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typos

release/4.3a0
Frank Dellaert 2010-03-11 13:45:39 +00:00
parent cf28e3ab04
commit b726e8e5e2
1 changed files with 72 additions and 83 deletions
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155
doc/LieGroups.lyx
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@ -419,9 +419,9 @@ The solution to this trivial differential equation is, with
\begin_inset Formula $x$
\end_inset
-position f the robot,
-position of the robot,
\begin_inset Formula \[
x=x_{0}+v_{x}t\]
x_{t}=x_{0}+v_{x}t\]
\end_inset
@ -431,13 +431,13 @@ A similar story holds for translation in the
direction, and in fact for translations in general:
\begin_inset Formula \[
(x,\, y,\,\theta)=(x_{0}+v_{x}t,\, y_{0}+v_{y}t,\,\theta_{0})\]
(x_{t},\, y_{t},\,\theta_{t})=(x_{0}+v_{x}t,\, y_{0}+v_{y}t,\,\theta_{0})\]
\end_inset
Similarly for rotation we have
\begin_inset Formula \[
(x,\, y,\,\theta)=(x_{0},\, y_{0},\,\theta_{0}+\omega t)\]
(x_{t},\, y_{t},\,\theta_{t})=(x_{0},\, y_{0},\,\theta_{0}+\omega t)\]
\end_inset
@ -488,7 +488,7 @@ Robot moving along a circular trajectory.
However, if we combine translation and rotation, the story breaks down!
We cannot write
\begin_inset Formula \[
(x,\, y,\,\theta)=(x_{0}+v_{x}t,\, y_{0}+v_{y}t,\,\theta_{0}+\omega t)\]
(x_{t},\, y_{t},\,\theta_{t})=(x_{0}+v_{x}t,\, y_{0}+v_{y}t,\,\theta_{0}+\omega t)\]
\end_inset
@ -499,7 +499,7 @@ The reason is that, if we move the robot a tiny bit according to the velocity
, we have (to first order)
\begin_inset Formula \[
(x_{t+\delta},\, y_{t+\delta},\,\theta_{t+\delta})=(x_{0}+v_{x}\delta,\, y_{0}+v_{y}\delta,\,\theta_{0}+\omega\delta)\]
(x_{\delta},\, y_{\delta},\,\theta_{\delta})=(x_{0}+v_{x}\delta,\, y_{0}+v_{y}\delta,\,\theta_{0}+\omega\delta)\]
\end_inset
@ -530,6 +530,43 @@ If rotation and translation commuted, we could do all rotations before leaving
\end_layout
\begin_layout Standard
\begin_inset Float figure
placement h
wide false
sideways false
status collapsed
\begin_layout Plain Layout
\align center
\begin_inset Graphics
filename /Users/dellaert/borg/gtsam/doc/images/n-steps.pdf
\end_inset
\begin_inset Caption
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:n-step-program"
\end_inset
Approximating a circular trajectory with
\begin_inset Formula $n$
\end_inset
steps.
\end_layout
\end_inset
\end_layout
\end_inset
To make progress, we have to be more precise about how the robot behaves.
Specifically, let us define composition of two poses
\begin_inset Formula $T_{1}$
@ -574,52 +611,7 @@ R=\left[\begin{array}{cc}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Float figure
placement h
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset Graphics
filename images/n-steps.pdf
\end_inset
\begin_inset Caption
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:n-step-program"
\end_inset
Approximating a circular trajectory with
\begin_inset Formula $n$
\end_inset
steps.
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Now, a
Now a
\begin_inset Quotes eld
\end_inset
@ -638,7 +630,7 @@ T(\delta)=\left[\begin{array}{ccc}
0 & 0 & 1\end{array}\right]=I+\delta\left[\begin{array}{ccc}
0 & -\omega & v_{x}\\
\omega & 0 & v_{y}\\
0 & 0 & 1\end{array}\right]\]
0 & 0 & 0\end{array}\right]\]
\end_inset
@ -655,7 +647,7 @@ Let us define the
\xihat\define\left[\begin{array}{ccc}
0 & -\omega & v_{x}\\
\omega & 0 & v_{y}\\
0 & 0 & 1\end{array}\right]\]
0 & 0 & 0\end{array}\right]\]
\end_inset
@ -715,13 +707,17 @@ The series can be similarly defined for square matrices,and the final result
\begin_inset Formula $ $
\end_inset
as the matrix exponential of
as the
\emph on
matrix exponential
\emph default
of
\begin_inset Formula $\xihat$
\end_inset
:
\begin_inset Formula \[
T(t)=e^{t\xihat}\define\lim_{n\rightarrow\infty}\left(I+\frac{t}{n}\xihat\right)^{n}=\sum_{k=0}^{\infty}\frac{\left(t\xihat\right)^{k}}{k!}\]
T(t)=e^{t\xihat}\define\lim_{n\rightarrow\infty}\left(I+\frac{t}{n}\xihat\right)^{n}=\sum_{k=0}^{\infty}\frac{t^{k}}{k!}\xihat^{k}\]
\end_inset
@ -746,8 +742,8 @@ special Euclidean group
\end_inset
.
