1545 lines
27 KiB
Plaintext
1545 lines
27 KiB
Plaintext
#LyX 1.6.5 created this file. For more info see http://www.lyx.org/
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\lyxformat 345
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\begin_document
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\begin_header
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\textclass article
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\use_default_options false
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\begin_modules
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theorems-std
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\end_modules
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\language english
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\rightmargin 1in
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\bottommargin 1in
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\secnumdepth 3
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\tocdepth 3
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\paragraph_separation indent
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\defskip medskip
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\quotes_language english
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\papercolumns 1
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\papersides 1
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\paperpagestyle default
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\tracking_changes false
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\output_changes false
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\author ""
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\author ""
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\end_header
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\begin_body
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\begin_layout Title
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Lie Groups for Beginners
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\end_layout
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\begin_layout Author
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Frank Dellaert
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\end_layout
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\begin_layout Standard
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\begin_inset CommandInset include
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LatexCommand include
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filename "macros.lyx"
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\end_inset
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\end_layout
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\begin_layout Section
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Basic Lie Group Concepts
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\end_layout
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\begin_layout Subsection
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A Manifold and a Group
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\end_layout
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\begin_layout Standard
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A Lie group
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\begin_inset Formula $G$
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\end_inset
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is a manifold that possesses a smooth group operation.
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Associated with it is a Lie Algebra
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\begin_inset Formula $\gg$
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\end_inset
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which, loosely speaking, can be identified with the tangent space at the
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identity and completely defines how the groups behaves around the identity.
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There is a mapping from
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\begin_inset Formula $\gg$
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\end_inset
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back to
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\begin_inset Formula $G$
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\end_inset
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, called the exponential map
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\begin_inset Formula \[
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\exp:\gg\rightarrow G\]
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\end_inset
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and a corresponding inverse
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\begin_inset Formula \[
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\log:G\rightarrow\gg\]
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\end_inset
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that maps elements in G to an element in
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\begin_inset Formula $\gg$
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\end_inset
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.
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\end_layout
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\begin_layout Subsection
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Lie Algebra
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\end_layout
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\begin_layout Standard
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The Lie Algebra
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\begin_inset Formula $\gg$
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\end_inset
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is called an algebra because it is endowed with a binary operation, the
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Lie bracket
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\begin_inset Formula $[X,Y]$
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\end_inset
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, the properties of which are closely related to the group operation of
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\begin_inset Formula $G$
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\end_inset
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.
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For example, in matrix Lie groups, the Lie bracket is given by
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\begin_inset Formula $[A,B]\define AB-BA$
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\end_inset
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.
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The relationship with the group operation is as follows: for commutative
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Lie groups vector addition
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\begin_inset Formula $X+Y$
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\end_inset
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in
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\begin_inset Formula $\gg$
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\end_inset
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mimicks the group operation.
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For example, if we have
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\begin_inset Formula $Z=X+Y$
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\end_inset
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in
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\begin_inset Formula $\gg$
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\end_inset
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, when mapped backed to
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\begin_inset Formula $G$
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\end_inset
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via the exponential map we obtain
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\begin_inset Formula \[
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e^{Z}=e^{X+Y}=e^{X}e^{Y}\]
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\end_inset
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However, this does
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\emph on
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not
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\emph default
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hold for non-commutative Lie groups:
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\begin_inset Formula \[
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Z=\log(e^{X}e^{Y})\neq X+Y\]
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\end_inset
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Instead,
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\begin_inset Formula $Z$
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\end_inset
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can be calculated using the Baker-Campbell-Hausdorff (BCH) formula:
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\begin_inset Foot
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status collapsed
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\begin_layout Plain Layout
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http://en.wikipedia.org/wiki/Baker–Campbell–Hausdorff_formula
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\end_layout
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\end_inset
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\begin_inset Formula \[
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Z=X+Y+[X,Y]/2+[X-Y,[X,Y]]/12-[Y,[X,[X,Y]]]/24+\ldots\]
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\end_inset
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For commutative groups the bracket is zero and we recover
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\begin_inset Formula $Z=X+Y$
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\end_inset
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.
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For non-commutative groups we can use the BCH formula to approximate it.
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\end_layout
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||
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\begin_layout Subsection
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Exponential Coordinates
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||
\end_layout
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\begin_layout Standard
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For
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\begin_inset Formula $n$
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\end_inset
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-dimensional matrix Lie groups, the Lie algebra
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\begin_inset Formula $\gg$
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\end_inset
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is isomorphic to
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\begin_inset Formula $\mathbb{R}^{n}$
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||
\end_inset
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||
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, and we can define the mapping
|
||
\begin_inset Formula \[
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\hat{}:\mathbb{R}^{n}\rightarrow\gg\]
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\end_inset
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||
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||
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\begin_inset Formula \[
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\hat{}:x\rightarrow\xhat\]
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\end_inset
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which maps
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||
\begin_inset Formula $n$
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\end_inset
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-vectors
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||
\begin_inset Formula $x\in$
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||
\end_inset
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\begin_inset Formula $\Rn$
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\end_inset
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to elements of
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\begin_inset Formula $\gg$
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\end_inset
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.
