RU2492499C1 - Method of determining location of object with use of global navigation satellite systems and system for its implementation - Google Patents

Abstract

FIELD: instrument engineering.

SUBSTANCE: corresponding system (dwg. 3) and the sequence of operations for work on two orbit groups (dwg. 4) are proposed. In addition, a special-purpose control unit is introduced, in which in the system of navigation equations the fifth time parameter is considered responsible for the divergence of hours of the navigation receiver from the system time of the second orbit group. At that a conversion to a certain linear vector space is carried out, in which the solution of navigation equations system is determined in the parametric form relatively two unknowns, with subsequent determination of these unknowns by calculation of the roots of a polynomial equation.

EFFECT: enhanced functionality.

3 cl, 6 dwg

Description

The present invention relates to the field of satellite navigation and geodesy.

Global navigation satellite systems (GNSS) allow you to determine the coordinates (x, y, z) of an object (phase center of the antenna located on the object of the navigation receiver) from the measured values of the pseudorange to the satellites and their ephemeris (spatial coordinates of the satellites given at strictly defined points in time).

The geometric distance between the object and the j-th satellite is:

$$Dj=\sqrt{{(x_j-x)}^2+{(y_j-y)}^2+{(z_j-z)}^2}(\mathrm{one})$$

where {x, y, z} are the desired coordinates of the object at the time of signal reception,

{x _j , y _j , z _j ] - coordinates of the j-th satellite at the moment of signal emission.

A receiver at an object measures the time it takes for a signal to pass from the moment of its emission to the moment of reception, which, multiplied by the speed of light c, equals the so-called “pseudorange” ρ _j . Due to the fact that the signal propagates in the atmosphere, and not in vacuum, the measured time includes atmospheric delays, errors associated with signal re-reflection, etc. In addition, the time stamp transmitted by the satellite in the emitted signal is shifted relative to the system time GNSS due to the instability of the frequency of the on-board generator. An even greater difference in the pseudorange ρ _j from the value of D _j is associated with the departure of the clock of the navigation receiver relative to the system time of the orbital grouping.

, а время приема (reception) приемником - t^r. Тогда выражение для результата измерения псевдодальности, который не был обработан или скорректирован (далее просто псевдодальность) можно представить в виде:We denote the moment of emission of the signal by the j-th satellite $$t_j^e$$

and the time of reception by the receiver is t ^r . Then the expression for the pseudorange measurement result that has not been processed or adjusted (hereinafter simply pseudorange) can be represented as:

$${\tilde{\rho }}_j=c\cdot (t^r-t_j^e)+c\cdot (\delta t^r-\delta t_j^e)+c\cdot T_{\mathrm{but}tm}+{\epsilon }_j^r,$$

or subject to (1),

$${\tilde{\rho }}_j=\sqrt{{(x_j-x)}^2+{(y_j-y)}^2+{(z_j-z)}^2}+c\cdot (\delta t^r-\delta t_j^e)+c\cdot T_j^{\mathrm{but}tm}+{\epsilon }_j^r,(2)$$

where δt ^r is the deviation of the receiver timeline from the GNSS system timeline at the time the receiver receives the signal,

- отклонение временной шкалы j-го спутника от шкалы системного времени ГНСС в момент излучения сигнала, $$\delta t_j^e$$

- deviation of the time scale of the j-th satellite from the GNSS system time scale at the time of signal emission,

- задержка, возникающая при распространении сигнала j-го спутника в атмосфере, $$T_j^{\mathrm{but}tm}$$

- delay arising from the propagation of the signal of the j-th satellite in the atmosphere,

- инструментальная погрешность оценки псевдодальности до j-го спутника в приемнике в сумме с некомпенсированными ошибками в момент приема сигнала. $${\epsilon }_j^r$$

- instrumental error in estimating the pseudorange to the jth satellite in the receiver in total with uncompensated errors at the time of signal reception.

, задержку $$T_j^{атм}$$

и записать уравнение для скорректированной псевдодальностиу j-го спутника:Using the ephemeris information and the corresponding mathematical models, it is possible to partially compensate and take into account the deviation in expression (2) $$\delta t_j^e$$

delay $$T_j^{\mathrm{but}tm}$$

and write the equation for the corrected pseudorange of the jth satellite:

$${\rho }_j=\sqrt{{\left(x_j-x\right)}^2+{\left(y_j-y\right)}^2+{\left(z_j-z\right)}^2}+a+{\epsilon }_j^r,\left(3\right)$$

Where $$a=c\cdot \delta t^r;\left(\mathrm{four}\right)$$

a is a time parameter characterizing the discrepancy between the receiver timeline and the satellite constellation system timeline.

Expression (3) is called the navigation equation. It has four unknowns (x, y, z, δt ^r ).

обычно входит в невязку решения системы навигационных уравнений и подлежит минимизации.In this case, the error $${\epsilon }_j^r$$

usually included in the residual of solving the system of navigation equations and should be minimized.

Thus, to determine the location of an object, at least four equations are necessary, i.e. four simultaneously visible satellites. To increase the reliability and accuracy of determining the coordinates of the object, modern navigation receivers are made two system, i.e. simultaneously receiving signals of two (GLONASS and GPS) orbital groups.

), используя только линейные члены, то есть:Usually, this system is linearized, expanding into a Taylor series, in the vicinity of the first approximation (x ₀ , y ₀ , z ₀ , $$\delta t_0^r$$

) using only linear terms, i.e.:

$${\rho }_j\left(x,y,z,\delta t^r\right)={\rho }_j\left(x_0,y_0,z_0,\delta t_0^r\right)+{\rho }_j^{\text{'}\text{'}}\left(x,y,z,\delta t^r\right)\cdot \left[\begin{array}{c}\Delta x\\ \Delta y\\ \Delta z\\ \Delta \delta t^r\end{array}\right],\left(5\right)$$

Where $$\left[\begin{array}{c}\Delta x\\ \Delta y\\ \Delta z\\ \Delta \delta t^r\end{array}\right]=\left[\begin{array}{c}x\\ y\\ z\\ \delta t^r\end{array}\right]-\left[\begin{array}{c}{x}_0\\ y_0\\ z_0\\ \delta t_0^r\end{array}\right]$$

There are various analytical (non-iterative) methods for calculating the coordinates of an object using 4 satellites: methods proposed by Krause [1] in 1987, Kleusberg [2] in 1994, Grafared and Shan [3] in 1996. The disadvantages of these methods are the impossibility the use of excess satellites in them.

Closest to the claimed method is the method proposed in 1985 by Bancroft [4]. It is intended to solve an excess system of navigation equations in the case of one orbital constellation, but is not applicable when using signals from two constellations at the same time. This is due to the fact that the receiver clock moves away from the system time of the GLONASS and GPS orbital groups. In addition, in user navigation receivers, usually the local oscillator frequencies in the GLONASS and GPS channels have independent frequency drifts. Therefore, five quantities become unknown: the coordinates of the object x, y, z and the times δt ₁ and δt _{2 of the} receiver’s clock departure (or a ₁ , a ₂ determined by formula 4) in the GLONASS and GPS channels.

The urgent task is to determine the location of an object when some of the navigation satellites belong to the GLONASS orbital constellation, and the other belongs to GPS (for example, there are three GLONASS satellites and two GPS).

