Multi-UAV Doppler Frequency Shift Positioning Method Based on Position Error

Abstract

This invention discloses a multi-UAV Doppler frequency shift positioning method based on position error, mainly addressing the problems of neglecting position error leading to deteriorated positioning accuracy and high complexity and computational burden in solving positioning equations in existing technologies. The implementation steps of this invention are as follows: acquiring Doppler frequency shift observations and distance change rate observations for each UAV; generating a joint coarse estimation vector of the radiation source's position and velocity vectors; constructing a coefficient matrix, observation matrix, and observation vectors; obtaining a joint precise estimation vector of the radiation source's position and velocity vectors; and calculating the estimated vectors of the radiation source's position and velocity vectors. This invention reduces the computational burden of equations while considering position error, achieving better positioning accuracy and speed, and can be used for locating moving radiation sources.

Description

Technical Field

[0001] This invention belongs to the field of radiation source localization technology, and further relates to a multi-UAV Doppler frequency shift localization method based on position error, which is particularly suitable for localization scenarios where the position error of UAVs is large. Background Technology

[0002] Passive positioning is a target detection technology that uses the signals emitted by a radiation source itself for location without emitting electromagnetic waves. In recent years, with the continuous development of UAV technology, the increased payload and range of UAVs have made passive positioning possible. Depending on the parameters observed by the positioning system, common passive positioning methods include angle-based cross-location, time-difference positioning, frequency-difference positioning, received signal strength positioning, and combined time-frequency-difference positioning. Among these, the combined time-frequency-difference positioning method is commonly used for passive positioning of moving radiation sources. However, this method requires the simultaneous observation of two parameters, with time-difference and frequency-difference observations supporting each other, placing high demands on the performance of the measurement equipment. Therefore, achieving radiation source positioning using only frequency observations in the absence of time-difference observations has enormous development potential and significant engineering value.

[0003] The Information Engineering University of the Strategic Support Force of the Chinese People's Liberation Army disclosed a closed-loop solution method for locating a stationary radiation source using a single moving UAV under the condition of measurement error in the carrier frequency of the transmitted signal in its patent application "A Single-Platform Doppler Two-Stage Closed-Loop Positioning Method with the Presence of Prior Error in Signal Carrier Frequency" (Patent Application No.: 202110927203.5, Authorization Announcement No.: CN 113835061 B). This method addresses the nonlinear relationship between the arrival frequency observation and the radiation source position vector in the positioning equation by constructing two sets of pseudo-linear observation equations and obtaining corresponding closed-loop solutions for each. High-precision positioning of the radiation source is achieved through two-stage calculation. However, this method still has shortcomings. It ignores the impact of UAV position error on positioning accuracy. The position and velocity parameters of the UAV itself are measured by navigation equipment and will also have certain errors. In scenarios where the UAV position error is large, if this error is ignored, it will lead to a deterioration in positioning accuracy.

[0004] In their paper "Doppler Frequency Shift Based SourceLocalization in Presence of Sensor Location Errors" (IEEE Access, 2018, 6: 59752-59760), Deng Lijuan et al. proposed a radiation source localization method using Doppler frequency shift observations under sensor position errors. This method considers the impact of sensor position errors on positioning accuracy, establishes a Doppler frequency shift observation model incorporating sensor position errors, and then follows a two-step weighted least squares framework to solve the corresponding pseudo-linear equations in two stages, ultimately obtaining accurate estimates of the radiation source motion parameters. This method exhibits good positioning accuracy and sensor position error handling capabilities. However, a drawback of this method is that it requires a large number of Doppler frequency shift observations for localization. Since there is a one-to-one correspondence between Doppler frequency shift observations and positioning equations, an increase in the number of Doppler frequency shift observations inevitably leads to an increase in the number of positioning equations, increasing the difficulty and computational burden of solving the system of positioning equations. Conversely, if the number of Doppler frequency shift observations is insufficient, positioning deviations will occur, severely affecting positioning accuracy. Summary of the Invention

[0005] The purpose of this invention is to address the shortcomings of the existing methods by proposing a multi-UAV Doppler frequency shift positioning method based on position error, which solves the problems of poor positioning accuracy caused by ignoring position error and the high complexity and computational cost of solving the positioning equation.

