## 3 LoMoB Localization Scheme
### Location Estimation
#### 3.3.1 Obtaining potential locations  
After estimating the distances between a sensor node and projected beacon points, the following equations need to be solved for localization:  
\[\begin{cases}\sqrt{(x-x^{\prime}_{1})^{2}+(y-y^{\prime}_{1})^{2}}=r^{\prime}_ {1}\\ \sqrt{(x-x^{\prime}_{2})^{2}+(y-y^{\prime}_{2})^{2}}=r^{\prime}_{2}\\ \vdots\\ \sqrt{(x-x^{\prime}_{N})^{2}+(y-y^{\prime}_{N})^{2}}=r^{\prime 2}_{N}\end{cases} \Leftrightarrow \begin{cases}(x-x^{\prime}_{1})^{2}+(y-y^{\prime}_{1})^{2}=r^{ \prime 2}_{1}\\ (x-x^{\prime}_{2})^{2}+(y-y^{\prime}_{2})^{2}=r^{\prime 2}_{2}\\ \vdots\\ (x-x^{\prime}_{N})^{2}+(y-y^{\prime}_{N})^{2}=r^{\prime 2}_{N}\end{cases} \tag{2}\]  
where \((x,y)\) is the location of a sensor node, \((x^{\prime}_{i},y^{\prime}_{i})\) is the location of the \(i^{th}\) projected beacon point, \(N\) is the number of the projected beacon points, and \(r^{\prime}_{i}\) is the estimated distance between a sensor node and the \(i^{th}\) projected beacon point.  
Let us explain how to estimate the location of a sensor node based on the bilateration method with Figures 5 and 6. When \(N\) projected beacon points are obtained, \(N\) circles can be drawn satisfying Equation (2); _i.e._, four circles with centers \(BP^{\prime}_{1}\), \(BP^{\prime}_{2}\), \(BP^{\prime}_{3}\), and \(BP^{\prime}_{4}\) and radii \(r^{\prime}_{1}\), \(r^{\prime}_{2}\), \(r^{\prime}_{3}\), and \(r^{\prime}_{4}\) can be drawn, as shown in Figure 5. The points of intersection of the \(N\) circles that are near the sensor node are obtained as potential locations for the sensor node's estimated location, as shown in Figure 5; some or all of the potential locations may overlap. However, because a potential location can be either of the two intersection points of two circles, a decision process is needed; here, because the intersection points are obtained using just two of many projected beacon points, this method is called bilateration [23]. One of the two intersection points of the two circles with centers at two projected beacon points is closer to the circles with centers at the other projected beacon points than the other intersection point; _i.e._, \(C_{12}(1)\) is closer to the two circles with centers \(BP^{\prime}_{3}\) and \(BP^{\prime}_{4}\) than \(C_{12}(2)\) in Figure 6. When \(C_{jk}(1)\) and \(C_{jk}(2)\) are the intersection points of the circles with centers \(BP_{j}^{\prime}\) and \(BP_{k}^{\prime}\) and the following inequality is satisfied, \(C_{jk}(1)\) is selected as the potential location \(\hat{P}_{jk}\); otherwise, \(C_{jk}(2)\) is selected.  
\[\sum_{i=1,i\
eq j,k}^{N}\|\|BP_{i}^{\prime}-C_{jk}(1)\|-r_{i}^{\prime}|<\sum_{i =1,i\
eq j,k}^{N}\|\|BP_{i}^{\prime}-C_{jk}(2)\|-r_{i}^{\prime}| \tag{3}\]  
Figure 5: Estimation of the sensor node location using projected beacon points.  
Figure 6: Selection of a potential location between two intersection points.  
In rare cases, the number of intersection points between two circles can be one or zero. If there is one intersection point, this single point becomes the potential location. If there is no intersection point, which occurs when the distance between the two beacon points \(BP^{\prime}_{j}\) and \(BP^{\prime}_{k}\) is larger than \(r^{\prime}_{j}+r^{\prime}_{k}\), the point that divides the line segment \(BP^{\prime}_{j}BP^{\prime}_{k}\) joining the points \(BP^{\prime}_{j}\) and \(BP^{\prime}_{k}\) into a \(r^{\prime}_{j}\):\(r^{\prime}_{k}\) ratio becomes the potential location. In these two cases, a decision process is not needed.