In the spreading algorithm explained here, the basic assumption is that all the
robots are within the region at first and the total formation expands from inside the
circular region. Hence, a controlling force at the boundary of the region is required to
prevent the robots from moving outside the fixed region. When the region is a circle, it is
easily possible, again due to its symmetry, to have a threshold for IF, which can act as a
force at the boundary. The threshold, thus, eliminates the need to have any extrinsic force
applied at the boundary, an unrealistic assumption in some applications.
3.2.1 Threshold Estimation
As the robots are spreading themselves inside a circular region, the threshold is the
total IF experienced by a robot from all other robots, when it reaches the boundary.
Hence, it becomes necessary to know, given a particular number of robots, their most
uniform distribution over the circle. Spreading a particular number of points over a disk
is a complex optimization problem and is difficult to solve.
3.2.1.1 Sphere packing problem
It was thought that the problem at hand might be equivalent to the famous sphere
packing problem. In sphere packing, a particular number of small circles are distributed
uniformly over a bigger circle with the goal of maximizing the radius of the small circles
[Spe02]. This enables us to consider the center of the small circles as the positions of the
robots. Further investigation revealed that this solution does not give the most uniform
distribution of robots possible. This can be explained better with figure 3.1. For a
particular number of circles (N = 30), the centers have been distributed uniformly. But in
other cases (N = 31 and 32), it is clear that the centers of the circles are not uniformly
distributed.