中南大学 蔡自兴，谢 斌 zxcai, xiebin@mail.csu.edu.cn 2010 机器人学基础 第四章 机器人动力学 Fundamentals of Robotics Ch.4 Manipulator Dynamics 中南大学 蔡自兴，谢 斌 zxcai, xiebin@mail.csu.edu.cn 2010 1 Fundamentals of Robotics

Contents  Introduction to Dynamics  Rigid Body Dynamics  Lagrangian Formulation  Newton-Euler Formulation  Articulated Multi-Body Dynamics 2 Ch.4 Manipulator Dynamics

Ch.4 Manipulator Dynamics Introduction 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. 3 Ch.4 Manipulator Dynamics

Ch.4 Manipulator Dynamics Two methods for formulating dynamics model： Newton-Euler dynamic formulation Newton's equation along with its rotational analog, Euler's equation, describe how forces, inertias, and accelerations relate for rigid bodies, is a "force balance" approach to dynamics. Lagrangian dynamic formulation Lagrangian formulation is an "energy-based" approach to dynamics. Ch.4 Manipulator Dynamics

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, Τ, calculate the resulting motion of the manipulator, . This is useful for simulating the manipulator. Inverse Dynamics: given a trajectory point, , find the required vector of joint torques, Τ. This formulation of dynamics is useful for the problem of controlling the manipulator. Ch.4 Manipulator Dynamics

Contents  Introduction to Dynamics  Rigid Body Dynamics  Lagrangian Formulation  Newton-Euler Formulation  Articulated Multi-Body Dynamics 6 Ch.4 Manipulator Dynamics

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 corresponding velocity, Fi stand for corresponding torque or force on the ith coordinate. Kinetic Energy Potential Energy 7 4.1 Dynamics of a Rigid Body

4.1.1 Kinetic and Potential Energy of a Rigid Body 4.1 Dynamics of a Rigid Body 4.1.1 Kinetic and Potential Energy of a Rigid Body 图4.1 一般物体的动能与位能 8 4.1 Dynamics of a Rigid Body

is a generalized coordinate 4.1.1 Kinetic and Potential Energy of a Rigid Body is a generalized coordinate ① Kinetic Energy due to (angular) velocity ② Kinetic Energy due to position (or angle) ③ Dissipation Energy due to (angular) velocity ④ Potential Energy due to position ⑤ External Force or Torque ① ② ③ ④ ⑤ 9 4.1 Dynamics of a Rigid Body

4.1.1 Kinetic and Potential Energy of a Rigid Body x0 and x1 are both generalized coordinates Written in Matrices form: 10 4.1 Dynamics of a Rigid Body

4.1.1 Kinetic and Potential Energy of a Rigid Body Kinetic and Potential Energy of a 2-links manipulator 图4.2 二连杆机器手（1） Kinetic Energy K1 and Potential Energy P1 of link 1 11 4.1 Dynamics of a Rigid Body

4.1.1 Kinetic and Potential Energy of a Rigid Body Kinetic Energy K2 and Potential Energy P2 of link 2 where 12 4.1 Dynamics of a Rigid Body

4.1.1 Kinetic and Potential Energy of a Rigid Body Total Kinetic and Potential Energy of a 2-links manipulator are 13 4.1 Dynamics of a Rigid Body

Contents  Introduction to Dynamics  Rigid Body Dynamics  Lagrangian Formulation  Newton-Euler Formulation  Articulated Multi-Body Dynamics 14 Ch.4 Manipulator Dynamics

4.1.2 Two Solutions for Dynamic Equation Lagrangian Formulation Lagrangian Function L of a 2-links manipulator: 15 4.1 Dynamics of a Rigid Body

4.1.2 Two Solutions for Dynamic Equation Lagrangian Formulation Dynamic 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 16 4.1 Dynamics of a Rigid Body

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 Equation Lagrangian Formulation Dynamic Equations: Written in Matrices Form: 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 17 4.1 Dynamics of a Rigid Body

4.1.2 Two Solutions for Dynamic Equation Lagrangian Formulation Dynamic 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 Written in Matrices Form: 18 4.1 Dynamics of a Rigid Body

4.1.2 Two Solutions for Dynamic Equation Lagrangian Formulation Dynamic 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) Written in Matrices Form: 19 4.1 Dynamics of a Rigid Body

重力项(gravity)：关节i,j处的重力 4.1.2 Two Solutions for Dynamic Equation Lagrangian Formulation Dynamic Equations: Written in Matrices Form: 重力项(gravity)：关节i,j处的重力 20 4.1 Dynamics of a Rigid Body

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。 Lagrangian Formulation of Manipulator Dynamics 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。 21 4.1 Dynamics of a Rigid Body

