A geometric method for A-optimal anchor deployment in target localization

Abstract This paper investigates the relationship between the Cramér–Rao lower bound (CRLB) and the geometric area or volume formed by anchors in target localization. For time-of-arrival (TOA), received signal strength (RSS), and joint TOA–RSS measurements, a unified formulation of the Fisher information matrix (FIM) is established by accounting for distance-dependent measurement means and variances. Under equidistant deployment and identical measurement conditions, the FIM can be decomposed into a scalar measurement coefficient and a geometric information matrix determined by the anchor direction vectors. In two-dimensional scenarios, the regular polygon maximizes the polygon area and attains the minimum trace of the CRLB. In three-dimensional scenarios, the general A-optimal condition is characterized by equal eigenvalues of the geometric information matrix. For the special case of four anchors with the target located strictly inside their convex hull, the regular tetrahedral configuration simultaneously attains the minimum trace of the CRLB and the maximum Euclidean tetrahedral volume. Monte Carlo simulations and indoor UWB experiments support the theoretical analysis. Compared to smaller-area configurations with the same number of anchors, the larger-area configurations reduced the experimental RMSE by up to 72.3%.