540 lines
No EOL
18 KiB
TeX
540 lines
No EOL
18 KiB
TeX
\documentclass[12pt,dvips]{article}
|
|
|
|
\usepackage[T1]{fontenc}
|
|
\usepackage[isolatin]{inputenc}
|
|
\usepackage{a4}
|
|
\usepackage{url}
|
|
\usepackage{afterpage}
|
|
\usepackage{latexsym}
|
|
\usepackage{graphicx}
|
|
\usepackage{longtable}
|
|
\usepackage{amsmath}
|
|
\usepackage[rflt]{floatflt}
|
|
\usepackage{fancybox}
|
|
\usepackage{bbm}
|
|
\usepackage{cite}
|
|
\usepackage{picins}
|
|
|
|
\usepackage[dvips]{epsfig}
|
|
|
|
\parindent0mm
|
|
\parskip5pt plus1pt minus2pt
|
|
\textwidth16cm
|
|
\textheight22cm
|
|
\topmargin-1cm
|
|
\oddsidemargin0cm
|
|
|
|
\renewcommand\floatpagefraction{1.0}
|
|
\renewcommand\topfraction{1.0}
|
|
\renewcommand\bottomfraction{1.0}
|
|
\renewcommand\textfraction{0.0}
|
|
\def\dbltopfraction{1.0}
|
|
\def\bottomfraction{1.0}
|
|
\def\dblfloatpagefraction{1.0}
|
|
|
|
|
|
\setcounter{secnumdepth}{5}
|
|
%% \setcounter{section}{-1}
|
|
|
|
\newcommand{\OO}{{\cal O}}
|
|
\newcommand{\bc}{\begin{center}}
|
|
\newcommand{\ec}{\end{center}}
|
|
|
|
\newcommand{\VNull}{\mathbf{0}}
|
|
%\newcommand{\transp}{^{\textsf{T}}}
|
|
\newcommand{\transp}{^T}
|
|
\renewcommand{\phi}{\varphi}
|
|
\newcommand{\verf}[1]{\textsf{#1}}
|
|
\renewcommand{\v}[1]{\text{\boldmath $#1$}}
|
|
\newcommand{\V}[1]{\text{\boldmath $#1$}}
|
|
\newcommand{\m}[1]{\v{#1}} % Format "Matrix"
|
|
\newcommand{\M}[1]{\v{#1}} % Format "Matrix"
|
|
\newcommand{\sR}{\ensuremath{{\cal R}}} % Referenz-Scan
|
|
\newcommand{\sS}{\ensuremath{{\cal S}}} % aktueller Scan
|
|
\renewcommand{\O}{{\cal O}}
|
|
\newcommand{\norm}[1]{\left | \left | #1 \right | \right |}
|
|
|
|
\newcommand{\Vector}[2]{\begin{pmatrix} #1 \\ #2 \end{pmatrix}}
|
|
\newcommand{\VEctor}[3]{\begin{pmatrix} #1 \\ #2 \\ #3 \end{pmatrix}}
|
|
\newcommand{\hist}[1]{{\cal #1}}
|
|
\newcommand{\trace}[1]{\mbox{tr}\left( #1 \right)}
|
|
|
|
\newcommand{\rVEctor}[3]{
|
|
\left(\!\!\!\begin{array}{l}
|
|
#1 \\ #2 \\ #3
|
|
\end{array}\!\!\right)}
|
|
\newenvironment{rmatrix}
|
|
{\left(\!\!\!\begin{array}{rr}}
|
|
{\end{array}\!\!\right)}
|
|
\newenvironment{Rmatrix}
|
|
{\left(\!\!\!\begin{array}{rrr}}
|
|
{\end{array}\!\!\right)}
|
|
\newcommand{\filter}[3]{\ensuremath{\bigl[\,#1,\;#2,\;#3\,\bigr]}}
|
|
\newcommand{\R}{\mathbbm{R}}
|
|
\newcommand{\N}{\mathbbm{N}}
|
|
%\newcommand{\R}{\mathbb{R}}
|
|
%\newcommand{\N}{\mathbb{N}}
|
|
\newcommand{\grad}{\ensuremath{^{\circ}}}
|
|
\newcommand{\textmenge}[2]{\{\, #1 \,| \; #2 \}}
|
|
\newcommand{\menge}[2]{\left\{\, #1 \,\left| \; #2 \right.\right\} }
|
|
\newcommand{\Menge}[2]{\bigl\{\, #1 \,| \; #2 \bigr\} }
|
|
\newcommand{\fueralle}[2]{\forall \, #1\!: \; #2}
|
|
\newcommand{\esgibt}[2]{\exists \, #1\!: \; #2}
|
|
\newcommand{\Widehat}[1]{\widehat{#1\hspace{+.5ex}}\hspace{-.5ex}}