[0006] The specific approach to achieving the objective of this invention is as follows: This invention constructs an observation matrix and observation vector using the UAV position vector containing position errors. Then, it uses the weighted least squares method to obtain a joint coarse estimation vector of the radiation source position vector and velocity vector. Next, it uses the constraint relationships between the elements of the joint coarse estimation vector of the radiation source position vector and velocity vector to correct the joint coarse estimation vector of the radiation source position vector and velocity vector, thus solving the problem in the prior art where ignoring the UAV position error leads to a sharp deterioration in positioning accuracy. When jointly estimating the radiation source position vector and velocity vector, none of the elements of the joint coarse estimation vector and the joint accurate estimation vector of the radiation source position vector and velocity vector involve the UAV position vector containing position errors. This makes the form simpler and the solution more convenient, overcoming the shortcomings of the prior art where the Doppler frequency shift positioning equation has high solution complexity and large computational load.

[0007] The specific steps of this invention include the following:

[0008] Step 1: Obtain the Doppler frequency shift observation and distance change rate observation for each UAV:

[0009] Step 1.1: Measure the Doppler frequency shift of a single moving radiation source signal arriving at the UAV when the UAV is stationary.

[0010] Step 1.2: Calculate the distance change rate observation for each UAV;

[0011] Step 1.3: Construct an observation matrix of N2-1 rows and 8 columns by taking the distance change rate observations of all UAVs, the X-axis component of the UAV position vector containing position error, and the Y-axis component of the UAV position vector containing position error. Here, the values ​​of N2 and N1 are equal, and N1 represents the total number of UAVs.

[0012] Step 1.4: Construct an observation vector containing N3-1 elements from all UAV distance change rate observations and all UAV position vectors containing position errors, where N3 and N1 have corresponding equal values.

[0013] Step 2: Generate a joint coarse estimate vector of the radiation source's position and velocity vectors:

[0014] Step 2.1: Calculate the joint initial estimate vector of the radiation source position vector and velocity vector;

[0015] Step 2.2: Construct a coefficient matrix of N2-1 rows and N2 columns based on the joint initial estimate vector of the radiation source position vector and velocity vector;

[0016] Step 2.3: Construct an N2-row, 2N2-column partial derivative matrix based on the joint initial estimate vector of the radiation source position vector and velocity vector;

[0017] Step 2.4: Calculate the covariance matrix of the UAV position measurement error and the Doppler frequency shift observation error;

[0018] Step 2.5: Calculate the weighted matrix of the joint coarse estimate vector of the radiation source position vector and velocity vector;

[0019] Step 2.6: Calculate the joint coarse estimate vector of the radiation source position vector and velocity vector;

[0020] Step 3: Construct the coefficient matrix, observation matrix, and observation vector based on the joint coarse estimate vector of the radiation source position vector and velocity vector:

[0021] Step 3.1: Construct an 8-row, 8-column coefficient matrix based on the joint coarse estimate vector of the radiation source position vector and velocity vector;

[0022] Step 3.2: Construct an 8-row, 4-column observation matrix based on the joint coarse estimate vector of the radiation source position vector and velocity vector;

[0023] Step 3.3: Construct an 8-row, 1-column observation vector, wherein the values ​​of the 1st to 5th elements of the observation vector correspond to the values ​​of the 1st to 5th elements of the joint coarse estimation vector of the radiation source position vector and velocity vector, respectively; the value of the 6th element of the observation vector corresponds to the square of the 6th element of the joint coarse estimation vector of the radiation source position vector and velocity vector, respectively; and the values ​​of the 7th to 8th elements of the observation vector correspond to the values ​​of the 1st to 5th elements of the joint coarse estimation vector of the radiation source position vector and velocity vector, respectively.

