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Added refs, included macros.lyx, and added quite a bit about dexp.

release/4.3a0
Frank Dellaert 2015-12-20 16:59:21 -08:00
parent 3dbb69dcbd
commit fd539b137d
4 changed files with 495 additions and 395 deletions
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331
doc/LieGroups.lyx
View File

@ -76,335 +76,10 @@ Frank Dellaert
\end_layout
\begin_layout Standard
\begin_inset Note Comment
status open
\begin_inset CommandInset include
LatexCommand include
filename "macros.lyx"
\begin_layout Plain Layout
Derivatives
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\deriv}[2]{\frac{\partial#1}{\partial#2}}
{\frac{\partial#1}{\partial#2}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\at}[2]{#1\biggr\rvert_{#2}}
{#1\biggr\rvert_{#2}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\Jac}[3]{ \at{\deriv{#1}{#2}} {#3} }
{\at{\deriv{#1}{#2}}{#3}}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Note Comment
status open
\begin_layout Plain Layout
Lie Groups
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\xhat}{\hat{x}}
{\hat{x}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\yhat}{\hat{y}}
{\hat{y}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\Ad}[1]{Ad_{#1}}
{Ad_{#1}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\AAdd}[1]{\mathbf{\mathop{Ad}}{}_{#1}}
{\mathbf{\mathop{Ad}}{}_{#1}}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\define}{\stackrel{\Delta}{=}}
{\stackrel{\Delta}{=}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\gg}{\mathfrak{g}}
{\mathfrak{g}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\Rn}{\mathbb{R}^{n}}
{\mathbb{R}^{n}}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Note Comment
status open
\begin_layout Plain Layout
SO(2), 1
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\Rtwo}{\mathfrak{\mathbb{R}^{2}}}
{\mathfrak{\mathbb{R}^{2}}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\SOtwo}{SO(2)}
{SO(2)}
\end_inset
\begin_inset FormulaMacro
\newcommand{\sotwo}{\mathfrak{so(2)}}
{\mathfrak{so(2)}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\that}{\hat{\theta}}
{\hat{\theta}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\skew}[1]{[#1]_{+}}
{[#1]_{+}}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Note Comment
status open
\begin_layout Plain Layout
SE(2), 3
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\SEtwo}{SE(2)}
{SE(2)}
\end_inset
\begin_inset FormulaMacro
\newcommand{\setwo}{\mathfrak{se(2)}}
{\mathfrak{se(2)}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\Skew}[1]{[#1]_{\times}}
{[#1]_{\times}}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Note Comment
status open
\begin_layout Plain Layout
SO(3), 3
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\Rthree}{\mathfrak{\mathbb{R}^{3}}}
{\mathfrak{\mathbb{R}^{3}}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\SOthree}{SO(3)}
{SO(3)}
\end_inset
\begin_inset FormulaMacro
\newcommand{\sothree}{\mathfrak{so(3)}}
{\mathfrak{so(3)}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\what}{\hat{\omega}}
{\hat{\omega}}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Note Comment
status open
\begin_layout Plain Layout
SE(3),6
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\Rsix}{\mathfrak{\mathbb{R}^{6}}}
{\mathfrak{\mathbb{R}^{6}}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\SEthree}{SE(3)}
{SE(3)}
\end_inset
\begin_inset FormulaMacro
\newcommand{\sethree}{\mathfrak{se(3)}}
{\mathfrak{se(3)}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\xihat}{\hat{\xi}}
{\hat{\xi}}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Note Comment
status open
\begin_layout Plain Layout
Aff(2),6
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\Afftwo}{Aff(2)}
{Aff(2)}
\end_inset
\begin_inset FormulaMacro
\newcommand{\afftwo}{\mathfrak{aff(2)}}
{\mathfrak{aff(2)}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\aa}{a}
{a}
\end_inset
\begin_inset FormulaMacro
\newcommand{\ahat}{\hat{a}}
