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A Novel Testing Platform for Characterizing Cervical Spine Biomechanics

Permanent Link: http://ufdc.ufl.edu/UFE0044937/00001

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Title: A Novel Testing Platform for Characterizing Cervical Spine Biomechanics
Physical Description: 1 online resource (87 p.)
Language: english
Creator: Hill, Ira J
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: biomechanics -- robotics -- spine
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: The biomechanics community needs a better understanding of cervical spine dynamics. It is inherently difficult to make experimental in-vivo measurements of cervical spine kinematics and current in-vitro methods and results with cadaveric specimens vary wildly. Previous methods have included custom cable-pulley systems, material tensile testers, parallel robots, and other platforms that are intended to apply pure moment loading to specimens. I am proposing a new spine testing paradigm using a six degree of freedom serial manipulator for characterizing biomechanical properties, including range of motion (ROM), instantaneous axis of rotation (IAR), and other general spinal properties. This tool will be able to test individual functional spinal units (FSU) as well as multi-body segments in a modular test design. With this method, the various changes in kinematics can be characterized under a wide range of testing conditions, i.e. different implants, disk degeneration, ligament detachment, and other conditions. Another advantage of this test design is the ability to re-create complex physiological motion; which is clinically beneficial for studying specific motions. Previous studies are limited to analyzing pure flexion-extension, lateral bending, and axial rotation to their extreme values. This new system can perform motions such as flexion with axial rotation under a compressive load, representing common neck motion. With the ability to create richer data, the fundamental understanding of the cervical spine will be improved. Furthermore, these data can be used to calibrate and validate finite element (FE) and multi-body dynamic models. This will allow clinicians and researchers to better understand the cervical spine and provide information to help treat various injuries and diseases.
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 Ira J Hill.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Banks, Scott Arthur.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0044937:00001

Permanent Link: http://ufdc.ufl.edu/UFE0044937/00001

Material Information

Title: A Novel Testing Platform for Characterizing Cervical Spine Biomechanics
Physical Description: 1 online resource (87 p.)
Language: english
Creator: Hill, Ira J
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: biomechanics -- robotics -- spine
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: The biomechanics community needs a better understanding of cervical spine dynamics. It is inherently difficult to make experimental in-vivo measurements of cervical spine kinematics and current in-vitro methods and results with cadaveric specimens vary wildly. Previous methods have included custom cable-pulley systems, material tensile testers, parallel robots, and other platforms that are intended to apply pure moment loading to specimens. I am proposing a new spine testing paradigm using a six degree of freedom serial manipulator for characterizing biomechanical properties, including range of motion (ROM), instantaneous axis of rotation (IAR), and other general spinal properties. This tool will be able to test individual functional spinal units (FSU) as well as multi-body segments in a modular test design. With this method, the various changes in kinematics can be characterized under a wide range of testing conditions, i.e. different implants, disk degeneration, ligament detachment, and other conditions. Another advantage of this test design is the ability to re-create complex physiological motion; which is clinically beneficial for studying specific motions. Previous studies are limited to analyzing pure flexion-extension, lateral bending, and axial rotation to their extreme values. This new system can perform motions such as flexion with axial rotation under a compressive load, representing common neck motion. With the ability to create richer data, the fundamental understanding of the cervical spine will be improved. Furthermore, these data can be used to calibrate and validate finite element (FE) and multi-body dynamic models. This will allow clinicians and researchers to better understand the cervical spine and provide information to help treat various injuries and diseases.
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 Ira J Hill.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Banks, Scott Arthur.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0044937:00001


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ANOVELTESTINGPLATFORMFORCHARACTERIZINGCERVICALSPINEBIOMECHANICSByIRAJ.HILLADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013IraJ.Hill 2

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Idedicatethisdissertationtomypatientwife,myparents,andallmyfriendsandfamilywhohavehelpedmereachwhereIamtoday. 3

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ACKNOWLEDGMENTS IwouldliketothankmyadvisorandmentorDr.ScottBankswhohasbeeninstrumentalinmyresearchandgraduatecareer.IamalsogratefultomycommitteemembersDr.Fregly,Dr.Crane,Dr.Horodyski,andDr.ConradwhoprovidedwonderfulinsightandspecialthankstoDr.Kojiwhohasbeeninstrumentalinmyexperiments. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 12 LiteratureReview ................................... 12 ClinicalImportance ............................... 12 CervicalSpineAnatomy ............................ 13 ExperimentalTestingMethods ........................ 16 SpecicAim ..................................... 19 DissertationOverview ................................ 20 2ANOVELSPINE-TESTINGPLATFORM ...................... 22 SpecicAim ..................................... 22 PolhemusMotionCaptureSystem ......................... 22 MitsubishiPA10-6CE ................................. 24 ControlSchemes ................................... 27 Conclusion ...................................... 32 3SERIALMANIPULATORFUNCTIONALCALIBRATION ............. 33 SpecicAim ..................................... 33 Introduction ...................................... 33 Methods ........................................ 35 ExperimentalSetupandProcedure ...................... 36 MathematicalModel .............................. 38 OptimizationTechniques ............................ 41 ExperimentalAssessment ........................... 42 Results ........................................ 42 Discussion ...................................... 45 Conclusion ...................................... 47 4ROBUSTANALYTICALINVERSEKINEMATICSFORALMOST-SPECIAL6RMANIPULATORS ................................... 49 SpecicAim ..................................... 49 Introduction ...................................... 49 Methods ........................................ 51 5

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TheLeeandLiangInverseKinematicsAlgorithm .............. 51 NumericalStability ............................... 53 Newton'sMethod ................................ 56 SensitivityAnalysis .................................. 58 Results ........................................ 59 Conclusion ...................................... 61 5TESTINGMETHODOLOGYTOIDENTIFYPASSIVEPROPERTIES ...... 64 SpecicAim ..................................... 64 Introduction ...................................... 64 Methods ........................................ 66 PolhemusPilotStudy ................................ 70 Camera-BasedPilotStudy ............................. 74 Discussion ...................................... 76 Conclusion ...................................... 78 6CONCLUSION .................................... 79 APPENDIX:DERIVATIONOFNEWTON'SMETHODFORINVERSEKINEMATICS 80 REFERENCES ....................................... 82 BIOGRAPHICALSKETCH ................................ 87 6

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LISTOFTABLES Table page 1-1Overviewoftestmethodsusedinliterature .................... 20 2-1JR3loadcellspecications ............................. 32 3-1Geometricparameterchangesandjointstiffnessesfromoptimizedrobotmodel 43 4-1SpecialcongurationsimilartoPuma560manipulator .............. 50 4-2NominalPA10-6CEDHparameters ........................ 59 4-3ExampleHammersleysamplesets100(left)and800(right) ........... 60 4-4ProblematicDHparametersets313(left)and907(right) ............ 61 5-1Intactmulti-segmentprotocolunderforcecontrol ................. 69 5-2Sequentialsectioningprotocolunderdisplacementcontrol ............ 69 7

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LISTOFFIGURES Figure page 1-1Typicalcervicalspineinjuries ............................ 13 1-2Cervicalspineanatomy ............................... 14 1-3Cervicalspineligaments ............................... 15 1-4Idealizedexion-extensioncurveforcervicalspine ................ 16 1-5PA10-6CEserialmanipulatorwithcadavercaninecervicalspine ........ 21 2-1CustomcalibrationstandforPolhemusmarkers .................. 24 2-2PolhemusvalidationresultswiththePA10-6CE .................. 25 2-3PA10-6CEjointconguration ............................ 26 2-4Velocity-basedforcecontrolpseudo-code ..................... 30 3-1TheFaroArmandPA10-6CErobotswererigidlycoupledattheirendeffectorsduringstaticanddynamictrials ........................... 38 3-2Measurementlocationsforstaticcalibrationwereevenlydistributedacrosstheworkingvolume ................................. 39 3-3Examplecadavericcervicalspinetestsetupthatmotivatesthecalibration ... 40 3-4Staticcalibrationresultsshowasignicantreductioninmeanandpeakpositionerrors ......................................... 43 3-5Staticvalidationresultsshowsignicantreductionsinendeffectorpositioningerrorsthatareindependentoftheappliedload .................. 44 3-6Functionalvalidationwithlumbarspinetrajectoriesshowsaverageendeffectorpositionerrorsof0.353mm ............................. 45 4-1Sensitivityresultsfor1,000DHparametervariationsofthePA10-6CE ..... 60 5-1ComputedtomographyscanofC3vertebralbodyusedforanatomyregistration 67 5-2Three-dimensionalreconstructionthatisneededforanalysis .......... 67 5-3CoordinatesystemregistrationforPolhemustrackers .............. 68 5-4Coordinatesystemregistrationwithreectivemarkers .............. 68 5-5ThedistortionofeffectsofthePolhemussystemforC3-C6 ........... 71 5-6ExperimentalsetupwithPolhemusmotioncapturesystem ............ 72 8

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5-7Flexion-extensionresultscomparedagainsttheliterature ............ 72 5-8Lateral-exionresultscomparedagainsttheliterature .............. 73 5-9C4-C5destabilizationeffectscomparedagainstthenominalmotion ...... 73 5-10Fiber-glasspostsinsertedwithreectivemarkers ................. 74 5-11Markertriadswithreectivemarkersrigidlyattachedtospinespecimen .... 75 5-12Experimentalsetupwithcamera-basedmotioncapture ............. 76 5-13TherelativerotationsbetweenC3-C4inlateralbending ............. 76 5-14TherelativerotationsbetweenC5-C6inlateralbending ............. 77 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyANOVELTESTINGPLATFORMFORCHARACTERIZINGCERVICALSPINEBIOMECHANICSByIraJ.HillMay2013Chair:ScottBanksMajor:MechanicalEngineeringThebiomechanicsandmedicalcommunitiesdeserveabetterunderstandingofcervicalspinepropertiestoallowimprovementofcurrenttreatmentmethodsforspinaltraumaanddisease.Itisinherentlydifculttomakeexperimentalin-vivomeasurementsofcervicalspinekinematicsandcurrentin-vitromethodsandresultswithcadavericspecimenscanhavesignicantvariations.Previousmethodshaveincludedcustomcable-pulleysystems,materialtensiletesters,parallelrobots,andotherplatformsintendedtoapplypuremomentloadingtospecimens.Iproposeanewspinetestingparadigmusingasixdegreesoffreedomserialmanipulatorforcharacterizingbiomechanicalproperties,includingrangeofmotion(ROM),neutralzone(NZ),andothergeneralspinalproperties.Thistoolwillbeabletotestindividualfunctionalspinalunits(FSU)aswellasmulti-bodysegmentsinamodularframework.Withthismethod,thevariouschangesinkinematicscanbecharacterizedunderawiderangeoftestingconditions,e.g.,differentimplants,diskdegeneration,ligamentdetachment,andotherconditions.Anotheradvantageofthistestdesignistheabilitytore-createcomplexphysiologicalmotion;whichisclinicallybenecialforstudyingspecicmotions.Previousstudieswerelimitedtoanalyzingexion-extension,lateralbending,andaxialrotationtotheirextremevalues.Thisnewsystemcanperformmotionssuchasexionwithaxialrotationunderacompressiveload,representingcommonneckmotion.Withtheabilitytocreatericherdata,thefundamentalunderstandingofthecervicalspinewillbe 10

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improved.Furthermore,thesedatacanbeusedtocalibrateandvalidateniteelement(FE)andmulti-bodydynamicmodels. 11

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CHAPTER1INTRODUCTIONThefunctioningofthecervicalspineisofgreatinteresttoresearchersandcliniciansbecauseofitsstructuralcomplexity,nonlinearbehavior,andphysicalimportancetothehumanbody[ 1 ].Itisdifculttocharacterizein-vivoandcurrentin-vitromethodshavevariouslimitations.Manyoftheexperimentalmethodsinliteratureusecadavericspecimenandhavevariousstrengthsandweaknesseselucidatingthetruemechanicalpropertiesofthecervicalspine;thesemethodsrangefromcustomcable-pulleysystemstoothermechanisms.Iproposeexpandingupontraditionaltestingmethodswiththeabilitytocompletelycharacterizethepassiveelementsofthecervicalspineusingasixdegreesoffreedom(DOF)serialmanipulator,sixaxisforce-torquesensor,andmotioncapturesystemfortesting.Thedatacollectedwill,byitself,furthertheunderstandingofcervicalspinekinematics,butcanalsobeusedtoimproveandvalidatesimulationmodels.Iwillalsoshowthatthisworkcanbeexpandedtootherjoints,includingthekneeandshoulder.LiteratureReview ClinicalImportance SpinemusculoskeletaldisordersaresomeofthemostcommoninjuriesinAmericans.In200613.2millionpatientssufferedfromcervicalspinepain[ 2 ].Thetotalestimatedcostforspinalinjuriesfrom2002-2004was$193.9billion,makingtheseinjuriesanimportantmedicalandnancialconcern[ 2 ].Itiscrucialtonotonlyidentifythesetypesofdisordersbutalsoquantifytheiroriginandpossibletreatmentmethods.ExamplesoftwospinaldisordersareshowninFig 1-1 .Traditionally,cervicalspinedisordersaredividedintothreegeneralgroups: Cervicaldiscdisorders:discdisplacement,herniations,degeneration Cervicalinjuries/trauma:sprains,strains,fractures Cervicaldisorders:spondylosis,osteophytes,andotherpaincausingdiseases 12

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Cliniciansandphysicianshavehadvaryingsuccesswithtreatmentsfordisordersofcervicalspine.Seventy-vepercentoftreatmentoptionsarenon-surgicalandincludephysicaltherapy,medication,andotherclassicalmethods[ 2 ].Forpatientswhoseconditionpersist,surgeryisalastresortduetorisksandcomplications.Foreitherscenario,itisnotalwaysclearhowsuccessfulthetreatmentwillbeforaspecicpatient.Experimentalmethodsareneededtoprovideinsightaboutaparticulartreatmentmethodforpatients.Theinherentchallengeisdevelopingthesenewexperimentalmethodsforthiscomplexclinicalproblem. AMagneticresonanceimageofher-niatedcervicalspineatC4-C5andC5-C6levels[ 3 ] BXrayof48yearoldmalewithspondylosisandosteophytes[ 4 ]Figure1-1. Typicalcervicalspineinjuries CervicalSpineAnatomy Thecervicalspineisanatomicallyimportantandcomplex.Itisphysicallyimportantbecauseithousesthespinalcordandotherneuralstructureswhilesupportingtheheadandhasanintricaterangeofmotion.Theanatomyofthecervicalspineisworthreviewingbecauseitprovidesinsightintounderstandingthedynamicresponsetoloadsandshapeshowexperimentaldataarecollected.Figure 1-2 belowshowsthebasicstructureofthecervicalspine.Thecervicalspinebeginsatthebaseoftheskullandconnectstotheuppervertebraeofthoracicspine,T1.Itiscomposedofsevenvertebrae,C1throughC7, 13

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Figure1-2. Cervicalspineanatomy[ 5 ] whereC1andC2areknownastheatlasandaxisrespectively.C1isdistinctfromtheothervertebraebecauseitdoesnothaveatraditionalvertebralbody.C2isalsouniquebecauseoftheodontoidprocess,thebonyprojectionthatextendsthroughC1.Theremainingsectionssharesimilarcharacteristicsandcanbegroupedintofunctionalspinalunits(FSUs),whicharecomprisedofasuperiorandinferiorvertebralbody,intervertebraldisc,andligaments[ 6 ].EachFSUhastenspinalligamentsattachedwhichhelpdeterminethenaturalmotionofthecervicalspine.Theligamentsarepassiveelementsthatbehavesimilartononlinearspringswithviscoelasticproperties[ 7 ].Theligamentsalsoservetoprotectthenervesthatrunthelengthofthecervicalspineandmechanicallylimittherangeofmotion(ROM)[ 1 6 ].Mostoftheligamentsconsistofacollagenousmaterialexceptfortheligamentumavumwhichismostlyelastin.Themoststructurallyimportantligamentsaretheanteriorandposteriorlongitudinalligaments,ligamentavum, 14

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interspinousandsuprapsinous[ 7 ].Ingeneral,theexactcontributionofindividualligamentsisnotcompletelyunderstood. Figure1-3. Cervicalspineligaments[ 8 ] Theanteriorandposteriorlongitudinalligamentsrunverticallyalongthevertebralbodiesandprovidesupportinextensionandexion,respectively.Theligamentumavumrunsparalleltothespinallongaxisandconnectsonelaminatothenext,providingsupportinexion.Thefacetcapsularligamentsconnectthearticularprocessesandrestricttorsionalmotion.Theintertransverseligamentconnectsonetransverseprocesstoanotherandlimitslateralexionandsmallrotations.Finally,theinterspinousandsupraspinousligamentsrestrictexion[ 6 ].Thereareseveralimportantparameterstoconsiderwhilecharacterizingthekinematicsandmotionpatternsofthecervicalspine.Thesystemisinherentlynonlinearandisresponsiblefortwodistinctregionsofmotion:theneutralandelasticzone[ 7 ].Theneutralzonehaslittleresistancetomotion;therebyallowinglargemotionsforsmallappliedloads.Conversely,theelasticzonehasastiff,nonlinearresistancetoappliedmotions.OtherimportantparametersincludetheROMofthespecimenaboutaparticularanatomicaldirection,suchasbendinginthecoronalplane.Thesevalues 15

