LEO satellite's position state given by TLE files has had its uncertainty analyzed based on the satellite's initial keplerian elements. In this research, the set of keplerian elements of a LEO satellite that minimizes its position state uncertainty has been calculated, which allows the improvement of a SOOP-based global positioning algorithm. In the first place, LEO satellite's dynamical evolution is studied to code a numerical propagator. Afterwards, a function Δ, whose output characterizes satellite's position state uncertainty , is implemented by making use of the numerical propagator. Finally, a genetic optimization algorithm has served to minimize Δ, and it converged to a single set of keplerian elements. The results show the existence of a Δ function minimum, which is related with the asymmetric gravitational potential of the Earth for being the strongest asymmetric force acting on the satellite.
selector algorithm, which would only take satellites whose orbit was similar to the calculated before, and use them in a global positioning algorithm to make it more precise.
Function Δ has been taken as the diference between the maximum and minimum distance from the receiver to the satellite, among all possible distances for all possible satellite’s state positions inside an uncertainty region. To calculate the objective orbit, it will be necessary to minimize Δ and, to aford that, a genetic algorithm has been used.
In order to obtain the uncertainty region where inside of it the position of a LEO satellite could be placed, it will be necessary to know the temporal evolution of the satellite position state. In this article, Newton’s second law of dynamics has been used, since the forces acting on the satellite are known.
The force acting on the satellite can be taken as a sum of forces, where each of these represents a diferent interaction. However, some interactions are stronger than others, which allows us to focus on higher orders of magnitude terms. A more detailed discussion about the following contents might be found in [8, 9, 10].
LEO satellites are strongly afected by gravitational interactions due to their proximity to the Earth. A stationary model of the gravity force between the Earth and the satellite can be derived from: Φ p, , q “ “0 ÿ“0 C psinpqqp cosp q ` sinp qq
8
C ÿ
which is a Legendre polynomial expansion potential expressed in geocentric coordinates, for radial
distance from the center of mass of the Earth, for latitude and for longitude. and are
coeficients that can be obtained from experimental measurements. are associated Legendre
polynomials of order and degree . is the universal gravitational constant, while and
are the mass and the radius respectively, and they are both referred to the Earth because of the symbol C.
It is also important to take into account the gravitational force produced by Solar System
bodies over the Earth and the satellite. The most relevant gravitational force is a consequence of the
Moon and the Sun, which are taken as punctual objects and with their positions known to avoid a
many-body problem. The acceleration over the satellite due to this interaction can be expressed as:
„
:
⃗` “ K
ˆ ⃗K ´ ⃗
‖⃗K ´ ⃗‖3 ´ ‖⃗K‖3
⃗K
̇
̇ȷ
(
For the sake of considering temporal variations of Earth’s gravitational potential, a study of
Earth tides can be carried out. In its simplest form, the acceleration associated with this interaction is:
⃗: “ 2234C5 p3 ´ 15 cos2 q ⃗ ` 6 cos
⃗
where 2 is a coeficient related to Earth’s elasticity, is the angle between ⃗ and ⃗, and subindex
means that the magnitude is referred to a certain body that produces the interaction, such as the Sun
and the Moon.
(
Last interaction taken into account is the radiation pressure related to the energy of photons absorbed by the satellite. The acceleration due to photons emitted by the Sun can be expressed as: ⃗: “ @ pAUq2 ‖p⃗⃗´´⃗⃗@@‖q3 (5) where P r0, 1s is a real value that takes values near 0 if few solar rays hit the satellite and near 1 if many solar rays hit it. is a reflectivity coeficient, is the cross-section area in the Sun’s direction, and @ “ @{ is related to solar flux ( @). AU are astronomical units.
Positioning based on SOOP precision depends on how precise the satellite position state can be calculated over time, which depends on Newton’s second law solver (usually known as propagator) accuracy and on initial conditions precision. In this work, focus has been set on the study of uncertainty due to initial conditions precision.
A common way to initialize a LEO satellite propagator is by making use of Two-Line Elements (TLE) ifles updated daily on the internet by the North American Aerospace Defense Command (NORAD). Those files contain the keplerian elements associated with a satellite at a specific epoch, apart from giving some information about atmospheric conditions. With that information, the initial position state of a LEO satellite can be obtained, but always with some uncertainty because the measurement of the keplerian elements can not be infinitely precise. These keplerian elements are: semi-major axis ( ), eccentricity (),inclination (), right ascension of the ascending node (Ω), argument of perigee () and the true anomaly ( ), which have been taken as in figure 1.
Initial uncertainty region of the position of a LEO satellite is not known. In this work, a spherical uncertainty region has been set at the initial instant corresponding to the newest TLE’s epoch. Propagating some points inside this spherical region over time, the uncertainty region at any moment can be calculated, and this region will be the largest 24 hours after the initial instant. After 24 hours, TLE file will update the satellite’s position and epoch, which can be set again as the initial instant.
