A method for fixing integer ambiguity based on complete integer least square estimation

Abstract

The present application belongs to the technical field of carrier phase measurement based on global navigation satellite system, and particularly relates to a method for fixing integer ambiguity based on complete integer least square estimation. The method comprises the following steps: a transformation matrix is constructed according to the nature of real parameters, and the real parameters are transformed into parameters with integer part and decimal part; complete integer least square estimation is performed on all the integer part parameters obtained by transformation; and the real estimation parameters and the integer estimation parameters are restored according to the estimation results. The present application unifies the two types of real variable and integer variable through a transformation matrix, and simultaneously estimates the real parameters and the integer parameters in the mixed integer least square model. Compared with the LAMBDA algorithm, the present application has smaller inversion error, and further improves the convergence time and positioning accuracy of satellite navigation and positioning. The idea of transforming real parameters into integer parameters and solving in the integer domain can also be used in other numerical calculation algorithms and related applications.

Description

Technical Field

[0001] This invention belongs to the field of carrier phase measurement technology based on Global Navigation Satellite System (GNSS), specifically involving a novel integer ambiguity fixing method based on fully integer least squares estimation. Background Technology

[0002] Satellite navigation and positioning is a crucial area in national infrastructure development. In multi-satellite navigation and positioning systems, such as the Global Positioning System (GPS), carrier phase measurements allow us to obtain multiple sets of distance relationships between ground receiving stations and satellites based on the phase difference between the electromagnetic signal transmissions of different satellites and positioning users. Since the signal phase changes periodically, the distance expression often contains an unknown quantity with an integer period, called integer ambiguity. Combining multiple relationships yields a set of linear equations that simultaneously include continuous and integer variables. Integer ambiguity represents the integer unknown corresponding to the first observation of the phase difference between the carrier phase and the reference phase. For accurate positioning, we need to solve for the integer ambiguity in the equations; this process is called integer ambiguity fixing.

[0003] When the error in a linear equation is uniform (such as Gaussian white noise), solving the linear equation can be transformed into a mixed-integer least squares problem. Therefore, academic research on fixed integer ambiguity mainly focuses on solving mixed-integer least squares problems. Current research on integer ambiguity estimation for GNSS models primarily focuses on improving the solution rate and model correction in the presence of additional information. However, some shortcomings still exist:

[0004] (1) Existing technologies do not make full use of observation information. In the LAMBDA algorithm, when constructing integer parameter estimates using maximum likelihood estimation, the error caused by numerical calculation will affect the observation information.

[0005] (2) Existing technologies have shortcomings in considering data processing under complex environmental conditions. For example, some studies treat the poor well-posedness of GNSS model data as an inverse problem, which reduces the accuracy of the final result while introducing regularization. Summary of the Invention

[0006] To address the shortcomings of existing technologies, the purpose of this invention is to propose a novel integer ambiguity fixing method based on fully integer least squares estimation with high accuracy, in order to solve the problem that the traditional LAMBDA algorithm only considers general and simple cases, and the fixing accuracy drops rapidly when the observation noise is slightly large.

[0007] The present invention provides a method for fixing integer ambiguity based on full integer least squares estimation. This method improves the accuracy of the final result by transforming the real-number domain to the integer domain, fully utilizing the limited real-number parameter information considered in existing technologies, and introducing virtually no new errors during parameter estimation. The method includes: constructing a transformation matrix based on the properties of real-number parameters to transform them into parameters with integer and fractional parts; performing full integer least squares estimation on all integer part parameters obtained from the transformation; restoring the real-number and integer estimated parameters based on the estimation results; and unifying the real-number and integer variable types through a single transformation matrix, while simultaneously estimating the real-number and integer parameters in the mixed integer least squares model. The specific steps are as follows:

[0008] (1) Obtain observation data. Calculate the coefficient matrix A corresponding to the real parameters, the coefficient matrix B corresponding to the integer parameters, and the observation result vector y based on prior information;

[0009] (2) Select the appropriate type of transformation matrix P based on the properties of the real parameters to be estimated. The transformation matrix P may have three forms: global scaling, block scaling, or a custom form that satisfies the approximation conditions.

