CN115436874A - Three-dimensional passive positioning method - Google Patents
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Abstract
The invention provides a three-dimensional passive positioning method, which is used for an N +1 hybrid base station positioning system to position a target and comprises the following steps: s1, obtaining TDOA and AOA of a target and performing time alignment; s2, constructing a geometric relation between the monitoring station, the reference station and the target position, establishing a linear equation of the TDOA and the AOA about the target position by utilizing the geometric relation and the definition of the TDOA and the AOA, and obtaining a linear matrix equation about the target position in parallel; and S3, solving an estimated value of the target position by a weighted least square method, and analyzing the influence of the measurement noise on the positioning result. According to the invention, the target can be positioned by only two base stations, and only one base station for measuring the TDOA of the target can observe the AOA of the target, so that a closed-form solution of the target position can be obtained, the calculation complexity is low, and the positioning performance can reach CRLB precision.
Description
Technical Field
The invention relates to the technical field of passive positioning, and particularly provides a three-dimensional passive positioning method applied to a scene that an N +1 hybrid base station positioning system positions a target, wherein N represents N monitoring stations, 1 represents a reference station, the monitoring stations and the reference station are collectively called base stations, the N +1 base stations are used for measuring time difference of arrival (TDOA), and only the reference station is used for measuring an angle of arrival (AOA).
Background
Passive target localization is a fundamental problem in passive detection systems such as radar, sonar, wireless sensor networks, photoelectric detection systems, etc. The positioning process is to obtain different types of measured values of the target through the stations with known positions and solve the position of the target by using a positioning algorithm. The object location problem is often difficult because the relationship between the object location and the TDOA and AOA measurements from it is non-linear and non-convex. Maximum Likelihood Estimation (MLE) is a general approach to solving such problems. MLE has progressive unbiasedness, and the performance of MLE can reach the Cramer-Lo lower bound (CRLB) when enough observation data are obtained. However, the MLE requires iterative computation, has high requirements for the accuracy of an iteration initial value, and is prone to local convergence. Therefore, it is more preferable to use a source location estimation algorithm with a closed-form solution.
In a small error range under the Gaussian noise condition, the performance of the existing closed solution positioning algorithm based on TDOA can reach CRLB precision. However, it requires a minimum of four base stations to participate in positioning, and when the number of base stations is small, these methods may generate ghost points. The existing positioning algorithm based on AOA utilizes a plurality of pairs of azimuth angles and pitch angles to carry out triangular positioning, and can determine the source position in a closed solution form. It is known that the more measurement values, the higher the positioning accuracy of the target. The existing positioning method based on mixed TDOA and AOA measurement has higher positioning accuracy, and the target can be positioned by only needing two stations at least. It requires that each base station measuring the target TDOA be able to observe the target AOA. In fact, due to multipath effects, non-line-of-sight propagation, etc., observers sometimes cannot obtain the AOA of the source using infrared sensors or antenna arrays. In addition, the feasibility of mass usage is low due to the high cost of the antenna array receivers and infrared sensors. These factors all contribute to the failure of the above algorithm. Therefore, the invention provides an effective three-dimensional passive positioning method, which only needs two base stations to participate in positioning and only needs one base station for measuring the TDOA of the target to observe the AOA of the target, so as to obtain a closed solution of the target position, namely an estimated value of the target position.
Disclosure of Invention
The invention provides a three-dimensional passive positioning method for solving the problems, which mainly utilizes the geometric relation between a base station and a target position, utilizes a measurement equation to establish a linear matrix equation between TDOA and AOA and an unknown target position, and solves the matrix equation by a weighted least square method to obtain an estimated value of the target position.
The three-dimensional passive positioning method is used for positioning a target by an N +1 hybrid base station positioning system, wherein N represents N monitoring stations, and is an integer greater than 0; 1 denotes a reference station, N +1 base stations are all used to determine time difference of arrival, i.e. TDOA, and only the reference station is used to determine angle of arrival, i.e. AOA, comprising the steps of:
s1, obtaining TDOA and AOA of a target and performing time alignment;
s2, constructing a geometric relation between the monitoring station, the reference station and the target position, establishing a linear equation of the TDOA and the AOA about the target position by utilizing the geometric relation and the definition of the TDOA and the AOA, and obtaining a linear matrix equation about the target position in parallel;
and S3, solving an estimated value of the target position by a weighted least square method, and analyzing the influence of the measurement noise on the positioning result.
