Single-beacon range circle positioning aided ins navigation method based on mahalanobis distance constraint

Abstract

The present application relates to a single-beacon distance circle positioning auxiliary INS navigation method based on Mahalanobis distance constraint, and belongs to the technical field of underwater auxiliary navigation research, comprising: constructing a single-beacon ranging reference map, and converting single-beacon ranging values to a two-dimensional plane; based on the converted single-beacon ranging values, constructing a distance circle with the single beacon as the center, and combining the INS calculated displacement vectors at the previous and subsequent moments to construct a target function model; determining a positioning solution based on the target function model, and when there are two positioning solutions, using a Mahalanobis distance cost function to select an optimal point to obtain a positioning position. The present application can achieve high-precision positioning in a ranging effective area and effectively suppresses the divergence of INS position errors.

Description

Technical Field

[0001] This invention relates to the field of underwater assisted navigation research technology, and in particular to a single beacon range circle positioning-assisted INS navigation method based on Mahalanobis distance constraints. Background Technology

[0002] With the increasing demand for underwater navigation in areas such as marine resource exploration and marine environmental monitoring, the development of navigation technologies for intelligent underwater platforms such as Autonomous Underwater Vehicles (AUVs) and Unmanned Underwater Vehicles (UUVs) has attracted widespread attention. Accurate and reliable underwater navigation technology is a core prerequisite and key guarantee for mission success during the autonomous operation of these underwater platforms. Inertial Navigation Systems (INS), with their advantages of strong autonomy and high stealth, have become the core equipment for underwater vehicle navigation. However, the working principle of INS dictates that its position error, without external information assistance, cannot meet the requirements for long-term, high-precision navigation. Therefore, underwater acoustic positioning systems have become the core link connecting underwater vehicles and absolute position references. Traditional multi-beacon underwater acoustic positioning systems, such as Long Baseline (LBL) and Short Baseline (SBL), form a geometric network by pre-deploying multiple underwater acoustic beacons with known locations in the operating area. The underwater vehicle can calculate its own position by measuring the distances to multiple beacons. While the aforementioned traditional methods can provide high positioning accuracy, their systems suffer from drawbacks such as complex deployment and high costs for beacon deployment and retrieval. To address the limitations of multi-beacon systems, single-beacon underwater acoustic positioning technology has gradually become a research hotspot. Its core idea is to deploy a single acoustic beacon in a localized sea area and effectively suppress INS position errors by measuring the distance between the beacon and the target area. Summary of the Invention

[0003] The purpose of this invention is to propose a single-beacon range circle positioning-assisted INS navigation method based on Mahalanobis distance constraints. This method transforms the ranging values ​​into a two-dimensional plane by constructing a single-beacon ranging reference map. Then, it uses INS to calculate displacement vectors and constructs a range circle to obtain the positioning solution. The positioning location is determined by the number of obtained positioning solutions. When the range circle has two intersection points, the optimal positioning location is determined using a Mahalanobis distance cost function. This method can achieve high-precision positioning within the effective ranging area and effectively suppresses INS position error divergence.

[0004] To achieve the above objectives, the present invention provides the following solution:

[0005] A single-beacon range-circle positioning-assisted INS navigation method based on Mahalanobis distance constraints includes:

[0006] Construct a single-beacon ranging reference map and transform the single-beacon ranging values ​​into a two-dimensional plane;

[0007] Based on the transformed single beacon ranging values, a distance circle centered on the single beacon is constructed, and the displacement vector is calculated by combining the INS values ​​at previous and subsequent time points to build an objective function model;

[0008] Based on the objective function model, a localization solution is determined. When two localization solutions exist, the optimal point is selected using the Mahalanobis distance cost function to obtain the localization location.

[0009] Optionally, constructing the single beacon ranging reference map includes:

[0010] A rectangular coordinate system is established with the beacon position as the origin. Measurement points are selected in the rectangular coordinate system, and the Euclidean distance from the measurement point position to the beacon position is obtained.

[0011] The depth difference is obtained based on the water depth at the beacon location and the water depth at the measuring point location;

[0012] Based on the difference between the Euclidean distance and the depth, the ranging information of the reference surface is obtained, and the single beacon ranging reference map is constructed.

[0013] Alternatively, methods for converting single beacon ranging values ​​to a two-dimensional plane include:

[0014] ;

[0015] in, The slant range is obtained from single-beacon underwater acoustic ranging. The depth difference between the beacon and the underwater body, Let the coordinates of the distance measurement point be set.

