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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
@@ Line 40: / Line 40: @@
  $R$  is in the Special Orthogonal Group  $SO(3)$
+** Rotation nasty to be represented **
+ $R_1 + R_2 \notin SO(3)$ , 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.
+ $\exp: so(3) \rightarrow SO(3)\quad \hat{w} -> e^{\hat{w}}$