A method for determining single epoch GNSS ambiguity precision factor with constraints

Abstract

The application discloses a kind of single epoch GNSS ambiguity precision factor determination methods with constraint, it is related to GNSS ambiguity estimation field;The method includes constructing additional constraint single epoch multi-frequency multi-system GNSS double difference mathematical model, obtains the analytic expression of variance-covariance matrix solution of ambiguity vector floating point solution after adding constraint, establishes the approximate formula of single epoch multi-frequency multi-system C-ADOP with constraint condition and the analytic expression of ADOP scaling factor non-difference non-inverse transformation, simultaneously C-ADOP approximate formula and ADOP scaling factor non-difference non-inverse transformation analytic expression, obtain the approximate formula of single epoch multi-frequency multi-system C-ADOP non-difference non-inverse transformation with constraint condition, and then calculate out the precision factor C-ADOP of single epoch multi-frequency multi-system GNSS ambiguity with constraint condition;The method improves the precision of single epoch GNSS ambiguity estimation with constraint condition and the precision of ambiguity fixing success rate estimation.

Description

Technical Field

[0001] This invention relates to the field of GNSS ambiguity estimation technology, specifically to a constrained single-epoch GNSS ambiguity accuracy factor determination method. Background Technology

[0002] Single-epoch positioning based on the Global Navigation Satellite System (GNSS) is the foundation for real-time positioning using GNSS, and the correct fixation of the single-epoch GNSS ambiguity vector is a prerequisite for real-time high-precision positioning using GNSS. Single-epoch GNSS relative positioning technology can achieve 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, the correct fixation of its ambiguity vector faces significant challenges. Adding external constraints or adding pre-fixed 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), and its approximate formula can accurately predict 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 prediction formulas / methods with constraints (such as known baseline length, known baseline vector, etc.) or with high-precision constraints for fixed ambiguity vector solutions. It is difficult to estimate the constrained ADOP value and ambiguity fixation success rate before single-epoch GNSS parameter calculation. Only the variance-covariance matrix of the floating-point solution of the ambiguity vector after calculation can be used for calculation, which will increase the complexity of the algorithm and is not conducive to improving the ambiguity fixation success rate by adjusting the added constraints before parameter calculation to achieve optimal relative positioning of single-epoch GNSS. Summary of the Invention

[0004] To address the shortcomings of existing technologies, which lack constrained GNSS ADOP prediction formulas / methods with high-precision constraints or fixed ambiguity vector solutions, making it difficult to estimate constrained ADOP values ​​and ambiguity fixation success rates before single-epoch GNSS parameter calculation, this invention proposes a constrained single-epoch GNSS ambiguity accuracy factor determination method, thereby solving the problems existing in the prior art.

[0005] A constrained method for determining the accuracy factor of single-epoch GNSS ambiguity includes the following steps:

[0006] 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 constraints is obtained using the least squares algorithm.

[0007] Based on the definition of the ambiguity accuracy factor 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 accuracy factor C-ADOP with constraints is established.

[0008] Based on the constrained single-epoch multi-frequency multi-system GNSS ambiguity accuracy factor C-ADOP approximation formula, an analytical expression for the difference-reduction and inverse transformation of the normal equation coefficient matrix with the first observation satellite as the master satellite is constructed in the ADOP scaling factor. Based on the difference-reduction and inverse transformation of the normal equation coefficient matrix with the constrained and unconstrained conditions, an analytical expression for the non-difference and non-inverse transformation of the constrained single-epoch multi-frequency multi-system GNSS ADOP scaling factor is established.

[0009] By combining the constrained approximate formula for the single-epoch multi-frequency multi-system GNSS ambiguity accuracy factor C-ADOP with the non-difference and non-inverse transformation analytical expression of the ADOP scaling factor, the constrained single-epoch multi-frequency multi-system GNSS ambiguity accuracy factor C-ADOP is determined.

[0010] Furthermore, the construction process of the single-epoch multi-frequency multi-system GNSS double-difference mathematical model specifically includes the following steps:

[0011] Multiple GNSS systems observed at the same time t Frequency One satellite, For the first k The number of observation satellites at a given frequency, and the fact that different GNSS systems select their master satellites within their respective systems and frequencies to form a double-difference mathematical model at the same or different frequencies, then the resulting multi-frequency, multi-system GNSS double-difference pseudorange observations... and carrier phase observations All dimensions are The mathematical models for the corresponding observations are expressed as follows:

[0012] ;

[0013] ;

[0014] in, and Represent Expectation and variance and Represent Expectation and variance; , , and Double-difference pseudorange observations of multi-frequency, multi-system GNSS Double-difference carrier phase observations Double-difference ambiguity vector and baseline vector Full-rank double-difference coefficient matrix and They correspond to the first k frequency , and , The dimension is N; , , and They are respectively with The corresponding number k The carrier wavelength and the dimension of the double-difference ambiguity vector at each frequency. for An identity matrix of order 1; , , Corresponding to multi-frequency, multi-system GNSS observations and The Middle k Cofactor array of double-difference observations at each frequency, standard deviation of unequal carrier phase, and standard deviation of unequal pseudorange. It is a symmetric positive definite matrix; , , , and These are respectively double-difference observations. and No. k Each frequency corresponds to The non-difference coefficient matrix and The difference coefficient matrix.