[0016] Optionally, based on the transformed single beacon ranging value, a distance circle centered on the single beacon is constructed, and the displacement vector is calculated by combining the INS values ​​at previous and subsequent time points. The objective function model includes:

[0017] Obtain the AUV after transformation to a two-dimensional plane at the current time. Compared to the previous moment Single beacon ranging value;

[0018] Based on the position of the INS on the AUV at the current moment and the previous moment, the dead reckoning displacement vector is obtained;

[0019] Construct a first distance circle and a second distance circle at the previous and current times, respectively, with the single beacon as the center and the distance between the AUV and the single beacon as the radius;

[0020] The first distance circle is translated according to the dead reckoning displacement vector;

[0021] Based on the coordinates of the intersection point generated by the first distance circle and the second distance circle after translation, a target function model is constructed.

[0022] Optionally, the objective function model is:

[0023] ;

[0024] ;

[0025] ;

[0026] ;

[0027] ;

[0028] ;

[0029] in, For the previous moment up to the current moment Dead reckoning displacement vector, , To unify the single beacon ranging values ​​from the previous moment and the current moment into the corresponding ranging values ​​in a two-dimensional plane, For the objective function, The position coordinates of a single beacon. for The coordinates of the position after the translation from the center of the circle at any given time. It is half the length of the chord connecting the intersection points of the circles. For the previous moment up to the current moment displacement vector magnitude, , Each is the previous moment up to the current moment The displacement vectors in the x and y directions, , These are the x and y coordinates of the beacon, respectively. This is the distance from the beacon's location to the midpoint of the line connecting the intersection points of the circles.

[0030] Optionally, the localization solution of the objective function model includes the following three cases:

[0031] ;

[0032] in, If the set is empty, then the distance circles have no intersection. The coordinates of the position 1 distance from the intersection point of the circles are, The coordinates are the distance from the intersection point of the circles, 2.

[0033] Optionally, determining the localization solution based on the objective function model includes:

[0034] When two distance circles are separate, there is no intersection point, and the distance circle positioning algorithm fails at that moment.

[0035] When two distance circles are tangent, they intersect at a point, which is the location.

[0036] When two distance circles intersect, there are two intersection points. At this point, it is necessary to perform constraint judgment on the two intersection points to obtain the optimal positioning solution.

[0037] Optionally, when two localization solutions exist, the optimal point is selected using the Mahalanobis distance cost function to obtain the localization location, including:

[0038] pass Position estimation of AUV at any time From Instagram arrive The displacement vector at time t is obtained. Estimated position of time :

[0039] ;

[0040] Estimated position The covariance matrix is:

[0041] ;

[0042] in, yes The posterior estimation error covariance matrix at time 1. INS displacement vector The error covariance matrix;

[0043] AUV in Location in real time The covariance matrix is:

[0044] ;

[0045] in, It is the covariance matrix of the input parameters. It is the Jacobian matrix based on the formula for the coordinates of the intersection point generated by the translated first and second distance circles, representing the sensitivity of the positioning position to each noisy input parameter. for The variance of the x-coordinate of the location at any given time. for The covariance between the x and y coordinates of the location at any given time. for The variance of the y-coordinate of the location at any given time;

[0046] Based on estimated location with location The Mahalanobis distance, representing the difference between two values, is used to measure the consistency between them; the difference vector... The covariance is the sum of the covariances of the two:

[0047] ;

[0048] Difference vector The squared Mahalanobis distance is defined as from Time position shift The cost of constantly determining location is as follows:

[0049] ;

[0050] This refers to the Mahalanobis distance cost function used to determine the intersection of two positioning points. The cost of the intersection of the distance circles is calculated using the Mahalanobis distance cost function, and the point with the minimum cost is the optimal positioning point.

[0051] The beneficial effects of this invention are:

[0052] This invention provides a single-beacon range-circle positioning-assisted INS navigation method based on Mahalanobis distance constraints. It can use a single acoustic beacon deployed in a local sea area to measure the distance between an underwater vehicle and the beacon, and then use this distance to assist the inertial navigation equipment onboard the underwater vehicle for high-precision navigation. This application only requires the underwater vehicle's own navigation equipment to provide a short-term relative position measurement result, and the positioning position can be obtained by measuring the distance to the single beacon. This method has advantages such as ease of implementation, low computational load, low deployment cost, and high accuracy. Attached Figure Description

[0053] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments 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.

