11.955 Tf 9.3 0 Td[(mtxCEg375 (6) whereFplanar2R21isaplanarvectorwhichconsistsofthersttwoelementsofF;ME;planaristhelastelementofME;aE;planar=[aE;x;aE;y]TisaplanarvectorwhichconsistsofthersttwoelementsofaE;isthelastelementof;!isthelastelementof!; pCE=[yCE;)]TJ /F7 11.955 Tf 9.3 0 Td[(xCE]T; pCE=[)]TJ /F7 11.955 Tf 9.29 0 Td[(yCE;xCE]forpCE=[xCE;yCE]T,andgp=[0;)]TJ /F7 11.955 Tf 9.3 0 Td[(g]Tisaplanarvectorforg.Thisconversioncanbeconrmedas 264100010375PCE;s=264100010375266664yCE)]TJ /F7 11.955 Tf 9.3 0 Td[(xCE0377775=264yCE)]TJ /F7 11.955 Tf 9.3 0 Td[(xCE375= pCE; 001PCE;saE=00126666400xCEaE;y)]TJ /F7 11.955 Tf 11.95 0 Td[(yCEaE;x377775=xCEaE;y)]TJ /F7 11.955 Tf 11.96 0 Td[(yCEaE;x=)]TJ /F7 11.955 Tf 9.3 0 Td[(yCExCE264aE;xaE;y375= pCEaE;planar; 107

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001PCE;s(PCE;s)=00126666400)]TJ /F7 11.955 Tf 9.3 0 Td[(y2CE)]TJ /F7 11.955 Tf 11.96 0 Td[(x2CE377775=)]TJ /F7 11.955 Tf 9.3 0 Td[((y2CE+x2CE)=)]TJ /F7 11.955 Tf 9.3 0 Td[(264xCEyCE375T264xCEyCE375=)]TJ /F7 11.955 Tf 9.3 0 Td[(pCETpCE; 264100010375!(!PCE;s)=264100010375266664)]TJ /F7 11.955 Tf 9.29 0 Td[(!2xCE)]TJ /F7 11.955 Tf 9.3 0 Td[(!2yCE0377775=)]TJ /F7 11.955 Tf 9.3 0 Td[(!2pCE; 001PCE;sg=00126666400)]TJ /F7 11.955 Tf 9.3 0 Td[(xCEg377775=)]TJ /F7 11.955 Tf 9.3 0 Td[(xCEg: Onceqisdenedas[xE;yE;E]T,whosecomponentsrepresentthepositionofthepointEinthexandydirectionandtheorientationofthetopbody,equation( 6 )canbere-writtenas Wplanar=Aq+b1: (6) whereAisa33matrixandb1isa31vector. AccelerationatthepointPionthetopbodycanbederivedas aPi;s=aE;planar+!(!pEPi;s)+pEPi;s: (6) 108

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TheprojectionofaPionaplaneperpendiculartothevectorni;siswrittenas aNi=(ni;saPi)ni;s=ni;s2aPi (6) whereniisaspatialcross-productoperatorforni;s.Then,theforce,FNcanbewrittenas FN=4i=1miaNi=4i=1mini2I3)]TJ /F3 11.955 Tf 9.29 0 Td[(pEPi264aE375+4i=1mini2!2pEPi;s (6) wheremiismassfortheparticleonthepointPiwhichcanbecalculatedfrommomentofinertiaandthedistancefromthepointtotheaxisofrotation,I3is33identitymatrixand!isaspatialcross-productoperatorfor!.Themoment,MENcanbealsowrittenas MEN=4i=1mipEPini2I3)]TJ /F3 11.955 Tf 9.3 0 Td[(pEPi264aE375+4i=1mipEPini2!2pEPi;s: (6) Throughtheconversionsimilartoequation( 6 ),bothequations( 6 )and( 6 )canbewrittenas 264FN;planarMEN;planar375=2644i=1miniTniI2)]TJ /F8 11.955 Tf 9.29 0 Td[(4i=1miniTni pEPi4i=1mi pEPiniTni)]TJ /F8 11.955 Tf 9.3 0 Td[(4i=1mi pEPiniTni pEPi375q+264)]TJ /F8 11.955 Tf 9.29 0 Td[(4i=1mi!2niTnipEPi)]TJ /F8 11.955 Tf 9.3 0 Td[(4i=1mi!2 pEPiniTnipEPi375: (6) Theleft-handsidetermoftheaboveequationcanbedenedasWN;planarandthiscanbewrittenas WN;planar=Dq+e1: (6) 109

