Trying to implement an Extended Information Filter and I have difficulty computing the Jabobians related to the below:
Given a $3 \times 3$ rotation matrix $\mathbf{R}$ and the SO(3) logarithmic map such that the vector $\phi$ is given by:
$\theta = \arccos(0.5tr(\mathbf{R})-0.5)$
$[\phi]_{\times} = 1/\theta (\mathbf{R} - \mathbf{R}^{T}) $
and functions: $h(\phi) = \mathbf{R}^{T}(\mathbf{p}-\mathbf{q})$ where $\mathbf{p}$ and $\mathbf{q}$ are position vectors. $g(\phi_{1}) = \mathbf{R}_{2}\mathbf{R}_{1}^{T}$
what are the derivatives:
$\frac{\partial}{\partial\phi}h(\phi)$ and $\frac{\partial}{\partial\phi}g(\phi_{1})$
Many thanks
You are representing the rotation $R$ as $R=e^{\theta \phi_\times}$ where $\phi_\times$ is an antisymmetric "cross product" matrix. Note that $\phi$ is normalized, so that differentiating is a little tricky (are you differentiating on the surface of a sphere, or not?).
I will give you an answer to a slightly different problem:
Consider a unnormalized vector $\bf r$ , you can think of in as $\theta \phi$. Now what is ${\partial \over {\partial \bf r}} e^{\bf{r}_\times}$? Call ${\bf{r}}_\times = A$ and we have ${\partial \over {\partial \bf r}} e^{A}=\int_0^1 dt e^{tA}{\partial A \over \partial \bf r} e^{(1-t)A}$. You will find that ${\partial A \over \partial \bf r}$ are the generators of $SO(3)$