Pseudo-satellite system geometric accuracy factor calculation method based on geometric configuration

Abstract

The invention discloses a pseudo-satellite system geometric accuracy factor calculation method based on geometric configuration, and belongs to the field of multi-pseudo-satellite navigation and communication. The method specifically comprises the following steps: aiming at a pseudo satellite navigation positioning system formed by n randomly distributed pseudo satellites, improving calculation of geometric accuracy factors of the pseudo satellite navigation positioning system; then, respectively expressing and expanding tr [(GTG) *] and GTG of the geometric accuracy factor by utilizing a determinant; respectively expressing and expanding tr [(GTG) *] and GTG by utilizing a matrix theory, a vector and analytic geometry, and finally calculating a final improved GDOP; and for the pseudo satellite system with any geometric configuration, substituting the coordinates of each pseudo satellite under the station center coordinate system into the final GDOP formula, and changing the station center coordinates of the pseudo satellites to change the angle relational expressions between the line and the line and between the line and the plane in the geometric configuration of the pseudo satellite system, so that the GDOP value of the pseudo satellite system is minimum. Therefore, geometric layout optimization of the pseudo-satellite system can be realized. According to the method, the geometric features of the geometric precision factors are displayed more conveniently and clearly.

Description

Technical Field

[0001] The invention belongs to the field of multi-pseudo-satellite navigation and communication, and in particular relates to a method for calculating geometrical dilution of precision of a pseudo-satellite system based on geometric configuration. Background Art

[0002] With the development of high-altitude airships and long-flight UAV technologies, the use of these flying platforms as pseudo-satellite navigation and communication systems has been increasingly studied and applied in military and social life. Pseudo-satellite systems can simulate satellite signals through ground transmitting stations to provide users with positioning services, especially in environments where satellite signals are limited or unavailable. However, the geometric layout of pseudo-satellite systems has a crucial impact on positioning accuracy.

[0003] In general satellite navigation and positioning systems, when measuring distances, including calculation errors, at least four or more pseudo-satellites are required to provide basic positioning and navigation services. However, due to poor geometric distribution in some cases, such as satellites being almost collinear or satellites being too concentrated, the columns of the observation matrix will be very close to linear correlation.

[0004] This will lead to the following disadvantages of the general geometric dilution of precision (GDOP) calculation method for inverting the geometric matrix: 1) The amount of calculation is large; 2) The traditional GDOP calculation formula simply expresses the GDOP by solving the trace of the inverse matrix of the geometric matrix, which cannot better show the relationship between the geometric dilution of precision and the geometric configuration; 3) The traditional GDOP calculation formula cannot directly solve the pseudo-satellite layout configuration that minimizes the GDOP under different numbers of pseudo-satellites, that is, the navigation system positioning accuracy is optimized.

[0005] In addition, when the matrix inversion method is directly applied to calculate GDOP, since it involves matrix inversion operations, when the geometric matrix has singularity, it will cause numerical instability in the GDOP calculation process and even cause the calculation process to diverge. Summary of the invention

[0006] In order to solve the above problems, the present invention proposes a method for calculating the geometric dilution of precision of a pseudolite system based on geometric configuration; this method can more clearly link the geometric configuration with the geometric dilution of precision and more conveniently and clearly display the geometric characteristics of the geometric dilution of precision.

[0007] The specific steps are as follows:

[0008] Step 1: for a pseudo-satellite navigation and positioning system composed of n (n≥4) pseudo-satellites distributed randomly, improve the calculation of its geometric precision factor;

[0009] The improved calculation formula is:

[0010]

[0011] tr(·) represents the matrix trace, G T represents the transposed matrix, (G T G) * G T The adjoint matrix of G; |G T G|means G T The determinant of G; G is the transformed geometric measurement matrix:

[0012]

[0013] E i represents the altitude angle of pseudo-satellite i relative to the user position O; D i represents the azimuth of pseudolite i relative to the user position O in the X-axis direction;

[0014] (m xi , m yi ,m zi ) is the direction cosine of the unit vector pointing from the user position O to the ith pseudo-satellite, i.e., the position information of the coordinate point of pseudo-satellite i in the station-centered coordinate system established with the user position O as the origin;

[0015] Step 2: Use the determinant to calculate the geometric precision factor tr[(G T G) * ] and |G T G|represent and expand them respectively;

