BDS high-precision relative positioning method with baseline length constraint

Abstract

This paper proposes a baseline-length-constrained high-precision relative positioning method for BeiDou, comprising the following steps: First, establishing a double-difference relative positioning model using the pseudorange and carrier phase raw observations of the reference station and the rover station; Second, obtaining a strictly baseline-length-constrained objective function based on the double-difference relative positioning model according to the least squares criterion; Third, employing an integer ambiguity search space amplification strategy to achieve effective ambiguity search, and accurately estimating the rover station's position relative to the reference station. This invention utilizes the often underutilized baseline length information of dual antennas in real-world scenarios to improve the relative positioning solution performance, exhibiting higher reliability compared to the traditional unconstrained case. The method employs an integer ambiguity search space amplification strategy, thereby extending the standard LAMBDA algorithm to baseline-length-constrained relative positioning, achieving effective integer ambiguity search.

Description

Technical Field

[0001] This invention relates to satellite navigation technology, specifically to a method for improving the relative positioning calculation performance of satellite navigation systems such as BeiDou by utilizing baseline length prior constraint information. Background Technology

[0002] Relative positioning technology is used to solve the three-dimensional position information of a mobile vehicle relative to a reference station. It focuses more on the relative position between the two rather than their absolute positions, and plays an important role in applications such as autonomous aerial refueling and fully automatic landing.

[0003] Relative positioning technology based on BeiDou carrier phase observations can provide measurement accuracy from centimeters to millimeters. The key lies in the rapid and reliable resolution of integer ambiguities contained in the carrier phase observations. Typically, a mobile vehicle and a reference station each have an antenna. The core of satellite-based relative positioning is to perform inter-station and inter-satellite differential processing on the raw observation data from both the mobile and reference stations, establishing an observation model that includes the relative position parameters between them. Under the premise of correctly resolving the integer ambiguity parameters, an accurate estimation of the relative position between the mobile and reference stations can be achieved.

[0004] With the rapid development of BeiDou-3, BeiDou-based relative positioning applications are becoming increasingly widespread, and scenarios with two or more antennas mounted on carriers are becoming more common. If the pre-measureable baseline length constraint information is rigorously integrated into the relative positioning observation model, it can not only effectively improve the strength of the observation model and thus enhance the reliability of integer ambiguity resolution, but also improve the accuracy of relative positioning to a certain extent. However, in most application scenarios, the two antennas on the carrier usually only serve as backups for each other, and the baseline length information between the antennas is not fully and effectively utilized, thus leaving considerable room for improvement in the reliability of relative positioning. Furthermore, because the carrier is in real-time motion rather than stationary, the baseline length constraint information of multiple antennas changes from linear to nonlinear. When this is integrated into the observation model, the form of the objective function will differ from the traditional unconstrained case, and the shape of the integer ambiguity search space determined by the objective function will also change significantly, making traditional integer ambiguity resolution methods difficult to use for parameter solving under constrained conditions. Therefore, how to fully integrate prior baseline length constraint information into the observation model to improve the reliability of parameter estimation, and how to design a reasonable and effective search strategy for integer ambiguity under a new objective function to achieve effective search for integer ambiguity, have become the difficulties and challenges in realizing baseline length-constrained relative positioning technology. Summary of the Invention

[0005] To address the problems existing in the prior art and improve the reliability of relative positioning calculations, this invention fully utilizes the baseline length information between the two antennas of the mobile carrier to provide a baseline length-constrained BeiDou high-precision relative positioning method, which specifically includes the following steps:

[0006] Step 1: Establish a double-difference relative positioning model using the pseudorange and carrier phase raw observations of the reference station and the rover station.

[0007] Assuming that the two antennas m1 and m2 on the mobile carrier and the base station antenna r simultaneously observe s+1 satellites at frequency point f, then for the first baseline formed by the first mobile station receiver m1 and the base station receiver r... The following observation equations exist.

