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Vestibular Dynamic Inclinometer and Measurement of Inclination Parameters

Permanent Link: http://ufdc.ufl.edu/UFE0043745/00001

Material Information

Title: Vestibular Dynamic Inclinometer and Measurement of Inclination Parameters
Physical Description: 1 online resource (83 p.)
Language: english
Creator: Vikas, Vishesh
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: inclinometer -- robotics -- sensor -- vestibular
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: A human body displays a remarkable quality of maintaining both static and dynamic equilibrium for a rigid body in unstable equilibrium (modeled as an inverted pendulum). Biologically, the sensing of inclination is done by the human vestibular system. Here, a novel design of a sensor motivated by the human vestibular system is presented. The sensor, called the Vestibular Dynamic Inclinometer(VDI), measures the dynamic inclination parameters - inclination angle, angular velocity, angular acceleration and magnitude of the acceleration of surface of contact (e.g., gravity). The VDI uses two dual-axis linear MEMS accelerometers and one single axis MEMS gyroscope to measure the inclination parameters for a four degree-of-freedom robot. The concept of the Dynamic Equilibrium Axis (DEA) is introduced. The DEA is the axis along which the robot is at equilibrium. The direction of the DEA is parallel to the direction of the resultant acceleration of the surface of contact. Two control strategies - torque control and acceleration control are simulated to explain this concept. The DEA attempts to explain the reason why humans lean forward while accelerating (sprinting) and backward while decelerating (sudden stopping). The VDI is used to measure joint parameters - joint angle, angular velocities and angular accelerations, for links joined by revolute joint. Here, the VDI is a contactless sensor withe flexible point of application. The VDI sensor is extended for inclination parameter measurement of five degree-of-freedom robot. The new sensor is called the planar Vestibular Dynamic Inclinometer (pVDI). The pVDI consists of four dual-axis linear MEMS accelerometers and one tri-axial MEMS gryoscope. The orientation of the linear accelerometers is different from the intuitive analogous VDI. This is due to the coupling in the kinematics. Similar to the VDI, the pVDI is used to measure joint parameters with links joined by universal hooke joint. The concept of the DEA is also extended. The novelty of the VDI and the pVDI lies in the fact that the measurements are independent of drift or integration errors, acceleration of surface of contact (e.g., gravitational acceleration), independent of the dynamics of the robot, and require significantly less computational burden (closed form solution). The inclination angle obtained from the VDI and the pVDI is from the DEA. As the goal of balancing tasks is to bring the robot to equilibrium, the inclination parameters obtained from the VDI and the pVDI are suitable candidates for control applications. The application of the VDI and the pVDI is foreseen in gait analysis, industrial robotics, humanoid robotics, etc.
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 Vishesh Vikas.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Crane, Carl D.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-06-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043745:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043745/00001

Material Information

Title: Vestibular Dynamic Inclinometer and Measurement of Inclination Parameters
Physical Description: 1 online resource (83 p.)
Language: english
Creator: Vikas, Vishesh
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: inclinometer -- robotics -- sensor -- vestibular
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: A human body displays a remarkable quality of maintaining both static and dynamic equilibrium for a rigid body in unstable equilibrium (modeled as an inverted pendulum). Biologically, the sensing of inclination is done by the human vestibular system. Here, a novel design of a sensor motivated by the human vestibular system is presented. The sensor, called the Vestibular Dynamic Inclinometer(VDI), measures the dynamic inclination parameters - inclination angle, angular velocity, angular acceleration and magnitude of the acceleration of surface of contact (e.g., gravity). The VDI uses two dual-axis linear MEMS accelerometers and one single axis MEMS gyroscope to measure the inclination parameters for a four degree-of-freedom robot. The concept of the Dynamic Equilibrium Axis (DEA) is introduced. The DEA is the axis along which the robot is at equilibrium. The direction of the DEA is parallel to the direction of the resultant acceleration of the surface of contact. Two control strategies - torque control and acceleration control are simulated to explain this concept. The DEA attempts to explain the reason why humans lean forward while accelerating (sprinting) and backward while decelerating (sudden stopping). The VDI is used to measure joint parameters - joint angle, angular velocities and angular accelerations, for links joined by revolute joint. Here, the VDI is a contactless sensor withe flexible point of application. The VDI sensor is extended for inclination parameter measurement of five degree-of-freedom robot. The new sensor is called the planar Vestibular Dynamic Inclinometer (pVDI). The pVDI consists of four dual-axis linear MEMS accelerometers and one tri-axial MEMS gryoscope. The orientation of the linear accelerometers is different from the intuitive analogous VDI. This is due to the coupling in the kinematics. Similar to the VDI, the pVDI is used to measure joint parameters with links joined by universal hooke joint. The concept of the DEA is also extended. The novelty of the VDI and the pVDI lies in the fact that the measurements are independent of drift or integration errors, acceleration of surface of contact (e.g., gravitational acceleration), independent of the dynamics of the robot, and require significantly less computational burden (closed form solution). The inclination angle obtained from the VDI and the pVDI is from the DEA. As the goal of balancing tasks is to bring the robot to equilibrium, the inclination parameters obtained from the VDI and the pVDI are suitable candidates for control applications. The application of the VDI and the pVDI is foreseen in gait analysis, industrial robotics, humanoid robotics, etc.
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 Vishesh Vikas.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Crane, Carl D.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-06-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043745:00001


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VESTIBULARDYNAMICINCLINOMETER ANDMEASUREMENTOFINCLINATIONPARAMETERS By VISHESHVIKAS ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2011

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c r 2011VisheshVikas 2

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v‡t XmhAkAys yrkoEVsmB. EnEvrm ^ k zmdvsvrkAyrq svrdA; Designisafunnyword.Somepeoplethinkdesignmeanshowitlooks.Bu tofcourse,if youdigdeeper,it'sreallyhowitworks. -SteveJobs 3

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ACKNOWLEDGMENTS Iwouldliketoexpressprofoundgratitudetomyadvisor,Prof.Car lCrane,forhis invaluablesupport,encouragement,supervisionandusefulsugg estionsthroughoutthis researchwork.Hismoralsupportandcontinuousguidanceenable dmetocompletemy worksuccessfully.IwouldliketothankProf.JohnSchueller,Dr.Wa rrenDixon,Prof. PaulGaderandProf.WilliamHagerforservingonmycommittee. Iamespeciallyindebtedtomyparents,Dr.OmVikasandMrs.Pramod Kumari Sharma,fortheirloveandsupporteversincemychildhood.Iwishto thankmybrother, Pranav,foralltheloveandencouragement.Myfamilyhasbeenmyu ltimatesupportand Icouldnothavedonethiswithoutthem. Iwouldliketothankwonderfulandinspiringteacherswhohavetaug htmeoverthe years:duringmyhighschool,duringmyundergraduatestudiesatI IT(esp.Prof.Naresh Chandiramani),duringmyinternshipatINRIALorraine(Prof.Fran coisCharpillet) andgraduatestudieshereatUFL(esp.Dr.AnilRao,Prof.BabaV emuri,Dr.Prabir BarooahandProf.JayGopalakrishnan).IwouldalsoliketothankSh annonRidgewayfor allthehelpandmanywonderfuldiscussionsaboutroboticsandlife. IthankmyfellowstudentsattheCenterforIntelligentMachinesan dRobotics.From them,Ilearnedagreatdealandfoundgreatfriendships.Iwoulda lsoliketothankthe MAEgraduatestudentsNitinSharma,ShubhenduBhasin,RyanChilt on,JonathonJeske andAnubiMoseswithwhomIhaveenjoyedexcitingdiscussions.Tha nkstoRakesh Mahadevapuram,NicoleKanizay,SreenivasVedantam,ArindamBan erjee,Mayank Srivastava,PrashantAnandkrishnanandVinayPandeyforbeings uchgreatfriends.Ialso thankRaghavArasforalltheexcitingdiscussionsandinspirational talks.Specialthanks tomydearest,oldestfriendsPiyushGrover,PraveenSharmaand RashiArora. Finally,Iwouldliketothankthealmightly,withoutwhomnoneofthiswou ldhave beenpossible,forblessingmewithsuchagreatopportunity. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 LISTOFSYMBOLS .................................... 10 ABSTRACT ........................................ 11 CHAPTER 1INTRODUCTION .................................. 13 2THEVESTIBULARDYNAMICINCLINOMETER(VDI) ............ 18 2.1ProblemFormulation .............................. 18 2.2SensorDesign .................................. 19 2.3DynamicEquilibriumAxis ........................... 22 2.4MathematicalManipulations .......................... 24 2.5SensorSimulation ................................ 25 2.5.1ControlStrategy ............................. 27 2.5.2TorqueControl ............................. 28 2.5.3AccelerationControlorPosturalThrottle ............... 30 2.6SensorExperiment ............................... 34 3ESTIMATINGJOINTPARAMETERSUSINGTHEVDI ............ 40 3.1ProblemDenition ............................... 40 3.2ParameterMeasurement ............................ 41 3.3BaseLinkparameters .............................. 43 3.4Inter-linkparameters .............................. 44 3.5Example ..................................... 45 4THEPLANARVESTIBULARDYNAMICINCLINOMETER(PVDI) ..... 52 4.1ProblemDenition ............................... 52 4.2IntermediateCoordinateSystemandRotationMatricies .......... 53 4.3SensorDesign .................................. 55 4.4DynamicEquilibriumAxisforplanarmotionofthebase .......... 56 4.5KinematicAnalysisandMathematicalManipulations ............ 58 4.6InclinationMeasurement-Closedformsolution ............... 59 4.7SensorSimulation ................................ 61 4.8SensorExperiment ............................... 61 5

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5ESTIMATINGJOINTPARAMETERSUSINGPVDI .............. 66 5.1ProblemDenition ............................... 66 5.2ParameterMeasurement ............................ 67 5.3BaseLinkparameters .............................. 69 5.4Inter-linkparameters .............................. 70 6CONCLUSIONANDFUTUREWORK ...................... 72 APPENDIX:AUTOCALIBRATIONOFMEMSACCELEROMETERS ....... 74 A.1ProblemStatement ............................... 74 A.2Autocalibration ................................. 75 A.3MisalignmentComputation .......................... 76 REFERENCES ....................................... 77 BIOGRAPHICALSKETCH ................................ 83 6

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LISTOFTABLES Table page 2-1Parametersofthesystemusedforsimulation ................... 33 7

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LISTOFFIGURES Figure page 2-1Modelofrobotwithlabeledcoordinatesystems .................. 19 2-2Schematicdrawingofhumanvestibularsystem .................. 19 2-3Vestibularorganconsistsofsemicircularcanals(ducts)andot olithorgans .... 22 2-4Detailviewofsensor ................................. 22 2-5Mechanicsofotolithorgans ............................. 23 2-6Inclinationangle vsobserved .......................... 26 2-7Angularacceleration vsobserved ........................ 26 2-8Netaccelerationexperiencedbybody,~ gvsobserved~ g .............. 27 2-9Angularvelocity_ vsobserved_ fromgyroscopeandangularacceleration .... 27 2-10Angularvelocity_ vsobserved_ fromgyroscopeanddirectlyfromaccelerometers 27 2-11Robotmodelwithmotioninknowndirectionascontrolparamet er ........ 31 2-12ControlTorqueStrategy-theDEAisalongvertical ................ 34 2-13Accelerationcontrolstrategy-theDEAisnotalongthevert ical ......... 35 2-14ExperimentalsetupoftheVDI ........................... 36 2-15Misalignmentoflinearaccelerometeraxes ..................... 36 2-16Plotof fromtheVDIvs fromtheencoderwhenbaseisstatic ........ 38 2-17Plotofangularvelocity_ fromtheVDIvs_ fromtheencoder .......... 38 2-18Plotofangularacceleration fromtheVDIvs fromtheencoder ........ 39 2-19Plotofabsoluteinclinationangle vsanglefromtheencoder .......... 39 3-1Linksi jjoinedatpointO i jhavingjointangle i j ................ 41 3-2LocationofVDIonalink .............................. 42 3-3TheVDIsensorfunction ............................... 43 3-4Baselinkbwithoneendincontactwithgroundsurface ............. 43 3-5Linksi jjoinedatpointO i jwithjointangle i jbetweenthem ......... 44 3-6Slidercrankmechanism ............................... 45 8

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3-7Plotofestimated r1vstrue r1 ............................ 46 3-8PlotofestimatedN!1vstrueN!1 .......................... 46 3-9PlotofestimatedN1vstrueN1 .......................... 47 3-10Plotofestimated 12vstrue 12 ........................... 47 3-11Plotofestimated_12vstrue_12 ........................... 48 3-12Plotofestimated12vstrue12 ........................... 48 3-13PlotofestimatedjointaccelerationN a O 12vstruejointacceleration ....... 49 3-14Plotofvariationofstandarddeviationofinclinationmeasured .......... 50 3-15Plotofvariationofstandarddeviationofangularvelocity ............ 50 3-16Plotofvariationofstandarddeviationofangularacceleration .......... 50 3-17Plotofvariationofstandarddeviationofresultantaccelerat ion ......... 51 4-1Modelofrobot .................................... 53 4-2DenitionofIntermediatecoordinatesystem .................... 53 4-3Sensordenition ................................... 57 4-4Simpliedproblemstatement ............................ 58 4-5Plotsofangles .................................. 62 4-6Plotsofangularvelocities_, ............................ 62 4-7Plotsofangularaccelerations, .......................... 62 4-8Plotofestimated~ gvstrue~ g ............................ 63 4-9ExperimentalsetupofthepVDI ........................... 64 4-10Plotof fromthepVDIvs fromtheencoder. .................. 65 5-1Linksi jjoinedatpointO i jhavingjointangles i j i jbetweenthem. ..... 67 5-2LocationofpVDIonalink ............................. 68 5-3ThepVDIsensorfunction .............................. 69 5-4Baselinkbwithoneendincontactwithgroundsurface ............. 69 5-5Linksi jjoinedatpointO i jwithEuler1-2jointangles i j i jbetweenthem 70 A-1Misalignmentofaccelerometers ........................... 75 9