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helpdescribethedisplacementextremesforparticularloadingconditions.Finally,theinstantaneousaxisofrotation(IAR)isthenaturalpointaboutwhichthecervicalspinebendsandrotates.Withinhealthycervicalspines,thispointisthoughttobeconstrainedtoasmallregionbutmaytendtomovemoreindegeneratingorinjuredcervicalspines[ 6 ].Anidealizedmoment-rotationcurveisgiveninFigure 1-4 tohelpillustratetheseproperties. Figure1-4. Idealizedexion-extensioncurveforcervicalspine ExperimentalTestingMethods Therehavebeennumerousstudiesaboutthemechanicsofthespine,recentworkdatingbacktoHirschandNachemson[ 9 ]in1953andeveninearlierworkbeginningin1866.Previousworkoftenfocusedonaspecicaspectofthespine,suchastheintervertebraldiscs[ 10 11 ]orligaments[ 12 ].Othersstudiedthemechanicsofthespineunderdifferentloadingconditions,includingBallandMeijers[ 13 ],WhiteandHirsch[ 14 ],andFarfanetal.[ 15 ].Thecentralthemewasabetterunderstandingofinjury 16

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mechanismsinthespinewhichwouldinuencetreatmentdecisions.Theexperimentalmethods,includingthetestingrigandmotioncapturesystemvaried.Lysel(1969)[ 16 ]andPanjabiandWhite(1975)[ 17 ]weretheinitialresearcherstoexperimentallystudythecervicalspineanditscomponents.ForPanjabietal.,theirgoalwastostudyclinicalstabilitydenedastheabilityofthespinetolimititspatternsofmotionunderphysiologicalloads,soasnottodamageorirritatethespinalcordornerves[ 17 ].Theyexperimentallytestedeightcadavercervicalspinesthatweresplitintoseventeenspinalsegments:C2)]TJ /F3 11.955 Tf 12.58 0 Td[(C3,C4)]TJ /F3 11.955 Tf 12.58 0 Td[(C5,C6)]TJ /F3 11.955 Tf 12.58 0 Td[(C7.Eachsegmenthadbeenfrozenat)]TJ /F3 11.955 Tf 9.3 0 Td[(20C,andcheckedforsignicantdefectsbeforetesting.Acustomexperimentalapparatuswasusedwithasingleweightandpulleysystemwithlineardisplacementgauges.ForeachFSU,horizontaltranslationandinplanerotationofthesuperiorbodyweremeasuredwithrespecttothexedinferiorbody.Theseparameterswerecalculatedfromtherstthreemeasurementspheres;thefourthwasforredundancy.Foreachtestingschedule,theresearchersusedloadsof25%ofbodyweightandsequentiallyremovedcomponentsineitherananteriortoposteriororposteriortoanteriorsequence.Inthismanner,theauthorswereabletoquantifythefollowing: Inexionandextension,onlyincrementalchangesinligamentsegmentationoccuruntiladiscretebreakingpoint Anteriorligamentsprovidestructuralstabilityinextension,whiletheposteriorstabilizeexionTheinitialworkofPanjabiandWhitewasexpandedoverthenextseveraldecadesbytheirownresearchgroupandseveralothers.Thekeychangeinsubsequentworkwastheexperimentaltestingproceduresused.Severalgroupsusedcable-pulleysystemssimilartoPanjabietal.,withslightadditions[ 18 19 ].Wheeldonetal.usedthismethodtogeneratedataforyoung,healthycadavericspecimen,whichisoftenusedasthestandardtocalibratecervicalspinesimulationmodels[ 20 ].Thissystemwasabletostaticallyapplyaloadtoarigidlinkconnectedtothesuperiorendofthespineby 17

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changingtheamountofdeadweightandthedistanceatwhichitwasapplied.Eguizabaletal.expandedonthissystembyaddingatensilematerialtestingmachine(MTS858MiniBionixII,MTSCorporation,EdenPrairie,MN)withaxedandslidingringcablesystem[ 21 ].Thesemachinesprovideacontrollablemethodforapplyinglargetensileandcompressiveloadsinsteadofdeadweights.Thenextsignicantchangeinexperimentaltestingwastheuseof6-DOFparallel[ 22 24 ]andserial[ 25 27 ]manipulators.Bothtypesofmanipulatorsareuniquelysuitedforpositionandforcecontrolofspecimensduetotheeasynaturetoimplementcontrolschemes.Parallelmanipulators,suchastheStewart/Goughplatform,areabletoapplyhighloadswithrepeatabilitybutaredifculttocontrolandcalibrateforbiomechanicaltesting[ 23 ].Alternatively,serialmanipulatorshavesmallerallowablepayloadsbutareexcellentatcreatingcomplex,physiologicalloadingpatterns.Manipulatorsarealsocapableofreplayingpreviousmotionpatterns,whichisnecessaryinexperimentallyevaluatingpassiveproperties.Stokesetal.andWalkeretal.bothconstructedtheirowncustomStewart/Goughplatformsforspinalmotiontesting[ 22 23 ].Ingeneral,bothdesignshadsixlinearactuatorsconnectingatopmovingplatetoaxedbottomplate.Thelinearactuatorswereconnectedbyeitheruniversalorsphericaljoints,allowingthetopplatetohave6-DOFbychangingtheactuatorlengths.Steppermotorswithleadscrewsproducedlinearactuationandwerecoupledwithlinearencodersforpositionmeasurement.Withthissetup,theloadcellandspecimenwereattachedtothebottomofthetopplateandrigidlyconnectedbacktothebaseplate.Usingkinematicsandgeometry,forcesandmomentswereappliedtothespecimenwithclosed-loopcontrol.Therehasalsobeenextensiveworkinbiomechanicaltestingusingserialmanipulators.Fujieetal.isoftencitedasthebasisforusingserialmanipulatorsfortheirworkonsynovialjointkinematicswithacustom6-DOFmanipulatorcreatedfromthecommercialMitsubishiRVmanipulator(RV-MIS-P2,Nagoya,Aichi,Japan). 18

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Theresearchersaddedanextratranslationaljointatthebaseforthesixthdegreeoffreedom.Ithadamaximumpayloadof200Nwithrandompositioningerrorof0.05mm[ 25 ].Theothersignicantcontributionwasthedevelopmentofanewhybrid-controlmethodforapplyingdesiredloadstospecimenwhilemaintainingphysiologicalmotion.Essentially,themanipulatorisoperatedinpositioncontrolandmakessmallincrementalmotions.Aftereachmovement,aninstantaneousstiffnessmatrixisdevelopedbasedontheresultingforcesandmoments.Finally,thenextincrementalchangeispredictedtoeliminateresidualloaderrors[ 26 ].Typically,oneaxishasadesiredmomenttarget,whiletheothersareregulatedtowardszero.ThiscontrolalgorithmwasthenexpandedtothePUMA(Unimate,StabuiInc,Duncan,SC),apopularopenplatformserialmanipulator.ThePUMAhassixrevolutejointsandeachlinkisdrivenbypermanentmagnetservomotors.Positionandvelocityarecalculatedfromon-boardpotentiometersandincrementalopticalencoders.Furthermore,therearegearedmechanicaldrive-trainsanddiscbrakesateachjoint.ThePUMAhasarandompositioningerrorof0.02mmintranslationand0.02inrotation[ 27 ].Gilbertsonetal.performedastudyonlumbarspineFSUs(L1)]TJ /F3 11.955 Tf 12.9 0 Td[(L2,L3)]TJ /F3 11.955 Tf 12.91 0 Td[(L4),examiningthepropertiesofthedisc,ligaments,andfacetjointsinexionandextension.Theyconcludedthatserialmanipulatorsarewellsuitedforinterrogatingthenonlinear,complexpropertiesofcadaverspinespecimens[ 27 ]. .SpecicAim Thisdissertationoutlinesthedesignofacervicalspinetestingplatformwithaserialmanipulatorthatcancollectrelevantphysiologicaldataforthecervicalspine.Theplatformwillhavetheabilitytodothetraditionalexion-extension,lateralbending,andaxialrotationseeninliteratureaswellascustomtrajectoriesthatwillhelpcreatesignicantlyricherdataforunderstandingthemechanicsofcervicalspine.Theproposedideasmakeseveraladvancementsovercurrenttestingtechniques, 19

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Table1-1. Overviewoftestmethodsusedinliterature TestDesignAdvantagesDisadvantages Fixedcableandweights[ 28 30 ]EasytestsetupNotwellsuitedforcomplexmotionsandloadsSlidingcableandweights[ 21 ]EasytestsetupNotwellsuitedforcomplexmotionsandloadsTensiletestingrigs[ 31 ]HighloadcapacityChallengingtoapplyloadsotherthancompressionandtensionStewart/Goughplatform[ 22 24 ]HighloadcapacityLimitedrangeofmotionSerialmanipulator[ 25 27 ]Excellentposition/forcecontrolLowloadcapacity includingpreviousdesignsbasedonotherserialmanipulators.Thisresearchwilldirectlyinuencethegeneralknowledgeaboutthekinematicsofcervicalspinewhilealsoprovidingtangibleimprovements,suchascreatingbettersimulationmodelsandimprovingtreatmentmethods.Speciccontributionsofthisresearchinclude: 1. Robotictestingframework-Developanewtestingplatformthatcanextendcurrentmethodsforin-vitrotestingofthecervicalspinewithcomplexloadingpatterns 2. Passivepropertycharacterization-Performadetailedstudyofhowtheindividualspinalligamentsinuencecervicalspinekinematicsforavarietyofmotions 3. Experimentaldatavalidation-UsecollecteddatatocalibrateandvalidatesimulationmodelsDissertationOverview Thisdissertationisorganizedintochaptersbasedonthetechnicalcontributionsofthiswork.Chapter2discussesthehardwareandsoftwareconsiderationsforusingaserialmanipulatortoperformbiomechanicaltesting,includingcontrollerdesign.Chapters3,4,and5areselfcontainedexcerptsthatcoverimportantkeyresearchtopics.Chapter3detailsadynamiccalibrationmethodforreducingpositionandorientationerrorsatthetheendeffector,allowingthemanipulatoritselfto 20

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Figure1-5. PA10-6CEserialmanipulatorwithcadavercaninecervicalspine serveasanaccuratemeasurementdevice.Chapter4providesinsightintoaninterestingengineeringchallengeofperforminginversekinematicsforcalibratedserialmanipulators,whichhasbeentraditionallyoverlookedinliterature.Chapter5illustrateshowtheseconceptscanbeappliedforbiomechanicaltestingofthecervicalspineandprovidesafullanalysisforanintacthumanspecimen. 21

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CHAPTER2ANOVELSPINE-TESTINGPLATFORMSpecicAim Manyofthecurrentmethodstoperformbiomechanicaltestingofthecervicalspinearewellestablishedbutusecumbersometestingmethodswithvariouslimitations.Thefocusofthischapteristooutlinethenecessarytechnologicalimprovementstocollectricherexperimentaldata.ThekeyadvancespresentedareanalternativemotioncapturemethodandusingtheMitsbuishiPA10-6CEserialmanipulatortoapplythenecessaryloadsandmotionstothecadavericspecimen.Itisimportanttonotethatotherstudieshaveusedserialmanipulatorsinsuchexperimentsbeforebutwithvariouslevelsofsuccessatactuallyapplyingpuremomentloadinginvariousanatomicaldirections.ExamplesincludetheworkofFujieetal.[ 25 ]whousedacustombuiltmanipulatorandRudyetal.[ 26 27 ]withthePUMAUltimatemanipulator.Furthermore,otherresearchershaveusedlargeindustrialmanipulatorsthataredifculttomanageinthesetypesofexperiments,includingtheFANUC(FANUCRoboticsAmericaInc.RochesterHills,MI)andKUKA(KUKARoboticsCorporation,SterlingHeights,MI)asdescribedbyDarcyetal.[ 32 ].Thischapterwillrstoutlinetheinherentchallengesofinvitrotestingofthecervicalspineandthendiscussparticularsolutionstocollectimprovedexperimentaldata.PolhemusMotionCaptureSystem Previously,researchershaveusedtechniquesincludinglineargauges[ 17 ]andcamera-basedmotioncapturesystems[ 20 ];Panjabietal.[ 18 ]givesanoverviewofthesemethods.Themotiondataareanintegralparttointerpretingthepropertiesofthespecimenalongwiththecollectedforce-torquedata.Camera-basedmotioncaptureisapopularandwellestablishedchoice,butithassomepracticallimitationsforspinetesting.Asignicantfootprintisneededforthecamerasandthereliabilityofthedatadependsonthecamera'sabilitytotrackcloselyplacedreectivemarkers.Furthermore, 22

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theaccuracyofrecreatingmarkerpositionsdependsonthecamerasusedandsizeoftheviewingarea.Asanalternative,wehavechosentouseanA/Cmagneticeldbasedmotiontrackingsystem,thePolhemusLiberty(Polhemus,Colchester,VT).Thesystemisconvenienttouse,compact,andhassensorresolutioncomparabletocamerabasedmotioncapture(0.762mmand6.98e)]TJ /F7 7.97 Tf 6.59 0 Td[(3radRMSerror).ThePolhemussystemcanmakedeterministic,dynamicmeasurementsfromuptoeightsensors,providingpositionandorientationdataduringtesting.Thereforeasensorcanberigidlyxedtoeachvertebralbodyofthespecimen,measuringallsixdegreesoffreedom.Thisdatacoupledwiththeloadcellinformationisusedtoreconstructforce-deectioncurvesforanalysis.AnimportanttechnicalaspectofthePolhemussystemisthatisthesensormeasurementsaresusceptibletometaldistortion.Metalnearthesourceorreceiverswillinterferewiththemagneticeldmeasurementsandpreventaccuratereconstructionofmarkerinformation.Ageneralguideisthatallmetalshouldbeatleastthreetimesthedistanceofthefarthestsensorfromthereceiverbutdoesnotguaranteeerrorfreemeasurements.ThePolhemussystemincludesanelectroniccontrolsboxthatindicatesifmetaldistortionlevelsareabnormal,butunfortunatelythisisapoorpracticalindicatorofperformanceandcaremustbetakentopreventmetaldistortion.Figure 2-1 showsacalibrationstandwherethemarkersandtransmitterarerigidlyplacedatknowndistances.Differentamountsofmetalcanthenbeintroducedinthescenetomonitorifthecorrectpositionsarestillmeasured.Evensmallamountsofmetalcancausecentimetersofmeasurementerror!Unfortunately,robotictestingwillalwaysinherentlyhavesomemetal,includinganymetalneededformountingandinterfacingwithspecimens(screws,connectors,etc.).Toensurethatthemeasurementsareaccurateandfreefromdistortion,avalidationprocedureisneeded.Tovalidatethemotioncapturesystem,asinglesensorwasmovedknowndistancesinsidethetestingareawhilerigidlyattachedtotheendeffector.Theonlymetalpresentwasthemanipulatoritself,80/20mountingbrackets,andafewmetalfasteningscrews. 23

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Figure2-1. CustomcalibrationstandforPolhemusmarkers Thismetalcontentrepresentsthetotalmetalneededforspinetesting,therebyprovidesanupperboundforthemaximumdistortionpossible.APVCpipewasmountedinBondoR(3M)andrigidlyattachedtotheendeffectorofthemanipulator,replicatingthespecimenmountingprocess.AsinglesensorwasthenxedwithziptiestothetopofthePVCclosesttothemanipulator,representinguppermostvertebralbody.Finally,themanipulatorwastranslatedknowndistancesintheworkspace,andmeasurementsweretakenateachlocationbythePA10-6CEandPolhemussystem.Eventhoughthemeasurementsweretakenintwodifferentcoordinatesystems,theabsoluterelativetransformationshouldbethesamefromlocationtolocationifthereisminimalmetaldistortion.Figure 2-2 showsthereisgoodagreementbetweentothetwosystems,withanaverageerrorlessthan300microns.Thisgivescondencethateveninthepresenceofthemanipulator,thePolhemussystemworkswithinexpectederrortolerances.Furthermore,thisvalidationstudycanbeeasilyrepeatediftheamountofmetalneededchanges.MitsubishiPA10-6CE Thisserialmanipulatorprovidesmanyadvantagesovertraditionaltestingmethodsbutcreatesothertechnicalchallengesbecauseofitsrelativecomplexity.Thissame 24

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Figure2-2. PolhemusvalidationresultswiththePA10-6CE designcanbeutilizedforallcervicalspinetestingscenariosandcanperformarichgamutoftests,includingcombinedloadingalongarbitrarypathswithdifferentloadingrequirements.ThemanipulatoritselfhassixrevolutejointspoweredbyACservomotorswithharmonicdrivetransmissionswitha50to1gearratio.Also,eachjointhason-boardencodersforpositionmeasurement.Torquesareappliedtoeachjointthroughaservocontrollerwhichacceptsinputsoveraber-opticcableusingtheARCNETcommunicationprotocol.Variouscontrolschemescanbeimplementedbysendingeitherdesiredjointvelocityortorquecommandstotheservocontroller.UsingacustomLabVIEW(NationalInstrumentsCorp,Austin,TX)framework,variousbiomechanicaltestingprotocolscanbeimplementedinreal-timewithclosed-loopfeedbackfromjoint 25

