1、1,中南大学 蔡自兴，谢 斌 zxcai, 2010,机器人学基础 第四章 机器人动力学,1,Ch.4 Manipulator Dynamics,Fundamentals of Robotics,Fundamentals of Robotics,2,Contents, Introduction to Dynamics Rigid Body Dynamics Lagrangian Formulation Newton-Euler Formulation Articulated Multi-Body Dynamics,2,Ch.4 Manipulator Dynamics,3,3,Ch.4 Ma
2、nipulator Dynamics,Introduction,Ch.4 Manipulator Dynamics,Manipulator 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.,4,Ch.4 Manipulator Dynamics,Two
3、methods for formulating dynamics model： Newton-Euler dynamic formulation Newtons 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 formulation Lagrangian
4、formulation is an “energy-based“ approach to dynamics.,Ch.4 Manipulator Dynamics,5,Ch.4 Manipulator Dynamics,There 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
5、useful for simulating the manipulator. Inverse Dynamics: given 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,6,Contents, Introduction to Dynamics Rigid Body Dynamics
6、Lagrangian Formulation Newton-Euler Formulation Articulated Multi-Body Dynamics,6,Ch.4 Manipulator Dynamics,7,7,4.1 Dynamics of a Rigid Body 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 correspondi
7、ng velocity, Fi stand for corresponding torque or force on the ith coordinate.,4.1 Dynamics of a Rigid Body,Kinetic Energy,Potential Energy,8,4.1.1 Kinetic and Potential Energy of a Rigid Body,8,图4.1 一般物体的动能与位能,4.1 Dynamics of a Rigid Body,4.1 Dynamics of a Rigid Body,9,9,is a generalized coordinate
8、 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 Torque,4.1.1 Kinetic and Potential Energy of a Rigid Body,4.1 Dynamics of a Rigid Body, ,10,10,x0 and x1 are both genera
9、lized coordinates,4.1.1 Kinetic and Potential Energy of a Rigid Body,4.1 Dynamics of a Rigid Body,Written in Matrices form:,11,11,Kinetic and Potential Energy of a 2-links manipulator,Kinetic Energy K1 and Potential Energy P1 of link 1,图4.2 二连杆机器手（1）,4.1.1 Kinetic and Potential Energy of a Rigid Bod
10、y,4.1 Dynamics of a Rigid Body,12,12,Kinetic Energy K2 and Potential Energy P2 of link 2,where,4.1.1 Kinetic and Potential Energy of a Rigid Body,4.1 Dynamics of a Rigid Body,13,Total Kinetic and Potential Energy of a 2-links manipulator are,13,4.1.1 Kinetic and Potential Energy of a Rigid Body,4.1
11、Dynamics of a Rigid Body,14,Contents, Introduction to Dynamics Rigid Body Dynamics Lagrangian Formulation Newton-Euler Formulation Articulated Multi-Body Dynamics,14,Ch.4 Manipulator Dynamics,15,15,Lagrangian FormulationLagrangian Function L of a 2-links manipulator:,4.1 Dynamics of a Rigid Body,4.1
12、.2 Two Solutions for Dynamic Equation,16,16,4.1.2 Two Solutions for Dynamic Equation,Lagrangian FormulationDynamic Equations:,Written in Matrices Form:,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 Body,17,Written in Matrices Form:,17,Lagrangian FormulationDynamic Equations:,耦合
13、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 Equation,4.1 Dynamics of a Rigid Body,18,Written in Matrices Form:,18,Lagrangian FormulationDynamic Equations:,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 Equation,4.1 Dynamics of a
14、 Rigid Body,19,Written in Matrices Form:,19,Lagrangian FormulationDynamic Equations:,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 Equation,4.1 Dynamics of a Rigid Body,20,Written in Matrices Form:,20,Lagrangian FormulationDynamic Equations:,重
15、力项(gravity)：关节i,j处的重力,4.1.2 Two Solutions for Dynamic Equation,4.1 Dynamics of a Rigid Body,21,21,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、地面满载和在外空间负载的三种不同情况；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 Manipulator Dynamics,4.1 Dynamics of
16、a Rigid Body,22,22,表4.1给出这些系数值及其与位置 的关系。表4.1,Lagrangian 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 Body,23,Contents, Introduction to Dynamics Rigid Body Dynamics Lagrangian Formulation Newton-Euler Formulation Articulated Multi-Body Dynamics,23,Ch.4 Manipul
17、ator Dynamics,24,4.1 Dynamics of a Rigid Body,4.1.2 Two Solutions for Dynamic Equation,Newton-Euler Dynamic Formulation,Newtons Law,rate of change of the linear momentum is equal to the applied force,Linear Momentum,25,4.1 Dynamics of a Rigid Body,4.1.2 Two Solutions for Dynamic Equation,Newton-Eule
18、r Dynamic Formulation,Rotational Motion,Angular Momentum,26,4.1 Dynamics of a Rigid Body,4.1.2 Two Solutions for Dynamic Equation,Newton-Euler Dynamic Formulation,Rotational Motion,Angular Momentum,Inertia Tensor,27,Newton-Euler Dynamic Formulation,where m is the mass of a rigid body, represent iner
