Axis-Angle Parameterization

Every rotation is equivalent to a rotation around an axis (of rotation) by a certain amount of angular displacement. From this premise, the Rodrigues rotation formula or Axis-Angle Parameterization is derived. The Rodrigues formulation aims to address the intrinsic singularity issues present in the Tait-Bryan and Euler formulations (where different combinations of values represent the same rotation matrix), while also providing a geometric and concise representation of rotation.

The rotation formula proposed by Rodrigues is composed of a unit vector $\mathbf{k}$ and an angle $\vartheta$, which together allow for the representation of a rotation of points in space by an angle $\vartheta$, around the axis defined by the vector $\mathbf{k}$, with a positive direction according to the right-hand rule.

It is possible to convert an axis and angle into a rotation matrix using a compact equation proposed by Rodrigues:

$$\mathbf{R} = \mathbf{I} + \sin\vartheta [ \mathbf{k} ]_{\ti... ...- \cos \vartheta) (\mathbf{k} \mathbf{k}^{\top} - \mathbf{I} )$$ (A.12)

(this is one of the many representations available in the literature) which, when the terms are expanded, corresponds to the rotation matrix

$$\mathbf{R} = \begin{bmatrix} c+k_x^{2} (1-c)& k_x k_y (1-c... ...) & k_x s +k_y k_z (1-c) & c+k_z^{2} (1-c) \\ \end{bmatrix}$$ (A.13)

where $s=\sin\vartheta$ and $c=\cos\vartheta$. When $\vartheta = 0$, that is, in the absence of rotation, the matrix reduces to the identity.

The inverse formulation is also extremely compact and is given by:

$$\begin{array}{l} \vartheta = \cos^{-1} \left( \dfrac{ \tra... ...r_{13} - r_{31} \\ r_{21} - r_{12} \end{bmatrix} \end{array}$$ (A.14)

Since $\mathbf{k}$ and $\vartheta$ are essentially 4 parameters, a generic vector $\mathbf{w}=\vartheta \mathbf{k}$ is typically used to represent a rotation in the Rodrigues formulation, and the substitutions are made as follows:

$$\begin{array}{l} \mathbf{k} = \dfrac{\mathbf{w}}{\Vert \ma... ...{w} \Vert} \\ \vartheta = \Vert \mathbf{w} \Vert \end{array}$$ (A.15)

to accurately represent the transformation from $\mathbf{so}(3)$ to $SO(3)$.