November 7, 2019

GPS has already become an integral part of our lives, and you can see a few useful applications from these examples. GPS is really an interesting technology. It uses a system of 24 satellites continuously orbiting the earth, and requires at least four satellites to track your location; it uses an atomic clock, and the time error of your mobile phone is also a matter of great concern. Moreover, Albert Einstein’s theory of relativity plays an important role in GPS technology, finally, a real-life application for the theory of relativity!

A detailed webpage version of the video is given below.

Let’s put aside all these complications and understand the technology of GPS in a step by step and logical manner. Let’s assume that your friend wants to find out your location, and you have a mobile phone, which has an integrated GPS receiver. In GPS, an interesting mathematical technique called trilateration is used to locate someone’s position. Let’s first understand trilateration in a two dimensional way.

At least 2 satellites are required to find out your position in two dimensional trilateration (Fig:1A). Using some engineering techniques, the distance between the satellite and your GPS receiver is measured. We will see the techniques for doing this later. Now things are easy. The first satellite knows you are at a distance of R1. So, you should be somewhere on this circle. The second satellite knows you are at a distance of R2. So, you should be on this circle as well. This means, your actual location should satisfy both these circles, in short you should be on the intersection points. Now there is a small issue, there are two intersection points, so which is your final position? For this you take the earth’s surface as the third circle and eliminate the improbable solution (Fig:1B).

In the three dimensional world you can also use the same approach. Here, instead of 2 satellites, we need 3 satellites as shown in Fig:2A. In the three dimensional world the satellite knows you are somewhere on a sphere. With the use of a 2nd satellite, your position narrows down to a circle. Note that the intersection of two spheres gives a circle. Now, with the help of a third satellite you will be able to narrow down your location to just two points. Here the intersection of a circle and a sphere gives 2 points. Just like in the previous case, using the earth as the fourth surface, we find the correct point, the three spatial coordinates (Fig:2B).

Now, let’s see how the distance between you and the satellite is measured. All the satellites are equipped with a very accurate atomic clock (Fig:3A). The satellite sends an intermittent radio signal down to earth. This radio signal will contain the exact time the signal was sent and the position of the satellite. Assume the receiver also has a very accurate clock. The receiver on earth receives this signal. A typical smartphone GPS receiver is shown here in Fig:3B.

Since radio waves travel at the speed of light, your receiver receives the signal after a certain time duration. By finding out the difference between the sent and received times and multiplying it by the speed of the light you will be able to find out the distance between you and the satellites. Since the satellite has already sent you its coordinate you can easily build a sphere around the satellite center point and find out your position as explained before. One thing to note here is that the time measurement has to be very accurate. Even an error of microseconds will give an error in the range of kilometers, since the speed of light is so huge.

Distance = (t_{2} - t_{1}) × c

Here comes the main issue. Your receiver does not have a highly accurate clock. Your mobile phones or laptops work on crystal clocks they are not accurate when compared to atomic clocks. Having an atomic clock in a smartphone is simply impractical (Fig:5A). You can easily see how inaccurate your smartphone clock is, compared to an atomic clock, by checking the time settings. We call the difference between the actual time, and the time measured by your mobile phone, as time offset. This time offset will cause a huge error in GPS calculations. How do we overcome this issue? The good news is that the time offset of your smartphone with all three of the satellites is the same, since the satellites all keep the same time. The time offset value of your device becomes the new unknown. This means apart from the three spatial coordinates, we have to solve the time offset value of your receiver as well. We need an extra satellite measurement to solve this fourth unknown, and that is why we need four satellites to measure your location. This way we avoid the need of an atomic clock in your mobile device. If you check your current GPS constellation, it will be clear that at least 4 satellites can see your location at any point in time (Fig:5B).

Please hold on, this article is not yet over! We have one more issue to solve. Even with all these advanced technologies, this GPS system will not give you the right location. Here comes the importance of Einstein's Theory of Relativity. Time is not absolute, it depends upon many other factors. According to the Theory of Special Relativity, a fast-moving clock will slow down as shown in Fig:6A. The atomic clocks, which are moving at a speed of 14,000 kilometers per hour, will slow down by 7 micro seconds every day due to this. At an altitude of 20,000 km above the earth, the satellites experience one quarter of the earth’s gravity. Thus, according to Einstein's General Relativity Theory, the clocks will tick slightly faster, in this case around 45 microseconds every day(Fig:6B). This means a net 38 microseconds offset is created every day in the atomic clock. To compensate for this, a Theory of Relativity equation is integrated into the computer chips and adjusts the rate of the atomic clocks. Without this application of the Theory of Relativity, the GPS would have produced an error of 10 kms every day (Fig:6C).

GPS is a navigation system developed by the US Department of Defense and is completely free for the public. However, there are accurate alternatives available in many countries nowadays. Modern receivers simultaneously make use of GPS and other navigation systems to get the most accurate position.

##### GPS

##### GLONASS

##### GALILEO

##### BeiDou

##### NAVIC

Now, a quick question. Does GPS require an internet connection? GPS does not require an Internet or cell phone signal. However, with their help, GPS startup can be greatly speeded up. Satellite location information can be downloaded via the Internet rather than direct satellite downloads, which are very slow. Such GPS systems are known as Assisted GPS (Fig:7).

So, the next time you track your food delivery or navigate your car, please keep in mind how important the Theory of Relativity, developed by Einstein, and the other mathematical ideas are behind GPS.

This article is written by Sabin Mathew, an IIT Delhi postgraduate in mechanical engineering. Sabin is passionate about understanding the physics behind complex technologies and explaining them in simple words. He is the founder of Learn Engineering educational platform.

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