[0024] Step 4: Obtain the joint accurate estimate vector of the radiation source position vector and velocity vector:

[0025] Step 4.1: Calculate the covariance matrix of the joint coarse estimation vector error of the radiation source position vector and velocity vector;

[0026] Step 4.2: Calculate the weighted matrix of the joint accurate estimate vector of the radiation source position vector and velocity vector;

[0027] Step 4.3: Calculate the joint accurate estimate vector of the radiation source position vector and velocity vector;

[0028] Step 5: Calculate the estimated vectors of the radiation source position vector and the radiation source velocity vector according to the following formulas:

[0029]

[0030] in, The estimated vector representing the location vector of the radiation source. The estimated vector representing the velocity vector of the radiation source. The first and second elements represent the joint exact estimate vector of the radiation source's position and velocity vectors. The third and fourth elements represent the joint exact estimate vector of the radiation source's position and velocity vectors.

[0031] Compared with the prior art, the present invention has the following advantages:

[0032] First, because this invention constructs an observation matrix and observation vector using the UAV position vector containing position errors, and then uses the weighted least squares method to obtain a joint coarse estimation vector of the radiation source position vector and velocity vector, and then uses the constraint relationship between each element of the joint coarse estimation vector of the radiation source position vector and velocity vector to correct the joint coarse estimation vector of the radiation source position vector and velocity vector, it solves the problem that ignoring the UAV position error in the prior art will lead to a sharp deterioration in positioning accuracy, thus improving the accuracy of Doppler frequency shift positioning.

[0033] Second, because the elements of the joint coarse estimation vector and the joint precise estimation vector of the radiation source position vector and velocity vector do not involve the UAV position vector containing position errors when the present invention performs joint estimation of the radiation source position vector and velocity vector, the form is simpler and the solution is more convenient. This overcomes the shortcomings of the existing technology in that the Doppler frequency shift positioning equation has high complexity and large amount of computation, thus reducing the complexity and amount of computation of solving the Doppler frequency shift positioning equation. Attached Figure Description

[0034] Figure 1 This is a flowchart of the present invention;

[0035] Figure 2 This is a comparison of the root mean square error and the lower bound of Cramer-Rao when the variance of the Doppler frequency shift observation error is different in the simulation experiment of this invention.

[0036] Figure 3 This is a comparison of the root mean square error and the Cramer-Rao lower bound of the radiation source velocity estimation method using the method of this invention when the variance of the Doppler frequency shift observation error takes different values ​​in the simulation experiment of this invention.

[0037] Figure 4 This is a comparison chart of the root mean square error of radiation source position estimation and the lower bound of Cramer-Rao when the variance value of the position measurement error of the unmanned aerial vehicle in the simulation experiment of this invention is different.

[0038] Figure 5 This is a comparison chart of the root mean square error of the radiation source velocity estimation method and the Cramer-Rao lower bound when the variance value of the position measurement error of the unmanned aerial vehicle in the simulation experiment of this invention is different. Detailed Implementation

[0039] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0040] Reference Figure 1 The specific steps for implementing the present invention will be further described below with reference to the embodiments.

[0041] Step 1: Obtain the Doppler frequency shift observation and distance change rate observation for each UAV:

[0042] Step 1.1, Measure the Doppler frequency shift of a single moving radiation source signal reaching the UAV when the UAV is stationary. The observations are as follows:

[0043]

[0044] Among them, f if represents the Doppler frequency shift observations of the i-th UAV, i = 1, 2, ..., 10. c Let c represent the carrier frequency of the moving radiation source signal, c represent the signal propagation speed, and p represent the position vector of the radiation source, where p = [x, y]. T , where x represents the component of the radiation source's position vector on the X-axis, and y represents the component of the radiation source's position vector on the Y-axis, (·) T Indicates the transpose operation, s i Let represent the position vector of the i-th drone, and s i =[x i ,y i ] T x i Let y represent the component of the position vector of the i-th UAV on the X-axis. i Let represent the position vector of the i-th UAV on the Y-axis, and q represent the velocity vector of the radiation source, where q = [v x ,v y ] T v x v represents the component of the velocity vector of the radiation source on the X-axis. y Let ||·|| represent the y-component of the velocity vector of the radiation source, and ||·|| represent the 2-norm operation. Δf i This represents the Doppler frequency shift observation error of the i-th UAV;

[0045] Step 1.2, according to the formula Calculate the distance change rate observation for each UAV, where d i Represents the observation of the rate of change of distance for the i-th UAV;

[0046] Step 1.3: Construct a 9x8 observation matrix using the following format: all observations of the distance change rate of all UAVs, the X-axis components of all UAV position vectors including position errors, and the Y-axis components of all UAV position vectors including position errors.