It is called a Lie group because it is both a manifold, and its group operation
is smooth when operating on this manifold.
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.
The space of 2D twists, together with a special binary operation to be
defined below, is called the Lie algebra
\begin_inset Formula $\setwo$
@ -813,7 +809,7 @@ logarithm
\begin_inset Quotes erd
\end_inset
:
\begin_inset Formula \[
\log:G\rightarrow\gg\]
@ -855,8 +851,11 @@ The Lie Algebra
\end_inset
.
The relationship with the group operation is as follows: for commutative
Lie groups vector addition
\end_layout
\begin_layout Standard
The relationship of the Lie bracket to the group operation is as follows:
for commutative Lie groups vector addition
\begin_inset Formula $X+Y$
\end_inset
@ -1460,7 +1459,7 @@ T=\left[\begin{array}{cc}
0 & t\\
0 & 1\end{array}\right]\left[\begin{array}{cc}
R & 0\\
0 & k\end{array}\right]\]
0 & 1\end{array}\right]\]
\end_inset
@ -1770,18 +1769,11 @@ where
.
Hence, a slightly more efficient variant is
\end_layout
\begin_layout Standard
\begin_inset Formula \[
e^{\what}=\cos\theta I+\what sin\theta+\omega\omega^{T}(1\text{−}cos\theta)\]
\end_inset
\end_layout
\begin_layout Standard
Since
\begin_inset Formula $\SOthree$
\end_inset
@ -1935,7 +1927,7 @@ R\Skew{\omega}R^{T}=\Skew{R\omega}\label{eq:property1}\end{equation}
Hence, given property
\begin_inset CommandInset ref
LatexCommand eqref
reference "proof1"
reference "eq:property1"
\end_inset
@ -2521,7 +2513,8 @@ From this it can be gleaned that the groups
\bar no
\noun off
\color none
By choosing the generators carefully we maintain this subgroup hierarchy.
By choosing the generators carefully we maintain this hierarchy among the
associated Lie algebras.
In particular,
\begin_inset Formula $\setwo$
\end_inset
@ -2612,10 +2605,6 @@ a_{4}+a_{3} & -a_{5}-a_{6} & a_{2}\\
\end_inset
\end_layout
\begin_layout Standard
Note that
\begin_inset Formula $G_{5}$
\end_inset
@ -2634,10 +2623,10 @@ Note that
but without changing the determinant:
\begin_inset Formula \[
e^{xG_{5}}=\exp\left(\left[\begin{array}{ccc}
e^{xG_{5}}=\exp\left[\begin{array}{ccc}
x & 0 & 0\\
0 & -x & 0\\
0 & 0 & 0\end{array}\right]\right)=\left[\begin{array}{ccc}
0 & 0 & 0\end{array}\right]=\left[\begin{array}{ccc}
e^{x} & 0 & 0\\
0 & 1/e^{x} & 0\\
0 & 0 & 1\end{array}\right]\]
@ -2646,10 +2635,10 @@ e^{x} & 0 & 0\\
\begin_inset Formula \[
e^{xG_{6}}=\exp\left(\left[\begin{array}{ccc}
e^{xG_{6}}=\exp\left[\begin{array}{ccc}
0 & 0 & 0\\
0 & -x & 0\\
0 & 0 & x\end{array}\right]\right)=\left[\begin{array}{ccc}
0 & 0 & x\end{array}\right]=\left[\begin{array}{ccc}
1 & 0 & 0\\
0 & 1/e^{x} & 0\\
0 & 0 & e^{x}\end{array}\right]\]
@ -2696,10 +2685,10 @@ and hence
\color inherit
\begin_inset Formula \[
e^{xG_{5}}=\exp\left(\left[\begin{array}{ccc}
e^{xG_{5}}=\exp\left[\begin{array}{ccc}
x & 0 & 0\\
0 & 0 & 0\\
0 & 0 & -x\end{array}\right]\right)=\left[\begin{array}{ccc}
0 & 0 & -x\end{array}\right]=\left[\begin{array}{ccc}
e^{x} & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1/e^{x}\end{array}\right]\]
@ -2708,10 +2697,10 @@ e^{x} & 0 & 0\\
\begin_inset Formula \[
e^{xG_{6}}=\exp\left(\left[\begin{array}{ccc}
e^{xG_{6}}=\exp\left[\begin{array}{ccc}
0 & 0 & 0\\
0 & x & 0\\
0 & 0 & -x\end{array}\right]\right)=\left[\begin{array}{ccc}
0 & 0 & -x\end{array}\right]=\left[\begin{array}{ccc}
1 & 0 & 0\\
0 & e^{x} & 0\\
0 & 0 & 1/e^{x}\end{array}\right]\]