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In the case of matrix Lie groups, the elements
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\begin_inset Formula $\xhat$
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\end_inset
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of
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\begin_inset Formula $\gg$
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\end_inset
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are
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\begin_inset Formula $n\times n$
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||
\end_inset
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||
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matrices, and the map is given by
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||
\begin_inset Formula \begin{equation}
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||
\xhat=\sum_{i=1}^{n}x_{i}G^{i}\label{eq:generators}\end{equation}
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||
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\end_inset
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||
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where the
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\begin_inset Formula $G^{i}$
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\end_inset
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||
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are
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||
\begin_inset Formula $n\times n$
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||
\end_inset
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||
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matrices known as the Lie group generators.
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The meaning of the map
|
||
\begin_inset Formula $x\rightarrow\xhat$
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\end_inset
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||
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will depend on the group
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||
\begin_inset Formula $G$
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||
\end_inset
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||
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||
and will be very intuitive.
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||
\end_layout
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||
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\begin_layout Subsection
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||
The Adjoint Map
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||
\end_layout
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||
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||
\begin_layout Standard
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||
Below we frequently make use of the equality
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||
\begin_inset Foot
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status collapsed
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\begin_layout Plain Layout
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||
http://en.wikipedia.org/wiki/Exponential_map
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\end_layout
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\end_inset
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\begin_inset Formula \[
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ge^{\xhat}g^{-1}=e^{\Ad g{\xhat}}\]
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||
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\end_inset
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where
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\begin_inset Formula $\Ad g:\gg\rightarrow\mathfrak{\gg}$
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||
\end_inset
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||
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is a map parameterized by a group element
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||
\begin_inset Formula $g$
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||
\end_inset
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||
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.
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The intuitive explanation is that a change
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||
\begin_inset Formula $\exp\left(\xhat\right)$
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||
\end_inset
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||
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||
defined around the orgin, but applied at the group element
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||
\begin_inset Formula $g$
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||
\end_inset
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||
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, can be written in one step by taking the adjoint
|
||
\begin_inset Formula $\Ad g{\xhat}$
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||
\end_inset
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||
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of
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\begin_inset Formula $\xhat$
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||
\end_inset
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||
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.
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||
In the case of a matrix group the ajoint can be written as
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\begin_inset Foot
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status collapsed
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\begin_layout Plain Layout
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http://en.wikipedia.org/wiki/Adjoint_representation_of_a_Lie_group
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||
\end_layout
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||
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\end_inset
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||
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\begin_inset Formula \[
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\Ad T{\xhat}\define Te^{\xhat}T^{-1}\]
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||
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\end_inset
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||
and hence we have
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||
\end_layout
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||
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||
\begin_layout Standard
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||
\begin_inset Formula \[
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||
Te^{\xhat}T^{-1}=e^{T\xhat T^{-1}}\]
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||
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||
\end_inset
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where both
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||
\begin_inset Formula $T$
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||
\end_inset
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||
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||
and
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||
\begin_inset Formula $\xhat$
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||
\end_inset
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||
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||
are
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||
\begin_inset Formula $n\times n$
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||
\end_inset
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||
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matrices for an
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||
\begin_inset Formula $n$
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||
\end_inset
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||
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||
-dimensional Lie group.
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||
\end_layout
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||
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||
\begin_layout Subsection
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||
Actions
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||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The (usual) action of an
|
||
\begin_inset Formula $n$
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||
\end_inset
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||
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||
-dimensional matrix group
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||
\begin_inset Formula $G$
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||
\end_inset
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||
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||
is matrix-vector multiplication on
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||
\begin_inset Formula $\mathbb{R}^{n}$
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||
\end_inset
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||
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||
,
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||
\begin_inset Formula \[
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||
q=Tp\]
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||
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||
\end_inset
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||
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with
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||
\begin_inset Formula $p,q\in\mathbb{R}^{n}$
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||
\end_inset
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||
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and
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||
\begin_inset Formula $T\in GL(n)$
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||
\end_inset
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||
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.
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||
\end_layout
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||
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||
\begin_layout Standard
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||
\begin_inset Newpage pagebreak
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||
\end_inset
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||
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||
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||
\end_layout
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||
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||
\begin_layout Section
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||
2D Rotations
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||
\end_layout
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||
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||
\begin_layout Standard
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||
We first look at a very simple group, the 2D rotations.
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||
\end_layout
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||
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||
\begin_layout Subsection
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||
Basics
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||
\end_layout
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||
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||
\begin_layout Standard
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||
The Lie group
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||
\begin_inset Formula $\SOtwo$
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\end_inset
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is a subgroup of the general linear group
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||
\begin_inset Formula $GL(2)$
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||
\end_inset
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||
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of
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\begin_inset Formula $2\times2$
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\end_inset
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||
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invertible matrices.
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Its Lie algebra
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\begin_inset Formula $\sotwo$
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\end_inset
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is the vector space of
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\begin_inset Formula $2\times2$
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\end_inset
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skew-symmetric matrices.
