1. I NTRODUCTION TO R OBOTICS CPSC - 460 Lecture 3A – Forward Kinematics

2. DH T ECHNIQUES A link j can be specified by two parameters, its length aj and its twist α j Joints are also described by two parameters. The link offset dj is the distance from one link coordinate frame to the next along the axis of the joint. The joint angle θ j is the rotation of one link with respect to the next about the joint axis.

3. DH T ECHNIQUES Link twist α i :the angle from the Z i-1 axis to the Z i axis about the X i axis. The positive sense for α is determined from z i-1 and z i by the right-hand rule. Joint angle θ i the angle between the X i-1 and X i axes about the Z i-1 axis.

4. DH T ECHNIQUES 4

5. The four parameters for each link a i : link length α i : Link twist d i : Link offset θ i : joint angle With the i th joint, a joint variable is q i associated where 5

6. T RANSFORMATION M ATRIX 6 Each homogeneous transformation A i is represented as a product of four basic transformations

7. T RANSFORMATION M ATRIX 7

8. The matrix A i is a function of only a single variable, as three of the above four quantities are constant for a given link, while the fourth parameter is the joint variable, depending on whether it is a revolute or prismatic link

9. DH N OTATION S TEPS 9

10. 10

11. DH N OTATION S TEPS From, the position and orientation of the tool frame are calculated.

12. T RANSFORMATION M ATRIX

13. E XAMPLE I - T WO L INK P LANAR A RM 13 Base frame O 0 All Z ‘s are normal to the page

14. E XAMPLE I - T WO L INK P LANAR A RM 14 Where (θ 1 + θ 2 ) denoted by θ 12 and

15. E XAMPLE 2 15

16. F ORWARD K INEMATICS OF E XAMPLE 2 16

17. E XAMPLE 3 - T HREE L INK C YLINDRICAL M ANIPULATOR 17

18. E XAMPLE 3 - T HREE L INK C YLINDRICAL M ANIPULATOR 18

19. E XAMPLE 3 - T HREE L INK C YLINDRICAL M ANIPULATOR 19

20. E XAMPLE 3 - T HREE L INK C YLINDRICAL M ANIPULATOR 20

21. E XAMPLE 4 – T HE S PHERICAL W RIST 21

22. E XAMPLE 4 – T HE S PHERICAL W RIST 22

23. E XAMPLE 4 – T HE S PHERICAL W RIST 23

24. E XAMPLE 4 – T HE S PHERICAL W RIST 24

25. E XAMPLE 5 - C YLINDRICAL M ANIPULATOR WITH S PHERICAL W RIST 25 derived in Example 2, and derived in Example 3.

26. E XAMPLE 5 - C YLINDRICAL M ANIPULATOR WITH S PHERICAL W RIST 26

27. E XAMPLE 5 - C YLINDRICAL M ANIPULATOR WITH S PHERICAL W RIST 27

28. E XAMPLE 5 - C YLINDRICAL M ANIPULATOR WITH S PHERICAL W RIST 28 Forward kinematics: 1. The position of the end-effector: (d x,d y,d z ) 2. The orientation {Roll, Pitch, Yaw }

29. R OTATION – R OLL, P ITCH, Y AW The rotation matrix for the following operations: X Y Z 29

30. E XAMPLE 4 T HE THREE LINKS CYLINDRICAL WITH S PHERICAL WRIST 30 How to calculate Compare the matrix R With the rotation part of