[0015] Furthermore, the constraints are general constraints or high-precision constraints; high-precision constraints are fixed solutions for ambiguity vectors, and general constraints are other constraints besides fixed solutions for ambiguity vectors.

[0016] Furthermore, the step of obtaining the variance-covariance matrix of the floating-point solution of the ambiguity vector after adding constraints, based on the single-epoch multi-frequency multi-system GNSS double-difference mathematical model with constraints, using the least squares algorithm, includes the following steps:

[0017] Introducing general constraints into the single-epoch multi-frequency multi-system GNSS double-difference mathematical model The single-epoch GNSS double-difference mathematical model with general constraints is obtained as follows:

[0018] ;

[0019] In the formula, for The corresponding baseline vector non-difference coefficient matrix, and They are respectively Standard deviation and cofactor matrix and It is a non-difference diagonal matrix;

[0020] The addition was obtained using the least squares method. The baseline vector after b and ambiguity vector Variance-covariance matrix of floating-point solutions and The analytical expressions are as follows:

[0021] ;

[0022] ;

[0023] Similarly, the high-precision constraint conditions for adding fuzzy vectors to the fixed solution are obtained. The variance-covariance matrix of the floating-point solution of the ambiguity vector The analytical expression is:

[0024] ;

[0025] In the formula, To add high-precision constraints The baseline vector after The variance-covariance matrix, To and corresponding A column-full-rank double-difference coefficient matrix , and Corresponding to high-precision constraint conditions The Middle g Cofactor array of double-difference carrier phase observations at each frequency and standard deviation of non-difference carrier phase.

[0026] Furthermore, the step of establishing an approximate formula for the single-epoch multi-frequency multi-system GNSS ambiguity accuracy factor C-ADOP based on the definition of the ambiguity accuracy factor ADOP and the variance-covariance matrix of the floating-point solution of the ambiguity vector after adding constraints specifically includes the following steps:

[0027] Define the ambiguity precision factor ADOP as:

[0028] ;

[0029] In the formula, for The determinant of ADOP, where N is the dimension of the single-epoch multi-frequency multi-system GNSS double-difference ambiguity vector, and the unit of ADOP is . ;

[0030] According to ADOP and Establish with general constraints C The approximate formula for single-epoch multi-frequency multi-system GNSS C-ADOP is expressed as:

[0031] ;

[0032] In the formula, For C-ADOP of single-epoch multi-frequency multi-system GNSS with general constraints, , For multi-frequency, multi-system GNSS observations, the first k The cofactor matrix of the double-difference observations of frequency; , ; For single-epoch multi-frequency multi-system GNSSADOP; coefficient matrix of unconstrained normal equations and constraint condition normal equation coefficient matrix All are positive definite matrices and all are Hermitian matrices; For general constraints Scaling factor for single-epoch multi-frequency multi-system GNSSADOP;

[0033] Similarly, the high-precision constraint conditions with fixed solutions and fuzzy vectors are obtained. The approximate formula for single-epoch multi-frequency multi-system GNSSC-ADOP is expressed as:

[0034] ;

[0035] In the formula, C-ADOP for single-epoch multi-frequency multi-system GNSS with high-precision constraints and fixed ambiguity vector solutions; To address 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 and the ADOP scaling factor without added constraints.

[0036] Furthermore, the step of establishing the non-difference, non-inverse transformation analytical expression for the single-epoch multi-frequency multi-system GNSS ADOP scaling factor with constraints based on the difference, differentiation, and inverse transformation analytical expression of the coefficient matrix of the normal equations with and without constraints includes the following steps:

[0037] According to the attached general constraints The single-epoch multi-frequency multi-system GNSS ADOP approximation formula, with unconstrained conditions, is used in the first... k The first observation satellite of the frequency is the master satellite for single-epoch multi-frequency multi-system GNSS. Analysis of difference and inverse transformation:

[0038] ;

[0039] In the formula, , Corresponding observations or The Middle k Frequency , , , , For the first k The weighting of observation satellites at each frequency, excluding the primary satellite. For the first k The sum of the weights of the observation satellites at each frequency, For the first k The weight of a reference satellite at a given frequency, The imaginary unit of complex numbers. ; ;

[0040] The non-difference, non-inverse transform analytical expression for the single-epoch multi-frequency multi-system GNSS ADOP scaling factor with general constraints is then obtained as follows:

[0041] ;

[0042] In the formula, It is the scaling factor for non-difference and non-inverse transform; Constraints The weight matrix; for general constraints, by adjusting the weight matrix... Taking the reciprocal of the diagonal elements yields the non-inverse transformation. ;

[0043] Similarly, the high-precision constraint condition is solved by fixing the fuzzy vector. g The analytical expression for the non-difference, non-inverse transform of the scaling factor for a single-epoch multi-frequency, multi-system GNSS ADOP system with high-precision constraints, based on the first observation satellite of the frequency as the master satellite, is as follows:

[0044] ;

[0045] In the formula, , To correspond to high-precision constraints The Middle Frequency , , , , To meet high-precision constraints g The frequency correspondence and the weights of the other observation satellites besides the first observation. To and The g The number of observation satellites corresponding to the frequency To and The g The sum of the weights of the observed satellites corresponding to the frequency. for The Middle g The rights of a reference satellite or the first observation satellite at a given frequency; ; It is the scaling factor for non-difference and non-inverse transformation.

