A bias-reduced multi-dimensional scaling analysis positioning method based on single station ranging

Abstract

This invention discloses a positioning method based on single-station ranging with bias reduction using multidimensional scaling analysis. The method first establishes a multidimensional similarity pseudolinear model based on the geometric relationship between the observation station and the target in the measured values; then, it sets an optimization function; and finally, it solves the model using Newton's iterative algorithm. This invention provides a general positioning method based on single-station ranging, achieving higher positioning accuracy compared to traditional pseudolinear and maximum likelihood methods.

Description

Technical Field

[0001] This invention belongs to the field of target detection and localization, and relates to a localization method based on single-station ranging with deviation reduction and multidimensional scaling analysis. Background Technology

[0002] Single-station ranging estimates the target's position based on distance information. It calculates the target's position by measuring the distance between the observation station and the target. Currently, single-station ranging positioning methods are mainly divided into two categories: non-analytical and pseudo-linear. Non-analytical methods yield a non-analytical target position, typically solved iteratively. While these methods offer high positioning accuracy, improper initial values ​​during iteration can lead to local optima or even iteration divergence. Furthermore, the iterative nature of these methods results in relatively high complexity. The other type of method yields an analytical target position. This involves analyzing the geometric relationship between the observation station and the target, transforming their non-linear relationship into a linear one. This pseudo-linear method obtains the target's position by solving the pseudo-linear expression. While this method has low complexity and provides an analytical solution, the linearization process leads to a loss of positioning accuracy, and no amount of optimization can reach the Cramer-Rao lower bound. Summary of the Invention

[0003] To address the shortcomings of existing technologies, this invention proposes a multidimensional scaling analysis positioning method based on single-station ranging to reduce deviation.

[0004] A positioning method based on single-station ranging with bias reduction using multidimensional scaling analysis, comprising the following steps:

[0005] The method specifically includes the following steps:

[0006] Step 1: Define the distance information between the observation station and the target at time m;

[0007] Step 2: Define the coordinate matrix as follows:

[0008]

[0009] Among them 1 M Let p represent an M×1 dimensional vector consisting entirely of 1s, where p = [p x ,p y ] T This represents the target's location information, where S represents the coordinate set matrix of M observation stations at different times:

[0010] S = [s1, s2, ..., s M ] T

[0011] sm =[s m,x ,s m,y ] T This represents the location information of the observation station at time m, where m = 1, 2, ..., M;

[0012] Defined scalar product matrix V = XX T Matrix V is a positive definite matrix of rank 2, and the elements of matrix V are represented as follows:

[0013]

[0014] Where d m With d n d represents the distance between the target and the observation station at time m and n. mn The distance between the observation station at time n and the observation station at time m is represented as:

[0015]

[0016] Where m, n = 1, 2, ..., M.

[0017] A multidimensional similarity pseudolinear model is defined:

[0018] Cz = 0 M

[0019] Where C = VA, [] + This represents the pseudo-inverse of a matrix, z = [1, p]. x ,p y ] T 0 M This represents an M×1 dimensional vector whose elements are all zero.

[0020] Step 3: In practice, because the measured values ​​are biased, the multidimensional similarity pseudolinear model is also biased, that is:

[0021]

[0022] Where the matrix Represents a scalar product matrix containing measurement noise. Performing a Taylor expansion and ignoring noise and higher-order terms, we obtain:

[0023]

[0024] Where matrix △V is a scalar product matrix. The compensation matrix for the noise components, matrix V o Let V represent a noise-free scalar matrix, where the expression for matrix ΔV is shown below:

[0025]

[0026] Right now:

[0027]

[0028] Right now:

[0029]

[0030] in Let q represent the unbiased measurement value without noise at the m-th time. m Indicates measurement error;

[0031] Step 4: Establish the optimization function expression, as shown below:

[0032]

[0033] in:

[0034]

[0035]

[0036]

[0037] Let C be a matrix containing measurement noise, where A M1 d represents the element in the M-th row and 1-th column of matrix A. m Let m represent the distance between the observation station and the target at the m-th time, where m = 1, 2, ..., M.

