Receiver Autonomous Integrity Monitoring Method Based on Bayesian Test

Abstract

The present invention relates to a receiver autonomous integrity monitoring method based on Bayesian test, belonging to the technical field of satellite navigation receivers. The present invention obtains the positioning estimation solution for the linearized pseudorange observation equation; projects the statistical model of the pseudorange observation noise into the positioning domain by using the multiple convolution formula to obtain the statistical model of the positioning error; proposes a fault detection model based on the positioning domain according to the variance inflation theory; for this fault detection model, constructs the Bayesian posterior probability odds ratio Ratio for fault detection based on the Bayesian test theory; gives the determination method of the key parameter of the Bayesian posterior probability odds ratio Ratio, i.e., the variance inflation coefficient k, according to the navigation continuity requirement; and finally uses the Bayesian posterior probability odds ratio Ratio for fault detection. The combination of the prior fault information, the real-time observation information and the positioning domain detection model in the present invention improves the detection performance of the faulty satellite and is applicable to the receiver autonomous integrity monitoring of global satellite navigation.

Description

Technical Field

[0001] The present invention relates to a receiver autonomous integrity monitoring method based on Bayesian test, and belongs to the technical field of satellite navigation receivers. Background Technique

[0002] The Global Navigation Satellite System (GNSS) is a radio navigation and positioning system that provides users with all-weather spatial coordinates, speeds, and time information at any location on the Earth's surface and in near-Earth space, and is widely used in fields such as civil aviation, urban transportation, and engineering construction.

[0003] For fields related to personal safety such as civil aviation and urban transportation, when the navigation information is incorrect, the wrong decisions made will directly endanger the lives of users and cause significant property losses. Therefore, the integrity monitoring of navigation performance is an essential link in the development of related systems. The integrity of a navigation system can be understood as the ability of the system to promptly warn users when it is unable to complete the navigation task and the navigation service is unavailable. Among them, Receiver Autonomous Integrity Monitoring (RAIM) uses the algorithms built into the receiver to perform consistency tests on the received redundant observation data, thereby completing fault detection. Due to its advantages such as rapid response, simple implementation, and no need for external equipment assistance, it has become a research hotspot in integrity monitoring technology.

[0004] According to different navigation basic positioning algorithms, the current RAIM algorithms are mainly divided into snapshot RAIM algorithms and filtering RAIM algorithms. The test statistics in these studies are basically constructed in the measurement domain, and when the test statistic exceeds the alarm threshold, an alarm is given to the user. However, for users, the information in the position domain is more important than the information in the measurement domain. At the same time, the original intention of RAIM is to promptly warn users when the integrity does not meet the requirements, rather than just informing users when the test statistic exceeds the threshold.

[0005] Traditional methods such as the pseudorange comparison method, least squares residual RAIM, and parity vector method are applicable to single satellite fault detection. Although they have the advantages of simple models, small computational complexity, and easy implementation, they have deficiencies such as low detection rate and insensitivity to positioning errors. At the same time, the existing fault detection methods based on machine learning and Bayesian test usually require a large amount of calculation or introduce too many hyperparameters, making it difficult to meet the actual usage requirements. Summary of the Invention

[0006] The object of the present invention is to solve the problem that the existing receiver autonomous integrity monitoring technology has a low detection rate, which may endanger the lives and safety of users. A receiver autonomous integrity monitoring method based on Bayesian test is provided. This method obtains the positioning estimation solution for the linearized pseudo-range observation equation; projects the statistical model of the pseudo-range observation noise into the positioning domain by using the multiple convolution formula to obtain the statistical model of the positioning error; proposes a fault detection model based on the positioning domain according to the variance inflation theory; for this fault detection model, constructs the Bayesian posterior probability odds ratio Ratio for fault detection based on the Bayesian test theory; gives the determination method of the key parameter of the Bayesian posterior probability odds ratio Ratio, i.e., the variance inflation coefficient k, according to the navigation continuity requirement; and finally uses the Bayesian posterior probability odds ratio Ratio for fault detection. The key points and the points to be protected in the present invention are the idea of constructing a fault detection model by using the variance inflation theory, the idea of constructing the Bayesian posterior probability odds ratio for fault detection for the proposed fault detection model based on the Bayesian test theory, the determination method of the variance inflation coefficient k, and the process design of applying the above three points to RAIM.

