A closed-form method for single-point positioning with six satellites in dual-GNSS constellations

With the impact of the Global Navigation Satellite System (GNSS), dual-GNSS constellations are playing an increasingly significant role in positioning, navigation and timing (PNT) applications. Aiming at improving from the existing method, i.e., linearization, of solving the single-point positioning problem under a dual-GNSS, this paper develops a closed-form method for solving PNT problems in the case of six satellites. This method reduces the positioning problem to a simple mathematical problem of finding solutions to a quadratic equation, thereby needing only one receiver clock bias (RCB) as variable. By solving the RCB, the positioning information in three dimensions is obtained by utilizing a linear equation. Compared with the existing method, the closed-form method requires no initial position or iterations. This method thus provides a direct solution to single-point positioning. Further, how to check the uniqueness and the validity of the solutions is also derived. Experimental results verify the validity, applicability and efficiency of the proposed method.