CN110133586A - TOA combined synchronization and localization method based on linearity correction - Google Patents
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Abstract
The invention discloses a kind of TOA combined synchronization and localization method based on linearity correction mainly solve the problems, such as that the destination node occurred in traditional arrival time TOA positioning is asynchronous with sensor anchor node and positioning accuracy is low.Its implementation is: 1) equation about target position and clock jitter is obtained according to TOA cover half type；2) by introducing auxiliary variable, the initial estimate of clock jitter and target position is found out according to weighted least-squares criterion；3) according to weighted least-squares criterion, the initial estimate deviation of clock jitter and target position is found out；4) the final estimated value of clock jitter and target position is obtained using initial estimate and estimated value deviation.The results showed that being compared with the traditional method, stability of the present invention is good, and calculation amount is low, in the case that target be located at sensor internal and it is two kinds external can approach Cramér-Rao lower bound, positioning accuracy with higher can be used for the location estimation of target signal source.
Description
Technical Field
The invention belongs to the technical field of radar signal processing, and particularly relates to a target positioning method which can be used for position estimation of a target signal source.
Background
The target positioning technology is widely applied to the fields of radar, navigation, target tracking, wireless communication and the like. The multi-station target positioning is an important positioning method, and the multi-station target positioning refers to that a processing center estimates the position of a target node by using target positioning parameter information received by a plurality of sensor anchor nodes. The current popular multi-station positioning technology is more, and can be divided into the following steps according to different positioning parameters: time of arrival (TOA) location techniques, time difference of arrival (TDOA) location techniques, angle of arrival (AOA) location techniques, and signal strength of arrival (RSS) location techniques. Among them, the TOA-based positioning technology has advantages of low implementation cost, high positioning accuracy, and the like, and thus has received wide attention.
The TOA positioning technique utilizes the propagation delay of the target signal to reach multiple sensors to determine the position of the target, and the calculation of the propagation delay usually requires that the transmission time of the target signal is known, so that the target and the sensors must be precisely synchronized. In practical applications, however, there is often a clock skew between the target and the sensor, especially for non-cooperative targets. Some studies show that even clock skew of nanometer level can seriously affect the positioning accuracy of the traditional TOA algorithm. For the above asynchronous problem, there are two main solutions at present: one is to subtract TOA data measured by the sensor anchor node, eliminate unknown clock bias and convert the TOA data into a TDOA location model. However, this method increases the degree of nonlinearity of the localization equation, and the subtraction process introduces colored noise, resulting in performance degradation; the other is to jointly estimate the clock bias and the target position by using TOA measurement data, and the joint estimation algorithm is more and more widely applied because the joint estimation algorithm can obtain higher positioning accuracy.
The joint synchronization and positioning problem is essentially a non-linear, non-convex parameter estimation problem. The maximum likelihood estimation MLE can be used for obtaining the progressive optimal solution of the nonlinear problem, but lattice point search is needed, and the calculation amount is large. The conventional method is to use a taylor series TS method to perform iterative solution, however, the method needs an iterative initial value, the convergence property of which depends too much on the selection of the initial value, and under the condition that the initial value is not selected well, the initial value easily falls into a local minimum point, even a divergence condition occurs.
Zhu S et al propose an algorithm in Joint synchronization and localization using TOAs, namely, interferometric based WLS solution, which converts the original Joint synchronization and positioning problem into the solution of quadratic equation, effectively avoids nonlinear operation, and obtains the closed solution of target position and clock bias. However, the algorithm is only suitable for the case that the target is located inside the sensor, when the target is located outside the sensor array, the quadratic equation can generate multiple solutions and even imaginary solutions, and the selection of the solutions can seriously affect the estimation performance of the algorithm.
Huang J et al, in An An effective closed-form solution for joint synchronization and localization using TOA, overcome the non-linearity problem by introducing auxiliary variables, and obtain An analytic solution of target parameter estimation by using a two-step weighted least square TSWLS algorithm.
