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

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

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

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

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

U SEFUL FORMULAS 6

Example 1 7

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

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

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

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

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

R OTATION WITH RESPECT TO C URRENT F RAME 13

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

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

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

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

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

A NSWER 19 X Y Z

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

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

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

E XAMPLE 8 Rotation about the X-axis by 23

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

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

Remember those double-angle formulas… 26

Review of matrix transpose Important property: 27

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

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