Navigation is an old technique that allows a person to locate a position on Earth. For this, one needs known reference points from which one’s own position can be derived. Satellite navigation provides moving references, i.e. satellites. The most prominent satellite navigation system is the Global Positioning System (GPS) (see corresponding section below). While few know how it actually works, most people know that is used in car navigation and in modern smartphones.
To derive one’s own location on Earth, the basic step is determining the distance to the satellites. From this, the desired location can be calculated. This technique is called trilateration. To measure the distance to those satellites, the constant speed of light is used. If one knows the duration a signal takes to travel from the source to the receiver, the known velocity directly yields the distance.
An analogue to this technique is estimating the distance to a thunderstorm by noting the time between a flash of lightning and the sound of thunder. Since lightning arrives almost instantly, the travel time of the following thunder can be converted into distance. The conversion is done by using the sound speed. Under normal circumstances, the sound of thunder travels 1 km in approximately 3 seconds (v = 343.2 m/s). If several observers calculate the interval between lightning and thunder from different positions, the location of the thunderstorm can be determined.
The challenge with satellite navigation is to exactly measure the signal travel time. Each transmitted signal contains a code that provides information about the corresponding satellite, its position and the time of transmission. Each satellite is equipped with an extremely exact and synchronised atomic clock. On receiving the signal, the clock inside the GPS receiver helps calculate the time the signal needed to travel.
Imagine a satellite sends a signal. After a given duration, the signal will have covered the same distance in all directions. Hence, it is transmitted on the surface of a sphere.
Figure 1: Configuration of two satellites. Position A on Earth receives the signals after 4 and 5 seconds, respectively. The two signals intersect at points A and B. Since only A is on Earth, B can be discarded (own work).
For the rest of this activity, the three-dimensional configuration of Earth and the satellites will be represented by simplified two-dimensional illustrations. The surface of the satellite signal wave front will be depicted by circles instead of spheres. Consequently, the intersections of two satellite signals are points instead of lines or arcs. This is done to improve the visibility of the illustrations.
The principle of positioning is shown in Figure 1. At an unknown position on Earth, the signals of the two satellites arrive at 4 and 5 seconds, respectively. Therefore, the location must be at one of the two intersections. Since only A is on Earth, position B can be discarded. However, this only works well, if the signal travel time is measured very accurately. Unfortunately, the clocks of the GPS receivers are usually quite inaccurate. If, for example, the receiving clock is 0.5 s fast, the measured intersection is A’ instead of A in Figure 2. The positioning is equally inaccurate. Therefore, at this step, the term pseudorange is introduced, which is the assumed distance to one of the satellites derived from the face values of time measurement.
Figure 2: The same configuration as in Figure 1, but this time with a receiving clock running 0.5 s fast. Therefore, the signal transmission times are measured to be 0.5 s longer. This results in an intersection at A' instead of A (own work).
This error can be reduced by adding another satellite (Figure 3). Again, the signal travel time is erroneously measured as 0.5 s later. This configuration is different from the previous situation with two satellites as it produces three intersections. It now becomes obvious that the measurement is inaccurate, and the true position must be surrounded by the apparent ones.
The GPS receiver notices the incompatible pseudoranges. It initiates an algorithm that changes the time of the receiver clock until a common intersection is found, or the difference between the pseudoranges is minimised. The GPS receiver clock can be corrected accordingly, and that speeds up the following positioning attempts. Each additional satellite improves the precision.
As mentioned before, this illustration uses a 2-D configuration. In reality, at least four satellites are needed to locate a position on Earth with acceptable accuracy.
Figure 3: A third satellite is added, whose signal travel time is again measured wrongly. This leads to three intersections A' surrounding the true position A (own work).
Modern applications of satellite navigation
Initially, satellite navigation was developed for military purposes. Today, it is crucial for many modern commercial businesses and transport services. Often, we do not even notice that satellite positioning is involved. Different navigational techniques that were introduced for a large variety of transport vessels have now been replaced by satellite navigation (cars, trucks, ships, airplanes). They play a role in services as simple as package tracking and as complex as helping businesses distribute their goods. Satellite navigation can also help save lives; for example, an autonomous emergency system can automatically transmit the location of a car that has been involved in an accident.
