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540 lines
18 KiB
TeX
540 lines
18 KiB
TeX
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%\renewcommand{\familydefault}{\sfdefault}
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%%----------------------------------------------------------------
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%%----------------------------------------------------------------
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\begin{document}
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{
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\title{\vspace*{-5mm}
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\bf 6D SLAM -- \\%[.5cm]
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Simultaneous 6 D.O.F. Localization \\and 3D Mapping\\%[1cm]
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}
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\author{
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\scalebox{.65}{\includegraphics{unilogo.eps}}\\[3ex]
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Andreas N\"uchter, Kai Lingemann, Joachim Hertzberg\\
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Department of Mathematics/Computer Science\\
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Institute of Computer Science\\
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Knowledge-Based Systems Research Group\\
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University of Osnabr\"uck\\
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{\small \texttt{http://www.informatik.uni-osnabrueck.de/kbs/}}
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}
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\maketitle
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\thispagestyle{empty}
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\vspace*{-12mm}
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\begin{center}
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\includegraphics[height=50mm]{stylish_scanner}
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\end{center}
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\vspace*{1mm}
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\begin{center}
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\textbf{Documentation}
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\end{center}
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\begin{quote}
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This document describes the algorithms for 6D SLAM --
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Simultaneous 6 D.O.F. Localization and 3D Mapping system. 6D SLAM
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with mobile robots considers six dimensions for the robot pose,
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namely the $x$, $y$ and $z$ coordinates and the roll, yaw and
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pitch angles. Robot motion and localization on natural surfaces,
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e.g., driving with a mobile robot outdoor, must necessarily
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regard these degrees of freedom.
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\end{quote}
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\newpage
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\setcounter{page}{1}
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\section{Range Image Registration and Robot Relocalization}
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Multiple 3D scans are necessary to digitalize environments
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without occlusions. To create a correct and consistent model, the
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scans have to be merged into one coordinate system. This process
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is called registration. If robot carrying the 3D scanner were
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precisely localized, the registration could be done directly
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based on the robot pose. However, due to the unprecise robot
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sensors, self localization is erroneous, so the geometric
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structure of overlapping 3D scans has to be considered for
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registration.
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The following method for registration of point sets is part of
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many publications, so only a short summary is given here. The
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complete algorithm was invented in 1992 and can be found, e.g.,
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in \cite{Besl_1992}. The method is called \emph{Iterative Closest
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Points (ICP) algorithm}.
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Given two independently acquired sets of 3D points, $M$ (model
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set, $|M| = N_m$) and $D$ (data set, $|D| = N_d$) which
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correspond to a single shape, we aim to find the transformation
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consisting of a rotation $\M R$ and a translation $\V t$ which
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minimizes the following cost function:
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\begin{equation}\label{DMin}
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E(\M R, \V t) =
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\sum_{i=1}^{N_m}\sum_{j=1}^{N_d}w_{i,j}\norm{\V m_{i}-(\M R
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\V d_j+\V t)}^2.
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\end{equation}
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$w_{i,j}$ is assigned 1 if the $i$-th point of $M$ describes the
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same point in space as the $j$-th point of $D$. Otherwise
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$w_{i,j}$ is 0. Two things have to be calculated: First, the
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corresponding points, and second, the transformation ($\M R$,
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$\V t$) that minimize $E(\M R, \V t)$ on the base of the
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corresponding points.
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The ICP algorithm calculates iteratively the point
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correspondences. In each iteration step, the algorithm selects
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the closest points as correspondences and calculates the
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transformation ($\M R, \V t$) for minimizing equation
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(\ref{DMin}). The assumption is that in the last iteration step
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the point correspondences are correct. Besl et al. prove that
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the method terminates in a minimum \cite{Besl_1992}. However,
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this theorem does not hold in our case, since we use a maximum
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tolerable distance $d_\text{max}$ for associating the scan
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data. Such a threshold is required, given that the 3D scans
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overlap only partially. Fig. \ref{samplematch} (top) shows three
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frames, i.e., iteration steps, of the ICP algorithm. The bottom
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part shows the start poses $(x,z,\theta_y)$ from which a correct
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matching is possible, here with only three degrees of freedom.