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Figure2-3. PA10-6CEjointconguration encodersalongwithexternalmeasurementdevices,suchasmotioncapturesystemsandforce-torquesensors.ThegeometricparametersforthemanipulatorcanbedescribedusingtheDenavit-Hartenberg(DH)convention,whichisanomenclaturethatlabelscoordinatesystemsateachjointofthemanipulator,includinglinklengths(aij),twistangles(ij),jointoffsets(Si),andjointangles(i).ArelativetransformfromlinktolinkisreadilyestablishedfromtheDHparameters,asshownin 2 .Systematicallydeningthetransformationsforalladjacentlinkswillrelateacoordinatesystemattheendeffectortothebaseofthemanipulator,denotedasF6T,(transformationfromthexedtosixthsystem).Thisisneededtodeterminethepositionandorientationoftheendeffectorinspacefromcurrentencodervalues,sincethereisnodirectmeasurement,knownasforwardkinematics.Chapter3willdetailthecalibrationprocedureneededtovalidatetheendeffectormeasurementisaccurateforbiomechanicaltesting.Furthermore,Chapter4detailshowthegeometriccalibrationresultscanbeusedinthenaccuratelypositioningthemanipulatorinadesiredlocation. 26

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ijT=266666664cj)]TJ /F3 11.955 Tf 9.3 0 Td[(sj0aijsjcijcjcij)]TJ /F3 11.955 Tf 9.3 0 Td[(sij)]TJ /F3 11.955 Tf 9.3 0 Td[(sijSjsjsijcjsijcijcijSj0001377777775 (2)F6T=F1T12T23T34T45T56T (2)FToolP=F6T6ToolP (2)ControlSchemes Performingbiomechanicaltestingusingserialmanipulatorsposesauniquecontrolschallenge.First,thecontrollermustbeabletorepeatablyfollowarbitrarytrajectorieswhilemaintainingglobalpositionalaccuracyandmeasuringtheresultingspecimenforces(pathfollowing).Second,itmustbeabletoapplypuremomentloadsaboutvariousanatomicaldirectionsofthespecimen,maintainingzeroforcesandmomentsabouttheunconstraineddegreesoffreedom(physiologicalloading).Forexampleinexion-extension,thereisaprescribedmomentinthesagitalplanetocreatemotion.Theremainingdirectionsareallowedtofreelyoatundernoloadandtherebyallowingthespecimentofollowitsnaturalmotion.Therstobjectiveissimplertoimplementregardlessoftheloadapparatusused,andthePA10-6CEisespeciallysuitedforpathfollowingsinceitisdesignedtobepreciseorrepeatable.Avarietyofcontrollersworkwellwithpropertuning,includingtraditionalPID(proportional,integral,derivative)control.Physiologicalloadingposesmorechallengesandseveraldesignapproacheshavebeenusedininvestigationsinanattempttoachievepuremomentloading.Theseincludepositioncontrol[ 23 25 ],loadcontrol[ 24 ],andhybridcontrolschemes[ 27 33 ].Inpositioncontrol,displacementsareappliedtothespecimen,andtheresultingforcesandmomentsaremonitoredtoachievethedesiredconditions.Alternatively,loadcontrol 27

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appliesthedesiredforceandmeasurestheresultingmotion;hybridcontrolcombinesbothofthesemethodsinanattempttousetheadvantagesofeach.Regardlessofthecontrollerchosen,thegoalistobestreplicatephysiologicalloadingconditions.Goeletal.[ 34 ]givesanoverviewofthebenetsofeachmethod.Independentofthechosencontrolscheme,thereisalsoadistinctionoflinearandnonlinearcontrolstheory.Lyapunovbasednonlinearcontrolsiswellsuitedforthesetypesofnonlinearproblemswithuncertaintiesbutcanbesubstantiallyhardertoimplementinpractice.TheLabVIEWframeworkhasseveralgeneralnonlinearcontrollersimplemented,includingRISE(RobustIntegralSignoftheError)andVSC(VariableStructureController).Initialtestingshowedthatforbiomechanicaltesting,theaddedcontrolbenetswerenegligibleinrelationtotheaddedcomplexity.Typicaljointtestingrequiresplacingpreciseloadsalongvariousdirections,buttestsarerelativelyslowwithloadingcyclesperformedoverminutes.Creativelinearcontrolschemescanreliablemeetthesedesignrequirements;thereforeatraditionalnonlinearcontrollerwasnotimplemented.Anotherconcernisthatseveralmethodsinliteraturerequirethecalculationofthespecimen'snon-constantstiffnessmatrix.Thisisdifcultinpracticebecausethereare36coefcientsthatdenethestiffnessmatrix,whichrelatestheforcesandmomentstodisplacementsandrotations[ 22 ].Thesecoefcientsaresusceptibletonoiseandarecomputationallyexpensivetocalculateinreal-time[ 24 ],causingtheteststobecomequasi-staticinsteadofcontinuous.Performingcontinuoustestisimportanttobetterreplicateinvivoconditionsforspinalmotion.Furthermore,thenonlinearpropertiesofthespinecancauselargeparameterchangesforsmallmotions.Forourtestingmethods,itisdesirabletoimplementacontrollerthatdoesnotdirectlydependoncalculatingthestiffnessmatrixwhichmaysufferfromspeedlimitations[ 24 ].Furthermore,thecontrollerperformanceisdependentontheerrorsinapproximatingstiffnessvalues. 28

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ThechosencontrolschemeforphysiologicalloadingisbasedontheworkofGoertzenetal.[ 24 ]whoimplementavelocity-basedforcecontrolmethod.Thismethodwaschosenbecauseitiscomputationallylightweightbuteffectiveinachievingunconstrainedmotionofthespine.Goertzenetal.implementedthisapproachforaparallelrobotbutthesamebasicprinciplesstillholdforserialmanipulators.Thekeyideaisthattheendeffectorjogsalongachangingvelocityvectorthatappliesdesiredforcesandmomentstothespecimen.Givenaspecicloadingscheme,suchasexionwithaxialrotation,thecontrollerisabletogenerateafeedforwardandfeedbackvelocityjoggingcommandthatresultsinphysiologicalmotion.Thefeedforwardcommandisessentiallyaninitialguessofhowforcesshouldbeappliedtothespecimenwhilethefeedbackallowsforconstantadjustmentsbasedonthecurrentloadingconditionsandspecimenproperties.Thefeedforwardcommandensuresthemanipulatorstartsalonganominaljogvectorbyapplyingdesiredmomentsabouttheinitialguessoftheinstantaneousaxisofrotation.Asthemanipulatorappliesthedesiredmoment,undesiredoffaxisforcesandmomentsarepreventedbythefeedbacktermswhichchangethejogvector.Combiningthesetwoterms,transformsthisgenericjogvectorintoaspecimenspecicloadingschemewithoutknowledgeofthespecimenproperties,includingthestiffnessmatrix.Pseudo-codefortheschemeisshowninFigure 2-4 below;itisdividedintothreesections:Moments,Forces,andControllerOutput. Readspecimenforcesandmoments//LogicforcontrollingmomentsappliedtothespecimenIfmomentaxisisconstrainedthenIfactualmoment
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SwitchtonextdesiredloadschemeEndifElseifmomentaxisisunconstrainedthenError=desired-actualRotationaljogaboutaxiswithk_p*errorLimitjogtomaxboundsEndIf//LogicforcontrollingforcesappliedtothespecimenError=desired-actualTranslationaljogalongaxiswithk+p*errorLimitjogtomaxbounds//AlgorithmforcalculatingcommandoutputtomanipulatorComputedesiredjogvectorbysummingCartesiancommandsComputejointvelocitiesusingjacobianUpdatemanipulatorjoggingcommand Figure2-4. Velocity-basedforcecontrolpseudo-code ItshouldbenotedthatthePA10-6CEdoesnothaveabuiltinjogfunctionlikemostparallelmanipulators,thereforeaspecicendeffectormotioniscreatedbycontrollingtheindividualjointvelocities.Thisisaccomplishedwithajacobian,whichisamatrixofpartialderivativethatmapsvectorfunctionsontoanotherdesignspace.Forserialmanipulators,thejacobiancanrelatejointspacevelocities(qi)totaskspacevelocitiesattheendeffector(xi)[ 35 ]. Jij=dfi dj(2) 30

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_x_!=J_Q(2)Bydenition,itaispositivedenite6bynmatrix,wherenisthenumberofjoints.Thejacobianisonlyafunctionofgeometricparametersandthecurrentjointangles,allowingittobecalculatedinreal-time.ThecurrentframeworkcalculatestheJacobianbasedonthemethodofPualandShimano[ 36 ].Essentially,amodiedforwardkinematicsprocedureisusedtocalculatetheindividualentriesofthejacobian.Itsinverseisneededtotransformthedesiredjogvectortoactualjointvelocities,whichisproblematicifthejacobianbecomessingularcausingarbitrarilyhighvelocities.Thesingularitiesoccuratcertaingeometriccongurationsandcanbesafelyavoidedbymonitoringtheconditionnumberofthejacobian:c(J)=jjJjjJ)]TJ /F7 7.97 Tf 6.58 0 Td[(1 (2)Inpractice,thisisnotanissuesincethesingularitiesofthePA10-6CEarenotinthetypicalworkspaceforbiomechanicaltesting.Thesemathematicalconceptsarebestillustratedwithapracticalexample.Forthetraditionalexion-extensiontest,amomentisappliedinthesagitalplaneofthespecimencausingtheprimarymotionanteriortoposterior.ThismomentshouldoccuratabouttheunknownIARofthespecimenwhilemaintainingoff-axisforcesandmomentsclosetozeroforphysiologicalloading.Usingthejacobian,thealgorithmcancreatethemotionneededbasedonaninitialguessoftheIAR,whichisthefeedforwardcontrolcomponent.Asthespecimenmovesinresponseandthedynamicpropertieschange,thefeedbackcomponentwilladjustthemotiontokeeptheloadingphysiologicalwithoutknowledgeofanyparticularspecimenproperties.Lastly,anintegralparttothisprocessistheabilitytomeasurethecurrentforcesandtorquesappliedtothespecimen.ApopularchoiceistheJR3loadcell(JR3Inc. 31

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Woodland,CA)whichprovidessixdegreeoffreedommeasurementsandisusedinthisstudy.Ithason-boardelectronicstoperformanalog-to-digitalconversion,signalamplication,andlow-passlteringtoprovideacleandatasignal.Dataisinternallycollectedat8kHzandcommunicatedtothehostovera2Mbpsserialdatastream;thecompletespecicationsareprovidedbelow. Table2-1. JR3loadcellspecications AxisCapacityResolutionRepeatability Fx200N0.05N0.2NFy200N0.05N0.2NFz400N0.2N0.8NMx20N-m0.005N-m0.02N-mMy20N-m0.005N-m0.02N-mMz20N-m0.005N-m0.02N-m Conclusion ThecapabilitiesofthePA10-6CEandcorrespondingtoolscreateaversatilebiomechanicaljointtestingplatformforcollectingrichexperimentaldata.Thedesigncanbeusedtoreplicatepreviousresultsinliteraturewithmoderateloadingconditionsandalsoextendpossibletestingscenarios.Thisplatformshouldhaveclearadvantagesoverpreviouscustomtestingjigs(outlinedinChapter1)becauseofitsabilitytoaccuratelyandrepeatedlycreatecomplexmotions.Furthermore,itwillalsobeshowntobemoreaccuratethantestingplatformswithotherserialmanipulatorsbecauseofpropercalibration.ThenextchapterdetailsimportanceandmethodologyforcalibratingthegeometricandstiffnessparametersofthePA10-6CEforaccurateeffectorpositioning.Havingthemanipulatoritselfbeanaccuratemeasurementdevicedeterminesifphysiologicalloadingisachieved.Subsequentchapterswillthenshowhowtousethecalibratedmanipulatortocollectnovelexperimentaldata. 32

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CHAPTER3SERIALMANIPULATORFUNCTIONALCALIBRATIONSpecicAim Serialmanipulatorsareoftenusedinbiomechanicaltestingofhumanjointsbecausetheyareprecise,repeatableinstrumentsthatcancreateinterestingloadingscenarios.Unfortunately,commercialserialmanipulatorsoftendonothaveacceptableglobalpositionalaccuracyduetomanufacturingtolerances,assemblyerrors,andothermechanicalimperfections.Numerouscalibrationmethodshavebeenreportedwhichcalibrategeometricandnon-geometricparameterstoreducestaticpositionerrorsunderconstantloadingconditions.However,themanipulator'sglobalaccuracyduringcontinuousmotionwithtime-varyingexternalloadingconditionsisoftennotaddressedbutisnecessaryforjointbiomechanicaltesting.UsingtheMitsubishiPA10-6CEasacasestudy,anovelfunctionalcalibrationprocedurewasdevelopedthatperformsbothstaticanddynamiccalibration.Thecalibrationusesoptimizationtechniquestopopulatea34-parametermodelthataccountsfortherobot'sgeometricandnon-geometricparametersandsignicantlyreducesthemean/peakstaticanddynamicpositionerrorsto0.368=0.67mmand0.353=0.81mm,respectively,whileexternallyloaded.Introduction Smallindustrialmanipulatorsareanattractivechoiceforperforminginvitrobiomechanicaltestingofhumanandanimalcadavericjoints.Theyprovideseveralkeyadvantagesovercustomrigs[ 17 20 ],includingmaterialtestingmachines[ 21 ],duetotheirabilitytoeasilyrecreatesixdegreesoffreedommotion.Othermethodsarecomparativelycumbersomeandoftenhaveverylimitedrangesofmotion,restrictingthescopeofpossibletests.Eventhoughindustrialmanipulatorsareuniquelysuitedforthesetypesoftests,theytypicallyhaveerrorsofamillimeterormoreinabsolutepositioningwhenexternallyloaded.Thesepositioningerrorsareproblematicforbiomechanicaltesting;forexample,absolutepositioningisrequiredtoorienttheend 33

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effectorrelativetotheanatomicalcoordinatesofthespecimen.Also,evenifthedesiredtrajectoriesarerstcreatedinthemanipulator'sowncoordinatesystem,absoluteaccuracyallowsthemanipulatortoalsoserveasasensorwhichcanaugmentmotioncapturedata.Thisisespeciallyusefulifthereislimitedspacetoattachmotioncapturemarkers;theendeffectoritselfcanprovidethemotiondata.Manipulatorcalibrationisnecessarysincethereisnotadirectmeasurementoftheendeffectorlocation;rather,itiscalculatedfromtheindividuallinklocationsusingforwardkinematics.Anyerrorsinthelinkpositionorgeometrycauserelativelylargeerrorsattheendeffector[ 37 ].Theseerrorsaredividedintovecategories:environmental,parametric,measurement,computational,andapplication[ 38 39 ].Parametricerrorsareofparticularimportanceandarethefocusofmostcurrentcalibrationroutines.Themanufacturertypicallyprovidesmeasurementsforgeometricparametersincludinglinklengths,twistangles,andjointoffsets,butthesemaybeinaccurate.Furthermore,jointexibilitymustalsobeconsideredforthesesmallerserialmanipulatorssincethejointsmayexhibitnon-lineardeectionunderload.Othernon-geometricerrorsarerelatedtothedynamicpropertiesofthemanipulator,includingfriction,hysteresis,andresonancevibration.Severalmethodshavebeenproposedtoaccountforthesenumeroussourcesoferror,anditisworthnotingthemostcommonsolutionmethods.Kwaketal.calibratedtheDaewooDR06industrialmanipulatorusingoptimizationtechniquesbutdidnotreportglobalpositioningaccuracyimprovements[ 40 ].Recently,NubiolaandBonevperformedaverydetailedcalibrationoftheABBIRB1600usingalasertrackerandasubstantialnumberofcalibrationpointstoreducemean/peakpositioningerrorsfrom0.968=2.158mmto0.364=0.696mm[ 41 ].Thismethodincorporated29designparameters,includingallDenavitHartenberg(DH)parameters,andastiffnessestimatebasedonthejointgravitytorquesfromstaticweights.Whilethisestimatehelpsreducestaticpositionerrors,itisnotbasedonfundamentaldynamicstheoryand 34

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maynotperformaswellfordynamictesting.NubiolaandBonevchosethismethodforitssimplicity,butitwillbeshownthatusingfundamentaldynamicsiscomputationallyefcientandsignicantlyreducesbothstaticanddynamicpositionerrors.Acommonthemetothesecalibrationmethodsisthatonlystaticendeffectorerrorswereconsidered.Biomechanicaltestingofcadavericjointsisinherentlydynamicthough,requiringenhancedcalibrationmethods.Notethatglobaldynamiccalibrationisfundamentallydifferentthanpathrepeatability,whichistheabilitytofollowthesamepathbetweenstaticjointcongurations[ 42 ].Whilepathrepeatabilityisimportant,itdoesnotguaranteeaccuracyinreferencetoaglobalcoordinatesystem.Asecondissueisthatjointexibilityisnotalwaysconsidered,oritiscrudelyapproximated.Theseapproximationstypicallyonlyconsidertheeffectsofgravityfromconstantexternalweights,whichisnotrepresentativeofbiomechanicaltesting.Externalloadsduringtestingareinarbitrarydirectionsanddynamicallychange.Forpropercalibration,externalforcesshouldbeaccountedforthroughthegoverningequationsofmotion.Thispaperaugmentscurrentcalibrationroutineswithanovelfunctionalcalibrationmethodforserialmanipulatorsusedinbiomechanicaltesting.Staticanddynamicdatawereusedinthecalibrationprocedure,andglobalandlocaloptimizationtechniqueswereemployedtopopulatea34-parametermodeltosignicantlyreduceglobalpositioningerrors.Thesuccessandrobustnessofthemodelwasquantiedagainstuntestedstaticanddynamicexperiments.Thispaperisorganizedintofoursections,beginningwithanoverviewoftheexperimentalmethodsincludingoptimizationconsiderationsinSectionII.SectionIIIdetailstheresultsfrommanipulatorcalibrationandvalidation.SectionIVprovidestheobservationsandconclusions.Methods Severalstaticanddynamicexperimentswereconductedtocollectarichsetofexperimentaldataforparameteroptimization.StaticdatawerecollectedbymovingthePA10-6CEwithanattachedcoordinatemeasuringmachinetopseudo-randomlocations 35