[0008]

[0009] All the baselines involved are short baselines less than 10 km, E(·) represents the expectation operator, and D(·) represents the variance operator. Let be the matrix of the i-th baseline observations, i = 1, 2, where Let be the pseudorange observation values ​​between the i-th rover station and the base station. Let i be the inter-station single-difference carrier phase observation value between the i-th mobile station and the base station. Let be the inter-station integer ambiguity between the i-th mobile station and the base station, belonging to the s-dimensional set of integers. First baseline Belongs to the three-dimensional real number set The baseline subscript "rm" represents the difference between the rover station's m-related terms and the base station's r-related terms, which is the coefficient matrix of integer ambiguities. Among them, diagonal array λ f Let I be the carrier wavelength at the f-th frequency point. s Represents an s-dimensional identity matrix; baseline vector coefficient matrix Unit vector of line-of-sight vector from the mobile station to the satellite The inter-satellite differential line-of-sight vector matrix contains... e represents the differential line-of-sight vector between two satellites with serial numbers a and b. f It is an f×1 dimensional column vector with all elements equal to 1, and its covariance matrix is... and These are the variances of the carrier phase observations and the pseudorange observations, respectively, where vec(·) is the vectorization operator;

[0010] Similarly, for the baseline from the second rover receiver m2 to the base station receiver... have

[0011]

[0012] in, As the second baseline, and These represent the observation matrix, integer ambiguity, and covariance matrix of the second rover station and the base station, respectively.

[0013] Since both equations (1) and (2) include observations from the base station, therefore and They are not independent of each other; assuming the rover and base station receivers have the same measurement accuracy for pseudorange or carrier phase, then... Inter-satellite single-difference observations y for reference station r r The covariance matrix;

[0014] Combine equations (1) and (2) to write as follows

[0015]

[0016] The above equation is the observation model for double-difference relative positioning; the composite matrix of the two baseline observations in the equation. Wavelength synthesis matrix in The block diagonalization matrix represents the new matrix generated with (A, A) as the diagonal elements, and the inter-satellite difference line-of-sight synthesis matrix. Integer ambiguity synthesis matrix Belongs to the set of 2s-dimensional integers Baseline Synthesis Matrix Belongs to the six-dimensional set of real numbers The covariance matrix was calculated using observations from two base stations. The value is a fixed value, representing the transformation matrix that converts the variance of unequal pseudorange and carrier phase observations into the variance of double-difference pseudorange and carrier phase observations; For the variance-covariance matrix of the standard double-difference observation model, we have In the formula, the subscript "rm" representing the difference between the rover and the base station is replaced with "0". Therefore, and These represent the variance of the carrier phase observation and the variance of the pseudorange observation in a typical dual-antenna relative positioning model, respectively.

[0017] Step 2: Based on the least squares criterion, obtain the objective function with strict baseline length constraints using the double-difference relative positioning model.

[0018] Assuming the mobile platform is a rigid carrier, and antennas m1 and m2 on the carrier are both securely mounted, with a straight-line distance of l between the antennas, then the baseline length between the two antennas is... The length will not change with the movement of the carrier, and this length can be accurately measured in advance; by applying the least squares criterion, equation (3) is transformed into a minimization problem.

[0019]

[0020] In the formula, yes Mahalanobis distance;

[0021] The following projection decomposition is performed on equation (4).

[0022]

[0023] In the formula, p is understood as a subscript. Solve the covariance matrix of the ambiguity floating-point solution. To design the projection matrix of the column vector space of matrix G, The baseline conditional solution given the ambiguity z. It is the baseline vector after the ambiguity is fixed. It is a floating-point solution with ambiguity. The covariance matrix of the baseline and the floating-point solution of the ambiguity. The variance-covariance matrix of the conditional baseline solution;

[0024] The above formula is equivalent to

[0025]

[0026] Equation (6) is the objective function for the strict baseline length constraint; in the equation, C is the baseline vector transformation matrix, which transforms the baseline vectors from two mobile stations into the baseline vector between the two mobile stations. I3 is a three-dimensional identity matrix. Indicates in Under the metric, find a satisfying The conditional vector makes its distance Recently, the addition of the baseline residual quadratic term has been shown to effectively improve the success rate of ambiguity resolution.

[0027] Step 3: Employ an integer ambiguity search space amplification strategy to achieve effective ambiguity search, and on this basis, accurately estimate the position information of the rover relative to the reference station.

[0028] According to equation (6), the fuzziness search space is defined as follows:

[0029]

[0030] In the formula, Ω(·) represents the fuzzy search space, and χ 2 For the search space threshold;

[0031] A strategy of expanding the fuzzy search space is adopted to make the initial χ 2 Gradually increase the value from a small positive value until the optimal ambiguity that satisfies equation (7) is found.