|
|
\newcommand{\textSF}[1]{\textsf{\small #1}}
|
|
\newcommand{\textSFsmall}[1]{\textsf{\scriptsize #1}}
|
|
\newcommand{\verfSmall}[1]{\textSFsmall{#1}}
|
|
\newcommand{\eps}{{\epsilon}}
|
|
\renewcommand{\epsilon}{\varepsilon}
|
|
\newcommand{\vR}{v_{\text{set}}}
|
|
\newcommand{\oR}{\omega_{\text{set}}}
|
|
\newcommand{\ATAN}{\text{atan2}}
|
|
|
|
%\renewcommand{\familydefault}{\sfdefault}
|
|
|
|
%%----------------------------------------------------------------
|
|
%%----------------------------------------------------------------
|
|
|
|
\begin{document}
|
|
|
|
{
|
|
\title{\vspace*{-5mm}
|
|
\bf 6D SLAM -- \\%[.5cm]
|
|
Simultaneous 6 D.O.F. Localization \\and 3D Mapping\\%[1cm]
|
|
}
|
|
\author{
|
|
\scalebox{.65}{\includegraphics{unilogo.eps}}\\[3ex]
|
|
Andreas N\"uchter, Kai Lingemann, Joachim Hertzberg\\
|
|
Department of Mathematics/Computer Science\\
|
|
Institute of Computer Science\\
|
|
Knowledge-Based Systems Research Group\\
|
|
University of Osnabr\"uck\\
|
|
{\small \texttt{http://www.informatik.uni-osnabrueck.de/kbs/}}
|
|
}
|
|
|
|
\maketitle
|
|
\thispagestyle{empty}
|
|
|
|
\vspace*{-12mm}
|
|
|
|
\begin{center}
|
|
\includegraphics[height=50mm]{stylish_scanner}
|
|
\end{center}
|
|
|
|
\vspace*{1mm}
|
|
|
|
\begin{center}
|
|
\textbf{Documentation}
|
|
\end{center}
|
|
|
|
\begin{quote}
|
|
This document describes the algorithms for 6D SLAM --
|
|
Simultaneous 6 D.O.F. Localization and 3D Mapping system. 6D SLAM
|
|
with mobile robots considers six dimensions for the robot pose,
|
|
namely the $x$, $y$ and $z$ coordinates and the roll, yaw and
|
|
pitch angles. Robot motion and localization on natural surfaces,
|
|
e.g., driving with a mobile robot outdoor, must necessarily
|
|
regard these degrees of freedom.
|
|
\end{quote}
|
|
|
|
\newpage
|
|
\setcounter{page}{1}
|
|
|
|
\section{Range Image Registration and Robot Relocalization}
|
|
|
|
Multiple 3D scans are necessary to digitalize environments
|
|
without occlusions. To create a correct and consistent model, the
|
|
scans have to be merged into one coordinate system. This process
|
|
is called registration. If robot carrying the 3D scanner were
|
|
precisely localized, the registration could be done directly
|
|
based on the robot pose. However, due to the unprecise robot
|
|
sensors, self localization is erroneous, so the geometric
|
|
structure of overlapping 3D scans has to be considered for
|
|
registration.
|
|
|
|
The following method for registration of point sets is part of
|
|
many publications, so only a short summary is given here. The
|
|
complete algorithm was invented in 1992 and can be found, e.g.,
|
|
in \cite{Besl_1992}. The method is called \emph{Iterative Closest
|
|
Points (ICP) algorithm}.
