# Global Positioning System (GPS)

Created | Updated Jan 28, 2002

Finding out where you are and finding the way to where you want to go have always been two basic and crucial skills for all travellers. The use of map, chronometer, compass and sextant have been taught to sailors and navigators for centuries. Today, these skills seem to become outdated, and the Global Positioning System (GPS) appears to be taking their place.

A very complete and accurate description of GPS can be found on the Trimble Website, where some of the research for this entry has been made.

### What is GPS?

GPS is a very sophisticated system consisting of three major segments:

**Satellites**: 24 NAVSTAR^{1}satellites are orbiting the earth, equipped with atomic clocks that will keep accurate time to within three nanoseconds (three billionths of a second, or 0.000000003 seconds). Light will travel about 1 metre in that time. The satellites are not geostationary. Their orbit level is approximately 20,200 km above the earth, and it takes them 12 hours to go around the Earth once.**Receivers**: These devices detect and process signals from the GPS satellites and display the desired information (exact position and time, speed and direction of movement). They fit easily into a car, plane or boat as they are no larger than a car radio. Hand held receivers are usually the size of a cellular phone.**Ground Control Stations**: These unmanned stations monitor the overall status of the system, especially the satellites. One of the most important jobs of the ground control stations is to check the satellites' positions. If they are where they are supposed to be, everything's fine. If they aren't, the offset is measured and sent to the satellites, which send it back to the receivers.

### How Does GPS work?

The basic principle behind GPS is just some very precisely calculated spatial geometry, or 'triangulation'. You need to know the exact distance from four known points to determine a fifth point. To understand that, let's add one dimension after another:

**One-dimensional**: If you know one point (base) on a straight line, and you choose a second point at random, you can measure the distance between these points. What this distance does not tell you is the direction; is the random point left or right from the base? If you know a second point (second base) and measure both distances, you'll have two possible random points for the first base and two for the second base, but only one of these points will be the same for both bases.**Two-dimensional**: Choose a random point on a piece of paper and start with one known point (first base). If your only information about the random point is its distance from the first base, you'll only know that it is on the perimeter of a circle with the distance as its radius and the first base as its centre. Add a second base and a second circle and you have two possible points (where the two circles intersect). The third circle will rule out one of these points.**Three-dimensional**: This is the 'highest' dimension we can visualize, but the basic principle is valid for any n-dimensional space^{2}. Take one random point (the position of your GPS receiver). Measure the distance to a known point (satellite one), and you'll know that you're on the surface of a sphere around the satellite. Add a second base (second satellite), measure the distance and your position is limited to the perimeter of the circle where those two spheres intersect. Add a third base (third satellite), measure the distance, and you'll get two possible positions, where all three spheres intersect. The fourth sphere will rule out one of these points, and you have an exact GPS position, a so called 'fix'.

This means that the positioning problem can be broken down to two tasks: knowing the exact positions of your bases (the satellites) and measuring the exact distance to each of them:

**Knowing the exact position**: The orbits of the satellites are very precise and very predictable. They have been sent into space according to the so called 'GPS Master Plan', which is also implemented in the receivers. Due to the feedback from the Ground Control Stations the satellites know their exact position at any given time. This position is continuously sent to the receivers.**Measuring the distance**: The satellites send a constant stream of highly accurate positioning and mega-accurate timing signals, along with a pseudo-random signal. The exact position of the satellite (synchronized by the ground control stations) is also encoded in the signal stream. The receiver compares the signals it gets from the satellites with the signals that it supposes it should get ('If I was at the satellite's place, what signal would I send right now?'). These will be slightly out of sync, as the satellites' signal needs some time to reach the receiver. If a satellite is exactly above the receiver (20,200 km high) it takes the signal 0.0591 seconds to reach it. Once this time is known, it's a simple multiplication of speed of light x time = distance^{3}.