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Figure6-15. Legfreebodydiagram Meanwhileequation( 6 )canbeconvertedintotheplanarformas Wplanar=264n1n2n3n4 pEP1n1 pEP2n2 pEP3n3 pEP4n4375266666664fP1fP2fP3fP4377777775+WN;planar=HfP+WN;planar (6) where pEPi=[)]TJ /F7 11.955 Tf 9.3 0 Td[(yEPi;xEPi]fori=1;:::;4andH2R34istheplanarformofHs. Substitutingequation( 6 )intoequation( 6 )andarrangingthemandequation( 6 ),theequationcanbeobtainedas Mtq+N=HfP (6) whereMt=A)]TJ /F7 11.955 Tf 11.96 0 Td[(D,N=b1)]TJ /F3 11.955 Tf 11.95 0 Td[(e1,andHisaplanarmatrixofHs. ThesimplemodelofthelegconnectorisshowninFigure 6-15 ,inwhichmpismassofthepiston,xaispistondisplacement,bpiseffectivedampingcoefcientofthepiston,kpiseffectivestiffnessofthepiston,kisstiffnessoftheactualspring,xsisspringdisplacement,faisactuationforceonthepiston,frisfrictionalforceinthepiston,andmassoftheactualspringisneglected. 110

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Themotionofthepistoncanbemodeledas mixai=)]TJ /F7 11.955 Tf 9.3 0 Td[(kpi(xai)]TJ /F7 11.955 Tf 11.96 0 Td[(xsi))]TJ /F7 11.955 Tf 11.96 0 Td[(bpi(_xai)]TJ /F8 11.955 Tf 13.98 0 Td[(_xsi))]TJ /F7 11.955 Tf 11.96 0 Td[(ki(xai)]TJ /F7 11.955 Tf 11.95 0 Td[(xsi))]TJ /F7 11.955 Tf 11.96 0 Td[(fri+fai (6) 0=)]TJ /F7 11.955 Tf 9.3 0 Td[(ki(xsi)]TJ /F7 11.955 Tf 11.95 0 Td[(xai))]TJ /F7 11.955 Tf 11.96 0 Td[(fpi (6) wherei=1;:::;4. Thisequationcanbere-writtenas Mpxa=)]TJ /F7 11.955 Tf 9.3 0 Td[(Kp(xa)]TJ /F3 11.955 Tf 11.95 0 Td[(xs))]TJ /F7 11.955 Tf 11.95 0 Td[(Bp(_xa)]TJ /F8 11.955 Tf 13.32 .43 Td[(_xs))]TJ /F3 11.955 Tf 11.96 0 Td[(fp)]TJ /F3 11.955 Tf 11.95 0 Td[(fr+fa (6) whereMp=diag(mi)2R44,Bp=diag(bpi)2R44,Kp=diag(kpi)2R44,K=diag(ki)2R44,andothersarethe41vectorswhichconsistofi-thcomponents,forexample,xa=[xa1;xa2;xa3;xa4]T. 6.5PoseControl Inthissection,thepose(positionandorientation)controloftheconnectordesignedinsection 6.3 isdiscussed.Asawayforposecontrol,kinestaticsapproachshowninsection 3 canbeused.Itisaquasi-staticcontrolthatcanachievecontrolifanepositioncontrolforthepistonisusedandadesiredpose,whosefrequencyisrelativelylower,isgivenbecausedynamicpropertiesoftheconnectorsuchasinertialforceandmomentcauseserrorinoperationthroughthismethod.Asanothermethod,two-stagecontrolisproposed.Thiscontrolconsistsoftwolevels;PID-computedtorquecontrolofthemotionofthetopplatformandadaptivecontrolofthepistoninthelegconnector.PIDcontroltracksadesiredposeusingthejointforcesalongthelineofthelegconnector,fp,whichistrackedbyadaptivecontrolwithestimationofuncertainparameterssuchasstiffness,dampingcoefcient,andfrictioncoefcientinthepiston. Thedynamicmodelforthetopplatformisrewrittenas Mt(ni(q))q+N(ni(q);_q)=H(ni(q))(fp(t))]TJ /F3 11.955 Tf 11.96 0 Td[(fd(t))fori=1;:::;4 (6) 111

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Figure6-16. Controlschematic wherefd2R4isdisturbance.Thecontrolobjectivetotrackadesiredtime-varyingpose(positionandorientation)ofthetopbody,isgivenas q)166(!qd whereqd(t)2R3. Thenaposetrackingerroranditsderivative,e1(t)2R3and_e1(t)2R3,canbewrittenas e1=qd)]TJ /F3 11.955 Tf 11.95 0 Td[(q_e1=_qd)]TJ /F8 11.955 Tf 13.65 .5 Td[(_q: (6) Thesecondderivativeoftheposetrackingerroriswrittenas e1=qd)]TJ /F8 11.955 Tf 12.35 .51 Td[(q=qd+M)]TJ /F9 7.97 Tf 6.59 0 Td[(1t(N)]TJ /F7 11.955 Tf 11.96 0 Td[(Hfp+Hfd): (6) 112