{\hat{a}}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Note Comment
status collapsed
\begin_layout Plain Layout
SL(3),8
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\SLthree}{SL(3)}
{SL(3)}
\end_inset
\begin_inset FormulaMacro
\newcommand{\slthree}{\mathfrak{sl(3)}}
{\mathfrak{sl(3)}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\hh}{h}
{h}
\end_inset
\begin_inset FormulaMacro
\newcommand{\hhat}{\hat{h}}
{\hat{h}}
\end_inset

137
doc/macros.lyx
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@ -1,42 +1,60 @@
#LyX 1.6.5 created this file. For more info see http://www.lyx.org/
\lyxformat 345
#LyX 2.0 created this file. For more info see http://www.lyx.org/
\lyxformat 413
\begin_document
\begin_header
\textclass article
\use_default_options true
\maintain_unincluded_children false
\language english
\language_package default
\inputencoding auto
\fontencoding global
\font_roman default
\font_sans default
\font_typewriter default
\font_default_family default
\use_non_tex_fonts false
\font_sc false
\font_osf false
\font_sf_scale 100
\font_tt_scale 100
\graphics default
\default_output_format default
\output_sync 0
\bibtex_command default
\index_command default
\paperfontsize default
\use_hyperref false
\papersize default
\use_geometry false
\use_amsmath 1
\use_esint 1
\use_mhchem 1
\use_mathdots 0
\cite_engine basic
\use_bibtopic false
\use_indices false
\paperorientation portrait
\suppress_date false
\use_refstyle 0
\index Index
\shortcut idx
\color #008000
\end_index
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\defskip medskip
\paragraph_indentation default
\quotes_language english
\papercolumns 1
\papersides 1
\paperpagestyle default
\tracking_changes false
\output_changes false
\author ""
\author ""
\html_math_output 0
\html_css_as_file 0
\html_be_strict false
\end_header
\begin_body
@ -62,14 +80,14 @@ Derivatives
\begin_inset FormulaMacro
\newcommand{\at}[2]{#1\biggr\rvert_{#2}}
{#1\biggr\rvert_{#2}}
\newcommand{\at}[1]{#1\biggr\vert_{\#2}}
{#1\biggr\vert_{\#2}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\Jac}[3]{ \at{\deriv{#1}{#2}} {#3} }
{\at{\deriv{#1}{#2}}{#3}}
{\at{\deriv{#1}{#2}}#3}
\end_inset
@ -107,6 +125,15 @@ Lie Groups
\end_inset
\end_layout
\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\AAdd}[1]{\mathbf{\mathop{Ad}}{}_{#1}}
{\mathbf{\mathop{Ad}}{}_{#1}}
\end_inset
\end_layout
\begin_layout Standard
@ -144,6 +171,12 @@ SO(2)
\end_layout
\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\Rone}{\mathfrak{\mathbb{R}}}
{\mathfrak{\mathbb{R}}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\Rtwo}{\mathfrak{\mathbb{R}^{2}}}
{\mathfrak{\mathbb{R}^{2}}}
@ -202,6 +235,12 @@ SE(2)
\end_inset
\begin_inset FormulaMacro
\newcommand{\Skew}[1]{[#1]_{\times}}
{[#1]_{\times}}
\end_inset
\end_layout
\begin_layout Standard
@ -243,7 +282,7 @@ SO(3)
\begin_inset FormulaMacro
\newcommand{\Skew}[1]{[#1]_{\times}}
\renewcommand{\Skew}[1]{[#1]_{\times}}
{[#1]_{\times}}
\end_inset
@ -288,6 +327,86 @@ SE(3)
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Note Comment
status open
\begin_layout Plain Layout
Aff(2),6
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\Afftwo}{Aff(2)}
{Aff(2)}
\end_inset
\begin_inset FormulaMacro
\newcommand{\afftwo}{\mathfrak{aff(2)}}
{\mathfrak{aff(2)}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\aa}{a}
{a}
\end_inset
\begin_inset FormulaMacro
\newcommand{\ahat}{\hat{a}}
{\hat{a}}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Note Comment
status collapsed
\begin_layout Plain Layout
SL(3),8
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\SLthree}{SL(3)}
{SL(3)}
\end_inset
\begin_inset FormulaMacro
\newcommand{\slthree}{\mathfrak{sl(3)}}
{\mathfrak{sl(3)}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\hh}{h}
{h}
\end_inset
\begin_inset FormulaMacro
\newcommand{\hhat}{\hat{h}}
{\hat{h}}
\end_inset
\end_layout
\end_body

396
doc/math.lyx
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@ -1237,21 +1237,28 @@ reference "eq:ApproximateObjective"
\end_inset
.