For obvious reasons it is crucial that the clocks in the satellites and the clocks in the receivers are absolutely in sync. The only way to achieve this seems to be to have an atomic clock in each receiver. As an atomic clock costs some $50,000 to $100,000 and is a really heavy object to transport, another solution had to be found: the receivers are equipped with standard quartz clocks. If they are not exactly on time (compared to the accuracy of an atomic clock), the receiver will get a perfect 'three-sphere-fix', but the accuracy of this fix will depend on the offset of the receiver's clock. The fourth satellite will show that there is no point in space where all four spheres overlap. The receiver will notice that its own clock is wrong and will start to fine tune it slightly until it gets a 'four-sphere-fix'. That means that it is in perfect sync with the satellites, and as an additional bonus the receiver will show the accurate time.

### Is there a way to explain that sphere's model more understandably?

Yes, there is. Soap bubbles. If you make one bubble (ie, one sphere), the position is anywhere on that bubble. Attach a second bubble to the first one, and you'll get a common circle. Attach a third one, and you'll get a common line with a starting point and an end point. Attach a fourth one, and those four bubbles will have exactly one common point. That's why it is possible to make a 'fix' with a minimum of three satellites, provided that your receiver is absolutely in sync with the satellites' clocks; the surface of the Earth will be the fourth sphere^{4}.

This example has one weak point; soap bubbles change their form and size to achieve a minimum energy status, satellites don't. Please understand it only as a visualization of why four spheres are needed to define one specific point in a three-dimensional space.

### How about the Accuracy?

The information provided by GPS can be lifesaving, but also deadly - just imagine a cruise missile flying through your door. That's why the so called 'selective availability' had been introduced. This means that regular GPS receivers provided an accuracy of about 30m-100m, whereas some special receivers were ten times more accurate. Selective availability was turned off in May 2000. Today all standard GPS receivers have an accuracy of approximately 3-10m in three dimensions.

It might be interesting to know that even today, as selective availability has been turned off, there is still a significant difference between the C/A-code and the P-code receivers. C/A-code means that the satellites do not send their exact position, thus creating some inaccuracy. This code is commonly available and is used by the standard GPS receivers. The P-code is encrypted and can be read only by US military and their allies. It is much more accurate.

GPS is still a military thing. The accuracy is regularly deteriorated in crisis areas.

### How can this accuracy be increased?

If you mount a GPS receiver to an exactly known place and plot its reading to a map, you'll notice that the GPS position roams around the real position^{5}. As you know exactly where your fixed GPS receiver is positioned, you can exactly determine the offset at every single moment. This offset is the same for any other GPS receiver nearby. If you can determine it, you can rule it out. If you know from your fixed receiver that the error was 5m to a heading of 195, just add 5m and a heading of 15 (ie, the opposite direction) to the reading of your mobile unit and you'll get a 'fix' that's accurate to within centimetres. The whole correction can be made automatically and is called **D**ifferential GPS or DGPS.

Another way to increase accuracy is to wait. If you have enough time, don't move and average the positions that your receiver gives you. After a couple of hours the average position may be accurate within centimeters. This is no solution if you need real-time navigation on moving objects, like planes, ships or cars, of course.

### Who Needs GPS?

Everybody who needs to know exact positions can take advantage from GPS. Navigational purposes (airplanes, ships, cars, bikes, hikers) are quite obvious, but even in modern farming some machines are equipped with GPS, allowing exact fertilizing without under- or overdoses. Modern container terminals use GPS to store and retrieve their containers. It has been said that 'Anything that happens, happens to happen somewhere'. If you have to know where this 'somewhere' is, you need GPS.

^{1}

**NAV**igation

**S**ystem with

**T**argetting

**A**nd

**R**anging.

^{2}In a six-dimensional space, we would need seven bases and seven distances from these bases to measure a random point.

^{3}Some factors are neglected in this entry: The speed of light will be slightly slower in the atmosphere, the signal from the satellites may not reach the receiver directly ('multipath error'), the satellite may have left its exact position due to the moon's or the earth's gravity, etc. The whole integrated system will rule out these error sources extensively.

^{4}To be exact, the surface of the Earth is not exactly spherical. Even if you ignore all the mountains and valleys, the earth would still be a 'rotational ellipsoid', which means it is flattened from pole to pole due to the centrifugal force of its own rotation. This is one of the factors that are taken into account by the receivers.

^{5}Take a laserpointer, hold it with your arm stretched out and try to aim at one point. The trembling of your muscles will cause the laser point to wander around your aim.