[0007] The object of the present invention is achieved by the following technical solutions:

[0008] A receiver autonomous integrity monitoring method based on Bayesian test, characterized in that it includes the following steps:

[0009] Step 1: Obtain the positioning estimation solution for the linearized pseudo-range observation equation;

[0010] Step 2: Project the statistical model of the pseudo-range observation noise into the positioning domain by using the multiple convolution formula to obtain the statistical model of the positioning error;

[0011] Step 3: Propose a fault detection model S based on the positioning domain according to the variance inflation theory j ;

[0012] Step 4: For this fault detection model, construct the Bayesian posterior probability odds ratio Ratio for fault detection based on the Bayesian test theory;

[0013] Step 5: Give the determination method of the key parameter of the Bayesian posterior probability odds ratio Ratio, i.e., the variance inflation coefficient k, according to the navigation continuity requirement.

[0014] Step 6: Use the Bayesian posterior probability odds ratio Ratio for fault detection.

[0015] A receiver autonomous integrity monitoring method based on Bayesian test, characterized in that the specific process is as follows:

[0016] Step 1: Obtain the positioning estimation solution for the linearized pseudorange observation equation. Solution 1: Use the weighted least squares algorithm to obtain the positioning estimation solution for the linearized pseudorange observation equation.

[0017] The linear pseudorange observation equation of the user receiver for the satellite is:

[0018] z = Hx + ε (1)

[0019] where z ∈ R n×1 , representing the offset vector between the pseudorange of the visible satellite and the estimated value, n is the number of visible satellites; x ∈ R m×1 is the deviation vector between the true value and the nominal value of the state variable, including the three-dimensional position of the receiver and the clock deviation of the satellite navigation system, m is the dimension of the state variable; H ∈ R n×m is the geometric observation matrix; ε = [ε1, ε2,..., ε n T represents the observation error vector, and ε i (i ∈ {1, 2,..., n}) are independent and follow Gaussian noise, σ obs,i represents the standard deviation of the observation error.

[0020] Based on the weighted least squares algorithm, the positioning estimation solution of x is

[0021]

[0022] where W is the weighted matrix related to the elevation angle of each satellite relative to the user and the user's ranging accuracy.

[0023] Solution 2: Use the Kalman filter algorithm to obtain the positioning estimation solution for the linearized pseudorange observation equation.

[0024] The state equation of the receiver is

[0025] x k = Ax k-1 + Bu k-1 + w k-1 (3)

[0026] where x is the state variable, including the three-dimensional position of the receiver and the clock deviation of the satellite navigation system; A and B are system parameters; u is the control quantity; w is the system noise; the subscript k of each symbol represents the current moment, and the subscript k - 1 represents the previous moment;

[0027] Meanwhile, the linear pseudorange observation equation of the receiver for the satellite is:

[0028] z k = Hx k + ε k (4)

[0029] Therefore, the time update equation of the Kalman filter is as follows

[0030]

[0031] where is the one-step prediction value of the state variable at the current moment; is 's covariance matrix; Q is the covariance matrix of the system noise.

[0032] The state update equation of the Kalman filter, that is, the positioning estimation solution is as follows:

[0033]

[0034] where K k is the gain matrix; R is the measurement noise covariance matrix; P k is the covariance matrix of the positioning estimation solution .