In recent years, some scholars introduce a convex optimization method, and convert an objective function into a convex function to perform optimization solution by using a semi-positive relaxation SDR technology. However, the objective function after relaxation approximation is no longer tight, so the method cannot obtain the optimal solution, and the optimization process is computationally intensive.
Disclosure of Invention
Aiming at the defects of the existing TOA positioning algorithm, a linear correction-based TOA joint synchronization and positioning method is provided to reduce the calculated amount, improve the positioning precision and realize the accurate positioning of the target outside the sensor array.
In order to solve the technical problems, the technical scheme of the invention is as follows:
(1) according to the TOA positioning model of the arrival time of the node anchor points of the plurality of mutually synchronized clocks, a TOA positioning equation set A of a target signal arrival sensor is obtained;
(2) introducing an auxiliary variable theta related to a target position coordinate x and a clock deviation tau, and converting a TOA positioning equation set A into a matrix equation B related to the auxiliary variable theta;
(3) performing weighted least square estimation on the matrix equation B to obtain an estimated value of the auxiliary variable thetaFrom the estimated valuesObtaining an initial estimation value of the target positionAnd initial estimate of clock bias
(4) Is provided withIs Δ x, letIs Δ τ, and these two deviations are substituted into equation set a in (1) to yield equation set C for Δ x and Δ τ:
wherein r isiRepresenting distance measurement data corresponding to the ith sensor, the distance measurement data being obtained by multiplying the speed of light by the arrival time, siIndicates the location of the ith sensor, | si-x||2Representing the distance of the target from the ith sensor, niRepresents the distance measurement error, i ═ 1,2||2Representing a two-norm operation, and T representing a transposition operation;
(5) introducing the auxiliary variables ξ for Δ x and Δ τ, a matrix equation D for the auxiliary variable ξ is derived from equation set C:
wherein,h2＝[h2,1,...,h2,i,...,h2,M]T，ξ＝[Δτ,ΔxT]T，
(6) performing weighted least squares estimation on the matrix equation D to obtain an estimated value of the auxiliary variable ξ
Wherein W is a weighting matrix, W is represented byCalculating to obtain;
(7) from the estimated valuesObtaining an estimate of DeltaxAnd an estimate of Δ τ
(8) Initial estimated valueAnd delta tau estimateSubtracting to obtain the clock deviation estimated value after deviation correctionInitial estimated valueEstimated value of Δ xSubtracting to obtain the target position estimated value after deviation correction
Compared with the prior art, the invention has the following advantages:
1. low calculation amount and stable performance
The traditional TOA positioning algorithm, such as the taylor series method, requires an iterative initial value, the convergence of which depends too much on the selection of the initial value, and under the condition that the initial value is not selected well, the initial value easily falls into a local minimum point, even a divergence condition occurs. The invention obtains the signal position and the clock deviation estimation through linear solving, effectively avoids nonlinear operation and reduces the calculated amount.
2. The estimation precision is high
According to the method, the initial value of the target parameter is obtained by using the weighted least square theory, and the deviation of the initial value is linearly corrected, so that compared with the traditional method, a closed-form solution can be obtained, the problem of root selection in the traditional closed-form solution method is solved, and the estimation accuracy can approach to the CrLB (CrLB) boundary under the two conditions that the target is positioned inside and outside the sensor array, wherein the CRLB is the best estimation accuracy which can be achieved theoretically.
Drawings
FIG. 1 is a flow chart of an implementation of the present invention;
FIG. 2 is a model diagram of a multi-sensor anchor node-based time-of-arrival location in the present invention;
FIG. 3 is a graph comparing the performance of locating objects within a sensor array using the method of the present invention and a prior art method;
FIG. 4 is a graph comparing the performance of locating an object outside of a sensor array using the method of the present invention and a prior art method.
Detailed Description
The present invention will be further described with reference to the accompanying drawings and examples, it being understood that the specific examples are set forth herein for the purpose of illustration only and are not intended to be limiting.