[0032] For the second term in equation (7), an optimal baseline term needs to be performed once for each set of integer ambiguity candidate vectors found. Solving An optimization algorithm is used to iteratively search for the optimal solution; the iteration stops when the convergence condition is met, and the ambiguity value obtained at this point is the optimal ambiguity. Once the ambiguity is correctly fixed, the solution contained within is obtained. The high-precision relative positioning baseline solution.

[0033] In one embodiment of the present invention, the steps of the ambiguity search space expansion strategy in the third step are as follows:

[0034] (1) Let the initial χ 2 It is a small positive value;

[0035] (2) If Ω0(χ 2 If ) is an empty set, then Ω(χ) 2 ) must also be an empty set, proceed to step (4); if Ω0(χ 2 If the result is not empty, proceed to step (3);

[0036] (3) Using the standard LAMBDA algorithm in The ambiguity is searched in the middle and it is determined whether the candidate ambiguity satisfies equation (7). If none of them are satisfied, the process is to proceed to step (4); otherwise, the process is to proceed to step (5).

[0037] (4) Increase χ by a certain step size 2 The value is then transferred to step (2);

[0038] (5) Select the ambiguity that minimizes equation (7) as the optimal ambiguity.

[0039] In one specific embodiment of the present invention, in step (4), the step size is 0.1.

[0040] In another specific embodiment of the present invention, in the third step, Newton's method or the multiplier method is used as the optimization algorithm.

[0041] The method of this invention utilizes the baseline length information of dual antennas, which is often underutilized in real-world scenarios, to improve the solution effect of relative positioning. Compared with the traditional unconstrained case, the solution has higher reliability.

[0042] The method of this invention adopts an integer ambiguity search space expansion strategy, thereby extending the standard LAMBDA algorithm to relative positioning with baseline length constraints, and realizing effective search of integer ambiguity. Attached Figure Description

[0043] Figure 1 A schematic diagram illustrating the relative positioning under the baseline length constraint of the mobile station is shown.

[0044] Figure 2 A schematic diagram illustrating the relationship between the baseline length constraint and the ambiguity search space in the unconstrained case is shown.

[0045] Figure 3 The results show the success rate of integer ambiguity resolution for unconstrained and baseline length-constrained relative positioning under different satellite count conditions;

[0046] Figure 4 The accuracy results of the relative positioning baseline solutions are shown for unconstrained and baseline length-constrained cases with different satellite counts. Detailed Implementation

[0047] The present invention will be further described below with reference to the embodiments and accompanying drawings.

[0048] The method of this invention is as follows: First, establish a double-difference relative positioning model using the pseudorange and carrier phase raw observations of the reference station and the rover station; Second, derive a strict baseline length constraint objective function based on the double-difference relative positioning model according to the least squares criterion; Third, adopt an integer ambiguity search space amplification strategy to achieve effective ambiguity search, and on this basis, accurately estimate the position information of the rover station relative to the reference station.

[0049] The specific steps for implementing the method of the present invention are as follows:

[0050] Step 1: Establish a double-difference relative positioning model using the pseudorange and carrier phase raw observations of the reference station and the rover station.

[0051] Figure 1 A relative positioning diagram with a baseline length constraint for the rover station is given. Assuming that antennas m1 and m2 on the rover and antenna r at the base station simultaneously observe s+1 satellites at frequency f, then for the first baseline formed by the first rover receiver m1 and the base station receiver r... (The baselines involved in this invention are all short baselines less than 10 km) The following observation equations apply.

[0052]

[0053] Where E(·) denotes the expectation operator and D(·) denotes the variance operator. Let be the matrix of the i-th baseline observations, i = 1, 2, where Let be the pseudorange observation values ​​(in meters) between the i-th rover station and the base station. Let i be the inter-station single-difference carrier phase observation value between the i-th mobile station and the base station. Let the inter-station integer ambiguity between the first rover and the base station belong to the s-dimensional integer set. First baseline Belongs to the three-dimensional real number set The baseline subscript "m" indicates the difference between the relevant terms of the rover station m and the relevant terms of the base station r, such as b. rm =b m -b r Etc., coefficient matrix of integer ambiguity Among them, diagonal array λ f Let I be the carrier wavelength at the f-th frequency point. s Represents an s-dimensional identity matrix; baseline vector coefficient matrix Unit vector of line-of-sight vector from the mobile station to the satellite The inter-satellite differential line-of-sight vector matrix contains... e represents the differential line-of-sight vector between two satellites with serial numbers a and b. f It is an f×1 dimensional column vector where all elements are 1. Since the observed value is at a single frequency point, e f It is a 1-dimensional vector with a value of 1, and its covariance matrix is... and denoted as the carrier phase observation variance and the pseudorange observation variance, respectively, where vec(·) is the vectorization operator.

