Mathematical minimum of Geometric Dilution of Precision (GDOP) for dual-GNSS constellations

Selecting optimal satellites for positioning calculation is a basic problem for the positioning, navigation and timing (PNT) applications with Global Navigation Satellite System (GNSS), and the Geometric Dilution of Precision (GDOP) is a key index to handle this problem. In general, the lower the GDOP values are, the more accurate the PNT solution is. Therefore, the minimum value of GDOP should be pursued. In this paper, we focused on the five-satellite as at least five satellites are required for dual-GNSS constellations. Utilizing the characteristics of matrix partial orders, the mathematical minimum of GDOP in the five-satellite case together with the optimal distribution of the five satellites has been theoretically derived. Furthermore, from a theoretical point of view, the detailed expressions of the impact of different constellational combinations of these satellites on the GDOP have been obtained. The results demonstrated that, for dual-GNSS, even if the geometric distribution of the five satellites is fixed, different constellational combinations of these satellites lead to different values of GDOP. This is different from the single-GNSS case.