|
|
|
|
Given two independently acquired sets of 3D points, $M$ (model
|
|
set, $|M| = N_m$) and $D$ (data set, $|D| = N_d$) which
|
|
correspond to a single shape, we aim to find the transformation
|
|
consisting of a rotation $\M R$ and a translation $\V t$ which
|
|
minimizes the following cost function:
|
|
\begin{equation}\label{DMin}
|
|
E(\M R, \V t) =
|
|
\sum_{i=1}^{N_m}\sum_{j=1}^{N_d}w_{i,j}\norm{\V m_{i}-(\M R
|
|
\V d_j+\V t)}^2.
|
|
\end{equation}
|
|
$w_{i,j}$ is assigned 1 if the $i$-th point of $M$ describes the
|
|
same point in space as the $j$-th point of $D$. Otherwise
|
|
$w_{i,j}$ is 0. Two things have to be calculated: First, the
|
|
corresponding points, and second, the transformation ($\M R$,
|
|
$\V t$) that minimize $E(\M R, \V t)$ on the base of the
|
|
corresponding points.
|
|
|
|
The ICP algorithm calculates iteratively the point
|
|
correspondences. In each iteration step, the algorithm selects
|
|
the closest points as correspondences and calculates the
|
|
transformation ($\M R, \V t$) for minimizing equation
|
|
(\ref{DMin}). The assumption is that in the last iteration step
|
|
the point correspondences are correct. Besl et al. prove that
|
|
the method terminates in a minimum \cite{Besl_1992}. However,
|
|
this theorem does not hold in our case, since we use a maximum
|
|
tolerable distance $d_\text{max}$ for associating the scan
|
|
data. Such a threshold is required, given that the 3D scans
|
|
overlap only partially. Fig. \ref{samplematch} (top) shows three
|
|
frames, i.e., iteration steps, of the ICP algorithm. The bottom
|
|
part shows the start poses $(x,z,\theta_y)$ from which a correct
|
|
matching is possible, here with only three degrees of freedom.
|
|
|
|
\begin{figure*}
|
|
\begin{center}
|
|
\includegraphics[width=50mm]{frame1_final}~~
|
|
\includegraphics[width=50mm]{frame2_final}~~
|
|
\includegraphics[width=50mm]{frame3_final} \\[1.5ex]
|
|
\caption{{Left: Initial odometry based pose of two 3D
|
|
scans. Middle: Pose after five ICP iterations. Right: final
|
|
alignment, pairwise matching.
|
|
\vspace*{-6mm}
|
|
}}\label{samplematch}
|
|
\end{center}
|
|
\end{figure*}
|
|
|
|
|
|
\subsection{Calculation of the rotation and translation}
|
|
|
|
In every iteration the optimal tranformation ($\M R$, $\V t$)
|
|
has to be computed. Eq. (\ref{DMin}) can be reduced to
|
|
\begin{eqnarray}
|
|
E(\M R, \V t) & \propto & \frac{1}{N} \sum_{i=1}^N
|
|
\norm{\V m_i - (\M R \V d_i + \V t)}^2,\label{DualDMin}
|
|
\end{eqnarray}
|
|
with $N = \sum_{i=1}^{N_m}\sum_{j=1}^{N_d}w_{i,j}$, since the
|
|
correspondence matix can be represented by a vector containing
|
|
the point pairs.
|
|
|
|
Four methods are known to minimize eq. (\ref{DualDMin})
|
|
\cite{Lorusso_1995}. The 6D SLAM system uses the following one,
|
|
based on singular value decomposition (SVD), is robust and easy
|
|
to implement, thus we give a brief overview of the SVD-based
|
|
algorithms. It was first published by Arun, Huang and Blostein
|
|
\cite{Arun_1987}. The difficulty of this minimization problem is
|
|
to enforce the orthonormality of matrix $\M R$. The first step of
|
|
the computation is to decouple the calculation of the rotation
|
|
$\M R$ from the translation $\V t$ using the centroids of the
|
|
points belonging to the matching, i.e.,
|
|
\begin{eqnarray}
|
|
\V c_m = \frac{1}{N} \sum_{i=1}^{N} \V m_{i}, \qquad \qquad \V c_d = \frac{1}{N}
|
|
\sum_{i=1}^{N} \V d_{j}\label{schwerpunkt2}
|
|
\end{eqnarray}
|
|
and
|
|
\begin{eqnarray}
|
|
M' &=& \{ \V m'_{i} = \V m_{i} - \V c_{m} \}_{1,\ldots,N}, \label{d_neu1}\\
|
|
\qquad
|
|
D' &=& \{ \V d'_{i}\ = \V d_{i}\, - \V c_{d} \}_{1,\ldots,N}\label{d_neu2}.
