GPS technology

In satellite navigation, GPS technology is the technical , geometric and electronic basis of the English Global Positioning System (GPS) and similar global navigation systems in conjunction with satellites .

Geometric basics

Spheres as geometric places

With the Global Positioning System, the location of the receiver - more precisely: the position of the receiving antenna - is determined by measuring the distance to several satellites that orbit the earth on high orbits. A satellite is not enough, because with a single distance measurement the receiver can be located anywhere on a spherical surface, the radius of which is the distance and the center of which is the current satellite position. Only further distance measurements narrow down the uncertainty of the location. If the recipient is z. B. on the surface of an idealized globe , its location can only be on a circle that forms the intersection of the distance sphere / globe.

When measuring distances to two satellites, the receiver can only be located in positions on both range spheres. The geometric location (the line of intersection) is again a circle, but depending on the satellite constellation now in a general spatial position. Theoretically, two distance measurements to a satellite are also possible, but such positionings are imprecise ( dragging cut ).

If a distance measurement to a third satellite is added, the receiver location is given by the intersection of the three spheres. Because only here does the receiver have the measured distances to all given spherical centers (satellite locations). Mathematically, this is solved in a Cartesian coordinate system :

In a geocentric defined 3D coordinate system is a point by specifying three coordinates ( , , ) uniquely determined. To determine the location coordinates of the receiver position, three equations are required. They result from the distances to the three centers of the sphere (the satellites). The solution of the equations is always possible in real cases (the theoretical second point of intersection is omitted because it is far from the earth's surface). If arbitrary locations and distances were specified, there would be no solution if the spherical surfaces did not intersect - which is always the case with real satellite navigation tasks . ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$

The implementation of this principle has two technical difficulties with GPS:

• The distance cannot be measured directly and
• the satellites move (at approx. 3.9 km / s), whereby the positions of the spheres' centers change continuously.

To solve the first difficulty, one measures the runtime of the radio signal, i. H. the short time it takes to travel from the satellite to the receiver. In space, the signal propagates at the speed of light in a vacuum , and slightly more slowly in the earth's atmosphere. The measuring principle is analogous to determining the distance during a thunderstorm : If you see lightning (the duration of the lightning itself is negligible), you count the seconds until the thunder can be heard. The duration of the thunder times the speed of sound gives the distance. The time of day lightning and thunder does not matter. ${\displaystyle c}$

With the GPS system, however, determining the transit time is more complicated because, on the one hand, the propagation speed is much higher. On the other hand, the receiver does not initially know when the received signal was sent. Since the transmitter and receiver of the radio waves cannot communicate with each other in the GPS system, it is a unidirectional (one-way) distance measurement. Technically, one speaks of pseudoranging because all runtimes are falsified by a constant amount.

The distance measurement is carried out in that the received signal "brings along" its transmission time as a code - namely as the GPS system time at the moment of transmission. If the receiver had the exact GPS time system, it could determine the transit time as the difference between the sending and receiving time. However, the receiver clock is initially not exactly synchronized with the GPS time. The difference is used as an additional unknown when determining the location. Because four unknowns (three position coordinates and a time constant) have to be determined, four equations are also required. This fourth equation requires adding the range to a fourth satellite.

False ranging

If the receiver clock is not yet synchronized, the calculated distances to the satellites are falsified by a small distance. If you take only three of these four distances, you can determine a possible (but incorrect) location of the recipient. However, it does not match the fourth distance - the less, the greater the “clock error” of the recipient. All four distances are only in harmony with the correct time of reception, for which the system has to be solved with all four equations.

Knowing the time of transmission is also important for another reason: Because the satellites are moving, their position at the time of transmission must be known. The satellite could not transmit its coordinates in that short moment. Therefore the equation of the path is used as a function of time to determine the position. The constants that the path equation contains are transmitted with the signal.

Depending on the number of known variables, the number of unknowns and thus the minimum number of satellites required is reduced . If the exact time is known, three unknowns remain for the location, which only requires three satellites. If the altitude is also known (e.g. altitude = 0 at sea level), only two unknowns and two satellites are necessary. If the exact geographic latitude were also known, only one unknown would be required, corresponding to a satellite.

