Robotic Arms and Matrices

By Chris Wong and Chris Marino

The Canadarm

•First operation 1998

•Used for assembly of the International

Space Station

•Composed of a series of arms of fixed length connected by rotating

joints

Key Concepts• Translation

• Rotation

• Homogeneous Coordinates

• Matrix Multiplication

Translation• Not a linear transformation

• Translation along vector V = [a,b] in R2

• Transformation represented by T(x) = x + V in R2

•

• Translation is caused by the position of the previous arm

Rotation• Rotation is a Linear Transformation

• Rotates any Vector about the origin•

Homogeneous Coordinates

• Represents vector in R2 as a vector in R3

• x = [x,y] in R2

• X = [x,y,1] in R3

• Rotation and Translation operations can thus be represented using homogeneous coordinates

Translation and Rotation in one

• Represented through Matrix Multiplication

• T R represents Translation by Rotation

• R T does not equal T R

Second Arm• To represent second arm’s movement

• Same as representing the first• 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• 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• T2 * T1