A cascade positioning method in a dual-satellite joint direction finding and positioning system

Abstract

The present invention discloses a cascade positioning method in a dual-satellite joint direction finding and positioning system, including: measuring the arrival time difference, arrival frequency difference and arrival angle information of the signal sent by the ground radiation source to the dual-satellite; establishing a time-frequency difference joint direction finding and positioning model of the skylark search algorithm and the Taylor cascade, measuring the arrival time difference, arrival frequency difference and arrival angle and other key parameters of the dual-satellite respectively, and establishing the earth surface equations together, thereby establishing a positioning equation group; deriving the skylark algorithm fitness function as a basis for measuring particle performance; improving the sparrow search algorithm into the skylark algorithm to avoid falling into a local extreme value, and improving the positioning accuracy and convergence speed; cascading the skylark algorithm with the Taylor series expansion method to improve the positioning accuracy. The present invention effectively avoids the problems of slow convergence speed and poor convergence accuracy of the sparrow algorithm, and improves the positioning accuracy and calculation efficiency of the algorithm.

Description

Technical Field

[0001] The present invention relates to a dual-star TDOA (Time Difference of Arrival)-FDOA (Frequency Difference of Arrival)-DOA (Direction of Arrival) joint positioning method, and in particular to a cascade positioning method in a dual-star joint direction finding positioning system, and belongs to the technical field of satellite communication positioning. Background Art

[0002] Satellite positioning technology is widely used in civil and defense fields such as signal search and rescue, military reconnaissance, ocean navigation, and navigation positioning due to its wide observation range, no weather influence, and high reliability. The radiation source can be located using the GPS (Global Positioning System) of the United States, GLONASS of Russia, Galileo of Europe, and Beidou of China. For satellite positioning of radiation sources, the key parameters to be used are Time Difference of Arrival (TDOA), Frequency Difference of Arrival (FDOA), and Direction of Arrival (DOA). Due to the long distance between the satellite and the radiation source, a slight parameter estimation deviation has a great influence on the positioning result, so the accuracy requirement of parameter estimation is also high. After obtaining the estimated parameters, the positioning equation is constructed using the known satellite coordinates and their velocities, and positioning can be achieved if the parameters are sufficient. A dual-star time-frequency difference positioning system is usually used, and major research directions at home and abroad are concentrated on this. However, there are relatively few studies on using dual-star TDOA (Time Difference of Arrival)-FDOA (Frequency Difference of Arrival)-DOA (Direction of Arrival) to achieve positioning.

[0003] For positioning using dual-star TDOA (Time Difference of Arrival)-FDOA (Frequency Difference of Arrival)-DOA (Direction of Arrival), the main problem is how to solve the nonlinear equations. The common method is to reduce the target equation to obtain an analytical solution in a specific case, but it is usually difficult to achieve. In addition, there is a method of using Taylor series expansion to calculate the position of the target point. This method requires an initial solution, and iterates from the initial solution until the iteration cutoff condition is met. However, when the initial solution is not selected reasonably or the measurement errors of TDOA (Time Difference of Arrival), FDOA (Frequency Difference of Arrival), and DOA (Direction of Arrival) are too large, the iterative algorithm will not converge and the optimal positioning result cannot be obtained. The use of simple swarm intelligence algorithms, such as particle swarm optimization (PSO), can be inaccurate because it is easy to fall into the problem of local optimal solutions. Another approach is to solve the slant range variables of the primary star and the radiation source by combining the slant range equation and the observation equation, and then obtain the radiation source position by inversely solving the slant range. However, this algorithm can only expand the precision positioning area and is not accurate enough in terms of positioning accuracy. Summary of the invention

[0004] The purpose of the present invention is to address the defects and shortcomings of the above-mentioned prior art, and propose a cascade positioning method in a dual-star joint direction finding and positioning system, and name it Sparrow Cascading Taylor algorithm (SCT) based on the Sparrow Taylor algorithm cascade positioning of the dual-star time-frequency difference joint direction finding and positioning system, through the Sparrow algorithm (i.e., the improved Sparrow Search Algorithm (SSA)) cascaded with the Taylor expansion algorithm, the positioning solution accuracy is effectively improved.

