Method and system for fixing ambiguity of pi-constrained navigation

Abstract

The application provides a PI-constrained navigation ambiguity fixing method and system, and relates to the technical field of satellite navigation. The method obtains array information of array satellite navigation, establishes a low-dimensional PI physical state joint observation model of multiple satellite additional phases generated by a small number of physical parameters through physical mapping, solves consistent PI physical states for each integer ambiguity candidate required by navigation calculation, projects the weighted distance of the integer translation PI physical manifold to the observation and the prior penalty to form a candidate score, adaptively performs full fixing or partial fixing operation according to each parameter quantity and error fixing risk corresponding to the candidate score, and keeps the coordinates consistent under the related integer reversible transformation. The application uses the common physical structure of multiple satellite additional phases to improve the correctness of ambiguity fixing, so that the array anti-interference receiver can still output reliable centimeter-level high-precision navigation results in a high-interference environment.

Description

Technical Field

[0001] This invention relates to the field of satellite navigation technology, and specifically to a PI-constrained navigation ambiguity fixing method and system. Background Technology

[0002] In high-precision satellite navigation, carrier phase observations, after inter-station and inter-satellite double-difference processing, can effectively eliminate common errors such as receiver clock bias, satellite clock bias, and most ionospheric and tropospheric delays, thus obtaining a high-precision observation equation with integer ambiguities as integer unknowns. Correctly fixing integer ambiguities is a key prerequisite for achieving centimeter-level or even millimeter-level positioning. The classic integer least squares method and its efficient implementation, LAMBDA (Least Squares Alternative Array), are discussed. The least squares ambiguity decorrelation adjustment algorithm (AMBiguityDecorrelation Adjustment) performs decorrelation and integer search on floating-point ambiguities and their covariances. It selects the candidate integer grid points that minimize the weighted residual sum of squares and confirms it with integrity methods such as ratio checks. It is widely used in scenarios such as real-time dynamic positioning, network RTK, precise single-point positioning, and multi-antenna orientation and attitude measurement.

[0003] In array-based anti-jamming receivers, antenna arrays typically employ adaptive spatial filtering, such as Power Inversion (PI), in the RF front-end or digital domain to suppress interference. In this specification, PI (Power Inversion) refers to an array adaptive spatial weighting criterion: while constraining the directional gain of the satellite signal, it creates deep nulls in the direction of interference. Its weights are usually obtained by solving the array received data covariance matrix according to criteria such as power inversion or minimum variance distortion-free response. Existing high-precision positioning processing workflows typically treat spatial processing and ambiguity fixing as independent steps: the spatial processor outputs a weighted composite carrier observation, and the high-precision navigation processor uses a standard integer least squares process to directly perform double-difference, floating-point calculations, and integer search on this carrier observation, adding phase compensation or constraint steps as needed to mitigate the impact of the added phase.

[0004] However, while array spatial weighting suppresses interference directional energy, it introduces additional carrier phases related to the satellite direction and varying with the spatial weights for satellite signals from different directions. For the standard integer least squares method, this additional phase constitutes an unmodeled systematic bias: if ignored, the floating-point ambiguity center will systematically deviate from the true integer, leading to a significant decrease in the probability of correct fixation and potentially introducing imperceptible positioning errors; if treated as a continuous unknown quantity independent of each satellite, the number of unknowns increases linearly with the number of satellites, and the additional phase and integer bias may be inseparable in the observation space, resulting in unpredictable or incorrectly fixed integer ambiguities; if untested hard constraints are used, true integers may be incorrectly excluded when the physical model mismatches. Therefore, how to fully utilize the common physical structure of the additional phases of multiple satellites under array anti-interference conditions to improve the accuracy and integrity of integer fixation, while avoiding the sacrifice of true integers due to constraint mismatches, has become a pressing technical problem to be solved in this field. Summary of the Invention

[0005] To address the shortcomings of existing technologies, this invention provides a PI-constrained navigation ambiguity fixing method and system, which solves the problems of decreased accuracy and increased error fixing rate of integer ambiguity caused by additional phase in spatial processing under array anti-interference reception conditions.

[0006] To achieve the above objectives, the present invention provides a method for fixing navigation ambiguity with PI constraints, comprising: Acquire carrier phase observations for array satellite navigation, as well as array information for determining additional phases for spatial processing; the array information includes at least one of actual spatial weights, array spatial manifold, channel state, co-source configuration phase measurements, and frequency residual recovery results; A joint observation model is established, which includes continuous navigation parameters, integer ambiguities, and low-dimensional PI physical states. The low-dimensional PI physical states consist of a small number of common physical parameters, and the spatial processing additional phase vectors of multiple navigation satellites are jointly generated by the low-dimensional PI physical states through physical mapping, rather than being unknowns that are independent for each satellite. Eliminate the influence of the continuous navigation parameters on integer decisions to obtain projected observations; Generate multiple integer ambiguity candidates; For each integer ambiguity candidate, within the feasible region of the low-dimensional PI physical state, a PI physical state compatible with the integer ambiguity candidate is solved. A candidate score is formed based on the weighted distance of the integer translation PI physical manifold corresponding to the integer ambiguity candidate observed by the projection, and the penalty of the PI physical state relative to its prior mean and prior covariance. The integer translation PI physical manifold is the image set formed by the spatial processing additional phase vector changing with the low-dimensional PI physical state, and the observation subset obtained after integer translation corresponding to the integer ambiguity candidate. Based on at least one of the candidate scores, the separation distance between different integer translation PI physical manifolds, the constraint fidelity probability, the fixed residual, and the risk of incorrect fixing, perform full fixing, partial fixing, floating-point holding, or exit PI constraints; wherein, the integer ambiguity determined by performing full fixing or partial fixing is used for subsequent navigation calculation.

[0007] In one embodiment of the present invention, the low-dimensional PI physical state includes at least one type of parameter selected from spatial interference parameters, array geometry and weighting parameters, channel calibration parameters, and phase recovery parameters; wherein, the spatial interference parameters include interference direction, interference distance, and interference intensity ratio; the array geometry and weighting parameters include array attitude and actual spatial weight parameters; the channel calibration parameters include channel amplitude and phase and relative time deviation; and the phase recovery parameters include phase anchor point and phase drift. The physical mapping is implemented using an analytical array model, a calibration lookup table model, an online identification model, or a physically constrained parameterized model. The prior mean and prior covariance of the low-dimensional PI physical state are provided using a frequency residual recovery model or a co-origin dual-configuration phase transfer model.

[0008] In one embodiment of the present invention, eliminating the influence of the continuous navigation parameters on integer decisions includes: Whitening is applied to the observation covariance, and an orthogonal complementary projection matrix of the column space of the design matrix for the continuous navigation parameters is constructed. This projection matrix is ​​then multiplied by the whitened observations to eliminate continuous components of position, baseline, clock error, tropospheric, or hardware bias; or, The continuous navigation parameters are jointly optimized within each integer ambiguity candidate to obtain a Schur complement score equivalent to orthogonal projection elimination.

[0009] In one embodiment of the present invention, the candidate score is represented as the minimum of the sum of the quadratic form of the projected observation residual and the quadratic form of the PI state prior; wherein, the inverse of the weighting matrix of the candidate score is the sum of the projected observation noise covariance, the physical mapping model error covariance, and the cross-correlation term; the PI state prior includes the prior mean and the prior covariance.

[0010] In one embodiment of the present invention, the step of solving for a PI physical state compatible with the integer ambiguity candidate within the feasible region of the low-dimensional PI physical state for each integer ambiguity candidate includes: Within the feasible region of the low-dimensional PI physical state, using the prior mean as the initial value, the PI physical state compatible with the integer ambiguity candidate is solved using Gauss-Newton iteration or normal equations with prior regularization. When the physical mapping can be locally linearized, the candidate scores can be equivalently reduced to a quadratic form with respect to integer candidates using the matrix inversion lemma, resulting in an equivalent weighted matrix.

[0011] In one embodiment of the present invention, the execution of fully fixed, partially fixed, floating-point hold, or exiting PI constraints includes: Calculate the score ratio between the best candidate and the second-best candidate and compare it with a preset threshold; Calculate the minimum weighted separation distance between the integer translation PI physical manifolds corresponding to different non-zero integer biases, and determine the upper bound of the error fixation probability or the sufficient mass condition for correct fixation based on the minimum weighted separation distance; When the score ratio does not exceed the preset threshold, or the error fixation probability exceeds the threshold, or the sufficient quality condition is not met, perform partial fixation, floating-point hold, or exit the PI constraint.

[0012] In one embodiment of the present invention, it further includes: When faced with at least one non-ideal condition such as multiple interferences, multiple array elements, finite interference intensity, near-field interference, weight randomness, or channel inconsistency, the generation of the integer ambiguity candidate and the solution of the PI physical state adopt joint optimization of the outer integer candidate and the inner low-dimensional continuous PI state. Whether to enable the corresponding PI state component is determined by the observability rank, the minimum eigenvalue of the post-information matrix, or the model residual. When the component is unidentifiable, fix it to the prior or remove it from the model.

[0013] In one embodiment of the present invention, it further includes: When the execution part is fixed, the subset of integers that can be fixed is selected in descending order of the minimum weighted separation distance. Priority is given to fixing integer combinations with large separation distances and weak coupling with the PI physical state, while dangerous integers with insufficient separation distances are kept as floating points. The fixed integer subset is used as a constraint to back-substitute, update the continuous navigation parameters and PI physical state, and then iteratively evaluate the remaining integers. When using LAMBDA decorrelation or other integer invertible transformations, the physical mapping, integer translation matrix, and covariance are transformed synchronously to maintain coordinate consistency.

