Constrained single-epoch GNSS ambiguity precision factor determination method

Abstract

The invention discloses a single-epoch GNSS ambiguity precision factor determination method with constraints, and relates to the field of GNSS ambiguity estimation. The method comprises the steps of constructing a single-epoch multi-frequency multi-system GNSS double-difference mathematical model with additional constraints, obtaining an ambiguity vector floating point solution variance-covariance matrix analytic expression after the constraints are added, and establishing a single-epoch multi-frequency multi-system C-ADOP approximation formula with constraint conditions and an ADOP scaling factor non-difference non-inverse transformation analytic expression. A C-ADOP approximation formula and an ADOP scaling factor non-difference non-inverse transformation analysis formula are combined, a single-epoch multi-frequency multi-system C-ADOP non-difference non-inverse transformation approximation formula with constraint conditions is obtained, and then a single-epoch multi-frequency multi-system GNSS ambiguity precision factor C-ADOP with the constraint conditions is calculated; according to the method, the single-epoch GNSS ambiguity estimation precision with constraint conditions added and the ambiguity fixing success rate estimation precision are improved.

Description

Technical Field

[0001] The present invention relates to the technical field of GNSS ambiguity estimation, and in particular to a method for determining a constrained single-epoch GNSS ambiguity dilution of precision. Background Art

[0002] Single-epoch positioning based on the Global Navigation Satellite System (GNSS) is the foundation of real-time positioning using GNSS. Correctly fixing the single-epoch GNSS ambiguity vector is a prerequisite for real-time, high-precision positioning using GNSS. Single-epoch GNSS relative positioning technology enables real-time ambiguity resolution, providing real-time location services for applications such as autonomous driving and shared vehicle management. However, due to the short observation time and lack of verification information in single-epoch GNSS positioning, correct fixation of the ambiguity vector presents significant challenges. Methods such as adding external constraints or pre-fixing some high-precision ambiguity vectors can improve the success rate of ambiguity fixation in single-epoch GNSS relative positioning. The ambiguity fixation success rate can be measured using the Ambiguity Dilution of Precision (ADOP), an approximate formula for which accurately estimates the ambiguity fixation success rate using GNSS observations before single-epoch GNSS parameter estimation.

[0003] However, there is currently a lack of single-epoch GNSS ADOP estimation formulas / methods with constraints (such as known baseline length, known baseline vector, etc.) or high-precision constraints for ambiguity vector fixation solutions. This makes it difficult to estimate the constrained ADOP value and ambiguity fixation success rate before single-epoch GNSS parameter solution. Calculations can only be performed using the variance-covariance matrix of the solved ambiguity vector floating-point solution. This increases the complexity of the algorithm and is not conducive to improving the ambiguity fixation success rate by adjusting the added constraints before parameter solution to achieve single-epoch GNSS optimal relative positioning. Summary of the Invention

[0004] In view of the shortcomings of the existing technology that there is a lack of single-epoch GNSS ADOP estimation formula / method with constraints or high-precision constraints on the ambiguity vector fixation solution, and it is difficult to estimate the constrained ADOP value and the ambiguity fixation success rate before the single-epoch GNSS parameter solution is solved, the present invention proposes a constrained single-epoch GNSS ambiguity precision dilution determination method, thereby solving the problems existing in the existing technology.

[0005] A method for determining a constrained single-epoch GNSS ambiguity dilution of precision comprises the following steps: By introducing constraints into the single-epoch multi-frequency multi-system GNSS double-difference mathematical model, a constrained single-epoch multi-frequency multi-system GNSS double-difference mathematical model is constructed. Based on the constrained single-epoch multi-frequency multi-system GNSS double-difference mathematical model, the variance-covariance matrix of the floating-point solution of the ambiguity vector after adding the constraints is obtained using the least squares algorithm. Based on the definition of the ambiguity dilution of precision (ADOP) and the variance-covariance matrix of the floating-point solution of the ambiguity vector after adding constraints, an approximate formula for the single-epoch multi-frequency multi-system GNSS ambiguity dilution of precision (C-ADOP) with constraints is established. Based on the constrained single-epoch multi-frequency multi-system GNSS ambiguity dilution of precision C-ADOP approximation formula, the de-differenced, differentiated, and inverse transformed analytical expression of the normal equation coefficient matrix with the constraints and unconstrained conditions of the first observed satellite as the primary satellite in the ADOP scaling factor is constructed. Based on the de-differenced, differentiated, and inverse transformed analytical expression of the normal equation coefficient matrix with the constraints and unconstrained conditions, the undifferenced, non-inverse transformed analytical expression of the constrained single-epoch multi-frequency multi-system GNSS ADOP scaling factor is established. The constrained single-epoch multi-frequency multi-system GNSS ambiguity dilution of precision C-ADOP is determined by combining the approximate formula of the constrained single-epoch multi-frequency multi-system GNSS ambiguity dilution of precision C-ADOP and the non-difference non-inverse transformation analytical formula of the ADOP scaling factor.

