Axis-Angle Representation of Rotation

An alternative representation to rotation matrices is the axis-angle representation in the form:

Where:

• $\hat{\omega} \in \mathbb{R}^3$: Rotation axis (unit vector: $\left\|\hat{\omega}\right\| = 1$)
• $\theta \in \mathbb{R}$: Rotation angle

Exponential Coordinates of Rotation

If a frame (initially coincident with the identity matrix $I$) rotates about $\hat{\omega}$ for $\theta$ units of time (i.e. $\hat{\omega}$ is integrated over that time interval) the final position would be expressed by the rotation matrix $R$.

Matrix Exponential (Rodrigues’ formula)

From the axis-angle $[\hat{\omega}]\theta =[\hat{\omega}\theta] \in so(3)$ representation the matrix exponential (rotation matrix) can be calculated:

With:

• $Rot(\hat{\omega}, \theta) \in SO(3)$: Matrix exponential of $\hat{\omega}\theta$
• $\hat{\omega} \in \mathbb{R}^3$: Rotation axis (unit vector)
• $\theta\in \mathbb{R}$: Rotation angle
• $[\hat{\omega}]\theta = [\hat{\omega}\theta] \in so(3)$: Skew-symmetric matrix form of axis-angle representation

The matrix exponential is used to construct a rotation matrix from a rotation axis $\hat{\omega}$ and an angle of rotaion $\theta$. The resulting rotation matrix represents the final orientation. The initial orientation is the identity matrix $I$.

Exponentiation integrates the angular velocity $\hat{\omega}$ for time $\theta$ seconds going from the identity matrix $I$ to the final rotation matrix $R$.

Exp: $[\hat{\omega}]\theta \in so(3) \rightarrow R \in SO(3)$

Matrix exponential is like integration

Rotation of a Vector

$e^{[\hat{\omega}]\theta}p$ has the effect of rotating $p \in \mathbb{R}^3$ about the fixed frame axis $\hat{\omega}$ by angle $\theta$.

Rotation of a Frame

$R' = e^{[\hat{\omega}] \theta} R = Rot(\hat{\omega}, \theta) R$ is the orientation achieved by rotating $R$ by $\theta$ about the axis $\hat{\omega}$ in the fixed frame.

$R'' = Re^{[\hat{\omega}] \theta} = R Rot(\hat{\omega}, \theta)$ is the orientation achieved by rotating $R$ by $\theta$ about the axis $\hat{\omega}$ in the body frame.

Logarithm

The inverse of the matrix exponential (matrix logarithm) takes a rotation matrix $R$ and returns the skew-symmetric representation of the exponential coordinates that achieve it starting from the identity orientation $I$.

It returns the angular velocity and the integration time that achieves the rotation matrix $R$.

The matrix logarithm is an algorithm that inverts Rodrigues’ formula:

1. if $R = I$ then $\theta = 0$ and $\hat{\omega}$ is undefined
2. else if $tr R = -1$ then $\theta = \pi$ and $\hat{\omega}$ is one of the following (choose a feasible solution):
   • $\hat{\omega} = \frac{1}{\sqrt{2(1 + r_{33})}}\begin{bmatrix} r_{13}\\r_{23}\\ 1 + r_{33}\end{bmatrix}$
   • $\hat{\omega} = \frac{1}{\sqrt{2(1 + r_{22})}}\begin{bmatrix} r_{12}\\1+r_{22}\\ r_{32}\end{bmatrix}$
   • $\hat{\omega} = \frac{1}{\sqrt{2(1 + r_{11})}}\begin{bmatrix} 1+ r_{11}\\r_{21}\\ r_{31}\end{bmatrix}$
   • Note: f $\hat{\omega}$ is a solution so is $-\hat{\omega}$ as well
3. otherwise $\theta = cos^{-1}(\frac{1}{2}(tr R - 1)) \in [0, \pi)$ and $[\hat{\omega}] = \frac{1}{2 sin \theta}(R - R^T)$

Every $R \in SO(3)$ satisfies one of the three cases of the algorithm. That means that for every $R$ there exists a matrix logarithm $[\hat{\omega}]\theta$ and thus also a set of exponential coordinates $\hat{\omega}\theta$ such that $R = e^{[\hat{\omega}]\theta}$.

Log: $R \in SO(3) \rightarrow [\hat{\omega}]\theta \in so(3)$

Matrix log is like differentiation

Literature

Notes taken from:

Modern Robotics: Mechanics, Planning, and Control by Kevin M. Lynch and Frank C. Park, Cambridge University Press, 2017