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Deningthecontrolforce fp=H+(Mt(qd)]TJ /F3 11.955 Tf 11.95 0 Td[(u)+N) (6) andadisturbancefunction wd=M)]TJ /F9 7.97 Tf 6.59 0 Td[(1tHfd; (6) equation( 6 )canbewrittenas e1=u+wd (6) whereH+ispseudoinverseofHandu(t)2R3iscontrolinput. PIDcontrollawcanbewrittenas u=)]TJ /F7 11.955 Tf 9.3 0 Td[(K1e1)]TJ /F7 11.955 Tf 11.95 0 Td[(K2_e1)]TJ /F7 11.955 Tf 11.96 0 Td[(K31 (6) where1=Re1dt. Substitutingequation( 6 )intoequation( 6 ),theclosed-looperrordynamicsystemiswrittenas e1+K1e1+K2_e1=wd)]TJ /F7 11.955 Tf 11.95 0 Td[(K31: (6) Equation( 6 )isanonhomogeneoussecond-orderdifferentialequationofe1withconstantcoefcients.Itssolutioncanbegivenasthesumofthehomogeneoussolutionobtainedfromtheleft-handsideofequation( 6 )andtheparticularsolutionobtainedfromtheright-handsideofequation( 6 ).Dependingonhowtochoosethecoefcientsofthedifferentialequation,K1andK2,andK3intheright-handside,e1cangotozero. Recallingthedynamicmodelforthelegconnector, fp(t)=K(xa(t))]TJ /F3 11.955 Tf 11.96 0 Td[(xs(t)); (6) Mpxa(t)=)]TJ /F7 11.955 Tf 9.3 0 Td[(Bp_xa(t))]TJ /F7 11.955 Tf 11.95 0 Td[(Kpxa(t))]TJ /F3 11.955 Tf 11.96 0 Td[(fp(t))]TJ /F3 11.955 Tf 11.95 0 Td[(fr(t)+fa(t): (6) 113

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Apositiontrackingerror,e2(t)2R4andalteredtrackingerror,r(t)2R4aredenedas e2=xa)]TJ /F3 11.955 Tf 11.95 0 Td[(xad; (6) r=_e2+1e2 (6) wherexad(t)2R4isadesiredpistonpositionwhichisdenedasK)]TJ /F9 7.97 Tf 6.58 0 Td[(1fp(t)+xs(t). Theopen-looptrackingerrorsystemforthelegconnectorcanbedevelopedas Mp_r=Mpe2+1Mp_e2=)]TJ /F7 11.955 Tf 9.3 0 Td[(Bp_xa)]TJ /F7 11.955 Tf 11.95 0 Td[(Kpxa)]TJ /F3 11.955 Tf 11.95 0 Td[(fp)]TJ /F3 11.955 Tf 11.95 0 Td[(fr+fa)]TJ /F7 11.955 Tf 11.96 0 Td[(Mpxad+1Mp_e2: (6) Inequation( 6 ),stiffness,dampingfactor,andthefrictionforce,whichismodeledasacoulombfriction,Frtanh(_xa(t))whereFr=diag(fri)9,forthepistonareunknownsandtheycanbewrittenas Y#=)]TJ /F7 11.955 Tf 9.3 0 Td[(Bp_xa)]TJ /F7 11.955 Tf 11.95 0 Td[(Kpxa)]TJ /F7 11.955 Tf 11.96 0 Td[(Frtanh(_xa)=)]TJ /F7 11.955 Tf 9.3 0 Td[(diag(_xa))]TJ /F7 11.955 Tf 9.3 0 Td[(diag(xa))]TJ /F7 11.955 Tf 9.3 0 Td[(diag(tanh(_xa))bp1;:::;bp4;kp1;:::;kp4;fr1;:::;fr4T (6) where#2R12consistsoftheconstantunknownparametersandY(xa;_xa)2R412isthedesiredregressionmatrix. Therefore,equation( 6 )canbere-writtenas Mp_r=Y#+e2)]TJ /F3 11.955 Tf 11.95 0 Td[(fp+fa)]TJ /F7 11.955 Tf 11.95 0 Td[(Mpxad+1Mp_e2: (6) Thecontrolinputforthepistonisdesignedas fa=)]TJ /F7 11.955 Tf 9.3 0 Td[(Y^#+fp+Mpxad)]TJ /F7 11.955 Tf 11.96 0 Td[(1Mp_e2)]TJ /F7 11.955 Tf 11.95 0 Td[(K4r)]TJ /F3 11.955 Tf 11.96 0 Td[(e2 (6) 9Thismodelisacoulombfrictionpartofthefrictionin[ 48 ]. 114

