Essay: Global Positioning System Principles

Abstract – The key problem in navigation now is to establish vehicle position (if cars, planes or ships), this operation is performed by several methods: (1) by observing the environment, (2) using the stars (celestial navigation),(3)estimate (measurement of speed and direction of travel) and (4) by electronic means. Electronic positioning processes are: (a) using radio waves (radio navigation, radio-positioning), (b) with radar and (c) inertial navigation systems (using gyroscopes, accelerometers, speedometers). Currently, all systems are used as complementary, not competing.
Keywords – Global Positioning System; Navigation; Satellites;

GPS is currently the only radio navigation system, fully functional , GPS NAVSTAR official name (Navigation Satellite Timing and Ranging Global Positioning System). The system was developed by the Ministry of Defence in the USA (United States Department of Defense – DoD). Although operating costs and replacement of satellites (about $ 400 million per year) are fully covered by the US, system is a public good – and can be used for free by anyone.
The GPS system was designed and built for military purposes and up through 1983 was only available to US army. There was a time when, under license, the system could be used in civilian applications, particularly in air transport. Since 1995, it was opened for civilian use, without restriction, but with lower performance than the military. After 2000-th, all restrictions were removed. Thus, before 2000-th, errors in positioning for civilians were about 100 m horizontal and about 156 m in altitude; after 2000-th, the maximum error is 20m, but may be less than 1-3mm. Additionally, the GPS system provides precise time with an error of 60 to 5 nano-seconds. GPS provider ensures for optimum operating conditions, the following performances: horizontal positioning error: ‘ 13m; vertical positioning error: ‘ 22m; error in determining the time: about 40ns.
GPS structure NAVSTAR GPS – consist in three distinct segments: space segment, control segment and user segment.
A. Space segment
It is composed from minimum 24 satellites in 6 orbits (4-5 satellites in orbit) inclined at 55 ?? to the equatorial plane at 20.183km altitude, with a revolution period of 11h and 56minutes. This constellation structure ensures virtually any point on Earth, visible satellites simultaneously 5-8 hours 24 /24. Each satellite is equipped with atomic clocks for accurately time measures.
B. Control segment
Consist of five ground stations for tracking and control. The main station is in Colorado Springs at the base of the US Air Force. Other stations are located in Hawaii, Kwajalien Atoll, Ascencion island and Diego Garcia island. They receive satellites signals, measure the characteristics and send them to the main station, which sends correction signals to each satellite.
C. User segment
It is build from receivers GPS signal positioned on the ground,on air (aircrafts) and on the sea (ships).GPS receivers ensure detection of radio signals, decoding the information and send data regarding: position (latitude, longitude and altitude in the WGS 84 system ), speed and time (current time).

Principle of Trilateration in the space – GPS satellites transmit very precise time informations so that a receiver can synchronize its clock with the satellite with a very small error. The received signal at a time contains even the time of emission (T’) and the receiver using its own clock sets the reception time (Tr). Knowing the speed of signal propagation (EM wave from RF) c can calculate the distance from the satellite S1 with known coordonate ( xs1, ys1, zs1), to the receiver R r1s =(Tr1-T’1) c = ‘ T1 c , so the receiver is on a sphere centered in the
S1(xs1, ys1,zs1)
and radius R r1s
In the same way we can determine which is the receiver sphere and position regarding the other two satellites
S2 (xs2, ys2, zs2) and
S3 (xs3, ys3, zs3) :
R rs2 = ‘ T2 c
R rs3= ‘ T3 c
Obviously, the receiver is at the intersection of the three spheres -( see figure 1), and to determine its coordinates R (xr, yr, zr) is a simple matter of trilateration.
Unfortunately, the procedure can not be applied in practice only in principle, due to errors – calculated radius is not “real”. Currently in use are two more acurate methods for GPS positioning: absolute positioning and differential positioning.

Trilateration positioning

A. Absolute positioning
Absolute Positioning involves a single passive receiver (not emiting ) and collection of data from several satellites (at least 4) to measure latitude, longitude and altitude – positioning in 3D (three dimensions). As any experimental measurement, GPS estimate position is a more or less accurate. The more satellites are available, the accuracy is better. Due to various errors, the receiver can calculate the approximate distance to the satellite called pseudo-distance or pseudo-range PSR:

PsR ‘ (Tem ‘ Tr m ) c

PsR ‘ (Tem ‘ Tr m )c ‘ (Te true ‘ Tr false) c ‘ ‘Te c ‘ Ra ‘ ‘Te c ,

and then for true Ra :

Ra ‘ PsR ‘ ‘te c

Te m , Tr m ‘ broadcasting time and reception of the measured signal ;
(??Tm = Te m ‘ Tr m ‘ measured propagation length);
Ra ‘ “true” receiver – satellite distance (and unknown);
??Te ‘ error in time propagation measurement;
c ‘ speed of light.


