A conditional equation for minimizing the GDOP of multi-GNSS constellation and its boundary solution with geostationary satellites

The Walker-delta constellation has been widely used in GNSS (Global Navigation Satellite System). As a key index to measure the positioning configuration, the GDOP minimization plays an important role in GNSS constellation design with a fixed number of satellites. In this paper, we analytically solve this criterion by revealing the geometry of GDOP minimization. Firstly, the graph composed of the GNSS constellation and the unknown point is established and the geometrical conditions for minimizing the GDOP are revealed by introducing two kinds of GDOP. As to the Walker-delta constellation applied in GNSS, a conditional equation is then given to analytically solve the GDOP minimization involved in multi-GNSS constellation optimization. It shows that: relative to the optimal inclination 54.75° for single GNSS constellation, the inclination of the inclined orbits from the multi-GNSS constellation mixed with a certain number of geostationary satellites should be increased to realize the GDOP minimization which is determined by the number of geostationary satellites and the number of inclined orbits. Ultimately, the multi-GNSS constellation design is performed to show the validation of the conditional equation.