|
|
\end{eqnarray}
|
|
|
|
|
|
After replacing (\ref{schwerpunkt2}), (\ref{d_neu1}) and
|
|
(\ref{d_neu2}) in the error function, $E(\M R,\V t)$
|
|
eq. (\ref{DualDMin}) becomes:
|
|
\begin{subequations}
|
|
\begin{eqnarray}
|
|
E(\M R, \V t)
|
|
\!\!\!\!\!&\propto&\!\!\!\!\! \frac{1}{N} \sum_{i=1}^{N}
|
|
\lvert\lvert{\V m'_{i}-\M R \V d'_i-\underbrace{(\V t-\V c_m+\M R
|
|
\V c_d)}_{= \tilde {\V t}}\lvert\lvert}^2
|
|
\nonumber \\
|
|
&=&\!\!\!\!\! \frac{1}{N}\sum_{i=1}^{N}\norm{\V m'_{i}-\M R \V d'_i}^2
|
|
\label{DMinpart1} \\
|
|
&&- \frac{2}{N} \tilde {\V t} \cdot \sum_{i=1}^{N} \left( \V
|
|
m'_{i}-\M R \V d'_i \right) \label{DMinpart2} \\
|
|
&&+ \frac{1}{N}
|
|
\sum_{i=1}^{N}\norm{\tilde {\V
|
|
t}}^2. \label{DMinpart3}\label{fehlerneu}
|
|
\end{eqnarray}
|
|
\end{subequations}
|
|
In order to minimize the sum above, all terms have to be
|
|
minimized. The second sum (\ref{DMinpart2}) is zero, since all
|
|
values refer to centroid. The third part (\ref{DMinpart3}) has
|
|
its minimum for $\tilde {\V t} = \VNull$ or
|
|
\begin{eqnarray}\label{translatB}
|
|
\V t = \V c_m - \M R \V c_d.
|
|
\end{eqnarray}
|
|
Therefore the algorithm has to minimize only the first
|
|
term, and the error function is expressed in terms of the
|
|
rotation only:
|
|
\begin{eqnarray}\label{DMinn}
|
|
E(\M R, \V t) \propto
|
|
\sum_{i=1}^{N}\norm{\V m'_{i}-\M R \V d'_i}^2.
|
|
\end{eqnarray}
|
|
|
|
\noindent \textit{Theorem:} The optimal rotation is calculated
|
|
by $\M R = \M V \M U^T$. Herby the matrices $\M V$ and $\M U$ are
|
|
derived by the singular value decomposition $\M H = \M U \M
|
|
\Lambda \M V^T$ of a correlation matrix $\M H$. This $3 \times 3$ matrix
|
|
$\M H$ is given by
|
|
\begin{eqnarray}
|
|
\M H = \sum_{i=1}^{N} \V m'^T_i \V d'_i
|
|
= \left(
|
|
\begin{array}{ccc}
|
|
S_{xx} & S_{xy} & S_{xz} \\
|
|
S_{yx} & S_{yy} & S_{yz} \\
|
|
S_{zx} & S_{zy} & S_{zz} \\
|
|
\end{array}
|
|
\right), \label{Korrelationsmatrix}
|
|
\end{eqnarray}
|
|
with $S_{xx} = \sum_{i=1}^{N} \ m'_{ix} d'_{ix}, \ S_{xy} =
|
|
\sum_{i=1}^{N} \ m'_{ix} d'_{iy}, \ \ldots \, $. The analogous
|
|
algorithm is derived directly from this theorem.
