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