[0046] Furthermore, it also includes obtaining the constrained single-epoch multi-frequency multi-system GNSS C-ADOP ambiguity accuracy factor approximation formula and the ADOP scaling factor non-difference non-inverse transformation analytical formula after simultaneously establishing the constrained single-epoch multi-frequency multi-system GNSS C-ADOP non-difference non-inverse transformation approximation formula; by analogy with the constrained single-epoch multi-frequency multi-system GNSS C-ADOP non-difference non-inverse transformation approximation formula, the constrained single-epoch single-frequency single-system GNSS C-ADOP non-difference non-inverse transformation approximation formula is obtained; and the constrained single-epoch single-frequency single-system GNSS C-ADOP ambiguity accuracy factor C-ADOP is determined using the constrained single-epoch single-frequency single-system GNSS C-ADOP non-difference non-inverse transformation approximation formula.

[0047] This invention provides a constrained method for determining the accuracy factor of single-epoch GNSS ambiguity, which has the following advantages:

[0048] This invention constructs a single-epoch multi-frequency multi-system GNSS double-difference mathematical model with added constraints, derives the variance-covariance matrix analytical expression of the floating-point solution of the ambiguity vector after adding constraints, establishes the single-epoch GNSS C-ADOP approximation formula with constraints and the AODP scaling factor non-difference non-inverse transformation analytical expression, and simultaneously obtains the single-epoch GNSS C-ADOP non-difference non-inverse transformation approximation formula with constraints. Using single-epoch GNSS observations and the single-epoch GNSS C-ADOP non-difference non-inverse transformation approximation formula with constraints, it achieves accurate estimation of the C-ADOP value after adding constraints before GNSS parameter calculation. This solves the problem that existing technologies can only accurately calculate the C-ADOP value with constraints after GNSS parameter calculation based on the ADOP definition. It reduces the complexity of existing technologies in improving the ADOP value of the ambiguity vector and fixing the success rate by adjusting constraints, and improves the ambiguity fixing success rate to achieve optimal relative positioning of single-epoch GNSS. Attached Figure Description

[0049] Figure 1 This is a flowchart of a method for determining the precision factor of single-epoch GNSS ambiguity with constraints in an embodiment of the present invention. Detailed Implementation

[0050] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments.

[0051] This invention proposes a method for determining the accuracy factor of single-epoch GNSS ambiguity with constraints. By constructing a single-epoch GNSS double-difference mathematical model with added general constraints, an approximate formula for single-epoch GNSS ADOP (ADOP with constraint, C-ADOP) with general constraints and an analytical expression for the non-difference, non-inverse transformation of the C-ADOP scaling factor are established. Using a fixed solution of the ambiguity vector (which can be considered as high-precision satellite observations) to replace the general constraints, a single-epoch GNSS double-difference mathematical model with added high-precision constraints of fixed solutions for single / multi-frequency single / multi-system ambiguity vectors is constructed. An approximate formula for single-epoch GNSS C-ADOP with fixed solutions of the ambiguity vector and its non-difference, non-inverse transformation scaling factor are established, forming an approximate formula for single-epoch GNSS C-ADOP with fixed solutions of the ambiguity vector or general constraints, thereby determining the accuracy factor of single-epoch GNSS ambiguity with constraints.

[0052] like Figure 1 As shown, the method specifically includes the following steps:

[0053] S1. 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 constraints is obtained using the least squares algorithm.

[0054] 1. Single-epoch GNSS positioning theory:

[0055] 1.1 Single-epoch GNSS double-difference / relative positioning model

[0056] If multiple GNSS systems are observed at the same time t Frequency satellites ( For the first k The number of observation satellites at each frequency, and the selection of master satellites within each system and frequency of different GNSS systems to form a double-difference mathematical model, then the resulting multi-frequency, multi-system GNSS double-difference pseudorange observations... and carrier phase observations All dimensions are The mathematical model (function model) of the corresponding observations and random models These can be represented as:

[0057] (1)

[0058] (2)

[0059] In the formula, and Represent Expectation and variance and Represent Expectation and variance; , , and Double-difference pseudorange observations of multi-frequency, multi-system GNSS Double-difference carrier phase observations Double-difference ambiguity vector and baseline vector Full-rank double-difference coefficient matrix , and They correspond to the first k frequency , , and The dimension of is N; , , and They are respectively with The corresponding number k The carrier wavelength and the dimension of the double-difference ambiguity vector at each frequency. for An identity matrix of order 1; , , and Corresponding to multi-frequency, multi-system GNSS observations and The Middle k Cofactor array of double-difference observations at each frequency, standard deviation of unequal carrier phase, and standard deviation of unequal pseudorange. It is a symmetric positive definite matrix; , , , and These are respectively double-difference observations. No. k Each frequency corresponds to The non-difference coefficient matrix and The difference coefficient matrix. Using the least squares method, according to equations (1) and (2), we can obtain and floating-point solution and and its variance-covariance matrix and :

[0060] (3)

[0061] (4)

[0062] In the formula, and All are symmetric positive definite matrices.

