mvg:lecture_03:start
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mvg:lecture_03:start [2018/11/29 13:23] – [Comments] admin | mvg:lecture_03:start [2018/12/23 18:45] (current) – [TRansformation between Lie Algebra and Lie Group] admin | ||
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$R$ is in the Special Orthogonal Group $SO(3)$ | $R$ is in the Special Orthogonal Group $SO(3)$ | ||
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+ | ** Rotation nasty to be represented ** | ||
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+ | $R_1 + R_2 \notin SO(3)$, not a linear space! | ||
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+ | $R_1 \cdot R_2 \in SO(3)$, not a linear space! | ||
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+ | R has 9 elements, choice is not independent. You have to guarantee: orthogonal and det(R) = 1 | ||
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+ | **Infinitesimal rotation** | ||
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+ | better representation of non-linear | ||
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+ | Group which is a smooth Manifold, is a Lie group. | ||
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+ | $x_{trans}(t) = R(t)x_{orig}$ | ||
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+ | $R(t)R(t)^T = I$, rotate forward and backward. You land where you started. | ||
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+ | $\frac{d}{dt}(RR^T) = \dot{R}R^T + R\dot{R^T} = 0$ | ||
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+ | $\dot{R}R^T = - (R\dot{R^T})^T$, | ||
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+ | $\dot{R}R^T = \hat{w} \Leftrightarrow \dot{R}(t)= \hat{w}R(t)$ | ||
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+ | $R(dr) = R(0) + dR = I + \hat{w}(0)dt$ | ||
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+ | ==== TRansformation between Lie Algebra and Lie Group ==== | ||
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+ | Lie Algebra: $so(3) = \{\hat{w} | w \in \mathbb{R}^3\}$ of skew symmetric matrices | ||
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+ | Lie Group: $SO(3)$, smooth manifold and a group | ||
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+ | 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. | ||
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+ | Lie Brackets: | ||
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+ | $\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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mvg/lecture_03/start.1543497790.txt.gz · Last modified: 2018/11/29 13:23 by admin