[0010] (3) Using the new algorithm, input A, B, y, and construct the transformation matrix P to obtain the real number estimated variables. and integer estimators Based on the transformation matrix P obtained in step (2), a new coefficient matrix G = [AP] is constructed using the formula. -1 [B], thus transforming the mixed integer least squares problem into a complete integer least squares problem, which can be solved using the LAMBDA algorithm or the traditional LLL algorithm;

[0011] (4) Using real numbers to estimate variables and integer estimators Perform location data calculations to complete the positioning; if the integer estimate variable... For integer ambiguity, further location information can be calculated based on it; if the real-valued estimated variable... The results contain location information, and the final location can be directly extracted from the results.

[0012] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0013] The method of this invention has a smaller inversion error compared to the LAMBDA algorithm. Using this algorithm, the convergence time and positioning accuracy of satellite navigation and positioning can be further improved. This idea of ​​transforming real-number parameters into integer parameters and solving them in the integer domain can also be used in other numerical computation algorithms and related applications. Attached Figure Description

[0014] Figure 1 This is an algorithm flowchart of a novel integer ambiguity fixing method based on complete integer least squares estimation provided by the present invention.

[0015] Figure 2 This is a schematic diagram illustrating a specific positioning process using the present invention.

[0016] Figure 3 This is a comparison chart of the calculation results of the present invention and the LAMBDA algorithm for a simulation dataset in the absence of multipath propagation.

[0017] Figure 4 This is a comparison chart of the calculation results of the present invention and the LAMBDA algorithm for a simulation dataset with multiple paths. Detailed Implementation

[0018] The embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0019] like Figure 2 As shown in the figure, an embodiment of the present invention discloses a process for specific location positioning using a novel integer ambiguity fixing method based on complete integer least squares estimation, which mainly includes the following steps:

[0020] Step S01: Obtain the observation data, and calculate the coefficient matrix A corresponding to the real parameters, the coefficient matrix B corresponding to the integer parameters, and the observation result vector y based on prior information. These data should satisfy a linear model that depends on both the real and integer parameters:

[0021] y = Ax + Bz + e, (1)

[0022] in, These are m×p and m×q dimensional coefficient matrices, respectively. For real-valued unknown parameters, For integer unknown parameters, (n=p+q) represents the random observation error.

[0023] Step S02: Based on the properties of the real parameters to be estimated, select a suitable type of transformation matrix P. The transformation matrix P may have three forms. For example... Figure 1 The judgment is as follows:

[0024] If it is necessary to scale the real number parameters as a whole, the scaling factor is k, and the form is:

[0025] P = kI, where I is the identity matrix; (2)

[0026] If block scaling of the real number parameter is required, the scaling factor for each block is k. i The format is:

[0027] P = diag(k1,…,k p (3)

[0028] If you need to construct the matrix yourself, it should satisfy the approximate conditions after the transformation:

[0029]

[0030] That is, the real parameter after matrix transformation should be close to the nearest integer lattice point. Generally speaking, to improve search efficiency and computational accuracy, ε should be as small as possible relative to the random observation error ‖e‖.

[0031] Step S03: Using the integer ambiguity fixing method based on complete integer least squares estimation, input A, B, y, and construct the transformation matrix P to obtain the real number estimated variables. and integer estimators This new algorithm is the core of this invention, such as Figure 1 As shown, the process includes the transformation matrix selection in step S02, and the restoration processes in steps S03 and S04. The specific solution process can be described as follows: First, input the parameters; second, based on the transformation matrix P obtained in step S02, construct a new coefficient matrix G = [AP] using the formula. -1 [,B], thus transforming the mixed integer least squares problem into a complete integer least squares problem:

[0032]

[0033] The problem can be rewritten as:

[0034]

[0035] in, This is an estimated solution to the least squares problem in the real number field.

[0036] Therefore, the LAMBDA algorithm can be used to solve it; in addition, this least squares form can also be regarded as the Closest Vector Problem (CVP) on the lattice G, which can be solved using Babai's nearest plane algorithm in discrete mathematics, or further transformed into the Shortest Vector Problem (SVP) on the lattice G, which can be solved using the traditional LLL algorithm.

[0037] Step S04: Estimate variables using real numbers and integer estimators The location data is calculated to complete the positioning. For example... Figure 1 As shown, according to the formula:

[0038] u = [Px, z], (7)

[0039] Estimated solution of the complete integer least squares problem It is divided into two parts. The first part is left-multiplied by the inverse P of the transformation matrix. -1 get The second part is

[0040] If integer estimator variables For integer ambiguity, further location information can be calculated, for example, using maximum likelihood estimation and least squares method.

