Gaussian mixture model based integrity threshold calculation method for underground ultra-wideband system

Abstract

The application discloses a kind of underground ultra-wideband system integrity threshold calculation method based on Gaussian mixture model, it is applicable to the new integrity evaluation system of underground large space.This method uses the hyperframe of Gaussian mixture model GMM to process the ultra-wideband ranging error sample based on TOA positioning, reflects the superiority of flexible Gaussian mixture model processing bimodal distribution, and fully considers the correlation between ranging error and distance under the environment of underground large space, introduces distance factor when establishing error model, clearly deduces the method for calculating protection level PL in position domain, continuously convolves double-component hyper GMM, obtains multi-component GMM, and the hyper nature still holds after convolution.This application solves the contradiction between traditional Gaussian hyper model and non-gaussian characteristic error under complex environment, tightens PL with small calculation cost, and improves system availability.

Description

Technical Field

[0001] This invention relates to the field of indoor positioning integrity technology, and in particular to a method for calculating the integrity threshold of underground ultra-wideband systems based on a Gaussian mixture model. Background Technology

[0002] With the development of navigation and positioning technology and the increasing demand for navigation applications, navigation and positioning are gradually expanding from aviation to various scenarios such as indoors and underground, moving towards greater accuracy and reliability. To address the issue of weak satellite signals in indoor / underground environments, various indoor positioning technologies have emerged, including Wi-Fi, Bluetooth, infrared, motion capture, and ultra-wideband (UWB). Among these, UWB positioning technology transmits data by sending and receiving extremely narrow pulses. Compared to traditional narrowband systems, it boasts advantages such as strong penetration, good resistance to multipath interference, and higher positioning accuracy, making it widely applicable in non-exposed spaces such as urban rail transit, tunnels, and large underground spaces. With the development of smart cities and new infrastructure, not only is there a need to introduce precise positioning technology for underground spaces, but higher requirements are also being placed on the reliability of positioning in non-exposed spaces, making research on the integrity of positioning in large underground spaces even more urgent.

[0003] The TOA (Time of Arrival) positioning principle of UWB systems in large underground spaces is similar to that of GNSS systems, but their error characteristics differ significantly. UWB systems in large underground spaces are greatly affected by multipath and non-line-of-sight (NLOS) propagation, especially under harsh underground conditions where terrain and electromagnetic environments are complex, resulting in ranging errors exhibiting distinct non-Gaussian characteristics. In contrast, GNSS systems generally use a Gaussian distribution to enclose errors. Furthermore, some error models established in the aviation field are no longer valid in underground environments. Therefore, the GNSS integrity framework is not entirely applicable to the integrity assessment system of large underground spaces. Summary of the Invention

[0004] The technical problem to be solved by this invention is to overcome the shortcomings of the prior art and provide a method for calculating the integrity threshold of underground ultra-wideband systems based on a Gaussian mixture model. Based on a novel UWB navigation and positioning platform, a Gaussian mixture model of TOA ranging error in underground large space environment is constructed, and a new method for calculating the integrity threshold of underground PNT network is proposed. This method greatly reduces the system conservatism and ensures the reliable assessment of the integrity service of underground hybrid PNT system.

[0005] To solve the above-mentioned technical problems, the present invention adopts the following technical solution:

[0006] This invention proposes a method for calculating the integrity threshold of underground ultra-wideband systems based on a Gaussian mixture model.

[0007] Considering the correlation between ranging error and distance in underground space environment, the ranging error is modeled, and a distance factor is introduced when establishing the ranging error model to obtain the normalized ranging error.

[0008] The probability density function PDF of the normalized ranging error is constructed into a two-component Gaussian mixture model (GMM). Based on this, the variance of the two components is further expanded to construct a superboundary model. Then, the distance factor is reintroduced into the mean and variance terms of the superboundary model to obtain the distance domain PDF.

[0009] Based on the distance domain PDF, the distance domain error of each base station is mapped to the location domain to obtain the corresponding location domain PDF. Then, continuous convolution is performed on it to obtain the final positioning error PDF.

[0010] The protection level in the XYZ axis direction is obtained by inverting the positioning error PDF. The horizontal protection level HPL is obtained based on the protection levels in the X and Y directions.

