# Hierarchical Task Assignment and Path Finding with Limited Communication for Robot Swarms
## 4 Experimental Evaluation
### Results
A more realistic setting is when communication is limited in range, due to limited emission power and imperfect receiver sensitivity (We use the following parameters for the communication model for Equations (1) and (2): \(P_{0}=-20\), \(\eta=5.6\), \(W=10\), \(\sigma=3.1\), \(FER_{S}=0.08\), \(\gamma=1\), \(N_{b}=-100\), and \(N_{th}=-100\)). We study the influence of limited communication by using three different sensitivities, \(S\in\{-120,-100,-80\}\) which substantially limit the range at which broadcast messages can be received (see Figure 4 for some examples). Note that both robots and tasks rely on the same communication model, meaning that tasks are known to a robot only when within range. Figure 6 shows the results for all environments and all approaches considered, while Table 2 reports the results of the statistical tests. Additionally, we also considered failures in robots happening at different rates (0.05 and 0.1 expected failures per time-step, corresponding to an average of 15 and 30 failing robots per simulation, respectively), which turn robots into static obstacles that need to be avoided, therefore influencing the overall task execution abilities of the system (see Figure 7 and Table 3 for the results of the statistical tests). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & & split16 & & & empty32 & \\ \hline \multirow{3}{*}{\(p\)-value} & **greedy** & **CNP** & **HTAPF** & **greedy** & **CNP** & **HTAPF** \\ \cline{2-8}  & median & 150 & 172 & 189.5 & 239 & 287 & 283.5 \\ \cline{2-8}  & Kruskal–Wallis & \(H=47.7\), \(p=1.1\)\(\times 10^{-4}\) & & \(H=43.8\), \(p=7.0\times 10^{-4}\) \\ \hline \multirow{2}{*}{\(p\)-value} & greedy & – & \(9.1\times 10^{-4}\) & \(6.7\times 10^{-8}\) & – & \(5.8\times 10^{-9}\) & \(2.1\times 10^{-4}\) \\  & CNP & – & – & \(3.6\times 10^{-2}\) & – & – & \(3.6\times 10^{-2}\) \\ \hline \multicolumn{8}{c}{facility} & \multicolumn{4}{c}{lab} \\ \hline  & **greedy** & **CNP** & **HTAPF** & **greedy** & **CNP** & **HTAPF** \\ \hline \multirow{3}{*}{\(p\)-value} & median & 204 & 211 & 208 & 225 & 262 & 275 \\  & Kruskal–Wallis & & \(H=4.7\), \(p=0.31\) & & \(H=62.0\), \(p=1.1\times 10^{-12}\) \\ \hline \multirow{2}{*}{\(p\)-value} & greedy & – & \(\times\) & \(\times\) & – & \(4.3\times 10^{-5}\) & \(1.4\times 10^{-12}\) \\  & CNP & – & – & \(\times\) & – & – & \(2.6\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results of the statistical tests on the simulations performed with unlimited communication and no robot failures. Figure 6: Number of accomplished tasks by the different approaches (greedy, CNP, and HTAPF) across the different environments, obtained with limited communication and no robot failures. Generally speaking, the value of HTAPF is confirmed practically everywhere. In the split16 environment, HTAPF proves robust against communication limitations, and always performs significantly better than both greedy and CNP, apart when a sensitivity of \(S=-120\) is considered where no significant difference was found between HTAPF and CNP (see Figure 6, top-left panel, and Table 2). The small dimensions and the limited number of walls entail that limitations in communication are not too evident, substantially confirming the baselines results. In addition, in case of failures, HTAPF presents a robust behavior, only marginally affected by the unavailable robots, but still outperforming the competitor approaches (Figure 7 top-left and Table 3). In the empty32 environment, limitations in communication have practically no impact, and we observe again that CNP and HTAPF are performing on par and significantly better