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Since
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\begin_inset Formula $\SOtwo$
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||
\end_inset
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||
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is a one-dimensional manifold,
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||
\begin_inset Formula $\sotwo$
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||
\end_inset
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||
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is isomorphic to
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||
\begin_inset Formula $\mathbb{R}$
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||
\end_inset
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and we define
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||
\begin_inset Formula \[
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\hat{}:\mathbb{R}\rightarrow\sotwo\]
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||
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\end_inset
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||
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||
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\begin_inset Formula \[
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\hat{}:\theta\rightarrow\that=\skew{\theta}\]
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\end_inset
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which maps the angle
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||
\begin_inset Formula $\theta$
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||
\end_inset
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||
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||
to the
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||
\begin_inset Formula $2\times2$
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||
\end_inset
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||
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skew-symmetric matrix
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||
\family roman
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||
\series medium
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||
\shape up
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||
\size normal
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||
\emph off
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||
\bar no
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||
\noun off
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||
\color none
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||
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||
\begin_inset Formula $\skew{\theta}$
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\end_inset
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||
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:
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||
\family default
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||
\series default
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||
\shape default
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||
\size default
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||
\emph default
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||
\bar default
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||
\noun default
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||
\color inherit
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||
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||
\begin_inset Formula \[
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||
\skew{\theta}=\left[\begin{array}{cc}
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||
0 & -\theta\\
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||
\theta & 0\end{array}\right]\]
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||
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||
\end_inset
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||
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||
The exponential map can be computed in closed form as
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||
\begin_inset Formula \[
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||
R=e^{\skew{\theta}}=\left[\begin{array}{cc}
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||
\cos\theta & -\sin\theta\\
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||
\sin\theta & \cos\theta\end{array}\right]\]
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||
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||
\end_inset
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||
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||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
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||
Actions
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||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In the case of
|
||
\begin_inset Formula $\SOtwo$
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||
\end_inset
|
||
|
||
the vector space is
|
||
\begin_inset Formula $\Rtwo$
|
||
\end_inset
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||
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||
, and the group action corresponds to rotating a point
|
||
\begin_inset Formula \[
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||
q=Rp\]
|
||
|
||
\end_inset
|
||
|
||
We would now like to know what an incremental rotation parameterized by
|
||
|
||
\begin_inset Formula $\theta$
|
||
\end_inset
|
||
|
||
would do:
|
||
\begin_inset Formula \[
|
||
q(\text{\theta})=Re^{\skew{\theta}}p\]
|
||
|
||
\end_inset
|
||
|
||
hence the derivative is:
|
||
\begin_inset Formula \[
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||
\deriv{q(\omega)}{\omega}=R\deriv{}{\omega}\left(e^{\skew{\theta}}p\right)=R\deriv{}{\omega}\left(\skew{\theta}p\right)=RH_{p}\]
|
||
|
||
\end_inset
|
||
|
||
Note that
|
||
\begin_inset Formula \begin{equation}
|
||
\skew{\theta}\left[\begin{array}{c}
|
||
x\\
|
||
y\end{array}\right]=\theta R_{\pi/2}\left[\begin{array}{c}
|
||
x\\
|
||
y\end{array}\right]=\theta\left[\begin{array}{c}
|
||
-y\\
|
||
x\end{array}\right]\label{eq:RestrictedCross}\end{equation}
|
||
|
||
\end_inset
|
||
|
||
which acts like a restricted
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
cross product
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
in the plane.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Newpage pagebreak
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
2D Rigid Transformations
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Basics
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The Lie group
|
||
\begin_inset Formula $\SEtwo$
|
||
\end_inset
|
||
|
||
is a subgroup of the general linear group
|
||
\begin_inset Formula $GL(3)$
|
||
\end_inset
|
||
|
||
of
|
||
\begin_inset Formula $3\times3$
|
||
\end_inset
|
||
|
||
invertible matrices of the form
|
||
\begin_inset Formula \[
|
||
T\define\left[\begin{array}{cc}
|
||
R & t\\
|
||
0 & 1\end{array}\right]\]
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $R\in\SOtwo$
|
||
\end_inset
|
||
|
||
is a rotation matrix and
|
||
\begin_inset Formula $t\in\Rtwo$
|
||
\end_inset
|
||
|
||
is a translation vector.
|
||
Its Lie algebra
|
||
\begin_inset Formula $\setwo$
|
||
\end_inset
|
||
|
||
is the vector space of
|
||
\begin_inset Formula $3\times3$
|
||
\end_inset
|
||
|
||
twists
|
||
\begin_inset Formula $\xihat$
|
||
\end_inset
|
||
|
||
parameterized by the
|
||
\emph on
|
||
twist coordinates
|
||
\emph default
|
||
|
||
\begin_inset Formula $\xi\in\Rthree$
|
||
\end_inset
|
||
|
||
, with the mapping
|
||
\begin_inset Formula \[
|
||
\xi\define\left[\begin{array}{c}
|
||
v\\
|
||
\omega\end{array}\right]\rightarrow\xihat\define\left[\begin{array}{cc}
|
||
\skew{\omega} & v\\
|
||
0 & 0\end{array}\right]\]
|
||
|
||
\end_inset
|
||
|
||
Note we think of robots as having a pose
|
||
\begin_inset Formula $(x,y,\theta)$
|
||
\end_inset
|
||
|
||
and hence I reserved the first two components for translation and the last
|
||
for rotation.
