机器人学基础-第4章-机器人动力学-蔡自兴

1、1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四章第四章 机器人动力学机器人动力学1Fundamentals of Robotics2Contents Introduction to Dynamics Rigid Body Dynamics Lagrangian Formulation Newton-Euler Formulation Articulated Multi-Body Dynamics2Ch.4 Manipulator Dynamics3 3Ch.4 Manipulator DynamicsIntroductionCh.4 Manipu

2、lator DynamicsManipulator Dynamics considers the forces required to cause desired motion. Considering the equations of motion arises from torques applied by the actuators, or from external forces applied to the manipulator.4Ch.4 Manipulator DynamicsTwo methods for formulating dynamics model：Newton-E

3、uler dynamic formulationNewtons equation along with its rotational analog, Eulers equation, describe how forces, inertias, and accelerations relate for rigid bodies, is a force balance approach to dynamics.Lagrangian dynamic formulationLagrangian formulation is an energy-based approach to dynamics.

4、Ch.4 Manipulator Dynamics5Ch.4 Manipulator DynamicsThere are two problems related to the dynamics of a manipulator that we wish to solve. Forward Dynamics: given a torque vector, , calculate the resulting motion of the manipulator, . This is useful for simulating the manipulator.Inverse Dynamics: gi

5、ven a trajectory point, , find the required vector of joint torques, . This formulation of dynamics is useful for the problem of controlling the manipulator.Ch.4 Manipulator Dynamics,and ,and 6Contents Introduction to Dynamics Rigid Body Dynamics Lagrangian Formulation Newton-Euler Formulation Artic

6、ulated Multi-Body Dynamics6Ch.4 Manipulator Dynamics7 74.1 Dynamics of a Rigid Body 刚体动力学刚体动力学Langrangian Function L is defined as:Dynamic Equation of the system (Langrangian Equation): where qi is the generalized coordinates, represent corresponding velocity, Fi stand for corresponding torque or fo

7、rce on the ith coordinate. 4.1 Dynamics of a Rigid BodyLPKniqLqLdtdiii, 2 , 1, F(4.1)(4.2)iq Kinetic EnergyPotential Energy84.1.1 Kinetic and Potential Energy of a Rigid Body82211001122KMMxx0011201)(21gxgxxxMMkP2101()2Dcxx01FxFx W图4.1 一般物体的动能与位能4.1 Dynamics of a Rigid Body4.1 Dynamics of a Rigid Bod

8、y9 9 is a generalized coordinate Kinetic Energy due to (angular) velocity Kinetic Energy due to position (or angle) Dissipation Energy due to (angular) velocity Potential Energy due to position External Force or Torque010,xx11111xWxPxDxKxKdtd4.1.1 Kinetic and Potential Energy of a Rigid Body4.1 Dyna

9、mics of a Rigid Body 1010 x0 and x1 are both generalized coordinatesFgxxxxx1010111)()(MkcM Fgxxxxx0010100)()(MkcM 1111000000McckkMcckkxxxFxxxF4.1.1 Kinetic and Potential Energy of a Rigid Body4.1 Dynamics of a Rigid BodyWritten in Matrices form:1111Kinetic and Potential Energy of a 2-links manipulat

10、or Kinetic Energy K1 and Potential Energy P1 of link 1 211 111 11111111,cos2Km vvdPm ghhd 22111111111,cos2Km dPm gd 图4.2 二连杆机器手（1）4.1.1 Kinetic and Potential Energy of a Rigid Body4.1 Dynamics of a Rigid Body1212Kinetic Energy K2 and Potential Energy P2 of link 2 222222211212211212sinsincoscosvxyxdd

11、ydd 2222221,2Km vPmgywhere2222222112212212211 22211221211cos22coscosKm dm dm d dPm gdm gd 4.1.1 Kinetic and Potential Energy of a Rigid Body4.1 Dynamics of a Rigid Body13Total Kinetic and Potential Energy of a 2-links manipulator are13(4.3)21KKK2222121122122212211211()()22cos()mm dm dm d d 21PPP)cos