A typical implementation of the location system of the object, the most common at the present time, is described in patent US 4672382 "Positioning system" from 12/04/1985, selected as an analogue. The block diagram of the system is shown in figure 1, where:

1. Orbital constellation of navigation satellites;

2. Receiving antenna of navigation signals;

3. Block forming measurements;

4. Block conditions;

5. Block rough initial assessment;

6. Block iterative calculation of coordinates;

7. The block estimates the storage of coordinates in memory;

8. The control unit of the route.

In the unit of formation of measurements, the measurement (estimation) of pseudorange and the separation of the ephemeris of all satellites take place. In the block of iterative calculation of coordinates, navigation satellites are selected, the weights of each pseudorange estimate are weighted, satellite coordinates are calculated at the moment of pseudorange measurement, then a linearized system of navigation equations is solved by successive approximations (5) with subsequent smoothing and output of the calculated coordinates to the tool using this information , for example, in the route control unit.

The system of navigation equations (3), connecting the coordinates of the object x, y, z and the value δt with the current values of the coordinates of the satellites, is nonlinear. It is solved by an iterative method using linearization of equations in a neighborhood of a priori known coordinates of the object. If the number of satellites is more than four, the number of equations in the system becomes redundant. Using the least squares method, it boils down to four. As a rule, the criterion for convergence of an iterative algorithm for solving navigation equations is the minimization of a certain functional.

The sequence of operations for the analogue (figure 1) is shown in figure 2, where:

9. Reception of a navigation signal;

10. Primary digital processing of navigation signals (with the separation of ephemeris and service information) and the formation of pseudo-ranges;

11. Assigning weights to pseudorange, satellite selection, correction of pseudorange;

12. The forecast of the coordinates of the satellites on the actual ephemeris at the time moments of the emission of signals;

13. Conditions for the existence of the first approximation x, y, z, δt ^r ;

14. The calculation of the first approximation x, y, z, δt ^r ;

15. The solution of the linearized system of navigation equations, iteration of the calculation of the next x, y, z, δt ^r ;

16. The feasibility condition Δδt _r <ε, where ε is the allowable value of the residual in time between the receiver scale and the GNSS system time;

17. The norm of the residual of the original system should be less than a given value;

18. Smoothing the results with a Kalman filter;

19. Record of calculations and output x, y, z, δt ^r for the user.

The disadvantages of iterative methods for solving the system of navigation equations are:

- the need to set the initial approximation;

- the probability of convergence of calculations for a given accuracy;

- the calculation time (calculation volume) depends not only on the number of satellites, but also on the observation conditions.

The problem to which the invention is directed is to create a method and system for determining the location of an object using two orbital constellations of navigation satellites with an analytical assessment of navigation parameters.

The technical result of the claimed invention is the iteration-free obtaining of the optimal result of determining the location of an object using the least square method (OLS) for an excess of satellites of two orbital constellations in a fixed time.

The technical result is achieved by the fact that the method of determining the location of the object when using global navigation satellite systems is that:

- receive signals from the first GPS and the second GLONASS orbital constellations of navigation satellites;

- produce primary digital processing of navigation signals with the allocation of ephemeris and pseudorange;

- calculate the corrected pseudorange;

- predicting the coordinates of the satellites according to the actual ephemeris at time instants of radiation of navigation signals;

- determine the coordinates x, y, z, the time parameter a ₁ corresponding to the divergence of the clock of the navigation receiver from the system time of the first orbital group and the time parameter a ₂ corresponding to the divergence of the clock of the navigation receiver from the system time of the second orbital group using the least squares method, using information from all satellites with the same weight of the first and second orbital constellations, by analytically solving a system of nonlinear equations with minimizing residuals;

- check the validity of the condition a ₁ <ε, where ε is the permissible time difference between the scale of the navigation receiver and the system time of the global navigation satellite system (GNSS);

- bring the clock of the navigation receiver to the GNSS system time;

- smooth out the results of the determination by the Kalman filter;

- record the results of determining x, y, z, a ₁ , a ₂ and display them for the user.

A system for determining the location of an object using global navigation satellite systems, including:

- The first orbital constellation of GPS navigation satellites;

- the second orbital group GLONASS;

- a receiving antenna for navigation satellite signals from the first orbital constellation of GPS navigation satellites and the second GLONASS orbital constellation;

- unit for forming measurements;

- unit for calculating coordinates and time parameters;

- a unit for recording and outputting calculation results to a user;

moreover, the output of the receiving antenna of the navigation satellite signals is connected to the input of the unit for forming measurements, the input of the unit for calculating coordinates and time parameters is connected to the output of the unit for forming measurements, the input of the unit for outputting calculation results is connected to the output of the unit for calculating coordinates and time parameters, the output of the unit for recording and transmitting calculation results is the output of the system.

The unit for calculating coordinates and time parameters contains an arithmetic-logical device with the possibility of addition, multiplication and subtraction, the first input of which is connected to the output of the unit for forming measurements, the unit for calculating the complex conjugate matrix, the input for which is connected to the output of the unit for forming measurements, the unit for calculating scalar line-by-line products of a vector with a matrix and a matrix with a matrix, the input of which is connected to the output of the unit for forming measurements, the second and third inputs of the arithmetic-logical device inens with outputs of a complex conjugate matrix and a block for calculating scalar row-wise products of a vector with a matrix and a matrix with a matrix, respectively, a block for calculating the roots of a polynomial whose input is connected to the output of an arithmetic-logic device with the possibility of addition, multiplication and subtraction, block for calculating the state vector { x, y, z, a _1, a _2}, whose input is connected to the output of the polynomial roots calculation unit, calculation unit output state vector {x, y, z, a _1, a _2} is the output of the coordinate calculations and temporary parameter in.

A generalized block diagram of the inventive system for determining the location of the object when using global navigation satellite systems is shown in figure 3., where:

20. The first orbital constellation of GPS navigation satellites;

21. The second orbital constellation of GLONASS navigation satellites;

22. The receiving antenna of navigation signals;

23. Block forming measurements;

24. Block for calculating coordinates and time parameters;

25. A unit for recording and outputting calculation results to a user.

A block diagram of the sequence for determining coordinates for the claimed system operating on two GNSSs is shown in figure 4., where:

26. Reception of a navigation signal;

27. The primary digital processing of navigation signals (with the separation of ephemeris and service information) and the formation of pseudorange. Performing correction of pseudorange;

28. Prediction of satellite coordinates by actual ephemeris at time points of signal emission;

29. Determination of x, y, z, a ₁ , a ₂ by analytical solution of a system of nonlinear equations with minimization of residuals;

30. The feasibility condition is a ₁ <ε, where ε is the allowable value of the residual in time between the receiver scale and the GNSS system time;

31. Drive receiver clock to GNSS system time;

32. Smoothing the results with a Kalman filter;

33. Completion of the calculation and output x, y, z, a ₁ , a ₂ .

The figure 5 presents a block diagram of a unit for calculating coordinates and time parameters to obtain an analytical solution to the system of navigation equations, where:

34. Arithmetic-logical device (addition, multiplication, subtraction);

35. Block calculation complex conjugate matrix;

36. The unit for calculating the scalar product for pairs of vectors, the vector matrix (line by line), the matrix matrix (line by line);

37. Block for calculating the roots of a polynomial;

38. The unit for computing the state vector {x, y, z, a ₁ , a ₂ }.

In accordance with the proposed sequence (figure 4), the first two steps are the reception of the navigation satellite antenna 22 and the primary digital processing of the navigation signals (with the separation of ephemeris and service information) and the formation of pseudorange by the unit for forming measurements 23. In block 23, the correction is also carried out measured pseudorange ρ _j by the amount of atmospheric delay according to generally accepted models, taking into account the departure of the satellite clock about the system time. The final stage of data processing in block 23 is the prediction of the coordinates of the satellites according to the actual ephemeris at the time moments of the emission of signals.

At the next step, the determination of {x, y, z, a ₁ , a ₂ } is carried out by means of an analytical solution in the block of calculation of coordinates and time parameters 24.