[0047]

[0048] G1 is an observation matrix consisting of the distance change rate observations of all UAVs;

[0049] Step 1.4: Construct an observation vector with 9 elements from all UAV distance change rate observations and all UAV position vectors containing position errors, as follows:

[0050]

[0051] Where h1 represents the observation vector consisting of the distance change rate observations of all UAVs;

[0052] Step 2: Generate a joint coarse estimate vector of the radiation source's position and velocity vectors:

[0053] Step 2.1, calculate the joint initial estimate vector of the radiation source position vector and velocity vector according to the following formula:

[0054]

[0055] in, This represents the joint initial estimate vector of the radiation source's position and velocity vectors;

[0056] Step 2.2: Construct a coefficient matrix of 9 rows and 10 columns based on the joint initial estimate vector of the radiation source position vector and velocity vector, as follows:

[0057]

[0058] Where B1 represents a coefficient matrix consisting of observations of the distance change rate of all UAVs. Let represent the distance between the i-th UAV and the initial estimated location of the radiation source, and Let represent the initial estimation vector of the radiation source's location, and The first and second elements represent the joint initial estimate vector of the radiation source's position and velocity vectors;

[0059] Step 2.3: Construct a 10x20 partial derivative matrix F based on the joint initial estimate vector of the radiation source position vector and velocity vector, where F = blkdiag(ρ1, ρ2, ..., ρ M blkdiag(·) represents the block diagonal matrix operation.

[0060] Steps 2 and 4: Calculate the covariance matrix of the UAV position measurement error and the Doppler frequency shift observation error according to the following formula:

[0061]

[0062] Among them, Q ξ Q represents the covariance matrix of the UAV position measurement error and the Doppler frequency shift observation error. s The covariance matrix Q represents the position measurement error of the UAV. f The covariance matrix representing the Doppler frequency shift observation error;

[0063] Step 2.5, calculate the weighted matrix of the joint coarse estimate vector of the radiation source position vector and velocity vector according to the following formula:

[0064] W1=(B1Q ξ B1T ) -1

[0065] Where W1 represents the weighted matrix of the joint coarse estimate vector of the target position vector and velocity vector;

[0066] Step 2.6, calculate the joint coarse estimate vector of the radiation source position vector and velocity vector according to the following formula:

[0067]

[0068] in, This represents a joint coarse estimate vector of the radiation source's position and velocity vectors;

[0069] Step 3: Construct the coefficient matrix, observation matrix, and observation vector based on the joint coarse estimation vector of the radiation source position vector and velocity vector;

[0070] Step 3.1: Construct an 8x8 coefficient matrix based on the joint coarse estimate vector of the radiation source position vector and velocity vector, as follows:

[0071]

[0072] Where B2 represents a coefficient matrix composed of some elements from the joint coarse estimation vector of the radiation source position vector and velocity vector. express The k-th element, k = 1, 2, ..., 8 express The fifth element takes half the value. express The fourth element takes half the value;

[0073] Step 3.2: Construct an 8-row, 4-column observation matrix based on the joint coarse estimate vector of the radiation source position vector and velocity vector, as follows:

[0074]

[0075] Wherein, G2 represents the observation matrix composed of some elements of the joint coarse estimation vector of the radiation source position vector and velocity vector;

[0076] Step 3.3: Construct an 8-row, 1-column observation vector based on the joint coarse estimate vector of the radiation source position vector and velocity vector, as follows:

[0077]

[0078] Where h2 represents the observation vector consisting of all elements in the joint coarse estimate vector of the radiation source position vector and velocity vector. express The first to fifth elements, express The square of the value of the 6th element, express The 7th and 8th elements;

[0079] Step 4: Obtain the joint accurate estimate vector of the radiation source position vector and velocity vector:

[0080] Step 4.1, calculate the covariance matrix of the joint coarse estimation vector error of the radiation source position vector and velocity vector according to the following formula:

[0081]

[0082] in, The overall representation is the covariance matrix of the joint coarse estimation vector error of the radiation source position vector and velocity vector.

