机器人学基础-第7章-机器人轨迹规划-蔡自兴.ppt

中南大学蔡自兴 谢斌zxcai xiebin 2010 机器人学基础第七章机器人轨迹规划 1 Ch 7TrajectoryPlanningofRobots FundamentalsofRobotics Ch 7TrajectoryPlanningofRobots 2 Ch 7TrajectoryPlanningofRobots Ch 7TrajectoryPlanningofRobots 3 Ch 7TrajectoryPlanningofRobots 7 1GeneralConsiderationsinTrajectoryPlanning轨迹规划应考虑的问题 BasicProblem Movethemanipulatorarmfromsomeinitialpositiontosomedesiredfinalposition Maybegoingthroughsomeviapoints 4 7 1Generalconsiderations 7 1GeneralConsiderationsinTrajectoryPlanning Trajectory Timehistoryofposition velocityandaccelerationforeachDOFPathpoints Initial finalandviapointsConstraints Spatial time smoothness 5 7 1Generalconsiderations JointspaceEasytogothroughviapoints Solveinversekinematicsatallpathpoints NoproblemswithsingularitiesLesscalculationsCannotfollowstraightlineCartesianspaceWecantrackashape fororientation equivalentaxes Eulerangles Moreexpensiveatruntime afterthepathiscalculatedneedjointanglesinalotofpoints Discontinuityproblems 6 GeneralConsiderations SolutionSpace 7 1Generalconsiderations Cartesianplanningdifficulties 7 GeneralConsiderations SolutionSpace 7 1Generalconsiderations Initial A andGoal B Pointsarereachable butintermediatepoints C unreachable Ch 7TrajectoryPlanningofRobots 8 Ch 7TrajectoryPlanningofRobots Joint SpaceSchemesEachpathpointis converted intoasetofdesiredjointanglesbyapplicationoftheinversekinematics Asmoothfunctionisfoundforeachofthenjointswhichpassthroughtheviapointsandendatthegoalpoint Timerequiredforeachsegmentisthesameforeachjoint Thedeterminationofthedesiredjointanglefunctionforaparticularjointisindependentwithotherjoints 9 7 2InterpolatedCalculationofJointTrajectories关节轨迹的插值计算 7 2Joint SpaceSchemes Choiceofinterpolationfunctionisnotunique 10 Joint SpaceSchemes 7 2Joint SpaceSchemes Severalpossiblepathshapesforasinglejoint Somepossibleinterpolationfunctions CubicpolynomialsCubicpolynomialsforapathwithviapointsHigher orderpolynomialsLinearfunctionwithparabolicblendsLinearfunctionwithparabolicblendsforapathwithviapoints 11 Joint SpaceSchemes 7 2Joint SpaceSchemes Inmakingasinglesmoothmotion atleastfourconstraintsonareevident 12 7 2 1CubicPolynomials三次多项式插值 7 2Joint SpaceSchemes Combiningthefourconstraintsyieldsfourequationswithfourunknowns 13 7 2 1CubicPolynomials 7 2Joint SpaceSchemes Thesefourconstraintsuniquelyspecifyaparticularcubic 14 7 2 1CubicPolynomials Thejointvelocityandaccelerationalongthispathare 7 2Joint SpaceSchemes Eg 7 1Asingle linkrobotwitharotaryjointismotionlessat 15degrees Itisdesiredtomovethejointinasmoothmannerto 75degreesin3seconds Findthecoefficientsofacubicwhichaccomplishesthismotionandbringsthemanipulatortorestatthegoal Plottheposition velocity andaccelerationofthejointasafunctionoftime 15 7 2 1CubicPolynomials 7 2Joint SpaceSchemes Solution Plugging 0 15 f 75 tf 3into 7 6 wefind 16 7 2 1CubicPolynomials 7 2Joint SpaceSchemes Solution 17 7 2 1CubicPolynomials 7 2Joint SpaceSchemes Startsat15degreesandendsat75degrees Solution 18 7 2 1CubicPolynomials 7 2Joint SpaceSchemes Startsandendsatrest Solution 19 7 2 1CubicPolynomials 