<abstract xmlns="http://www.w3.org/1999/xhtml"><p>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.</p></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.

Some Improvements on Relativistic Positioning Systems

Departament d’Astronomia i Astrofísica

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"Some Improvements on Relativistic Positioning Systems"}

We make some considerations about Relativistic Positioning Systems (RPS). Four satellites are needed to position a user. First of all we define the main...

Some Improvements on Relativistic Positioning Systems

Published Online: Oct 03, 2018

Page range: 161 - 166

Received: Feb 24, 2018

Accepted: May 13, 2018

DOI: https://doi.org/10.21042/AMNS.2018.1.00012

Keywords
Astronomy and Astrophysics, Mathematical Physics, General Relativity, Relativistic Positioning Systems, Numerical Methods

© 2018 Màrius Josep Fullana i Alfonso et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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Some Improvements on Relativistic Positioning Systems

Published Online: Oct 03, 2018

Page range: 161 - 166

Received: Feb 24, 2018

Accepted: May 13, 2018

DOI: https://doi.org/10.21042/AMNS.2018.1.00012

KeywordsAstronomy and Astrophysics, Mathematical Physics, General Relativity, Relativistic Positioning Systems, Numerical Methods

© 2018 Màrius Josep Fullana i Alfonso et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Keywords
Astronomy and Astrophysics, Mathematical Physics, General Relativity, Relativistic Positioning Systems, Numerical Methods