[0011] As a further optimization of the integrity threshold calculation method for underground ultra-wideband systems based on Gaussian mixture models described in this invention, a distance factor α is obtained by fitting the relationship between ranging error and distance. When establishing the ranging error model, a distance factor is introduced to obtain a normalized ranging error γ that is independent of distance factors. The ranging error is defined as ε, and then the normalized ranging error...

[0012] As a further optimization of the integrity threshold calculation method for underground ultra-wideband systems based on Gaussian mixture models described in this invention, a probability density function PDF of the normalized ranging error γ is constructed using a two-component Gaussian mixture model:

[0013] f γ (x)=w1×G(u1,σ1)+w2×G(u2,σ2)

[0014] Among them, f γ (x) represents the PDF of the normalized ranging error γ, w1 and w2 are the weights of the first and second components respectively, u1 and u2 are the means of the first and second components respectively, σ1 and σ2 are the variances of the first and second components respectively, G() is the notation of the Gaussian distribution function, and x is the variable.

[0015] As a further optimization scheme for the integrity threshold calculation method of underground ultra-wideband system based on Gaussian mixture model described in this invention, w1, w2, u1, u2, σ1, and σ2 are given by the EM algorithm.

[0016] As a further optimization of the integrity threshold calculation method for underground ultra-wideband systems based on Gaussian mixture models described in this invention, the variance in the normalized ranging error PDF is expanded until it coincides with the median when constructing the superboundary model; after obtaining the two-component GMM by the EM algorithm, the variance is increased until the median of the superboundary CDF coincides with the median of the sample CDF, thus satisfying the conditions of the probability envelopes on both sides:

[0017]

[0018]

[0019] Among them, G o (x) represents the superboundary CDF, G a (x) represents the sample CDF, and x represents the variable;

[0020] The resulting superboundary model is obtained by updating σ1 and σ2 in the two-component GMM model with the values ​​after variance inflation.

[0021] As a further optimization of the integrity threshold calculation method for underground ultra-wideband systems based on Gaussian mixture models described in this invention, a distance factor α is reintroduced into the mean and variance terms of the super-boundary model to obtain the distance domain PDF, which is described as follows:

[0022]

[0023] f ε (x) represents the distance domain PDF, w1 and w2 are the weights of the first and second components, respectively, and u1 and u2 are the means of the first and second components, respectively. G represents the updated variances of the first and second components, respectively. G() is the notation for the Gaussian distribution function, and x is the variable.

[0024] As a further optimization of the method for calculating the integrity threshold of an underground ultra-wideband system based on a Gaussian mixture model as described in this invention,

[0025] The range domain error of each base station is mapped from the range domain to the location domain using a spatial geometric projection matrix. Specifically, this is achieved by introducing the spatial geometric projection matrix into the mean and variance terms of the range domain PDF to obtain the location domain PDF. The location domain PDFs of each base station in the X and Y directions are represented as follows:

[0026]

[0027]

[0028] Where i represents the i-th base station, f i Xf represents the position domain PDF in the X direction from the error source of the i-th base station. i Y The position domain PDF from the error source at the i-th base station is shown in the Y direction; w i,1 w represents the weight of the first group of the i-th base station. i,2 u represents the weight of the second component of the i-th base station; i,1 Let u represent the mean of the first group of the i-th base station. i,2 This represents the mean of the second group of the i-th base station; This represents the variance of the first group of the i-th base station. α represents the variance of the second component of the i-th base station; i S represents the distance factor of the i-th base station; 1,i S represents the first element of the spatial geometric projection matrix, i.e., the projection in the X direction. 2,i The second row and i-th element of the spatial geometric projection matrix represents the projection in the Y direction, N is the number of currently available base stations for the user, and G() is the notation for the Gaussian distribution function;

[0029] By performing continuous convolution integration on the location domain PDFs from N base stations, the final positioning error PDFs are expressed as follows in the X and Y directions:

[0030]

[0031]