|
|
\medskip
|
|
|
|
|
|
\textit{Proof:} Since rotation is length preserving, i.e.,
|
|
$\lvert\lvert\M R\V d'_i\lvert\lvert^2 = \lvert\lvert \V
|
|
d'_i\lvert \lvert^2$ the error function (\ref{DMinn}) is expanded
|
|
\begin{eqnarray*}
|
|
E(\M R, \V t) \propto
|
|
\sum_{i=1}^{N}\norm{\V m'_i}^2
|
|
- 2 \sum_{i=1}^{N} \V m'_{i} \cdot \M R
|
|
\V d'_i
|
|
+ \sum_{i=1}^{N} \norm{\V d'_i}^2.
|
|
\end{eqnarray*}
|
|
The rotation affects only the middle term, thus it is sufficient
|
|
to maximize
|
|
\begin{eqnarray}\label{max}
|
|
\sum_{i=1}^{N} \V m'_{i} \cdot \M R \V
|
|
d'_i
|
|
& = &
|
|
\sum_{i=1}^{N} \V {m'_{i}}^T \M R \V
|
|
d'_i.
|
|
\end{eqnarray}
|
|
Using the trace of a matrix, (\ref{max}) can be rewritten to
|
|
obtain
|
|
\begin{eqnarray*}
|
|
\trace{\sum_{i=1}^{N} \M R \V d'_i \V {m'_{i}}^T } =
|
|
\trace{\M R \M H},
|
|
\end{eqnarray*}
|
|
With $\M H$ defined as in (\ref{Korrelationsmatrix}). Now we have
|
|
to find the matrix $\M R$ that maximizes $\trace{\M R \M H}$.
|
|
|
|
Assume that the singular value decomposition of $\M H$ is
|
|
\begin{eqnarray*}
|
|
\M H = \M U \M \Lambda \M V^T,
|
|
\end{eqnarray*}
|
|
with $\M U$ and $\M V$ orthonormal $3 \times 3$ matrices and $\M
|
|
\Lambda$ a $3 \times 3$ diagonal matrix without negative
|
|
elements. Suppose
|
|
\begin{eqnarray*}
|
|
\M R = \M V \M U^T.
|
|
\end{eqnarray*}
|
|
$\M R$ is orthonormal and
|
|
\begin{eqnarray*}
|
|
\M R \M H & = & \M V \M U^T \M U \M \Lambda \M V^T \\
|
|
& = & \M V \M \Lambda \M V^T
|
|
\end{eqnarray*}
|
|
is a symmetric, positive definite matrix. Arun, Huang and
|
|
Blostein provide a lemma to show that
|
|
\begin{eqnarray*}
|
|
\trace{\M R \M H} \geq \trace{\M B \M R \M H}
|
|
\end{eqnarray*}
|
|
for any orthonormal matrix $\M B$. Therefore the matrix $\M R$ is
|
|
optimal. Prooving the lemma is straightforward using the
|
|
Cauchy-Schwarz \cite{Arun_1987}. Finally, the
|
|
optimal translation is calculated as (cf. eq. (\ref{DMinpart3})
|
|
and (\ref{translatB}))
|
|
\begin{eqnarray*}
|
|
\V t = \V c_m - \M R \V c_d.
|
|
\end{eqnarray*}
|
|
|
|
|
|
\section{ICP-based 6D SLAM}
|
|
|
|
To match two 3D scans with the ICP algorithm it is necessary to
|
|
have a sufficient starting guess for the second scan pose.