[0035] Step 2: Use the multiple convolution formula to project the statistical model of the pseudorange observation noise into the positioning domain to obtain the statistical model of the positioning error

[0036] According to the positioning estimation solution the estimation error μ of x is

[0037]

[0038] Expand μ into μ1, μ2, μ3, which respectively represent the positioning errors of the x-axis, y-axis, and z-axis:

[0039]

[0040] where a ji is the element in the j-th row and i-th column of matrix A.

[0041] Since ε i are independent of each other, the statistical model of the positioning error μ is obtained by using the statistical model of the observation noise ε through the multiple convolution formula j as follows:

[0042]

[0043] where σ1, σ2, σ3 respectively represent the standard deviations of the position errors projected onto the x-axis, y-axis, and z-axis.

[0044] Step 3: Propose a fault detection model S based on the positioning domain according to the variance inflation theory j

[0045] When the satellite navigation system fails, compared with the fault-free situation, the positioning error μ is regarded as a random variable with the same expectation but an inflated variance, denoted by μ f as follows:

[0046]

[0047] where k represents the variance inflation coefficient.

[0048] Therefore, a classification variable S j is constructed, and the fault detection model is given as follows:

[0049]

[0050] where S1, S2, and S3 respectively represent the classification variables on the three coordinate axes. When S j = kσ j , it indicates that there are faulty satellites in the satellite navigation system; when S j = σ j , it indicates that there are no faulty satellites in the satellite navigation system.

[0051] Step 4: For the fault detection model in Step 3, based on Bayesian testing, a Bayesian posterior probability odds ratio Ratio is constructed for fault detection

[0052] In the actual usage process, when the positioning system obtains n s positioning estimation solutions samples through continuous sampling, they can be used to estimate the sample standard deviation S of the corresponding positioning error samples m . According to Bayesian testing theory, the posterior probability that the satellite navigation system is fault-free is:

[0053]

[0054] where, for the sake of simplified expression, the subscripts of some symbols are omitted, that is, S is used to represent S j and σ is used to represent σ j ; P{S = σ} represents the prior probability that the satellite navigation system is fault-free; P{S m = y|S = σ} represents the likelihood probability that the satellite navigation system is fault-free; y is the specific value of the sample standard deviation S m in the actual sampling situation.

[0055] Similarly, the posterior probability that the satellite navigation system has a fault can be obtained as:

[0056]

[0057] where P{S = kσ} represents the prior probability that the satellite navigation system has a fault; P{S m= y|S = kσ} represents the likelihood probability of satellite navigation system failure.

[0058] Therefore, the posterior probability odds ratio of failure to non-failure is:

[0059]

[0060] The prior probabilities and likelihood probabilities in the case of non-failure and failure are as follows:

[0061] P{S = σ} = (1 - P sat ) n (15)

[0062]

[0063] where P sat is the prior probability of failure of each satellite; is the probability density function of the chi-square function; Δx is the differential element of the random variable.

[0064] Substituting equations (15), (16), (17), and (18) into equation (14), the specific expression of the Bayesian posterior probability odds ratio can be obtained:

[0065]

[0066] Step Five: Determine the adaptive variance inflation coefficient k

[0067] The false alarm rate P fa is defined as:

[0068] P fa = P{Ratio > 1|H0} (20)

[0069] where H0 represents the non-failure condition, which is consistent with the case of S = σ.

[0070] P fa 's maximum allowable value [P fa is derived from the continuity requirement, and the result is as follows:

[0071]

[0072] where P{H0} is consistent with P{S = σ}, representing the prior probability of the satellite navigation system being free of failure; C req is the continuity requirement index specified by the International Civil Aviation Organization (ICAO).

[0073] To meet the false alarm rate, for all S m < T fa , the receiver autonomous integrity monitoring method based on Bayesian testing should not detect a failure, that is, the following requirements should be met:

[0074]

[0075] Among them, represents the inverse cumulative distribution function of the chi-square distribution.

