The GPS

$Index$

GPS basics

References

Exercises

The GPS system

The Global Positioning System (GPS) is a worldwide radio-navigation system formed from a constellation of 24 satellites, each in its own orbit 11,000 nautical miles above the Earth, and five ground stations that make sure the satellites are working properly. The GPS satellites each take 12 hours to orbit the Earth.

GPS uses these satellites as reference points to calculate positions accurate to just a few meters. In fact, with advanced forms of GPS you can make measurements to better than a centimeter! Each satellite is equipped with an accurate clock to let it broadcast signals coupled with a precise time message. The ground unit receives the satellite signal, which travels at the speed of light. Even at this speed, the signal takes a measurable amount of time to reach the receiver. The difference between the time the signal is sent and the time it is received, multiplied by the speed of light, enables the receiver to calculate the distance to the satellite. Here's a graphic to illustrate the complete functionality.
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GPS receivers have been miniaturized to just a few integrated circuits and so are becoming very economical. And that makes the technology accessible to virtually everyone. These days GPS is finding its way into cars, boats, planes, construction equipment, movie making gear, farm machinery, even laptop computers. Soon GPS will become almost as basic as the telephone.

How does it work?

Here's how GPS works in five steps:

  1. The basis of GPS is "triangulation" from satellites.

  2. To "triangulate," a GPS receiver measures distance using the travel time of radio signals.

  3. To measure travel time, GPS needs very accurate timing.

  4. Along with distance, you need to know exactly where the satellites are in space.

  5. Finally you must correct for any delays the signal experiences as it travels through the atmosphere. You must also correct for the clock differences between the GPS receiver and the satellites.

The triangulation step is something like this. You have at hand your trusty GPS receiver. What the GPS can do is precisely measure your distance from any one of a number of satellites. From Satellite 1 $(S_{1})$ your device (GPS unit) measures your distance from the satellite to be 11,000 miles away, then you can only localize yourself on a circle on the earth that is the locus of all points 11,000 miles from $S_{1}.$ Now from From Satellite 2 ($S_{2})$ your GPS unit measures your distance from the satellite to be 12,000 miles away. Now you are localized to be on the intersection of the spheres centered at the satellites and of the respective radii. The intersection of the spheres is a circle. Next, obtaining the distance from a third satellite ($S_{3}$) allows us to locate in the intersection of the three spheres and this intersection is two points (A circle intersected by a sphere is two points, in general.) One of the points is where you are gps__8.png. The other point is rarely on the planet's surface and so by a little reckoning, you can eliminate this possibility and select your location. This is broad strokes is how its done. But there are fine details and they lie in computing those distances.


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Another view of the same situation shows the spheres centered at the satellites. In the picture below, The planet earth is the small circle toward the middle of the intersection of the spheres.
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Here's the easy way to see that observations from four satellites can give the exact position.
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*In each event, your position is on the locus of result points.

Now most GPS units search for and determine measurements from as many as six satellites. This can only improve the accuracy using least squares methods. Here is a link to further information on least squares.

How exactly is the distance computed?

We now know that if we have the locations of the satellites and distances from us, we can accurately determine our position. But we need to know how the distance is computed. In fact there are a couple of ways. Here are some methods to compute distance.

As mentioned above GPS methods are related to measuring light propagation time but not directly. How to measure distance by light? There are a couple of methods. First, distance can be measured directly by sending a pulse and measuring how it takes to travel between two points. This most common method is to reflect the signal and the time between when the pulse was transmitted and when the reflected signal returns. Such systems, called bi-directional, are used in radar and satellite laser ranging that require single millimeter accuracy. They require a clock capable of timing accuracy of $3x10^{-12}$seconds (3 picoseconds). The clock stability need is MATH. A clock with this longtime stability would gain or lose 0.03 seconds in a year. Such equipment is expensive, costing for satellites about $1m. More on this later.

Back to the GPS

If we know the transit time of a signal and the speed of propagation of the signal, then we can determine the distance or range. Since the GPS receiver clock is not perfectly sychronized with the satellite clock, the ranges are in error. For that reason they are called pseudoranges. We must determine the time offset between the clocks to accurately measure the distances.