Preferably, the geometric relationship is expressed as:
u-s 0 ＝r 0 b；
wherein the content of the first and second substances,
representing the three-dimensional spatial position of the object, is an unknown,
the dimensions are represented by a number of dimensions,
representing a unit vector directed from the reference station to the target, alpha and beta representing the azimuth and elevation angles, respectively, of the target relative to the reference station, r 0 ＝||u-s 0 | | is the distance between the reference station and the target,
i =0,1,2,.. N denotes the three-dimensional spatial position of N +1 base stations, i.e. s when i =0 0 Represents the three-dimensional spatial position of the reference station, when i =1,2,3, ·, Ν, i.e., s 1 ,s 2 ,s 3 ,...,s N Representing the three-dimensional spatial locations of the N monitoring stations.
Preferably, with the definition of TDOA: r is i,0 ＝r i -r 0 And a geometric relationship, representing TDOA as a linear equation:
wherein the content of the first and second substances, T is the transposed operation sign of the matrix, | s i I represents s i Euclidean norm of r i,0 Indicating a monitoring station s i And a reference station s 0 Difference in distance from the target, r i Indicating a monitoring station s i The distance to the target.
Preferably, the AOA is defined as:
wherein arctan represents the arctan function;
the AOA is expressed as a linear equation using its definition and geometric relationship:
γ T s 0 ＝γ T u；
wherein the content of the first and second substances,
preferably, the linear equation expressed by TDOA and the linear equation expressed by AOA are combined to obtain a linear matrix equation as follows:
h m ＝G m u；
wherein the content of the first and second substances,
preferably, the influence process of the analysis measurement noise on the positioning result is as follows: performing first-order Taylor expansion on each TDOA and AOA, removing noise items above a second order to obtain a measurement error, substituting the TDOA and the AOA containing the measurement error into a measurement equation to obtain an error component of the measurement equation, expressing the error component as an estimated deviation of a target position, and expressing the estimated deviation as:
wherein, the first and the second end of the pipe are connected with each other,
an estimate value representing a target position;
the covariance matrix of the target position, which is found from the estimated deviations, is expressed as:
wherein, the first and the second end of the pipe are connected with each other,
representing G affected by measurement noise m Δ m denotes the measurement noise, Q, following a Gaussian white distribution m A covariance matrix representing Δ m,
represents a weighting matrix, where H = diag ([ 2 r) 1 ,2r 2 ,...,2r N ,r 0 cosβ,r 0 ])。
Preferably, the estimated value of the target position is obtained by using a weighted least squares method as follows:
wherein argmin () is a mathematical function representing the value of the argument when the function value is minimized,
representing h affected by measurement noise m 。
Compared with the prior art, the invention can obtain the following beneficial effects:
the invention can position the target by only two base stations at least, and only one base station for measuring the TDOA of the target can observe the AOA of the target, so that the closed-form solution of the target position can be obtained, the calculation complexity is low, and the positioning performance can reach the CRLB precision.
Drawings
Fig. 1 is a schematic diagram of an N +1 hybrid base station positioning system for positioning a target according to an embodiment of the present invention;
FIG. 2 is a schematic diagram of a three-dimensional passive positioning method provided in accordance with an embodiment of the present invention;
FIG. 3 is a graph comparing the location performance of the present invention and the conventional algorithm affected by TDOA measurement noise;
fig. 4 is a comparison graph of the positioning performance of the invention and the traditional algorithm affected by the AOA measurement noise.
Detailed Description
Hereinafter, embodiments of the present invention will be described with reference to the accompanying drawings. In the following description, the same reference numerals are used for the same blocks. In the case of the same reference numerals, their names and functions are also the same. Therefore, detailed description thereof will not be repeated.
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in further detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not to be construed as limiting the invention.
The application scenario of the invention is an N +1 hybrid base station positioning system, wherein N represents N monitoring stations, and N is a positive integer greater than 0; reference station 1 indicates a reference station, and the monitoring station and the reference station are collectively called a base station. The three-dimensional spatial position of the base station is represented as
Wherein
Indicating latitude information, i.e. here s i When i =0, representing the three-dimensional space position of the reference station; when i is not equal to 0, representing the three-dimensional space position of the monitoring station; the target at an unknown position is represented as
Reference station s 0 And any monitoring station s i All for determining the TDOA value of the object, but only the reference station s 0 The AOA value of the target can be determined.