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LISTOFSYMBOLS,NOMENCLATURE,ORABBREVIATIONS DEADynamicEquilibriumAxispVDIplanarVestibularDynamicInclinometerVDIVestibularDynamicInclinometerA a BAccelerationofpointBinreferenceframeA 10

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy VESTIBULARDYNAMICINCLINOMETER ANDMEASUREMENTOFINCLINATIONPARAMETERS By VisheshVikas December2011 Chair:CarlCraneMajor:MechanicalEngineering Ahumanbodydisplaysaremarkablequalityofmaintainingbothstatica nd dynamicequilibriumforarigidbodyinunstableequilibrium(modeledasanin verted pendulum).Biologically,thesensingofinclinationisdonebythehumanv estibular system.Here,anoveldesignofasensormotivatedbythehumanv estibularsystemis presented.Thesensor,calledtheVestibularDynamicInclinometer (VDI),measuresthe dynamicinclinationparameters-inclinationangle,angularvelocity,an gularacceleration andmagnitudeoftheaccelerationofsurfaceofcontact(e.g.,gra vity).TheVDIuses twodual-axislinearMEMSaccelerometersandonesingleaxisMEMSgyr oscopeto measuretheinclinationparametersforafourdegree-of-freedo mrobot.Theconceptof theDynamicEquilibriumAxis(DEA)isintroduced.TheDEAistheaxisalon gwhich therobotisatequilibrium.ThedirectionoftheDEAisparalleltothedir ectionofthe resultantaccelerationofthesurfaceofcontact.Twocontrols trategies-torquecontroland accelerationcontrolaresimulatedtoexplainthisconcept.TheDEA attemptstoexplain thereasonwhyhumansleanforwardwhileaccelerating(sprinting)a ndbackwardwhile decelerating(suddenstopping).TheVDIisusedtomeasurejointp arameters-jointangle, angularvelocitiesandangularaccelerations,forlinksjoinedbyrevo lutejoint.Here,the VDIisacontactlesssensorwitherexiblepointofapplication. TheVDIsensorisextendedforinclinationparametermeasurement ofvedegree-of-freedom robot.ThenewsensoriscalledtheplanarVestibularDynamicInclinom eter(pVDI).The 11

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pVDIconsistsoffourdual-axislinearMEMSaccelerometersandone tri-axialMEMS gryoscope.Theorientationofthelinearaccelerometersisdieren tfromtheintuitive analogousVDI.Thisisduetothecouplinginthekinematics.Similartoth eVDI,the pVDIisusedtomeasurejointparameterswithlinksjoinedbyunivers alhookejoint.The conceptoftheDEAisalsoextended. ThenoveltyoftheVDIandthepVDIliesinthefactthatthemeasure mentsare independentofdriftorintegrationerrors,accelerationofsurfa ceofcontact(e.g., gravitationalacceleration),independentofthedynamicsofther obot,andrequire signicantlylesscomputationalburden(closedformsolution).The inclinationangle obtainedfromtheVDIandthepVDIisfromtheDEA.Asthegoalofb alancingtasksis tobringtherobottoequilibrium,theinclinationparametersobtained fromtheVDIand thepVDIaresuitablecandidatesforcontrolapplications.Theapp licationoftheVDIand thepVDIisforeseeningaitanalysis,industrialrobotics,humanoidr obotics,etc. 12

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CHAPTER1 INTRODUCTION Thehumanbodyhasbeenasourceofmotivationformechanicaldes ign.It hasinspireddesignsofalargenumberofsensorse.g.,vision,stereo vision,haptics, etc.Mechanically,thehumanbodydisplaysaremarkablequalityofma intaining staticequilibriumforabodythatisinastateofunstableequilibrium(bip edstance). Biologically,thesensingofinclinationofthehumanbodyisdoneusingth evestibular system.Itprovidesfeedbacktomaintainthebodyinanequilibriumpo sitionatalltimes. Thehumanbodyisabletomaintainequilibriumevenwhengravitychange s(e.g.,the moon,etc)andwhenthesurfaceofcontactisaccelerating(e.g.,a nacceleratingbus). Interestingly,insuchcircumstancestheequilibriumpositionoftheb odychangese.g., leaningforwardwhilesprintingandleaningbackwardwhiletryingtosto p. Inertialsensorsareaimedatmeasuringacceleration.Theretwot ypesofinertial sensors-accelerometersandgyroscopes.Sincethewidespread availabilityofmicro electromechanical(MEMS)sensors,inertialsensorshavebeenw idelyusedinamubulatory systemstostudyhumanmovement,navigationandestimatejointp arameters.Applications includegaitanalysis[ 66 ],[ 77 ],[ 76 ],[ 46 ],[ 47 ],[ 48 ],[ 52 ],researchinmotorandcontrol stability[ 50 ],loadestimation[ 69 ]andmonitoringactivitiesofdailyliving(ADL)and levelofactivity[ 12 ],[ 71 ].Fortheseapplications,estimationoftheorientationisessential e.g.,forloadestimationusinginversedynamicsusingaccelerometers andgyroscopes,the orientationandangularvelocityofasegmentneedtobeknown. Intheabsenceofnon-gravityacceleration,atri-axialaccelerom etercanbeusedas aninclinometer[ 76 ],[ 37 ],[ 31 ],[ 8 ].Thesedays,awidelycommonuseisinsmartphones whichusethistoswitchbetweenlandscapeandportraitmode.Fort hisstaticcase,the algorithmmeasurestheanglebetweenthesensorunitwithrespect tothegravity.This willbelessaccurateforrelativelylargenon-gravityaccelerations. Furthermore,linear accelerationdoesnotgivecompleteinformationabouttheorientat ion. 13

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Gyroscopesareusedtomeasureangularvelocity.Strapdowninte grationalgorithms [ 28 ],[ 11 ],[ 29 ]calculatethechangeinorientationbyintegratingtheangularveloc ity. Thewordstrapdownindicatesthattheangularvelocityisobtainedf romthegyroscope strappedontoanobject.However,smallerrorsinangularaccele ration(gyroscopesignal) willgiverisetolargeintegrationerrors.Moreover,formeasuringa bsoluteorientationand notchangeinorientation,areferenceorientationhastobeset. Orientationestimationisalsodonebyfusionofaccelerometerandgy roscopedata. SuchsensorsarecalledInertialMeasurementUnits(IMUs).IMUs areusedineld robotics[ 5 ],assessmentofhumanbalancing[ 6 ],spacenavigation[ 33 ],etc.Theusual practiceistousethreesingle-axisaccelerometersandthreesingle -axisgyroscopesaligned orthogonally[ 33 ],[ 68 ].AKalmanlter[ 30 ],[ 60 ]withknowledgeof(error)dynamics ofthesystemisappliedtominimizetheseerrors,emphasizingthecor rectnessoflinear accelerationswhenangularaccelerationsarelow,andtrustingthe gyroscopedatamore whenthemotionismoredynamic[ 39 ],[ 38 ],[ 61 ],[ 3 ],[ 51 ],[ 1 ],[ 36 ].Alterforestimating theorientationofhumanbodysegmentshasbeenresearched[ 2 ].Herethelterused theaccelerometerandmagnetometerreadingstoobtainthelowfr equencycomponentof theorientationandusedgyroscopeformeasuringthefastercha ngesintheorientation. Thismaybeproblematicastheuseofamagnetometerinthevicinityof ferromagnetic materialswillgivelargeerrors.Anothersensorunit[ 22 ]containingadual-axisruid inclinometer,adual-axiselectroniccompassandtri-axialgyroscop e,withaKalmanlter thatincorporatedcontinuousgyroscopeosetestimatehasbee nresearched. Manydierentprinciplesforinclinationestimationarecommonlyused[ 10 ],[ 80 ], [ 35 ].Otherapproachesinvolvetrackingthemovementofthependulum usingconductive, resistiveorcapacitivemeasurementsystems[ 7 ],[ 56 ].Gyroscope-freedesignsusing onlyaccelerometershavebeenexploredtodetectinertialsensing -angularvelocity andlinearacceleration[ 62 ],[ 15 ].However,inalltheapproaches,theintegrationerrors accumulate.Robotsfromcommercialcompanies(e.g.,Sony,Honda )usegyroscopes 14

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and/oraccelerometersintheiractivebalancing,butdetailsarenot providedbythe creatorsandquantitativecomparisontohumansarenotavailable. Inhumans,thebalancingmechanismisbasedonvisualandvestibular feedbackto maintainthebodyinanunstableequilibriumbipedstance.Thehumanbo dyenjoys stabilityevenduringlackofvisualfeedbackandawellknowncauseo finstabilityis failureofvestibularsensors[ 13 ].Thisclearlyindicatestheimportanceofthevestibular systemforhumanstancestabilization.Biomechanicsofthevestibu larsystemhasbeen investigatedindetail[ 57 ].Thehumanvestibularsystem-sensoranalogyisalwaysdrawn assinglegyroscopeandsingleaccelerometerbasedorusinganaddit ionalsensorlike magnetometera[ 44 ],[ 65 ],[ 39 ],[ 38 ],[ 61 ],[ 70 ],[ 53 ],[ 79 ].Acompletelydierentanalogyis drawnformeasurementofinclinationparametersforabodymodele dasaplanarinverted pendulum.Here,twodual-axislinearaccelerometersandasingle-ax isgyroscopeareused assensorsforparametermeasurement.Thesymmetricdesignof theearsmotivatedthe placementoftheaccelerometers-symmetricallyplacedacrossave rticalline.Thisnovel sensoriscalledtheVestibularDynamicInclinometer(VDI).Theprop osedsensorsare MEMSaccelerometersandgyroscopes[ 9 ]. Humanoidrobotsrequiretomaintaindynamicbalanceduringawalk.Th eZero MomentPoint(ZMP)[ 73 ],aconceptusedforequilibriumanalysisandmotionplanning ofbipedlocomotionhasbeenextensivelyresearched.TheZMPplays andimportantrole forgaitanalysis,synthesis,andcontrol.However,thehumanoidr obotsrunatmaximum speedof7km/h(Toyota'shumanoid,BostonDynamics'Petman)or 6km/h(Honda's Asimo).TheconceptoftheDynamicEquilibriumAxis(DEA)relatedtot hedynamic equilibriumofrobotsisintroduced.TheDEAisforeseentobehelpful inthetheoretical considerationandpracticaldevelopmentoffastspeedrunningro bots. Studiesshowthatcontributionsfromvisual,vestibular,jointangle proprioceptive, andforcehelpinhumanstancecontrol[ 27 ].Visionimprovesstancestability,butin principle,canbedispensedwith[ 45 ].Jointtorqueandgroundforcesensorscanbeapplied 15

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forsensingforce-relatedinformation.Inclinationparameters-in clinationangle,angular velocity,andangularaccelerationarealsoneededforclosed-loopf eedbackcontrolof manipulators,humanoidrobots,etc.Theconventionaljointangle measurementsensors aremagneticrotaryencodersandopticalrotaryencoders[ 16 ].Magneticencodersare low-cost,contactlessandreliablebutrequirespecialmagnetcoup lingalignmentand magneticshielding.Opticalencodersareveryaccurateandconta ctless,butareexpensive andaresensitivetoenvironmentalinruences(shock,vibration,e tc.).Thesesensors requireinstallmentatthejointcenterwhichmaybeproblematic(orim possible)for someapplications,e.g.,humankneejoints.Othercontactlessjoint angularmeasurement sensorsusemicroelectromechanicalsystem(MEMS)acceleromet ersandgyroscopes[ 9 ]. Unliketheconventionalsensors,theydonotrequiretightcoupling torelativemechanical movements,andthusaremorerexibleatthepointofinstallation,mo rereliable,andlast longer.Thecommon-mode-rejection(CMR)method[ 34 ],[ 24 ],[ 25 ],[ 77 ]usestwodual-axis accelerometersmountedonadjacentlinks,ideallyattachedtothe jointcenter.This methoddisplayslargeerrorsforrapidrotation.Non-idealplaceme ntoftheaccelerometers alsoleadstoerrors.Todealwiththis,thefollowingthreemethods[ 16 ]-CMRwith gyro-integration(CMRGI),CMRwithgyro-dierentiation(CMRGD) anddistributed CMR(DCMR)wereintroduced.TheCMRGIandCMRGDuseonedual-ax islinear accelerometerandonesingle-axisgyroscopeperlinkforjointangle estimation[ 16 ]. TheCMRGI[ 77 ],[ 41 ]integratesthegyroscopesignaltoobtainachangeinorientation Thismethodfacesproblemsduetointegrationerror/driftdueton oise.TheCMRGD [ 21 ]dierentiatesthegyroscopesignaltoobtainangularacceleratio n.Itusesthe angularaccelerationandangularvelocitytoobtaintheacceleration ofthejoint,then calculatesthejointangle.Duetonoise,thedierentiationofangula rvelocitytoangular accelerationhasundesirableerrors.TheDCMRmethod[ 16 ],[ 75 ]usestwodual-axislinear accelerometers(notsymmetricallyplaced)perlinktoestimatethej ointangle.Here,the dierenceofthetwoaccelerationsignalsperlinkisusedtodetermine thejointangle,thus, 16

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avoidingtheerrorsfacedbyotherCMRmethods.Anovelwayofus ingtheVDIsensors, oneoneachlink,tomeasuretheinclination,angularvelocity,andang ularaccelerationis presented.Measuringtheinclinationparametersforthebaselink( linkincontactwiththe ground)isalsodiscussed. Thehumanbodycanbevisualizedasavedegrees-of-freedominve rtedpendulum. TheVDIisextendedtoinclinationmeasurementofvedegrees-offreedomrobots.The modiedsensordesignistermedastheplanarVestibularDynamicInc linometer(pVDI) anddiersfromtheVDIinsensororientation.TheconceptoftheD EAisalsoextended here.TheapplicationofthepVDItomeasureinclinationparameters betweentwolinks joinedbyauniversaljointispresented. TheinclinationangleobtainedfromtheVDIandthepVDIisrelativetot heDynamic EquilibriumAxis(DEA),theequilibriumaxis/position.Astheinclinationisr elative toanequilibriumpoint,itisforeseenasamoresuitablecontrolfeedb ackcandidatefor robotbalancingandstabilizing.Theecacyofthesensorscomesfr omthefactthat theyaresimple,low-cost,requirelowcomputation,andprovidethe dynamicinclination angledirectly,withoutrequiringtointegratetheangularvelocityan dapplyingaKalman lterforsensorfusion.Also,theangularcalculationsarevalidforla rgeanglesandare independentoftheaccelerationofsurfaceofcontact(gravity, etc.).Thismakesthesensor veryusefulforrobotbalancinginvaryinggravitationalenvironme nts(spaceapplications) andacceleratingplatforms(running,walkingmotionofrobots,etc .).Applicationsofthe VDIandpVDItomeasurelinkparametersinlinkagemechanismsmakes themusefulfor gaitanalysis,bipedrobots,etc. 17

