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Consider:

Enter image description here

Is it because the 2nd and 3rd satellites' centres are collinear? Can a satellite overlap happen? If so then how is it avoided? If not avoided then what happen to the signal that is sent from such a satellite? This is just a 2D representation. The same concept can be applied in 3D.

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    $\begingroup$ detailed explanations of GPS system operation are trivial to find on the web. have you tried? $\endgroup$ Dec 4 '16 at 15:29
  • $\begingroup$ Usually one satellite is enough since it is moving and its orbit is known and since the observer usually has a fairly precise clock, two make traditional triangulation possible for a known surface, and location is usually known well enough to know altitude. Three satellites are enough to make triangulation in space. Even more satellites improve precision. I suggest you rewrite the question. $\endgroup$ Mar 15 at 19:29
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You can't ignore the difference between 2D and 3D in this situation. To a first approximation, all GPS satellites are located on the surface of a sphere. There is no way that any three points on a sphere can be colinear.

There is one situation that can be ambiguous, and that's when N satellites are equidistant from the receiver. However, in this case, one solution will be on the surface of the Earth, and the other solution will be thousands of miles straight up from there. It's relatively easy to pick the correct solution.

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how are three satellites sufficient to find a unique point in gps?

They are not. It takes four satellites minimum.

GPS needs to solve for the point in four dimensions, X, Y, Z and time. Four unknowns requires at least four inputs to solve. In the case of GPS these are the relative time offsets of the signals received at the coordinate in question.

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    $\begingroup$ That's a valid point, but not directly relevant to the actual question about the geometry. $\endgroup$
    – Dave Tweed
    Dec 4 '16 at 16:10
  • $\begingroup$ Can you show with a reference to precision and uncertainties that this is a relevant answer? Have you made assumptions about where the observing point is? Have you made assumptions about the Galilean component of the Lorentz transformations? $\endgroup$ Mar 15 at 19:36
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First, a GPS satellite cannot actually tell the receiver how far away it is. The receiver can detect the difference between the distance to one satellite A and another satellite B. As a result, four satellites are needed to find your position (however, if there are only three satellites then GPS can guess that you might be on the surface of the earth and calculate your location on that assumption. And if you are standing on the street, the location would be right. If you are on the tenth floor of a high building, you might be 100 metres off. If you are in a flying airplane, it's totally wrong).

Let's ignore that. If three satellites could tell you the distance from each satellite in 3D space, then there are usually two solutions to the equations. (Same in 2D space with two satellites). Typically, one solution will be close to the earth surface, and the other one won't. You pick the one close to the earth surface.

You could also do the same measurement again 100 ms later. So the two solutions move. The solution for the point where you really are moves with the speed of earth rotation. (When you sit still on your sofa, you are actually moving at about 1700 km/h or so; the number is from memory and may be wrong but it's roughly that). GPS can calculate how fast a solution moved relative to the earth surface; that speed will be low (say 190mph if you are travelling on Eurostar). The other incorrect solution will move at a massively different speed, mostly because the satellites themselves are moving at a high speed.

If you have another satellite (the third one in a 2D situation in your example), then the satellites are actually arranged in a pattern around the earth so the situation that you described will not happen. So all the circles will have one point in common. Actually, because all your data is slightly imprecise, all the circles will meet at slightly different points very close together. And you can mathematically find the point that is closest to all the circles.

And interestingly, it is illegal to build a GPS system that could be used to guide ballistic missiles. If a GPS system detects that it is at high altitude and/or moving at very high speed, as it would be typical for a ballistic missile, then it is not allowed to tell its location. You can bet that the second solution will be far from the earth surface and moving fast, so if it was the correct location it would be illegal for the GPS to report that location.

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  • $\begingroup$ Uhhuh, and if you did build and launch your own ICBM, violating a minor law would be the least of anyone's worries. $\endgroup$ Dec 5 '16 at 14:51
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3 satellites are sufficient for a 2D position fix where the receiver assumes that it is at a fixed height. Generally these are avoided whenever possible because a small error in this assumption can result in large position errors.

For a true 3D position fix 4 satellites are needed. As others have said this is required to solve for 4 unknowns, you need to know the exact time in order to calculate your location. Generally far more SVs are tracked and a least squares or similar optimization is used to find the best solution to the measurements.

You don't ever measure range to the satellites, you measure range differences between them, e.g. you know that you are nnn meters closer to SV 1 than SV 2. With 4 satellites you do actually end up with two possible solutions but one of them will be 10,000km above the earth and in a direction where none of the GPS signal is broadcast. The receivers are generally smart enough to ignore this possibility.

Since all the satellites are at the same altitude and arranged into a fixed number of orbital planes I don't think the geometry you describe is possible in the real world. If it was it would only remain true for a tiny fraction of a second, the satellites are moving at high speeds rather than the static points always show in illustrations.
The system is also tracking the satellites velocities relative to the receiver, even if the geometry you describe did somehow happen at the exact moment measurements were taken and only the 3/4 satellites involved were being tracked then you would still get the correct location, only one of the two positions would match both the measured ranges and the measured velocities.

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