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undervariousloadingconditionsinthePA10-6CE'sworkspace.Dynamictestswerebasedonfunctionalexamplesfrombiomechanicaltestingandincludedmotionsforinteractingwithacadavericspecimen;forexample,exion-extensionandlateralbendingtrajectoriesfromapreviouslyconductedhumancervicalspineexperiment.Duringthecontinuousmotion,dataweresampledfromalldevicesandtimesynchronized.Globalandlocaloptimizationtechniqueswerethenemployedtondtheoptimalcalibrationparametersthatreduceendeffectorpositionerrors.ExperimentalSetupandProcedure Lightcapetal.reportedstaticcalibrationofaMitsubishiPA10-6CEmanipulatorusingacoordinatemeasurementmachine(CMM)[ 43 ].Theirgantry-basedmachinehadexcellentaccuracy(measurementuncertaintyof12.1m)butwasnotsuitedfordynamicmeasurements.A30-parameterexibilitymodelbasedonthemanipulatordynamicswasoptimized,reducingthemean/peakpositionerrorswerefrom1.80=2.45mmto0.33=0.71mmwitha44Nstaticweight[ 43 ].OurstudyusestheFaroArm(FaroTechnologies,LakeMary,FL),asix-DOFpassivemanipulator,asthetrustedmeasurementsystem.Ithasahighermeasurementuncertainty(189m)thantheCMMbutcantakecontinuousmeasurements.Thetrade-offinaccuracyisnotdetrimentaltothecalibrationprocedure;Lightcapetal.showedthatthetheoreticalaccuracylimitofthePA10-6CEisapproximately200mduetotheresolutionofthejointencoders,whichislargerthantheuncertaintyofFaroArm.Therefore,theFaroArmservedasourtrustedmeasurementsystemandprovidesthereferencemeasurementforimprovingtheparametersofthePA10-6CE.TheFaroArmwasrigidlyattachedtothePA10-6CEthroughanaluminummountingplate(Fig. 3-1 ).Abenettothissetupisthatthelastjointaxes(S6)ofeachmanipulatorarealigned,eliminatingsomeoftheuncertaintyinquantifyingthecommonmeasurementpointinbothcoordinatesystems.Itisnotnecessary(orpractical)toregistermultiplepointsontheendeffectorofthePA10-6CEastraditionallydoneinother 36

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calibrationmethods.NubiolaandBonevshowedthatasinglecalibrationpointcommontobothtoolaxesisjustaseffectiveasregisteringmultiplepointsduringcalibration[ 41 ].ThereareseveralimportantfactorstoconsiderwhenusingtheFaroArmduringcontinuousmotionexperiments.First,sensordatafromtheFaroArmandPA10-6CEmustbesynchronizedtoacommontimebaseforanalysis.Synchronizationisnotnecessaryforstatictestsbutout-of-syncdynamicdatawillcomplicatethecalibrationprocedure.Second,thePA10-6CEisoperatedbyareal-timeembeddedcontrollerwithkHzdataacquisitionrates,whiletheFaroArm'son-boardcontrolleroperatesat100Hz.Toovercomethesechallenges,datawerecollectedfrombothsystems,andtheFaroArmdataweretimeshiftedtoaccountforcommunicationlatenciesandaslowersamplingfrequency.Thetimelagwasidentiedfromthemanufacturer'sestimatesandvalidatedthroughexperimentation.Withthetwomanipulatorsrigidlyxed,thePA10-6CEwasmovedthroughseveralstaticjointcongurationsthatspannedtherobotworkspace.Onehundred-sevencongurationswerechosenusingHammersleysamplingtechniquestopseudo-randomlysamplea0.4mby0.4mby0.35mcubecenteredinfrontofthemanipulator(Fig. 3-2 ).Hammersleyispreferredtorandomsamplingsinceitevenlyspansthedesignspace[ 44 ].Ateachstaticlocation,theFaroArmcontrolunitreportedthreetranslationsandthreerotationsgivenasroll,pitch,andyawvalues.ThePA10-6CEprovidedsensorfeedbackforjointangles,jointvelocity(backwardsdifferencewithltering),force-torquemeasurements,andjointtorques(inferredfrommotorcurrent).EachoftheHammersleylocationswasalsorepeatedwitha22NhangingweightattachedtotheendeffectorofthePA10-6CEtocompletethestaticcalibrationphase.Fordynamiccalibration,functionaltestsfrompreviouslyconductedhumanspineexperimentswerechosentoprovidedynamicexperimentalmotion(Fig. 3-3 ).Flexion-extensionandlateralbendingtrajectorieswerechosentorepresentnormaltestingconditions. 37

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Figure3-1. TheFaroArmandPA10-6CErobotswererigidlycoupledattheirendeffectorsduringstaticanddynamictrials MathematicalModel Amathematicalrobotmodelwasneededtoappropriatelyaccountforthemanipulatordynamicsandderiveacostfunctionforparameterestimation.Thedynamicsforarigid-link,exible-jointmanipulatoraregivenby[ 45 ]: M(q)q+C(q,_q)_q+G(q)+f(_q)+K(q)]TJ /F3 11.955 Tf 11.96 0 Td[(qm)=J(q)TF(3) Iqm+B_qm+K(qm)]TJ /F3 11.955 Tf 11.96 0 Td[(q)=(3)whereqandqmrepresentthelinkandmotorangle,respectively.TheinertiaandCorioliseffectsaremodeledasM(q)andC(q,_q)_q.ThecontributionofgravitytoeachlinkisgivenasG(q),andallfrictioncomponentsaremodeledinf(_q).Thejointstiffness,denotedbyK(q)]TJ /F3 11.955 Tf 12.45 0 Td[(qm),representsthenon-linearjointdeectionoftheharmonicdrive 38

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Figure3-2. Measurementlocationsforstaticcalibrationwereevenlydistributedacrosstheworkingvolume transmission.JTFrepresentsthetorquesateachjointfromexternalloadswhereJisthegeneralmanipulatorJacobian.Finally,themotorinertiaandmotorviscousfrictionaredenotedbyIandB.ThisformulationisnotthetraditionalEuler-Lagrangeformulationforarigid-link,rigid-jointmanipulatorwherethelinkandmotorpositionareequivalent(q=qm).Withexiblejointmanipulators,thelinkangleqistypicallynotmeasurable,sincetheencodersareattachedtothemotorsideofthejoint;therefore,itisdifculttodirectlyevaluateparametersinEq. 3 .Alternatively,qisrelatedtothemeasurablemotordynamicsandjointstiffness(Eq. 3 )andusedinforwardkinematicstobuildanoptimizationcostfunction.Onlyrelativelyslowjointvelocitiesareneededforbiomechanicaltesting,makingIqmnegligible.Othercalibrationmethodshavemodeled 39

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Figure3-3. Examplecadavericcervicalspinetestsetupthatmotivatesthecalibration jointdeectionsbyevaluatingG(q)andf(_q),bysubstitutingthemeasurableqmforq,eventhoughtheyaredistinctlydifferentquantities.Whilethesemethodshavehadsuccessforstaticcalibration,wefeelitismoreappropriatetorelatethejointstiffnesstothejointtorquesandmotordynamics.Therefore,anestimateforqisreadilyavailable: K(qm)]TJ /F3 11.955 Tf 11.96 0 Td[(q)=)]TJ /F3 11.955 Tf 11.96 0 Td[(B_qm(3) q=qm)]TJ /F3 11.955 Tf 14.78 8.09 Td[(1 K()]TJ /F3 11.955 Tf 11.95 0 Td[(B_qm)(3)whereKandBaredesignvariablesforparameteroptimization.TendesignvariablesareneededtorepresentKandBforjoints1)]TJ /F3 11.955 Tf 13.22 0 Td[(5;joint6isnotincludedsinceithasnegligibleeffectwhenexternallyloadedduetotheorientationofitsjointvectorattheendeffector. 40

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OptimizationTechniques Globalandlocaloptimizationtechniqueswereusedtoincreasethechanceofndingtheglobaloptimalmanipulatorparameterchanges.First,aglobalparticleswarmoptimizer(PSO)waschosenduetoitsrobustnesstolocalminimaandinsensitivitytoinitialconditions,makingitidealforndingparameterdeviationsfromexperimentaldata[ 46 ].Schutteetal.providedaparallelimplementationoftheiralgorithmintheCprogramminglanguage,whichwaseasilyadaptedtothisproblemandcomputationallyefcient.TwentysimulationparticleswereusedasrecommendedbySchutteetal..toexplorethedesignspaceandwererandomlyplacedusingvariousinitialseeds.ThePSOprovidedoptimizedmanipulatorparameterchangesbasedonaninitialestimateoftherobot-to-robottransformation,FAROBASET.Second,agradient-basedlocaloptimizer(Trust-region-reectivenonlinearconstrainedoptimizer,TheMathworks,Natick,MA)wasusedtoimprovethenalPSOsolutionandoptimizeFAROBASETthroughanestedtwostagealgorithm.Theouterstageimprovedthemanipulatorparameters,andthenestedstageimprovedFAROBASETbasedonthecurrentmanipulatorparameterestimates.ThelocalandglobaloptimizerssatisedthesamemathematicalequationwhichrelatesFaroArmmeasurementstothePA10-6CEjointinformation(Eq. 3 ).TheFaroArmprovideddirectCartesianendeffectormeasurementswhilethePA10-6CEendeffectorlocationwascalculatedfromforwardkinematics: FAROPi=FAROBASETBASEEET(aij,ij,Si,q)EEPi(3)wherePiisthemeasurementoftheendeffectoratframei.BASEEETrepresentsforwardkinematicsandrelatesapointlocatedintheendeffectorframetothemanipulator'sbasecoordinatesystem.Fromthisrelationship,theouterloopcostfunctionwasdevelopedtominimizepositionerrorswhilereducingmanipulatorparameterchanges(Eq. 3 ).Differentvalueswerechosentopenalizethepositionerrors(meters)andorientationerrors(radians)atsimilarscales. 41

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costouter=NXi=0norm(~xFARO)]TJ /F10 11.955 Tf 11.2 0 Td[(~xPA1O)+dis[X(aij)+X(Si)]+deg[X(ij)+X(i)](3)Forthelocaloptimizer,anadditionalcostfunctionwasneededinthenestedoptimizationtoimprovetheestimateforFAROBASET.ThisnestedcostfunctionalsominimizesthepositionerrorsbutbychangingtheindependentvariablesofFAROBASET. costnested=NXj=0norm(~xFARO)]TJ /F10 11.955 Tf 11.2 0 Td[(~xPA1O)(3)ExperimentalAssessment Toquantifytheoptimizationresultsandthedelityofthemodel,thecalibrationexperimentswererepeatedatnovellocationsunderuntestedloadingconditions.Forstaticvalidation,fty-eightadditionalpseudo-randomcongurationsweretestedwithhangingweightsof0,13.34and35.59N.Fordynamicvalidation,thefunctionaltrajectorieswererepeatedatdifferentloadingconditionsof13.34and35.59N.Thesevalidationtestsshowedtheimprovedabsoluteaccuracyofthemanipulatorwithdatanotusedforcalibration.Results Thenaloptimizedmanipulatorparameterchanges(Tab. 3-1 )reducedbothstaticanddynamicpositionerrors.Beforecalibration,mean/peakendeffectorpositionerrorsforonehundred-sevenjointcongurationswith0Nand22Nwere1.66/2.80mmand1.66/2.80mm.Theseerrorsweresignicantlyreducedaftercalibrationto0.384=0.787mmand0.384=0.681mm(Fig. 3-4 ).Forstaticvalidation,fty-eightstaticpositionswith0,13.34,and35.59Nhangingloadshadmean/peakpositionalerrorsof1.86=2.97mm,2.19=3.04mm,and2.71=3.59mm, 42

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Table3-1. Geometricparameterchangesandjointstiffnessesfromoptimizedrobotmodel Jointiaij(mm)ij(deg)Si(mm)i(deg)K(Nm=rad) 1)]TJ /F3 11.955 Tf 9.3 0 Td[(0.8340.228)]TJ /F3 11.955 Tf 42.53 0 Td[(0.04043.82x10420.352)]TJ /F3 11.955 Tf 9.3 0 Td[(0.249)]TJ /F3 11.955 Tf 9.29 0 Td[(0.3170.2827.18x1043)]TJ /F3 11.955 Tf 9.3 0 Td[(0.7380.1020.831)]TJ /F3 11.955 Tf 9.3 0 Td[(0.1762.16x10440.206)]TJ /F3 11.955 Tf 9.3 0 Td[(0.299)]TJ /F3 11.955 Tf 9.29 0 Td[(0.735)]TJ /F3 11.955 Tf 9.3 0 Td[(0.0102.04x1035)]TJ /F3 11.955 Tf 9.3 0 Td[(0.0740.231)]TJ /F3 11.955 Tf 9.29 0 Td[(0.614)]TJ /F3 11.955 Tf 9.3 0 Td[(0.3283.03x1036)]TJ 57.4 0 Td[()]TJ /F3 11.955 Tf 47.75 0 Td[(0.2880.233)]TJ ET q .398 w 66.41 -163.34 m 401.59 -163.34 l S Q q 388.7996 0 0 388.9061 39.6 -564.4 cm /Im12 Do Q BT /F1 11.955 Tf 0 -585.47 Td[(Figure3-4. Staticcalibrationresultsshowasignicantreductioninmeanandpeakpositionerrors 43

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respectively.Aftercalibration,theseerrorswerereducedto0.408=0.93mm,0.314=0.67mm,and0.383=0.67mm(Fig. 3-5 ). Figure3-5. Staticvalidationresultsshowsignicantreductionsinendeffectorpositioningerrorsthatareindependentoftheappliedload Similarresultswereachievedfordynamicvalidationtrialsusingfunctionaltrajectoriesatdifferentloadingconditions.Threeseparatetrajectoriesforexion-extensionandlateralbendingat0,13.34and35.59Nloadsshowedmean/peakdynamicerrorsaftercalibrationof0.353=3.456mm(Fig. 3-6 ).Highpeakerrorsresultedfromisolatedspikeswhichmostlikelywereduetopoormotorvelocityestimatesfromthebackwards 44

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differencecalculationorindividualmisaligneddatasamples.Iftheseisolatedspikesareignored,thepeakerroris0.8mm. Figure3-6. Functionalvalidationwithlumbarspinetrajectoriesshowsaverageendeffectorpositionerrorsof0.353mm Discussion Tosuccessfullycalibrateaserialmanipulator,theendeffectorerrorsmustbereducedandtheoptimizeddesignvariablesmustbephysicallyreasonable.Forexample,geometricchangesforlengthsandanglesshouldbenolargerthan1mmand0.5degreesforthePA10-6CEbasedonmanufacturingandassemblytolerances.Themaximumoptimizedparameterchangeswere0.834mmand0.328degrees,which 45

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fallwithintheexpectedrange.Parameterchangesfrompreviousstudieswereoftennotreportedmakingitdifculttoassessifothermethodsproducedphysicallymeaningfulresults.Furthermore,ourparameterestimatesforthejointstiffnessaresimilartopreviouslyreportedvaluesforthesamerobotdesign[ 43 ].Staticcalibrationreducedthemeanendeffectorpositionerrorto0.384mmunderloadwhichisslightlyhigherthanthemethodsofLightcapetal.(0.33mm)andNubiolaandBonev(0.364mm).TheindependentmeasuringsystemsusedinthosestudieshadsignicantlybetterresolutionthantheFaroArmbutonlymadestaticmeasurements.Theseslightlyhighercalibrationerrorsstillprovideexcellentresolutionforbiomechanicaltestingandmanyroboticsapplications.Thistrade-offinreferenceinstrumentsallowedfornoveldynamiccalibrationresults,whicharenecessaryforimprovedbiomechanicaltestingprocedures.Mooreetal.alsoproposedaserialmanipulatorcalibrationprocedureforbiomechanicaltestingusingthePUMA(Model762,Unimate)[ 47 ].Insteadofadjustingthegeometricandnon-geometricparametersofthemanipulator,anoffsetinthetoollocationwaschosentoaccountforallsourcesoferrorinthemanipulator,includingnon-geometricerrors.Thissimplicationonlyworksfortheparticularjointcongurationusedforthecalibration.Foreachnewjointcongurationandloadingcondition,aseparatetooloffsetisneeded,whichisnotpracticalforexperimentaltesting.Asingletooloffsetcannotaccuratelyaccountforthenon-linearjointdeectionofaexiblemanipulatorsuchasthePUMAorPA10-6CE.Theauthorsnotedtheseissuesandreportedmeancalibratedpositionerrorsofupto2.80.4mmneartheedgeofthecalibratedworkspaceusingsingleregistration[ 47 ].Theseerrorsareunacceptableformanyapplicationsandalsodonotconsiderdynamicmotions.Thecalibrationprocedurepresentedhereaddressestheseissuesandprovidesaconsistentandefcientmethodforaccuratelyusingserialmanipulatorsinbiomechanicaltestingthatresearcherscaneasilyadopt. 46

