# Effect of Formation Size on Flocking Formation Performance for the Goal Reach Problem
## 3 Results
### Experimental Results
#### 3.2.2 Experimental Results in an Environment with Obstacles  
Next, experiments simulating the flocking behavior of 10-100 agents in steps of 10, in an environment with obstacles are conducted. In all experiments, the flock was able to complete 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 4. The relationships are also illustrated in Figure 6. Each sub-figure shows the data results over all experiments for the number of agents (on the \(x\)-axis) versus the respective performance metric (on 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. From Table 4 and similar to the observation drawn from results of the experiments without obstacles, no relationship can be easily drawn between the flock size and the goal localization time. The scatter plot in (Figure 6a) shows that the lowest goal localization time was obtained in experiments with 50 agents. Considering the significance of the obtained function curves, interestingly, the \(p\)-value associated with each of the different functions was smaller than \(alpha\) except that for the inverse function. The highest \(R^{2}\) value was that for the cubic model (22.4%), closely followed by that for the quadratic model at 21.7%. The average distance to center of mass tends to decrease along with an increase in flock size as shown in Table 4. The pattern observed is similar to that illustrated by the results in experiments without obstacles. The \(p\)-value associated with all the curve estimation functions in (Figure 6b) supports rejecting the null hypothesis. The \(R^{2}\) value associated with the inverse function was 57.8%, while that associated with the cube function was found to be slightly higher at 58.7%. Furthermore, similar to the observations drawn from the experiments without obstacles, Table 4 displays a monotonic increase in the average value obtained for each of the performance measures separations, cohesion, and task completion time along with the increase in flock size. The table also shows a tendency of increase in the number of collisions and localization window with the increase in flock size. The \(p\)-value obtained for each of the curve estimates in each of (Figure 6c,d,f,g,i) provides strong evidence that the null hypothesis should be rejected. The \(R^{2}\) value associated with the cubic function in (Figure 6c) was 32.4% closely followed by that for the quadratic function (31.2%). Similarly, The \(R^{2}\) value associated with the cubic function and the quadratic function in (Figure 6i) was, equally, 72.8%. The highest \(R^{2}\) value for the models in each of (Figure 6d,f,g) was that associated with the exponential function, having considerable values of 85%, 70.6%, and 78.7%, respectively. In contrast, considering the number of alignment steps, the \(p\)-value obtained for each of the curve estimation values in Figure 6e does not provide sufficient evidence to reject the null hypothesis, except for the cubic and quadratic models. However, the \(R^{2}\) value