## 3 Simulation Results
### Performance of LASM(B) and LASM(P)  
The simulation process of LASM algorithm is shown in Figure 4. In this experiment, there are 40 anchor nodes (20%) in the network, with radio range \(R\) equaling to 1.5r. The distance error is 5%. The mass for each particle m=1, each spring constant k=2, \(\eta\)=2. The thresholds for calculation steps and total force are _lstep_=700, _Tforce_=1 respectively. The constant \(c\)=0.2. After the virtual position of each particle is initialized, the position error is about 2.4\(R\). Then these particles are attracted or repulsed by the springs between them. Their virtual positions change quickly. After 25 calculation steps, the total force for each particle is smaller than _Tforce_ (=1) and the position error is 0.2238. The result of position estimation for each node is shown in Figure 4(c). The blue circles represent the true positions of blind nodes; the red circles represent the true positions of anchor nodes; the red lines represent the position errors of the nodes; the two endpoints of the line represent the true position and estimated position of the node. Most of these nodes have good position estimations. However, a small number of nodes (about 20 nodes) have large position errors. These nodes are located in needless points (described in Section 2). After we reinitialize the positions of these needless nodes, and use LASM(B) again for one time, which is the process of Patch A, The position error decreases to 0.1014. The result of the position estimation for each node is shown in Figure 4(d). Most of the needless nodes in Figure 4(c) are located in the right positions now. Just a few of them are also located in needless points. We just reinitialize these nodes for one time. We can conclude that if we reinitialize these nodes for several times, the LASM(P) can get a even smaller localization error. In Figure 4(b) we set _Tforce_=0.1. The position error decreases slowly from step number 25 to 99 and then obtain a relative small position error 0.1562 in LASM(B). The result of position estimation for each node is shown in Figure 4(e). After one time reinitializing, the position error of LASM(P) decreases to 0.0476. Just one node is in needless position in Figure 4(f). Though it can get a smaller position estimation error by reinitializing several times in LASM(P), the number of calculation steps increases correspondingly. This will increase communication cost and energy consumption. In order to decrease them, we just use one time reinitializing in LASM(P) in the follow experiments and set _Tforce_=0.1. The robustness of the LASM algorithm is tested in Figure 5. 100 random examples are generated. In each example the true and virtual positions of nodes are randomly deployed in the square. The other parameter values are the same as those in Figure 4(b). Figures 5(a,b) show general calculation steps the LASM(B and P) have used. In LASM(B), all calculation steps are smaller than 300, and the number of average calculation steps is smaller than 100. This means that the number of calculation steps is very stable, which doesn't change a lot in different examples. In LASM(P), the number of average steps is about 160. This means that in LASM(P), after reinitializing the needless nodes once, the number of average calculation steps to use LASM(B) again is about 60 (160-100). The  
Figure 4: The simulation process and results of LASM. (a) the position estimation error changes in simulation process, _Tforce_=1. (b) _Tforce_=0.1. (c) The result of position estimation in LASM(B), _Tforce_=1. (d) LASM(P), _Tforce_=1. (e) LASM(B), _Tforce_=0.1. (e) LASM(P), _Tforce_=0.1. Convergence Speed increased after reinitializing again. Figures 5(c,d) show the position estimation errors the LASM(B and P) have obtained. In LASM(B), the largest position error is about 0.5, while the average position error is about 0.2. This also shows that the LASM(B) is very robust. In LASM(P), the average position error is smaller than 0.1. It shows that the LASM(P) can obtain a relatively accurate localization.