Accuracy evaluation method for underwater acoustic positioning system of irregular offshore station distribution

Abstract

The invention belongs to the field of underwater target navigation and positioning, and discloses a precision evaluation method of an underwater acoustic positioning system for irregular offshore station distribution, which comprises the following specific steps of: 1) setting initial conditions, selecting a single-point, track or region evaluation mode, and establishing theoretical values of array element positions of a target to be measured and a long baseline measurement array; 2) collecting and analyzing field information, estimating a station address error and a distance measurement error in combination with measurement equipment and offshore collected information, and obtaining an estimated maximum operating distance through a pull distance test; and 3) precision simulation evaluation: generating a site position with an error and a slope distance measurement value by using a Monte Carlo method, carrying out error transfer simulation and parameter solving under the action distance constraint condition, and carrying out precision estimation by taking a root-mean-square error as an index. The method has the advantages of being good in adaptability to complex and irregular offshore station distribution, accurate in error transmission and reproduction, wide in application scene and the like, and can provide technical support for underwater target high-quality positioning measurement.

Description

Technical Field

[0001] The present invention belongs to the field of underwater target navigation and positioning, and particularly relates to a method for evaluating the accuracy of an underwater acoustic positioning system with irregular sea stations. It is a technology for evaluating the positioning accuracy under various working conditions such as single point, trajectory, and area when using a long baseline underwater acoustic positioning system to measure underwater targets in a marine environment. Background Art

[0002] The long baseline underwater acoustic positioning system is a commonly used technical means for underwater target positioning, with advantages such as supporting redundant station design, high positioning accuracy, and robust measurement performance. However, it also has problems such as being highly restricted by the marine environment and the measurement performance being highly dependent on the station layout geometry. In engineering applications, it is very necessary to adopt a reliable accuracy evaluation technology to avoid the risk of measurement failure and ensure the high-quality implementation of measurement work.

[0003] During offshore operations, the array elements of the long baseline underwater acoustic system are pre-deployed in the operation sea area in a certain geometric configuration. After receiving the positioning acoustic signals provided by the target to be measured (which needs to be equipped with a cooperative acoustic beacon) and performing signal processing, measurement elements such as slant range and radial velocity are obtained. Then, the data information of each array element is aggregated to the data processing platform to complete the intersection calculation, and finally the positioning result is obtained. In the actual operation process, the achievable positioning accuracy is often not the calibration accuracy of the equipment when it leaves the factory. The limiting factors of the on-site environment and working conditions must be considered. When the working conditions are complex and the environment is harsh, it may lead to a decrease in accuracy, or even the situation where the target cannot be measured and the data is unavailable. In order to make the underwater acoustic positioning system reach the performance that meets the engineering requirements, it is necessary to pre-evaluate the measurement performance and accuracy in combination with the environmental characteristics of the operation sea area and the working conditions used.

[0004] There are mainly two aspects that affect the performance of the long baseline underwater acoustic positioning system in marine applications: one is the influence of the geometric station layout. There are usually certain undulations in the seabed topography of the operation sea area. When there are various types of array elements on the water surface and underwater, its geometric configuration may show obvious irregularities, which has an important impact on the positioning characteristics. Especially for the measurement of vertically moving targets, when using a geometric configuration with coplanar array elements, the vertical error will increase rapidly as the baseline increases; the other is the influence of the underwater acoustic channel. The sound speed gradient varies significantly in different sea areas, and even in the same sea area, there are often large seasonal changes. Especially in shallow sea areas, the strong attenuation acoustic channel under the thermocline condition and the surface channel under the mixed layer condition can cause the operating range of the underwater acoustic positioning system to differ by several times.

[0005] In recent years, with the continuous progress of underwater acoustic technology, technologies such as environment-adaptive underwater acoustic positioning signal processing, high-precision measurement, and precise calibration of underwater measurement stations have been successively developed, greatly improving the performance of underwater acoustic positioning systems. However, the complexity of environmental changes and the diversity of offshore working conditions cannot be solved solely by underwater acoustic technology. Especially in the case of irregular offshore station layout, conventional theoretical methods are difficult to effectively estimate the accuracy of underwater acoustic positioning systems. It is necessary to develop an accuracy evaluation technology for underwater acoustic positioning systems that meets offshore engineering applications by combining numerical simulation technology and on-site observation information, providing technical support for high-quality positioning measurement of underwater targets. Summary of the Invention

