N=30 N=31
Figure 3.1 Sphere packing problem N denotes number of small circles
3.2.1.2 Empirical estimation of threshold
Since an analytical expression for threshold is very difficult to find, the threshold
was empirically estimated as follows. The self-organization algorithm was applied on a
particular number of robots, which were initially positioned very close to each other. The
IF between the robots results in an expansion of the formation of robots. At every time
step, the approximate radius of the formation was calculated as the mean of the distance
from the center of the formation to all robots, which form the outer boundary of the
formation. This was continued till the estimated radius is equal to unity. Then the
threshold was calculated as the mean of IFs experienced by all the robots at the boundary.
The number of robots was varied and the process was repeated every time. A general
expression for threshold for any number of robots was then found by fitting an
exponential curve to the threshold data obtained from the above process, as shown in
figure 3.2.
The threshold for any radii can then be obtained just by dividing the above found
threshold for unit radius by the square of the radius. Thus, the general expression for the
threshold, for a particular number of robots N and radius r is
y =0.21128 A- N15107 /r2 (3.1)
N=32