[0014] The present invention also provides a PI-constrained navigation ambiguity fixing system, comprising: An array observation acquisition module is used to acquire carrier phase observations for array satellite navigation, as well as array information for determining additional phases for spatial processing; the array information includes at least one of actual spatial weights, array spatial manifold, channel state, co-source configuration phase measurement, and frequency residual recovery results; The physical state modeling module is used to establish a joint observation model that includes continuous navigation parameters, integer ambiguities, and low-dimensional PI physical states. The low-dimensional PI physical states consist of a small number of common physical parameters, and the spatial processing additional phase vectors of multiple navigation satellites are jointly generated by the low-dimensional PI physical states through physical mapping, rather than being unknowns that are independent for each satellite. A continuous parameter elimination module is used to eliminate the influence of the continuous navigation parameters on integer decisions and obtain projected observations; The candidate generation module is used to generate multiple integer ambiguity candidates. The state optimization module is used to solve for each integer ambiguity candidate within the feasible region of the low-dimensional PI physical state, finding a PI physical state compatible with that integer ambiguity candidate, and forming a candidate score based on the weighted distance of the integer translation PI physical manifold corresponding to the integer ambiguity candidate observed by the projection, and the penalty of the PI physical state relative to its prior mean and prior covariance; wherein, the integer translation PI physical manifold is the image set formed by the spatial processing additional phase vector changing with the low-dimensional PI physical state, and the observation subset obtained after integer translation corresponding to the integer ambiguity candidate; The risk monitoring module is used to perform full fixation, partial fixation, floating-point hold, or exit PI constraints based on at least one of the candidate scores, the separation distance between different integer translation PI physical manifolds, the constraint fidelity probability, the fixed residual, and the risk of incorrect fixation. The positioning and orientation solution module is used to perform integer ambiguity calculations determined by fully or partially fixed methods for subsequent navigation solutions.

[0015] The present invention also provides an electronic device, including a memory and a processor, wherein the memory is a computer-readable storage medium and stores program instructions; the processor is configured to execute the program instructions to perform the navigation ambiguity fixing method with PI constraints as described above.

[0016] This invention provides a method and system for fixing navigation ambiguities with PI constraints. Compared with existing technologies, it has the following advantages: By modeling the spatial processing additional phase of multiple satellites as a low-dimensional PI physical state jointly generated by a small number of common physical parameters, a joint observation model including continuous navigation parameters, integer ambiguities, and low-dimensional PI physical states is established. After eliminating the influence of continuous navigation parameters on integer decisions, multiple integer ambiguity candidates are generated. For each candidate, a compatible PI physical state is solved. A candidate score is formed by projecting the weighted distance of the integer translation PI physical manifold corresponding to the candidate, as well as the penalties of the prior mean and prior covariance of the PI state. Based on at least one of the candidate score, manifold separation distance, constraint fidelity probability, fixed residual, and error fixing risk, fully fixing, partially fixing, floating-point holding, or exiting the PI constraint is adaptively executed. Thus, the common physical structure of the additional phase of multiple satellites is used to restore integer separability. While significantly improving the success rate of integer fixing, the error fixing risk is effectively controlled through prior soft constraints and integrity checks, achieving both high fixing rate and high reliability. Improving the accuracy of ambiguity fixation and reducing the error rate enables the array anti-jamming receiver to still output reliable centimeter-level high-precision navigation results even in high-interference environments. Attached Figure Description

[0017] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0018] Figure 1 This is the overall flowchart of the present invention for fixing the integer ambiguity of PI constraints; Figure 2 A schematic diagram of integer-translated PI physical manifold and separation margin; Figure 3 A schematic diagram of continuous parametric whitening projection and orthogonal compensation elimination; Figure 4 A schematic diagram of optimizing the physical states of candidate inner PI (outer layer integer, inner layer continuous); Figure 5 A state machine diagram for constraining fidelity and error fixed risk control. Detailed Implementation

[0019] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention are described clearly and completely. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0020] This application provides a PI-constrained navigation ambiguity fixing method and system, which solves the problem that the satellite-direction-related additional carrier phase introduced by spatial processing (such as power inversion) under array anti-interference reception conditions destroys the correctness and integrity of integer ambiguity fixing. It addresses the problems of dimensional expansion and inseparability of integers caused by treating the additional phase of multiple satellites as independent unknowns on a satellite-by-satellite basis, the systematic shift of the ambiguity center caused by ignoring the additional phase, and the lack of a model mismatch protection mechanism in existing integer least squares methods that do not verify the compatibility of integer candidates with physically realizable additional phases at the candidate level. It achieves the technical effect of compressing the additional phase of multiple satellites into a low-dimensional PI physical state and performing candidate scoring and integrity verification at the level of integer translation physical manifold, thereby achieving both high fixing success rate and low error fixing rate.

[0021] To better understand the technical solution of this application, the above technical solution will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0022] Example 1: Complete Flowchart of the PI-Constrained Navigation Ambiguity Fixation Method Step S1: Obtain observation data and array information.

[0023] The system acquires carrier phase observations for array satellite navigation. These observations are double-differenced values, meaning that inter-station and inter-satellite differentials are performed on the carrier phase observations of multiple satellites to eliminate common errors such as receiver clock bias, satellite clock bias, and most ionospheric and tropospheric delays. Simultaneously, it acquires array information for determining additional phases for space processing. This array information includes at least one of the following: actual spatial weights (i.e., the spatial weighting vector currently used for array adaptive space filtering, such as power inversion weights, minimum variance distortion-free response weights, or linearly constrained minimum variance weights; the actual spatial weights are complex weight vectors with dimensions equal to the number of array elements), array spatial manifold (i.e., the set of array steering vectors determined by the geometric positions of array elements, element radiation patterns, and channel amplitude and phase characteristics), channel states (including amplitude gain, phase delay, and group delay of each channel), co-source configuration phase measurements (i.e., the phase difference measured for the same satellite signal when the same array uses different spatial weight configurations), and frequency residual recovery results (i.e., the phase recovery value obtained by integrating the residual frequency error output from frequency tracking loops such as frequency-locked loops).

[0024] Step S2: Establish a joint observation model.

[0025] A joint observation model is established, comprising continuous navigation parameters, integer ambiguities, and a low-dimensional PI physical state. The continuous navigation parameters include at least one of baseline components, position increments, clock errors, tropospheric delay residuals, or hardware biases. The integer ambiguities are independent double-difference integer ambiguity vectors. The low-dimensional PI physical state consists of a small number of common physical parameters, with a dimension much smaller than the number of visible satellites. The spatial processing additional phase vectors of multiple navigation satellites are jointly generated from this low-dimensional PI physical state through physical mapping, rather than being independent unknowns for each satellite.

[0026] Specifically, if the pre-whitening carrier observation vector after double-difference processing or linearization of continuous parameters is l, then the observation model can be expressed as: (1) In equation (1) above, x represents the continuous navigation parameter increment, including position increment, baseline increment, clock error increment, tropospheric residual increment, etc., and H x Design its matrix; N is an independent double-difference integer ambiguity vector ( , (representing the dimension of the integer ambiguity vector). This is the corresponding wavelength-related design matrix (i.e., the integer ambiguity design matrix). For low-dimensional PI physical states, the values ​​are taken within the feasible region. ; The additional phase vector is a double-difference vector generated by physical mapping of actual spatial weights, array manifold, interference field, or channel state; v represents the observation noise with zero mean and covariance R. The core of this model lies in the additional phase. No longer arbitrary vectors that are independent of each star, but rather low-dimensional physical states. The deterministic mapping reduces high-dimensional unknowns to a small number of physical parameters.

[0027] It should be noted that, in this embodiment of the invention, the carrier observation vector is... The symbol (lowercase L) represents the vector; the identity matrix is ​​represented by I (uppercase i). The two can be confused in certain fonts, so please be sure to distinguish them according to the context: l is the vector symbol, and I is the matrix symbol.

[0028] Simultaneously, prior information about the low-dimensional PI physical state is obtained, including the prior mean. and prior covariance The prior mean and prior covariance can be provided by a frequency residual recovery model or a co-origin dual-configuration phase transfer model, or determined by the estimation results of the previous epoch, an array calibration model, or prior information about the interference field. Furthermore, the low-dimensional PI physical state is also subject to the feasible region. Constraints. Optionally, feasible region. Take box-type constraint form: (2) In the above formula (2), For the i-th PI physical state component, Let be the prior center of the i-th PI physical state component. Let be the allowable half-width of the i-th component. This is the component index (applicable only to this formula). Let be the physical state dimension of PI. The information sources include: the search range of the direction of arrival of the interference field, the physical constraints of the actual spatial weight magnitude, and the angular range of the array attitude prior. The intersection of the three is used to determine the result, and the tightest one is selected.

[0029] Step S3: Eliminate the influence of continuous navigation parameters on integer decisions.

[0030] Since the determination of integer ambiguity should not depend on the specific values ​​of continuous navigation parameters, this step separates the influence of continuous navigation parameters from the integer determination. Specifically, either of the following two equivalent methods can be used: Method 1 (Weighted Projection Elimination): To facilitate subsequent measurement using Euclidean norm and decoupling of noise correlation, observations are whitened. The purpose of whitening is to make the noise of different observation components independent and of equal variance, thus simplifying the subsequent weighted least squares problem into a regular least squares problem, providing a unified metric for comparing biases in different observation directions. In engineering, this can be understood as normalizing each observation component according to its noise level: components with higher noise levels have lower weights, and components with lower noise levels have higher weights. Covariance physical mapping is then performed. The symmetric positive definite square root of the physical mapping, and in terms of the physical mapping Multiplying each quantity by the physical mapping, we obtain the whitening quantity defined by equation (3): (3) For the sake of brevity, the overline of the symbols in this embodiment (e.g., () represents the amount after whitening treatment, i.e., left multiplication. The result after that, The inverse of the symmetric positive definite square root of the observed noise covariance R is represented by a wavy line (e.g.) The expression represents the quantity after orthogonal projection. After whitening, the whitening noise... Satisfying unit covariance, i.e. It indicates that the covariance matrix of the whitened noise vector is the identity matrix, where Represents mathematical expectation (statistical average); Let I represent the product of column vectors and row vectors, and let I be the identity matrix. This degenerates weighted least squares into ordinary least squares, providing a unified metric basis for subsequent orthogonal projection and norm comparisons.

