3-D Rigid Body Localization Using 1-D AOA: Boundary Condition Analysis and Generic Majorization-Minimization Framework

This paper focuses on the problem of using one-dimensional (1-D) angle-of-arrival (AOA) measurements, also referred to as space angles (SA), from linear arrays to achieve three-dimensional (3-D) rigid body localization (RBL). We address the constrained weighted least squares (CWLS) and maximum likelihood estimation (MLE) problems for SA RBL. We first establish a boundary condition for SA RBL, which determines the minimum number of sensors and anchors for SA RBL. Subsequently, we employ the semidefinite relaxation (SDR) technique to relax the feasible set of the CWLS problem, converting it into a convex semidefinite program (SDP) to achieve a coarse suboptimal solution. Since the SDR technique is limited to bilinear structures, it cannot tackle the MLE problem for SA RBL (given the inverse cosine term in the MLE objective). To handle the MLE problem for SA RBL and to obtain an exact solution for the CWLS problem, we propose a novel, generic majorization-minimization (MM) framework capable of finding a stationary point solution under the special orthogonal group constraint. We establish surrogate functions for the MLE and CWLS objectives and, as a by-product, derive upper bounds for the inverse cosine, squared inverse cosine, and nonhomogeneous Rayleigh quotient functions. Simulation results demonstrate that using the coarse solution obtained from SDR as the initial point, the proposed MM-based CWLS and MLE algorithms can achieve Cramér–Rao lower bound (CRLB) performance under low noise conditions, with the MLE algorithm exhibiting the lowest bias.