[0016] According to the transformed geometric observation matrix G, we get:

[0017]

[0018] Thus we get |G T G|:

[0019]

[0020] At the same time, the adjoint matrix is ​​further obtained as:

[0021]

[0022] That is, we get tr[(G T G) * ]:

[0023] tr[(G T G) * ]=B11+B22+B33+B44

[0024]

[0025] Step 3: Use matrix theory, vectors and analytic geometry to solve tr[(G T G) * ] is expressed and expanded to obtain:

[0026]

[0027] V1 represents the volume of the parallelepiped formed by the three edges connecting any three pseudo-satellites and user O (i.e., the origin) in the station-centered coordinate system established with user O as the origin; It represents the sum of the squares of the areas of the triangles formed by the projection points of any three pseudo-satellites in the XOY plane; It represents the sum of the squares of the areas of the triangles formed by the projection points of any three pseudo-satellites in the YOZ plane; It represents the sum of the squares of the triangle areas formed by the projection points of any three pseudo-satellites in the XOZ plane. The specific calculation formula is:

[0028]

[0029] 1≤i<j<k≤n; i, j, k represent any three different pseudolites in the pseudolites navigation and positioning system;

[0030] Step 4: Use matrix theory, vectors and analytic geometry to solve |G T G| is represented and expanded; we get:

[0031] |G T G|=6∑m xi m yj m zk V2

[0032] in,

[0033] i, j, l, l represent pseudo-satellites in the pseudo-satellite navigation and positioning system; V2 represents the volume of the tetrahedron formed by the four pseudo-satellites i, j, l, l in the station-center coordinate system.

[0034] Step 5: Use the above representation and expand tr[(G T G) * ] and |G T G|Conversion formula, calculate the final GDOP;

[0035]

[0036] Step 6. For pseudo-satellite systems of arbitrary geometric configuration, the coordinates of each pseudo-satellite station center coordinate system are substituted into the final GDOP formula to obtain the geometric precision factor of the pseudo-satellite system. Changing the pseudo-satellite station center coordinates will change the angle relationship between lines and lines, and lines and surfaces in the geometric configuration of the pseudo-satellite system, so that the GDOP value of the pseudo-satellite system is minimized, and the geometric layout optimization of the pseudo-satellite system can be achieved.

[0037] The advantages of the present invention are:

[0038] 1) A method for calculating the geometric dilution of precision of a pseudolite system based on geometric configuration solves the problem of large amount of calculation when the traditional GDOP calculation method uses matrix calculation; the present invention can solve the evaluation of positioning accuracy (GDOP) through the geometric configuration of the pseudolite group, i.e., the surface area and volume, and better demonstrates the influence of the spatial geometric configuration of the pseudolite system on the GDOP;

[0039] 2) A method for calculating the geometric dilution of precision of a pseudolite system based on geometric configuration, which minimizes the GDOP value of the pseudolite system by bringing the coordinates of each pseudolite in the station center coordinate system into the improved GDOP calculation method, and can obtain the angle relationship between lines and lines, and between lines and planes, thereby solving the optimal layout configuration under different numbers of pseudolites;

[0040] 3) A method for calculating the geometric dilution of precision of a pseudo-satellite system based on geometric configuration improves and transforms the traditional calculation formula, and through matrix transformation, represents it through determinants and vectors respectively, and links the GDOP with the geometric structure of the pseudo-satellite. This overcomes the problems of the traditional method, such as the large amount of calculation and divergence of calculation results, the inability to better show the relationship between the geometric dilution of precision and the geometric configuration, and the need to involve matrix inversion operations. When the geometric matrix has singularity, it will lead to numerical instability in the GDOP calculation process and even cause the calculation process to diverge. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 It is a flow chart of a method for calculating geometric dilution of precision of a pseudo-satellite system based on geometric configuration of the present invention;

[0042] Figure 2 It is a schematic diagram of the volume of a tetrahedron formed by calculating the coordinate points of any four pseudo-satellites in the station-center coordinate system of the present invention;

[0043] Figure 3 It is a schematic diagram of pseudolites selected by the present invention for verifying the feasibility of the improved novel GDOP calculation method;

[0044] Figure 4 It is the user GDOP variation trend diagram of two different methods of the present invention;

[0045] Figure 5 This is a rendering of the simulation of the region using the original calculation method of the present invention;

[0046] Figure 6 This is a diagram showing the effect of simulating a region using an improved GDOP calculation method in the present invention;

[0047] Figure 7 It is a schematic diagram of GDOP in a specific area of ​​a regular tetrahedron configuration of the present invention;

[0048] Figure 8 Schematic diagram of GDOP simulation of any regular tetrahedron configuration 1 of the present invention;

[0049] Fig. 9 It is a schematic diagram of the GDOP simulation of any regular tetrahedral configuration 2 of the present invention. DETAILED DESCRIPTION

[0050] The present invention will be further described in detail below with reference to the accompanying drawings and implementation examples.

