# Effect of Formation Size on Flocking Formation Performance for the Goal Reach Problem
## 3 Results
### Experimental Results
#### 3.2.1 Experimental Results in an Environment without Obstacles  
First, we simulated the flocking behavior of 10-100 agents in steps of 10, in an environment without obstacles. In all experiments, the flock completed the flocking task successfully by reaching the goal zone area. Recorded simulation outputs (performance metrics) for each flock size, averaged over twenty runs of each experiment, are shown in Table 3. The relationship is also illustrated in Figure 5. Each sub-figure shows the data results over all experiments for the number of agents (at the \(x\)-axis) versus the respective performance metric (at the \(y\)-axis) as points distributed on a plane. Moreover, the best-fit curves for each of the nine flocking performance measures against the flock size, using six functions (linear, inverse, logarithmic, quadratic, cubic, and exponential) are shown. The detailed model summary and parameter estimates for the curve fitting obtained are found in Appendix B.  
Localization time: Table 3 shows that the average goal zone localization time does not exhibit any monotonic relation along with an increase in the flock size. The distribution of data points in the scatter plot shown in Figure 5a supports this observation. Figure 5a also shows curve estimates for each of the six functions (linear, inverse, logarithmic, quadratic, cubic, and exponential) relating the flock size to the goal localization time. The \(p\)-value obtained in each of these models was larger than \(\alpha=0.05\) (Appendix B); thus, the null hypothesis cannot be rejected, meaning that there exists no association between the changes in flock size and shifts in goal localization time. An exception is the model associated with the exponential function, in which the \(p\)-value detected was \(0.011<\alpha\); however, the \begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Flock Size**} \\ \hline
**Performance Metric** & **10** & **20** & **30** & **40** & **50** \\ \hline Localization Time & 673.20 & 594.96 & 673.19 & 564.29 & 661.26 \\ Distance to center of mass & 6.25 & 5.63 & 5.17 & 5.03 & 4.96 \\ Collisions & 0.00 & 0.00 & 0.01 & 0.01 & 0.03 \\ Separations & 1.85 & 6.98 & 12.58 & 18.46 & 34.80 \\ Alignments & 706.40 & 616.23 & 690.57 & 580.90 & 668.76 \\ Cohesions & 25.22 & 32.85 & 34.41 & 35.83 & 42.83 \\ Task completion time & 0.30 & 0.51 & 0.88 & 1.06 & 1.65 \\ Speed matching time & 82.30 & 68.6 & 33.00 & 54.85 & 34.50 \\ Localization window & 66.45 & 58.65 & 58.65 & 60.60 & 66.75 \\ \hline \hline \multicolumn{5}{c}{**Flock Size**} \\ \hline
**Performance Metric** & **60** & **70** & **80** & **90** & **100** \\ \hline Localization Time & 627.81 & 615.74 & 688.05 & 654.63 & 800.21 \\ Distance to center of mass & 4.94 & 4.93 & 4.92 & 4.95 & 4.98 \\ Collisions & 0.07 & 0.10 & 0.12 & 0.40 & 0.68 \\ Separations & 53.55 & 77.70 & 122.61 & 175.68 & 293.34 \\ Alignments & 606.85 & 578.00 & 609.39 & 530.52 & 571.07 \\ Cohesions & 46.33 & 53.06 & 63.70 & 80.56 & 112.82 \\ Task completion time & 1.81 & 2.16 & 2.91 & 3.56 & 4.38 \\ Speed matching time & 30.75 & 38.25 & 43.80 & 39.90 & 33.00 \\ Localization window & 66.00 & 78.30 & 89.45 & 104.15 & 130.35 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance metric results averaged over twenty simulations for flock sizes in 10–100, in an environment without obstacles. associated \(R^{2}\) value indicates that the only 3.2% of the variation in the localization time is explained by the flock size. Distance to center of mass: The average distance to center of mass values tend to decrease along with the increase in the flock size (Table 3 and Figure 5b). The flocking behavior is exhibited here, where a larger flock size enables more members to be contained in a flocking zone, allowing the average distance to the center of mass to be reduced. Figure 5b also shows curve estimates for each of the six functions relating the flock size to the distance to center of mass.