Space target positioning precision evaluation method and device based on observation matrix condition number

Abstract

The invention provides a space target positioning precision evaluation method and device based on an observation matrix condition number. The method comprises the following steps: obtaining a positioning method of distance measurement positioning and time difference positioning according to a TOA positioning system and a TDOA positioning system; analyzing the ranging positioning and the time difference positioning to obtain a measurement equation; determining a space target positioning error transfer model according to the measurement equation; and measuring the positioning precision of the space target by using the space target positioning error transfer model. Compared with a traditional geometric accuracy factor evaluation method, the evaluation method based on the condition number of the observation matrix has higher accuracy in the aspect of positioning accuracy representation, and the change trend of the positioning accuracy generated along with the movement of the target can be more intuitively explained. The method is of great significance in the aspects of high-precision measurement equipment identification, observation station layout geometric configuration optimization and the like.

Description

Technical Field

[0001] The present invention relates to the technical field of positioning, and in particular to a method and device for evaluating the positioning accuracy of space targets based on the condition number of an observation matrix. Background Art

[0002] In recent years, with the continuous development of wireless positioning technology, a relatively mature space target positioning system has gradually taken shape. Among them, common positioning systems include Time of Arrival (TOA) and Time Difference of Arrival (TDOA), and the corresponding positioning methods are ranging positioning methods and time difference positioning methods. The mathematical models of these positioning methods essentially intersect through the spatial geometry formed between the measurement stations and the space targets to solve the position information of the space targets.

[0003] Whether it is the ranging positioning method or the time difference positioning method, the positioning accuracy of space targets is affected by various factors such as time synchronization, station location error, and the geometric configuration of the measurement station layout. Among them, the quality of the geometric configuration of the measurement station layout is the key to the positioning accuracy of space targets. Currently, the commonly used Geometric Dilution of Precision (GDOP) is used to evaluate the quality of the geometric configuration of the measurement station layout, and based on this, the location selection of the measurement station layout is designed with the optimization objective function to achieve the optimization of the geometric configuration of the measurement station layout. At the same time, from the perspective of the positioning error transfer relationship, GDOP is the magnification factor between the observation error and the positioning error, so it is often used as a method for evaluating the positioning accuracy of space targets under the geometric configuration of this measurement station layout.

[0004] The advantage of the positioning accuracy evaluation method based on GDOP is that the solution is simple and fast. When analyzing the influence of errors on the positioning accuracy, GDOP is the upper bound of the magnification factor of the observation error, rather than the supremum. The value of the magnification factor of the observation error is less than GDOP, indicating that under the known geometric configuration of the measurement station layout, GDOP is a rough measure index for the positioning accuracy of space targets.

[0005] In summary, the following problems exist in the prior art: how to more accurately evaluate the positioning accuracy of space targets. Summary of the Invention

[0006] The purpose of the present invention is to solve the problem of how to more accurately evaluate the positioning accuracy of space targets.

[0007] To this end, on the one hand, an embodiment of the present invention provides a method for evaluating the positioning accuracy of space targets based on the condition number of an observation matrix, and the method includes the following steps:

[0008] Obtain the positioning methods of ranging positioning and time difference positioning according to the TOA positioning system and the TDOA positioning system;

[0009] Analyze the ranging positioning and the time difference positioning to obtain a measurement equation;

[0010] Determine a spatial target positioning error transfer model according to the measurement equation;

[0011] Use the spatial target positioning error transfer model to measure the positioning accuracy of the spatial target.

[0012] On the other hand, an embodiment of the present invention provides a spatial target positioning accuracy evaluation device based on the condition number of an observation matrix, including:

[0013] An initial unit for obtaining positioning methods for ranging positioning and time difference positioning according to the TOA positioning system and the TDOA positioning system;

[0014] An analysis unit for analyzing the ranging positioning and the time difference positioning to obtain a measurement equation;

[0015] A construction unit for determining a spatial target positioning error transfer model according to the measurement equation;

[0016] A positioning unit for using the spatial target positioning error transfer model to measure the positioning accuracy of the spatial target.

[0017] The above technical solution has the following beneficial effects: By using the properties of vector and matrix norms for contraction transformation in a normed linear space, the present invention derives three evaluation methods for the positioning accuracy of spatial targets, namely the relative condition number, the absolute condition number, and the volume condition number. Theoretical analysis shows that the spatial target positioning accuracy evaluation method based on the condition number of the observation matrix is the supremum of the measurement error magnification factor, and from the perspective of the solid geometry formed between the target and the measurement stations, this evaluation method can more intuitively explain the change process of the measurement station layout geometry during the movement of the target. Finally, through numerical comparison experiments, the rationality of the theoretical derivation is further verified. Compared with the traditional geometric dilution of precision evaluation method, the evaluation method based on the condition number of the observation matrix has higher accuracy in representing the positioning accuracy and can more intuitively explain the change trend of the positioning accuracy with the movement of the target. This is of great significance for the identification of high-precision measurement equipment and the optimization of the geometric configuration of measurement station layout. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 is a flowchart of the spatial target positioning accuracy evaluation method based on the condition number of the observation matrix provided by an embodiment of the present invention;