|
||
|
||
\family roman
|
||
\series medium
|
||
\shape up
|
||
\size normal
|
||
\emph off
|
||
\bar no
|
||
\noun off
|
||
\color none
|
||
The Lie group generators are
|
||
\begin_inset Formula \[
|
||
G^{x}=\left[\begin{array}{ccc}
|
||
0 & 0 & 1\\
|
||
0 & 0 & 0\\
|
||
0 & 0 & 0\end{array}\right]\mbox{ }G^{y}=\left[\begin{array}{ccc}
|
||
0 & 0 & 0\\
|
||
0 & 0 & 1\\
|
||
0 & 0 & 0\end{array}\right]\mbox{ }G^{\theta}=\left[\begin{array}{ccc}
|
||
0 & -1 & 0\\
|
||
1 & 0 & 0\\
|
||
0 & 0 & 0\end{array}\right]\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\family default
|
||
\series default
|
||
\shape default
|
||
\size default
|
||
\emph default
|
||
\bar default
|
||
\noun default
|
||
\color inherit
|
||
Applying the exponential map to a twist
|
||
\begin_inset Formula $\xi$
|
||
\end_inset
|
||
|
||
yields a screw motion yielding an element in
|
||
\begin_inset Formula $\SEtwo$
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula \[
|
||
T=\exp\xihat\]
|
||
|
||
\end_inset
|
||
|
||
A closed form solution for the exponential map is in the works...
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
The Adjoint Map
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The adjoint is
|
||
\begin_inset Formula \begin{eqnarray}
|
||
\Ad T{\xihat} & = & T\xihat T^{-1}\nonumber \\
|
||
& = & \left[\begin{array}{cc}
|
||
R & t\\
|
||
0 & 1\end{array}\right]\left[\begin{array}{cc}
|
||
\skew{\omega} & v\\
|
||
0 & 0\end{array}\right]\left[\begin{array}{cc}
|
||
R^{T} & -R^{T}t\\
|
||
0 & 1\end{array}\right]\nonumber \\
|
||
& = & \left[\begin{array}{cc}
|
||
\skew{\omega} & -\skew{\omega}t+Rv\\
|
||
0 & 0\end{array}\right]\nonumber \\
|
||
& = & \left[\begin{array}{cc}
|
||
\skew{\omega} & Rv-\omega R_{\pi/2}t\\
|
||
0 & 0\end{array}\right]\label{eq:adjointSE2}\end{eqnarray}
|
||
|
||
\end_inset
|
||
|
||
From this we can express the Adjoint map in terms of plane twist coordinates:
|
||
\begin_inset Formula \[
|
||
\left[\begin{array}{c}
|
||
v'\\
|
||
\omega'\end{array}\right]=\left[\begin{array}{cc}
|
||
R & -R_{\pi/2}t\\
|
||
0 & 1\end{array}\right]\left[\begin{array}{c}
|
||
v\\
|
||
\omega\end{array}\right]\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Actions
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The action of
|
||
\begin_inset Formula $\SEtwo$
|
||
\end_inset
|
||
|
||
on 2D points is done by embedding the points in
|
||
\begin_inset Formula $\mathbb{R}^{3}$
|
||
\end_inset
|
||
|
||
by using homogeneous coordinates
|
||
\begin_inset Formula \[
|
||
\hat{q}=\left[\begin{array}{c}
|
||
q\\
|
||
1\end{array}\right]=\left[\begin{array}{cc}
|
||
R & t\\
|
||
0 & 1\end{array}\right]\left[\begin{array}{c}
|
||
p\\
|
||
1\end{array}\right]=T\hat{p}\]
|
||
|
||
\end_inset
|
||
|
||
Analoguous to
|
||
\begin_inset Formula $\SEthree$
|
||
\end_inset
|
||
|
||
, we can compute a velocity
|
||
\begin_inset Formula $\xihat\hat{p}$
|
||
\end_inset
|
||
|
||
in the local
|
||
\begin_inset Formula $T$
|
||
\end_inset
|
||
|
||
frame:
|
||
\begin_inset Formula \[
|
||
\xihat\hat{p}=\left[\begin{array}{cc}
|
||
\skew{\omega} & v\\
|
||
0 & 0\end{array}\right]\left[\begin{array}{c}
|
||
p\\
|
||
1\end{array}\right]=\left[\begin{array}{c}
|
||
\skew{\omega}p+v\\
|
||
0\end{array}\right]\]
|
||
|
||
\end_inset
|
||
|
||
By only taking the top two rows, we can write this as a velocity in
|
||
\begin_inset Formula $\Rtwo$
|
||
\end_inset
|
||
|
||
, as the product of a
|
||
\begin_inset Formula $2\times3$
|
||
\end_inset
|
||
|
||
matrix
|
||
\begin_inset Formula $H_{p}$
|
||
\end_inset
|
||
|
||
that acts upon the exponential coordinates
|
||
\begin_inset Formula $\xi$
|
||
\end_inset
|
||
|
||
directly:
|
||
\begin_inset Formula \[
|
||
\skew{\omega}p+v=v+R_{\pi/2}p\omega=\left[\begin{array}{cc}
|
||
I_{2} & R_{\pi/2}p\end{array}\right]\left[\begin{array}{c}
|
||
v\\
|
||
\omega\end{array}\right]=H_{p}\xi\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Newpage pagebreak
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
3D Rotations
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Basics
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The Lie group
|
||
\begin_inset Formula $\SOthree$
|
||
\end_inset
|
||
|
||
is a subgroup of the general linear group
|
||
\begin_inset Formula $GL(3)$
|
||
\end_inset
|
||
|
||
of
|
||
\begin_inset Formula $3\times3$
|
||
\end_inset
|
||
|
||
invertible matrices.