Assuming we have the distances from four satellites and we know MATH, $i=1,2,3,4$ are the exact postions of the satellites, we must then solve the following system equations where the .
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where $c$ is the speed of light and $t_{B}$ is the receiver clock offset time. The receiver clock offset is the difference between GPS time and internal receiver time. Obviously, a key portion of all this is that there is just one clock offset time. This means the all the satellites must have perfectly synchronized clocks, and this is just one of the tasks of the control sites. The unknowns above are $x,~y,~z,$ and $t_{B}$. Let us reiterate: if the clocks of the GPS receiver and the satellite were perfectly synchronized, the time offset would be zero.
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The GPS receiver is not just a fine electronic mechanism, it can do a whole lot of mathematics as well. Wow and they put it in such a small package!

Still more math

Just how is that system solved? It is multivariate and nonlinear. There are numerous methods that have been designed for just such systems. Most notably is the famous Newton's method. We have provided a link to the basics of the Newton's method for functions of one variable. In a nutshell... To solve the equation MATH, make an initial guess $x_{0}$, compute MATH. Then replace $x_{0}$ in the expression MATH with $x_{1}$ and compute $x_{2}$ and continue in this manner. That is, we compute
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and continue to do so until the value of MATH is sufficiently close to zero.


In the general multivariate case the same equation works, but has a slightly different look. Let
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be the sytem to be solved., where $\QTR{bf}{f}$ and $\QTR{bf}{x}$ are column vectors MATH and MATH Each function $f_{i}$ MATH is a function of each of the $n$ variables. Define the $n\times n$ Jacobian matrix
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If $\QTR{bf}{x}_{0}$ is some starting vector, then the Newton iterations are
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where MATH is the inverse matrix of MATH

Convergence problems for Newton's method are legend. However, if we have a good starting value, then Newton's method often converges rapidly. You can see more about this in our one dimensional treatment.

Sources of position location error

There are many sources of error in GPS measurement. Among them are the atmosphere, ionosphere, satellite orbit errors, receiver noise, multipath ambiguities, and satellite clock errors.

A number of other corrections and tricks are required to obtain precise distances. Sometimes GPS units use dual frequency transmission, in part because ionospheric errors that are inherent in all observations can be modelled and significantly reduced by combining satellite observations made on two different frequencies and observations on two frequencies allow for faster ambiguity resolution times.

Until recently, another source of error was intentionally created. However, as of August 2000, the Selective Availability of the signal, an intentional degradation of the signal, was turned off. Therefore, accuracies of the horizontal position is in the 5-7 meter range.

Optical systems

With optical systems, a flat reflector does not work because of the obvious need that the light signal be exactly perpendicular to the ranging mirror. So, what is commonly used is a corner cube reflector, as shown below.
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An alternative method to measure distance is to measure the phase difference between the incoming and outgoing continuous wave. Such a device is called an interferometer.

The mathematics behind this is elementary trigonometry. Suppose the outgoing signal is given by
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and also the $\pi /2$ lagged signal MATH. The incoming signal is of the same frequency but out of phase. Thus we receive the signal
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When the signal returned it is multiplied (beating) by the outgoing signal to obtain
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We apply the trig identities
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to obtain
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The terms $\cos 4\pi \omega t$ and $\sin 4\pi \omega t$ oscillate at twice the frequency of the original signal. By averaging over product over a period long compared to $1/\omega $ we obtain zero. The remaining terms are the sine and cosine of the phase.

Do we have the distance now? Not quite. If the distance is less than 1 wavelength, then the answer is unique. If the distance is more than 1 wavelength, then we need to number of integer cycles. Surveying instruments use this and make phase difference measurements at multiple frequencies, then solve the resulting system of equations to determine the distance.

References

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Exercises

  1. Show that the corner cube reflector reflects light by $\pi $ radians when the corner is $\pi /2$ radians. What is the angle of reflection when the angle of the corner is $\phi $ radians?

  2. Suppose you have the phase angle pertaining to exactly two different frequency reflections. How can this help you better obtain the distance between the GPS and the satellite?

  3. Explain why solving the system
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    given the positions of and distances from three satellites does not yield the unique, exact GPS receiver position. There are after all, three equations and three unknowns. Note there is no time offset in this situation.

  4. One way to solve the system above is by squaring both sides and computing differences of pairs of equations and solving the resulting system. Show that with four satellite distances, we can convert the system to a linear system for the three unknowns $x,~y,$ and $z.$
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    What is the linear system?