Referring to fig. 1, the implementation steps of the linear correction based TOA joint synchronization and positioning method of the present invention are as follows:
step 1, obtaining a TOA positioning equation set A of the arrival time of a target signal to each sensor according to a TOA positioning model of the arrival time of a plurality of mutually synchronized sensor node anchor points of a clock.
(1a) Referring to fig. 2, assume that M sensor anchor nodes with mutually synchronized clocks are distributed in a two-dimensional plane, and a coordinate of a target node to be measured is set to x ═ x, y]TSetting the target node and the sensor network to be asynchronous and the clock deviation to be tau, wherein the coordinates of the anchor nodes are respectively si＝[xi,yi]TI 1, 2.. multidot.m, x is the abscissa of the target position, y is the ordinate of the target position, xiIs the abscissa, y, of the ith sensor positioniT represents the transpose operation as the ordinate of the ith sensor position;
(1b) according to the TOA positioning principle, the time of the target signal reaching the ith sensor is calculated:
wherein, tiRepresents the time of arrival of the target signal at the ith sensor, | si-x||2Representing the distance, Δ t, of the target to the ith sensoriFor time measurement error, i ═ 1, 2., M, c is the speed of light, | · | | luminance2Representing the operation of taking two norms, and M represents the number of sensors;
(1c) will be a formula<1>Is multiplied by the speed of light c and let c tau be tau in the derivation process to obtain the measured distance riThe system of equations of (1):
ri＝cti＝||si-x||2+τ+ni,<2>
wherein n isi＝cΔtiRepresenting the distance measurement error, niObedience mean 0 and varianceI 1,2, M, the clock offset τ representing the distance quantity;
(1d) will be a formula<2>The clock bias τ in (1) is shifted to the left of the equation and then both sides are squared simultaneously, ignoring the second order error termThe TOA positioning equation set A can be obtained through arrangement:
and 2, converting the TOA positioning equation set A into a matrix equation B related to the auxiliary variable theta.
(2a) Introducing intermediate parameters:
the defining variable η ═ xTx-τ2；
Introducing auxiliary variables theta, theta ═ tau, xT,η]T；
Introducing an auxiliary matrix G1，G1Is a matrix of M × 4, G1Ith action of
Introducing an auxiliary vector h1，
Let the data error be n is a distance measurement error vector, n ═ n1,n2,...,nM]TB is a diagonal matrix, B ═ diag (| | s)1-x||2,||s2-x||2,...,||sM-x||2) Diag (·) denotes a diagonalization operation;
(2b) rewriting the formula <3> into a matrix form according to the parameters to obtain a matrix equation B:
step 3Obtaining an estimated value of the auxiliary variable theta from the matrix equation BFrom the estimated valueObtaining an initial estimation value of the target positionAnd initial estimate of clock bias
(3a) A weighting matrix W is introduced and,Qnto measure the covariance matrix of the error vector n,IMis an M by M identity matrix, (.)-1Denotes the inverse operation, E [. cndot]Indicating a desired operation;
(3b) assuming that the variables in θ are independent of each other, for the formula<4>Performing weighted least square solution to obtain the estimated value of the auxiliary variable theta
Wherein, the calculation of W is related to the real target position x, and x is not known a priori, so that the initial estimation value of the auxiliary variable theta is obtained by firstly changing W to I
(3c) By estimation of the auxiliary variable thetaObtaining an initial estimation value of the target positionAnd initial estimate of clock bias
Wherein,express getThe first element of (a) is,express getThe second and third elements of (1).
And 4, correcting the deviation of the estimated value by utilizing the TOA positioning equation set A.
Due to the ongoing formula<5>When solving for the weighted least squares, it is assumed that the variables in θ are independent of each other, but as can be seen from the definition of η, η is closely related to x and τ, so the formula<6>The obtained estimated valueAndand is not accurate and needs to be corrected.
