Axis-Angle

Definition

Rotation can be respresented by a unit vector, \(\hat{\mathbf{e}} \in \mathbb{R}^3\) and a rotation angle \(\theta \in \mathbb{R}\). Let \(\boldsymbol{\rho} = \theta \hat{\mathbf{e}} \in \mathbb{R}^3\), with \(\left[ \boldsymbol{\rho} \right]_\times \in \mathfrak{so}(3)\).

Given frames \(F_\alpha\) and \(F_\beta\), \(\boldsymbol{\rho}_{\beta \alpha} = \theta_{\beta \alpha} \hat{\mathbf{e}}_{\beta \alpha}\) denotes the rotation vector from \(F_\beta\) to \(F_\alpha\). It does not matter in which frame \(\hat{\mathbf{e}}\) is expressed in since \(\mathbf{R} \hat{\mathbf{e}} = \hat{\mathbf{e}}\).

Axis-Angle Representation
Rodrigues' Formula	\(\begin{align*} \mathbf{R}^{\alpha}_{\beta} &= \exp{\left( \left[ \boldsymbol{\rho}_{\beta \alpha} \right]_\times \right)} \\ &= \mathbf{I} + \sin \theta \left[ \hat{\mathbf{e}} \right]_\times + \left(1 - \cos \theta \right) \left[ \hat{\mathbf{e}} \right]^2_\times \\ &= \cos \theta \mathbf{I} + \left(1 - \cos \theta \right) \hat{\mathbf{e}} \hat{\mathbf{e}}^T + \sin \theta \left[ \hat{\mathbf{e}} \right]_\times \end{align*}\)
Logarithmic Mapping	\(\begin{align*} \left[ \boldsymbol{\rho}_{\beta \alpha} \right]_\times &= \ln \left( \mathbf{R}^\alpha_{\beta} \right) = \sum^{\infty}_{k=0} \frac{(-1)^k}{k + 1} \left( \mathbf{R}^{\alpha}_{\beta} - \mathbf{I} \right)^{k + 1} \end{align*}\)
Closed-Form Rotation Angle	\(\begin{align*} \theta_{\beta \alpha} &= \text{arccos} \left( \frac{\text{tr} \left( \mathbf{R}^{\alpha}_{\beta}\right) - 1}{2} \right) \\ &= \text{arccos} \left( \frac{1}{2} \left( \mathbf{R}^{\alpha}_{\beta \ 1,1} + \mathbf{R}^{\alpha}_{\beta \ 2, 2} + \mathbf{R}^{\alpha}_{\beta \ 1,1} - 1\right) \right) \end{align*}\)
Closed-Form Rotation Vector	\(\begin{align*} \boldsymbol{\rho}_{\alpha \beta} &= \frac{\theta_{\beta \alpha}}{2 \text{sin} (\theta_{\beta \alpha})} \left[ \begin{array}{c} \mathbf{R}^{\beta}_{\alpha \ 2,3} - \mathbf{R}^{\beta}_{\alpha \ 3,2} \\ \mathbf{R}^{\beta}_{\alpha \ 3,1} - \mathbf{R}^{\beta}_{\alpha \ 1,3} \\ \mathbf{R}^{\alpha}_{\beta \ 1,2} - \mathbf{R}^{\alpha}_{\beta \ 2,1} \end{array} \right]. \end{align*}\)
Infinitesimal Rotations	\(\begin{align*} \mathbf{R^{\alpha}_{\beta}}= e^{\left[\boldsymbol{\rho}_{\beta \alpha} \right]_{\times}} \approx \sum^{\infty}_{k=0} \frac{\left[\boldsymbol{\rho}_{\beta \alpha} \right]^k_{\times}}{k!} \approx \mathbf{I}_3 + \left[\boldsymbol{\rho}_{\beta \alpha} \right]_{\times} \end{align*}\)

Note that rotation vector and Euler angles are identical for small perturbation.

Proofs

Rodrigues' Formula

Let \(F_\alpha\) and \(F_\beta\) be two frames. A rotation can be represented by a rotation axis \(\hat{\mathbf{e}}\) and an angle \(\theta\) (from \(F_\beta\) to \(F_\alpha\)), or equivalently by a 3D vector \(\boldsymbol{\rho} = \theta \hat{\mathbf{e}}\). The rotation matrix can be computed via exponential mapping:

\[ \begin{align} \mathbf{R}^\alpha{\beta} =e^{\left[\boldsymbol{\rho} \right]_\times} &= e^{ \left[ \theta \hat{\mathbf{e}} \right]_\times} = \sum^{\infty}_{k = 0} \frac{1}{k!} \left(\theta \left[\hat{\mathbf{e}} \right]_\times \right)^k \\ &= \mathbf{I} + \theta \left[ \hat{\mathbf{e}} \right]_\times + \frac{\theta^2}{2} \left[ \hat{\mathbf{e}} \right]^2_\times + \frac{\theta^3}{3!} \left[ \hat{\mathbf{e}} \right]^2_\times + \ldots \\ &= \mathbf{I} + \left( \theta - \frac{\theta^3}{3!} + \ldots \right) \left[ \hat{\mathbf{e}} \right]_\times + \left( \frac{\theta^2}{2} - \frac{\theta^4}{4!} + \ldots \right) \left[ \hat{\mathbf{e}} \right]^2_\times \\ &= \mathbf{I} + \sin \theta \left[ \hat{\mathbf{e}} \right]_\times + \left(1 - \cos \theta \right) \left[ \hat{\mathbf{e}} \right]^2_\times \\ &= \cos \theta \mathbf{I} + \left(1 - \cos \theta \right) \hat{\mathbf{e}} \hat{\mathbf{e}}^T + \sin \theta \left[ \hat{\mathbf{e}} \right]_\times, \end{align} \]

or simply:

\[ \mathbf{R} = \cos \theta \mathbf{I} + \left(1 - \cos \theta \right) \hat{\mathbf{e}} \hat{\mathbf{e}}^T + \sin \theta \left[ \hat{\mathbf{e}} \right]_{\times}. \label{rodrigues_formula} \]

References

1. Groves, P., Principles of GNSS, Inertial, and Multisensor Integrated Navigation Systems, Second Edition
2. Sola, J., Quaternion Kinematics for the Error-State Kalman Filter
3. Ma., Y., An Invitation to 3-D Vision: From Images to Models, 2001