Optimal TOA-Based Sensor-Anchor-Source Geometries and Estimation Bounds for Simultaneous Sensor and Source Localization

This paper focuses on optimal time-of-arrival (TOA) sensor placement for simultaneous sensor and source localization (SSSL) with the help of selected anchors at known positions in the environment. Firstly, the problem of sensor placement for SSSL is analyzed and formulated as an optimization task based on the approximate Cramér-Rao lower bound (CRLB), which is an approximation of the intractable true CRLB. Secondly, by minimizing the trace of the approximate CRLB, the optimal accuracy bounds for the estimated sensor and source positions are derived, which can serve as a useful evaluation metric for other studies. Thirdly, a systematic solution including both the analytical and algebraic methods is proposed to obtain the optimal sensor-anchor-source geometries for achieving the approximate bounds simultaneously. Significantly, the analytical sensor placement approach can quickly offer an optimal placement for some special cases, and the algebraic algorithm can provide a (sub-)optimal solution numerically for the general case. Furthermore, theoretical guidance for placing anchors in the localization area is provided. Finally, the theoretical findings and proposed algorithms are verified by computer simulations and experimental studies, demonstrating that the optimized sensor positions yield accurate performance. The results in this paper can be utilized as an evaluation tool and a performance improvement guidance for practical SSSL problems.