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CHAPTER2 THEVESTIBULARDYNAMICINCLINOMETER(VDI) 2.1ProblemFormulation ArobotismodeledasaninvertedpendulumasshowninFigure 2-1 .Itisdesired tosensetheinclinationandangularvelocityoftherobotindependen tofaccelerationof surfaceofcontact(pointO).Therigidbodyismodeledasarodwithmassm,centerof massCalongtherodatadistancer Cfromthebaseoftherod,momentofinertiaI B Cat pointCandangulardampingcoecientK d.LetpointObethebaseoftherigidbodyin contactwithaplatform.LetNrepresenttheinertialreferenceframeandBrepresentthe referenceframexedontherigidbody.Letgbethegravitationalacceleration,abethe accelerationofpointOwithrespecttotheearthreferenceframeand~ gbetheresultantof theprevioustwomentionedaccelerations.Thus~ g = g + a(2{1) Acoordinatesystemxedintheinertialreferenceframewithorigin atO,Y-axisparallel totheground,Z-axisintotheplaneofthepaper,andxedinrefer enceframeNis denedwith f^x ^y ^zg astheorthonormalbasis.AnothercoordinatesystemwithoriginOxedinthebodyreferenceframeB,with f^e r ^e, ^e zg orthogonalbasisvectorsinthe radial,tangentialdirections,andintotheplaneofthepaperrespe ctively,isdened.The DynamicEquilibriumCoordinatesystemisdenedtobexedintheinert ialreference frameNwithoriginatpointO.Thevectors f^e x ^e y ^e zg formasetoforthonormalbasis vectorssuchthat^e xisparalleltovector~ gand^e zisintotheplaneofthepaper.Let be theanglebetweengand~ ginaclockwisedirection.Let betheanglebetween~ gandrigid body,asshowninFigure 2-1 .TheangularvelocitybetweencoordinatesystemNandBis denotedbyN!BwhichmaybewrittenasN!B = _^e z(2{2) 18

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OP C B er ^eq ^ r C l L R e x e y ^ ^ Gq f f N X Y a g g ~ Figure2-1.Modelofrobotwithlabeledcoordinatesystemsindiere ntreferenceframes. NotethatpointOmayalsoexperiencenon-gravityacceleration(a) Figure2-2.Schematicdrawingofhumanvestibularsystem PointPisatadistancelradiallyalongthebody,whereas,theaccelerometersR Lare locatedatadistanced=2oneithersideofpointPinatangentialdirectionofthebodyas showninFigure 2-4 .LetN a L=RdenotetheaccelerationsensedbytheL=Raccelerometer. 2.2SensorDesign Formeasurementofthespatialorientationofthebody,humansp ossessvestibular organs(Figure 2-2 ),theequivalentbiologicalinertialmeasurementunit.Thevestibula r organslieintheinnerear,areencapsulatedbybone,thus,areae ctedonlybyforceelds suchasgravitationalforceelds[ 44 ],[ 78 ].Theyconsistoftwomainreceptorsystems[ 42 ] 19

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forinertialsensing-semicircularcanals(ducts)andotolithorgans .Thereceptorcellsof theotolithsandsemicircularcanalssendsignalsthroughthevestibu larnerveberstothe neuralstructuresthatcontroleyemovements,posture,and balance. Thesemicircularcanalsaretheprimarysystemsresponsibleforsen singangular accelerationoftheheadandtransmittingtheinformationtothebr ainstem.Eachcanal iscomprisedofacirculartubecontainingruidcontinuity,interrupte dattheampulla (thatcontainsthesensoryepithelium)byawatertight,elasticmem branecalledthe cupula.Thethreesemicircularcanals,arrangedinorthogonalplan es,takeadvantageof endolymphruiddynamicstosenseangularmotion.Angularmotionsen sationrelieson inertialforces,causedbyheadaccelerations,togenerateendo lymphruidrowwithinthe toroidalsemicircularcanals[ 14 ].Thesiteofmechanotransductionwithinthesemicircular canalsislocalizedtothecristaampullaris,whichisacrestlikeridgeinthe ampullarywall thatprotrudesintothelumenoftheampulla.Asensoryepitheliumre sidesonthesurface ofthecrista,whichisencasedbythecupula(Figure 2-3A ).Duringrotation(butnot translation),ruidinertiaexertspressureonneuralreceptorsa ndleadstoaneuralsignal. Duetomechanicalfactors(suchasruidadhesion),thecanalscod eangularacceleration duringlowfrequencyrotationandangularvelocityinthemid-tohighfrequencyrange [ 44 ]asillustratedinFigure 2-3A .Theangularvelocitysignalfromthesemicircularcanals containnoise,that,aftervelocity-positionintegration,haveruc tuatingdrifts[ 43 ],[ 42 ]. Researchhasbeendonetobuildmacromechanicalmodelofthesem icircularcanals[ 20 ], [ 19 ],[ 58 ].Here,consideringthereliabilityofmicroelectromechanicalsensor smotivates ustoassumethesemicircularcanalstobeanalogoustoaMEMSgyros copesensor. TheMEMSgyroscopesignalalsocontainsnoise,but,isnotintegrat edtoobtainthe inclination. Theotolithorganscompriseofutricleandsacculeoftheinnerear.T heotolithorgans areratlayeredstructuresthatliebetweentheendolymphaticruid andthemembranous labyrinthsubstrate(Figure 2-3B ).Thebiomechanicalresponseoftheotolithorgansis 20

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criticaltotheabilitytosensethedirectionofgravito-inertialacce leration.Theutricleis sensitivetoachangeinhorizontallinearaccelerationandthesaccu leissensitivetothe verticallinearacceleration[ 57 ].Here,adual-axisaccelerometer(MEMS)isassumedtobe analogoustotheotolithorgans. Eachvestibularorganis,thus,assumedtobeanalogoustoadual-a xisaccelerometer andasingle-axisgyroscope.Humanearsareplacessymmetricallyab outtheaxisof symmetry(alongwhichthenoselies),thus,providingmotivationtod esignasensor thathasvestibular-analogousaccelerometer-gyroscopesymme tricallyplacedabouta symmetricalline.Thegyroscopereadingsforboththegyroscope swillbetheoretically thesame(astheyareattachedtothesamerigidbody).However, theaccelerometer readingsforbothaccelerometerswillbedierent.ThedesigninFigu re 2-4 isproposed whichhastwodual-axisaccelerometers(L R)symmetricallyplacedacrossavertical lineandonesingle-axisgyroscope(G).ThissensoriscalledtheVestibularDynamic Inclinometer(VDI).Throughouttheliterature,thehumanvestib ularsystem-sensor analogyisdrawnasasinglegyroscopeandsingleaccelerometer.Ast helinearaccelerations ineachofthevestibularorgansaredierent,suchanalogyleadsto lossofcritical information. Thevestibularsystemiscapableofsensingtheangularacceleration fromthe dierenceofthelinearaccelerationreadingsoftheleftandrightve rticalotoliths (Figure 2-5A ).Thedierencebetweenleftandrighthorizontalotolithscanalso sense themagnitudeoftheangularvelocitybutisnotabletodeterminethe direction(Figure 2-5B ).Furtheranalysiswillconrmthatthedierencebetweenhorizon tal(orradial) otoliths(accelerometerreadings)yieldangularacceleration.Also, thedierencebetween vertical(ortangential)otoliths(accelerometerreadings)yieldsm agnitude,andnot directionofangularvelocity. 21

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(A)Angularacceleration causesviscousrowofendolymphthroughthesemicircularcanalandintotheampulla,derectingthecupulatoregisteraccumulatedangularvelocity (B)Dampedbyruid endolymph,otolithorgansderectfromequilibriumpositiontoregister linear acceleration Figure2-3.Vestibularorganconsistsofsemicircularcanals(ducts )andotolithorgans P B e r e q ^ ^ G d/2 L R d/2 Figure2-4.Detailviewofsensor-accelerometersR LandgyroscopeG 2.3DynamicEquilibriumAxis AtanypointMonthebody,performingakinematicanalysis[ 59 ]toobtain acceleration(N a M)yieldsN a M = N a O + NBr O!M + N!B N!Br O!M (2{3) 22

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(A)Thedierencebetweencorrespondingleft andrightsacculeregisters angular acceleration (B)Thedierencebetweencorrespondingleft andrightutriclecanregisterthemagnitudeofangularvelocity butnot thedirection. Figure2-5.Mechanicsofotolithorgans whereMmaybe fL R Cg andNBistheangularvelocitybetweencoordinatesystemNandB.Itisknownthatr O!C = r C ^e r r O!L=R = l ^e rd 2 ^eN a O = ~g = ~ g ^e x =~ g cos^e r + ~ g sin^e (2{4) whereN a Odenotestheaccelerationsensedbytheaccelerometerwhenitispla cedatpointO.Therefore,N a C = ~g + r C^e _2 ^e r (2{5) N a L = l _2d 2 ~ g cos ^e r +l + d 2 _2~ g sin ^e (2{6) N a R = l _2 + d 2 ~ g cos ^e r +l d 2 _2~ g sin ^e. (2{7) LetthereactionforcesatpointObeF R.ApplicationofEuler'srstandsecondlaw[ 59 ] aboutthecenterofmassCgivesm~g + r C ( ^e _2 ^e r )= F R .(2{8)I B CNC =K d _^e z + (r C ^e r )F R .(2{9) 23

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Thus, I B C + mr 2 C=K d _+ m ~ gr C sin (2{10) FromEquation 2{10 ,theequilibriumposition( = 0, = 0)ofthesystemis = 0. Theaimoftheproblemistobringthebodytoitsequilibriumpositionwhich isno moretheabsoluteverticalposition(i.e.,directionparalleltothegra vityvectorg).The newequilibriumpositionisdenedastheDynamicEquilibriumAxis(DAE)wh ichis paralleltotheresultingaccelerationonthebody~ g,inclinedatangle tog.TheDynamic EquilibriumAxisisdependentontheresultantaccelerationactingont hebody,andthus istime-varying(moreprecisely,accelerationvarying).Whenthebo dyisaccelerating, is positive,whenitsde-accelerating, isnegative.Thisfactcanbeobservedwhenhumans leanforward(changeequilibriumaxis)whentryingtoaccelerate(sp rint)andbend backwardswhileattemptingtode-accelerate.Boththecasesdisp layhowtheequilibrium axis(DEA)changeswhenaccelerationisexperiencedbythebody. 2.4MathematicalManipulations InEquations 2{6 and 2{7 thekinematicanalysisofthebodyisperformed.Itis importanttoobservethatallthecalculationsareindependentoft hedynamicsofthe body,i.e.,controltorque,externalforce,etc.Fourreadingsa reobtainedfromthetwo accelerometers(radialandtangentialcomponentsinEquations 2{6 2{7 ).Calculatingthe dierenceandmeanofthereadingsandcallingthem 1 ,2yields 1 = F a LF a R (2{11) 2 = 1 2F a L + F a R (2{12) 1 = d ^e r + d _2 ^e (2{13) 2 = l _2~ g cos ^e r +l ~ g sin ^e (2{14) 24

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ir ,i representtheradial(^e r)andtangential(^e )componentsof i,where,i = 1, 2.Thus, theangularacceleration,velocityandinclinationcanbeobtainedas=1 r d (2{15) sin= l d1 r 2 g (2{16) cos= l d1+2 r g (2{17) tan= l d1 r 2 l d1+2 r (2{18) ~ g = l d1 r 2 sin= l d1+2 r cos. (2{19) Itisimportanttoobservethattan (Equation 2{18 )isindependentoftheresultant acceleration,~ g,ofsurfaceofcontact.Inclination canbeuniquelydeterminedas~ g>0. Itisalsopossibletodeterminetheresultingaccelerationmagnitude~ gfromEquation 2{19 Obtainingtheangularvelocity(_ )isalittletricky,asthesensormathematicsisableto obtainthemagnitudeoftheangularacceleration,butnotthedirec tion.Angularvelocity canbeobtainedbyintegrationofangularacceleration(Equation 2{20 ),directlyfrom accelerometers(Equation 2{21 )orfromthegyroscopereadings(Equation 2{2 )._I =Zdt (2{20) _otolith = sign Zdt s 1 d (2{21) CalculationofangularvelocityfromEquation 2{20 ispronetoaccumulationoferror (drift)whenthebodyisinstateofequilibriumandEquation 2{21 willresultin magnicationofcontributionofnoiseasasquarerootisinvolved.Fo rallthesereasons, thereadingfromthegyroscopeisusedtocalculatetheangularvelo city. 2.5SensorSimulation Simultaneousanalysisofsensorsimulationandexperimentisperform ed.Similar resultsfrombothanalysiswillconrmtheaccuracyofthesimulation modelandallow 25

