I am working on a project where given satellite positions in the sky, I am to find my location on earth. The assumption is that there are 24 satellites orbiting on the earth but only a few will be above the horizon that can pinpoint your location, and a spherical Earth.
Suppose there are 9 such satellites above the horizon. You can use a minimization problem to locate your position using all 9 satellites, but I am trying to use just 4 satellites using the following formula,
||x_si-x||=c||t-ts||
where x_si is the satellite location, x is my location on earth, t is the time I receive the signal, and ts is the time the signal is sent.
My problem is that when I solve this for the following set of data,
12122.917273538935 2.605234313778725E7 2986153.9652697924 4264669.833325115
12122.918115974104 -1.718355633086311E7 -1.8640834276186436E7 7941901.319733662
12122.91517247339 1.8498279256616846E7 -1.4172390064384513E7 -1.2758766855293432E7
12122.889110685195 -1.7119296100655418E7 -1.2232039954541935E7 2.9907940364210207E7
where the columns are ts (time signal sent from satellite), xs, ys, zs (cartesian coordinates) respectively, I get the wrong answer. A minimization formula finds the right answer... I solved this so many ways that I'm convinced using 4 satellites may not be sufficient for this case.
The location that should be found is (-1795225.28989696,-4477174.36119832,4158593.45315397), and the location that I am finding using a system of 4 equations with 4 unknowns is (2323927.24786382,-4226971.28381194,4158593.4469918).
Something I noticed is that the z-values are the same, and the magnitude of (x^2+y^2)^(1/2) is also pretty much equal for my answer and the correct answer. I do get the correct value for t as well, which is 12123.0 (t is the time I received the signal on earth). I am new to GPS and hoping someone can enlighten me as to why this might be occurring.
أكثر...
Suppose there are 9 such satellites above the horizon. You can use a minimization problem to locate your position using all 9 satellites, but I am trying to use just 4 satellites using the following formula,
||x_si-x||=c||t-ts||
where x_si is the satellite location, x is my location on earth, t is the time I receive the signal, and ts is the time the signal is sent.
My problem is that when I solve this for the following set of data,
12122.917273538935 2.605234313778725E7 2986153.9652697924 4264669.833325115
12122.918115974104 -1.718355633086311E7 -1.8640834276186436E7 7941901.319733662
12122.91517247339 1.8498279256616846E7 -1.4172390064384513E7 -1.2758766855293432E7
12122.889110685195 -1.7119296100655418E7 -1.2232039954541935E7 2.9907940364210207E7
where the columns are ts (time signal sent from satellite), xs, ys, zs (cartesian coordinates) respectively, I get the wrong answer. A minimization formula finds the right answer... I solved this so many ways that I'm convinced using 4 satellites may not be sufficient for this case.
The location that should be found is (-1795225.28989696,-4477174.36119832,4158593.45315397), and the location that I am finding using a system of 4 equations with 4 unknowns is (2323927.24786382,-4226971.28381194,4158593.4469918).
Something I noticed is that the z-values are the same, and the magnitude of (x^2+y^2)^(1/2) is also pretty much equal for my answer and the correct answer. I do get the correct value for t as well, which is 12123.0 (t is the time I received the signal on earth). I am new to GPS and hoping someone can enlighten me as to why this might be occurring.
أكثر...