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In the following, we will review with some details two of the most important
models based on self-propelled particles and briey mention some of their general-
izations. Starting from the consideration that the underlying interactions between
animals/particles are time-dependent and can be described by a time-variant net-
work, we present some recent results and we focus on possible applications in au-
tonomous robotics. We will not discuss another aspect of collective motion, i.e. the
behavior of human crowds, even if a lot of interesting works have been developed in
this context, e.g. the simulation of pedestrians escaping in a panic situation [Hel-
bing and Molnar (1995); Helbing et al. (2000)] or the study of movement pattern
through the analysis of data coming from the mobile telephone network [Gonzalez
et al. (2008)]. This is beyond the scope of this Chapter.
The eld of collective motion models is in rapid evolution. In fact, only recently,
theoretical considerations are beginning to be accompanied by experimental ndings
which were not possible in the past (for instance tracking individuals in a ock or in a
school through extremely fast image processing algorithms). Technological advances
are on one hand validating many aspects of these models, and on the other hand
adding new valuable information which rene and add new hypotheses. In this con-
text, the data acquisition phase is of crucial importance, and the related techniques
still reveal aws, leading to controversial conclusion, as pointed out in [Buchanan
(2008)]. To mention some examples, advanced image processing techniques have
been recently developed for the characterization of schools of Atlantic bluen tuna
[Newlands and Porcelli (2008)], the analysis of ocks of starlings [Ballerini et al.
(2008)], and the evaluation of the trajectories of sh populations of giant danios
(danio aequipinnatus) in a laboratory tank [S.V. Viscido and Grunbaum (2004);
Grunbaum et al. (2005)].
17.2. Mathematical Models for Collective Motion
In this section we review two important models able to generate self-organized mo-
tion of autonomous agents. The rst one, introduced by Vicsek and co-workers [Vic-
sek et al. (1995)], is considered by many as an ancestor of a class of collective motion
models. This model has inspired many scientists for modications and generaliza-
tions, as regards for example the second model which we will present, introduced
by Couzin and co-workers [Couzin et al. (2002, 2005)], which is more oriented to
modeling the behavior of animal groups on the move.
The Vicsek's model is a manifest example of complexity: a complex behavior
can emerge from a population in which each element's life is governed by simple
rules.
In particular, as Vicsek states in [Vicsek et al. (1995)], the only rule of the
model is that at each time step a given particle driven with a constant absolute
velocity assumes the average direction of motion of the particles in its neighborhood
of radius r with some random perturbation added. The model is discrete in time