[0031] The determination of ambiguity should not depend on the specific values ​​of continuous navigation parameters. To address this ambiguity, the decision should be based on the values ​​of the continuous parameters. The influence of integer decisions is removed, and then a whitening continuous parameter design matrix is ​​constructed. x Orthogonal complement projection matrix of column space: (4) Where I is the identity matrix, and the superscript † denotes the Moore-Penrose generalized inverse, used to accommodate the case of rank deficiency in the design matrix. In this embodiment, the superscript T denotes the transpose of a matrix or vector. Projection matrix It is a symmetric idempotent matrix that satisfies (here) Brief Notes ),and .by Multiplying the whitened observation by the left yields the projected observation, which eliminates continuous degrees of freedom: (5) in, For orthogonal complementary projection matrices, The whitened observation vector, Let N be the projected integer design matrix, and N be the independent double-difference integer ambiguity vector. Add phase to the projected PI. This is noise after projection. Because... Idempotent, the projected observation no longer contains anything that can be obtained from... Zhang Cheng's continuous components mean that integer decisions are no longer affected by the values ​​of continuous parameters.

[0032] Method 2 (Schur complement elimination): Jointly minimize the continuous parameters under each integer candidate. Given N and ζ, the least squares subproblem with respect to x is an unconstrained quadratic programming problem, and its optimal residual is precisely the projection of the whitened observations onto the orthogonal complement of the column space, i.e.: (6) in, The whitened observation vector, continuous navigation parameters In this context, q represents the dimension of the continuous navigation parameters. Let x represent the square of the Euclidean norm. Equation (5) shows that first minimizing x (Schur complement elimination) is different from first minimizing x. The residuals are then recalculated after projection, yielding identical candidate scores. Therefore, in this embodiment, the "weighted projection elimination," "Schur complement elimination," and "candidate intra-joint optimization elimination" can be implemented interchangeably, all yielding equivalent projected observations and candidate scores for integer decision-making.

[0033] Step S4: Generate integer ambiguity candidates.

[0034] Based on the floating-point ambiguity and its covariance matrix, LAMBDA decorrelation is employed in conjunction with integer search, spherical decoding, branch and bound, or enumeration methods to generate multiple integer ambiguity candidates. Specifically, with the floating-point ambiguity as the center, the search neighborhood radius is determined according to the confidence ellipsoid of the floating-point covariance. Within this neighborhood, a set of integer candidates satisfying the condition of having the smallest weighted residual sum of squares is enumerated. The generated candidate set should contain several integer vectors with the best scores for subsequent PI physical manifold scoring. When using LAMBDA decorrelation, the integer invertible transformation matrix needs to be recorded so that the candidates and covariance can be transformed back to the original ambiguity space later.

[0035] This step establishes the correspondence between candidate scores and mathematical expressions. The integer-shifted PI physical manifold refers to the subset of observations obtained by shifting the additional phase image set spanned by the low-dimensional PI physical states according to the integer corresponding to the candidate, given an integer ambiguity candidate. If the candidate is a true integer, the projected observations, after deducting its integer contribution, should fall near this subset of observations. Shifting this image set according to the integer candidate yields the subset of observations compatible with that candidate. The "weighted distance" measures the deviation of the projected observations from this subset, and the "prior penalty" measures the deviation of the solved PI state from its prior mean; the sum of these two measures constitutes the candidate score.

[0036] Step S5: Solving and scoring the candidate intra-PI physical state.

[0037] This step is crucial. Define the physical manifold spanned by the projected PI physical states: (7) Among them, PI physical manifold It is the physical state of PI Traverse its feasible region hour The image set, whose dimension is determined by the number of physical parameters, is usually much smaller than the number of satellites, and is therefore a low-dimensional subset of the observation space. For the nth integer candidate, the entire manifold is translated. This yields a subset of observations compatible with the candidate: (8) in, For the integer translation of the PI physical manifold corresponding to the nth integer candidate, given the PI prior (i.e., the prior mean) with prior covariance The score of the nth integer candidate is defined as the score of the projected observation of the translated manifold. The minimum sum of weighted distance and prior penalty: (9) in, The score for the integer candidate n. for The square of the weighted norm, It is the inverse of the prior covariance. For Let W be the quadratic norm of the weight matrix. In equation (9), the weight matrix W is determined by the projected noise covariance, the model error covariance of the PI physical mapping, and the prior recovery error covariance of the PI: (10) in, For the projection noise covariance, The model error covariance for PI physical mapping. For cross-related terms ( and Between), W is the inverse of the sum of the above. Based on this, the system selects the integer candidate with the smallest score that passes the integrity check, rather than just comparing the Euclidean or Mahalanobis distances from the floating-point solution to the integer grid points, thereby explicitly checking the compatibility of "integer candidate superposition with physically realizable additional phase" with real array observations at the candidate level.

[0038] The inner optimization problem of equation (9), namely the specific method for solving the PI physical state ζ that is compatible with candidate n, will be elaborated in detail in the following embodiment five.

[0039] Please see the appendix Figure 2 As shown, this is a schematic diagram of the integer translation PI physical manifold and separation margin in this invention. The figure uses observation subspace direction 1 as the horizontal axis and observation subspace direction 2 as the vertical axis, illustrating the translation physical manifolds corresponding to two different integer candidates: solid curves... dashed curve ,in The image set with added phase is projected when the low-dimensional PI physical state traverses the feasible region (Equation (7)). Different integer candidates cause the entire manifold to be translated to different positions (Equation (8)). The positions of integer grid points are marked in the figure. , And the projected observation point y. The system compares the weighted distances from the projected observation point y to each translated manifold. , Candidate scores are generated (Equation (9)). The smaller the score, the better the compatibility between the integer candidate and the PI physical manifold. The minimum distance between two translated manifolds is denoted as 2. ,in The minimum weighted separation distance (Equation (24)) is used to quantify the separability between different integer translation manifolds and is an important basis for integrity testing and selection of some fixed subsets. It should be noted that the projected observation point y is the result of the actual observation after projection, and generally will not fall exactly on a certain translation manifold, but there is a certain weighted distance between it and the manifold. That is, the integer candidate n makes the PI manifold translate as a whole. , It represents the minimum weighted half distance between different translated manifolds.

[0040] Step S6: Integrity check and fixed decision.

[0041] Based on at least one of the candidate scores, the separation distance between different integer-translated PI physical manifolds, the constraint fidelity probability, the fixed residual, and the risk of incorrect fixing, the PI constraint is executed as fully fixed, partially fixed, floating-point held, or exited. Specific criteria and decision logic will be detailed in Example Six.

[0042] Step S7: Output a fixed result for navigation solution.

[0043] The integer ambiguities determined by fully or partially fixing are used for subsequent navigation solutions, including position, baseline, attitude, and PI state estimation. Simultaneously, model mismatch is continuously monitored, and the fixed residuals, PI state innovations, and other monitored parameters are fed back to the model update and prior update in step S2, forming a closed loop.

[0044] It should be noted that correctly fixing the integer ambiguity is a necessary prerequisite for achieving centimeter-level high-precision positioning: if it is not fixed and floating-point is maintained, the millimeter-level measurement accuracy of the carrier phase cannot be converted into an equivalent distance constraint, and the positioning accuracy can only reach the decimeter level; if it is incorrectly fixed, the navigation solution will carry an imperceptible systematic deviation, the harm of which is greater than maintaining floating-point. Therefore, this invention prioritizes correctness in the fixing decision, and ensures a reliable balance between "high precision" and "reliability" in the output results through integrity checks and risk control mechanisms.

[0045] Please see the appendix Figure 1 As shown, this is the overall flowchart for PI-constrained integer ambiguity fixing. It illustrates the seven main steps from data acquisition to final localization and orientation: S1: Acquire carrier phase observations and code observations for array satellite navigation, as well as at least one of the following array information: actual spatial weights, array spatial manifold, channel status, co-source configuration phase measurements, and frequency residual recovery results.

[0046] S2: Establish a joint observation model that includes continuous navigation parameters, integer ambiguities, and low-dimensional PI physical states, and obtain the prior mean and prior covariance of the PI states.

[0047] S3: By eliminating the influence of continuous navigation parameters on integer decisions through whitening projection, Schur complement elimination, or candidate intra-component joint optimization, projected observations are obtained.

[0048] S4: Generate an integer ambiguity candidate set using LAMBDA decorrelation, spherical decoding, branch and bound, or enumeration methods.

[0049] S5: For each integer candidate, solve the PI physical state that is compatible with the integer ambiguity candidate in the feasible region, calculate the weighted distance and prior penalty of the integer translation PI physical manifold observed by projection, and form the candidate score J(n).

[0050] S6: Make an integrity decision based on at least one of the following: score ratio, manifold separation distance, constraint fidelity probability, fixed residual, and error fixing risk, and execute full fixation, partial fixation, floating-point hold, or exit PI constraint.

[0051] S7: Output the integer ambiguity determined by the fully or partially fixed operation to the positioning and orientation solution, and continuously monitor the model mismatch. Feed back the monitoring quantities such as the fixed residual and PI state innovation to S2 to update the model and prior, forming a closed loop.

[0052] Example 2: The composition and physical mapping of low-dimensional PI physical states This embodiment provides a detailed description of the specific composition of low-dimensional PI physical states and the implementation method of physical mapping.

