Abstract
Abstract 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.
Highlights
This paper gives some considerations about Relativistic Positioning Systems (RPS)
Errors increase as the user spacecraft separates from the emission satellite region, since the tetrahedron volume decreases
3) If a spacecraft carries devices to get the unit vectors n1A at emission times τA, the tetrahedron volume (|J|) may be estimated for any set of four visible satellites and, sets leading to excessively small |J| values should not be used for positioning
Summary
This paper gives some considerations about Relativistic Positioning Systems (RPS). First of all we define the main concepts. 3) If a spacecraft carries devices to get the unit vectors at emission times, the tetrahedron volume may be estimated for any set of four visible satellites and, sets leading to excessively small Jacobian values should not be used for positioning. 4) Only configurations of four satellites leading to big enough Jacobian values will be considered for positioning to get small U and S-errors Each of these configurations gives a position, and the distribution of positions will be used to get the most probable position and the probability of any other location. This positioning should be compared with those obtained by current methods. 6) a proposal to position a user-spacecraft moving far from Earth, with suitable devices (autonomous), should be presented
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