[0019] Figure 2 is a structural schematic diagram of the spatial target positioning accuracy evaluation device based on the condition number of the observation matrix provided by an embodiment of the present invention;

[0020] Figure 3It is a schematic diagram of the first ranging, velocity measurement and positioning system based on TOA technology;

[0021] Figure 4 It is a schematic diagram of the second ranging, velocity measurement and positioning system based on TOA technology;

[0022] Figure 5 It is a schematic diagram of the first time difference and frequency difference positioning system based on TDOA technology;

[0023] Figure 6 It is a schematic diagram of the second time difference and frequency difference positioning system based on TDOA technology;

[0024] Figure 7 It is a schematic diagram of the first spatial geometry formed by a space target and a measurement station;

[0025] Figure 8 It is a schematic diagram of the second spatial geometry formed by a space target and a measurement station;

[0026] Figure 9 It is a schematic diagram of GDOP and ACN of a space target under the first ranging and positioning system;

[0027] Figure 10 It is a schematic diagram of GDOP and ACN of a space target under the second ranging and positioning system;

[0028] Figure 11 It is a schematic diagram of GDOP and CAN of a space target under the first time difference positioning system;

[0029] Figure 12 It is a schematic diagram of GDOP and ACN of a space target under the second time difference positioning system;

[0030] Figure 13 It is a diagram of the relative magnitude relationship of GDOP and ACN in the first three-dimensional space;

[0031] Figure 14 It is a diagram of the relative magnitude relationship of GDOP and ACN in the second three-dimensional space;

[0032] Figure 15 It is a diagram of the changing trend of positioning accuracy during the movement of the first space target;

[0033] Figure 16 It is a diagram of the changing trend of positioning accuracy during the movement of the second space target. Specific implementation mode

[0034] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0035] In the embodiments of the present invention, as Figure 1 , a method for evaluating the positioning accuracy of a space target based on the condition number of an observation matrix is provided. The method includes the following steps:

[0036] S101: Obtain positioning methods for ranging positioning and time difference positioning according to the TOA positioning system and the TDOA positioning system;

[0037] Ranging positioning and time difference positioning are two different intersection positioning methods in the time measurement positioning system. Among them, the ranging positioning method requires three or more measurement stations to achieve space target positioning according to the spherical intersection principle. The schematic diagram of the principle is as Figure 3 , Figure 4 shown.

[0038] In the figure, each measurement station forms spheres with radii R1, R2, and R3 with itself as the origin and the space target. The three measurement stations respectively form three spheres with the space target, and intersect at a point in three-dimensional space, which is the position of the target.

[0039] In the ranging positioning system, the measurement equation is as follows:

[0040]

[0041] where X = [x, y, z] is the position information of the space target, that is, the parameter to be estimated for the space target position. X 0i = [x 0i , y 0i , z 0i is the coordinate of the i-th measurement station, and R i is the distance from the measurement station to the target to be measured.

[0042] S102: Analyze the ranging positioning and the time difference positioning to obtain a measurement equation;

[0043] S103: Determine the space target positioning error transfer model according to the measurement equation;

[0044] S104: Use the space target positioning error transfer model to measure the positioning accuracy of the space target. By using the properties of vector and matrix norms for contraction transformation in a normed linear space, three evaluation methods for the positioning accuracy of space targets, namely relative condition number, absolute condition number, and volume condition number, are derived.

[0045] The measurement equation includes:

[0046] Y = f R (X);

[0047] Wherein, is the ranging elements obtained from m different measurement stations; f R is the measurement function under the ranging positioning system and the time difference positioning system.

[0048] The spatial target positioning error transfer model is specifically:

[0049]

[0050] Where ΔR = [ΔDR1, ΔDR2, …, ΔDR m T is the observation error of m measurement stations; ΔX = [Δx, Δy, Δz] T is the spatial target position estimation error, ΔX station = [ΔX 01 ,..., ΔX 0m T is the station site error of m measurement stations, ΔX station0 = [Δx 00 , Δy 00 , Δz 00 T is the station site error of the main observation station.

[0051] Using the spatial target positioning error transfer model to measure the positioning accuracy of the spatial target specifically includes:

[0052] Determine the relative condition number, the absolute condition number, and the volume condition number;

[0053] Determine the supremum of the measurement error magnification factor;

[0054] Determine the positioning accuracy evaluation equation according to the relative condition number, the absolute condition number, the volume condition number, and the supremum of the measurement error magnification factor;

[0055] Use the positioning accuracy evaluation equation to measure the positioning accuracy of the spatial target.

[0056] The positioning accuracy evaluation equation is specifically:

[0057]

[0058] Wherein, is the error transfer matrix when it is the position estimation error, that is, the positioning error ΔX is obtained by performing matrix multiplication of the error transfer matrix with the measurement error and the station site error; λ1, λ2, λ3 are J​​​R All the singular values of the matrix; J R is the Jacobian matrix of the measurement equation.