[0076] Therefore, in order to meet the false alarm rate, the minimum inflation coefficient k needs to satisfy:

[0077]

[0078] By iteratively solving Equation (23), the minimum value k of the inflation coefficient k can be determined min . On the premise of meeting the false alarm rate, in order to maximize the detection ability, during actual use, let the inflation coefficient k be taken as k min .

[0079] Step 6: Use the Bayesian posterior probability odds Ratio for fault detection

[0080] Input the actually sampled result S m and the obtained inflation coefficient k into the Bayesian posterior probability odds Ratio, that is, Equation (19).

[0081] When Ratio ≥ 1, there are faulty satellites in the satellite navigation system, and a fault occurs;

[0082] When Ratio < 1, there are no faulty satellites in the satellite navigation system, and no fault occurs;

[0083] Beneficial effects:

[0084] 1. The receiver autonomous integrity monitoring method based on Bayesian test proposed by the present invention focuses on position domain information that is more suitable for application in integrity monitoring. For users, the more direct and important information is the positioning information calculated based on the observation information, rather than the observation information itself. Therefore, from this perspective, it is more reasonable to carry out integrity monitoring in the positioning domain, and the corresponding integrity monitoring method is also more sensitive to positioning faults.

[0085] 2. Compared with traditional methods such as the least squares residual RAIM, the receiver autonomous integrity monitoring method based on Bayesian test proposed by the present invention can significantly improve the correct detection ability of the system and reduce the risk of missed detection. Especially when a satellite with a large slope fails, the effect is particularly significant. Compared with traditional methods, the receiver autonomous integrity monitoring method based on Bayesian test has a higher correct detection rate and is more sensitive to satellite faults. When a satellite with a large slope fails, using the traditional least squares RAIM method is prone to missed detection, but the proposed method further enhances the fault detection ability for this situation.

[0086] 3. The proposed receiver autonomous integrity monitoring method based on Bayesian test in the present invention has better universality. During the actual use of a navigation satellite system, any satellite may malfunction, and users cannot determine in advance which satellite will malfunction. Therefore, fault detection should consider all possible satellite faults. Compared with traditional methods, to achieve a 100% correct detection rate for all single-fault satellite cases, the minimum pseudorange fault bias corresponding to the receiver autonomous integrity monitoring method based on Bayesian test is smaller.

[0087] 4. At the same time, compared with existing methods based on machine learning and Bayesian test, the receiver autonomous integrity monitoring method based on Bayesian test does not have a large number of hyperparameters. Meanwhile, its key parameter - the variance inflation factor k can be determined by continuity requirements and the number of visible satellites, greatly reducing the computational amount and improving the feasibility of the method. BRIEF DESCRIPTION OF THE DRAWINGS

[0088] Figure 1 Flowchart of satellite fault detection for the receiver autonomous integrity monitoring method based on Bayesian test;

[0089] Figure 2 Graph of fault detection results for the receiver autonomous integrity monitoring method based on Bayesian test. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0090] To better illustrate the purpose, content, and advantages of the present invention, the following further elaborates on the detailed implementation manners in combination with the embodiments and drawings of the present invention.

[0091] Embodiment 1

[0092] The process of applying the receiver autonomous integrity monitoring method based on Bayesian test proposed in the present invention to detect faulty satellites is shown in Figure 1 . This embodiment simulates the process of a civil aviation aircraft using a GNSS receiver to receive data for navigation and positioning and perform fault detection. The fault detection performance of the proposed receiver autonomous integrity monitoring method based on Bayesian test is evaluated through experiments and compared with the traditional least squares RAIM.

[0093] Since there is relatively little real fault data in the satellite navigation system, and accidents caused by faults often make it difficult to recover the data for subsequent research. Therefore, in this embodiment, without changing the GNSS navigation data, pseudorange fault biases of 0 - 100 m with a step size of 5 m are artificially injected into the real observation data to construct fault data. Meanwhile, without loss of generality, all visible satellites are respectively set as faulty satellites, and the fault detection rates of the receiver autonomous integrity monitoring method based on Bayesian test and the least squares RAIM method are calculated and compared in each case.