[0083] Step 4.2, calculate the weighted matrix of the joint exact estimate vector of the radiation source position vector and velocity vector according to the following formula:

[0084]

[0085] Where W2 represents the weighted matrix of the exact estimated vectors of the radiation source's position and velocity vectors;

[0086] Step 4.3, calculate the joint exact estimate vector of the radiation source position vector and velocity vector according to the following formula:

[0087]

[0088] in, The vector representing the joint accurate estimate of the target's position and velocity vectors;

[0089] Step 5: Calculate the estimated vectors of the radiation source position vector and the radiation source velocity vector according to the following formulas:

[0090]

[0091] in, The estimated vector representing the location vector of the radiation source. The estimated vector representing the velocity vector of the radiation source. The first and second elements represent the joint exact estimate vector of the radiation source's position and velocity vectors. The third and fourth elements represent the joint exact estimate vector of the radiation source's position and velocity vectors.

[0092] The effects of the present invention will be further explained below with reference to simulation experiments.

[0093] 1. Simulation experimental conditions:

[0094] The hardware platform for the simulation experiment of this invention is: an Intel i7 7700 CPU with a main frequency of 3.6GHz and 16GB of memory.

[0095] The software platform for the simulation experiment of this invention is: Windows 10 operating system and MATLAB R2020a.

[0096] The simulation experiment of this invention is conducted in a two-dimensional plane, assuming that 10 UAVs participate in the localization of the moving radiation source, and the position vector of the radiation source is [145, 735]. T The velocity vector of the radiation source is [85, 55]. T The unit of position is meters (m), the unit of velocity is m / s, the carrier frequency of the radiation source is 9 GHz, and the signal propagation speed is 3 × 10⁻⁶. 8 m / s. The position coordinates of each UAV are shown in the table below.

[0097] Table 1. List of UAV location coordinates

[0098] UAV serial number 1 2 3 4 5 6 7 8 9 10 <![CDATA[x i ]]> -445 -580 625 65 585 630 40 -310 -225 -245 <![CDATA[y i ]]> 500 -485 300 520 640 -595 -460 385 -200 -745

[0099] Assume the covariance matrix of the Doppler frequency shift observation error is Q. f =σ f 2 I 10 , σ f 2 I represents the variance of the Doppler frequency shift observation error. 10 It is a 10x10 identity matrix. The covariance matrix of the UAV position measurement error is Q. s =σ s 2 I 20 , σ s 2 I represents the variance of the position measurement error of the UAV. 20 It is a 20x20 identity matrix. In this experiment, the estimation accuracy of the radiation source position vector and velocity vector is measured by the root mean square error (RMSE), which is defined as follows: and in, and Let p° represent the position and velocity estimation vectors of the radiation source obtained in the i-th Monte Carlo simulation experiment, q° represent the true position of the radiation source, ||·||2 represent the 2-norm operation, and L represent the number of simulation experiments. In this experiment, L is set to 1000.

[0100] 2. Simulation content and result analysis:

[0101] There are two simulation experiments for this invention.

[0102] Simulation Experiment 1:

[0103] The simulation experiment 1 of this invention is conducted in σ s 2 The value is -60dB, σ f 2 When the values ​​are different, the root mean square error of the position vector estimation and velocity vector estimation of the radiation source located by the method of this invention is compared with the Cramer-Rao lower bound, and the comparison diagram is as follows. Figure 2 and Figure 3 As shown. Figure 2 This is a comparison chart of the root mean square error of the radiation source position vector estimation error and the Cramero lower bound of the radiation source position vector estimation error using the method of this invention. Figure 2 The X-axis coordinate in the figure represents the variance of the Doppler frequency shift measurement error, in dB. Figure 2 The Y-axis coordinate in the figure represents the root mean square error of the radiation source position vector estimation error, in dB. Figure 2 The curve marked with a square in the middle represents the Cramer-Rao lower bound curve for the estimation error of the radiation source position vector. Figure 2 The curve marked with a diamond represents the root mean square error of the estimation error of the radiation source location vector located using the method of this invention. Figure 3 This is a comparison chart of the root mean square error of the radiation source velocity vector estimation error and the Cramer-Rao lower bound of the radiation source velocity vector estimation error, obtained using the method of this invention. Figure 3 The X-axis coordinate in the figure represents the variance of the Doppler frequency shift observation error, in dB. Figure 3 The Y-axis coordinate in the figure represents the root mean square error of the radiation source velocity vector estimation error, in dB. Figure 3 The curve marked with a square in the middle represents the Cramer-Rao lower bound curve for the estimation error of the radiation source velocity vector. Figure 3 The curve marked with a diamond represents the root mean square error of the velocity vector estimation error of the radiation source located using the method of this invention.