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and^#(t)2R12isaparameterestimatevectorgeneratedfromthefollowinggradientupdatelaw _^#(t)=)]TJ /F7 11.955 Tf 27.61 0 Td[(YTr (6) whereK42R44and)]TJ /F2 11.955 Tf 10.64 0 Td[(2R1212aregainmatrices. Equation( 6 )canbenallywrittenas Mp_r=Y~#)]TJ /F7 11.955 Tf 11.96 0 Td[(K4r)]TJ /F3 11.955 Tf 11.96 0 Td[(e2 (6) where~#(t)2R12istheparameterestimationerrorvectordenedas~#=#)]TJ /F8 11.955 Tf 13.16 3.15 Td[(^#. Thestabilityoftheadaptivecontrollerforthelegconnector,giveninequation( 6 ),canbeexaminedthroughthefollowingtheorem. Theorem6.1. Theproposedadaptivecontrollergiveninequation( 6 )ensuresasymptotictrakinginthesensethat limt!1e2(t)=0 (6) wheree2(t)wasdenedinequation( 6 ). Proof. ToproveTheorem6.1,anonnegativefunctionV(t)2Risdenesas V=1 2rTMpr+1 2~#T)]TJ /F13 7.97 Tf 7.31 4.94 Td[()]TJ /F9 7.97 Tf 6.59 0 Td[(1~#+1 2eT2e2 (6) where)]TJ /F2 11.955 Tf 10.63 0 Td[(2R1212. Takingthetimederivativeofequation( 6 )andusingequation( 6 )and( 6 ),and( 6 ),thefollowingexpressioncanbeobtained _V=rTMp_r)]TJ /F8 11.955 Tf 13.16 3.15 Td[(~#T)]TJ /F13 7.97 Tf 7.31 4.93 Td[()]TJ /F9 7.97 Tf 6.59 0 Td[(1_^#+eT2_e2=rT(Y~#)]TJ /F7 11.955 Tf 11.95 0 Td[(K4r)]TJ /F3 11.955 Tf 11.95 0 Td[(e2))]TJ /F8 11.955 Tf 13.17 3.16 Td[(~#TYTr+eT2(r)]TJ /F7 11.955 Tf 11.96 0 Td[(1e2)=)]TJ /F3 11.955 Tf 9.29 0 Td[(rTK4r)]TJ /F7 11.955 Tf 11.96 0 Td[(1eT2e20: (6) Equations( 6 )and( 6 )canbeusedtoprovethatr(t);e2(t)2L1andthatr(t);e2(t)2L2.SincetheregressionmatrixYismadeupofboundedarguments,Y2L1,thenfa2L1from 115

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Table6-3. Simulationparametersandinitialvalues Par.DescriptionValue mtmassofthetopplatform0:05kgmpimassofthepiston0:01kgItmomentofinertia1:3e)]TJ /F8 11.955 Tf 11.96 0 Td[(7kgcm2kispringstiffness[1;1;1;1]TN/cmk0ispringfreelength[4;4;4;4]TcmLipistonlength2[0;5]cmbpidampingcoeff.ofthepiston[5;6;7;8]TNs/cmkpistiffnessofthepiston[50;60;70;80]TN/cmfrifrictionmagnitude[0:001;0:001;0:001;0:001]TNq0initialposition/orientation[3:7;15:7;5]Tcm,cm,deg_q0initialvelocity031cm/s,cm/s,deg/sqddesiredposition/orientation[6sin(t);12sin(0:4t);5sin(0:2t)]Tcm,cm,degxa0initialpistondisplacement[0:01;0:01;0:01;0:01]Tcm_xa0initialpistonvelocity041cm_q0initialvelocity031cm/s,cm/s,deg/s#0initialestimates0121fddisturbance0:01sin(2t)+0:005sin(10t)N equation( 6 ).Equation( 6 )canbealsousedtoprovethat_e2(t)2L1.Basedonthefactsthate2(t);_e2(t)2L1ande2(t)2L2,Barbalat'sLemmacanbeinvokedtoprovethat limt!1e2(t)=0: (6) Inordertodemonstratetheperformanceofthecontrollerdesignedabove,asimulationiscarriedout.TheparametersusedinthesimulationareshowninTable 6-3 .ThegainsusedinthesimulationareK1=35I3,K2=10:5I3,K3=0:005I3,K4=2:5I4,1=2,and)-313(=1:2I12whereInmeansidentitymatrixwithnndimension. ThesimulationresultisshowninFigure 6-17 to 6-23 .InFigure 6-17 and 6-18 ,theend-effectoroftheconnectortracksthedesiredpositionandvelocity,andthetrackingerrorseemsverysmallafter1second.Figure 6-19 and 6-20 showpistondisplacementconvergestothedesiredvalueandtheerror,e2becomessmall.InFigure 6-21 ,parameterestimatesconverge.Figure 6-22 and 6-23 showcontrolinputsineachlevelcontrol. 116

