# Temporal Logic Planning and Receding Horizon Control for Signal Source Localization
## 5 Simulation Results  
Algorithm 1 (RHC-TLP) is implemented based on Matlab. The searching environment is 10 \(\times\) 10 m and is divided into 0.25 \(\times\) 0.25 m small areas. The robot detects the signal strength at a time interval of 1 second where the observed noise \(R\) is 5. Moreover, the simulation parameters are shown in Table 1. The initial position and velocity are (\(-4.5\), 4.5 m) and (0, 0 m/s), respectively. In order to show the effectiveness of RHC-TLP, the RHC approach in [9] is used as a comparison algorithm. The main reasons are the following. The first reason is that the other signal source searching algorithms, including the RHC approach, do not deal with the case with complicated task specifications. Because of the no-planning part, the complex task is hard to finish. The other reason is that the RHC approach can deal with uncertain problems in the search process well and can be used as a typical search algorithm. Hence, the RHC approach is utilized as a comparison algorithm.  
In the following, we consider two cases. In Case 1, we put a different number of signal sources in different regions. For Region A, the source number is 1 and the position is (\(-2.5\), 2). For Region B, the source number is 1 and the position is (2, 3). For Region C, the source number is 2 and the corresponding positions are (4, \(-4\)) and (2, \(-2\)). For Region D, the source number is 1 and the position is (\(-2\), \(-4\)). In Case 2, we place the same number of signal sources in different regions. The number of signal sources in each region is 2. For Region A, the positions are (\(-1\), 3) and (\(-2.5\), 2). For Region B, the positions are (4, \(1\)) and (1, 4). For Region C, the positions are (4, \(-2\)) and (2, \(-4\)). For Region D, the positions are (\(-4\), \(-1\)) and (\(-2\), \(-4\)). For the two cases, the green five-pointed star indicates the real signal sources and the red dot indicates the predicted signal sources. The red circle is the starting position, and the red star is the ending position. The black blocks refer to the obstacles.  
From Figure 3, the mobile robot can effectively locate all the signal sources and does not collide with the obstacles. Although there are some corner-like obstacles, the mobile robot can avoid the obstacles. If there is no signal source in a certain area, the mobile robot reaches the default node and then moves to the next area. For Case 2, shown in Figure 4, like Case 1, the mobile robot can effectively locate all the signal sources and does not collide with any obstacles. Table 2 gives the localization time and path length.  
From Tables 2 and 3, it can be seen that, with the increase in the number of signal sources, the localization time and path length obtained by the RHC-TLP approach also increase. In addition, from the standard error, the fluctuation range of the error is relatively small, which indicates that the proposed RHC-TLP approach is relatively stable. Moreover, in contrast to the RHC approach, the proposed RHC-TLP approach can obtain a lesser  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline The control parameters \(\alpha\) and \(\beta\) & 2, 1 \\ The safety distance \(d\) & 0.3 m \\ The detection radius \(r\) & 0.5 m \\ The prediction horizon \(N\) & 6 \\ The RSS model parameters \(c\), \(\alpha\), \(d_{a}\), \(\varphi\) and \(\lambda\) & 0.001, 1.96, 0.2, 40\({}^{\circ}\), 2 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The simulation parameters.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Algorithms** & **Localization Time** & **Path Length** \\ \hline RHC-TLP & 117.74 (1.08) & 29.53 (0.23) \\ RHC & 259 (4.2) & 37.39 (0.81) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Average (Standard error) results for localization time (s) and path length (m) in light of 30 runs for Case 1.