To minimize the number of calculations in the receiver, its timeline must be tied to the system time of one of the GNSS. Figure 4 shows an example with reference to the time scale of the first GNSS. After calculating the solution of the system of navigation equations, if the receiver time deviation is more than acceptable, it is necessary to bring the receiver clock to the scale of the selected GNSS and recalculate the solution once taking into account the corrected position of the satellites.

At the last stages of processing, if necessary, the information is smoothed by the Kalman filter and the obtained values {x, y, z, a ₁ , a ₂ } are output to the recording and transmission unit of the calculation results, which is capable of recording data and transmitting it to the user.

The following describes the data processing algorithm in the computer.

To understand the sequence of data processing, it is necessary to compose a system of navigation equations of a certain type, taking into account five unknowns.

For the orbital constellation at number 21 (Fig. 3.) (GLONASS), the equation for the j-th satellite in accordance with (3) has the form:

$$\sqrt{{\left(x_j-x\right)}^2+{\left(y_j-y\right)}^2+{\left(z_j-z\right)}^2}+a_{\mathrm{one}}={\rho }_j,\left(6\right)$$

where [x, y, z] are the desired coordinates of the receiver,

[x _j , y _j , z _j ) - coordinates of the j-th satellite,

ρ _j is the corrected pseudorange to the j-th satellite,

a ₁ - time parameter characterizing the divergence of the receiver timeline from the system time of the first orbital grouping, multiplied by the speed of light.

For the orbital constellation at number 20 (FIG. 2) (GPS), the equation for the gth satellite in accordance with (6) has the form:

$$\sqrt{{\left(x_g-x\right)}^2+{\left(y_g-y\right)}^2+{\left(z_g-z\right)}^2}+a_2={\rho }_g,\left(7\right)$$

where {x _g , y _g , z _g } are the coordinates of the gth satellite,

ρ _j is the corrected pseudorange to the gth satellite,

a ₂ - time parameter characterizing the divergence of the receiver timeline from the system time of the second orbital grouping, multiplied by the speed of light.

After transferring the time parameter to the right side of equations (6), (7) and squaring we get:

Further, it is necessary to take into account that from each exhaust gas a different number of satellites can be observed, let the number of satellites = n _{1 be} observed in the GLONASS exhaust gas, and n ₂ , the total number of satellites n = n ₁ + n ₂ in GPS. The deviation of the receiver scale from GLONASS and GPS will be a ₁ and a _2, respectively. As a result, we obtain a system of n equations of the form (8).

и $$\overrightarrow{t}=\{t_1,t_{\mathrm{2,}}{t}_3,t_4,t_5\}$$

скалярным произведением $$\langle \overrightarrow{r},\overrightarrow{t}\rangle$$

будет называться произведение вида:To solve this system, it is necessary to go to the five-dimensional linear space of vectors L (R), such that for two vectors $$\overrightarrow{r}=\{r_{\mathrm{one}},r_2{\mathrm{<figure-callout\; id="3"\; label="r"\; filenames="00000077.png,00000079.png"\; state="\{\{state\}\}">r</figure-callout>}}_3,r_{\mathrm{four}},r_5\}$$

and $$\overrightarrow{t}=\{t_{\mathrm{one}},t_2{\mathrm{<figure-callout\; id="3"\; label="t"\; filenames="00000077.png,00000079.png"\; state="\{\{state\}\}">t</figure-callout>}}_3,t_{\mathrm{four}},t_5\}$$

scalar product $$〈\overrightarrow{r},\overrightarrow{t}〉$$

will be called a product of the form:

$$〈\overrightarrow{r},\overrightarrow{t}〉=r_{\mathrm{one}}{t}_{\mathrm{one}}+r_2{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="00000079.png"\; state="\{\{state\}\}">t</figure-callout>}}_2+r_3{t}_3-r_{\mathrm{four}}{t}_{\mathrm{four}}-r_5{t}_5-r_{\mathrm{four}}{t}_5-r_5{t}_{\mathrm{four}}\left(9\right)$$

We define the following desired 5-dimensional vector:

Solution of a system of equations of the form (8).

We define the following 5-dimensional vectors:

.Where the index k denotes the dimension of the unit vector $${\overrightarrow{e}}_k$$

.

.Then the system will be solved with respect to the desired vector $$\overrightarrow{u}$$

.

For the selected vectors, in accordance with (8), we have:

при любом j

for any j

, при любом j $$〈{\overrightarrow{s}}_j,{\overrightarrow{s}}_j〉={\mathrm{<figure-callout\; id="2"\; label="x\; j"\; filenames="00000079.png"\; state="\{\{state\}\}">x\; </figure-callout>}}_j^{}+{\mathrm{<figure-callout\; id="2"\; label="y\; j"\; filenames="00000079.png"\; state="\{\{state\}\}">y\; </figure-callout>}}_j^{}+z_j^2-{\mathrm{<figure-callout\; id="2"\; label="p\; j"\; filenames="00000079.png"\; state="\{\{state\}\}">p\; </figure-callout>}}_j^{}$$

for any j

In this case, the system of navigation equations for GLONASS and GPS can be written in the following form:

The system of equations (14) can be rewritten in matrix form. For this, it is necessary to go over to linear algebra operations, replacing the scalar product and arithmetic operations with vectors in each separate equation of the system by matrix-vector operations.

We introduce the following notation:

; $$\text{}B=\left[\begin{array}{ccccc}{x}_1& y_1& z_1& -p_1& 0\\ \dots & \dots & \dots & \dots & \dots \\ x_n& y_n& z_n& 0& -p_n\end{array}\right]$$

; $$B^{*}=\left[\begin{array}{ccc}{x}_1& \dots & x_n\\ y_1& \dots & y_n\\ z_1& \dots & z_n\\ p_1& \dots & 0\\ 0& \dots & p_n\end{array}\right]$$

; $$\Lambda =\frac{〈\overrightarrow{u},\overrightarrow{u}〉}{2}$$

; $$\text{A.}B=\left[\begin{array}{ccccc}{x}_{\mathrm{one}}& y_{\mathrm{one}}& z_{\mathrm{one}}& -p_{\mathrm{one}}& 0\\ ...& ...& ...& ...& ...\\ x_n& y_n& z_n& 0& -p_n\end{array}\right]$$

; $$B^{*}=\left[\begin{array}{ccc}{x}_{\mathrm{one}}& ...& x_n\\ y_{\mathrm{one}}& ...& y_n\\ z_{\mathrm{one}}& ...& z_n\\ p_{\mathrm{one}}& ...& 0\\ 0& ...& p_n\end{array}\right]$$

;

; $$D=diag\left(B{B}^{*}\right)=\left[\begin{array}{c}{x}_1^2+y_1^2+z_1^2-p_1^2\\ \cdots \\ x_n^2+y_n^2+z_n^2-p_n^2\end{array}\right]$$

$$P=\left[\begin{array}{ccc}{p}_{\mathrm{one}}& ...& 0\\ 0& ...& 0\\ ...& ...& ...\\ 0& ...& 0\\ 0& ...& p_n\end{array}\right]$$

; $$D=diag\left(B{B}^{*}\right)=\left[\begin{array}{c}{x}_{\mathrm{one}}^2+y_{\mathrm{one}}^2+z_{\mathrm{one}}^2-p_{\mathrm{one}}^2\\ \cdots \\ x_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="00000079.png"\; state="\{\{state\}\}">n</figure-callout>}}^2+{\mathrm{<figure-callout\; id="2"\; label="y\; n"\; filenames="00000079.png"\; state="\{\{state\}\}">y\; </figure-callout>}}_n^{}+z_n^2-p_n^2\end{array}\right]$$

Λ is a scalar quantity,

B and B ^* are the matrix of the system and its complex conjugate matrix,

D is a vector composed of significant elements of the diagonal matrix, with elements of the main diagonal of the product BB ^* .