[0063] 1.2 Single-Epoch ADOP Theory

[0064] The Ambiguity Dilution of Precision (ADOP) is an easily calculated scalar factor primarily used to measure the accuracy of ambiguity estimation and the success rate of ambiguity fixation. Its definition is as follows:

[0065] (5)

[0066] In the formula, for The determinant of ADOP is given by N, where N is the dimension of the single-epoch multi-frequency multi-system GNSS double-difference ambiguity vector, and ADOP is measured in cycles. ADOP is related to the success rate of ambiguity fixation. The relationship is:

[0067] (6)

[0068] In the formula, and These are the success rates of sequential floor function and integer least squares, respectively. The standard normal distribution function is used. When the ambiguity vector dimension is fixed, the smaller the ADOP value, the higher the ambiguity vector accuracy and the greater the success rate of fixing the ambiguity vector. Single-epoch multi-frequency multi-system GNSS ADOP ( The approximate formula is:

[0069] (7)

[0070] In the formula, , , , For multi-frequency, multi-system GNSS double-difference observations and The Middle k The ratio of the weighted sum to the weighted product of all observed satellites corresponding to each frequency. For the first k The frequency of s The rights of a single observation satellite. Specifically, for a single-epoch, single-frequency, single-system GNSS, in equation (8) Can be written as :

[0071] (8)

[0072] In the formula, and These represent the wavelength, standard deviation of the unequal carrier phase, standard deviation of the unequal pseudorange, number of observed satellites, and dimension of the ambiguity vector corresponding to the frequencies used in a single-frequency, single-system GNSS. For single-frequency single-system GNSS s The right to observe satellites.

[0073] Common constraints used in equations (1) and (2) include known baseline length, known baseline vector, and fixed ambiguity vector solution. Since the fixed ambiguity vector solution (which can be used as a high-precision pseudorange observation) is consistent with the type of single-epoch GNSS observation, the fixed ambiguity vector solution will differ from other constraints in constructing the constrained single-epoch GNSS ADOP approximation formula. Therefore, this invention refers to the fixed ambiguity vector solution as the high-precision constraint condition of the fixed ambiguity vector solution, and refers to other constraints besides the fixed ambiguity vector solution as general constraints.

[0074] 2. Single-epoch GNSS ADOP with general constraints.

[0075] 2.1 Mechanism of the Influence of General Constraints on Single-Epoch GNSS Parameter Estimation

[0076] The known baseline length and known baseline vector are used as general constraints. The linearized function model and the stochastic model in matrix form can be written as:

[0077] (9)

[0078] In the formula, Constraints The corresponding baseline vector non-difference coefficient matrix, and These are the constraints. Standard deviation and cofactor matrix and It is a non-differenced diagonal matrix. If the general constraints are... Adding these to equations (1) and (2), equations (1) and (2) can be written as:

[0079] (10)

[0080] The least squares method can be used to obtain the general constraints. The baseline vector after and ambiguity vector Variance-covariance matrix of floating-point solutions and Analytical expression:

[0081] (11)

[0082] (12)

[0083] Using matrix inversion formulas, the floating-point solutions of baseline vectors before and after adding constraints are obtained. The formula for the change of the variance-covariance matrix can be written as:

[0084] (13)

[0085] In the formula, , Considering equation (13), the floating-point solution of the ambiguity vector before and after adding general constraints. The formula for the change of the variance-covariance matrix can be written as:

[0086] (14)

[0087] In the formula, .because and Both are symmetric positive definite matrices, according to the definitions of positive definite and positive semi-definite matrices (i.e., for... n Square Array and any n 3D column vector If all have ,but It is a positive definite matrix; if all of them have ,but (a positive semi-definite matrix) It is a positive definite matrix and and It is a positive definite or positive semi-definite matrix. Equations (13) and (14) show that, compared with equations (1) and (2), the baseline vector floating-point solution calculated by equation (10) is... Floating-point solution of ambiguity vector It achieves higher accuracy by adding general constraints to the single-epoch GNSS double-difference mathematical model. Can improve parameters and The accuracy of the estimate.

[0088] S2. Based on the definition of the ambiguity accuracy factor ADOP and the variance-covariance matrix of the floating-point solution of the ambiguity vector after adding constraints, establish an approximate formula for the single-epoch multi-frequency multi-system GNSS ambiguity accuracy factor C-ADOP with constraints.

[0089] According to the definition of ADOP (5) and The analytical expression (12) is accompanied by general constraints. The approximate formula for single-epoch multi-frequency multi-system GNSS ADOP (ADOP with constraint, C-ADOP) can be expressed as:

[0090] (15)

[0091] In the formula, C-ADOP is designed for single-epoch, multi-frequency, multi-system GNSS. Coefficient matrix of unconstrained normal equations and constraint condition normal equation coefficient matrix All are positive definite matrices and are Hermitian matrices. Let and Each is a matrix sum matrix Eigenvalues ​​arranged in ascending order and all of them are positive, using the Peter-Weyl theorem for two Hermitian matrices (i.e., ... and ) and Eigenvalues ​​and Hermitian matrix Relationship between eigenvalues Equation (16) holds true:

[0092] (16)

[0093] therefore, This is the scaling factor for single-epoch multi-frequency multi-system GNSS ADOP. From equations (15) and (16), it can be seen that the higher the accuracy of the added constraints (i.e., ...), the better. The smaller the value, the less fuzzy vector. The higher the accuracy and fixation success rate, the better.