[0038] The target localization problem of a multidimensional similar pseudolinear model is transformed into a constrained overall least squares problem. However, in the complex domain, this constrained problem has been transformed into an unconstrained optimization problem. Since the variables in the target localization problem are all real numbers, the following unconstrained optimization expression is obtained in the real domain:

[0039]

[0040] Among them W p =G1+p x G2+p y G3.

[0041] Step 5: Solve the multidimensional similar pseudolinear model using the least squares objective estimation method.

[0042] Target location information:

[0043]

[0044] Where p LS Let A1 represent the target estimate under the multidimensional similarity pseudolinear least squares method, and A2 represent the first column of matrix A. This represents a scalar product matrix containing measurement noise.

[0045] Step Six: Establish the iterative formula for the Newton-Raphson iteration algorithm, and apply it to the multidimensional similar pseudolinear least squares method.

[0046] The lower target estimate is set as the initial value for the Newton-Raphson iterative algorithm:

[0047]

[0048] Where p n Represents the target position information in Newton's nth iteration; gradient vector Hessian matrix

[0049] in

[0050] Step 7: Iterate the Newton algorithm multiple times and output the target position information.

[0051] Preferably, the distance information between the observation station and the target at time m is defined by the following expression:

[0052]

[0053] However, in practice, measurements contain noise, so the above formula becomes:

[0054]

[0055] Let m represent the actual measured value at time m, where the measurement noise covariance matrix is:

[0056]

[0057] In the above expression, Q represents a diagonal element that is all σ. 2 It is an M×M dimensional diagonal matrix, where q represents the noise vector.

[0058] Preferably, the number of iterations in step seven is 5-8.

[0059] To address the target localization problem using single-station ranging, a deviation-reducing multidimensional scaling analysis localization method based on single-station ranging is proposed. This method first establishes a multidimensional similarity pseudo-linear model based on the geometric relationship between the observation station and the target in the measured values; then, an optimization function is set; and finally, the model is solved using Newton's iterative algorithm.

[0060] The advantages of this invention compared to existing technologies are: compared to existing single-station target positioning methods, this invention has lower complexity and higher positioning accuracy. Attached Figure Description

[0061] Figure 1 This is a network diagram of the sensor and the target;

[0062] Figure 2 This is a flowchart of the present invention;

[0063] Figure 3 The motion trajectory of the sensor and the position of the target in the simulation;

[0064] Figure 4 A comparison of the root mean square error (RMSE) of target localization for various single-station localization methods as signal and noise levels change;

[0065] Figure 5 A comparison graph of the bias of target localization (BIAS) as measurement noise changes for various single-station localization methods. Detailed Implementation

[0066] The present invention will be further analyzed below with reference to specific embodiments.

[0067] like Figure 2 As shown, a multidimensional scaling analysis-based positioning method for reducing deviations from single-station ranging is proposed. This method first establishes a multidimensional similarity pseudo-linear model based on the geometric relationship between the observation station and the target in the measured values; then, an optimization function is set; and finally, the model is solved using Newton's iterative algorithm.

[0068] Stap1: as Figure 1 As shown, the distance information between the observation station and the target at time m is defined, and its expression is as follows:

[0069]

[0070] Where m = 1, 2, ..., M, d m s represents the distance between the observation station and the target at the m-th time. m =[s m,x ,s m,y ] T p represents the location information of the observation station at the m-th time, p = [p x ,p y ] T This indicates the location information of the target.

[0071] However, in practice, measurements contain noise, so the above formula becomes:

[0072]

[0073] in This represents the unbiased measurement value without noise at the m-th time. q represents the actual measured value at time m. m Represents the measurement error, where the measurement noise covariance matrix is:

[0074] Q = E(qq) T )=diag(σ 2 ,σ 2 ,...,σ 2 (2)

[0075] In the above expression, Q represents a diagonal element that is all σ. 2 It is an M×M dimensional diagonal matrix, where q represents the noise vector.