[0005] The technical solution adopted by the present invention to solve the technical problem is: a cascade positioning method in a dual-satellite joint direction finding and positioning system, the method comprising the following steps:

[0006] Step 1: Measure the TDOA (Time Difference of Arrival), FDOA (Frequency Difference of Arrival) and DOA (Direction of Arrival) parameters of the signal sent by the ground radiation source to the two observation satellites;

[0007] Step 2: Establish the TDOA (Time Difference of Arrival)-FDOA (Frequency Difference of Arrival)-DOA (Direction of Arrival) positioning model of SCT (Sparrow Cascading Taylor algorithm), select two satellites to perform TDOA (Time Difference of Arrival) and FDOA (Frequency Difference of Arrival) measurements respectively, and a main satellite measures the DOA (Direction of Arrival) information of the target source, and according to the measured time difference and frequency difference of the signal reaching the two satellites and the receiving angle information, jointly establish the earth surface equations to establish the positioning equation group;

[0008] Step 3: Derive the fitness function of the Skylark algorithm as a basis for measuring the current particle performance;

[0009] Step 4: Improve the sparrow algorithm. First, in the sparrow position update strategy, the sine-cosine SCA algorithm idea, Levy flight strategy, and random walk strategy are introduced to improve it. The skylark search algorithm is used to iteratively update the individual position and obtain the solution corresponding to the optimal fitness value, thereby improving the algorithm's ability to find the best solution and escape from the local extreme value.

[0010] Step 5: Use the Skylark and Taylor expansion methods in cascade to further improve positioning accuracy;

[0011] Step 6: Under the same conditions, the radiation source is located respectively using the PSO (Particle Swarm Optimization) algorithm and the SCT (Sparrow Cascading Taylor) algorithm positioning technology of the dual-satellite time-frequency difference joint direction finding positioning system, and the simulation results are compared.

[0012] Furthermore, the step 1 includes measuring the TDOA (Time Difference of Arrival) and FDOA (Frequency Difference of Arrival) of the signal sent by the ground radiation source to the two observation satellites, as well as the azimuth and elevation angles measured by the master satellite;

[0013] The dual-satellite time-frequency difference joint direction-finding positioning technology uses the TDOA (Time Difference of Arrival) / FDOA (Frequency Difference of Arrival) generated by the movement of two satellites, one of which is a main satellite with a direction-finding function, relative to the radiation source, and the azimuth and pitch angles measured by the main satellite. The corresponding relationship between the position and speed of the radiation source is used to calculate the position of the target.

[0014] The dual-satellite time-frequency difference joint direction-finding system consists of a main satellite with direction-finding function, an auxiliary satellite, a receiving station and a radiation source. For a radiation source stationary on the ground or in the air, the time difference and frequency difference generated by the two satellites are combined with the azimuth and elevation angles measured by the main satellite to obtain the positioning solution through four parameters. Therefore, the time difference between the radiation source signal sent by the satellite and the arrival of the two satellites is recorded as t 21 , the expression is as follows:

[0015]

[0016] where r 21 is the distance difference between the radiation source and the two satellites, c is the speed of light;

[0017] The frequency f of the signal sent by the radiation source measured by the satellite 21 The relationship between the actual frequency f of the ground radiation source signal is:

[0018]

[0019] where v 21 is the radial velocity difference between the radiation source and the two satellites, f is the actual signal frequency of the radiation source,

[0020] In the satellite direction finding model, the target source DOA (Direction of Arrival) observation is obtained in the satellite body coordinate system. Assume that the radiation source position in the satellite body coordinate system is x T,b =[x t,b ,y t,b ,z t,b ] T, T is the transpose of the matrix. The arrival angles of the radiation source signal measured at a certain moment are the azimuth angle α and the elevation angle β. The angle of the radiation source signal satisfies the following equation:

[0021]

[0022]

[0023] Among them, α and β are the azimuth and elevation angles of the radiation source signal arrival angle, respectively, and x t,b ,y t,b and z t,b is the coordinate of the radiation source position of satellite 1 in the stellar coordinate system.