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\begin{figure*}
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\begin{center}
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\includegraphics[width=50mm]{frame1_final}~~
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\includegraphics[width=50mm]{frame2_final}~~
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\includegraphics[width=50mm]{frame3_final} \\[1.5ex]
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\caption{{Left: Initial odometry based pose of two 3D
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scans. Middle: Pose after five ICP iterations. Right: final
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alignment, pairwise matching.
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\vspace*{-6mm}
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}}\label{samplematch}
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\end{center}
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\end{figure*}
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\subsection{Calculation of the rotation and translation}
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In every iteration the optimal tranformation ($\M R$, $\V t$)
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has to be computed. Eq. (\ref{DMin}) can be reduced to
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\begin{eqnarray}
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E(\M R, \V t) & \propto & \frac{1}{N} \sum_{i=1}^N
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\norm{\V m_i - (\M R \V d_i + \V t)}^2,\label{DualDMin}
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\end{eqnarray}
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with $N = \sum_{i=1}^{N_m}\sum_{j=1}^{N_d}w_{i,j}$, since the
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correspondence matix can be represented by a vector containing
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the point pairs.
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Four methods are known to minimize eq. (\ref{DualDMin})
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\cite{Lorusso_1995}. The 6D SLAM system uses the following one,
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based on singular value decomposition (SVD), is robust and easy
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to implement, thus we give a brief overview of the SVD-based
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algorithms. It was first published by Arun, Huang and Blostein
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\cite{Arun_1987}. The difficulty of this minimization problem is
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to enforce the orthonormality of matrix $\M R$. The first step of
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the computation is to decouple the calculation of the rotation
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$\M R$ from the translation $\V t$ using the centroids of the
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points belonging to the matching, i.e.,
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\begin{eqnarray}
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\V c_m = \frac{1}{N} \sum_{i=1}^{N} \V m_{i}, \qquad \qquad \V c_d = \frac{1}{N}
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\sum_{i=1}^{N} \V d_{j}\label{schwerpunkt2}
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\end{eqnarray}
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and
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\begin{eqnarray}
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M' &=& \{ \V m'_{i} = \V m_{i} - \V c_{m} \}_{1,\ldots,N}, \label{d_neu1}\\
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\qquad
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D' &=& \{ \V d'_{i}\ = \V d_{i}\, - \V c_{d} \}_{1,\ldots,N}\label{d_neu2}.
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\end{eqnarray}
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After replacing (\ref{schwerpunkt2}), (\ref{d_neu1}) and
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(\ref{d_neu2}) in the error function, $E(\M R,\V t)$
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eq. (\ref{DualDMin}) becomes:
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\begin{subequations}
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\begin{eqnarray}
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E(\M R, \V t)
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\!\!\!\!\!&\propto&\!\!\!\!\! \frac{1}{N} \sum_{i=1}^{N}
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\lvert\lvert{\V m'_{i}-\M R \V d'_i-\underbrace{(\V t-\V c_m+\M R
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\V c_d)}_{= \tilde {\V t}}\lvert\lvert}^2
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\nonumber \\
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&=&\!\!\!\!\! \frac{1}{N}\sum_{i=1}^{N}\norm{\V m'_{i}-\M R \V d'_i}^2
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\label{DMinpart1} \\
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&&- \frac{2}{N} \tilde {\V t} \cdot \sum_{i=1}^{N} \left( \V
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m'_{i}-\M R \V d'_i \right) \label{DMinpart2} \\
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&&+ \frac{1}{N}
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\sum_{i=1}^{N}\norm{\tilde {\V
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t}}^2. \label{DMinpart3}\label{fehlerneu}
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\end{eqnarray}
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\end{subequations}
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In order to minimize the sum above, all terms have to be
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minimized. The second sum (\ref{DMinpart2}) is zero, since all
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values refer to centroid. The third part (\ref{DMinpart3}) has
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its minimum for $\tilde {\V t} = \VNull$ or
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\begin{eqnarray}\label{translatB}
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\V t = \V c_m - \M R \V c_d.
