## 3 Proposed Method
### The Case of Non-Ideal Signal  
In practical cases, radio coverage is not able to achieve a perfect disk shape because environmental factors, such as temperature, moisture, and obstacle, may influence the broadcast radio signal of the mobile anchor. In this section, the impact of the unstable fluctuation of radio signals on the positioning of unknown nodes will be considered. Fluctuations in radio signal cause the communication range of the mobile anchor to be shorter than the theoretical value \(R\). Let the error be \(\varepsilon\). The actual distance will fall in the range \(\{R\)-\(\varepsilon\), \(R\}\) when the mobile anchor first detects the signal of an unknown node. The positioning results obtained by using triangulation to compute the location of unknown nodes become a region, instead of a point. The ISP algorithm is used to apply the range-free positioning algorithm coverage area pruning positioning system (CAPPS) [40] to compute the positioning area instead of using triangulation on the basis of three reference points. All neighboring-position-aware nodes of an unknown node will serve as reference points in CAPPS. In addition, the mobile anchor will record all the locations of all unknown node radio signals as they appear and disappear. Figure 4 illustrates the main idea of the CAPPS. The example and the explanation were extracted from the content provided in [40]. Let \(\Omega\) be the sensor nodes that detect the target, \(\Omega_{c}\) be the set of vertices comprising the convex hull of the set \(\Omega\), and \(T_{g}\) be the centroid of the coordinates of the sensor nodes in the set \(\Omega\). We have sets \(\Omega=\{r,s,t,u,v,w,x,y,z\}\) and \(\Omega_{c}=\{u,v,w,x,y,z\}\) shown in Figure 4a. The set \(\Omega\) is employed to compute \(T_{g}\). Its location is represented by the circle located in the positioning area shown in Figure 4b. For the sensor nodes in \(\Omega_{c}\), their one-hop neighbors are \(\Pi_{u}=\{a,b,c,d\}  
For the sensor node \(u\), its sorted one-hop neighbors are stored in the set \(\Pi^{*}_{u}=\{d,a,\)\(b,\)\(c\}\). Let \(A_{ud}\) be the intersection area of the two circles centered at sensor nodes \(u\) and \(d\), each with radius \(D^{*}=\left|\overline{ud}\right|\). There is a sensor node \(c\) enclosed in \(A_{ud}\), as shown in Figure 4c. Therefore, the sensor node \(d\) is removed from the set \(\Pi^{*}_{u}\). Next, the intersection area \(A_{ua}\) of the circles centered at sensor nodes \(u\) and \(a\), each with radius \(D^{*}=\left|\overline{ua}\right|\), is evaluated as shown in Figure 4d. The sensor node \(a\) is removed because the sensor node \(b\) lies within the area \(A_{ua}\). The sensor node \(b\) is also removed for the same reason, as shown in Figure 4e. Figure 4: Positioning procedure of CAPPS (Reprinted with permission from Ref. [40]. 2018, [PERSON], [PERSON]; [PERSON]) **(a)** The coverage areas of the non-target-detecting sensor nodes [_c, i, m, p_]. **(b)** The sensor nodes in the set \(\Omega\) are used to compute \(T_{g}\), illustrated by the circle located at the positioning area. The one-hop neighbors of the sensor node \(u\) are identified, and the sorted sequence is stored in the set \(\Pi^{*}_{u}=\{d,a,\)\(b,\)\(c\}\). **(c)** The first sensor nodes \(d\) of the set \(\Pi^{*}_{u}\) are verified. The sensor node \(c\) is enclosed within the intersection area \(A_{ud}\). Thus, the sensor node \(d\) is removed from the set \(\Pi^{*}_{u}\). **(d–f)** The other sensor nodes \(a,\)\(b\), and \(c\) in the set \(\Pi^{*}_{u}\) are verified. **(g)** The set of remaining sensor nodes is used for pruning the positioning area \(\Gamma=\{c,\)\(e,\)\(i,\)\(k,\)\(m,\)\(p\}\). **(h)** The pruned positioning area obtained by applying the sensor nodes in the set \(\Gamma\). Finally, the sensor node \(c\) is checked, and no sensor node is found to be enclosed in the area \(A_{uc}\). Thus, the sensor node \(c\) is kept in the set \(\Pi^{*}_{u}\) in order to prune the positioning area, as shown in Figure 4\(f\).