19、tia 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）,（Newton Equation）,4.1 Dynamics of a Rigid Body,4.1.2 Two Solutions for Dynamic Equation,28,例1. 求解下图所示的1自由度机械手的运动方
20、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,4.1 Dynamics of a Rigid Body,4.1.2 Two Solutions for Dynamic Equation,29,该式即为1自由度机械手的欧拉运动方程式。,由欧拉运动方程式,4.1 Dynamics of a Rigid Body,4.1.2 Two Solutions for Dynamic Equation,30,30,Langrangian Function L is defined as:D
21、ynamic Equation of the system (Langrangian Equation):where qi is the generalized coordinates, represent corresponding velocity, Fi stand for corresponding torque or force on the ith coordinate.,4.1 Dynamics of a Rigid Body,Kinetic Energy,Potential Energy,4.1.2 Two Solutions for Dynamic Equation,31,例
22、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。,解：假设为广义坐标，则有,由拉格朗日运动方程,4.1 Dynamics of a Rigid Body,4.1.2 Two Solutions for Dynamic Equation,32,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的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。,
23、4.1 Dynamics of a Rigid Body,4.1.2 Two Solutions for Dynamic Equation,33,对多自由度的机械手，拉格朗日法可以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。 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，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 Body,4.1.2 Two Solutions for Dynamic Equation,34,Conte
24、nts, Introduction to Dynamics Rigid Body Dynamics Lagrangian Formulation Newton-Euler Formulation Articulated Multi-Body Dynamics,34,Ch.4 Manipulator Dynamics,35,4.2 Dynamic Equation of a Manipulator 机械手的动力学方程,Forming dynamic equation of any manipulator described by a series of A-matrices: (1) Compu
25、ting the Velocity of any given point; (2) Computing total Kinetic Energy; (3) Computing total Potential Energy; (4) Forming Lagrangian Function of the system; (5) Forming Dynamic Equation through Lagrangian Equation.,35,4.2 Dynamic Equation of a Manipulator,36,36,4.2.1 Computation of Velocity 速度的计算,
26、Velocity of point P on link-3:Velocity of any given point on link-i:,图4.4 四连杆机械手,4.2 Dynamic Equation of a Manipulator,37,37,4.2.1 Computing the Velocity,Acceleration of point P:,图4.4 四连杆机械手,4.2 Dynamic Equation of a Manipulator,38,38,Square of velocityThe trace of an square matrix is defined to be
27、the sum of the diagonal elements.,4.2.1 Computing the Velocity,图4.4 四连杆机械手,4.2 Dynamic Equation of a Manipulator,39,39,Square of velocity of any given point:,4.2.1 Computing the Velocity,图4.4 四连杆机械手,4.2 Dynamic Equation of a Manipulator,40,40,Computing the Kinetic Energy令连杆3上任一质点P的质量为dm，则其动能为：,图4.4
28、四连杆机械手,4.2 Dynamic Equation of a Manipulator,4.2.2 Computation of Kinetic and Potential Energy 动能和位能的计算,41,41,4.2.2 Computation of Kinetic and Potential Energy,Kinetic Energy of any particle on link-i with position vector ir :Kinetic Energy of link-3:,4.2 Dynamic Equation of a Manipulator,42,42,Kine
29、tic Energy of any given link-i:Total Kinetic Energy of the manipulator:,4.2.2 Computation of Kinetic and Potential Energy,4.2 Dynamic Equation of a Manipulator,43,43,Computing the Potential EnergyPotential Energy of a object (mass m) at h height:so the Potential Energy of any particle on link-i with
30、 position vector ir :where,4.2.2 Computation of Kinetic and Potential Energy,4.2 Dynamic Equation of a Manipulator,44,44,Potential Energy of any particle on link-i with position vector ir :Total Potential Energy of the manipulator:,4.2.2 Computation of Kinetic and Potential Energy,4.2 Dynamic Equati
31、on of a Manipulator,45,45,Lagrangian Function,4.2 Dynamic Equation of a Manipulator,4.2.3 Forming the Dynamic Equation 动力学方程的推导,46,46,4.2.3 Forming the Dynamic Equation,Derivative of Lagrangian function,4.2 Dynamic Equation of a Manipulator,47,47,According to Eq.(4.18), Ii is a symmetric matrix, lea
32、d to,4.2.3 Forming the Dynamic Equation,4.2 Dynamic Equation of a Manipulator,48,48,4.2.3 Forming the Dynamic Equation,4.2 Dynamic Equation of a Manipulator,49,49,4.2.3 Forming the Dynamic Equation,4.2 Dynamic Equation of a Manipulator,50,50,Dynamic Equation of a n-link manipulator:,4.2.3 Forming th
33、e Dynamic Equation,注意：上述惯量项与重力项在机械手的控制中特别重要，它们将直接影响到机械手系统的稳定性和定位精度。只有当机械手高速运动时，向心力和哥氏力才变得重要。,4.2 Dynamic Equation of a Manipulator,51,51,4.3 Summary 小结,Two methods to form dynamic equation of a rigid body: Lagrangian Equation (Energy-based) Newton-Euler Equation (Force-balance) Summarize steps to form Lagrangian Equation of n-link manipulators: Computing the Velocity of any given point; Computing total Kinetic Energy; Computing total Potential Energy; Forming Lagrangian Function of the system; Forming Dynamic Equation through Lagrangian Equation.,4.3 Summary,
