SE(3) Special Euclidian Group
Source: Forster 2017 – IMU Preintegration
Group of rigid motion in 3D.
Consists of
- a rotation in SO(3)
- a translation in $\mathbb{R}^3$
$$\begin{aligned} \text{SE}(3) \dot{=} \left\lbrace \left(\mathbf{R}, \mathbf{p} \right) : \mathbf{R} \in \text{SO}(3), \mathbf{p} \in \mathbb{R}^3 \right\rbrace \end{aligned}$$
$$\begin{aligned}
\mathbf{T}_1\mathbf{T}_2 &= \left(
\mathbf{R}_1\mathbf{R}_2,
\mathbf{R}_1\mathbf{p}_2 + \mathbf{p}_1
\right)\
\mathbf{T}_1^{-1} &= \left(
\mathbf{R}_1^\text{T}, -\mathbf{R}_1^{\text{T}}\mathbf{p}_1
\right)
\end{aligned}$$