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Figure6-17. Positionandorientation Figure6-18. Velocity 117

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Figure6-19. Pistondisplacement Figure6-20. Pistondisplacementerror 118

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Figure6-21. Parameterestimates Figure6-22. Intermediatecontrolinputatthejoint 119

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Figure6-23. Controlinputatthepiston 120

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CHAPTER7CONCLUSION Thisresearchpresentedatensegrity-basedcompliantmechanism.Aplanarandspatialcompliantmechanismsweredesignedbythetensegrityconcept.Theproblemsforthekinestaticanalysisandstiffnesssynthesiswerethenperformedforbothmechanisms.Inthekinestaticproblem,theforcesinthespringsandthelengthsofthelegpistonsweredeterminedthatwouldpositionandorientthetopbodyasdesiredsothatitisinequilibriumwhenaknownexternalwrenchisapplied.Thespringconstants,springfreelengths,andtotaldesiredpotentialenergyinthesystemweregiven.Itwasshownthatamaximumoftworealsolutionscouldexist.Inthestiffnesssynthesisproblem,thespringconstantsandpistonlengthsweretobedeterminedtopositionandorientthetopbodyasspeciedwhileinequilibriumunderagivenappliedexternalwrench.Alsothedesiredstiffnessmatricesforthemechanismswerespecied.Auniquesolutionisobtainedunlesscertaingeometricconditionsoccurwhichcausethematricestobecomesingular.Intheanalyses,itwasconsideredthatthefreelengthsofthespringsarenon-zeros,thejointsonthetopandbasebodiesarearbitrarilylocated,andthetopandbasebodieshavearbitraryshapes.Thevelocityandaccelerationanalyseswerealsoperformedanditwasfoundthatmagnitudesofvelocityandaccelerationofinthejointscoulddeterminebythegivenvelocityandaccelerationstates.ThedynamicequationwasderivedusingtheLagrangeformulawhenthegeneralizedcoordinatesarechosenastask-spacevariables,buttheresultequationisverycomplexandlong,comparedtothegeneralGough-Stewartplatform.Asanapplicationproblem,aplanartensegrity-basedmechanismwasconsideredasaconnectorofthegroundvehicletypedmodularrobot,whosemissionisassignedtoformagivenpathconguration.Inordertodesignaspecicmechanism,acostfunctionrelatedtotheworkspaceandsingularityissetupandtheoptimizationproblemwassolvedbyusingthegeneticalgorithm.Itisconrmedthatthemechanismdesignedhasawideworkspaceandraresingulararea.Athree-dimensionalmodelforthedesignedmechanismwasmadeandsimulatedincommercialprograms.Thesimplied 121

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dynamicmodelforcontrolwasderivedandtheposecontrollerbasedontwo-levelPIDandadaptivecontrolwasdesignedandsimulatedtoverifyvalidityofthecontroldesign. Thisresearchcouldbeextendedtosomepotentialresearchtopics.Thealgebraicsolutionfortheforwardanalysisisstillopenwhilenumericalsolutioncanbefoundbyvariousmethods.Ifthisproblemissolved,thenthedynamicequationcouldbeeasilyderivedthatwasdiscussedinthisresearch.Itcouldbeafutureworktomanufactureanactualmechanismwhereactuatorsandsensorsareembeddedandverifytheproposedapplicationanditstaskperformance.Downsizingandrealizingamotorizedpistonwouldbeachallenge.Aspatialmechanismcanbealsodesignedandusedinthesamesystem,butunderdifferentconditionsuchasunevenground.Forthecontrol,kinestaticcontrolinventedbyGrifscanbeused.Ifthecontactforcebetweenthemodulesisrequiredtokeeptheconnection,explicitkinestaticcontrolwithadesiredpositionandwrenchinputsisagoodmethodforusingtheproblemsdiscussedinthisresearch,kinestaticandstiffnesssynthesisproblems. 122

