G4S_2.0 is a new project funded by the Italian Space Agency which aims to perform measurements in the field of Fundamental Physics with two satellites, DORESA and MILENA, of the Galileo-FOC constellation. These satellites are characterized by the high eccentricity of their orbits and the accuracy of their atomic clocks. For these characteristics, they have recently been used to improve a previous measurement of gravitational redshift (GRS) by Gravity Probe-A in 1980 ([1]). GRS, which is a local position-invariance test, is only one of the predictions of General Relativity (GR) that can be tested with the Galileo constellation. In particular, the G4S_2.0 project aims to provide a new measurement of GRS and to measure relativistic precessions of the elliptical orbits. These results will place new constraints on possible alternative theories of gravitation, both metric and non-metric in their structure. Furthermore, constraints on the presence of Dark Matter in our Galaxy can be placed by analyzing the data of the constellation's atomic clocks. In this framework a fundamental point is obtaining a satellite orbit solution precise as far as possible. For this purpose, we focus firstly on the precise orbit determination and on a dynamic model for the non-conservative forces acting on these satellites. In particular, the model manages the perturbing effects produced by the direct solar radiation pressure (the major perturbation), the Earth's infrared radiation and the Earth-albedo. The results of G4S_2.0 project will extend the number of tests of Einstein's Theory of GR that can be achieved with Galileo satellites. [1] Vessot R.F.C. et al., (1980) Test of relativistic gravitation with a spaceborne hydrogen maser. Phys Rev Lett 45(26):2081–2084. https://doi. org/ 10. 1103/ Phys. Rev. Lett. 45. 2081.