Periodic solutions of the non-autonomous lagrange systems

This paper presents a robust distributed localization approach which employs weighted least squares scaling with soft constrains. It incorporates the a priori deployment constraints, i.e., minimum and maximum node separation among 2-hop nodes, into localization as soft constraints and penalizes pairs of 2-hop nodes whose assigned coordinates violate the minimum and maximum constraints. Combined with applying consistency checking and statistical filtering of ranging measurements, we improve the location estimates compared with classical least squares scaling. For received signal strength based range measurements, extensive simulation results confirm that this localization scheme outperforms classical least squares scaling and is resilient against large ranging errors and sparse range measurements, which are common in wireless sensor network deployments. Gained a theorem about existent periodic solutions of the non-autonomous lagrange systems with minimal action principle.