SO(3) 3D rotation group

Parent: Rotations / SO(3) group index
See also:  Orientation parametrisations , Linearisation of an orientation , Solà 2017 quaternion kinematics for eskf

Source: MKok 2017

• All orthogonal matrices with dim 3x3 have the property
  $$RR^\text{T} = R^\text{T}R = I_3$$
• They are part of the orthogonal group O(3)
• If, additionally, $\det R = 1$, then the matrix belongs to SO(3) and is a rotation matrix

Source:  http://en.wikipedia.org/wiki/3D_rotation_group

The SO(3) group

• Group of all rotations about the origin of the 3D space (Euclidian space, $\mathbb{R}^3$)
• Has a natural structure as a smooth  manifold .
  • Group operations on the manifold are smoothly differentiable
  • Is therefore a Lie group
• Compact, dim = 3

Rotations in general

• Rotations are linear transformations of $\mathbb{R}^3$
• Can be represented as matrices using an orthonormal basis of $\mathbb{R}^3$
• These matrices are called ‘special orthogonal matrices’, i.e. SO(3)

Source:  Forster 2017 IMU Preintegration
See: Lie group

Uncertainty description in SO(3)

• Define a distribution in the tangent space (Lie algebra)
• Map the distribution in the tangent space to SO(3) via the exponential map
• We get the random variable $\tilde{R} \in \text{SO}(3)$ $$\begin{aligned} \tilde{R} &= \mathbf{R}~ \text{Exp}(\varepsilon) \quad \varepsilon \sim \mathcal{N} (0, \Sigma) \end{aligned}$$

Distribution $p(\tilde{R})$ of the random variable:

When $\Sigma$ is small, the normalisation factor can be approximated to

If the normalisation factor beta is approximated as a constant, the negative log-likelihood of a rotation R given its measurement is