[0054] Figure 1 This is a single beacon ranging reference diagram according to an embodiment of the present invention;

[0055] Figure 2 This is a schematic diagram of single beacon distance circle positioning according to an embodiment of the present invention;

[0056] Figure 3 This is a flowchart of the distance circle positioning algorithm according to an embodiment of the present invention;

[0057] Figure 4 The experimental vessel and its mounted equipment are embodiments of the present invention.

[0058] Figure 5 This refers to the beacon installation location in an embodiment of the present invention.

[0059] Figure 6 The test trajectory is shown in the embodiment of the present invention;

[0060] Figure 7 This is a schematic diagram of the 9th positioning point in the positioning area according to an embodiment of the present invention;

[0061] Figure 8 The distance circle positioning trajectory is shown in this embodiment of the invention.

[0062] Figure 9 This refers to the eastward position error in an embodiment of the present invention.

[0063] Figure 10 This refers to the northward position error in an embodiment of the present invention.

[0064] Figure 11 This refers to the radial position error in an embodiment of the present invention. Detailed Implementation

[0065] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and 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.

[0066] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0067] Single-beacon ranging-assisted INS navigation is an important direction in the field of underwater assisted navigation research. This embodiment constructs a single-beacon ranging reference map to transform ranging values ​​into a two-dimensional plane. Then, it uses INS to calculate displacement vectors and constructs a distance circle to obtain a positioning solution. The positioning position is determined by the number of obtained positioning solutions. When the distance circle has two intersection points, the optimal positioning position is determined using the Mahalanobis distance cost function. Through the above steps, the design of a single-beacon distance circle positioning-assisted INS navigation algorithm based on Mahalanobis distance constraints is completed. This algorithm only needs to combine the distance measurements of the preceding and following sampling points with the INS position information to achieve high-precision positioning and has single-point iterative real-time matching navigation capabilities.

[0068] Based on this, this embodiment proposes a single-beacon range circle positioning-assisted INS navigation method based on Mahalanobis distance constraints, including:

[0069] Construct a single-beacon ranging reference map and transform the single-beacon ranging values ​​into a two-dimensional plane;

[0070] Based on the transformed single beacon ranging values, a distance circle centered on the single beacon is constructed, and the displacement vector is calculated by combining the INS values ​​at previous and subsequent time points to build an objective function model;

[0071] Based on the objective function model, a localization solution is determined. When two localization solutions exist, the optimal point is selected using the Mahalanobis distance cost function to obtain the localization location.

[0072] Furthermore, constructing the single beacon ranging reference map includes:

[0073] A rectangular coordinate system is established with the beacon position as the origin. Measurement points are selected in the rectangular coordinate system, and the Euclidean distance from the measurement point position to the beacon position is obtained.

[0074] The depth difference is obtained based on the water depth at the beacon location and the water depth at the measuring point location;

[0075] Based on the difference between the Euclidean distance and the depth, the ranging information of the reference surface is obtained, and the single beacon ranging reference map is constructed.

[0076] Specifically, in this embodiment, constructing a single beacon ranging reference map to convert ranging values ​​to a two-dimensional plane includes:

[0077] Single-beacon distance circle positioning uses concentric circles centered on the coordinates of a single beacon. Considering that single-beacon positioning is generally calculated based on a two-dimensional plane, this paper unifies the distance measurement information of all distance measurement points to the same reference plane. (Plane) Then perform distance circle positioning, with the single beacon position as the coordinate origin, as shown in the distance measurement reference diagram. Figure 1 As shown.

[0078] Set the distance measuring point The position coordinates are Unify it to The horizontal distance of the plane is:

[0079] (1);

[0080] In the formula, The slant range is obtained from single-beacon underwater acoustic ranging. The depth difference between the beacon and the underwater body can be obtained using a high-precision pressure depth measurement sensor. Equation (1) is the result of any measurement point on the track. exist Projection on a plane The Euclidean distance to the beacon coordinates is used to convert all subsequent actual distance measurements into two-dimensional planar distance measurements before calculation.