[0076] Stap2: Define the coordinate matrix as follows:

[0077]

[0078] Among them 1 M Let S be an M×1 dimensional vector consisting entirely of 1s, and let S be the set matrix of coordinates of the M observation stations at different times.

[0079] S = [s1, s2, ..., s M ] T

[0080] Defined scalar product matrix V = XX T It is not difficult to see that matrix V is a positive definite matrix with rank 2, and each element of matrix V can be represented as:

[0081]

[0082] Where d m With d n d represents the distance between the target and the observation station at time m and n. mn The distance between the observation station at time n and the observation station at time m is represented as:

[0083]

[0084] Where m, n = 1, 2, ..., M.

[0085] A multidimensional similarity pseudolinear model is defined:

[0086] Cz = 0 M

[0087] Where C = VA, [] + This represents the pseudo-inverse of a matrix, z = [1, p].x ,p y ] T 0 M This represents an M×1 dimensional vector whose elements are all zero.

[0088] Stap3: In practice, because the measured values ​​are biased, the multidimensional similarity pseudolinear model is also biased, that is:

[0089]

[0090] Where the matrix Represents a scalar product matrix containing measurement noise. Performing a Taylor expansion and ignoring noise and higher-order terms yields:

[0091]

[0092] Where matrix △V is a scalar product matrix. The compensation matrix for the noise components, matrix V o Let V represent a noise-free scalar matrix, where the expression for matrix ΔV is shown below:

[0093]

[0094] Right now:

[0095] It is not difficult to see:

[0096]

[0097] Right now:

[0098]

[0099] Stap4: Create the optimized function expression as shown below:

[0100]

[0101] in:

[0102]

[0103]

[0104]

[0105] This represents the C matrix containing measurement noise.

[0106] The target localization problem of a multidimensional similar pseudolinear model is transformed into a constrained overall least squares problem. However, in the complex domain, this constrained problem has been transformed into an unconstrained optimization problem. Since the variables in the target localization problem are all real numbers, and in the real domain, I can also obtain the following unconstrained optimization expression:

[0107]

[0108] Among them W p =G1+p x G2+p y G3.

[0109] Stap5: Utilizes the least squares target estimation method to solve the multidimensional similarity pseudolinear model and obtain the target's location information.

[0110]

[0111] Where p LS Let A1 represent the target estimate under the multidimensional similarity pseudolinear least squares method, and A2 represent the first column of matrix A. This represents a scalar product matrix containing measurement noise.

[0112] Stap6: Establish the iterative formula for Newton's iteration algorithm, and set the target estimate under the multidimensional similar pseudolinear least squares method as the initial value for Newton's iteration algorithm:

[0113]

[0114] Where p n Represents the target position information in Newton's nth iteration; gradient vector Hessian matrix

[0115] in

[0116] Stap7: Performs 5 to 8 iterations of the Newton algorithm and outputs the target position information.

[0117] like Figure 3 As shown, this is a simulation diagram of the sensor's motion trajectory and the target's position; Figure 4 The graph shown is a comparison of the root mean square error (RMSE) of target localization for various single-station localization methods under varying signal and noise levels (Monte Carlo 5000 runs).

[0118] The following is the RMSE expression:

[0119]

[0120] Where L represents the number of simulations. This represents the estimation result obtained for the l-th (l=1,2,…,L) target localization.

[0121] like Figure 4 As shown, Cramer-Rao Lower Bound (CRLB) analysis determines a lower bound for the variance of the unbiased estimator of the target under this feature, and its calculation expression is as follows:

[0122]

[0123] Among them, the Fischer information matrix

[0124] like Figure 5 As shown: Figure 5 Comparison of target localization bias (BIAS) for various single-station localization methods as measurement noise changes (Monte Carlo 5000 times):

[0125] The following is the BIAS expression:

[0126]

[0127] Where L represents the number of simulations. This represents the estimation result obtained for the l-th (l=1,2,…,L) target localization.