7 2Joint SpaceSchemes Accelerationprofileislinear Ifwecometorestateachpointuseformulafrompreviousslideorcontinuousmotion nostops needvelocitiesatintermediatepoints InitialConditions 20 7 2 2Cubicpolynomialswithviapoints过路径点的三次多项式插值 7 2Joint SpaceSchemes Solutions Howtospecifyvelocityattheviapoints TheuserspecifiesthedesiredvelocityateachviapointintermsofaCartesianlinearandangularvelocityofthetoolframeatthatinstant ThesystemautomaticallychoosesthevelocitiesattheviapointsbyapplyingasuitableheuristicineitherCartesianspaceorjointspace averageof2sidesetc Thesystemautomaticallychoosesthevelocitiesattheviapointsinsuchawayastocausetheaccelerationattheviapointstobecontinuous 21 7 2Joint SpaceSchemes 7 2 2Cubicpolynomialswithviapoints Higherorderpolynomialsaresometimesusedforpathsegments Forexample ifwewishtobeabletospecifytheposition velocity andaccelerationatthebeginningandendofapathsegment aquinticpolynomialisrequired 22 7 2 3Higher orderpolynomials高阶多项式插值 7 2Joint SpaceSchemes Wheretheconstraintsaregivenas 23 7 2 3Higher orderpolynomials 7 2Joint SpaceSchemes Solutiontotheseequations 24 7 2 3Higher orderpolynomials 7 2Joint SpaceSchemes Linearinterpolation Straightline Note Althoughthemotionofeachjointinthisschemeislinear theend effectoringeneraldoesnotmoveinastraightlineinspace 25 7 2 4Linearfunctionwithparabolicblends用抛物线过渡的线性插值 7 2Joint SpaceSchemes Discontinuousvelocity cannotbecontrolled Tocreateasmoothpathwithcontinouspositionandvelocity westartwiththelinearfunctionbutaddaparabolicblendregionateachpathpoint Constantaccelerationisusedduringtheblendportiontochangevelocitysmoothly 26 7 2 4Linearfunctionwithparabolicblends 7 2Joint SpaceSchemes Assumethattheparabolicblendsbothhavethesameduration andthereforethesameconstantacceleration moduloasign Therearemanysolutionstotheproblem buttheanswerisalwayssymmetricaboutthehalfwaypoint 27 7 2 4Linearfunctionwithparabolicblends 7 2Joint SpaceSchemes Thevelocityattheendoftheblendregionmustequalthevelocityofthelinearsection 28 7 2 4Linearfunctionwithparabolicblends 7 2Joint SpaceSchemes Lett 2th combining 7 13 and 7 14 29 7 2 4Linearfunctionwithparabolicblends Theaccelerationchosenmustbesufficientlyhigh toensuretheexistenceofasolution 7 2Joint SpaceSchemes Belowshowsasetofjointspaceviapointsforsomejoints Linearfunctionsconnecttheviapoints andparabolicblendregionsareaddedaroundeachviapoint 30 7 2 5Linearfunctionwithparabolicblendsforapathwithviapoints 过路径点的用抛物线过渡的线性插值 7 2Joint SpaceSchemes Multi segmentlinearpathwithblends Given positionsdesiredtimedurationsthemagnitudesoftheaccelerationsCompute blendstimesstraightsegmenttimesslopes velocities signedaccelerations 31 7 2Joint SpaceSchemes 7 2 5Linearfunctionwithparabolicblendsforapathwithviapoints Insidesegment 32 7 2Joint SpaceSchemes 7 2 5Linearfunctionwithparabolicblendsforapathwithviapoints Firstsegment 33 7 2Joint SpaceSchemes 7 2 5Linearfunctionwithparabolicblendsforapathwithviapoints Lastsegment 34 7 2Joint SpaceSchemes 7 2 5Linearfunctionwithparabolicblendsforapathwithviapoints Togothroughtheactualviapoints Introduce