On solving the multi-constellation pseudorange equations

Future Global Navigation Satellite System (GNSS) receivers may employ a combination of different navigation satellite constellations to provide positioning, navigation, and timing information. As different constellations may not be time-synchronized, the multi-constellation navigation problem involves the determination of position of the receiver and clock bias with respect to each constellation. Even though iterative and linearization method may remain applicable for terrestrial users when the initial position error is not too significant, the fact that the multi-constellation navigation equations are augmented with additional unknowns demands some investigation into the resulting solvability. In this paper, a spectral decomposition approach is proposed to characterize all admissible solutions of a multi-constellation navigation problem. The solvability of the problem is addressed, all potential solutions are determined, and the navigation errors are analyzed. By applying the proposed algorithm to envisioned future multi-constellation navigation problems, some potential pitfalls in combined GPS+QZSS and GPS+Galileo navigation are identified.