Using Lie theory to solve a problem in SE(2)

Asked May 05 '22 at 17:32

Active May 05 '22 at 17:32

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For background, I have little to no experience with Lie theory but from what I do know it feels like this problem could be easily solved using it, so I'm curious to see how I can get to the solution using the tools provided by Lie theory.

Given: $$ P_0,P_1 \in SE(2) \\ f_0,f_1 \in se(2) \\ s_0,s_1 \in \mathbb{R} $$ Find $s_0,s_1$ such that: $$ P_0 exp(s_0 f_0) = P_1 exp(s_1 f_1) $$ Basically I have two SE(2) poses with associated tangent vectors, and I want to find the scaling factors on those vectors such that the resulting poses after applying the exponential map have the same position. I think my formulation is a bit wrong below as that would require the headings of the resulting poses to be equal as well, but I'm not entirely sure how to represent this.

Does Lie theory provide the tools to easily solve this kind of problem? Or should I just resort to thinking about this geometrically? I'm struggling to find information on the kinds of operations I can do. For example, can I just invert $P_0$ and take the $log$ of both sides? Does $log(P_0^{-1} P_1 exp(s_1 f_1))$ equal $log(P_0^{-1} P_1) log(exp(s_1 f_1))$?

Thanks for any help!

asked May 05 '22 at 17:32

japata

What are SE(2) and se(2)? – Jean Marie May 05 '22 at 17:45
1

Ah sorry I should have defined that. $SE(2)$ is the set of 2D rigid transformation matrices, so a rotation and a translation, and $se(2)$ is it's associated Lie algebra. – japata May 05 '22 at 17:57
What do you call a "pose" ? The position of an object, for example the right triangle $3-4-5$ ? – Jean Marie May 05 '22 at 20:45
A pose represents a 2D position and a 2D rotation – japata May 06 '22 at 06:04
See pages 13-16 of this document – Jean Marie May 06 '22 at 12:07

Using Lie theory to solve a problem in SE(2)

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