[0094] According to the matrix To ensure the independence of the double-difference master satellite (i.e., the matrix is ​​independent of the selection of the master satellite), and to facilitate calculation in practice and reduce algorithm complexity and improve computational efficiency, it is stipulated that the first observation satellite is used as the master satellite to form the double-difference mathematical model; at the same time, in order to use the observation information to realize the early estimation of the ambiguity vector ADOP, according to formula (15), the following is given: The first observation satellite of the frequency is the master satellite for single-epoch multi-frequency multi-system GNSS. Analysis of difference and inverse transformation:

[0095] (17)

[0096] In the formula, , Corresponding observations (or ) k Frequency , , , , For the first k The weighting of observation satellites at each frequency, excluding the primary satellite (i.e., the first observation satellite). For the first k The sum of the weights of the observation satellites at each frequency, For the first k The weight of the reference satellite (or the first observation satellite) at each frequency, i The imaginary unit of complex numbers. Based on equations (15) and (17), a single-epoch multi-frequency multi-system with general constraints is... Approximation formulas for non-difference and inverse transforms and scaling factors for single-epoch multi-frequency multi-system GNSS ADOP The analytical expressions for the non-difference and non-inverse transformations can be written as follows:

[0097] (18)

[0098] In the formula, Constraints The weight matrix. For general constraints, by adjusting the weight matrix... Taking the reciprocal of the diagonal elements yields the non-inverse transformation. According to equations (15) and (18), a single-epoch single-frequency single-system with general constraints is... Approximate formulas for non-difference and inverse transforms and scaling factors for single-epoch, single-frequency, single-system GNSS ADOP. The analytical expression for the non-difference, non-inverse transformation can be written as:

[0099] (19)

[0100] In the formula, , , , For the weights of all observation satellites in a single-frequency, single-system GNSS system other than the first observation satellite, For the correspondence with single-frequency single-system GNSS The non-difference coefficient matrix, It is the sum of the observation satellite weights for a single-epoch, single-frequency, single-system GNSS. The rights are for a single-epoch, single-frequency, single-system GNSS reference satellite (or the first observation satellite).

[0101] Single-epoch GNSS ADOP with high-precision constraints based on fixed ambiguity vector solutions: Once the ambiguity vector is successfully fixed, the distance between the ground receiver and the observed satellite (satellite-to-ground distance) based on the fixed ambiguity vector solution can be considered as a high-precision satellite-to-ground distance constraint. Unlike general constraints such as known baseline vectors and known baseline lengths, the single-epoch GNSS ADOP scaling factor differs from equations (18) and (19) when the fixed ambiguity vector solution is used as a constraint because it is consistent with the GNSS observation type (i.e., the fixed ambiguity vector solution can be considered as a high-precision pseudorange observation). Furthermore, the fixed ambiguity vector solution also includes ambiguity vector fixed solutions for single-frequency single-system GNSS and multi-frequency multi-system GNSS. Therefore, this invention will further provide a specific form of the analytical expression for single-epoch GNSS ADOP with fixed ambiguity vector solutions.

[0102] Assuming a single-epoch multi-frequency multi-system GNSS double-difference carrier phase observation The corresponding double-difference ambiguity vector Fixed as ,but This can be viewed as a constraint condition for high-precision pseudorange observations. If Includes Given several frequencies, and specifying that a master satellite is selected within each system and frequency to form a double-difference mathematical model, then... The double-difference function model and the stochastic model can be written as:

[0103] (20)

[0104] In the formula, , To and The corresponding number g A fixed solution for the ambiguity vector at a given frequency provides high-precision constraint conditions. and They are respectively with The corresponding fixed solution of the double-difference ambiguity vector and the double-difference carrier phase observation; To and corresponding A column-full-rank double-difference coefficient matrix , , , and These are respectively related to high-precision constraint conditions No. g Each frequency corresponds to The non-difference coefficient matrix and The difference coefficient matrix; , , and They are respectively with The corresponding number g The carrier wavelength and the dimension of the double-difference ambiguity vector at each frequency; , and Corresponding to high-precision constraint conditions The Middle g The cofactor array of double-difference carrier phase observations at each frequency and the standard deviation of non-difference carrier phase. 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 equation (10) can be written as:

[0105] (twenty one)

[0106] By analogy with equation (12), we obtain the conditions for adding high-precision constraints. The variance-covariance matrix of the floating-point solution of the ambiguity vector The analytical expression is:

[0107] (twenty two)

[0108] In the formula, To add high-precision constraints The baseline vector after The variance-covariance matrix.