Propagation speed To determine the exact location, it must be taken into account that there is no vacuum between the satellite and the receiver. The signal speed is slower in the earth's atmosphere. The satellite distance is therefore calculated too large when using the vacuum speed of light and the actual transit time must be corrected accordingly, taking into account the vertical angle of the radio link. Since the delay occurs mainly in the ionosphere , it is called ionospheric correction. The one in the troposphere ( tropospheric delay ) is also a few meters. The technique of location determination also includes the consideration of avoidable and unavoidable signal interference.

Electronic basics

Description of the GPS signal

Combined LFSR for generating the C / A code (gold code). The two selectors in the lower area of ​​the screen set the code phase position and are set differently for each satellite ("GPS SAT number").

The GPS provides two different signals: the civilian L1 signal with a carrier wave of 1575.42 MHz and the encryptable L2 signal with a carrier wave of 1227.60 MHz. The C / A code explained in more detail below is transmitted in the L1 signal and the encrypted and not publicly known P code in the L2 signal. Each of these two signals consists of three parts: carrier wave, code and navigation data.

Each satellite uses the C / A data to communicate its orbit data, the ephemeris , and the time of transmission to the receiver. In addition, it must be possible for the receiver to determine which satellite the received signal comes from - even though all satellites use the same transmission frequency. The CDMA modulation method is used for the GPS signal to meet all of these requirements . The times at which a signal part starts are always relevant, with different signal parts having different durations - but all signal parts are present at the same time.

The reference time for all signals is Sunday 00:00. The number of the week is also included in the transmitted messages. However, this information is not relevant for determining the location. The next signal unit is the subframe . A subframe has a duration of 6 s and at the beginning carries a message stating how often 6 s have passed since 0 o'clock on Sunday at the beginning of the transmission of the next subframe. The next smaller unit is the word . A subframe consists of 10 words. This smaller word unit consists of 30 message bits. The messages are carried directly by 24 bits; at the end of each word there are six parity bits in order to control error-free reception. The individual words do not carry any information about their transmission time, the transmission time of the beginning of the word results from counting their position in the subframe. Without counting, the transmission time of the beginning of the word would be ambiguous by 0.6 s.

Each message bit lasts 0.02 s (= 20 ms) and of course does not contain any further information. The exact timing of its beginning is given by counting within the word. Without counting, the transmission time of the start of each message bit would be ambiguous by 20 ms. Each message bit is divided into 20 code blocks. These 20 code blocks with a duration of 1 ms are identical and consist of a sequence of 1023 chips each . To distinguish them from the message bits, the bits of the CDMA code blocks are called chips . Since each code block lasts 1 ms and cannot carry a message, its time must be determined again by counting within its bit, word or subframe. Without counting, the transmission time of the beginning of each code block would be ambiguous by 1 ms.

A sequence of 1023 chips can be formed in 2 ¹⁰²³ (approx. 8.99 x 10 ³⁰⁷ ) ways. A sequence from the gold codes is used so that the selected sequence is unique for each satellite despite the many possibilities . Gold codes are 2 ⁿ  - 1 bits in length . n is an integer that was chosen to be 10 for GPS. The two generator polynomials of degree 10 have the following form:

${\displaystyle G_{1}=x^{10}+x^{3}+1}$

${\displaystyle G_{2}=x^{10}+x^{9}+x^{8}+x^{6}+x^{3}+x^{2}+1}$

They are combined as shown in the adjacent illustration. The specific code sequence is set by a fixed code phase on the second generator polynomial; this is shown schematically in the figure by the variable taps on the lower LFSR. In total, 2 ¹⁰  - 1 = 1023 code phases are possible, but not all code phases are used in practice. In the underlying method GPS-ICD-200 , 32 code phases were selected, with one code phase each being assigned to a specific GPS satellite and representing the GPS PRN number . The 32 code phases were selected to have minimal cross-correlation with one another .

An identical gold code sequence is generated in the receiver for each satellite received. First of all, the code sequence received and the code sequence generated in the receiver itself have no temporal relationship. To establish this temporal relationship, both sequences are multiplied with one another and the multiplication results are added. This procedure is known as autocorrelation . If the time difference is varied, the sum changes. The sum becomes a maximum when the sequences coincide in time. The code sequences were chosen with GPS in such a way that it is ensured that the maximum occurs only with the correct code sequence and with the correct time shift (uniqueness). The code sequence generated in the receiver can be adapted to the received code sequence sent by the satellite with a time error of less than 1 ns. The exact transmission time is known from the counting in the signal units and the position of the current time in the code block.