12、(cos)(21221121gdmgdmm(4.4)4.1.1 Kinetic and Potential Energy of a Rigid Body4.1 Dynamics of a Rigid Body14Contents Introduction to Dynamics Rigid Body Dynamics Lagrangian Formulation Newton-Euler Formulation Articulated Multi-Body Dynamics14Ch.4 Manipulator Dynamics1515Lagrangian Formulation Lagrang

13、ian Function L of a 2-links manipulator: PKL)2(21)(21222121222212121dmdmm)cos(cos)()(cos2122112121212212gdmgdmmddm(4.5)niqLqLdtdiii, 2 , 1, F4.1 Dynamics of a Rigid Body4.1.2 Two Solutions for Dynamic Equation16164.1.2 Two Solutions for Dynamic EquationLagrangian Formulation Dynamic Equations:211221

14、2212121211122221222211122111212221121121DDDDDDDDDDDDDDTT (4.10)111dLLTdt222dLLTdtWritten in Matrices Form:(4.6)(4.7)有效惯量(effective inertial)：关节i的加速度在关节i上产生的惯性力4.1 Dynamics of a Rigid Body17Written in Matrices Form:17Lagrangian Formulation Dynamic Equations:2112212212121211122221222211122111212221121

15、121DDDDDDDDDDDDDDTT (4.10)111dLLTdt222dLLTdt(4.6)(4.7)耦合惯量(coupled inertial)：关节i,j的加速度在关节j,i上产生的惯性力4.1.2 Two Solutions for Dynamic Equation4.1 Dynamics of a Rigid Body18Written in Matrices Form:18Lagrangian Formulation Dynamic Equations:2112212212121211122221222211122111212221121121DDDDDDDDDDDDDDTT

16、(4.10)111dLLTdt222dLLTdt(4.6)(4.7)向心加速度(acceleration centripetal)系数关节i,j的速度在关节j,i上产生的向心力4.1.2 Two Solutions for Dynamic Equation4.1 Dynamics of a Rigid Body19Written in Matrices Form:19Lagrangian Formulation Dynamic Equations:2112212212121211122221222211122111212221121121DDDDDDDDDDDDDDTT (4.10)111dL

17、LTdt222dLLTdt(4.6)(4.7)哥氏加速度(Coriolis accelaration)系数：关节j,k的速度引起的在关节i上产生的哥氏力(Coriolis force)4.1.2 Two Solutions for Dynamic Equation4.1 Dynamics of a Rigid Body20Written in Matrices Form:20Lagrangian Formulation Dynamic Equations:2112212212121211122221222211122111212221121121DDDDDDDDDDDDDDTT (4.10)1

18、11dLLTdt222dLLTdt(4.6)(4.7)重力项(gravity)：关节i,j处的重力4.1.2 Two Solutions for Dynamic Equation4.1 Dynamics of a Rigid Body2121对上例指定一些数字，以估计此二连杆机械手在静止和固定重力负载下的 T1 和 T2 的数值。取 d1=d2=1，m1=2，计算m2=1,4和100（分别表示机械手在地面空载地面空载、地面满载地面满载和在外空间负在外空间负载载的三种不同情况；在外空间由于失重而允许有较大的负载）三个不同数值下各系数的数值。Lagrangian Formulation of Ma

19、nipulator Dynamics4.1 Dynamics of a Rigid Body2222表4.1给出这些系数值及其与位置 的关系。 表4.1 222cos11D12D22D1I2ILagrangian Formulation of Manipulator Dynamics注意：有效惯量的变化将对机械手的控制产生显著影响！4.1 Dynamics of a Rigid Body23Contents Introduction to Dynamics Rigid Body Dynamics Lagrangian Formulation Newton-Euler Formulation A

20、rticulated Multi-Body Dynamics23Ch.4 Manipulator Dynamics24Fm a4.1 Dynamics of a Rigid Body4.1.2 Two Solutions for Dynamic EquationNewton-Euler Dynamic FormulationNewtons Lawrate of change of the linear momentum is equal to the applied forcedmvFdtLinear Momentummv25pviiiim4.1 Dynamics of a Rigid Bod