[0054] Similarly, for the baseline from the second rover receiver m2 to the base station receiver... have

[0055]

[0056] in, As the second baseline, and These represent the observation matrix, integer ambiguity, and covariance matrix of the second rover station and the base station, respectively.

[0057] Since both equations (1) and (2) include observations from the base station, therefore and They are not independent of each other. Assuming the mobile station and base station receivers have the same measurement accuracy for pseudorange or carrier phase, then... (in the formula) Inter-satellite single-difference observations y for reference station r r (covariance matrix).

[0058] Combine equations (1) and (2) to write as follows

[0059]

[0060] The above equation is the observation model for double-difference relative positioning. The composite matrix of the two baseline observations is given by the equation. Wavelength synthesis matrix in The block diagonalization matrix represents the new matrix generated with (A, A) as the diagonal elements, and the inter-satellite difference line-of-sight synthesis matrix. Integer ambiguity synthesis matrix Belongs to the set of 2s-dimensional integers Baseline Synthesis Matrix Belongs to the six-dimensional set of real numbers The covariance matrix was calculated using observations from two base stations. Here The value is a fixed value, representing the transformation matrix that converts the variance of unequal pseudorange and carrier phase observations into the variance of double-difference pseudorange and carrier phase observations. For the variance-covariance matrix of the standard double-difference observation model, we have In the formula, the subscript "rm" representing the difference between the rover and the base station is replaced with "0". Therefore, and These represent the variance of the carrier phase observation and the variance of the pseudorange observation in a typical dual-antenna relative positioning model, respectively.

[0061] Step 2: Based on the least squares criterion, obtain the objective function with strict baseline length constraints using the double-difference relative positioning model.

[0062] Assuming the mobile platform is a rigid carrier, and antennas m1 and m2 on the carrier are both securely mounted, with a straight-line distance of l between the antennas, then the baseline length between the two antennas is... This length will not change with the movement of the carrier, and it can be accurately measured in advance. Applying the least squares criterion, equation (3) can be transformed into solving the following minimization problem.

[0063]

[0064] In the formula, yes Mahalanobis distance.

[0065] Equation (4) can be decomposed by projection as follows:

[0066]

[0067] In the formula, p can be understood as a subscript. Solve the covariance matrix of the ambiguity floating-point solution. To design the projection matrix of the column vector space of matrix G, The baseline conditional solution given the ambiguity z. It is the baseline vector after the ambiguity is fixed. It is a floating-point solution with ambiguity. The covariance matrix of the baseline and the floating-point solution of the ambiguity. Let be the variance-covariance matrix of the conditional baseline solution.

[0068] The above formula is equivalent to

[0069]

[0070] Equation (6) is the objective function for the strict baseline length constraint. In the equation, C is the baseline vector transformation matrix, which transforms the baseline vectors from two mobile stations into the baseline vector between the two mobile stations. I3 is a three-dimensional identity matrix, in which the additional baseline residual quadratic term (i.e., the second term in equation (6)) will effectively improve the success rate of ambiguity resolution. This is because, when baseline length constraint information is added, for equation (6), Indicates in Under the metric, find a satisfying The conditional vector makes its distance Recently, in this case, even if the erroneous ambiguity has the smallest quadratic form of the ambiguity residual (i.e., the first term in equation 6), due to... This has a high probability of making the baseline residual quadratic form (i.e., the second term in equation 6) very large. Therefore, only the correct ambiguity vector is most likely to minimize the sum of the ambiguity and the baseline residual quadratic forms.

[0071] Step 3: Employ an integer ambiguity search space amplification strategy to achieve effective ambiguity search, and on this basis, accurately estimate the position information of the rover relative to the reference station.

