Unified Constrained Geometric Configuration Optimization for Source Localization Systems: A Riemannian Manifold-Based Approach

Abstract

Time of arrival (TOA), time difference of arrival (TDOA), received signal strength (RSS), received signal strength difference (RSSD) and angle of arrival (AOA) are commonly used techniques for source localization. The positioning accuracy of these systems depends heavily on the geometric configuration of sensors, which is typically constrained by practical conditions. This paper presents a unified framework for optimizing sensor geometry across all five localization systems, explicitly incorporating both distance and angle constraints on sensor positions. First, the Cramér-Rao lower bounds (CRLBs) of these systems are transformed to obtain a unified expression. Based on this expression, a unified constrained geometric configuration optimization problem is formulated. The problem is then simplified into a compact form by replacing the sensor-target angles with an orientation matrix. Subsequently, a Riemannian manifold-based constrained geometric configuration optimization algorithm (RM-CGCOA) is proposed to optimize the sensor-target distances and the orientation matrix. This algorithm casts the orientation matrix onto a product manifold of unit circles. An adaptive pullback is further proposed to strictly enforce angle-related inequality constraints, ensuring that all iterates remain feasible when updating the orientation matrix via the Riemannian gradient. Within RM-CGCOA, analytical optimal distances are derived for TOA, TDOA, RSS and AOA, whereas for RSSD, the distances are updated using projected gradient descent (PGD) together with the orientation matrix. Experimental results demonstrate that the proposed RM-CGCOA consistently achieves a significantly lower position error bound (PEB) compared with the existing strategies and yields a similar PEB to the near-optimal search algorithm, but with a much faster running time.