In order to solve Δ function, whose output is the diference between the minimal and maximal distance from the receiver to the possible position state of the LEO satellite, the function has been implemented in MATLAB. The code calculates the diference between distances after obtaining a set of position states where the satellite could be 24 hours after the lecture of the TLE file. To calculate the set of states at a certain time, a numerical orbit propagator has also been implemented based on the force model explained in section 2.1.
In the first place, two functions ( r_moon and r_sun) have been coded to give the position
state of the Moon and Sun at a certain time, and to use that information in equations: (
The terrestrial gravity term, equation (
Finally, to solve Newton’s equations with the total force calculated, the Runge-Kutta method is implemented by using ode45 function in MATLAB.
To reach the objective of this work, and find the orbit that minimizes Δ it has been used a genetic algorithm (table 2), because it is a robust and simple method to calculate good solutions for optimization problems of non-linear functions [11]. It has been taken order and degree 4 for gravitysphericalharmonic function since convergence time increases strongly with these parameters. Also, to characterize LEO satellite’s allowed positions (they are confined into low Earth orbit region), some keplerian element’s restrictions have been taken into account by considering these satellites safely (in terms of drag function’s altitude interval defined) between a 450 and a 1.450 kilometers altitude (table 3).
With the selected configuration, genetic algorithm took 27 hours to converge to a set of keplerian elements that are Δ function’s inputs and they define a LEO satellite’s orbit. Convergence and orbit solution are shown in figures 2 and 3.
Objective function’s uni-dimensional variations might be studied by calculating Δ function’s value for each keplerian element variation. In figures from 4 to 9 it can be seen that Δ becomes minimum for only one or two keplerian element’s values. The fact of this solution’s existence is due to the asymmetric Earth’s gravitational potential, because this is the strongest interaction with the satellite that changes with its angular position.
The eccentricity behavior (figure 5) is the only one afected by the orbit’s mathematical geometry. Since Δ calculates a distance in a straight line and not in a curved line, for higher values of eccentricity, the ellipse starts having sides where two points are separated by longer curved line distances than straight line distances (figure 10).
After this work, the existence of a LEO orbit that minimizes the lack of precision in the receiver’s position due to initial conditions’ uncertainty has been proven. Nevertheless, the precision taken for the terrestrial gravity term is not enough to study other perturbations’ contributions to Δ variation. The future objective of this research is to try to enhance terrestrial gravity precision without increasing convergence time for the genetic algorithm. To achieve that, other orbit propagators will be used, such as SGP4 model or optimized numerical propagators. Moreover, other algorithms will be executed to minimize Δ, for example, simulated annealing or tabu search.
This work has been supported by project PID2021-122642OB-C42 funded by MCIN/AEI/10.13039/501100011033/ and project GR24102, funded by the Regional Government of Extremadura and the European Regional Development Fund (ERDF).
During the preparation of this work, the authors used X-GPT-4 and Gramby in order to: Grammar and spelling check. Further, the authors used X-AI-IMG for figure 10 in order to: Generate images. After using these tools, the authors reviewed and edited the content as needed and take full responsibility for the publication’s content. [4] J. Saroufim, S. Hayek, S. Kozhaya, Z. M. Kassas, Improved leo pnt accuracy enabled by long baseline ephemeris corrections, in: Proceedings of the 37th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2024), 2024, pp. 1219–1229. [5] N. Khairallah, Z. M. Kassas, Ephemeris tracking and error propagation analysis of leo satellites with application to opportunistic navigation, IEEE Transactions on Aerospace and Electronic Systems 60 (2023) 1242–1259. [6] F. Caldas, C. Soares, Precise and eficient orbit prediction in leo with machine learning using exogenous variables, in: 2024 IEEE Congress on Evolutionary Computation (CEC), IEEE, 2024, pp. 1–8. [7] F. Van Diggelen, A-GPS: Assisted GPS, GNSS, and SBAS, 2009. [8] O. Montenbruck, E. Gill, Satellite Orbits, volume 1, 2000. doi:10.1007/978-3-642-58351-3. [9] G. Seeber, Satellite Geodesy, De Gruyter, Berlin, New York, 2003. URL: https://doi.org/10.1515/ 9783110200089. doi:doi:10.1515/9783110200089. [10] J. Mena, J. Berrios, Geodesia superior, Centro Nacional de Información Geográfica, 2008. URL: https://books.google.es/books?id=DfdlQwAACAAJ. [11] D. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison Wesley series in artificial intelligence, Addison-Wesley, 1989. URL: https://books.google.es/books?id= 2IIJAAAACAAJ.