PseudoViaPoints Usesufficientlyhighacceleration 35 7 2Joint SpaceSchemes 7 2 5Linearfunctionwithparabolicblendsforapathwithviapoints Ch 7TrajectoryPlanningofRobots 36 Ch 7TrajectoryPlanningofRobots WhenpathshapesaredescribedintermsoffunctionsofCartesianpositionandorientation wecanalsospecifythespatialshapeofthepathbetweenpathpoints Themostcommonpathshapeisastraightline butcircular sinusoidal orotherpathshapescouldbeused Cartesianschemesaremorecomputationallyexpensivetoexecutesinceatruntime inversekinematicsmustbesolvedatthepathupdaterate 7 3Cartesian SpaceSchemes 37 7 3Cartesian SpaceSchemes Descriptionofatask 7 3Cartesian SpaceSchemes 38 7 3Cartesian SpaceSchemes CartesianstraightlinemotionMovefrompointPitoPi 1 whichdescribedbyrelativehomogenoustransformation 7 3Cartesian SpaceSchemes 39 7 3Cartesian SpaceSchemes Inordertoensurecontinuousvelocitiesintrajectory asplineoflinearfunctionswithparabolicblendsisalwaysused Duringthelinearportionofeachsegment sinceallthreecomponentsofpositionchangeinalinearfashion theend effectorwillmovealongalinearpathinspace 7 3Cartesian SpaceSchemes 40 7 3Cartesian SpaceSchemes Cartesianplanningdifficulties 1 3 41 Initial A andGoal B Pointsarereachable butintermediatepoints C unreachable 7 3Cartesian SpaceSchemes 7 3Cartesian SpaceSchemes 42 Approachingsingularitiessomejointvelocitiesgoto causingdeviationfromthepath 7 3Cartesian SpaceSchemes 7 3Cartesian SpaceSchemes Cartesianplanningdifficulties 2 3 43 Startpoint A andgoalpoint B arereachableindifferentjointspacesolutions Themiddlepointsarereachablefrombelow 7 3Cartesian SpaceSchemes 7 3Cartesian SpaceSchemes Cartesianplanningdifficulties 3 3 Ch 7TrajectoryPlanningofRobots 44 Ch 7TrajectoryPlanningofRobots 7 4PathGenerationatReal Time Atruntimethepathgeneratorroutineconstructsthetrajectory usuallyintermsof andfeedsthisinformationtothemanipulator scontrolsystem Thispathgeneratorcomputesthetrajectoryatthepathupdaterate 7 4PathGenerationatRunTime 45 7 4 1Generationofjointspacepaths Inthecaseofcubicsplines thepathgeneratorsimplycomputes 7 3 and 7 4 astisadvanced Whentheendofonesegmentisreached anewsetofcubiccoefficientsisrecalled tissetbacktozero andthegenerationcontinues 7 4PathGenerationatRunTime 46 Inthecaseoflinearsplineswithparabolicblends thevalueoftime t ischeckedoneachupdatetodeterminewhetherwearecurrentlyinthelinearortheblendportionofthesegment Inthelinearportion thetrajectoryforeachjointiscalculatedas 7 4PathGenerationatRunTime 47 7 4 1Generationofjointspacepaths Inthecaseoflinearsplineswithparabolicblends thevalueoftime t ischeckedoneachupdatetodeterminewhetherwearecurrentlyinthelinearortheblendportionofthesegment Intheblendregion thetrajectoryforeachjointiscalculatedas 7 4PathGenerationatRunTime 48 7 4 1Generationofjointspacepaths Inthecaseoflinearsplinewithparabolicblendspath Rewrite 7 45 and 7 46 withthesymbolXrepresentingacomponentoftheCartesianpositionandorientationvector Inthelinearportionofthesegment eachdegreeoffreedominXiscalcuatedas 7 4PathGenerationatRunTime 49 7 4 2GenerationofCartesianspacepaths Inthecaseof