Cost Functions¶
Cost functions for optimization
Useful equations :
\[\mathrm{Exp}(\phi + \delta \phi) \approx \mathrm{Exp}(\phi)\mathrm{Exp}(J_{r}(\phi)\delta \phi)\]
\[\mathrm{Log}(\mathrm{Exp}(\phi)\mathrm{Exp}(\delta \phi)) \approx \phi + J_{r}^{-1}(\phi)\delta \phi\]
\[\begin{split}\begin{align}
&R \mathrm{Exp}(\phi)R^{T} = exp(R[\phi]_{\times}R^{T}) = \mathrm{Exp}(R\phi) \\
&\Leftrightarrow \quad \mathrm{Exp}(\phi)R = R \mathrm{Exp}(R^{T}\phi)
\end{align}\end{split}\]