[0094] In the embodiment, a high-precision T300 GNSS receiver located in Beijing (116.315253°E, 39.959792°N, 58m) is used to collect real satellite navigation files and observation data. The collection time is from 00:00:00 to 23:59:59 on April 10, 2021, the sampling frequency is 1 Hz, and there are a total of 86,400 epochs.

[0095] Step 1: Obtain the positioning estimation solution for the linearized pseudorange observation equation

[0096] The linear pseudorange observation equation of the receiver for the satellite is:

[0097] z = Hx + ε (24)

[0098] where z ∈ R n×1 , representing the offset vector between the visible satellite pseudorange and the estimated value, n is the number of visible satellites; x ∈ R m×1 is the deviation vector between the true value and the nominal value of the state variable, including the three-dimensional position of the receiver and the clock deviation of the satellite navigation system, m is the dimension of the state variable; H ∈ R n×m is the geometric observation matrix; ε = [ε1, ε2,..., ε n T represents the observation error vector, and ε i (i ∈ {1, 2,..., n}) are independent and follow Gaussian noise, σ obs,i represents the standard deviation of the observation error.

[0099] Based on the weighted least squares algorithm, the positioning estimation solution of x is

[0100]

[0101] where W is the weighted matrix related to the elevation angle of each satellite and the user range accuracy.

[0102] Step 2: Use the multiple convolution formula to project the statistical model of the pseudorange observation noise into the positioning domain to obtain the statistical model of the positioning error

[0103] The estimation error μ of x is

[0104]

[0105] Expand μ into μ1, μ2, μ3, which respectively represent the positioning errors on the x-axis, y-axis, and z-axis.

[0106]

[0107] where a ji is the element in the j-th row and i-th column of matrix A. ​

[0108] Since ε i are independent of each other, the positioning error μ is obtained through the multiple convolution formula using the statistical model of the observation noise ε j The statistical model is as follows:

[0109]

[0110] where σ1, σ2, and σ3 represent the standard deviations of the position error projected onto the x-axis, y-axis, and z-axis, respectively.

[0111] Step 3: Propose a fault detection model based on the positioning domain according to the variance inflation theory

[0112] When a satellite navigation system fails, compared with the fault-free situation, the positioning error μ is regarded as a random variable with the same expectation but an inflated variance, denoted by μ f which is expressed as:

[0113]

[0114] where k represents the variance inflation coefficient.

[0115] Therefore, construct the classification variable S j , and the fault detection model is given as follows:

[0116]

[0117] where S1, S2, and S3 represent the classification variables on the three coordinate axes. When S j = kσ j , it indicates that there are faulty satellites in the satellite navigation system; when S j = σ j , it indicates that there are no faulty satellites in the satellite navigation system.

[0118] Step 4: For the fault detection model in Step 3, construct the Bayesian posterior probability odds ratio Ratio for fault detection based on Bayesian testing

[0119] In the actual use process, when the positioning system obtains n s positioning estimation solutions samples through continuous sampling, they can be used to estimate the sample standard deviation S of the corresponding positioning error samples m . According to the Bayesian testing theory, the posterior probability of the satellite navigation system being fault-free is:

[0120]

[0121] where, for simplicity of expression, the subscripts of some symbols are omitted, that is, S is used to represent S j , and σ is used to represent σj ; P{S = σ} represents the prior probability that the satellite navigation system is fault - free; P{S m = y|S = σ} represents the likelihood probability that the satellite navigation system is fault - free; y is the specific value of the sample standard deviation S m under the actual sampling situation.

[0122] Similarly, the posterior probability that the satellite navigation system has a fault can be obtained as follows:

[0123]

[0124] Among them, P{S = kσ} represents the prior probability that the satellite navigation system has a fault; P{S m = y|S = kσ} represents the likelihood probability that the satellite navigation system has a fault.