[0104] Depend on Figure 2 As can be seen from the above, the root mean square error of the radiation source location estimation using the method of this invention is within σ. f 2 In the case of smaller values, the lower bound of Cramer-Rao's approximation of the position estimate increases with σ. f 2 As the σ increases, the root mean square error of the radiation source location estimation using the method of this invention also gradually increases, from σ f 2Starting at 0 dB, the root mean square error of the radiation source location estimation error using the method of this invention gradually moves away from the Cramer-Rao lower bound of the location estimation. The results show that the method of this invention has good location estimation accuracy; when the Doppler frequency shift measurement error is small, the location estimation accuracy basically converges to the Cramer-Rao lower bound.

[0105] Depend on Figure 3 As can be seen from this, the root mean square error of the velocity estimation of the radiation source located using the method of this invention is within σ. f 2 In the case of smaller values, the lower bound of the velocity estimation approximates the Cramer-Rao bound, as σ... f 2 As the magnitude of the error increases, the root mean square error of the radiation source velocity estimation using the method of this invention also gradually increases, from σ f 2 Starting at 0 dB, the root mean square error of the velocity estimation error of the radiation source located using the method of this invention gradually moves away from the Cramer-Rao lower bound of the velocity estimation. The results show that the method of this invention has good velocity estimation accuracy; when the Doppler frequency shift measurement error is small, the velocity estimation accuracy basically converges to the Cramer-Rao lower bound.

[0106] Simulation Experiment 2:

[0107] Simulation experiment 2 of this invention is conducted in σ f 2 The value is -10dB, σ s 2 When the values ​​are different, the root mean square error of the position vector estimation and velocity vector estimation of the radiation source located by the method of this invention is compared with the Cramer-Rao lower bound, and the comparison diagram is as follows. Figure 4 and Figure 5 As shown. Figure 4 This is a comparison chart of the root mean square error of the radiation source position vector estimation error and the Cramero lower bound of the radiation source position vector estimation error using the method of this invention. Figure 4 The X-axis coordinate in the figure represents the variance of the Doppler frequency shift measurement error, in dB. Figure 4 The Y-axis coordinate in the figure represents the root mean square error of the radiation source position vector estimation error, in dB. Figure 4 The curve marked with a square in the middle represents the Cramer-Rao lower bound curve for the estimation error of the radiation source position vector. Figure 4 The curve marked with a diamond represents the root mean square error of the estimation error of the radiation source location vector located using the method of this invention. Figure 5 This is a comparison chart of the root mean square error of the radiation source velocity vector estimation error and the Cramer-Rao lower bound of the radiation source velocity vector estimation error, obtained using the method of this invention. Figure 5 The X-axis coordinate in the figure represents the variance of the UAV position measurement error, in dB. Figure 5The Y-axis coordinate in the figure represents the root mean square error of the radiation source velocity vector estimation error, in dB. Figure 5 The curve marked with a square in the middle represents the Cramer-Rao lower bound curve for the estimation error of the radiation source velocity vector. Figure 5 The curve marked with a diamond represents the root mean square error of the velocity vector estimation error of the radiation source located using the method of this invention.

[0108] Depend on Figure 4 As can be seen from the above, the root mean square error of the radiation source location estimation using the method of this invention is within σ. s 2 When the value is small, it approaches the lower bound of Cramer-Rao for position estimation; as σ increases... s 2 As the σ increases, the root mean square error of the radiation source location estimation using the method of this invention also gradually increases, from σ s 2 Starting at 0 dB, the root mean square error of the radiation source location estimation error using the method of this invention gradually moves away from the Cramer-Rao lower bound of the location estimation. The results show that the method of this invention has good location estimation accuracy, and when the variance of the UAV location measurement error is small, the location estimation accuracy basically converges to the Cramer-Rao lower bound.

