第7章-轨迹规划课件

第6章机器人轨迹规划6.1GeneralConsiderationsinRobotTrajectoryPlanning6.2InterpolatedCalculationofJointTrajectories6.3PlanningofCartesianPathTrajectories6.4RealTimeGenerationofPlanningTrajectories6.5Summary6.1GeneralConsiderationsinTrajectoryPlanning轨迹规划应考虑的问题BasicProblem: Movethemanipulatorarmfromsomeinitialpositiontosomedesiredfinalposition(Maybegoingthroughsomeviapoints).Trajectory:Timehistoryofposition,velocityandaccelerationforeachDOFPathpoints:Initial,finalandviapointsConstraints:Spatial,time,smoothness轨迹规划应考虑的问题JointspaceEasytogothroughviapoints (Solveinversekinematicsatallpathpoints)NoproblemswithsingularitiesLesscalculationsCannotfollowstraightlineGeneralConsiderations-SolutionSpaceCartesianspaceWecantrackashape (fororientation:equivalentaxes,Eulerangles,…)Moreexpensiveatruntime (afterthepathiscalculatedneedjointanglesinalotofpoints)DiscontinuityproblemsGeneralConsiderations-SolutionSpaceCartesianplanningdifficulties:GeneralConsiderations-SolutionSpace Initial(A)andGoal(B)Pointsarereachable,butintermediatepoints(C)unreachable.Joint-SpaceSchemesEachpathpointis"converted"intoasetofdesiredjointanglesbyapplicationoftheinversekinematics.Asmoothfunctionisfoundforeachofthenjointswhichpassthroughtheviapointsandendatthegoalpoint.Timerequiredforeachsegmentisthesameforeachjoint.Thedeterminationofthedesiredjointanglefunctionforaparticularjointisindependentwithotherjoints.7.2InterpolatedCalculationofJoint

Trajectories

关节轨迹的插值计算Choiceofinterpolationfunctionisnotunique!Joint-SpaceSchemes

Severalpossiblepathshapesforasinglejoint.Somepossibleinterpolationfunctions:CubicpolynomialsCubicpolynomialsforapathwithviapointsHigher-orderpolynomialsLinearfunctionwithparabolicblendsLinearfunctionwithparabolicblendsforapathwithviapointsJoint-SpaceSchemes

Inmakingasinglesmoothmotion,atleastfourconstraints

:7.2.1Cubic

Polynomials三次多项式插值Thesefourconstraintsuniquelyspecifyaparticularcubic:

Thejointvelocityandaccelerationalongthispathare:Cubic

PolynomialsCubic

Polynomials-4个系数Eg.7.1

Asingle-linkrobotwitharotaryjointismotionlessatθ=15degrees.Itisdesiredtomovethejointinasmoothmannertoθ=75degreesin3seconds.Findthecoefficientsofacubicwhichaccomplishesthismotionandbringsthemanipulatortorestatthegoal.Plottheposition,velocity,andaccelerationofthejointasafunctionoftime.ExampleExample关节运动轨迹-位置Startsat15degreesandendsat75degrees!Startsandendsatrest!关节运动轨迹-速度Accelerationprofileislinear!关节运动轨迹-关节加速度Ifwecometorestateachpoint useformulafrompreviousslideorcontinuousmotion(nostops) needvelocitiesatintermediatepoints:InitialConditions:6.2.2Cubic

polynomialswithviapoints

过路径点的三次多项式插值DesiredVelocityatthevapointsTheuserspecifiesthedesiredvelocityateachviapointintermsofaCartesianlinearandangularvelocityofthetoolframeatthatinstant.ThesystemautomaticallychoosesthevelocitiesattheviapointsbyapplyingasuitableheuristicineitherCartesianspaceorjointspace(averageof2sidesetc.).Thesystemautomaticallychoosesthevelocitiesattheviapointsinsuchawayastocausetheaccelerationattheviapointstobecontinuous.1用Jacob矩阵计算如果Jacob矩阵奇异，需用户指定速度用户负担2启发式方法Higherorderpolynomialsaresometimesusedforpathsegments.Forexample,ifwewishtobeabletospecifytheposition,velocity,andaccelerationatthebeginningandendofapathsegment,aquinticpolynomialisrequired:6.2.3Higher-order

polynomials

高阶多项式插值Wheretheconstraintsaregivenas:Higher-order

polynomials

高阶多项式插值Solutiontotheseequations:Higher-order

polynomials

高阶多项式插值Linearinterpolation(Straightline):Note:Althoughthemotionofeachjointinthisschemeislinear,theend-effectoringeneraldoesnotmoveinastraightlineinspace.6.2.4Linearfunctionwithparabolicblends