[0072] According to equation (6), the fuzziness search space is defined as follows:

[0073]

[0074] In the formula, Ω(·) represents the fuzzy search space, and χ 2 This is the search space threshold. It is related to the standard ellipsoidal space in the unconstrained case. Unlike other equations, due to the addition of a baseline residual quadratic term, the ambiguity search space defined in the above equation is no longer ellipsoidal, as shown below. Figure 2 As shown, its relationship with the unconstrained case is as follows: Ω0(χ 2The ambiguity candidate vectors in (7) do not necessarily satisfy the inequality in (7). Therefore, the standard least squares reduced correlation adjustment (LAMBDA) algorithm, which is applicable to unconstrained cases, cannot be directly used for ambiguity search in the above equation. (LAMBDA is an abbreviation for Least-squares Ambiguity Decorrelation Adjustment, a classic method proposed by PJG Teunissen for estimating integer ambiguity in satellite navigation. See the reference: [Teunissen PJ G. The least-squares ambiguity decorrelation adjustment: a Method for Fast GPS Ambiguity Estimation[J]. Journal of Geodesy, 1995, 70: 65-82.])

[0075] Note Figure 2 This also means that if for Ω0(χ) 2 If we iterate through the fuzzy candidate vectors in (7), we can definitely find the optimal solution that satisfies the inequality in (7). The advantage of doing this is that the standard LAMBDA algorithm can be used to search for new objective functions; however, it also brings a problem, namely, χ 2 The choice of objective function can no longer be obtained using the common methods in the standard LAMBDA algorithm. This is because the new objective function contains nonlinear constraints. Optimal baseline term It cannot be calculated analytically; instead, it requires a search method. Essentially, it's a nonlinear constrained optimization problem, which involves significant computation in practice. In the standard LAMBDA algorithm, the initial choice of χ... 2 It is often more than the χ corresponding to the optimal ambiguity. 2 This will result in too many candidate ambiguity vectors. As can be seen from equation (7), during the ambiguity search process, a conditionally optimal baseline term needs to be performed for each candidate ambiguity encountered. The search for ambiguities is performed, and most of these candidate ambiguities are excluded by equation (7) due to the large baseline residual quadratic form generated by ambiguity errors. Therefore, the standard LAMBDA algorithm will have problems of large computational cost and low efficiency when used for ambiguity search of new objective functions, which will seriously affect the real-time performance of relative positioning. In this case, χ 2 The choice of χ becomes particularly important. Based on the above reasons, a strategy of expanding the fuzzy search space is adopted, allowing the initial χ... 2 The search space is gradually increased from a small positive value until the optimal ambiguity satisfying equation (7) is obtained. The main steps of this ambiguity search space expansion strategy are as follows:

[0076] (1) Let the initial χ 2 It is a small positive value.

[0077] (2) If Ω0(χ 2 If ) is an empty set, then Ω(χ) 2 ) must also be an empty set, proceed to step (4); if Ω0(χ 2 If the result is not empty, proceed to step (3).

[0078] (3) Using the standard LAMBDA algorithm in The ambiguity is searched and it is determined whether the candidate ambiguity satisfies equation (7). If none of them are satisfied, the process proceeds to step (4); otherwise, the process proceeds to step (5).

[0079] (4) Increase χ in increments of, for example, 0.1. 2 The value is then transferred to step (2).

[0080] (5) Select the ambiguity that minimizes equation (7) as the optimal ambiguity.

[0081] For the second term in equation (7), each time a set of integer ambiguity candidate vectors is found (the integer ambiguity parameters obtained by searching using the LAMBDA algorithm are a vector, i.e., a set of vectors), an optimal baseline term needs to be performed. Solving Due to the existence of nonlinear baseline constraints, it is impossible to derive the analytical expression for this term. In this case, optimization algorithms such as Newton's method (a method proposed by Newton to solve equations approximately in the real and complex number domains through iteration) and the multiplier method ("Hestenes M R. Multiplier and gradient methods[J]. Journal of Optimization Theory and Applications, 1969, 4(5): 303-320.") can be used to iteratively search for the optimal solution. When the convergence condition is met, the iteration stops, and the ambiguity value obtained at this time is the optimal ambiguity. When the ambiguity is correctly fixed, the ambiguity contained in the equation can be obtained. The high-precision relative positioning baseline solution.