[0053] The low-dimensional PI physical state includes at least one type of parameter selected from spatial interference parameters, array geometry and weighting parameters, channel calibration parameters, and phase recovery parameters. Wherein: The spatial interference parameters include interference direction (i.e., the azimuth and elevation angles at which the interference signal arrives at the array), interference distance (i.e., the distance from the near-field interference source to the array reference point), and interference intensity ratio (i.e., the power ratio of the interference signal to noise or satellite signal). For far-field interference, the interference distance can be considered infinite and is not treated as an independent parameter; for near-field interference, the distance parameter is an important physical quantity affecting wavefront curvature and additional phase distribution.

[0054] The array geometry and weighting parameters include array attitude (i.e., the rotation angle of the array coordinate system relative to the navigation coordinate system, including roll, pitch and yaw angles) and actual spatial weight parameters (i.e., the real part, imaginary part or polar coordinate representation of the adaptive weight vector used by the current spatial filter, which can be directly taken from the output of algorithms such as power inversion or minimum variance distortionless response).

[0055] The channel calibration parameters include channel amplitude and phase (i.e., the amplitude gain and phase delay of each RF channel) and relative time deviation (i.e., the relative time offset between channels caused by hardware delay differences).

[0056] The phase recovery parameters include the phase anchor point (i.e., the initial phase offset of a certain reference channel or reference frequency) and the phase drift (i.e., the slowly varying amount of phase drift over time, usually modeled as a low-order polynomial or random walk process).

[0057] All of the above parameters fall under the category of "a small number of common physical parameters". Their total dimension p is much smaller than the number of visible satellites n. Therefore, the additional phase of multiple satellites can be compressed from high-dimensional independent unknowns into a low-dimensional mapping of common physical states.

[0058] The physical mapping refers to the mapping from low-dimensional PI physical states. To double-difference additional phase vector The deterministic functional relationship can be achieved using any of the following methods: (1) Analytical Array Model: Based on the phase center position of the array elements, the spatial weight vector, and the satellite direction unit vector, the additional phase angle of each satellite is directly calculated according to the array signal processing theory, and then obtained through double-difference linear transformation. Taking a two-element uniform linear array as an example (see Example 8 below), the PI additional phase of the j-th satellite on a single array element is the phase angle of its weighted array response: (11) Where w is the PI weight vector. Let be the array steering vector in the direction of the j-th satellite. By performing a double difference between the reference and non-reference satellites, we obtain the double-difference additional phase vector: (12) Where D is the double difference operator. Add phase stacking to the individual differences of each star.

[0059] (2) Calibration lookup table model: Under the conditions of dark room or known interference field, the additional phase under different interference directions and different weight states is measured in advance to form a lookup table. When running online, the corresponding additional phase can be obtained by interpolation based on the current PI state.

[0060] (3) Online identification model: Using real-time observation data, the mapping relationship between the PI physical state and the additional phase is identified online, such as by updating the mapping parameters through least squares fitting or recursive estimation.

[0061] (4) Physically constrained parameterized models: Approximate the additional phase mapping with a neural network or other parameterized functions, but apply physical constraints (such as satisfying array manifold structure, reciprocity or power constraints) during training to ensure that the output conforms to physical laws.

[0062] Based on this, the prior mean and prior covariance (i.e., PI state prior) are provided using a frequency residual recovery model or a co-source dual-configuration phase transfer model. The frequency residual recovery model is based on the residual frequency error output of the frequency-locked loop, and the phase recovery value is obtained through integration. Its covariance is adaptively calibrated with changes in the carrier-to-noise ratio. A simplified method is as follows: (13) in, The standard deviation of phase recovery varies with the carrier-to-noise ratio. It decreases and then increases. For phase recovery variance, The identity matrix is ​​used. The dual-configuration phase transfer model utilizes the phase measurement difference of the same satellite signal under two different spatial weight configurations of the same array to construct the transfer relationship of the additional phase, thereby obtaining the prior information of the PI state. Regardless of the source used, verifiable observation residuals and uncertainties should be provided to support the construction and risk estimation of the weighted matrix of Equation (10).

[0063] Example 3: Implementation of Continuous Parameter Elimination and Whitening Projection This embodiment details the specific method for eliminating the influence of continuous navigation parameters.

[0064] Whitening: Take the symmetric positive definite square root of the observation noise covariance matrix R. ,by Multiply the observation vector and the design matrix by left. The whitened noise satisfies unit covariance, causing weighted least squares to degenerate into ordinary least squares, providing a unified metric basis for subsequent orthogonal projection and norm comparisons. When the observation noise covariance is ill-conditioned or rank-deficient, it can be approximated using truncated singular value decomposition or diagonal loading. This ensures numerical stability.

[0065] Construction of the orthogonal complement projection matrix: Construct the orthogonal complement projection matrix of the column space of the continuous parameter design matrix according to equation (4). When the design matrix column is full rank, the Moore-Penrose generalized inverse degenerates into the ordinary inverse; when there is a rank deficiency (such as when the geometry of a certain satellite is poor, causing the design matrix to be less than rank), the generalized inverse ensures that the projection matrix is ​​still defined. After projection, the observation vector y no longer contains components that can be spanned by continuous parameters.

[0066] Please see the appendix Figure 3 This is a schematic diagram of continuous parameter whitening projection and orthogonal complement elimination in this invention. The gray plane at the bottom of the diagram represents the whitening column space of the continuous navigation parameter design matrix. x ), a vector diagonally upwards This is the whitened observation vector. This vector can be decomposed into two orthogonal components: a horizontal component lying in the plane. x x The continuous components that can be explained by the continuous navigation parameters x; and the component perpendicular to the plane y = P⊥ That is, projection observation. The projection matrix is ​​constructed according to equation (4), i.e. ,in This represents the Moore-Penrose generalized inverse. Through this orthogonal complementary projection, components in the whitened observations that can be spanned by continuous navigation parameters are completely eliminated, resulting in a projected observation y containing only integer ambiguity and the PI-added phase. This ensures that subsequent integer decisions are not affected by the specific values ​​of the continuous navigation parameters. In other words, after whitening, [the whitened observation will...]. Projected onto span ( x The orthogonal complement of ) eliminates the representable components of the continuous navigation parameter x, resulting in an observation y containing only integers and the PI additional phase.

[0067] Equivalence with Intra-Candidate Joint Optimization: This invention can also avoid explicitly constructing the projection and instead perform joint minimization of continuous parameters under each integer candidate. Given candidates n and ζ, minimization of x is an unconstrained quadratic programming problem, and its optimal residual is equal to the residual after projection (see Equation (5)). Therefore, the "projection elimination", "Schur complement elimination", and "intra-candidate joint optimization" can be implemented interchangeably to obtain completely equivalent candidate scores. In engineering implementation, for scenarios with a large number of candidates, explicit projection is preferred to reuse pre-computation; for scenarios with low continuous parameter dimensions (such as those containing only position increments), intra-candidate joint optimization is also competitive.

[0068] Example 4: Specific Composition and Soft Constraint Characteristics of Candidate Scores This embodiment details the mathematical form of the candidate scores and the method for determining the weighting matrix.

[0069] As shown in equation (9), the candidate score consists of the sum of two items: the first item is the quadratic form of the projected observation residual. The first term measures the weighted bias between projected observations and model predictions under the current integer candidate and PI state assumptions; the second term is the PI state prior quadratic form. This measures the degree to which the desired PI state deviates from its prior mean. Minimizing the sum of the two terms achieves the optimal trade-off between "data fitting" and "prior constraints".

[0070] The inverse of the weighting matrix W is the sum of the projected observation noise covariance, the physical mapping model error covariance, and the cross-correlation term. Specifically: Projected observation noise covariance It is obtained by transforming the whitened unit covariance using a projection matrix; PI physical mapping model error covariance It is used to characterize the deviation between the physical mapping (such as the analytical array model or the calibration lookup table model) and the actual additional phase. It can be estimated by the statistical analysis of the fixed residual sample or by the propagation of the channel / interference uncertainty according to the physical mapping. Cross-related terms This is used to characterize the correlation between projection noise and PI model error; when the two are uncorrelated, it can be taken as a zero matrix.

[0071] PI state prior is determined by prior mean and prior covariance The structure embodies the "soft constraint" characteristic of this invention. In simple terms, soft constraints allow the PI state to freely adjust within its prior uncertainty range to adapt to the observed data: the greater the prior uncertainty (larger covariance), the wider the range of activity of the PI state and the stronger its adaptability to the observed data; the more certain the prior (smaller covariance), the more the PI state is restricted to the vicinity of the prior mean. This characteristic ensures that when the physical model mismatches, overly strong prior constraints will not incorrectly exclude true integer candidates—the prior covariance provides a reasonable range of activity for the PI state, and the constraint only tightens when the data is consistent with the prior.

[0072] When prior covariance When the prior value approaches infinity (i.e., no prior information), the prior penalty term in equation (9) approaches zero, and the candidate score is determined solely by the data fitting term, with the PI state being entirely data-driven. When the prior covariance approaches zero (i.e., the prior is very precise), the PI state is pinned to the prior mean, equivalent to a hard-fixed constraint. Thus, this invention unifies the two extreme cases of "ignoring the additional phase" and "hard-fixed additional phase" in a continuously adjustable manner. This characteristic ensures that when the physical model mismatches, overly strong prior constraints will not incorrectly exclude true integer candidates—the prior covariance provides a reasonable range of motion for the PI state, and the constraint will only tighten when the data is consistent with the prior.

[0073] During the solution process, the low-dimensional PI physical state must satisfy the constraint of the feasible region Ω. For the box constraint (Equation (2)), the out-of-bounds components are treated by projection clipping in each iteration, that is, the updated solution is projected back to the box boundary along each component; or the logarithmic barrier term can be applied to the objective function to ensure that the iterative solution always falls inside Ω, thereby ensuring feasibility and numerical stability.

[0074] Example 5: Specific Method for Solving Candidate Inner PI States This embodiment details the specific algorithm for solving the PI physical state that is compatible with each integer ambiguity candidate.

