1. Robotic Arms and Matrices By Chris Wong and Chris Marino

2. The Canadarm First operation 1998 First operation 1998 Used for assembly of the International Space Station Used for assembly of the International Space Station Composed of a series of arms of fixed length connected by rotating joints Composed of a series of arms of fixed length connected by rotating joints

3. Key Concepts Translation Translation Rotation Rotation Homogeneous Coordinates Homogeneous Coordinates Matrix Multiplication Matrix Multiplication

4. Translation Not a linear transformation Not a linear transformation Translation along vector V = [a,b] in R 2 Translation along vector V = [a,b] in R 2 Transformation represented by T(x) = x + V in R 2 Transformation represented by T(x) = x + V in R 2 Translation is caused by the position of the previous arm Translation is caused by the position of the previous arm

5. Rotation Rotation is a Linear Transformation Rotation is a Linear Transformation Rotates any Vector about the origin Rotates any Vector about the origin

6. Homogeneous Coordinates Represents vector in R 2 as a vector in R 3 Represents vector in R 2 as a vector in R 3 x = [x,y] in R 2 x = [x,y] in R 2 X = [x,y,1] in R 3 X = [x,y,1] in R 3 Rotation and Translation operations can thus be represented using homogeneous coordinates Rotation and Translation operations can thus be represented using homogeneous coordinates

7. Translation and Rotation in one Represented through Matrix Multiplication Represented through Matrix Multiplication T  R represents Translation by Rotation T  R represents Translation by Rotation R  T does not equal T  R R  T does not equal T  R

8. Second Arm To represent second arm’s movement To represent second arm’s movement Same as representing the first Same as representing the first Give each arm its own coordinate system Give each arm its own coordinate system a and b are the x and y coordinates of the origin of the second arm with respect to the origin of the first arm a and b are the x and y coordinates of the origin of the second arm with respect to the origin of the first arm This new origin is obtained when by taking the components of the first arm when it is rotated about an angle theta This new origin is obtained when by taking the components of the first arm when it is rotated about an angle theta Now combining movements of the first and second arm Now combining movements of the first and second arm T 2 * T 1 T 2 * T 1