  A local positional system (LPS) is proposed, in which particles are launched
at given velocities, and a sensor system measures the trajectory of particles
in the platform frame. These measurements allow us to restore the position and
orientation of the platform in the frame of the rotating Earth, without solving
navigation equations. If there are also velocimeters installed on the platform,
then one can restore the velocity and angular rate of the platform rotation in
respect to the Earth. Instead of navigational equations, it is necessary to
obtain the classical trajectory of a particle in the field of a rotating
gravity source. Taking into account the gravity-gradient, Coriolis, and
centrifugal forces, the exact expression for this trajectory is derived, which
can be widely used in atomic interferometry. A new iterative method for
restoring the orientation of the platform without using gyroscopes is proposed.
The simulation allowed us to determine the conditions under which the
LPS-navigation error per hour is about 10m.