[0006] Furthermore, the process of constructing the single-epoch multi-frequency multi-system GNSS double-difference mathematical model specifically includes the following steps: Multiple GNSS systems observed at the same time t frequency satellites, For the k The number of observation satellites of different frequencies, and the same frequency or different frequencies of different GNSS systems all select the main satellite in their respective systems and frequencies to form a double-difference mathematical model, then the multi-frequency multi-system GNSS double-difference pseudo-range observation quantity is and carrier phase observations The dimensions of , and the mathematical models of the corresponding observation quantities are expressed as follows: ; ; in, and Respectively represent The expectation and variance of and Respectively represent The expectation and variance of 、 、 and They are double-difference pseudo-range observations of multi-frequency multi-system GNSS , double-difference carrier phase observation , double difference ambiguity vector and the baseline vector The column-full rank double difference coefficient matrix and Corresponding to k Frequency 、 and , The dimension is N; , , and They are respectively The corresponding k The carrier wavelength and double-difference ambiguity vector dimensions of the frequencies, for The unit matrix of order; , , Corresponding to multi-frequency and multi-system GNSS observations and Middle k The double-difference observation cofactor matrix of the frequency, the standard deviation of the undifferenced carrier phase and the standard deviation of the undifferenced pseudorange, is a symmetric positive definite matrix; , , , and are respectively the double difference observations and No. k The frequency corresponding to The non-difference coefficient matrix and The difference coefficient matrix of .

[0007] Furthermore, the constraint condition is a general constraint condition or a high-precision constraint condition; the high-precision constraint condition is a fixed solution of the ambiguity vector, and the general constraint condition is other constraint conditions except the fixed solution of the ambiguity vector.

[0008] Furthermore, the method of obtaining the variance-covariance matrix of the floating-point solution of the ambiguity vector after adding the constraints by using the least squares algorithm based on the single-epoch multi-frequency multi-system GNSS double-difference mathematical model with the constraints includes the following steps: Introducing general constraints into the single-epoch multi-frequency multi-system GNSS double-difference mathematical model , we get the single epoch GNSS double-difference mathematical model with general constraints: ; Where, for The corresponding baseline vector non-difference coefficient matrix, and They are The standard deviation and cofactor matrix of is a non-difference diagonal matrix; Using the least squares method to obtain the addition The baseline vector after b and the ambiguity vector variance-covariance matrix of the floating-point solution and The analytical expressions are: ; ; Similarly, we get the high-precision constraint condition of adding ambiguity vector to fix the solution The variance-covariance matrix of the floating-point solution of the ambiguity vector after The analytical expression is: ; Where, To add high-precision constraints The baseline vector after The variance-covariance matrix of For Corresponding The column-full rank double difference coefficient matrix, , and Corresponding to high-precision constraints Middle g The cofactor matrix of the double-difference carrier phase observations at each frequency and the standard deviation of the undifferenced carrier phase.

[0009] Furthermore, the method establishes an approximate formula for the single-epoch multi-frequency multi-system GNSS ambiguity dilution of precision C-ADOP with constraints based on the definition of the ambiguity dilution of precision ADOP and the variance-covariance matrix of the ambiguity vector floating-point solution after adding constraints, specifically comprising the following steps: The ambiguity dilution of precision ADOP is defined as: ; Where, for The determinant of , N is the dimension of the single epoch multi-frequency multi-system GNSS double-difference ambiguity vector, and the unit of ADOP is ; According to ADOP and , establish a general constraint C The approximate formula of single-epoch multi-frequency multi-system GNSS C-ADOP is expressed as: ; Where, is the C-ADOP for single-epoch multi-frequency multi-system GNSS with general constraints, , It is the first in the multi-frequency and multi-system GNSS observation k The frequency double-difference observation cofactor matrix; , ; is the single epoch multi-frequency multi-system GNSS ADOP; the coefficient matrix of the unconstrained condition equation and the constraint condition equation coefficient matrix are all positive definite matrices and are all Hermitian matrices; For general constraints Scaling factor for single-epoch multi-frequency multi-system GNSS ADOP; Similarly, we get the high-precision constraint condition with the fixed solution of the ambiguity vector The approximate formula of the single-epoch multi-frequency multi-system GNSSC-ADOP is expressed as: ; Where, C-ADOP for single-epoch multi-frequency multi-system GNSS with high-precision constraints for ambiguity vector fixation solutions; For high-precision constraints Scaling factor for single-epoch multi-frequency multi-system GNSS ADOP; , The C-ADOP approximation formula is used to characterize the relationship between the single-epoch multi-frequency multi-system GNSS ADOP approximation formula without adding constraints and the ADOP scaling factor.

[0010] Furthermore, the undifferenced, differentiated, and inverse transformed analytical expression of the normal equation coefficient matrix based on the constraints and non-constraints is used to establish a undifferenced, non-inverse transformed analytical expression of the single-epoch multi-frequency, multi-system GNSS ADOP scaling factor with constraints, including the following steps: According to the general restrictions The single-epoch multi-frequency multi-system GNSS ADOP approximate formula is based on the non-constrained condition k The frequency of the first observation satellite is the main satellite of the single epoch multi-frequency multi-system GNSS De-differentiation and de-inverse transformation analytical formula: ; Where, , Corresponding observations or Middle k frequency , , , , For the k The frequency of the observation satellite other than the main satellite, For the k The sum of the observation satellite weights of the frequencies, For the k The weight of the reference satellite with a frequency of is the imaginary unit of the complex number, ; ; The non-difference and non-inverse transformation formula of the single-epoch multi-frequency multi-system GNSS ADOP scaling factor with general constraints is obtained as follows: ; Where, is the non-difference non-inverse transformation scaling factor; For constraints The weight matrix of Take the inverse of the diagonal elements to obtain the non-inverse transformation ; Similarly, the fuzzy vector is used to fix the high-precision constraint condition. g The analytical expression of the non-difference non-inverse transformation of the single-epoch multi-frequency multi-system GNSS ADOP scaling factor with high-precision constraints for the first observation satellite of the frequency as the main satellite is: ; Where, , To correspond to high-precision constraints Middle frequency , , , , For high precision constraints g The frequencies correspond to the weights of the satellites observing except the first one. For No. g The number of observation satellites corresponding to the frequency, For No. gThe sum of the observation satellite weights corresponding to the frequency, for Middle g The weight of the reference satellite or the first observation satellite of a frequency; ; is the non-difference non-inverse transformation scaling factor.