(4a) The true values of the clock bias τ and the target position x can be expressed as:
wherein Δ x isIs an estimated deviation of Δ τIs estimated deviation of
(4b) Will be a formula<7>Substituting into TOA positioning equation set A, neglecting second order deviation term delta xTΔ x and Δ τ2The available equation set C:
wherein, i is 1, 2.
Step 5, a matrix equation D for the auxiliary variables ξ is obtained from equation set C.
(5a) Introducing intermediate parameters:
introducing an auxiliary variable ξ ═ Δ τ, Δ xT]T；
Introducing an auxiliary matrix G2，Wherein
Introducing an auxiliary vector h2，h2＝[h2,1,...,h2,i,...,h2,M]TWherein
(5b) Rewriting the formula <8> into a matrix form to obtain a matrix equation D:
and 6, obtaining an estimated value of the auxiliary variable ξ through a matrix equation D.
(6a) By preliminary estimate of xCalculating the weight of weighted least squares: w ═ (BQ)nBT)-1Wherein
(6b) performing weighted least square estimation on the matrix equation D according to the weight W of the weighted least square to obtain an estimated value of the auxiliary variable ξ
Step 7, obtaining a preliminary estimate from the auxiliary variable ξAndthe estimated deviation of (2).
Based on estimated values of auxiliary variables ξTo obtainDeviation estimate ofAnddeviation estimate of
Wherein,express getThe first element of (a) is,express getThe second and third elements of (1).
And 8, obtaining target position estimation and clock deviation estimation after deviation correction according to the preliminary estimation value and the estimation deviation.
Will be a formula<11>Substitution formula<7>Obtaining the target position after deviation correctionAnd clock skew estimation
The positioning of the target position is thus completed and the clock offset between the target and the sensor is obtained.
The effect of the present invention will be further explained with the simulation experiment.
Test conditions are as follows:
and (3) system model: the simulation experiment of the invention is carried out in a two-dimensional environment, and the two-dimensional plane is assumed to have 8 receiving sensors, and the coordinates of the receiving sensors are respectively as follows: s1＝[50,50]T，s2＝[50,-50]T，s3＝[-50,50]T，s4＝[-50,-50]T，s5＝[50,0]T，s6＝[0,50]T，s7＝[-50,0]T，s8＝[0,-50]TIn meters.
The distance measurement error of TOA data received by the sensor is assumed to be zero in mean value and zero in varianceOf a covariance matrix ofIMThe matrix is a unit matrix of M × M, wherein M represents the number of sensors. In this experiment, the accuracy of the estimation of the target position and the clock offset is measured by the root mean square error RMSE, which is defined asWherein,representing the estimated value of the first time parameter; l represents the number of monte carlo simulation experiments, and L is 5000 in the experiment. Performing iterative operation on ML algorithm by utilizing a Ritaylor series method in an experiment(6) Initial estimation value of target position obtained in formulaAs an iteration initial value.
Experimental conditions 1: in the case of an object inside the sensor array, the coordinates of the object position are [30,20 ]]TThe clock deviation tau between the target and the sensor is assumed to follow a uniform distribution between-10 m and 10m, i.e. tau-U (-10m,10m), the measurement error varianceFrom 10-1.5Change to 102.5。
Experimental conditions 2: in the case of an object outside the sensor array, the coordinates of the object position are [150,20 ]]TThe clock deviation tau between the target and the sensor is assumed to follow a uniform distribution between-10 m and 10m, i.e. tau-U (-10m,10m), the measurement error varianceFrom 10-2.0Change to 102.0。
Second, test contents and results
Experiment one: target position and clock bias estimation was performed using the MMA algorithm, WLS algorithm, ML algorithm, TSWLS algorithm and the method of the present invention under experimental condition 1, and the results are shown in fig. 3, in which:
FIG. 3(a) is a test result of the performance of each algorithm on target position estimation as a function of measurement error;
fig. 3(b) is a test result of performance of each algorithm on clock bias estimation as a function of measurement error.