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\end{eqnarray}
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Therefore the algorithm has to minimize only the first
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term, and the error function is expressed in terms of the
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rotation only:
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\begin{eqnarray}\label{DMinn}
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E(\M R, \V t) \propto
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\sum_{i=1}^{N}\norm{\V m'_{i}-\M R \V d'_i}^2.
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\end{eqnarray}
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\noindent \textit{Theorem:} The optimal rotation is calculated
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by $\M R = \M V \M U^T$. Herby the matrices $\M V$ and $\M U$ are
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derived by the singular value decomposition $\M H = \M U \M
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\Lambda \M V^T$ of a correlation matrix $\M H$. This $3 \times 3$ matrix
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$\M H$ is given by
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\begin{eqnarray}
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\M H = \sum_{i=1}^{N} \V m'^T_i \V d'_i
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= \left(
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\begin{array}{ccc}
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S_{xx} & S_{xy} & S_{xz} \\
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S_{yx} & S_{yy} & S_{yz} \\
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S_{zx} & S_{zy} & S_{zz} \\
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\end{array}
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\right), \label{Korrelationsmatrix}
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\end{eqnarray}
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with $S_{xx} = \sum_{i=1}^{N} \ m'_{ix} d'_{ix}, \ S_{xy} =
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\sum_{i=1}^{N} \ m'_{ix} d'_{iy}, \ \ldots \, $. The analogous
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algorithm is derived directly from this theorem.
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\medskip
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\textit{Proof:} Since rotation is length preserving, i.e.,
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$\lvert\lvert\M R\V d'_i\lvert\lvert^2 = \lvert\lvert \V
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d'_i\lvert \lvert^2$ the error function (\ref{DMinn}) is expanded
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\begin{eqnarray*}
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E(\M R, \V t) \propto
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\sum_{i=1}^{N}\norm{\V m'_i}^2
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- 2 \sum_{i=1}^{N} \V m'_{i} \cdot \M R
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\V d'_i
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+ \sum_{i=1}^{N} \norm{\V d'_i}^2.
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\end{eqnarray*}
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The rotation affects only the middle term, thus it is sufficient
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to maximize
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\begin{eqnarray}\label{max}
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\sum_{i=1}^{N} \V m'_{i} \cdot \M R \V
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d'_i
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& = &
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\sum_{i=1}^{N} \V {m'_{i}}^T \M R \V
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d'_i.
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\end{eqnarray}
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Using the trace of a matrix, (\ref{max}) can be rewritten to
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obtain
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\begin{eqnarray*}
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\trace{\sum_{i=1}^{N} \M R \V d'_i \V {m'_{i}}^T } =
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\trace{\M R \M H},
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\end{eqnarray*}
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With $\M H$ defined as in (\ref{Korrelationsmatrix}). Now we have
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to find the matrix $\M R$ that maximizes $\trace{\M R \M H}$.
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Assume that the singular value decomposition of $\M H$ is
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\begin{eqnarray*}
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\M H = \M U \M \Lambda \M V^T,
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\end{eqnarray*}
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with $\M U$ and $\M V$ orthonormal $3 \times 3$ matrices and $\M
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\Lambda$ a $3 \times 3$ diagonal matrix without negative
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elements. Suppose
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\begin{eqnarray*}
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\M R = \M V \M U^T.
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\end{eqnarray*}
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$\M R$ is orthonormal and
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\begin{eqnarray*}
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\M R \M H & = & \M V \M U^T \M U \M \Lambda \M V^T \\
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& = & \M V \M \Lambda \M V^T
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\end{eqnarray*}
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is a symmetric, positive definite matrix. Arun, Huang and
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Blostein provide a lemma to show that
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\begin{eqnarray*}
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\trace{\M R \M H} \geq \trace{\M B \M R \M H}
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\end{eqnarray*}
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for any orthonormal matrix $\M B$. Therefore the matrix $\M R$ is
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optimal. Prooving the lemma is straightforward using the
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Cauchy-Schwarz \cite{Arun_1987}. Finally, the
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optimal translation is calculated as (cf. eq. (\ref{DMinpart3})
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and (\ref{translatB}))
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\begin{eqnarray*}
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\V t = \V c_m - \M R \V c_d.
