1. K INEMATICS P OSE ( POSITION AND ORIENTATION ) OF A R IGID B ODY University of Bridgeport 1 Introduction to ROBOTICS

2. Representing Position (2D) (“column” vector) A vector of length one pointing in the direction of the base frame x axis A vector of length one pointing in the direction of the base frame y axis 2

3. Representing Position: vectors The prefix superscript denotes the reference frame in which the vector should be understood Same point, two different reference frames 3

4. Representing Position: vectors (3D) right-handed coordinate frame 4 A vector of length one pointing in the direction of the base frame x axis A vector of length one pointing in the direction of the base frame y axis A vector of length one pointing in the direction of the base frame z axis

5. The rotation matrix :To specify the coordinate vectors for the fame B with respect to frame A 5 θ: The angle between and in anti clockwise direction

6. U SEFUL FORMULAS 6

7. Example 1 7

8. B ASIC R OTATION M ATRIX Rotation about x-axis with 8

9. B ASIC R OTATION M ATRICES Rotation about x-axis with Rotation about y-axis with Rotation about z-axis with 9

10. E XAMPLE 2 A point is attached to a rotating frame, the frame rotates 60 degree about the OZ axis of the reference frame. Find the coordinates of the point relative to the reference frame after the rotation. 10

11. E XAMPLE 3 A point is the coordinate w.r.t. the reference coordinate system, find the corresponding point w.r.t. the rotated OUVW coordinate system if it has been rotated 60 degree about OZ axis. 11

12. C OMPOSITE R OTATION M ATRIX A sequence of finite rotations rules: if rotating coordinate OUVW is rotating about principal axis of OXYZ frame, then Pre-multiply the previous (resultant) rotation matrix with an appropriate basic rotation matrix [rotation about fixed frame] if rotating coordinate OUVW is rotating about its own principal axes, then post-multiply the previous (resultant) rotation matrix with an appropriate basic rotation matrix [rotation about current frame] 12

13. R OTATION WITH RESPECT TO C URRENT F RAME 13

14. E XAMPLE 4 Find the rotation matrix for the following operations: 14 Post-multiply if rotate about the current frame Pre-multiply if rotate about the fixed frame

15. E XAMPLE 5 Find the rotation matrix for the following operations: 15 Pre-multiply if rotate about the fixed frame Post-multiply if rotate about the current frame

16. E XAMPLE 6 Find the rotation matrix for the following operations: 16 Pre-multiply if rotate about the fixed frame Post-multiply if rotate about the current frame

17. E XAMPLE 6 Find the rotation matrix for the following operations: 17

18. Q UIZ Description of Roll Pitch Yaw Find the rotation matrix for the following operations: 18 X Y Z

19. A NSWER 19 X Y Z

20. H OMOGENEOUS T RANSFORMATION Special cases 1. Translation 2. Rotation 20

21. E XAMPLE 7 Translation along Z-axis with h: 21 O h O

22. E XAMPLE 7 Translation along Z-axis with h: 22

23. E XAMPLE 8 Rotation about the X-axis by 23

24. H OMOGENEOUS T RANSFORMATION Composite Homogeneous Transformation Matrix Rules: Transformation (rotation/translation) w.r.t fixed frame, using pre-multiplication Transformation (rotation/translation) w.r.t current frame, using post-multiplication 24

25. E XAMPLE 9 Find the homogeneous transformation matrix (H) for the following operations: 25

26. Remember those double-angle formulas… 26

27. Review of matrix transpose Important property: 27

28. and matrix multiplication… Can represent dot product as a matrix multiply: 28

29. HW Problems 2.10, 2.11, 2.12, 2.13, 2.14,2.15, 2.37, and 2.39 Quiz next class 29