mvg:lecture_03:start
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mvg:lecture_03:start [2018/11/29 14:04] – [TRansformation between Lie Algebra and Lie Group] admin | mvg:lecture_03:start [2018/12/23 18:45] (current) – [TRansformation between Lie Algebra and Lie Group] admin | ||
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The Lie algebra $so(3)$ is the tangent space at the identity of the rotation group $SO(3)$ | The Lie algebra $so(3)$ is the tangent space at the identity of the rotation group $SO(3)$ | ||
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+ | algebra over a field K is a vector space over K with multiplication on the space V. Not commutative. | ||
Lie Brackets: | Lie Brackets: | ||
- | $\left[\hat{w}, | + | $\left[\hat{w}, |
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+ | Two Lie groups we will study: $so(3), se(3)$ | ||
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+ | === Exponential Map === | ||
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+ | $\left\{ \dot{R}(t)=\hat{w}R(t)\right. $ | ||
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+ | $R(0) = I$ | ||
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+ | How does the rotation matrix change from one time within a little time step? | ||
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+ | $R(t) = e^{\hat{w}t} = \sum \ldots $ expansion. That solves the diff. eqn. above. | ||
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+ | $\exp: so(3) \rightarrow SO(3)\quad \hat{w} -> e^{\hat{w}}$ | ||
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- | Two Le groups we will study: $so(3), se(3)$ | ||
mvg/lecture_03/start.1543500249.txt.gz · Last modified: 2018/11/29 14:04 by admin