2D geometrical performance for localization algorithms from 3D perspective

In conventional wireless positioning systems, the mobile target and anchors are not with the same height due to the infrastructure deployment. However, many localization algorithms ignore the relative differences in height and assume the distance obtained is in the 2D field (defined as 2D-ranging). In this paper, we attempt to analyze the impact of the relative height between mobile target and anchor for the estimation performance. We assume the range measurement is based on the 3D distance (defined as 3D-ranging) in a small playing field, where the relative height difference can not be ignored in the range calculation, but the position estimation is drawn on the 2D playing field. Cramér-Rao lower bounds (CRLBs) for both line-of-sight (LOS) and non-line-of-sight (NLOS) noise environment are derived. The analytical results indicate that the geometric shapes of optimal mean square error (MSE) for unbiased estimator change, if a relative height difference exists. Thus, conventional GDOP or CRLB-based location algorithm and anchor selection method are unreliable without considering the relative height difference. Besides, we evaluate spatial position error distribution (SPED) of several commonly used localization algorithms, such as linear-least-square (LLS), non-linear-least-square (NLLS), min-max algorithm, Geo-n algorithm. The estimation accuracy of the above algorithms are degraded comparing with 2D ranging case in SPED. Therefore, based on our work, the relative height should be considered in the localization algorithm design in order to obtain an accurate estimation.