机器人学基础-第4章-机器人动力学-蔡自兴.

1中南大学蔡自兴，谢斌zxcai,[e-mail]2010机器人学基础第四章机器人动力学1FundamentalsofRobotics2ContentsIntroductiontoDynamicsRigidBodyDynamicsLagrangianFormulationNewton-EulerFormulationArticulatedMulti-BodyDynamics2Ch.4ManipulatorDynamics33Ch.4ManipulatorDynamicsIntroductionCh.4ManipulatorDynamicsManipulatorDynamicsconsiderstheforcesrequiredtocausedesiredmotion.Consideringtheequationsofmotionarisesfromtorquesappliedbytheactuators,orfromexternalforcesappliedtothemanipulator.4Ch.4ManipulatorDynamicsTwomethodsforformulatingdynamicsmodel：Newton-EulerdynamicformulationNewton'sequationalongwithitsrotationalanalog,Euler'sequation,describehowforces,inertias,andaccelerationsrelateforrigidbodies,isaforcebalanceapproachtodynamics.LagrangiandynamicformulationLagrangianformulationisanenergy-basedapproachtodynamics.Ch.4ManipulatorDynamics5Ch.4ManipulatorDynamicsTherearetwoproblemsrelatedtothedynamicsofamanipulatorthatwewishtosolve.ForwardDynamics:givenatorquevector,Τ,calculatetheresultingmotionofthemanipulator,.Thisisusefulforsimulatingthemanipulator.InverseDynamics:givenatrajectorypoint,,findtherequiredvectorofjointtorques,Τ.Thisformulationofdynamicsisusefulfortheproblemofcontrollingthemanipulator.Ch.4ManipulatorDynamics,,and,,and6ContentsIntroductiontoDynamicsRigidBodyDynamicsLagrangianFormulationNewton-EulerFormulationArticulatedMulti-BodyDynamics6Ch.4ManipulatorDynamics774.1DynamicsofaRigidBody刚体动力学LangrangianFunctionLisdefinedas:DynamicEquationofthesystem(LangrangianEquation):whereqiisthegeneralizedcoordinates,representcorrespondingvelocity,Fistandforcorrespondingtorqueorforceontheithcoordinate.4.1DynamicsofaRigidBodyLPKniqLqLdtdiii,2,1,F(4.1)(4.2)iqKineticEnergyPotentialEnergy84.1.1KineticandPotentialEnergyofaRigidBody82211001122KMMxx0011201)(21gxgxxxMMkP2101()2Dcxx01FxFxWFFx0x1M0kcM1图4.1一般物体的动能与位能4.1DynamicsofaRigidBody4.1DynamicsofaRigidBody99isageneralizedcoordinate①KineticEnergydueto(angular)velocity②KineticEnergyduetoposition(orangle)③DissipationEnergydueto(angular)velocity④PotentialEnergyduetoposition⑤ExternalForceorTorque010,xx11111xWxPxDxKxKdtd4.1.1KineticandPotentialEnergyofaRigidBodyFFx0x1M0kcM14.1DynamicsofaRigidBody①②③④⑤1010x0andx1arebothgeneralizedcoordinatesFgxxxxx1010111)()(MkcMFgxxxxx0010100)()(MkcM1111000000McckkMcckkxxxFxxxF4.1.1KineticandPotentialEnergyofaRigidBodyFFx0x1M0kcM14.1DynamicsofaRigidBodyWritteninMatricesform:1111KineticandPotentialEnergyofa2-linksmanipulatorKineticEnergyK1andPotentialEnergyP1oflink121111111111111,,,cos2KmvvdPmghhd22111111111,cos2KmdPmgd图4.2二连杆机器手（1）xyθ1θ2T2T1d1d2m2m1(x1,y1)g(x2,y2)4.1.1KineticandPotentialEnergyofaRigidBody4.1DynamicsofaRigidBody1212KineticEnergyK2andPotentialEnergyP2oflink2xyθ1θ2T2T1d1d2m2m1(x1,y1)g(x2,y2)222222211212211212sinsincoscosvxyxddydd2222221,2KmvPmgywhere222222211221221221122211221211cos22coscosKmdmdmddPmgdmgd4.1.1KineticandPotentialEnergyofaRigidBody4.1DynamicsofaRigidBody13TotalKineticandPotentialEnergyofa2-linksmanipulatorare13(4.3)21KKK2222121122122212211211()()22cos()mmdmdmdd21PPP)cos(cos)(21221121gdmgdmm(4.4)4.1.1KineticandPotentialEnergyofaRigidBodyxyθ1θ2T2T1d1d2m2m1(x1,y1)g(x2,y2)4.1DynamicsofaRigidBody14ContentsIntroductiontoDynamicsRigidBodyDynamicsLagrangianFormulationNewton-EulerFormulationArticulatedMulti-BodyDynamics14Ch.4ManipulatorDynamics1515LagrangianFormulationLagrangianFunctionLofa2-linksmanipulator:PKL)2(21)(21222121222212121dmdmm)cos(cos)()(cos2122112121212212gdmgdmmddm(4.5)xyθ1θ2T2T1d1d2m2m1(x1,y1)g(x2,y2)niqLqLdtdiii,2,1,F4.1DynamicsofaRigidBody4.1.2TwoSolutionsforDynamicEquation16164.1.2TwoSolutionsforDynamicEquationLagrangianFormulationDynamicEquations:2112212212121211122221222211122111212221121121DDDDDDDDDDDDDDTT(4.10)xyθ1θ2T2T1d1d2m2m1(x1,y1)g(x2,y2)111dLLTdt222dLLTdtWritteninMatricesForm:(4.6)(4.7)有效惯量(effectiveinertial)：关节i的加速度在关节i上产生的惯性力4.1DynamicsofaRigidBody17WritteninMatricesForm:17LagrangianFormulationDynamicEquations:2112212212121211122221222211122111212221121121DDDDDDDDDDDDDDTT(4.10)xyθ1θ2T2T1d1d2m2m1(x1,y1)g(x2,y2)111dLLTdt222dLLTdt(4.6)(4.7)耦合惯量(coupledinertial)：关节i,j的加速度在关节j,i上产生的惯性力4.1.2TwoSolutionsforDynamicEquation4.1DynamicsofaRigidBody18WritteninMatricesForm:18LagrangianFormulationDynamicEquations:2112212212121211122221222211122111212221121121DDDDDDDDDDDDDDTT(4.10)xyθ1θ2T2T1d1d2m2m1(x1,y1)g(x2,y2)111dLLTdt222dLLTdt(4.6)(4.7)向心加速度(accelerationcentripetal)系数关节i,j的速度在关节j,i上产生的向心力4.1.2TwoSolutionsforDynamicEquation4.1DynamicsofaRigidBody19WritteninMatricesForm:19LagrangianFormulationDynamicEqua