[0075] The inner layer of equation (9) is a nonlinear weighted least squares problem with respect to ζ. When When ζ is differentiable, it can be estimated in the current time. Perform a first-order Taylor expansion at this point: (14) in, for about exist Jacobian matrix at the location, This is the PI physical state estimate for the k-th iteration. for right The partial derivative (Jacobi). Let the current residual be: (15) in, Let the current residual be the residual of the k-th iteration. Candidates are integers. for exist The value at the position is used to substitute equation (14) into the inner target of equation (9) and adjust the increment. Setting the gradient to zero, we obtain the Gauss-Newton normal equation: (16) in, For the PI physical state increment of the k-th iteration, the coefficient matrix on the left side of equation (16) is the regularized information matrix, whose positive definiteness is determined by the inverse of the prior covariance. It is guaranteed that as long as the prior covariance is bounded, the prior information term is always positive definite, therefore the coefficient matrix is ​​always positive definite, the increment at each step exists and is unique, and the iteration is well-posed. Based on this, iterative updates are performed. Until convergence, the PI physical state compatible with the integer candidate is obtained. Regarding convergence, given bounded residuals and Lipschitz continuity of the Jacobian, the region of attraction can be entered with the prior mean as the initial value, and the iterative increment satisfies: (17) In the formula, The increment for the k-th iteration. This is the contraction factor. This ensures that the inner layer converges to a compatible PI physical state within a finite number of steps, and that the scores of each candidate in the outer layer... It is comparable, providing a stable input for the ratio test of equation (20) and the correct determination of equation (19) in subsequent embodiments.

[0076] Please see the appendix Figure 4This is a nested flowchart of the optimization of candidate inner PI physical states in this invention, illustrating the two-layer optimization structure of outer integer candidate search and inner continuous PI state solution. The outer layer (dashed box) traverses each candidate n in the integer candidate set, translating the PI physical manifold by ∠Nn; the inner layer (solid box) performs Gaussian-Newton iteration for each integer candidate, using the prior mean ζ0 of the PI states as the initial value: calculating the current residual. (Equation (15)) Solve the normal equation with prior regularization. (Equation (16)), update state This process continues until a compatible PI physical state is obtained. After the inner layer converges, the score of the integer candidate is output. (Equation (19)). The figure also reflects the characteristics of the aforementioned convergence analysis that "the region of attraction can be entered with the prior mean as the initial value", and the characteristic that the complexity of each iteration is determined by the low-dimensional PI state dimension p rather than the high-dimensional integer dimension.

[0077] Analytical Jacobian is used when it is analytically given; otherwise, automatic differentiation or numerical difference is used. Optionally, projection clipping (box constraint) is performed on out-of-bounds ζ components in each iteration to ensure that ζ always lies within the feasible region Ω.

[0078] exist Locally linearizable (i.e., within the prior neighborhood) When the value is approximately constant (J), the inner problem has a closed-form solution: (18) in, For closed-form solutions of PI physical states compatible with integer candidate n (applicable only to this formula). To locally linearize the Jacobian matrix, take the prior mean. The value is obtained at the given location, or approximately constant in its neighborhood (applicable only to this equation). Substituting equation (18) back into equation (9) and applying the matrix inversion lemma (Woodbury identity) to eliminate ζ, the candidate score can be transformed into a quadratic form with respect to integer candidates n: (19) in, Let n be the linearized residual corresponding to the integer candidate n, where the equivalent weighting matrix is: (20) in, For the locally linearized Jacobian matrix, Equation (20) shows that the net effect of the PI soft constraint is equivalent to “compressing” the original weighting matrix W along the tangent space of the PI manifold (spanned by the column space of J): reducing the weights in the directions where PI is interpretable and retaining the weights in the directions where PI is not interpretable. When the prior covariance (Without prior knowledge) Weeff degenerates into a weighted sum of the orthogonal complements of the J-column space, meaning the PI state is entirely determined by the data; when In the case of (strong prior), Weff→W, meaning the PI state is pinned to the prior. Thus, this invention unifies the two extreme cases of "ignoring the additional phase" and "hard-fixing the additional phase" in a continuously adjustable manner.

[0079] The physical meaning of Equation (20) can be intuitively understood as follows: In the direction that the PI additional phase can explain (i.e., along the PI manifold tangent space direction), the model's tolerance for observation bias in this direction increases, and the weights in the corresponding direction in the weighting matrix decrease; in the direction that the PI additional phase cannot explain (i.e., the direction perpendicular to the manifold tangent space), the model still strictly requires that the integer assumptions be consistent with the observations, and the weights are retained. Thus, the PI soft constraint is not simply "relaxing" or "tightening" the overall constraint, but rather releasing degrees of freedom only in physically interpretable directions, while maintaining the original decision strength in other directions.

[0080] The algorithm complexity for solving the inner PI state is: (twenty one) in, For the number of outer candidates, Let m be the number of inner-layer iterations, p be the dimension of the observed data after whitening, and p be the dimension of the PI physical state. Since p is much smaller than the integer ambiguity and the dimension of the continuous navigation parameters, the inner-layer solution is a low-dimensional dense computation. The array manifold, orthogonal projection operator, and equivalent weighting matrix can all be pre-computed and reused, with the additional overhead being only proportional to the low-dimensional p, and the real-time performance is controllable.

[0081] Example 6: Integrity Inspection, Separation Distance, and Error Fixation Risk Control This embodiment details the integrity criteria and calculation process upon which the execution of fully fixed, partially fixed, floating-point hold, or exit PI constraints is based.

[0082] Global separability condition: Define the difference set of the PI physical manifold Let the set of differences between any two points on the manifold be: (twenty two) Under noise-free ideal conditions, the translation manifolds corresponding to two different integer candidates do not coincide if and only if the intersection of the manifold difference set and the translation of the non-zero integer lattice is empty, i.e., the following globally separable condition is satisfied: (twenty three) in, Describing the intersection of sets, when equation (23) holds, any non-zero integer bias. None of them can be absorbed by the degrees of freedom of the PI physical manifold (the meaning of the symbol z applies only to this formula). It represents the set of n-dimensional integer lattice points excluding the zero vector (where n is the fuzziness dimension). Since the set is empty, integer ambiguity is globally identifiable under PI constraints.

[0083] Minimum weighted separation distance: To quantify the degree of separability, a minimum weighted separation distance is defined between different integer translation manifolds. (twenty four) in, Let z denote the infimum, and z denote the non-zero integer bias vector. This represents two independent PI physical state variables. The following numerical algorithm can be used to calculate: Step 1: Using the floating-point solution as the center, enumerate the set {n} of integer candidate grid points within its weighted neighborhood. The neighborhood radius is determined by the confidence ellipsoid of the floating-point covariance. The confidence ellipsoid is the ellipsoidal search range centered on the floating-point solution, defined by the floating-point covariance matrix. Its size is determined by the chi-square distribution quantiles (e.g., taking a 95% confidence level), ensuring that true integers fall within this range with a very high probability. A neighborhood radius that is too small may miss true candidates, while a radius that is too large will increase unnecessary computation. In engineering, it can be adaptively adjusted according to computational resources and real-time requirements.

[0084] Step 2: For any candidate pair (n, n′), find the two closest pairs of points on the translated PI manifold under the weighted metric, i.e., with respect to ζ and ζ. For continuous nonlinear least squares, the weighted nearest point pairs are solved using Gauss-Newton iteration; Step 3: Calculate the weighted half-distance of the candidate pair. ,in Let be the difference vector between the two nearest points in the observation subspace; Step 4: Find the minimum value among all nearest neighbor candidate pairs. ; Step 5: When expanding the candidate neighborhood no longer reduces (relative change less than threshold) The iteration terminates when ) and outputs. .

[0085] Correctly fixed sufficient mass conditions: based on Give a sufficient condition for correct fixation: (25) in, For model error margin, The optimal integer candidate is selected based on the candidate score. For the true integer ambiguity vector, "Indicates that it implies (sufficient conditions)".

[0086] Specifically, the intuitive meaning of this condition is: when the total level of observation noise and model error does not exceed half the shortest distance between two different integer translation manifolds, the noise disturbance is insufficient to cause the projected observation to "jump" from the manifold corresponding to the true integer to the incorrect integer manifold. Therefore, the candidate with the smallest score must be the true integer. Conversely, if the noise or model error level exceeds half the distance, there is a possibility that the score of the incorrect candidate is lower than or equal to the score of the true candidate. In this case, the system should not be forcibly fixed.

[0087] Among them, model error margin Phase residual covariance can be added by PI The trace is conservatively determined: (26) In the formula, tr(·) is the trace operator. The model error covariance for the PI physical mapping is defined in equation (10), where c is a conservative coefficient (typically taken as...). (corresponding to approximately 2σ~3σ confidence level). The larger the value, the more conservative the judgment and the lower the error fixation rate; in engineering, c can be adjusted online according to the load-to-noise ratio and residual monitoring quantity to balance the fixation rate and reliability.

[0088] The proof of equation (25) above is as follows: Let the real integer be... The true PI status is Then the scores of the true candidates satisfy For any erroneous integer candidate By the definition of the triangle inequality and equation (24), the square root of the candidate score is not less than .when At that time, the lower bound of the incorrect candidate scores is not less than the upper bound of the true candidate scores, therefore If the condition holds true for all erroneous candidates, meaning the smallest score must be a true integer, then the proposition is proved. When this sufficient condition is not met, the system transitions to either fixed-point or floating-point operations.

[0089] Ratio test: In the optimal candidate and the second best candidate Between these criteria, the scoring ratio is used as one of the integrity criteria: (27) in, Let J(·) be the ratio of the scores of the best candidate to the second-best candidate, and let J(·) be the candidate scoring function. This represents the optimal candidate (the one with the lowest score). This represents the second-best candidate (the one with the second-lowest score), and γ is the preset threshold (typically taken as...). ). This indicates that the optimal candidate has sufficient discriminative margin compared to the second-best candidate, and the confidence level for implementing fully fixed or partially fixed constraints is high; otherwise, switch to floating-point hold or exit PI constraints.