[0006] Aiming at the problems that the measurement performance of long baseline underwater acoustic positioning systems under irregular station layout geometries is strongly correlated with the station layout geometry and is greatly affected by the environment, the present invention proposes an accuracy evaluation method for long baseline underwater acoustic positioning systems under irregular offshore station layout conditions. It uniformly models underwater targets and measurement arrays under irregular station layout geometries, estimates site errors and ranging errors by combining external field information, then realizes error propagation simulation through the Monte Carlo method, and obtains the accuracy evaluation result with the root mean square error as the index. This method provides various application modes such as single point, trajectory, and area working conditions, has the characteristics of good adaptability to complex irregular offshore station layouts, accurate reproduction of error propagation, and wide application scenarios. It solves the multi-dimensional positioning accuracy evaluation problem of long baseline underwater acoustic positioning systems under irregular station layout geometries, and has certain reference significance for pre-detecting the measurement performance and defects of underwater acoustic positioning systems when used under different geometric station layout conditions, and then carrying out optimized design and effectiveness evaluation, and can provide technical support for high-quality positioning measurement of underwater targets.

[0007] The technical solution of the present invention is as follows:

[0008] An accuracy evaluation method for an underwater acoustic positioning system with irregular offshore station layout. First, set the initial conditions. For underwater targets in different states, select one of the single point evaluation mode, trajectory evaluation mode, or area evaluation mode, and establish the true value of the simulation model and the theoretical value of the positions of the elements of the long baseline measurement array. Secondly, carry out on-site information collection and analysis. Combine the measurement equipment and test information under offshore conditions to estimate the site error and ranging error, and obtain the estimated maximum operating distance through a ranging test. Thirdly, carry out accuracy simulation evaluation. Simulate and generate site positions and slant range measurement values with errors according to the theoretical values, and use the nonlinear least squares method for iterative calculation and solution to obtain large sample simulated measurement values, and then conduct statistical calculations with the root mean square error (RMSE) as the index to obtain the accuracy evaluation results in the three modes of single point evaluation, trajectory evaluation, or area evaluation.

[0009] The specific steps are as follows:

[0010] Step 1: Set the initial conditions

[0011] Set the elevation zero point where the initial position of the target theory is located as the origin O, establish a survey station coordinate system, with the OX axis pointing east, the OY axis pointing north, and the OZ axis vertically upward.

[0012] Step (1.1) Establish an evaluation model

[0013] For underwater targets in different states, three evaluation models are provided:

[0014] Single-point evaluation model: Evaluate the underwater point target, whose position is fixed and the target parameters are constant values, expressed as X in the measurement coordinate system P =[x P , y P , z P ;

[0015] Trajectory evaluation model: Evaluate the underwater target trajectory, whose position changes with time and the target parameters are sequences of positions changing with time, expressed as where l represents the time node serial number of the trajectory sequence, l = 1, 2,..., N L , N L is the number of sequence samples;

[0016] Area evaluation model: Evaluate the coverage area where the underwater target activities are located. Its position is allocated in a grid according to the operation sea area, but does not change with time. The target parameters are matrices, expressed as where m and n respectively represent the grid point labels in the horizontal and vertical directions, m = 1, 2,..., M A , n = 1, 2,..., N A , M A , N A respectively represent the number of grids in the horizontal and vertical directions.

[0017] Step (1.2) Establish measurement geometry

[0018] According to the layout design of the stations, the long baseline measurement array with a certain geometric configuration can be expressed in the measurement coordinate system as where j is the element serial number, j = 1, 2,..., N S , N S is the total number of elements, N S ≥ 3; the geometric configuration of the measurement array is defined by the elements and constrained by the effective action distance of each element, while the measurement geometry is jointly constituted by the relative positions of the target and the measurement array.

