Elliptic Localization of a Moving Object in the Absence of Transmitter Position and Synchronization

In this paper, we address the problem of elliptic localization of a moving object with a constant velocity, when the transmitter position and the synchronization between the transmitter and receivers are not available. To solve this challenging problem, we propose a two-step optimization method having a simple optimization problem solved in each step. Specifically, a generalized trust region subproblem (GTRS) is formulated using the direct path measurements in the first step to estimate the transmitter position and the range offset caused by the synchronization error. The GTRS can be solved efficiently and its global solution can be guaranteed. In the second step, a non-convex constrained weighted least squares (CWLS) problem is formulated using the indirect path measurements and the transmitter position and range offset estimate provided in the first step. The CWLS problem is generally difficult to solve. To address this issue, we relax it into a convex semidefinite programming (SDP) problem by making use of semidefinite relaxation (SDR). The mean square error analysis is conducted to show that the proposed two-step method is able to reach the Cramer-Rao lower bound performance for the initial object position and velocity estimations, under small Gaussian noise. Simulation results confirm the good performance of the proposed method.