Optimal Sensor Placement for 3-D Time-of-Arrival Target Localization

Abstract

This paper focuses on finding the optimal sensor placement strategies for circular time-of-arrival (TOA) localization in the three-dimensional (3D) space. The A-optimality criterion, minimizing the trace of the inverse Fisher information matrix (FIM), is applied to determine optimal sensor placements under the Gaussian measurement noise. A comprehensive analysis of optimal sensor-target geometries is provided in the general circumstance with no restriction on the number of sensors, sensor elevation angle and sensor-target range. The analysis is divided into two detailed sub-cases with uniform sensor measurement noise variance. The reachable minimum trace of Cramer-Rao lower bound (CRLB) $=\frac{9\sigma ^2}{4N}$ is derived. Two new closed-form solutions, i.e., the resistor network and special solution methods, are developed to quickly determine the optimal geometries. Furthermore, the theoretical smallest tr(CRLB) $=\frac{9}{4} (\sum _{i=1}^N\frac{1}{\sigma _{i}^{2}})^{-1}$ of using different noise variances is also presented. The analytical results are verified by simulation examples.