|
||
Its Lie algebra
|
||
\begin_inset Formula $\sothree$
|
||
\end_inset
|
||
|
||
is the vector space of
|
||
\begin_inset Formula $3\times3$
|
||
\end_inset
|
||
|
||
skew-symmetric matrices.
|
||
The exponential map can be computed in closed form using Rodrigues' formula.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Since
|
||
\begin_inset Formula $\SOthree$
|
||
\end_inset
|
||
|
||
is a three-dimensional manifold,
|
||
\begin_inset Formula $\sothree$
|
||
\end_inset
|
||
|
||
is isomorphic to
|
||
\begin_inset Formula $\Rthree$
|
||
\end_inset
|
||
|
||
and we define the map
|
||
\begin_inset Formula \[
|
||
\hat{}:\Rthree\rightarrow\sothree\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula \[
|
||
\hat{}:\omega\rightarrow\what=\Skew{\omega}\]
|
||
|
||
\end_inset
|
||
|
||
which maps 3-vectors
|
||
\begin_inset Formula $\omega$
|
||
\end_inset
|
||
|
||
to skew-symmetric matrices
|
||
\begin_inset Formula $\Skew{\omega}$
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula \[
|
||
\Skew{\omega}=\left[\begin{array}{ccc}
|
||
0 & -\omega_{z} & \omega_{y}\\
|
||
\omega_{z} & 0 & -\omega_{x}\\
|
||
-\omega_{y} & \omega_{x} & 0\end{array}\right]=\omega_{x}G^{x}+\omega_{y}G^{y}+\omega_{z}G^{z}\]
|
||
|
||
\end_inset
|
||
|
||
where the
|
||
\begin_inset Formula $G^{i}$
|
||
\end_inset
|
||
|
||
are the generators for
|
||
\begin_inset Formula $\SOthree$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula \[
|
||
G^{x}=\left(\begin{array}{ccc}
|
||
0 & 0 & 0\\
|
||
0 & 0 & -1\\
|
||
0 & 1 & 0\end{array}\right)\mbox{}G^{y}=\left(\begin{array}{ccc}
|
||
0 & 0 & 1\\
|
||
0 & 0 & 0\\
|
||
-1 & 0 & 0\end{array}\right)\mbox{ }G^{z}=\left(\begin{array}{ccc}
|
||
0 & -1 & 0\\
|
||
1 & 0 & 0\\
|
||
0 & 0 & 0\end{array}\right)\]
|
||
|
||
\end_inset
|
||
|
||
corresponding to a rotation around
|
||
\begin_inset Formula $X$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $Y$
|
||
\end_inset
|
||
|
||
, and
|
||
\begin_inset Formula $Z$
|
||
\end_inset
|
||
|
||
, respectively.
|
||
The Lie bracket
|
||
\begin_inset Formula $[x,y]$
|
||
\end_inset
|
||
|
||
corresponds to the cross product
|
||
\begin_inset Formula $x\times y$
|
||
\end_inset
|
||
|
||
in
|
||
\begin_inset Formula $\Rthree$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For every
|
||
\begin_inset Formula $3-$
|
||
\end_inset
|
||
|
||
vector
|
||
\begin_inset Formula $\omega$
|
||
\end_inset
|
||
|
||
there is a corresponding rotation matrix
|
||
\begin_inset Formula \[
|
||
R=e^{\Skew{\omega}}\]
|
||
|
||
\end_inset
|
||
|
||
and this is defines the canonical parameterization of
|
||
\begin_inset Formula $\SOthree$
|
||
\end_inset
|
||
|
||
, with
|
||
\begin_inset Formula $\omega$
|
||
\end_inset
|
||
|
||
known as the canonical or exponential coordinates.