用抛物线过渡的线性插值Discontinuousvelocity-cannotbecontrolled!Tocreateasmoothpathwithcontinouspositionandvelocity,westartwiththelinearfunctionbutaddaparabolicblendregionateachpathpoint.Constantaccelerationisusedduringtheblendportiontochangevelocitysmoothly.LinearfunctionwithparabolicblendsAssumethattheparabolicblendsbothhavethesameduration,andthereforethesameconstantacceleration(moduloasign).Therearemanysolutionstotheproblem-buttheanswerisalwayssymmetricaboutthehalfwaypoint.LinearfunctionwithparabolicblendsThevelocityattheendoftheblendregionmustequalthevelocityofthelinearsection:LinearfunctionwithparabolicblendsLett=2th，combining(7.13)

and(7.14)Theaccelerationchosenmustbesufficientlyhigh,toensuretheexistenceofasolution:LinearfunctionwithparabolicblendsBelowshowsasetofjointspaceviapointsforsomejoints.Linearfunctionsconnecttheviapoints,andparabolicblendregionsareaddedaroundeachviapoint.6.2.5Linearfunctionwithparabolicblends

forapathwithviapoints过路径点的用抛物线过渡的线性插值Multi-segmentlinearpathwithblends.Given:positionsdesiredtimedurationsthemagnitudesoftheaccelerationsCompute:blendstimesstraightsegmenttimesslopes(velocities)signedaccelerationsLinearfunctionwithparabolicblends

forapathwithviapointsInsidesegment:Linearfunctionwithparabolicblends

forapathwithviapointsFirstsegment:Linearfunctionwithparabolicblends

forapathwithviapointsLastsegment:Linearfunctionwithparabolicblends

forapathwithviapointsTogothroughtheactualviapoints:Introduce“PseudoViaPoints”UsesufficientlyhighaccelerationLinearfunctionwithparabolicblends

forapathwithviapointsWhenpathshapesaredescribedintermsoffunctionsofCartesianpositionandorientation,wecanalsospecifythespatialshapeofthepathbetweenpathpoints.Themostcommonpathshapeisastraightline;butcircular,sinusoidal,orotherpathshapescouldbeused.Cartesianschemesaremorecomputationallyexpensivetoexecutesinceatruntime,inversekinematicsmustbesolvedatthepathupdaterate.6.3Cartesian-SpaceSchemesDescriptionofataskCartesian-SpaceSchemesCartesianstraightlinemotion MovefrompointPitoPi+1,whichdescribedbyrelativehomogenoustransformation:Cartesian-SpaceSchemesInordertoensurecontinuousvelocitiesintrajectory,asplineoflinearfunctionswithparabolicblendsisalwaysused.Duringthelinearportionofeachsegment,sinceallthreecomponentsofpositionchangeinalinearfashion,theend-effectorwillmovealongalinearpathinspace.Cartesian-SpaceSchemesCartesian-SpaceSchemes Initial(A)andGoal(B)Pointsarereachable,butintermediatepoints(C)unreachable.GeometricProblemswithCartesianPath

Approachingsingularities

somejointvelocities

goto∞causingdeviation

fromthepath.GeometricProblemswithCartesianPath

Startpoint(A)and

goalpoint(B)are

reachableindifferent

jointspacesolutions

(Themiddlepoints

arereachablefrom

below.)GeometricProblemswithCartesianPath6.4

PathGenerationatReal-TimeAtruntimethepathgeneratorroutineconstructsthetrajectory,usuallyintermsof

,andfeedsthisinformationtothemanipulator'scontrolsystem.Thispathgeneratorcomputesthetrajectoryatthepathupdaterate.Inthecaseofcubicsplines,thepathgeneratorsimplycomputes(7.3)and(7.4)astisadvanced.Whentheendofonesegmentisreached,anewsetofcubiccoefficientsisrecalled,tissetbacktozero,andthegenerationcontinues.6.4.1Path

GenerationinJointSpaceInthecaseoflinearsplineswithparabolicblends,thevalueoftime,t,ischeckedoneachupdatetodeterminewhetherwearecurrentlyinthelinearortheblendportionofthesegment.Inthelinearportion,thetrajectoryforeachjointiscalculatedasPath

GenerationinJointSpaceInthecaseoflinearsplineswithparabolicblends,thevalueoftime,t,ischeckedoneachupdatetodeterminewhetherwearecurrentlyinthelinearortheblendportionofthesegment.Intheblendregion,thetrajectoryforeachjointiscalculatedasPath

GenerationinJointSpaceInthecaseoflinearsplinewithparabolicblendspath.Rewrite(7.45)and(7.46)withthesymbolXrepresentingacomponentoftheCartesianpositionandorientationvector.Inthelinearportionofthesegment,eachdegreeoffreedominXiscalcuatedas6.4.2

GenerationofCartesianspacepathsInthecaseoflinearsplinewithparabolicblendspath.Rewrite(7