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0 0.5 1 1.5 2 2.5 3 3.5 4 80 60 40 20 0 20 40 60 timedegreesq !vs!observed! q q observed! q Figure2-6.Inclinationangle vsobserved 0 0.5 1 1.5 2 2.5 3 3.5 4 80 60 40 20 0 20 40 60 80 timedegrees/sec 2Angular!Acceleration!vs!Observed!Angular!Acceleration Angular!Acceleration Observed!Angular!Acceleration Figure2-7.Angularacceleration vsobserved furtheranalysisusingthesimulationmodel.Simulationanalysisofthea bovestated approachisperformedinMATLAB R r .Theparametersforsimulationsaretakentobe analogoustothehumanbodywithl = 2 m d = 0.25 m.Thesensornoisewasmodeled asstationarywhitenoisewhichisconstantthroughoutthefreque ncyspectrumbasedon thespecicationdatasheetofaccelerometerADXL213(noiseden sityof160g= p Hzrms) andgyroscopeADXRS450(noisedensityof0.015 o=sec= p Hz)whicharemanufactured bythecompanyAnalogDevices.Forfuturesimulations,sensornois eismodeledina similarfashion.Figure 2-6 showsgoodestimationofinclinationangle purelyfrom kinematiccalculationsandwithouttheuseofsystemdynamics.Thea ngularacceleration andmagnitudeofnetaccelerationactingonthebodyarealsowelles timatedasshown inFigure 2-7 and 2-8 .Figures 2-10 2-9 provewhycalculationoftheangularvelocity fromEquations 2{20 2{21 iserroneous.Figure 2-10 comparestheangularvelocity estimatedusingEquation 2{21 tothatfromgyroscope.Figure 2-9 showstheconceptof erroraccumulation(drift)duetointegrationwhentheangularvelo cityiscalculatedusing Equation 2{20 26

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0 0.5 1 1.5 2 2.5 3 3.5 4 8 10 12 14 16 18 20 22 24 26 28 timem/sec 2Resultant acceleration on body Net Acceleration on body Observed Net Acceleration Figure2-8.Netaccelerationexperiencedbybody,~ gvsobserved~ g 0 0.5 1 1.5 2 2.5 3 3.5 4 1 0 1 timedegrees/secAngular!Velocity!from!Gyroscope Angular!Velocity Observed!Angular!Velocity 0 0.5 1 1.5 2 2.5 3 3.5 4 1 0 1 timedegrees/secAngular!Velocity!from!Integration Angular!Velocity Observed!Angular!Velocity Figure2-9.Angularvelocity_ vsobserved_ fromgyroscopeandobserved_ from Equation 2{20 .Accumulationoferror(ordrift)canbeviewed 2.5.1ControlStrategy ToanalyzetheconceptoftheDEA,itisproposedtodesignacontro lstrategyto bringthebodytoequilibrium.Thecontrolstrategiescanbetorque application(analogous totorqueonhumanbodyforequilibriumviahip,ankle)oracceleration /de-accelerationof thebody(analogoustotheleaningofthehumanbodywhileaccelerat ingorde-accelerating). Theformerisreferredtoastorquecontrolandthelatterasacc elerationcontrolor posturalthrottling. 0 0.5 1 1.5 2 2.5 3 3.5 4 100 0 100 timedegrees/secAngular!Velocity!from!Gyroscope Angular!Velocity Observed!Angular!Velocity 0 0.5 1 1.5 2 2.5 3 3.5 4 100 0 100 timedegrees/secAngular!Velocity!from!Accelerometers Angular!Velocity Observed!Angular!Velocity Figure2-10.Angularvelocity_ vsobserved_ fromgyroscopeandobserved_ from Equation 2{21 .Itcanbeobservedthattheerrorismagnied 27

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Itisdesiredtodesignacontrollerforvaryingweight,unknownmome ntofinertia, anddampingcoecient.Forthesereasons,Lyaponovbasednonlinearcontrollersare proposed.Thegoalistobringtherigidbodybacktoequilibrium.Toac hievethecontrol purpose,atrackingerrore2 R isdenedase, d = (2{22) wherethedesiredangleofinclination d2 R iszeroforalltime.Forstabilityanalysisof thesystem,lteredtrackingerrorr2 R isdenedasr,_ e +e =_ (2{23) where 2 R isapositiverealconstant. 2.5.2TorqueControl Inthiscase,thetorqueisappliedonthebodyandtheequationsofm otionchange fromEquation 2{10 toI B O + K d _+ m ~ gr C sin= T C(2{24) whereI B O =I B C + mr 2 C andT Cisthecontroltorque.Y2 R13isdenedasY, _, ~ g sin, .(2{25) ThecontroltorqueisdesignedasT C = Y ^ kr(2{26) wherek2 R isapositiveconstantandtheupdatelawfor^ 2 R31is_ ^ = Y T r(2{27) where2 R33positivedeniteadaptationgainmatrixand kareconstrainedas follows k>1 2 .(2{28) 28

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Theorem2.1. ThecontrollergiveninEquations 2{26 and 2{27 withgainconditionsgiven inEquation 2{28 ensuresthatthetrackingerrorisregulatedasfollowslim t!1 jje ( t )jj= 0, lim t!1 jj_ e ( t )jj= 0thus,assuringglobalasymptoticstability. Proof. Dierentiatingthelteredtrackingerroryields_ r = e +_ e = _.(2{29) MultiplyingEquation 2{29 byI B OandsimplifyingusingEquation 2{24 I B O r = K d _+ m ~ gr sin I B O_ T C (2{30) I B O r = Y T C (2{31) where2 R31isunknown,yetdeterministicanddenedas, K d m I B OT .(2{32)~ 2 R31isdenedas~ = ^ .(2{33) CombiningEquation 2{26 2{31 2{33 givesI B O r = Y ~ kr .(2{34) LetVdenoteacontinuouslydierentiablepositivedeniteradiallyunbound edLyapunov functioncandidatedenedasV,1 2 I B O r 2 + 1 2 e 2 + 1 2 ~ T 1 ~ .(2{35) Dierentiating 2{35 andsimplifyingyields_ V =kr 2 e 2 + er + Y ~ r + ~ T 1 ~ (2{36) 29

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usingtheArithmeticMean-GeometricMeaninequalityerr 2 + e 2 2 .(2{37)isunknownanddeterministic.Thus,_ ~ =_ ^ .SimplifyingEquation 2{36 using 2{37 and 2{27 gives_ V k1 2r 2 1 2e 2 .(2{38) UsingtheconstraintsgiveninEquation 2{28 ,theexpressioninEquation 2{38 isupper boundedbyacontinuous,negativesemi-denitefunction.Byusing Barbalat'sLemma[ 32 ]lim t!1 jje ( t )jj= 0, lim t!1 jj_ e ( t )jj= 08 2 R.(2{39) 2.5.3AccelerationControlorPosturalThrottle Inthiscase,thecontrolandmotionlooksasshowninFigure 2-11 .Thecontrol ofthebodyisaforceinaspecicdirection(^ y),andthusitissafetoassumethatthe accelerationintheotherdirection(^ x)isknown/calibrated(usuallygravitational).Thus,~ g = g ^ x + a h ^ y .(2{40) Let = (+),betheinclinationofthebodyrelativetotheabsolutegravityg.Now, theequationsofmotionchangetoI B O + mr C cosa h + K d mr C sing = 0 (2{41) mr C cos+ ma hmr C sin_2F C = 0 (2{42) = tan1a h g= cos1g ~ g (2{43) whereF Cisthecontrolforceand~ g =p g 2 + a 2 h.DeningY2 R14asY, _, _,gs,_+ rc s s_ .(2{44) 30

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O C B N X Y a h g F D F C Figure2-11.Motioninknowndirectionascontrolparameter.Here =+ ,inclination ofthebodyfromtheabsolutegravityg. ThecontrolforceisdesignedasF C =1 cos Y ^ + kr (2{45) wherek2 R isapositiveconstantandtheupdatelawfor^ 2 R41is_ ^ = Y T r(2{46) where2 R44isapositivedeniteadaptationgainmatrixand kareconstrainedas follows k>1 2 .(2{47) Itisworthwhiletomentionthatthesystembecomesuncontrollablew hen ==2,i.e., whentherigidbodyisparalleltothedirectionofacceleration.There fore,cos>0forall controllablecases. Theorem2.2. ThecontrollergiveninEquations 2{45 and 2{46 withgainconditionsgiven inEquation 2{47 ensuresthatthetrackingerrorisregulatedasfollowslim t!1 jje ( t )jj= 0, lim t!1 jj_ e ( t )jj= 0thus,assuringglobalasymptoticstability. 31

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Proof. Theproofproceedsonsimilarlinesastheproofforthepreviousthe orem.Solving Equations 2{41 2{42 for anda hgivesA =K d mr 2 c sc_2 + mgr c s r C cF C(2{48)Aa h = I B O m F C + I B O r C s_2 + K d r C c_ mr 2 C scg(2{49) whereA =I B C + mr 2 C s 2 .ItcanbeobservedthatA>08 .Calculatingthefollowing expressionusingtheexpressionforthelteredtrackingerroras denedinEquation 2{23 givesA r + Ar r C = K d r C I B C r C_ mgs+ cF C + mr C_+ rc s s_A r + Ar r C = Y + cF C(2{50) where2 R41isunknown,yetdeterministicandisdenedas =K d r C I B C r C m mr CT .(2{51)~ 2 R41isdenedas~ = ^ .(2{52) CombiningEquations 2{45 2{50 2{52 givesA r + Ar r C = Y ~ kr .(2{53) LetVdenoteacontinuouslydierentiablepositivedeniteradiallyunbound edLyapunov functioncandidatedenedasV,1 2 r C Ar 2 + 1 2 e 2 + 1 2 ~ T 1 ~ .(2{54) Dierentiating 2{54 ,usingEquation 2{53 andsimplifyingyields_ V =kr 2 e 2 + er + Y ~ r + ~ T 1 ~ .(2{55) 32

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Asisunknownanddeterministic,_ ~ =_ ^ .SimplifyingEquation 2{55 using 2{37 and 2{46 gives_ V k1 2r 2 1 2e 2 .(2{56) UsingtheconstraintsgiveninEquation 2{47 ,theexpressioninEquation 2{56 isupper boundedabycontinuous,negativesemi-denitefunction.Byusing Barbalat'sLemma[ 32 ]lim t!1 jje ( t )jj= 0, lim t!1 jj_ e ( t )jj= 08 2 R.(2{57) SimulationswereperformedinMATLAB R r .Thesensornoisewasmodeledas explainedinSection 2.5 .Thefrequencyofoperationofthesensorsisassumedtobe10 Hz.Valuesofotherparametersaremotivatedbymodelingofthehuma nbodyas aninvertedpendulum.ThevaluesforsimulationaredisplayedinTable 2-1 .Forthe Parameter Value Parameter Value m 85 kg l 2 m d 0.25 m L 1.8 m R 1.25 m I O mL 2=3 K d 2N m s rad1 g 9.81 m=s 2 Table2-1.Parametersofthesystemusedforsimulation simulationofthetorquecontrolstrategy,thegainsareassumed tobe = 4, k = 2, = 10 3 I 33.Thelearningofparametersisoneonline.Angularresponseandcon trol torquecomparisonsfordierentinitialinclinationsareshowninFigur e 2-12A and 2-12B respectively.ItisimportanttoobserveinFigure 2-12A thattherigidbodybalancesto 0.5 oinarelativelyquicktimeandremainsinthatvicinitythereafter.Also,t hetorque requiredtomaintaintherigidbodyatthatinclinationisrelativelysmallw henthefoot ismodeledasapoint.Inreallife,thefootisnotapointofcontact,r atherasurfaceof contact,thusassuringbetterbalance.Thepeaktorquegenera tedisalittlehighbutthe resultsareveryencouraging. 33

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0 1 2 3 4 5 6 7 8 9 10 1 0 1 2 3 4 5 6 7 8 Time!(sec)Angle!(degrees)Angle!Response!over!time q (0)!=!7 o q (0)!=!8 o q (0)!=!9 o q (0)!=!10 o (A)AngularResponsefordierent initialinclinations,zeroinitialvelocity(_(0) = 0) 0 1 2 3 4 5 6 7 8 9 10 600 400 200 0 200 400 600 800 Time!(sec)Torque!(Nm)Control!Torque!over!time q (0)!=!7 o q (0)!=!8 o q (0)!=!9 o q (0)!=!10 o (B)ControlTorquefordierentinitial inclinations,zeroinitialvelocity(_(0) = 0) Figure2-12.ControlTorqueStrategy-theDEAisalongvertical Forthesimulationoftheaccelerationcontrolstrategy,thegains areassumedas = 1, k = 4, = 25 I 44.Angularresponse( ,not ,as istheabsolutepositionofthe body),controlforceandhorizontalaccelerationcomparisonsf ordierentinitialinclination areshowninFigures 2-13A 2-13B and 2-13C respectively.Thehorizontalacceleration increasesaccordinglytobalancetherobotatthedesiredDEA.The peakcontrolforce seemshigherthannormal(gaitforcesfora85kghuman),but,th etransientresponse looksgoodinarelativelylargeneighborhood.Forthetorquecontro lstrategy,theDEA isalignedwiththevertical(along g ).However,fortheaccelerationcontrolstrategy,the DEAisnotalignedwiththeverticalastheaccelerationactingontheb odyisnomorethe gravitationalacceleration.Thissimulationhelpstoexplaintheleaning forwardofathletes whentheysprint/accelerate. 2.6SensorExperiment Theexperimentwassetwithtwodual-axislinearMEMSaccelerometer s(ADXL320) symmetricallyplacedaboutthecenter-lineoftheinvertedpendulum asshownin Figure 2-14 .Thesingle-axisMEMSgyroscope(ADRS613)wasalsostrappedont othe invertedpendulumasshowninFigure 2-14 .AlinearMEMSaccelerometer(ADXL320) wasxedtothebaseoftheinvertedpendulumtosensetheacceler ationofthebase (Figure 2-14 ).InterfacingoftheanalogvoltagesignalwasdoneusingNI-DAQm xand 34