Fig. 1 shows a mathematical model for positioning an object by an N +1 hybrid base station positioning system according to an embodiment of the present invention.
As shown in FIG. 1, X, Y, Z represents the corresponding coordinate axes of a coordinate system, O is the origin of coordinates, r i Denotes an arbitrary base station s i Position in three-dimensional space with the target
Represents the distance of the reference station from the target when i = 0; when i ≠ 0, it represents the distance between the monitoring station and the target, s i And s j Representing the three-dimensional spatial positions of any two monitoring stations, and alpha and beta representing the azimuth angle and the elevation angle of the target relative to the reference station respectively.
It should be noted that: in this embodiment, a method of a numerical simulation experiment is adopted for verification, and all the steps are successfully simulated on Matlab2019 a.
It should be noted that: in the N +1 hybrid base station positioning system in this embodiment, the value of N is 7, and the position coordinates of 8 base stations used for simulation are as follows:
it should be noted that: the true position of the target is
The data is used for comparing and calculating with the estimated value of the target position which is finally obtained, and drawing a curve to explain the effect, and the data is not needed to participate in the operation in the process of solving the estimated value of the target position and is regarded as an unknown number.
Fig. 2 shows a flow of a three-dimensional passive positioning method provided according to an embodiment of the present invention.
As shown in fig. 1 and fig. 2, the three-dimensional passive positioning method comprises the following steps:
s1, establishing a 7+1 hybrid base station positioning system to position a target, initializing system parameters, wherein any two base stations can cooperatively measure the TDOA of the target, but only a reference station can measure the AOA of the target, acquire the TDOA and the AOA of the target, and perform time alignment.
S2, constructing a geometric relation between the monitoring station, the reference station and the target position, establishing a linear equation of the TDOA and the AOA about the target position by utilizing the geometric relation and the definition of the TDOA and the AOA, and obtaining a linear matrix equation about the target position by parallelly connecting the linear equations of the TDOA and the AOA about the target position.
The geometrical relationship is represented as:
u-s 0 ＝r 0 b；
wherein the content of the first and second substances,
representing a unit vector, r, pointed at the target by a reference station 0 ＝||u-s 0 | | is the distance between the reference station and the target,
i =0,1,2, 7 denotes the three-dimensional spatial position of 7+1 base stations, i.e. s when i =0 0 Represents the three-dimensional spatial position of the reference station, when i =1,2,3, ·, Ν, i.e., s 1 ,s 2 ,s 3 ,...,s 7 Representing the three-dimensional spatial locations of 7 monitoring stations.
With the definition of TDOA: r is i,0 ＝r i -r 0 And geometric relationships, representing TDOA as a linear squareThe process is as follows:
where T is the transpose of the matrix, | s i | | denotes s i Euclidean norm r of i,0 Indicating a monitoring station s i And the distance between the target and the reference station s 0 Difference in distance from the target, r i Indicating a monitoring station s i The distance to the target.
AOA is defined as:
wherein arctan represents the arctan function;
the definition and geometric relationship of the AOA are used to express the AOA as a linear equation as follows:
γ T s 0 ＝γ T u；
wherein
α and β represent the azimuth and elevation angles, respectively, of the target relative to the reference station.
The linear equation of TDOA and the linear equation of AOA are combined to obtain a measurement equation as follows:
h m ＝G m u；
wherein the content of the first and second substances,
s3, because the influence of the measurement noise on the positioning result exists in the real positioning process, the process of analyzing the influence of the measurement noise on the positioning result is as follows: performing first-order Taylor expansion on each TDOA and AOA, removing noise items above a second order to obtain a measurement error, substituting the TDOA and the AOA containing the measurement error into a measurement equation to obtain an error component of the measurement equation, and expressing the estimation deviation of the target position by using the error component, wherein the estimation deviation is expressed as:
wherein the content of the first and second substances,
representing an estimate of the target position.