[0032] Among them, f X Let f be the localization error PDF in the X direction after N consecutive convolutions. Y The localization error PDF in the Y direction after N consecutive convolutions, where ki equals 1 or 2, indicating that the component comes from the first or second component; w i,ki μ represents the weight of the first or second component from the i-th base station. i,ki This represents the mean of the first or second component from the i-th base station; Let G(x) represent the variance of the first or second component from the i-th base station; G() is the notation for the Gaussian distribution function. The symbol for convolution;

[0033] Inverting the positioning error PDF yields the protection level PL. The protection levels in the X and Y directions are expressed as follows:

[0034]

[0035]

[0036] Where XPL represents the protection level in the X direction and YPL represents the protection level in the Y direction; This represents the inverse of the positioning error PDF in the X direction. This represents the inverse of the positioning error PDF in the Y direction; P IR It is the probability of integrity risk;

[0037] The Horizontal Protection Level (HPL) is the horizontal component of the Position Domain (PL). This invention considers planar positioning and only takes into account the Horizontal Protection Level (HPL). The method for mapping HPL to the position domain is as follows:

[0038]

[0039] Compared with the prior art, the present invention, employing the above technical solution, has the following technical effects:

[0040] (1) This invention mainly focuses on the integrity analysis of multi-base station TOA positioning mode. Based on the GNSS integrity research and the new UWB platform in underground large space, it proposes an integrity threshold algorithm strategy to reduce conservatism.

[0041] (2) Using Gaussian mixture model to approximate the non-Gaussian distribution of observation data error can improve PDF accuracy. It focuses on the tail while tightening the core. The two-component GMM model can more accurately enclose the distribution of the core and tail.

[0042] (3) The two-component mixed distribution integrity threshold calculation method based on GMM, although increasing the computational load, only tightens the protection level with a small computational cost, and the PL is reduced by about 30% compared with the calculation of a single Gaussian superboundary. It should be noted that when the number of base stations is too large, the number of terms in the location domain error PDF will increase exponentially, and with the increase of convolution, the problem caused by the non-zero mean of the Gaussian distribution will become more serious, and the mean will shift to the tail after multiple convolutions. Attached Figure Description

[0043] Figure 1 This is a flowchart of the present invention. Detailed Implementation

[0044] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0045] Figure 1 This is a flowchart of the present invention, a method for calculating the integrity threshold of an underground ultra-wideband system based on a Gaussian mixture model.

[0046] Considering the correlation between ranging error and distance in underground space environment, the ranging error is modeled, and a distance factor is introduced when establishing the ranging error model to obtain the normalized ranging error.

[0047] The probability density function PDF of the normalized ranging error is constructed into a two-component Gaussian mixture model (GMM). Based on this, the variance of the two components is further expanded to construct a superboundary model. Then, the distance factor is reintroduced into the mean and variance terms of the superboundary model to obtain the distance domain PDF.

[0048] Based on the distance domain PDF, the distance domain error of each base station is mapped to the location domain to obtain the corresponding location domain PDF. Then, continuous convolution is performed on it to obtain the final positioning error PDF.

[0049] The protection level in the XYZ axis direction is obtained by inverting the positioning error PDF. The horizontal protection level HPL is obtained based on the protection levels in the X and Y directions.

[0050] As a further optimization of the integrity threshold calculation method for underground ultra-wideband systems based on Gaussian mixture models described in this invention, a distance factor α is obtained by fitting the relationship between ranging error and distance. When establishing the ranging error model, a distance factor is introduced to obtain a normalized ranging error γ that is independent of distance factors. The ranging error is defined as ε, and then the normalized ranging error...

[0051] As a further optimization of the integrity threshold calculation method for underground ultra-wideband systems based on Gaussian mixture models described in this invention, a probability density function PDF of the normalized ranging error γ is constructed using a two-component Gaussian mixture model:

[0052] f γ (x)=w1×G(u1,σ1)+w2×G(u2,σ2)

[0053] Among them, f γ (x) represents the PDF of the normalized ranging error γ, w1 and w2 are the weights of the first and second components respectively, u1 and u2 are the means of the first and second components respectively, σ1 and σ2 are the variances of the first and second components respectively, G() is the notation of the Gaussian distribution function, and x is the variable.