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Conclusion IthasbeenshownthatpositionerrorsofthePA10-6CEaredramaticallyreducedforbothstaticanddynamicmotions.Asecondpassivemanipulatorwasusedforcalibrationthatallowedindependent,dynamicmeasurementsofthetruePA10-6CEendeffectorlocation.Thiscalibrationissignicantlydifferentfrompreviousworkbyincorporatingdynamicinformationintothecalibrationprocedure,andallowingthe34-parametermodeltoaccountforbothstaticanddynamicmotions.Thisisanecessarysteptoappropriatelyuseserialmanipulatorsinbiomechanicaltesting,wherethemanipulatorisrequiredtocontinuouslyapplymotionsorforcestoaspecimeninaglobalcoordinatesystem.Equallyasimportant,acalibratedmanipulatorcanserveasanadditionalpositionmeasurementforreconstructingcadavericexperimentaldata.Furthermore,thecalibrationproceduretakesadvantageofglobalandlocaloptimizationtechniquestondanoptimalsolution.Theadditionoftheglobaloptimizerensureslocalminimaareavoidedandphysicallymeaningfulmanipulatorparametersarefound.Apotentialimprovementtothismethodisreducingthepeakerrorsinthedynamicvalidation.Thestiffnessmodeldependsonthemotorvelocitywhichisnotdirectlymeasurable;insteadthetraditionalbackwardsdifferenceisemployedtocalculatevelocityfrompositionandtimedata.Thisprocesscanbeinherentlynoisyforcomplexdynamicsystems,contributingtothespikesinthedynamicerrors.Furthermore,thejointstiffnesswaslinearlyapproximatedeventhoughnonlinearitiesexistintheharmonicdrivesystem[ 48 ].Also,orientationerrorswerenotincludedintheoptimizationduetoaninitialchallengeinrecoveringfulldatasetsfromtheFaroArm.Futureworkwillinvestigateahigherorderfrictionmodelalongwithquantifyingtheperformancefororientationerrors.Despitetheselimitations,thecalibrationprocedurepresentedsignicantlyreducedendeffectorpositionerrors.Weconcludethatthiscalibrationwillimprovetheuseofserialmanipulatorsfortestingjointsandprovidebetterexperimental 47

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dataforthedevelopmentandvalidationofcomputationalmodelsandimprovedmedicaldevices. 48

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CHAPTER4ROBUSTANALYTICALINVERSEKINEMATICSFORALMOST-SPECIAL6RMANIPULATORSSpecicAim Thispaperinvestigatestheinversekinematicsfor6Rserialmanipulatorsthatarenumericallyclosetospecialcongurationsthroughasensitivityanalysis.Numerousinversekinematicsalgorithmshavebeenreportedandtypicallyhavebeenvalidatedagainstmanipulatorswithspecialgeometryorarbitrarygeometricparameters.Theseexamplesignoreanimportantclassofmanipulatorsthatarenominallyinaspecialcongurationbutrequiregeometriccalibrationtoreduceglobalpositioningerrorsthatarisefrommanufacturingandassemblytolerances.Calibrationchangesthenominalgeometricparameters,sothemanipulatorfailstoqualifyasaspecialconguration.UsingthePA10-6CEmanipulatorasacasestudy,1,000pseudo-randomsetsofgeometricvariationsweregeneratedfromthenominalmanufactureparametersandevaluatedfor121,680commonendeffectorposesusinganaugmentedLeeandLiangapproach.Thealgorithmhadasuccessrateof99.89%ofndingallpossiblejointsolutionswithendeffectorerrorslessthan0.1mmand0.01degrees.Introduction Analyticinversekinematicssolutionsforsix-revolute(6R)serialmanipulatorsarewellestablishedintheliterature.Theobjectiveofeachmethodistoprovideallpossiblesetsofjointsolutionsthatpositiontheendeffectorofthemanipulatorinadesiredpositionandorientation[ 49 ].SolutionsreportedbyDuffyandCrane[ 50 ],LeeandLiang[ 51 ],andRaghavanandRoth[ 52 ]arewellknownandhighlycited.Regardlessoftheapproach,eachofthesemethodsderivesa16thorderpolynomialwhoserealrootscorrespondtothetan-halfangleofasinglejoint.Theremainingvejointanglesarethendeterminedfromothergeometricrelationshipsusingtheserealroots.Achallengeforeachoftheseapproachesisposedbyserialmanipulatorswithspecialcongurationsthathaveparallelorintersectingjoints.Thesecongurations 49

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Table4-1. SpecialcongurationsimilartoPuma560manipulator Jointiaij(mm)ij(deg)Si(mm) 1variable90.0)]TJ /F3 11.955 Tf -182.47 -18.78 Td[(2variable0.0variable3variable90.0=270.00.040.090.0variable50.090.00.06)]TJ 70.6 0 Td[()]TJ /F3 11.955 Tf 21.25 0 Td[(variable causethelossofdegreesoffreedomorconstraintequationsrequiredtoformthe16thordergoverningpolynomial.Therefore,special-casesolutionsarerequiredwhenspecialcongurationsareencountered.Forexample,Rothetal.explainedtheadjustmentsrequiredtotheiralgorithmwhenthelastthreejointaxes(S4,S5andS6)ofthemanipulatorintersect[ 53 ].CraneandDuffyprovideathoroughanalysisofcommonspecialgeometriesincommercialmanipulators,includingtheUnimationPuma560,G.E.P60,andCincinnatiMilicronT3-776[ 49 ].ThespecialcongurationofthePumamanipulatorisofparticularinterest,sincemanycommonmanipulatorshavethisdesign,includingmanymodelsfromKUKA,ABB,Fanuc,andMitsubishi.Table 4-1 illustratesthiscongurationwherejointaxesS2andS3areparallel,andS4,S5andS6intersect.Importantly,machiningandassemblytolerancesfornominallyspecialmanipulatorcongurationsresultinrealrobotsthatarenot-quite-special,andrequirecalibrationtoadjustgeometricparametersforimprovedglobalpositioningaccuracy.Thesecalibratedmanipulatorsnolongerexactlysatisfythespecialcongurationrequirementsandareproblematicforspecial-caseanalyticinversekinematicsolutions[ 43 ].ThispaperinvestigatesandvalidatesarobustnumericalapproachtousingLeeandLiang'sinversekinematicssolutionmethodformanipulatorsnumericallyclosetospecialcongurations.TheMitsubishiPA10-6CEisusedasacasestudyandhasthenominalspecialcongurationofTable 4-1 50

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Thispaperisorganizedintofoursections,beginningwithanoverviewofanenhancedLeeandLiangalgorithminSectionII.SectionIIIdetailsanextensivenumericalsensitivityanalysisusedtovalidatetheenhancedmethod.SectionIVpresentsresults,andSectionVpresentsdiscussionandconclusions.Methods TheLeeandLiangInverseKinematicsAlgorithm TounderstandhowtheLeeandLianganalyticalsolutionisaffectedbyvaryinggeometricparametersclosetoaspecialconguration,itisnecessarytooutlinethesolutionprocesspresentedin[ 49 ].LeeandLiangforma16thorderinput-outputpolynomialthatonlyisafunctionofasingleunknownjointangle1,insteadofallsixunknowns.Therealvaluedrootsofthispolynomialrepresentthenumberofsolutionsetsandcorrespondtothetan-halfangleof1.Forexample,thePA10-6CEtypicallyhaseightsolutionsetsforagivenendeffectorpose.Theserealvaluesarethenusedinothergeometricrelationshipstosolvefortheveremainingunknownjointangles.Therestofthissectionhighlightsthekeypartsofthealgorithmandimprovementsforrobustnesstovariousmanipulatorcongurations.Regardlessofthenumericalvaluesforaparticularspatialmanipulator,fourequationscanbecreatedthatdependonlyon1byexploitingthelinktolinkgeometrictransformationsofthemanipulator.Theseequationsaregivenintheformof: (aix2j+bixj+di)xk+(eix2j+fixj+gi)=0i=1..4(4)whereeachcoefcientisquadraticinthetan-halfangleandarederivedthroughcomplexgeometricmanipulationdetailedin[ 49 51 ].ThefullderivationforthesegeometricrelationshipsisnotdetailedherebutcanbecompactlyexpressedinmatrixformwherethenumericalentriesarethecoefcientsofEq. 4 .LeeandLiang'salgorithmbeginsbydevelopingAb=T3a,whereA2<16x16onlyconsistsofknowngeometricparameters.Vectorsb2<16x1anda2<18x1containsinesandcosinesof 51

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theunknownjointangles.Thesevectorsareparticularlyimportantsincetheirnumericalvaluesrepresentparticularjointsolutions.T32<16x18isamathematicallyconvenientgroupingoftermsinthecreationofa. b=[c4c5x6,c4s5x6,c4x6,s4c5x6,s4s5x6,s4x6,c5x6,s5x6,c4c5,c4s5,c4,s4c5,s4s5,s4,c5,s5]T(4) a=[c1c2x6,s1c2x6,c2x6,c1s2x6,s1s2x6,s2x6,c1x6,s1x6,x6,c1c2,s1c2,c2,c1s2,s1s2,s2,c1,s1,1]T(4)Thisrelationshipcanbefurtherpartitionedintoothernon-squareTimatriceswhichalsoconsistsonlyofknowngeometricparameters.GiventhatT2b=T1a,wecanformthefollowingrelationshipswhereT12<4x18andT22<4x16:b=A)]TJ /F7 7.97 Tf 6.58 0 Td[(1T3a (4)[T2A)]TJ /F7 7.97 Tf 6.59 0 Td[(1T3]a=T1a (4)Ta=0,T=T1)]TJ /F3 11.955 Tf 11.95 0 Td[(T2A)]TJ /F7 7.97 Tf 6.59 0 Td[(1T3 (4)andtheentriesofthematrixT2<4x18formthecoefcientsofEq. 4 .Finally,therequired16thorderinput-outputpolynomialisformedfromEq. 4 byevaluatingitsdeterminant.Thereisonlyasolutiontothissystemofequationsifallarelinearlydependent,makingthedeterminantinEq. 4 zero.Thisdeterminantrepresentsthe16thorderinput-outputpolynomialofthetan-halfangleof1. 00aibidieifigiaieibidi0figi0=0(4) 52

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Theremainingsetsofjointanglesarethenfoundbysubstitutingthecorrespondingnumericalvalueof1intoEqs. 4 4 tosolvefor2and6,wherexiisthetan-halfangleofi.Next,numericalvaluesfor4and5correspondtotheentriesofvectorbusingthesineandcosinevaluestouniquelydetermineeachangle.Lastly,thesolutionfor3requiresevaluatingoneofthefundamentalsineandcosinelawsusingthenumericalvaluesfromtheotherangles[ 50 ]. x2=jabegjjadgbj+jadgfjjabedj jabefjjadgbj)-222(jadgejjabedj(4) x6=jabefjjadgfj+jadgejjabegj jabefjjadgbj)-222(jadgejjabedj(4) 5=atan2(b16,b15);(4) 4=atan2(b14,b11);(4)NumericalStability Severalimportantaspectsofthisalgorithmcanbedeterminedfromtheformoftheseequations.TheinverseofmatrixAisrequiredwhichisafunctionofthemanipulatorgeometricparameters,butAbecomessingularifthelinksofthemanipulatorareinaspecialconguration.Thissimplyindicatesthatthegeneralsolutionisnotneededbutratheroneofthespecialsolutionscorrespondingtotheparticulargeometry.AlternativelyifAisfullrankandsufcientlyfarfrombeingsingular,LeeandLiang'soriginalsolutionprocesscontinuesasdetailedabove.Thefocusofthepresenteffortishandlingthein-betweencaseofgeneralandspecialgeometrywhenAmaystillbeinvertedbutisill-posedandclosetosingular.Itwillbeshownthatthismethodcanbeaugmentedtoaccountforthesescenariosaswell,makingtheapproachrobusttovariationsinmanipulatorgeometry. 53

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Thestconsiderationformaintainingnumericalstabilityistheoatingprecisionusedinthealgorithm.BothMorganetal.[ 54 ]andCannyetal.[ 55 ]pointouttheimportanceofthedatatypeusedinimplementinginversekinematicroutinesinsoftware.Doubleprecisionoatingpointnumbersarecommontomostprogramminglanguagesandprovideroughly16accuratedecimaldigits.Whenthemanipulatoristrulyspecialorgeneral,doubleprecisionhassufcientaccuracysincethemathematicalequationsarewellposed.Formanipulatorsin-betweenthesecases,itisdesirabletomaintainmorenumericalprecisionwithoutsacricingcomputationalspeed.Commonalternativesarethe80-bitlongdoubleand128-bitquadprecisiondatatypesprovidedbyGCCcompilersandothers[ 56 ].Longdoublesareanexcellentcompromisesincetheyprovideapproximately19digitsofaccuracywithoutspeedpenaltiesonmoderncomputerhardware[ 56 ].Quadprecisionprovidesroughly34digitsofaccuracybutisoften10to20timesslowerandiscurrentlynotpracticalformanyroboticsapplicationsthatrequireinversekinematicsontheorderofmilliseconds[ 55 ].WeadaptedLeeandLiang'salgorithminthisworktotakeadvantageoflongdoubledatatypesacrossdifferenthardwareplatforms.Asecondconsiderationformaintainingnumericalstabilityistheimplementationforndingcomplexrootsofthegoverninginput-outputpolynomial.Insteadofusingtraditionalrootndingtechniquesthatsufferfromtheirownnumericalissues,itisbettertocastthepolynomialasaneigenvalueproblemusingitscompanionmatrix[ 57 ].Thecompanionmatrixplacesthecoefcientsofthepolynomialinthelastcolumnofasemi-triangularmatrix,whoseeigenvaluescorrespondtotherootsoftheoriginalpolynomial[ 57 ].Eigenvaluestechniquescanthenbeusedtondtherootsoftheoriginalpolynomial[ 57 ]. 54

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x16+c15x15+c14x14+...+c2x2+c1x+c0 (4)companion=26666666666400...0)]TJ /F3 11.955 Tf 9.3 0 Td[(c010...0)]TJ /F3 11.955 Tf 9.3 0 Td[(c101...0)]TJ /F3 11.955 Tf 9.3 0 Td[(c2.....00...1)]TJ /F3 11.955 Tf 9.3 0 Td[(c15377777777775 (4)Thereareseverallinearalgebratechniquesforcalculatingeigenvalues,andQRdecompositionisarobustchoicewhichcanbeeasilyimplementedinsoftwarewithlittlecomputationalcost[ 58 ].TheEigenLinearAlgebrasoftwarepackage[ 59 ]performsQRdecompositionusinganexplicitFrancisdoubleshiftforfastconvergence.Inthismethod,thecompanionmatrixCmayberewrittenasC=QRwhereQisorthogonalandRisuppertriangular.SuccessivetransformationsmaybeappliedtoCwhilepreservingtheoriginaleigenvaluesusingtheQRtransformation,Ci+1=QTCiQ[ 58 ].Usuallyafterafewtransformations,Cbecomesadiagonalmatrixwhosenon-zerovaluesaretherootsoftheinput-outputequation[ 59 ].Afteraccuratelyndingtherealrootsoftheinput-outputequation,therearestillotherpossiblesourcesofnumericalerror.Theseerrorscanfurtherbereducedbyensuringthatthederivedgeometricrelationshipsremainnumericallyconsistentthroughouttheanalysis.Forexample,thevectorahasonly1,2and6asunknownsandmustsatisfyTa=~0.Afterndingtheserstthreejointangles,evaluatingthisexpressionshowsthecurrentnumericalerrorbeforecontinuingthesolutionprocess.InmostscenariostheL2normofTaisapproximately1e)]TJ /F7 7.97 Tf 6.59 0 Td[(20orsmallerindicatingthesolutionprocessisnumericallyconsistentthusfar.WhenAisill-posed,thenormissignicantlylargerandindicatesthenalsolutionwillhaveendeffectorerrorsofseveralmillimetersormore.Thishappensbecausesmallerrorsinacanresultinndingthe 55

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wrongnumericalvaluesfor4,5and6whennumericallyevaluatingtheentriesinbusingAb=T3a.Asimplebutimportantstepistoadjustthevectoraifitbecomesnumericallyinconsistentduringtheanalysis.GiventhatTashouldbeequaltoanullvector,wecanperformthefollowingadjustment:T(a+a)=~0 (4)aadj=a+a (4)whereaistakenasthesmallestsolutiontothisoverdeterminedequation.Byslightlyperturbinga,wealsoneedtoupdatethecurrentnumericalvaluesfor1,2and6.Itisimportanttoemphasizethatthisstepdoesnoteliminatenumericalerror,butratherpreventsitfromcompoundingandcausingacompletelywrongsolution.TstillcontainstheinverseofAwhichmaybeill-posed.Withoutthisprecautionthegeometricequationsmaybecomenumericallyinconsistent,causingnonsensicaljointsolutionsthatarenoteasilyimproved.1=atan2(a17,a16) (4)2=atan2(a15,a12) (4)6=2.0atan(a9) (4)Newton'sMethod Evenwiththenumericalconsiderationsthusfar,therearesituationswherethejointanglesolutionsdonotmeetthedesiredendeffectorerrortolerances.Asmentionedin[ 55 ],Newton'smethodcanquicklyimprovethenumericalprecisiontoadesiredamountinfewsteps.Ingeneral,Newton'smethodisusedtocalculaterootsofapolynomialorfunctionusinggradientstoconvergetothesolution[ 55 ].Thismethodispractical 56