[0059] Analysis of the positioning principle:

[0060] Range-based positioning and time-difference positioning are two different intersection positioning methods in the time measurement positioning system. Among them, the range-based positioning method requires three or more measurement stations. According to the spherical intersection principle, the spatial target positioning is realized. The schematic diagram of the principle is as Figure 3 、 Figure 4 shown.

[0061] In the figure, each measurement station forms spheres with radii R1, R2, and R3 with itself as the origin and the spatial target. The three measurement stations form three spheres with the spatial target respectively, and they intersect at a point in the three-dimensional space, which is the position of the target.

[0062] Under the range-based positioning system, the measurement equation is as follows:

[0063]

[0064] where X = [x, y, z] is the position information of the spatial target, that is, the parameter to be estimated for the spatial target position. X 0i =[x 0i , y 0i , z 0i is the coordinate of the i-th measurement station, and R i is the distance from the measurement station to the target to be measured.

[0065] The time-difference positioning method requires four or more measurement stations, and a master station must be set. According to the hyperboloid intersection principle, the spatial target positioning is realized. The schematic diagram of the principle is as Figure 5 、 Figure 6 shown.

[0066] In the figure, Station0 is the master measurement station, and the other measurement stations are auxiliary measurement stations. During the measurement process, the master measurement station forms hyperboloids with the auxiliary measurement stations respectively. In the three-dimensional space, multiple hyperboloids intersect, and the solutions that do not conform to the actual situation are excluded, which is the position of the target.

[0067] Under the time-difference positioning system, the measurement equation is as follows

[0068]

[0069] Compared with the range-based positioning system, the time-difference positioning system needs to set a master measurement station X 00 =[x 00 , y 00 , z 00 to form the hyperboloid intersection positioning.

[0070] For the convenience of subsequent discussion, whether it is the ranging and positioning system or the time difference positioning system, the mathematical models of both systems can be summarized as follows:

[0071] Y = f R (X) (4)

[0072] Among them, the ranging elements obtained from m different measurement stations are f R is the measurement function under the ranging and positioning system and the time difference positioning system.

[0073] Error propagation relationship:

[0074] For theoretical accuracy analysis, to give the theoretical calculation formula for the error of the target motion state quantity, it is necessary to perform error propagation analysis on equation (4). The basic method used is to perform a first-order Taylor expansion on the observation equations (2) and (3).

[0075] The spatial target positioning error transfer equation under the ranging and positioning system is:

[0076]

[0077] Among them, ΔR = [ΔR1, ΔR2,..., ΔR m T is the observation error of m measurement stations, J R is the Jacobian matrix of the measurement equation, ΔX = [Δx, Δy, Δz] T is the spatial target position estimation error, ΔX station = [ΔX 01 ,..., ΔX 0m T is the station site error of m measurement stations.

[0078] The spatial target positioning error transfer equation under the time difference positioning system is:

[0079]

[0080] Among them, ΔR = [ΔDR1, ΔDR2,..., ΔDR m T is the observation error of m measurement stations; J R is the Jacobian matrix of the measurement equation, ΔX = [Δx, Δy, Δz] T is the spatial target position estimation error, ΔX station = [ΔX 01 ,..., ΔX 0m T is the station site error of m measurement stations, ΔX station0 = [Δx 00 , Δy 00 , Δz​​​​00 T is the site error of the main observation station.

[0081] Further transform equations (5) and (6), and the position estimation error solved under the ranging and positioning system is

[0082]

[0083] The position estimation error of the space target solved under the time difference positioning system is:

[0084]

[0085] By observing equations (7) and (8), it can be found that the observation matrix (J T J) -1 J T is the magnification factor of the observation error, where the observation error includes random errors in the measurement process and systematic errors such as the site error of the measurement station. Since this observation matrix is determined by the geometric configuration of the measurement station layout, the accuracy of this positioning system can be measured by the observation matrix (J T J) -1 J T . Based on the above error propagation relationship, a method for evaluating the positioning accuracy of space targets can be given.

[0086] Method for evaluating the positioning accuracy of space targets:

[0087] In a normed linear space, use different norm definitions to calculate the matrix (J T J) -1 J T , deduce the commonly used accuracy evaluation methods based on GDOP, and the accuracy evaluation methods based on the absolute condition number and volume condition number, and compare the magnitude relationships of different space target positioning accuracy evaluation methods, and explain the relative advantages and disadvantages of different accuracy evaluation methods.

[0088] Positioning accuracy evaluation index based on the geometric dilution of precision GDOP:

[0089] First, give the definitions of the commonly used matrix norms and vector norms, where ||·|| F is the Frobenius norm (hereinafter referred to as the F-norm). Let λ max be the largest singular value of the matrix . The commonly used norm definitions for the vector matrix are:

[0090]

[0091] Under the ranging and positioning system, according to equation (7) above, using the norm definition, it can be obtained that:​

[0092]

[0093] According to the compatibility of matrix norm and vector norm, it can be known that:

[0094]

[0095] According to the definition of GDOP, (11) can be transformed into the following formula:

[0096] ||ΔX||2 ≤ GDOP·||ΔR + AΔX Station ||2 (12)

[0097] where the norm is λ i is the singular value of matrix J R .

