机器人学基础-第7章-机器人轨迹规划-蔡自兴.ppt

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机器人学基础-第7章-机器人轨迹规划-蔡自兴.ppt

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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,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-SpaceSchemesEachpathpointisconvertedintoasetofdesiredjointanglesbyapplicationoftheinversekinematics.Asmoothfunctionisfoundforeachofthenjointswhichpassthroughtheviapointsandendatthegoalpoint.Timerequiredforeachsegmentisthesameforeachjoint.Thedeterminationofthedesiredjointanglefunctionforaparticularjointisindependentwithotherjoints.,9,7.2InterpolatedCalculationofJointTrajectories关节轨迹的插值计算,7.2JointSpaceSchemes,Choiceofinterpolationfunctionisnotunique!

10,Joint-SpaceSchemes,7.2JointSpaceSchemes,Severalpossiblepathshapesforasinglejoint.,Somepossibleinterpolationfunctions:

CubicpolynomialsCubicpolynomialsforapathwithviapointsHigher-orderpolynomialsLinearfunctionwithparabolicblendsLinearfunctionwithparabolicblendsforapathwithviapoints,11,Joint-SpaceSchemes,7.2JointSpaceSchemes,Inmakingasinglesmoothmotion,atleastfourconstraintsonareevident:

12,7.2.1CubicPolynomials三次多项式插值,7.2JointSpaceSchemes,Combiningthefourconstraintsyieldsfourequationswithfourunknowns:

13,7.2.1CubicPolynomials,7.2JointSpaceSchemes,Thesefourconstraintsuniquelyspecifyaparticularcubic:

14,7.2.1CubicPolynomials,Thejointvelocityandaccelerationalongthispathare:

7.2JointSpaceSchemes,Eg.7.1Asingle-linkrobotwitharotaryjointismotionlessat=15degrees.Itisdesiredtomovethejointinasmoothmannerto=75degreesin3seconds.Findthecoefficientsofacubicwhichaccomplishesthismotionandbringsthemanipulatortorestatthegoal.Plottheposition,velocity,andaccelerationofthejointasafunctionoftime.,15,7.2.1CubicPolynomials,7.2JointSpaceSchemes,Solution:

Plugging0=15,f=75,tf=3into(7.6),wefind,16,7.2.1CubicPolynomials,7.2JointSpaceSchemes,Solution:

17,7.2.1CubicPolynomials,7.2JointSpaceSchemes,Startsat15degreesandendsat75degrees!

Solution:

18,7.2.1CubicPolynomials,7.2JointSpaceSchemes,Startsandendsatrest!

Solution:

19,7.2.1CubicPolynomials,7.2JointSpaceSchemes,Accelerationprofileislinear!

Ifwecometorestateachpointuseformulafrompreviousslideorcontinuousmotion(nostops)needvelocitiesatintermediatepoints:

InitialConditions:

20,7.2.2Cubicpolynomialswithviapoints过路径点的三次多项式插值,7.2JointSpaceSchemes,Solutions:

Howtospecifyvelocityattheviapoints:

TheuserspecifiesthedesiredvelocityateachviapointintermsofaCartesianlinearandangularvelocityofthetoolframeatthatinstant.ThesystemautomaticallychoosesthevelocitiesattheviapointsbyapplyingasuitableheuristicineitherCartesianspaceorjointspace(averageof2sidesetc.).Thesystemautomaticallychoosesthevelocitiesattheviapointsinsuchawayastocausetheaccelerationattheviapointstobecontinuous.,21,7.2JointSpaceSchemes,7.2.2Cubicpolynomialswithviapoints,Higherorderpolynomialsaresometimesusedforpathsegments.Forexample,ifwewishtobeabletospecifytheposition,velocity,andaccelerationatthebeginningandendofapathsegment,aquinticpolynomialisrequired:

22,7.2.3Higher-orderpolynomials高阶多项式插值,7.2JointSpaceSchemes,Wheretheconstraintsaregivenas:

23,7.2.3Higher-orderpolynomials,7.2JointSpaceSchemes,Solutiontotheseequations:

24,7.2.3Higher-orderpolynomials,7.2JointSpaceSchemes,Linearinterpolation(Straightline):

Note:

Althoughthemotionofeachjointinthisschemeislinear,theend-effectoringeneraldoesnotmoveinastraightlineinspace.,25,7.2.4Linearfunctionwithparabolicblends用抛物线过渡的线性插值,7.2JointSpace

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