[0109] Based on equations (15) and (22), the high-precision constraints for multi-frequency, multi-system GNSS are attached. Single-epoch multi-frequency multi-system Approximate formula and ADOP scaling factor They can be written as:

[0110] (twenty three)

[0111] In the formula, , .

[0112] S3. Based on the constrained single-epoch multi-frequency multi-system GNSS ambiguity accuracy factor C-ADOP approximation formula, construct the difference-reduction and inverse-reduction transformation analytical expression of the normal equation coefficient matrix of the ADOP scaling factor with the first observation satellite as the main satellite under constraints or unconstraints. Then, establish the non-difference and non-inverse transformation analytical expression of the constrained single-epoch multi-frequency multi-system GNSS ADOP scaling factor.

[0113] If the ambiguity vector of multi-frequency, multi-system GNSS is fixed, the coefficient matrix of the high-precision constraint condition normal equation is solved. The first constraint g (g =1,2,…, If the first observation satellite of a frequency is the primary satellite, then analogous to equations (17) and (18), a single-epoch multi-frequency multi-system GNSS high-precision constraint condition is attached. Approximate formulas for non-difference and inverse transforms and scaling factors for single-epoch multi-frequency multi-system GNSS ADOP. The analytical expressions for non-difference and non-inverse transformations can be written as follows:

[0114] (twenty four)

[0115] In the formula, , To correspond to high-precision constraints The Middle g Frequency , , , , To meet high-precision constraints The g The frequency correspondence and the weights of the other observation satellites besides the first observation. To and The g The number of observation satellites corresponding to the frequency To and The g The sum of the weights of the observed satellites corresponding to the frequency. for The Middle g The rights of a reference satellite (or the first observation satellite) at a given frequency.

[0116] When the ambiguity vector fixed solution corresponds to a single-frequency, single-system GNSS, the high-precision constraint condition of the ambiguity vector fixed solution in equation (20) Can be written as :

[0117] (25)

[0118] In the formula, For high-precision constraints of single-frequency, single-system GNSS. , and These are, respectively, double-difference carrier phase observations, carrier wavelength, and other parameters for high-precision solutions to fixed-constraint conditions in single-epoch, single-frequency, single-system GNSS ambiguity vector solutions. The full-rank double-difference coefficient matrix, the fixed solution of the ambiguity vector and the column full-rank double-difference coefficient matrix. Dimensions and They are respectively 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 condition corresponds to a single-frequency single-system GNSS, the single-epoch multi-frequency multi-system GNSS with the high-precision constraint condition of the fixed solution of the single-frequency single-system GNSS ambiguity vector is attached. and single-epoch multi-frequency multi-system GNSS ADOP scaling factor The analytical expressions for non-difference and non-inverse transformations can be written as follows:

[0119] (26)

[0120] In the formula, , , , To meet the high-precision constraints of single-epoch single-frequency single-system GNSS The corresponding weights of observation satellites other than the primary satellite, To and The corresponding number of observation satellites, To and corresponding The non-difference coefficient matrix, , To and The weight of the corresponding reference satellite (or the first observation satellite).

[0121] When equations (1) and (2) and the high-precision constraint condition for fixed ambiguity vector solution are all oriented towards single-epoch single-frequency single-system GNSS, the high-precision constraint condition for fixed ambiguity vector solution in equation (25) is... The accuracy is consistent with that of single-frequency, single-system GNSS double-difference carrier phase observations (i.e., both are accurate). )and :

[0122] (27)

[0123] In the formula, , For high-precision constraints of single-epoch single-frequency single-system GNSS carrier wavelength, It is the double-difference ambiguity vector of single-epoch, single-frequency, single-system GNSS carrier phase observations.

[0124] According to equations (23) and (26), when both the ambiguity vector fixed solution and the GNSS observations are oriented towards single-epoch single-frequency single-system GNSS, the high-precision constraint condition of the single-frequency single-system GNSS ambiguity vector fixed solution is attached. Single epoch single frequency single system Approximate formulas for non-difference and inverse transforms and scaling factors for single-epoch, single-frequency, single-system GNSSADOP. The analytical expressions for non-difference and non-inverse transformations can be written as follows:

[0125] (28)

[0126] S4. By combining the constrained single-epoch multi-frequency multi-system GNSS C-ADOP approximation formula and the scaling factor non-difference non-inverse transformation analytical formula, the constrained single-epoch multi-frequency multi-system GNSS C-ADOP non-difference non-inverse transformation approximation formula is obtained; the constrained single-epoch multi-frequency multi-system GNSS ambiguity accuracy factor C-ADOP is determined through the constrained single-epoch multi-frequency multi-system GNSS C-ADOP non-difference non-inverse transformation approximation formula.