[0011] Furthermore, the method also includes obtaining a constrained single-epoch multi-frequency multi-system GNSS C-ADOP non-difference non-inverse transformation approximate formula after jointly combining the constrained single-epoch multi-frequency multi-system GNSS ambiguity precision dilution C-ADOP approximate formula and the non-difference non-inverse transformation analytical expression of the ADOP scaling factor; obtaining a constrained single-epoch single-frequency single-system GNSS C-ADOP non-difference non-inverse transformation approximate formula by analogy derivation of the constrained single-epoch multi-frequency multi-system GNSS C-ADOP non-difference non-inverse transformation approximate formula; and determining the constrained single-epoch single-frequency single-system GNSS ambiguity precision dilution C-ADOP through the constrained single-epoch single-frequency single-system GNSS C-ADOP non-difference non-inverse transformation approximate formula.

[0012] The present invention provides a constrained single-epoch GNSS ambiguity dilution of precision determination method, which has the following beneficial effects: The present invention constructs a single-epoch multi-frequency multi-system GNSS double-difference mathematical model with added constraints, obtains the variance-covariance matrix analytical expression of the floating-point solution of the ambiguity vector after adding the constraints, establishes a single-epoch GNSS C-ADOP approximate formula with constraints and an analytical expression of the AODP scaling factor non-difference non-inverse transformation, and jointly obtains the single-epoch GNSS C-ADOP non-difference non-inverse transformation approximate formula with constraints. The single-epoch GNSS observation quantity and the single-epoch GNSS C-ADOP non-difference non-inverse transformation approximate formula with constraints are used to accurately estimate the C-ADOP value after adding the constraints before GNSS parameter solution. This solves the problem in the prior art that the C-ADOP value with constraints can only be accurately calculated after GNSS parameter solution according to the ADOP definition, reduces the complexity of the prior art in improving the ambiguity vector ADOP value and the fixation success rate by adjusting the constraints, and improves the ambiguity fixation success rate to achieve single-epoch GNSS optimal relative positioning. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] Figure 1 Flowchart of a method for determining a constrained single-epoch GNSS ambiguity dilution of precision in an embodiment of the present invention. DETAILED DESCRIPTION

[0014] The technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments.

[0015] The present invention proposes a method for determining the single-epoch GNSS ambiguity dilution of precision with constraints. By constructing a single-epoch GNSS double-difference mathematical model with general constraints, an approximate formula for the single-epoch GNSS ADOP with constraint (C-ADOP) with general constraints and an analytical expression for the undifferenced, non-inverse transformation of the C-AODP scaling factor are established. By using an ambiguity vector fixed solution (which can be regarded as a high-precision satellite observation) to replace the general constraints, a single-epoch GNSS double-difference mathematical model with high-precision constraints for the single- / multi-frequency, single- / multi-system ambiguity vector fixed solution is constructed. An approximate formula for the single-epoch GNSS C-ADOP with the ambiguity vector fixed solution and its undifferenced, non-inverse transformation scaling factor are established, forming an approximate formula for the single-epoch GNSS undifferenced, non-inverse transformation C-ADOP with the ambiguity vector fixed solution or general constraints, thereby determining the single-epoch GNSS ambiguity dilution of precision with constraints.

[0016] like Figure 1 As shown, the method specifically includes the following steps: S1. By introducing constraints into the single-epoch multi-frequency multi-system GNSS double-difference mathematical model, a single-epoch multi-frequency multi-system GNSS double-difference mathematical model with constraints is constructed; based on the single-epoch multi-frequency multi-system GNSS double-difference mathematical model with constraints, the variance-covariance matrix of the floating-point solution of the ambiguity vector after adding the constraints is obtained using the least squares algorithm.

