Kinematics and Representing Position in Robotics

1. Kinematics Pose (position and orientation) of a Rigid Body 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 : The angle between and in anti clockwise direction

6. The rotation matrix 6

7. Useful formulas 7

8. Example 1 8

9. Example 1 9

10. Example 1 10 Another Solution

11. Basic Rotation Matrix Rotation about x-axis with 11

12. Basic Rotation Matrices Rotation about x-axis with Rotation about y-axis with Rotation about z-axis with 12

13. Example 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. 13

14. Example 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. 14

15. Composite Rotation Matrix A sequence of finite rotations matrix multiplications do not commute 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] 15

16. Rotation with respect to Current Frame 16

17. 17 Example 4 Find the rotation matrix for the following operations: Post-multiply if rotate about the current frame Pre-multiply if rotate about the fixed frame

18. 18 Example 5 Find the rotation matrix for the following operations: Pre-multiply if rotate about the fixed frame Post-multiply if rotate about the current frame

19. 19 Example 6 Find the rotation matrix for the following operations: Pre-multiply if rotate about the fixed frame Post-multiply if rotate about the current frame

20. Example 6 Find the rotation matrix for the following operations: 20

21. Quiz Description of Roll Pitch Yaw Find the rotation matrix for the following operations: X Y Z 21

22. Answer X Y Z 22

23. Coordinate Transformations p osition vector of P in { B } is transformed to position vector of P in { A } description of frame{ B } as seen from an observer in { A } Rotation of { B } with respect to { A } Translation of the origin of { B } with respect to origin of { A } 23

24. Homogeneous Representation Coordinate transformation from { B } to { A } Can be written as Position vector (3*1) Rotation matrix (3*3) 24

25. Homogeneous Representation P osit ion vector of the origin of frame B wrt frame A (3*1) Rotation matrix (3*3) 25

26. Homogeneous Transformation Special cases 1. Translation 2. Rotation 26

27. Example 7 Translation along Z-axis with h: O h O 27

28. Example 7 Translation along Z-axis with h: 28

29. Example 8 Rotation about the X-axis by 29

30. Homogeneous Transformation 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 30

31. Example 9 Find the homogeneous transformation matrix (H) for the following operations: 31

32. Remember those double-angle formulas 32

33. Review of matrix transpose Important property: 33

34. and matrix multiplication Can represent dot product as a matrix multiply: 34

35. HW Problems 2.10, 2.11, 2.12, 2.13, 2.14 ,2.15, 2.22, 2.24, 2.37, and 2.39 Quiz next class 35