[0024] Furthermore, the step 2 includes establishing a positioning equation group of the SCT (SparrowCascading Taylor) algorithm positioning technology of the dual-satellite time-frequency difference joint direction finding positioning system;

[0025] Assume that the true position of the target point to be estimated is u o =(x o ,y o ,z o ) T , the true positions and velocities of the two satellites are There are errors in the satellite position and speed. For the convenience of expression, in Therefore, the satellite random error at this moment is Δβ=β-β 0 ,in Define Δβ to have a mean of zero and a covariance matrix of Q β A Gaussian random vector,

[0026] Satellite 1 is selected as the reference satellite, and the arrival time difference between satellite 2 and satellite 1 is recorded as t 21 , the expression is as follows:

[0027]

[0028] Where c is the signal propagation speed, so r 21 =ct 21 , is the Range Difference of Arrival (RDOA), r i o (i=1,2) is the distance between the actual position of the i-th satellite and the actual position of the desired positioning target point. The calculation formula is as follows:

[0029]

[0030] is the TDOA (Time Difference of Arrival) measurement noise,

[0031] The arrival frequency difference between satellite 2 and satellite 1 is f 21 , the calculation formula is as follows:

[0032]

[0033] Where f is the signal frequency sent by the radiation source, This is the radial velocity difference between the main satellite and the target point at this time, is the actual radial velocity of the ith satellite and the radiation point without error, and its expression is as follows:

[0034]

[0035] The FDOA (Frequency Difference of Arrival) measurement noise is

[0036] In the satellite direction finding model, the DOA (Direction of Arrival) observation of the target source is obtained in the satellite body coordinate system. Assume that the radiation source position in the satellite body coordinate system is x T,b =[x t,b ,y t,b ,z t,b ] T There will be errors in the satellite position. At a certain moment, the arrival angles of the target source signal measured by the satellite array antenna are the azimuth angle α and the elevation angle θ respectively. Here, the measurement noise of the azimuth angle and the elevation angle are η6 and η7, α o =α+η6,θ o =θ+η7, then the angular relationship between the satellite and the target source signal satisfies the following equation:

[0037]

[0038]

[0039] Among them, α o and θ o are the true values ​​of the azimuth and elevation angles of the radiation source signal arrival angle, and is the coordinate of the real position of the radiation source of satellite 1 in the stellar coordinate system.

[0040] At this time, the vector formed by the observation quantity is δ = [r 21 ,f 21 ,α,θ] T , the measurement error is expressed as Δδ, which has a mean of zero and a covariance matrix of Q α A Gaussian random variable, so δ = δ o +Δδ, where

[0041] The vector m formed by the observed quantity tf =[r 21 ,f 21 ,α,β] T , and the positioning equations constructed with the WGS-84 earth ellipsoid model are expressed as follows:

[0042]

[0043] where r 21 is the arrival distance difference observation, f 21 is the arrival frequency difference observation, α is the azimuth observation, β is the elevation observation, A=6378137, is the major semi-axis of the WGS-84 ellipsoid, e 2 =0.00669437999013, which is the square of the first eccentricity.

[0044] Furthermore, the step 3 includes deriving a formula for the fitness function of the Skylark algorithm of the technology according to the SCT (SparrowCascading Taylor) positioning model of the dual-satellite time-frequency difference joint direction finding positioning system as a basis for measuring the current particle performance.

[0045] For r 21 , f 21 , and move the terms on both sides of the equal sign to obtain the following expression:

[0046]

[0047]

[0048] Assuming that the target point to be measured is on the surface of the earth, the WGS-84 equation expressing the earth's surface is introduced and transformed to obtain the following expression:

[0049]

[0050] The following expression is obtained:

[0051]

[0052]

[0053] Since the true position and speed of the satellite are unknown, we use s, Instead of the satellite's true position and speed s o , Replace the above with Formula 15 and The true position and velocity of the satellite in formula 16 are expressed as follows:

[0054] r 21 -||u o -s2||+||u o -s1||=η4 Formula 17

[0055]

[0056] For α and θ, we have:

[0057] α-α o =η6 Formula 19

[0058] θ-θ o =η7 Formula 20

[0059] Combined with the above Formula 14, r 21 -||u o -s2||+||u o -s1||=η4 Formula 17, Formula 18, α-α o =η6 Formula 19, θ-θ o =η7 Formula 20, construct the fitness value calculation function required by SCT (Sparrow Cascading Taylor, Skylark Cascading Taylor algorithm), let

[0060] ε=[η3,η4,η5,η6,η7] T Performing the 2-norm operation on it will give the fitness function, which is expressed as follows:

[0061] fit(u)=||ε|| Formula 21

[0062] Furthermore, the step 4 includes improving the sparrow algorithm to avoid being easily trapped in a local extreme value and improve the positioning accuracy and convergence speed;