[0090] Upper bound on fixed error probability: Further upper bounds are given for the fixed error risk: (28) in, For a fixed probability of error, This represents the upper bound relation, and z represents the non-zero integer bias vector. Describes the set of n-dimensional integer lattice space excluding the zero vector. It is the right-tail function of the standard normal distribution. For the separation half distance corresponding to the non-zero integer bias z, Let be the standard deviation of the equivalent noise after whitening. This formula shows that the error fixing probability is controlled by the separation margin of each non-zero integer direction through a Gaussian tail-weighted sum: the larger the separation margin, the smaller the error fixing probability. When When the preset threshold is exceeded, the system cancels the fixed state and reverts to a partially fixed or floating-point state.

[0091] Post-fixation residual monitoring: After full or partial fixation is completed, continuously calculate monitoring quantities such as post-fixation carrier residual, code carrier consistency, PI state innovation, and position or attitude innovation, and compare them with preset thresholds. When the monitoring quantities show an abnormal increase or exhibit unmodeled structural deviations, it indicates that the PI physical model may be mismatched. The system should reduce the PI constraint strength or exit PI constraints, and fall back to floating point or partial fixation. Optionally, cross-validation can be performed using independent frequency points or independent array observations to further reduce the risk of incorrect fixation.

[0092] Please see the appendix Figure 5 This is a state machine diagram of constraint fidelity and error fixation risk control in this invention, showing the system's transition relationship between four working states: floating hold, partial fix, fully fixed, and exit-constraint.

[0093] The transition conditions between each state are as follows: FLOAT→PARTIAL: The manifold separation margin of the candidate integer subset is sufficient to satisfy the sufficient quality condition of equation (25), and the part of the subset is fixed.

[0094] PARTIAL→FLOAT: If the amount of innovation monitored after fixing exceeds the limit (such as PI state innovation, position / posture innovation exceeding the threshold), it will fall back to floating point and hold.

[0095] PARTIAL → FIXED: Rating Ratio If ≥γ (Equation (27)) and the residual test is passed after fixation, it is upgraded to full fixation.

[0096] FIXED → PARTIAL: Fixed probability of error P FF If the preset threshold (Equation (28)) is exceeded, the value will be reduced to a partially fixed value.

[0097] PARTIAL→EXIT-CONSTRAINT: If a PI prior mismatch is detected or the residuals after fixing exhibit unmodeled structural biases, exit the PI constraint.

[0098] EXIT-CONSTRAINT→FLOAT: After being processed by protective measures such as prior covariance expansion, it re-enters the floating-point hold state.

[0099] The dashed box on the right lists at least one of the protection mechanisms used to meet the goal of "ensuring the retention probability of true integer candidates", including: prior covariance inflation, robust loss function (such as Huber kernel), mixture model, suspicious constraint elimination, true integer retention probability threshold, etc., corresponding to the various protection measures described in Example 7.

[0100] The bottom section explains the monitoring metrics used for fixed-point continuous monitoring, including carrier residuals, code-carrier consistency, PI state innovation, integer ratio, position or attitude innovation, and at least two of the following: independent frequency point or independent array observations; and defines the error fixing probability P. FF The calculation method and the conditions for triggering rollback / exit, i.e., the fixed error probability P. FF ≤ It is a simplified version of equation (28); when P FF When the threshold is exceeded or the monitored quantity exceeds the limit, a rollback / exit from the transfer is triggered; and sufficient quality conditions for correct fixation are given: when At that time, the candidate integer with the smallest score must be equal to the true integer. (Equation (25)), where Let the whitening projection residual norm be . For model error margin, This is the minimum weighted separation distance.

[0101] Example 7: Observability Management and Joint Optimization under Non-ideal Conditions This embodiment illustrates an observability management method for PI states under non-ideal conditions such as multiple interferences, multiple array elements, finite interference intensity, near-field interference, random weights, and inconsistent channels.

[0102] Under non-ideal conditions, some components of the low-dimensional PI physical state may be unidentifiable or ill-conditioned. For example, when multiple interference sources are highly correlated in space, there is coupling between the parameters of each interference direction, and the observation data may not be sufficient to uniquely determine all parameters; or, when the channel amplitude and phase error and the interference direction parameters are estimated simultaneously, the two may be linearly correlated, leading to a singular information matrix.

[0103] To address this, the present invention employs a hierarchical architecture that jointly optimizes outer-layer integer candidates and inner-layer low-dimensional continuous PI states: the outer layer generates and ranks integer candidates, while the inner layer performs continuous nonlinear least squares with respect to the PI physical state ζ. During the inner-layer solution process, the identifiability of each PI state component is determined online using an observability criterion, and the corresponding component is then activated accordingly.

[0104] The observability criterion adopts at least one of the following: (1) Observability Rank: Construct the inner Jacobian matrix J and calculate the linear independence of each column from the corresponding PI state component. If the corresponding column of a component is linearly or approximately linearly correlated with the other columns, then the component is unidentifiable under the current observation conditions. Specifically, this can be achieved through the singular value decomposition of J: when the minimum singular value is below a threshold, the direction spanned by the corresponding right singular vector is unidentifiable. The observability rank criterion answers the question: whether the current observation data contains enough information to uniquely determine a certain PI state parameter. Intuitively, if the combination of two different parameters produces almost the same additional phase vector in the observation space (i.e., the corresponding columns of the Jacobian matrix are approximately linearly correlated), then these two parameters cannot be distinguished under the current observation conditions, and forcibly estimating them simultaneously will lead to ill-conditioned solutions. By checking the rank or singular values ​​of the Jacobian matrix, such unidentifiable directions can be identified and excluded from the estimation range.

[0105] (2) Minimum eigenvalue of the post-information matrix: Calculate the post-information matrix corresponding to the equivalent weighted matrix. The smallest eigenvalue ,Right now: (29) in, To locally linearize the Jacobian matrix, For the equivalent weighted matrix, It is the inverse of the prior covariance. This is the identifiability threshold. When... When the value is below the identifiability threshold τ (see Equation (29)), the corresponding PI state component is determined to be unidentifiable. This criterion comprehensively considers the amount of data information (from...). Representation) and prior information (by The combined effect of characterization.

[0106] (3) Model residuals: The fixed residuals are compared with the preset residual thresholds. If the residuals show a structural pattern related to a certain PI state component, the component may be incorrectly modeled or the current value is inaccurate; if the residuals do not improve significantly after adding the component, the information corresponding to the component is insufficient or the component has no actual impact.

[0107] When a PI state component is determined to be unidentifiable, the system performs one of the following actions: fixes the component to its prior mean; or removes the component from the model, retaining only the identifiable components for subsequent scoring and solution. The decision should be based on the component's influence on candidate scores: if the component has a small influence on the additional phase (Jacobi norm below a threshold), removing it will have a limited impact on model accuracy; if the influence is large and it is unidentifiable, it is fixed to the prior and the model error covariance in that direction is increased accordingly to reflect uncertainty.

[0108] When the fixed residuals exhibit unmodeled structural biases, the system triggers protection mechanisms, including at least one of the following: prior covariance inflation, use of robust loss functions (such as Huber kernels), construction of a mixture model, elimination of suspicious constraints, expansion of the candidate set, independent observation cross-validation, and monitoring of the fixed residuals. The common goal of these protection mechanisms is to ensure that true integer candidates are within the candidate set, rather than simply maximizing the candidate compression rate.

[0109] Example 8: Partially Fixed Strategy and Coordinate Consistency This embodiment details a fixed implementation method and a coordinate consistency preservation method under integer reversible transformation.

[0110] Partially fixed subset selection strategy: Partial fixation is used in scenarios where the separation distance is insufficient or the partial integer direction is unreliable. When performing partial fixation, follow these steps: (1) Sort the candidate integer subsets in descending order of minimum weighted separation distance. For a subset consisting of several components of an integer vector, calculate the manifold separation margin (i.e., the minimum separation distance that can be achieved in the direction of the subset when other components are fixed).

[0111] (2) Prioritize fixing integer combinations with large separation distances and weak coupling with the PI physical state. The "weak coupling with the PI physical state" is measured by the norm of the Jacobian matrix J in the direction of the subset: if the projection norm of J in the direction of a certain integer subset is small, it means that the ambiguity estimation of the subset is less affected by the PI state error, and fixing the subset is safer.

[0112] (3) Dangerous integers with insufficient separation distance are kept as floating points and not fixed. The criteria for determining the dangerous direction include: the separation margin in that direction does not meet the sufficient condition of equation (25), or the upper bound of the corresponding error fixation probability exceeds the threshold.

[0113] (4) Use the fixed integer subset as a constraint to back-substitute and update the continuous navigation parameters and PI physical state. After the fixed subset passes the integrity test, it is added as an equality constraint to the normal equation of equation (16) to tighten the floating-point solution covariance of the remaining integers and improve the estimation of the PI physical state.

[0114] (5) Iteratively evaluate the remaining integers. Based on the updated floating-point solution covariance and PI state estimate, recalculate the separation margin and score of the remaining integer subset, and repeat the above subset selection and fixing process until it is no longer possible to fix safely.

[0115] Coordinate consistency maintenance: When using an integer invertible matrix Z to perform a downcorrelation transformation on the ambiguity, let the transformed ambiguity z = Z If N is the coordinate system, then the PI bias, candidate translation matrix, and covariance should be updated synchronously according to the same coordinate transformation. (30) in," "Represents coordinate transformation" mapped to the symbol "". This represents the covariance matrix of the floating-point solution of the ambiguity after transformation. Let represent the covariance matrix of the floating-point solution of ambiguity before transformation. Equation (30) ensures that the PI physical mapping and the integer lattice are described in the same coordinate system, avoiding model inconsistencies caused by only transforming the integers without simultaneously transforming the PI physical mapping. The linear transformation of reference satellite switching and multi-frequency combination is also applied synchronously according to the same principle. , This is correlated with the corresponding covariance. In this way, candidate scores and separation distance calculations are performed within the same coordinate frame, ensuring consistency in decision-making.