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0 2 4 6 8 10 12 14 16 18 20 26 28 30 32 34 36 38 40 42 44 Time (sec)b (degrees)b Response over time q (0) = 15 o q (0) = 20 o q (0) = 25 o q (0) = 30 o (b=q+f) Absolute angle (A)AngularResponse( )fordierent initialinclinations,zeroinitialvelocity(_(0) = 0) 0 2 4 6 8 10 12 14 16 18 20 0 200 400 600 800 1000 1200 1400 1600 1800 Time (sec)Force (N)Control Force over time q (0) = 15 o q (0) = 20 o q (0) = 25 o q (0) = 30 o (B)ControlForcefordierentinitial inclinations,zeroinitialvelocity(_(0) = 0) 0 2 4 6 8 10 12 14 16 18 20 5 0 5 10 15 20 25 30 Time!(sec)a h !(m/sec 2 )Horizontal!Acceleration!over!time q (0)!=!15 o q (0)!=!20 o q (0)!=!25 o q (0)!=!30 o (C)HorizontalAcceleration(a h) responsefordierentinitialinclinations,zeroinitialvelocity(_(0) = 0) Figure2-13.Accelerationcontrolstrategy-theDEAisnotalongt hevertical LabVIEW R r .Magneticencoders(USDigitalMA3-A10-125-N)werexedatthe joints. Themeasurementoftheencoderanglewasassumedtobethegrou ndtruth. Symmetricplacementofthelinearaccelerometerswaseasilyachieve d.However, thealignmentoftheaccelerometerstothe`theoretical'radialand tangentialdirections inthebodycoordinatesystem(^e r ^e )wasachallenge.Tocompensateforthecoupling duetomis-alignment,assuminglinearityoftheaccelerometer,thev oltage-acceleration relationshipforaparticularaccelerometercanbemodeledasfollows 264N a r N a 375=264x 1 x 2 x 3 x 4375 264v r v 375+264x 5 x 6375 (2{58) 35

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MEMS Linear Accelerometers MEMS Gyroscope 25 cm Magnetic Encoder MEMS Accelerometer (Base Acceleration) 56 cm Figure2-14.ExperimentalsetupoftheVestibularDynamicInclinom eter(VDI).Thetwo symmetricallyplacedlinearMEMSaccelerometersaremarkedinboxes .The MEMSgyroscopeismarkedbycircle.Themagneticencoderandanaccelerometerxedtothebaseareindicatedbyarrows. P B N a Rr N a Lr N a L q N a R q v L q v R q v Rr v Lr e q e r ^ ^ G d/2 L Rd/2 Figure2-15.Themisalignmentoflinearaccelerometeraxesisshown. Itisdesiredto obtaintheaccelerationsinthe^e r ^e directions-N a Lr N a Rr N a L, N a R .The linearaccelerometersvoltagesignalsreadlinearaccelerationsinthemisalignedaxes(dottedlines)-v Lr v Rr v L, v R 36

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whereN a r N a representtheradial,tangentialcomponentsoftheacceleration ofthe linearaccelerometerinthebodycoordinatesystem(^e r ^e )andv r v representvoltage readingsofthelinearaccelerometercorrespondingtoacceleratio ncomponentsinthe misalignedaxesasshowninFigure 2-15 .Itisdesiredtocalibratefor(ornd)X 61 = [ x 1 x 2 x 3 x 4 x 5 x 6 ] T.RewritingEquation 2{58 264v r v0 0 1 0 0 0 v r v0 1375X =264N a r N a 375 (2{59) Letv r i v, i N a r i N a, ibethesetofvoltageandaccelerationsignalsforsample/readingi. Formsamples/readingsV 2 m6 =2666666666666664v r ,1 v,1 0 0 1 0... ... ... ... ... ...v r m v, m 0 0 1 0 0 0 v r ,1 v,1 0 1... ... ... ... ... ...0 0 v r m v, m 0 13777777777777775A 2 m1 =2666666666666664N a r ,1...N a r m N a,1...N a, m3777777777777775 (2{60) ItisdesiredtosolveforXfromthefollowinglinearequationV X = A(2{61) TheleastsquaressolutiontotheequationcanbeobtainedusingNor malEquations, QRFactorization,orSingularValueDecomposition(SVD)[ 67 ].Forthisexperiment,8 samples/readings(m = 8)weretakenforcalibrationofeachlinearaccelerometer.Amore generalapproachtocalibratelinearMEMSaccelerometersisgivenin Appendix 6 Therstexperimentwasperformedkeepingthebasexed,i.e.,N a O = g ^ E 1(Figure 2-1 ).Thereadingsfromtheencoderareassumedtobethegroundtr uthfor theinclinationangle.Theplotsofcomparisonofinclinationangle,angu larvelocityand angularaccelerationareshowninFigures 2-16 2-17 2-18 .Thenextexperimentwas 37

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0 200 400 600 800 1000 1200 1400 80 60 40 20 0 20 40 60 q degTime!(1!unit!=!0.02!sec) Encoder VDI Figure2-16.Inclinationanglemeasurementwhenthebaseisstatic( noaccelerationalong thesurfaceofcontact).Plotof fromtheVDIvs fromtheencoder. 0 200 400 600 800 1000 1200 1400 150 100 50 0 50 100 150 q deg/secTime!(1!unit!=!0.02!sec) Encoder VDI Figure2-17.Plotofangularvelocity_ fromtheVDIvs_ fromtheencoder.The gyroscopeusedfortheexperimenthadlimitofsensingangularveloc ity between 75 deg=secasgivenonthespecicationsheet. performedbyacceleratingthebaseasshowninFigure 2-11 .Asmentionedintheearlier section,thiseortisdonetoanalyzetheconceptoftheDEA.Thea ngle oftheDEA iscalculatedbyassuminggravitationalaccelerationas9.81 m=sec 2andacceleration obtainedfromthebaseaccelerometer(Figure 2-14 )usingEquation 2{43 .Themeasureof absoluteinclinationangle forthecasewhenthebaseisacceleratingiscomparedwith theencoderangleinFigure 2-19 .Theexperimentalresultsconrmthesimulationresults andareveryencouraging.TheconceptofDEAisalsoexperimentally observed. 38

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0 200 400 600 800 1000 1200 1400 400 200 0 200 400 600 q .. deg/sec 2Time!(1!unit!=!0.02!sec) Encoder VDI Figure2-18.Plotofangularacceleration fromtheVDIvs fromtheencoder. 0 200 400 600 800 1000 -80 -60 -40 -20 0 20 40 b = q + f degTime (1 unit = 0.02 sec) Encoder VDI Figure2-19.Plotofabsoluteinclinationangle fromtheVDI-baseaccelerometer combinationvsanglefromtheencoder. 39

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CHAPTER3 ESTIMATINGJOINTPARAMETERSUSINGTHEVDI Inclinationparameters-inclinationangle,angularvelocityandangula racceleration areneededforclosed-loopfeedbackcontrolofmanipulators,hu manoidrobots.The bodymotioncharacteristicsneedtobeevaluatedduringtherehab ilitationprocessof disabledpeople[ 64 ],[ 54 ],[ 81 ].Thecurrentmethodformotionsensingistousecamera basedmotioncapturesystems[ 49 ],[ 17 ],[ 55 ].Thistechniqueisobtrusiveandexpensive. Itisalsodiculttobeintegratedintoamodernmedicalsystemssuch asaportable medicaldeviceorpoint-of-care(POC)medicationsystem[ 81 ].Recentadvancesin micro-electro-mechanical-systems(MEMS)withhighaccuracy,hig hreliability,and multiplefunctionalitieshaveprovidedapowerfultoolsetforbodym otionsensing[ 4 ],[ 63 ], [ 74 ],[ 40 ],[ 26 ],[ 79 ].AnovelwayofusingtheVDIsensors,oneoneachlink,tomeasure theinclinationparametersispresented.TheVDIsensoriscontact -lessandrexible,thus, portableandnon-obtrusive.Measuringtheinclinationparameters forthebaselink(link incontactwiththeground)isalsodiscussed.Thedesiredapplication forthesensor aregaitanalysis,fallevaluation,sportsmedicine,balanceprosthe sis,remotepatience surveillance,etc. 3.1ProblemDenition Amechanismcomprisedofaseriesoflinkagesjoinedbyrevolutejoint sisgiven.The linkagesaremodeledasrigidbodieswithknownlinklengthsandtheVest ibularDynamic Inclinometer(VDI)sensorislocatedatsomeconvenientdistancea longthelinejoiningthe twolinkjointsorthepointofcontactandajointincaseofthebaselin k.Thebaselink isdenedasalinkthathasajointatoneendofthelinkandtheothere ndisincontact withthebase(orground,whichisassumedtobestationary)surfa ceasshowninFigure 3-4 .LetO i jrepresentthejointjoininglinkitolinkj.ThelinejoiningjointsO h i(O bforbaselink)andO i j(O b hforthebaselink)isreferredtoasthelinkvectorofthelink. LettheVDIsensorbelocatedatadistancer ialongthelinkvectoroflinkiasshownin 40

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VDI i l i r i O h,i Lieirei q O i,j P i VDIj Ljejrej q rj lj q i,j O j,k Figure3-1.Linksi jjoinedatpointO i jhavingjointangle i j Figure 3-1 .Thejointangle i jbetweenlinksiandjisdenedastheangleofrotation betweentherespectivelinkcoordinatesystems.Thelinklengthoflin kiisgivenbyl i. LetNrepresenttheinertialreferenceframeandL irepresentthereferenceframexed onthelinki.ThelinkicoordinatesystemwithoriginatpointO p iisxedinthelinkireferenceframeL i,withe i =fe ir e i, e izg orthogonalbasisvectorssuchthate irisalong thelinkvectoroflinkiande izisintotheplaneofpaperasshowninFigure 3-1 .The jointangle i jistheanglebetweene irande jr.Forthecaseofthebaselink,theoriginof thecoordinatesystemliesatthepointofcontactO bandtheanglebetweenthevertical (directionofgravitationalacceleration)andthelinkisreferredto asthebaseangle( rb)as showninFigure 3-4 Itisdesiredtoobtainthebaseangle( rb),jointangles( i js),angularvelocities(N!is) andtheangularaccelerations(Nis)ofthelinkswithgivenlinklengths(l is)andlocationof theVDIsensors(r is). 3.2ParameterMeasurement ForthecaseofalinkijoinedtolinkshandjasshowninFigure 3-2 ,theinclination parametersmeasuredbytheVDIwillbetheinclinationfromtheDEA i,angular velocity_i,angularaccelerationiandmagnitudeofacceleration~ g iactingatthepointof (virtual)contactO h igiventhelocationoftheVDI,r i.ForapointQonlinki,giventhe 41

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VDI i l i r i O h,i Lieirei q q i O i,j g i ~ g j ~ f i P i Figure3-2.TheVestibularDynamicInclinometer(VDI)sensorisloca tedonlinkiof lengthl iatpointP iwhichisatdistancer ifrompointO h i displacementvectorfrompointO h itopointQasr O h,i!Q,theaccelerationofpointQcan bewrittenasN a Q = N a O h,i + Nir O h,i!Q + N!i N!ir O h,i!Q (3{1) TheresultantaccelerationofpointO i jcanbedeterminedinthefollowingmannerN a O i,j = N a P i + ( l ir i )Nie ir + N!i( N!ie ir ) (3{2) TheaccelerationofpointP iintheL icoordinatesystem,N a i P i,isthemeanoftheleftand rightaccelerometerreadingsoftheVDIandisanobservablequant ity.Theaccelerationof pointO i jintheL icoordinatesystemcanbewrittenasN a i O i,j = N a i P i + ( l ir i )264 _2 i i375 fe ir ,e i g= ~g j .(3{3) Let ibetheangleoftheresultantaccelerationatpointO i jwithe ir.Itisworthwhileto indicatethattheresultantaccelerationofpointO i jwillbeparalleltotheDEAforthe nextlinkj.Forthisreason,N a O i,jisreferredtoas~g jinFigure 3-2 .Theangle icanbe uniquelydeterminedas i = atan2N a i O i,j ,N a i O i,jr (3{4) 42

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VDI i l i r i ~ input output ... q i f i q i q i g i Figure3-3.TheVDI isensorfunctioncancalculatetheinclinationparametersand accelerationofjointpointsasoutputsgiventheinputsofl iandr i VDI b l b r b O b Lbebreb q g b O b,h g Figure3-4.Baselinkbwithoneendincontactwithgroundsurface Where N a i O i,j ,N a i O i,jrdenotethetangentialandradialcomponentsofvectorN a i O i,j. TheoutputsoftheVDI iaresummarizedinFigure 3-3 withtheparametersillustrated inFigure 3-2 .Thedetailedexplanationofhowtoobtaintheoutputparametersw as discussedearlier. 3.3BaseLinkparameters GiventhebaselinkbasinFigure 3-4 ,theparameterstobeestimatedareobtained fromtheVDI b(Figure 3-3 ).Asshowninthegure,the\ground"surfaceisassumed tobeatrest(onlygravitationalacceleration).Inthiscase,theD EAcoincideswiththe vertical,i.e.,directionofgravitationalacceleration(g).Theinclinationparameters(base angle,angularvelocityofthelink,andangularacceleration)areobt ainedinthefollowing manner rb =b (3{5) 43

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VDI i l i r i O h,i Lieirei q q i O i,j g i ~ f i P i VDIj Ljejrej q rj lj q j q i,j ~g j O j,k Figure3-5.Linksi jjoinedatpointO i jwithjointangle i jbetweenthem N!b = _b e z (3{6) Nb = b e z (3{7) wheree zistheunitvectorintotheplaneofpaper. 3.4Inter-linkparameters Fortwolinksi jjoinedatpointO i jasshowninFigure 3-5 ,itisdesiredtondthe inclinationparameters.TheDEAforlinkjisparalleltotheresultantaccelerationat pointO i j(~g j).Theanglebetweene irand~g jisgivenby i.Thevalueof iisdetermined fromVDI iand jisobtainedfromVDI j(Figure 3-3 ).So,thejointangleisestimatedas follows i j =i +j(3{8) TheotherinclinationparametersareestimatedasfollowsN!i = _i e z N!j = _j e z(3{9)Ni = i e z Nj = j e z(3{10) 44