[0041] If the real estimator variable If the result contains location information, the final location can be directly extracted from the result.

[0042] To verify the effectiveness of this invention, a comparative test was conducted between this invention and the traditional LAMBDA algorithm on ambiguity simulation datasets generated with different parameters. The simulation datasets used the GPS system, with each dataset containing 184 data points, and each 10 epochs forming an ambiguity relation equation. All simulation data included ionospheric delay but excluded hardware biases such as DCB. For the multipath effect in GPS, the datasets were divided into two groups for comparison. The first group of datasets did not simulate multipath effects, only considering pseudorange noise and phase noise; specific noise levels and comparison results are detailed below. Figure 3 The second dataset has a fixed pseudorange, and the noise follows a set pattern. Phase noise obeys Considering the multipath effect in the 0–5 m range, the specific multipath occurrence rate, number of satellites, and comparison results are shown in [link to relevant documentation]. Figure 4 .

[0043] from Figure 3 and Figure 4 As can be seen, this invention offers a significant improvement over the traditional LAMBDA algorithm. In simulated datasets without multipath effects, this invention can substantially improve the accuracy of ambiguity fixation; in simulated datasets containing multipath effects, it can also partially improve the accuracy of ambiguity fixation. Both results demonstrate that this invention has certain advantages over the traditional LAMBDA algorithm and is of great significance for increasing the accuracy of satellite positioning.

Claims

1. A method for fixing integer ambiguity based on complete integer least squares estimation, characterized in that, Construct a transformation matrix based on the properties of real parameters to transform them into parameters with integer and fractional parts; perform full integer least squares estimation on all integer part parameters obtained from the transformation; restore the real and integer estimated parameters based on the estimation results; unify the two types of real and integer variables through a single transformation matrix, and simultaneously estimate the real and integer parameters in the mixed integer least squares model; the specific steps are as follows: Step S01: Obtain the observation data and calculate the coefficient matrix corresponding to the real parameters based on the prior information. Integer parameter corresponding coefficient matrix and observation vector These data satisfy a linear model that depends on both real and integer parameters: ， （1） in, They are respectively and dimensional coefficient matrix, For real-valued unknown parameters, For integer unknown parameters, This is due to random observation error. ; Step S02: Select a suitable transformation matrix based on the properties of the real parameters to be estimated. Type; Step S03: Using the integer ambiguity fixing method based on complete integer least squares estimation, input... And construct the transformation matrix. To obtain real-valued estimated variables and integer estimators Specifically, first, input parameters; second, the transformation matrix obtained based on the judgment in step S02. Construct a new coefficient matrix using the formula. This transforms the mixed integer least squares problem into a complete integer least squares problem: ， （2） The problem can be rewritten as: ， （3） in, This is an estimated solution to the least squares problem in the real number field; Therefore, the LAMBDA algorithm can be used to solve it; or, the least squares form can be viewed as a lattice. The problem of finding the nearest vector on a lattice can be solved using Babai's nearest plane algorithm in discrete mathematics, or further transformed into a lattice problem. The shortest vector problem on the vector plane is solved using the LLL algorithm. Step S04: Estimate variables using real numbers and integer estimators Location data is calculated to complete the positioning; specifically, according to the formula: ， （4） Estimated solution of the complete integer least squares problem It is divided into two parts. The first part is left-multiplied by the inverse of the transformation matrix. get The second part is ; If integer estimator variables If the ambiguity is integer, then the location information is further calculated based on it; If the real estimator variable If the result contains location information, the final location can be directly extracted from the result.

2. The integer ambiguity fixing method according to claim 1, characterized in that, In step S02, the transformation matrix The type has three possible forms: If we scale the real number parameters as a whole, the scaling factor is... The format is: ,in For a unit array; (5) If the real number parameter is scaled in blocks, the scaling factor for each block is... The format is: ； （6） If we construct the matrix ourselves, then the approximate conditions after the transformation will be satisfied: ， （7） That is, the real parameter after matrix transformation should be close to the nearest integer lattice point.

3. The integer ambiguity fixing method according to claim 1, characterized in that, In step S02, problem (3) is solved using the LAMBDA algorithm, or the least squares form is considered as a lattice. The problem of finding the nearest vector on a lattice can be solved using Babai's nearest plane algorithm in discrete mathematics, or further transformed into a lattice problem. The shortest vector problem on the vector plane is solved using the LLL algorithm.

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