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APPENDIXDERIVATIONOFTHEDYNAMICEQUATION Thersttermintheleft-handsideofequation( 5 )areobtainedforeachsub-term,EKifori=1;:::;5andEPinthisappendix.Thesecondtermforkineticenergyisnotwrittenherebecauseequationsareextremelylong. ThepartialderivativeofEK1withrespectto_qcanbecalculatedas @EK1 @_q1=mt(_q1+v15_q5+v16_q6)@EK1 @_q2=mt(_q2+v24_q4+v25_q5+v26_q6)@EK1 @_q3=mt(_q3+v34_q4+v35_q5+v36_q6)@EK1 @_q4=mtf(v24_q2+v34_q3+(v224+v234)_q4+(v34v35+v24v25)_q5+(v24v26+v34v36)_q6@EK1 @_q5=mtfv15_q1+v25_q2+v35_q3+(v24v25+v34v35)_q4+(v215+v225+v235)_q5+(v15v16+v25v26+v35v36)_q6g@EK1 @_q6=mtfv16_q1+v26_q2+v36_q3+(v24v26+v34v36)_q4+(v15v16+v25v26+v35v36)_q5+(v216+v226+v236)_q6g (A) Time-differentiatingthis,theequationforthekineticenergybythelinearmotionofthetopbodycanbecalculatedas d dt@EK1 @_q1=mtq1+mtv15q5+mtv16q6+mt_v15_q5+mt_v16_q6d dt@EK1 @_q2=mtq2+mtv24q4+mtv25q5+mtv26q6+mt_v24_q4+mt_v25_q5+mt_v26_q6d dt@EK1 @_q3=mt_q3+mtv34q4+mtv35q5+mtv36q6+mt_v34_q4+mt_v35_q5+mt_v36_q6 123

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d dt@EK1 @_q4=mtv24q2+mtv34q3+mt(v224+v234)q4+mt(v34v35+v24v25)q5+mt(v24v26+v34v36)q6+mt_v24_q2+mt_v34_q3+mt(2_v24v24+2_v34v34)_q4+mt(_v34v35+v34_v35+_v24v25+v24_v25)_q5+mt(_v24v26+v24_v26+_v34v36+v34_v36)_q6d dt@EK1 @_q5=mtv15q1+mtv25q2+mtv35q3+mt(v24v25+v34v35)q4+mt(v215+v225+v235)q5+mt(v15v16+v25v26+v35v36)q6mt_v15_q1+mt_v25_q2+mt_v35_q3++mt(_v24v25+v24_v25+_v34v35+v34_v35)_q4+mt(2_v15v15+_v25v25+_v35v35)_q5+mt(_v15v16+v15_v16+_v25v26+v25_v26+_v35v36+v35_v36)_q6d dt@EK1 @_q6=mtv16q1+mtv26q2+mtv36q3+mt(v24v26+v34v36)q4+mt(v15v16+v25v26+v35v36)q5+mt(v216+v226+v236)q6mt_v16_q1+mt_v26_q2+mt_v36_q3++mt(_v24v26+v24_v26+_v34v36+v34_v36)_q4+mt(_v15v16+v15_v16+_v25v26+v25_v26+_v35v36+v35_v36)_q5+mt(2_v16v16+2_v26v26+2_v36v36)_q6: (A) ThepartialderivativeofEK2withrespectto_qcanbecalculatedas @EK2 @_qi=mpu _luC LtUCx)]TJ /F7 11.955 Tf 13.15 8.09 Td[(luC L2tUCx_Lt+luC Ltd dtUCx!+@ @_qi _luC LtUCx)]TJ /F7 11.955 Tf 13.15 8.09 Td[(luC L2tUCx_Lt+mpuluC Ltd dtUCx! 124

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+mpu _luC LtUCy)]TJ /F7 11.955 Tf 13.15 8.09 Td[(luC L2tUCy_Lt+luC Ltd dtUCy!+mpu@ @_qi _luC LtUCy)]TJ /F7 11.955 Tf 13.15 8.09 Td[(luC L2tUCy_Lt+luC Ltd dtUCy!+mpu _luC LtUCz)]TJ /F7 11.955 Tf 13.16 8.08 Td[(luC L2tUCz_Lt+luC Ltd dtUCz!+mpu@ @_qi _luC LtUCz)]TJ /F7 11.955 Tf 13.15 8.08 Td[(luC L2tUCz_Lt+luC Ltd dtUCz! (A) wherei=1;:::;6. Time-differentiatingthis,theequationforthekineticenergybythelinearmotionoftheupperpistoncanbecalculatedas d dt@EK2 @_qi=mpu luC LtUCx+(luC Lt+_luC Lt)]TJ /F7 11.955 Tf 13.15 8.09 Td[(luC L2t)]TJ /F7 11.955 Tf 13.16 8.09 Td[(luC L2t_Lt)_UCx+(luC Lt)]TJ /F7 11.955 Tf 13.15 8.09 Td[(luC L2t)]TJ /F8 11.955 Tf 14.38 11.25 Td[(_luC L2t_Lt)]TJ /F7 11.955 Tf 13.15 8.09 Td[(luC L2tLt+2luC L3t_Lt)UCx!+mpud dt@ @_qi _luC LtUCx)]TJ /F7 11.955 Tf 13.15 8.09 Td[(luC L2tUCx_Lt+luC Lt_UCx!+mpu luC LtUCy+(luC Lt+_luC Lt)]TJ /F7 11.955 Tf 13.16 8.09 Td[(luC L2t)]TJ /F7 11.955 Tf 13.15 8.09 Td[(luC L2t_Lt)_UCy+(luC Lt)]TJ /F7 11.955 Tf 13.15 8.09 Td[(luC L2t)]TJ /F8 11.955 Tf 14.38 11.24 Td[(_luC L2t_Lt)]TJ /F7 11.955 Tf 13.15 8.09 Td[(luC L2tLt+2luC L3t_Lt)UCy!+mpud dt@ @_qi _luC LtUCy)]TJ /F7 11.955 Tf 13.15 8.09 Td[(luC L2tUCy_Lt+luC Lt_UCy!+mpu luC LtUCz+(luC Lt+_luC Lt)]TJ /F7 11.955 Tf 13.15 8.09 Td[(luC L2t)]TJ /F7 11.955 Tf 13.15 8.09 Td[(luC L2t_Lt)_UCz+(luC Lt)]TJ /F7 11.955 Tf 13.15 8.09 Td[(luC L2t)]TJ /F8 11.955 Tf 14.38 11.24 Td[(_luC L2t_Lt)]TJ /F7 11.955 Tf 13.15 8.09 Td[(luC L2tLt+2luC L3t_Lt)UCz!+mpud dt@ @_qi _luC LtUCz)]TJ /F7 11.955 Tf 13.15 8.08 Td[(luC L2tUCz_Lt+luC Lt_UCz! (A) wherei=1;:::;6. 125

