机器人学基础-第7章-机器人轨迹规划-蔡自兴.ppt

机器人学基础-第7章-机器人轨迹规划-蔡自兴.ppt

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中南大学中南大学蔡自兴，谢蔡自兴，谢斌斌zxcai,2010机器人学基础机器人学基础第七章第七章机器人轨迹规划机器人轨迹规划1Ch.7TrajectoryPlanningofRobotsFundamentalsofRoboticsCh.7TrajectoryPlanningofRobots2Ch.7TrajectoryPlanningofRobots7.1GeneralConsiderationsinRobotTrajectoryPlanning7.2InterpolatedCalculationofJointTrajectories7.3PlanningofCartesianPathTrajectories7.4RealTimeGenerationofPlanningTrajectories7.5SummaryCh.7TrajectoryPlanningofRobots37.1GeneralConsiderationsinRobotTrajectoryPlanning7.2InterpolatedCalculationofJointTrajectories7.3PlanningofCartesianPathTrajectories7.4RealTimeGenerationofPlanningTrajectories7.5SummaryCh.7TrajectoryPlanningofRobots7.1GeneralConsiderationsinTrajectoryPlanning轨迹规划应考虑的问题轨迹规划应考虑的问题BasicProblem:

Movethemanipulatorarmfromsomeinitialpositiontosomedesiredfinalposition（Maybegoingthroughsomeviapoints）.47.1Generalconsiderations7.1GeneralConsiderationsinTrajectoryPlanningTrajectory:

Timehistoryofposition,velocityandaccelerationforeachDOFPathpoints:

Initial,finalandviapointsConstraints:

Spatial,time,smoothness57.1GeneralconsiderationsJointspaceEasytogothroughviapoints（Solveinversekinematicsatallpathpoints）NoproblemswithsingularitiesLesscalculationsCannotfollowstraightlineCartesianspaceWecantrackashape（fororientation:

equivalentaxes,Eulerangles,）Moreexpensiveatruntime（afterthepathiscalculatedneedjointanglesinalotofpoints）Discontinuityproblems6GeneralConsiderations-SolutionSpace7.1GeneralconsiderationsCartesianplanningdifficulties:

7GeneralConsiderations-SolutionSpace7.1GeneralconsiderationsInitial（A）andGoal（B）Pointsarereachable,butintermediatepoints（C）unreachable.Ch.7TrajectoryPlanningofRobots87.1GeneralConsiderationsinRobotTrajectoryPlanning7.2InterpolatedCalculationofJointTrajectories7.3PlanningofCartesianPathTrajectories7.4RealTimeGenerationofPlanningTrajectories7.5SummaryCh.7TrajectoryPlanningofRobotsJoint-SpaceSchemesEachpathpointisconvertedintoasetofdesiredjointanglesbyapplicationoftheinversekinematics.Asmoothfunctionisfoundforeachofthenjointswhichpassthroughtheviapointsandendatthegoalpoint.Timerequiredforeachsegmentisthesameforeachjoint.Thedeterminationofthedesiredjointanglefunctionforaparticularjointisindependentwithotherjoints.97.2InterpolatedCalculationofJointTrajectories关节轨迹的插值计算关节轨迹的插值计算7.2JointSpaceSchemesChoiceofinterpolationfunctionisnotunique!

10Joint-SpaceSchemes7.2JointSpaceSchemesSeveralpossiblepathshapesforasinglejoint.Somepossibleinterpolationfunctions:

CubicpolynomialsCubicpolynomialsforapathwithviapointsHigher-orderpolynomialsLinearfunctionwithparabolicblendsLinearfunctionwithparabolicblendsforapathwithviapoints11Joint-SpaceSchemes7.2JointSpaceSchemesInmakingasinglesmoothmotion,atleastfourconstraintsonareevident:

127.2.1CubicPolynomials三次多项式插值三次多项式插值7.2JointSpaceSchemesCombiningthefourconstraintsyieldsfourequationswithfourunknowns:

137.2.1CubicPolynomials7.2JointSpaceSchemesThesefourconstraintsuniquelyspecifyaparticularcubic:

147.2.1CubicPolynomialsThejointvelocityandaccelerationalongthispathare:

7.2JointSpaceSchemesEg.7.1Asingle-linkrobotwitharotaryjointismotionlessat=15degrees.Itisdesiredtomovethejointinasmoothmannerto=75degreesin3seconds.Findthecoefficientsofacubicwhichaccomplishesthismotionandbringsthemanipulatortorestatthegoal.Plottheposition,velocity,andaccelerationofthejointasafunctionoftime.157.2.1CubicPolynomials7.2JointSpaceSchemesSolution:

Plugging0=15，f=75，tf=3into（7.6）,wefind167.2.1CubicPolynomials7.2JointSpaceSchemesSolution:

177.2.1CubicPolynomials7.2JointSpaceSchemesStartsat15degreesandendsat75degrees!

Solution:

187.2.1CubicPolynomials7.2JointSpaceSchemesStartsandendsatrest!

Solution:

197.2.1CubicPolynomials7.2JointSpaceSchemesAccelerationprofileislinear!

Ifwecometorestateachpointuseformulafrompreviousslideorcontinuousmotion（nostops）needvelocitiesatintermediatepoints:

InitialConditions:

207.2.2Cubicpolynomialswithviapoints过路径点的三次多项式插值过路径点的三次多项式插值7.2JointSpaceSchemesSolutions:

Howtospecifyvelocityattheviapoints:

TheuserspecifiesthedesiredvelocityateachviapointintermsofaCartesianlinearandangularvelocityofthetoolframeatthatinstant.ThesystemautomaticallychoosesthevelocitiesattheviapointsbyapplyingasuitableheuristicineitherCartesianspaceorjointspace（averageof2sidesetc.）.Thesystemautomaticallychoosesthevelocitiesattheviapointsinsuchawayastocausetheaccelerationattheviapointstobecontinuous.217.2JointSpaceSchemes7.2.2CubicpolynomialswithviapointsHigherorderpolynomialsaresometimesusedforpathsegments.Forexample,ifwewishtobeabletospecifythepositi