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1 2 O 12 q 12 g 1 O 1 g O 2 Figure3-6.Slidercrankmechanismwithbaseangle r1andjointangle 12.Simulations areperformedwithlinklengthsl 1 = 0.3 m,l 2 = 0.6 mandsensorlocationsofr 1 = 0.25 m,r 2 = 0.5 m. 3.5Example Aslidercrankmechanism,showninFigure 3-6 withlinklengthsl 1 = 0.3 m,l 2 = 0.6 mandsensorlocationsofr 1 = 0.25 m,r 2 = 0.5 m,issimulatedinMATLAB R r .The sensornoisewasmodeledasmentionedinSection 2.5 .Thefrequencyofoperationof thesensorsisassumedtobe10 Hz.Thelocationofthelinearaccelerometersinthe VDIwasassumedtobe10cmapart.Theplatformisassumedtobestationary.The simulationresultsareveryencouragingandshowninFigures 3-7 through 3-13 .The estimatesfor r1 ,12 N!1 _12 N1 12,gandtheresultantaccelerationofthejointshave standarddeviationsof0.31deg,0.10deg,0.15deg/sec,0.15deg/sec,11.89deg/sec2,11.5deg/sec2,0.05m/sec2and0.11m/sec2respectively.Thepropagationoferrorisnot observedinsimulationsandtheestimatesdonot`drift'overtime. Itisobservedthattheratioofthedistancebetweenthelinearacc elerometersin theVDI(d)andthelocationoftheVDI(r)aectthestandarddeviationerrorsfor inclinationparameters(baseangle,jointangle,etc.).Forthepres entsimulation,theplots ofthevariationofstandarddeviationofangle,angularvelocity,an gularaccelerationand resultantaccelerationmeasuredareshowninFigures 3-14 through 3-17 .Asitcanbe observed,achangeind=rratiohasaneectoninclination(angular),angularacceleration andresultantaccelerationmeasurementerror(standarddeviat ion).Theideald=rratio maybe1(whichmaynotbepracticallypossible),butaratioof0.4orab ovemaygivea satisfactorymeasurementerrorforcommonpracticalapplicatio ns.Asthesimulationswere 45

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0 0.5 1 1.5 2 2.5 3 3.5 4 200 150 100 50 0 50 100 150 200 timedegreesBase!Angle!( g 1 ) g 1 observed! g 1 Figure3-7.Plotofestimated r1vstrue r1 0 0.5 1 1.5 2 2.5 3 3.5 4 -300 -200 -100 0 100 200 300 timedegrees/secBase Angular Velocity Angular Velocity Observed Angular Velocity Figure3-8.PlotofestimatedN!1vstrueN!1 doneusingwhitenoisealongthewholefrequencyspectrum,theres ultsfromsimulations areveryconservativeandareexpectedtoimprovewhenputtopr actice. 46

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0 0.5 1 1.5 2 2.5 3 3.5 4 -500 -400 -300 -200 -100 0 100 200 300 400 500 timedegrees/sec 2Base Angular Acceleration Angular Acceleration Observed Angular Acceleration Figure3-9.PlotofestimatedN1vstrueN1 0 0.5 1 1.5 2 2.5 3 3.5 4 -150 -100 -50 0 50 100 150 200 250 timedegreesJoint Angle vs observed Joint Angle Joint Angle observed Joint Angle Figure3-10.Plotofestimated 12vstrue 1247

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0 0.5 1 1.5 2 2.5 3 3.5 4 -400 -300 -200 -100 0 100 200 300 400 timedegrees/secJoint Angular Velocity Joint Angular Velocity Observed Joint Angular Velocity Figure3-11.Plotofestimated_12vstrue_12 0 0.5 1 1.5 2 2.5 3 3.5 4 -800 -600 -400 -200 0 200 400 600 timedegrees/sec 2Joint Angular Acceleration vs Observed Angular Acceleration Joint Angular Acceleration Joint Observed Angular Acceleration Figure3-12.Plotofestimated12vstrue1248

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0 0.5 1 1.5 2 2.5 3 3.5 4 2 4 6 8 10 12 14 timem/sec 2Resultant Acceleration of joint Acceleration observed Acceleration Figure3-13.PlotofestimatedjointaccelerationN a O 12vstruejointacceleration 49

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Ratio of d/rStandart deviation (degrees)Angular sensitivity g 1 q 12 Figure3-14.Plotofvariationofstandarddeviationofinclinationmea suredwithchangeind=rratio. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.13 0.135 0.14 0.145 0.15 0.155 0.16 0.165 0.17 0.175 Ratio of d/rStandart deviation (degrees/sec)Angular velocity sensitivity g 1 q 12 Figure3-15.Plotofvariationofstandarddeviationofangularveloc itywithchangeind=rratio. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 35 40 45 Ratio of d/rStandart deviation (degrees/sec 2 )Angular acceleration sensitivity g 1 q 12 .. .. Figure3-16.Plotofvariationofstandarddeviationofangularacce lerationwithchangeind=rratio. 50

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Ratio of d/rStandart deviation (m/sec 2 )Resultant Acceleration sensitivity at O 1 at O 1,2 Figure3-17.Plotofvariationofstandarddeviationofresultantac celerationwithchange ind=rratio. 51

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CHAPTER4 THEPLANARVESTIBULARDYNAMICINCLINOMETER(PVDI) TheplanarVestibularDynamicInclinometer(pVDI)sensorextends theVDIforve degrees-of-freedommotionoftherobotbase.Theconceptoft heDynamicEquilibrium Axis(DEA),theaxisalongwhichtherobotisatequilibriumispreserved fromthe VDIdesignandisdiscussed.TheinclinationangleobtainedfromthepV DIisrelative totheDEA,andthusismoresuitableasacontrolinputratherthan theinclination anglerelativetotheabsolutegravityvector(asobtainedfromoth erinertialunits).The inclinationangleobtainedisindependentofaccelerationofthesurfa ceofcontact.A closedformsolutionforinclinationmeasurementfromsensorreadin gsisdiscussed. 4.1ProblemDenition ArobotismodeledasaninvertedpendulumasshowninFigure 4-1 .Therobot hasvedegrees-of-freedom(planarmotionofthebaseandtwod egrees-of-freedomserial chain).Itisdesiredtosensetheinclination,angularvelocity,angula racceleration,and magnitudeofaccelerationactingonthebody.Therigidbodyismodele dasarodwith massm,centerofmassCalongtherodatadistancer Cfromthebaseoftherod,moment ofinertiaI B CatpointCandangulardampingcoecientK d.LetpointObeatthebase oftherigidbodyincontactwithabaseplatformwhichisconstrainedt omoveinaplane. LetNrepresenttheinertialreferenceframeandBrepresentthereferenceframexedon therigidbody.Letgbethegravitationalaccelerationonthebody,abetheacceleration ofpointOwithrespecttotheearth(inertial)referenceframeand~ gbetheresultantof theprevioustwomentionedaccelerations.Thus~ g = g + a(4{1) Acoordinatesystemxedintheinertialreferenceframewithorigin atO,Z-axis orthogonaltosurfaceofmotion,andxedinreferenceframeNisdenedwith f^ X ^ Y ^ Zg astheorthonormalbasis.BodycoordinatesystemwithoriginOisxedinthebody 52

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X Y Z e 3 e 2 e 1 g E 3 ~ g g a ~ O C P l Figure4-1.Modelofrobot.PointPdenotesthelocationofthesensor,pointCdenotes thecenterofmassand~gdenotestheresultantaccelerationactingonthebody. E1E2E3e b e c e a q q O e 3ye 2 e 1y Figure4-2.DenitionofIntermediatecoordinatesystemdenedb yorthogonalbasis f^e a ^e b ^e cg referenceframeB,withe =f^e 1 ^e 2 ^e 3g orthogonalbasisvectorswith^e 3alongthe vectorjoiningpointOtopointC.TheDynamicEquilibriumCoordinatesystemis denedtobexedintheinertialreferenceframeNwithoriginatpointO.ThevectorsE =f^ E 1 ^ E 2 ^ E 3g formasetoforthonormalbasisvectorssuchthat^ E 3isparalleltovector~ g.PointPisapointonthebodyatadistancelfrompointOasshowninFigure 4-1 4.2IntermediateCoordinateSystemandRotationMatricies Thebodypossessestwoangulardegrees-of-freedom,i.e.,aminimu moftworotations arerequiredtoalignthebodycoordinatesystemtotheDynamicEqu ilibriumCoordinate 53

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system.Anintermediatecoordinatesystemisdenedwithoriginatp ointOand orthogonalbasisI =f^e a ^e b ^e cg obtainedbyrotationoftheDynamicEquilibrium coordinatesystemwithanglean alongthe^ E 1axisasshowninFigure 4-2 .The intermediatecoordinatesystemalignswiththebodycoordinatesys temwhenrotated withanangle aboutaxis^e b.TheangularvelocityofthebodyreferenceframeBinthe inertialreferenceframeN,N!BmaybewrittenasN!B = _^ E 1 + ^e 2(4{2) ItisworthwhiletomentionthattheunitvectorsE 1 e 2aretherotationaxesoftheHooke jointatpointO.TheangularaccelerationofreferenceframeBinreferenceframeN,NB, canbecalculatedusingthetransporttheorem[ 59 ]NB = ^ E 1 + ^e 2 +N!B( ^e 2 ) (4{3) Rotationmatrices[ 18 ]totransformfrombasisItobasisEandfrombasisetobasisIare denedbyI E Rande I RrespectivelyasI E R = R 1 () =2666641 0 0 0 cs0sc 377775 (4{4) e I R = R 2 ( ) =266664c 0s 0 1 0 s 0 c 377775 (4{5) e E R = e I R I E R (4{6) 54

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wheres j = sin( j ), c j = cos( j ), j =, .LetN!B eandNB ebethematrixrepresentationof theangularvelocityandaccelerationrepresentedinthebasise.ThusN!B e =266664_c _s 377775 (4{7) NB e =266664c _s s + _c 377775. (4{8) Asthesimplerotationsrequiredtoreachthenalcoordinatesyst emarerotationof aboutaxise 1(R 1 ())androtationof aboutaxise b(R 2 ( )),forfuturereferences, the combinationisdenotedas1-2Eulerangles.Assumingthattherobot has motors/angularactuatorsthatallowthemovementoftherobota longdirectionsE 1ande 2=e b(equivalenthooke-joint),itisdesiredtondangles and asgiveninFigure 4-2 thatrealigntherobotalongtheDEA. 4.3SensorDesign Thesensoranalogyofthehumanvestibularsystemhasbeendonea sfollowsdual-axisaccelerometersareassumedtobeanalogoustotheotolit hsorgansandthe semicircularcanalsarevisualizedasone-dimensionalgyroscope.As therobot/human experiencesplanarmotion,itisnaturaltodrawadeeperanalogywit hthehuman vestibularsystem.Thehumanvestibularsystemconsistsofthree semicircularcanals thatareapproximatelyorthogonaltoeachother.Thismotivates theuseofone3-axis gyroscopeinthedesignofthenewsensor.Asthelinearacceleratio nissensedby stimulationofhaircellsduetothemovementofviscousruidinthesacc uleandutricle, oneotolithsorganisassumedtobeanalogoustotwolinearaccelerom etersplacedatsome distance. 55

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TheplanarVestibularDynamicInclinometer(pVDI)isdesignedbystr ategicallyand symmetricallyplacingfour2-axisaccelerometersandone3-axisgyr oscopeasshownin Figure 4-3 .Thehollowarrowsgivethedirectionsofthedual-axislinearaccelero meters. UnliketheVDIin[ 72 ],thedirectionsofthedual-axislinearaccelerometersaredierent ThelocationofthesensorisatpointPwhichisatdistancelfrompointOinthe^e 3direction.r P = r O + l ^e 3 .(4{9) ThegyroscopeisplacedatpointP.LinearaccelerometersL 1andR 1areplaced symmetricallyatadistanced 1=2alongthe^e 1directionaboutpointPandmeasure theaccelerationalong^e 3and^e 2.Similarly,linearaccelerometersL 2andR 2areplaced symmetricallyatdistanced 2=2along^e 2directionaboutpointPandmeasurethe accelerationalong^e 3and^e 1asshowninFigure 4-3 .Thusr OL 1 = l ^e 3d 1 2 ^e 1 (4{10) r OR 1 = l ^e 3 + d 1 2 ^e 1 (4{11) r OL 2 = l ^e 3d 2 2 ^e 2 (4{12) r OR 2 = l ^e 3 + d 2 2 ^e 2 (4{13) 4.4DynamicEquilibriumAxisforplanarmotionofthebase ForpointConthebody,performingakinematicanalysistoobtainthelinear accelerationinreferenceframeN(N a C)yieldsN a C = N a O + NBr OC + N!B N!Br OC (4{14) wherer OCdenotesthevectorfrompointOtopointCandN a O = ~ g = ~ g ^e 3(4{15) 56

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e1e2e3 R 2 L 2 d 1 /2 d 1 /2 d 2 /2 d 2 /2 G P R 1 L 1 Figure4-3.SensorismadeofaccelerometersL 1 R 1 L 2 R 2andgyroscopeG.Hollow arrowsindicatetheorientationofthedual-axislinearacceleromete rsatthe respectivelocations LetthereactingforcesactingonthebodyatthepointofcontactObeF R.Applicationof Euler'srstandsecondlawaboutcenterofmassCgivesmN a C = F R (4{16) I C BNB =K d N!B + r COF R (4{17) Atequilibrium = = 0and = = 0.Therefore,fromEquations 4{2 4{3 4{14 4{15 4{16 and 4{17 ,theequilibriumpositionis = = 0,i.e.,r COisparallel to~ g.Theaxisparallelto^ E 3iscalledtheDynamicEquilibriumAxis(DEA)forplanar motionofthebase.Whenthebodyisalignedalongthisaxis,itisinequilibr ium.The DynamicEquilibriumAxis(DEA)isparalleltotheresultingaccelerationa ctingonthe body~g,thus,istime-varying(moreprecisely,accelerationvarying).This issimilarto theDEAthathasbeendiscussedearlier.Theobjectiveforrobotb alancingapplications istobringtherobottoequilibrium,i.e.,toaligntherobotalongtheDEA. Therobot possessesvedegrees-of-freedom-twotranslational(oneles sduetosurfacecontact)and threerotational.Theanalysisaboveshowsthattheequilibriumposit ionfortherobot isnotapointorasurface,butanaxiscalledtheDynamicEquilibriumAxis (DEA). 57

