1. Dynamics

2.  relationship between the joint actuator torques and the motion of the structure  Derivation of dynamic model of a manipulator  Simulation of motion  Design of control algorithms  Analysis of manipulator structures  Method based on Lagrange formulation

3. Lagrange Formulation  Generalized coordinates  n variables which describe the link positions of an n- degree-of-mobility manipulator  The Lagrange of the mechanical system

4. Lagrange Formulation  The Lagrange of the mechanical system Function of generalized coordinates Kinetic energy Potential energy

5. Lagrange Formulation  The Lagrange’s equations  Generalized force  Given by the nonconservative force  Joint actuator torques, joint friction torques, joint torques induced by interaction with environment

6. Lagrange Formulation Example 4.1 Rotor inertia Reduction gear ratio Stator is fixed on the previous link Actuation torque Viscous friction Initial position Generalized coordinate? Kinetic energy? Potential energy?

7. Lagrange Formulation Example 4.1  Generalized coordinate: theta  Kinetic energy  Potential energy

8. Lagrange Formulation Example 4.1  Lagrangian of the system

9. Lagrange Formulation Example 4.1  Contributions to the generalized force  Dynamic of the model  Relations between torque and joint position, velocity and acceleration

10. Mechanical Structure  Joint actuator torques are delivered by the motors  Mechanical transmission  Direct drive

11. Computation of Kinetic Energy  Consider a manipulator with n rigid links Kinetic energy of link i Kinetic energy of the motor actuating joint i. The motor is located on link i-1

12. Kinetic Energy of Link  Kinetic energy of link i is given by

13. Kinetic Energy of Link  Kinetic energy of a rigid body (appendix B.3) translationalrotational

14. Kinetic Energy of Link  Translational  Centre of mass

15.  Rotational  Inertia tensor

16.  Inertia tensor is constant when referred to the link frame (frame parallel to the link frame with origin at centre of mass) Constant inertia tensor Rotation matrix from link i frame to the base frame

17. Kinetic Energy of Link  Express the kinetic energy as a function of the generalized coordinates of the system, that are the joint variables

18.  Apply the geometric method for Jacobian computation to the intermediate link

19.  The kinetic energy of link i is

20. Kinetic Energy of Motor  Assume that the contribution of the stator is included in that of the link on which such motor is located  The kinetic energy to rotor i

21.  On the assumption of rigid transmission  According to the angular velocity composition rule Angular position of the rotor

23. attention  Kinetic energy of rotor

24. Kinetic Energy of Manipulator

25. Computation of Potential Energy  Consider a manipulator with n rigid links

26. Equations of Motion

28.  For the acceleration terms  For the quadratic velocity terms  For the configuration-dependent terms

29. Joint Space Dynamic Model Viscous friction torques Coulomb friction torques Actuation torques Force and moment exerted on the environment Multi-input-multi-output; Strong coupling; Nonlinearity