机器人学基础_第7章_机器人轨迹规划.ppt

中南大学蔡自兴，谢斌zxcai,xiebin2010,机器人学基础第七章机器人轨迹规划,1,Ch.7TrajectoryPlanningofRobots,FundamentalsofRobotics,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,7.2JointSpaceSchemes,Discontinuousvelocity-cannotbecontrolled!,Tocreateasmoothpathwithcontinouspositionandvelocity,westartwiththelinearfunctionbutaddaparabolicblendregionateachpathpoint.Constantaccelerationisusedduringtheblendportiontochangevelocitysmoothly.,26,7.2.4Linearfunctionwithparabolicblends,7.2JointSpaceSchemes,Assumethattheparabolicblendsbothhavethesameduration,andthereforethesameconstantacceleration(moduloasign).Therearemanysolutionstotheproblem-buttheanswerisalwayssymmetricaboutthehalfwaypoint.,27,7.2.4Linearfunctionwithparabolicblends,7.2JointSpaceSchemes,Thevelocityattheendoftheblendregionmustequalthevelocityofthelinearsection:,28,7.2.4Linearfunctionwithparabolicblends,7.2JointSpaceSchemes,Lett=2th，combining(7.13)and(7.14),29,7.2.4Linearfunctionwithparabolicblends,Theaccelerationchosenmustbesufficientlyhigh,toensuretheexistenceofasolution:,7.2JointSpaceSchemes,Belowshowsasetofjointspaceviapointsforsomejoints.Linearfunctionsconnecttheviapoints,andparabolicblendregionsareaddedaroundeachviapoint.,30,7.2.5Linearfunctionwithparabolicblendsforapathwithviapoints,过路径点的用抛物线过渡的线性插值,7.2JointSpaceSchemes,Multi-segmentlinearpathwithblends.,Given:positionsdesiredtimedurationsthemagnitudesoftheaccelerationsCompute:blendstimesstraightsegmenttimesslopes(velocities)signedaccelerations,31,7.2JointSpaceSchemes,7.2.5Linearfunctionwithparabolicblendsforapathwithviapoints,Insidesegment:,32,7.2JointSpaceSchemes,7.2.5Linearfunctionwithparabolicblendsforapathwithviapoints,Firstsegment:,33,7.2JointSpaceSchemes,7.2.5Linearfunctionwithparabolicblendsforapathwithviapoints,Lastsegment:,34,7.2JointSpaceSchemes,7.2.5Linearfunctionwithparabolicblendsforapathwithviapoints,Togothroughtheactualviapoints:Introduce“PseudoViaPoints”Usesufficientlyhighacceleration,35,7.2JointSpaceSchemes,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,Approachingsingularitiessomejointvelocitiesgotocausingdeviationfromthepath.,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,andfeedsthisinformationtothemanipulatorscontrolsystem.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,Inthecaseo