[0128] This invention proposes a multidimensional scaling analysis-based positioning method for reducing deviations in single-station ranging: Multidimensional Scaling Constrained Total Least Squares (MDS-CTLS), Constrained Weighted Least Squares Calibration (CWLSC), Multidimensional Scaling Weighted Least Squares (MDS-WLS), and Multidimensional Scaling Total Least Squares (MDS-TLS).

[0129] This invention is compared with:

[0130] MDS-CTLS MDS-WLS MDS-TLS CWLSC Distance measurement accuracy high Low Low high Implementation complexity Low Low Low high

Claims

1. A positioning method based on single-station ranging with bias reduction using multidimensional scaling analysis, characterized in that: The method specifically includes the following steps: Step 1: Define the distance information between the observation station and the target at time m; Step 2: Define the coordinate matrix as follows: Among them 1 M Let p represent an M×1 dimensional vector consisting entirely of 1s, where p = [p x ,p y ] T This represents the target's location information, where S represents the coordinate set matrix of M observation stations at different times: S=[s1,s2,...,s M ] T s m =[s m,x ,s m,y ] T This represents the location information of the observation station at time m, where m = 1, 2, ..., M; Defined scalar product matrix V = XX T Matrix V is a positive definite matrix of rank 2, and the elements of matrix V are represented as follows: Where d m With d n d represents the distance between the target and the observation station at time m and n. mn The distance between the observation station at time n and the observation station at time m is represented as: Where m,n=1,2,...,M; A multidimensional similarity pseudolinear model is defined: Cz=0 M Where C = VA, [ ] + This represents the pseudo-inverse of a matrix, z = [1, p]. x ,p y ] T 0 M Represents an M×1 dimensional vector whose elements are all zero; Step 3: In practice, because the measured values ​​are biased, the multidimensional similarity pseudolinear model is also biased, that is: Where the matrix Represents a scalar product matrix containing measurement noise. Performing a Taylor expansion and ignoring noise and higher-order terms, we obtain: Where matrix △V is a scalar product matrix. The compensation matrix for the noise components, matrix V o Let V represent a noise-free scalar matrix, where the expression for matrix ΔV is shown below: Right now: Right now: in Let q represent the unbiased measurement value without noise at the m-th time. m Indicates measurement error; Step 4: Establish the optimization function expression, as shown below: in: Let C be a matrix containing measurement noise, where A M1 d represents the element in the M-th row and 1-th column of matrix A. m Let m represent the distance between the observation station and the target at the m-th time, where m = 1, 2, ..., M; The target localization problem of the multidimensional similar pseudolinear model is transformed into a constrained global least squares problem, resulting in the following unconstrained optimization expression: Among them W p =G1+p x G2+p y G3; Step 5: Solve the multidimensional similarity pseudolinear model using the least squares target estimation method to obtain the target's location information. Where p LS Let A1 represent the target estimate under the multidimensional similarity pseudolinear least squares method, and A2 represent the first column of matrix A. Represents a scalar product matrix containing measurement noise; Step Six: Establish the iterative formula for Newton's iteration algorithm, and set the target estimate under the multidimensional similar pseudolinear least squares method as the initial value for Newton's iteration algorithm: Where p n Represents the target position information in Newton's nth iteration; gradient vector Hessian matrix in Step 7: Iterate the Newton algorithm multiple times and output the target position information.

2. The positioning method based on single-station ranging with deviation reduction and multidimensional scaling analysis according to claim 1, characterized in that: The distance information between the observation station and the target at time m is defined as follows: However, in practice, measurements contain noise, so the above formula becomes: Let m represent the actual measured value at time m, where the measurement noise covariance matrix is: Q=E(qq T )=diag(σ 2 ,s 2 ,...,s 2 ) (2); In the above expression, Q represents a diagonal element that is all σ. 2 It is an M×M dimensional diagonal matrix, where q represents the noise vector.

3. The positioning method based on single-station ranging with deviation reduction and multidimensional scaling analysis according to claim 1, characterized in that: The number of iterations in step seven is 5-8.

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