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onlyiftheinitialguessisclosetotherealsolution,whichisoftenthecaseusingtheaugmentedstepsintheLeeandLiangalgorithmdespitebeingclosetoaspecialconguration.Newton'supdatemethodisgivenas: qi+1=qi+J)]TJ /F7 7.97 Tf 6.59 0 Td[(1f(q)(4)Thismethodisadaptedtoserialmanipulatorsbyformulatingsixequationsthatrelatethesixjointanglestoendeffectorspace.Fromforwardkinematics,thesefunctionsarereadilyavailable: F6T=F1T12T23T34T45T56T=F6A(q1,q2,q3,q4,q5,q6)(4)Thematricesontheleftandrighthandsideoftheequationareboth2<4x4transformations,yielding12equationsbut6areredundantintherotationcomponent,R.UsingEulersequences,therotationcomponentonbothsidescanbetransformedintoroll(),pitch(),andyaw()valuesfordirectcomparison.Thisconversionisgivenas:=atan2()]TJ /F3 11.955 Tf 9.29 0 Td[(R2,3,R3,3) (4)=atan2(R1,3,(cos()R3,3)]TJ /F3 11.955 Tf 11.95 0 Td[(sin()R2,3)) (4)=atan2()]TJ /F3 11.955 Tf 9.3 0 Td[(R1,2,R1,1) (4)Thesixgoverningequationscannowbedenedasfollows: 57

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f1:F6T(1,4))]TJ /F7 7.97 Tf 11.96 4.93 Td[(F6A(1,4)=0f2:F6T(2,4))]TJ /F7 7.97 Tf 11.96 4.94 Td[(F6A(2,4)=0f3:F6T(3,4))]TJ /F7 7.97 Tf 11.96 4.94 Td[(F6A(3,4)=0f4:lhs)]TJ /F10 11.955 Tf 11.96 0 Td[(rhs=0f5:lhs)]TJ /F10 11.955 Tf 11.96 0 Td[(rhs=0f6:lhs)]TJ /F10 11.955 Tf 11.95 0 Td[(rhs=0 (4)Finally,theJacobiancanalsobecalculatedsymbolicallybytaking36partialderivatives: J=2666666666666664@f1 @q1@f1 @q2@f1 @q3@f1 @q4@f1 @q5@f1 @q6@f2 @q1@f2 @q2@f2 @q3@f2 @q4@f2 @q5@f2 @q6@f3 @q1@f3 @q2@f3 @q3@f3 @q4@f3 @q5@f3 @q6@f4 @q1@f4 @q2@f4 @q3@f4 @q4@f4 @q5@f4 @q6@f5 @q1@f5 @q2@f5 @q3@f5 @q4@f5 @q5@f5 @q6@f6 @q1@f6 @q2@f6 @q3@f6 @q4@f6 @q5@f6 @q63777777777777775(4)SuccessiveiterationsofEq. 4 typicallyreduceresidualsolutionerrorsquadratically[ 59 ].SensitivityAnalysis Previouspaperstypicallyhaveevaluatedinversekinematicalgorithmsbyanalyzingamanipulatorinonlyafewcongurations[ 54 55 ].Thisapproachdoesnotprovideinformationaboutmanipulatorsin-betweenspecialandgeneralcongurations.Furthermore,therearenumerousdeviationsawayfromspecialgeometrythatarisefromgeometriccalibration.AmoreappropriatevalidationtechniqueistouseMonte-Carloliketoolstoanalyzeawiderangeofgeometricparametersdeviations.Thegoalistoevaluateawiderangeofgeometricparametersnumericallyclosetothisspecial 58

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Table4-2. NominalPA10-6CEDHparameters Jointiaij(mm)ij(deg)Si(mm) 10.090.0)]TJ /F3 11.955 Tf -161.43 -18.78 Td[(2450.00.00.030.090.00.040.090.0480.050.090.00.06)]TJ 51.66 0 Td[()]TJ /F3 11.955 Tf 38.41 0 Td[(70.0 congurationoverthenormalworkingvolumeofthemanipulator.Startingfromthenominalmanufacturer'sgeometricparameterslistedinTable 4-2 ,Hammersleyquasi-randomsamplingtechniqueswereusedtogenerategeometricparametervariations[ 44 60 ].ForthePA10-6CE,themanufacturingandassemblytolerancesmaycauseerrorsinDHparametersapproximately1mmforlinklengthsandjointoffsetsand0.5degreesfortwistangles.TheserangesserveastheboundsfortheHammersleysamplingmethod,representingpossiblevaluesthatmightresultfromageometriccalibration.Onethousandgeometricgeometricparametersetsweregeneratedtothoroughlyexplorethecalibrationspace,examplesetsareshowninTable 4-3 .Foreachgeometricparameterset,121,680endeffectorposesweretestedthatuniformlyspanthenormalworkspaceofthemanipulatorincludingdifferentendeffectororientations.Eachposewaschosentohave8setsofjointsolutions,soover973millionsolutionswereanalyzed.Eachsolutionwascheckedusingforwardkinematicsagainstthedesiredendeffectorposeandonlysolutionswitherrorslessthan0.1mmand0.01degreeswereconsideredvalid.Results Figure 4-1 showsthattheaugmentedLeeandLiangmethodperformswellforndingjointsolutionswithinstricttolerances.Themeansuccessrateis99.89%overallone-thousandperturbedsetsofmanipulatorgeometry,withaworstcaseof93.69%forasinglegeometricperturbation.ItisimportanttoclarifythattheLeeandLiangmethod 59

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Table4-3. ExampleHammersleysamplesets100(left)and800(right) Jointiaij(mm)ij(deg)Si(mm) 1)]TJ /F3 11.955 Tf 9.3 0 Td[(0.703)]TJ /F3 11.955 Tf 9.3 0 Td[(0.088)]TJ /F3 11.955 Tf -178.32 -18.78 Td[(2)]TJ /F3 11.955 Tf 9.3 0 Td[(0.417)]TJ /F3 11.955 Tf 9.3 0 Td[(0.335)]TJ /F3 11.955 Tf 9.3 0 Td[(0.93630.799)]TJ /F3 11.955 Tf 9.3 0 Td[(0.2230.4674)]TJ /F3 11.955 Tf 9.3 0 Td[(0.096)]TJ /F3 11.955 Tf 9.3 0 Td[(0.271)]TJ /F3 11.955 Tf 9.3 0 Td[(0.2895)]TJ /F3 11.955 Tf 9.3 0 Td[(0.120)]TJ /F3 11.955 Tf 9.3 0 Td[(0.1730.4086)]TJ 57.3 0 Td[()-1638()]TJ /F3 11.955 Tf 38.19 0 Td[(0.743 Jointiaij(mm)ij(deg)Si(mm) 1)]TJ /F3 11.955 Tf 9.3 0 Td[(0.9630.451)]TJ /F3 11.955 Tf -178.33 -18.78 Td[(2)]TJ /F3 11.955 Tf 9.3 0 Td[(0.3360.281)]TJ /F3 11.955 Tf 9.3 0 Td[(0.9643)]TJ /F3 11.955 Tf 9.3 0 Td[(0.791)]TJ /F3 11.955 Tf 9.3 0 Td[(0.3830.18740.2370.3320.60750.0470.1140.2746)]TJ 57.3 0 Td[()-1639()]TJ /F3 11.955 Tf 38.19 0 Td[(0.942 Figure4-1. Sensitivityresultsfor1,000DHparametervariationsofthePA10-6CE returns100%ofsolutionsusingthenominalgeometricparametersshowninTable 4-2 .Thesensitivityanalysisprovesthatifthegeometricparametersaremodiedduringgeometriccalibration(whichisacommonscenario),thealgorithmisstillpracticaltouse.Itisconstructivetolookattheworstperformingcasestobetterunderstandpotentiallimitationsusingthismethod.Fromtheone-thousandgeometricparametervariationstested,only1%hadasuccessratelowerthan99%.Hammersleysets313and907werethemostproblematic, 60

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Table4-4. ProblematicDHparametersets313(left)and907(right) Jointiaij(mm)ij(deg)Si(mm) 10.2230.119)]TJ /F3 11.955 Tf -178.32 -18.78 Td[(20.5450.0060.3983)]TJ /F3 11.955 Tf 9.3 0 Td[(0.1690.018)]TJ /F3 11.955 Tf 9.3 0 Td[(0.71540.610)]TJ /F3 11.955 Tf 9.3 0 Td[(0.3920.26750.277)]TJ /F3 11.955 Tf 9.3 0 Td[(0.217)]TJ /F3 11.955 Tf 9.3 0 Td[(0.0706)]TJ 57.3 0 Td[()]TJ /F3 11.955 Tf 38.19 0 Td[(0.324 Jointiaij(mm)ij(deg)Si(mm) 10.6390.101)]TJ /F3 11.955 Tf -178.33 -18.78 Td[(20.2900.001)]TJ /F3 11.955 Tf 9.3 0 Td[(0.0973)]TJ /F3 11.955 Tf 9.3 0 Td[(0.2790.2620.5904)]TJ /F3 11.955 Tf 9.3 0 Td[(0.443)]TJ /F3 11.955 Tf 9.3 0 Td[(0.212)]TJ /F3 11.955 Tf 9.3 0 Td[(0.0705)]TJ /F3 11.955 Tf 9.3 0 Td[(0.730)]TJ /F3 11.955 Tf 9.3 0 Td[(0.3960.0626)]TJ 57.3 0 Td[()-1639()]TJ /F3 11.955 Tf 38.19 0 Td[(0.387 withmeansof96.07%and93.69%respectively;theirperturbedgeometricparametersarelistedinTable 4-4 .Forthesescenarios,thenumericalstabilitytoolswerenotalwayssufcientinreducingnumericalerrorwithintolerance.Analyzingtheseerrorsshowthatx)]TJ /F3 11.955 Tf 12.22 0 Td[(y)]TJ /F3 11.955 Tf 12.22 0 Td[(zpositioningerrorswereoftenzerobuttheorientationerrorsweregreaterthan0.01degrees,typicallyuptoafewdegrees.Eveninthesenon-successfulinstances,thealgorithmndsjointssolutionsthatareclosetothetruevalue,buthaverelativelysmallresidualendeffectororientationerrors.Furthermore,thistypicallyoccursforonlyonesolutionoutofthepossibleeight.ItisinterestingtonotethatNewton'smethodwouldseemidealforthesesituations,sinceerrorsaretypicallyreducedquadratically.ThispropertyisdependentontheJacobianinverseprovidingadequatesensitivitytorelatejointanglestoendeffectorpose,whichisnotguaranteed.Furthermore,Newton'smethodmightalsoconvergetoaduplicatesolutionwhenthejointanglesforagivenendeffectorposearecloselyspaced.Lastly,therearesomerareoccasionswhenonlycomplexrootsarefoundfortheinput-outputequation,correspondingtonosolutionsinsteadofthedesired8solutions.Thesensitivitystudyshowsthatthesepotentialpitfallsarerareexceptions,occurringinonly0.048%casesoutofthe973milliontested.Conclusion ThispaperhasshownthattheaugmentedLeeandLianginversekinematicalgorithmworkswellevenwhenthemanipulatorisnumericallyclosetoaspecial 61

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conguration.TheMitsubishiPA10-6CEhasanominalspecialcongurationmatchingmanycommercialserialmanipulators,soourresultsapplybroadlyforcommonroboticarms.OnethousandgeometricparametersetvariationsweregeneratedusingHammersleysamplingtechniquestorepresentsmallgeometricchangestypicallyproducedbymanipulatorcalibration.Despitethenumericalchallengestheseincrementalparametricchangescause,theaugmentedLeeandLiangalgorithmisabletosuccessfullyreturn99.89%ofallpossiblejointsolutionswithinstricterrortolerances(0.1mmand0.01degrees).ThecomputationtimefortheaugmentedLeeandLiangmethodisontheorderofonemillisecondonmodernhardware(AMDAthlonIIX4635)makingthisalgorithmpracticallyusefulindiverseroboticsapplications.Furthermore,thesameperformanceismaintainedonembeddedhardware(533MHzPowerPC)ifconstantalgorithmparametersareprecomputedforaparticularmanipulator.ThreespecicchangesweremadetotheclassicLeeandLiangsolutioninordertoachievegoodnumericalstability.First,QRdecompositionforeigenvalueswasemployedtosolvefortherootsoftheinput-outputpolynomial.Second,numericalprecisionroutineswereaddedtothealgorithmtoimprovetheoverallaccuracyofthejointsolutions.Third,anacceleratedNewton'smethodwasconstructedforimprovingtheaccuracyofthesolutionstotherequirederrorthreshold.AllofthesetechniqueswereimplementedusingtheC++languageandopen-sourcesoftwarepackages[ 59 ],allowingforeasyimplementationonwidevarietyofsystemsandarchitectures,includingembeddedroboticcontrollers.Thealgorithmiseasilyimplementedforothermanipulatorsandisfreelyavailable(http://code.google.com/p/drics/).ItisimportanttonotethisaugmentedLeeandLiangmethodisusefulonlyifseveralofthegeometricparameterschangefromthespecialconguration.Forexample,changingonlya45inTable 4-1 wouldbeproblematic.Thespecialtechniquesoutlinedin[ 49 ]wouldhaveabiaserrorinthesolutionswhiletheaugmentedalgorithmwouldnotndasolutionasmatrixAbecomessingular.Ifthesesingle-parameterchangesoccur, 62

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theycouldbehandledonanindividualbasisbyextendingspecialsolutiontechniques.However,itistheauthors'experiencethatgeometriccalibrationincrementallychangesallthegeometryparameters,sothisissuedoesnotoccurinpractice.Weconcludethemethodpresentedhereprovidesanaccurateandgeneraltechniquefortheanalyticalinversesolutionofspatialmanipulatorsclosetospecialcongurations. 63

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CHAPTER5TESTINGMETHODOLOGYTOIDENTIFYPASSIVEPROPERTIESSpecicAim Identifyingthepassivepropertiesofthecervicalspineisclinicallyimportantforimprovedtreatmentmethodsforcervicalspineinjuries,diseases,andotherdisorders.Clinicalstudiescanprovidethenecessaryexperimentaldatatoimprovethecurrentunderstandingofspinebiomechanics,buttheseexperimentsarechallengingtoimplement.Currentmethodsarecumbersomeandofteninvolvecustomrigswithweightsoraloadingframethatlimittheabilitytofullycharacterizethespine'spassiveproperties.Serialmanipulatorsareanattractivealternativethatcanrecreatetheclassicaltestingmotionswhilealsoperformingcomplexandarbitrarytrajectoriesnottestedbefore.Theseadditionalmotionswillprovideuniqueexperimentaldatathatareclinicallyinterestingandnecessaryforidentifyingpassiveproperties.Furthermore,thesenoveldatacancalibrateandvalidateniteelementmodelsforexploringcervicalspinetreatmentmethods.Thischapterdetailsseveralimportanttestingconsiderationsforevaluatingthepassivepropertiesofcervicalspines.Introduction Traditionalcervicalspineexperimentsonlyincludethreecommonloadingconditions:exion-extension,lateralbending,andaxialrotation.Thesemotionswereoftenimplementedwithcustomrigs[ 17 20 ],includingmaterialtestingmachines[ 21 ],thatwerenotwellsuitedforcreatingphysiologicalloading,anddidnotfullycapturethepassivepropertiesofthecervicalspine,whichareprimarilyafunctionoftheligaments.Morecomplexmotionsareneededtofullytesttheoperationalenvelopeofeachligament.Also,thesemotionsshouldbephysiologicalwheretheappliedmomentisconstrainedbutoff-axisforcesandmomentsareunconstrainedallowingthespinetonaturallyrespond.PreviouschaptersoutlinedhowthePA10-6CEserialmanipulatoris 64