[0081] Furthermore, based on the transformed single beacon ranging values, a distance circle centered on the single beacon is constructed, and the displacement vector is calculated by combining the INS values ​​at previous and subsequent time points. The objective function model includes:

[0082] Obtain the AUV after transformation to a two-dimensional plane at the current time. Compared to the previous moment Single beacon ranging value;

[0083] Based on the position of the INS on the AUV at the current moment and the previous moment, the dead reckoning displacement vector is obtained;

[0084] Construct a first distance circle and a second distance circle at the previous and current times, respectively, with the single beacon as the center and the distance between the AUV and the single beacon as the radius;

[0085] The first distance circle is translated according to the dead reckoning displacement vector;

[0086] Based on the coordinates of the intersection point generated by the first distance circle and the second distance circle after translation, a target function model is constructed.

[0087] Specifically, in this embodiment, the modeling of the single beacon distance circle positioning algorithm includes:

[0088] like Figure 2 As shown, suppose the AUV is at the current time Compared to the previous moment The position coordinates are respectively , The distance measurement values ​​of the single beacon and the single beacon are respectively , Equation (1) is used to unify the single beacon ranging value to the corresponding ranging value in the two-dimensional plane. , Then use the distance measurement value to perform the calculation. For the previous moment up to the current moment The dead reckoning displacement vector, by... Distance vector calculated from the distance circle translation dead reckoning at a given time. ,and The distance circle at a given time produces two intersection points. By further constraining and judging these intersection points and selecting the optimal point, the corrected positioning position of the INS at this time can be obtained.

[0089] Let the coordinates of the acoustic beacon be... ,but The coordinates of the intersection point generated by the time-distance circles can be expressed by the following formula:

[0090] (2);

[0091] in, , , , , .

[0092] Equation (2) is the objective function for the coordinates of the intersection point of the single beacon distance circle, and its solution includes the following three cases:

[0093] (3);

[0094] (1) When the distance circles are disjoint, there is no intersection point, and the distance circle positioning algorithm fails at that moment. When there is no intersection, the algorithm uses the distance circle positioning coordinates from the previous moment and shifts them to the dead reckoning displacement vector from the previous moment to the current moment to obtain the positioning result for that moment. ;

[0095] (2) When the distance circles are tangent, they intersect at a point, which is the positioning point;

[0096] (3) When the distance circles intersect, there are two intersection points. At this time, it is necessary to perform constraint judgment on the two intersection points to obtain the optimal positioning solution.

[0097] Due to the continuous underwater movement of the AUV and the minimal positional drift of the INS within a short period, cases where single-beacon range-circle positioning has no intersection or only one intersection are very rare. This algorithm typically includes two circular intersections within the effective range of the beacon; therefore, it is necessary to select the optimal point using a cost function to obtain the positioning location. This invention constructs a Mahalanobis distance cost function to select the optimal point, which can then be used to correct the INS position in local sea areas.

[0098] Furthermore, the selection of the optimal point based on the Mahalanobis distance constraint includes:

[0099] Based on the above analysis, when there are two intersection points in the single beacon distance circle, this paper proposes a cost function based on Mahalanobis distance as a constraint for the optimal point. This cost function, through uncertainty propagation theory, integrates INS cumulative error and acoustic ranging error, providing a statistically optimal metric for selecting positioning trajectory points. The following details the construction process of the Mahalanobis distance cost function and its final mathematical analytical form.

[0100] pass Position estimation of AUV at any time From Instagram arrive The displacement vector at time t is used to calculate the estimated position at the current time. :

[0101] (4);

[0102] The covariance matrix of the above equation is:

[0103] (5);

[0104] in, yes The posterior estimation error covariance matrix at time 1. INS displacement vector The error covariance matrix.

[0105] Location It is obtained by solving the nonlinear function of equation (2), and its uncertainty comes from noisy input. , and beacon location It can be considered to have no error.

[0106] According to the first-order error propagation theory, AUVs in The covariance matrix of the location at time step is:

[0107] (6);

[0108] (7);

[0109] (8);

[0110] In the formula, It is the covariance matrix of the input parameters, which is a diagonal matrix since the input errors are uncorrelated; It is the Jacobian matrix of equation (2), which represents the sensitivity of the positioning position to each noisy input parameter.