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e 3 e 2 e 1 O C P l E 3 E 1 E 2 g ~ Figure4-4.E 3isdenedtobeparalleltoresultantaccelerationvector~g.AxesE 1ande 2areparalleltothetwohooke-jointaxes. Aligningtherobotalongtheaxisrequiresonetomovetherobotabou tanequivalent hooke-joint(tworotationaldegrees-of-freedom).Theorient ationofthebodyaboutthe DEAisirrelevantforrobotequilibrium,thus,requiringonlytwoindepe ndentparameters (equivalenthooke-joint)toaligntherobotalongtheDEA.Itshould alsobeobservedthat whentherobotisnotincontactwiththegroundandexperiencesfr eefall,theconcept oftheDEAceasestoexistastheaccelerationexperiencedbythelin earaccelerometerat pointOiszero,i.e.,~g = 0.Theoretically,itreinforcestheconceptoftheDEAasthe conceptof`equilibriumposition'ceasestoexistinzerogravity. 4.5KinematicAnalysisandMathematicalManipulations AccelerationofanypointQonthebodycanbewrittenasN a Q = N a O + ( NBr OQ ) + N!B( N!Br OQ )(4{18) whereQmaybe fL 1 R 1 L 2 R 2g .Gyroscopeandaccelerometerssensetheangularvelocity andlinearaccelerationsinthebasise.Thefollowingquantitiesaredenedintermsofthe 58

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measuredaccelerometerandgyroscopedataas 1 = N a R 1N a L 1 d 1 (4{19) 2 = N a R 2N a L 2 d 2 (4{20) 3 = N a R 2 + N a L 2 2 l = N a R 1 + N a L 1 2 l (4{21) 4 = N!B e (4{22) SimplifyingthequantitiesusingEquations 4{10 4{18 yields 1 = NB^e 1 + N!B N!B^e 1 (4{23) 2 = NB^e 2 + N!B N!B^e 2 (4{24) 3 = N a O l + NB^e 3 + N!B N!B^e 3. (4{25) 4.6InclinationMeasurement-Closedformsolution Representingequations 4{22 4{25 inthebasise,i.e.,thebasisinwhichthesensor readingsareactuallyobserved,isaccomplishedbyusingEquations 4{7 4{8 and 4{15 to obtain 1 =266664 2_2 s 2 s + 2 _c + _2 c s 377775 (4{26) 2 =266664 s _2 c 377775 (4{27) 3 =266664 (~ gs c)=l + _2 c s (~ gs)=lc + 2 _s 2 + (~ gc c)=l_2 c 2 377775 (4{28) 59

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4 =266664_c _s 377775. (4{29) Thecomponentsofthevectors i i = 1, 2, 3, 4,willbereferredtoas ij j = 1, 2, 3. Thevaluesthatarenotobservedduetosensorplacementare 11 ,22(strikedoutinthe equations).Themeanofreadingsoflinearaccelerometers fR 1 L 1g and fR 2 L 2g provide f 32 ,33g and f 31 ,33g respectively,thus,allthreecomponentsof 3areobserved. Theterms ijaremanipulatedtogivesevenequationsinsevenunknowns( _, , ~ g) = 13 +4143 (4{30) =42 (4{31) c =23 (4{32) ~ gs c= l ( 31 13 + 24143 ) (4{33) ~ gs= l (32 +2324243 ) (4{34) _c =41 (4{35) _s =43 (4{36) Otherequationsthatfalloutares = 21 =(12 + 24142 ) (4{37) ~ gc c= l (33 +2 42 +2 41 ) (4{38) ThefunctiongetVal (,1 ,2 )solvesthepairofequationsa sin() =1 a cos() =2fora,given 2( =2,=2) getVal (,1 ,2 ) = sign( c)sign(2 )q 2 1 +2 2(4{39) 60

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Algorithm1 Measurementofinclinationparametersfromsensordata Require: f 12 ,13g f 21 ,23g 3 ,4Let 1 ,2 ,3aredenedasfollows 1 = l ( 31 13 + 24143 )2 = l (33 +2 42 +2 41 )3 = l (32 +2324243 )Require: j 2 1 +2 2j >1and j 2 1 +2 2 +3 3j >2 ( 13 +4143 ) 42 tan1 1 2 2( =2,=2) getVal ( ,43 ,41 ) getVal 12 21 2 4143,23 Let 4 = getVal ( ,1 ,2 ) atan2(3 ,4 ) 2( ,] ~ g getVal (,3 ,4 ) 4.7SensorSimulation SimulationswereperformedinMATLAB R r .Thesensornoisewasmodeledas mentionedinSection 2.5 .Thefrequencyofoperationofthesensorsisassumedtobe10 Hz.Itshouldbeindicatedthatwhen~ giszero,theconceptoftheDEAceasestoexist. Insuchcases,themeasurementforinclinationparametersisinvalid .Thecase ==2isnotpossiblease b=e 2alignswithe 3aftertherstrotationandnopossiblerotation alonge 2canaligne cwithe 3.So,thepossibilityofestimationofinclinationparameters existsonlywhen~ gc 6= 0.Thesimulationresultsareverygoodandencouraging.One setofsimulationresultsareshowninFigures 4-5A 4-5B 4-6A 4-6B 4-7A 4-7B and 4-8 Theestimatesfor _, and~ ghaveastandarddeviationsof0.46deg,0.45deg,0.15deg/sec,0.15rad/sec,4.5 rad=sec 2,4.6 rad=sec 2and0.05 m=sec 2respectively.The estimatesdonot`drift'overtimeasitdoesnotinvolveintegrationof thequantities. 4.8SensorExperiment Theexperimentwassetwithfourdual-axislinearMEMSacceleromete rs(ADXL335) symmetricallyplacedaboutthecenter-lineoftheinvertedpendulum asshowninFigure 61

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0 1 2 3 4 5 6 40 30 20 10 0 10 20 30 40 Comparison of q Time (sec)q (deg) True q Observed q (A)Plotofestimated vstrue 0 1 2 3 4 5 6 50 40 30 20 10 0 10 20 30 40 50 Comparison of y Time (sec)y (deg) True y Observed y (B)Plotofestimated vstrue Figure4-5.Plotsofangles 0 1 2 3 4 5 6 40 30 20 10 0 10 20 30 40 Comparison of q Time (sec)q (deg/sec) True q Observed q . (A)Plotofestimated_ vstrue_ 0 1 2 3 4 5 6 50 40 30 20 10 0 10 20 30 40 50 Comparison of Time (sec)y (deg/sec) True y Observed y y . (B)Plotofestimated_ vstrue_ Figure4-6.Plotsofangularvelocities_, 0 1 2 3 4 5 6 50 40 30 20 10 0 10 20 30 40 50 Comparison of q Time (sec)q (deg/sec 2 ) True q Observed q .. .. .... (A)Plotofestimated vstrue 0 1 2 3 4 5 6 60 40 20 0 20 40 60 Comparison of y Time (sec)y (deg/sec 2 ) True y Observed y .. .. .... (B)Plotofestimated vstrue Figure4-7.Plotsofangularaccelerations, 62

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0 1 2 3 4 5 6 6 7 8 9 10 11 12 Estimation of g Time (sec)g (m/s 2 ) True g Estimated g ~ ~ ~~ Figure4-8.Plotofestimated~ gvstrue~ g 4-9 .Thesingle-axisMEMSgyroscope(LPY503AL)wascombinedwithadu al-axis gyroscope(LPY503AL)tomakeatriaxialgyroscopeandstrappe dontotheinverted pendulumasshowninFigure 4-9 .Interfacingoftheanalogvoltagesignalwasdoneusing NI-DAQmxandLabVIEW R r .Magneticencoders(USDigitalMA3-A10-125-N)werexed atthejoints.Themeasurementoftheencoderanglewasassumed tobethegroundtruth. Theaccelerometerswereautocalibratedandthemis-alignmentoft heaccelerometersto the`theoretical'radialandtangentialdirectionsinthebodycoord inatesystemewasdone usingthetechniquegiveninAppendix 6 Theexperimentwasperformedkeepingthebasexed,i.e.,N a O = g ^ E 1(Figure 4-1 ).Thereadingsfromtheencoderareassumedtobegroundtruth forinclination angles( ).Theplotsofcomparisonofinclinationangle,angularvelocityandan gular accelerationareshowninFigures 4-10A 4-10B 4-10A 4-10B 4-10 4-11A .Thelaginthe encoderangularvelocityandangularaccelerationsignalappearsd uetodierentiationof theencodersignalandthensmoothingit.Theexperimentalresult sconrmthesimulation resultsandareveryencouraging.TheconceptoftheDEAisalsoex perimentally observed. 63

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MEMS Linear Accelerometers MEMS Gyrocope 50 cm 22 cm Magnetic Encoders 22 cm Figure4-9.ExperimentalsetupoftheplanarVestibularDynamicIn clinometer(pVDI). ThefoursymmetricallyplacedlinearMEMSaccelerometersaremarke din boxes.TheMEMSgyroscopeismarkedbycircle.Themagneticencod ersare placedtomeasuretheEulerangles. 0 100 200 300 400 500 600 700 -50 -40 -30 -20 -10 0 10 20 q degTime (1 unit = 0.02 sec) Encoder pVDI (A)Comparisonof fromthepVDIvs fromthe encoder. 0 100 200 300 400 500 600 700 -40 -30 -20 -10 0 10 20 30 40 y degTime (1 unit = 0.02 sec) Encoder pVDI (B)Comparisonof fromthepVDIvs fromthe encoder. 64

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0 100 200 300 400 500 600 700 60 40 20 0 20 40 60 80 q deg/secTime!(1!unit!=!0.02!sec) Encoder pVDI (A)Plotof_ fromthepVDIvs_ fromtheencoder. 0 100 200 300 400 500 600 700 80 60 40 20 0 20 40 60 80 y deg/secTime!(1!unit!=!0.02!sec) Encoder pVDI (B)Plotof_ fromthepVDIvs_ fromtheencoder. 0 100 200 300 400 500 600 700 300 200 100 0 100 200 300 400 q .. deg/sec 2Time!(1!unit!=!0.02!sec) Encoder pVDI Figure4-10.Plotof fromthepVDIvs fromtheencoder. 0 100 200 300 400 500 600 700 400 300 200 100 0 100 200 300 400 y .. deg/sec 2Time!(1!unit!=!0.02!sec) Encoder pVDI (A)Plotof fromthepVDIvs fromtheencoder. 65

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CHAPTER5 ESTIMATINGJOINTPARAMETERSUSINGPVDI TheuseofVDIformeasuringlinkparametersoftwolinksjoinedbyre volutejoints (e.g.,kneejoint)hasbeendiscussedinChapter 3 .TheapplicationofpVDItomeasure linkparametersoftwolinksjoinedbyuniversal/Hookejointispresen ted.Thecore conceptofobtainingresultantaccelerationatthejointtoobtaint hejointanglesis extended.TheapproachremainssimilartothatpresentedinChapt er 3 5.1ProblemDenition TheproblemdenitionproceedsinasimilarfashionaspresentedinCha pter 3.1 .A mechanismcomprisedofaseriesoflinkagesjoinedbyuniversal/Hook ejointsisgiven.The linkagesaremodeledasrigidbodieswithknownlinklengthsandtheplana rVestibular DynamicInclinometer(pVDI)sensorislocatedatsomeconvenientd istancealongtheline joiningthetwolinkjointsorthepointofcontactandajointincaseof thebaselink.The baselinkisdenedasalinkthathasajointatoneendofthelinkandthe otherendis incontactwiththebase(orground,whichisassumedtobestationa ry)surfaceasshown inFigure 5-4 .LetO i jrepresentthejointjoininglinkitolinkj.ThelinejoiningjointsO h i(O bforbaselink)andO i j(O b hforthebaselink)isreferredtoasthelinkvectorof thelink.LettheVDIsensorbelocatedatadistancer ialongthelinkvectoroflinkias showninFigure 3-1 .Thejointangles i j i jbetweenlinksiandjaredenedas1-2 Eulerrotationangles(Figure 4-2 )betweentherespectivelinkcoordinatesystems.The linklengthoflinkiisgivenbyl i. LetNrepresenttheinertialreferenceframeandL irepresentthereferenceframexed onthelinki.ThelinkicoordinatesystemwithoriginatpointO p iisxedinthelinkireferenceframeL i,withe i =fe i1 e i2 e i3g orthogonalbasisvectorssuchthate i 3isalong thelinkvectoroflinkiande i2isalongthesecondrotationaxisofthehookjoint(T). Thejointangles i j i jaretheEuler1-2anglesbetweene i3ande j3.Forthecaseofthe baselink,theoriginofthecoordinatesystemliesatthepointofcont actO bandtheEuler 66

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pVDI i l i r i O h,i Liei3ei1 ei2 O i,j P i pVDIj Ljej3ej2ej 1 rj lj {q i,j ,y i,j } O j,k T L i e i1 e j1 e j2 e j3 e i2 e i3 L j O ij Figure5-1.Linksi jjoinedatpointO i jhavingjointangles i j i jbetweenthem. 1-2anglesbetweenthevertical(directionofgravitationalaccele ration)andthelinkare referredtoasthebaseangles( rb ,b)asshowninFigure 5-4 Itisdesiredtoobtainthebaseangles( rb ,b),jointangles( i js, i js),angular velocities(N!is)andtheangularaccelerations(Nis)ofthelinkswithgivenlinklengths(l is) andlocationoftheVDIsensors(r is). 5.2ParameterMeasurement ForthecaseofalinkijoinedtolinkshandjasshowninFigure 5-2 ,theinclination parametersmeasuredbythepVDIwillbetheEuler1-2inclinationang lesfromtheDEA i i,angularvelocities_i i,angularaccelerationi iandmagnitudeofacceleration~ g iactingatthepointof(virtual)contactO h igiventhelocationofthepVDI,r i.Fora pointQonlinki,giventhedisplacementvectorfrompointO h itopointQasr O h,i!Q,the accelerationofpointQcanbewrittenasN a Q = N a O h,i + Nir O h,i!Q + N!i N!ir O h,i!Q (5{1) TheresultantaccelerationofpointO i jcanbedeterminedinthefollowingmannerN a O i,j = N a P i + ( l ir i )Nie i3 + N!i( N!ie i3 ) (5{2) 67