GPS and Galileo
Global systems of satellites used for positioning and navigation are called global navigation satellite systems (GNSS). The most renowned GNSS is the GPS (Global Positioning System), or officially Navstar GPS. It was developed by the US military in the 1970s. Currently, it consists of 32 satellites, of which at least 24 are always operational. They are orbiting Earth at an altitude of 20200 km and on 6 trajectories inclined with respect to each other.
Although it was originally designed for military purposes, its full capabilities have been available for general and civil use since 2000. Nevertheless, the USA holds the right to artificially reduce the accuracy of the GPS for tactical reasons at any time. Since many civilian applications rely on fully working services of the GPS, they are potentially in constant danger to fail.
This is one reason, and not necessarily the least, why the European Union (EU) decided to develop their own GNSS called Galileo, to be controlled by civilian authorities. However, it is also intended to serve tactical, security and defence purposes. Galileo will be compatible with other GNSS like the US-based GPS, the Russian GLONASS as well as the Chinese Beidou.
At the same time, the USA is currently working on a third generation of the GPS. The new satellites will be equipped with an additional signal band that makes them compatible with the Galileo system. Moreover, they will lack the option to intentionally reduce the accuracy of the positioning.
In 2017, the fleet of 30 Galileo satellites, of which 6 are spares, is still incomplete. They will be in three orbits inclined with respect to one another and at an altitude of 23222 km. First services were initiated in 2016. Full operations are expected to start in 2020.
Figure 4: Illustration of the satellite orbits of the Galileo GNSS (Credit: ESA/P. Carril).
The accuracy of Galileo is expected to be of the order of 4 m without any additional corrections. This is about 3 times better than what the GPS can achieve. Galileo can reach higher accuracies and precisions by including terrestrial reference stations (differential GPS). In this way, Galileo will be accurate enough for applications in aerial, marine and land navigation. In common car navigation devices, it will be possible to determine the driving lane.
Figure 5: Computer image of a Galileo satellite (Credit: ESA/P. Carril).
Radio signals are electromagnetic waves that travel at the speed of light. In vacuum conditions, its value is 299792458 m/s, according to the BIPM (Bureau International des Poids et Mesures).
Graphical approximation of centre of mass
At the end of this activity, the students have to determine their location inside an area of intersection constructed by arcs drawn with compasses. A real GPS receiver runs an interpolation algorithm to determine the location and the clock correction. The students will use a simpler approach that approximates the correct result.
Given is an irregularly shaped area confined by four lines (Figure 6). The first step is to determine the bisectors of each of the four lines. In the second step, the opposing bisectors are connected with lines. This results in two crossing lines whose intersection is an approximation of the centre of mass.
Figure 6: Illustration of how to approximate the centre of mass of an irregular tetragon. The shape is confined by four lines (left). After finding the bisectors of each line, the opposing bisectors are connected with a line (right). The intersection is an approximation of the centre of mass (own work).
An average value is defined as a number that is the minimum of the sum of differences between the average and the individual values. In statistics, it is used to calculate a representative number when many values that scatter around the true value.
If t stands for a given value of a time measurement, we can indicate a series of measurements by adding an index, like e.g. t1, t2, t3, … which corresponds to the first, second and third value, respectively. In order to calculate the average of a series of measurements, one has to calculate the sum of the individual values and divide it by the number of measurements. For three temperature measurements, the average would be calculated as follows:
Or in general:
t ̅ is the symbol for the average of times measured, and n is a natural number that corresponds to the number of measurements.
Navigational technique to determine a position by measuring the distances to at least three reference points.
The calculated distance to the reference points used in trilateration without applying the necessary corrections. The pseudorange is purely based on the face values of the underlying measurement.
The mid-point of a straight line.