[0051] In the field of satellite navigation technology, the geometric dilution of precision (GDOP) plays a key role in satellite selection, positioning accuracy assessment, and identification of system anomalies. When implementing a satellite navigation system, the GDOP value is usually calculated using matrix inversion techniques.

[0052] The present invention proposes a method for calculating the geometric precision factor of a pseudo-satellite system based on geometric configuration, and uses the volume and area of ​​the geometric body formed by the pseudo-satellite to calculate the geometric precision factor: In the traditional geometric precision factor calculation formula, it is necessary to solve the observation matrix A and further obtain the geometric matrix H = (G T G) -1 , and then transform the geometric matrix to get tr[(G T G) * ] and det(G T G), by expressing the transformed matrix and the geometric configuration of pseudolites (such as area, volume, etc.) with vector matrices and determinants, the geometric dilution of precision of the pseudolites system is obtained, which overcomes the separation between the geometric configuration of pseudolites layout and the geometric dilution of precision (GDOP) calculation formula in traditional calculation methods, aiming to optimize the layout of pseudolites system and improve positioning accuracy.

[0053] like Figure 1 As shown, the specific steps are as follows:

[0054] Step 1: Improve the calculation of the geometric precision factor for a satellite navigation and positioning system composed of n (n≥4) randomly distributed pseudo-satellites;

[0055] The traditional geometric observation matrix is:

[0056]

[0057] By improving it, let m xi =cosE i dxD i 、m yi =cosE i sI i 、m zi = sinE i ;

[0058] E i represents the altitude angle of pseudo-satellite i relative to the user position O; D i represents the azimuth of pseudolite i relative to the user position O on the X-axis;

[0059] (m xi , m yi ,m zi ) is the direction cosine of the unit vector pointing from the user's approximate position O to the ith pseudo-satellite, i.e., the position information of the coordinate point of pseudo-satellite i in the station-centered coordinate system established with the user's position O as the origin;

[0060] Get the transformed geometric observation matrix:

[0061]

[0062] The improved calculation formula is:

[0063]

[0064] tr(·) represents the matrix trace, G T represents the transposed matrix, (G T G) * G T The adjoint matrix of G; |G T G|means G T The determinant of G; g is the transformed geometric measurement matrix:

[0065] Step 2: Use the determinant to calculate the geometric precision factor tr[(G T G) * ] and |G T G|represent and expand them respectively;

[0066] According to the transformed geometric observation matrix G, we get:

[0067]

[0068] Thus we get |G T G|:

[0069]

[0070] At the same time, the adjoint matrix is ​​further obtained as:

[0071]

[0072] For the adjoint matrix (G T G) * Find its trace and we get tr[(G T G) * ]:

[0073] tr[(G T G) * ]=B11+B22+B33+B44

[0074]

[0075] Step 3: Use matrix theory, vectors and analytic geometry to solve tr[(G T G) * ] to represent and expand;

[0076] Because tr[(G T G) * ]=B11+B22+B33+B44; after matrix transformation, we get:

[0077]

[0078]

[0079] 1≤i<j<k≤n; i, j, k represent any three different pseudolites in the pseudolites navigation and positioning system;

[0080] Through vector and analytical analysis, we can know that:

[0081]

[0082] Thus, the area of ​​the triangle formed by the three pseudo-satellites i, j, and k can be further obtained:

[0083]

[0084] That is to say

[0085] Therefore, the geometric meanings of B11, B22, and B33 after conversion are:

[0086]

[0087] It represents the sum of the squares of the areas of the triangles formed by the projection points of any three pseudo-satellites in the XOY plane; It represents the sum of the squares of the areas of the triangles formed by the projection points of any three pseudo-satellites in the YOZ plane; It represents the sum of the squares of the areas of the triangles formed by the projection points of any three pseudo-satellites in the XOZ plane;