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\end{eqnarray*}
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\section{ICP-based 6D SLAM}
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To match two 3D scans with the ICP algorithm it is necessary to
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have a sufficient starting guess for the second scan pose.
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\begin{itemize}
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\item
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Extrapolate the odometry readings to all six degrees of freedom
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using previous registration matrices. The change of the robot
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pose $\Delta \M P$ given the odometry information
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$(x_n,z_n,\theta_{y,n})$, $(x_{n+1},z_{n+1},\theta_{y,n+1})$
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and the registration matrix $\M R({\theta_{x,{n}}},
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{\theta_{y,{n}}}, {\theta_{z,{n}}})$ is calculated by solving:
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\begin{small}
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\begin{eqnarray}
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\left(
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\begin{array}{c}
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x_{n+1} \\
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y_{n+1} \\
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z_{n+1} \\
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{\theta_{x,{n+1}}} \\
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{\theta_{y,{n+1}}} \\
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{\theta_{z,{n+1}}} \\
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\end{array}
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\right)
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=
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\left(
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\begin{array}{c}
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x_{n} \\
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y_{n} \\
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z_{n} \\
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{\theta_{x,{n}}} \\
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{\theta_{y,{n}}} \\
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{\theta_{z,{n}}} \\
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\end{array}
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\right)
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+
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\left(
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\begin{array}{ccc|ccc}
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& & & & & \\
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& \M R({\theta_{x,{n}}},{\theta_{y,{n}}}, {\theta_{z,{n}}}) & & & \M 0 &
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\\
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& & & & & \\
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\hline
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& & & 1 & 0 & 0 \\
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& \M 0 & & 0 & 1 & 0 \\
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& & & 0 & 0 & 1 \\
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\end{array}
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\right)
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\cdot
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\underbrace{\left(
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\begin{array}{c}
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\Delta x_{n+1} \\
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\Delta y_{n+1} \\
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\Delta z_{n+1} \\
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\Delta {\theta_{x,{n+1}}} \\
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\Delta {\theta_{y,{n+1}}} \\
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\Delta {\theta_{z,{n+1}}} \\
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\end{array}
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\right).}_{\Delta \V P} \label{PosUpdate6D}
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\label{extrapol}
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\end{eqnarray}
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\end{small}
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Therefore, calculating $\Delta \V P$ requires a matrix
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inversion. Finally, the 6D pose $\M P_{n+1}$ is calculated by
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\vspace*{-2mm}
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\begin{small}
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\begin{eqnarray*}\label{inital6DPose}
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\M P_{n+1} = \Delta \M P \cdot \M P_{n}
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\end{eqnarray*}
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\end{small}
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using the poses' matrix representations.\\[-1.5ex]
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|
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\end{itemize}
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\section{Variable Correspondences}
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\begin{tabular}{lll}
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$(\M R, \V t)$ & \texttt{double alignxf[16]} & Transformation
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Matrix \\
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$\V m_i$, $\V d_i$ & \texttt{class PtPair} & Point Pair\\
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$\V c_m$, $\V c_d$ & \texttt{double cm[3], cd[3]} & Centroids\\
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$\V m'_i$, $\V d'_i$ & \texttt{double** m, d} & Centered
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Point Pairs\\
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$\M H$, $\M U$, $\M \Lambda$, $\M V$ & \texttt{Matrix} & SVD
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|
Matrices\\
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|
$\M R$ & \texttt{double transMat[16]} & Pose as Matrix\\
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|
$(x_n, y_n, z_n)$ & \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}
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|
\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} |