[0019] Step Two: On-site information collection and analysis

[0020] Step (2.1) Estimate the station site error

[0021] Under offshore operation conditions, for the array elements carried by the surface platform, they are installed in a strapdown manner with the satellite navigation and positioning equipment, and the station error is directly determined by the accuracy of the satellite navigation and positioning equipment; for the array elements carried by the underwater fixed duty platform, they are calibrated on-site using shipborne calibration equipment, and the station error is determined by the accuracy of the shipborne calibration equipment. The station error of each array element in the three directions of X, Y, and Z is expressed as

[0022] Step (2.2) Estimate the ranging error

[0023] Under offshore operation conditions, a set of slant range observation samples is collected under typical measurement distance conditions using the positioning acoustic beacon and receiving array elements of the hydroacoustic positioning system carried by an underwater fixed standby platform or a surface ship platform. N R is the number of sequence samples; at the same time, according to the position information of the underwater fixed guard platform and the surface ship platform, the calculated samples with the same sequence length and time synchronization with the slant range observation samples are obtained. in is the acoustic beacon position at the kth moment, is the receiving array element position at the kth moment; according to the slant range observation sample and calculate samples The residual estimate of the ranging error σ R , the calculation formula is

[0024]

[0025] Step (2.3) Estimate the maximum range

[0026] The distance test was carried out using the surface ship platform and the underwater fixed guard platform to obtain the farthest distance at which the receiving array element can robustly detect the positioning sound signal and obtain correct ranging information, which is the maximum effective distance of the array element, denoted as R. max .

[0027] Step 3: Accuracy simulation evaluation

[0028] Step (3.1) Simulate and generate station location and slant range measurements

[0029] In the simulation scenario, the theoretical value of the target position of the single-point evaluation, trajectory evaluation or regional evaluation mode is taken as the true value, and the Monte Carlo method is used for large sample sampling. The site error and ranging error are substituted in a normal distribution manner to obtain the simulated site location and slant range measurement value.

[0030] First, according to the true value of the station location According to normal distribution Generate a set of samples of Nsam The observed site location value sequence, where the site location of the k-th sampling and the j-th array element is expressed as

[0031] Secondly, according to the normal distribution Generate a set of observed slant range sequences with a sample number of N sam where is the current target position X t =[x t , y t , z t and the direct slant range value between the array element j, X t represents a single-point evaluation mode (X P ), a trajectory evaluation mode or a regional evaluation mode in a single value.

[0032] Step (3.2) Effective array element screening

[0033] Using as the criterion to determine the effective array elements, the number of effective array elements is obtained as N V , N V ≤N s ; For the k-th sampling, based on the observed site location value sequence , the effective array elements are reorganized according to the sequence of serial numbers j = 1, 2,..., N V to obtain the observed site location value sequence with effective action distance constraints

[0034]

[0035] Step (3.3) Positioning solution

[0036] For the k-th sampling, define the measured target position at the i-th iteration as The measured target position at the (i + 1)-th iteration is N ite is the preset maximum number of iterations. According to the prior information, set the initial estimation error of the target position and generate a set of observed slant range sequences with a sample number of N sam which is the initial value of the target position for the k-th sampling.

[0037] For the k-th sampling, from and ​​​Drive, and iterative calculation is performed using the non - linear least - squares method. The recursive calculation formula is

[0038]

[0039] Where:

[0040]

[0041] When meets the preset accuracy, stop the iteration. At this time, is the measured value of the k - th sampling, denoted as

[0042] Step (3.4) Error analysis and evaluation

[0043] For the single - point evaluation mode, only calculate the true value of a single position; for the trajectory evaluation mode or the area evaluation mode, repeat the steps described in step (3.2) and step (3.3) for point - by - point calculation. After completing all N sam times of sampling calculations, statistical calculations are performed using the root - mean - square error as an index to obtain the accuracy evaluation results in the X, Y, and Z directions under different modes. The calculation formula is

[0044]

[0045] Advantages of the present invention:

[0046] This method fully considers the possible impacts of environments such as irregular station - layout geometry and complex underwater acoustic channels on the measurement performance, and realizes the accurate simulation of the random error transfer process. This method provides diverse application modes such as single - point, trajectory, and area working conditions, and has the characteristics of good adaptability to complex and irregular station - layouts at sea, accurate reproduction of error transfer, and wide application scenarios. It solves the problem of evaluating the positioning accuracy of diverse underwater targets such as single - point, trajectory, and area under irregular station - layout geometric conditions, helps engineering personnel to pre - master the measurement performance and defects under different geometric station - layout conditions, and then carry out optimization design and accuracy evaluation, avoid the risk of ineffective operations at sea, and thus achieve high - quality positioning measurement of underwater targets under marine conditions, providing technical support for high - quality positioning measurement of underwater targets. Brief description of the drawings

[0047] Figure 1 is the basic process of the method proposed by the present invention.