The covariance matrix for the target position from the estimated deviations is expressed as:
wherein the content of the first and second substances,
representing G affected by measurement noise m Δ m denotes the measurement noise, Q, following a Gaussian white distribution m Covariance matrix representing Δ m, W = (HQ) m H T ) -1 Is a weighting matrix in which
Based on the analysis, TDOA and AOA white noises with the same distribution are respectively introduced as measurement noises by Meng Daka Row method in the simulation process, and covariance matrixes of the TDOA and AOA white noises are respectively
And
wherein, I respectively represents an identity matrix, the corner marks represent the dimension of the identity matrix,
and
the variance is indicated. At this time, the TDOA measured value and AOA measured value of the observed target of each base station obtained by initializing the system parameters are respectively
The measurement equation after introducing white noise of TDOA and AOA with the same distribution is as follows:
respectively substituting the measured values of the targets observed by each base station into the vector
Sum matrix
In the expression (c), the following formula is obtained:
first, using the estimated deviation of the target position and the covariance matrix, a weight matrix W = (TQ) is calculated m T T ) -1 Wherein T = diag ([ 2 r) 1 ,2r 2 ,...,2r 7 ,r 0 cosβ,r 0 ]) Measuring the covariance matrix of the noise
Since the matrix T depends on the true position u of the target, we can first use the identity matrix instead of W or let W = Q m -1 To solveAn initial position estimation value of the target is obtained, then a more accurate weighting matrix W is calculated by utilizing the initial position estimation value, and then a final solution, namely the estimation value of the target position is obtained.
Solving the estimate of the target position by a weighted least squares method is as follows:
it should be noted that: the estimated value of the target position, the deviation of the estimated value of the target position and the covariance matrix are directly generated by simulation software. Since a single experiment is accidental and is not convincing, a large amount of simulation operations are required, and the number of estimated values of target positions affected by noise is huge, which cannot be listed one by one, and the effectiveness of the present invention will be illustrated by the following performance comparison, i.e., fig. 3 and 4.
The results obtained by the invention are verified by the following steps:
in this embodiment, the traditional least square method (LSE), the IMLE and the operation time of the present invention are compared (the IMLE refers to the maximum likelihood estimation of iteration with the real target position as the initial value), and the computation complexity of these algorithms is roughly estimated by the running time of the Intel Core-i7 CPU, and the specific data are shown in the following table:
the three-dimensional passive positioning method provided by the invention can obtain an accurate weighting matrix only by iterating once to twice, and the calculation complexity of the method is slightly higher than LSE but still far less than IMLE. Notably, the number of IMLE iterations averaged 11.2.
The following is a comparison of the positioning performance of the present example for LSE, IMLE and the present invention:
FIG. 3 shows the positioning performance of the present invention compared to the conventional algorithm affected by TDOA measurement noise.
Fig. 4 shows the positioning performance of the present invention compared to the conventional algorithm affected by AOA measurement noise.
In fig. 3 and 4, x represents a performance curve for locating a target by using the LSE method,
a performance curve representing the localization of an object using the LSE method is shown, a performance curve representing the localization of an object using the method of the invention is shown, and a curve representing the estimated deviation data for the localization of an object using the method of the invention is shown.
As shown in FIG. 3, the abscissa of the coordinate axis is 10 times the TDOA standard deviation σ r The unit of the natural logarithm of (1) is degree; the ordinate is a natural log value of 10 times the root mean square error and variance, in meters, and is used to describe the positioning performance of each positioning method affected by the noise of TDOA measurements. The selection of logarithmic operations herein may represent a comparison of the performance of different algorithms with greater dynamic range errors.
Viewing fig. 3, it can be seen that: when the TDOA measurement noise is small, the performance of the method and the IMLE of the invention is superior to that of the LSE, and the CRLB precision can be achieved. Compared with IMLE, the method of the invention has smaller estimation deviation and is approximate to unbiased estimation. As the measurement noise increases, the root mean square error of the method of the present invention increases, and the IMLE threshold effect occurs when the TDOA measurement noise standard deviation is about 90 meters, i.e., the abscissa. This may be due to irregular viewing error planes near the true target location, which are susceptible to noise interference. But the process of the invention still has better performance at this time.
As shown in FIG. 4, the abscissa of the coordinate axis is 10 times the AOA standard deviation
The unit of the natural logarithm of (1) is degree; the ordinate is a natural logarithm of 10 times the root mean square error and variance, in meters, and is used to describe the positioning performance of each positioning method affected by the AOA measurement noise.
Viewing fig. 4, it can be seen that: when AOA measurement noise is small, the performance of the method and IMLE of the invention is superior to LSE, and CRLB precision can be achieved. Compared with IMLE, the method of the invention has smaller estimation deviation and is approximate to unbiased estimation. Meanwhile, the number of the AOA measurement values is far less than that of the TDOA measurement values, so that the AOA measurement values have lower weight to the result, and the positioning performance of each positioning method is less influenced by AOA measurement noise.