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idealforperformingtheseparticularbiomechanicaltestsduetoitsabilitytoaccuratelyandrepeatedlyapplythesecomplexmotions.Previousstudiestypicallyonlyreportedrelativesegmentrotationstodescribethespine'spassiveresponse,whicharethetraditionalmoment-rotationcurvesfoundintheliterature.Totheauthor'sknowledge,relativesegmenttranslationsarenotconsideredeventhoughthesedataareclinicallyimportantforquantifyingpassivepropertiesandbuildingbetterniteelementmodels.Apossibleexplanationisthatsegmentrotationsarerelativelystraightforwardtoidentifyusingstandardmotioncapturetechniques,butsegmenttranslationsrequirederivinganatomicalmotionfrommotioncapturedata.Forexample,Wheeldonetal.performedseveralstudiesonlyreportingsegmentrotationscalculatedfromreectivemarkersplacedoneachvertebralbodyusingcamera-basedmotioncapture[ 20 ].Markersweregroupedinsetsofthreetodenearigidbody,andtherelativerotationduringloadingwasidentiedusingstandardmathematicaltechniques.Thesegmenttranslationscouldnotbeidentiedwithoutrelatingmarkerpositionstoeachvertebralbody.Furthermore,onlythethreebasicmotionsweretested.Yoganandanetal.providesanoverviewofpreviousexperimentalresultsandhowmethodologiesdiffer[ 61 ].Maximumappliedmomentsrangedfrom0.3Nm(Goeletal.)to4.5Nm(Wenetal.).Testswereeitherperformedstaticallyordynamically,andthepurityofthemomentsappliedwasnotalwaysidentied.Thischapteroutlinesdifferentexperimentaltechniquesforevaluatingpassiveproperties,includingpreviouslyunreportedsegmenttranslations.First,theelectromagneticPolhemusmotioncapturesystemwasevaluated,whichisconvenientandaccuratebutsuffersfrommetaldistortion.Thislimitationmotivatedtheuseofcamera-basedmotioncapturewithreectivemarkers.Finally,severalpracticalconsiderationsforrobotictestingofspinesareconsidered. 65

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Methods UsingthePA10-6CEserialmanipulator,twopilotstudieswereconductedwithhealthycervicalspines.EachspecimenwasratedbyaclinicianbaseduponthegradingscaledevelopedbyCusicketal.[ 62 ].Thisratingscalehasametricof0-3thatindicatessuchfactorsasdiscnarrowing,osteophytes,facetarthrosis,andotherhealthconditions.Bothspecimenstestedhadnomajordefectsandwere53and60yearoldmales,respectively.EachspinalcolumnhadC1toT1vertebralbodiesincludingtheligamentsandsofttissue.Afterspecimenpreparation,motioncapturemarkerswereattachedtoC3-C6vertebralbodies.Polhemusmotiontrackerswereusedintherstpilotstudy,andthreereectivemarkerswereattachedtoeachvertebralbodyinthesecondstudy.Computedtomography(CT)scansweretakenwiththemarkersattachedtodeterminethetransformationbetweenmarkersandtheirrespectivevertebralbody(Fig. 5-1 ).WithouttheCTscans,itisdifculttocorrectlydeterminethetranslationalmotionofthevertebralbodies,whichhasnotbeenpreviouslyreported.CTscanswerealsousedforthree-dimensionalanatomyreconstructionwhichvisualizetheexperimentalmotions.Scansweretakenat0.7mmsliceswitha0.31x0.31x0.8mmvoxelsize,yieldingaresolutionof3.18pixels mm.AllsliceswereimportedintotheITK-SNAP[ 63 ]softwarepackageforanalysis,whichprovidestoolsforsegmentation(identifyingthree-dimensionalstructuresfromtwo-dimensionalimages).Segmentationswereperformedmanuallyinsteadofusingautomatictoolsduetotherelativelycomplexspineanatomythatisdifculttocorrectlyprocessautomatically.Foreachspecimen,thenalreconstructionshowsthemotioncapturemarkersandC3-C6(Fig. 5-2 ).CoordinateaxeswereassignedtoeachbodybyimportingthevertebralmodelsintoGeomagic(GeomagicStudio,Morrisville,NC).ConsistentlandmarkswerechosenbasedontheWhiteandPanjabiconventionforallvertebraetosystematicallycreatecoordinatesystemsforeachvertebralbody[ 64 ].TheXaxispointsfromlefttoright 66

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Figure5-1. ComputedtomographyscanofC3vertebralbodyusedforanatomyregistration Figure5-2. Three-dimensionalreconstructionthatisneededforanalysis paralleltothepedicles.TheZaxisrunsfromposteriortoanteriorpassingthroughthecenterofthespinousprocess.Lastly,theYaxisisperpendiculartothersttwoaxesandpointsfrominferiortosuperior.WiththisnomenclatureexionoccursaboutthepositiveXaxis,rightlateralbendingaboutthepositiveZaxis,andaxialrotationabouttheYaxis.Coordinatesystemswerealsoassignedtothegroupingsofmotioncapturemarkers,eitherPolhemustrackers(Fig. 5-3 )orreectivemarkers(Fig. 5-4 ).Aftercreatingallcoordinatesystems,thetransformsbetweenthemotioncapturemarkersandvertebralbodieswasdevelopedusingmatrixmanipulation:TrackerBodyT=ImageTrackerT)]TJ /F7 7.97 Tf 6.59 0 Td[(1ImageBodyT. 67

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Thistransformwasthenappliedtoallcollectedexperimentalmotioncapturedatatodetermineanatomicalsegmentrotationsandtranslations. AC4vertebralbodywithcoordinatesystem BPolhemusmotioncapturemarkerwithcoordinatesystemFigure5-3. CoordinatesystemregistrationforPolhemustrackers Figure5-4. Coordinatesystemregistrationwithreectivemarkers Beforeexperimentallytestingeachspecimen,theinferiorandsuperiorendswererigidlyxedinBondoR(3M)pottingmaterial.C1-C2werexedalongwithC7-T1toprovidearigidsurfacetoxthespecimentotherobotandground.Thesedetailsareimportantforcomparingresultsagainsttheliterature;Wheeldonetal.hypothesized 68

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thatdifferentxationmethodsmayresultindifferentboundaryconditionsthataffectexperimentalresults,includingrangeofmotionandneutralzone[ 20 ].Afterrigidlyxingeachspecimen,theyweremechanicallytestedfollowingtheexperimentalprotocoldevelopedbyDr.RullkoetterattheUniversityofDenver.Theprotocolprovidesausefulcalibrationbaselineoftraditionalloadingscenariosalongwithmorecomplexmotionsforimprovedcalibration.First,alltestswereperformedfortheintactspine(Table 5-1 ),andthenrepeatedaftersectioningligamentsattheC4-C5level(Table 5-2 ).ThePA10-6CEserialmanipulatorisuniquelysuitedforreplayingexperimentaltrajectoriesfoundinTable 5-1 .It'sabilitytoquicklyandaccuratelyrecreatearbitrarymotionsiscrucialtothesuccessofthesequentialsectioningprotocol. Table5-1. Intactmulti-segmentprotocolunderforcecontrol TestTestDescriptionMomentLimits 1Flexion-Extension(FE)2Nm2LateralBending(LB)2Nm3AxialRotation(AR)2Nm4FEwithRightLB1.4Nm5FEwithLeftLB1.4Nm6FEwithAR1.4Nm7LBwithAR1.4Nm8FEwith90Npreload2Nm Table5-2. Sequentialsectioningprotocolunderdisplacementcontrol TestTestDescriptionDisplacementmotion 10SupraspinousLig.FE,AR11InterspinousLig.FE,AR12AnteriorLongitudinalLig.FE,AR13FacetCapsulesFE,LB,AR14EntireFacetFE,LB,AR 69

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PolhemusPilotStudy Intherstpilotstudy,thePolhemusmotioncapturesystemhadsignicantdifcultycapturingvertebralbodytranslationsduetometaldistortion.Metaldistortionisawellknownproblemwithelectromagneticsystems,buttypicallysmallamountsofmetalhavenegligibleeffects.Thesourcesofmetalincludedthescrewsusedtosecurelyattachthetrackerstoboneandtheroboticsystemitself.Thedistortioneffectsarenonlinearandseveralmillimetersinmagnitude,whichisrelativelylargewhenmodelingspinalmotion.Thisisespeciallyeasytonoticewhenanimatingthespinalanatomy(Fig. 5-5 ).Therawmotioncapturedataneedsacorrectionfactortoaccountforthemetaldistortionbutthisbiasfactorisnotalwayssufcientduetothenon-linearityofthenoise.ThemostproblematicsensorswereS3andS6(attachedtoC3andC6respectively)andrequiredthefollowingbiascorrectionsforreliablerotationinformation:TS3)]TJ /F7 7.97 Tf 6.59 0 Td[(Bias=2666666641000.007501)]TJ /F3 11.955 Tf 9.3 0 Td[(0.00436000.00436100001377777775 (5)TS6)]TJ /F7 7.97 Tf 6.59 0 Td[(Bias=2666666641000.00500.98480.17360.0070)]TJ /F3 11.955 Tf 9.3 0 Td[(0.17360.9848)]TJ /F3 11.955 Tf 9.3 0 Td[(0.0120001377777775 (5)Despitethemetaldistortionproblems,thedatacollectedfromtherstpilotstudyisusefulforcomparingbasicresultsagainsttheliterature.ThesedatawerecomparedagainsttheworkofWheeldonetal.andYoganandanetal.,whousedasimilartestingprotocolforexion-extensionandlateralbending[ 65 ].Alsonotethatthepurityoftheappliedmomentwasensuredbyincorporatingtheoff-axisforcesandmomentsinto 70

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Figure5-5. ThedistortionofeffectsofthePolhemussystemforC3-C6 thecontrolalgorithm.Forexion-extension,themax/minoff-axisforcesandmomentswere2.051=)]TJ /F3 11.955 Tf 12.46 0 Td[(3.368Nand0.129=)]TJ /F3 11.955 Tf 12.46 0 Td[(0.075179Nm.Inlateralbending,thephysiologicalloadingwasmaintainedwithmax/minoff-axisforcesandmomentsof4.255=)]TJ /F3 11.955 Tf 12.35 0 Td[(3.600Nand0.255=)]TJ /F3 11.955 Tf 12.96 0 Td[(0.229Nm.Overall,thedataagreereasonablywellwiththeliterature.Yoganandanetal.onlyreportedonesideoflateral-exion,andthecollecteddatawasadjustedappropriatelyforcomparison[ 65 ].Inthesecondhalfoftheexperimentalprocedure,theligamentslistedinTable 5-2 weresectionedtodestabilizethespine,andtheresultingchangeinloadsweremeasured.Themotionfromintactexion-extensionforcecontrolwassavedandreconstructedusingcubicsplines.Thereforethesameloadingmotionwasrepeatedlyappliedaftersectioningeachligament.Itwasexpectedthatdestabilizingthespinewillhaveequivalentorevenlargersegmentmotionsatadecreasedappliedload,whichisbestillustratedattheC4-C5levelwheretheinjurywasapplied(Fig. 5-9 ). 71

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Figure5-6. ExperimentalsetupwithPolhemusmotioncapturesystem Figure5-7. Flexion-extensionresultscomparedagainsttheliterature 72

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Figure5-8. Lateral-exionresultscomparedagainsttheliterature Figure5-9. C4-C5destabilizationeffectscomparedagainstthenominalmotion 73

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Camera-BasedPilotStudy DuetothelimitationsofthePolhemussystem,camera-basedmotioncaptureisabetteralternative.Metaldoesnotaffectthissystem,butrathercamerasmustbeabletoidentifyclustersofreectivemarkers.TheEaglefour-camerasystem(MotionAnalysisCorporation,SantaRosa,CA)trackedtwelvemarkersattachedtoC3-C6vertebralbodiesforcalculatingsegmentrotationsandtranslations.Thereareseveralkeyconsiderationswhenattachingthereectivemarkerstothespecimen.Threemarkersareneededtoidentifyeachvertebralbody,andthemakerscannotbecollinear.Furthermore,themarkerscannotmoveorrotaterelativetotheboneduringtesting.Duetothelimitedspaceoncervicalspinespecimens,threeber-glasspostswereinsertedanteriorlyinthevertebralbodieswithmarkersattachedtotheend( 5-10 ).Thepostswerestaggeredatdifferentanglestopreventthemarkersfrombeingcollinearandrigidlysecuredwithapresst. Figure5-10. Fiber-glasspostsinsertedwithreectivemarkers Whilethissetupwasconvenient,itwasimpracticalduetomotioncapturelimitations.Eachcamerainthemotioncapturesystemmustbeabletodistinguishthe 74

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reectivemarkersatapixellevel.Markerscanbephysicallyseparatedbutstillappeartobetouchingintheviewofthecamerawhichpreventedaccuratemeasurements.Evenreducingthemarkersizetoavemmdiameterdidnotalleviatetheproblembecausethemarkerstendedtobunchtogetherduringtesting.Thesolutionwastogroupthemarkersontriadsthatbranchawayfromthevertebralbodies,creatingclearviewsforthecameras(Fig. 5-11 ). Figure5-11. Markertriadswithreectivemarkersrigidlyattachedtospinespecimen Despitetheinitiallimitationswiththepostsystem,reasonabledatawerecollectedforintactlateralbending(Fig. 5-12 ).Thelateralbendingmotionhadtheleastmarkerdistortion,althoughthedatastillrequiredsignicantltering(lowpassbutterworth)andsmoothing.Thedatasufferedfromcentriodjumpsandmissingdatafrommarkerdropoutandwasmanuallycorrectedtoprovidethebestestimateofthetruemarkertrajectories.ThesedatahadsimilartrendstothepreviouslycollecteddatawiththePolhemussystem,providingmorecondenceintherobotictestingframework. 75

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Figure5-12. Experimentalsetupwithcamera-basedmotioncapture Figure5-13. TherelativerotationsbetweenC3-C4inlateralbending Discussion Therewereseveralimportantengineeringlessonslearnedfromthersttwopilotstudiesinadditiontoevaluatingdifferentmotioncapturesystems.Animportantconsiderationforrobotictestingisensuringthemobilityrangeofeachspecimenfallswithintheworkingenvelopeoftherobot.Thiswaschallengingduetodifferentcharacteristicsofthecadaverspecimens;forexamplehealthierspinesmayhaveasmallerneutralzone.Specimencurvatureofthespinalcolumnmayalsoshiftthe 76

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Figure5-14. TherelativerotationsbetweenC5-C6inlateralbending workingregionofthespecimen.Toaccountforthesevariousproperties,agimbalbasecanprovideextradegreesoffreedomforrigidlyorientingspecimensdespitetheirnominalcurvature.Ideally,thesuperiorendofthespecimenshouldnaturallyrestverticallyintheworkspace.Thespecimensinthersttwopilotstudiesdidnothavesignicantcurvaturebutotherspecimensexaminedhadsignicantcurvatureof30degreesormore.Furthermore,theanatomicaldirectionofthespecimenwiththelargestrangeofmotionshouldalwaysbeorientedparalleltotherobotbase,whichiseasilydeterminedduringspecimenpreparation.Thisorientationavoidsproblematicregionsthattherobotcannotreachnearitsbase.Anotherconsiderationwastheinitialestimateoftheinstantaneousaxisofrotation.Withthecurrentframework,therobotneededaninitialpointtoapplydesiredmoments.Thispointcontinuouslyupdatesduringtestingbasedonforcefeedbackfromthespecimen,butareasonableinitialestimateisrequired.AsdiscussedinChapter2,theaxisofrotationisbuiltintotheJacobianformulation.Itismathematicallyconvenienttoassumethecenterofrotationliesalongtheaxisoftheendeffectoroftherobot,butthisisuntrueforspecimenswithsignicantnominalcurvature.Ageneralsolutionistoadd 77

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anadditionaltransformtotheJacobianformulationwhichrelatesthecenterofrotationofthespecimentotherobot.Conclusion TheseresultsshowthattheexperimentalproceduredevelopedaroundthePA10-6CEserialmanipulatoriswellsuitedforcervicalspineexperimentaltestingandreasonablyagreeswithpreviousstudies,includingtheworkofWheeldonetal.andYoganandanetal.Puremomentloadingwasapplieddynamicallyandoff-axisforcesandmomentswerequantied.Thepilotstudiesinvestigatedtwoseparatemotioncapturetechniques,theelectromagneticPolhemusandcamera-basedwithreectivemarkers.Bothsystemspresentchallengesforspinetestingbutthecamera-basedmotioncapturehasclearandimplementablesolutions.Thesepilotstudiesillustratethecapabilitiesofthetestingsystemandshowthatnoveldatacanbecollectedwiththeappropriateadjustments.Subsequenttestingwillperformtraditionalloadingscenariosaswellasuniqueloadingtrajectoriestocalibrateniteelementmodels.Ultimately,thegoalistohavevalidatedniteelementmodelsthatcanbeusedinapredictivemannerforimprovingtreatmentmethods.Thesepilotstudieshaveshownthattheframeworkpresentedcanhelpachievethatgoal. 78

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CHAPTER6CONCLUSIONThisdissertationhasoutlinedseveralimportantfactorsfordevelopinganewroboticsystemforcervicalspinetesting.Thissystemiseasilyextendedtootherhumanjointsaswellascaninejoints.Thekeycontributionsareoutlinedbelow. Designofatestingplatformforthecervicalspine,whichcanbeextendedtootherjoints Dynamiccalibrationmethodsforserialmanipulatorsoveralargeworkingvolume Generalinversekinematicsimprovementsandvalidation Noveltestingprotocolsforevaluatingpassiveproperties,includinglocalstiffnessanddegeneration 79