[0048] Figures 2(a) to 2(c) is the distribution diagram of the target and the measurement stations, where Fig. 2(a) represents a point target, Fig. 2(b) represents a trajectory target, and Fig. 2(c) represents an area target.

[0049] Figures 3(a) and 3(b) are schematic diagrams of on-site ranging sample collection and ranging test. Among them, Figure 3(a) is the ranging sample collection scenario, and Figure 3(b) is the ranging test scenario.

[0050] Figure 4 It is a schematic diagram of single-point measurement for underwater positioning of point targets.

[0051] Figures 5(a) and 5(b) are the underwater target positioning results in the single-point evaluation mode. Among them, Figure 5(a) is the three-dimensional positioning result, and Figure 5(b) is the positioning result on the XOY plane.

[0052] Figures 6(a) and 6(b) are respectively the position sequence and positioning error curve of an underwater moving vehicle in the trajectory evaluation mode. Among them, Figure 6(a) is the position sequence in the X, Y, and Z directions. The solid line represents the true value, and the dashed line represents the measured value. Figure 6(b) is the RMSE sequence in the X, Y, and Z directions.

[0053] Figures 7(a), 7(b), and 7(c) are respectively the positioning error distribution diagrams in the X, Y, and Z directions in the area evaluation mode, presented in the form of contour lines. The scale unit of the contour line is m. Detailed implementation manners

[0054] The following further describes the specific implementation manners of the method proposed by the present invention in combination with the technical solutions and the drawings.

[0055] A method for evaluating the accuracy of an underwater acoustic positioning system with irregular sea station layout according to the present invention mainly includes the following steps Figure 1 as shown below.

[0056] (1) Set initial conditions

[0057] Consider the following working conditions: Using ocean buoys and submersible buoys as the carrier platforms for the elements of the long baseline underwater acoustic positioning system, positioning and measuring the on-duty platforms or underwater vehicles (equipped with positioning acoustic beacons) within a limited area, and respectively using three evaluation modes of single-point evaluation, trajectory evaluation, and area evaluation for accuracy evaluation.

[0058] Combined with different evaluation modes, establish the measurement geometry:

[0059] The ocean buoys are arranged in a regular quadrilateral pattern, and the elements are evenly distributed on a circle with a radius of 200 m. The baseline direction is parallel or perpendicular to the X-axis, and the depth of the receiving element is 5 m. The positions of each element are (j = 1, 2, 3, 4); The submersible buoys are arranged in a regular quadrilateral pattern, and the elements are evenly distributed on a circle with a radius of 200 m. The baseline direction forms a 45° angle with the X-axis, and the depth of the receiving element is 100 m. The positions of each element are

[0060]

[0061] For the single-point evaluation mode (see Fig. 2(a)), the positioning performance of the operating platform in the on-duty state is mainly examined, and the theoretical value (true value) of the position of the target to be measured is set to [0, 0, -50] (m).

[0062] For the trajectory evaluation mode (see Fig. 2(b)), the positioning performance of the underwater vehicle in the maneuvering state is mainly examined, and the theoretical value (true value) of the trajectory of the target to be measured is set to a trajectory moving in the positive X-axis direction, with an average depth of 50 m and a navigation distance of 30 m. The number of samples in the data sequence is 101.

[0063] For the area evaluation mode (see Fig. 2(c)), the positioning performance of the underwater vehicle at different positions within an 800 m × 800 m area is mainly examined. The theoretical value (true value) of the target grid points is set such that the range along the X-axis is from -800 m to 800 m, with an interval of 20 m, and the range along the Y-axis is from -800 m to 800 m, with an interval of 20 m. The number of grid points is 41 × 41.

[0064] (2) Combine on-site calibration and testing to obtain simulation parameters

[0065] For the site error of the ocean buoy, it is given by the accuracy of the strap-down installed satellite navigation and positioning equipment. The site errors in the X, Y, and Z directions are respectively For the site error of the submersible buoy, it is given by the accuracy of the shipborne calibration equipment. The site errors in the X, Y, and Z directions are respectively σ Sx = 0.3 (m), σ Sy = 0.3 (m), σ Sz = 0.3 (m), j = 5, 6, 7, 8.

[0066] In the operation sea area, use the 1# surface ship and the 2# surface ship to carry the receiving array element and the positioning acoustic beacon respectively. The relative distance between the two ships is kept within the range of 300 - 500 m, and their positions are given by the strap-down satellite navigation and positioning equipment. Conduct on-site ranging sample collection (see Fig. 3(a)). According to 300 groups of ranging samples, obtain the ranging error σ R = 0.5 (m).