[0082] Example: Relative positioning solution with baseline constraints

[0083] NovAtel GPS-702-GGL and GPS-703-GGG antennas were connected to NovAtel OEM628 and OEM638 receivers, respectively, with a fixed baseline length of 3.3 meters between the two antennas, serving as the mobile terminal. Another GPS-702-GGL antenna was connected to an OEM719 receiver, serving as the reference station. The mobile station and reference station were approximately 6 meters apart. During the experiment, the BeiDou satellite with the highest elevation angle was used as the reference satellite, the data sampling frequency was 1Hz, and the data acquisition time was approximately 1 hour.

[0084] The collected raw data were processed for different satellite countable cases (i.e., 5, 6, 7, 8, 9, and 10 satellites). Unconstrained and baseline-length-constrained relative positioning methods were used respectively. Integer ambiguity was resolved using only one epoch (comprehensive resolution using only one epoch of data; compared to multi-epoch resolution, single-epoch relative positioning bypasses cycle slip problems and is unaffected by historical data), and the ambiguity resolution success rate was statistically analyzed. Then, using the correct ambiguity, the baseline resolution accuracy was calculated for different satellite countable cases using both unconstrained and baseline-length-constrained relative positioning methods. The results are analyzed below.

[0085] Figure 3 The results show the ambiguity resolution success rate under different numbers of observed satellites. It can be seen that the ambiguity resolution success rate increases with the number of satellites. Compared to the unconstrained case, adding a baseline length constraint improves the ambiguity resolution success rate to varying degrees, with the improvement reaching its maximum of approximately 30% in the case of 7 satellites. When the number of satellites reaches 10, the ambiguity can be correctly fixed for each epoch.

[0086] Figure 4 The accuracy results of the baselines calculated under different satellite numbers in the Northeastern Sky (ENU) coordinate system are presented. It can be seen that when the baseline length constraint is added, the accuracy of the baseline solution under different combinations of satellite numbers is improved to a certain extent compared with the solution under the unconstrained case. In addition, the accuracy of the baseline solution also increases with the increase of the number of satellites.

[0087] The method of this invention utilizes the baseline length information of dual antennas, which is often underutilized in real-world scenarios, to improve the solution effect of relative positioning. Compared with the traditional unconstrained case, the solution has higher reliability.

[0088] The method of this invention adopts an integer ambiguity search space expansion strategy, thereby extending the standard LAMBDA algorithm to relative positioning with baseline length constraints, and realizing effective search of integer ambiguity.