P is the diagonal matrix, with elements p _j on the main diagonal.

We also introduce the notation of a composite vector of the following form:

, например, $$\left[\begin{array}{c}{\overrightarrow{e}}_{n_1}\\ 0\cdot {\overrightarrow{e}}_{n_2}\end{array}\right]=\left[\begin{array}{c}1\\ \dots \\ 1\\ 0\\ \dots \\ 0\end{array}\right]\begin{array}{c}\begin{array}{l}\\ \\ \end{array}\}\\ \begin{array}{l}\\ \\ \end{array}\}\end{array}\begin{array}{c}\begin{array}{l}{n}_1-элементов\\ \\ \\ n_2-элементов\end{array}\\ \end{array}$$

. $$\left[\begin{array}{c}{\overrightarrow{e}}_{k_{\mathrm{one}}}\\ {\overrightarrow{e}}_{k_2}\end{array}\right]$$

, eg, $$\left[\begin{array}{c}{\overrightarrow{e}}_{n_{\mathrm{one}}}\\ 0\cdot {\overrightarrow{e}}_{n_2}\end{array}\right]=\left[\begin{array}{c}\mathrm{one}\\ ...\\ \mathrm{one}\\ 0\\ ...\\ 0\end{array}\right]\begin{array}{c}\begin{array}{l}\\ \\ \end{array}\}\\ \begin{array}{l}\\ \\ \end{array}\}\end{array}\begin{array}{c}\begin{array}{l}{n}_{\mathrm{one}}-\mathrm{uh}lement\mathrm{about}\mathrm{at}\\ \\ \\ n_2-\mathrm{uh}lement\mathrm{about}\mathrm{at}\end{array}\\ \end{array}$$

.

Using the introduced notation, we obtain the following form of the system of navigation equations (14):

,

,

after expressing the generalized Lorentz product by matrix operations, we obtain:

при количестве спутников, равном количеству неизвестных (пять), или по методу наименьших квадратов при избыточном количестве спутников. Примечательно, что если количество орбитальных группировок равно единице, то a ₂=n₂=0 и система уравнений сводится к системе уравнений по методу Банкрофта, как к частному случаю решения.This system of equations is solved uniquely with respect to an unknown vector $$\overrightarrow{u}$$

with the number of satellites equal to the number of unknowns (five), or by the least squares method with an excess of satellites. It is noteworthy that if the number of orbital groupings is equal to unity, then a ₂ = n ₂ = 0 and the system of equations reduces to a system of equations according to the Bancroft method, as a special case of a solution.

The solution of the system will be as follows:

$$\overrightarrow{u}=B^{+}\left(\Lambda {\overrightarrow{e}}_n+\frac{\mathrm{one}}{2}\left[\begin{array}{c}{a}_2^2\cdot {\overrightarrow{e}}_{n_{\mathrm{one}}}\\ a_{\mathrm{one}}^2\cdot {\overrightarrow{e}}_{n_2}\end{array}\right]+a_{\mathrm{one}}{a}_2{\overrightarrow{e}}_n+\frac{D}{2}\right),\left(16\right)$$

where B ⁺ = (B ^T B) ^-1 B ^T

. Можно заметить, что В⁺ в данном случае имеет размерность 5×n, а слагаемые в скобках являются векторами размерности n. a ₁ ; a ₂ - parameters when searching for a solution, and Λ we find based on its definition $$\Lambda =\frac{\mathrm{one}}{2}〈\overrightarrow{u},\overrightarrow{u}〉$$

. You may notice that B ⁺ in this case has a dimension of 5 × n, and the terms in brackets are vectors of dimension n.

и Λ. Шестое уравнение, позволяющее выразить Λ через параметры a ₁, а ₂ имеет вид:The product B ⁺ by any term in brackets gives a vector of dimension 5. Thus (16) is a system of five equations with six unknowns $$\overrightarrow{u}=\left[\begin{array}{c}x\\ y\\ z\\ a_{\mathrm{one}}\\ a_2\end{array}\right]$$

and Λ. The sixth equation, allowing to express Λ in terms of the parameters a ₁ , and ₂ has the form:

; $$\Lambda =\frac{\mathrm{one}}{2}〈\overrightarrow{u},\overrightarrow{u}〉$$

;

$$\begin{array}{l}\Lambda =\frac{\mathrm{one}}{2}〈B^{+}\left(\Lambda {\overrightarrow{e}}_n+\frac{\mathrm{one}}{2}\left[\begin{array}{c}{a}_2^2\cdot {\overrightarrow{e}}_{n_{\mathrm{one}}}\\ a_{\mathrm{one}}^2\cdot {\overrightarrow{e}}_{n_2}\end{array}\right]+a_{\mathrm{one}}{a}_2{\overrightarrow{e}}_n+\frac{D}{2}\right),B^{+}(\Lambda {\overrightarrow{e}}_n+\frac{\mathrm{one}}{2}\left[\begin{array}{c}{a}_2^2\cdot {\overrightarrow{e}}_{n_{\mathrm{one}}}\\ a_{\mathrm{one}}^2\cdot {\overrightarrow{e}}_{n_2}\end{array}\right]+\\ a_{\mathrm{one}}{a}_2{\overrightarrow{e}}_n+\frac{D}{2})〉.\left(17\right)\end{array}$$

Using the linearity property of the selected scalar product of vectors, as a result of transformations we get:

$$\begin{array}{l}\Lambda =\frac{\mathrm{one}}{2}〈B^{+}\left(\Lambda {\overrightarrow{e}}_n+\frac{\mathrm{one}}{2}\left[\begin{array}{c}{a}_2^2\cdot {\overrightarrow{e}}_{n_{\mathrm{one}}}\\ a_{\mathrm{one}}^2\cdot {\overrightarrow{e}}_{n_2}\end{array}\right]+a_{\mathrm{one}}{a}_2{\overrightarrow{e}}_n+\frac{D}{2}\right),B^{+}\Lambda {\overrightarrow{e}}_n〉\\ +\frac{\mathrm{one}}{2}〈B^{+}\left(\Lambda {\overrightarrow{e}}_n+\frac{\mathrm{one}}{2}\left[\begin{array}{c}{a}_2^2\cdot {\overrightarrow{e}}_{n_{\mathrm{one}}}\\ a_{\mathrm{one}}^2\cdot {\overrightarrow{e}}_{n_2}\end{array}\right]+a_{\mathrm{one}}{a}_2{\overrightarrow{e}}_n+\frac{D}{2}\right),\frac{\mathrm{one}}{2}B{\text{A.}}^{+}\left[\begin{array}{c}{a}_2^2\cdot {\overrightarrow{e}}_{n_{\mathrm{one}}}\\ a_{\mathrm{one}}^2\cdot {\overrightarrow{e}}_{n_2}\end{array}\right]〉\\ +\frac{\mathrm{one}}{2}〈B^{+}\left(\Lambda {\overrightarrow{e}}_n+\frac{\mathrm{one}}{2}\left[\begin{array}{c}{a}_2^2\cdot {\overrightarrow{e}}_{n_{\mathrm{one}}}\\ a_{\mathrm{one}}^2\cdot {\overrightarrow{e}}_{n_2}\end{array}\right]+a_{\mathrm{one}}{a}_2{\overrightarrow{e}}_n+\frac{D}{2}\right),a_{\mathrm{one}}{a}_2{B}^{+}{\overrightarrow{e}}_n〉\\ +\frac{\mathrm{one}}{2}〈B^{+}\left(\Lambda {\overrightarrow{e}}_n+\frac{\mathrm{one}}{2}\left[\begin{array}{c}{a}_2^2\cdot {\overrightarrow{e}}_{n_{\mathrm{one}}}\\ a_{\mathrm{one}}^2\cdot {\overrightarrow{e}}_{n_2}\end{array}\right]+a_{\mathrm{one}}{a}_2{\overrightarrow{e}}_n+\frac{D}{2}\right),\frac{\mathrm{one}}{2}{B}^{+}D〉;\left(\mathrm{eighteen}\right)\end{array}$$

