李群李代数求导-常用求导公式

参考

A micro Lie theory for state estimation in robotics
manif issues 116

常用求导公式

Operation	左雅克比	右雅克比
${\mathcal{X}}^{-1}$	${\mathbf{J}}_{\mathcal{X}}^{{\mathcal{X}}^{-1}}=-\mathbf{I}$	${\mathbf{J}}_{\mathcal{X}}^{{\mathcal{X}}^{-1}}=-{\mathbf{Ad}}_{\mathcal{X}}$
$\mathcal{X}\circ \mathcal{Y}$	${\mathbf{J}}_{\mathcal{X}}^{\mathcal{X}\circ \mathcal{Y}}=\mathbf{I}\mid {\mathbf{J}}_{\mathcal{Y}}^{\mathcal{X}\circ \mathcal{Y}}={\mathbf{Ad}}_{\mathcal{X}}$	${\mathbf{J}}_{\mathcal{X}}^{\mathcal{X}\circ \mathcal{Y}}={\mathbf{Ad}}_{\mathcal{Y}}^{-1}\mid {\mathbf{J}}_{\mathcal{Y}}^{\mathcal{X}\circ \mathcal{Y}}=\mathbf{I}$
$Exp(\mathbf{\tau })$	${\mathbf{J}}_{\mathbf{\tau }}^{Exp(\mathbf{\tau })}={\mathbf{J}}_l(\mathbf{\tau })$	${\mathbf{J}}_{\mathbf{\tau }}^{Exp(\mathbf{\tau })}={\mathbf{J}}_r(\mathbf{\tau })$
$Log(\mathcal{X})$	${\mathbf{J}}_{\mathcal{X}}^{Log(\mathcal{X})}={\mathbf{J}}_l^{-1}(\mathbf{\tau })$	${\mathbf{J}}_{\mathcal{X}}^{Log(\mathcal{X})}={\mathbf{J}}_r^{-1}(\mathbf{\tau })$
Plus	${\mathbf{J}}_{\mathcal{X}}^{\mathbf{\tau }\oplus \mathcal{X}}={\mathbf{Ad}}_{Exp(\mathbf{\tau })}\mid {\mathbf{J}}_{\mathbf{\tau }}^{\mathbf{\tau }\oplus \mathcal{X}}={\mathbf{J}}_l(\mathbf{\tau })$	${\mathbf{J}}_{\mathcal{X}}^{\mathcal{X}\oplus \mathbf{\tau }}={\mathbf{Ad}}_{Exp(\mathbf{\tau })}^{-1}\mid {\mathbf{J}}_{\mathbf{\tau }}^{\mathcal{X}\oplus \mathbf{\tau }}={\mathbf{J}}_r(\mathbf{\tau })$
Minus	${\mathbf{J}}_{\mathcal{X}}^{\mathcal{X}\ominus \mathcal{Y}}=-{\mathbf{J}}_r^{-1}(\mathbf{\tau })\mid {\mathbf{J}}_{\mathcal{Y}}^{\mathcal{X}\ominus \mathcal{Y}}={\mathbf{J}}_l^{-1}(\mathbf{\tau })$	${\mathbf{J}}_{\mathcal{X}}^{\mathcal{X}\ominus \mathcal{Y}}={\mathbf{J}}_r^{-1}(\mathbf{\tau })\mid {\mathbf{J}}_{\mathcal{Y}}^{\mathcal{X}\ominus \mathcal{Y}}=-{\mathbf{J}}_l^{-1}(\mathbf{\tau })$

公式中的伴随矩阵

对于SO3:
$$\mathbf{A}{\mathbf{d}}_{\mathbf{R}}=\mathbf{R}$$
对于SE3:
$$\begin{array}{rl}\mathbf{M}& =\left[\begin{array}{cc}\mathbf{R}& \mathbf{t}\\ 0& 1\end{array}\right]\\ \mathbf{A}{\mathbf{d}}_{\mathbf{M}}& =\left[\begin{array}{cc}\mathbf{R}& {\lfloor \mathbf{t}\rfloor }_{\times }\mathbf{R}\\ 0& \mathbf{R}\end{array}\right]\end{array}$$

公式中的左右雅克比 ${\mathbf{J}}_r$、${\mathbf{J}}_l$

对于SO3:

对于SE3: