Single epoch gnss arbitrary direction positioning accuracy factor one-step prediction map method

Abstract

The application discloses a single-epoch GNSS arbitrary direction positioning precision factor one-step prediction map method, comprising the following steps: obtaining a double-difference observation cofactor matrix and a double-difference coefficient matrix according to GNSS observations, and obtaining a weighted coefficient matrix in a geocentric geodetic coordinate system by using a least square method; establishing a station-centered rectangular coordinate system with a mobile station as an origin, and converting the double-difference coefficient matrix and the weighted coefficient matrix in the geocentric geodetic coordinate system to the station-centered rectangular coordinate system; calculating the plane coordinates of the station-centered rectangular coordinates of a grid point; calculating a double-difference coefficient vector and a double-difference cofactor in a station-centered polar coordinate system; reconstructing and adding the weighted coefficient matrix in the station-centered rectangular coordinate system after any grid point, and calculating the arbitrary direction positioning precision factor value at the grid point; and constructing the arbitrary direction positioning precision factor one-step prediction map by using a bilinear interpolation method; and the application provides important method support for quickly and efficiently selecting GNSS observation satellites with the minimum positioning precision factor value and analyzing the GNSS positioning precision influence mechanism.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of satellite positioning technology, and particularly relates to a one-step prediction graph method for positioning precision factor of GNSS in any direction. BACKGROUND

[0002] The existing global satellite navigation system (GNSS) includes the Beidou satellite navigation system of China, the global positioning system of the United States, the global navigation satellite system of Russia and the Galileo system of the European Union. The satellites in orbit and available for positioning services total 120, and about 35 satellites can be observed at the same epoch in a certain position on the earth. Although a large number of observation satellites improve the precision and reliability of GNSS positioning, the high-dimensional matrix and ambiguity vector formed by a large number of observation satellites also lead to time-consuming single-epoch positioning. The positioning dilution of precision (PDOP) is a scalar factor that is easy to calculate and used to measure the three-dimensional positioning precision of GNSS. It also includes scalar factors used to measure two-dimensional positioning precision and one-dimensional positioning precision, i.e. horizontal dilution of precision (HDOP), east-west dilution of precision (EDOP), south-north dilution of precision (SDOP) and vertical dilution of precision (VDOP). In order to improve the GNSS positioning efficiency on the basis of ensuring a certain positioning precision, scholars have proposed a GNSS satellite selection method based on positioning precision factor. However, due to the lack of a method for quickly determining the distribution of observation satellites with the smallest positioning precision factor (PDOP, HDOP or VDOP), the existing GNSS satellite selection method and process is extremely time-consuming. For example, it takes 3268760 calculations of positioning precision factors of satellites to select 15 observation satellites with the smallest positioning precision factor from 25 observation satellites. In addition, the positioning precision factor of GNSS relative positioning is a function of the weighted double-difference coefficient matrix. The existing method for analyzing the influence of the positioning precision factor value of the weighted double-difference coefficient matrix is insufficient, and it is difficult to intuitively reveal the influence mechanism of the spatial distribution of observation satellites on GNSS positioning precision. SUMMARY

[0003] In view of the above technical deficiencies, the purpose of the present application is to provide a single epoch GNSS arbitrary direction positioning accuracy factor one-step prediction map method, which can quickly and efficiently select GNSS observation satellites with the smallest positioning accuracy factor value and provide important method support for analyzing the GNSS positioning accuracy influence mechanism.

[0004] To solve the above technical problems, the present application adopts the following technical scheme: the present application provides a single epoch GNSS arbitrary direction positioning accuracy factor one-step prediction map method, comprising:

[0005] Step S1: according to GNSS observations, a single epoch double difference mathematical model in the geocentric geodetic coordinate system is constructed, double difference observation cofactor matrix and double difference coefficient matrix in the geocentric geodetic coordinate system are obtained, and the weighted coefficient matrix in the geocentric geodetic coordinate system is obtained by using the least square method;

[0006] Step S2: a station-centered rectangular coordinate system with the mobile station as the origin is established, and the conversion matrix of the geocentric geodetic coordinate system and the station-centered rectangular coordinate system is constructed; the double difference coefficient matrix and the weighted coefficient matrix in the geocentric geodetic coordinate system are converted to the station-centered rectangular coordinate system;

[0007] Step S3: according to the station-centered rectangular coordinate system, a station-centered polar coordinate system representing the point position height angle is established; the station-centered polar coordinate system is gridized by equally dividing the polar angle and the polar radius length which are higher than the cutoff height angle, and the plane coordinates of the grid point station-centered rectangular coordinates are calculated;

[0008] Step S4: according to the station-centered polar coordinates and the height angle of the grid point and the reference satellite, the double difference coefficient vector and the double difference cofactor in the station-centered polar coordinate system of the grid point are calculated;

[0009] Step S5: combining the double difference observation cofactor matrix in step S1 and the double difference coefficient matrix in the station-centered rectangular coordinate system in step S2, the weighted coefficient matrix in the station-centered rectangular coordinate system after adding any grid point is reconstructed, and the arbitrary direction positioning accuracy factor value at the grid point is calculated;

[0010] Step S6: according to the plane coordinates of the station-centered rectangular coordinates of the grid point and the arbitrary direction positioning accuracy factor value at the grid point, the arbitrary direction positioning accuracy factor one-step prediction map is constructed by using the bilinear interpolation method.

