# An Obstacle-Tolerant Path Planning Algorithm for Mobile-Anchor-Node-Assisted Localization
## 3 Overview  
### Preliminary  
Table 2 presents the terminologies used in this paper.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Terminologies** & **Definition and Description** \\ \hline \multirow{4}{*}{Unknown (Ordinary) Node} & The ordinary sensor nodes are used to finish the sensing data collection. \\  & The ordinary nodes are assumed to have less battery energy and keep silent \\  & state unless the broadcast beacon packet from MAN wakes them up, then \\  & estimate their coordinates. \\ \hline \multirow{2}{*}{Mobile Anchor Node (MAN)} & The node with more energy, larger communication and mobility, is aware of \\  & its position by GPS or manually placed to give help to locate the \\  & ordinary nodes. \\ \hline MANAL & Mobile-anchor-node-assisted localization. \\ \hline Beacon Packet (packet) & A packet with position information broadcast by MAN. \\ \hline Beacon Point & The location where the mobile anchor node broadcasts packets. \\ \hline Basic Square & The minimum sensing field in OTPP. \\ \hline Sub Square & The basic unit of basic square \\ \hline \(k\)-covered area & An area can receive \(k\) non-collinear beacon packets from the MAN \\ \hline OTPP & Obstacle-tolerant path planning \\ \hline \hline \end{tabular}
\end{table}
Table 2: Terminologies.  
**Definition 1**.: **Basic square.** _According to References [7; 8], any \(N\times N\) region can be partitioned into \(k\) basic squares. Each basic square must be divided into four sub-squares, \(Sq_{h}\), with a side length of \(d\); where \(h\in\{1,\,2,\) 3, 4}, as shown in Figure 2._  
The locations of the beacon points of a MAN in a basic square are denoted by \(c_{0}\), \(c_{1}\), \(c_{2}\), \(c_{3}\), and \(c_{4}\), in Figure 2. Lemma 1 ensures that all the unknown nodes in a basic square can receive at least three position packets from the MAN.  
**Lemma 1**.: _An unknown node, \(s_{i}\), (\(i\in\{1,2,\,\ldots,\,n\}\)), in a basic square can receive at least three position packets from the MAN if and only if \(CR\geq\sqrt{(5/2)}d\), where CR represents the communication range of the MAN [7; 8]._  
Proof.: This lemma originates from reference [7] where the detailed proof can be found. Here, we give a brief proof. Based on the localization method, an unknown node, \(s_{1}\), inside \(Sq_{3}\) can be localized only if it can receive three non-collinear position packets from the MAN. As \(CR\geq\sqrt{(5/2)}d\), \(dist(c_{0},s_{1})\leq\sqrt{(5/2)}d\), \(dist(c_{3},s_{1})\leq\sqrt{(5/2)}d\), and \(dist(c_{4},s_{1})\leq\sqrt{(5/2)}d,s_{1}\) can receive the position packets broadcast by the MAN at locations \(c_{1}\), \(c_{3}\), and \(c_{4}\), thereby being localized, as illustrated in Figure 2. Similarly, all the unknown nodes \(s_{i}\) (\(i\in\{1,\,2,\,..,\,n\}\)) that are inside \(Sq_{1}\), \(Sq_{2}\), and \(Sq_{4}\) can be localized.  
The farthest location from the center of the basic square is the vertex of the square. Assume the unknown node \(s_{1}\) located at the vertex of the square can receive at least three position packets from the three nearest beacon points \(c_{1}\), \(c_{3}\), and \(c_{4}\), the communication range \(CR\) must be larger than \(\text{Max}[dist(c_{0},\,s_{1})]\), \(dist(c_{3},\,s_{1})\), \(dist(c_{4},\,s_{1})]\) and \(\text{Max}[dist(c_{1},\,s_{1})]\), \(dist(c_{3},\,s_{1})\), \(dist(c_{4},\,s_{1})]=\sqrt{(5/2)}d\) so \(CR\geq\sqrt{(5/2)}d\).