In a system Oxyz, the positions of the 4 satellites Sk(xsk, ysk, zsk) are known with an little error; receiver position R(xr, yr, zr) is unknown (see figure 2). True distance is:


B. Differential GPS Positioning
There are plenty applications where several meters errors (inevitable to GPS absolute positioning) are unacceptable. To drastically reduce the errors to less than 1 cm, differential GPS positioning (Differential GPS – DGPS) can be used.
The principle of differential GPS presume the existence of an reference station (SRef) – (see figure 2), placed in a very accurately determined point and known position. SRef determines its position using 4 satellites with an known error; in fact, determine the error in pseudo-radius calculated for each satellite (??xsref k, ??ysref k, ??zsref k, – and differences between PsRk and the true known value). User Receiver (UR) determine its position using the same 4 satellites as SRef but with a unknown error.
Comparing several sequences in the (UR) and SRef measurement results, corrected values can be obtained based on the observation that errors in the determination of (UR) and PsRk are very close to those of SRef and can be used to correct the results from the (UR).
Obviously UR and SRef can’t be too far (maximum 200km) and should communicate one with each other.
Differential GPS can be achieved by two techniques:
DGPS based on propagation time measurement (error falls below 1m);
DGPS based on measuring the phase of the carrier signal (error falls below 1cm).
DGPS applications may vary, depending on the size of the reference station coverage: local DGPS; regional DGPS and wide area DGPS.
The reference station computer is able to : determine time travel for the wave from the known position of the receiver and satellite positions received from them (in the messages); time travel measure from any GPS receiver; propagation errors are calculated for each satellite in the vision area; report common errors for GPS receivers that request that.
Usually, systems are working on a post-processing frame: satellite data are collected and processed in plants, after which a link between SRef and UR is established for necessary data correction and sending to a computer located at the user (UR).
Software (programs) of company (manufacturer of GPS receivers) are usualy used by the computer. The use of DGPS systems is expensive (software and access to reference stations is expensive) and not always justified.

C. Errors in establishing the right position using GPS sistem
When GPS position is determined, so-called “dilution of precision” errors are inevitable. Among the most important causes of errors are:
‘ Satellite time – although satellites have atomic clocks with very good stability (error is about 1-2 to 10’? parts per day) with regular synchronization, time errors about 10ns / day, in positioning about 3m are common .
‘ The position of satellites (ephemeris) – although it is always determined by the control stations, position is known with errors of order 1 to 5m due to the impossibility of determining movement effect for satellites.
‘ Speed of light – changes in the atmosphere and ionosphere can not be regarded as constant. The speed of light varies depending on air density and the content of particles with electric charge. Moreover, if the angle of entry into the atmosphere is different from 90??, refraction occurs, altitude is changed and the index of refraction varies with the density of the atmosphere. Currently, satellite message include a model of refraction and correction can be made to reduce the error by 66%.
‘ Multi-path propagation due to reflection can produce substantial errors. In present, these errors are virtually eliminated. Based on measurement of the 2 gaps, reflected wave reaches a lower level and slightly later than the direct wave , reflected signal can be removed.
‘ Propagation time measurement – is based on the received signal moment for running a level transition – this is done with an error of 10 – 20ns (magnitude of the signal period) and consequently a error position of 3 to 6m . The error in the determination of level transition is more sensitive to: noise , reflections and refractions of the signal.
‘ The satellite orbit geometry – if the 4 satellites are close, error in positioning is large; there is a “geometric dilution of precision” (GDOP – Geometric Dilution of Precision). Receiver position is determined based on distance to the satellite calculation. These distances (rays) are determined with errors, so between a minimum and a maximum. As a result, the receiver position will be in a limited volume of minimum and maximum values of determined 3-rays.
‘ The error due to ambiguity in establishing the phase arises because of the way in which receiver time is known. Due to the nature of the carrier (a sinusoidal signal), shift phase can be located at different time, which means the satellite location may vary with distances by 1 to 4 wavelengths (20 to 100cm). To eliminate the ambiguity various ways are used : supplimentary information (sent encrypted); causing changes in satellite position or using more satellites than is strictly necessary. US Federal Aviation Administration measurements, in long journey, for the whole constellation of satellites visible by receptors on various distributions (in 2004 year) indicated the following average values of errors:
‘ error in vertical position (68.3% from results): 12,8m
‘ error in vertical position (95.5% from results): 25,6m
‘error in horizontal positioning (68.3% from results):10,2m
‘error in horizontal positioning (95.5% from results):20,4m

reflected wave

direct wave
GPS reciver

fig. 3

direct and reflected wave for GPS satellite reciver


The Global Positioning System is managed by the Ministry of Defense of the United States and is based on between 24 and 32 satellites located in a “constellation” orbiting the planet at medium earth orbit. This implies a distance of at least 20,000 km above the earth’s surface, but not higher than the orbit used by TV satellites, communication satellites and internet, or weather satellites, the so-called reporting orbit, which is at an altitude of approximately 35,000 km. If the number of satellites of the Global Positioning System is higher, the better the accuracy of the received data. If one satellite fails, or sends wrong signals, the reciver can use the information provided by another satellite in the constellation. For example, emergency services can use a GPS device not only to find the fastest route to an incident, but also to signal the location of the accident so that other teams can quickly find the place. This function is particularly useful for search and rescue crews at sea and be useful in extreme weather conditions, when every minute counts. Scientists and engineers, can use GPS devices for scientific experiments and monitoring geological activity such as earthquakes or volcanic eruptions. They can use strategically positioned GPS devices to help them in tracking climate change and other phenomena. GPS can now be used to create highly accurate maps. Airlines, shipping companies, freight terrestrial carriers, and drivers everywhere use GPS to find vehicles or to follow the fastest route that can take you from point A to point B.


[1] “Electronic Navigation Systems”, Laurie Tetley, David Calcutt Butterwoth & Heinemann, Oxford, 1988, ISBN 0 7506 51385
[2] “An Introduction to the Global Positioning System. What It Is and How It Work”, Gregory T. Frenh, GeoReserch Tnc, Bethesda, USA, 1996, ISBN 0-9655723-0-7
[3] “Global Position Systems, Inertial Navigation and Integration” Mohinder S. Grewal, Lawrance R. Weill, Angus P. Andrews, J. Wiley, 2007, ISBN 13 978-0-470-04190-1

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