1. The Forward Kinematics of Manipulators Sebastian van Delden USC Upstate svandelden@uscupstate.edu

2. Kinematics Kinematics is the science of motion that treats the subject without regard to the forces that cause it. Forward Kinematics of Manipulators:  compute position and orientation of the end effector given a set of joint angles.  EASY(er) Inverse Kinematics of Manipulators:  given the position and orientation of the end effector, calculate all possible joint angles that could achieve this position and orientation.  HARD  Stäubli RX60s – As many as 8 possible ways to achieve end effector pose

3. Joint Types

4. Link Descriptions N dof manipulators have N joints and N-1 rigid links that connect them. Link 0 is the non-moving link at the base Link 1 is the first moving link, etc… Our Stäubli RX60s have 6 revolute joints.

5. Link Descriptions cont…

6. Link Length, a, is the distance between joints. Link Twist, , is the angle difference between axes of two joints.  Picture a vector pointing from joint i-1 to joint i and then rotate joint i until it is inline with joint i-1. Joints axes are vectors that run though the center of the joint. End points are given a value of 0.

7. Link Description Example…

8. Link-Connection Description Describes the joint. Link Offset, d, and the distance between a i-1 and a i. Joint Angle, θ, is the angle between a i-1 and a i. End points are given a value of 0.

9. Link-Connection Description cont…

10. Link parameters For revolute joints, θ varies. For prismatic joints, d varies. The pose of a manipulator can be defined in terms of these link and link-connection parameters. A Denavit-Hartenburg Table (DH Table) can be used to represent all of these parameters.

11. Attaching Frames to Links

12. Z axes are coincident with the joint axes. X axes point along a to the next joint, or if Zs intersect then X is normal to this plane. Y can then be automatically determined. Each joint has a frame attached to it, frames {0}, {1}, {2}, {3}, etc..  {0} is attached to the non-moving base and is usually aligned initially with {1}. To calculate forward kinematics, we just need to find the transformations that describe the chain frames.

13. Summary of parameters

14. Summary of Link-Frame Attachment

15. Example Attach frames to the following manipulators and populate the DH Table.

16. Example cont…

17. The DH Table

18. Example 2

19. Example 2 cont…

21. Another Example Frames can be assigned in different ways! Consider this RRR machine

22. Another Example cont… Two possible frame assignments!

23. Another Example cont… Two more assignments!

24. Derivation of Link Transformation We need to derive the transformation from frame {i} to frame {i-1}. This transformation can be broken down into sub- problems and intermediate frames {P}, {Q}, and {R}.  To get from frame {i-1} to {R}, rotate by  degrees around X.  To get from {R} to {Q}, translate by a mm.  To get from {Q} to {P}, rotate by the θ degrees around Z.  To get from {P} to frame {i}, translate by d mm.

25. {P}, {Q}, and {R}

26. Deriving the Link Transformation So the transformation from {i-1} to {i} is: Which can also be written as:

27. Deriving the Link Transformation Multiplying out, we get: Multiply all link transformation together to compute forward kinematics.

28. Example: PUMA 560

30. Multiplying out the transformations:

31. Another Example: RRR

32. Example Cont…

33. Prismatic Joint Example: A RPR Machine

34. Example Cont…

35. Example Cont… Note that a “d” varies…

36. Another Prismatic Example: RPR

37. Example Cont… d 2, θ 1 and θ 2 are variable…

38. Another Prismatic Example: RRP

39. Example Cont…