As can be seen from fig. 3, the root mean square error of the MMA algorithm always deviates from the cramer-circle CRLB, mainly because the MMA algorithm uses a convex relaxation technique, resulting in that the constraint condition of the objective function is not tight, resulting in performance loss and only obtaining a sub-optimal solution. The estimation performance of the WLS algorithm, the ML algorithm, the TSWLS algorithm and the method approaches to CRLB when the measurement error is small, however, the root mean square error of each algorithm is increased along with the increase of the measurement error, wherein the WLS algorithm has a threshold effect earlier than the algorithm of the invention, and the root mean square error of the ML algorithm is increased sharply, because when the measurement error is large, the iteration initial value deviates far from the true value, so that the algorithm is locally converged or even diverged.
Meanwhile, the performance of the method is superior to that of the TSWLS algorithm when the measurement error is large, and the main reason is that the method not only utilizes the functional relation among variables in the second step of deviation correction process, but also fully utilizes the original positioning equation of the system, thereby improving the tolerance to the measurement error. The method of the invention keeps the estimation root mean square error to the minimum all the time, and the estimation performance is superior to the traditional method.
Experiment two: target position and clock bias estimation was performed using the MMA algorithm, WLS algorithm, ML algorithm, TSWLS algorithm and the method of the present invention under experimental condition 2, and the results are shown in fig. 4, in which:
FIG. 4(a) is a test result of the performance of each algorithm on target position estimation as a function of measurement error;
fig. 4(b) is a test result of performance of each algorithm on clock bias estimation as a function of measurement error.
A similar conclusion to the experiment one can be drawn from fig. 4. It is noted that under the conditions of the present experiment, the root mean square error of the WLS algorithm always deviates from the CRLB, and when the measurement error is large, even a divergence phenomenon occurs, mainly because when the target is outside the sensor array, an imaginary solution occurs in the root solving process of the binomial equation of the WLS algorithm, resulting in a severe degradation of the algorithm performance. The performance of the invention is similar to that of the TSWLS algorithm, but according to the partial enlarged images in the figures 4(a) and 4(b), the invention is still slightly superior to the TSWLS algorithm when the measurement error is larger, but the performance of all the algorithms is reduced along with the increase of the measurement error, and the root mean square error of the invention is always kept lowest and has higher precision.
The above is only a specific example of the present invention, and does not constitute any limitation to the present invention, and it should still be obvious to those skilled in the art that modifications can be made to the technical solutions described in the foregoing embodiments, or equivalent replacements can be made to some technical features, and any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (7)
1. The TOA joint synchronization and positioning method based on linear correction is characterized by comprising the following steps:
(1) according to the TOA positioning model of the arrival time of the node anchor points of the plurality of mutually synchronized clocks, a TOA positioning equation set A of a target signal arrival sensor is obtained;
(2) introducing an auxiliary variable theta related to a target position coordinate x and a clock deviation tau, and converting a TOA positioning equation set A into a matrix equation B related to the auxiliary variable theta;
(3) performing weighted least square estimation on the matrix equation B to obtain an auxiliary variable thetaIs estimated value ofFrom the estimated valuesObtaining an initial estimation value of the target positionAnd initial estimate of clock bias
(4) Is provided withIs Δ x, letIs Δ τ, and these two deviations are substituted into equation set a in (1) to yield equation set C for Δ x and Δ τ:
wherein r isiRepresenting distance measurement data corresponding to the ith sensor, the distance measurement data being obtained by multiplying the speed of light by the arrival time, siIndicates the location of the ith sensor, | si-x||2Representing the distance of the target from the ith sensor, niRepresents the distance measurement error, i ═ 1, 2.., M represents the number of sensors, | · tory |, counting2Representing a two-norm operation, and T representing a transposition operation;
(5) introducing the auxiliary variables ξ for Δ x and Δ τ, a matrix equation D for the auxiliary variable ξ is derived from equation set C:
wherein,h2＝[h2,1,...,h2,i,...,h2,M]T，ξ＝[Δτ,ΔxT]T，
(6) performing weighted least squares estimation on the matrix equation D to obtain an estimated value of the auxiliary variable ξ
Wherein W is a weighting matrix, W is represented byCalculating to obtain;
(7) from the estimated valuesObtaining an estimate of DeltaxAnd an estimate of Δ τ
(8) Initial estimated valueAnd delta tau estimateSubtracting to obtain the clock deviation estimated value after deviation correctionInitial estimated valueEstimated value of Δ xSubtracting to obtain the target position estimated value after deviation correction
2. The method according to claim 1, wherein the TOA positioning model in (1) is obtained by assuming that M sensor anchor nodes with mutually synchronized clocks are distributed in a two-dimensional plane, and setting coordinates of a target node to be measured as x ═ x, y]TAssuming that the target node is asynchronous with the sensor network and the clock deviation is tau, the coordinates of the anchor nodes are si＝[xi,yi]TI 1, 2.. multidot.m, x is the abscissa of the target position, y is the ordinate of the target position, xiIs the abscissa, y, of the ith sensor positioniIs the ordinate of the ith sensor position.