In addition to this C / A code based on a Gold sequence, the navigation message is also modulated into the L1 signal at 50 bit / s. It consists of a 50 Hz signal and contains data such as satellite orbits, clock corrections and other system parameters (e.g. the status of the satellites, i.e. whether they are OK or faulty). This data is continuously transmitted from each satellite. From this the GPS receiver receives its date, the approximate time and the positions of the satellites.

Modulation of the GPS signal on the carrier For transmission, the binary signal is modulated on a carrier frequency whose phase is rotated by 180 ° depending on the signal value (0 or 1). With a 0 the carrier frequency is sent directly, with a 1 the carrier frequency shifted by 180 °.

Determination of the transmission times

The receiver generates its own code sequence for each satellite and shifts these code sequences so that each code sequence corresponds at most to the associated satellite. This means that the temporal position of each satellite signal at the receiving location is known.

The conditions at the receiver when receiving the satellite signals

Of the satellite signals received at the same time, only the signals from two satellites are shown in the picture (red and green). One bit of the GPS message is 20 code blocks long (each of which is 1023 chips ( CDMA code bits) long). Only five code blocks are shown here. For the consideration, it is not important at what time the recipient carries out the evaluation (magenta), nor what absolute recipient time is present (labeling on the blue time axis). For the evaluation, only the time of the start of a code block in the satellite needs to be known, and the receiver only needs to measure the time between the evaluation time and the start of a code block. The transmission time of the code block is determined by evaluating the satellite message.

After determining the position of the time of reception within a code block, the position of the code block on Sunday midnight must now be determined. To do this, the recipient must always determine the next larger unit from the unit code block backwards:

• First, it is determined for which successive code blocks the sequence changes between ones (with zeros in between) and zeros (with ones in between). This change is the beginning of a message bit. This means that the distance to Sunday midnight is determined with an uncertainty of multiples of 20 ms.
  A change does not have to take place after 20 ms at the latest, as there can be several consecutive zeros or ones, so that such a change does not occur every 20 ms. But in a duration of 0.6 s (duration of a word length) there are certainly several bit changes because of the parity bits.
  After the bit boundary is recognized, the bits can be received and stored. In this bit sequence, the position of the subframes must be determined, each of which is 300 bits long.
• Then the position of a subframe is identified. For this, the receiver needs twice the subframe length, i.e. at least 600 received bits, which it receives in 12 s.
  The receiver takes a block of 600 bits from the stored bit sequence and searches for the bit sequence 10001011 in this block. This bit sequence is contained in the first word of each subframe. This first word is called the TLM word (= Telemetry Word). The bit sequence that is searched for, which is contained in the TLM word, is referred to as a preamble. With the identification of this bit sequence, the beginning of each subframe and each word is known, since all words are each 30 bits long. However, since the bit sequence 10001011 can also appear in the rest of the message, two more checks must be carried out to check whether it is really the beginning of the TLM word. Firstly: if the bit sequence found is actually the preamble, the 6 parity bits at the end of each word are correct, and secondly, a new subframe must come every 300 bits (6 s), which always begins with a TLM word. If the check is successful, the beginning of each subframe in the receiver time is known. The transmission time of the beginning of each frame in the satellite is still uncertain in multiples of 6 s.
  If the receiver detects a possibly incorrect identification, a search is made in the 600-bit block for another 10001011 sequence and a check is made to see whether it is the preamble.
  The HOW (hand over word) following the TLM word is evaluated to identify how often 6 s have passed since midnight on Sunday. The first 17 bits of the HOW indicate the time of the beginning of the following subframe as an integer count of 6 s since 0 o'clock on Sunday. This means that in the receiver time it is always known when each transmission takes place in the GPS time, since the procedure just described takes place with every received signal from a satellite.

Determination of the satellite locations at the time of transmission

The orbit parameters of each satellite are contained in the satellite message. With them and the transmission time, the satellite location can be calculated for each transmission time.