|
|
|
|
\begin{itemize}
|
|
\item
|
|
Extrapolate the odometry readings to all six degrees of freedom
|
|
using previous registration matrices. The change of the robot
|
|
pose $\Delta \M P$ given the odometry information
|
|
$(x_n,z_n,\theta_{y,n})$, $(x_{n+1},z_{n+1},\theta_{y,n+1})$
|
|
and the registration matrix $\M R({\theta_{x,{n}}},
|
|
{\theta_{y,{n}}}, {\theta_{z,{n}}})$ is calculated by solving:
|
|
|
|
\begin{small}
|
|
\begin{eqnarray}
|
|
\left(
|
|
\begin{array}{c}
|
|
x_{n+1} \\
|
|
y_{n+1} \\
|
|
z_{n+1} \\
|
|
{\theta_{x,{n+1}}} \\
|
|
{\theta_{y,{n+1}}} \\
|
|
{\theta_{z,{n+1}}} \\
|
|
\end{array}
|
|
\right)
|
|
=
|
|
\left(
|
|
\begin{array}{c}
|
|
x_{n} \\
|
|
y_{n} \\
|
|
z_{n} \\
|
|
{\theta_{x,{n}}} \\
|
|
{\theta_{y,{n}}} \\
|
|
{\theta_{z,{n}}} \\
|
|
\end{array}
|
|
\right)
|
|
+
|
|
\left(
|
|
\begin{array}{ccc|ccc}
|
|
& & & & & \\
|
|
& \M R({\theta_{x,{n}}},{\theta_{y,{n}}}, {\theta_{z,{n}}}) & & & \M 0 &
|
|
\\
|
|
& & & & & \\
|
|
\hline
|
|
& & & 1 & 0 & 0 \\
|
|
& \M 0 & & 0 & 1 & 0 \\
|
|
& & & 0 & 0 & 1 \\
|
|
\end{array}
|
|
\right)
|
|
\cdot
|
|
\underbrace{\left(
|
|
\begin{array}{c}
|
|
\Delta x_{n+1} \\
|
|
\Delta y_{n+1} \\
|
|
\Delta z_{n+1} \\
|
|
\Delta {\theta_{x,{n+1}}} \\
|
|
\Delta {\theta_{y,{n+1}}} \\
|
|
\Delta {\theta_{z,{n+1}}} \\
|
|
\end{array}
|
|
\right).}_{\Delta \V P} \label{PosUpdate6D}
|
|
\label{extrapol}
|
|
\end{eqnarray}
|
|
\end{small}
|
|
|
|
Therefore, calculating $\Delta \V P$ requires a matrix
|
|
inversion. Finally, the 6D pose $\M P_{n+1}$ is calculated by
|
|
\vspace*{-2mm}
|
|
\begin{small}
|
|
\begin{eqnarray*}\label{inital6DPose}
|
|
\M P_{n+1} = \Delta \M P \cdot \M P_{n}
|
|
\end{eqnarray*}
|
|
\end{small}
|
|
using the poses' matrix representations.\\[-1.5ex]
|
|
|
|
|
|
\end{itemize}
|
|
|
|
\section{Variable Correspondences}
|
|
|
|
\begin{tabular}{lll}
|
|
$(\M R, \V t)$ & \texttt{double alignxf[16]} & Transformation
|
|
Matrix \\
|
|
$\V m_i$, $\V d_i$ & \texttt{class PtPair} & Point Pair\\
|
|
$\V c_m$, $\V c_d$ & \texttt{double cm[3], cd[3]} & Centroids\\
|
|
$\V m'_i$, $\V d'_i$ & \texttt{double** m, d} & Centered
|
|
Point Pairs\\
|
|
$\M H$, $\M U$, $\M \Lambda$, $\M V$ & \texttt{Matrix} & SVD
|
|
Matrices\\
|
|
$\M R$ & \texttt{double transMat[16]} & Pose as Matrix\\
|
|
$(x_n, y_n, z_n)$ & \texttt{double rPos[3]} & Position of $n$-th 3D Scan\\
|
|
$(\theta_{x,n}, \theta_{y,n}, \theta_{z,n})$ & \texttt{double
|
|
rPostheta[3]} & Rotation of $n$-th 3D Scan\\
|
|
\end{tabular}
|
|
|
|
\section{File Formats and Units}
|
|
|
|
\begin{figure}
|
|
\begin{center}
|
|
\includegraphics[width=3.2cm]{coordinate_system_white}
|
|
\caption{Left handed coordinate system.}\label{coord}
|
|
\end{center}
|
|
\end{figure}
|
|
The coordinate system is left handed, with the $y$ axis pointing
|
|
upwards, and the depth axis $z$ (cf. Figure~\ref{coord}). Input
|
|
and output files are:
|
|
|
|
\footnotetext[1]{stating
|
|
with \texttt{XXX} = 000, 001,\dots, until no more files are
|
|
found in the specified directory.}
|
|
|
|
\begin{enumerate}
|
|
\item The 3D scan files (\texttt{scanXXX.3d})\footnotemark[1]\
|
|
have to be of the following structure:\\ The first line
|
|
contains the scan's resolution (w x b), followed by lines of
|
|
data points (x, y, z).