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ThepartialderivativeofEK3withrespectto_qcanbecalculatedas @EK3 @_qi=mpl )]TJ /F7 11.955 Tf 13.15 8.09 Td[(llC L2tLCx_Lt+llC Ltd dtLCx!+mpl@ @_qi )]TJ /F7 11.955 Tf 13.15 8.09 Td[(llC L2tLCx_Lt+llC Ltd dtLCx!+mpl )]TJ /F7 11.955 Tf 13.15 8.09 Td[(llC L2tLCy_Lt+llC Ltd dtLCy!+mpl@ @_qi )]TJ /F7 11.955 Tf 13.15 8.09 Td[(llC L2tLCy_Lt+llC Ltd dtLCy!+mpl )]TJ /F7 11.955 Tf 13.15 8.09 Td[(llC L2tLCz_Lt+llC Ltd dtLCz!+mpl@ @_qi )]TJ /F7 11.955 Tf 13.15 8.09 Td[(llC L2tLCz_Lt+llC Ltd dtLCz! (A) wherei=1;:::;6. Time-differentiatingthis,theequationforthekineticenergybythelinearmotionofthelowerpistoncanbecalculatedas d dt@EK3 @_qi=mpl llC LtLCx)]TJ /F8 11.955 Tf 11.95 0 Td[((llC L2t_Lt+llC L2t)_LCx+(2llC L3t_Lt)]TJ /F7 11.955 Tf 13.15 8.08 Td[(llC L2tLt)LCx!+mpld dt@ @_qi )]TJ /F7 11.955 Tf 13.15 8.09 Td[(llC L2tLCx_Lt+llC Ltd dtLCx!+mpl llC LtLCy)]TJ /F8 11.955 Tf 11.95 0 Td[((llC L2t_Lt+llC L2t)_LCy+(2llC L3t_Lt)]TJ /F7 11.955 Tf 13.15 8.09 Td[(llC L2tLt)LCy!+mpld dt@ @_qi )]TJ /F7 11.955 Tf 13.15 8.09 Td[(llC L2tLCy_Lt+llC Ltd dtLCy!+mpl llC LtLCz)]TJ /F8 11.955 Tf 11.95 0 Td[((llC L2t_Lt+llC L2t)_LCz+(2llC L3t_Lt)]TJ /F7 11.955 Tf 13.15 8.09 Td[(llC L2tLt)LCz!+mpld dt@ @_qi )]TJ /F7 11.955 Tf 13.15 8.09 Td[(llC L2tLCz_Lt+llC Ltd dtLCz! (A) wherei=1;:::;6. 126

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Equation( 5 )canbere-writtenas EK4=w1_q24+w2_q25+w3_q26+w4_q4_q5+w5_q4_q6+w6_q5_q6 (A) ThepartialderivativeofEK4withrespectto_qcanbecalculatedas @EK4 @_q1=0@EK4 @_q2=0@EK4 @_q3=0@EK4 @_q4=2w1_q4+w4_q5+w5_q6@EK4 @_q5=2w2_q5+w4_q4+w6_q6@EK4 @_q6=2w3_q6+w5_q4+w6_q5 (A) Time-differentiatingthis,theequationforthekineticenergybytherotationalmotionofthetopbodycanbecalculatedas d dt@EK4 @_q1=0d dt@EK4 @_q2=0d dt@EK4 @_q3=0d dt@EK4 @_q4=2w1q4+w4q5+w5q6+2_w1_q4+_w4_q5+_w5_q6d dt@EK4 @_q5=2w2q5+w4q4+w6q6+2_w2_q5+_w4_q4+_w6_q6d dt@EK4 @_q6=2w3q6+w5q4+w6q5+2_w3_q6+_w5_q4+_w6_q5: (A) 127

