1. Kinematics Pose (position and orientation) of a Rigid Body
Introduction to ROBOTICS
Kinematics Pose (position and orientation) of a Rigid Body
University of Bridgeport

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

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

4. Representing Position: vectors (3D)
right-handed coordinate frame
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
θ: The angle between and
in anti clockwise direction

6. The rotation matrix

7. Useful formulas

8. Example 1

9. Example 1

10. Example 1
Another Solution

11. Basic Rotation Matrix
Rotation about x-axis with

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

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.

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.

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]

16. Rotation with respect to Current Frame

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

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. 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:

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

22. Answer X Y Z

23. Coordinate Transformations
position 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}

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

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

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

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

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

29. Example 8
Rotation about the X-axis by

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

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

32. Remember those double-angle formulas…

33. Review of matrix transpose
Important property:

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

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