|
|
%
|
|
\item The pose files (\texttt{scanXXX.pose})\footnotemark[1]\ associated with each
|
|
3D scan contain information of the estimated pose of the
|
|
respective scan as given by, e.g., odometry. The first line
|
|
contains the 3 translatorial positions ($x$, $y$, $z$), the
|
|
second the rotations pitch, yaw and roll ($\theta_x$,
|
|
$\theta_y$, $\theta_z$ around the respective axis) in deg.\\
|
|
Values that are not estimated by the robot (odometry) can be
|
|
set to 0 and are extrapolated as described by Eq.~\eqref{extrapol}.
|
|
%
|
|
\item The SLAM program generated files \texttt{scanXXX.frames},
|
|
consisting of the transformations computed from the scan
|
|
matching. Each line contains a $4 \times 4$ OpenGL-style
|
|
matrix. The very last matrix is the final transformation for
|
|
registering the scan into the common coordinate system.
|
|
|
|
The matrix is stored in the following format:
|
|
\begin{quote}
|
|
$(R[1,1], R[1,2], 0, R[1,3], 0, R[2,1], R[2,2], R[2,3], 0,
|
|
R[3,1], R[3,2], R[3,3], 0,$\\$t[1], t[2], t[3], 1),$
|
|
\end{quote}
|
|
with $R[x,y]$ the $(x,y)$-th entry of the rotation matrix $\M R$, and
|
|
$t[x]$ the $x$-th translation component of $\V t$.
|
|
\end{enumerate}
|
|
|
|
\section{Requirements}
|
|
|
|
All executables can be compiled and used both with Linux and Windows.
|
|
|
|
\begin{description}
|
|
\item[Linux:] The system was developed and tested under Linux 9.1 with
|
|
the g++ compiler version 3.3.3. As additional library,
|
|
\texttt{OpenGL} and \texttt{glut} have to be installed, which
|
|
should be included in your Linux distribution (tested with
|
|
freeglut 2.2.0-78).\\
|
|
To compile, type in \texttt{make} in the main directory. The
|
|
executables are generated in the \texttt{./bin} directory.
|
|
\item[Windows:]
|
|
The system was tested with the C++ compiler from Microsoft Visual
|
|
Studio.NET 2005.
|
|
To compile, load the respective project file (\texttt{.sln})
|
|
from the directory \texttt{.$\backslash$Visual\_Studio\_Projects$\backslash$}.
|
|
The executables are generated in the respective \texttt{Debug}
|
|
or \texttt{Release} directories, depending on your compiler settings.\\
|
|
Precompiled versions can be found in the \texttt{./bin}
|
|
directory, too. If moving the executables, take care about the \texttt{glut}
|
|
directory as well.
|
|
\end{description}
|
|
|
|
\section{Usage}
|
|
|
|
For a detailed explanation about the programs' usage, just start
|
|
the respective binary. Both applcations can be configured by a
|
|
set of command line parameters, which are explained when starting
|
|
the program as mentioned above.
|
|
|
|
Especially, take care of the reduction parameters
|
|
\texttt{-r}/\texttt{-R} of the SLAM system: Without using one of
|
|
those, the registration is being slowed down tremendously due to
|
|
taking \emph{all} data points as input. Other potentially
|
|
critical parameters are the maximal distance of points that may
|
|
form corresponding point pairs (matrix entries $w_{ij}$,
|
|
parameter \texttt{-d}), as well as the maximal range distance of
|
|
points used for scan matching or displaying (\texttt{-m}),
|
|
especially used for eliminating outliers (i.e., data points with
|
|
the maximal range distance of the range finder).
|
|
|
|
\bibliographystyle{plain}
|
|
\bibliography{diss,paper,diplom}
|
|
|
|
\end{document} |