Let $A$ and $B$ be two rigid transformations in 3D space that transform things from global to local coordinates.

Let their relative transformation be expressed by $W=A*B^{-1}$.

$W$ can also be expressed by an element of $se3$: $\omega=ln(W)$ where $ln()$ is defined to be the inverse of the exponential map.

I'd like to know the closed form solutions for computing the 6x6 Jacobians

$\dfrac{\delta\omega}{\delta\alpha}$ and $\dfrac{\delta\omega}{\delta\beta}$

where $\alpha$ and $\beta$ are the six degrees of freedom of motion of $A$ and $B$ respectively.

In plain English: How does $\omega$ change when I wiggle $A$ and $B$?

It might be that there's a very simple solution to this and I'm just not seeing it...


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