[0067] In the operation sea area, use the 1# surface ship and the 2# surface ship to carry the receiving array element and the positioning acoustic beacon respectively. Their positions are given by the strap-down satellite navigation and positioning equipment. Through the distance pulling test of the two ships (see Fig. 3(b)), obtain the maximum operating distance R at which the receiving array element can robustly detect the positioning acoustic signal and obtain high-precision ranging information max = 400 (m).

[0068] (3) Precision simulation evaluation

[0069] According to the true value of the station layout position According to the normal distribution Generate a sequence of observed site location values with a sample size of N sam

[0070] For the single-point evaluation mode, the target true value is X t = X P According to the normal distribution Generate a sequence of observed slant ranges with a sample size of N sam For the k-th sampling, use as the criterion to determine the effective array elements, and calculate the measurement value of this sampling according to the method described in (3.3) in the invention content (See the measurement schematic diagram in Figure 4 ) sam After completing N

[0071] For the trajectory evaluation mode, the target true value is At each sample point, according to the normal distribution Generate a sequence of observed slant ranges with a sample size of N sam For the k-th sampling, use as the criterion to determine the effective array elements, and calculate the measurement value of this sampling according to the method described in step (3.3) in the invention content After completing N sam times of sampling calculations, according to the method described in step (3.4) in the invention content, obtain the RMSE calculation values in the X, Y, and Z directions of a single grid point; after traversing all sample points of the sequence, obtain the error distribution results in the X, Y, and Z directions, as shown in Fig. 6(a) and Fig. 6(b), and the average errors of the entire sequence in the X, Y, and Z directions are 0.29m, 0.30m, and 0.87m respectively

[0072] For the area evaluation mode, the target true value is At each grid point, according to the normal distribution Generate a sequence of observed slant ranges with a sample size of N sam For the k-th sampling, use as the criterion to determine the effective array elements, and calculate the measurement value of this sampling according to the method described in step (3.3) in the invention content After completing N sam ​​​​After the secondary sampling calculation, the RMSE calculated values in the X, Y, and Z directions of a single grid point are obtained according to the method described in step (3.4) of the invention content. After traversing all grid points, the error distribution results in the X, Y, and Z directions of the entire area are obtained, as shown in FIGS. 7(a), 7(b), and 7(c).

[0073] Thus, auxiliary decision-making information can be provided for relevant applications based on the results obtained from different evaluation modes. According to the single-point evaluation results, the accurate position and error of the underwater manned platform can be determined to support operations such as regular inspection, maintenance, and recovery of underwater facilities; according to the trajectory evaluation results, the trajectory and error range of the underwater vehicle can be identified to support the design, implementation of test measurements, and evaluation of the motion performance of the vehicle; according to the area evaluation results, the positioning error distribution characteristics of the measurement array coverage area can be provided to support the optimization design, plan formulation, and operation implementation of the underwater acoustic positioning system at sea.

