1. Dynamics of Serial Manipulators
Professor Nicola Ferrier
ME Room 2246,
ME 439
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
ME 439
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
ME 439
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
ME 439
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
ME 439
Professor N. J. Ferrier

6. Inertia
ME 439
Professor N. J. Ferrier

7. Equations of Motion Newton-Euler approach Lagrangian (energy methods)
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)
ME 439
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
ME 439
Professor N. J. Ferrier

9. Equations of Motion Lagrangian Dynamics, continued ME 439
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
ME 439
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)
ME 439
Professor N. J. Ferrier

12. Notation The following are w.r.t. reference frame R: ME 439
Professor N. J. Ferrier

13. Force on Isolated Link
ME 439
Professor N. J. Ferrier

14. Torque on Isolated Link
ME 439
Professor N. J. Ferrier

15. Force-torque balance on manipulator
Applied by actuators in z direction
external
ME 439
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
ME 439
Professor N. J. Ferrier

17. Force/Torque on link n
ME 439
Professor N. J. Ferrier

18. Newton’s Law
ME 439
Professor N. J. Ferrier

19. Newton-Euler Algorithm
ME 439
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
ME 439
Professor N. J. Ferrier

21. “Change of coordinates” for force/torque
ME 439
Professor N. J. Ferrier

22. Recursive Newton-Euler Algorithm
ME 439
Professor N. J. Ferrier

23. Two-link manipulator
ME 439
Professor N. J. Ferrier

24. Two link planar arm DH table for two link arm L1 L2 x0 x1 x2 2 Z0 1
Var  d  a 1 1 L1 2 2 L2
ME 439
Professor N. J. Ferrier

25. Forward Kinematics: planar 2-link arm
ME 439
Professor N. J. Ferrier

26. Forward Kinematics: planar 2-link manipulator
ME 439
Professor N. J. Ferrier

27. Forward Kinematics: planar 2-link manipulator
w.r.t. base frame {0}
ME 439
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
ME 439
Professor N. J. Ferrier

29. Forward Kinematics: planar 2-link manipulator
w.r.t. base frame {0}
ME 439
Professor N. J. Ferrier

30. Point Mass model for two link planar arm
DH table for two link arm m1 m2
ME 439
Professor N. J. Ferrier

31. Dynamic Model of Two Link Arm w/point mass
ME 439
Professor N. J. Ferrier

32. General Form Coriolis & centripetal terms Inertia (mass) Joint torques
Gravity terms
Joint accelerations
ME 439
Professor N. J. Ferrier

33. General Form: No motion
No motion so
Joint torques required to hold manipulator in a static position (i.e. counter gravitational forces)
Gravity terms
ME 439
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
ME 439
Professor N. J. Ferrier

35. Manipulator motion from input torques
Integrate to get
ME 439
Professor N. J. Ferrier

36. Dynamic Model of Two Link Arm w/point mass
ME 439
Professor N. J. Ferrier

37. Dynamics of 2-link – point mass
ME 439
Professor N. J. Ferrier

38. Dynamics in block diagram of 2-link (point mass)
ME 439
Professor N. J. Ferrier

39. Dynamics of 2-link – slender rod
ME 439
Professor N. J. Ferrier