21、y4.1.2 Two Solutions for Dynamic EquationNewton-Euler Dynamic FormulationRotational MotionppiiiimAngular Momentum:densityimdv26I4.1 Dynamics of a Rigid Body4.1.2 Two Solutions for Dynamic EquationNewton-Euler Dynamic FormulationRotational MotionppVdvAngular MomentumppVdvInertia Tensor27Newton-Euler

22、Dynamic FormulationccFvmNIIcc)(where m is the mass of a rigid body, represent inertia tensor, FC is the external force on the center of gravity, N is the torque on the rigid body, vC represent the translational velocity, while is the angular velocity.（Euler Equation）3 3CIR（Newton Equation）4.1 Dynami

23、cs of a Rigid Body4.1.2 Two Solutions for Dynamic Equation28例1. 求解下图所示的1自由度机械手的运动方程式，在这里，由于关节轴制约连杆的运动，所以可以将运动方程式看作是绕固定轴的运动。1自由度机械手解：假设绕关节轴的惯性矩为 I，取垂直纸面的方向为 z 轴，则有 II000000000IIcos00cmgLN4.1 Dynamics of a Rigid Body4.1.2 Two Solutions for Dynamic Equation29该式即为1自由度机械手的欧拉运动方程式。coscmgLI 由欧拉运动方程式()CCI I

24、 N II000000000IIcos00cmgLN4.1 Dynamics of a Rigid Body4.1.2 Two Solutions for Dynamic Equation3030Langrangian Function L is defined as:Dynamic Equation of the system (Langrangian Equation): where qi is the generalized coordinates, represent corresponding velocity, Fi stand for corresponding torque o

25、r force on the ith coordinate. 4.1 Dynamics of a Rigid BodyLPKniqLqLdtdiii, 2 , 1, F(4.1)(4.2)iq Kinetic EnergyPotential Energy4.1.2 Two Solutions for Dynamic Equation31例例2.通过拉格朗日运动方程式求解之前推导的1自由度机械手。221IK sincmgLP 21sin2cLKPImgLILcoscmgLLcoscImgL解：假设为广义坐标，则有 ddLLtqq由拉格朗日运动方程4.1 Dynamics of a Rigid B

26、ody4.1.2 Two Solutions for Dynamic Equation32我们研究动力学的重要目的之一是为了对机器人的运动进行有效控制，以实现预期的轨迹运动。常用的方法有牛顿欧拉法、拉格朗日法等。牛顿欧拉动力学法是利用牛顿力学牛顿力学的刚体力学刚体力学知识导出逆动力学的递推计算公式，再由它归纳出机器人动力学的数学模型机器人矩阵形式的运动学方程；拉格朗日法是引入拉格朗日方程拉格朗日方程直接获得机器人动力学方程的解析公式，并可得到其递推计算方法。4.1 Dynamics of a Rigid Body4.1.2 Two Solutions for Dynamic Equation3

27、3对多自由度的机械手，拉格朗日法可以直接推导直接推导运动方程式，但随着自由度的增多演算量将大量增加大量增加。与此相反，牛顿欧拉法着眼于每一个连杆的运动，即便对于多自由度的机械手其计算量也不增加计算量也不增加，因此算法易于编程。由于推导出的是一系列公式的组合，要注意惯性矩阵等的选惯性矩阵等的选择和求解择和求解问题。进一步的问题请参考相关文献资料。 4.1 Dynamics of a Rigid Body4.1.2 Two Solutions for Dynamic Equation34Contents Introduction to Dynamics Rigid Body Dynamics La

28、grangian Formulation Newton-Euler Formulation Articulated Multi-Body Dynamics34Ch.4 Manipulator Dynamics354.2 Dynamic Equation of a Manipulator 机械手的动力学方程机械手的动力学方程Forming dynamic equation of any manipulator described by a series of A-matrices:(1) Computing the Velocity of any given point;(2) Computin

29、g total Kinetic Energy;(3) Computing total Potential Energy;(4) Forming Lagrangian Function of the system;(5) Forming Dynamic Equation through Lagrangian Equation.354.2 Dynamic Equation of a Manipulator36364.2.1 Computation of Velocity 速度的计算速度的计算Velocity of point P on link-3:Velocity of any given po