[0116] Example 9: Specific numerical calculation example of a two-element single-interference scenario The following example, a two-element uniform linear array, illustrates the implementation process of the above method with specific numerical simulation examples. Let the element spacing be... The PI weights are determined according to the minimum variance distortion-free criterion. ,in The array covariance matrix, Let be the array steering vector in the direction of the j-th satellite. The PI-added phase of the j-th satellite on a single array element is the phase angle of its weighted array response: (11) in, The argument (phase angle) is a complex number. It is the PI weight vector (complex array weighted vector). for The conjugate transpose of is used to perform a double difference between the reference star and the non-reference star to obtain a double-difference additional phase vector, where This is a double difference operator (applicable only to this expression). Add phase stacking to each star's single difference: (12) The simulation parameters are as follows: The geometry of the satellite and the reference star is shown in Table 1.

[0117] Table 1. Satellite and Reference Star Geometry (Simulation Example) The single interference direction angle is 120.0° azimuth and 25.0° elevation, with an interference-to-signal ratio (JNR) of 35dB. Substituting the geometry and interference into equations (11) and (12), the double-difference additional phase and the floating-point ambiguity and covariance calculated from the code / carrier are shown in Table 2.

[0118] Table 2. Double-difference additional phase and floating-point ambiguity (simulation example) In this scenario, all interference sources can be approximated as a single dominant far-field interference. Therefore, the low-dimensional PI physical state degenerates into the two-dimensional direction of arrival angle of the interference, i.e., the physical state dimension p = 2, which is much lower than the dimension of the double-difference ambiguity. The double-difference PI additional phase image set is generated by the unit direction vector of the single interference and the known satellite geometry, and spans a two-dimensional low-dimensional surface in the observation space as ζ changes. For each integer candidate, the surface is translated globally according to Equation (8) to form a candidate-specific physical manifold.

[0119] The system first obtains a floating-point baseline using the code and carrier, and then eliminates the projection of the continuous baseline according to Equation (4). Subsequently, LAMBDA generates dozens to hundreds of double-difference integer candidates based on the floating-point ambiguity and its covariance. For each candidate, Gauss-Newton iteration is performed with the prior mean as the initial value according to Equation (16), which usually converges within 2 to 3 steps. After obtaining the PI physical state (i.e., the interference direction parameter) compatible with the candidate, the candidate score is calculated according to Equation (9). The integer candidates generated by LAMBDA and scored according to Equation (9) are shown in Table 3, along with their weighted distance and score J(n).

[0120] Table 3 Integer Candidate Scores (Simulation Example) In this example, the prior covariance is large, and the prior penalty is approximately zero, equal to the weighted distance. In practice, a non-zero prior penalty may be included; specifically, the optimal candidate ( The ratio of scores of 3, 6) to the second-best candidate =2.55 / 0.62≈4.11, exceeding the threshold (taking γ=3), and the residual test after fixing passed, therefore the double-difference ambiguity is fixed at ( 3, 6), complete the initial fixation in a two-element single-interference scenario. Comparing the three methods (standard LAMBDA, two-stage PI compensation + LAMBDA, and this invention) under the same simulation configuration, the fixation success rate of this invention is improved from 82.4% to 98.6%, the error fixation rate is reduced from 3.10% to 0.15%, and the average convergence time is shortened from 6.8 s to 2.3 s (simulation example data, used to illustrate the relative trend).

[0121] In summary, this invention models the multi-star additional phase introduced by array spatial processing as a low-dimensional PI physical state generated by a small number of common physical parameters, and establishes a complete technical link of "low-dimensional physical state—integer translation physical manifold—candidate scoring—integrity verification—adaptive decision-making," achieving reliable fixation of integer ambiguity under array anti-interference reception conditions. Compared with existing standard LAMBDA methods and two-stage PI compensation methods, this invention achieves significant improvements in three key indicators: fixation success rate, error fixation rate, and average convergence time. The above embodiments are only used to illustrate the technical solution and effects of this invention, and are not intended to limit the scope of protection of this invention. Those skilled in the art can make several modifications or equivalent substitutions to the array configuration, PI state composition and parameterization method, physical mapping modeling means, candidate generation and search strategy, threshold values ​​and decision criteria, and the selection order of some fixed subsets in each embodiment without departing from the concept of this invention. These modifications or substitutions all fall within the scope of protection defined by the claims of this invention.

[0122] This application also provides a PI-constrained navigation ambiguity fixing system, including: An array observation acquisition module is used to acquire carrier phase observations for array satellite navigation, as well as array information for determining additional phases for spatial processing; the array information includes at least one of actual spatial weights, array spatial manifold, channel state, co-source configuration phase measurement, and frequency residual recovery results; The physical state modeling module is used to establish a joint observation model that includes continuous navigation parameters, integer ambiguities, and low-dimensional PI physical states. The low-dimensional PI physical states consist of a small number of common physical parameters, and the spatial processing additional phase vectors of multiple navigation satellites are jointly generated by the low-dimensional PI physical states through physical mapping, rather than being unknowns that are independent for each satellite. A continuous parameter elimination module is used to eliminate the influence of the continuous navigation parameters on integer decisions and obtain projected observations; The candidate generation module is used to generate multiple integer fuzzyness candidates. The state optimization module is used to solve for each integer ambiguity candidate within the feasible region of the low-dimensional PI physical state, finding a PI physical state compatible with that integer ambiguity candidate, and forming a candidate score based on the weighted distance of the integer translation PI physical manifold corresponding to the integer ambiguity candidate observed by the projection, and the penalty of the PI physical state relative to its prior mean and prior covariance; wherein, the integer translation PI physical manifold is the image set formed by the spatial processing additional phase vector changing with the low-dimensional PI physical state, and the observation subset obtained after integer translation corresponding to the integer ambiguity candidate; The risk monitoring module is used to perform full fixation, partial fixation, floating-point hold, or exit PI constraints based on at least one of the candidate scores, the separation distance between different integer translation PI physical manifolds, the constraint fidelity probability, the fixed residual, and the risk of incorrect fixation. The positioning and orientation solution module is used to perform integer ambiguity calculations determined by fully or partially fixed methods for subsequent navigation solutions.

[0123] Specifically, this embodiment provides a navigation ambiguity fixing system with PI constraints, belonging to the same inventive concept as the aforementioned method embodiment, and adopting a modular architecture to implement each step of the method. The system includes an array observation acquisition module, a physical state modeling module, a continuous parameter elimination module, a candidate generation module, a state optimization module, a risk monitoring module, and a positioning and orientation calculation module. Each module is used to perform the processing corresponding to steps S1 to S7 in the method embodiment: the array observation acquisition module acquires carrier phase observations and array information; the physical state modeling module establishes a joint observation model and determines the prior mean and prior covariance of the low-dimensional PI physical state; the continuous parameter elimination module eliminates the influence of continuous navigation parameters and obtains projected observations; the candidate generation module generates multiple integer ambiguity candidates; the state optimization module solves for compatible PI physical states for each candidate and forms a candidate score; the risk monitoring module performs full fixing, partial fixing, floating-point retention, or exit from PI constraints based on at least one of the score, separation distance, constraint fidelity probability, fixed residual, and error fixing risk; the positioning and orientation calculation module uses the integer ambiguities determined by performing full fixing or partial fixing for subsequent navigation calculations. The specific processing flow, calculation formulas and criteria of each module are completely consistent with the aforementioned method implementation, and will not be repeated here.

[0124] An electronic device provided in this application includes a memory and a processor; the memory is a computer-readable storage medium used to store computer programs; the processor is used to implement the above-mentioned PI-constrained navigation ambiguity fixing method when executing the computer program.

[0125] In this embodiment, the beneficial effects of the electronic device and the computer-readable storage medium are similar to those of the above-described PI-constrained navigation ambiguity fixing method, and will not be repeated here.

[0126] The present invention describes electronic devices that can serve as servers or clients of this application, which are examples of hardware devices applicable to various aspects of this application. Electronic devices are intended to represent various forms of digital electronic computer devices. Electronic devices can also represent various forms of mobile devices and other similar computing devices. The components shown herein, their connections and relationships, and their functions are merely examples and are not intended to limit the implementation of the present application described and / or claimed herein.

[0127] Electronic devices include a computing unit that can perform various appropriate actions and processes based on a computer program stored in read-only memory (ROM) or loaded from a storage unit into random access memory (RAM). The RAM can also store various programs and data required for device operation. The computing unit, ROM, and RAM are interconnected via a bus. Input / output (I / O) interfaces are also connected to the bus.

[0128] Those skilled in the art will understand that all or part of the processes in the methods of the above embodiments can be implemented by a computer program instructing related hardware. The computer program can be stored in a computer-readable storage medium, and when executed, it can include the processes of the embodiments of the methods described above. The storage medium can be a read-only memory (ROM) or a random access memory (RAM), etc. The integrated units described above can be implemented in hardware or as software functional units.

