  We present a variant of a Global Navigation Satellite System called a
Relativistic Positioning System (RPS), which is based on emission coordinates.
We modelled the RPS dynamics in a space-time around Earth, described by a
perturbed Schwarzschild metric, where we included the perturbations due to
Earth multipoles (up to the 6th), the Moon, the Sun, Venus, Jupiter, solid
tide, ocean tide, and Kerr rotation effect. The exchange of signals between the
satellites and a user was calculated using a ray-tracing method in the
Schwarzschild space-time. We find that positioning in a perturbed space-time is
feasible and is highly accurate already with standard numerical procedures: the
positioning algorithms used to transform between the emission and the
Schwarzschild coordinates of the user are very accurate and time efficient --
on a laptop it takes 0.04 s to determine the user's spatial and time
coordinates with a relative accuracy of $10^{-28}-10^{-26}$ and
$10^{-32}-10^{-30}$, respectively.
