  In a time-of-arrival (TOA) or pseudorange based positioning system, user
location is obtained by observing multiple anchor nodes (AN) 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
sub-optimal positioning accuracy. In this paper, 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 (WLS) method is used to optimize the positioning
accuracy. We prove that the positioning error of the new method reaches
Cramer-Rao Lower Bound (CRLB) in far field conditions with small measurement
noise. Simulations on 2D and 3D 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. 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.