[0011] Preferably, in step S1:

[0012] It is specified that the GNSS receiver of the mobile station and the GNSS receiver of the reference station observe the same satellite at the same time, and one of the observed satellites is the reference satellite, and based on the GNSS observations, one double difference pseudorange observation equation and one double difference carrier phase observation equation are formed respectively. m n m n ​​​a double-difference carrier phase observation equation; and a double-difference random model of GNSS observations, to jointly form a single-epoch double-difference mathematical model in the Earth-Centered Earth-Fixed coordinate system:

[0013] ;

[0014] ;

[0015] wherein, and are the expectation and variance of the double-difference pseudo-range observation respectively, and are the expectation and variance of the double-difference carrier phase observation respectively; and are the double-difference pseudo-range observation vector and the double-difference carrier phase observation vector respectively; and are the baseline vector in the Earth-Centered Earth-Fixed coordinate system and the n dimension double-difference ambiguity vector respectively; is the order double-difference coefficient matrix of , is the carrier wavelength, is the order unit matrix; is the double-difference observation cofactor matrix; and are the standard deviation of the undifferenced pseudo-range observation and the standard deviation of the undifferenced carrier phase observation respectively; , is the weight of the j th observed satellite, is the weight of the reference satellite, re is an integer and .

[0016] Preferably, in step S1, obtaining the weighted coefficient matrix in the Earth-Centered Earth-Fixed coordinate system by the least square method comprises:

[0017] using the least square method, obtaining the float solution of the baseline vector b and the ambiguity vector a and the variance-covariance matrix and of the baseline vector and the ambiguity vector:

[0018] ;

[0019] wherein, is the weighted coefficient matrix in the Earth-Centered Earth-Fixed coordinate system;

[0020] based on the above formula, obtaining the fixed solution of the baseline vector and its variance-covariance matrix :

[0021] ;

[0022] wherein, is the fixed solution of ambiguity vector, is the cross-covariance matrix of the float solution of ambiguity vector and the float solution of baseline vector; and are functions of the weighted coefficient matrix with respect to the Earth-Centered Earth-Fixed coordinate system.

[0023] Preferably, in step S2, establishing the station-centered rectangular coordinate system with the GNSS rover receiver as the coordinate origin O comprises:

[0024] taking the GNSS rover receiver as the coordinate origin O;

[0025] taking the direction of the ellipsoidal normal passing through the point O as the U-axis, wherein upward is positive and downward is negative;

[0026] taking the direction perpendicular to the U-axis and pointing to the short semi-axis of the reference ellipsoid as the N-axis, wherein northward is positive and southward is negative;

[0027] taking the direction perpendicular to the N-axis and the U-axis and pointing to the long semi-axis of the reference ellipsoid as the E-axis, wherein eastward is positive and westward is negative;

[0028] forming the station-centered rectangular coordinate system.

[0029] Preferably, in step S2, constructing the conversion matrix of the Earth-Centered Earth-Fixed coordinate system and the station-centered rectangular coordinate system comprises:

[0030] translating and rotating the coordinate origin O of the Earth-Centered Earth-Fixed coordinate system around the Z-axis, and then converting the Earth-Centered Earth-Fixed coordinate system XYZ into the station-centered rectangular coordinate system ENU, wherein the conversion formula and the conversion matrix are respectively:

[0031] ;

[0032] ;

[0033] wherein, and are the coordinates X, Y and Z of the Earth-Centered Earth-Fixed coordinate system respectively, , and are the coordinates E, N and U of the station-centered rectangular coordinate system respectively; R is the conversion matrix from the Earth-Centered Earth-Fixed coordinate system to the station-centered rectangular coordinate system, and are the geodetic longitude and the geodetic latitude of the GNSS rover receiver respectively.

[0034] Preferably, in step S2, converting the double difference coefficient matrix and weighted coefficient matrix in the geocentric Earth-fixed coordinate system to the station-centered rectangular coordinate system includes:

[0035] By transforming the matrix R The double difference coefficient matrix in the geocentric-fixed coordinate system B Convert to double difference coefficient matrix in station-centered rectangular coordinate system :

[0036] ;

[0037] According to the formula Japanese style The weighted coefficient matrix in the geocentric-fixed coordinate system Convert to weighted coefficient matrix in station-centered rectangular coordinate system :

[0038] ;

[0039] The specific formulas for calculating the positioning accuracy factor values ​​in each direction are as follows:

[0040] The formula for calculating the Positioning Accuracy Factor (PDOP) is:

[0041] ;

[0042] In the formula, for traces; if let ,in for The Middle x line, number y Column elements, x =1, 2, 3, y =1, 2, 3, then ;

[0043] The formulas for calculating the horizontal positioning accuracy factor (HDOP), vertical positioning accuracy factor (VDOP), east-west positioning accuracy factor (EDOP), and north-south positioning accuracy factor (SDOP) are as follows:

[0044] .

[0045] Preferably, step S3 includes:

[0046] Step S3.1: Based on the station-centered rectangular coordinate system, establish a station-centered polar coordinate system that can represent the elevation angle of the point:

[0047] The origin O of the station-centered rectangular coordinate system is taken as the pole;

[0048] The horizontal plane formed by the EON plane is used as the reference plane;

[0049] Draw a ray from point O along a direction parallel to the positive N-axis as the polar axis, with a length of... The edge of the polar radius is the polar radius on the plane. The angle by which the polar radius on the plane is rotated clockwise in the reference plane with the polar axis as the starting direction is the polar angle. Among them, polar angle Values ​​range from 0° to 360°;

[0050] Establish a station-centered polar coordinate system under a station-centered rectangular coordinate system;

[0051] Starting from the reference plane EON and along a direction perpendicular to the reference plane, the length... The angle by which the polar radius in three-dimensional space is rotated counterclockwise is the elevation angle. elevation angle Values ​​range from 0° to 90°; establish the polar radius length in three-dimensional space. With elevation angle The relationship between them is:

[0052] ;

[0053] In the formula, For dimensionless length; when hour, =0; when hour, =90; will have the same elevation angle And the length is The circle formed by rotating the polar radius of a point around the U-axis in three-dimensional space is projected perpendicularly onto the reference plane EON, forming a station-centered polar coordinate system that can characterize the elevation angle of the point. The length of the polar radius on the plane in the station-centered polar coordinate system is then... and elevation angle The relationship can be represented as:

[0054] ;

[0055] In the formula, Dimensionless length; polar radius length on a plane The polar radius length in three-dimensional space The projected length on the reference plane EON and Then, after this projection, the spatial position of a point can be represented using coordinates in the station-centered polar coordinate system. The elevation angle is ;

[0056] A station-centered polar coordinate system is formed to characterize the elevation angle of a point.

[0057] Step S3.2: Grid the geocentric polar coordinate system by dividing the polar angle above the cut-off height angle and the polar radius length into equal parts, and calculate the plane coordinates of each grid point based on the geocentric rectangular coordinate system:

[0058] In the area of , the polar angle and the polar radius length on the plane are divided into K parts respectively to form a grid composed of KxK grid points, so as to grid the geocentric polar coordinate system; wherein is the cut-off height angle, the polar angle is between , and the polar radius length on the plane is between ;

[0059] According to the coordinates of each grid point based on the geocentric polar coordinate system , the plane coordinates E and N of each grid point based on the geocentric rectangular coordinate system are calculated:

[0060] ;

[0061] In the formula, and are the polar radius length i , the polar angle and the coordinates E and N of the geocentric rectangular coordinate system of the i-th grid point, respectively, wherein, =1, 2,…, KxK. i

[0062] Preferably, in step S4, the double-difference coefficient vector and the double-difference cofactor of the grid point in the geocentric polar coordinate system are calculated according to the geocentric polar coordinate system and the elevation angle of the grid point and the reference satellite, including:

[0063] According to the geocentric polar coordinates of the i-th grid point i , the baseline vector non-difference coefficient vector of the i-th grid point in the geocentric polar coordinate system is obtained by using trigonometric functions, wherein =1, 2,…, KxK: i i In the formula, ,

[0064] and are the non-difference coefficients of the baseline vector E, N and U directions of the i-th grid point in the geocentric polar coordinate system, respectively.

[0065] i

[0066] ​​​​​​The coordinates of the selected reference satellite in the station-centered polar coordinate system are defined as follows: , and These are the polar radius and polar angle in the reference satellite station's polar coordinate system, respectively. For the elevation angle of the reference satellite, the baseline vector in the station-centered polar coordinate system of the reference satellite is the non-difference coefficient vector. for:

[0067] ;

[0068] In the formula, and These are the non-difference coefficients in the E, N, and U directions of the baseline vector in the polar coordinate system of the reference satellite station, respectively;

[0069] The double difference coefficient vector in the polar coordinate system of the grid point station center is obtained according to the above formula:

[0070] ;

[0071] In the formula, For the first i The double-difference coefficient vector in the polar coordinate system of each grid point, where... i =1, 2,…, K×K;

[0072] according to i The height angle of each grid point Obtain the i Double difference cofactor of each grid point ,in, i =1, 2,…, K×K:

[0073] .

[0074] Preferably, in step S5, reconstructing the weighted coefficient matrix in the station-centered rectangular coordinate system after adding any grid point, and calculating the positioning accuracy factor value in any direction at the grid point includes:

[0075] Reconstruct the double-difference coefficient matrix and double-difference observation cofactor matrix after adding any grid point or any observation satellite:

[0076] ;

[0077] In the formula, To add the first i The double difference coefficient matrix of the rectangular coordinate system and polar coordinate system of the back station center of each grid point; To add the first i The cofactor matrix of double-difference observations after each grid point For the original observation and the first i individual grid pointsn a vector of the mutual correlation factors, a vector of 1s, n a vector of 1s, i =1, 2,…, K×K.

[0078] the weighted coefficient matrix based on the geocentric rectangular coordinate system and the polar coordinate system after adding any one grid point or any one observation satellite is obtained based on the formula and

[0079] ;

[0080] wherein, is the weighted coefficient matrix of the geocentric rectangular coordinate system and the polar coordinate system after adding the kth grid point, wherein, i =1, 2,…, K×K. i Let

[0081] wherein is the element in the ith row and the jth column in x y =1, 2,3, x =1, 2, 3, the positioning accuracy factor PDOP value, the horizontal positioning accuracy factor HDOP value, the vertical positioning accuracy factor VDOP value, the east-west positioning accuracy factor EDOP value and the south-north positioning accuracy factor SDOP value after adding any one grid point or any one observation satellite are respectively calculated according to the formula of the directional positioning accuracy factor value, that is, y

[0082] ;

[0083] wherein, and are respectively the positioning accuracy factor PDOP value, the horizontal positioning accuracy factor HDOP value, the vertical positioning accuracy factor VDOP value, the east-west positioning accuracy factor EDOP value and the south-north positioning accuracy factor SDOP value after adding the kth grid point, wherein, i =1, 2,…, K×K. i Preferably, in step S6, the one-step prediction map of the directional positioning accuracy factor is constructed by using the bilinear interpolation method, which includes:

[0084] the plane coordinates E and N of the four grid points of each grid in the geocentric polar coordinate system under the geocentric rectangular coordinate system and the directional positioning accuracy factor values corresponding to the four grid points;

[0085]

[0086] ​​​​​​The bilinear interpolation method is used to interpolate the plane coordinates E and N of any position in each grid in the station-centered rectangular coordinate system and the arbitrary direction positioning precision factor value corresponding to the position, different colors are given to different arbitrary direction positioning precision factor values, and a one-step prediction map of the station-centered polar coordinate system GNSS arbitrary direction positioning precision factor is formed.

[0087] The application has the advantages that: the application constructs a single-element GNSS double-difference mathematical model according to pseudo-range and carrier phase observations, obtains a double-difference observation cofactor matrix and a double-difference coefficient matrix in the geocentric geodetic coordinate system; the double-difference coefficient matrix in the geocentric geodetic coordinate system is converted into a double-difference coefficient matrix in the station-centered rectangular coordinate system through the establishment of a coordinate system conversion matrix; the station-centered polar coordinate system is constructed according to the station-centered rectangular coordinate system and is gridded uniformly, the station-centered rectangular coordinate system plane coordinates, double-difference cofactors and double-difference coefficient vectors in the station-centered polar coordinate system of each grid point are calculated; the weighted coefficient matrix after adding any grid point is calculated, the arbitrary direction positioning precision factor values of each grid point are obtained, according to the original double-difference observation cofactor matrix and the double-difference coefficient matrix in the station-centered rectangular coordinate system, and the double-difference coefficient matrix and the double-difference observation cofactor matrix of the station-centered rectangular coordinate system and the polar coordinate system after adding any observation satellite or any grid point; the one-step prediction map of the arbitrary direction positioning precision factor value is formed by using the bilinear interpolation method according to the plane coordinates in the station-centered rectangular coordinate system and the arbitrary direction positioning precision factor values of each grid point, which provides important method support for quickly and efficiently selecting GNSS observation satellites with the minimum positioning precision factor value and analyzing the GNSS positioning precision influence mechanism. BRIEF DESCRIPTION OF DRAWINGS

[0088] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the following will briefly introduce the drawings needed to be used in the embodiments or prior art description. Obviously, the drawings in the following description only show some embodiments of the present application, and for those skilled in the art, other drawings can also be obtained without creative labor.

[0089] Figure 1 The flowchart provided by the present application is shown in the figure;

[0090] Figure 2 The relative relationship between the geocentric geodetic coordinate system XYZ and the station-centered rectangular coordinate system ENU is shown in the figure;

[0091] Figure 3 The GNSS station-centered polar coordinate system representing different elevation angles is shown in the figure;

[0092] Figure 4 The gridding of the GNSS station-centered polar coordinate system is shown in the figure;

[0093] Figure 5 One-step prediction map of PDOP based on single epoch Beidou satellite navigation system;

[0094] Figure 6 One-step prediction map of HDOP based on single epoch Beidou satellite navigation system;

[0095] Figure 7 One-step prediction map of EDOP based on single epoch Beidou satellite navigation system;

[0096] Figure 8 One-step prediction map of SDOP based on single epoch Beidou satellite navigation system;

[0097] Figure 9 One-step prediction map of VDOP based on single epoch Beidou satellite navigation system. DETAILED DESCRIPTION

[0098] The technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, rather than all the embodiments of the present application. Based on the embodiments in the present application, all other embodiments obtained by a person of ordinary skill in the art without creative work are within the protection scope of the present application.

[0099] Embodiment: As shown in the figure, the present application provides a one-step prediction map method of single epoch GNSS positioning precision factor in any direction, which specifically comprises: Figure 1

[0100] Step S1: according to GNSS observation, a single epoch double difference mathematical model under the Earth-centered Earth-fixed coordinate system is constructed, double difference observation cofactor matrix and double difference coefficient matrix under the Earth-centered Earth-fixed coordinate system are obtained, and the weighted coefficient matrix under the Earth-centered Earth-fixed coordinate system is obtained by using the least square method.

[0101] Specifically, it is assumed that the mobile station GNSS receiver and the reference station GNSS receiver observe to n satellites at the same time, one of the n satellites is selected as the reference satellite, then the GNSS observation and the three-dimensional space coordinates of the GNSS receiver and the observation satellite based on the Earth-centered Earth-fixed coordinate system are used to form n-1 double difference pseudo-range observation equations and n double difference carrier phase observation equations respectively; and the double difference random model of the GNSS observation is combined to form a single epoch double difference mathematical model under the Earth-centered Earth-fixed coordinate system. m n m n

[0102] (1);

[0103] ​​​​​ (2);

[0104] wherein, and are the expectation and variance of the double-difference pseudorange observation respectively, and are the expectation and variance of the double-difference carrier phase observation respectively; and are the double-difference pseudorange observation vector and the double-difference carrier phase observation vector respectively; and are the baseline vector and n a dimensional double-difference ambiguity vector in the Earth-Centered Earth-Fixed coordinate system respectively; is a dimensional double-difference coefficient matrix in the Earth-Centered Earth-Fixed coordinate system, is the carrier wavelength, is a dimensional identity matrix; is the double-difference observation cofactor matrix; and are the standard deviation of the undifferenced pseudorange observation and the standard deviation of the undifferenced carrier phase observation respectively; , is the weight of the j th observed satellite, is the weight of the reference satellite, re is an integer and .

[0105] According to equation (1) and equation (2), the float solutions of the baseline vector b and the ambiguity vector a and their variance-covariance matrices and can be obtained by using the least squares method:

[0106] (3);

[0107] wherein, is the weighted coefficient matrix in the Earth-Centered Earth-Fixed coordinate system. According to equation (3), the fixed solutions of the baseline vector and its variance-covariance matrix can be obtained by using the least squares ambiguity decorrelation adjustment method:

[0108] (4);

[0109] wherein, is the fixed solution of the ambiguity vector, ​​is the cross-covariance matrix of the ambiguity vector float solution and the baseline vector float solution. As can be seen from equation (3) and equation (4), both the baseline vector float solution and the baseline vector fixed solution are related to the weighted coefficient matrix in the geocentric geodetic coordinate system .

[0110] Step S2: Establish a station-centered rectangular coordinate system with the flow station as the origin, construct the conversion matrix of the geocentric geodetic coordinate system and the station-centered rectangular coordinate system; convert the double-difference coefficient matrix and the weighted coefficient matrix in the geocentric geodetic coordinate system to the station-centered rectangular coordinate system.

[0111] Specifically, it includes:

[0112] Step S2.1: Establish a station-centered rectangular coordinate system with the flow station as the origin.

[0113] Take the GNSS flow station receiver as the coordinate origin O; take the direction of the ellipsoid normal passing through point O as the U axis, wherein upward is positive and downward is negative; take the direction perpendicular to the N axis and the U axis and pointing to the reference ellipsoid short semi-axis as the N axis, wherein north is positive and south is negative; take the direction perpendicular to the N axis and the U axis and pointing to the reference ellipsoid long semi-axis as the E axis, wherein east is positive and west is negative; construct a station-centered rectangular coordinate system based on the GNSS flow station receiver, as shown in Figure 2 .

[0114] Step S2.2: Construct the conversion matrix of the geocentric geodetic coordinate system and the station-centered rectangular coordinate system.

[0115] As shown in Figure 2 , the geocentric geodetic coordinate system coordinate origin O is translated and rotated around the Z axis, and the geocentric geodetic coordinate system XYZ is converted into the station-centered rectangular coordinate system ENU, wherein the conversion formula and the conversion matrix are respectively:

[0116] (5)

[0117] (6)

[0118] In the formula, and are the coordinates X, Y and Z in the geocentric geodetic coordinate system, and are the coordinates E, N and U in the station-centered rectangular coordinate system; R is the conversion matrix from the geocentric geodetic coordinate system to the station-centered rectangular coordinate system, and are the geodetic longitude and geodetic latitude of the flow station GNSS receiver.

[0119] Step S2.3: Convert the double-difference coefficient matrix in the geocentric geodetic coordinate system to the double-difference coefficient matrix in the station-centered rectangular coordinate system.

[0120] with the conversion matrix R , the double-difference coefficient matrix in the geocentric geodetic coordinate system in formula (1) can be converted into the double-difference coefficient matrix in the geocentric rectangular coordinate system B :

[0121] (7);

[0122] According to formula and formula (7), the weighted coefficient matrix in the geocentric geodetic coordinate system is converted into the weighted coefficient matrix in the geocentric rectangular coordinate system :

[0123] (8);

[0124] wherein, the positioning accuracy factor PDOP is an easily calculated scalar factor, and the value size can reflect the positioning accuracy, that is, the smaller the value is, the higher the positioning accuracy is, and the specific formula is as follows:

[0125] (9);

[0126] In the formula, is the trace of ; if , wherein is the element in the i-th row and the j-th column of , x =1, 2, 3, y =1, 2, 3, then x ; y

[0127] Then the calculation formula of the horizontal positioning accuracy factor HDOP value, the vertical positioning accuracy factor VDOP value, the east-west positioning accuracy factor EDOP value and the south-north positioning accuracy factor SDOP value can be written as:

[0128] (10).

[0129] Step S3: According to the geocentric rectangular coordinate system, a geocentric polar coordinate system representing the point height angle is established; the grid point geocentric rectangular coordinate plane coordinates are calculated by dividing the polar angle and polar radius length grid geocentric polar coordinate system higher than the cutoff height angle.

[0130] Specifically, it includes:

[0131] Step S3.1: According to the geocentric rectangular coordinate system, a geocentric polar coordinate system of point height angle is established:

[0132] ​​​With the origin O of the station-centered rectangular coordinate system as the pole; with the horizontal plane formed by the EON plane as the reference plane; and drawing a ray from point O along the positive direction parallel to the N-axis as the polar axis, with a length... The edge of the polar radius is the polar radius on the plane. The angle by which the polar radius on the plane is rotated clockwise in the reference plane with the polar axis as the starting direction is the polar angle. Among them, polar angle Values ​​range from 0° to 360°. Establish a station-centered polar coordinate system within a station-centered rectangular coordinate system, such as... Figure 4 As shown;

[0133] Because GNSS satellites operate in three-dimensional space, in addition to two-dimensional polar coordinates, the elevation angle of the point must also be considered. Therefore, it is stipulated that a certain length is drawn from the reference plane EON as the starting direction and along a direction perpendicular to the reference plane. The angle by which the polar radius in three-dimensional space is rotated counterclockwise (i.e., towards the U-axis) is the elevation angle. elevation angle Values ​​range from 0° to 90°; establish the polar radius length in three-dimensional space. With elevation angle The relationship between them is:

[0134] (11);

[0135] In the formula, For dimensionless length; when hour, =0; when hour, =90; will have the same elevation angle And the length is The circle formed by rotating the polar radius around the U-axis in three-dimensional space is projected perpendicularly onto the reference plane EON, forming a station-centered polar coordinate system that can characterize the elevation angle of a point. The relationship between the polar radius length and the elevation angle on the plane in the station-centered polar coordinate system can then be expressed as:

[0136] (12);

[0137] In the formula, Dimensionless length; polar radius length on a plane The polar radius length in three-dimensional space The projected length on the reference plane EON and Then, after this projection, the spatial position of a point can be represented using coordinates in the station-centered polar coordinate system. The elevation angle is ;

[0138] Step S3.2: By equally dividing the polar angle and polar radius length above the cutoff elevation angle, the station-centered polar coordinate system is gridded, and the planar coordinates of each grid point based on the station-centered rectangular coordinate system are calculated.

[0139] From equation (12), it can be seen that the sum of the polar radius projection length of a certain elevation angle and the dimensionless elevation angle is 90°. Therefore, after this projection, the spatial position of a point can be represented as follows: Figure 3 The coordinates of the station-centered polar coordinate system shown are expressed as follows: The elevation angle is .

[0140] against Figure 2 middle The area (of which) (The cutoff elevation angle), obtained by considering the polar angle of the station's polar coordinate system. and the polar radius length on the plane Divide each part into K equal parts to form a grid consisting of K×K grid points, thus gridding the station-centered polar coordinate system. The pink solid line represents the cutoff elevation angle. Projection onto the reference plane EON. Wherein, "the polar radius length on the plane..." "Dividing into K equal parts" is equivalent to "dividing the elevation angles into K equal parts". Divide into K equal parts.

[0141] Since the station-centered polar coordinate system is constructed based on the station-centered rectangular coordinate system, the coordinates of each grid point based on the station-centered polar coordinate system can be used to determine the coordinates of the grid points. Calculate the planar coordinates E and N of each grid point based on the station-centered rectangular coordinate system:

[0142] (13);

[0143] In the formula, and The first i Polar radius length of each grid point on the plane Polar angle and the coordinates E and N based on the station-centered rectangular coordinate system ,in, i =1, 2,…, K×K.

[0144] Step S4: Calculate the double difference coefficient vector and double difference cofactor in the grid point polar coordinate system based on the grid point and the reference satellite's station-centered polar coordinates and elevation angle.

[0145] Specifically, according to the first i The polar coordinates of the station center of each grid point The first trigonometric function can be used to obtain the second trigonometric function. i Baseline vector and non-difference coefficient vector in polar coordinate system of each grid point ,in,i =1,2,…, K×K:

[0146] (14)

[0147] where, and are the undifferenced coefficients of the baseline vector E, N, U direction of the geodetic polar coordinate system of the i-th grid point, respectively. i

[0148] The coordinates of the selected reference satellite in the station-centered polar coordinate system in formula (1) are defined as , and are the polar radius length and polar angle of the reference satellite in the station-centered polar coordinate system, respectively, is the elevation angle of the reference satellite, and the undifferenced coefficient vector of the baseline vector of the reference satellite in the station-centered polar coordinate system is : (15);

[0149] where, and are the undifferenced coefficients of the baseline vector E, N, U direction of the reference satellite station-centered polar coordinate system, respectively.

[0150] According to formula (14) and formula (15), the double-difference coefficient vector in the station-centered polar coordinate system of the grid point can be obtained as

[0151] (16);

[0152] where, is the double-difference coefficient vector in the station-centered polar coordinate system of the i-th grid point, wherein, i =1, 2,…, K×K. i According to the elevation angle of the i-th grid point

[0153] , the double-difference correlation factor of the i-th grid point i can be obtained, wherein, =1, 2,…, K×K: i i (17)。

[0154]

[0155] Step S5: Combine the double-difference observation correlation factor matrix in step S1 and the double-difference coefficient matrix in the station-centered rectangular coordinate system in step S2 to reconstruct the weighted coefficient matrix in the station-centered rectangular coordinate system after adding any grid point. According to formula (9) and formula (10), the positioning accuracy factor value in any direction at the grid point is calculated.

[0156] ​​​​Specifically, according to equation (13), the constructed station-centered rectangular coordinate system and station-centered polar coordinate system have the same reference on the plane EON. Therefore, according to the first equation... i The polar coordinates of the station center of each grid point and elevation angle ,in, i =1, 2,…, K×K, using equations (2), (7), (16), and (17), the double-difference coefficient matrix and the double-difference observation cofactor matrix after adding any grid point (or any observation satellite) can be reconstructed:

[0157] (18);

[0158] In the formula, To add the first i The double difference coefficient matrix of the rectangular coordinate system and polar coordinate system of the back station center of each grid point; To add the first i The cofactor matrix of double-difference observations after each grid point For the original observation and the first i individual grid points n dimensional cofactor vector, For elements all of 1 n A column vector of dimension, where, i =1, 2,…, K×K.

[0159] According to equation (18), the weighting coefficient matrix of the station-centered rectangular coordinate system and polar coordinate system after adding any grid point or any observation satellite can be obtained:

[0160] (19);

[0161] In the formula, To add the first i The weighted coefficient matrix based on the station-centered rectangular coordinate system and polar coordinate system after each grid point, where, i =1, 2,…, K×K.

[0162] make ,in for The Middle x line, number y Column elements, x =1, 2, 3, y=1, 2, 3, the positioning accuracy factor values of each direction are calculated according to the formula (9) and the formula (10), and the positioning accuracy factor PDOP value, the horizontal positioning accuracy factor HDOP value, the vertical positioning accuracy factor VDOP value, the east-west positioning accuracy factor EDOP value and the south-north positioning accuracy factor SDOP value after adding any grid point or any observation satellite are respectively:

[0163]

[0164] In the formula, and are respectively the PDOP, HDOP, VDOP, EDOP and SDOP values after adding the first i grid point, wherein, i =1, 2, …, K×K.

[0165] Step S6: according to the plane coordinates of the station-centered rectangular coordinates of the grid points and the arbitrary direction positioning accuracy factor values at the grid points, a one-step prediction map of the arbitrary direction positioning accuracy factor is constructed by using the bilinear interpolation method.

[0166] Specifically, it includes:

[0167] (1) the plane coordinates E and N of the K×K grid points in the station-centered rectangular coordinate system are calculated by using the formula (5);

[0168] (2) the arbitrary direction positioning accuracy factor values (PDOP, HDOP, VDOP, EDOP or SDOP) after adding the first i grid point are calculated by using the formula (12) to the formula (20), wherein, i =1, 2, …, K×K;

[0169] (3) according to the plane coordinates E and N of the four grid points in the station-centered rectangular coordinate system and the arbitrary direction positioning accuracy factor values corresponding to the four grid points in the station-centered polar coordinate system, the plane coordinates E and N of any position in each grid in the station-centered rectangular coordinate system and the arbitrary direction positioning accuracy factor value corresponding to the position are interpolated by using the bilinear interpolation method, and different colors are given to different arbitrary direction positioning accuracy factor values, so as to form a one-step prediction map of the arbitrary direction positioning accuracy factor in the station-centered polar coordinate system, as shown in the formula (21) to the formula (25), which are respectively the PDOP, HDOP, EDOP, SDOP and VDOP one-step prediction maps of the single-element Beidou satellite navigation system, and the cut-off elevation angles are all Figures 5 to 9 . .

[0170] ​Obviously, many modifications and variations of the present application are possible in light of the above teachings. It is, therefore, to be understood that within the scope of the appended claims and their equivalents, the application can be practiced otherwise than as specifically described.

