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\hypersetup{
  pdfauthor = {Carlo Pinciroli},
  pdftitle = {Self-Organizing Grid Formation for Mobile Robots with Minimal Assumptions},
  pdfsubject = {},
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\title{%
  \LARGE\bf Self-Organizing Grid Formation for Mobile Robots\\with Minimal Assumptions
  \thanks{C.~Pinciroli and M.~Dorigo are with IRIDIA, CoDE,
    Universit\'e  Libre de Bruxelles, Brussels, Belgium. C.~Ampatzis
    is with the European Space Agency, Noordwijk, The
    Netherlands. L.~Isella is with the Joint Research Centre, Ispra, Italy.}}
\author{Carlo Pinciroli, Christos Ampatzis, Lorenzo Isella, Marco Dorigo}

\maketitle

\begin{abstract}
  
\end{abstract}

\section{Introduction}
\label{sec:introduction}
When dealing with a multiagent system, one the first and most
important problems is its deployment. In many applications, the agents
need to form a particular lattice to perform their work. In sensors
networks, for example, a specific topology of connections among the
agents is often desired to ensure optimal sensing and communication
coverage~\cite{Brown07}. In swarm robotics, lattice formation
techniques offer useful tools to achieve effective autonomous
self-assembly~\cite{Christensen2007} and coordinated
motion~\cite{Turgut2008}.

In this paper, we focus on a scalable system to place robots in a grid
lattice.

\section{Grid Formation with Mobile Robots}
\label{sec:gridformation}
\begin{itemize}
  \item Some words about related works in forming grids.
  \item Why this work is better: it is based just on simple positional
    information, available on nearly any robot (proximity sensors or
    camera); and on 1-bit messages exchanged by the robots to inform
    the neighbourhood about their current color.
\end{itemize}

\section{Color-Based Grid Formation}
\label{sec:method}
The method proposed in this paper is inspired by salt crystal structure.
 The salt unit cell  exhibits a cubic symmetry, where  each
 positive (negative)
ion, Na$^+$ (Cl$^-$), is surrounded by six negative (positive)
ions.
The NaCl crystal is held together by the electrostatic interaction
between the opposite-charged ions. The magnitude of the pair-wise force acting between
any two ions is given by Coulomb law 
$$
|F_{\rm ion-to-ion}| \sim r^{-2},
$$
where $r$ is the ion-to-ion distance. In
particular, the force is repulsive for ions having the same charge and
attractive otherwise, hence ions of the same kind are kept as far
from each other as possible in the NaCl unit cell. 


Two main ingredients: colors and color interaction rules.
\begin{itemize}
\item How does it generalise? Is the interaction between the robots
  tunable? What does it aim at? Do we carry out some new simulations
  or do we sell the method by validating it against something already
  known? Is validation against some known results in literature
  possible?
\item Is the interaction between the robots pair-wise, hence the
total ``force'' acting on a robot is the sum of all the interactions
between the given robot and all the other robots in the ensemble?
I think that an equation/formula quantitatively explaining this may be
necessary.
\end{itemize}
\section{Experimental Evaluation}
\label{sec:experiments}
The experiments to assess the quality of the system. Scalability,
precision, convergence.

Do we have experiments here? Performed in your lab and with a
significant number of interaction robots?


\section{Conclusions and Future Work}
\label{sec:conclusions}
Final words about this paper.

\section*{Acknowledgments}
Carlo Pinciroli acknowledges support from the \emph{SWARMANOID}
project funded by the Future and Emerging Technologies programme
(IST-FET) of the European Commission (grant IST-022888). Marco Dorigo
acknowledges support from the F.R.S-FNRS of the French Community of
Belgium of which he is Research Director. Christos Ampatzis
acknowledges$\dots$.

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