diff --git a/doc/common_macros.tex b/doc/common_macros.tex new file mode 100644 index 000000000..73648c6a4 --- /dev/null +++ b/doc/common_macros.tex @@ -0,0 +1,124 @@ +\global\long\def\Vector#1{{\bf #1}} + \global\long\def\Matrix#1{{\bf #1}} + + +\global\long\def\eq#1{equation (\ref{eq:=0000231})} + + +\global\long\def\eye#1{\Vector{I_{#1}}} + + +\global\long\def\leftsparrow#1{\stackrel{#1}{\leftarrow}} + \global\long\def\rightsparrow#1{\stackrel{#1}{\rightarrow}} + \global\long\def\chain{\mathcal{M}} + + +\global\long\def\define{\stackrel{\Delta}{=}} + + +\global\long\def\argmin#1{\mathop{\textrm{argmin \,}}_{#1}} + + +\global\long\def\Norm#1{\Vert#1\Vert} + \global\long\def\SqrNorm#1{\Vert#1\Vert^{2}} + \global\long\def\Ltwo#1{\mathcal{L}^{2}\left(#1\right)} + + +\global\long\def\Normal#1#2#3{\mathcal{N}(#1;#2,#3)} + + +\global\long\def\LogNormal#1#2#3{ (#1-#2)^{T} #3^{-1} (#1-#2) } + + +\global\long\def\SqrMah#1#2#3{\Vert{#1}-{#2}\Vert_{#3}^{2}} + + +\global\long\def\SqrZMah#1#2{\Vert{#1}\Vert_{#2}^{2}} + + +\global\long\def\Info#1#2#3{\mathcal{N}^{-1}(#1;#2,#3)} + + +\providecommand{\half}{\frac{1}{2}} + +\global\long\def\Mah#1#2#3{\Vert{#1}-{#2}\Vert_{#3}} + \global\long\def\MahDeriv#1#2#3#4{\biggl(\deriv{#2}{#4}\biggr)^{T} #3^{-1} (#1-#2)} + + +\global\long\def\argmin#1{\mathop{\textrm{argmin \,}}_{#1}} + \global\long\def\argmax#1{\mathop{\textrm{argmax \,}}_{#1}} + + + + +\global\long\def\deriv#1#2{\frac{\partial#1}{\partial#2}} + + +\global\long\def\at#1#2{#1\biggr\rvert_{#2}} + + +\global\long\def\Jac#1#2#3{ \at{\deriv{#1}{#2}} {#3} } + + + + +\global\long\def\Rone{\mathbb{R}} +\global\long\def\Pone{\mathbb{P}} + + +\global\long\def\Rtwo{\mathbb{R}^{2}} +\global\long\def\Ptwo{\mathbb{P}^{2}} + + +\global\long\def\Stwo{\mathbb{S}^{2}} + \global\long\def\Complex{\mathbb{C}} + + +\global\long\def\Z{\mathbb{Z}} + \global\long\def\Rplus{\mathbb{R}^{+}} + + +\global\long\def\SOtwo{SO(2)} +\global\long\def\sotwo{\mathfrak{so(2)}} +\global\long\def\skew#1{[#1]_{+}} + + +\global\long\def\SEtwo{SE(2)} +\global\long\def\setwo{\mathfrak{se(2)}} +\global\long\def\Skew#1{[#1]_{\times}} + + +\global\long\def\Rthree{\mathbb{R}^{3}} +\global\long\def\Pthree{\mathbb{P}^{3}} + + +\global\long\def\SOthree{SO(3)} +\global\long\def\sothree{\mathfrak{so(3)}} + + +\global\long\def\Rsix{\mathbb{R}^{6}} +\global\long\def\SEthree{SE(3)} +\global\long\def\sethree{\mathfrak{se(3)}} + + +\global\long\def\Rn{\mathbb{R}^{n}} + + +\global\long\def\Afftwo{Aff(2)} +\global\long\def\afftwo{\mathfrak{aff(2)}} + + +\global\long\def\SLthree{SL(3)} +\global\long\def\slthree{\mathfrak{sl(3)}} + + + + +\global\long\def\stirling#1#2{\genfrac{\{}{\}}{0pt}{}{#1}{#2}} + + +\global\long\def\matlabscript#1#2{\begin{itemize}\item[]\lstinputlisting[caption=#2,label=#1]{#1.m}\end{itemize}} + + +\global\long\def\atan{\mathop{atan2}} + diff --git a/doc/gtsam.lyx b/doc/gtsam.lyx new file mode 100644 index 000000000..38b601d23 --- /dev/null +++ b/doc/gtsam.lyx @@ -0,0 +1,3760 @@ +#LyX 2.1 created this file. For more info see http://www.lyx.org/ +\lyxformat 474 +\begin_document +\begin_header +\textclass article +\begin_preamble +\usepackage{times} +\usepackage{listings} + +\usepackage[noend]{algpseudocode} +\usepackage[usenames,dvipsnames]{color} +% This is the color used for MATLAB comments below +\definecolor{MyDarkGreen}{rgb}{0.0,0.4,0.0} + +% For faster processing, load Matlab syntax for listings +\lstloadlanguages{Matlab}% +\lstset{language=Matlab, % Use MATLAB + frame=single, % Single frame around code + basicstyle=\small\ttfamily, % Use small true type font + keywordstyle=[1]\color{Blue}\bf, % MATLAB functions bold and blue + keywordstyle=[2]\color{Purple}, % MATLAB function arguments purple + keywordstyle=[3]\color{Blue}\underbar, % User functions underlined and blue + identifierstyle=, % Nothing special about identifiers + % Comments small dark green courier + commentstyle=\usefont{T1}{pcr}{m}{sl}\color{MyDarkGreen}, + stringstyle=\color{Purple}, % Strings are purple + showstringspaces=false, % Don't put marks in string spaces + tabsize=5, % 5 spaces per tab + % + %%% Put standard MATLAB functions not included in the default + %%% language here + morekeywords={normpdf,normcdf,Pose2,Pose2SLAM}, + % + %%% Put MATLAB function parameters here + morekeywords=[2]{on, off, interp}, + % + %%% Put user defined functions here + morekeywords=[3]{FindESS, homework_example, gtsamSharedNoiseModel_Sigmas}, + % + morecomment=[l][\color{Blue}]{...}, % Line continuation (...) like blue comment + numbers=left, % Line numbers on left + firstnumber=1, % Line numbers start with line 1 + numberstyle=\tiny\color{Blue}, % Line numbers are blue + stepnumber=1 % Line numbers go in steps of 1 + } +\end_preamble +\use_default_options false +\maintain_unincluded_children false +\language english +\language_package default +\inputencoding auto +\fontencoding T1 +\font_roman ae +\font_sans default +\font_typewriter default +\font_math auto +\font_default_family rmdefault +\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 10 +\spacing onehalf +\use_hyperref false +\papersize custom +\use_geometry true +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 0 +\use_package esint 0 +\use_package mathdots 1 +\use_package mathtools 0 +\use_package mhchem 1 +\use_package stackrel 0 +\use_package stmaryrd 0 +\use_package undertilde 0 +\cite_engine natbib +\cite_engine_type authoryear +\biblio_style plainnat +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification true +\use_refstyle 0 +\index Index +\shortcut idx +\color #008000 +\end_index +\paperwidth 7.44in +\paperheight 9.68in +\leftmargin 1in +\topmargin 1in +\rightmargin 1in +\bottommargin 1in +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation indent +\paragraph_indentation default +\quotes_language english +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Title +Factor Graphs and GTSAM: +\begin_inset Newline newline +\end_inset + +A Hands-on Introduction +\end_layout + +\begin_layout Author +Frank Dellaert +\begin_inset Newline newline +\end_inset + +Technical Report number GT-RIM-CP&R-2014-XXX +\end_layout + +\begin_layout Date +September 2014 +\end_layout + +\begin_layout Standard +\begin_inset CommandInset include +LatexCommand input +filename "../common_macros.tex" + +\end_inset + + +\end_layout + +\begin_layout Section* +Overview +\end_layout + +\begin_layout Standard +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 + 3.0 library and beyond. +\end_layout + +\begin_layout Standard + +\series bold +Factor graphs +\series default + are graphical models +\begin_inset CommandInset citation +LatexCommand citep +key "Koller09book" + +\end_inset + + that are well suited to modeling complex estimation problems, such as Simultane +ous Localization and Mapping (SLAM) or Structure from Motion (SFM). + You might be familiar with another often used graphical model, Bayes networks, + which are directed acyclic graphs. + A +\series bold +factor graph, +\series default +however, is a +\emph on +bipartite +\emph default + graph consisting of factors connected to variables. + The +\series bold +variables +\series default + represent the unknown random variables in the estimation problem, whereas + the +\series bold +factors +\series default + represent probabilistic constraints on those variables, derived from measuremen +ts or prior knowledge. + In the following sections I will illustrate this with examples from both + robotics and vision. +\end_layout + +\begin_layout Standard +The GTSAM toolbox (GTSAM stands for +\begin_inset Quotes eld +\end_inset + +Georgia Tech Smoothing and Mapping +\begin_inset Quotes erd +\end_inset + +) toolbox is a BSD-licensed C++ library based on factor graphs, developed + at the Georgia Institute of Technology by myself, many of my students, + and collaborators. + It provides state of the art solutions to the SLAM and SFM problems, but + can also be used to model and solve both simpler and more complex estimation + problems. + It also provides a MATLAB interface which allows for rapid prototype developmen +t, visualization, and user interaction. +\end_layout + +\begin_layout Standard +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 + connected. + This is exploited by the algorithms implemented in GTSAM to reduce computationa +l complexity. + Even when graphs are too dense to be handled efficiently by direct methods, + GTSAM provides iterative methods that are quite efficient regardless. +\end_layout + +\begin_layout Standard +You can download the latest version of GTSAM at +\begin_inset Flex URL +status open + +\begin_layout Plain Layout + +http://tinyurl.com/gtsam +\end_layout + +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset CommandInset toc +LatexCommand tableofcontents + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Newpage pagebreak +\end_inset + + +\end_layout + +\begin_layout Section +Factor Graphs +\end_layout + +\begin_layout Standard +Let us start with a one-page primer on factor graphs, which in