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pVDI i l i r i O h,i Liei3ei 1ei 2 {q i ,y i } O i,j g i ~ g j ~ {f i ,d i } P i Figure5-2.TheplanarVestibularDynamicInclinometer(pVDI)sens orislocatedonlinkioflengthl iatpointP iwhichisatdistancer ifrompointO h i TheaccelerationofpointP iintheL icoordinatesystem,N a i P i,isthemeanoftheleftand rightaccelerometerreadingsoftheVDIandisanobservablequant ity.Theaccelerationof pointO i jintheL icoordinatesystemcanbewrittenasN a i O i,j = N a i P i + ( l ir i )266664 (~ gs c)=l + _2 c s (~ gs)=lc + 2 _s 2 + (~ gc c)=l_2 c 2 377775e i = ~g j .(5{3) Thecalculationof~g jcanbedoneasfollow~g j = r i3 + ( l ir i )26666424143 13 23 + 24243 2 41 2 42377775e i(5{4) Let( i ,i)betheEuler1-2anglesoftheresultantaccelerationatpointO i jfrome i 3.It isworthwhiletoindicatethattheresultantaccelerationofpointO i jwillbeparalleltothe DEAforthenextlinkj.Forthisreason,N a O i,jisreferredtoas~g jinFigure 5-2 .Theangle icanbeuniquelydeterminedas = sin1~ g j 1 jj~g jjj ,= atan2 c~ g j 2 jj~g jjj, c~ g j 3 jj~g jjj (5{5) 68

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pVDI i l i r i ~ input output {q i ,y i },{f i ,d i } {q i ,y i } .. .. {q i ,y i } ,g i Figure5-3.ThepVDI isensorfunctioncancalculatetheinclinationparametersand accelerationofjointpointsasoutputsgiventheinputsofl iandr i pVDIb lb rb O b Lbeb3eb 1 {g b ,j b } O b,h g Figure5-4.Baselinkbwithoneendincontactwithgroundsurface where~ g jk k = 1, 2, 3arethekthcomponentsof~g j.TheoutputsofthepVDI iare summarizedinFigure 5-3 withtheparametersillustratedinFigure 5-2 .Detailtoobtain theoutputparametershavebeendiscussedearlier. 5.3BaseLinkparameters GiventhebaselinkbasinFigure 5-4 ,theparameterstobeestimatedareobtained fromthepVDI b(Figure 5-3 ).Asshowninthegure,the\ground"surfaceisassumed tobeatrest(onlygravitationalacceleration).Inthiscase,theD EAcoincideswiththe vertical,i.e.,directionofgravitationalacceleration(g).Theinclinationparameters(base angles,angularvelocityofthelink,andangularacceleration)areob tainedinthefollowing 69

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pVDI i l i r i O h,i Liei3ei1ei2 {q i ,y i } O i,j g i ~ {f i ,d i } P i pVDIj Ljej3ej1ej2 rj lj{q j ,y j } {q i,j ,y i,j } ~g j O j,k T T Figure5-5.Linksi jjoinedatpointO i jwithEuler1-2jointangles i j i jbetweenthem manner rb =b ,'b =b (5{6) N!b =266664_b c b b _b s b377775e b (5{7) Nb =266664b c b_b s b b b b s b + _b c b b377775e b (5{8) 5.4Inter-linkparameters Fortwolinksi jjoinedatpointO i jasshowninFigure 5-5 ,itisdesiredtondthe inclinationparameters.TheDEAforlinkjisparalleltotheresultantaccelerationat pointO i j(~g j).The1-2Euleranglesbetweene i3and~g jaregivenby i ,i.Thevalues of i ,iaredeterminedfromVDI iand j jareobtainedfrompVDI j(Figure 5-3 ).Lete i e j Rrepresenttherotationmatrix[ 18 ]totransformfrombasise itobasise j.So,thejoint 70

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anglesareestimatedasfollowse i e j R = R 1 (i ) R 2 (i ) R 1 (j ) R 2 ( j ) =266664c ij 0 s ij si j s i j ci jsi j c i jci j s i j si j ci j c i j377775 (5{9) i j = atan2e i e j R 32 e i e j R 22 (5{10) i j = atan2e i e j R 13 e i e j R 11 (5{11) whereR i ()istherotationmatrixrepresentingrotationaboutithEuleraxisbyangle TheotherinclinationparametersareestimatedasfollowsN!i =266664_i c i i _i s i377775e i (5{12) Ni =266664i c i_i s i i i i s i + _i c i i377775e i (5{13) 71

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CHAPTER6 CONCLUSIONANDFUTUREWORK TheVestibularDynamicInclinometer(VDI)andplanarVestibularDyn amic Inclinometer(pVDI)arehumanvestibularsystemmotivatedinclinat ionmeasurement sensors.Theymeasuretheinclinationparameters-angularinclinat ion,angularvelocity, angularaccelerationandmagnitudeofaccelerationofthesurface ofcontact.TheVDI andpVDImeasuretheinclinationparametersforfourdegrees-of -freedomrobotandve degrees-of-freedomrobotsrespectively.Theinclinationmeasur ementsareindependentof theaccelerationofthesurfaceofcontact(gravity,etc.),anyd rift/integrationerrorsand validforlargeinclinationangles. ThedesignoftheVDIsensorconsistsoftwosymmetricallyplaceddu al-axisMEMS linearaccelerometersandonesingleaxisMEMSgyroscope.While,for thepVDIsensor, thedesignconsistsofstrategicallyandfoursymmetricallyplaceddu al-axisMEMSlinear accelerometersandonetri-axialMEMSgyroscope.Theorientatio nofthesymmetrically placeddual-axislinearaccelerometersforthepVDIisdierentfrom thatoftheanalogous accelerometersinthecaseoftheVDI. TheDynamicEquilibriumAxis(DEA)istheaxisalongwhichtherobotisat equilibrium.TheDEA,whichisparalleltothedirectionoftheresultant accelerationof theplatform/surfaceofcontact(gravity,etc),actsasthere ferenceformeasurement.The DEAceasestoexistwhentheresultantaccelerationofthecontac tplatform/surfaceis zero,i.e.,zerogravity(e.g.,free-fall).TheDEAhelpstoexplainthele aningwhenhumans accelerateandbendingbackwhentheydecelerate. Theinclinationmeasurementsarevalidforenvironmentswithvarying gravity(space applications)andacceleratingplatforms(running,walkingmotionof robots,etc.).The sensoroutputsareidealcontrolinputsforbalancingofrobotsa sthegoalistobringthe robotstotheequilibriumposition(i.e.,alignitalongtheDEA).Thesemak estheVDI andpVDIidealforbalancingofrobots,humanoids,etc.Duetoeco nomicaladvantages 72

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ofMEMSsensors,theapplicationsofVDIandpVDIextendtojointa nglemeasurement, humanmovementstudiese.g.,gaitanalysis,armmovementaboutelbo w,etc.Thejoint anglemeasurementofbipedrobotsisalsoapotentialapplication. Astheresultsfromthesimulationareveryconservative,thefutu reworkrequiresto attempttoshrinkthetotalsizeofthesensorsanddevelopcalibra tiontechniquestodeal witherrorsindistancemeasurements.Algorithmtocorrectsenso rdistancemisalignment (asymmetry)needstoberesearched.Itisalsodesiredtoapplyth esensorstomeasure inclinationparametersofspatialmanipulators(allrevolutejoints) 73

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APPENDIX:AUTOCALIBRATIONOFMEMSACCELEROMETERS Anautocalibrationprocedurebasedontheassumptionthatastat icinertialsensor onlyexperiencesforceduetogravity[ 23 ].Theresultantoftheaccelerationvector measuredbythesensormustequalg = 9.81 m=s 2.Thismethodrequiresmeasuring outputsignals(V)oftheMEMSaccelerometerinninedierentdirect ions.Astheadopted accelerometersaredesignedtoprovideratio-metricoutput,letV T = [ v x v y v z ]represent theratioofoutputvoltagetopowersupplyvoltageV CC(i.e.v i = V i=V CC i = x y z). A.1ProblemStatement LettheaccelerationvectorbedenedasA T = [ a x a y a z ]representedinthesensor coordinatesystem.Theinertialcoordinatesystemisxedininertia lreferenceframeNwithorthogonalbasis fX Y Zg andoriginatpointO.Thethesensorcoordinatesystem isxedinthesensorreferenceframewithorthonormalbasis fe 1 e 2 e 3g andoriginat pointOasshowninFigure A-1 .ThesensorismodeledasfollowsA = S ( VO )(A{1) wherethesensitivitySandbiasOareS =266664S xx S xy S xz S yx S yy S yz S zx S zy S zz377775, O =266664O x O y O z377775 (A{2) Afterimposingthesymmetryconstraint(S xy = S yx S xz = S zx S zy = S yz,thenine unknownparametersarewrittenasX 91 = [ x 1 x 2 ,, x 9 ] = [ S xx S yy S zz S xy S yz S zx O x O y O z ]. ItisdesiredtondtheunknownparametersXandmisalignmentangles (Figure A-1 ). 74

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j r e 1 Y e 3 Z X e 2 g (gravity) O FigureA-1.Thesensorcoordinatesystemhasconstantmisalignme ntfromtheinertial coordinatesystemthatcanbequantizedby ,' A.2Autocalibration Theautocalibrationprocedureusestheconceptthatinstaticorie ntations,the resultantaccelerationexperiencedbythesensorisonlyduetogra vityi.e.g = 9.81 m=sec 2. jjAjj=q a 2 x + a 2 y + a 2 z = g(A{3) LetA idenotecorrespondingaccelerometerreadingwhenthesensorisin theith conguration.ForNcongurations,deningtheleastsquareser rorE E ( X ) = 1 N NXi =1 jjA ijj2g 2= NXi =1 jjS ( V iO )jj2g 2 (A{4) Autocalibrationofthesensorwillbethesolutiontothemathematica lproblemarg min X E ( X )(A{5) 75

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whereE ( X )isanon-linearfunctioninunknownparameters.Startingfrominitial guessof thesensorparametersasgivenbythemanufacturers,thesolut ionisiterativelyupdatedasX k +1 = X k H1 ( X k )J ( X k )(A{6) forkthiteration,whereH ( X ), J ( X )aretheHessianmatrixandJacobianvectorforthe errorE,respectively.J ( X ) = @E @x 1 ,,@E @x 9, H ( X ) =h ij =@2 E @x i@x j (A{7) isstepparameterwhichislessthan1andcomputedoneachiteration usinglinesearch procedure.Theterminationprocedureischosenasmax m =1,,9x k +1 mx k m ( x k +1 mx k m )=2 < (A{8) where issomethreshold. A.3MisalignmentComputation Themisalignmentanglescanbecalculatedwhenthesensoriskeptrat ontheground. Thentheaccelerationexperiencedbythesensorwillbe 266664a x a y a z377775 fe 1 e 2 3g= gE z = g266664 ss'scc' 377775 fe 1 e 2 3g (A{9) So, = sin1a y g,'= atan2 (sa x ca z )(A{10) 76

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REFERENCES [1] Algrain,M.C.andSaniie,J.\Estimationof3Dangularmotionusinggyro scopesand linearaccelerometers." IEEETransactionsonAerospaceandElectronicSystems 27 (1991).6:910{920. [2] Bachmann,E.R. InertialAndMagneticTrackingOfLimbSegmentOrientation For InsertingHumansIntoSyntheticEnvironments .StormingMedia,2000. [3] Baerveldt,A.J.andKlang,R.\Alow-costandlow-weightattitudees timation systemforanautonomoushelicopter." IEEEInternationalConferenceonIntelligent EngineeringSystems .1997,391{395. [4] Barbour,N.andSchmidt,G.\Inertialsensortechnologytrends ." IEEESensors Journal 1(2001).4:332{339. [5] Barshan,B.andDurrant-Whyte,H.F.\Inertialnavigationsyste msformobile robots." IEEETransactionsonRoboticsandAutomation 11(1995).3:328{342. [6] Baselli,G.,Legnani,G.,Franco,P.,Brognoli,F.,Marras,A.,Quaranta ,F.,and Zappa,B.\Assessmentofinertialandgravitationalinputstothe vestibularsystem." JournalofBiomechanics 34(2001).6:821{826. [7] Baxter,L.K. CapacitiveSensors:DesignandApplications(IEEEPressSe rieson ElectronicsTechnology) .Wiley-IEEEPress,1996. [8] Bernmark,E.andWiktorin,C.\Atriaxialaccelerometerformeasu ringarm movements." AppliedErgonomics 33(2002).6:541{547. [9] Bernstein,J.\AnoverviewofMEMSinertialsensingtechnology." Sensors-The JournalofAppliedSensingTechnology 20(2003).2:14{21. [10] Billat,S.,Glosch,H.,Kunze,M.,Hedrich,F.,Frech,J.,Auber,J.,Lang ,W., Sandmaier,H.,andWimmer,W.\Convection-basedmicromachinedinc linometer usingSOItechnology." The14thIEEEInternationalConferenceonMicroElectro MechanicalSystems,2001 .IEEE,2001,159{161. [11] Bortz,J.E.\Anewmathematicalformulationforstrapdowninertia lnavigation." IEEETransactionsonAerospaceandElectronicSystems (1971).1:61{66. [12] Bouten,C.V.C.,Koekkoek,K.T.M.,Verduin,M.,Kodde,R.,andJanssen ,J.D. \Atriaxialaccelerometerandportabledataprocessingunitforth eassessmentof dailyphysicalactivity." IEEETransactionsonBiomedicalEngineering 44(1997).3: 136{147. [13] Brandt,T. Vertigo:ItsMultisensorySyndromes .SpringerVerlag,1999. [14] Camis,M.andCreed,R.S.\Thephysiologyofthevestibularapparat us." The AmericanJournaloftheMedicalSciences 180(1930).6:849. 77

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BIOGRAPHICALSKETCH VisheshVikaswasborninNewDelhi,Indiain1983.Hereceivedhisbache lor'sdegree inmechanicalengineeringfromIndianInstituteofTechnology,Guw ahatiin2005.After thatheworkedatAutonomousIntelligentMAchines(MAIA)Lab,LO RIA.Hejoinedthe CenterofIntelligentMachinesandRobotics(CIMAR),Universityof Florida,Gainesville inJanuary2007.HereceivedhisMSinmechanicalengineeringinsumme r2009.He completedhisPhDinmechanicalengineeringwithminorsinmathematics andcomputer scienceinfall2011.Hisresearchinterestsincluderobotics,artic ialintelligence,machine learning,optimalestimation,nonlinearcontrol,humanoidrobots,e nergyecientdevices andmulti-agentsystems. 83