## 3 Proposed Method
### Mobile Anchor Route Scheduling Method (MARS)  
In this section, the proposed mobile anchor route scheduling (MARS) method based on the ISP mechanism is described in detail. MARS includes a projection mechanism and works in cooperation with the ISP algorithm to improve on the SCAN method. MARS consists of a movement phase and a projection phase. The mobile anchor receives messages from the detected unknown nodes and records the area covered by its radio signal during the movement phase. Let \(L_{u}\) and \(L_{d}\) be the upper and lower boundaries of the covered area. In the beginning, the mobile anchor moves from left to right, as in the SCAN method. When the mobile anchor reaches the right boundary of the deployment region, the \(L_{u}\) of the covered area corresponds to the upper boundary of the deployment region. The \(L_{d}\) of the covered area is a line parallel to \(L_{u}\). The distance between \(L_{u}\) and \(L_{d}\) is \(2R\). During the projection phase, the unknown nodes detected during the movement phase are projected vertically onto the line \(L_{d}\). Because the mobile anchor always employs a straight movement trajectory when using the SCAN method, this results in the collinear problem of the three reference points, whereby two possible points are determined by the mobile anchor in the straight-line segment of the movement trajectory for each detected unknown node. MARS projects these two possible points vertically onto the \(L_{d}\). When the mobile anchor finishes a line segment of the SCAN movement trajectory and reaches the right boundary of the deployment area, the mobile anchor shifts to the next line segment, which is also line \(L_{d}\), but in the reverse direction. If there are any projected points on \(L_{d}\), the mobile anchor will move directly to the closest projection point, instead of shifting vertically to the next parallel line segment, as it does when using the SCAN method. If there are no projection points on \(L_{d}\), this indicates that there are no unknown nodes in the current scanned area. The mobile anchor does not need to scan overlapping coverage to perform positioning. Rather, it is able to directly shift to the next line segment of the SCAN movement trajectory. Figure 2: Three cases in which the ISP algorithm is able to directly locate an unknown node: (**a**) the unknown node has three position-aware neighbors; (**b**) the unknown node has two position-aware neighbors; (**c**) the unknown node has one position-aware neighbor. Figure 3 presents a small example of the application of MARS. \(M\) is the mobile anchor, and \(\{x,y,e,s,w\}\) are unknown nodes, as shown in Figure 3a. \(\{E_{2},E_{9},E_{a},E_{3},E_{4},E_{b},E_{c},E_{5},E_{6},E_{d}\}\) is the traveling trajectory of \(M\) when using the SCAN method. The mobile anchor using MARS starts moving along the line segment \(\{E_{2},E_{9}\}\). When \(M\) reaches location \(x_{1}\), it detects the signal of unknown nodes \(x\) and \(y\). Subsequently, their signal vanishes at \(x_{2}\) and \(y_{2}\), respectively. When \(M\) reaches location \(E_{9}\), it finds that both \(x\) and \(y\) have only two reference points. \(M\) performs the projection phase from here to obtain the locations \(x^{\prime\prime}\) and \(y^{\prime\prime}\) on the line segment \(\{E_{a},E_{3}\}\). Then, \(M\) moves toward the closest projection location, \(y^{\prime\prime}\). At location \(y_{3}\), \(M\) detects the signal of the unknown node \(y\) and determines the actual position of node \(y\). Figure 3: MARS with ISP algorithm: (**a**) the projection of MARS in the first line segment; (**b**) the unknown node has one position-aware neighbor; (**c**) the unknown node has two position-aware neighbors. As \(M\) moves continuously toward the location \(y\)\" on the line segment \(\{E_{a},E_{3}\}\), shown in Figure 3b, it detects two unknown nodes, \(s\) and \(e\), at the locations \(s_{1}\) and \(e_{1}\), respectively. Both nodes \(s\) and \(e\) report that they have a position-aware neighbor \(y\).