[0098] Similarly, in the time difference positioning system, it can be obtained that:

[0099] ||ΔX||2 ≤ GDOP·||(ΔR + AΔX Station - BΔX Station0 )||2 (13)

[0100] where the norm is λ i is the singular value of matrix J DR .

[0101] Positioning accuracy evaluation index based on absolute condition number:

[0102] Relative condition number:

[0103] Before introducing the concept of absolute condition number, first introduce the concept and mathematical meaning of the common relative condition number (also known as condition number). From equations (5)(6), it can be seen that the relationship between measurement element error and positioning error is linear, and it is a linear equation with coefficient matrix J R or J DR . Thus, the stability of the linear equation can be characterized by the sensitivity of error propagation, and this characterization quantity can be represented by the relative condition number.

[0104] In the ranging positioning system, if all singular values of matrix J R are λ1 ≥ λ2 ≥ λ3 > 0, according to the definition of the condition number of the matrix, the relative condition number is expressed as:

[0105]

[0106] where \(RCN = \lambda_1 / \lambda_3\) is called the Relative Conditional Number.

[0107] Similarly, in the time difference location system, it can be obtained that

[0108]

[0109] The relative conditional number \(RCN\) is the magnification factor of the relative measurement error and is an important indicator to measure the accuracy stability of the location system. However, in the actual application process, since the relative error has no dimension and what is more concerned about is the accuracy evaluation index of the location system, the following concept of the absolute conditional number is introduced.

[0110] Absolute Conditional Number:

[0111] The idea of the absolute conditional number of a matrix was mentioned by Alan Turing in the literature and is used to analyze the rounding error in the matrix operation process. Considering the one-dimensional function \(y = f(x)\) in the linear space, when the independent variable \(x\) with error is obtained * , rather than the true value \(x\), the influence on \(y\) can be expressed as follows

[0112]

[0113] Using the Lagrange mean value theorem, the following formula can be further obtained

[0114] \(e(y)=f'(\xi)(x\) * \(-x)\ (17)\)

[0115] where \(\xi\) is a point between \(x\) * and \(x\). When \(y = f(x)\) is in the multi-dimensional normed linear space, equation (16) can be expressed as

[0116] \(\|e(y)\|_2=\|f'(\xi)(x\) * \(-x)\|_2\ (18)\)

[0117] According to the definition of the normed linear space, equation (18) satisfies the triangle inequality

[0118] \(\|e(y)\|_2\leq\|f'(\xi)\|_2\|(x\) * \(-x)\|_2\ (19)\)

[0119] That is, \(\|f'(\xi)\|_2\) is the magnification factor of the independent variable error generated by the independent variable. Applied to the location accuracy evaluation index, in the 2-norm space, the ranging location system equation (7) is

[0120]

[0121] Matrix \(J\) RThe 2-norm of is the square root of the largest singular value of the matrix Then the 2-norm of the matrix is the square root of the largest singular value λ Rmax , that is

[0122]

[0123] Then equation (20) can be expressed as

[0124] ||ΔX||2 ≤ λ Rmax ||ΔR + AΔX Station ||2 (22)

[0125] λ Rmax is called the absolute condition number (ACN - Absolute Conditional Number).

[0126] Similarly, in the time difference location system, equation (22) can be written as

[0127] ||ΔX||2 ≤ λ DRmax ||ΔR + AΔX Station - BΔX Station0 ||2 (23)

[0128] where λ DRmax is the square root of the largest singular value.

[0129] Next, analyze the main performances of the two space target location accuracy evaluation methods of GDOP and ACN. Taking the ranging location system as an example, according to the above derivation process, GDOP is the F-norm of the matrix and ACN is the 2-norm of the matrix Due to the properties of different norms, a proof is given that GDOP is the upper bound of the observation error magnification factor and ACN is the supremum of the observation error magnification factor.

[0130] Considering the ranging location system, if the singular values of the matrix J R are λ1 ≥ λ2 ≥ λ3, then let the matrix vector m is the number of measurement stations, then equation (10) can be transformed into

[0131]

[0132] According to the Cauchy-Schwarz inequality, equation (24) can be transformed into

[0133]

[0134] Continuing the above derivation, given the matrix 2-norm ||·||2 and the induced norm of the vector 2-norm, according to the property of the compatibility between the induced norm and the vector norm, it can be known that:

[0135]

[0136] Solve for ACN according to equation (26):

[0137]

[0138] where the condition for "=" to hold is ||ΔR + AΔX Station ||2 = 1.

[0139] From the definitions of the F-norm and the 2-norm, it can be known that:

[0140]

[0141] Obviously, GDOP > ACN. Therefore, the geometric dilution of precision GDOP is the upper bound of the measurement error magnification factor, that is, the error magnification factor will never be greater than GDOP; the absolute condition number ACN is the supremum, that is, the error magnification factor will never exceed ACN, and will reach ACN in specific cases. This special case is when "=" holds in equation (27).

