PRÁCTICA GPS (2017)

Pràctica Inglés
Universidad Universidad Politécnica de Cataluña (UPC)
Grado Ingeniería de Aeronavegación - 3º curso
Asignatura Navegación, cartografía y cosmografía
Año del apunte 2017
Páginas 11
Fecha de subida 25/06/2017
Descargas 3
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GROUND TRACK OF GPS SATELLITES PRACTICE 1 Air Navigation, Cartography and Cosmology Alba Martín Anna Reig 6GX31 17.04.2017 Table of contents 1.
Introduction ......................................................................................................................3 1.1 Aim of the practice .................................................................................................3 1.2 Almanac explanation .............................................................................................3 2.
ECEF coordinates .............................................................................................................5 3.
LLA coordinates ...............................................................................................................7 4.
Complete satellite ground track .................................................................................8 5.
All satellites ground track in the next hour ............................................................9 6.
Conclusions ....................................................................................................................11 1. Introduction 1.1 Aim of the practice The main objective of this project is to plot the satellite ground track for any given satellite or constellation of it. To do so, real-time ephemerides are available on Internet for known Keplerian elements, what makes it easy to work with GPS constellation.
1.2 Almanac explanation First it is crucial to understand ephemerides and Keplerian elements. Satellite ephemerides are on GPS almanac, which includes information of the entire GPS satellite constellation for every orbit. The files used can be found on http://celestrak.com/GPS/almanac/Yuma/2017/.
Every almanac contains the following data and Keplerian elements about each satellite that must be defined and correctly understand: 1.
ID: PRN (Pseudo Random Noise) ID of the SVN (Space Vehicle NAVSTAR) 2.
Health: Indicates the operational state of the SV (Space Vehicle). The value “000” means that the SV is usable.
3.
Eccentricity: It shows the amount of the orbit deviation from a circular orbit. It is also the distance between the foci divided by the length of the semi-major axis (the NAVSTAR orbits are almost circular).
4.
Time of Applicability (ToA): The time instant (seconds within the GPS week) for which the almanac has been computed (ephemeris epoch).
5.
Orbital Inclination: The angle of the SV orbit plane with respect to the equator (GPS is at approx. 55 degrees) in radians. The SV ground track will not rise above approximately 55 degrees of latitude.
6.
Rate of Right Ascension: Rate of change of the angle of right ascension in rad/s.
7.
SQRT(A) Square Root of Semi-Major Axis: The semi-major axis is the distance from the centre of the ellipse to either the point of apogee or the point of perigee.
8.
Right Ascension at GPS week epoch: Geographic longitude (in radians, with respect to Greenwich meridian) of the ascending node of the orbit at the GPS week epoch (Saturday to Sunday midnight).
9.
Argument of Perigee at ToA: An angular measurement (in radians) along the orbital path measured from the ascending node to the point of perigee. It is measured in the direction of the SV's motion.
10.
Mean Anomaly at ToA: Angle (in radians) travelled past the perigee at ToA.
When the SV has passed perigee and heading towards apogee, the mean anomaly is positive. After the point of apogee, the mean anomaly value will be negative to the point of perigee.
11.
Af(0): SV clock bias in seconds 12.
Af(1): SV clock Drift in seconds per seconds 13.
Week: GPS week number (0000-1023). Counter increased (module 1024) every week since 0h UTC of Jan 6th, 1980 (the GPS epoch = 2444244.5 JD). First roll-out happened at 23:59:47 (UTC) on Aug 21, 1999 (roll epoch = 2451412.499850 JD).
Once defined all the parameters, the almanac.m function, already created, will download current GPS almanac and create a matrix within the ephemerides.
2. ECEF coordinates ECEF (Earth-Centred, Earth-Fixed) coordinates is a geographic and Cartesian coordinate system that represents positions as X, Y and Z coordinates. Its axes are aligned with the International Reference Pole and International Reference Meridian that are fixed with respect to the earth. In other words, it has its origin at the Earth’s centre of mass with its x-axis extend through the intersection of the prime meridian (0º longitude) and the equator (0º Figure 1 ECEF Coordinates system latitude). The z-axis extends through the true north pole (i.e., coincident with the Earth spin axis). The y-axis completes the righthanded coordinate system, passing through the equator and 90º longitude.
Consequently, the created function ECEF.m in Matlab will be able to compute ECEF coordinates of any satellite at a given time, and will have the possibility of plotting ground track in three dimensions in meters for a given period of time.
