We make some considerations about Relativistic Positioning Systems (RPS). Four satellites are needed to position a user. First of all we define the main concepts. Errors should be taken into account. Errors depend on the Jacobian transformation matrix. Its Jacobian is proportional to the tetrahedron volume whose vertexes are the four tips of the receiver-satellite unit vectors. If the four satellites are seen by the user on a circumference in the sky, then, the Jacobian and the tetrahedron volume vanish. The users we consider are spacecraft. Spacecraft to be positioned cannot be close to a null Jacobian satellites-user configuration. These regions have to be avoided choosing an appropriate set of four satellites which are not seen too close to the same circumference in the sky. Errors also increase as the user spacecraft separates from the emission satellite region, since the tetrahedron volume decreases.We propose a method to autonomously potion a user-spacecraft which can test our method. This positioning should be compared with those obtained by current methods. Finally, a proposal to position a user-spacecraft moving far from Earth, with suitable devices (autonomous), is presented.

Uncertainties in the satellite world lines lead to dominant positioning errors. In the present work, using the approach presented in Puchades and Sáez (Astrophys. Space Sci. 352, 307–320, 2014), a new analysis of these errors is developed inside a great region surrounding Earth. This analysis is performed in the framework of the so-called Relativistic Positioning Systems (RPS). Schwarzschild metric is used to describe the satellite orbits corresponding to the Galileo Satellites Constellation. Those orbits are circular with the Earth as their centre. They are defined as the nominal orbits. The satellite orbits are not circular due to the perturbations they have and to achieve a more realistic description such perturbations need to be taken into account. In Puchades and Sáez (Astrophys. Space Sci. 352, 307–320, 2014) perturbations of the nominal orbits were statistically simulated. Using the formula from Coll et al. (Class. Quantum Gravity. 27, 065013, 2010) a user location is determined with the four satellites proper times that the user receives and with the satellite world lines. This formula can be used with any satellite description, although photons need to travel in a Minkowskian space-time. For our purposes, the computation of the photon geodesics in Minkowski space-time is sufficient as demonstrated in Puchades and Sáez (Adv. Space Res. 57, 499–508, 2016). The difference of the user position determined with the nominal and the perturbed satellite orbits is computed. This difference is defined as the U-error. Now we compute the perturbed orbits of the satellites considering a metric that takes into account the gravitational effects of the Earth, the Moon and the Sun and also the Earth oblateness. A study of the satellite orbits in this new metric is first introduced. Then we compute the U-errors comparing the positions given with the Schwarzschild metric and the metric introduced here. A Runge-Kutta method is used to solve the satellite geodesic equations. Some improvements in the computation of the U-errors using both metrics are introduced with respect to our previous works. Conclusions and perspectives are also presented.