[0088] because That is Geometric analysis of:

[0089] Let vector From the knowledge of vectors and analytic geometry, we can get: The volume of the parallelepiped formed by these three vectors

[0090] and

[0091] Therefore, it can be inferred

[0092] Then for B44:

[0093]

[0094] V1 represents the volume of the parallelepiped formed by the three edges connecting any three pseudo-satellites and user O (i.e. the origin) in the station-centered coordinate system established with user O as the origin; therefore:

[0095]

[0096] Step 4: Use matrix theory, vectors and analytic geometry to solve |G T G| to represent and expand;

[0097] because

[0098] Change according to the properties of the determinant:

[0099] Step 1: Right |G T The first column of G| is expanded into n items;

[0100] Step 2: Extract common factors from the first column of the expanded n determinants;

[0101] Sequentially |G T Repeat the first and second steps for the second, third and fourth columns of G|;

[0102] Inferred:

[0103]

[0104] From the knowledge of vectors and analytic geometry, we can get:

[0105] like Figure 2 As shown, V2 represents the volume of the tetrahedron formed by the four pseudo-satellites i, j, k, and l in the station center coordinate system;

[0106] V 以DC、DB、DA为三边棱所作的平行六面体 The volume of the parallelepiped with D, C, B, and A as vertices and the three edges DC, DB, and DA is calculated as follows:

[0107]

[0108] The observation matrix: and From the knowledge of vectors and analytic geometry, we know that:

[0109] Therefore, we can know that:

[0110] get:

[0111] |G T G|=6∑m xi m yj m zk V2

[0112] in,

[0113] l represents the pseudo-satellite in the pseudo-satellite navigation and positioning system;

[0114] Step 5: Use the tr[(G T G) * ] and |G T G|Conversion formula, calculate the final improved GDOP;

[0115]

[0116] Step 6. For pseudo-satellite systems of arbitrary geometric configuration, the coordinates of each pseudo-satellite station center coordinate system are substituted into the final GDOP formula to obtain the geometric precision factor of the pseudo-satellite system. Changing the pseudo-satellite station center coordinates will change the angle relationship between lines and lines, and lines and surfaces in the geometric configuration of the pseudo-satellite system, so that the GDOP value of the pseudo-satellite system is minimized, and the geometric layout optimization of the pseudo-satellite system can be achieved.

[0117] For a pseudo-satellite system of any geometric configuration, its volume and surface area can be directly calculated geometrically to obtain the GDOP value; by substituting the coordinates of each pseudo-satellite station center coordinate system into the final GDOP formula, the geometric precision factor of the pseudo-satellite system can be obtained; by changing the pseudo-satellite station center coordinates and thus changing the angle relationship between lines and lines, and lines and surfaces in the geometric configuration of the pseudo-satellite system, the GDOP value of the pseudo-satellite system can be minimized, thus realizing the geometric layout optimization of the pseudo-satellite system.

[0118] The present invention converts the traditional calculation formula by matrix transformation and the determinant of tr(A T A) * 、|A T A| is used to represent it, and its geometric meaning is found out, thus realizing the combination and representation of geometric configuration and geometric precision factor. For the improved calculation formula, by calculating the geometric configuration of the pseudo-satellite system, such as volume, surface area, etc., the numerical value of the geometric precision dilution factor of each pseudo-satellite can be solved, so that the receiving user can make corrections, thereby improving the positioning accuracy; 1) It avoids the single and complex matrix inverse matrix and trace solution in the traditional GDOP calculation formula; the change of GDOP can be reflected by the change of geometric configuration; 2) combined with the geometric layout configuration of the pseudo-satellite system, the geometric precision dilution factor under the configuration can be directly reflected and calculated, and then the pseudo-satellite layout can be adjusted. By combining the improved GDOP calculation method with the geometric configuration of the pseudo-satellite system, the layout of the pseudo-satellite system can be optimized, and the optimal configuration of the pseudo-satellite system at any time can be obtained;

[0119] The feasibility and advantages of improving the new GDOP calculation method are demonstrated below:

[0120] The pseudo-satellite base stations are 1, 2, 3, 4, and 5 respectively; the pseudo-satellite schematic diagram is as follows Figure 3 As shown, its position coordinates are:

[0121] X = (25000 50000 50000 25000 37500) Unit: m

[0122] Y=(25000 25000 50000 50000 37500)Unit: m

[0123] Z=(20000 20000 20000 20000 25000)Unit: m

[0124] The user locations are M1-M9;

[0125] When users are located in different locations, the GDOP calculation results of two different methods are as follows:

[0126] M1=(25000 50000 50000 25000 37500)

[0127] M2=(25000 50000 50000 25000 37500)

[0128] M3=(25000 50000 50000 25000 37500)

[0129] M4=(25000 50000 50000 25000 37500)

[0130] M5=(25000 50000 50000 25000 37500)

[0131] M6=(25000 50000 50000 25000 37500)

[0132] M7=(25000 50000 50000 25000 37500)

[0133] M8=(25000 50000 50000 25000 37500)

[0134] M9=(25000 50000 50000 25000 37500)

[0135] As shown in Table 1,

[0136] Table 1

[0137]

[0138] like Figure 4 As shown in the figure, the user GDOP change trend diagram of two different methods, the pseudo-satellite base stations are A, B, C, and D respectively; their location coordinates are as follows:

[0139] A=(37500 50000 20000)Unit: m

[0140] B=(25000 25000 20000)Unit: m

[0141] C=(50000 25000 20000)Unit: m

[0142] D = (37500 50000 25000) Unit: m

[0143] Two methods are used to simulate the region and the simulation results are as follows: Figure 5 and Figure 6As shown in the figure, the results show that the two methods are completely consistent in evaluating the GDOP index, so the new improved method is established;

[0144] Example:

[0145] This embodiment selects four pseudolites S1, S2, S3, and S4, and uses an improved GDOP calculation method combined with mathematical derivation to optimize the layout of the pseudolites configuration;

[0146] Specifically, after unitization and coordinate transformation, the four pseudo-satellites are located on a unit sphere with user O as the center in the station center coordinate system of user O; the specific coordinates of pseudo-satellite i are:

[0147] m xi =cosE i dxD i

[0148] m yi =cosE i sI i

[0149] m zi = sinE i

[0150]

[0151] E1, E2, E3, and E4 represent the altitude angles of S1, S2, S3, and S4 to the user O, respectively;

[0152] D1, D2, D3, and D4 represent the azimuths of the X-axis of S1, S2, S3, and S4, respectively;

[0153] The coordinates of the four pseudo-satellites S1, S2, S3, and S4 in the station center coordinate system established by user O. The specific form is:

[0154] A=(cosE1cosD1cosE1sinD1sinE1)

[0155] B=(cosE2cosD2cosE2sinD2sinE2)

[0156] C=(cosE3cosD3cosE3sinD3sinE3)

[0157] D=(cosE4cosD4cosE4sinD4sinE4)

[0158] make And it is unit length;

[0159] From vectors and analytic geometry we know that:

[0160]

[0161] For N, we have:

[0162] For the three points ABC:

[0163] ABC three points:

[0164] ABC three points:

[0165] ABD three points:

[0166] ABD three points:

[0167] ABD three points:

[0168] ACD three points:

[0169] ACD three points:

[0170] ACD three points:

[0171] BCD three points:

[0172] BCD three points:

[0173] BCD three points: Substituting the above contents into M, N, and T respectively, we can obtain:

[0174] m xi =cosE i dxD i

[0175] m yi =cosE i sI i

[0176] m zi = sinE i

[0177] get:

[0178] M=cosE1cosD1cosE2sinD2sinE3|sinα3*cosθ1|

[0179] +cosE1cosD1cosE2sinD2sinE4|sinα5*cosθ2|+cosE1cosD1cosE3sinD3sinE4|sinα6*cosθ3|+cosE2cosD2cosE3sinD3sinE4|sinα6*cosθ4|

[0180] N = 2 (sinα1 2 +sinα2 2 +sinα3 2 +sinα4 2 +sinα5 2 +sinα6 2 )

[0181] T=6(cosE1cosD1cosE3sinD3sinE4|sinα6*cosθ3|-cosE2cosD2cosE3sinD3sinE4|sinα6*cosθ4|+cosE1cosD1cosE2sinD2sinE3|sinα3*

[0182] cosθ1|-cosE1cosD1cosE2sinD2sinE4|sinα5*cosθ2|)