In particular, the notion of an exponential map allows us to define an
incremental transformation as tracing out a geodesic curve on the group
manifold along a certain
In particular, the notion of an exponential map allows us to define a mapping
from
\series bold
tangent vector
local coordinates
\series default
\begin_inset Formula $\xi$
\end_inset
back to a neighborhood in
\begin_inset Formula $G$
\end_inset
around
\begin_inset Formula $a$
\end_inset
,
\begin_inset Formula
\[
a\oplus\xi\define a\exp\left(\hat{\xi}\right)
\]
\begin{equation}
a\oplus\xi\define a\exp\left(\hat{\xi}\right)\label{eq:expmap}
\end{equation}
\end_inset
@ -1263,11 +1270,12 @@ with
\begin_inset Formula $n$
\end_inset
-dimensional Lie group,
-dimensional Lie group.
Above,
\begin_inset Formula $\hat{\xi}\in\mathfrak{g}$
\end_inset
the Lie algebra element corresponding to the vector
is the Lie algebra element corresponding to the vector
\begin_inset Formula $\xi$
\end_inset
@ -1305,7 +1313,7 @@ For the Lie group
\end_inset
is denoted as
\begin_inset Formula $\omega$
\begin_inset Formula $\omega t$
\end_inset
and represents an angular displacement.
@ -1314,17 +1322,17 @@ For the Lie group
\end_inset
is a skew symmetric matrix denoted as
\begin_inset Formula $\Skew{\omega}\in\sothree$
\begin_inset Formula $\Skew{\omega t}\in\sothree$
\end_inset
, and is given by
\begin_inset Formula
\[
\Skew{\omega}=\left[\begin{array}{ccc}
\Skew{\omega t}=\left[\begin{array}{ccc}
0 & -\omega_{z} & \omega_{y}\\
\omega_{z} & 0 & -\omega_{x}\\
-\omega_{y} & \omega_{x} & 0
\end{array}\right]
\end{array}\right]t
\]
\end_inset
@ -1334,12 +1342,136 @@ Finally, the increment
\end_inset
corresponds to an incremental rotation
\begin_inset Formula $R\oplus\omega=Re^{\Skew{\omega}}$
\begin_inset Formula $R\oplus\omega t=Re^{\Skew{\omega t}}$
\end_inset
.
\end_layout
\begin_layout Subsection
Local Coordinates vs.
Tangent Vectors
\end_layout
\begin_layout Standard
In differential geometry,
\series bold
tangent vectors
\series default
\begin_inset Formula $v\in T_{a}G$
\end_inset
at
\begin_inset Formula $a$
\end_inset
are elements of the Lie algebra
\begin_inset Formula $\mathfrak{g}$
\end_inset
, and are defined as
\begin_inset Formula
\[
v\define\Jac{\gamma(t)}t{t=0}
\]
\end_inset
where
\begin_inset Formula $\gamma$
\end_inset
is some curve that passes through
\begin_inset Formula $a$
\end_inset
at
\begin_inset Formula $t=0$
\end_inset
, i.e.
\begin_inset Formula $\gamma(0)=a$
\end_inset
.
In particular, for any fixed local coordinate
\begin_inset Formula $\xi$
\end_inset
the map
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:expmap"
\end_inset
can be used to define a
\series bold
geodesic curve
\series default
on the group manifold defined by
\begin_inset Formula $\gamma:t\mapsto ae^{\widehat{t\xi}}$
\end_inset
, and the corresponding tangent vector is given by
\begin_inset Formula
\begin{equation}
\Jac{ae^{\widehat{t\xi}}}t{t=0}=a\xihat\label{eq:tangent-vector}
\end{equation}
\end_inset
This defines the mapping between local coordinates
\begin_inset Formula $\xi\in\Rn$
\end_inset
and actual tangent vectors
\begin_inset Formula $a\xihat\in g$
\end_inset
: the vector
\begin_inset Formula $\xi$
\end_inset
defines a direction of travel on the manifold, but does so in the local
coordinate frame
\begin_inset Formula $a$
\end_inset
.
