Global Overview

Global Overview

Coordinates and Frames

Grégoire Passault

Conventions

Let's call the columns of :

A rotation matrix is 9 numbers with 6 constraints,
thus 3 degrees of freedom

We call such a representation an
implicit representation

Consider:

The transpose of is its inverse

Homogeneous coordinates

We can pack and in a 4x4 matrix:

Any frame change can be now expressed as a
(4x4) matrix multiplication

No need to mix multiplications and sums
Frame change can be pre-multiplied together
Any translation or rotation can be expressed in such a matrix

About notations

We then have:

See how the "cancels" together:
we call it the subscript cancellation rule.

Similarly:

Since it takes a point from to ,
then from to ,
then from to .

What is the inverse of ?

In the 2D case, all rotation matrices are of the form:

In the 3D case, we can define 3 matrices that rotates about axises:

Transformations and notation in code

# A point expressed in the robot frame
P_robot

# Transformation matrix taking a point in robot to the world
T_world_robot

# The subscript cancellation rule applies
P_world = T_world_robot @ P_robot

# The chaining applies
T_world_effector = T_world_robot @ T_robot_effector

Motion

a first step toward kinematics

In the first case, we apply the motion,
then take the point to the world, it is applied locally (intrinsic)

In the second case, we take the point to the world,
then apply the motion, it is apply in the world (extrinsic)