Using algebraic transformations, we obtain an expression of the form:

$${\mathrm{<figure-callout\; id="2"\; label="Q"\; filenames="00000079.png"\; state="\{\{state\}\}">Q</figure-callout>}}_2{\mathrm{<figure-callout\; id="2"\; label="\Lambda "\; filenames="00000079.png"\; state="\{\{state\}\}">\Lambda </figure-callout>}}^2+Q_{\mathrm{one}}\Lambda +Q_0=0\left(\mathrm{twenty}\right)$$

where Q ₂ , Q ₁ , Q ₀ have the following form:

; $${\mathrm{<figure-callout\; id="2"\; label="Q"\; filenames="00000079.png"\; state="\{\{state\}\}">Q</figure-callout>}}_2=\frac{\mathrm{one}}{2}〈B^{+}{\overrightarrow{e}}_n,B^{+}{\overrightarrow{e}}_n〉$$

;

The solutions of the quadratic equation with parameters (20) have the following form:

$${\mathrm{<figure-callout\; id="1"\; label="\Lambda "\; filenames="00000077.png,00000079.png"\; state="\{\{state\}\}"><figure-callout\; id="2"\; label="\Lambda "\; filenames="00000079.png"\; state="\{\{state\}\}">\Lambda </figure-callout></figure-callout>}}_{\mathrm{1,2}}=\frac{-Q_{\mathrm{one}}\pm \sqrt{Q_{\mathrm{one}}^2-\mathrm{four}{Q}_2{Q}_0}}{2{\mathrm{<figure-callout\; id="2"\; label="Q"\; filenames="00000079.png"\; state="\{\{state\}\}">Q</figure-callout>}}_2}$$

$$\begin{array}{l}{Q}_{\mathrm{one}}^2=\frac{\mathrm{one}}{\mathrm{four}}{〈}^{B^{+}}+a_{\mathrm{one}}{a}_2〈B^{+}\left[\begin{array}{c}{a}_2^2\cdot {\overrightarrow{e}}_{n_{\mathrm{one}}}\\ a_{\mathrm{one}}^2\cdot {\overrightarrow{e}}_{n_2}\end{array}\right],B^{+}{\overrightarrow{e}}_n〉〈B^{+}{\overrightarrow{e}}_n,B^{+}{\overrightarrow{e}}_n〉\\ +\frac{\mathrm{one}}{2}〈B{\text{A.}}^{+}\left[\begin{array}{c}{a}_2^2\cdot {\overrightarrow{e}}_{n_{\mathrm{one}}}\\ a_{\mathrm{one}}^2\cdot {\overrightarrow{e}}_{n_2}\end{array}\right],B^{+}{\overrightarrow{e}}_n〉〈B^{+}D,B^{+}{\overrightarrow{e}}_n〉-〈B{\text{A.}}^{+}\left[\begin{array}{c}{a}_2^2\cdot {\overrightarrow{e}}_{n_{\mathrm{one}}}\\ a_{\mathrm{one}}^2\cdot {\overrightarrow{e}}_{n_2}\end{array}\right],B^{+}{\overrightarrow{e}}_n〉\\ +a_{\mathrm{one}}^2{a}_{{}^2}^2{〈}^{B^{+}}+a_{\mathrm{one}}{a}_2〈B^{+}{\overrightarrow{e}}_n,B^{+}{\overrightarrow{e}}_n〉〈B^{+}D,B^{+}{\overrightarrow{e}}_n〉\\ -2a_{\mathrm{one}}{a}_2〈B^{+}{\overrightarrow{e}}_n,B^{+}{\overrightarrow{e}}_n〉+\frac{\mathrm{one}}{\mathrm{four}}{〈}^{B^{+}}-〈B^{+}D,B^{+}{\overrightarrow{e}}_n〉+\mathrm{one};\end{array}$$

$$\begin{array}{l}\mathrm{four}{Q}_2{Q}_0=\frac{\mathrm{one}}{\mathrm{four}}〈B^{+}{\overrightarrow{e}}_n,B^{+}{\overrightarrow{e}}_n〉〈B^{+}\left[\begin{array}{c}{a}_2^2\cdot {\overrightarrow{e}}_{n_{\mathrm{one}}}\\ a_{\mathrm{one}}^2\cdot {\overrightarrow{e}}_{n_2}\end{array}\right],B^{+}\left[\begin{array}{c}{a}_2^2\cdot {\overrightarrow{e}}_{n_{\mathrm{one}}}\\ a_{\mathrm{one}}^2\cdot {\overrightarrow{e}}_{n_2}\end{array}\right]〉\\ +a_{\mathrm{one}}{a}_2〈B^{+}{\overrightarrow{e}}_n,B^{+}{\overrightarrow{e}}_n〉〈B^{+}{\overrightarrow{e}}_n,B^{+}\left[\begin{array}{c}{a}_2^2\cdot {\overrightarrow{e}}_{n_{\mathrm{one}}}\\ a_{\mathrm{one}}^2\cdot {\overrightarrow{e}}_{n_2}\end{array}\right]〉\\ +\frac{\mathrm{one}}{2}〈B^{+}{\overrightarrow{e}}_n,B^{+}{\overrightarrow{e}}_n〉〈B^{+}D,B^{+}\left[\begin{array}{c}{a}_2^2\cdot {\overrightarrow{e}}_{n_{\mathrm{one}}}\\ a_{\mathrm{one}}^2\cdot {\overrightarrow{e}}_{n_2}\end{array}\right]〉a_{\mathrm{one}}^2{a}_{{}^2}^2{〈}^{B^{+}}\\ +a_{\mathrm{one}}{a}_2〈B^{+}{\overrightarrow{e}}_n,B^{+}{\overrightarrow{e}}_n〉〈B^{+}D,B^{+}{\overrightarrow{e}}_n〉+\frac{\mathrm{one}}{\mathrm{four}}〈B^{+}{\overrightarrow{e}}_n,B^{+}{\overrightarrow{e}}_n〉〈B^{+}D,B^{+}D〉;\end{array}$$

