A Closed-Form Localization Method Utilizing Pseudorange Measurements From Two Nonsynchronized Positioning Systems

In a time of arrival (TOA) or pseudorange-based positioning system, user location is obtained by observing multiple anchor nodes (ANs) at known positions. Utilizing more than one positioning systems, e.g., combining global positioning system (GPS) and BeiDou navigation satellite system (BDS), brings better positioning accuracy. However, ANs from two systems are usually synchronized to two different clock sources. Different from single-system localization, an extra user-to-system clock offset needs to be handled. Existing dual-system methods either have high computational complexity or suboptimal positioning accuracy. In this article, we propose a new closed-form dual-system localization (CDL) approach that has low complexity and optimal localization accuracy. We first convert the nonlinear problem into a linear one by squaring the distance equations and employing intermediate variables. Then, a weighted least-squares (WLSs) method is used to optimize the positioning accuracy. We prove that the positioning error of the new method reaches Cramér–Rao lower bound (CRLB) in far-field conditions with small measurement noise. Simulations on 2-D and 3-D positioning scenes are conducted. Results show that, compared with the iterative approach, which has high complexity and requires a good initialization, the new CDL method does not require initialization and has lower computational complexity with comparable positioning accuracy. The numerical results verify the theoretical analysis on positioning accuracy, and show that the new CDL method has superior performance over the state-of-the-art closed-form method. Experiments using real GPS and BDS data verify the applicability of the new CDL method and the superiority of its performance in the real world.