|
||
It is equivalent to the axis-angle representation for rotations, where
|
||
the unit vector
|
||
\begin_inset Formula $\omega/\left\Vert \omega\right\Vert $
|
||
\end_inset
|
||
|
||
defines the rotation axis, and its magnitude the amount of rotation
|
||
\begin_inset Formula $\theta$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
The Adjoint Map
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For rotation matrices
|
||
\begin_inset Formula $R$
|
||
\end_inset
|
||
|
||
we can prove the following identity (see
|
||
\begin_inset CommandInset ref
|
||
LatexCommand vref
|
||
reference "remove"
|
||
|
||
\end_inset
|
||
|
||
):
|
||
\begin_inset Formula \begin{equation}
|
||
R\Skew{\omega}R^{T}=\Skew{R\omega}\label{eq:property1}\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Hence, given property
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "remove"
|
||
|
||
\end_inset
|
||
|
||
, the adjoint map for
|
||
\begin_inset Formula $\sothree$
|
||
\end_inset
|
||
|
||
simplifies to
|
||
\begin_inset Formula \[
|
||
\Ad R{\Skew{\omega}}=R\Skew{\omega}R^{T}=\Skew{R\omega}\]
|
||
|
||
\end_inset
|
||
|
||
and this can be expressed in exponential coordinates simply by rotating
|
||
the axis
|
||
\begin_inset Formula $\omega$
|
||
\end_inset
|
||
|
||
to
|
||
\begin_inset Formula $R\omega$
|
||
\end_inset
|
||
|
||
.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
As an example, to apply an axis-angle rotation
|
||
\begin_inset Formula $\omega$
|
||
\end_inset
|
||
|
||
to a point
|
||
\begin_inset Formula $p$
|
||
\end_inset
|
||
|
||
in the frame
|
||
\begin_inset Formula $R$
|
||
\end_inset
|
||
|
||
, we could:
|
||
\end_layout
|
||
|
||
\begin_layout Enumerate
|
||
First transform
|
||
\begin_inset Formula $p$
|
||
\end_inset
|
||
|
||
back to the world frame, apply
|
||
\begin_inset Formula $\omega$
|
||
\end_inset
|
||
|
||
, and then rotate back:
|
||
\begin_inset Formula \[
|
||
q=Re^{\Skew{\omega}}R^{T}\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Enumerate
|
||
Immediately apply the transformed axis-angle transformation
|
||
\begin_inset Formula $\Ad R{\Skew{\omega}}=\Skew{R\omega}$
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula \[
|
||
q=e^{\Skew{R\omega}}p\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Actions
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In the case of
|
||
\begin_inset Formula $\SOthree$
|
||
\end_inset
|
||
|
||
the vector space is
|
||
\begin_inset Formula $\Rthree$
|
||
\end_inset
|
||
|
||
, and the group action corresponds to rotating a point
|
||
\begin_inset Formula \[
|
||
q=Rp\]
|
||
|
||
\end_inset
|
||
|
||
We would now like to know what an incremental rotation parameterized by
|
||
|
||
\begin_inset Formula $\omega$
|
||
\end_inset
|
||
|
||
would do:
|
||
\begin_inset Formula \[
|
||
q(\omega)=Re^{\Skew{\omega}}p\]
|
||
|
||
\end_inset
|
||
|
||
hence the derivative is:
|
||
\begin_inset Formula \[
|
||
\deriv{q(\omega)}{\omega}=R\deriv{}{\omega}\left(e^{\Skew{\omega}}p\right)=R\deriv{}{\omega}\left(\Skew{\omega}p\right)=RH_{p}\]
|
||
|
||
\end_inset
|
||
|
||
To calculate
|
||
\begin_inset Formula $H_{p}$
|
||
\end_inset
|
||
|
||
we make use of
|
||
\begin_inset Formula \[
|
||
\Skew{\omega}p=\omega\times p=-p\times\omega=\Skew{-p}\omega\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Newpage pagebreak
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
3D Rigid Transformations
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The Lie group
|
||
\begin_inset Formula $\SEthree$
|
||
\end_inset
|
||
|
||
is a subgroup of the general linear group
|
||
\begin_inset Formula $GL(4)$
|
||
\end_inset
|
||
|
||
of
|
||
\begin_inset Formula $4\times4$
|
||
\end_inset
|
||
|
||
invertible matrices of the form
|
||
\begin_inset Formula \[
|
||
T\define\left[\begin{array}{cc}
|
||
R & t\\
|
||
0 & 1\end{array}\right]\]
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $R\in\SOthree$
|
||
\end_inset
|
||
|
||
is a rotation matrix and
|
||
\begin_inset Formula $t\in\Rthree$
|
||
\end_inset
|
||
|
||
is a translation vector.
|
||
Its Lie algebra
|
||
\begin_inset Formula $\sethree$
|
||
\end_inset
|
||
|
||
is the vector space of
|
||
\begin_inset Formula $4\times4$
|
||
\end_inset
|
||
|
||
twists
|
||
\begin_inset Formula $\xihat$
|
||
\end_inset
|
||
|
||
parameterized by the
|
||
\emph on
|
||
twist coordinates
|
||
\emph default
|
||
|
||
\begin_inset Formula $\xi\in\Rsix$
|
||
\end_inset
|
||
|
||
, with the mapping
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "Murray94book"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula \[
|
||
\xi\define\left[\begin{array}{c}
|
||
\omega\\
|
||
v\end{array}\right]\rightarrow\xihat\define\left[\begin{array}{cc}
|
||
\Skew{\omega} & v\\
|
||
0 & 0\end{array}\right]\]
|
||
|
||
\end_inset
|
||
|
||
Note we follow Frank Park's convention and reserve the first three components
|
||
for rotation, and the last three for translation.
