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 $W=AB^{-1}$. $W$ can also be expressed by an element of SE(3): $\omega=\ln(W)$ 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

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

where $\alpha$ and $\beta$ are the six degrees of freedom 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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