[0127] In summary, after introducing the high-precision constraint condition for the fixed solution of the ambiguity vector in (1), this invention calculates the single-epoch GNSS accuracy using equation (7) or equation (8) based on the single-epoch multi-frequency multi-system or single-frequency single-system GNSS observations. value or Value; based on GNSS observations, the non-difference, non-inverse transform scaling factor of single-epoch GNSS ADOP is calculated using the second equation of equation (24), the second equation of equation (26), or the second equation of equation (28). or The calculated single-epoch GNSS or NAND-NOT-NOT-NOT-Scaling Factor or Multiplying them yields the single-epoch GNSS C-ADOP value with the single-epoch ambiguity vector fixed solution constraint. (2) When facing general constraints such as known baseline length and known baseline vector, the single-epoch GNSS C-ADOP value 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 Value; based on GNSS observations, calculate the non-difference, non-inverse transform scaling factor of single-epoch GNSS ADOP using the second equation of equation (18) or the second equation of equation (19). or The calculated single-epoch GNSS or NAND-NOT-NOT-NOT-Scaling Factor or Multiplying them yields the single-epoch GNSS C-ADOP value with other general constraints.

[0128] This invention constructs a single-epoch multi-frequency multi-system GNSS double-difference mathematical model with added constraints, derives the variance-covariance matrix analytical expression of the floating-point solution of the ambiguity vector after adding constraints, establishes the single-epoch GNSS C-ADOP approximation formula with constraints and the AODP scaling factor non-difference non-inverse transformation analytical expression, and simultaneously obtains the single-epoch GNSS C-ADOP non-difference non-inverse transformation approximation formula with constraints. Using single-epoch GNSS observations and the single-epoch GNSS C-ADOP non-difference non-inverse transformation approximation formula with constraints, it achieves accurate estimation of the ADOP value after adding constraints before GNSS parameter calculation. This solves the problem that existing technologies can only accurately calculate the ADOP value with constraints based on the ADOP definition after GNSS parameter calculation, reducing the complexity of improving the ADOP value of the ambiguity vector and fixing the success rate by adjusting constraints, and improving the ambiguity fixing success rate to achieve optimal relative positioning of single-epoch GNSS.

[0129] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. A method for determining the precision factor of single-epoch GNSS ambiguity with constraints, characterized in that, Includes 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 constraints is obtained using the least squares algorithm. Based on the definition of the ambiguity accuracy factor 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 accuracy factor C-ADOP with constraints is established. Based on the constrained single-epoch multi-frequency multi-system GNSS ambiguity accuracy factor C-ADOP approximation formula, an analytical expression for the difference-reduction and inverse transformation of the normal equation coefficient matrix with the first observation satellite as the master satellite is constructed in the ADOP scaling factor. Based on the difference-reduction and inverse transformation of the normal equation coefficient matrix with the constrained and unconstrained conditions, an analytical expression for the non-difference and non-inverse transformation of the constrained single-epoch multi-frequency multi-system GNSS ADOP scaling factor is established. By combining the constrained approximate formula for the single-epoch multi-frequency multi-system GNSS ambiguity accuracy factor C-ADOP with the non-difference and non-inverse transformation analytical expression of the ADOP scaling factor, the constrained single-epoch multi-frequency multi-system GNSS ambiguity accuracy factor C-ADOP is determined.

2. The method for determining the precision factor of a constrained single-epoch GNSS ambiguity according to claim 1, characterized in that, 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 One satellite, For the first k The number of observation satellites at a given frequency, and the fact that different GNSS systems select their master satellites within their respective systems and frequencies to form a double-difference mathematical model at the same or different frequencies, then the resulting multi-frequency, multi-system GNSS double-difference pseudorange observations... and carrier phase observations All dimensions are The mathematical models for the corresponding observations are expressed as follows: ； ； in, and Represent Expectation and variance and Represent Expectation and variance; , , and Double-difference pseudorange observations of multi-frequency, multi-system GNSS Double-difference carrier phase observations Double-difference ambiguity vector and baseline vector Full-rank double-difference coefficient matrix and They correspond to the first k frequency , , and , The dimension is N; , , and They are respectively with The corresponding number k The carrier wavelength and the dimension of the double-difference ambiguity vector at each frequency. for An identity matrix of order 1; , , Corresponding to multi-frequency, multi-system GNSS observations and The Middle k Cofactor array of double-difference observations at each frequency, standard deviation of unequal carrier phase, and standard deviation of unequal pseudorange. It is a symmetric positive definite matrix; , , , and These are respectively double-difference observations. and No. k Each frequency corresponds to The non-difference coefficient matrix and The difference coefficient matrix.

3. The method for determining the precision factor of constrained single-epoch GNSS ambiguity according to claim 2, characterized in that, The constraints are either general constraints or high-precision constraints; high-precision constraints are solutions with a fixed ambiguity vector, while general constraints are other constraints besides solutions with a fixed ambiguity vector.

4. The method for determining the precision factor of a constrained single-epoch GNSS ambiguity according to claim 3, characterized in that, The process of obtaining the variance-covariance matrix of the floating-point solution of the ambiguity vector after adding constraints, based on the single-epoch multi-frequency multi-system GNSS double-difference mathematical model with constraints, using the least squares algorithm, includes the following steps: Introducing general constraints into the single-epoch multi-frequency multi-system GNSS double-difference mathematical model The single-epoch GNSS double-difference mathematical model with general constraints is obtained as follows: ； In the formula, for The corresponding baseline vector non-difference coefficient matrix, and They are respectively Standard deviation and cofactor matrix and It is a non-difference diagonal matrix; The addition was obtained using the least squares method. The baseline vector after b and ambiguity vector Variance-covariance matrix of floating-point solutions and The analytical expressions are as follows: ； ； Similarly, the high-precision constraint conditions for adding fuzzy vectors to the fixed solution are obtained. The variance-covariance matrix of the floating-point solution of the ambiguity vector The analytical expression is: ； In the formula, To add high-precision constraints The baseline vector after The variance-covariance matrix, To and corresponding A column-full-rank double-difference coefficient matrix , and Corresponding to high-precision constraint conditions The Middle g Cofactor array of double-difference carrier phase observations at each frequency and standard deviation of non-difference carrier phase.