[0054] As a further optimization scheme for the integrity threshold calculation method of underground ultra-wideband system based on Gaussian mixture model described in this invention, w1, w2, u1, u2, σ1, and σ2 are given by the EM algorithm.

[0055] As a further optimization of the integrity threshold calculation method for underground ultra-wideband systems based on Gaussian mixture models described in this invention, the variance in the normalized ranging error PDF is expanded until it coincides with the median when constructing the superboundary model; after obtaining the two-component GMM by the EM algorithm, the variance is increased until the median of the superboundary CDF coincides with the median of the sample CDF, thus satisfying the conditions of the probability envelopes on both sides:

[0056]

[0057]

[0058] Among them, G o (x) represents the superboundary CDF, G a (x) represents the sample CDF, and x represents the variable;

[0059] The resulting superboundary model is obtained by updating σ1 and σ2 in the two-component GMM model with the values ​​after variance inflation.

[0060] As a further optimization of the integrity threshold calculation method for underground ultra-wideband systems based on Gaussian mixture models described in this invention, a distance factor α is reintroduced into the mean and variance terms of the super-boundary model to obtain the distance domain PDF, which is described as follows:

[0061]

[0062] f ε (x) represents the distance domain PDF, w1 and w2 are the weights of the first and second components, respectively, and u1 and u2 are the means of the first and second components, respectively. G represents the updated variances of the first and second components, respectively. G() is the notation for the Gaussian distribution function, and x is the variable.

[0063] As a further optimization of the method for calculating the integrity threshold of an underground ultra-wideband system based on a Gaussian mixture model as described in this invention,

[0064] The range domain error of each base station is mapped from the range domain to the location domain using a spatial geometric projection matrix. Specifically, this is achieved by introducing the spatial geometric projection matrix into the mean and variance terms of the range domain PDF to obtain the location domain PDF. The location domain PDFs of each base station in the X and Y directions are represented as follows:

[0065]

[0066]

[0067] Where i represents the i-th base station, f i X f represents the position domain PDF in the X direction from the error source of the i-th base station. i Y The position domain PDF from the error source at the i-th base station is shown in the Y direction; w i,1 w represents the weight of the first group of the i-th base station. i,2 u represents the weight of the second component of the i-th base station; i,1Let u represent the mean of the first group of the i-th base station. i,2 This represents the mean of the second group of the i-th base station; This represents the variance of the first group of the i-th base station. α represents the variance of the second component of the i-th base station; i S represents the distance factor of the i-th base station; 1,i S represents the first element of the spatial geometric projection matrix, i.e., the projection in the X direction. 2,i The second row and i-th element of the spatial geometric projection matrix represents the projection in the Y direction, N is the number of currently available base stations for the user, and G() is the notation for the Gaussian distribution function;

[0068] By performing continuous convolution integration on the location domain PDFs from N base stations, the final positioning error PDFs are expressed as follows in the X and Y directions:

[0069]

[0070]

[0071] Among them, f X Let f be the localization error PDF in the X direction after N consecutive convolutions. Y The localization error PDF in the Y direction after N consecutive convolutions, where ki equals 1 or 2, indicating that the component comes from the first or second component; w i,ki μ represents the weight of the first or second component from the i-th base station. i,ki This represents the mean of the first or second component from the i-th base station; Let G(x) represent the variance of the first or second component from the i-th base station; G() is the notation for the Gaussian distribution function. The symbol for convolution;

[0072] Inverting the positioning error PDF yields the protection level PL. The protection levels in the X and Y directions are expressed as follows:

[0073]

[0074]

[0075] Where XPL represents the protection level in the X direction and YPL represents the protection level in the Y direction; This represents the inverse of the positioning error PDF in the X direction. This represents the inverse of the positioning error PDF in the Y direction; P IR It is the probability of integrity risk;

[0076] The Horizontal Protection Level (HPL) is the horizontal component of the Position Domain (PL). This invention considers planar positioning and only takes into account the Horizontal Protection Level (HPL). The method for mapping HPL to the position domain is as follows:

[0077]

[0078] The core idea of ​​this invention is to propose a novel integrity threshold calculation method for underground large-space UWB platforms based on traditional GNSS integrity analysis methods. Following a progression from simple to complex and from bottom to top, the system is programmed according to the theoretical architecture. The specific implementation method is as follows:

[0079] 1. First, be familiar with the theoretical foundations of traditional GNSS integrity, including fault detection and troubleshooting, boundary overflow models, and PL calculations; be familiar with various integrity control methods in the aviation field, such as the integrity information provided by the three known standardized augmentations in ABAS (Airborne Augmentation System), GBAS (Ground Augmentation System), and SBAS (Satirical Augmentation System); be familiar with ultra-wideband (UWB) systems, mainly including ranging noise models, including non-line-of-sight, multipath, and white noise, and understand TOA-based UWB indoor positioning and navigation technology; be familiar with the error characteristics of underground large-space environments; be familiar with probability density function models, such as Gaussian distribution, Pareto distribution, and Gaussian mixture distribution; be familiar with model parameter estimation algorithms, such as the EM algorithm; and be familiar with two different ways in which position domain errors are generated depending on the order of mapping and boundary overflow.

[0080] 2. In the error probability statistics module, an independent TOA ranging error model is constructed for the ranging signal of each UWB base station. The selection of the distance factor is given by the relationship between the fitting error and the distance; the weights, mean, and variance of each component in the two-component Gaussian mixture model are all given by the EM algorithm. After obtaining the two-component GMM by the EM algorithm, the variance is increased until the median of the superboundary CDF coincides with the median of the sample CDF. The parameter update of the superboundary model includes two variance updates.

[0081] 3. When calculating the position error PE, observations are first generated based on the base station location and the position information of the rover with a known trajectory. After adding noise, the position coordinates are recalculated using the least squares method. The error between the generated position and the accurate position is the position error PE. In actual positioning, the true position and position error are unknown. Therefore, a protection level (PL) is introduced to measure the position error. In this invention, to measure whether the PL can effectively enclose the PE, the PE is generated through simulation using positioning points with known coordinate information.

[0082] 4. When calculating the protection level PL, the distance factor is reintroduced and the variance is updated, and the normalized error PDF is converted into the distance domain error PDF. Then, the distance domain error of each base station is mapped to the location domain to obtain the location domain PDF of each base station. The location domain PDF of each base station is continuously convolved to give the expression of the positioning error PDF. Finally, the inverse of the positioning error PDF is obtained to get PL.

[0083] The principle of this invention is as follows: Based on traditional GNSS integrity analysis methods, a new integrity threshold calculation method adapted to novel UWB platforms in large underground spaces is proposed. A Gaussian Mixture Model (GMM) superboundary framework is used to process ultra-wideband ranging error samples based on TOA positioning, leveraging the advantages of the flexible GMM in handling bimodal distribution problems. Furthermore, the correlation between ranging error and distance is fully considered in the large underground space environment. A distance factor is introduced when establishing the error model. Continuous convolution of the two-component superboundary GMM yields a multi-component GMM, and the superboundary property remains valid after convolution. This resolves the conflict between computational load and model accuracy, tightening the PL (Property Parameter) with relatively low computational cost.

[0084] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for calculating the integrity threshold of an underground ultra-wideband system based on a Gaussian mixture model, characterized in that, Considering the correlation between ranging error and distance in underground space environment, the ranging error is modeled, and a distance factor is introduced when establishing the ranging error model to obtain the normalized ranging error. The probability density function PDF of the normalized ranging error is constructed into a two-component Gaussian mixture model (GMM). Based on this, the variance of the two components is further expanded to construct a superboundary model. Then, the distance factor is reintroduced into the mean and variance terms of the superboundary model to obtain the distance domain PDF. Based on the distance domain PDF, the distance domain error of each base station is mapped to the location domain to obtain the corresponding location domain PDF. Then, continuous convolution is performed on it to obtain the final positioning error PDF. The protection level in the XYZ axis direction is obtained by inverting the positioning error PDF, and the horizontal protection level HPL is obtained based on the protection levels in the X and Y directions. When constructing the super-boundary model, the variance in the normalized ranging error PDF is expanded until it coincides with the median; after obtaining the two-component GMM by the EM algorithm, the variance is increased until the super-boundary CDF coincides with the median of the sample CDF, thus satisfying the conditions of the probability envelopes on both sides: ； in, Indicates superboundary CDF, Let x represent the sample CDF; The resulting superboundary model is the variance of the first component in the two-component GMM model. Variance of the second component Update to the value after variance inflation.