Time determination expiry

First, the receiver shifts the code block generated in the receiver itself in the receiver's own time by one chip (in the range of 1 ms) until the correlation maximum occurs. The shift takes place with a delay locked loop . This means that the start of each code block in the receiver time is known (corresponds to the arrival time), and the transmission time is uncertain in multiples of 1 ms.

With the CA code used in the civil sector, 20 code blocks with 1023 chips each (CDMA code bits) are sent per 20 ms. With bit-precise correlation, the measurement of the time difference can therefore be accurate to 1/1023 ms (9.775 · 10 ⁻⁷  s). With an assumed propagation speed of 299,792,458 m / s ( speed of light in a vacuum), the maximum accuracy of the distance measurement would be 293 m. Modern GPS receivers can determine any point in time between the beginning (rising edge) and the end (falling edge) of a bit with an accuracy of 1% of the bit length. This means that the distance s to the satellite can be determined with an accuracy of 2.93 m: s = 1% 299,792,458 m / s 1/1023 10 ⁻³  s. For exclusively military use, the encrypted P (Y) signal with a chip rate that is 10 times higher is used, which means that distances can be measured with an accuracy of approx. 30 cm.

The receiver is now looking for bit changes in the navigation message. The duration of one bit is 20 ms, which results in a transmission rate of 50 bits per second. If 0 bits or 1 bits follow one another, no change can be detected. However, since the navigation message certainly does not only consist of consecutive 0 or consecutive 1, a bit change occurs at some point. The time at which this bit change occurs is therefore known in the receiver time. Because of the 20 ms time grid, the transmission time of each bit change is only uncertain in multiples of 20 ms.

The receiver now saves the received bit sequence. In the receiver time, the start of each bit is known exactly (to less than 1 ns).

In the received bit stream, the receiver takes a block of at least 330 bits (= 10 + 1 words = 6.6 s) and searches in the block for the bit sequence 10001011 (more precisely also for 01110100, because initially it is not yet known what 0 and what is 1). This bit sequence is the preamble in the TLM word (telemetry word). With the identification of this bit sequence, the beginning of each TLM word is known, since all TLM words are each 30 bits long. However, since the bit sequence 10001011 can also appear in the rest of the message, two further checks must be carried out to check whether it really is the beginning of the TLM word: First, if the bit sequence found is actually the preamble, are the 6 parity bits at the end of each TLM word are correct, and secondly, a new subframe must come every 300 bits (6 s), which always begins with the TLM word. If the check is successful, the beginning of each subframe in the receiver time is known. The transmission time of each subframe start is therefore only uncertain in multiples of 6 s.

If the receiver detects a possibly incorrect identification, a further 10001011 sequence is searched for in the 330-bit block and a check is made to determine whether it is the preamble.

Now the HOW word ( Hand over Word ) following the TLM word is evaluated. The first 17 bits indicate the time of the beginning of the following subframe as an integer count of 6 s since 0 o'clock on Sunday.

The received bits were transmitted synchronously to the time frame of the satellite time (internal time of each individual satellite). Since the satellite time is not exactly identical to the GPS system time, correction values ​​for time determination in subframe 1 are also transmitted in the navigation message. The exact GPS time of the transmission can be calculated using the time correction values.

This means that in the receiver time it is always known when each transmission takes place in the GPS time, since the procedure just described takes place for each received satellite.

Now the transmission times of all received satellites are calculated for this receiver time at any given receiver time (the receiver will of course start immediately when all necessary information has arrived). ${\displaystyle t_{0}^{*}}$

With the help of the additional information in the messages ( ephemeris data), the three-dimensional position of the satellite at the respective transmission time is calculated for each satellite.

This means that everything necessary for this arbitrarily selected receiver time is known: the locations of the satellites and the transmission times in GPS system time and in receiver time. The difference between the time of sending and receiving results in the runtime, which is still error-prone because the receiver clock is not synchronized with the GPS system time. The time error of the receiver clock is the same for all satellites and therefore does not apply to differences between the transmission times of the received satellites.