|
||
Hence, with this parameterization, the generators for
|
||
\begin_inset Formula $\SEthree$
|
||
\end_inset
|
||
|
||
are
|
||
\begin_inset Formula \[
|
||
G^{1}=\left(\begin{array}{cccc}
|
||
0 & 0 & 0 & 0\\
|
||
0 & 0 & -1 & 0\\
|
||
0 & 1 & 0 & 0\\
|
||
0 & 0 & 0 & 0\end{array}\right)\mbox{}G^{2}=\left(\begin{array}{cccc}
|
||
0 & 0 & 1 & 0\\
|
||
0 & 0 & 0 & 0\\
|
||
-1 & 0 & 0 & 0\\
|
||
0 & 0 & 0 & 0\end{array}\right)\mbox{ }G^{3}=\left(\begin{array}{cccc}
|
||
0 & -1 & 0 & 0\\
|
||
1 & 0 & 0 & 0\\
|
||
0 & 0 & 0 & 0\\
|
||
0 & 0 & 0 & 0\end{array}\right)\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula \[
|
||
G^{4}=\left(\begin{array}{cccc}
|
||
0 & 0 & 0 & 1\\
|
||
0 & 0 & 0 & 0\\
|
||
0 & 0 & 0 & 0\\
|
||
0 & 0 & 0 & 0\end{array}\right)\mbox{}G^{5}=\left(\begin{array}{cccc}
|
||
0 & 0 & 0 & 0\\
|
||
0 & 0 & 0 & 1\\
|
||
0 & 0 & 0 & 0\\
|
||
0 & 0 & 0 & 0\end{array}\right)\mbox{ }G^{6}=\left(\begin{array}{cccc}
|
||
0 & 0 & 0 & 0\\
|
||
0 & 0 & 0 & 0\\
|
||
0 & 0 & 0 & 1\\
|
||
0 & 0 & 0 & 0\end{array}\right)\]
|
||
|
||
\end_inset
|
||
|
||
Applying the exponential map to a twist
|
||
\begin_inset Formula $\xi$
|
||
\end_inset
|
||
|
||
yields a screw motion yielding an element in
|
||
\begin_inset Formula $\SEthree$
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula \[
|
||
T=\exp\xihat\]
|
||
|
||
\end_inset
|
||
|
||
A closed form solution for the exponential map is given in
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "page 42"
|
||
key "Murray94book"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
The Adjoint Map
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The adjoint is
|
||
\begin_inset Formula \begin{eqnarray*}
|
||
\Ad T{\xihat} & = & T\xihat T^{-1}\\
|
||
& = & \left[\begin{array}{cc}
|
||
R & t\\
|
||
0 & 1\end{array}\right]\left[\begin{array}{cc}
|
||
\Skew{\omega} & v\\
|
||
0 & 0\end{array}\right]\left[\begin{array}{cc}
|
||
R^{T} & -R^{T}t\\
|
||
0 & 1\end{array}\right]\\
|
||
& = & \left[\begin{array}{cc}
|
||
\Skew{R\omega} & -\Skew{R\omega}t+Rv\\
|
||
0 & 0\end{array}\right]\\
|
||
& = & \left[\begin{array}{cc}
|
||
\Skew{R\omega} & t\times R\omega+Rv\\
|
||
0 & 0\end{array}\right]\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
From this we can express the Adjoint map in terms of twist coordinates (see
|
||
also
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "Murray94book"
|
||
|
||
\end_inset
|
||
|
||
and FP):
|
||
\begin_inset Formula \[
|
||
\left[\begin{array}{c}
|
||
\omega'\\
|
||
v'\end{array}\right]=\left[\begin{array}{cc}
|
||
R & 0\\
|
||
\Skew tR & R\end{array}\right]\left[\begin{array}{c}
|
||
\omega\\
|
||
v\end{array}\right]\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Actions
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The action of
|
||
\begin_inset Formula $\SEthree$
|
||
\end_inset
|
||
|
||
on 3D points is done by embedding the points in
|
||
\begin_inset Formula $\mathbb{R}^{4}$
|
||
\end_inset
|
||
|
||
by using homogeneous coordinates
|
||
\begin_inset Formula \[
|
||
\hat{q}=\left[\begin{array}{c}
|
||
q\\
|
||
1\end{array}\right]=\left[\begin{array}{cc}
|
||
R & t\\
|
||
0 & 1\end{array}\right]\left[\begin{array}{c}
|
||
p\\
|
||
1\end{array}\right]=T\hat{p}\]
|
||
|
||
\end_inset
|
||
|
||
We would now like to know what an incremental rotation parameterized by
|
||
|
||
\begin_inset Formula $\xi$
|
||
\end_inset
|
||
|
||
would do:
|
||
\begin_inset Formula \[
|
||
\hat{q}(\xi)=Te^{\xihat}\hat{p}\]
|
||
|
||
\end_inset
|
||
|
||
hence the derivative (following the exposition in Section
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "sec:Derivatives-of-Actions"
|
||
|
||
\end_inset
|
||
|
||
):
|
||
\begin_inset Formula \[
|
||
\deriv{\hat{q}(\xi)}{\xi}=T\deriv{}{\xi}\left(\xihat\hat{p}\right)=TH_{p}\]
