Latest changes

release/4.3a0
Frank Dellaert 2010-06-24 19:35:56 +00:00
parent 676a74a0ac
commit 760f61ce4b
2 changed files with 151 additions and 24 deletions

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@ -123,6 +123,12 @@ Lie Groups
\end_inset
\begin_inset FormulaMacro
\newcommand{\AAdd}[1]{\mathbf{\mathop{Ad}}{}_{#1}}
{\mathbf{\mathop{Ad}}{}_{#1}}
\end_inset
\end_layout
\begin_layout Standard
@ -583,7 +589,7 @@ T_{1}T_{2}=(x_{1},\, y_{1},\,\theta_{1})(x_{2},\, y_{2},\,\theta_{2})=(x_{1}+\co
\end_inset
This is a bit clumsy, so we resort to a trick: embed the 2D poses in the
space of
\begin_inset Formula $3\times3$
\end_inset
@ -697,7 +703,7 @@ e^{x}=\lim_{n\rightarrow\infty}\left(1+\frac{x}{n}\right)^{n}=\sum_{k=0}^{\infty
\end_inset
The series can be similarly defined for square matrices,and the final result
The series can be similarly defined for square matrices, and the final result
is that we can write the motion of a robot along a circular trajectory,
resulting from the 2D twist
\begin_inset Formula $\xi=(v,\omega)$
@ -742,8 +748,9 @@ special Euclidean group
\end_inset
.
It is called a Lie group because it is both a manifold and a group, and
its group operation is smooth when operating on this manifold.
It is called a Lie group because it is simultaneously a topological group
and a manifold, which implies that the multiplication and the inversion
operations are smooth.
The space of 2D twists, together with a special binary operation to be
defined below, is called the Lie algebra
\begin_inset Formula $\setwo$
@ -778,7 +785,7 @@ A Lie group
\begin_inset Formula $G$
\end_inset
is a manifold that possesses a smooth group operation.
is a a group and a manifold that possesses a smooth group operation.
Associated with it is a Lie Algebra
\begin_inset Formula $\gg$
\end_inset
@ -896,8 +903,15 @@ Instead,
\begin_inset Formula $Z$
\end_inset
can be calculated using the Baker-Campbell-Hausdorff (BCH) formula:
\begin_inset Foot
can be calculated using the Baker-Campbell-Hausdorff (BCH) formula
\begin_inset CommandInset citation
LatexCommand cite
key "Hall00book"
\end_inset
\begin_inset Note Note
status collapsed
\begin_layout Plain Layout
@ -906,7 +920,7 @@ http://en.wikipedia.org/wiki/BakerCampbellHausdorff_formula
\end_inset
:
\begin_inset Formula \[
Z=X+Y+[X,Y]/2+[X-Y,[X,Y]]/12-[Y,[X,[X,Y]]]/24+\ldots\]
@ -929,7 +943,7 @@ For
\begin_inset Formula $n$
\end_inset
-dimensional matrix Lie groups, the Lie algebra
-dimensional matrix Lie groups, as a vector space the Lie algebra
\begin_inset Formula $\gg$
\end_inset
@ -937,7 +951,7 @@ For
\begin_inset Formula $\mathbb{R}^{n}$
\end_inset
, and we can define the wedge operator
, and we can define the hat operator
\begin_inset CommandInset citation
LatexCommand cite
after "page 41"
@ -990,7 +1004,7 @@ where the
\begin_inset Formula $n\times n$
\end_inset
matrices known as the Lie group generators.
matrices known as Lie group generators.
The meaning of the map
\begin_inset Formula $x\rightarrow\xhat$
\end_inset
@ -1003,12 +1017,92 @@ where the
\end_layout
\begin_layout Subsection
The Adjoint Map
Tangent Spaces and the Tangent Bundle
\end_layout
\begin_layout Standard
Below we frequently make use of the equality
\begin_inset Foot
The tangent space
\begin_inset Formula $T_{g}G$
\end_inset
at a group element
\begin_inset Formula $g\in G$
\end_inset
is the vector space of tangent vectors at
\begin_inset Formula $g$
\end_inset
.
The tangent bundle
\begin_inset Formula $TG$
\end_inset
is the set of all tangent vectors
\begin_inset Formula \[
TG\define\bigcup_{g\in G}T_{g}G\]
\end_inset
A vector field
\begin_inset Formula $X:G\rightarrow TG$
\end_inset
assigns to each element
\begin_inset Formula $g$
\end_inset
as single tangent vector
\begin_inset Formula $x\in T_{g}G$
\end_inset
.
\end_layout
\begin_layout Subsection
The Adjoint Map and Adjoint Representation
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Plain Layout
We do not fully understand this yet!!!
\end_layout
\end_inset
\end_layout
\begin_layout Standard
The
\emph on
adjoint map
\emph default
\begin_inset Formula $\AAdd g$
\end_inset
maps a group element
\begin_inset Formula $a\in G$
\end_inset
to its conjugate
\begin_inset Formula $gag^{-1}$
\end_inset
by
\begin_inset Formula $g$
\end_inset
.
\end_layout
\begin_layout Standard
Below we frequently make use of the equality
\begin_inset Note Note
status collapsed
\begin_layout Plain Layout
@ -1019,7 +1113,7 @@ http://en.wikipedia.org/wiki/Exponential_map
\begin_inset Formula \[
ge^{\xhat}g^{-1}=e^{\Ad g{\xhat}}\]
\AAdd ge^{\xhat}=ge^{\xhat}g^{-1}=e^{\Ad g{\xhat}}\]
\end_inset
@ -1031,8 +1125,17 @@ where
\begin_inset Formula $g$
\end_inset
, and is called the
\emph on
adjoint representation
\emph default
.
The intuitive explanation is that a change
\begin_inset Note Note
status open
\begin_layout Plain Layout
The intuitive explanation is that a change
\begin_inset Formula $\exp\left(\xhat\right)$
\end_inset
@ -1049,8 +1152,16 @@ where
\end_inset
.
In the case of a matrix group the ajoint can be written as
\begin_inset Foot
\end_layout
\end_inset
\end_layout
\begin_layout Standard
In the case of a matrix group the adjoint can be written as
\begin_inset Note Note
status collapsed
\begin_layout Plain Layout
@ -1307,13 +1418,9 @@ all
\end_inset
which we can write in terms of
\begin_inset Formula $\omega$
\end_inset
as
i.e.,
\begin_inset Formula \[
\Ad R\omega=\omega\]
\Ad R\what=\what\]
\end_inset
@ -2292,6 +2399,8 @@ The action of
\begin_inset Formula \[
\hat{q}=\left[\begin{array}{c}
q\\
1\end{array}\right]=\left[\begin{array}{c}
Rp+t\\
1\end{array}\right]=\left[\begin{array}{cc}
R & t\\
0 & 1\end{array}\right]\left[\begin{array}{c}
@ -2372,6 +2481,24 @@ v\end{array}\right]\]
\end_inset
The inverse action
\begin_inset Formula $T^{-1}p$
\end_inset
is
\begin_inset Formula \[
\hat{q}=\left[\begin{array}{c}
q\\
1\end{array}\right]=\left[\begin{array}{c}
R^{T}(p-t)\\
1\end{array}\right]=\left[\begin{array}{cc}
R^{T} & -R^{T}t\\
0 & 1\end{array}\right]\left[\begin{array}{c}
p\\
1\end{array}\right]=T^{-1}\hat{p}\]
\end_inset
\end_layout

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