$$Q_{\mathrm{one}}^2-\mathrm{four}{Q}_2{Q}_0$$

$$\begin{array}{l}=\frac{\mathrm{one}}{\mathrm{four}}{〈}^{B^{+}}+\frac{\mathrm{one}}{2}〈B^{+}\left[\begin{array}{c}{a}_2^2\cdot {\overrightarrow{e}}_{n_{\mathrm{one}}}\\ a_{\mathrm{one}}^2\cdot {\overrightarrow{e}}_{n_2}\end{array}\right],B^{+}{\overrightarrow{e}}_n〉〈B^{+}D,B^{+}{\overrightarrow{e}}_n〉\\ -〈B{\text{A.}}^{+}\left[\begin{array}{c}{a}_2^2\cdot {\overrightarrow{e}}_{n_{\mathrm{one}}}\\ a_{\mathrm{one}}^2\cdot {\overrightarrow{e}}_{n_2}\end{array}\right],B^{+}{\overrightarrow{e}}_n〉-2a_{\mathrm{one}}{a}_2〈B^{+}{\overrightarrow{e}}_n,B^{+}{\overrightarrow{e}}_n〉+\frac{\mathrm{one}}{\mathrm{four}}{〈}^{B^{+}}\\ -〈B^{+}D,B^{+}{\overrightarrow{e}}_n〉+\mathrm{one}-\frac{\mathrm{one}}{\mathrm{four}}〈B^{+}{\overrightarrow{e}}_n,B^{+}{\overrightarrow{e}}_n〉〈B^{+}\left[\begin{array}{c}{a}_2^2\cdot {\overrightarrow{e}}_{n_{\mathrm{one}}}\\ a_{\mathrm{one}}^2\cdot {\overrightarrow{e}}_{n_2}\end{array}\right],B^{+}\left[\begin{array}{c}{a}_2^2\cdot {\overrightarrow{e}}_{n_{\mathrm{one}}}\\ a_{\mathrm{one}}^2\cdot {\overrightarrow{e}}_{n_2}\end{array}\right]〉\\ -\frac{\mathrm{one}}{2}〈B^{+}{\overrightarrow{e}}_n,B^{+}{\overrightarrow{e}}_n〉〈B^{+}D,B^{+}〉\left[\begin{array}{c}{a}_2^2\cdot {\overrightarrow{e}}_{n_{\mathrm{one}}}\\ a_{\mathrm{one}}^2\cdot {\overrightarrow{e}}_{n_2}\end{array}\right]〉\\ -\frac{\mathrm{one}}{\mathrm{four}}〈B^{+}{\overrightarrow{e}}_n,B^{+}{\overrightarrow{e}}_n〉〈B^{+}D,B^{+}D〉.\end{array}$$

The results obtained are substituted in (16):

.

.

Generally speaking, it is necessary to solve only the last two equations, since the first three (x, y, z) are already expressed in terms of the parameters ( a ₁ , a ₂ ).

The last two equations must be substituted in the literal expression Q ₀ , Q ₁ , Q ₂ , transfer irrationality to the left side, and the parameters to the right and square both sides. After all the transformations, a subsystem with two unknowns and two equations should be obtained, having decided which parameter values can be put in the first three equations.

$$\left[\begin{array}{c}x\\ y\\ z\\ a_{\mathrm{one}}\\ a_2\end{array}\right]=\frac{-Q_{\mathrm{one}}\pm \sqrt{Q_{\mathrm{one}}^2-\mathrm{four}{Q}_2{Q}_0}}{2Q_2}{B}^{+}{\overrightarrow{e}}_n+\frac{\mathrm{one}}{2}{B}^{+}\left[\begin{array}{c}{a}_2^2\cdot {\overrightarrow{e}}_{n_{\mathrm{one}}}\\ a_{\mathrm{one}}^2\cdot {\overrightarrow{e}}_{n_2}\end{array}\right]+a_{\mathrm{one}}{a}_2{B}^{+}{\overrightarrow{e}}_n+\frac{\mathrm{one}}{2}{B}^{+}D$$

With a total number of satellites n, the dimension of the B ⁺ matrix will be 5 × n. To perform further calculations, we introduce the following notation:

. $$B^{+}=\left[\begin{array}{ccc}{b}_{\mathrm{eleven}}& ...& b_{n\mathrm{one}}\\ ...& ...& ...\\ b_{51}& ...& b_{5n}\end{array}\right]$$

.

The last two last equations of system (16) in this case have the form:

и помножим на него уравнение (21), после чего складываем с (22). При этом иррациональность исключается и остается квадратное уравнение содержащее a ₁ и а ₂. Необходимо решить его, например, относительно a₁, выразив его через а ₂ и подставив в (22).We denote $$K=-{\displaystyle \sum {}_{i=\mathrm{one}}^n{b}_{5i}/}{\displaystyle \sum {}_{i=\mathrm{one}}^n{b}_{\mathrm{four}i}}$$

and multiply equation (21) by it, and then add it to (22). In this case, irrationality is eliminated and the quadratic equation containing a ₁ and a ₂ remains. It is necessary to solve it, for example, with respect to a ₁ , expressing it through a ₂ and substituting it in (22).

After substitution, we obtain the equation:

We obtain the quadratic equation from which we find a ₁ as a function of a ₂ . We introduce the following notation

; $$\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="00000079.png"\; state="\{\{state\}\}">W</figure-callout>}2=\left({\displaystyle \sum _{i=n_{\mathrm{one}}+\mathrm{one}}^n{b}_{5i}}+K{\displaystyle \sum _{i=n_{\mathrm{one}}+\mathrm{one}}^n{b}_{\mathrm{four}i}}\right)$$

;

W1 = -2K

The solution is as follows:

. $$a_{\mathrm{one}}=\frac{-w_{\mathrm{one}}\pm \sqrt{w_{\mathrm{one}}^2-\mathrm{four}{w}_2{w}_0}}{2{\mathrm{<figure-callout\; id="2"\; label="w"\; filenames="00000079.png"\; state="\{\{state\}\}">w</figure-callout>}}_2}$$

.

We substitute the found value into expression (22):

$$\begin{array}{l}\mp \sqrt{Q_{\mathrm{one}}^2-\mathrm{four}{Q}_2{Q}_0}{\displaystyle \sum _{i=\mathrm{one}}^n{b}_{5i}}\\ =-2Q_2{a}_2-Q_{\mathrm{one}}{\displaystyle \sum _{i=\mathrm{one}}^n{b}_{5i}}+Q_2{a}_2^2{\displaystyle \sum _{i=\mathrm{one}}^{n_{\mathrm{one}}}{b}_{5i}}\\ +Q_2{\left(\frac{-W_{\mathrm{one}}\pm \sqrt{\text{A.}{W}_{\mathrm{one}}^2-\mathrm{four}\text{<figure-callout id="2" label="A. W" filenames="00000079.png" state="{{state}}">A. </figure-callout>}{W}_{}{}_2\text{A.}{W}_0}}{2W_2}\right)}^2{\displaystyle \sum _{i=n_{\mathrm{one}}+\mathrm{one}}^n{b}_{5i}}\\ +2Q_2{a}_2\frac{-W_{\mathrm{one}}\pm \sqrt{W_{\mathrm{one}}^2-\mathrm{four}{W}_2{W}_0}}{2{\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="00000079.png"\; state="\{\{state\}\}">W</figure-callout>}}_2}{\displaystyle \sum _{i=\mathrm{one}}^n{b}_{5i}}\\ +{\mathrm{<figure-callout\; id="2"\; label="Q"\; filenames="00000079.png"\; state="\{\{state\}\}">Q</figure-callout>}}_2{\displaystyle \sum _{i=\mathrm{one}}^n\left({\mathrm{<figure-callout\; id="2"\; label="x\; i"\; filenames="00000079.png"\; state="\{\{state\}\}">x\; </figure-callout>}}_i^{}+y_i^2+z_i^2-p_i^2\right)b_{5i}.\left(23\right)}\end{array}$$

, а также зависимости W₀, W₁, W₂ и возводя обе части уравнения в квадрат получим выражение вида:Substituting in (23) in symbolic form the values of Q ₂ and $$Q_{\mathrm{one}}^2-\mathrm{four}{Q}_2{Q}_0$$

, as well as the dependences W ₀ , W ₁ , W ₂ and squaring both sides of the equation into a square, we obtain an expression of the form:

$$S_8{a}_2^8+S_7{a}_2^7+S_6{a}_2^6+S_5{a}_2^5+S_{\mathrm{four}}{a}_2^{\mathrm{four}}+S_3{a}_2^3+S_2{a}_2^2+S_{\mathrm{one}}{a}_2+S_0=0.\left(2\mathrm{four}\right)$$

The coefficients of polynomial (24) do not contain unknowns and are calculated using the input data. In general, the coefficients are calculated, but due to their bulkiness are not given. Using environment mat. MATLAB / Maple simulation, the coefficients of the last equation can be obtained in symbolic form for the purpose of further implementation of the algorithm in low-level languages (C / C ++).