Claims

1. A method for evaluating the accuracy of an underwater acoustic positioning system with irregular offshore station layout, characterized in that, The specific steps are as follows: Step 1: Set initial conditions Set the elevation zero point where the initial position of the target theory is located as the origin O, establish a survey station coordinate system, with the OX axis pointing east, the OY axis pointing north, and the OZ axis vertically upward; Step (1.1) Establish an evaluation mode For underwater targets in different states, three evaluation modes are provided: Single-point evaluation mode: It is used to evaluate underwater point targets. Their positions are fixed, and the target parameters are constant values, which are represented as X in the measurement coordinate system P = [x P , y P , z P ; Trajectory evaluation mode: For evaluating the trajectory of an underwater target, whose position changes with time, and the target parameter is the sequence of positions over time, expressed in the measurement coordinate system as where l represents the time node serial number of the trajectory sequence, l = 1, 2, …, N L , N L is the number of sequence samples; Area evaluation mode: Evaluate the coverage area of underwater target activities. Its position is grid-allocated according to the operation sea area, but does not change with time. The target parameters are matrices and are represented in the measurement coordinate system as where m and n respectively represent the grid point labels in the horizontal and vertical directions, m = 1, 2, …, M A and n = 1, 2, …, N A , M A , N A respectively represent the number of grids in the horizontal and vertical directions; Step (1.2) Establish a measurement geometry According to the station layout design, a long baseline measurement array with a certain geometric configuration is represented in the measurement coordinate system as where j is the element number, j = 1, 2, …, N S , N S being the total number of elements, N S ≥ 3; the geometric configuration of the measurement array is defined by the elements and constrained by the effective action distance of each element, while the measurement geometry is jointly constituted by the relative positions of the target and the measurement array; Step 2: On-site information collection and analysis Step (2.1) Estimate the station site error Under offshore operation conditions, for the array elements carried by the surface platform, they are installed in a strapdown manner with the satellite navigation and positioning equipment, and the station error is directly determined by the accuracy of the satellite navigation and positioning equipment; for the array elements carried by the underwater fixed duty platform, they are calibrated on-site using shipborne calibration equipment, and the station error is determined by the accuracy of the shipborne calibration equipment. The station error of each array element in the three directions of X, Y, and Z is expressed as Step (2.2) Estimate the ranging error Under offshore operation conditions, a set of slant range observation samples is collected under the measurement distance conditions using the positioning acoustic beacon and receiving array elements of the hydroacoustic positioning system carried by an underwater fixed standby platform or a surface ship platform. k=1,2,…,N R , N R is the number of sequence samples; at the same time, according to the position information of the underwater fixed guard platform and the surface ship platform, the calculated samples with the same sequence length and time synchronization with the slant range observation samples are obtained. in is the acoustic beacon position at the kth moment, is the receiving array element position at the kth moment; according to the slant range observation sample and calculate samples The residual estimate of the ranging error σ R , the calculation formula is Step (2.3) Estimate the maximum operating distance The distance test was carried out using the surface ship platform and the underwater fixed guard platform to obtain the farthest distance at which the receiving array element can robustly detect the positioning sound signal and obtain correct ranging information, which is the maximum effective distance of the array element, denoted as R. max ; Step 3: Precision simulation evaluation Step (3.1) Simulate and generate the station site position and slant range measurement values In the simulation scenario, take the theoretical target position value of the single-point evaluation, trajectory evaluation or area evaluation mode as the true value, use the Monte Carlo method for large-sample sampling, substitute the station site error and ranging error in the form of a normal distribution, and obtain the simulated station site position and slant range measurement values; First, according to the true value of the station layout position generate a set of observed values of the station location with a sample size of N in accordance with the normal distribution sam where the observed value of the station location for the k-th sampling and array element j is expressed as k = 1, 2, …, N sam and j = 1, 2, …, N S ; Secondly, according to the normal distribution generate a set of observed slant range sequences with the number of samples being N sam , where j = 1, 2, …, N S , and is the slant range direct value between the current target position X t = [x t , y t , z t and the array element j; X t represents a single value in the single-point evaluation mode (X P ) defined in step (1.1), the trajectory evaluation mode or the area evaluation mode ; Step (3.2) Effective array element screening Taking as the criterion for determining effective array elements, the number of effective array elements is obtained as N V , N V ≤ N s ; For the k-th sampling, based on the observed value sequence at the site location, the effective array elements are reorganized according to the sequence of serial numbers j = 1, 2, …, N V to obtain the observed value sequence of the site location with effective action distance constraint k = 1, 2, …, N sam , j = 1, 2, …, N V ; Step (3.3) Positioning solution For the k-th sampling, define the measured value of the target position at the i-th iteration as the measured value of the target position at the (i + 1)-th iteration as i = 0, 1, 2, …, N ite , N ite is the preset maximum number of iterations; according to the prior information, set the initial estimation error of the target position and generate a set of observed slant range sequences with a sample number of N in accordance with the normal distribution sam which is the initial value of the target position for the k-th sampling; k = 1, 2, …, N sam ​ For the k-th sampling, driven by and , iterative calculation and solution are performed using the non-linear least squares method, and the recursive calculation formula is Where: When meets the preset accuracy, stop the iteration. At this time, is the measured value of the k-th sampling, denoted as Step (3.4) Error analysis and evaluation For the single-point evaluation mode, only the case of calculating the true value at a single position is required; for the trajectory evaluation mode or the area evaluation mode, repeat the steps described in step (3.2) and step (3.3) for point-by-point calculation; after completing all N sam times of sampling calculations, statistical calculations are performed using the root mean square error as an index to obtain the accuracy evaluation results in the X, Y, and Z directions under different modes. The calculation formula is

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