[0017] 1. Single epoch GNSS positioning theory: 1.1. Single epoch GNSS double-difference / relative positioning model If multiple GNSS systems are observed at the same time t frequency Satellites ( For the k The number of observation satellites of different frequencies), and the same frequency or different frequencies of different GNSS systems all select the main satellite in their respective systems and frequencies to form a double-difference mathematical model, then the multi-frequency multi-system GNSS double-difference pseudo-range observation quantity is and carrier phase observations The dimensions of , the mathematical model of the corresponding observation (function model and random models ) can be expressed as: (1) (2) Where, and Respectively represent The expectation and variance of and Respectively represent The expectation and variance of 、 、 and They are double-difference pseudo-range observations of multi-frequency multi-system GNSS , double-difference carrier phase observation , double difference ambiguity vector and the baseline vector The column-full rank double difference coefficient matrix , and Corresponding to k Frequency 、 、 and , the dimension is N; , , and They are respectively The corresponding k The carrier wavelength and double-difference ambiguity vector dimensions of the frequencies, for The unit matrix of order; , , and Corresponding to multi-frequency and multi-system GNSS observations and Middle k The double-difference observation cofactor matrix of the frequency, the standard deviation of the undifferenced carrier phase and the standard deviation of the undifferenced pseudorange, is a symmetric positive definite matrix; , , , and are respectively the double difference observations No. k The frequency corresponding to The non-difference coefficient matrix and The differential coefficient matrix of . Using the least squares method, according to formula (1) and formula (2), we can get and Floating-point solution of and and its variance-covariance matrix and : (3) (4) Where, and are all symmetric positive definite matrices.

[0018] 1.2 Single epoch ADOP theory Ambiguity Dilution of Precision (ADOP) is an easily calculated scalar factor used to measure the accuracy of ambiguity estimation and the success rate of ambiguity fixation. It is defined as follows: (5) Where, for The determinant of , N is the dimension of the single epoch multi-frequency multi-system GNSS double-difference ambiguity vector, and the unit of ADOP is cycle. ADOP and ambiguity fixation success rate The relationship is: (6) Where, and are the sequential rounding success rate and integer least squares success rate, is the standard normal distribution function. When the ambiguity vector dimension is constant, the smaller the ADOP value, the higher the ambiguity vector accuracy and the greater the success rate of ambiguity vector fixation. Single epoch multi-frequency multi-system GNSS ADOP ( )The approximate formula is: (7) Where, , , , Multi-frequency multi-system GNSS double-difference observations and Middle k The ratio of the sum of the weights of all observed satellites corresponding to a frequency to the product of their weights, For the k Frequency s In particular, for a single-epoch single-frequency single-system GNSS, the Can be written as : (8) Where, and are the wavelength, undifferenced carrier phase standard deviation, undifferenced pseudorange standard deviation, number of observed satellites and ambiguity vector dimension corresponding to the frequency used by single-frequency single-system GNSS, It is the first single-frequency single-system GNSS s The right of an observation satellite.

[0019] Commonly used constraints in equations (1) and (2) include known baseline length, known baseline vector, and ambiguity vector fixation. Because the ambiguity vector fixation (which can be used as a high-precision pseudorange observation) is consistent with the type of single-epoch GNSS observation, the ambiguity vector fixation and other constraints differ in constructing the constrained single-epoch GNSS ADOP approximation formula. Therefore, in this disclosure, the ambiguity vector fixation is referred to as the ambiguity vector fixation high-precision constraint, and all other constraints other than the ambiguity vector fixation are referred to as general constraints.

[0020] 2. Single epoch GNSS ADOP with general constraints.

[0021] 2.1. The impact of general constraints on single-epoch GNSS parameter estimation Taking known baseline length, known baseline vector, etc. as general constraints The linearized function model and random model matrix form can be written as: (9) Where, For constraints The corresponding baseline vector non-difference coefficient matrix, and Constraints The standard deviation and cofactor matrix of is a non-difference diagonal matrix. If the general constraint Adding to equations (1) and (2), equations (1) and (2) can be written as: (10) Using the least squares method, we can add the general constraint The baseline vector after and the ambiguity vector variance-covariance matrix of the floating-point solution and Analytical formula: (11) (12) Using the matrix inversion formula, add the baseline vector floating point solution before and after the constraints The variance-covariance matrix change formula can be written as: (13) Where, , Taking into account equation (13), the floating point solution of the ambiguity vector before and after adding the general constraint conditions is The variance-covariance matrix change formula can be written as: (14) Where, .because and are all symmetric positive definite matrices. According to the definition of positive definite and semi-positive definite matrices (i.e. for n Order Matrix and any n dimensional column vector , if both have ,but is a positive definite matrix; if ,but is a positive semidefinite matrix), is a positive definite matrix and and is a positive definite or semi-positive definite matrix. Equations (13) and (14) show that compared with Equations (1) and (2), the floating-point solution of the baseline vector solved by Equation (10) is and ambiguity vector float solution Higher accuracy, that is, adding general constraints to the single-epoch GNSS double-difference mathematical model Parameters can be improved and The precision of the estimate.

[0022] S2. Based on the definition of the ambiguity dilution of precision (ADOP) and the variance-covariance matrix of the floating-point solution of the ambiguity vector after adding constraints, an approximate formula for the single-epoch multi-frequency multi-system GNSS ambiguity dilution of precision (C-ADOP) with constraints is established.

[0023] According to the definition of ADOP (5) and The analytical formula (12) with the general constraints The approximate formula of single-epoch multi-frequency multi-system GNSS ADOP (ADOP with constraint, C-ADOP) can be expressed as: (15) Where, C-ADOP for single-epoch multi-frequency multi-system GNSS. . Unconstrained condition equation coefficient matrix and the constraint condition equation coefficient matrix are all positive definite matrices and are all Hermitian matrices. Let and The matrices and matrix The eigenvalues ​​are arranged in ascending order and their values ​​are all positive. With the help of Peter-Weyl theorem, the two Hermitian matrices (i.e. and ) and Eigenvalues ​​and Hermitian matrices The relationship between the eigenvalues ​​of , formula (16) holds: (16) therefore, is the scaling factor of the single-epoch multi-frequency multi-system GNSS ADOP. From Equations (15) and (16), we can see that the higher the accuracy of the added constraints (i.e. The smaller the ambiguity vector The higher the accuracy and fixation success rate.