\end_layout
\begin_layout Example
Assume a rigid body's attitude is described by
\begin_inset Formula $R_{b}^{n}(t)$
\end_inset
, where the indices denote the navigation frame
\begin_inset Formula $N$
\end_inset
and body frame
\begin_inset Formula $B$
\end_inset
, respectively.
An extrinsically calibrated gyroscope measures the angular velocity
\begin_inset Formula $\omega^{b}$
\end_inset
, in the body frame, and the corresponding tangent vector is
\begin_inset Formula
\[
\dot{R}_{b}^{n}(t)=R_{b}^{n}(t)\widehat{\omega^{b}}
\]
\end_inset
\end_layout
\begin_layout Subsection
Derivatives
\end_layout
@ -1352,7 +1484,7 @@ reference "def:differentiable"
\end_inset
to map exponential coordinates
to map local coordinates
\begin_inset Formula $\xi$
\end_inset
@ -1368,7 +1500,7 @@ reference "def:differentiable"
\begin_inset Formula $Df_{a}$
\end_inset
locally approximates a function
approximates the function
\begin_inset Formula $f$
\end_inset
@ -1378,6 +1510,10 @@ reference "def:differentiable"
to
\begin_inset Formula $\Reals m$
\end_inset
in a neighborhood around
\begin_inset Formula $a$
\end_inset
:
@ -1455,41 +1591,6 @@ derivative
.
\end_layout
\begin_layout Standard
Note that the vectors
\begin_inset Formula $\xi$
\end_inset
can be viewed as lying in the tangent space to
\begin_inset Formula $G$
\end_inset
at
\begin_inset Formula $a$
\end_inset
, but defining this rigorously would take us on a longer tour of differential
geometry.
Informally,
\begin_inset Formula $\xi$
\end_inset
is simply the direction, in a local coordinate frame, that is locally tangent
at
\begin_inset Formula $a$
\end_inset
to a geodesic curve
\begin_inset Formula $\gamma:t\mapsto ae^{\widehat{t\xi}}$
\end_inset
traced out by the exponential map, with
\begin_inset Formula $\gamma(0)=a$
\end_inset
.
\end_layout
\begin_layout Subsection
Derivative of an Action
\begin_inset CommandInset label
@ -3000,7 +3101,7 @@ f(ge^{\xhat})=f(ge^{\xhat}g^{-1}g)=f(e^{\Ad g\xhat}g)
\end_layout
\begin_layout Subsection
Derivative of the Exponential and Logarithm Map
Derivative of the Exponential Map
\end_layout
\begin_layout Theorem
@ -3064,17 +3165,196 @@ For
\begin_inset Formula $\xi\neq0$
\end_inset
, things are not simple, see .
, things are not simple.
As with pushforwards above, we will be looking for an
\begin_inset Formula $n\times n$
\end_inset
\begin_inset Flex URL
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\uuline off
\uwave off
\noun off
\color none
Jacobian
\begin_inset Formula $f'(\xi)$
\end_inset
such that
\family default
\series default
\shape default
\size default
\emph default
\bar default
\strikeout default
\uuline default
\uwave default
\noun default
\color inherit
\begin_inset Formula
\begin{equation}
f\left(\xi+\delta\right)\approx f\left(\xi\right)\exp\left(\widehat{f'(\xi)\delta}\right)\label{eq:push_exp}
\end{equation}
\end_inset
Differential geometry tells us that for any Lie algebra element
\begin_inset Formula $\xihat\in\mathfrak{g}$
\end_inset
there exists a
\emph on
linear
\emph default
map
\begin_inset Formula $d\exp_{\xihat}:T_{\xihat}\mathfrak{g}\rightarrow T_{\exp(\xihat)}G$
\end_inset
, which is given by
\begin_inset Foot
status collapsed
\begin_layout Plain Layout
See
\begin_inset Flex URL
status open
http://deltaepsilons.wordpress.com/2009/11/06/helgasons-formula-for-the-differenti
al-of-the-exponential/
\begin_layout Plain Layout
http://deltaepsilons.wordpress.com/2009/11/06/
\end_layout
\end_inset
or
\begin_inset Flex URL
status open
\begin_layout Plain Layout
https://en.wikipedia.org/wiki/Derivative_of_the_exponential_map
\end_layout
\end_inset
.
