Exponential map

Parents: Quaternion index , Rotations / SO(3) group index

Notation

Variables

$$\begin{alignedat}{3} &\phi &&\in \mathbb{R}^3\
&\phi^\wedge &&\in \mathfrak{so}(3)\
&\mathbf{R} &&\in \text{SO}(3)\
\end{alignedat}$$

Functions

$$\begin{alignedat}{3} \text{(skew) } \wedge &:& \mathbb{R}^3 &&\rightarrow \mathfrak{so}(3) &\
\exp &:& ~&&\mathfrak{so}(3) &\rightarrow \text{SO}(3)\
\text{Exp} &:& ~\mathbb{R}^3 && &\rightarrow \text{SO}(3) \end{alignedat}$$

Thus, $$\begin{alignedat}{3} &\exp(\phi^\wedge) = \text{Exp}(\phi) = \mathbf{R} \end{alignedat}$$

Source:  Forster 2017 IMU Preintegration

At the identity

Maps an element of the Lie algebra ($\phi^\wedge \in \mathfrak{so}(3)$, a skew symmetric matrix) to a rotation

First order approximation $$ \exp(\phi^\wedge) \approx \mathbf{I} + \phi^\wedge$$

Some properties of the exponential map

• Perturbations, first order approximation with the right Jacobian of SO(3)

• $J_r(\phi) = \mathbf{I}$ for very small angles

• Following the Adjoint expression

• difference between Exp and exp? — I think exp acts on the skew-sym matrix; Exp acts on the corresp. rotation vector

Source:  MKok 2017

Approximations for small perturbations

$$ \begin{aligned} \exp_q(\eta) &\approx \left( \begin{array}{c}1 \ \eta\end{array} \right)\
\exp_R(\eta) &\approx \mathbf{I}_3 + \left[ \eta \times\right] \end{aligned} $$

Note: in this source, $\exp_q$ and $\exp_R$ equivalent to the $\text{Exp}$ notation in other sources. So here, $\eta$ is implicitly converted either to $\left[ \eta \times\right]$ or $\eta/2$ in an intermediate step.