Dynamics of Serial Manipulators

1. Dynamics of Serial Manipulators Professor Nicola Ferrier ME Room 2246, 265-8793 ferrier@engr.wisc.edu Professor N. J. Ferrier

2. Dynamic Modeling • For manipulator arms: • Relate forces/torques at joints to the motion of manipulator + load • External forces usually only considered at the end-effector • Gravity (lift arms) is a major consideration Professor N. J. Ferrier

3. Dynamic Modeling • Need to derive the equations of motion • Relate forces/torque to motion • Must consider distribution of mass • Need to model external forces Professor N. J. Ferrier

4. Manipulator Link Mass • Consider link as a system of particles • Each particle has mass, dm • Position of each particle can be expressed using forward kinematics Professor N. J. Ferrier

5. Manipulator Link Mass • The density at a position x is r(x), • usually r is assumed constant • The mass of a body is given by • where is the set of material points that comprise the body • The center of mass is Professor N. J. Ferrier

6. Inertia Professor N. J. Ferrier

7. Equations of Motion • Newton-Euler approach • P is absolute linear momentum • F is resultant external force • Mo is resultant external moment wrt point o • Ho is moment of momentum wrt point o • Lagrangian (energy methods) Professor N. J. Ferrier

8. Equations of Motion • Lagrangian using generalized coordinates: • The equations of motion for a mechanical system with generalized coordinates are: • External force vector • ti is the external force acting on the ith general coordinate Professor N. J. Ferrier

9. Equations of Motion • Lagrangian Dynamics, continued Professor N. J. Ferrier

10. Equations of Motions • Robotics texts will use either method to derive equations of motion • In “ME 739: Advanced Robotics and Automation” we use a Lagrangian approach using computational tools from kinematics to derive the equations of motion • For simple robots (planar two link arm), Newton-Euler approach is straight forward Professor N. J. Ferrier

11. Manipulator Dynamics • Isolate each link • Neighboring links apply external forces and torques • Mass of neighboring links • External force inherited from contact between tip and an object • D’Alembert force (if neighboring link is accelerating) • Actuator applies either pure torque or pure force (by DH convention along the z-axis) Professor N. J. Ferrier

12. Notation The following are w.r.t. reference frame R: Professor N. J. Ferrier

13. Force on Isolated Link Professor N. J. Ferrier

14. Torque on Isolated Link Professor N. J. Ferrier

15. Force-torque balance on manipulator Applied by actuators in z direction external Professor N. J. Ferrier

16. Newton’s Law • A net force acting on body produces a rate of change of momentum in accordance with Newton’s Law • The time rate of change of the total angular momentum of a body about the origin of an inertial reference frame is equal to the torque acting on the body Professor N. J. Ferrier

17. Force/Torque on link n Professor N. J. Ferrier

18. Newton’s Law Professor N. J. Ferrier

19. Newton-Euler Algorithm Professor N. J. Ferrier

20. Newton-Euler Algorithm • Compute the inertia tensors, • Working from the base to the end-effector, calculate the positions, velocities, and accelerations of the centroids of the manipulator links with respect to the link coordinates (kinematics) • Working from the end-effector to the base of the robot, recursively calculate the forces and torques at the actuators with respect to link coordinates Professor N. J. Ferrier

21. “Change of coordinates” for force/torque Professor N. J. Ferrier

22. Recursive Newton-Euler Algorithm Professor N. J. Ferrier

23. Two-link manipulator Professor N. J. Ferrier

24. Two link planar arm DH table for two link arm L2 L1 x0 x1 x2 2 1 Z2 Z0 Z1 Professor N. J. Ferrier

25. Forward Kinematics: planar 2-link arm Professor N. J. Ferrier

26. Forward Kinematics: planar 2-link manipulator Professor N. J. Ferrier

27. Forward Kinematics: planar 2-link manipulator w.r.t. base frame {0} Professor N. J. Ferrier

28. Forward Kinematics: planar 2-link manipulator position vector from origin of frame 0 to c.o.m. of link 1 expressed in frame 0 position vector from origin of frame 1 to c.o.m. of link 2 expressed in frame 0 position vector from origin of frame 0 to origin of frame 1 expressed in frame 0 position vector from origin of frame 1 to origin of frame 2 expressed in frame 0 Professor N. J. Ferrier

29. Forward Kinematics: planar 2-link manipulator w.r.t. base frame {0} Professor N. J. Ferrier

30. Point Mass model for two link planar arm DH table for two link arm m1 m2 Professor N. J. Ferrier

31. Dynamic Model of Two Link Arm w/point mass Professor N. J. Ferrier

32. General Form Coriolis & centripetal terms Inertia (mass) Joint torques Gravity terms Joint accelerations Professor N. J. Ferrier

33. General Form: No motion No motion so Gravity terms Joint torques required to hold manipulator in a static position (i.e. counter gravitational forces) Professor N. J. Ferrier

34. Independent Joint Control revisited • Called “Computed Torque Feedforward” in text • Use dynamic model + setpoints (desired position, velocity and acceleration from kinematics/trajectory planning) as a feedforward term Professor N. J. Ferrier

35. Manipulator motion from input torques Integrate to get Professor N. J. Ferrier

36. Dynamic Model of Two Link Arm w/point mass Professor N. J. Ferrier

37. Dynamics of 2-link – point mass Professor N. J. Ferrier

38. Dynamics in block diagram of 2-link (point mass) Professor N. J. Ferrier

39. Dynamics of 2-link – slender rod Professor N. J. Ferrier