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dellaert committed Mar 24, 2022
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98 changes: 81 additions & 17 deletions doc/gtsam.lyx
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Expand Up @@ -150,18 +150,14 @@ filename "common_macros.tex"

\end_layout

\begin_layout Section*
Overview
\end_layout

\begin_layout Standard
\begin_layout Abstract
In this document I provide a hands-on introduction to both factor graphs
and GTSAM.
This is an updated version from the 2012 TR that is tailored to our GTSAM
4.0 library and beyond.
\end_layout

\begin_layout Standard
\begin_layout Abstract

\series bold
Factor graphs
Expand Down Expand Up @@ -202,7 +198,7 @@ ts or prior knowledge.
robotics and vision.
\end_layout

\begin_layout Standard
\begin_layout Abstract
The GTSAM toolbox (GTSAM stands for
\begin_inset Quotes eld
\end_inset
Expand All @@ -223,7 +219,7 @@ Georgia Tech Smoothing and Mapping
y.
\end_layout

\begin_layout Standard
\begin_layout Abstract
GTSAM exploits sparsity to be computationally efficient.
Typically measurements only provide information on the relationship between
a handful of variables, and hence the resulting factor graph will be sparsely
Expand All @@ -234,8 +230,11 @@ l complexity.
GTSAM provides iterative methods that are quite efficient regardless.
\end_layout

\begin_layout Standard
You can download the latest version of GTSAM from GitHub at
\begin_layout Abstract
You can download the latest version of GTSAM from GitHub at
\end_layout

\begin_layout Abstract
\begin_inset Flex URL
status open

Expand Down Expand Up @@ -1365,14 +1364,18 @@ where
\end_inset

is the measurement,
\begin_inset Formula $q$
\begin_inset Formula $q\in SE(2)$
\end_inset

is the unknown variable,
\begin_inset Formula $h(q)$
\end_inset

is a (possibly nonlinear) measurement function, and
is a
\series bold
measurement function
\series default
, and
\begin_inset Formula $\Sigma$
\end_inset

Expand Down Expand Up @@ -1633,7 +1636,7 @@ Many of our users, when attempting to create a custom factor, are initially
\begin_inset Formula $2\times3$
\end_inset

diagonal matrix.
identity matrix.
This
\emph on
would
Expand All @@ -1647,7 +1650,7 @@ would
such that
\begin_inset Formula
\[
h(qe^{\hat{\xi}})\approx h(q)+H\xi
h(q\exp\hat{\xi})\approx h(q)+H\xi
\]

\end_inset
Expand All @@ -1664,7 +1667,8 @@ where
\series bold
exponential map
\series default
for the variable we want to update, In this case
for the variable we want to update.
In this case
\begin_inset Formula $q\in SE(2)$
\end_inset

Expand All @@ -1689,7 +1693,7 @@ The exponential map for
\[
\exp\hat{\xi}\approx\left[\begin{array}{ccc}
1 & -\delta\theta & \delta x\\
\delta\theta & 1 & \delta x\\
\delta\theta & 1 & \delta y\\
0 & 0 & 1
\end{array}\right]
\]
Expand All @@ -1709,7 +1713,7 @@ h(qe^{\hat{\xi}})\approx h\left(\left[\begin{array}{ccc}
0 & 0 & 1
\end{array}\right]\left[\begin{array}{ccc}
1 & -\delta\theta & \delta x\\
\delta\theta & 1 & \delta x\\
\delta\theta & 1 & \delta y\\
0 & 0 & 1
\end{array}\right]\right)=\left[\begin{array}{c}
q_{x}+\cos(q_{\theta})\delta x-\sin(q_{\theta})\delta y\\
Expand All @@ -1726,6 +1730,66 @@ which then explains the Jacobian
.
\end_layout

\begin_layout Standard
Lie groups are very relevant in the robotics context, and you can read more
here:
\end_layout

\begin_layout Itemize
\begin_inset Flex URL
status open

\begin_layout Plain Layout

https://github.com/borglab/gtsam/blob/develop/doc/LieGroups.pdf
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Flex URL
status open

\begin_layout Plain Layout

https://github.com/borglab/gtsam/blob/develop/doc/math.pdf
\end_layout

\end_inset


\end_layout

\begin_layout Standard
In some cases you want to go even beyond Lie groups to a looser concept,

\series bold
manifolds
\series default
, because not all unknown variables behave like a group, e.g., the space of
3D planes, 2D lines, directions in space, etc.
For manifolds we do not always have an exponential map, but we have a retractio
n that plays the same role.
Some of this is explained here:
\end_layout

\begin_layout Itemize
\begin_inset Flex URL
status open

\begin_layout Plain Layout

https://gtsam.org/notes/GTSAM-Concepts.html
\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Using Custom Factors
\end_layout
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