Solving-the-Inverse-Kinematic-Robotics-Problem机器人逆

SolvingtheInverseKinematicRoboticsProblem:AComparisonStudyoftheDenavit-HartenbergMatrixandGroebnerBasisTheoryExceptwherereferenceismadetotheworkofothers,theworkdescribedinthisdissertationismyownorwasdoneincollaborationwithmyadvisorycommittee.Thisdissertationdoesnotincludeproprietaryorclassi¯edinformation.KimberlyKendricksCerti¯cateofApproval:MichelSmithProfessorandChairmanMathematicsandStatisticsOvertounJenda,ChairProfessorMathematicsandStatisticsRichardZalikProfessorMathematicsandStatisticsJoeF.PittmanInterimDeanGraduateSchoolSolvingtheInverseKinematicRoboticsProblem:AComparisonStudyoftheDenavit-HartenbergMatrixandGroebnerBasisTheoryKimberlyKendricksADissertationSubmittedtotheGraduateFacultyofAuburnUniversityinPartialFul¯llmentoftheRequirementsfortheDegreeofDoctorofPhilosophyAuburn,AlabamaAugust4,2007SolvingtheInverseKinematicRoboticsProblem:AComparisonStudyoftheDenavit-HartenbergMatrixandGroebnerBasisTheoryKimberlyKendricksPermissionisgrantedtoAuburnUniversitytomakecopiesofthisdissertationatitsdiscretion,upontherequestofindividualsorinstitutionsandattheirexpense.Theauthorreservesallpublicationrights.SignatureofAuthorDateofGraduationiiiVitaKimberlyD.KendrickswasbornonAugust27,1980inDayton,Ohio.Aftercompletinghighschool,sheattendedtheUniversityofPittsburghinPittsburgh,Pennsylvania,whereshereceivedaDualB.S.inMathematicsandBusiness.Then,shecontinuedherpassionformathematicsatAuburnUniversityinAuburn,AlabamawhereshereceivedherMaster'sinAppliedMathematics,andsincethen,hasservedasagraduateteachingassistantintheDepartmentofMathematics.ivDissertationAbstractSolvingtheInverseKinematicRoboticsProblem:AComparisonStudyoftheDenavit-HartenbergMatrixandGroebnerBasisTheoryKimberlyKendricksDoctorofPhilosophy,August4,2007(M.A.,AuburnUniversity,2006)(DualB.S.,UniversityofPittsburgh,2003/2004)90TypedPagesDirectedbyOvertounJendaTheaimistoanalyzetwoverydi®erentmethodsforsolvingtheinversekinematicroboticsproblem.The¯rstmethod,calledtheEngineeringApproach,usestheDenavit-HartenbergMatrix,andthesecondmethod,calledtheMathematician'sApproach,usesGroebnerBasisTheoryandMAGMA,amathematicscomputersoftwarepackage.Witheachapproach,wewillsolvetheinversekinematicsroboticsproblemforvariousrobotmanipulators.Bycomparingeachmethod,thispaperwilldemonstratethatGroebnerBasisTheoryismoreadvantageous,andfurthermore,morebene¯cialtothe¯eldofmathematicsandroboticsengineering.vStylemanualorjournalusedJournalofApproximationTheory(togetherwiththestyleknownas\auphd).BibliographyfollowsvanLeunen'sAHandbookforScholars.ComputersoftwareusedThedocumentpreparationpackageTEX(speci¯callyLATEX)togetherwiththedepartmentalstyle-¯leauphd.sty.viTableofContentsListofFiguresix1Introduction12TheGMFRoboticsA-510RobotManipulator52.1Introduction....................................52.2De¯nitions.....................................52.3Properties.....................................63TheInverseKinematicRoboticsProblem83.1De¯nitions.....................................83.2ForwardKinematicRoboticsProblem.....................83.3InverseKinematicRoboticsProblem......................94TheEngineeringApproach114.1Introduction....................................114.2HomogeneousCoordinatesandMatrices....................114.2.1De¯nitions................................114.2.2Remarks..................................124.3Transformations.................................124.3.1TranslationalTransformations......................134.3.2RotationalTransformations.......................164.4TheDenavit-HartenbergMatrix.........................184.4.1Introduction...............................184.4.2DerivationoftheDenavit-HartenbergMatrix.............194.5AKinematicModelfortheGMFRoboticsA510Robot...........264.6SolvingtheInverseKinematicRoboticsProblem...............284.6.1CaseExamples..............................344.7Summary:EngineeringApproach........................414.7.1ResultsfortheGMFRoboticsA510Robot..............414.7.2Di±culties.................................435TheMathematician'sApproach445.1Introduction....................................445.2GroebnerBasisTheory..............................445.2.1TerminologyandNotation........................455.3AlgorithmforComputingaGroebnerBasis..................46vii5.3.1S-PolynomialsandBuchberger'sAlgorithm..............475.4MAGMA:AlgebraicComputerSoftware....................505.5SolvingtheInverseKinematicsRoboticsProblem...............515.5.1Introduction...............................515.5.2AlgebraicModelfortheGMFRoboticsA-510Robot.........515.5.3CalculatingAGroebnerBasisWithMAGMA.............565.5.4CaseExamples..............................585.6Summary:Mathematician'sApproach.....................606MethodAnalysis626.1GMFRoboticsA-510Robot...........................626.2TheJacobianMatrixandSingularities.....................646.2.1TheJacobianforaRobotManipulator.................656.2.2FindingtheJacobianfortheGMFRoboticsA-510Robot......676.2.3Singularities................................706.2.4FindingSingularitiesfortheGMFA-510Robot...........716.