\end_layout
\end_inset
\begin_inset Formula
\begin{equation}
d\exp_{\xihat}\hat{x}=\exp(\xihat)\frac{1-\exp(-ad_{\xihat})}{ad_{\xihat}}\hat{x}\label{eq:dexp}
\end{equation}
\end_inset
with
\begin_inset Formula $\hat{x}\in T_{\xihat}\mathfrak{g}$
\end_inset
and
\begin_inset Formula $ad_{\xihat}$
\end_inset
itself a linear map taking
\begin_inset Formula $\hat{x}$
\end_inset
to
\begin_inset Formula $[\xihat,\hat{x}]$
\end_inset
, the Lie bracket.
The actual formula above is not really as important as the fact that the
linear map exists, although it is expressed directly in terms of tangent
vectors to
\begin_inset Formula $\mathfrak{g}$
\end_inset
and
\begin_inset Formula $G$
\end_inset
.
Equation
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:dexp"
\end_inset
is a tangent vector, and comparing with
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:tangent-vector"
\end_inset
we see that it maps to local coordinates
\begin_inset Formula $y$
\end_inset
as follows:
\begin_inset Formula
\[
\yhat=\frac{1-\exp(-ad_{\xihat})}{ad_{\xihat}}\hat{x}
\]
\end_inset
which can be used to construct the Jacobian
\begin_inset Formula $f'(\xi)$
\end_inset
.
\end_layout
\begin_layout Example
For
\begin_inset Formula $\SOthree$
\end_inset
, the operator
\begin_inset Formula $ad_{\xihat}$
\end_inset
is simply a matrix multiplication when representing
\begin_inset Formula $\sothree$
\end_inset
using 3-vectors, i.e.,
\begin_inset Formula $ad_{\xihat}x=\xihat x$
\end_inset
, and the
\begin_inset Formula $3\times3$
\end_inset
Jacobian corresponding to
\begin_inset Formula $d\exp$
\end_inset
is
\begin_inset Formula
\[
f'(\xi)=\frac{I_{3\times3}-\exp(-\xihat)}{\xihat}=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(k+1)!}\xihat^{k}
\]
\end_inset
which, similar to the exponential map, has a simple closed form expression
for
\begin_inset Formula $\SOthree$
\end_inset
.
@ -3097,7 +3377,7 @@ Retractions
\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\retract}{\mathcal{R}}
\renewcommand{\retract}{\mathcal{R}}
{\mathcal{R}}
\end_inset
@ -6797,7 +7077,7 @@ Then
\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
bibfiles "/Users/dellaert/papers/refs"
bibfiles "refs"
options "plain"
\end_inset

26
doc/refs.bib Normal file
View File

@ -0,0 +1,26 @@
@article{Iserles00an,
title = {Lie-group methods},
author = {Iserles, Arieh and Munthe-Kaas, Hans Z and
N{\o}rsett, Syvert P and Zanna, Antonella},
journal = {Acta Numerica 2000},
volume = {9},
pages = {215--365},
year = {2000},
publisher = {Cambridge Univ Press}
}
@book{Murray94book,
title = {A mathematical introduction to robotic manipulation},
author = {Murray, Richard M and Li, Zexiang and Sastry, S
Shankar and Sastry, S Shankara},
year = {1994},
publisher = {CRC press}
}
@book{Spivak65book,
title = {Calculus on manifolds},
author = {Spivak, Michael},
volume = {1},
year = {1965},
publisher = {WA Benjamin New York}
}