[0125] Therefore, the posterior probability odds ratio of fault and fault - free is:

[0126]

[0127] The prior probabilities and likelihood probabilities in the fault - free and faulty cases are specifically as follows:

[0128] P{S = σ}=(1 - P sat ) n (34)

[0129]

[0130] Among them, P sat is the prior probability of failure of each satellite; is the probability density function of the chi - square function; Δx is the differential element of the random variable.

[0131] Substituting equations (15), (16), (17) and (18) into equation (14), the specific expression of the Bayesian posterior probability odds ratio can be obtained:

[0132]

[0133] Step Five: Determine the adaptive variance inflation coefficient

[0134] The false - alarm rate P fa is defined as:

[0135] P fa = P{Ratio > 1|H0} (39)

[0136] Among them, H0 represents the fault - free condition, which is consistent with the case of S = σ.

[0137] P fa The maximum allowable value of [P faDerived from the continuity requirement, the results are as follows:

[0138]

[0139] Among them, P{H0} represents the prior probability that the satellite navigation system is fault-free; C req is the continuity requirement index specified by the International Civil Aviation Organization (ICAO).

[0140] For all S m <T fa , the receiver autonomous integrity monitoring method based on Bayesian test should not detect a fault, that is, it should meet the following requirements:

[0141]

[0142] Among them, represents the inverse cumulative distribution function of the chi-square distribution.

[0143] Therefore, in order to meet the false alarm rate, the minimum inflation coefficient k needs to satisfy:

[0144]

[0145] By iteratively solving Equation (23), the minimum value k min of the inflation coefficient k can be determined. On the premise of meeting the false alarm rate, in order to maximize the detection effect, during actual use, let the inflation coefficient k be taken as k min .

[0146] Step Six: Use the Bayesian posterior probability odds ratio for fault detection

[0147] Input the actually sampled result S m and the obtained inflation coefficient k into the Bayesian posterior probability odds ratio Ratio.

[0148] When Ratio≥1, there is a faulty satellite in the satellite navigation system and a fault occurs;

[0149] When Ratio<1, there is no faulty satellite in the satellite navigation system and no fault occurs;

[0150] The results of the fault detection rate in this embodiment are shown in Figure 2 . Among them, the correct detection rate corresponding to the proposed receiver autonomous integrity monitoring method based on Bayesian test is represented by a solid line with a circle; the correct detection rate corresponding to the traditional least squares RAIM is represented by a dashed line with a triangle; the correct detection rate curves corresponding to the same faulty satellite are drawn in the same color.

[0151] As the pseudorange deviation increases, the correct detection rates of different detection methods also increase accordingly and finally reach 100%. However, the minimum pseudorange fault deviation corresponding to the 100% correct detection rate in all fault satellite cases achieved by the proposed receiver autonomous integrity monitoring method based on Bayesian test is less than that of the traditional least squares RAIM, indicating that the proposed receiver autonomous integrity monitoring method based on Bayesian test has better performance.

[0152] Compared with the traditional least squares RAIM, the proposed receiver autonomous integrity monitoring method based on Bayesian test has better detection performance, especially when large slope satellites fail. As shown in Table 1, G17 and G27 are large slope satellites, which are prone to higher missed detection risks. Taking the failure of satellite G27 as an example, the minimum pseudorange fault deviation corresponding to the 100% detection rate achieved by the least squares RAIM method is 50m, while the receiver autonomous integrity monitoring method based on Bayesian test can achieve a 100% fault detection rate when the pseudorange fault deviation reaches 20m.