[0183] In order to find the geometric configuration layout with the best positioning accuracy of the four pseudo-satellite system, that is, the smallest geometric precision factor, it is only necessary to make Minimum;

[0184] Optimal configuration solution process:

[0185] (1) Analyze spatial geometric structure

[0186] The four pseudo-satellites S1, S2, S3, and S4 are converted into vectors in space through their altitude angles and azimuth angles. The specific form is:

[0187] A=(cosE1cosD1cosE1sinD1sinE1)

[0188] B=(cosE2cosD2cosE2sinD2sinE2)

[0189] C=(cosE3cosD3cosE3sinD3sinE3)

[0190] D=(cosE4cosD4cosE4sinD4sinE4)

[0191] The vectors formed by these pseudo-satellites are:

[0192] vector

[0193] (2) Analyze M, N, and T in the formula

[0194] 1. M is a comprehensive quantity related to the altitude angle, azimuth angle and vector angle of the pseudo satellite and the angle between the vector and the plane. Formulas such as cosE1cosD1cosE2sinD2sinE3, etc., reflect the vector The geometric relationship between them.

[0195] 2. N is the sum of the squares of the angles between the vectors.

[0196] 3. T contains the geometric information of the surface composed of vectors, and the angle between the vector and the plane is expressed by terms such as sinα6*cosθ3. By studying the geometric configuration, we can find In this case, the geometric configuration is usually a structure with strong symmetry, because the symmetric structure can make the angle between the vectors reach a certain optimal relationship.

[0197] (3) Inferred geometric configuration: regular tetrahedron

[0198] According to the above analysis, the regular tetrahedron is a geometric configuration with very high symmetry. In the regular tetrahedron:

[0199] 1. The angles between the vectors are equal, that is, α1=α2=α3=α4=α5=α6;

[0200] 2. The angle between the vector and the plane may also maintain symmetry;

[0201] In the configuration of a regular tetrahedron, the distances and angles between the four points (pseudo-satellites) in space are evenly distributed, which may make minimum, while T remains relatively large, which eventually makes Minimum.

[0202] (4) Analysis of the proof process

[0203] 1. Symmetry analysis of a regular tetrahedron: All sides of a regular tetrahedron are equal in length, and all faces are equilateral triangles.

[0204] The angles α between the vectors are equal, so sinα1=sinα2=sinα3=sinα4=sinα5=

[0205] sinα 6. At this point, the N term reaches its minimum value.

[0206] By N = 2 (sin α1 2 +sinα2 2 +sinα3 2 +sinα4 2 +sinα52 +sinα6 2 )middle

[0207]

[0208] If and only if sinα1=

[0209] The minimum value is obtained when sinα2=sinα3=sinα4=sinα5=sinα6;

[0210] 2. Simplification of M terms: Due to the symmetry of the regular tetrahedron, the angle between vectors and the angle between the vector and the plane are also equal. Therefore, the terms in M ​​are symmetrical, the absolute value parts will be aligned, and unnecessary difference terms will be eliminated, simplifying the entire expression.

[0211] 3. Simplification of T term: In the regular tetrahedral configuration, the angle between the vector and the plane is also symmetrical, so the T term is also simplified and maintains a larger value.

[0212] 4. Final extreme value determination: By calculating M+N and T under the regular tetrahedron configuration, it can be determined that this configuration can make Minimum.

[0213] (5) Proof of the optimal configuration of the four pseudo-satellites

[0214] 1. Symmetry analysis of regular tetrahedron:

[0215] In the tetrahedron configuration, the four pseudo-satellites A, B, C, and D are distributed in space so that the geometric figure they form satisfies the following characteristics:

[0216] 1. The angles between all vectors are equal, that is: α1=α2=α3=α4=α5=α6

[0217] 2. Due to the symmetry of the regular tetrahedron, these angles are known constants and satisfy the internal angle relationship of the regular tetrahedron. The internal angle of the regular tetrahedron (that is, the angle between every two sides) is:

[0218]

[0219] This is the angle between all the faces of a regular tetrahedron. In this case, the angle α between all the vectors i are all equal to a fixed value α0.

[0220] Therefore, in the tetrahedral configuration, all α i are all equal and depend on the inherent symmetry of the regular tetrahedron. This symmetry allows the N term to be simplified.