The 3D positions of the satellites and the transmission times in the GPS time are now inserted into the GPS basic equations and the system of equations is solved. The result is the corresponding reception time in GPS time and the recipient coordinates. ${\displaystyle t_{0}}$

Location and time of the recipient

The receiver location is determined with the now known transmission times and locations of the received signals. The reception times do not have to be known either in the GPS system time or in the receiver time. During synchronization, the recipient time is assigned to the moment of reception . In the system of equations, these data are sufficient to determine the receiver coordinates and the time of reception in the GPS system time. ${\displaystyle t_{0}^{*}}$

After solving the basic GPS equations, the receiver knows its coordinates (more precisely: the coordinates of the receiving antenna) and the time of reception in the GPS time. With this he can in principle set his receiver clock with the difference to the GPS system time (synchronize), but this is only necessary if the GPS receiver is to serve as the time standard . The GPS time and the UTC time were identical when GPS was started up. The number of additional leap seconds that have passed since then are transmitted in the satellite messages. ${\displaystyle t_{0}}$ ${\displaystyle t_{0}^{*}-t_{0}}$

Signal interference

Ionospheric correction

When passing through the ionosphere, radio waves experience a delay that (in contrast to light waves) depends on the electron content (TEC) along the signal path and on the frequency used. It is generally a few meters, but in extreme cases up to 50 meters. Since the TEC is difficult to model, two frequencies are used and the correction is determined by their small difference in transit time . In the GPS system, the two frequencies do not differ much (L1 frequency = 1575.42 MHz, L2 frequency = 1227.60 MHz), but the delay has the square of the quotient and can therefore be determined with sufficient accuracy.

The refraction in the troposphere , on the other hand, is calculated using meteorological models (from the course of pressure, temperature and air humidity).

Consideration of unavoidable signal interference

Theoretically, the time could be determined more precisely by counting the bits in the code blocks. This is not possible with the received signal because the individual chips in the received code sequence are superimposed with a lot of noise . The cause is the low transmission power of the satellites, the required large receiver bandwidth and the unknown direction of the transmitter (therefore a large satellite antenna cannot be used).

Because of the limited power supply options, the transmission power of the GPS satellites, similar to that of television satellites, is only low (half of the Astra satellites, i.e. around 50 watts). Because of the rapid sequence of characters (approx. Every µs), the receiver must have a large bandwidth. This is why the received signal has such a high proportion of noise that the searched signals are drowned out in the noise.

The correlation is therefore usually carried out in such a way that the noisy signal is multiplied by the receiver's own code sequence. The required summation takes place with the multiplied signal. This is a constant signal and therefore it can be summed over a longer period of time. During this long time, the noise averages itself - and only a small amount remains. As a result, the correlated signal stands out from the noise. Because all satellites transmit on the same frequency, the signals also run through the same receiver components for a very long time in the receiver, so the transit times in the receiver do not interfere with the determination of the transit time differences.

Consideration of avoidable signal interference

These disorders mainly belonging multipath ( English multipath ), which is a mixture of direct satellite signal reflected Umwegesignalen the vicinity. Such reflections of the radio waves occur primarily on buildings and metallic surfaces with a corresponding direction of reflection, sometimes also through wet leaves.

Multipath effects can be a few centimeters to a decimeter, which is far superior to the accuracy of precise GPS positioning. They become noticeable in small, semi-periodic jumps in the displayed position and cannot be avoided entirely; They can be reduced by a suitable installation location for the antenna and a downward shielding base plate on which the antenna sits.

GPS positioning

The simultaneous signal propagation time measurement to four satellites provides four determinants of which theoretically three spatial coordinates , , and time to be determined. If the reference system is known, for GPS it is the WGS84 , terrestrial coordinates latitude and longitude are calculated from it. Without correction data for the geoid , the absolute height error is several hundred meters. ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$ ${\displaystyle t}$

In practice, a 4-channel GPS receiver cannot be used because of the high level of signal noise. Instead, the signals from all received satellites are evaluated. A Kalman filtering optimizes the solution of the error-prone, overdetermined system of equations.

Solving method for 4 equations with 4 unknowns

The GPS receiver is located at a location with the coordinates at the GPS system time and receives the radio signals from the four satellites at the system times . The simplification is that the speed of propagation is assumed to be constant and the direction of propagation to be straight. ${\displaystyle t_{0}}$ ${\displaystyle x_{0},~y_{0},~z_{0}}$ ${\displaystyle t_{n}(n=1..4)}$

A multichannel receiver can receive all satellites at the same time because they transmit continuously. Single-channel devices receive the signals one after the other and correct the time differences accordingly (positive reception time or negative transmission time). At least 4 satellites are required to determine the position of the receiver. If more than 4 satellites are received, little changes in the calculation process, but the equations are overdetermined because there are more equations than unknowns. Overdetermined systems of equations are treated according to the methods of equalization calculation.