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $\xihat\hat{p}$
|
||
\end_inset
|
||
|
||
corresponds to a velocity in
|
||
\begin_inset Formula $\mathbb{R}^{4}$
|
||
\end_inset
|
||
|
||
(in the local
|
||
\begin_inset Formula $T$
|
||
\end_inset
|
||
|
||
frame):
|
||
\begin_inset Formula \[
|
||
\xihat\hat{p}=\left[\begin{array}{cc}
|
||
\Skew{\omega} & v\\
|
||
0 & 0\end{array}\right]\left[\begin{array}{c}
|
||
p\\
|
||
1\end{array}\right]=\left[\begin{array}{c}
|
||
\omega\times p+v\\
|
||
0\end{array}\right]\]
|
||
|
||
\end_inset
|
||
|
||
Notice how velocities are anologous to points at infinity in projective
|
||
geometry: they correspond to free vectors indicating a direction and magnitude
|
||
of change.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
By only taking the top three rows, we can write this as a velocity in
|
||
\begin_inset Formula $\Rthree$
|
||
\end_inset
|
||
|
||
, as the product of a
|
||
\begin_inset Formula $3\times6$
|
||
\end_inset
|
||
|
||
matrix
|
||
\begin_inset Formula $H_{p}$
|
||
\end_inset
|
||
|
||
that acts upon the exponential coordinates
|
||
\begin_inset Formula $\xi$
|
||
\end_inset
|
||
|
||
directly:
|
||
\begin_inset Formula \[
|
||
\omega\times p+v=-p\times\omega+v=\left[\begin{array}{cc}
|
||
-\Skew p & I_{3}\end{array}\right]\left[\begin{array}{c}
|
||
\omega\\
|
||
v\end{array}\right]=H_{p}\xi\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Newpage pagebreak
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section*
|
||
Appendix: Proof of Property
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "remove"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
We can prove the following identity for rotation matrices
|
||
\begin_inset Formula $R$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula \begin{eqnarray}
|
||
R\Skew{\omega}R^{T} & = & R\Skew{\omega}\left[\begin{array}{ccc}
|
||
a_{1} & a_{2} & a_{3}\end{array}\right]\nonumber \\
|
||
& = & R\left[\begin{array}{ccc}
|
||
\omega\times a_{1} & \omega\times a_{2} & \omega\times a_{3}\end{array}\right]\nonumber \\
|
||
& = & \left[\begin{array}{ccc}
|
||
a_{1}(\omega\times a_{1}) & a_{1}(\omega\times a_{2}) & a_{1}(\omega\times a_{3})\\
|
||
a_{2}(\omega\times a_{1}) & a_{2}(\omega\times a_{2}) & a_{2}(\omega\times a_{3})\\
|
||
a_{3}(\omega\times a_{1}) & a_{3}(\omega\times a_{2}) & a_{3}(\omega\times a_{3})\end{array}\right]\nonumber \\
|
||
& = & \left[\begin{array}{ccc}
|
||
\omega(a_{1}\times a_{1}) & \omega(a_{2}\times a_{1}) & \omega(a_{3}\times a_{1})\\
|
||
\omega(a_{1}\times a_{2}) & \omega(a_{2}\times a_{2}) & \omega(a_{3}\times a_{2})\\
|
||
\omega(a_{1}\times a_{3}) & \omega(a_{2}\times a_{3}) & \omega(a_{3}\times a_{3})\end{array}\right]\nonumber \\
|
||
& = & \left[\begin{array}{ccc}
|
||
0 & -\omega a_{3} & \omega a_{2}\\
|
||
\omega a_{3} & 0 & -\omega a_{1}\\
|
||
-\omega a_{2} & \omega a_{1} & 0\end{array}\right]\nonumber \\
|
||
& = & \Skew{R\omega}\label{remove}\end{eqnarray}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $a_{1}$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $a_{2}$
|
||
\end_inset
|
||
|
||
, and
|
||
\begin_inset Formula $a_{3}$
|
||
\end_inset
|
||
|
||
are the
|
||
\emph on
|
||
rows
|
||
\emph default
|
||
of
|
||
\begin_inset Formula $R$
|
||
\end_inset
|
||
|
||
.
|
||
Above we made use of the orthogonality of rotation matrices and the triple
|
||
product rule:
|
||
\begin_inset Formula \[
|
||
a(b\times c)=b(c\times a)=c(a\times b)\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset CommandInset bibtex
|
||
LatexCommand bibtex
|
||
bibfiles "/Users/dellaert/papers/refs"
|
||
options "plain"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_body
|
||
\end_document
|