In summary, the comparison of the elapsed operation time and the comparison of the performance affected by the noise are enough to show that the method of the present invention has the effectiveness, and at the same time, has the advantage over the conventional method.
While embodiments of the present invention have been shown and described above, it should be understood that the above embodiments are exemplary and should not be taken as limiting the invention. Variations, modifications, substitutions and alterations of the above-described embodiments may be made by those of ordinary skill in the art without departing from the scope of the present invention.
The above embodiments of the present invention should not be construed as limiting the scope of the present invention. Any other corresponding changes and modifications made according to the technical idea of the present invention should be included in the protection scope of the claims of the present invention.
Claims (7)
1. A three-dimensional passive positioning method is applied to an N +1 hybrid base station positioning system to position a target and comprises N monitoring stations and a reference station, wherein N is an integer greater than 0; n +1 base stations are all used to determine the time difference of arrival, i.e. TDOA, and only the reference station is used to determine the angle of arrival, i.e. AOA, characterized by the steps of:
s1, acquiring the TDOA and the AOA of a target, and performing time alignment;
s2, constructing a geometric relation among the monitoring station, the reference station and a target position, establishing a linear equation of the TDOA and the AOA about the target position by utilizing the geometric relation and the definition of the TDOA and the AOA, and obtaining a linear matrix equation about the target position in parallel;
and S3, solving the estimation value of the target position through a weighted least square method, and analyzing the influence of the measurement noise on the positioning result.
2. A three-dimensional passive localization method according to claim 1, characterized in that said geometrical relationship is represented by:
u-s 0 ＝r 0 b；
wherein the content of the first and second substances,
representing the three-dimensional spatial position of the object, is an unknown,
the dimensions are represented by a number of dimensions,
representing a unit vector directed towards the target by said reference station, alpha and beta representing the azimuth and elevation angles, respectively, of the target with respect to the reference station, r 0 ＝||u-s 0 | | is the distance between the reference station and the target,
represents the three-dimensional spatial position of N +1 base stations, i.e. s, when i =0 0 Represents the three-dimensional spatial position of the reference station when i =1,2,3, ·, Ν, i.e., s 1 ,s 2 ,s 3 ,...,s N Representing the three-dimensional spatial locations of the N monitoring stations.
3. The three-dimensional passive location method of claim 2, wherein with the definition of TDOA: r is i,0 ＝r i -r 0 And a geometric relationship, representing the TDOA as a linear equation:
wherein the content of the first and second substances, T is the transposed operation sign of the matrix, | s i I represents s i Euclidean norm r of i,0 Representing the monitoring station s i And said reference station s 0 Difference in distance from the target, r i Representing the monitoring station s i The distance to the target.
4. A three-dimensional passive localization method according to claim 3, characterized in that AOA is defined as:
wherein arctan represents the arctan function;
the AOA is expressed as a linear equation using its definition and geometric relationship:
γ T s 0 ＝γ T u；
wherein the content of the first and second substances,
5. the three-dimensional passive location method of claim 4, wherein the linear matrix equation obtained by combining the linear equation represented by the TDOA and the linear equation represented by the AOA is as follows:
h m ＝G m u；
wherein the content of the first and second substances,
6. a three-dimensional passive localization method according to claim 5, characterized in that the influence of analytical measurement noise on the localization result is analyzed as follows: performing first-order Taylor expansion on each TDOA and AOA, removing noise terms above the second order to obtain a measurement error, substituting the TDOA and the AOA containing the measurement error into the measurement equation to obtain an error score of the measurement equationA quantity representing the error component as an estimated deviation of a target position, the estimated deviation being represented as:
wherein the content of the first and second substances,
an estimate value representing a target position;
the covariance matrix of the target position, which is found from the estimated deviations, is expressed as:
wherein the content of the first and second substances,
representing G affected by measurement noise m Δ m denotes the measurement noise, Q, following a Gaussian white distribution m Covariance matrix representing Δ m, W = (HQ) m H T ) -1 Represents a weighting matrix, where H = diag ([ 2 r) 1 ,2r 2 ,...,2r N ,r 0 cosβ,r 0 ])。
7. A three-dimensional passive localization method according to claim 6, wherein the estimate of the target position is obtained using a weighted least squares method as follows:
wherein argmin () is a mathematical function representing the value of the argument when the function value is minimized,
representing h affected by measurement noise m 。
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