[0142] Similarly, the proof under the time difference positioning system is similar, and the conclusion is the same.

[0143] Volume condition number:

[0144] From the above (28), it can be seen that both methods for evaluating the precision of space targets are related to the singular values of the matrix J R and the singular values of the matrix J R are in turn related to the space geometry formed by the space target and the measurement stations. Finding a method to measure such a relationship can clearly reflect the change in the positioning precision during the movement of the space target, thus leading to a method for evaluating the positioning precision of space targets based on the volume conditional number (Volume Conditional Number, VCN).

[0145]

[0146] To make the spatio-temporal relationship of target positioning more explicit, it is set that the number of measurement stations is 3 in the ranging positioning system and 4 in the time difference positioning system, with 1 being the main measurement station. Then the space geometry formed by the space target and the measurement stations is as follows Figure 7 、 Figure 8 as shown.

[0147] In the above figure, the left figure is a spatial geometric schematic diagram of a space target and a measurement station under the ranging and positioning system, and the right figure is a spatial geometric schematic diagram of a space target and a measurement station under the time difference positioning system. In the figure, the hemisphere represents a unit sphere, that is, a sphere with a radius of unit 1. And from equations (28) and (29), it can be seen that under the observation conditions shown in the figure, the change trend of VCN is consistent with that of ACN and GDOP.

[0148] According to the calculation formula of the volume of a tetrahedron, if four points P i =(x i ,y i ,z i )(i = 0, 1, 2, 3) in space, let

[0149]

[0150] After transforming the determinant (30) and calculating according to the algebraic cofactor expansion of the first column, we get

[0151]

[0152] According to the geometric meaning of the vector mixed product, the volume V i of the parallelepiped with vectors P p P0 (i = 1, 2, 3) as adjacent sides is V i =|D|. At the same time, the volume i of the tetrahedron formed by four points P i =(x i ,y In the ranging and positioning system, according to equations (3) and (5), the Jacobian matrix in equation (29) is the tetrahedron V1 with unit 1 side length formed by Target and S1, S2, S3. The volume V1 of this tetrahedron is equal to the reciprocal of VCN. Then, during the movement, the change in the volume V1 of the tetrahedron is the change in the positioning accuracy. It can be seen from the figure that during the descent of the space target, the volume V1 of the tetrahedron first increases and then decreases, and VCN first decreases and then increases, that is, the ranging and positioning accuracy first becomes higher and then lower.

[0153] Similarly, in the time difference positioning system, the volume V2 of the tetrahedron formed by S0 and S1, S2, S3 is exactly equal to the reciprocal of VCN. That is, during the descent of the target, the volume V2 of the tetrahedron continuously increases and VCN continuously decreases, that is, the time difference positioning accuracy continuously becomes higher.

[0154] In the embodiment of the present invention, as Figure 2 , a spatial target positioning accuracy evaluation device based on the condition number of the observation matrix is also provided, including:

[0155] Initial unit 21, for obtaining ranging positioning and time difference positioning methods according to TOA positioning system and TDOA positioning system;

[0156] Analysis unit 22, for analyzing the ranging positioning and the time difference positioning to obtain a measurement equation;

[0157] Construction unit 23, for determining a spatial target positioning error transfer model according to the measurement equation;

[0158] Positioning unit 24, for measuring the positioning accuracy of a spatial target by using the spatial target positioning error transfer model.

[0159] The measurement equation includes:

[0160] Y = f R (X);

[0161] Wherein, are ranging elements obtained from m different measurement stations; f R is a measurement function under the ranging positioning system and the time difference positioning system.

[0162] The spatial target positioning error transfer model is specifically:

[0163]

[0164] Where ΔR = [ΔDR1, ΔDR2,..., ΔDR m T is the observation error of m measurement stations; ΔX = [Δx, Δy, Δz] T is the spatial target position estimation error, ΔX station = [ΔX 01 ,..., ΔX 0m T is the station site error of m measurement stations, ΔX station0 = [Δx 00 , Δy 00 , Δz 00 T is the station site error of the main observation station.

[0165] Measuring the positioning accuracy of a spatial target by using the spatial target positioning error transfer model specifically includes:

[0166] Determining the relative condition number, the absolute condition number, and the volume condition number;

[0167] Determining the supremum of the measurement error magnification factor;

[0168] Determining a positioning accuracy evaluation equation according to the relative condition number, the absolute condition number, the volume condition number, and the supremum of the measurement error magnification factor;​​​

[0169] Use the positioning accuracy evaluation equation to measure the positioning accuracy of space targets.

[0170] The positioning accuracy evaluation equation is specifically as follows:

[0171]

[0172] where is the error transfer matrix at the position estimation error, that is, the positioning error ΔX is obtained by performing matrix multiplication of the error transfer matrix with the measurement error and the station location error; λ1, λ2, and λ3 are all singular values of the J R matrix; J R is the Jacobian matrix of the measurement equation.

[0173] The working principle of the space target positioning accuracy evaluation device based on the condition number of the observation matrix is the same as that of the space target positioning accuracy evaluation method based on the condition number of the observation matrix, and will not be elaborated here.