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ThepartialderivativeofEK4withrespectto_qcanbecalculatedas @EK5 @_qi=(Iuxx+Ilxx)f0!1s1x+1!2 Lt(s1yUCz)]TJ /F7 11.955 Tf 11.96 0 Td[(s1zUCy)g+(Iuxx+Ilxx)@ @_qif0!1s1x+1!2 Lt(s1yUCz)]TJ /F7 11.955 Tf 11.95 0 Td[(s1zUCy)g+(Iuyy+Ilyy)f0!1s1y+1!2 Lt(s1zUCx)]TJ /F7 11.955 Tf 11.95 0 Td[(s1xUCz)g+(Iuzz+Ilzz)@ @_qif0!1s1z+1!2 Lt(s1xUCy)]TJ /F7 11.955 Tf 11.96 0 Td[(s1yUCx)g+(Iuxy+Ilxy)@ @_qif0!1s1x+1!2 Lt(s1yUCz)]TJ /F7 11.955 Tf 11.95 0 Td[(s1zUCy)gf0!1s1y+1!2 Lt(s1zUCx)]TJ /F7 11.955 Tf 11.95 0 Td[(s1xUCz)g+(Iuxy+Ilxy)f0!1s1x+1!2 Lt(s1yUCz)]TJ /F7 11.955 Tf 11.95 0 Td[(s1zUCy)g@ @_qif0!1s1y+1!2 Lt(s1zUCx)]TJ /F7 11.955 Tf 11.95 0 Td[(s1xUCz)g+(Iuxz+Ilxz)@ @_qif0!1s1x+1!2 Lt(s1yUCz)]TJ /F7 11.955 Tf 11.95 0 Td[(s1zUCy)gf0!1s1z+1!2 Lt(s1xUCy)]TJ /F7 11.955 Tf 11.95 0 Td[(s1yUCx)g+(Iuxz+Ilxz)f0!1s1x+1!2 Lt(s1yUCz)]TJ /F7 11.955 Tf 11.96 0 Td[(s1zUCy)g@ @_qif0!1s1z+1!2 Lt(s1xUCy)]TJ /F7 11.955 Tf 11.95 0 Td[(s1yUCx)g+(Iuyz+Ilyz)@ @_qif0!1s1y+1!2 Lt(s1zUCx)]TJ /F7 11.955 Tf 11.95 0 Td[(s1xUCz)gf0!1s1z+1!2 Lt(s1xUCy)]TJ /F7 11.955 Tf 11.95 0 Td[(s1yUCx)g+(Iuyz+Ilyz)f0!1s1y+1!2 Lt(s1zUCx)]TJ /F7 11.955 Tf 11.96 0 Td[(s1xUCz)g@ @_qif0!1s1z+1!2 Lt(s1xUCy)]TJ /F7 11.955 Tf 11.95 0 Td[(s1yUCx)g (A) wherei=1;:::;6. Time-differentiatingthis,theequationforthekineticenergybytherotationalmotionofthetopbodycanbecalculatedbutitistoolongtowriteinthisdissertation. 128

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ThepartialderivativeofEPwithrespectto_qiszerovector,02R6becauseEPisnotafunctionof_q.ThepartialderivativeofEPwithrespecttoqcanbeobtainedas @EP @q=EPy@ @qUCy Lt+EPx@ @qUCx Lt+k 2@ @q2 (A) where EPy=(mulpu+mllpl)gs1xEPx=(mulpu+mllpl)gs1y: (A) 129

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BIOGRAPHICALSKETCH YoungjinMoonwasborninBusan,SouthKorea,in1973.HestudiedcontrolengineeringinPusanNationalUniversity,wherehereceivedBachelorofScienceandMasterofSciencedegreesin1996and1998,respectively.HethenworkedatTanktechCo.Ltd.forveyears.HejoinedtheUniversityofFloridain2006topursueaPh.D.degreeinmechanicalengineering.HehasworkedwithDr.ScottBanksatOrthopedicBiomechanicsLaboratoryfortwoandhalfyears,andthenstartedworkingwithDr.CarlD.CraneIIIattheCenterforIntelligentMachinesandRoboticsin2009.Hisresearchinterestsincluderobotkinematicsanddynamics,screwtheory,tensegrity,positionand/orforcecontrol,andsynergisticsystems. 137