Claims

1. A single-epoch GNSS all-direction positioning precision factor one-step prediction map method, characterized in that, Comprise: Step S1: according to GNSS observation, construct the single epoch double difference mathematical model under the earth fixed coordinate system, obtain the double difference observation cofactor matrix and the double difference coefficient matrix under the earth fixed coordinate system, obtain the weighted coefficient matrix under the earth fixed coordinate system by using the least square method; Step S2: establish the station heart rectangular coordinate system with the mobile station as the origin, construct the conversion matrix of the earth fixed coordinate system and the station heart rectangular coordinate system;The double difference coefficient matrix and the weighted coefficient matrix under the earth fixed coordinate system are converted to the station heart rectangular coordinate system; Step S3: according to the station heart rectangular coordinate system, establish the station heart polar coordinate system representing the height angle of point position;The grid point station heart rectangular coordinate plane coordinate is calculated by dividing the polar angle and polar radius length above the cutoff height angle and griding the station heart polar coordinate system, which comprises: Step S3.1: according to the station heart rectangular coordinate system, establish the station heart polar coordinate system which can represent the height angle of point position: Take the origin O of the station heart rectangular coordinate system as the pole; Take the horizontal plane formed by the EON plane as the reference plane; A polar axis is drawn from point O in the positive direction of the N-axis to a length of The polar radius on the plane is the side of the polar triangle with a length of The polar angle is the angle in the reference plane that the polar radius on the plane is turned in the clockwise direction from the polar axis with a value of 0° to 360°; Form the station heart polar coordinate system of the station heart rectangular coordinate system; Starting from the reference plane EON and along a direction perpendicular to the reference plane, the length... The angle by which the polar radius in three-dimensional space is rotated counterclockwise is the elevation angle. elevation angle Values ​​range from 0° to 90°; establish the polar radius length in three-dimensional space. With elevation angle The relationship between them is: ； In the formula, is a dimensionless length; when , ; when , ; the three-dimensional space with the same height angle and length is rotated around the U-axis for one revolution, and the vertical projection of the circle formed to the reference plane EON forms the station polar coordinate system that can represent the point height angle, then the relationship between the polar radius length and the height angle on the plane in the station polar coordinate system can be expressed as: ； where is the dimensionless length; the polar radius length on the plane is the polar radius length in three-dimensional space is the projected length on the reference plane EON and ; then the spatial position of a point after the projection is expressed by the coordinates of the station-centered polar coordinate system as where the altitude angle is ; Form the station heart polar coordinate system which can represent the height angle of point position; Step S3.2: the station heart polar coordinate system is gridded by dividing the polar angle and polar radius length above the cutoff height angle, and the plane coordinate of each grid point based on the station heart rectangular coordinate system is calculated: In the area, by the polar angle and the polar radius length on the plane are divided into K parts respectively, forming a grid composed of K×K grid points, in order to grid the polar coordinate system; wherein is the cut-off height angle, the polar angle between , the polar radius length on the plane between ; calculating plane coordinates E and N of each grid point based on the geocentric rectangular coordinate system according to the coordinates of each grid point based on the geocentric polar coordinate system , calculating plane coordinates E and N of each grid point based on the geocentric rectangular coordinate system according to the coordinates of each grid point based on the geocentric polar coordinate system ； wherein, and are the polar radius length, the polar angle and the coordinates E and N on the plane of the Kthgrid point, respectively, i i wherein,​​​​​ Step S4: according to the station heart polar coordinate and height angle of the grid point and reference satellite, the double difference coefficient vector and double difference cofactor under the station heart polar coordinate system are calculated, which comprises: According to the first i grid point, the polar coordinates of the station center , the baseline vector non-difference coefficient vector of the first i grid point in the polar coordinate system of the station center is obtained by using trigonometric functions , wherein i =1, 2,…, K×K: ； In the formula, , and The first i The non-difference coefficients of the baseline vectors E, N, and U directions in the polar coordinate system of each grid point station; The coordinates of the selected reference satellite in the station-centered polar coordinate system are defined as , and are the polar radius length and the polar angle of the reference satellite in the station-centered polar coordinate system, respectively, is the elevation angle of the reference satellite, then the baseline vector undifferenced coefficient vector of the reference satellite in the station-centered polar coordinate system is . ； wherein, , and are the undifferenced coefficients of the baseline vector E, N, U in the reference satellite station body-fixed polar coordinate system. The double difference coefficient vector under the station heart polar coordinate system is obtained according to the above formula: ； In the formula, For the first i The double-difference coefficient vector in the polar coordinate system of each grid point, where... i =1, 2,…, K×K; According to i the elevation angle of the grid point obtain the double difference cofactor of the i grid point wherein i = 1, 2, …, K x K: ； Step S5: combining the double difference observation cofactor matrix in step S1 and the double difference coefficient matrix under the station heart rectangular coordinate system in step S2, the weighted coefficient matrix under the station heart rectangular coordinate system after adding any grid point is reconstructed, and the directional positioning accuracy factor value at the grid point is calculated; Step S6: according to the plane coordinate of the station heart rectangular coordinate of the grid point and the directional positioning accuracy factor value at the grid point, the directional positioning accuracy factor one step prediction map is constructed by using the bilinear interpolation method.

2. A single epoch GNSS all-direction positioning precision factor one-step prediction map method according to claim 1, characterized in that, In step S1: Setting the flow station GNSS receiver and reference station GNSS receiver observe at the same time m The GNSS observation data of the two satellites, and one of the observation satellites is a reference satellite, based on the GNSS observation data, respectively form n = m -1 double difference pseudorange observation equation and n Double difference carrier phase observation equation; combined with the double difference stochastic model of GNSS observation data to form a single epoch double difference mathematical model in the geocentric coordinate system: ； where and are the mean and variance of the double-difference pseudorange observation, respectively, and are the mean and variance of the double-difference carrier phase observation, respectively; and are the double-difference pseudorange observation vector and the double-difference carrier phase observation vector, respectively; and are the baseline vector in the Earth-Centered Earth-Fixed (ECEF) coordinate system and n is the is the order double-difference coefficient matrix in the ECEF coordinate system, , is the carrier wavelength, is the order identity matrix; is the double-difference observation cofactor matrix; and are the standard deviation of the undifferenced pseudorange observation and the undifferenced carrier phase observation, respectively; , , , is the weight of the j th observed satellite, is the weight of the reference satellite, re is an integer and .