[0174] Simulation experiment:

[0175] In this part, we verify the correctness of the above theoretical derivation through two groups of comparative experiments. Experiment 1 is to compare the advantages and disadvantages of two accuracy evaluation methods based on the geometric dilution of precision GDOP and the absolute condition number ACN under the ranging positioning system and the time difference positioning system; Experiment 2 is to verify that the method based on the volume condition number VCN can accurately reflect the change trend of the positioning accuracy of the space target in the ranging positioning and time difference positioning systems during the movement process, and further prove the advantages and disadvantages of the two accuracy evaluation methods of GDOP and ACN for space targets at different altitudes.

[0176] Experiment 1: Assume that the movement trajectory of the space target is a uniform linear motion. Four stations are evenly arranged within a range of 24 km × 16 km, with one station located at the geometric center and serving as the master station in the time difference positioning system. Evaluate the two methods from the perspective of the space target. When the space target is at an altitude of 1.5 km, the positioning accuracies of the time difference and ranging positioning systems represented by the two methods are as Figure 9 、 Figure 10 shown.

[0177] In the ranging and positioning system, the x and y axes in the above figure represent the observation range of 24 km × 16 km, and the contour values represent the values of GDOP and ACN on the horizontal plane at an altitude of 1.5 km. It can be seen from the contour values in the above figure that within the same horizontal plane of altitude, the value of GDOP is greater than that of ACN; it can be seen from the thermal distribution in the above figure that the ACN contour lines are denser than those of GDOP. Based on the above simulation results, the size relationship between GDOP and ACN in the ranging and positioning system is verified, and the spatial target evaluation method based on ACN is more accurate than GDOP.

[0178] As Figure 11 , Figure 12 shown, in the time difference positioning system, the x and y axes in the above figure represent the observation range of 24 km × 16 km, and the contour values represent the values of GDOP and ACN on the horizontal plane at an altitude of 1.5 km. Similarly, based on the above simulation results, the size relationship between GDOP and ACN in the time difference positioning system is verified, and the spatial target evaluation method based on ACN is more accurate than GDOP.

[0179] To more intuitively verify the size relationship between ACN and GDOP, the two methods are observed and evaluated from the perspective of the measurement stations. At this time, the positioning accuracies of the two methods for the time difference and ranging and positioning systems are as Figure 13 , Figure 14 shown.

[0180] The x and y axes in the above figure represent the observation range of 24 km × 16 km, and the curved surfaces represent the values of GDOP and ACN at different altitudes within the observation range. The left figure represents the ranging and positioning system, and the right figure represents the time difference positioning system. It can be seen from observing the above figure that the value of GDOP is always greater than that of ACN, further verifying that GDOP is the upper bound of the observation error magnification factor and ACN is the least upper bound of the observation error magnification factor, and the spatial target evaluation method based on ACN is more accurate.

[0181] Experiment 2: Assume that the spatial target moves on the central line of the station layout geometry. Within the range of 24 km × 16 km, 3 stations are evenly arranged in the ranging and positioning system, and 4 stations are evenly arranged in the time difference positioning system, with the main station arranged at the geometric center of the station layout. Verify that the volume condition number VCN can accurately reflect the changing trends of GDOP and ACN. The experimental results are as Figure 15 , Figure 16 shown.

[0182] In the left figure of the drawing, it shows the changing trends of the positioning accuracies of GDOP, ACN, and VCN when the space target is at different heights under the ranging and positioning system. The abscissa x-axis represents the moving height of the space target, and the ordinate y-axis represents the values of different accuracy evaluation methods. It can be seen that as the moving height of the space target continuously increases, the values of GDOP and ACN first decrease and then increase, and the relative magnitude relationship between the GDOP and ACN values is always maintained, which is the same as the changing trend of VCN. Moreover, the relative magnitude relationship between the GDOP and ACN values is always maintained, verifying the correctness of the above conclusion.

[0183] In the right figure of the drawing, it shows the changing trends of the positioning accuracies of GDOP, ACN, and VCN when the space target is at different heights under the time difference positioning system. The abscissa x-axis represents the moving height of the space target, and the ordinate y-axis represents the values of different accuracy evaluation methods. It can be seen that as the moving height of the space target continuously increases, the values of GDOP and ACN continuously increase, which is the same as the changing trend of VCN. Moreover, the relative magnitude relationship between the GDOP and ACN values is always maintained, verifying the correctness of the above conclusion.

[0184] The present invention derives three evaluation methods for the positioning accuracy of space targets, namely the relative condition number, the absolute condition number, and the volume condition number, by using the properties of vector and matrix norms for contraction transformation in a normed linear space. Theoretical analysis shows that the evaluation method for the positioning accuracy of space targets based on the condition number of the observation matrix is the supremum of the measurement error magnification factor, and from the perspective of the solid geometry formed between the target and the measurement stations, this evaluation method can more intuitively explain the changing process of the geometric layout of the measurement stations during the movement of the target. Finally, through numerical comparison experiments, the rationality of the theoretical derivation is further verified. Compared with the traditional geometric dilution of precision evaluation method, the evaluation method based on the condition number of the observation matrix has higher accuracy in representing the positioning accuracy and can more intuitively explain the changing trend of the positioning accuracy with the movement of the target. This is of great significance for aspects such as the identification of high-precision measurement equipment and the optimization of the geometric configuration of measurement station layout.