[0109] Depend on Figure 5 As can be seen from this, the root mean square error of the velocity estimation of the radiation source located using the method of this invention is within σ. s 2 When the value is small, it approaches the Cramer-Rao lower bound for velocity estimation; as σ increases... s 2 As the magnitude of the error increases, the root mean square error of the radiation source velocity estimation using the method of this invention also gradually increases, from σ s 2 Starting at 0 dB, the root mean square error of the velocity estimation error of the radiation source located using the method of this invention gradually moves away from the Cramer-Rao lower bound of the velocity estimation. The results show that the method of this invention has good velocity estimation accuracy, and when the variance of the UAV position measurement error is small, the velocity estimation accuracy basically converges to the Cramer-Rao lower bound.

Claims

1. A multi-UAV Doppler frequency shift positioning method based on position error, characterized in that, The observation matrix and observation vector are constructed using the UAV position vector that includes position error; the specific steps of this method are as follows: Step 1: Obtain the Doppler frequency shift observation and distance change rate observation for each UAV: Step 1.1: Measure the Doppler frequency shift of a single moving radiation source signal arriving at the UAV when the UAV is stationary. Step 1.2: Calculate the distance change rate observation for each UAV; Step 1.3, measure the distance change rate of all UAVs and the position vectors of all UAVs including position errors. The components on the axis, all UAV position vectors including position errors, are in Components on the axis, construction An observation matrix with 8 rows and 8 columns, where, and The values ​​are equal. Indicates the total number of drones; Step 1.4: Construct a system containing the distance change rate observations of all UAVs and the position vectors of all UAVs including position errors. An observation vector of n elements, where, and The values ​​are equal; Step 2: Generate a joint coarse estimate vector of the radiation source's position and velocity vectors: Step 2.1: Calculate the joint initial estimate vector of the radiation source position vector and velocity vector; Step 2.2: Construct the initial estimation vector based on the joint initial estimation vector of the radiation source position vector and velocity vector. OK The coefficient matrix of the columns; Step 2.3: Construct the initial estimation vector based on the joint initial estimate vector of the radiation source position vector and velocity vector. OK The partial derivative matrix of the column; Step 2.4: Calculate the covariance matrix of the UAV position measurement error and the Doppler frequency shift observation error; Step 2.5: Calculate the weighted matrix of the joint coarse estimate vector of the radiation source position vector and velocity vector; Step 2.6: Calculate the joint coarse estimate vector of the radiation source position vector and velocity vector; Step 3: Construct the coefficient matrix, observation matrix, and observation vector based on the joint coarse estimate vector of the radiation source position vector and velocity vector: Step 3.1: Construct an 8-row, 8-column coefficient matrix based on the joint coarse estimate vector of the radiation source position vector and velocity vector; Step 3.2: Construct an 8-row, 4-column observation matrix based on the joint coarse estimate vector of the radiation source position vector and velocity vector; Step 3.3: Construct an 8-row, 1-column observation vector, wherein the values ​​of the 1st to 5th elements of the observation vector correspond to the values ​​of the 1st to 5th elements of the joint coarse estimation vector of the radiation source position vector and velocity vector, respectively; the value of the 6th element of the observation vector corresponds to the square of the 6th element of the joint coarse estimation vector of the radiation source position vector and velocity vector, respectively; and the values ​​of the 7th to 8th elements of the observation vector correspond to the values ​​of the 1st to 5th elements of the joint coarse estimation vector of the radiation source position vector and velocity vector, respectively. Step 4: Obtain the joint accurate estimate vector of the radiation source position vector and velocity vector: Step 4.1: Calculate the covariance matrix of the joint coarse estimation vector error of the radiation source position vector and velocity vector; Step 4.2: Calculate the weighted matrix of the joint accurate estimate vector of the radiation source position vector and velocity vector; Step 4.3: Calculate the joint accurate estimate vector of the radiation source position vector and velocity vector; Step 5: Calculate the estimated vectors of the radiation source position vector and the radiation source velocity vector according to the following formulas: ； in, The estimated vector representing the location vector of the radiation source. The estimated vector representing the velocity vector of the radiation source. The first and second elements represent the joint exact estimate vector of the radiation source's position and velocity vectors. The third and fourth elements represent the joint exact estimate vector of the radiation source's position and velocity vectors.