As exposed before, the orbits are characterized by the six Keplerian parameters.
Using these parameters, the ECEF coordinates will be computed for any satellite at current time without taking into account the slow changes in the elements. To obtain the mentioned coordinates, the following steps must be done: 1.
Computation of the time between the ToA of the ephemerides and the current time: 𝑡𝑘 = 𝑡 − 𝑡0 2.
Getting the semi-major axis: 𝑎 = (√𝑎) 3.
2 Getting the mean motion (corresponding to the angular speed): 𝑛=√ 𝐺·𝑀 𝑎3 Where G in the gravitational constant and M the earth mass.
4.
Computation of the mean anomaly: 𝑀𝑘 = 𝑀0 + 𝑛 · 𝑡𝑘 5.
Iteratively for Ek , find the eccentric anomaly: 𝑀𝑘 = 𝐸𝑘 − 𝑒 · sin⁡(𝐸𝑘 ) 𝐸𝑘 (𝑛) = 𝑀𝑘 + 𝑒 · sin⁡(𝐸𝑘 (𝑛 − 1)) 6.
Computation of the true anomaly without ambiguity (converting sinus and cosine into complex numbers and using then the function angle): sin(𝑣𝑘 ) = √1 − 𝑒 2 · ⁡cos(𝑣𝑘 ) = 7.
sin(𝐸𝑘 ) , 1 − 𝑒 · cos(𝐸𝑘 ) cos(𝐸𝑘 ) − 𝑒 ⁡ 1 − 𝑒 · cos(𝐸𝑘 ) Computation of the argument of latitude: 𝑢𝑘 = 𝑣𝑘 + 𝜔 8.
Find the orbit radius, that corresponds to the distance to Earth center): 𝑟𝑘 = 𝑎 · (1 − 𝑒 · cos(𝐸𝑘 )) 9.
Current longitude of the ascending node: Ω𝑘 = Ω0 + Ω̇0 · 𝑡𝑘 − Ω̇𝑒 · 𝑡 Where Ω̇𝑒 = 7.2921151467 · 10−5 𝑟𝑎𝑑/𝑠 10.
Computation of the X coordinate within the orbital plane: 𝑋𝑝 = 𝑟𝑘 · cos(𝑢𝑘 ) 11.
Computation of the Y coordinate within the orbital plane: 𝑌𝑝 = 𝑟𝑘 · sin(𝑢𝑘 ) Once done step eleven, it is time to compute the X, Y and Z coordinates for ECEF.
12.
ECEF X-coordinate: 𝑋 = 𝑋𝑝 · cos(Ω𝑘 ) − 𝑌𝑝 · cos(𝑖0 ) · sin⁡(Ω𝑘 ) 13.
ECEF Y-coordinate: 𝑌 = 𝑋𝑝 · sin(Ω𝑘 ) + 𝑌𝑝 · cos(𝑖0 ) · cos⁡(Ω𝑘 ) 14.
ECEF Z-coordinate: 𝑍 = 𝑌𝑝 · sin⁡(𝑖0) This type of system is useful to know where we are in the Earth or the coordinates of a point fixed on the surface of the earth but if one wants to plot the satellite ground track it is necessary to convert them into LLA coordinates as the ECEF coordinates moves with the Earth unlike the system of LLA coordinates.
Figure 2 ECEF Coordinates representation 3. LLA coordinates LLA (Latitude, Longitude and Altitude) coordinates are chosen such that one of the numbers represents a vertical position, and two of the numbers represent horizontal position. Enables every location on Earth to be specified by a set of numbers, letters or symbols.
Consequently, the created function LLA.m in Matlab will be able to compute Figure 3 LLA Coordinates system LLA coordinates of any satellite at a given time, and will have the possibility of plotting ground track using degrees for the latitude and the longitude and meters for the height for a given period of time.
As it is a two dimensions’ representation, the altitude will not be considered in the plots.
So that, once having the conversion to ECEF Coordinates, they must be converted to LLA following the algorithm shown below so to have a plot in degrees representing the ground track of a certain satellite correctly.
1.
𝑟 = √𝑋 2 + 𝑌 2 2.
𝐸2 = 𝑎2 − 𝑏 2 3.
𝐹 = 54 · 𝑏 2 · 𝑍 2 4.
𝐺 = 𝑟 2 + (1 − 𝑒 2 ) · 𝑍 2 − 𝑒 2 · 𝐸 2 5.