3. The method of claim 1, wherein the TOA localization equation set a in (1) is expressed as follows:
where i 1, 2.
4. The method of claim 1, wherein the matrix equation B in (2) is expressed as follows:
wherein G is1Is an M × 4 matrix with the ith actionAuxiliary variable θ ═ τ, xT,η]TThe variable η ═ xTx-τ2，Equation Right sideRepresenting data error, n is a distance measurement error vector, n ═ n1,n2,...,nM]TB is a diagonal matrix, B ═ diag (| | s)1-x||2,||s2-x||2,...,||sM-x||2) And diag (·) denotes a diagonalization operation.
5. The method of claim 1, wherein the weighted least squares estimation in (3) is performed according to a matrix equation B to obtain an estimate of the auxiliary variable θThe method is carried out by the following formula:
wherein G is1Is an M × 4 matrix with the ith action For the intermediate variables, W is the weighting matrix, W ═ I, and I is the identity matrix.
6. The method of claim 1, wherein (3) is based on an estimate of an auxiliary variable θObtaining an initial estimation value of the target positionAnd initial estimate of clock biasThe method is carried out by the following formula:
wherein,express getThe first element of (a) is,express getThe second and third elements of (1).
7. The method of claim 1, wherein (a), (b), and (c)7) Estimate of medium Δ xAnd an estimate of Δ τThe method is carried out by the following formula:
wherein,express getThe first element of (a) is,express getThe second and third elements of (1).
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN111487586A (en) * | 2020-04-22 | 2020-08-04 | 中国民航大学 | Positioning precision improving method based on distributed passive positioning technology |
| CN115426615A (en) * | 2022-07-21 | 2022-12-02 | 中山大学 | High-precision clock calibration and positioning direction-finding method based on 5G millimeter wave communication |
| CN115508774A (en) * | 2022-10-12 | 2022-12-23 | 中国电子科技集团公司信息科学研究院 | Time difference positioning method and device based on two-step weighted least square and storage medium |
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| 田强: "基于线性校正的TOA联合同步与定位算法", 《系统工程与电子技术》 * |
Cited By (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN111487586A (en) * | 2020-04-22 | 2020-08-04 | 中国民航大学 | Positioning precision improving method based on distributed passive positioning technology |
| CN111487586B (en) * | 2020-04-22 | 2023-06-02 | 中国民航大学 | Positioning precision improving method based on distributed passive positioning technology |
| CN115426615A (en) * | 2022-07-21 | 2022-12-02 | 中山大学 | High-precision clock calibration and positioning direction-finding method based on 5G millimeter wave communication |
| CN115508774A (en) * | 2022-10-12 | 2022-12-23 | 中国电子科技集团公司信息科学研究院 | Time difference positioning method and device based on two-step weighted least square and storage medium |
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Application publication date: 20190816 |