To find the roots of the eighth degree polynomial, it is advisable to use the companion matrix method [5].

Next, using the obtained real values of a ₂ calculate a ₁ . And using the obtained parameters, get the solution x, y, z.

Thus, the flowchart for solving a system of nonlinear equations for two orbital groups, taking into account redundancy, can be represented in the form shown in Fig. 6., where:

39. Formation of matrix B and B ⁺ for two exhaust gas;

40. Calculation of auxiliary factors;

41. The calculation of the coefficients of the eighth order;

42. Solution of the equation by the method of the accompanying matrix, screening of roots a _{2 of} high order;

43. Calculation of the time a ₁ , as well as Q ₀ , Q ₁ , Q ₂ ;

44. Calculation of the coordinates of the receiver x, y, z knowing a ₁ , and ₂ , Q ₀ , Q ₁ , Q ₂ ;

45. Elimination of unnecessary decision groups {x, y, z, a ₁ , a ₂ } based on physical meaning;

46. Operational control of the quality of the decision on the residuals of the original system.

The unit for calculating coordinates and time parameters shown in FIG. 4 may have two different implementations:

- a partial hardware implementation where a block with optimized vector computing (for example, a graphics processor) and a standard arithmetic-logic device are used;

- fully software model of the device.

To analyze the quality of work of the proposed method, in terms of determining navigation parameters, it makes sense to compare it with the Bancroft method, since the classical iterative approach in the case of one grouping will give a similar result with a good initial approximation and a certain amount of iteration.

Consider the quality of the method described above and compare with the Bancroft method in two cases:

1) the number of satellites is equal to the number of unknowns (five);

2) excessive constellation.

As a preliminary stage in the preparation of the initial data, it is necessary to bring the coordinate grids and timelines into one coordinate-time space, choosing, for example, the GPS timeline and WGS-84 coordinates as the basis. Calculate at the current time (the moment of measuring pseudorange) the coordinates of the GPS and GLONASS satellites in WGS-84. The big problem in this case is the transition matrix from PZ-90 to WGS-84, since there is no single standard. But for the method of solving a system of nonlinear equations, this feature does not matter. Since, from a mathematical point of view, the system is solved and the solution should make the original system identical with five satellites and have a minimum of least squares error in case of redundancy. Next, it is necessary to compile matrix B for GPS and GLONASS satellites and separately for each grouping. Take the “hard” case, a mixed system in which there are only 3 GPS satellites and 2 GLONASS satellites. In this case, the classical Bancroft method will not be able to give a result separately for each of the groups, therefore, solutions will be obtained on a constellation of 4 GPS satellites (including the indicated 3) and on a constellation of 4 GLONASS satellites (including the indicated 2), then a solution with averaged coordinates x, y, z for GPS and GLONASS systems, and time corrections a ₁ for the GPS system and a ₂ for the GLONASS system are also added. The discrepancies when substituting the indicated solution vector will be compared with the discrepancies obtained by the generalized method for two orbital groups.

The initial matrices for solving by the classical Bancroft method have the following form (x, y, z and adjusted pseudorange to the satellite):

$$\begin{array}{l}{B}_{GPS}\\ =\left(\begin{array}{cccc}\mathrm{one}\mathrm{four}20373\mathrm{four}.262\mathrm{four}86\mathrm{one}0& \mathrm{one}6\mathrm{four}\mathrm{one}0\mathrm{four}03.\mathrm{four}2003980& \mathrm{one}`5203757.\mathrm{four}53\mathrm{four}2960& 20736033.6268737\\ \mathrm{one}\mathrm{four}003986.\mathrm{four}9\mathrm{one}\mathrm{four}0\mathrm{one}\mathrm{one}0& 786893\mathrm{four}.5900\mathrm{four}587& 2\mathrm{one}20\mathrm{four}297.5932570& 202779\mathrm{one}\mathrm{four}.\mathrm{one}273996\\ -\mathrm{four}505835.583997\mathrm{one}3& 230\mathrm{one}2353\mathrm{four}.0938\mathrm{four}\mathrm{one}30& \mathrm{one}203\mathrm{four}030.80\mathrm{four}92650& 23090637.2\mathrm{one}26\mathrm{one}\mathrm{four}0\\ -\mathrm{one}\mathrm{one}0203\mathrm{one}2.9\mathrm{four}58078& \mathrm{one}\mathrm{one}\mathrm{one}32699.387258\mathrm{one}0& 2\mathrm{one}685\mathrm{one}52/80\mathrm{four}28200& 232852\mathrm{one}9/\mathrm{four}30\mathrm{four}809\end{array}\right)\end{array}$$

The rows of the _GPS matrix B correspond to the system numbers of the GPS satellites 20, 23, 30, and 31, respectively.

$$\begin{array}{l}{B}_{GLONASS}\\ =\left(\begin{array}{cccc}\mathrm{one}0827\mathrm{four}85.5\mathrm{one}9767& \mathrm{one}5696365.750\mathrm{one}692& \mathrm{one}6966205.66\mathrm{four}9\mathrm{one}59& \mathrm{one}957303\mathrm{one}.8\mathrm{four}23675\\ -556\mathrm{four}69\mathrm{four}.6\mathrm{four}30556\mathrm{one}& 905\mathrm{one}6\mathrm{four}\mathrm{four}.93869\mathrm{one}72& 23\mathrm{one}73\mathrm{four}7\mathrm{four}.\mathrm{four}59267\mathrm{four}& 2095\mathrm{one}263.2\mathrm{one}79756\\ 2\mathrm{one}908277.725\mathrm{four}9\mathrm{one}\mathrm{one}& \mathrm{one}2882877.6\mathrm{one}\mathrm{one}\mathrm{four}2\mathrm{four}\mathrm{one}& -\mathrm{<figure-callout\; id="2"\; label="one"\; filenames="00000079.png"\; state="\{\{state\}\}">one</figure-callout>}288000.62667727& 22807392.\mathrm{one}252677\\ \mathrm{one}658\mathrm{four}65\mathrm{one}.966\mathrm{four}5\mathrm{one}9& -\mathrm{one}5022\mathrm{one}86.\mathrm{one}37\mathrm{one}066& \mathrm{one}23\mathrm{one}76\mathrm{one}9.\mathrm{one}579296& 23\mathrm{one}38068.\mathrm{four}029622\end{array}\right)\end{array}$$

The rows of matrix B _GL0NASS correspond to the system numbers of GLONASS satellites 3, 13, 15, and 20, respectively.