no way replaces + the excellent and detailed reviews by +\begin_inset CommandInset citation +LatexCommand citet +key "Kschischang01it" + +\end_inset + + and +\begin_inset CommandInset citation +LatexCommand citet +key "Loeliger04spm" + +\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/hmm.pdf + scale 60 + BoundingBox 40bp 37bp 400bp 150bp + clip + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +\begin_inset CommandInset label +LatexCommand label +name "fig:unrolledHMM" + +\end_inset + +An HMM, unrolled over three time-steps, represented by a Bayes net. + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +noindent +\end_layout + +\end_inset + + Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:unrolledHMM" + +\end_inset + + shows the +\series bold +Bayes network +\series default + for a hidden Markov model (HMM) over three time steps. + In a Bayes net, each node is associated with a conditional density: the + top Markov chain encodes the prior +\begin_inset Formula $P(X_{1})$ +\end_inset + + and transition probabilities +\begin_inset Formula $P(X_{2}|X_{1})$ +\end_inset + + and +\begin_inset Formula $P(X_{3}|X_{2})$ +\end_inset + +, whereas measurements +\begin_inset Formula $Z_{t}$ +\end_inset + + depend only on the state +\begin_inset Formula $X_{t}$ +\end_inset + +, modeled by conditional densities +\begin_inset Formula $P(Z_{t}|X_{t})$ +\end_inset + +. + Given known measurements +\begin_inset Formula $z_{1}$ +\end_inset + +, +\begin_inset Formula $z_{2}$ +\end_inset + + and +\begin_inset Formula $z_{3}$ +\end_inset + + we are interested in the hidden state sequence +\begin_inset Formula $(X_{1},X_{2},X_{3})$ +\end_inset + + that maximizes the posterior probability +\begin_inset Formula $P(X_{1},X_{2},X_{3}|Z_{1}=z_{1},Z_{2}=z_{2},Z_{3}=z_{3})$ +\end_inset + +. + Since the measurements +\begin_inset Formula $Z_{1}$ +\end_inset + +, +\begin_inset Formula $Z_{2}$ +\end_inset + +, and +\begin_inset Formula $Z_{3}$ +\end_inset + + are +\emph on +known +\emph default +, the posterior is proportional to the product of six +\series bold +factors +\series default +, three of which derive from the the Markov chain, and three likelihood + factors defined as +\begin_inset Formula $L(X_{t};z)\propto P(Z_{t}=z|X_{t})$ +\end_inset + +: +\begin_inset Formula +\[ +P(X_{1},X_{2},X_{3}|Z_{1},Z_{2},Z_{3})\propto P(X_{1})P(X_{2}|X_{1})P(X_{3}|X_{2})L(X_{1};z_{1})L(X_{2};z_{2})L(X_{3};z_{3}) +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +vspace{-3mm} +\end_layout + +\end_inset + + +\begin_inset Float figure +placement H +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center +\begin_inset Graphics + filename images/hmm-FG.pdf + scale 60 + BoundingBox 30bp 40bp 340bp 130bp + clip + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +\begin_inset CommandInset label +LatexCommand label +name "fig:HMM-FG" + +\end_inset + +An HMM with observed measurements, unrolled over time, represented as a + factor graph. +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +noindent +\end_layout + +\end_inset + + This motivates a different graphical model, a +\series bold +factor graph +\series default +, in which we only represent the unknown variables +\begin_inset Formula $X_{1}$ +\end_inset + +, +\begin_inset Formula $X_{2}$ +\end_inset + +, and +\begin_inset Formula $X_{3}$ +\end_inset + +, connected to factors that encode probabilistic information on them, as + in Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:HMM-FG" + +\end_inset + +. + To do maximum a-posteriori (MAP) inference, we then maximize the product + +\begin_inset Formula +\[ +f(X_{1},X_{2},X_{3})=\prod f_{i}(\mathcal{X}_{i}) +\] + +\end_inset + +i.e., the value of the factor graph. + It should be clear from the figure that the connectivity of a factor graph + encodes, for each factor +\begin_inset Formula $f_{i}$ +\end_inset + +, which subset of variables +\begin_inset Formula $\mathcal{X}_{i}$ +\end_inset + + it depends on. + In the examples below, we use factor graphs to model more complex MAP inference + problems in robotics. +\end_layout + +\begin_layout Section +\begin_inset CommandInset label +LatexCommand label +name "sec:Robot-Localization" + +\end_inset + +Modeling Robot Motion +\end_layout + +\begin_layout Subsection +Modeling with Factor Graphs +\end_layout + +\begin_layout Standard +Before diving into a SLAM example, let us consider the simpler problem of + modeling robot motion. + This can be done with a +\emph on +continuous +\emph default + Markov chain, and provides a gentle introduction to GTSAM. +\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/FactorGraph.pdf + scale 80 + BoundingBox 40bp 585bp 300bp 625bp + clip + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +\begin_inset CommandInset label +LatexCommand label +name "fig:OdometryFG" + +\end_inset + +Factor graph for robot localization. +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +noindent +\end_layout + +\end_inset + + The factor graph for a simple example is shown in Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:OdometryFG" + +\end_inset + +. + There are three variables +\begin_inset Formula $x_{1}$ +\end_inset + +, +\begin_inset Formula $x_{2}$ +\end_inset + +, and +\begin_inset Formula $x_{3}$ +\end_inset + + which represent the poses of the robot over time, rendered in the figure + by the open-circle variable nodes. + In this example, we have one +\series bold +unary factor +\series default + +\begin_inset Formula $f_{0}(x_{1})$ +\end_inset + + on the first pose +\begin_inset Formula $x_{1}$ +\end_inset + + that encodes our prior knowledge about +\begin_inset Formula $x_{1}$ +\end_inset + +, and two +\series bold +binary factors +\series default + that relate successive poses, respectively +\begin_inset Formula $f_{1}(x_{1},x_{2};o_{1})$ +\end_inset + + and +\begin_inset Formula $f_{2}(x_{2},x_{3};o_{2})$ +\end_inset + +, where +\begin_inset Formula $o_{1}$ +\end_inset + + and +\begin_inset Formula $o_{2}$ +\end_inset + + represent odometry measurements. +\end_layout + +\begin_layout Subsection +Creating a Factor Graph +\end_layout + +\begin_layout Standard +The following C++ code, included in GTSAM as an example, creates the factor + graph in Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:OdometryFG" + +\end_inset + +: +\end_layout + +\begin_layout Standard +\begin_inset CommandInset include +LatexCommand lstinputlisting +filename "Code/OdometryExample.cpp" +lstparams "aboveskip=10pt,basicstyle={\\ttfamily\\small},caption={Excerpt from examples/OdometryExample.cpp},captionpos=b,frame=single,identifierstyle={\\bfseries},label={listing:OdometryExample},language={C++},numbers=left" + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +noindent +\end_layout + +\end_inset + + Above, line 2 creates an empty factor graph. + We then add the factor +\begin_inset Formula $f_{0}(x_{1})$ +\end_inset + + on lines 5-8 as an instance of +\series bold +\emph on +PriorFactor +\series default +\emph default +, a templated class provided in the slam subfolder, with +\series bold +\emph on +T=Pose2 +\series default +\emph default +. + Its constructor takes a variable +\series bold +\emph on +Key +\series default +\emph default + (in this case 1), a mean of type +\series bold +\emph on +Pose2, +\series default +\emph default + created on Line 5, and a noise model for the prior density. + We provide a diagonal Gaussian of type +\series bold +\emph on +noiseModel::Diagonal +\series default +\emph default + by specifying three standard deviations in line 7, respectively 30 cm. +\begin_inset space ~ +\end_inset + +on the robot's position, and 0.1 radians on the robot's orientation. + Note that the +\series bold +\emph on +Sigmas +\series default +\emph default + constructor returns a shared pointer, anticipating that typically the same + noise models are used for many different factors. +\end_layout + +\begin_layout Standard +Similarly, odometry measurements are specified as +\series bold +\emph on +Pose2 +\series default +\emph default + on line 11, with a slightly different noise model defined on line 12-13. + We then add the two factors +\begin_inset Formula $f_{1}(x_{1},x_{2};o_{1})$ +\end_inset + + and +\begin_inset Formula $f_{2}(x_{2},x_{3};o_{2})$ +\end_inset + + on lines 14-15, as instances of yet another templated class, +\series bold +\emph on +BetweenFactor +\series default +\emph default +, again with +\series bold +\emph on +T=Pose2 +\series default +\emph default +. +\end_layout + +\begin_layout Standard +When running the example ( +\emph on +make OdometryExample.run +\emph default + on the command prompt), it will print out the factor graph as follows: +\family typewriter +\size small + +\begin_inset CommandInset include +LatexCommand verbatiminput +filename "Code/OdometryOutput1.txt" + +\end_inset + + +\end_layout + +\begin_layout Subsection +Factor Graphs versus Values +\end_layout + +\begin_layout Standard +At this point it is instructive to emphasize two important design ideas + underlying GTSAM: +\end_layout + +\begin_layout Enumerate +The factor graph and its embodiment in code specify the joint probability + distribution +\begin_inset Formula $P(X|Z)$ +\end_inset + + over the +\emph on +entire +\emph default + trajectory +\begin_inset Formula $X\define\{x_{1},x_{2},x_{3}\}$ +\end_inset + + of the robot, rather than just the last pose. + This +\emph on +smoothing +\emph default + view of the world gives GTSAM its name: +\begin_inset Quotes eld +\end_inset + +smoothing and mapping +\begin_inset Quotes erd +\end_inset + +. + Later in this document we will talk about how we can also use GTSAM to + do filtering (which you often do +\emph on +not +\emph default + want to do) or incremental inference (which we do all the time). +\end_layout + +\begin_layout Enumerate +A factor graph in GTSAM is just the specification of the probability density + +\begin_inset Formula $P(X|Z)$ +\end_inset + +, and the corresponding +\series bold +\emph on +FactorGraph +\series default +\emph default + class and its derived classes do not ever contain a +\begin_inset Quotes eld +\end_inset + +solution +\begin_inset Quotes erd +\end_inset + +. + Rather, there is a separate type +\series bold +\emph on +Values +\series default +\emph default + that is used to specify specific values for (in this case) +\begin_inset Formula $x_{1}$ +\end_inset + +, +\begin_inset Formula $x_{2}$ +\end_inset + +, and +\begin_inset Formula $x_{3}$ +\end_inset + +, which can then be used to evaluate the probability (or, more commonly, + the error) associated with particular values. +\end_layout + +\begin_layout Standard +The latter point is often a point of confusion with beginning users of GTSAM. + It helps to remember that when designing GTSAM we took a functional approach + of classes corresponding to mathematical objects, which are usually immutable. + You should think of a factor graph as a +\emph on +function +\emph default + to be applied to values -as the notation +\begin_inset Formula $f(X)\propto P(X|Z)$ +\end_inset + + implies- rather than as an object to be modified. +\end_layout + +\begin_layout Subsection +Non-linear Optimization in GTSAM +\end_layout + +\begin_layout Standard +The listing below creates a +\series bold +\emph on +Values +\series default +\emph default + instance, and uses it as the initial estimate to find the maximum a-posteriori + (MAP) assignment for the trajectory +\begin_inset Formula $X$ +\end_inset + +: +\end_layout + +\begin_layout Standard +\begin_inset CommandInset include +LatexCommand lstinputlisting +filename "Code/OdometryOptimize.cpp" +lstparams "aboveskip=10pt,basicstyle={\\ttfamily\\small},caption={Excerpt from examples/OdometryExample.cpp},captionpos=b,frame=single,identifierstyle={\\bfseries},label={listing:OdometryOptimize},language={C++},numbers=left" + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +noindent +\end_layout + +\end_inset + + Lines 2-5 in Listing +\begin_inset CommandInset ref +LatexCommand ref +reference "listing:OdometryOptimize" + +\end_inset + + create the initial estimate, and on line 8 we create a non-linear Levenberg-Mar +quardt style optimizer, and call +\series bold +\emph on +optimize +\series default +\emph default + using default parameter settings. + The reason why GTSAM needs to perform non-linear optimization is because + the odometry factors +\begin_inset Formula $f_{1}(x_{1},x_{2};o_{1})$ +\end_inset + + and +\begin_inset Formula $f_{2}(x_{2},x_{3};o_{2})$ +\end_inset + + are non-linear, as they involve the orientation of the robot. + This also explains why the factor graph we created in Listing +\begin_inset CommandInset ref +LatexCommand ref +reference "listing:OdometryExample" + +\end_inset + + is of type +\series bold +\emph on +NonlinearFactorGraph +\series default +\emph default +. + The optimization class linearizes this graph, possibly multiple times, + to minimize the non-linear squared error specified by the factors. +\end_layout + +\begin_layout Standard +The relevant output from running the example is as follows: +\family typewriter +\size small + +\begin_inset CommandInset include +LatexCommand verbatiminput +filename "Code/OdometryOutput2.txt" + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +noindent +\end_layout + +\end_inset + + It can be seen that, subject to very small tolerance, the ground truth + solution +\begin_inset Formula $x_{1}=(0,0,0)$ +\end_inset + +, +\begin_inset Formula $x_{2}=(2,0,0)$ +\end_inset + +, and +\begin_inset Formula $x_{3}=(4,0,0)$ +\end_inset + + is recovered. +\end_layout + +\begin_layout Subsection +\begin_inset CommandInset label +LatexCommand label +name "sub:Full-Posterior-Inference" + +\end_inset + +Full Posterior Inference +\end_layout + +\begin_layout Standard +GTSAM can also be used to calculate the covariance matrix for each pose + after incorporating the information from all measurements +\begin_inset Formula $Z$ +\end_inset + +. + Recognizing that the factor graph encodes the +\series bold +posterior density +\series default + +\begin_inset Formula $P(X|Z)$ +\end_inset + +, the mean +\begin_inset Formula $\mu$ +\end_inset + + together with the covariance +\begin_inset Formula $\Sigma$ +\end_inset + + for each pose +\begin_inset Formula $x$ +\end_inset + + approximate the +\series bold +marginal posterior density +\series default + +\begin_inset Formula $P(x|Z)$ +\end_inset + +. + Note that this is just an approximation, as even in this simple case the + odometry factors are actually non-linear in their arguments, and GTSAM + only computes a Gaussian approximation to the true underlying posterior. +\end_layout + +\begin_layout Standard +The following C++ code will recover the posterior marginals: +\end_layout + +\begin_layout Standard +\begin_inset CommandInset include +LatexCommand lstinputlisting +filename "Code/OdometryMarginals.cpp" +lstparams "aboveskip=10pt,basicstyle={\\ttfamily\\small},caption={Excerpt from examples/OdometryExample.cpp},captionpos=b,frame=single,identifierstyle={\\bfseries},label={listing:OdometryMarginals},language={C++},numbers=left" + +\end_inset + +The relevant output from running the example is as follows: +\size footnotesize + +\begin_inset CommandInset include +LatexCommand verbatiminput +filename "Code/OdometryOutput3.txt" + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +noindent +\end_layout + +\end_inset + + What we see is that the marginal covariance +\begin_inset Formula $P(x_{1}|Z)$ +\end_inset + + on +\begin_inset Formula $x_{1}$ +\end_inset + + is simply the prior knowledge on +\begin_inset Formula $x_{1}$ +\end_inset + +, but as the robot moves the uncertainty in all dimensions grows without + bound, and the +\begin_inset Formula $y$ +\end_inset + + and +\begin_inset Formula $\theta$ +\end_inset + + components of the pose become (positively) correlated. +\end_layout + +\begin_layout Standard +An important fact to note when interpreting these numbers is that covariance + matrices are given in +\emph on +relative +\emph default + coordinates, not absolute coordinates. + This is because internally GTSAM optimizes for a change with respect to + a linearization point, as do all nonlinear optimization libraries. +\end_layout + +\begin_layout Section +Robot Localization +\end_layout + +\begin_layout Subsection +Unary Measurement Factors +\end_layout + +\begin_layout Standard +In this section we add measurements to the factor graph that will help us + actually +\emph on +localize +\emph default + the robot over time. + The example also serves as a tutorial on creating new factor types. +\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/FactorGraph2.pdf + scale 80 + BoundingBox 70bp 550bp 300bp 630bp + clip + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +\begin_inset CommandInset label +LatexCommand label +name "fig:LocalizationFG" + +\end_inset + +Robot localization factor graph with unary measurement factors at each time + step. +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +noindent +\end_layout + +\end_inset + + In particular, we use +\series bold +unary measurement factors +\series default + to handle external measurements. + The example from Section +\begin_inset CommandInset ref +LatexCommand ref +reference "sec:Robot-Localization" + +\end_inset + + is not very useful on a real robot, because it only contains factors correspond +ing to odometry measurements. + These are imperfect and will lead to quickly accumulating uncertainty on + the last robot pose, at least in the absence of any external measurements + (see Section +\begin_inset CommandInset ref +LatexCommand ref +reference "sub:Full-Posterior-Inference" + +\end_inset + +). + Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:LocalizationFG" + +\end_inset + + shows a new factor graph where the prior +\begin_inset Formula $f_{0}(x_{1})$ +\end_inset + + is omitted and instead we added three unary factors +\begin_inset Formula $f_{1}(x_{1};z_{1})$ +\end_inset + +, +\begin_inset Formula $f_{2}(x_{2};z_{2})$ +\end_inset + +, and +\begin_inset Formula $f_{3}(x_{3};z_{3})$ +\end_inset + +, one for each localization measurement +\begin_inset Formula $z_{t}$ +\end_inset + +, respectively. + Such unary factors are applicable for measurements +\begin_inset Formula $z_{t}$ +\end_inset + + that depend +\emph on +only +\emph default + on the current robot pose, e.g., GPS readings, correlation of a laser range-finde +r in a pre-existing map, or indeed the presence of absence of ceiling lights + (see +\begin_inset CommandInset citation +LatexCommand citet +key "Dellaert99b" + +\end_inset + + for that amusing example). +\end_layout + +\begin_layout Subsection +Defining Custom Factors +\end_layout + +\begin_layout Standard +In GTSAM, you can create custom unary factors by deriving a new class from + the built-in class +\series bold +\emph on +NoiseModelFactor1 +\series default +\emph default +, which implements a unary factor corresponding to a measurement likelihood + with a Gaussian noise model, +\begin_inset Formula +\[ +L(q;m)=exp\left\{ -\frac{1}{2}\SqrMah{h(q)}{m}{\Sigma}\right\} \define f(q) +\] + +\end_inset + +where +\begin_inset Formula $m$ +\end_inset + + is the measurement, +\begin_inset Formula $q$ +\end_inset + + is the unknown variable, +\begin_inset Formula $h(q)$ +\end_inset + + is a (possibly nonlinear) measurement function, and +\begin_inset Formula $\Sigma$ +\end_inset + + is the noise covariance. + Note that +\begin_inset Formula $m$ +\end_inset + + is considered +\emph on +known +\emph default + above, and the likelihood +\begin_inset Formula $L(q;m)$ +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Plain Layout + of +\begin_inset Formula $q$ +\end_inset + + given +\begin_inset Formula $m$ +\end_inset + + +\end_layout + +\end_inset + + will only ever be evaluated as a function of +\begin_inset Formula $q$ +\end_inset + +, which explains why it is a unary factor +\begin_inset Formula $f(q)$ +\end_inset + +. + It is always the unknown variable +\begin_inset Formula $q$ +\end_inset + + that is either likely or unlikely, given the measurement. + +\end_layout + +\begin_layout Standard + +\series bold +Note: +\series default +many people get this backwards, often misled by the conditional density + notation +\begin_inset Formula $P(m|q)$ +\end_inset + +. + In fact, the likelihood +\begin_inset Formula $L(q;m)$ +\end_inset + + is +\emph on +defined +\emph default + as any function of +\begin_inset Formula $q$ +\end_inset + + proportional to +\begin_inset Formula $P(m|q)$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Listing +\begin_inset CommandInset ref +LatexCommand vref +reference "listing:LocalizationFactor" + +\end_inset + + shows an example on how to define the custom factor class +\series bold +\emph on +UnaryFactor +\series default +\emph default + which implements a +\begin_inset Quotes eld +\end_inset + +GPS-like +\begin_inset Quotes erd +\end_inset + + measurement likelihood: +\end_layout + +\begin_layout Standard +\begin_inset CommandInset include +LatexCommand lstinputlisting +filename "Code/LocalizationFactor.cpp" +lstparams "aboveskip=10pt,basicstyle={\\ttfamily\\small},caption={Excerpt from examples/LocalizationExample.cpp},captionpos=b,frame=single,identifierstyle={\\bfseries},label={listing:LocalizationFactor},language={C++},numbers=left" + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +noindent +\end_layout + +\end_inset + + In defining the derived class on line 1, we provide the template argument + +\series bold +\emph on +Pose2 +\series default +\emph default + to indicate the type of the variable +\begin_inset Formula $q$ +\end_inset + +, whereas the measurement is stored as the instance variables +\series bold +\emph on +mx_ +\series default +\emph default + and +\series bold +\emph on +my_ +\series default +\emph default +, defined on line 2. + The constructor on lines 5-6 simply passes on the variable key +\begin_inset Formula $j$ +\end_inset + + and the noise model to the superclass, and stores the measurement values + provided. + The most important function to has be implemented by every factor class + is +\series bold +\emph on +evaluateError +\series default +\emph default +, which should return +\begin_inset Formula +\[ +E(q)\define h(q)-m +\] + +\end_inset + +which is done on line 12. + Importantly, because we want to use this factor for nonlinear optimization + (see e.g., +\begin_inset CommandInset citation +LatexCommand citealt +key "Dellaert06ijrr" + +\end_inset + + for details), whenever the optional argument +\begin_inset Formula $H$ +\end_inset + + is provided, a +\series bold +\emph on +Matrix +\series default +\emph default + reference, the function should assign the +\series bold +Jacobian +\series default + of +\begin_inset Formula $h(q)$ +\end_inset + + to it, evaluated at the provided value for +\begin_inset Formula $q$ +\end_inset + +. + This is done for this example on line 11. + In this case, the Jacobian of the 2-dimensional function +\begin_inset Formula $h$ +\end_inset + +, which just returns the position of the robot, +\begin_inset Formula +\[ +h(q)=\left[\begin{array}{c} +q_{x}\\ +q_{y} +\end{array}\right] +\] + +\end_inset + + with respect the 3-dimensional pose +\begin_inset Formula $q=\left(q_{x},q_{y},q_{\theta}\right)$ +\end_inset + +, yields the following simple +\begin_inset Formula $2\times3$ +\end_inset + + matrix: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +H=\left[\begin{array}{ccc} +1 & 0 & 0\\ +0 & 1 & 0 +\end{array}\right] +\] + +\end_inset + + +\end_layout + +\begin_layout Subsection +Using Custom Factors +\end_layout + +\begin_layout Standard +The following C++ code fragment illustrates how to create and add custom + factors to a factor graph: +\end_layout + +\begin_layout Standard +\begin_inset CommandInset include +LatexCommand lstinputlisting +filename "Code/LocalizationExample2.cpp" +lstparams "aboveskip=10pt,basicstyle={\\ttfamily\\small},caption={Excerpt from examples/LocalizationExample.cpp},captionpos=b,frame=single,identifierstyle={\\bfseries},label={listing:LocalizationExample2},language={C++},numbers=left" + +\end_inset + + +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +noindent +\end_layout + +\end_inset + + In Listing +\begin_inset CommandInset ref +LatexCommand vref +reference "listing:LocalizationExample2" + +\end_inset + +, we create the noise model on line 2-3, which now specifies two standard + deviations on the measurements +\begin_inset Formula $m_{x}$ +\end_inset + + and +\begin_inset Formula $m_{y}$ +\end_inset + +. + On lines 4-6 we create +\series bold +\emph on +shared_ptr +\series default +\emph default + versions of three newly created +\series bold +\emph on +UnaryFactor +\series default +\emph default + instances, and add them to graph. + GTSAM uses shared pointers to refer to factors in factor graphs, and +\series bold +\emph on +boost::make_shared +\series default +\emph default + is a convenience function to simultaneously construct a class and create + a +\series bold +\emph on +shared_ptr +\series default +\emph default + to it. + +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +and on lines 4-6 we add three newly created +\series bold +\emph on +UnaryFactor +\series default +\emph default + instances to the graph. +\end_layout + +\end_inset + + We obtain the factor graph from Figure +\begin_inset CommandInset ref +LatexCommand vref +reference "fig:LocalizationFG" + +\end_inset + +. + +\family typewriter +\size small + +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +The relevant output from running the example is as follows: +\family typewriter +\size small + +\begin_inset CommandInset include +LatexCommand verbatiminput +filename "Code/LocalizationOutput4.txt" + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Subsection +Full Posterior Inference +\end_layout + +\begin_layout Standard +The three GPS factors are enough to fully constrain all unknown poses and + tie them to a +\begin_inset Quotes eld +\end_inset + +global +\begin_inset Quotes erd +\end_inset + + reference frame, including the three unknown orientations. + If not, GTSAM would have exited with a singular matrix exception. + The marginals can be recovered exactly as in Section +\begin_inset CommandInset ref +LatexCommand ref +reference "sub:Full-Posterior-Inference" + +\end_inset + +, and the solution and marginal covariances are now given by the following: +\size footnotesize + +\begin_inset CommandInset include +LatexCommand verbatiminput +filename "Code/LocalizationOutput5.txt" + +\end_inset + + +\end_layout + +\begin_layout Standard +Comparing this with the covariance matrices in Section +\begin_inset CommandInset ref +LatexCommand ref +reference "sub:Full-Posterior-Inference" + +\end_inset + +, we can see that the uncertainty no longer grows without bounds as measurement + uncertainty accumulates. + Instead, the +\begin_inset Quotes eld +\end_inset + +GPS +\begin_inset Quotes erd +\end_inset + + measurements more or less constrain the poses evenly, as