[0111] Mahalanobis distance measures a random vector Relative to its expectations Covariance The degree of deviation is defined as:

[0112] (9);

[0113] Its square Obeying the degree of freedom The chi-square distribution is used. Therefore, it is necessary to measure the estimated location of INS. with location The consistency between them. Considering that both have uncertainty, the most suitable measure is the Mahalanobis distance of their differences, the difference vector. The covariance is the sum of the covariances of the two:

[0114] (10);

[0115] Therefore, the difference vector The squared Mahalanobis distance is defined as from Time position shift The cost of constantly determining location is as follows:

[0116] (11);

[0117] The above formula is the Mahalanobis distance cost function for determining the intersection of two positioning points. The cost of the intersection points is calculated using the Mahalanobis distance cost function, and the point with the minimum cost is the optimal positioning point.

[0118] In summary, the entire process of the single-beacon range-circle positioning-assisted INS navigation algorithm based on Mahalanobis distance constraints is as follows: Figure 3 As shown:

[0119] Experimental verification and analysis:

[0120] To verify the effectiveness of the algorithm, this invention conducts lake-based experimental verification.

[0121] This invention selected a lake area to conduct a surface experiment on a single-beacon ranging circular positioning algorithm. The track reference was obtained by high-precision GPS measurement and can be regarded as a real track. The installation of each device at the test vessel and beacon location is as follows: Figure 4 , Figure 5 As shown. Let the starting point of the experimental vessel be (0,0,0), and a single beacon be fixed on the lake bottom with coordinates (103.26m, -3773.14m, -2.5m). Since the beacon's deployment depth is known and measured by a high-precision depth sensor, the actual distance between the experimental vessel and the beacon can be converted to a two-dimensional plane for calculation using a distance reference map.

[0122] Table 1 shows the parameter indicators of each experimental device. Figure 6 The data includes the INS track, GPS track, and single beacon position during the experiment. The entire voyage of the test vessel lasted approximately 24 hours. After the inertial navigation system was activated, the vessel docked at the pier for 11 hours before commencing its voyage. Figure 6It can be seen that INS calculations will produce positional errors during docking. The underwater acoustic rangefinder uses the GPS PPS signal switched to synchronous mode for distance measurement, with a measurement period of 1 second. Due to the limited depth of the lake test area, the depth difference between the underwater end and the surface end of the rangefinder is small, resulting in the farthest distance measured by the rangefinder being approximately 758m, and the effective distance measurement period being 44005s~44665s.

[0123] Table 1 Parameters of each test device

[0124]

[0125] Experimental analysis revealed that the sampling interval of the ranging value has a certain impact on the single beacon distance circle algorithm. The experimental results are shown in Table 2. Table 2 shows that the distance circle algorithm has the highest positioning accuracy and the smallest standard deviation when the sampling interval is about 60s. Figure 7 The 9th location point in the positioning area is displayed.

[0126] Table 2. Statistics on Sampling Interval and Algorithm Positioning Error

[0127]

[0128] By setting an appropriate sampling interval through experiments, the final positioning trajectory of the distance circle was obtained as follows: Figure 8 As shown, within the single beacon range circle positioning area, the positional errors of the positioning trajectory in the east and north directions are respectively as follows: Figure 9 , Figure 10 Radial position error such as Figure 11 .

[0129] Table 3 Comparison of INS and Positioning Errors (Unit: meters)

[0130]

[0131] Table 3 shows the error and standard deviation statistics of INS and positioning trajectory within the positioning area. As can be seen from Table 3, when the mean radial error of INS is 1424.62m, the position error after single beacon distance circle positioning correction is 48.94m, and the positioning accuracy is improved by 96.56%.

[0132] Analysis of the lake trial experiments shows that, by selecting an appropriate sampling interval, the single-beacon range circle positioning method can achieve high-precision local positioning within the effective range of the beacon. Theoretically, range circle positioning can also result in cases with no intersection points or only one intersection point. In the aforementioned lake trial experiments, the range circles all had two intersection points, so cases with no intersection points or only one intersection point are very rare. If such cases occur in practical applications, they can be addressed by... Figure 2 In terms of process handling, this situation has a negligible impact on navigation.

