From 476eb9c0607f7884529ca10502972b0dbe15b72e Mon Sep 17 00:00:00 2001 From: Frank Dellaert Date: Thu, 24 Mar 2022 13:23:10 -0400 Subject: [PATCH] Addressed comments --- doc/gtsam.lyx | 98 +++++++++++++++++++++++++++++++++++++++++--------- doc/gtsam.pdf | Bin 1620931 -> 1621850 bytes 2 files changed, 81 insertions(+), 17 deletions(-) diff --git a/doc/gtsam.lyx b/doc/gtsam.lyx index 482aaac25..705a84911 100644 --- a/doc/gtsam.lyx +++ b/doc/gtsam.lyx @@ -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 @@ -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 @@ -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 @@ -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 @@ -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 @@ -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 @@ -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 @@ -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 @@ -1689,7 +1693,7 @@ The exponential map for \[ \exp\hat{\xi}\approx\left[\begin{array}{ccc} 1 & -\delta\theta & \delta x\\ -\delta\theta & 1 & \delta x\\ +\delta\theta & 1 & \delta y\\ 0 & 0 & 1 \end{array}\right] \] @@ -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\\ @@ -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 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