expected. +\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 Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center +\begin_inset Graphics + filename images/Odometry.pdf + width 80text% + BoundingBox 70bp 310bp 525bp 500bp + clip + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Odometry marginals +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\align center +\begin_inset space ~ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\align center +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center +\begin_inset Graphics + filename images/Localization.pdf + width 80text% + BoundingBox 70bp 310bp 525bp 500bp + clip + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Localization Marginals +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +\begin_inset CommandInset label +LatexCommand label +name "fig:CompareMarginals" + +\end_inset + +Comparing the marginals resulting from the +\begin_inset Quotes eld +\end_inset + +odometry +\begin_inset Quotes erd +\end_inset + + factor graph in Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:OdometryFG" + +\end_inset + + and the +\begin_inset Quotes eld +\end_inset + +localization +\begin_inset Quotes erd +\end_inset + + factor graph in Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:LocalizationFG" + +\end_inset + +. +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +It helps a lot when we view this graphically, as in Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:CompareMarginals" + +\end_inset + +, where I show the marginals on position as covariance ellipses that contain + 68.26% of all probability mass. + For the odometry marginals, it is immediately apparent from the figure + that (1) the uncertainty on pose keeps growing, and (2) the uncertainty + on angular odometry translates into increasing uncertainty on y. + The localization marginals, in contrast, are constrained by the unary factors + and are all much smaller. + In addition, while less apparent, the uncertainty on the middle pose is + actually smaller as it is constrained by odometry from two sides. +\end_layout + +\begin_layout Standard +You might now be wondering how we produced these figures. + The answer is via the MATLAB interface of GTSAM, which we will demonstrate + in the next section. +\end_layout + +\begin_layout Standard +\begin_inset Newpage pagebreak +\end_inset + + +\end_layout + +\begin_layout Section +\begin_inset CommandInset label +LatexCommand label +name "sec:Pose2SLAM" + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "sec:WithMarginals" + +\end_inset + +PoseSLAM +\end_layout + +\begin_layout Subsection +Loop Closure Constraints +\end_layout + +\begin_layout Standard +The simplest instantiation of a SLAM problem is +\series bold +PoseSLAM +\series default +, which avoids building an explicit map of the environment. + The goal of SLAM is to simultaneously localize a robot and map the environment + given incoming sensor measurements +\begin_inset CommandInset citation +LatexCommand citep +key "DurrantWhyte06ram" + +\end_inset + +. + Besides wheel odometry, one of the most popular sensors for robots moving + on a plane is a 2D laser-range finder, which provides both odometry constraints + between successive poses, and loop-closure constraints when the robot re-visits + a previously explored part of the environment. +\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/FactorGraph3.pdf + scale 80 + BoundingBox 40bp 585bp 330bp 710bp + clip + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +\begin_inset CommandInset label +LatexCommand label +name "fig:Pose2SLAM" + +\end_inset + +Factor graph for PoseSLAM. +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +noindent +\end_layout + +\end_inset + + A factor graph example for PoseSLAM is shown in Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:Pose2SLAM" + +\end_inset + +. + The following C++ code, included in GTSAM as an example, creates this factor + graph in code: +\end_layout + +\begin_layout Standard +\begin_inset CommandInset include +LatexCommand lstinputlisting +filename "Code/Pose2SLAMExample.cpp" +lstparams "aboveskip=10pt,basicstyle={\\ttfamily\\small},caption={Excerpt from examples/Pose2SLAMExample.cpp},captionpos=b,frame=single,identifierstyle={\\bfseries},label={listing:Pose2SLAMExample},language={C++},numbers=left" + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +noindent +\end_layout + +\end_inset + + As before, lines 1-4 create a nonlinear factor graph and add the unary + factor +\begin_inset Formula $f_{0}(x_{1})$ +\end_inset + +. + As the robot travels through the world, it creates binary factors +\begin_inset Formula $f_{t}(x_{t},x_{t+1})$ +\end_inset + + corresponding to odometry, added to the graph in lines 6-12 (Note that + M_PI_2 refers to pi/2). + But line 15 models a different event: a +\series bold +loop closure +\series default +. + For example, the robot might recognize the same location using vision or + a laser range finder, and calculate the geometric pose constraint to when + it first visited this location. + This is illustrated for poses +\begin_inset Formula $x_{5}$ +\end_inset + + and +\begin_inset Formula $x_{2}$ +\end_inset + +, and generates the (red) loop closing factor +\begin_inset Formula $f_{5}(x_{5},x_{2})$ +\end_inset + +. +\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 figures/PoseSLAM/example.pdf + width 80text% + BoundingBox 30bp 170bp 610bp 630bp + clip + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +\begin_inset CommandInset label +LatexCommand label +name "fig:example" + +\end_inset + +The result of running optimize on the factor graph in Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:Pose2SLAM" + +\end_inset + +. +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +We can optimize this factor graph as before, by creating an initial estimate + of type +\series bold +\emph on +Values +\series default +\emph default +, and creating and running an optimizer. + The result is shown graphically in Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:example" + +\end_inset + +, along with covariance ellipses shown in green. + These covariance ellipses in 2D indicate the marginal over position, over + all possible orientations, and show the area which contain 68.26% of the + probability mass (in 1D this would correspond to one standard deviation). + The graph shows in a clear manner that the uncertainty on pose +\begin_inset Formula $x_{5}$ +\end_inset + + is now much less than if there would be only odometry measurements. + The pose with the highest uncertainty, +\begin_inset Formula $x_{4}$ +\end_inset + +, is the one furthest away from the unary constraint +\begin_inset Formula $f_{0}(x_{1})$ +\end_inset + +, which is the only factor tying the graph to a global coordinate frame. +\end_layout + +\begin_layout Standard +The figure above was created using an interface that allows you to use GTSAM + from within MATLAB, which provides for visualization and rapid development. + We discuss this next. +\end_layout + +\begin_layout Subsection +Using the MATLAB Interface +\end_layout + +\begin_layout Standard +A large subset of the GTSAM functionality can be accessed through wrapped + classes from within MATLAB +\begin_inset Foot +status open + +\begin_layout Plain Layout +GTSAM also allows you to wrap your own custom-made classes, although this + is outside the scope of this manual +\end_layout + +\end_inset + +. + The following code excerpt is the MATLAB equivalent of the C++ code in + Listing +\begin_inset CommandInset ref +LatexCommand ref +reference "listing:Pose2SLAMExample" + +\end_inset + +:` +\begin_inset CommandInset include +LatexCommand lstinputlisting +filename "Code/Pose2SLAMExample.m" +lstparams "aboveskip=10pt,basicstyle={\\ttfamily\\small},caption={Excerpt from examples/matlab/Pose2SLAMExample.m},captionpos=b,frame=single,identifierstyle={\\bfseries},label={listing:Pose2SLAMExample-MATLAB},language={Matlab},numbers=left" + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +noindent +\end_layout + +\end_inset + + Note that the code is almost identical, although there are a few syntax + and naming differences: +\end_layout + +\begin_layout Itemize +Objects are created by calling a constructor instead of allocating them + on the heap. +\end_layout + +\begin_layout Itemize +Namespaces are done using dot notation, i.e., +\series bold +\emph on +noiseModel::Diagonal::SigmasClasses +\series default +\emph default + becomes +\series bold +\emph on +noiseModel.Diagonal.Sigmas +\series default +\emph default +. +\end_layout + +\begin_layout Itemize + +\series bold +\emph on +Vector +\series default +\emph default + and +\series bold +\emph on +Matrix +\series default +\emph default + classes in C++ are just vectors/matrices in MATLAB. +\end_layout + +\begin_layout Itemize +As templated classes do not exist in MATLAB, these have been hardcoded in + the GTSAM interface, e.g., +\series bold +\emph on +PriorFactorPose2 +\series default +\emph default + corresponds to the C++ class +\series bold +\emph on +PriorFactor +\series default +\emph default +, etc. +\end_layout + +\begin_layout Standard +After executing the code, you can call +\emph on +whos +\emph default +on the MATLAB command prompt to see the objects created. + Note that the indicated +\emph on +Class +\emph default + corresponds to the wrapped C++ classes: +\size small + +\begin_inset CommandInset include +LatexCommand verbatiminput +filename "Code/whos.txt" + +\end_inset + + +\size default + +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +noindent +\end_layout + +\end_inset + + In addition, any GTSAM object can be examined in detail, yielding identical + output to C++: +\size small + +\begin_inset CommandInset include +LatexCommand verbatiminput +filename "Code/print.txt" + +\end_inset + + +\size default + +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +noindent +\end_layout + +\end_inset + + And it does not stop there: we can also call some of the functions defined + for factor graphs. + E.g., +\end_layout + +\begin_layout Standard + +\size small +\begin_inset CommandInset include +LatexCommand verbatiminput +filename "Code/calls.txt" + +\end_inset + + +\size default + +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +noindent +\end_layout + +\end_inset + + computes the sum-squared error +\begin_inset Formula $\frac{1}{2}\sum_{i}\SqrMah{h_{i}(X_{i})}{z_{i}}{\Sigma}$ +\end_inset + + before and after optimization. +\end_layout + +\begin_layout Subsection +Reading and Optimizing Pose Graphs +\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/w100-result.pdf + width 80text% + BoundingBox 30bp 170bp 610bp 630bp + clip + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +\begin_inset CommandInset label +LatexCommand label +name "fig:w100" + +\end_inset + +MATLAB plot of small Manhattan world example with 100 poses (due to Ed Olson). + The initial estimate is shown in green. + The optimized trajectory, with covariance ellipses, in blue. +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +The ability to work in MATLAB adds a much quicker development cycle, and + effortless graphical output. + The optimized trajectory in Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:w100" + +\end_inset + + was produced by the code below, in which +\emph on +load2D +\emph default + reads TORO files. + To see how plotting is done, refer to the full source code. +\begin_inset CommandInset include +LatexCommand lstinputlisting +filename "Code/Pose2SLAMExample-graph.m" +lstparams "aboveskip=10pt,basicstyle={\\ttfamily\\small},caption={Excerpt from examples/matlab/Pose2SLAMExample-graph.m},captionpos=b,frame=single,identifierstyle={\\bfseries},label={listing:Pose2SLAMExample-graph},language={Matlab},numbers=left" + +\end_inset + + +\end_layout + +\begin_layout Subsection +PoseSLAM in 3D +\end_layout + +\begin_layout Standard +PoseSLAM can easily be extended to 3D poses, but some care is needed to + update 3D rotations. + GTSAM supports both +\series bold +quaternions +\series default + and +\begin_inset Formula $3\times3$ +\end_inset + + +\series bold +rotation matrices +\series default +to represent 3D rotations. + The selection is made via the compile flag GTSAM_USE_QUATERNIONS. +\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/sphere2500-result.pdf + width 70text% + BoundingBox 60bp 150bp 610bp 610bp + clip + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +\begin_inset CommandInset label +LatexCommand label +name "fig:w100-1" + +\end_inset + +3D plot of sphere example (due to Michael Kaess). + The very wrong initial estimate, derived from odometry, is shown in green. + The optimized trajectory is shown red. + Code below: +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset CommandInset include +LatexCommand lstinputlisting +filename "Code/Pose3SLAMExample-graph.m" +lstparams "aboveskip=10pt,basicstyle={\\ttfamily\\small},caption={Excerpt from examples/matlab/Pose3SLAMExample-graph.m},captionpos=b,frame=single,identifierstyle={\\bfseries},label={listing:Pose3SLAMExample-graph-1},language={Matlab},numbers=left" + +\end_inset + + +\end_layout + +\begin_layout Section +\begin_inset CommandInset label +LatexCommand label +name "sec:Landmark-based-SLAM" + +\end_inset + +Landmark-based SLAM +\end_layout + +\begin_layout Subsection +Basics +\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/FactorGraph4.pdf + scale 80 + BoundingBox 50bp 590bp 290bp 710bp + clip + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +\begin_inset CommandInset label +LatexCommand label +name "fig:SLAM" + +\end_inset + +Factor graph for landmark-based SLAM +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +noindent +\end_layout + +\end_inset + + In +\series bold +landmark-based SLAM +\series default +, we explicitly build a map with the location of observed landmarks, which + introduces a second type of variable in the factor graph besides robot + poses. + An example factor graph for a landmark-based SLAM example is shown in Figure + +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:SLAM" + +\end_inset + +, which shows the typical connectivity: poses are connected in an odometry + Markov chain, and landmarks are observed from multiple poses, inducing + binary factors. + In addition, the pose +\begin_inset Formula $x_{1}$ +\end_inset + + has the usual prior on it. +\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 figures/PlanarSLAM/example.pdf + scale 47 + BoundingBox 90bp 220bp 520bp 555bp + clip + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +\begin_inset CommandInset label +LatexCommand label +name "fig:PlanarSLAMExample" + +\end_inset + +The optimized result along with covariance ellipses for both poses (in green) + and landmarks (in blue). + Also shown are the trajectory (red) and landmark sightings (cyan). +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +The factor graph from Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:SLAM" + +\end_inset + + can be created using the MATLAB code in Listing +\begin_inset CommandInset ref +LatexCommand ref +reference "listing:PlanarSLAMExample" + +\end_inset + +. + As before, on line 2 we create the factor graph, and Lines 8-18 create + the prior/odometry chain we are now familiar with. + However, the code on lines 20-26 is new: it creates three +\series bold +measurement factors +\series default +, in this case +\begin_inset Quotes eld +\end_inset + +bearing/range +\begin_inset Quotes erd +\end_inset + + measurements from the pose to the landmark. +\end_layout + +\begin_layout Standard +\begin_inset CommandInset include +LatexCommand lstinputlisting +filename "Code/PlanarSLAMExample.m" +lstparams "aboveskip=10pt,basicstyle={\\ttfamily\\small},caption={Excerpt from examples/matlab/PlanarSLAMExample.m},captionpos=b,frame=single,identifierstyle={\\bfseries},label={listing:PlanarSLAMExample},language={Matlab},numbers=left" + +\end_inset + + +\end_layout + +\begin_layout Subsection +Of Keys and Symbols +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +noindent +\end_layout + +\end_inset + + The only unexplained code is on lines 4-6: here we create integer keys + for the poses and landmarks using the +\series bold +\emph on +symbol +\series default +\emph default + function. + In GTSAM, we address all variables using the +\series bold +\emph on +Ke +\emph default +y +\series default + type, which is just a typedef to +\series bold +\emph on +size_t +\series default +\emph default + +\begin_inset Foot +status open + +\begin_layout Plain Layout +a 32 or 64 bit integer, depending on your platform +\end_layout + +\end_inset + +. + The keys do not have to be numbered continuously, but they do have to be + unique within a given factor graph. + For factor graphs with different types of variables, we provide the +\series bold +\emph on +symbol +\series default +\emph default + function in MATLAB, and the +\series bold +\emph on +Symbol +\series default +\emph default + type in C++, to help you create (large) integer keys that are far apart + in the space of possible keys, so you don't have to think about starting + the point numbering at some arbitrary offset. + To create a a +\emph on +symbol key +\emph default + you simply provide a character and an integer index. + You can use base 0 or 1, or use arbitrary indices: it does not matter. + In the code above, we we use 'x' for poses, and 'l' for landmarks. +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +, and use the resulting keys +\series bold +\emph on +i1 +\series default +\emph default +, +\series bold +\emph on +i2 +\series default +\emph default +, +\series bold +\emph on +i3 +\series default +\emph default +, +\series bold +\emph on +j1 +\series default +\emph default +, and +\series bold +\emph on +j2 +\series default +\emph default + to create the factors in the correct way. +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +The optimized result for the factor graph created by Listing +\begin_inset CommandInset