30、int on link-i: ppppTTdtddtdrrrv333300)()(rrviijjjiqqTdtd1(4.15)图4.4 四连杆机械手4.2 Dynamic Equation of a Manipulator37374.2.1 Computing the VelocityAcceleration of point P:30033331()()pppipjiTdddTqdtdtdtqavrrpkjjkkjpjiiqqqqTqdtdqTrr33131323313pkjjkkjpjiiqqqqTqqTrr33131323313 图4.4 四连杆机械手4.2 Dynamic Equati

31、on of a Manipulator3838Square of velocityThe trace of an square matrix is defined to be the sum of the diagonal elements. )()()()()(000020TpppppTracevvvvv31313333)()(jkTpkkpjjqqTqqTTracerr31333133)( )(jkjkTTpkpjqqqTqTTracerr4.2.1 Computing the Velocity图4.4 四连杆机械手4.2 Dynamic Equation of a Manipulator

32、3939Square of velocity of any given point:ijikTikkiijjiqqTqqTTracedtdr1122rrvijkkTkiTikiikiqqqTqTTrace11rr(4.16)4.2.1 Computing the Velocity图4.4 四连杆机械手4.2 Dynamic Equation of a Manipulator4040Computing the Kinetic Energy 令连杆3上任一质点P的质量为dm，则其动能为：dmvdKp2321dmqqqTrrqTTracejkiTkkTppi31313333)(2131313333)

33、(21jkiTkkTTppiqqqTrdmrqTTrace图4.4 四连杆机械手4.2 Dynamic Equation of a Manipulator4.2.2 Computation of Kinetic and Potential Energy 动能和位能的计算 41414.2.2 Computation of Kinetic and Potential EnergyKinetic Energy of any particle on link-i with position vector ir :Kinetic Energy of link-3:dmqqqTqTTracedKijkji

34、kkTiTijjii1121rrijkjikkTiTTiijiqqqTdmqTTrace11)(21rr3333333311link3link31()2TTppjkjkjkTTKdKTracedmq qqq rr4.2 Dynamic Equation of a Manipulator4242Kinetic Energy of any given link-i:Total Kinetic Energy of the manipulator:linkiiiKdK ijikkjkiijiqqqTIqTTrace1121(4.17)ninjkiikkTiijiniiqqqTIqTraceTKK111

35、121(4.19)4.2.2 Computation of Kinetic and Potential Energy4.2 Dynamic Equation of a Manipulator4343Computing the Potential EnergyPotential Energy of a object (mass m) at h height:so the Potential Energy of any particle on link-i with position vector ir :whererdmTrdmdPiiTTigg0mghP 1 ,zyxTgggglink lin

36、k link TiTiiiiiiiiPdPT rdmTrdm ggiiiTiiiiiTrTmrmTgg4.2.2 Computation of Kinetic and Potential Energy4.2 Dynamic Equation of a Manipulator4444Potential Energy of any particle on link-i with position vector ir :Total Potential Energy of the manipulator: rdmTrdmdPiiTTigg0niniiaiiPPPP11)(niiiiTirTm1g (4

37、.21)4.2.2 Computation of Kinetic and Potential Energy4.2 Dynamic Equation of a Manipulator4545Lagrangian FunctionPKLt,2121112111niniiiiTiiainiijikkjkTiiiirTmqIqqqTIqTTraceg2 , 1n (4.22)4.2 Dynamic Equation of a Manipulator4.2.3 Forming the Dynamic Equation 动力学方程的推导动力学方程的推导46464.2.3 Forming the Dynamic EquationDerivative of Lagrangian function nikikkTiipipqqTIqTTraceqL11211112TniiiijappijipTTTraceIqI qqqnp, 2 , 14.2 Dynamic Equation of a Manipulator4747According to Eq.(4.18), Ii is a symmetric matrix, lead toTTTTiiiiiiiiijkkjkjTTTTTTTraceITraceITraceIqqqqqqnipapkikpTiikipqIqqTIqTTraceqL11