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Page 7 -- Definition: Coordinate Transformation

The pose of a rigid body is defined by its position and attitude.
POSITION: origin of local coordinate system with respect to global coordinates
ATTITUDE: orientation of local coordinate system with respect to global axes

A coordinate transformation can describe:
(1) the relationship between coordinates in a global or local reference frame, and
(2) the position and attitude of a moving body at each instant

Diagram

where:
G: global coordinate system
L: local coordinate system
RG: vector from global origin to point P on the local body (equivalent to the global coordinates of P)
RO: vector from the global origin to the local origin
RL: vector from the local origin to point P (equivalent to the local coordinates of P)


Over time, we can track displacement and rotation of the body using vectors which describe the position and orientation of the body in local and global coordinates. The derivatives can be calculated for linear and angular velocities and accelerations for complete kinematic description.

The best way to execute a coordinate transformation is using a 4x4 transformation matrix. This matrix contains a position vector, which gives the location of a local body relative to a global coordinate frame, and a rotation submatrix, which relates the attitude of the local axes to the global axes.

The next page describes in more detail the structure and manipulation of 4x4 transformation matrices.