𝑠 = √1 + 𝑐 + √𝑐 2 + 2 · 𝑐 6.
𝑃= 7.
𝑄 = √1 + 2 · 𝑒 4 · 𝑃 8.
𝑟0 = − 9.
𝑈 = √𝑍 2 + (𝑟 − 𝑟0 · 𝑒 2 )2 3 10.
11.
12.
13.
𝐹 2 1 𝑠 3·(𝑠+ +1) ·𝐺 2 𝑃·𝑒 2 ·𝑟 1+𝑄 1 1 𝑃·(1−𝑒 2 )·𝑍 2 2 𝑄 𝑄(1+𝑄) + √ · 𝑎2 · (1 + ) − ⁡ 𝑏 2 ·𝑍 𝑌 𝜆 = 𝑎𝑟𝑐𝑡𝑔 ( ) 𝑋 Where 𝑒 and 𝑎 are given by WGS-84 and have the value of: 𝑎·𝑉 ℎ = 𝑈 · (1 − 𝜙 = 𝑎𝑟𝑐𝑡𝑔 ( 2 14.
𝑉 = √𝑍 2 · (1 − 𝑒 2 ) + (𝑟 − 𝑟0 · 𝑒 2 )2 𝑍0 = 1 − · 𝑃 · 𝑟2 𝑏2 𝑎·𝑉 𝑎 = 6378137𝑚 ) 𝑍+𝑒 ′2 ·𝑍0 𝑟 𝑒 2 = 0.00669437999014 ) 𝑐 𝑐 𝑎 𝑏 And also: 𝑎2 = 𝑏 2 + 𝑐 2 , 𝑒 = → 𝑒 ′ = 4. Complete satellite ground track The complete satellite ground track has been obtained by creating a loop in the Matlab program called Main_One_Satellite.m. This loop iterates for each 300 seconds until it reaches 24 hours and, at each iteration, it computes the ECEF coordinates for the chosen satellite and then with these obtained data, the LLA coordinates. As the period of the satellite orbit is ½ of a sideral day (12h), it takes two consecutive orbits (periods) for the satellite to travel a complete ground track.
After selecting the satellite 12 of the 31 available and by means of LLA coordinates the latitude and longitude can be plotted with in a world map in the background, has shown below.
Figure 4 Ground track of the satellite number 12 5. All satellites ground track in the next hour To know the current subsatellite point of all the GPS satellites and their corresponding ground track for the next hour, it is needed to develop a new Matlab program called Main_All_Satellits.m. This program has a loop that computes the ECEF coordinates and then the LLA coordinates for each time interval until it reaches 1 hour for a certain satellite as the Main_One_Satellite.m but also has another loop which englobes the previous one and computes the next 60 minutes for the 31 available satellites.
Figure 5 GPS satellite ground tracks in the next hour Figure 6 Online satellites ground track for past hour The resulting plot must be validated to assure their reliability. In the Web-page http://www.nstb.tc.faa.gov/rt_waassatellitestatus.htm it is possible to download the updated map on real-time but taking into account that this plot (Figure 6) corresponds to the past hour, while the one extracted from the developed program Main_All_Satellits.m is computing the ground track for the next hour. Looking to both figures 6 and 7, it can be appreciated the match between the profile followed by each satellite in the past hour and the profile that each satellite will perform in the next hour.
6. Conclusions Computing the GPS satellites position is a hard task that requires the use of many equations involving the Keplerian elements. By means of the provided information of the ephemerides, it is needed to change the coordinates system to ECEF coordinates and then to LLA coordinates to obtain a better and more understanding representation of the satellites positions and ground tracks.
Once developed the functions changing the coordinates system and the almanac, it is just needed some looping iterations to represent the ground tracks of the desired satellites.
If one analyses Figure 4, the satellite is covering different areas of the earth, both the northern hemisphere and the south hemisphere. Even though, this is not enough to cover the whole surface, that’s why it is needed at least four satellites to give the position of a device in a precise way. As it has been seen, all over the world, there are 31 satellites that are always covering the GPS service.
In figure 5, it is possible to verify how, thanks to these 31 satellites which compose the GPS system, the covered Earth area is much greater. Although it is only possible to see their positions in the next hour, it can be seen how the orbits represented on the map of the earth would weave a kind of net, covering almost the whole terrestrial surface. This fact explains the efficiency of the GPS system as well as the fact that sometimes, when someone tries to get GPS positioning, the device takes a while as it tries to get information from 4 of the 31 satellites.
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