2. The method for calculating the integrity threshold of an underground ultra-wideband system based on a Gaussian mixture model according to claim 1, characterized in that, The distance factor is obtained by fitting the relationship between the ranging error and the distance. In establishing the ranging error model, a distance factor is introduced to obtain a normalized ranging error that is independent of the distance factor. The ranging error is defined as Then the normalized ranging error .

3. The method for calculating the integrity threshold of an underground ultra-wideband system based on a Gaussian mixture model according to claim 1, characterized in that, Construct the probability density function PDF of the normalized ranging error γ using a two-component GMM: ； in, PDF of the normalized ranging error γ , These are the weights of component 1 and component 2, respectively. , These are the mean values ​​of component 1 and component 2, respectively. , Let G be the variances of the first and second components, respectively, and let G() be the notation for the Gaussian distribution function. For variables.

4. The method for calculating the integrity threshold of an underground ultra-wideband system based on a Gaussian mixture model according to claim 3, characterized in that, , , , , The results are provided using the EM algorithm.

5. The method for calculating the integrity threshold of an underground ultra-wideband system based on a Gaussian mixture model according to claim 1, characterized in that, Reintroducing distance factors into the mean and variance terms of the superboundary model. The distance domain PDF is obtained, and it is described as follows: ； For distance domain PDF, , These are the weights of component 1 and component 2, respectively. , These are the mean values ​​of component 1 and component 2, respectively. , Let G() represent the updated variances of the first and second components, respectively, and let G() be the notation for the Gaussian distribution function. For variables.

6. The method for calculating the integrity threshold of an underground ultra-wideband system based on a Gaussian mixture model according to claim 1, characterized in that, The range domain error of each base station is mapped from the range domain to the location domain using a spatial geometric projection matrix. Specifically, this is achieved by introducing the spatial geometric projection matrix into the mean and variance terms of the range domain PDF to obtain the location domain PDF. The location domain PDFs of each base station in the X and Y directions are represented as follows: ； ； in, Indicates the first One base station, Indicates from the The location domain PDF of each base station error source in the X direction. Indicates from the The location domain PDF of each base station error source in the Y direction; Indicates the first The weights of the first group of each base station, Indicates the first The weights of the second group of each base station; Indicates the first The mean of the first group of each base station, Indicates the first The mean of the second group of each base station; Indicates the first The variance of the first group of each base station, Indicates the first The variance of the second component of each base station; Indicates the first Distance factor of each base station; The first row of the spatial geometric projection matrix represents the first line of the projection matrix. Each element represents a projection in the X direction. The second row of the spatial geometric projection matrix represents the first row of the second row. The elements represent the projection in the Y direction, N is the number of base stations currently available to the user, and G() is the notation of the Gaussian distribution function; By performing continuous convolution integration on the location domain PDFs from N base stations, the final positioning error PDFs are expressed as follows in the X and Y directions: ； ； in, PDF represents the localization error in the X direction after N consecutive convolutions. PDF represents the localization error in the Y direction after N consecutive convolutions. An equal value of 1 or 2 indicates that the component comes from component 1 or component 2; Indicates from the The weights of the first or second component of each base station; Indicates from the The mean of the first or second component of each base station; Indicates from the The variance of the first or second component of each base station; G() is the notation for the Gaussian distribution function. The symbol for convolution; Inverting the positioning error PDF yields the protection level PL. The protection levels in the X and Y directions are expressed as follows: ； ； Where XPL represents the protection level in the X direction and YPL represents the protection level in the Y direction; This represents the inverse of the positioning error PDF in the X direction. This represents the inverse of the positioning error PDF in the Y direction; It is the probability of integrity risk; The Horizontal Protection Level (HPL) is the horizontal component of the Position Domain (PL). This invention considers planar positioning and only takes into account the Horizontal Protection Level (HPL). The method for mapping HPL to the position domain is as follows: 。

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