The four assumed satellites emit their signals at system time at the locations ; the index runs from 1 to 4. The signal propagates at the speed of light . The basic equations result from equating the 4 distances between the satellites and the receiver in Cartesian coordinates and the 4 distances from the transit time measurement , i.e. H. the time difference between transmission and reception multiplied by the speed of light. In order not to need roots , the equations are written in square form: ${\displaystyle t_{0}}$ ${\displaystyle x_{n},~y_{n},~z_{n}}$ ${\displaystyle n}$ ${\displaystyle c}$

${\displaystyle {\begin{matrix}(x_{1}-x_{0})^{2}+(y_{1}-y_{0})^{2}+(z_{1}-z_{0})^{2}=[c(t_{1}-t_{0})]^{2}\quad (1)\\(x_{2}-x_{0})^{2}+(y_{2}-y_{0})^{2}+(z_{2}-z_{0})^{2}=[c(t_{2}-t_{0})]^{2}\quad (2)\\(x_{3}-x_{0})^{2}+(y_{3}-y_{0})^{2}+(z_{3}-z_{0})^{2}=[c(t_{3}-t_{0})]^{2}\quad (3)\\(x_{4}-x_{0})^{2}+(y_{4}-y_{0})^{2}+(z_{4}-z_{0})^{2}=[c(t_{4}-t_{0})]^{2}\quad (4)\\\end{matrix}}}$

The solution to this system of equations provides the transmission time and the 3 coordinates , and . ${\displaystyle t_{0}}$ ${\displaystyle x_{0}}$ ${\displaystyle y_{0}}$ ${\displaystyle z_{0}}$

Note: In the literature, the system of equations is often solved iteratively and the iteration method is interpreted physically. For is used for this . The iteration of the system of equations then takes place with different starting values. Mostly an approximate location is assumed for the recipient; In the first step, the transit times for this starting point are calculated and its coordinates are improved. This means that more precise transit times are determined and, with these, improved coordinates. Another iteration method takes approximately 70 ms as the starting value for the runtimes. In the further course of the calculation, the two iteration methods no longer differ. As a rule, the solution is sufficiently accurate after 3 to 4 iterations. ${\displaystyle t_{i}-t_{0}}$ ${\displaystyle \Delta t_{i}=t_{i}-t_{0}}$ ${\displaystyle \Delta t_{i}}$ ${\displaystyle \Delta t_{i}}$

These iteration methods are not necessary. As shown below, a closed solution is also possible.

When multiplied, equations (1) to (4) result:

${\displaystyle {\begin{matrix}x_{1}^{2}-2x_{1}x_{0}+x_{0}^{2}+y_{1}^{2}-2y_{1}y_{0}+y_{0}^{2}+z_{1}^{2}-2z_{1}z_{0}+z_{0}^{2}=\\c^{2}t_{1}^{2}-2c^{2}t_{1}t_{0}+c^{2}t_{0}^{2}\qquad \qquad (5)\\\\x_{2}^{2}-2x_{2}x_{0}+x_{0}^{2}+y_{2}^{2}-2y_{2}y_{0}+y_{0}^{2}+z_{2}^{2}-2z_{2}z_{0}+z_{0}^{2}=\\c^{2}t_{2}^{2}-2c^{2}t_{2}t_{0}+c^{2}t_{0}^{2}\qquad \qquad (6)\\\\x_{3}^{2}-2x_{3}x_{0}+x_{0}^{2}+y_{3}^{2}-2y_{3}y_{0}+y_{0}^{2}+z_{3}^{2}-2z_{3}z_{0}+z_{0}^{2}=\\c^{2}t_{3}^{2}-2c^{2}t_{3}t_{0}+c^{2}t_{0}^{2}\qquad \qquad (7)\\\\x_{4}^{2}-2x_{4}x_{0}+x_{0}^{2}+y_{4}^{2}-2y_{4}y_{0}+y_{0}^{2}+z_{4}^{2}-2z_{4}z_{0}+z_{0}^{2}=\\c^{2}t_{4}^{2}-2c^{2}t_{4}t_{0}+c^{2}t_{0}^{2}\qquad \qquad (8)\\\end{matrix}}}$