[0024] According to the matrix The double difference of the main satellite is independent (that is, the matrix is ​​independent of the selection of the main satellite). In order to facilitate the calculation and reduce the complexity of the algorithm to improve the calculation efficiency in practice, it is stipulated that the first observation satellite is used as the main satellite to form a double difference mathematical model. At the same time, in order to use the observation information to realize the advance estimation of the ambiguity vector ADOP, according to formula (15), the first observation satellite is used as the main satellite to form a double difference mathematical model. The frequency of the first observation satellite is the main satellite of the single epoch multi-frequency multi-system GNSS De-differentiation and de-inverse transformation analytical formula: (17) Where, , Corresponding observations (or ) k frequency , , , , For the k The frequency weights of the observation satellites other than the main satellite (i.e. the first observation satellite), For the k The sum of the observation satellite weights of the frequencies, For the k The weight of the reference satellite (or the first observation satellite) with the same frequency, i is the imaginary unit of the complex number, According to equations (15) and (17), the single-epoch multi-frequency multi-system with general constraints is Non-difference non-inverse transformation approximation formula and single-epoch multi-frequency multi-system GNSS ADOP scaling factor The analytical expressions of the non-difference and non-inverse transformation can be written as: (18) Where, For constraints For general constraints, by Take the inverse of the diagonal elements to obtain the non-inverse transformation According to equations (15) and (18), the single-epoch single-frequency single-system with general constraints is Non-difference non-inverse transformation approximation formula and single-epoch single-frequency single-system GNSS ADOP scaling factor The analytical expression of the non-difference and non-inverse transformation can be written as: (19) Where, , , , is the weight of the observation satellites other than the first observation satellite in the single-frequency single-system GNSS. Corresponding to single-frequency single-system GNSS The non-difference coefficient matrix of is the sum of the observation satellite weights of a single-epoch single-frequency single-system GNSS, is the weight of a single-epoch, single-frequency, single-system GNSS reference satellite (or the first observation satellite).

[0025] Single-epoch GNSS ADOP with high-precision constraints for ambiguity vector fixation: When the ambiguity vector is successfully fixed, the distance between the ground receiver and the observation satellite (satellite-to-ground distance) based on the ambiguity vector fixation solution can be regarded as a high-precision satellite-to-ground distance constraint. Unlike general constraints such as known baseline vectors and known baseline lengths, because the ambiguity vector fixation solution is consistent with the GNSS observation type (i.e., the ambiguity fixation solution can be regarded as a high-precision pseudorange observation), when the ambiguity vector fixation solution is used as a constraint, the single-epoch GNSS ADOP scaling factor is different from Equations (18) and (19). In addition, the ambiguity vector fixation solution also includes ambiguity vector fixation solutions for single-frequency single-system GNSS and multi-frequency multi-system GNSS. Therefore, the present invention further provides a specific analytical form of the single-epoch GNSS ADOP with ambiguity vector fixation solution.

[0026] Assume that the double-difference carrier phase observations of single-epoch multi-frequency multi-system GNSS are The corresponding double difference ambiguity vector Fixed to ,but It can be regarded as a high-precision pseudo-range observation constraint. Including frequencies and it is stipulated that the main satellite is selected in each system and each frequency to form a double difference mathematical model, then The double difference function model and random model can be written as: (20) Where, , For The corresponding g The ambiguity vector of each frequency is fixed to solve the high-precision constraint condition. and Respectively The corresponding double-difference ambiguity vector fixed solution and double-difference carrier phase observation; For Corresponding The column-full rank double difference coefficient matrix, , , , and and high-precision constraints respectively No. g The frequency corresponding to The non-difference coefficient matrix and The difference coefficient matrix of ; , , and They are respectively The corresponding g The carrier wavelength and double-difference ambiguity vector dimensions of the frequencies; , and Corresponding to high-precision constraints Middle g The cofactor matrix of the double-difference carrier phase observations and the standard deviation of the non-difference carrier phase are used. The high-precision constraint condition (20) of the fixed solution of the multi-frequency multi-system GNSS ambiguity vector is used to replace the general constraint condition in the single-epoch GNSS double-difference mathematical model (10). C , then formula (10) can be written as: (twenty one) By analogy with formula (12), we can add high-precision constraints The variance-covariance matrix of the floating-point solution of the ambiguity vector after The analytical expression is: (twenty two) Where, To add high-precision constraints The baseline vector after The variance-covariance matrix of .

[0027] According to equations (15) and (22), with the multi-frequency multi-system GNSS high-precision constraint conditions Single epoch multi-frequency multi-system Approximate formula and ADOP scaling factor They can be written as: (twenty three) Where, , .

[0028] S3. Based on the C-ADOP approximate formula of the single-epoch multi-frequency multi-system GNSS ambiguity precision coefficient matrix with constraints, construct the de-differenced, differentiated, and inverse transformed analytical expression of the normal equation coefficient matrix with constraints or non-constraints based on the first observed satellite as the main satellite in the ADOP scaling factor, and then establish the undifferenced, non-inverse transformed analytical expression of the single-epoch multi-frequency multi-system GNSS ADOP scaling factor with constraints.