Claims

1. A baseline-length-constrained BeiDou high-precision relative positioning method, characterized in that, Specifically, the following steps are included: Step 1: Establish a double-difference relative positioning model using the pseudorange and carrier phase raw observations of the reference station and the rover station. Assuming that the two antennas m1 and m2 on the mobile carrier and the base station antenna r simultaneously observe s+1 satellites at frequency point f, then for the first baseline formed by the first mobile station receiver m1 and the base station receiver r... The following observation equations exist. All the baselines involved are short baselines less than 10 km, E(·) represents the expectation operator, and D(·) represents the variance operator. Let be the matrix of the i-th baseline observations, i = 1, 2, where Let be the pseudorange observation values ​​between the i-th rover station and the base station. Let i be the inter-station single-difference carrier phase observation value between the i-th mobile station and the base station. Let the inter-station integer ambiguity between the first rover and the base station belong to the s-dimensional integer set. First baseline Belongs to the three-dimensional real number set The baseline subscript "rm" represents the difference between the rover station's m-related terms and the base station's r-related terms, which is the coefficient matrix of integer ambiguities. Among them, diagonal array λ f Let I be the carrier wavelength at the f-th frequency point. s Represents an s-dimensional identity matrix; baseline vector coefficient matrix Unit vector of line-of-sight vector from the mobile station to the satellite The inter-satellite differential line-of-sight vector matrix contains... e represents the differential line-of-sight vector between two satellites with serial numbers a and b. f It is an f×1 dimensional column vector with all elements equal to 1, and its covariance matrix is... and These are the variances of the carrier phase observations and the pseudorange observations, respectively, where vec(·) is the vectorization operator; Similarly, for the baseline from the second rover receiver m2 to the base station receiver... have in, As the second baseline, and These represent the observation matrix, integer ambiguity, and covariance matrix of the second rover station and the base station, respectively. Since both equations (1) and (2) include observations from the base station, therefore and They are not independent of each other; assuming the rover and base station receivers have the same measurement accuracy for pseudorange or carrier phase, then... Inter-satellite single-difference observations y for reference station r τ The covariance matrix; Combine equations (1) and (2) to write as follows The above equation is the observation model for double-difference relative positioning; the composite matrix of the two baseline observations in the equation. Wavelength synthesis matrix in The block diagonalization matrix represents the new matrix generated with (A, A) as the diagonal elements, and the inter-satellite difference line-of-sight synthesis matrix. Integer ambiguity synthesis matrix Belongs to the set of 2s-dimensional integers Baseline Synthesis Matrix Belongs to the six-dimensional set of real numbers The covariance matrix was calculated using observations from two base stations. The value is a fixed value, representing the transformation matrix that converts the variance of unequal pseudorange and carrier phase observations into the variance of double-difference pseudorange and carrier phase observations; For the variance-covariance matrix of the standard double-difference observation model, we have In the formula, the subscript "rm" representing the difference between the rover and the base station is replaced with "0". Therefore, and These represent the variance of the carrier phase observation and the variance of the pseudorange observation in a typical dual-antenna relative positioning model, respectively. Step 2: Based on the least squares criterion, obtain the objective function with strict baseline length constraints using the double-difference relative positioning model. Assuming the mobile platform is a rigid carrier, and antennas m1 and m2 on the carrier are both securely mounted, with a straight-line distance of l between the antennas, then the baseline length between the two antennas is... The length will not change with the movement of the carrier, and this length can be accurately measured in advance; by applying the least squares criterion, equation (3) is transformed into a minimization problem. In the formula, yes Mahalanobis distance; The following projection decomposition is performed on equation (4). In the formula, p is understood as a subscript. Solve the covariance matrix of the ambiguity floating-point solution. To design the projection matrix of the column vector space of matrix G, The baseline conditional solution given the ambiguity z. It is the baseline vector after the ambiguity is fixed. It is a floating-point solution with ambiguity. The covariance matrix of the baseline and the floating-point solution of the ambiguity. The variance-covariance matrix of the conditional baseline solution; The above formula is equivalent to Equation (6) is the objective function for the strict baseline length constraint; in the equation, C is the baseline vector transformation matrix, which transforms the baseline vectors from two mobile stations into the baseline vector between the two mobile stations. I3 is a three-dimensional identity matrix. Indicates in Under the metric, find a satisfying The conditional vector makes its distance Recently, the addition of the baseline residual quadratic term has been shown to effectively improve the success rate of ambiguity resolution. Step 3: Employ an integer ambiguity search space amplification strategy to achieve effective ambiguity search, and on this basis, accurately estimate the position information of the rover relative to the reference station. According to equation (6), the fuzziness search space is defined as follows: In the formula, Ω(·) represents the fuzzy search space, and χ 2 For the search space threshold; A strategy of expanding the fuzzy search space is adopted to make the initial χ 2 Gradually increase the value from a small positive value until the optimal ambiguity that satisfies equation (7) is found. For the second term in equation (7), an optimal baseline term needs to be performed once for each set of integer ambiguity candidate vectors found. Solving An optimization algorithm is used to iteratively search for the optimal solution; the iteration stops when the convergence condition is met, and the ambiguity value obtained at this point is the optimal ambiguity. Once the ambiguity is correctly fixed, the solution contained within is obtained. The high-precision relative positioning baseline solution.

2. The BeiDou high-precision relative positioning method with baseline length constraint as described in claim 1, characterized in that, In the third step, the steps of the fuzziness search space expansion strategy are as follows: (1) Let the initial χ 2 It is a small positive value; (2) If Ω0(χ 2 If ) is an empty set, then Ω(χ) 2 ) must also be an empty set, proceed to step (4); if Ω0(χ 2 If the result is not empty, proceed to step (3); (3) Using the standard LAMBDA algorithm in The ambiguity is searched in the middle and it is determined whether the candidate ambiguity satisfies equation (7). If none of them are satisfied, the process is to proceed to step (4); otherwise, the process is to proceed to step (5). (4) Increase χ by a certain step size 2 The value is then transferred to step (2); (5) Select the ambiguity that minimizes equation (7) as the optimal ambiguity.

3. The BeiDou high-precision relative positioning method with baseline length constraint as described in claim 2, characterized in that, In step (4), the step size is 0.

1.

4. The BeiDou high-precision relative positioning method with baseline length constraint as described in claim 1, characterized in that, In the third step, Newton's method or the multiplier method is used as the optimization algorithm.

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