[0133] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made by those skilled in the art to the technical solutions of the present invention without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A single-beacon range-circle positioning-assisted INS navigation method based on Mahalanobis distance constraints, characterized in that, include: Construct a single beacon ranging reference map and transform the single beacon ranging values ​​into a two-dimensional plane; Based on the transformed single beacon ranging value, a distance circle centered on the single beacon is constructed, and the displacement vector is calculated by combining the INS values ​​at previous and subsequent time points to construct the objective function model; Based on the objective function model, a localization solution is determined. When there are two localization solutions, the Mahalanobis distance cost function is used to select the optimal point to obtain the localization location. Based on the transformed single-beacon ranging values, a distance circle centered on the single beacon is constructed. The displacement vector is then calculated using the INS values ​​at preceding and following times. The objective function model includes: Obtain the AUV after transformation to a two-dimensional plane at the current time. Compared to the previous moment Single beacon ranging value; Based on the position of the INS on the AUV at the current moment and the previous moment, the dead reckoning displacement vector is obtained; Construct a first distance circle and a second distance circle at the previous and current times, respectively, with the single beacon as the center and the distance between the AUV and the single beacon as the radius; The first distance circle is translated according to the dead reckoning displacement vector; Based on the coordinates of the intersection point generated by the translated first distance circle and the second distance circle, a target function model is constructed; The objective function model is as follows: ； ； ； ； ； ； in, For the previous moment up to the current moment The dead reckoning displacement vector, , To unify the single beacon ranging values ​​from the previous moment and the current moment into the corresponding ranging values ​​in a two-dimensional plane, Let be the objective function. These are the position coordinates of a single beacon. for The coordinates of the position after the translation from the center of the circle. It is half the length of the chord connecting the intersection points of the circles. For the previous moment up to the current moment The displacement vector magnitude, , Each is the previous moment up to the current moment of x axis, y Displacement vector in the axial direction, , beacons x , y The position coordinates of the axis The distance from the beacon location to the midpoint of the line connecting the intersection points of the circles; Determining the localization solution based on the objective function model includes: When two distance circles are separate, there is no intersection point, and the distance circle positioning algorithm fails at that moment. When two distance circles are tangent, they intersect at a point, which is the location. When two distance circles intersect, there are two intersection points. At this point, it is necessary to perform constraint judgment on the two intersection points to obtain the optimal positioning solution. When two localization solutions exist, the optimal point is selected using the Mahalanobis distance cost function to obtain the localization location, including: pass Position estimation of AUV at any time From Instagram arrive The displacement vector at time t is obtained. Estimated position of time : ； Estimated position The covariance matrix is: ； in, yes The posterior estimation error covariance matrix at time 1. INS displacement vector The error covariance matrix; AUV in Location in real time The covariance matrix is: ； in, It is the covariance matrix of the input parameters. It is a Jacobian matrix based on the formula for the coordinates of the intersection points generated by the translated first and second distance circles, representing the sensitivity of the positioning position to each noisy input parameter. for Location in real time x The variance of coordinates for Location in real time x coordinates and y Covariance between coordinates for Location in real time y The variance of the coordinates; Based on estimated location with location The Mahalanobis distance, representing the difference between two values, is used to measure the consistency between them; the difference vector... The covariance is the sum of the covariances of the two: ； Difference vector The squared Mahalanobis distance is defined as from Time position shift The cost of constantly determining the location is as follows: ； This refers to the Mahalanobis distance cost function used to determine the intersection of two positioning points. The cost of the intersection of the distance circles is calculated using the Mahalanobis distance cost function, and the point with the minimum cost is the optimal positioning point.

2. The single-beacon range circle positioning-assisted INS navigation method based on Mahalanobis distance constraints according to claim 1, characterized in that, Constructing the single beacon ranging reference map includes: Establish a rectangular coordinate system with the beacon position as the origin, select a measuring point in the rectangular coordinate system, and obtain the Euclidean distance from the measuring point position to the beacon position; The depth difference is obtained based on the water depth at the beacon location and the water depth at the measuring point location; Based on the difference between the Euclidean distance and the depth, the ranging information of the reference surface is obtained, and the single beacon ranging reference map is constructed.

3. The single-beacon range circle positioning-assisted INS navigation method based on Mahalanobis distance constraints according to claim 1, characterized in that, Methods for converting single beacon ranging values ​​to a two-dimensional plane include: ； in, The slant range is obtained from single-beacon underwater acoustic ranging. The depth difference between the beacon and the underwater body, Let the coordinates of the distance measurement point be set.

4. The single-beacon range circle positioning-assisted INS navigation method based on Mahalanobis distance constraints according to claim 1, characterized in that, The localized solution of the objective function model includes the following three cases: ； in, If the set is empty, then the distance circles have no intersection. The coordinates of the position 1 distance from the intersection point of the circles are: The coordinates are the distance from the intersection point of the circles, 2.

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