# Effect of Formation Size on Flocking Formation Performance for the Goal Reach Problem
## Appendix B
\begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline \multicolumn{10}{c}{**Localization Time**} \\ \hline
**Equation** & \(\mathbf{R^{2}}\) & \multicolumn{6}{c}{**Model Summary**} & \multicolumn{6}{c}{**Parameter Estimates**} \\ \hline  & **F** & \(\mathbf{df_{1}}\) & \(\mathbf{df_{2}}\) & **Sig** & **Constant** & \(\mathbf{b_{1}}\) & \(\mathbf{b_{2}}\) & \(\mathbf{b_{3}}\) \\ \hline Linear & 0.074 & 15.81 & 1 & 198 & \(<\)0.001 & 567.876 & 4.90 & & \\ Logarithmic & 0.025 & 5.126 & 1 & 198 & \(0.025\) & 386.60 & 118.218 & & \\ Inverse & 0.003 & 0.526 & 1 & 198 & \(0.469\) & 867.027 & \(-1012.61\) & & \\ Quadratic & 0.217 & 27.329 & 2 & 197 & \(<\)0.001 & 1160.904 & \(-24.752\) & 0.270 & \\ Cubic & 0.224 & 18.87 & 3 & 196 & \(<\)0.001 & 950.885 & \(-6.124\) & \(-0.134\) & 0.002 \\ Exponential & 0.104 & 23.030 & 1 & 198 & \(<\)0.001 & 529.631 & 0.006 & & \\ \hline \hline \multicolumn{10}{c}{**Distance to Center of Mass**} \\ \hline
**Equation** & \(\mathbf{R^{2}}\) & \multicolumn{6}{c}{**Model Summary**} & \multicolumn{6}{c}{**Parameter Estimates**} \\ \hline  & **F** & \(\mathbf{df_{1}}\) & \(\mathbf{df_{2}}\) & **Sig** & **Constant** & \(\mathbf{b_{1}}\) & \(\mathbf{b_{2}}\) & \(\mathbf{b_{3}}\) \\ \hline Linear & 0.305 & 86.780 & 1 & 198 & \(<\)0.001 & 5.647 & \(-0.009\) & & \\ Logarithmic & 0.476 & 179.599 & 1 & 198 & \(<\)0.001 & 6.967 & \(-0.481\) & & \\ Inverse & 0.578 & 271.302 & 1 & 198 & \(<\)0.001 & 4.724 & 14.011 & & \\ Quadratic & 0.529 & 110.645 & 2 & 197 & \(<\)0.001 & 6.342 & -0.044 & 0.000 & \\ Cubic & 0.587 & 92.824 & 3 & 196 & \(<\)0.001 & 6.911 & \(-0.095\) & 0.001 & \(-6.64\times 10^{6}\) \\ Exponential & 0.332 & 98.279 & 1 & 198 & \(<\)0.001 & 5.605 & \(-0.002\) & & \\ \hline \multicolumn{10}{c}{**Collisions per Member**} \\ \hline
**Equation** & \(\mathbf{R^{2}}\) & \multicolumn{6}{c}{**Model Summary**} & \multicolumn{6}{c}{**Parameter Estimates**} \\ \hline  & **F** & \(\mathbf{df_{1}}\) & \(\mathbf{df_{2}}\) & **Sig** & **Constant** & \(\mathbf{b_{1}}\) & \(\mathbf{b_{2}}\) & \(\mathbf{b_{3}}\) \\ \hline Linear & 0.223 & 56.737 & 1 & 198 & \(<\)0.001 & \(-0.296\) & 0.012 & & \\ Logarithmic & 0.141 & 32.570 & 1 & 198 & \(<\)0.001 & \(-1117\) & 0.384 & & \\ Inverse & 0.071 & 15.029 & 1 & 198 & \(<\)0.001 & 0.555 & \(-7.166\) & & \\ Quadratic & 0.312 & 44.759 & 2 & 197 & \(<\)0.001 & 0.348 & \(-0.021\) & 0.000 & \\ Cubic & 0.324 & 31.293 & 3 & 196 & \(<\)0.001 & \(-0.023\) & 0.012 & 0.000 & \(4.32\times 10^{6}\) \\ Exponential & & & & & & & & \\ \hline \hline \multicolumn{10}{c}{**Separations per Member**} \\ \hline
**Equation** & \(\mathbf{R^{2}}\) & \multicolumn{6}{c}{**Model Summary**} & \multicolumn{6}{c}{**Parameter Estimates**} \\ \hline  & **F** & \(\mathbf{df_{1}}\) & \(\mathbf{df_{2}}\) & **Sig** & **Constant** & \(\mathbf{b_{1}}\) & \(\mathbf{b_{2}}\) & \(\mathbf{b_{3}}\) \\ \hline Linear & 0.536 & 228.356 & 1 & 198 & \(<\)0.001 & \(-144.410\) & 5.184 & & \\ Logarithmic & 0.357 & 109.964 & 1 & 198 & \(<\)0.001 & \(-525.907\) & 174.826 & & \\ Inverse & 0.187 & 45.561 & 1 & 198 & \(<\)0.001 & \(288.693\) & \(-3345.39\) & & \\ Quadratic & 0.700 & 229.466 & 2 & 197 & \(<\)0.001 & 105.097 & \(-7.291\) & 0.113 & \\ Cubic & 0.728 & 174.652 & 3 & 196 & \(<\)0.001 & \(-61.395\) & 7.476 & \(-0.207\) & 0.002 \\ Exponential & 0.850 & 1124.172 & 1 & 198 & \(<\)0.001 & 2.364 & 0.055 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Model summary and parameter estimates for the curve fitting obtained in experiments with obstacles for each of the nine flocking performance measures, showing five equations (linear, inverse, logarithmic, quadratic, cubic, and exponential) where the independent variable is the flock size and the dependent variable is each performance measure, respectively.