5. The method for determining the precision factor of a constrained single-epoch GNSS ambiguity according to claim 4, characterized in that, The step of establishing an approximate formula for the single-epoch multi-frequency multi-system GNSS ambiguity accuracy factor C-ADOP based on the definition of the ambiguity accuracy factor ADOP and the variance-covariance matrix of the floating-point solution of the ambiguity vector after adding constraints includes the following steps: Define the ambiguity precision factor ADOP as: ； In the formula, for The determinant of ADOP, where 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 with general constraints C The approximate formula for single-epoch multi-frequency multi-system GNSS C-ADOP is expressed as: In the formula, For C-ADOP of single-epoch multi-frequency multi-system GNSS with general constraints, , For multi-frequency, multi-system GNSS observations, the first k The cofactor matrix of the double-difference observations of frequency; , ; For single-epoch multi-frequency multi-system GNSSADOP; coefficient matrix of unconstrained normal equations and constraint condition normal equation coefficient matrix All are positive definite matrices and all are Hermitian matrices; For general constraints Scaling factor for single-epoch multi-frequency multi-system GNSSADOP; Similarly, the high-precision constraint conditions with fixed solutions and fuzzy vectors are obtained. The approximate formula for single-epoch multi-frequency multi-system GNSS C-ADOP is expressed as: In the formula, C-ADOP for single-epoch multi-frequency multi-system GNSS with high-precision constraints and fixed ambiguity vector solutions; To address 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 and the ADOP scaling factor without added constraints.

6. The method for determining the precision factor of a constrained single-epoch GNSS ambiguity according to claim 5, characterized in that, The process of establishing the non-difference and non-inverse transformation analytical expression for the single-epoch multi-frequency multi-system GNSS ADOP scaling factor with constraints, based on the difference-reduction and inverse-reduction transformation analytical expression of the coefficient matrix of the normal equations with and without constraints, includes the following steps: According to the attached general constraints The single-epoch multi-frequency multi-system GNSS ADOP approximation formula, with unconstrained conditions, is used in the first... k The first observation satellite of the frequency is the master satellite for single-epoch multi-frequency multi-system GNSS. Analysis of difference and inverse transformation: ； In the formula, , Corresponding observations or The Middle k Frequency , , , , For the first k The weighting of observation satellites at each frequency, excluding the primary satellite. For the first k The sum of the weights of the observation satellites at each frequency, For the first k The weight of a reference satellite at a given frequency, The imaginary unit of complex numbers. ; ; Then according to After removing the difference and inverse transform analytical expressions, the non-difference and non-inverse transform analytical expressions of the single-epoch multi-frequency multi-system GNSS ADOP scaling factor with general constraints are obtained as follows: In the formula, It is the scaling factor for non-difference and non-inverse transform; Constraints The weight matrix; for general constraints, by adjusting the weight matrix... Taking the reciprocal of the diagonal elements yields the non-inverse transformation. ; Similarly, the high-precision constraint condition is solved by fixing the fuzzy vector. g The analytical expression for the non-difference, non-inverse transform of the scaling factor for a single-epoch multi-frequency, multi-system GNSS ADOP system with high-precision constraints, based on the first observation satellite of the frequency as the master satellite, is as follows: ； In the formula, , To correspond to high-precision constraints The Middle Frequency , , , , To meet high-precision constraints The g The frequency correspondence and the weights of the other observation satellites besides the first observation. To and The g The number of observation satellites corresponding to the frequency To and The g The sum of the weights of the observed satellites corresponding to the frequency. for The Middle g The rights of a reference satellite or the first observation satellite at a given frequency; ; It is the scaling factor for non-difference and non-inverse transformation.

7. The method for determining the precision factor of a constrained single-epoch GNSS ambiguity according to claim 1, characterized in that, This also includes obtaining the constrained single-epoch multi-frequency multi-system GNSS C-ADOP ambiguity accuracy factor approximation formula and the ADOP scaling factor non-difference non-inverse transformation analytical formula by simultaneously establishing the constrained single-epoch multi-frequency multi-system GNSS C-ADOP non-difference non-inverse transformation approximation formula; obtaining the constrained single-epoch single-frequency single-system GNSS C-ADOP non-difference non-inverse transformation approximation formula by analogy with the constrained single-epoch multi-frequency multi-system GNSS C-ADOP non-difference non-inverse transformation approximation formula; and determining the constrained single-epoch single-frequency single-system GNSS ambiguity accuracy factor C-ADOP by using the constrained single-epoch single-frequency single-system GNSS C-ADOP non-difference non-inverse transformation approximation formula.

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