[0029] If the multi-frequency multi-system GNSS ambiguity vector is fixed, the high-precision constraint condition method equation coefficient matrix With the constraint condition g ( g =1,2,…, ) frequency is the main satellite, then by analogy with Equations (17) and (18), the single-epoch multi-frequency multi-system GNSS with high-precision constraints is Non-difference non-inverse transformation approximation formula and single-epoch multi-frequency multi-system GNSS ADOP scaling factor The analytical expressions of non-difference and non-inverse transformation can be written as: (twenty four) Where, , To correspond to high-precision constraints Middle g frequency , , , , For high precision constraints No. g The frequencies correspond to the weights of the satellites observing except the first one. For No. g The number of observation satellites corresponding to the frequency, For No. g The sum of the observation satellite weights corresponding to the frequency, for Middle g The weight of the reference satellite (or the first observing satellite) with the same frequency.

[0030] When the ambiguity vector fixed solution corresponds to a single-frequency single-system GNSS, the high-precision constraint of the ambiguity vector fixed solution in equation (20) is Can be written as : (25) Where, For high-precision constraints for single-frequency single-system GNSS, , and They are double-difference carrier phase observations, carrier wavelength, and The column rank double difference coefficient matrix, the ambiguity vector fixed solution and Dimensions, and They are The cofactor matrix of the double-difference carrier phase observations and the standard deviation of the non-difference carrier phase. According to Equations (23) and (24), when the ambiguity vector fixed solution constraint corresponds to a single-frequency single-system GNSS, the single-epoch multi-frequency multi-system with the high-precision constraint of the single-frequency single-system GNSS ambiguity vector fixed solution is and single epoch multi-frequency multi-system GNSS ADOP scaling factors The analytical expressions of non-difference and non-inverse transformation can be written as: (26) Where, , , , Constraints for high precision of single epoch single frequency single system GNSS The corresponding weights of observation satellites other than the main satellite, For The corresponding number of observation satellites, For Corresponding The non-difference coefficient matrix of , For The weight of the corresponding reference satellite (or the first observing satellite).

[0031] When Equations (1) and (2) and the high-precision constraints of the ambiguity vector fixed solution are all oriented to single-epoch single-frequency single-system GNSS, the high-precision constraints of the ambiguity vector fixed solution in Equation (25) are The accuracy is consistent with the single-frequency single-system GNSS double-difference carrier phase observation accuracy (i.e. both are )and : (27) Where, , For single epoch single frequency single system GNSS high precision constraints The carrier wavelength, is the double-difference ambiguity vector of the single-epoch single-frequency single-system GNSS carrier phase observation.

[0032] According to equations (23) and (26), when both the ambiguity vector fixed solution and the GNSS observations are oriented to the single-epoch single-frequency single-system GNSS, the high-precision constraint condition of the single-frequency single-system GNSS ambiguity vector fixed solution is Single-epoch single-frequency single system Non-difference non-inverse transformation approximation formula and single-epoch single-frequency single-system GNSS ADOP scaling factor The analytical expressions of non-difference and non-inverse transformation can be written as: (28) S4. Combine the constrained single-epoch multi-frequency multi-system GNSS C-ADOP approximate formula and the scaling factor undifferenced non-inverse transformation analytical formula to obtain the constrained single-epoch multi-frequency multi-system GNSS C-ADOP undifferenced non-inverse transformation approximate formula; determine the constrained single-epoch multi-frequency multi-system GNSS ambiguity dilution of precision C-ADOP through the constrained single-epoch multi-frequency multi-system GNSS C-ADOP undifferenced non-inverse transformation approximate formula.

[0033] In summary, after introducing (1) the high-precision constraint condition for the fixed solution of the ambiguity vector, the present invention calculates the single-epoch GNSS using equation (7) or equation (8) based on the single-epoch multi-frequency multi-system or single-frequency single-system GNSS observations. Value or According to the GNSS observations, the non-difference non-inverse transformation scaling factor of the single epoch GNSS ADOP is calculated using the second formula of formula (24), the second formula of formula (26), or the second formula of formula (28). or ; The calculated single epoch GNSS or and non-difference non-inverse transform scaling factors or The single-epoch GNSS C-ADOP value with the single-epoch ambiguity vector fixed solution constraint is obtained by multiplication. (2) When facing general constraints such as known baseline length and known baseline vector, the single-epoch GNSS C-ADOP is calculated using Equation (7) or Equation (8) based on the single-epoch multi-frequency multi-system or single-frequency single-system GNSS observations. or According to the GNSS observations, the non-difference non-inverse transformation scaling factor of the single epoch GNSS ADOP is calculated using the second formula of formula (18) or the second formula of formula (19). or ; The calculated single epoch GNSS or and non-difference non-inverse transform scaling factors or The multiplication results in a single-epoch GNSS C-ADOP value with other general constraints.