Now the 4th equation is subtracted from the first 3. This means that all square unknowns are omitted:

${\displaystyle {\begin{matrix}x_{1}^{2}-x_{4}^{2}-2(x_{1}-x_{4})x_{0}+y_{1}^{2}-y_{4}^{2}-2(y_{1}-y_{4})y_{0}+z_{1}^{2}-z_{4}^{2}-2(z_{1}-z_{4})z_{0}=\\c^{2}t_{1}^{2}-c^{2}t_{4}^{2}-2c^{2}(t_{1}-t_{4})t_{0}\qquad \qquad (9)\\\\x_{2}^{2}-x_{4}^{2}-2(x_{2}-x_{4})x_{0}+y_{2}^{2}-y_{4}^{2}-2(y_{2}-y_{4})y_{0}+z_{2}^{2}-z_{4}^{2}-2(z_{2}-z_{4})z_{0}=\\c^{2}t_{2}^{2}-c^{2}t_{4}^{2}-2c^{2}(t_{2}-t_{4})t_{0}\qquad \qquad (10)\\\\x_{3}^{2}-x_{4}^{2}-2(x_{3}-x_{4})x_{0}+y_{3}^{2}-y_{4}^{2}-2(y_{3}-y_{4})y_{0}+z_{3}^{2}-z_{4}^{2}-2(z_{3}-z_{4})z_{0}=\\c^{2}t_{3}^{2}-c^{2}t_{4}^{2}-2c^{2}(t_{3}-t_{4})t_{0}\qquad \qquad (11)\\\end{matrix}}}$

This is rearranged:

${\displaystyle {\begin{matrix}2(x_{1}-x_{4})x_{0}+2(y_{1}-y_{4})y_{0}+2(z_{1}-z_{4})z_{0}=\\x_{1}^{2}-x_{4}^{2}+y_{1}^{2}-y_{4}^{2}+z_{1}^{2}-z_{4}^{2}-(c^{2}t_{1}^{2}-c^{2}t_{4}^{2})+2c^{2}(t_{1}-t_{4})t_{0}\quad (12)\\\\2(x_{2}-x_{4})x_{0}+2(y_{2}-y_{4})y_{0}+2(z_{2}-z_{4})z_{0}=\\x_{2}^{2}-x_{4}^{2}+y_{2}^{2}-y_{4}^{2}+z_{2}^{2}-z_{4}^{2}-(c^{2}t_{2}^{2}-c^{2}t_{4}^{2})+2c^{2}(t_{2}-t_{4})t_{0}\quad (13)\\\\2(x_{3}-x_{4})x_{0}+2(y_{3}-y_{4})y_{0}+2(z_{3}-z_{4})z_{0}=\\x_{3}^{2}-x_{4}^{2}+y_{3}^{2}-y_{4}^{2}+z_{3}^{2}-z_{4}^{2}-(c^{2}t_{3}^{2}-c^{2}t_{4}^{2})+2c^{2}(t_{3}-t_{4})t_{0}\quad (14)\\\end{matrix}}}$

By reducing to 3 equations for still 4 unknowns, this system of equations is initially underdetermined. It can therefore be interpreted as a functional dependence of the searched coordinates on a variable . Since we are dealing with a linear system of equations, this dependency is also linear, i.e. H. it has the shape ${\displaystyle t_{0}}$

${\displaystyle {\begin{matrix}x_{0}=x_{00}+x_{0t}t_{0}&&y_{0}=y_{00}+y_{0t}t_{0}&&z_{0}=z_{00}+z_{0t}t_{0}\qquad (15)\\\end{matrix}}}$

with corresponding constants , etc. If you insert these expressions into the 3 equations and thus eliminate the coordinates you are looking for, apart from the variables they only contain constant values. These 3 equations must therefore apply separately for the constant terms and the prefactors of . This gives the two systems of equations ${\displaystyle x_{00}}$ ${\displaystyle x_{0t}}$ ${\displaystyle t_{0}}$ ${\displaystyle t_{0}}$

