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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 Robotics1Contents Introduction to Dynamics Rigid Body Dynamics Lagrangian Formulation Newton-Euler Formulation Articulated Multi-Body Dynamics2Ch.4 Manipulator Dynamics23Ch.4 Manipulator DynamicsIntroductionCh.4 Manipulat
2、or 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.3Ch.4 Manipulator DynamicsTwo methods for formulating dynamics model：Newton-Euler
3、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.Ch.4 Manipu
4、lator Dynamics4Ch.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:given a trajectory po
5、int,find the required vector of joint torques,.This formulation of dynamics is useful for the problem of controlling the manipulator.Ch.4 Manipulator Dynamics5Contents Introduction to Dynamics Rigid Body Dynamics Lagrangian Formulation Newton-Euler Formulation Articulated Multi-Body Dynamics6Ch.4 Ma
6、nipulator Dynamics674.1 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 force on the ith coordinate.4.1 Dynamics
7、of a Rigid BodyKinetic EnergyPotential Energy74.1.1 Kinetic and Potential Energy of a Rigid Body8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 Body89 is a generalized coordinate Kinetic Energy due to(angular)velocity Kinetic Energy due to position(or angle)Dissipation Energy due
8、 to(angular)velocity Potential Energy due to position External Force or Torque4.1.1 Kinetic and Potential Energy of a Rigid Body4.1 Dynamics of a Rigid Body 910 x0 and x1 are both generalized coordinates4.1.1 Kinetic and Potential Energy of a Rigid Body4.1 Dynamics of a Rigid BodyWritten in Matrices
9、 form:1011Kinetic 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 Body4.1 Dynamics of a Rigid Body1112Kinetic Energy K2 and Potential Energy P2 of link 2 where4.1.1 Kinetic and Potential Ene
10、rgy of a Rigid Body4.1 Dynamics of a Rigid Body12Total Kinetic and Potential Energy of a 2-links manipulator are134.1.1 Kinetic and Potential Energy of a Rigid Body4.1 Dynamics of a Rigid Body13Contents Introduction to Dynamics Rigid Body Dynamics Lagrangian Formulation Newton-Euler Formulation Arti
11、culated Multi-Body Dynamics14Ch.4 Manipulator Dynamics1415Lagrangian Formulation Lagrangian Function L of a 2-links manipulator:4.1 Dynamics of a Rigid Body4.1.2 Two Solutions for Dynamic Equation15164.1.2 Two Solutions for Dynamic EquationLagrangian Formulation Dynamic Equations:Written in Matrices
12、 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 Body16Written in Matrices Form:17Lagrangian Formulation Dynamic Equations: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 Body17Written in Matrices Form:18Lagrangian
13、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4.1.2 Two Solutions for Dynamic Equation4.1 Dynamics of a Rigid Body18Written in Matrices Form:19Lagrangian 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)4.1.2 Two
14、 Solutions for Dynamic Equation4.1 Dynamics of a Rigid Body19Written in Matrices Form:20Lagrangian Formulation Dynamic Equations: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 Body2021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
15、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 Manipulator Dynamics4.1 Dynamics of a Rigid Body2122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09018027010-1064242101111164242323地面满载09018027010-1018102108404444418102102626外空间负载09018027010-104022022202200100010
16、0100100100100402202220221022102Lagrangian 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 Body22Contents Introduction to Dynamics Rigid Body Dynamics Lagrangian Formulation Newton-Euler Formulation Articulated Multi-Body Dynamics23Ch.4 Manipulator Dynamics234.1 Dy
17、namics 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 forceLinear Momentum244.1 Dynamics of a Rigid Body4.1.2 Two Solutions for Dynamic EquationNewton-Euler Dynamic FormulationRotational
18、MotionAngular Momentum254.1 Dynamics of a Rigid Body4.1.2 Two Solutions for Dynamic EquationNewton-Euler Dynamic FormulationRotational MotionAngular MomentumInertia Tensor26Newton-Euler Dynamic Formulationwhere m is the mass of a rigid body,represent inertia tensor,FC is the external force on the ce
19、nter 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 Body4.1.2 Two Solutions for Dynamic Equation27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
20、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 Body4.1.2 Two Solutions for Dynamic Equation28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 Body4.1.2 Two Solutions for Dynamic Equation2930Langrangian Function L is defined as:Dynamic Equation of the system(Langrangian Equation):where qi is the
21、 generalized coordinates,represent corresponding velocity,Fi stand for corresponding torque or force on the ith coordinate.4.1 Dynamics of a Rigid BodyKinetic EnergyPotential Energy4.1.2 Two Solutions for Dynamic Equation30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。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 B
22、ody4.1.2 Two Solutions for Dynamic Equation31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
23、2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 Equation33Contents Introduction to Dynamics Rigid Body Dynamics Lag
24、rangian Formulation Newton-Euler Formulation Articulated Multi-Body Dynamics34Ch.4 Manipulator Dynamics344.2 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)Computing t
25、otal 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 Manipulator35364.2.1 Computation of Velocity 速度的计算速度的计算Velocity of point P on link-3:Velocity of any given point on
26、 link-i:图4.4 四连杆机械手4.2 Dynamic Equation of a Manipulator36374.2.1 Computing the VelocityAcceleration of point P:图4.4 四连杆机械手4.2 Dynamic Equation of a Manipulator3738Square of velocityThe trace of an square matrix is defined to be the sum of the diagonal elements.4.2.1 Computing the Velocity图4.4 四连杆机械
27、手4.2 Dynamic Equation of a Manipulator3839Square 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 Manipulator3940Computing 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 四连杆机械手4.2 Dynamic Equation of a Manipulator4.2.2 Computation of Kinetic and Potenti
28、al Energy 动能和位能的计算 40414.2.2 Computation of Kinetic and Potential EnergyKinetic Energy of any particle on link-i with position vector ir:Kinetic Energy of link-3:4.2 Dynamic Equation of a Manipulator4142Kinetic Energy of any given link-i:Total Kinetic Energy of the manipulator:4.2.2 Computation of K
29、inetic and Potential Energy4.2 Dynamic Equation of a Manipulator4243Computing 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:where4.2.2 Computation of Kinetic and Potential Energy4.2 Dynamic Equation of a
30、Manipulator4344Potential 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 Energy4.2 Dynamic Equation of a Manipulator4445Lagrangian Function 4.2 Dynamic Equation of a Manipulator4.2.3 Forming the Dynamic Equ
31、ation 动力学方程的推导动力学方程的推导45464.2.3 Forming the Dynamic EquationDerivative of Lagrangian function 4.2 Dynamic Equation of a Manipulator4647According to Eq.(4.18),Ii is a symmetric matrix,lead to4.2.3 Forming the Dynamic Equation4.2 Dynamic Equation of a Manipulator4748 4.2.3 Forming the Dynamic Equation
32、4.2 Dynamic Equation of a Manipulator48494.2.3 Forming the Dynamic Equation4.2 Dynamic Equation of a Manipulator4950Dynamic Equation of a n-link manipulator:4.2.3 Forming the Dynamic Equation注意：注意：上述惯量项惯量项与重力项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稳定性和定位精度定位精度。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
33、 of a Manipulator50514.3 Summary 小结小结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 Summary51