[0034] The present invention constructs a single-epoch multi-frequency multi-system GNSS double-difference mathematical model with added constraints, obtains the variance-covariance matrix analytical expression of the floating-point solution of the ambiguity vector after adding the constraints, establishes a single-epoch GNSS C-ADOP approximate formula with constraints and an analytical expression of the AODP scaling factor non-difference non-inverse transformation, and jointly obtains the single-epoch GNSS C-ADOP non-difference non-inverse transformation approximate formula with constraints. The single-epoch GNSS observation quantity and the single-epoch GNSS C-ADOP non-difference non-inverse transformation approximate formula with constraints are used to accurately estimate the ADOP value after adding the constraints before GNSS parameter solution. This solves the problem in the prior art that the ADOP value with constraints can only be accurately calculated after GNSS parameter solution according to the ADOP definition, reduces the complexity of the prior art in improving the ambiguity vector ADOP value and the fixation success rate by adjusting the constraints, and improves the ambiguity fixation success rate to achieve single-epoch GNSS optimal relative positioning.

[0035] The above description is only a preferred specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any technician familiar with the technical field, within the technical scope disclosed by the present invention, who makes equivalent replacements or changes based on the technical solution and inventive concept of the present invention, should be covered by the scope of protection of the present invention.

Claims

1. A constrained single-epoch GNSS ambiguity DOP determination method, characterized in that: The following steps are involved: By introducing constraints into the single-epoch multi-frequency multi-system GNSS double-difference mathematical model, a constrained single-epoch multi-frequency multi-system GNSS double-difference mathematical model is constructed. Based on the constrained single-epoch multi-frequency multi-system GNSS double-difference mathematical model, the variance-covariance matrix of the floating-point solution of the ambiguity vector after adding the constraints is obtained using the least squares algorithm. Based on the definition of the ambiguity dilution of precision (ADOP) and the variance-covariance matrix of the floating-point solution of the ambiguity vector after adding constraints, an approximate formula for the single-epoch multi-frequency multi-system GNSS ambiguity dilution of precision (C-ADOP) with constraints is established. Based on the constrained single-epoch multi-frequency multi-system GNSS ambiguity dilution of precision C-ADOP approximation formula, the de-differenced, differentiated, and inverse transformed analytical expression of the normal equation coefficient matrix with the constraints and unconstrained conditions of the first observed satellite as the primary satellite in the ADOP scaling factor is constructed. Based on the de-differenced, differentiated, and inverse transformed analytical expression of the normal equation coefficient matrix with the constraints and unconstrained conditions, the undifferenced, non-inverse transformed analytical expression of the constrained single-epoch multi-frequency multi-system GNSS ADOP scaling factor is established. The constrained single-epoch multi-frequency multi-system GNSS ambiguity dilution of precision C-ADOP is determined by combining the approximate formula of the constrained single-epoch multi-frequency multi-system GNSS ambiguity dilution of precision C-ADOP and the non-difference non-inverse transformation analytical formula of the ADOP scaling factor.

2. The method for determining the constrained single-epoch GNSS ambiguity dilution of precision according to claim 1, wherein: The construction process of the single-epoch multi-frequency multi-system GNSS double-difference mathematical model specifically includes the following steps: Multiple GNSS systems observed at the same time t frequency satellites, For the k The number of observation satellites of different frequencies, and the same frequency or different frequencies of different GNSS systems all select the main satellite in their respective systems and frequencies to form a double-difference mathematical model, then the multi-frequency multi-system GNSS double-difference pseudo-range observation quantity is and carrier phase observations The dimensions of , and the mathematical models of the corresponding observation quantities are expressed as follows: ； ； in, and Respectively represent The expectation and variance of and Respectively represent The expectation and variance of 、 、 and They are double-difference pseudo-range observations of multi-frequency multi-system GNSS , double-difference carrier phase observation , double difference ambiguity vector and the baseline vector The column-full rank double difference coefficient matrix and Corresponding to k Frequency 、 、 and , The dimension is N; , , and They are respectively The corresponding k The carrier wavelength and double-difference ambiguity vector dimensions of the frequencies, for The unit matrix of order; , , Corresponding to multi-frequency and multi-system GNSS observations and Middle k The double-difference observation cofactor matrix of the frequency, the standard deviation of the undifferenced carrier phase and the standard deviation of the undifferenced pseudorange, is a symmetric positive definite matrix; , , , and are respectively the double difference observations and No. k The frequency corresponding to The non-difference coefficient matrix and The difference coefficient matrix of .

3. The method for determining the constrained single-epoch GNSS ambiguity dilution of precision according to claim 2, wherein: The constraint conditions are general constraint conditions or high-precision constraint conditions; the high-precision constraint conditions are ambiguity vector fixed solutions, and the general constraint conditions are other constraint conditions except for the ambiguity vector fixed solutions.

4. The method for determining the constrained single-epoch GNSS ambiguity dilution of precision according to claim 3, wherein: The method comprises the following steps: obtaining a variance-covariance matrix of a floating-point solution of an ambiguity vector after adding constraints by using a least squares algorithm based on a single-epoch multi-frequency multi-system GNSS double-difference mathematical model with constraints, wherein the variance-covariance matrix of the floating-point solution of the ambiguity vector after adding constraints is obtained. Introducing general constraints into the single-epoch multi-frequency multi-system GNSS double-difference mathematical model , we get the single epoch GNSS double-difference mathematical model with general constraints: ； Where, for The corresponding baseline vector non-difference coefficient matrix, and They are The standard deviation and cofactor matrix of is a non-difference diagonal matrix; Using the least squares method to obtain the addition The baseline vector after b and the ambiguity vector variance-covariance matrix of the floating-point solution and The analytical expressions are: ； ； Similarly, we get the high-precision constraint condition of adding ambiguity vector to fix the solution The variance-covariance matrix of the floating-point solution of the ambiguity vector after The analytical expression is: ； Where, To add high-precision constraints The baseline vector after The variance-covariance matrix of For Corresponding The column-full rank double difference coefficient matrix, , and Corresponding to high-precision constraints Middle g The cofactor matrix of the double-difference carrier phase observations at each frequency and the standard deviation of the undifferenced carrier phase.