The obtained solutions (see table 1, all units in meters)

Table 1. GPS GLONASS x = 2846250.36421181 x = 2846135.03277489 y = 2201677.85798526 y = 2201641.31968233 z = 5248681.95352191 z = 5248736.41078770 a1 = -18.33113694 A2 = -129.08714075

Thus, we can take as a basis the solution obtained by the Bancroft method of the form:

$$\left[\begin{array}{c}x\\ y\\ z\\ a_{\mathrm{one}}\\ a_2\end{array}\right]=\left[\begin{array}{c}28\mathrm{four}6\mathrm{one}92.698\mathrm{four}9335\\ 220\mathrm{one}659.58883379\\ 52\mathrm{four}8709.\mathrm{one}82\mathrm{one}5\mathrm{four}80\\ -\mathrm{<figure-callout\; id="8"\; label="one"\; filenames="00000077.png"\; state="\{\{state\}\}">one</figure-callout>}8.33\mathrm{one}\mathrm{one}369\mathrm{four}\\ -\mathrm{<figure-callout\; id="2"\; label="one"\; filenames="00000079.png"\; state="\{\{state\}\}">one</figure-callout>}29.087\mathrm{one}\mathrm{four}075\end{array}\right]$$

The solution vector according to the generalized method for two exhaust gases containing 3 GPS (23, 30, 31) +2 GLONASS (15, 20):

$$\left[\begin{array}{c}x\\ y\\ z\\ a_{\mathrm{one}}\\ a_2\end{array}\right]=\left[\begin{array}{c}28\mathrm{four}6250.\mathrm{four}9\mathrm{four}35830\\ 220\mathrm{one}7\mathrm{four}\mathrm{four}.95376056\\ 52\mathrm{four}8766.3\mathrm{four}853599\\ 66.898\mathrm{one}05\mathrm{four}5\\ 7.36727\mathrm{one}\mathrm{one}5\end{array}\right]$$

It can be seen that the solutions are close, and in a ₁ and ₂ in the second case, there were errors of coordinate transformation, residual vectors and their norms for a mixed initial system have the following values (see Table 2):

Table 2. Bancroft Generalized method Discrepancies 15.41122860 0.0000000149 by satellites -9.89679900 0.0000000112 meters -46.55330177 0.0000000149 -64.55570337 0.0000000149 -19.80951382 0.0000000075 The norm of the residual vector, meters 84.03878972 0.0000000291

Obviously, in the case of the Bancroft method, residuals close to zero are obtained only on the first three equations (GPS), since the coordinates and time correspond to the solution only by GPS, and only the time correction is taken from the GLONASS solution.

In the case of the generalized method, residuals close to zero were obtained, but not zero, since this is due to the accuracy of the discharge grid. The norm of the residual vector for typical cases of satellite arrangement, as a rule, does not exceed 10 ^-6 m, in double-precision floating-point calculations. This accuracy is sufficient for high precision applications. However, based on the values of the corrections characterizing the time in the solution vector, one should seriously study the values of the transition matrices in a single coordinate-time space, since this will be a source of errors that takes time corrections away from the true values.

In case of redundancy, the situation with the Bancroft method is even more complicated:

The obtained solutions (see table 3, all units in meters)

Table 3. GPS GLONASS x = 2 846 251.46141769 x = 2846072.63300129 y = 2 201709.04664132 y = 2201646.99218817 z = 5 248717.76349039 z = 5248658.14254719 a1 = 18.79691716 A2 = -171.02294209

Thus, we can take as a basis the solution obtained by the Bancroft method of the form:

$$\left[\begin{array}{c}x\\ y\\ z\\ a_{\mathrm{one}}\\ a_2\end{array}\right]=\left[\begin{array}{c}28\mathrm{four}6\mathrm{one}62.0\mathrm{four}7209\mathrm{four}9\\ 220\mathrm{one}678.0\mathrm{one}92\mathrm{one}\mathrm{four}75\\ 52\mathrm{four}8687.9530\mathrm{one}879\\ \mathrm{one}8.7969\mathrm{one}7\mathrm{one}6\\ -\mathrm{one}7\mathrm{one}.0229\mathrm{four}209\end{array}\right]$$

The solution vector according to the generalized method for two exhaust gases containing an excessive GPS / GLONASS grouping:

$$\left[\begin{array}{c}x\\ y\\ z\\ a_{\mathrm{one}}\\ a_2\end{array}\right]=\left[\begin{array}{c}29\mathrm{four}6220.99\mathrm{one}9\mathrm{one}026\\ 220\mathrm{one}70\mathrm{one}.\mathrm{four}665\mathrm{four}892\\ 52\mathrm{four}8758.75958\mathrm{four}39\\ 3\mathrm{four}.26225\mathrm{one}55\\ -69.59068525\end{array}\right]$$

The norms of the residual vectors for the mixed initial system have the following values (see table 4):

Table 4. Bancroft Generalized method The norm of the residual vector, meters 205.35810742 108.57034441

As you can see, the generalized method works efficiently and can be recommended for calculating the coordinates of points with minimizing the error vector by least squares method over the entire mixed GLONASS / GPS orbital group, which cannot be achieved using the classical Bancroft method.

Bibliography

1. Krause, L.O. (1987): A Direct Solution to GPS-Type Navigation Equations. IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-23, No. 2, pp. 225-232

2. Kleusberg, A. (1994): Die direkte Lösung des räumlichen Hyperbelschnitts. Zeitschrift fur Vermessungswesen, Vol. 119, No. 4, pp. 188-192

3. Grafarend, E.W. and Chan, J. (1996): A Closed-form Solution of the Nonlinear Pseudo-Ranging Equations (GPS), ARTIFICIAL SATELLITES Planetary Geodesy, No. 28, pp. 133-147

4. Bancroft, S. (1985): An Algebraic Solution of the GPS Equations. IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-21, No. 6, pp. 56-59

5. Maintenance matrix method: http://en.wikipedia.org/wiki/Companion_matrix

Claims (3)

1. The method of determining the location of the object when using global navigation satellite systems, which consists in the fact that:
receive signals from the first GPS and the second GLONASS orbital constellations of navigation satellites;
primary digital processing of navigation signals with the allocation of ephemeris and pseudorange;
corrected pseudo-ranges are calculated;
predicting the coordinates of the satellites according to the actual ephemeris at the time moments of the emission of navigation signals;
determine the coordinates x, y, z, the time parameter a ₁ corresponding to the divergence of the clock of the navigation receiver from the system time of the first orbital group, and the time parameter a ₂ , corresponding to the divergence of the clock of the navigation receiver from the system time of the second orbital group using the least squares method, using information from all satellites with the same weight of the first and second orbital constellations, by analytically solving a system of nonlinear equations with minimizing discrepancies;
check the validity of the condition a ₁ <ε, where ε is the permissible time difference between the scale of the navigation receiver and the system time of the global navigation satellite system (GNSS);
bring the navigation receiver clock to the GNSS system time;
smooth the results of the Kalman filter determination;
record the results of determining x, y, z, and ₁ , and ₂ and display them for the user. 2. The system for determining the location of the object when using global navigation satellite systems, including:
first orbital constellation of GPS navigation satellites;
the second GLONASS orbital group;
a receiving antenna for navigation satellite signals from the first orbital constellation of GPS navigation satellites and the second GLONASS orbital constellation;
unit for forming measurements;
block for calculating coordinates and time parameters;
a unit for recording and outputting calculation results to a user;
moreover, the output antenna of the navigation satellite signals is connected to the input of the unit for forming measurements, the input of the unit for calculating coordinates and time parameters is connected to the output of the unit for forming measurements, the input of the unit for recording and outputting calculation results is connected to the output of the unit for calculating coordinates and time parameters, the output of the unit for recording and output calculation results is the output of the system. 3. The system according to claim 2, characterized in that the unit for calculating coordinates and time parameters contains an arithmetic logic device with the possibility of addition, multiplication and subtraction operations, the first input of which is connected to the output of the measurement unit, the complex conjugate matrix calculation unit, the input which is connected to the output of the unit for forming measurements, the unit for calculating scalar line-by-line products of a vector with a matrix and matrix with a matrix, whose input is connected to the output of the unit for forming measurements, the second and third inputs a the logic-logical device is connected to the outputs of the complex conjugate matrix and the block for calculating scalar row-wise products of a vector with a matrix and the matrix with a matrix, respectively, the block for calculating the roots of a polynomial whose input is connected to the output of the arithmetic-logical device with the possibility of addition, multiplication and subtraction, block computing the state vector {x, y, z, a ₁ , a ₂ }, the input of which is connected to the output of the block for calculating the roots of the polynomial, the output of the state vector computation unit {x, y, z, a ₁ , a ₂ } is the output of the calculating block tracking coordinates and time parameters.

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