[0221] N-term simplification:

[0222] N = 2 (sinα1 2 +sinα2 2 +sinα3 2 +sinα4 2 +sinα5 2 +sinα6 2 )

[0223] Since each α i Equal, that is, α i =α0, therefore:

[0224] N=12sinα0 2

[0225] Will Substituting into the calculation, we get:

[0226]

[0227] Therefore, the N term in the regular tetrahedral configuration is a definite constant.

[0228] 2. Simplification of M item:

[0229] The expression of the M term involves the altitude angle E i 、Azimuth D i , the angle between the vectors, and the angle between the vector and the plane. Since the altitude angle and azimuth angle of the regular tetrahedron are evenly distributed under the symmetrical structure, the angles between the vectors are equal, so the multiple results in the formula can be simplified.

[0230] Derive the M term:

[0231] M=cosE1cosD1cosE2sinD2sinE3|sinα3*cosθ1|

[0232] +cosE1cosD1cosE2sinD2sinE4|sinα5*cosθ2|+cosE1cosD1cosE3sinD3sinE4|sinα6*

[0233] cosθ3|+cosE2cosD2cosE3sinD3sinE4|sinα6*cosθ4|

[0234] Considering the symmetry between each pseudo-satellite in the regular tetrahedron, all angles are equal, so each term can be expressed as the same function form. Due to the angular symmetry, similar terms cancel each other out or simplify each other, and finally M can be simplified to:

[0235] M=k1·sinα·cosθ

[0236] Where k1 is a constant derived from symmetry, α is the angle between pseudolites, and θ is the angle between the vector and the plane. In the tetrahedral configuration, the angles are evenly distributed, so that the M term also reaches the minimum value.

[0237] 3. Impact of T-term

[0238] The T term expression is as follows:

[0239] T=6(cosE1cosD1cosE3sinD3sinE4|sinα6*cosθ3|-cosE2cosD2cosE3sinD3sinE4|sinα6*cosθ4|+cosE1cosD1cosE2sinD2sinE3|sinα3*

[0240] cosθ1|-cosE1cosD1cosE2sinD2sinE4|sinα5*cosθ2|)

[0241] Similarly, due to the symmetry of the regular tetrahedron, all angles θ1 = θ2 = θ3 = θ4 are equal, and the geometric relationship of all pseudo-satellite pairs remains consistent, so the multiple angle terms in the T term can be simplified to a constant term. The final T term expression can be simplified to:

[0242] T=k2·sinα·cosθ

[0243] Where k2 is a constant derived from symmetry. Since the tetrahedron configuration maintains maximum symmetry, T reaches a larger value.

[0244] 4. Extreme value determination

[0245] Combining the derivation results of M, N, and T, we can conclude that:

[0246] (1) In the tetrahedral geometry, M+N reaches its minimum value, so all α i are equal, and the symmetry of the angles simplifies the complex terms.

[0247] (2) At the same time, since the T term is a function related to the angle between vectors and the altitude angle, the symmetry of the regular tetrahedron ensures that its value is large.

[0248] therefore, It is smallest in the tetrahedral configuration.

[0249] 5. Simulation demonstration

[0250] Assuming that the simulation area is a square area with a side length of 75 km, the regular tetrahedron configuration and other tetrahedron configurations are simulated within this area. Figure 7 , Figure 8 and Fig. 9 It can be seen that compared with other tetrahedral configurations, the regular tetrahedral structure under four pseudo-satellites has a larger range of GDOP < 10 in the simulation area, that is, the positioning accuracy is more accurate. The simulation proves that the regular tetrahedral configuration is the optimal configuration layout under four pseudo-satellites.

[0251] Note: The blank area in the simulation diagram indicates GDOP≥30, which has no effect on the experimental conclusion and is not discussed here.

[0252] in conclusion:

[0253] The layout of the pseudolite system is optimized by using an improved geometric dilution of precision (GDOP) calculation method combined with vector analysis, analytic geometry theory and geometric configuration principles. This method allows the pseudolite layout in the system to be optimized under different numbers of pseudolite configurations, thereby finding the optimal geometric configuration for different numbers of pseudolites.

[0254] Through this process, the positioning accuracy of the navigation system is significantly improved, ensuring that users can obtain more accurate navigation and positioning services under conditions of different numbers of pseudo-satellites.

