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--- mvg:lecture_03:start [2018/11/29 13:24] – [Comments] admin
+++ mvg:lecture_03:start [2018/12/23 18:45] (current) – [TRansformation between Lie Algebra and Lie Group] admin
@@ Line 43: / Line 43: @@
 ** Rotation nasty to be represented **
-$R_1 + R_2 \notin SO(3)$
+$R_1 + R_2 \notin SO(3)$, not a linear space!
+$R_1 \cdot R_2 \in SO(3)$, not a linear space!
+R has 9 elements, choice is not independent. You have to guarantee: orthogonal and det(R) = 1
+**Infinitesimal rotation**
+better representation of non-linear
+Group which is a smooth Manifold, is a Lie group.
+$x_{trans}(t) = R(t)x_{orig}$
+$R(t)R(t)^T = I$, rotate forward and backward. You land where you started.
+$\frac{d}{dt}(RR^T) = \dot{R}R^T + R\dot{R^T} = 0$
+$\dot{R}R^T = - (R\dot{R^T})^T$, is skew symmetric!
+$\dot{R}R^T = \hat{w} \Leftrightarrow \dot{R}(t)= \hat{w}R(t)$
+$R(dr) = R(0) + dR = I + \hat{w}(0)dt$
+==== TRansformation between Lie Algebra and Lie Group ====
+Lie Algebra: $so(3) = \{\hat{w} | w \in \mathbb{R}^3\}$ of skew symmetric matrices
+Lie Group: $SO(3)$, smooth manifold and a group
+The Lie algebra $so(3)$ is the tangent space at the identity of the rotation group $SO(3)$
+algebra over a field K is a vector space over K with multiplication on the space V. Not commutative.
+Lie Brackets:
+$\left[\hat{w},\hat{v}\right] \identical \hat{w}\hat{v}-\hat{v}\hat{w} \in so(3)$
+Two Lie groups we will study: $so(3), se(3)$
+=== Exponential Map ===
+$\left\{ \dot{R}(t)=\hat{w}R(t)\right. $
+$R(0) = I$
+How does the rotation matrix change from one time within a little time step?
+$R(t) = e^{\hat{w}t} = \sum \ldots $ expansion. That solves the diff. eqn. above.
+$\exp: so(3) \rightarrow SO(3)\quad \hat{w} -> e^{\hat{w}}$