${\displaystyle {\begin{matrix}2(x_{1}-x_{4})x_{00}+2(y_{1}-y_{4})y_{00}+2(z_{1}-z_{4})z_{00}=\\x_{1}^{2}-x_{4}^{2}+y_{1}^{2}-y_{4}^{2}+z_{1}^{2}-z_{4}^{2}-(c^{2}t_{1}^{2}-c^{2}t_{4}^{2})\qquad \qquad (16)\\\\2(x_{2}-x_{4})x_{00}+2(y_{2}-y_{4})y_{00}+2(z_{2}-z_{4})z_{00}=\\x_{2}^{2}-x_{4}^{2}+y_{2}^{2}-y_{4}^{2}+z_{2}^{2}-z_{4}^{2}-(c^{2}t_{2}^{2}-c^{2}t_{4}^{2})\qquad \qquad (17)\\\\2(x_{3}-x_{4})x_{00}+2(y_{3}-y_{4})y_{00}+2(z_{3}-z_{4})z_{00}=\\x_{3}^{2}-x_{4}^{2}+y_{3}^{2}-y_{4}^{2}+z_{3}^{2}-z_{4}^{2}-(c^{2}t_{3}^{2}-c^{2}t_{4}^{2})\qquad \qquad (18)\\\\2(x_{1}-x_{4})x_{0t}+2(y_{1}-y_{4})y_{0t}+2(z_{1}-z_{4})z_{0t}=2c^{2}(t_{1}-t_{4})\qquad (19)\\2(x_{2}-x_{4})x_{0t}+2(y_{2}-y_{4})y_{0t}+2(z_{2}-z_{4})z_{0t}=2c^{2}(t_{2}-t_{4})\qquad (20)\\2(x_{3}-x_{4})x_{0t}+2(y_{3}-y_{4})y_{0t}+2(z_{3}-z_{4})z_{0t}=2c^{2}(t_{3}-t_{4})\qquad (21)\\\end{matrix}}}$

There are 2 systems of linear equations with the same coefficient matrix. Your solutions are now inserted into one of the output equations. Thus, after the reduction to 3 equations, a fourth independent equation is added, and the temporary underdetermination is eliminated. One obtains z. B. by inserting it into the fourth output equation

${\displaystyle {\begin{matrix}(x_{4}-x_{00}-x_{0t}t_{0})^{2}+(y_{4}-y_{00}-y_{0t}t_{0})^{2}+(z_{4}-z_{00}-z_{0t}t_{0})^{2}=[c(t_{4}-t_{0})]^{2}\\\end{matrix}}}$

The terms are multiplied out and ordered by powers of : ${\displaystyle t_{0}}$

${\displaystyle {\begin{aligned}A&:=(x_{4}-x_{00})^{2}+(y_{4}-y_{00})^{2}+(z_{4}-z_{00})^{2}-[ct_{4}]^{2}\\B&:=(x_{4}-x_{00})x_{0t}+(y_{4}-y_{00})y_{0t}+(z_{4}-z_{00})z_{0t}-c^{2}t_{4}\\C&:=x_{0t}^{2}+y_{0t}^{2}+z_{0t}^{2}-c^{2}\end{aligned}}}$

With the constants , and we get a quadratic equation for ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle C}$ ${\displaystyle t_{0}}$

${\displaystyle Ct_{0}^{2}-2Bt_{0}+A=0}$

with the two solutions:

${\displaystyle t_{0}={\frac {2B\pm {\sqrt {4B^{2}-4AC}}}{2C}}}$

With realistic output data, there are always two real solutions. One of them can be ruled out as unrealistic, either because it lies before the times of transmission of the satellites or leads to positions that are above the satellite orbits. This means that all values ​​are known for specifying the coordinates.

${\displaystyle {\begin{aligned}x_{0}&=x_{00}+x_{0t}t_{0}\\y_{0}&=y_{00}+y_{0t}t_{0}\\z_{0}&=z_{00}+z_{0t}t_{0}\end{aligned}}}$

The angle of the connecting lines between the point to be measured and the reference points must be as large as possible. If the angles are too small, exact position determination is not possible - see the accuracy parameters PDOP and GDOP .

Individual evidence

1. ↑ Arnic Research Corporation: Navstar GPS Space Segment navigation user interfaces . December 7, 2004 ( uscg.gov [PDF]).