Angle Axis Representation

Resources:

• page=180

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Can be summarized by Rodrigues rotation formula

$${\mathbf{v}}^{\prime }=(1-\cos \alpha )(\mathbf{n}\cdot \mathbf{v})\mathbf{n}+\cos \alpha \mathbf{v}-\sin \alpha (\mathbf{v}\times \mathbf{n})$$

Where:

• $\mathbf{v}$ is the vector in the body frame of the vehicle
• ${\mathbf{v}}^{\prime }$ is the same vector in the world frame
• $\mathbf{n}$ is a tree-dimensional vector (the axis)
• $\alpha$ is a scalar angle

Which again can be written as

$$R(\alpha )=\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right]$$

Around the $z$-axis if $\mathbf{n}=[0,0,1{]}^T$, or generally as

Where $c=\cos$, and $S$ is the Skew-Symmetric Matrix.

Why use angle axis representation?

An alternative representation to avoid singularity issues that arises when using Euler Angles.

• Useful for a fighter jet since it can be any rotation, but for a ship you can put the singularity point when it is upside down.