[0185] It should be understood that the specific order or hierarchy of steps in the disclosed process is an example of an exemplary method. Based on design preferences, it should be understood that the specific order or hierarchy of steps in the process can be rearranged without departing from the scope of the present disclosure. The appended method claims present the elements of various steps in an exemplary order and are not intended to be limited to the specific order or hierarchy described.

[0186] In the above detailed description, various features are grouped together in a single embodiment to simplify the disclosure. This method of disclosure should not be interpreted as reflecting an intention that the embodiments of the claimed subject matter require more features than are clearly stated in each claim. On the contrary, as reflected in the appended claims, the invention is in a state of having less than all the features of the disclosed individual embodiments. Therefore, the appended claims are hereby expressly incorporated into the detailed description, with each claim standing on its own as a separate preferred embodiment of the invention.

[0187] The disclosed embodiments are described above to enable any person skilled in the art to implement or use the present invention. Various modifications of these embodiments are obvious to those skilled in the art, and the general principles defined herein may also be applied to other embodiments without departing from the spirit and scope of the present disclosure. Therefore, the present disclosure is not limited to the embodiments given herein, but is consistent with the broadest scope of the principles and novel features disclosed in this application.

[0188] The above description includes examples of one or more embodiments. Of course, it is impossible to describe all possible combinations of components or methods for the purpose of describing the above embodiments, but it should be recognized by those skilled in the art that the various embodiments may be further combined and arranged. Therefore, the embodiments described herein are intended to cover all such changes, modifications and variations that fall within the scope of protection of the appended claims. In addition, with respect to the term "comprising" used in the specification or claims, the word is covered in a manner similar to the term "including", just as "including," is explained as a transitional word in the claims. In addition, any term "or" used in the specification of the claims is intended to mean "non-exclusive or".

[0189] Those skilled in the art may also understand that the various illustrative logical blocks, units, and steps listed in the embodiments of the present invention may be implemented by electronic hardware, computer software, or a combination of the two. In order to clearly demonstrate the interchangeability of hardware and software, the various illustrative components, units, and steps described above have generally described their functions. Whether such functions are implemented by hardware or software depends on the specific application and the design requirements of the entire system. Those skilled in the art may use various methods to implement the described functions for each specific application, but such implementation should not be understood as exceeding the scope of protection of the embodiments of the present invention.

[0190] In the embodiments of the present invention, the various illustrative logical blocks or units can be implemented or operated with the described functions by a general-purpose processor, a digital signal processor, an application specific integrated circuit (ASIC), a field programmable gate array or other programmable logic device, discrete gate or transistor logic, discrete hardware components, or any combination of the above designs. The general-purpose processor can be a microprocessor. Optionally, the general-purpose processor can also be any conventional processor, controller, microcontroller or state machine. The processor can also be implemented by a combination of computing devices, such as a digital signal processor and a microprocessor, multiple microprocessors, one or more microprocessors combined with a digital signal processor core, or any other similar configuration.

[0191] The steps of the methods or algorithms described in the embodiments of the present invention can be directly embedded in hardware, software modules executed by a processor, or a combination of the two. The software modules can be stored in a RAM memory, a flash memory, a ROM memory, an EPROM memory, an EEPROM memory, a register, a hard disk, a removable disk, a CD-ROM, or any other form of storage medium in the art. Exemplarily, the storage medium can be connected to the processor so that the processor can read information from the storage medium and write information to the storage medium. Optionally, the storage medium can also be integrated into the processor. The processor and the storage medium can be disposed in an ASIC, and the ASIC can be disposed in a user terminal. Optionally, the processor and the storage medium can also be disposed in different components of the user terminal.

[0192] In one or more exemplary designs, the functions described in embodiments of the present invention may be implemented in hardware, software, firmware, or any combination of the three. If implemented in software, these functions may be stored on a computer-readable medium or transmitted as one or more instructions or code on a computer-readable medium. A computer-readable medium includes both computer storage media and communication media that facilitate transfer of a computer program from one place to another. The storage media may be any available media that can be accessed by a general-purpose or special-purpose computer. By way of example, and not limitation, such computer-readable media can include RAM, ROM, EEPROM, CD-ROM or other optical disk storage, magnetic disk storage or other magnetic storage devices, or any other medium that can be used to carry or store program code in the form of instructions or data structures and that can be accessed by a general-purpose or special-purpose computer, or a general-purpose or special-purpose processor. Additionally, any connection is properly termed a computer-readable medium. For example, if the software is transmitted from a website, server, or other remote source using a coaxial cable, fiber optic cable, twisted pair, DSL, or wireless means such as infrared, radio, and microwave, it is included in the definition of computer-readable medium. Disk and disc include compact disc, laser disc, optical disc, DVD, floppy disk, and Blu-ray disc, where disks usually reproduce data magnetically, while discs usually reproduce data optically with lasers. Combinations of the above should also be included within the scope of computer-readable media.