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APPENDIX:DERIVATIONOFNEWTON'SMETHODFORINVERSEKINEMATICS Theinversekinematicssolutionpresentedwilloccasionallyfailtondasolutionsetqthatplacestheendeffectoratthedesiredpositionandorientationwithinanacceptabletolerance.TheseerrorscanbequicklyxedthoughusingtheiterativeNetwon'smethod.Newton'smethodisanalgorithmforndingtherootsofasetofmulti-dimensionalfunctions,f1..fn.Themethodisdenedasxi+1=xi)]TJ /F10 11.955 Tf 11.96 0 Td[(J)]TJ /F7 7.97 Tf 6.59 0 Td[(1xi (A)wherexiisacolumnvectorofthecurrentrootestimates,JistheJacobian,isaconvergencemodier,andxi+1istheupdatedrootapproximation.Thevectorxisofsizen=6representingthesixdegreesoffreedomofthemanipulator.TheentriesoftheJacobianarethepartialderivativesofthevaluefunctionsfiwithrespecttoeachdegreeoffreedomqi.TheinverseoftheJacobianguidestheiterativesolutionprocesstoabetterrootapproximation,typicallyinonlyafewiterations.Usingthesolutionsetfoundwithinversekinematics(whichhassomenominalerror)astheinitialguess,thepositioningerrorcanbequicklyreduced.Thesixequationsneededtorelatejointanglestotaskspacepositionandorientationarederivedfromforwardkinematics.GiventhedesiredtransformF6T,asystemofequationsisdenedusingthejointangles:F6T=F6A(q1,q2,q3,q4,q5,q6) (A)Thematricesontheleftandrighthandsideseachhave12uniqueentriesbutonly6areindependent.Theupper3by3ofanytransformationmatrixisarotationmatrix,R,whichcanbebrokenintothreerotationangles.Usingeulersequences,Ronboth 80

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sidescanbetransformedintoroll,pitch,andyawvaluesfordirectcomparison,whichisa1-2-3Eulersequence.Thisconversionisgivenas:roll=atan2()]TJ /F3 11.955 Tf 9.3 0 Td[(R(2,3),R(3,3)) (A)pitch=atan2(R(1,3),cos(roll)R(3,3))]TJ /F3 11.955 Tf 11.96 0 Td[(sin(roll)R(2,3)) (A)yaw=atan2()]TJ /F3 11.955 Tf 9.3 0 Td[(R(1,2),R(1,1)) (A)InorderforF6TandF6Atobeequivalent,theroll,pitch,andyawvaluesmustequalalongwiththetranslationalcomponentsofthetransforms.TakingthesesixcomparisonsandsettingthemequaltozeroyieldssixgoverningequationsthatareneededtoformtheJacobian.Partialderivativesofeachfunctionf1-f6istakenwithrespecttoq1-q6.f1:F6T(1,4))]TJ /F7 7.97 Tf 11.95 4.93 Td[(F6A(1,4)=0 (A)f2:F6T(2,4))]TJ /F7 7.97 Tf 11.95 4.94 Td[(F6A(2,4)=0 (A)f3:F6T(3,4))]TJ /F7 7.97 Tf 11.95 4.94 Td[(F6A(3,4)=0 (A)f4:F6T)]TJ /F10 11.955 Tf 12.62 0 Td[(>roll)]TJ /F7 7.97 Tf 11.95 4.93 Td[(F6A)]TJ /F10 11.955 Tf 12.62 0 Td[(>roll=0 (A)f5:F6T)]TJ /F10 11.955 Tf 12.62 0 Td[(>pitch)]TJ /F7 7.97 Tf 11.95 4.94 Td[(F6A)]TJ /F10 11.955 Tf 12.62 0 Td[(>pitch=0 (A)f6:F6T)]TJ /F10 11.955 Tf 12.62 0 Td[(>yaw)]TJ /F7 7.97 Tf 11.95 4.94 Td[(F6A)]TJ /F10 11.955 Tf 12.62 0 Td[(>yaw=0 (A)Fromhere,Newton'smethodcanbeiterativelyevaluatedtoimprovethejointsolutionsuntiltheyreachthenecessarytolerances,whichisevaluatedusingforwardkinematics. 81

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REFERENCES [1] Huelke,D.,andNusholtz,G.,1986.Cervicalspinebiomechanics:areviewoftheliterature.Journaloforthopaedicresearch,4(2),pp.232. [2] Bone,andDecade,J.,2011.Theburdenofmusculoskeletaldiseases.fromhttp://www.boneandjointburden.org/. [3] Mayeld-Clinic,2011.Herniatedcervicaldisc.fromhttp://www.mayeldclinic.com/Images/PE-HerniatedCervical. [4] Bone,andSpine,2011.Spondylosisandsevereanteriorosteophytes.fromhttp://boneandspine.com/wp-content/uploads/2010/05/. [5] Emedicine,2011.Cervicalspineanatomy.fromhttp://emedicine.medscape.com/article/1264065-overviewa04. [6] Kelley,B.S.,2009.SpineBiomechanics. [7] Stokes,I.,andIatridis,J.,2005.Biomechanicsofthespine.BasicOrthopaedicBiomechanicsandMechano-Biology.VCMowandR.Huiskes,Eds.Lippincott,WilliamsandWilkins,Philadelphia. [8] SpineUniverse,2011.Cervicalspineligaments.fromhttp://www.spineuniverse.com/conditions/neck-pain/. [9] Hirsch,C.,andNachemson,A.,1954.Newobservationsonthemechanicalbehavioroflumbardiscs..ActaOrthopaedicaScandinavica,23(4),p.254. [10] Horton,W.,1958.Furtherobservationsontheelasticmechanismoftheintervertebraldisc.JournalofBoneandJointSurgery-BritishVolume,40(3),p.552. [11] Galante,J.,1967.Tensilepropertiesofthehumanlumbarannulusbrosus. [12] Tkaczuk,H.,1968.Tensilepropertiesofhumanlumbarlongitudinalligaments..ActaOrthopaedicaScandinavica,pp.Suppl. [13] Ball,J.,andMeijers,K.,1964.Oncervicalmobility.AnnalsoftheRheumaticDiseases,23(6),p.429. [14] WhiteIII,A.,andHirsch,C.,1971.Thesignicanceofthevertebralposteriorelementsinthemechanicsofthethoracicspine.ClinicalOrthopaedicsandRelatedResearch,81,p.2. [15] Farfan,H.,Cossette,J.,Robertson,G.,Wells,R.,andKraus,H.,1970.Theeffectsoftorsiononthelumbarintervertebraljoints:theroleoftorsionintheproductionofdiscdegeneration..TheJournalofboneandjointsurgery.Americanvolume,52(3),p.468. 82

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[16] Lysell,E.,1969.Motioninthecervicalspine.anexperimentalstudyonautopsyspecimens..ActaOrthopaedicaScandinavica,pp.Suppl. [17] Panjabi,M.,WhiteIII,A.,andJohnson,R.,1975.Cervicalspinemechanicsasafunctionoftransectionofcomponents.Journalofbiomechanics,8(5),pp.327. [18] Panjabi,M.,Krag,M.,andGoel,V.,1981.Atechniqueformeasurementanddescriptionofthree-dimensionalsixdegree-of-freedommotionofabodyjointwithanapplicationtothehumanspine.JournalofBiomechanics,14(7),pp.447. [19] Goel,V.,Clark,C.,McGowan,D.,andGoyal,S.,1984.Anin-vitrostudyofthekinematicsofthenormal,injuredandstabilizedcervicalspine.Journalofbiomechanics,17(5),pp.363. [20] Wheeldon,J.,Pintar,F.,Knowles,S.,andYoganandan,N.,2006.Experimentalexion/extensiondatacorridorsforvalidationofniteelementmodelsoftheyoung,normalcervicalspine.Journalofbiomechanics,39(2),pp.375. [21] Eguizabal,J.,Tufaga,M.,Scheer,J.,Ames,C.,Lotz,J.,andBuckley,J.,2010.Puremomenttestingforspinalbiomechanicsapplications:Fixedversusslidingringcable-driventestdesigns.Journalofbiomechanics,43(7),pp.1422. [22] Stokes,I.,Gardner-Morse,M.,Churchill,D.,andLaible,J.,2002.Measurementofaspinalmotionsegmentstiffnessmatrix.Journalofbiomechanics,35(4),pp.517. [23] Walker,M.,andDickey,J.,2007.Newmethodologyformulti-dimensionalspinaljointtestingwithaparallelrobot.MedicalandBiologicalEngineeringandComput-ing,45(3),pp.297. [24] Goertzen,D.,andKawchuk,G.,2009.Anovelapplicationofvelocity-basedforcecontrolforuseinroboticbiomechanicaltesting.Journalofbiomechanics,42(3),pp.366. [25] Fujie,H.,Mabuchi,K.,Savio,L.,Livesay,G.,Arai,S.,andTsukamoto,Y.,1993.Theuseofroboticstechnologytostudyhumanjointkinematics:anewmethodology.Journalofbiomechanicalengineering,115,p.211. [26] Rudy,T.,Livesay,G.,Woo,S.,andFu,F.,1996.Acombinedrobotic/universalforcesensorapproachtodetermineinsituforcesofkneeligaments.Journalofbiomechanics,29(10),pp.1357. [27] Gilbertson,L.,Doehring,T.,andKang,J.,2000.Newmethodstostudylumbarspinebiomechanics:Delineationofinvitroload-displacementcharacteristicsbyusingarobotic/UFStestingsystemwithhybridcontrol*.OperativeTechniquesinOrthopaedics,10(4),pp.246. 83

PAGE 84

[28] AcostaJr,F.,Buckley,J.,Xu,Z.,Lotz,J.,andAmes,C.,2008.Biomechanicalcomparisonofthreexationtechniquesforunstablethoracolumbarburstfractures.JournalofNeurosurgery:Pediatrics,8(4). [29] Crawford,N.,Brantley,A.,Dickman,C.,andKoeneman,E.,1995.Anapparatusforapplyingpurenonconstrainingmomentstospinesegmentsinvitro.Spine,20(19),p.2097. [30] Melcher,R.,Puttlitz,C.,Kleinstueck,F.,Lotz,J.,Harms,J.,andBradford,D.,2002.Biomechanicaltestingofposterioratlantoaxialxationtechniques.Spine,27(22),p.2435. [31] Kallemeyn,N.,Gandhi,A.,Kode,S.,Shivanna,K.,Smucker,J.,andGrosland,N.,2010.ValidationofaC2-C7cervicalspineniteelementmodelusingspecimen-specicexibilitydata.MedicalEngineering&Physics,32(5),pp.482. [32] Darcy,S.,Gil,J.,Woo,S.,andDebski,R.,2009.Theimportanceofpositionandpathrepeatabilityonforceatthekneeduringsix-dofjointmotion.Medicalengineering&physics,31(5),pp.553. [33] Raibert,M.,andCraig,J.,1981.Hybridposition/forcecontrolofmanipulators.JournalofDynamicSystems,Measurement,andControl,102(127),pp.126. [34] Goel,V.,Wilder,D.,Pope,M.,andEdwards,W.,1995.ControversyBiomechanicalTestingoftheSpine:Load-ControlledVersusDisplacement-ControlledAnalysis.Spine,20(21),p.2354. [35] An,C.,Atkeson,C.,andHollerbach,J.,1988.Model-basedcontrolofarobotmanipulator,Vol.214.MITpressCambridge,MA. [36] Paul,R.,andShimano,B.,1978.Kinematiccontrolequationsforsimplemanipulators.InDecisionandControlincludingthe17thSymposiumonAdaptiveProcesses,1978IEEEConferenceon,Vol.17,IEEE,pp.1398. [37] Judd,R.,andKnasinski,A.,1990.Atechniquetocalibrateindustrialrobotswithexperimentalverication.RoboticsandAutomation,IEEETransactionson,6(1),pp.20. [38] Karan,B.,andVukobratovic,M.,1994.Calibrationandaccuracyofmanipulationrobotmodelsanoverview.MechanismandMachineTheory,29(3),pp.479. [39] AndrewLiou,Y.,Lin,P.,Lindeke,R.,andChiang,H.,1993.Tolerancespecicationofrobotkinematicparametersusinganexperimentaldesigntechniquethetaguchimethod.Roboticsandcomputer-integratedmanufacturing,10(3),pp.199. [40] Jang,J.,Kim,S.,andKwak,Y.,2001.Calibrationofgeometricandnon-geometricerrorsofanindustrialrobot.Robotica,19(03),pp.311. 84

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[41] Nubiola,A.,andBonev,I.,2012.Absolutecalibrationofanabbirb1600robotusingalasertracker.RoboticsandComputer-IntegratedManufacturing. [42] Riley,D.,1987.Robotcalibrationandperformancespecicationdetermination.SocietyofManufacturingEngineers. [43] Lightcap,C.,Hamner,S.,Schmitz,T.,andBanks,S.,2008.ImprovedpositioningaccuracyofthePA10-6CErobotwithgeometricandexibilitycalibration.Robotics,IEEETransactionson,24(2),pp.452. [44] Hammersley,J.,1960.Montecarlomethodsforsolvingmultivariableproblems.AnnalsoftheNewYorkAcademyofSciences,86(3),pp.844. [45] Spong,M.,1987.Modelingandcontrolofelasticjointrobots.JournalofDynamicSystems,Measurement,andControl,109(4),pp.310. [46] Schutte,J.,ByungKoh,I.,Reinbolt,J.,Haftka,R.,George,A.,andFregly,B.,2005.Evaluationofaparticleswarmalgorithmforbiomechanicaloptimization.Journalofbiomechanicalengineering,127(3),p.465. [47] Moore,S.,Thomas,M.,Woo,S.,Gabriel,M.,Kilger,R.,andDebski,R.,2006.Anovelmethodologytoreproducepreviouslyrecordedsix-degreeoffreedomkinematicsonthesamediarthrodialjoint.Journalofbiomechanics,39(10),pp.1914. [48] Tuttle,T.,1992.Understandingandmodelingthebehaviorofaharmonicdrivegeartransmission.Tech.rep.,MASSACHUSETTSINSTOFTECHCAMBRIDGEARTIFICIALINTELLIGENCELAB. [49] CraneIII,C.,andDuffy,J.,1998.Kinematicanalysisofrobotmanipulators.CambridgeUniversityPress. [50] Duffy,J.,andCrane,C.,1980.Adisplacementanalysisofthegeneralspatial7-link,7rmechanism.MechanismandMachineTheory,15(3),pp.153. [51] Lee,H.,andLiang,C.,1988.Displacementanalysisofthegeneralspatial7-link7rmechanism.MechanismandMachineTheory,23(3),pp.219. [52] Raghavan,M.,andRoth,B.,1990.Kinematicanalysisofthe6rmanipulatorofgeneralgeometry.InProc.FifthInt.SymposiumonRoboticsResearch,MITPress,Cambridge. [53] Raghavan,M.,andRoth,B.,1993.Inversekinematicsofthegeneral6rmanipulatorandrelatedlinkages.Journalofmechanicaldesign,115(3),pp.502. [54] Wampler,C.,andMorgan,A.,1991.Solvingthe6rinversepositionproblemusingageneric-casesolutionmethodology.MechanismandMachineTheory,26(1),pp.91. 85

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[55] Manocha,D.,andCanny,J.,1994.Efcientinversekinematicsforgeneral6rmanipulators.RoboticsandAutomation,IEEETransactionson,10(5),pp.648. [56] Stroustrup,B.,1997.TheC++programminglanguage.Addison-WesleyLongmanPublishingCo.,Inc. [57] Horn,R.,andJohnson,C.,1990.Matrixanalysis.Cambridgeuniversitypress. [58] Trefethen,L.,andBauIII,D.,1997.Numericallinearalgebra.No.50.SocietyforIndustrialMathematics. [59] Guennebaud,G.,andJacob,B.,2012.Eigenc++library,version3.1. [60] Kocis,L.,andWhiten,W.,1997.Computationalinvestigationsoflow-discrepancysequences.ACMTransactionsonMathematicalSoftware(TOMS),23(2),pp.266. [61] Yoganandan,N.,Stemper,B.,Pintar,F.,Baisden,J.,Shender,B.,andPaskoff,G.,2008.Normativesegment-specicaxialandcoronalangulationcorridorsofsubaxialcervicalcolumninaxialrotation.Spine,33(5),pp.490. [62] Cusick,J.F.,Yoganandan,N.,Pintar,F.,andGardon,M.,1996.Cervicalspineinjuriesfromhigh-velocityforces:apathoanatomicandradiologicstudy.Journalofspinaldisorders,9(1),pp.1. [63] Yushkevich,P.,Piven,J.,Hazlett,H.,Smith,R.,Ho,S.,Gee,J.,andGerig,G.,2006.User-guided3dactivecontoursegmentationofanatomicalstructures:signicantlyimprovedefciencyandreliability.Neuroimage,31(3),pp.1116. [64] White,A.,andPanjabi,M.,1990.Clinicalbiomechanicsofthespine,Vol.446.LippincottPhiladelphia. [65] Yoganandan,N.,Pintar,F.,Stemper,B.,Wola,C.,Shender,B.,andPaskoff,G.,2007.Level-dependentcoronalandaxialmoment-rotationcorridorsofdegeneration-freecervicalspinesinlateralexion.TheJournalofBone&JointSurgery,89(5),pp.1066. 86

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BIOGRAPHICALSKETCH IraJeromeHillreceivedhisB.S.inmechanicalengineeringfromtheUniversityofPittsburghin2008asaNationalMeritScholar.HebeganhisgraduatecareerattheUniversityofFloridathatsameyearasaNationalScienceFoundationBridgetotheDoctorateFellow.HecompletedhismastersworkintheTribologyLaboratoryin2010beforecompletinghisPh.D.researchinroboticsandbiomechanicsunderDr.ScottBanks. 87