Spatial Algebra

# Axis-Angle Representation (Rotation Vector)

First introduction here through Steven M. Lavalle VR series. Read more into the modern robotics book, chapter 3.

Axis-Angle Representation are implemented with rotation vectors. This vector is parallel with the axis of rotation, and the length is equal to the angle of rotation.

For a rotation axis with unit-length vector and the angle , then the rotation vector is represented by

Euler's Rotation Theorem

All 3D rotations have an axis-angle representation.

At first, I thought these were the same as Euler Angles.

But NO! With Euler Angles, we have 3 rotation angles expressed relative to 3 fixed axes , , and -axis (pitch, roll and yaw).

On the other hand, angle-axis representation, expresses a rotation as a single rotation by an angle around a specific axis. Thus, only a 3D vector here is needed to describe the rotation.

Because of Eulerâs Rotation Theorem, we know that the 3 rotations can simply collapse to 1 rotation around a given axis: any sequence of rotations in 3D space is equivalent to a pure rotation about a single fixed axis.

### Rotation Vector to Rotation Matrix

Rodrigues' Formula

The conversion from rotation vector to rotation matrix is shown by Rodriguesâ formula. The result of the conversion is the follow

where

• is the rotation matrix
• the rotation axis has unit length vector and angle
• is the Skew-symmetric operator

Since the rotation axis does not change after the rotation, we have Therefore, the axis is the eigenvector corresponding to the matrix âs Eigenvalue 1.