CN114025304B - Parameter fine-tuning processing method for positioning singular or approximate singular problem - Google Patents
Parameter fine-tuning processing method for positioning singular or approximate singular problem Download PDFInfo
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04W—WIRELESS COMMUNICATION NETWORKS
- H04W4/00—Services specially adapted for wireless communication networks; Facilities therefor
- H04W4/02—Services making use of location information
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- G01S5/00—Position-fixing by co-ordinating two or more direction or position line determinations; Position-fixing by co-ordinating two or more distance determinations
- G01S5/02—Position-fixing by co-ordinating two or more direction or position line determinations; Position-fixing by co-ordinating two or more distance determinations using radio waves
- G01S5/0278—Position-fixing by co-ordinating two or more direction or position line determinations; Position-fixing by co-ordinating two or more distance determinations using radio waves involving statistical or probabilistic considerations
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Abstract
The invention discloses a parameter fine-tuning processing method for positioning singular or approximate singular problems, which comprises the following steps: after a positioning solution singularity or approximate singularity problem occurs by directly using a least square method, counting the total number n of the current communicable base stations; if n is less than or equal to the number of the empirical base stations, judging the topological constraint relationship, and finding m base stations which do not meet the topological constraint relationship; if n-m is more than or equal to the minimum base station number, performing position estimation based on a least square method; if n-m and n are both smaller than the minimum base station number, performing dimensionality reduction estimation, otherwise, selecting q base stations from the m base stations, fine-tuning local coordinates of the q base stations, and performing position estimation based on a least square method, wherein q is less than or equal to m, and n-m + q is greater than or equal to the minimum base station number; and if n is larger than the number of the empirical base stations, fine adjustment is carried out on all coordinates of all current communicable base stations, and then position estimation is carried out based on a least square method. The invention can obtain credible and more accurate positioning solution.
Description
Technical Field
The invention belongs to the technical field of microwave and millimeter wave multi-base station communication and positioning, and particularly relates to a parameter fine-tuning processing method for positioning singular or approximate singular problems.
Background
A User Equipment (UE) may communicate with a plurality of Base Stations (BSs) to obtain time, distance, or angle related information, and may further perform positioning by a geometric positioning method, such as time of arrival (TOA), time difference of arrival (TDOA), angle of arrival (AOA). The Least Square Method (LSM) is the most common tool for solving a set of over-determined equations, however, when the LSM performs position estimation, matrix inversion operation is involved, and the situation of positioning solution singularity or approximate singularity occurs. Aiming at the problem of LSM positioning solution singularity or approximate singularity, an effective processing means is lacked at present, so that the positioning result is unreliable when the LSM is singularity or approximate singularity.
In order to solve the above problem, it is necessary to provide a method for solving the problem of locating solution singularities or near singularities.
Disclosure of Invention
In order to solve the technical problems in the background art, the present invention aims to provide a parameter fine tuning processing method for positioning singular or near singular problems.
In order to achieve the purpose, the invention adopts the technical scheme that: a parameter fine-tuning processing method for positioning singular or approximate singular problems comprises the following steps:
1) after a positioning solution singularity or approximate singularity problem occurs by directly using a Least Square Method (LSM), counting the total number n of the current communicable base stations;
2) comparing the total number n of the current communicable base stations with the number of experience base stations, if n is less than or equal to the number of experience base stations, judging the topological constraint relationship of all the current communicable base stations, and finding m base stations which do not meet the topological constraint relationship, so that the position estimation of a user with n-m base stations can be determined; if n-m is larger than or equal to the currentDimensional position estimation minimum number of base stations, i.e. coefficient matrix tr (A) formed by n-m base stations T A) Performing Least Square Method (LSM) position estimation to obtain a non-singular and credible positioning solution if the sequence is full; if n-m is less than the minimum base station number of the current dimension position estimation and n is less than the minimum base station number of the current dimension position estimation, the coefficient matrix tr (A) formed by n-m base stations T A) If the rank is not full and n is less than the minimum base station number of the current dimension position estimation, performing dimension reduction estimation and re-entering the topological constraint relation judgment and the subsequent steps; if n-m is less than the minimum base station number of the current dimension position estimation and n is more than or equal to the minimum base station number of the current dimension position estimation, the coefficient matrix tr (A) formed by the n-m base stations T A) If the rank is not full and n is larger than or equal to the minimum base station number estimated at the current dimension position, q base stations are selected from m base stations, q is not larger than m, and n-m + q is not smaller than the minimum base station number estimated at the current dimension position, one or more coordinates of the q base stations are subjected to fine adjustment, and then Least Square Method (LSM) position estimation is carried out to obtain a non-singular and credible positioning solution;
if n is larger than the number of the empirical base stations, fine adjustment is carried out on all coordinates of all current communicable base stations, and then Least Square Method (LSM) position estimation is carried out, so that a non-singular and credible positioning solution is obtained.
Further, the number of the empirical base stations is 3 times of the minimum number of the current dimension position estimation base stations.
Further, the minimum number of base stations for estimating the current dimension position specifically is:
the Least Square Method (LSM) based on time of arrival (TOA) is used for estimating the minimum base station number of the three-dimensional position of a plurality of base stations to be 4, estimating the minimum base station number of the two-dimensional position to be 3 and estimating the minimum base station number of the one-dimensional position to be 2;
the method comprises the following steps that a Least Square Method (LSM) based on time difference of arrival (TDOA) is adopted, the minimum base station number of multi-base station three-dimensional position estimation is 5, the minimum base station number of two-dimensional position estimation is 4, and the minimum base station number of one-dimensional position estimation is 3;
the minimum base station number of the multi-base station three-dimensional position estimation (LSM) based on the arrival angle (AOA) is 3, and the minimum base station number of the two-dimensional position estimation is 2.
Further, the topological constraint relationship is specifically as follows:
when a Least Square Method (LSM) multi-base station three-dimensional position estimation based on time of arrival (TOA) is carried out, every 4 base stations are not coplanar; each 3 base stations are not collinear during the two-dimensional position estimation; each 2 base stations are not in common point when one-dimensional position estimation is carried out;
when a Least Square Method (LSM) multi-base station three-dimensional position estimation is carried out on the basis of time difference of arrival (TDOA), every 5 base stations are not coplanar; each 4 base stations are not collinear during the two-dimensional position estimation; when the one-dimensional position estimation is carried out, each 3 base stations are not concurrent, and the user position can only be estimated in a nonsingular way in a line segment formed by the 3 base station points (excluding line segment end points);
when a Least Square Method (LSM) based on an arrival angle (AOA) is used for estimating the three-dimensional position of a plurality of base stations, a user and any 3 base stations form vectors which are pairwise subjected to hadamard product, and the obtained three vectors are not coplanar (except that the user is right above any base station); the user is not collinear with any two base stations in the two-dimensional position estimation.
Further, the dimension reduction estimation comprises reducing three dimensions to two dimensions and reducing two dimensions to one dimension.
Further, the fine tuning of one or more coordinates of q base stations specifically includes: randomly superposing the small offset sigma on one or more components in one or more coordinates of q base stations to form a coefficient matrix tr (A) of n-m + q base stations T A) The full rank.
Further, the fine-tuning all the coordinates of all the current communicable base stations specifically includes: randomly superimposing different bias small quantities sigma on all components of all coordinates of all current communicable base stations.
Further, the value of the bias small amount σ needs to satisfy that σ does not exceed 1% of the original coordinate, and meanwhile, σ is at least one order of magnitude higher than the data truncation error.
The invention has the beneficial effects that: (1) aiming at the problems of the singularity and the approximate singularity of the LSM positioning solution, the method analyzes the reason and is closely related to the condition that the base stations do not meet geometric topology station distribution constraint or the number of the base stations participating in positioning is insufficient, and provides a new solution for the problems of the singularity and the approximate singularity of the LSM positioning solution; (2) a dimensionality reduction estimation strategy is provided, namely when a limited base station participates in positioning solution operation and a singular solution appears, the singular problem in three-dimensional position estimation is relieved by reducing the position estimation dimensionality (the lower the position estimation dimensionality is, the less the requirement on the number of the base stations is); (3) aiming at the problems that the number of the participating base stations is moderate and the positioning solution is singular or approximate singular, the base stations contributing to the singularity are found according to a certain topological constraint relation, the coordinates of the base stations are changed by a proper small amount sigma, the LSM parameters with small changes are further obtained, and the LSM is utilized to carry out approximate positioning solution estimation; (4) aiming at the situation that the number of the base stations participating in positioning calculation is large or even large, in order to avoid the steep increase of algorithm complexity caused by topological constraint relation judgment, the positions of all the base stations participating in positioning calculation are subjected to random small-amount sigma superposition to obtain LSM parameters, and then LSM is used for positioning solution estimation. (5) A sigma value-taking principle is given, namely sigma does not exceed 1% of the original coordinate of the base station, and simultaneously sigma is at least one order of magnitude higher than the data truncation error, so that the balance of calculation precision and singular problems is realized; (6) the strategies of topological constraint relation judgment, dimension reduction estimation, base station coordinate fine adjustment and the like provided by the invention form a relatively perfect parameter fine adjustment method, the efficiency of processing the positioning solution singularity and approximate singularity problem is improved, and the method has relatively high theoretical research value and engineering practice significance.
Drawings
FIG. 1 is a schematic diagram of the TOA positioning principle;
FIG. 2 is a schematic diagram of three-dimensional coordinates of 4 base stations and users;
FIG. 3 is a schematic diagram of TDOA location principles;
FIG. 4 is a schematic diagram of three-dimensional coordinates of 5 base stations and a user;
FIG. 5 is a schematic diagram of AOA positioning principle;
FIG. 6 is a schematic three-dimensional coordinate diagram of 3 base stations and users;
fig. 7 is a schematic three-dimensional coordinate diagram of 100 base stations and users.
Detailed Description
For a better understanding of the present invention, the following examples are given to illustrate the present invention and not to limit the scope of the present invention.
Example 1
A parameter fine-tuning processing method for positioning singular or approximate singular problems comprises the following steps:
1) after the problem of positioning solution singularity or approximate singularity occurs by directly using a Least Square Method (LSM), the total number n of the current communicable base stations and the index i and the position information (x) of each base station are counted i ,y i ,z i )；
2) Comparing the total number n of the current communicable base stations with the number of experience base stations, if n is less than or equal to the number of experience base stations, judging the topological constraint relationship of all the current communicable base stations, and finding m base stations which do not meet the topological constraint relationship, so that the position estimation of a user with n-m base stations can be determined; if n-m is more than or equal to the minimum base station number of the current dimension position estimation, the coefficient matrix tr (A) formed by n-m base stations T A) Performing Least Square Method (LSM) position estimation to obtain a non-singular and credible positioning solution if the rank is full; if n-m is less than the minimum base station number of the current dimension position estimation and n is less than the minimum base station number of the current dimension position estimation, the coefficient matrix tr (A) formed by n-m base stations T A) If the rank is not full and n is less than the minimum base station number of the current dimension position estimation, performing dimension reduction estimation and re-entering the topological constraint relation judgment and the subsequent steps; if n-m is less than the minimum base station number of the current dimension position estimation and n is more than or equal to the minimum base station number of the current dimension position estimation, the coefficient matrix tr (A) formed by the n-m base stations T A) If the rank is not full and n is larger than or equal to the minimum base station number estimated at the current dimension position, q base stations are selected from m base stations, q is not larger than m, and n-m + q is not smaller than the minimum base station number estimated at the current dimension position, one or more coordinates of the q base stations are subjected to fine adjustment, and then Least Square Method (LSM) position estimation is carried out to obtain a non-singular and credible positioning solution;
if n is larger than the number of the empirical base stations, fine adjustment is carried out on all coordinates of all current communicable base stations, and then Least Square Method (LSM) position estimation is carried out, so that a non-singular and credible positioning solution is obtained.
Further, the number of the empirical base stations is 3 times of the minimum number of the base stations of the current dimension position estimation.
Further, the minimum number of base stations for estimating the current dimension position specifically is:
the minimum base station number of multi-base station three-dimensional position estimation based on the Least Square Method (LSM) of the time of arrival (TOA) is 4, the minimum base station number of two-dimensional position estimation is 3, and the minimum base station number of one-dimensional position estimation is 2;
the method comprises the following steps that a Least Square Method (LSM) based on time difference of arrival (TDOA) is adopted, the minimum base station number of multi-base station three-dimensional position estimation is 5, the minimum base station number of two-dimensional position estimation is 4, and the minimum base station number of one-dimensional position estimation is 3;
the minimum base station number of the multi-base station three-dimensional position estimation (LSM) based on the arrival angle (AOA) is 3, and the minimum base station number of the two-dimensional position estimation is 2.
Further, the topological constraint relationship is specifically as follows:
when a Least Square Method (LSM) multi-base station three-dimensional position estimation based on time of arrival (TOA) is carried out, every 4 base stations are not coplanar; each 3 base stations are not collinear during the two-dimensional position estimation; each 2 base stations are not in common point when one-dimensional position estimation is carried out;
when a Least Square Method (LSM) multi-base station three-dimensional position estimation is carried out on the basis of time difference of arrival (TDOA), every 5 base stations are not coplanar; each 4 base stations are not collinear during the two-dimensional position estimation; when the one-dimensional position estimation is carried out, each 3 base stations are not concurrent, and the user position can only be estimated in a nonsingular way in a line segment formed by the 3 base station points (excluding line segment end points);
when a Least Square Method (LSM) based on an arrival angle (AOA) is used for estimating the three-dimensional position of a plurality of base stations, a user and any 3 base stations form vectors which are pairwise subjected to hadamard product, and the obtained three vectors are not coplanar (except that the user is right above any base station); the user is not collinear with any two base stations in the two-dimensional position estimation.
Further, the dimension reduction estimation comprises reducing three dimensions to two dimensions and reducing two dimensions to one dimension.
Further, the fine tuning of one or more coordinates of q base stations specifically includes: randomly superposing the small offset sigma on one or more components in one or more coordinates of q base stations to form a coefficient matrix tr (A) of n-m + q base stations T A) The full rank.
Further, the fine tuning all coordinates of all current communicable base stations specifically includes: randomly superimposing different bias small quantities sigma on all components of all coordinates of all current communicable base stations.
Further, the value of the bias small amount σ needs to satisfy that σ does not exceed 1% of the original coordinate, and meanwhile, σ is at least one order of magnitude higher than the data truncation error. The sigma value is good in balance between the positioning accuracy and the singularity problem, namely in the aspect of accuracy, sigma does not exceed 1% of an original coordinate, otherwise, the larger sigma causes larger positioning solution estimation errors, meanwhile, sigma is at least one order of magnitude higher than data truncation errors, otherwise, the smaller sigma is covered by the data truncation errors and the singularity problem cannot be solved, and the sigma value principle specifies that sigma cannot be too large or too small.
The topology constraint relation judgment means that base stations i are traversed one by one, i is the label of each base station, the TOA three-dimensional position estimation is taken as an example, i is 1, if any 3 of n-i base stations are not coplanar with the base station i, the base stations i are continuously traversed i +1, and the operation is repeated in the rest n-i base stations until the number of the rest base stations is 3; during the traversal, if 3 base stations in the n-i base stations are coplanar with the base station i, the base station i is temporarily removed from the base stations participating in the user positioning, and the total number of the base stations i is counted to be m, so that the position estimation of the user participating in the n-m base stations can be determined. Whether 4 base stations are coplanar or not can be determined by selecting one base station as a reference point, wherein the reference point points to the coordinates of the other 3 base stations respectively, so that 3 vectors are formed, then, whether 3 rows and 3 columns formed by the 3 vectors are 0 or not is examined, and if the 3 rows and the 3 columns are zero, the base stations are coplanar or not.
Example 2
A theoretical solution for estimating the user position when a plurality of base stations are given is deduced based on a least square method, which is as follows:
the definition coefficient matrix a is shown in equation (1),
R(A)＝{Ax|x∈R n },A∈R n×n (1)
the matrix constraint form of the least squares is shown in equation (2),
since the 2-norm is equivalent to the 2-norm squared constraint on the state quantity x, equation (2) can be changed to equation (3),
the partial derivative is calculated for the state quantity x in the formula (3), the optimal solution is obtained when the derivative is zero, the formula (4) is obtained,
further obtaining the state quantity x as shown in the formula (5),
x＝(A T A) -1 A T b (5)
wherein x is a state quantity, A is a coefficient matrix, and b is an offset; a T Representing pairwise and transposed -1 Indicating a pair inversion.
As is apparent from the definition of the formula (5) and the coefficient matrix A, the condition that the state quantity x has a solution is shown in the formula (6),
tr(A T A)＝n (6)
formula (6) illustrates if A T A full rank, at which time A T A has an inverse matrix, is not singular, and has a solution to the state quantity x; otherwise if A T And if the A does not have the inverse matrix, the equation is singular, and the state quantity x has no solution.
FIG. 1 is a schematic diagram of the TOA positioning principle; LSM multi-base station position estimation based on TOA is as follows:
setting n total BS participating in positioning, BS i The signal one-way propagation time with the UE is t i The propagation speed of the electromagnetic wave in free space is c, i belongs to [1n ]]Indicating the number of BS, BS i Distance d from UE i Is represented by the formula (7),
d i ＝ct i (7)
let BS i Has the coordinates of (x) i ,y i ,z i ) The coordinate of the UE is (x, y, z), then the BS i The distance from the UE is represented by equation (8),
BS i the distance to the origin is given by the equation (9),
the distance of the UE from the origin is equation (10),
substituting the formula (7) into the formula (8), squaring two sides and finishing to obtain a formula (11),
the best solution can be obtained by combining the formula (5) as shown in the formula (12),
wherein x is a state quantity including a position quantity (x, y, z), and r is a modulus of the position quantity (x, y, z); a is a coefficient matrix and b is an offset, both of which contain the base station coordinates (x) i ,y i ,z i ) Information of (i ∈ [1n ]) of]Denotes the number of the base station, c is the propagation velocity of the electromagnetic wave in free space, t i Is the signal one-way propagation time between the base station i and the user; r is i Is the base station i position quantity (x) i ,y i ,z i ) The modulus value of (a).
Estimating position in three dimensionsFor example, as shown in fig. 2, a schematic three-dimensional coordinate diagram of 4 base stations and users is shown, which is used to verify the influence of the base station with the least TOA and its topological relation on singular or near singular positioning solutions. Generating 4 coplanar base stations according to geometrical topological constraint relation, and enabling the coordinates of Z ≡ 10 and 4 base stations to be (40,80,10), (90,90,10), (130,10,10), (50,20,10) respectively for convenience of demonstration; the coordinates of the user are (30,40, 50). Then, based on the TOA method, as shown in equation (12), position estimation is performed, at which time tr (A) T A) Not full rank, matrix near singular, solution (30,40, 87.5); the method provided by the invention is adopted to change the z-axis of one coordinate (40,80,10) of the base station into (40,80,10.001) after one sigma is superposed to be 0.001, the original base station is replaced by the base station, and the position calculation is carried out based on the formula (12), wherein at the moment, tr (A) is trthe base stations are not coplanar T A) The problem of full rank, singular solution or approximate singular solution disappears, and the correct solution of the user is obtained as (30,40, 50); if σ is 0.00001, the z-axis of the coordinates (40,80,10) is also superimposed by one σ, the z-axis is replaced by the original base station, and the position solution is performed based on the equation (12) again, at this time, because the value of σ is too small, the positioning solution still has an approximate singular problem, and the solution is (30,40, 26.912).
Example 3
FIG. 3 is a schematic diagram of TDOA positioning principle; the LSM multi-base station position estimation based on TDOA is as follows:
BS i denotes the ith base station, BS j Denotes the jth base station, i, j ∈ [1n ]]Indicating the number of BS, in BS j Is a reference base station. Let t ij Signals sent for the UE arrive at the BS i And BS j Time difference of (1), then BS i And BS j Is d distance difference ij As shown in formula (13)
d ij ＝ct ij ＝d i -d j (13)
Formula (14) is obtained from formula (8) and formula (9),
united vertical type (13), formula (14) de formula (15)
The optimal solution can be obtained by combining the formula (5) as shown in the formula (16),
without loss of generality, if the base stations participating in the solution are sequentially used as reference base stations, equation (16) can be expanded to equation (17),
wherein i, j ∈ [1n ]]A serial number indicating a base station, x is a state quantity including a position quantity (x, y, z), and d j Distance of the jth base station to the position quantity (x, y, z); a is a coefficient matrix and b is an offset, both of which contain the base station coordinates (x) i ,y i ,z i )、(x j ,y j ,z j ) C is the propagation velocity of the electromagnetic wave in free space, r i Is the base station i position quantity (x) i ,y i ,z i ) Modulus value of r j Is the base station j position quantity (x) j ,y j ,z j ) A modulus value of (d); t is t ij Is the time difference of the propagation of the electromagnetic wave from the position quantity (x, y, z) to the base stations i and j.
Taking three-dimensional position estimation as an example, as shown in fig. 4, a schematic diagram of three-dimensional coordinates of 5 base stations and a user is shown, which is used to verify the influence of the TDOA minimum base station and its topological relation on positioning solution singularity or approximate singularity. Generating 5 coplanar base stations according to geometrical topological constraint relation, and enabling the coordinates of Z ≡ 10 and 5 base stations to be (40,80,10), (90,90,10), (130,10,10), (50,20,10), (35,65 and 10) respectively for convenience of demonstration; the coordinates of the user are (30,40, 50). Then, based on the TDOA method, as shown in equation (17), a position estimation is performed, at which time tr (A) T A) The rank is not full, and the matrix is singular and has no solution; by adopting the method pair provided by the inventionAfter the z-axis of one coordinate (40,80,10) of the base station is superposed by one sigma to be 0.001, the z-axis is changed into (40,80,10.001), the base station is replaced by the original base station, and the position is solved based on the formula (17), wherein tr (A) is satisfied because the topological constraint relationship that TDOA is not coplanar every 5 base stations is satisfied at the moment T A) Full rank, singular solution problem disappears, getting the correct solution (30,40,50) of the user; if σ is 0.00001, similarly, superimposing one σ on the z-axis of the coordinates (40,80,10), replacing the z-axis with the original base station, and performing position calculation based on equation (17), at this time, because the value of σ is too small, the positioning solution still has an approximate singular problem, the solution is (30,40,50.000025), and the smaller the σ, the positioning solution is converted from the approximate singular problem to the singular problem, and the larger the positioning error is.
Example 4
FIG. 5 is a schematic diagram of AOA positioning principle, which shows BS i Relative position and direction relation with the UE; the AOA-based LSM multi-base station position estimation is as follows:
let BS i The azimuth angle of the signal between the UE and the UE is psi i ∈(-π,π]The pitch angle is theta i ∈(-π,π]，BS i At a distance d from the UE i As shown in FIG. 4, the following relationships are satisfied as shown in the formulas (18), (19) and (20),
x-x i ＝d i sinθ i cosψ i (18)
y-y i ＝d i sinθ i sinψ i (19)
z-z i ＝d i cosθ i (20)
the formula (21) can be obtained by combining the vertical type (18) and the formula (19),
xsinψ i -ycosψ i ＝x i sinψ i -y i cosψ i (21)
the optimal solution can be obtained by combining the formula (5) as shown in the formula (22),
further, z is obtained from the formula (20) and the formula (22), and is represented by the formula (23)
Wherein m' is the number of base stations corresponding to the pitch angle of 0 (the base stations and the user are on the same height plane); m is the number of base stations corresponding to a pitch angle of non-0 (the base stations and the users are in different height planes), and n is the total number of the base stations; d is a radical of i (z＝z i ) Denotes d i Projection modulus on the horizontal plane; the problem addressed in this way is that the estimation of the UE vertical distance z couples in errors of x, y; but has the advantages that the three-dimensional position estimation can be obtained only by two BSs and knowing the direction relation between the BS and the UE; it is noted that in order to make a or b physically meaningful when ψ takes any value, the elements of a or b should be avoided to be represented by tangent or cotangent functions; while in order to avoid singularities of A or b, | ψ i Not all of them are pi/2.
Another method for implementing the LSM multi-base station position estimation based on AOA is given below:
vertical combination of (18) - (20) with elimination of d i While preserving the relevant terms of x, y, z, equation (24),
the optimal solution can be obtained by combining the formula (5) as shown in the formula (25),
in the AOA three-dimensional position estimation, in order to ensure that A or b has physical significance when psi and theta take any values, elements of A or b are not expressed by tangent or cotangent functions; while in order to avoid singularities of A or b, | ψ i Not all of pi/2, and | θ i Not all is pi/2; equation (25) can estimate three-dimensional position simultaneously, but requires at least 3 BSs, which is complex compared to equations (22) and (23)The degree of impurity is higher. While equation (25) has poor compatibility and cannot pass the pair θ i Or ψ i The assignment of (a) degrades it into a 2-dimensional or even a 1-dimensional position estimate.
Wherein i ∈ [1n ]]Indicating the base station number, x is the state quantity containing the position quantity (x, y, z), A is the coefficient matrix, b is the offset, ψ i Is the ith base station (x) i ,y i ,z i ) Azimuth of signal, theta, from the user position quantity (x, y, z) i Is the ith base station (x) i ,y i ,z i ) Pitch angle from the user position quantity (x, y, z).
Taking three-dimensional position estimation as an example, as shown in fig. 6, a schematic diagram of three-dimensional coordinates of 3 base stations and a user is shown, and is used for verifying the influence of the minimum base station of AOA and the topological relation thereof on positioning solution singularity or approximate singularity. Wherein, the sub-graph (a) is a three-dimensional view of coordinates of a user and a base station; and the sub-graphs (b), (c) and (d) are the projections of the three-dimensional coordinates of the user and the base station on an XOY plane, an XOZ plane and a YOZ plane respectively. The coordinates of the user are (30,40, 50); giving the coordinates of 3 base stations as (50.9424,21.9885,78.3622), (67.4857,7.7604,31.2019) and (12.3241,55.2022,70.2657), forming vectors by the 3 base stations to be hadamard products pairwise, making the three vectors coplanar, estimating the position based on the AOA method as shown in formula (25), and at the moment tr (A) T A) The non-full rank, matrix approximation singular, estimates for user position the result is (103.4474,36.3443, 50.0000). The method provided by the invention is adopted to change the y axis of one coordinate (50.9424,21.9885,78.3622) of the base station into (50.9424,21.9895,78.3622) after one sigma is superposed on the y axis of 0.001, the base station is replaced by the original base station, and the position calculation is carried out based on the formula (25), wherein tr (A) is tr (A) because the topological constraint relation during the AOA three-dimensional position estimation is satisfied at the moment T A) With full rank, the approximate singular solution problem disappears, resulting in the correct solution for the user (29.9998,40.0001, 50.0000). If σ is small, it is concluded to be consistent with examples 2 and 3.
Example 5
As shown in fig. 7, a schematic diagram of three-dimensional coordinates of 100 base stations and users is shown, where the number of base stations is large, fine tuning needs to be performed on coordinates of all communicable base stations, and the conclusion is consistent with embodiments 2 to 4 and is not repeated.
Example 6
Performing dimension reduction estimation based on TOA, TDOA and AOA, the steps are the same as those of the embodiments 2-5; except that the geometric topological constraint relationship needs to be adjusted correspondingly, and the obtained conclusion is consistent with the embodiment 2 to the embodiment 5. The method further verifies the effectiveness of the LSM positioning and singular problem solving.
The above description is only a specific embodiment of the present invention, and not all embodiments, and any equivalent modifications of the technical solutions of the present invention, which are made by those skilled in the art through reading the present specification, are covered by the claims of the present invention.
Claims (4)
1. A parameter fine-tuning processing method for positioning singular or approximate singular problems is characterized by comprising the following steps:
1) after a positioning solution singularity or approximate singularity problem occurs by directly using a least square method, counting the total number n of the current communicable base stations;
2) comparing the total number n of the current communicable base stations with the number of experience base stations, if n is less than or equal to the number of experience base stations, judging the topological constraint relationship of all the current communicable base stations, and finding m base stations which do not meet the topological constraint relationship, so that the position estimation of a user with n-m base stations can be determined; if n-m is larger than or equal to the minimum base station number of the current dimension position estimation, namely a coefficient matrix tr (A) formed by n-m base stations T A) Performing position estimation by a least square method to obtain a non-singular and credible positioning solution if the rank is full; if n-m is less than the minimum base station number of the current dimension position estimation and n is less than the minimum base station number of the current dimension position estimation, namely a coefficient matrix tr (A) formed by n-m base stations T A) If the rank is not full and n is less than the minimum base station number of the current dimension position estimation, performing dimension reduction estimation and re-entering the topological constraint relation judgment and the subsequent steps; if n-m is less than the minimum base station number of the current dimension position estimation and n is more than or equal to the minimum base station number of the current dimension position estimation, the coefficient matrix tr (A) formed by the n-m base stations T A) If the rank is not full and n is more than or equal to the minimum base station number of the current dimension position estimation, then m bases are usedSelecting q base stations from the station, wherein q is less than or equal to m, and n-m + q is greater than or equal to the minimum base station number of the current dimension position estimation, finely adjusting one or more coordinates of the q base stations, and then performing least square method position estimation to obtain a non-singular and credible positioning solution;
if n is larger than the number of the empirical base stations, fine tuning is carried out on all coordinates of all current communicable base stations, then least square method position estimation is carried out, and non-singular and credible positioning solutions are obtained;
the number of the empirical base stations is 3 times of the minimum number of the base stations estimated by the current dimension position;
the minimum number of base stations for estimating the current dimension position specifically is as follows:
the minimum base station number of the multi-base station three-dimensional position estimation based on the least square method of the arrival time is 4, the minimum base station number of the two-dimensional position estimation is 3, and the minimum base station number of the one-dimensional position estimation is 2;
the minimum base station number of the multi-base station three-dimensional position estimation based on the least square method of the arrival time difference is 5, the minimum base station number of the two-dimensional position estimation is 4, and the minimum base station number of the one-dimensional position estimation is 3;
the minimum base station number of the three-dimensional position estimation of the multiple base stations is 3 based on the least square method of the arrival angle, and the minimum base station number of the two-dimensional position estimation is 2;
the topological constraint relationship is specifically as follows:
when the three-dimensional position estimation of the multiple base stations is carried out based on the least square method of the arrival time, every 4 base stations are not coplanar; each 3 base stations are not collinear during the two-dimensional position estimation; each 2 base stations are not in common point when one-dimensional position estimation is carried out;
when the three-dimensional position of the multiple base stations is estimated by the least square method based on the arrival time difference, every 5 base stations are not coplanar; each 4 base stations are not collinear during the two-dimensional position estimation; when the one-dimensional position estimation is carried out, the 3 base stations are not in common point, and the user position can only be estimated in a nonsingular way in a line segment which is formed by the 3 base station points and does not comprise line segment end points;
forming vectors pairwise by a user except for a user right above any base station and any 3 base stations to be subjected to hadamard product during the three-dimensional position estimation of the multiple base stations based on the least square method of arrival angles, wherein the obtained three vectors are not coplanar; the user and any two base stations are not collinear during two-dimensional position estimation;
the fine tuning of one or more coordinates of q base stations specifically includes: randomly superposing the small offset sigma on one or more components in one or more coordinates of q base stations to form a coefficient matrix tr (A) of n-m + q base stations T A) The full rank.
2. The method of claim 1, wherein the dimensionality reduction comprises a two-dimensional reduction from three dimensions to one dimension.
3. The method for fine-tuning parameters for solving singular or near singular problems according to claim 1, wherein the fine-tuning all coordinates of all current communicable base stations specifically comprises: randomly superimposing different bias small quantities sigma on all components of all coordinates of all current communicable base stations.
4. The method for fine-tuning parameters for solving singular or near singular problems according to any of claims 1-3, wherein the bias small quantity σ is taken to satisfy the condition that σ does not exceed 1% of the original coordinates, and σ is at least one order of magnitude higher than the data truncation error.
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