5. The method for determining the constrained single-epoch GNSS ambiguity dilution of precision according to claim 4, wherein: According to the definition of the ambiguity dilution of precision ADOP and the variance-covariance matrix of the ambiguity vector floating-point solution after adding the constraint conditions, an approximate formula of the single-epoch multi-frequency multi-system GNSS ambiguity dilution of precision C-ADOP with constraints is established, which specifically includes the following steps: The ambiguity dilution of precision ADOP is defined as: ； Where, for The determinant of , N is the dimension of the single epoch multi-frequency multi-system GNSS double-difference ambiguity vector, and the unit of ADOP is ; According to ADOP and , establish a general constraint C The approximate formula of single-epoch multi-frequency multi-system GNSS C-ADOP is expressed as: Where, is the C-ADOP for single-epoch multi-frequency multi-system GNSS with general constraints, , It is the first in the multi-frequency and multi-system GNSS observation k The frequency double-difference observation cofactor matrix; , ; is the single epoch multi-frequency multi-system GNSS ADOP; the coefficient matrix of the unconstrained condition equation and the constraint condition equation coefficient matrix are all positive definite matrices and are all Hermitian matrices; For general constraints Scaling factor for single-epoch multi-frequency multi-system GNSS ADOP; Similarly, we get the high-precision constraint condition with the fixed solution of the ambiguity vector The approximate formula of single-epoch multi-frequency multi-system GNSS C-ADOP is expressed as: Where, C-ADOP for single-epoch multi-frequency multi-system GNSS with high-precision constraints for ambiguity vector fixation solutions; For high-precision constraints Scaling factor for single-epoch multi-frequency multi-system GNSS ADOP; , The C-ADOP approximation formula is used to characterize the relationship between the single-epoch multi-frequency multi-system GNSS ADOP approximation formula without adding constraints and the ADOP scaling factor.

6. The method for determining the constrained single-epoch GNSS ambiguity dilution of precision according to claim 5, wherein: The undifferenced, differentiated, and inverse transformed analytical expressions of the normal equation coefficient matrix based on the constraints and non-constraints are used to establish the undifferenced, non-inverse transformed analytical expression of the single-epoch multi-frequency, multi-system GNSS ADOP scaling factor with constraints, including the following steps: According to the general restrictions The single-epoch multi-frequency multi-system GNSS ADOP approximate formula is based on the non-constrained condition k The frequency of the first observation satellite is the main satellite of the single epoch multi-frequency multi-system GNSS De-differentiation and de-inverse transformation analytical formula: ； Where, , Corresponding observations or Middle k frequency , , , , For the k The frequency of the observation satellite other than the main satellite, For the k The sum of the observation satellite weights of the frequencies, For the k The weight of the reference satellite with a frequency of is the imaginary unit of the complex number, ; ; According to By removing the difference and inverse transform analytical expressions, we obtain the undifference and inverse transform analytical expressions of the single-epoch multi-frequency multi-system GNSS ADOP scaling factor with general constraints: Where, is the non-difference non-inverse transformation scaling factor; For constraints The weight matrix of Take the inverse of the diagonal elements to obtain the non-inverse transformation ; Similarly, the fuzzy vector is used to fix the high-precision constraint condition. g The analytical expression of the non-difference non-inverse transformation of the single-epoch multi-frequency multi-system GNSS ADOP scaling factor with high-precision constraints for the first observation satellite of the frequency as the main satellite is: ； Where, , To correspond to high-precision constraints Middle frequency , , , , For high precision constraints No. g The frequencies correspond to the weights of the satellites observing except the first one. For No. g The number of observation satellites corresponding to the frequency, For No. g The sum of the observation satellite weights corresponding to the frequency, for Middle g The weight of the reference satellite or the first observation satellite of a frequency; ; is the non-difference non-inverse transformation scaling factor.

7. The method for determining the constrained single-epoch GNSS ambiguity dilution of precision according to claim 1, wherein: The invention also includes obtaining a constrained single-epoch multi-frequency multi-system GNSS C-ADOP non-difference non-inverse transformation approximate formula after jointly combining the constrained single-epoch multi-frequency multi-system GNSS ambiguity precision factor C-ADOP approximate formula and the non-difference non-inverse transformation analytical expression of the ADOP scaling factor; obtaining a constrained single-epoch single-frequency single-system GNSS C-ADOP non-difference non-inverse transformation approximate formula by analogy derivation of the constrained single-epoch multi-frequency multi-system GNSS C-ADOP non-difference non-inverse transformation approximate formula; and determining the constrained single-epoch single-frequency single-system GNSS ambiguity precision factor C-ADOP through the constrained single-epoch single-frequency single-system GNSS C-ADOP non-difference non-inverse transformation approximate formula.

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