Axis-Angle Representation (Rotation Vector)

Covered to understand Rodrigues’ Rotation Formula in Mechanization.

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 n and the angle $\theta$, the rotation vector is represented by $\theta$n

Euler's Rotation Theorem

All 3D rotations have an axis-angle representation.

There is a difference in comparison with Euler Angles.

• With Euler Angles, we have 3 rotation angles expressed relative to 3 fixed axes $x,y,z$ (pitch, roll, yaw)
• 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.

Rotation Vector to Rotation Matrix

See Rodrigues’ Rotation Formula.