[0063] The position update strategy of the skylark finder adopts the sine-cosine SCA algorithm idea and introduces a nonlinear sine learning factor; the learning factor formula and the improved finder position formula are expressed as follows:

[0064] ω=ω min +(ω max-ω min )·sin(tπ / iter max ) Formula 22

[0065]

[0066] Where r1 is a random number in [0, 2π], r2 is a random number in [0, 2], ω is the learning factor, It indicates the position of the i-th individual in the j-th dimension when the number of iterations is t, R2∈[0,1] and ST∈[0.5,1] represent the warning value and safety value respectively;

[0067] Combine the Levy flight strategy and the random walk strategy to fully explore the search space while maintaining the search for the local optimal solution;

[0068] The Levy flight strategy is introduced into the Skylark joiner update formula, and the improved formula is expressed as follows:

[0069]

[0070] Where d is the vector dimension,

[0071] The calculation formula of the Levi flight strategy is as follows:

[0072]

[0073]

[0074] Where Γ(x)=(x-1)!, Q is a random number that obeys the normal distribution, r3,r4∈[0,1], and ξ is 1.5. The position update formula of the skylark joiner after introducing the random walk strategy is expressed as follows:

[0075]

[0076] in, and They represent two random solutions when the number of iterations is t, τ is the scaling factor, and they obey the uniform distribution of (0,1);

[0077] The current optimal value is obtained through the position of the skylark obtained in this round, and it is recorded. Then the optimal value of this round is compared with the global optimal value. If the current optimal value is better than the global optimal value, it is replaced. Otherwise, the global optimal value is retained and the next round is entered until the maximum number of iterations is reached to output the optimal value.

[0078] Furthermore, the step 6 includes further improving the positioning accuracy by cascading the Skylark algorithm and the Taylor algorithm;

[0079] The Taylor series method iterates the precise target point through precise initial values. When the initial values ​​are selected reasonably, accurate results are obtained, and the results of the Skylark algorithm are used as its initial values.

[0080] Let the solution of the above Skylark algorithm be expressed as the initial position Let G be the Jacobian matrix, and according to the best linear unbiased estimate, we get:

[0081]

[0082]

[0083] Where: W is the covariance matrix of the observation error, ρ = [Δx, Δy, Δz],

[0084] Finally, the initial value is continuously iterated according to the set threshold value until it is smaller than the threshold value to obtain the radiation source position.

[0085] Beneficial effects:

[0086] 1. The SCT (Sparrow Cascading Taylor) algorithm cascade positioning technology of the dual-satellite time-frequency difference positioning system proposed in the present invention, when only two satellites are available, uses the arrival time difference and arrival frequency difference of the signal sent by the radiation source to the two satellites and the angle information to construct an equation, and then combines it with the earth surface equation to construct a positioning equation group, and uses the method proposed in the present invention to solve it. Compared with the commonly used solution scheme, it has higher accuracy.

[0087] 2. The present invention improves the sparrow algorithm and then cascades it with the Taylor expansion algorithm to avoid the problem of Taylor's initial solution, make up for the shortcomings of the ordinary sparrow algorithm, and improve the positioning accuracy of the algorithm.

[0088] 3. The present invention optimizes the accuracy of the solution through the SCT (Sparrow Cascading Taylor) algorithm. Compared with the PSO (Particle Swarm Optimization) algorithm, the SCT (Sparrow Cascading Taylor) algorithm does not require complex parameters similar to those in the PSO (Particle Swarm Optimization) algorithm, and the algorithm complexity is greatly reduced. Since the SSA algorithm mainly searches in the divided subspace, it pays more attention to the local optimal solution, overcomes the limitations of the PSO (Particle Swarm Optimization) algorithm, and its convergence speed is also faster than that of the PSO (Particle Swarm Optimization) algorithm.

[0089] 4. The present invention improves the ordinary sparrow algorithm by adding the Levy flight strategy to improve the situation of falling into the local optimal solution. After the solution obtained by the skylark algorithm is used as the initial solution of the Taylor algorithm for the second iteration. First, compared with the particle swarm algorithm, the skylark algorithm has the obvious advantage of not being easily trapped in the local optimal solution and the parameter complexity is not high. Secondly, the skylark algorithm is improved on the basis of the sparrow algorithm by adding the Levy flight strategy to improve the algorithm's optimization ability and the ability to jump out of the local extreme value. Then, the Taylor series expansion algorithm is used for secondary iteration to improve the accuracy of the solution. BRIEF DESCRIPTION OF THE DRAWINGS

[0090] Figure 1 The figure is a flow chart of the method of the present invention.