ref +LatexCommand ref +reference "listing:PlanarSLAMExample" + +\end_inset + + is shown in Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:PlanarSLAMExample" + +\end_inset + +, and it is readily apparent that the landmark +\begin_inset Formula $l_{1}$ +\end_inset + + with two measurements is better localized. + In MATLAB we can also examine the actual numerical values, and doing so + reveals some more GTSAM magic: +\end_layout + +\begin_layout Standard + +\size small +\begin_inset CommandInset include +LatexCommand verbatiminput +filename "Code/PlanarSLAMExample.txt" + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +noindent +\end_layout + +\end_inset + + Indeed, the keys generated by symbol are automatically detected by the + +\series bold +\emph on +print +\series default +\emph default + method in the +\series bold +\emph on +Values +\series default +\emph default + class, and rendered in human-readable form +\begin_inset Quotes eld +\end_inset + +x1 +\begin_inset Quotes erd +\end_inset + +, +\begin_inset Quotes eld +\end_inset + +l2 +\begin_inset Quotes erd +\end_inset + +, etc, rather than as large, unwieldy integers. + This magic extends to most factors and other classes where the +\series bold +Key +\series default + type is used. +\end_layout + +\begin_layout Subsection +A Larger Example +\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/littleRobot.pdf + width 90text% + BoundingBox 0bp 200bp 612bp 600bp + clip + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +\begin_inset CommandInset label +LatexCommand label +name "fig:littleRobot" + +\end_inset + +A larger example with about 100 poses and 30 or so landmarks, as produced + by gtsam_examples/PlanarSLAMExample_graph.m +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +GTSAM comes with a slightly larger example that is read from a .graph file + by PlanarSLAMExample_graph.m, shown in Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:littleRobot" + +\end_inset + +. + To not clutter the figure only the marginals are shown, not the lines of + sight. + This example, with 119 (multivariate) variables and 517 factors optimizes + in less than 10 ms. +\end_layout + +\begin_layout Subsection +A Real-World Example +\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 figures/PlanarSLAM/Victoria.pdf + width 90text% + BoundingBox 0bp 0bp 420bp 180bp + clip + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +\begin_inset CommandInset label +LatexCommand label +name "fig:Victoria-1" + +\end_inset + +Small section of optimized trajectory and landmarks (trees detected in a + laser range finder scan) from data recorded in Sydney's Victoria Park (dataset + due to Jose Guivant, U. + Sydney). +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +A real-world example is shown in Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:Victoria-1" + +\end_inset + +, using data from a well known dataset collected in Sydney's Victoria Park, + using a truck equipped with a laser range-finder. + The covariance matrices in this figure were computed very efficiently, + as explained in detail in +\begin_inset CommandInset citation +LatexCommand citep +key "Kaess09ras" + +\end_inset + +. + The exact covariances (blue, smaller ellipses) obtained by our fast algorithm + coincide with the exact covariances based on full inversion (orange, mostly + hidden by blue). + The much larger conservative covariance estimates (green, large ellipses) + were based on our earlier work in +\begin_inset CommandInset citation +LatexCommand citep +key "Kaess08tro" + +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Newpage pagebreak +\end_inset + + +\end_layout + +\begin_layout Section +Structure from Motion +\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/cube.pdf + width 80text% + BoundingBox 60bp 90bp 612bp 380bp + clip + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +\begin_inset CommandInset label +LatexCommand label +name "fig:SFMExample" + +\end_inset + +An optimized +\begin_inset Quotes eld +\end_inset + +Structure from Motion +\begin_inset Quotes erd +\end_inset + + with 10 cameras arranged in a circle, observing the 8 vertices of a +\begin_inset Formula $20\times20\times20$ +\end_inset + + cube centered around the origin. + The camera is rendered with color-coded axes, (RGB for XYZ) and the viewing + direction is is along the positive Z-axis. + Also shown are the 3D error covariance ellipses for both cameras and points. +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +noindent +\end_layout + +\end_inset + + +\series bold + Structure from Motion +\series default + (SFM) is a technique to recover a 3D reconstruction of the environment + from corresponding visual features in a collection of +\emph on +unordered +\emph default + images, see Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:SFMExample" + +\end_inset + +. + In GTSAM this is done using exactly the same factor graph framework, simply + using SFM-specific measurement factors. + In particular, there is a +\series bold +projection factor +\series default + that calculates the reprojection error +\begin_inset Formula $f(x_{i},p_{j};z_{ij},K)$ +\end_inset + + for a given camera pose +\begin_inset Formula $x_{i}$ +\end_inset + + (a +\series bold +\emph on +Pose3 +\series default +\emph default +) and point +\begin_inset Formula $p_{j}$ +\end_inset + + (a +\series bold +\emph on +Point3 +\series default +\emph default +). + The factor is parameterized by the 2D measurement +\begin_inset Formula $z_{ij}$ +\end_inset + + (a +\series bold +\emph on +Point2 +\series default +\emph default +), and known calibration parameters +\begin_inset Formula $K$ +\end_inset + + (of type +\series bold +\emph on +Cal3_S2 +\series default +\emph default +). + The following listing shows how to create the factor graph: +\begin_inset CommandInset include +LatexCommand lstinputlisting +filename "Code/SFMExample.m" +lstparams "aboveskip=10pt,basicstyle={\\ttfamily\\small},caption={Excerpt from examples/matlab/SFMExample.m},captionpos=b,frame=single,identifierstyle={\\bfseries},label={listing:SFMExample},language={Matlab},numbers=left" + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +noindent +\end_layout + +\end_inset + + In Listing +\begin_inset CommandInset ref +LatexCommand ref +reference "listing:SFMExample" + +\end_inset + +, assuming that the factor graph was already created, we add measurement + factors in the double loop. + We loop over images with index +\begin_inset Formula $i$ +\end_inset + +, and in this example the data is given as two cell arrays: Z{i} specifies + a set of measurements +\begin_inset Formula $z_{k}$ +\end_inset + + in image +\begin_inset Formula $i$ +\end_inset + +, and +\begin_inset Formula $J\{i\}$ +\end_inset + + specifies the corresponding point index. + The specific factor type we use is a +\series bold +\emph on +GenericProjectionFactorCal3_S2 +\series default +\emph default +, which is the MATLAB equivalent of the C++ class +\series bold +\emph on +GenericProjectionFactor +\series default +\emph default +, where +\series bold +\emph on +Cal3_S2 +\series default +\emph default + is the camera calibration type we choose to use (the standard, no-radial + distortion, 5 parameter calibration matrix). + As before landmark-based SLAM (Section +\begin_inset CommandInset ref +LatexCommand ref +reference "sec:Landmark-based-SLAM" + +\end_inset + +), here we use symbol keys except we now use the character 'p' to denote + points, rather than 'l' for landmark. +\end_layout + +\begin_layout Standard +Important note: a very tricky and difficult part of making SFM work is (a) + data association, and (b) initialization. + GTSAM does neither of these things for you: it simply provides the +\begin_inset Quotes eld +\end_inset + +bundle adjustment +\begin_inset Quotes erd +\end_inset + + optimization. + In the example, we simply assume the data association is known (it is encoded + in the J sets), and we initialize with the ground truth, as the intent + of the example is simply to show you how to set up the optimization problem. +\end_layout + +\begin_layout Section +iSAM: Incremental Smoothing and Mapping +\end_layout + +\begin_layout Standard +GTSAM provides an incremental inference algorithm based on a more advanced + graphical model, the Bayes tree, which is kept up to date by the +\series bold +iSAM +\series default + algorithm (incremental Smoothing and Mapping, see +\begin_inset CommandInset citation +LatexCommand citet +key "Kaess08tro,Kaess12ijrr" + +\end_inset + + for an in-depth treatment). + For mobile robots operating in real-time it is important to have access + to an updated map as soon as new sensor measurements come in. + iSAM keeps the map up-to-date in an efficient manner. + +\end_layout + +\begin_layout Standard +Listing +\begin_inset CommandInset ref +LatexCommand ref +reference "listing:iSAMExample" + +\end_inset + + shows how to use iSAM in a simple visual SLAM example. + In line 1-2 we create a +\series bold +\emph on +NonlinearISAM +\series default +\emph default + object which will relinearize and reorder the variables every 3 steps. + The corect value for this parameter depends on how non-linear your problem + is and how close you want to be to gold-standard solution at every step. + In iSAM 2.0, this parameter is not needed, as iSAM2 automatically determines + when linearization is needed and for which variables. +\end_layout + +\begin_layout Standard +The example involves eight 3D points that are seen from eight successive + camera poses. + Hence in the first step -which is omitted here- all eight landmarks and + the first pose are properly initialized. + In the code this is done by perturbing the known ground truth, but in a + real application great care is needed to properly initialize poses and + landmarks, especially in a monocular sequence. +\end_layout + +\begin_layout Standard +\begin_inset Newpage pagebreak +\end_inset + + +\begin_inset CommandInset include +LatexCommand lstinputlisting +filename "Code/VisualISAMExample.cpp" +lstparams "aboveskip=10pt,basicstyle={\\ttfamily\\small},caption={Excerpt from examples/VisualISAMExample.cpp},captionpos=b,frame=single,identifierstyle={\\bfseries},label={listing:iSAMExample},language={Matlab},numbers=left" + +\end_inset + + +\end_layout + +\begin_layout Standard +The remainder of the code illustrates a typical iSAM loop: +\end_layout + +\begin_layout Enumerate +Create factors for new measurements. + Here, in lines 9-18, a small +\series bold +\emph on +NonlinearFactorGraph +\series default +\emph default + is created to hold the new factors of type +\series bold +\emph on +GenericProjectionFactor +\series default +\emph default +. +\end_layout + +\begin_layout Enumerate +Create an initial estimate for all newly introduced variables. + In this small example, all landmarks have been observed in frame 1 and + hence the only new variable that needs to be initialized at each time step + is the new pose. + This is done in lines 20-22. + Note we assume a good initial estimate is available as +\emph on +initial_x[i] +\emph default +. +\end_layout + +\begin_layout Enumerate +Finally, we call +\emph on +isam.update() +\emph default +, which takes the factors and initial estimates, and incrementally updates + the solution, which is available through the method +\emph on +isam.estimate() +\emph default +, if desired. +\end_layout + +\begin_layout Standard +\begin_inset Newpage pagebreak +\end_inset + + +\end_layout + +\begin_layout Section +More Applications +\end_layout + +\begin_layout Standard +While a detailed discussion of all the things you can do with GTSAM will + take us too far, below is a small survey of what you can expect to do, + and which we did using GTSAM. +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Float figure +wide false +sideways false +status collapsed + +\begin_layout Plain Layout +\align center +\begin_inset Graphics + filename figures/SAM/FactorGraph.pdf + scale 80 + BoundingBox 40bp 600bp 300bp 792bp + clip + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +\begin_inset CommandInset label +LatexCommand label +name "fig:Filtering" + +\end_inset + +Factor graph explanation of Filtering. +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +The factor graph for landmark-based SLAM is shown in Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:Filtering" + +\end_inset + +. +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Subsection +Conjugate Gradient Optimization +\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/Beijing.pdf + width 70text% + BoundingBox 100bp 240bp 500bp 550bp + clip + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +\begin_inset CommandInset label +LatexCommand label +name "fig:Beijing" + +\end_inset + + A map of Beijing, with a spanning tree shown in black, and the remaining + +\emph on +loop-closing +\emph default + constraints shown in red. + A spanning tree can be used as a +\emph on +preconditioner +\emph default + by GTSAM. +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +GTSAM also includes efficient preconditioned conjugate gradients (PCG) methods + for solving large-scale SLAM problems. + While direct methods, popular in the literature, exhibit quadratic convergence + and can be quite efficient for sparse problems, they typically require + a lot of storage and efficient elimination orderings to be found. + In contrast, iterative optimization methods only require access to the + gradient and have a small memory footprint, but can suffer from poor convergenc +e. + Our method, +\emph on +subgraph preconditioning +\emph default +, explained in detail in +\begin_inset CommandInset citation +LatexCommand citet +key "Dellaert10iros,Jian11iccv" + +\end_inset + +, combines the advantages of direct and iterative methods, by identifying + a sub-problem that can be easily solved using direct methods, and solving + for the remaining part using PCG. + The easy sub-problems correspond to a spanning tree, a planar subgraph, + or any other substructure that can be efficiently solved. + An example of such a subgraph is shown in Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:Beijing" + +\end_inset + +. +\end_layout + +\begin_layout Subsection +Visual Odometry +\end_layout + +\begin_layout Standard +A gentle introduction to vision-based sensing is +\series bold +Visual Odometry +\series default + (abbreviated VO, see e.g. + +\begin_inset CommandInset citation +LatexCommand citet +key "Nister04cvpr2" + +\end_inset + +), which provides pose constraints between successive robot poses by tracking + or associating visual features in successive images taken by a camera mounted + rigidly on the robot. + GTSAM includes both C++ and MATLAB example code, as well as VO-specific + factors to help you on the way. +\end_layout + +\begin_layout Subsection +Visual SLAM +\end_layout + +\begin_layout Standard + +\series bold +Visual SLAM +\series default + (see e.g., +\begin_inset CommandInset citation +LatexCommand citet +key "Davison03iccv" + +\end_inset + +) is a SLAM variant where 3D points are observed by a camera as the camera + moves through space, either mounted on a robot or moved around by hand. + GTSAM, and particularly iSAM (see below), can easily be adapted to be used + as the back-end optimizer in such a scenario. +\end_layout + +\begin_layout Subsection +Fixed-lag Smoothing and Filtering +\end_layout + +\begin_layout Standard +GTSAM can easily perform recursive estimation, where only a subset of the + poses are kept in the factor graph, while the remaining poses are marginalized + out. + In all examples above we explicitly optimize for all variables using all + available measurements, which is called +\series bold +Smoothing +\series default + because the trajectory is +\begin_inset Quotes eld +\end_inset + +smoothed +\begin_inset Quotes erd +\end_inset + + out, and this is where GTSAM got its name (GT +\emph on +Smoothing +\emph default + and Mapping). + When instead only the last few poses are kept in the graph, one speaks + of +\series bold +Fixed-lag Smoothing +\series default +. + Finally, when only the single most recent poses is kept, one speaks of + +\series bold +Filtering +\series default +, and indeed the original formulation of SLAM was filter-based +\begin_inset CommandInset citation +LatexCommand citep +key "Smith87b" + +\end_inset + +. +\end_layout + +\begin_layout Subsection +Discrete Variables and HMMs +\end_layout + +\begin_layout Standard +Finally, factor graphs are not limited to continuous variables: GTSAM can + also be used to model and solve discrete optimization problems. + For example, a Hidden Markov Model (HMM) has the same graphical model structure + as the Robot Localization problem from Section +\begin_inset CommandInset ref +LatexCommand ref +reference "sec:Robot-Localization" + +\end_inset + +, except that in an HMM the variables are discrete. + GTSAM can optimize and perform inference for discrete models. +\end_layout + +\begin_layout Section* +Acknowledgements +\end_layout + +\begin_layout Standard +GTSAM was made possible by the efforts of many collaborators at Georgia + Tech and elsewhere, including but not limited to Doru Balcan, Chris Beall, + Alex Cunningham, Alireza Fathi, Eohan George, Viorela Ila, Yong-Dian Jian, + Michael Kaess, Kai Ni, Carlos Nieto, Duy-Nguyen Ta, Manohar Paluri, Christian + Potthast, Richard Roberts, Grant Schindler, and Stephen Williams. + In addition, Paritosh Mohan helped me with the manual. + Many thanks all for your hard work! +\end_layout + +\begin_layout Standard +\begin_inset Newpage pagebreak +\end_inset + + +\begin_inset CommandInset bibtex +LatexCommand bibtex +bibfiles "../refs" +options "apalike" + +\end_inset + + +\end_layout + +\end_body +\end_document diff --git a/doc/images/Beijing.pdf b/doc/images/Beijing.pdf new file mode 100644 index 000000000..57f11822f Binary files /dev/null and b/doc/images/Beijing.pdf differ diff 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