Diagnosis and solution of the ill-posed problem in WSNs node localization

Node localization technology is one of the important support technologies in wireless sensor networks (WSNs) applications. In node localization, the distances between nodes can be obtained from the received signal strength indicator (RSSI). According to the geometric relationships between the nodes, the coordinates of the unknown nodes can be determined using the least square estimate (LSE). But in experiments and applications, it is discovered that different spatial positions of reference nodes sometimes lead to large errors. In this paper, this phenomenon is studied and it is found that WSNs node localization is an ill-posed problem, which means that the results of localization are unstable and significantly affected by the ranging errors. The causes and mechanisms of the ill-posed problem are analyzed. It is proposed that the ill-posed problem is diagnosed using the condition number method. If the ill-posed degree is weak, the node can be localized using LS estimate, but if the ill-posed degree is serious, the ridge estimate method can be used to solve the ill-posed problem. Test results indicate that the ridge estimate method dilutes the influence of ill-posed problem, reduces the location errors to about 3 meters and improves the localization stability.