The Geese shown above in effect form a 1dimensional "string formation". The picture in the middle
is a caricature of a 2dimensional artificial vehicle formation. Schools of fish appear to be 3dimensional
formations. If you've seen videos of such formations on something like the Discovery Channel, you probably
have observed that fish schools appear to swim in much tighter formations than the somewhat meandering string
formations of Geese. In other words, fish schools are more "coherent" than Geese formations, and it turns out
that this is a fundamental pattern. 3dimensional formations are more coherent than 1dimensional ones when
agents can only use local feedback to control their relative motion.
It is also intuitively clear that the larger
the formation is, the less likely it is to be coherent.
These observations raise several
questions:
 How does one quantify "coherence", i.e. how closely does a formation resemble a rigid lattice?
We refer to these as macroscopic performance measures.
 How does one capture the local behavior of vehicles, such as the tendency to run into each other?
We refer to these as microscopic performance measures.
 How do these measures scale with formation size, and how do they depend on the network topology
and underlying spatial dimension?
 How does all this depend on whether vehicles have local feedback versus global feedback of things
like position errors and velocity errors?
The answers to these questions are provided in the paper below. It turns out that
a common
phenomenon appears where a higher spatial dimension implies a more favorable
scaling of coherence measures, with a dimensions of 3 being necessary to achieve
coherence in consensus and vehicular formations under certain conditions.