3. A single epoch GNSS all-direction positioning precision factor one-step prediction map method according to claim 2, characterized in that, In step S1, the weighted coefficient matrix under the earth fixed coordinate system is obtained by using the least square method Comprise: Using the least squares method, the baseline vector b and the float solution for the ambiguity vector a and and its variance-covariance matrix and are obtained.​ ； In the formula, is a weighted coefficient matrix in the geocentric geodetic coordinate system; Based on the above formula, the fixed solution of the baseline vector is obtained by using the least squares ambiguity reduction adjustment method and the variance-covariance matrix : ； where is the fixed ambiguity vector solution, is the float ambiguity vector solution and the cross-covariance matrix of the float baseline vector solution; and are functions of the weighted coefficient matrix with respect to the Earth-Centered Earth-Fixed coordinate system.

4. A single epoch GNSS all-direction positioning precision factor one-step prediction map method according to claim 3, characterized in that, In step S2, the station heart rectangular coordinate system with the mobile station as the origin comprises: Take the GNSS mobile station receiver as the coordinate origin O; Take the direction of the ellipsoid normal through point O as the U axis, wherein positive is upward and negative is downward; Take the direction perpendicular to the N axis and the U axis and pointing to the reference ellipsoid long half axis as the E axis, wherein positive is east and negative is west; Form the station heart rectangular coordinate system. In step S2, the conversion matrix of the earth fixed coordinate system and the station heart rectangular coordinate system comprises:

5. A single epoch GNSS all-direction positioning precision factor one-step prediction map method according to claim 4, characterized in that, After translating the earth fixed coordinate system coordinate origin O and rotating around the Z axis, the earth fixed coordinate system XYZ is converted into the station heart rectangular coordinate system ENU, wherein the conversion formula and the conversion matrix are respectively: In step S2, the double difference coefficient matrix and the weighted coefficient matrix under the earth fixed coordinate system are converted to the station heart rectangular coordinate system, which comprises: ； wherein, , and are the coordinates X, Y and Z of the geocentric geodetic coordinate system, , and are the coordinates E, N and U of the topocentric rectangular coordinate system; R is the conversion matrix from the geocentric geodetic coordinate system to the topocentric rectangular coordinate system, and are the geodetic longitude and geodetic latitude of the rover GNSS receiver.

6. A single-epoch GNSS all-direction positioning precision factor one-step prediction map method according to claim 5, characterized in that, ​ By means of a conversion matrix R The double-difference coefficient matrix in the geocentric geodetic coordinate system B is converted into a double-difference coefficient matrix in the geocentric geodetic coordinate system : ； According to the formula and the formula , the weighted coefficient matrix in the geocentric geodetic coordinate system is converted into the weighted coefficient matrix in the geocentric rectangular coordinate system : ； The formula for calculating the directional positioning accuracy factor value is specifically: The formula for calculating the positioning accuracy factor PDOP value is: ； wherein is trace; if wherein is the element in the i-th row and j-th column of the matrix x , the element in the i-th row and j-th column of the matrix y , the element in the i-th row and j-th column of the matrix x = 1, 2, 3, y = 1, 2, 3, then ; The formulas for calculating the horizontal positioning accuracy factor HDOP value, the vertical positioning accuracy factor VDOP value, the east-west positioning accuracy factor EDOP value, and the south-north positioning accuracy factor SDOP value are respectively: 。 7. A single-epoch GNSS all-direction positioning precision factor one-step prediction map method according to claim 1, characterized in that, In step S5, the weighted coefficient matrix in the geocentric rectangular coordinate system after adding any grid point is reconstructed in combination with the double-difference observation cofactor matrix in step S1 and the double-difference coefficient matrix in the geocentric rectangular coordinate system in step S2, and the directional positioning accuracy factor value at the grid point is calculated, including: The double-difference coefficient matrix and the double-difference observation cofactor matrix after adding any grid point or any observation satellite are reconstructed: ； In the formula, To add the first i The double difference coefficient matrix of the rectangular coordinate system and polar coordinate system of the back station center of each grid point; To add the first i The cofactor matrix of double-difference observations after each grid point For the original observation and the first i individual grid points n dimensional cofactor vector, For elements all of 1 n A column vector of dimension, where, i =1, 2,…, K×K; Based on the formula and The weighted coefficient matrix of the station-centered rectangular coordinate system and the polar coordinate system after adding any grid point or any observation satellite is obtained: ； In the formula, To add the first i The weighted coefficient matrix of the station-centered rectangular coordinate system and polar coordinate system after each grid point, where... i =1, 2,…, K×K; Let where is the x element y in the i-th row and j-th column of matrix A, x =1, 2, 3, y =1, 2, 3, then according to the directional positioning accuracy factor value calculation formula, the positioning accuracy factor PDOP value, the horizontal positioning accuracy factor HDOP value, the vertical positioning accuracy factor VDOP value, the east-west positioning accuracy factor EDOP value and the south-north positioning accuracy factor SDOP value after adding any grid point or any observation satellite are respectively: ； In the formula, , , , and Add the first one respectively i The values ​​of the positioning accuracy factor PDOP, horizontal positioning accuracy factor HDOP, vertical positioning accuracy factor VDOP, east-west positioning accuracy factor EDOP, and north-south positioning accuracy factor SDOP after each grid point are as follows: i =1, 2,…, K×K.

8. A single-epoch GNSS all-direction positioning precision factor one-step prediction map method according to claim 1, characterized in that, In step S6, the one-step prediction map of the directional positioning accuracy factor is constructed by using the bilinear interpolation method, including: According to the plane coordinates E and N in the geocentric rectangular coordinate system of each grid of the four grid points in the geocentric polar coordinate system and the directional positioning accuracy factor values corresponding to the four grid points; The plane coordinates E and N in the geocentric rectangular coordinate system of any position in each grid and the directional positioning accuracy factor value corresponding to the position are interpolated by using the bilinear interpolation method, different colors are given to different directional positioning accuracy factor values, and the one-step prediction map of the directional positioning accuracy factor in the geocentric polar coordinate system is formed.

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