注意：有效惯量的变化将对机械手的控制产生显著影响！ 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！ 表4.1给出这些系数值及其与位置 的关系。 表4.1 负载 地面空 载 0 90 180 270 1 -1 6 4 2 3 地面满载 18 10 8 外空间负载 402 202 200 100 102 22 4.1 Dynamics of a Rigid Body

Contents  Introduction to Dynamics  Rigid Body Dynamics  Lagrangian Formulation  Newton-Euler Formulation  Articulated Multi-Body Dynamics 23 Ch.4 Manipulator Dynamics

4.1.2 Two Solutions for Dynamic Equation Newton-Euler Dynamic Formulation Newton’s Law rate of change of the linear momentum is equal to the applied force Linear Momentum 4.1 Dynamics of a Rigid Body

4.1.2 Two Solutions for Dynamic Equation Newton-Euler Dynamic Formulation Rotational Motion Angular Momentum 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 4.1 Dynamics of a Rigid Body

4.1.2 Two Solutions for Dynamic Equation Newton-Euler Dynamic Formulation （Newton Equation） （Euler Equation） 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 ω is the angular velocity. 4.1 Dynamics of a Rigid Body

4.1.2 Two Solutions for Dynamic Equation 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。 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 1自由度机械手 4.1 Dynamics of a Rigid Body

4.1.2 Two Solutions for Dynamic Equation 由欧拉运动方程式 该式即为1自由度机械手的欧拉运动方程式。 4.1 Dynamics of a Rigid Body

4.1.2 Two Solutions for Dynamic Equation 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 force on the ith coordinate. Kinetic Energy Potential Energy 30 4.1 Dynamics of a Rigid Body

4.1.2 Two Solutions for Dynamic Equation 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。 解：假设θ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 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。 4.1 Dynamics of a Rigid Body

4.1.2 Two Solutions for Dynamic Equation 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

Contents  Introduction to Dynamics  Rigid Body Dynamics  Lagrangian Formulation  Newton-Euler Formulation  Articulated Multi-Body Dynamics 34 Ch.4 Manipulator Dynamics

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) Computing 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

4.2.1 Computation of Velocity 速度的计算 图4.4 四连杆机械手 Velocity of point P on link-3: Velocity of any given point on link-i: 36 4.2 Dynamic Equation of a Manipulator

4.2.1 Computing the Velocity 图4.4 四连杆机械手 Acceleration of point P: 37 4.2 Dynamic Equation of a Manipulator

4.2.1 Computing the Velocity 图4.4 四连杆机械手 Square of velocity The trace of an square matrix is defined to be the sum of the diagonal elements. 38 4.2 Dynamic Equation of a Manipulator

4.2.1 Computing the Velocity 图4.4 四连杆机械手 Square of velocity of any given point: 39 4.2 Dynamic Equation of a Manipulator

4.2.2 Computation of Kinetic and Potential Energy 动能和位能的计算 图4.4 四连杆机械手 Computing the Kinetic Energy 令连杆3上任一质点P的质量为dm，则其动能为： 40 4.2 Dynamic Equation of a Manipulator

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: 41 4.2 Dynamic Equation of a Manipulator

Kinetic Energy of any given link-i: 4.2.2 Computation of Kinetic and Potential Energy Kinetic Energy of any given link-i: Total Kinetic Energy of the manipulator: 42 4.2 Dynamic Equation of a Manipulator

Computing the Potential Energy 4.2.2 Computation of Kinetic and Potential Energy Computing the Potential Energy Potential Energy of a object (mass m) at h height: so the Potential Energy of any particle on link-i with position vector ir : where 43 4.2 Dynamic Equation of a Manipulator

Potential Energy of any particle on link-i with position vector ir : 4.2.2 Computation of Kinetic and Potential Energy Potential Energy of any particle on link-i with position vector ir : Total Potential Energy of the manipulator: 44 4.2 Dynamic Equation of a Manipulator

4.2.3 Forming the Dynamic Equation 动力学方程的推导 Lagrangian Function 45 4.2 Dynamic Equation of a Manipulator

4.2.3 Forming the Dynamic Equation Derivative of Lagrangian function 46 4.2 Dynamic Equation of a Manipulator

4.2.3 Forming the Dynamic Equation According to Eq.(4.18), Ii is a symmetric matrix, lead to 47 4.2 Dynamic Equation of a Manipulator

4.2.3 Forming the Dynamic Equation 48 4.2 Dynamic Equation of a Manipulator

4.2.3 Forming the Dynamic Equation 49 4.2 Dynamic Equation of a Manipulator

4.2.3 Forming the Dynamic Equation Dynamic Equation of a n-link manipulator: 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。 50 4.2 Dynamic Equation of a Manipulator

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. 51 4.3 Summary