2. The multi-UAV Doppler frequency shift positioning method based on position error according to claim 1, characterized in that, The Doppler frequency shift observation described in step 1.1 is as follows: ； in, Indicates the first Doppler frequency shift observations from a single drone , The carrier frequency represents the signal from the moving radiation source. Indicates the speed of signal propagation. Let represent the position vector of the radiation source, and , The position vector of the radiation source is in Components on the axis, The position vector of the radiation source is in Components on the axis, This indicates the transpose operation. Indicates the first The position vectors of the drones, and , Indicates the first The position vector of the drone is Components on the axis, Indicates the first The position vector of the drone is Components on the axis, Let represent the velocity vector of the radiation source, and , The velocity vector of the radiation source is in Components on the axis, The velocity vector of the radiation source is in Components on the axis, This represents the 2-norm operation. Indicates the first Doppler shift observation error of the drone.

3. The multi-UAV Doppler frequency shift positioning method based on position error according to claim 2, characterized in that, The calculation of the distance change rate observation for each UAV in step 1.2 is based on the formula... Of those obtained, Indicates the first Observations on the rate of change of distance for each UAV.

4. The multi-UAV Doppler frequency shift positioning method based on position error according to claim 3, characterized in that, The joint initial estimate vector of the radiation source position vector and velocity vector mentioned in step 2.1 is derived from the formula... Of those obtained, This represents the joint initial estimate vector of the radiation source's position and velocity vectors. This represents the observation matrix consisting of observations of the rate of change of distance for all UAVs. This represents the observation vector consisting of the distance change rate observations of all UAVs.

5. The multi-UAV Doppler frequency shift positioning method based on position error according to claim 4, characterized in that, The covariance matrix of the UAV position measurement error and Doppler frequency shift observation error mentioned in step 2.4 is obtained by the following formula: ； in, The covariance matrix representing the position measurement error and the Doppler frequency shift observation error of the UAV. Represents the partial derivative matrix. The covariance matrix represents the position measurement error of the UAV. The covariance matrix represents the Doppler frequency shift observation error.

6. The multi-UAV Doppler frequency shift positioning method based on position error according to claim 5, characterized in that, The weighting matrix of the joint coarse estimate vector of the radiation source position vector and velocity vector mentioned in step 2.5 is obtained by the following formula: ； in, The weighted matrix representing the joint coarse estimate of the target position and velocity vectors. This represents a coefficient matrix composed of observations of the rate of change of distance for all UAVs.

7. The multi-UAV Doppler frequency shift positioning method based on position error according to claim 6, characterized in that, The joint coarse estimate vector of the radiation source position vector and velocity vector mentioned in step 2.6 is obtained by the following formula: ； in, This represents a joint coarse estimate vector of the radiation source's position and velocity vectors.

8. The multi-UAV Doppler frequency shift positioning method based on position error according to claim 7, characterized in that, The covariance matrix of the joint coarse estimation vector error of the radiation source position vector and velocity vector mentioned in step 4.1 is obtained by the following formula: ； in, The overall representation is the covariance matrix of the joint coarse estimation error of the radiation source position vector and velocity vector.

9. The multi-UAV Doppler frequency shift positioning method based on position error according to claim 8, characterized in that, The weighting matrix of the joint accurate estimation vector of the radiation source position vector and velocity vector mentioned in step 4.2 is obtained by the following formula: ； in, The weighted matrix representing the exact estimates of the radiation source's position and velocity vectors. This represents a coefficient matrix composed of some elements from the joint coarse estimate vector of the radiation source's position and velocity vectors.

10. The multi-UAV Doppler frequency shift positioning method based on position error according to claim 9, characterized in that, The joint accurate estimate vector of the radiation source position vector and velocity vector mentioned in step 4.3 is obtained by the following formula: ； in, This represents the joint accurate estimate vector of the target's position and velocity vectors. This represents the observation matrix, composed of a subset of elements from the joint coarse estimate vector of the radiation source's position and velocity vectors. This represents the observation vector, which consists of all elements in the joint coarse estimate vector of the radiation source's position and velocity vectors.

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