[0153] Table 1 Characteristic slopes of all visible satellites

[0154] satellite G01 G07 G08 G14 G17 G21 G27 G28 G30 slope 0.481 0.121 0.616 0.314 1.724 0.232 1.077 0.178 0.675

[0155] The above specific description further details the purpose, technical solution, and beneficial effects of the invention. It should be understood that the above is only a specific embodiment of the present invention and is not used to limit the protection scope of the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. Receiver autonomous integrity monitoring method based on Bayesian test, characterized in that: It includes the following steps: Step 1: Obtain the positioning estimation solution for the linearized pseudo-range observation equation; Step 2: Project the statistical model of the pseudo-range observation noise into the positioning domain by using the multiple convolution formula to obtain the statistical model of the positioning error; Step 3: Propose a fault detection model S based on the location domain according to the variance inflation theory j ; Step 4: For this fault detection model, based on the Bayesian test theory, construct the Bayesian posterior probability odds Ratio for fault detection; Step 5: According to the navigation continuity requirement, give the determination method of the key parameter - variance inflation coefficient k of the Bayesian posterior probability odds Ratio; Step 6: Use the Bayesian posterior probability odds Ratio for fault detection; The specific implementation method of Step 6 is: Use the Bayesian posterior probability odds Ratio for fault detection Input the actual sampling result S m = y and the obtained expansion coefficient k into the Bayesian posterior probability odds ratio Ratio, that is, Equation (19); When Ratio≥1, there is a faulty satellite in the satellite navigation system and a fault occurs; When Ratio<1, there is no faulty satellite in the satellite navigation system and no fault occurs.

2. The receiver autonomous integrity monitoring method based on Bayesian test according to claim 1, wherein: The specific implementation method of Step 1 is: Use the weighted least squares algorithm to obtain the positioning estimation solution for the linearized pseudo-range observation equation The linear pseudo-range observation equation of the user receiver for the satellite is: z=Hx+ε(1) where \(z\in R\) n×1 , represents the offset vector between the visible satellite pseudorange and the estimated value, \(n\) is the number of visible satellites; \(x\in R\) m×1 is the deviation vector between the true value and the nominal value of the state variable, including the three-dimensional position of the receiver and the clock deviation of the satellite navigation system, \(m\) is the dimension of the state variable; \(H\in R\) n×m is the geometric observation matrix; \(\varepsilon=[\varepsilon_1,\varepsilon_2,...,\varepsilon\) n T represents the observation error vector, and \(\varepsilon\) i (\(i\in\{1,2,...,n\}\)) are independent and follow Gaussian noise, \(\sigma\) obs,i represents the standard deviation of the observation error;​ Based on the weighted least squares algorithm, the positioning estimation solution of x is where W is the weighted matrix related to the elevation angle of each satellite relative to the user and the user's ranging accuracy.

3. The receiver autonomous integrity monitoring method based on Bayesian test according to claim 1, wherein: The specific implementation method of Step 1 is: Use the Kalman filter algorithm to obtain the positioning estimation solution for the linearized pseudo-range observation equation The state equation of the receiver is x k = Ax k-1 + Bu k-1 + w k-1 (3) where x is the state variable, including the three-dimensional position of the receiver and the clock bias of the satellite navigation system; A and B are system parameters, u is the control quantity; w is the system noise; the subscript k of each symbol represents the current moment, and the subscript k - 1 represents the previous moment; At the same time, the linear pseudo-range observation equation of the receiver for the satellite is: z k = Hx k + ε k (4) Therefore, the time update equation of the Kalman filter is as follows wherein, is the one-step predicted value of the state variable at the current moment; is 's covariance matrix; Q is the covariance matrix of the system noise; The state update equation of the Kalman filter, that is, the positioning estimation solution is as follows: Among them, K k is the gain matrix; R is the measurement noise covariance matrix; P k is the covariance matrix of the positioning estimation solution .

4. The receiver autonomous integrity monitoring method based on Bayesian test according to claim 1, characterized in that: The specific implementation method of Step 2 is: Project the statistical model of the pseudo-range observation noise into the positioning domain by using the multiple convolution formula to obtain the statistical model of the positioning error Solution based on positioning estimation The estimated error μ of x is obtained as Expand μ into μ1, μ2, μ3, which respectively represent the positioning errors on the x-axis, y-axis and z-axis: where a ji is the element in the j-th row and the i-th column of matrix A; Since ε i are independent of each other, the statistical model of the observation noise ε is used to obtain the statistical model of the positioning error μ j as follows: where σ1, σ2, σ3 respectively represent the standard deviations of the position errors projected onto the x-axis, y-axis and z-axis.

