Abig part of why we use Lie groups for robotics is because they enable efficient optimization for large scale robotics problems. The reason why Lie groups enable this is because we can represent objects like rotation matrices in lower dimensions and perform unconstrained optimization instead.
This post will dive into the mechanics of how Lie groups achieve this through the Lie Algebra and Exponential Map. I recommend reading my post on intuition for Lie Groups before reading this article.
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