Robust Time-of-Arrival-Based Localization With Unknown Transmission Time Using Convex Relaxation and Penalty Term

This paper considers the time-of-arrival (TOA) based source localization problems when the start transmission time is unknown for both cases of accurate and inaccurate sensor locations. For the case where the sensor locations are accurate, the weighted least squares (WLS) criterion is adopted, leading to a constrained weighted least squares (CWLS) problem. By using the convex relaxation technique, the CWLS problem is relaxed into a novel quasi-convex fractional programming (FP) problem, which is then transformed into an equivalent mixed semi-definite and second-order cone programming (SD/SOCP) problem. This SD/SOCP problem is further tightened by adding a series of second-order cone constraints and a penalty term. For the case where the sensor locations are inaccurate, Taylor series expansion is applied to formulate the new expression of the error term, resulting in the CWLS problem considering the sensor location errors. The solution of this problem can be approximately obtained through similar operations. The results from the simulations and experiments confirm the proposed method can achieve the Cramer-Rao lower bound (CRLB) accuracy and possess a good robustness against the selection of the penalty factor.