Anchor-Based Three-Dimensional Localization Using Range Measurements

Real-time high-accuracy position information is essential to a variety of applications. This paper discusses anchor-based three-dimensional (3D) localization problem using range measurements. Under zero-mean independent Gaussian ranging error model, Cramér-Rao bound (CRB) is derived and its relationship with geometric dilution of precision (GDOP) is given. Maximum likelihood (ML) and least square (LS) localization algorithms are given and Newton-Raphson (NR), Gauss-Newton (GN), and steepest decent (SD) methods are used to approximate the ML/LS estimation. Ranging error is related to signal-to-noise-ratio (SNR) in simulation examples, and we illustrate that the geometric relationship between agent and anchors can influence the localization performance, especially outside the cube bounded by anchors and in the direction from the cubic center to each anchor. We also intuitively show that GDOP is not a proper measurement for localization performance, when the differences on ranging error variances under different SNRs are nonnegligible. We demonstrate the feasibility of using NR, GN, and SD methods to approach the ML/LS estimation and show the relatively slow convergence speed feature of SD method. We illustrate ML estimator is optimal since its mean square error attains the CRB for high SNR conditions, while LS can only provide suboptimal localization accuracy. We also show that decreasing the number of anchors will degrade the localization performance and vice versa.