[0255] Specifically, the calculation method of GDOP was first optimized, taking into account the influence of the geometric distribution of pseudolites on positioning accuracy. On this basis, a mathematical model for optimizing pseudolites layout was established using the relevant knowledge of vector and analytic geometry. This model can find the pseudolites distribution scheme that minimizes the GDOP value through iterative calculation under the given number of pseudolites. In addition, through in-depth analysis of the geometric configuration, the spatial layout of pseudolites was further optimized to ensure the stability and reliability of the navigation system in complex environments.

[0256] Finally, through this series of theoretical derivation, demonstration and simulation, we successfully achieved layout optimization under different numbers of pseudo-satellites, and verified that the proposed optimal geometric configuration can effectively improve the positioning accuracy of the navigation system, providing a solid theoretical support for the actual deployment and application of pseudo-satellite systems.

Claims

1. A method for calculating the geometric dilution of precision of a pseudo-satellite system based on geometric configuration, characterized in that: The specific steps are as follows: Step 1: for a pseudo-satellite navigation and positioning system composed of n randomly distributed pseudo-satellites, improve the calculation of its geometric precision factor; n≥4； The improved calculation formula is: tr(·) represents the matrix trace, G T represents the transposed matrix, (G T G) * G T The adjoint matrix of G; |G T G|means G T The determinant of G; G is the transformed geometric measurement matrix: E i represents the altitude angle of pseudo-satellite i relative to the user position O; D i represents the azimuth of pseudo-satellite i relative to the user position O in the X-axis direction; (m xi , m yi ,m zi ) is the direction cosine of the unit vector pointing from the user's approximate position O to the ith pseudo-satellite, i.e., the position information of the coordinate point of pseudo-satellite i in the station-centered coordinate system established with the user's position O as the origin; Step 2: Use the determinant to calculate the geometric precision factor tr[(G T G) * ] and |G T G|represent and expand them respectively; According to the transformed geometric observation matrix G, we get: Thus we get |G T G|: At the same time, the adjoint matrix is ​​further obtained as: That is, we get tr[(G T G) * ]: tr[(G T G) * ]=B11+B22+B33+B44 Step 3: Use matrix theory, vectors and analytic geometry to solve tr[(G T G) * ] is expressed and expanded to obtain: V1 represents the volume of the parallelepiped formed by the three edges connecting any three pseudo-satellites and user O in the station-centered coordinate system established with user O as the origin; It represents the sum of the squares of the areas of the triangles formed by the projection points of any three pseudo-satellites in the XOY plane; It represents the sum of the squares of the areas of the triangles formed by the projection points of any three pseudo-satellites in the YOZ plane; It represents the sum of the squares of the triangle areas formed by the projection points of any three pseudo-satellites in the XOZ plane. The specific calculation formula is: 1≤i<j<k≤n; i, j, k represent any three different pseudolites in the pseudolites navigation and positioning system; Step 4: Use matrix theory, vectors and analytic geometry to solve |G T G| to represent and expand; get: |G T G|=6∑m xi m yj m zk V2 in, l represents the pseudo-satellite in the pseudo-satellite navigation and positioning system; V2 represents the volume of the tetrahedron formed by pseudo-satellites i, j, k, and l in the station center coordinate system; Step 5: Use the above representation and expand tr[(G T G) * ] and |G T G|Conversion formula, calculate the final improved GDOP; Step 6: For any pseudo-satellite system with any geometric configuration, the coordinates of each pseudo-satellite station center coordinate system are substituted into the final GDOP formula to obtain the geometric precision factor of the pseudo-satellite system; Changing the coordinates of the pseudo-satellite station center can change the angle relationship between lines and lines, and lines and surfaces in the pseudo-satellite system's geometric configuration, so that the GDOP value of the pseudo-satellite system is minimized, that is, the geometric layout optimization of the pseudo-satellite system can be achieved.

2. The method for calculating the geometric dilution of precision of a pseudo-satellite system based on geometric configuration according to claim 1, characterized in that: The geometric measurement matrix G of the conversion is obtained by improving the traditional geometric measurement matrix; specifically: The traditional geometric observation matrix is: By improving it, let m xi =cosE i dxD i 、m yi =cosE i sinD i 、m zi = sinE i ; They represent the position information of user O in the geocentric coordinate system respectively; xi, yi, zi represent the position information of pseudo-satellite i in the geocentric coordinate system respectively.

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