5. The receiver autonomous integrity monitoring method based on Bayesian test according to claim 1, characterized in that: The specific implementation method of Step 3 is as follows: According to the variance inflation theory, a fault detection model S based on the positioning domain is proposed j When the satellite navigation system fails, compared with the fault-free situation, the positioning error μ is regarded as a random variable with the same expectation but an inflated variance, denoted by μ f which is expressed as: where k represents the variance inflation coefficient; Therefore, construct the classification variable S j , and the fault detection model is given as follows: Among them, S1, S2, and S3 respectively represent classification variables on three coordinate axes. When S j = kσ j , it indicates that there is a faulty satellite in the satellite navigation system; when S j = σ j , it indicates that there is no faulty satellite in the satellite navigation system.

6. The receiver autonomous integrity monitoring method based on Bayesian test according to claim 1, characterized in that: The specific implementation method of Step 4 is: For the fault detection model in Step 3, based on the Bayesian test, construct the Bayesian posterior probability odds Ratio for fault detection In the actual usage process, when the positioning system obtains n s positioning estimation solutions samples through continuous sampling, it can be used to estimate the sample standard deviation S of the corresponding positioning error samples; according to the Bayesian test theory, the posterior probability of the satellite navigation system being fault-free is: m ; Among them, for the sake of simplifying the expression, the subscripts of some symbols are omitted, that is, S is used to represent S j , and σ is used to represent σ j ; P{S = σ} represents the prior probability that the satellite navigation system is fault-free; P{S m = y|S = σ} represents the likelihood probability that the satellite navigation system is fault-free; y is the specific value of the sample standard deviation S m in the actual sampling case; Similarly, the posterior probability of the satellite navigation system fault can be obtained as: Among them, P{S = kσ} represents the prior probability of satellite navigation system failure; P{S m = y|S = kσ} represents the likelihood probability of satellite navigation system failure; Therefore, the posterior probability odds ratio of fault to non-fault is: The prior probabilities and likelihood probabilities in the cases of non-fault and fault are specifically as follows: P{S = σ}=(1 - P sat ) n (15) Among them, P sat is the prior probability of each satellite failure; is the probability density function of the chi-square function; Δx is the differential element of the random variable; Substituting equations (15), (16), (17) and (18) into equation (14), the specific expression of the Bayesian posterior probability odds ratio can be obtained: .

7. The receiver autonomous integrity monitoring method based on Bayesian test according to claim 1, characterized in that: The specific implementation method of Step 5 is: Determine the adaptive variance inflation coefficient k False alarm rate P fa Is defined as: P fa = P{Ratio > 1|H0} (20) where H0 represents the non-fault condition, which is consistent with the case of S = σ; P fa The maximum allowable value of [P fa is derived from the continuity requirement, and the result is as follows: Among them, \(P\{H_0\}\) is consistent with \(P\{S = \sigma\}\), representing the prior probability that the satellite navigation system is fault-free; C req is the continuity requirement index specified by the International Civil Aviation Organization (ICAO); To meet the false alarm rate, for all S m <T fa , the receiver autonomous integrity monitoring method based on Bayesian test should not detect a fault, that is, meet the following requirements: wherein, represents the inverse cumulative distribution function of the chi-square distribution; Therefore, in order to meet the false alarm rate, the minimum inflation coefficient k needs to satisfy: By iteratively solving equation (23), the minimum value k of the expansion coefficient k can be determined min ; on the premise of satisfying the false alarm rate, in order to maximize the detection ability, in the actual use process, let the expansion coefficient k be taken as k min .

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