[0129] In summary, compared with existing technologies, it has the following beneficial effects: 1. By utilizing the common physical structure of the additional phases of multi-star PI, the arbitrary biases of each star are compressed into low-dimensional physical states, suppressing the expansion of unknowns and restoring integer separability. 2. Compare candidates in the correct integer space after eliminating continuous navigation parameters to improve the consistency between the decision model and real observations; 3. The array model, FLL frequency residual recovery, and homologous measurements can be uniformly connected in the form of soft constraints of "mean plus covariance", and the constraint strength can be adaptively adjusted according to the recovery quality. 4. Separability is explained and quantified by the separation distance of integer translation manifolds, providing a theoretical basis for the selection of fixed thresholds and partially fixed subsets; 5. Supports reordering after candidate generation, compatible with existing LAMBDA receivers, without needing to replace the existing integer search kernel; 6. By employing high-fidelity gating and partially fixed-point and floating-point hold, the risk of erroneous hard constraints caused by physical model mismatch is significantly reduced. 7. Applicable to various scenarios such as initial fixation, fixed hold, fast reconvergence, and dual-antenna / multi-antenna orientation and attitude integer estimation.

[0130] 8. Dimension of inner physical state It is much smaller than the dimension of integers and continuous parameters. The candidate intra-dimensional solution is a low-dimensional dense operation. Compared with pure LAMBDA, it only adds low-order overhead and can be implemented in real time on embedded and parallel hardware platforms. 9. Prior knowledge is accessed using a soft constraint of "mean plus covariance" and is adaptive to the recovery quality. When the recovery quality decreases, it automatically degenerates into pure data-driven, and when the recovery quality improves, it enhances the candidate separation, balancing robustness and fixed speed. 10. The identifiability of the PI state is determined online by using the minimum eigenvalue of the post-information matrix and the observability criterion. Unidentifiable components are automatically disabled or fixed a priori to avoid ill-conditioned solutions and further reduce the risk of incorrect fixation.

[0131] 11. Achieving centimeter-level high-precision navigation based on correct fixation, while avoiding the imperceptible risks of incorrect fixation. Millimeter-level accuracy of carrier phase observation depends on the correct fixation of integer ambiguity: if floating-point is maintained, the high-precision information of the carrier phase will be absorbed by the uncertainty of floating-point ambiguity, and the positioning accuracy will only reach the decimeter level; if incorrect fixation is achieved, the navigation solution will contain systematic deviations that are difficult to detect by conventional quality indicators, and their harm is far greater than maintaining floating-point. This invention significantly improves the fixation success rate and reduces the incorrect fixation rate, ensuring that users are in a reliable centimeter-level high-precision state more often, rather than degenerating to a decimeter-level floating-point solution or suffering the potential risks of incorrect fixation.

[0132] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for fixing navigation ambiguity with PI constraints, characterized in that, include: Acquire carrier phase observations for array satellite navigation, as well as array information for determining additional phases for space processing; The array information includes at least one of the following: actual spatial weights, array spatial manifold, channel status, phase measurement of co-source configuration, and frequency residual recovery results; A joint observation model is established, which includes continuous navigation parameters, integer ambiguities, and low-dimensional PI physical states. The low-dimensional PI physical states consist of a small number of common physical parameters, and the spatial processing additional phase vectors of multiple navigation satellites are jointly generated by the low-dimensional PI physical states through physical mapping, rather than being unknowns that are independent for each satellite. Eliminate the influence of the continuous navigation parameters on integer decisions to obtain projected observations; Generate multiple integer ambiguity candidates; For each integer ambiguity candidate, within the feasible region of the low-dimensional PI physical state, a PI physical state compatible with the integer ambiguity candidate is solved. A candidate score is formed based on the weighted distance of the integer translation PI physical manifold corresponding to the integer ambiguity candidate observed by the projection, and the penalty of the PI physical state relative to its prior mean and prior covariance. The integer translation PI physical manifold is the image set formed by the spatial processing additional phase vector changing with the low-dimensional PI physical state, and the observation subset obtained after integer translation corresponding to the integer ambiguity candidate. Based on at least one of the candidate scores, the separation distance between different integer translation PI physical manifolds, the constraint fidelity probability, the fixed residual, and the risk of incorrect fixing, perform full fixing, partial fixing, floating-point holding, or exit PI constraints; wherein, the integer ambiguity determined by performing full fixing or partial fixing is used for subsequent navigation calculation.

2. The method according to claim 1, characterized in that, The low-dimensional PI physical state includes at least one type of parameter selected from spatial interference parameters, array geometry and weighting parameters, channel calibration parameters, and phase recovery parameters; wherein, the spatial interference parameters include interference direction, interference distance, and interference intensity ratio; the array geometry and weighting parameters include array attitude and actual spatial weight parameters; the channel calibration parameters include channel amplitude and phase and relative time deviation; and the phase recovery parameters include phase anchor point and phase drift. The physical mapping is implemented using an analytical array model, a calibration lookup table model, an online identification model, or a physically constrained parameterized model. The prior mean and prior covariance of the low-dimensional PI physical state are provided using a frequency residual recovery model or a co-origin dual-configuration phase transfer model.

3. The method according to claim 1, characterized in that, Eliminating the influence of the continuous navigation parameters on integer decisions includes: Whitening is applied to the observation covariance, and an orthogonal complementary projection matrix of the column space of the design matrix for the continuous navigation parameters is constructed. This projection matrix is ​​then multiplied by the whitened observations to eliminate continuous components of position, baseline, clock error, tropospheric, or hardware bias; or, The continuous navigation parameters are jointly optimized within each integer ambiguity candidate to obtain a Schur complement score equivalent to orthogonal projection elimination.

4. The method according to claim 1, characterized in that, The candidate score is represented as the minimum of the sum of the quadratic form of the projected observation residual and the quadratic form of the PI state prior; wherein, the inverse of the weighting matrix of the candidate score is the sum of the projected observation noise covariance, the physical mapping model error covariance, and the cross-correlation term; the PI state prior includes the prior mean and the prior covariance.

5. The method according to claim 4, characterized in that, For each integer ambiguity candidate, solving for a PI physical state compatible with that integer ambiguity candidate within the feasible region of the low-dimensional PI physical state includes: Within the feasible region of the low-dimensional PI physical state, using the prior mean as the initial value, the PI physical state compatible with the integer ambiguity candidate is solved using Gauss-Newton iteration or normal equations with prior regularization. When the physical mapping can be locally linearized, the candidate scores can be equivalently reduced to a quadratic form with respect to integer candidates using the matrix inversion lemma, resulting in an equivalent weighted matrix.

6. The method according to claim 1, characterized in that, The execution of fully fixed, partially fixed, floating-point hold, or exit PI constraints includes: Calculate the score ratio between the best candidate and the second-best candidate and compare it with a preset threshold; Calculate the minimum weighted separation distance between the integer translation PI physical manifolds corresponding to different non-zero integer biases, and determine the upper bound of the error fixation probability or the sufficient mass condition for correct fixation based on the minimum weighted separation distance; When the score ratio does not exceed the preset threshold, or the error fixation probability exceeds the threshold, or the sufficient quality condition is not met, perform partial fixation, floating-point hold, or exit the PI constraint.

7. The method according to claim 1, characterized in that, Also includes: When faced with at least one non-ideal condition such as multiple interferences, multiple array elements, finite interference intensity, near-field interference, weight randomness, or channel inconsistency, the generation of the integer ambiguity candidate and the solution of the PI physical state adopt joint optimization of the outer integer candidate and the inner low-dimensional continuous PI state. Whether to enable the corresponding PI state component is determined by the observability rank, the minimum eigenvalue of the post-information matrix, or the model residual. When the component is unidentifiable, fix it to the prior or remove it from the model.

8. The method according to claim 6, characterized in that, Also includes: When the execution part is fixed, the subset of integers that can be fixed is selected in descending order of the minimum weighted separation distance. Priority is given to fixing integer combinations with large separation distances and weak coupling with the PI physical state, while dangerous integers with insufficient separation distances are kept as floating points. The fixed integer subset is used as a constraint to back-substitute, update the continuous navigation parameters and PI physical state, and then iteratively evaluate the remaining integers. When using LAMBDA decorrelation or other integer invertible transformations, the physical mapping, integer translation matrix, and covariance are transformed synchronously to maintain coordinate consistency.

9. A PI-constrained navigation ambiguity fixing system, characterized in that, include: The array observation acquisition module is used to acquire carrier phase observations for array satellite navigation, as well as array information for determining additional phases for space processing. The array information includes at least one of the following: actual spatial weights, array spatial manifold, channel status, phase measurement of co-source configuration, and frequency residual recovery results; The physical state modeling module is used to establish a joint observation model that includes continuous navigation parameters, integer ambiguities, and low-dimensional PI physical states. The low-dimensional PI physical states consist of a small number of common physical parameters, and the spatial processing additional phase vectors of multiple navigation satellites are jointly generated by the low-dimensional PI physical states through physical mapping, rather than being unknowns that are independent for each satellite. A continuous parameter elimination module is used to eliminate the influence of the continuous navigation parameters on integer decisions and obtain projected observations; The candidate generation module is used to generate multiple integer fuzzyness candidates. The state optimization module is used to solve for each integer ambiguity candidate within the feasible region of the low-dimensional PI physical state, finding a PI physical state compatible with that integer ambiguity candidate, and forming a candidate score based on the weighted distance of the integer translation PI physical manifold corresponding to the integer ambiguity candidate observed by the projection, and the penalty of the PI physical state relative to its prior mean and prior covariance; wherein, the integer translation PI physical manifold is the image set formed by the spatial processing additional phase vector changing with the low-dimensional PI physical state, and the observation subset obtained after integer translation corresponding to the integer ambiguity candidate; The risk monitoring module is used to perform full fixation, partial fixation, floating-point hold, or exit PI constraints based on at least one of the candidate scores, the separation distance between different integer translation PI physical manifolds, the constraint fidelity probability, the fixed residual, and the risk of incorrect fixation. The positioning and orientation solution module is used to perform integer ambiguity calculations determined by fully or partially fixed methods for subsequent navigation solutions.

10. An electronic device, characterized in that, include: The memory is a computer-readable storage medium that stores program instructions. A processor for executing the program instructions to perform the navigation ambiguity fixing method for PI constraints as described in any one of claims 1 to 8.

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