[0193] The specific embodiments described above have further elaborated on the objectives, technical solutions, and beneficial effects of the present invention. It should be understood that the above are only specific embodiments of the present invention and are not used to limit the protection scope of the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for evaluating the accuracy of spatial target positioning based on the condition number of the observation matrix, characterized in that: The method comprises the following steps: Based on the TOA positioning system and TDOA positioning system, the ranging positioning and time difference positioning methods are obtained; Analyze the distance measurement positioning and the time difference positioning to obtain a measurement equation; Determine a spatial target positioning error transmission model according to the measurement equation; The space target positioning error transfer model is used to measure the positioning accuracy of the space target.

2. The method for evaluating the spatial target positioning accuracy based on the observation matrix condition number according to claim 1, characterized in that: The measurement equation includes: Y=f R (X); in, is the distance measurement element obtained from m different measuring stations; f R It is the measurement function under the ranging positioning system and the time difference positioning system.

3. The method for evaluating the spatial target positioning accuracy based on the observation matrix condition number according to claim 2, characterized in that: The space target positioning error transmission model is specifically: Where ΔR=[ΔDR1,ΔDR2,…,ΔDR m ] T is the observation error of m stations; ΔX=[Δx,Δy,Δz] T is the spatial target position estimation error, ΔX station =[ΔX 01 ,...,ΔX 0m ] T is the site error of m stations, ΔX station0 =[Δx 00 ,Δy 00 ,Δz 00 ] T is the site error of the main observing station.

4. The method for evaluating the spatial target positioning accuracy based on the observation matrix condition number according to claim 3 is characterized in that: The using the space target positioning error transfer model to measure the positioning accuracy of the space target specifically includes: Determine relative condition number, absolute condition number, and volume condition number; Determine the supremum of the measurement error amplification factor; Determine a positioning accuracy evaluation equation according to the relative condition number, the absolute condition number, the volume condition number and the supremum of the measurement error magnification factor; The positioning accuracy evaluation equation is used to measure the positioning accuracy of the space target.

5. The method for evaluating the spatial target positioning accuracy based on the observation matrix condition number according to claim 4 is characterized in that: The positioning accuracy evaluation equation is specifically: in, The error transfer matrix when the position estimation error is the error transfer matrix, that is, the positioning error ΔX is obtained through the error transfer matrix It is obtained by matrix multiplication of the measurement error and the site error; λ1, λ2, λ3 are J R All singular values ​​of the matrix; J R is the Jacobian matrix of the measurement equation.

6. A space target positioning accuracy evaluation device based on the observation matrix condition number, characterized in that: include: An initial unit is used to obtain ranging positioning and time difference positioning methods based on the TOA positioning system and the TDOA positioning system; An analysis unit, used for analyzing the ranging positioning and the time difference positioning to obtain a measurement equation; A construction unit is used to determine a space target positioning error transfer model according to the measurement equation; The positioning unit is used to measure the positioning accuracy of the space target by using the space target positioning error transfer model.

7. The space target positioning accuracy evaluation device based on the observation matrix condition number according to claim 6 is characterized in that: The measurement equation includes: Y=f R (X); in, is the distance measurement element obtained from m different measuring stations; f R It is the measurement function under the ranging positioning system and the time difference positioning system.

8. The space target positioning accuracy evaluation device based on the observation matrix condition number according to claim 7 is characterized in that: The space target positioning error transmission model is specifically: Where ΔR=[ΔDR1,ΔDR2,…,ΔDR m ] T is the observation error of m stations; ΔX=[Δx,Δy,Δz] T is the spatial target position estimation error, ΔX station =[ΔX 01 ,...,ΔX 0m ] T is the site error of m stations, ΔX station0 =[Δx 00 ,Δy 00 ,Δz 00 ] T is the site error of the main observing station.

9. The space target positioning accuracy evaluation device based on the observation matrix condition number according to claim 8, characterized in that: The using the space target positioning error transfer model to measure the positioning accuracy of the space target specifically includes: Determine relative condition number, absolute condition number, and volume condition number; Determine the supremum of the measurement error amplification factor; Determine a positioning accuracy evaluation equation according to the relative condition number, the absolute condition number, the volume condition number and the supremum of the measurement error magnification factor; The positioning accuracy evaluation equation is used to measure the positioning accuracy of the space target.

10. The space target positioning accuracy evaluation device based on the observation matrix condition number according to claim 9, characterized in that: The positioning accuracy evaluation equation is specifically: in, The error transfer matrix when the position estimation error is the error transfer matrix, that is, the positioning error ΔX is obtained through the error transfer matrix It is obtained by matrix multiplication of the measurement error and the site error; λ1, λ2, λ3 are J R All singular values ​​of the matrix; J R is the Jacobian matrix of the measurement equation.

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