[0091] Figure 2 This is a dual-star TDOA (Time Difference of Arrival)-FDOA (Frequency Difference of Arrival)-DOA (Direction of Arrival) positioning model diagram of the present invention.

[0092] Figure 3 This is a schematic diagram of the dual-star TDOA (Time Difference of Arrival)-FDOA (Frequency Difference of Arrival)-DOA (Direction of Arrival) positioning principle of the present invention.

[0093] Figure 4This is a flow chart of the SCT (Sparrow Cascading Taylor) positioning method of the present invention.

[0094] Figure 5 It is a comparison chart of positioning performance curves of the SCT (Sparrow Cascading Taylor) algorithm of the present invention and the PSO (Particle Swarm Optimization) algorithm as the DOA (Direction of Arrival) measurement error changes.

[0095] Figure 6 It is a comparison chart of positioning performance curves of the SCT (Sparrow Cascading Taylor) algorithm of the present invention and the PSO (Particle Swarm Optimization) algorithm as the TDOA (Time Difference of Arrival) measurement error changes.

[0096] Figure 7 It is a comparison chart of positioning performance curves of the SCT (Sparrow Cascading Taylor) algorithm and the PSO (Particle Swarm Optimization) algorithm of the present invention as the FDOA (Frequency Difference of Arrival) measurement error changes.

[0097] Figure 8 It is the iteration graph of the SCT (Sparrow Cascading Taylor) and SSA (Sparrow Search Algorithm) algorithms of the present invention. DETAILED DESCRIPTION

[0098] The invention will be further described in detail below in conjunction with the accompanying drawings.

[0099] like Figure 1As shown, the present invention proposes a cascade positioning method in a dual-satellite joint direction finding positioning system. The method estimates the position of the radiation source by using the received signal TDOA (Time Difference of Arrival), FDOA (Frequency Difference of Arrival) and DOA (Direction of Arrival) information. Compared with the traditional positioning technology, the method has higher positioning solution accuracy, and specifically includes the following steps:

[0100] Step 1: Measure the TDOA (Time Difference of Arrival) and FDOA (Frequency Difference of Arrival) of the signal sent by the ground radiation source to the two observation satellites and receive the DOA (Direction of Arrival) information.

[0101] The dual-star TDOA (Time Difference of Arrival)-FDOA (Frequency Difference of Arrival)-DOA (Direction of Arrival) positioning technology uses the TDOA (Time Difference of Arrival) and FDOA (Frequency Difference of Arrival) generated by the movement of two satellites relative to the radiation source, and the correspondence between the DOA (Direction of Arrival) measured by one of the satellites with direction finding function and the position of the radiation source to calculate the position of the target.

[0102] The model of the dual-satellite time-frequency difference joint direction finding positioning system is as follows: Figure 2 As shown in the figure, it mainly consists of a main satellite with direction finding function, an auxiliary satellite, a receiving station and a radiation source. Figure 3As shown in the figure, for a stationary radiation source on the ground or in the air, the movement of the satellite will cause the carrier frequency reaching the two satellites to produce a Doppler frequency shift effect, so that there is a difference in the frequency received by the two satellites, FDOA (Frequency Difference of Arrival), and there is also a difference in the time when the signal reaches the two satellites, TDOA (Time Difference of Arrival). The direction finding information DOA (Direction of Arrival) obtained by the interferometer direction finding function of the main satellite can be combined to obtain the radiation source position. Therefore, the time difference measured by the satellite for the radiation source to send the signal to the two satellites is recorded as t 21 , the expression is as follows:

[0103]

[0104] where r 21 is the distance difference between the radiation source and the two satellites, c is the speed of light;

[0105] The frequency f of the signal sent by the radiation source measured by the satellite 21 The relationship between the actual frequency f of the ground radiation source signal is:

[0106]

[0107] where v 21 is the radial velocity difference between the radiation source and the two satellites, f is the actual signal frequency of the radiation source,

[0108] In the satellite direction finding model, the target source DOA (Direction of Arrival) observation is obtained in the satellite body coordinate system. Assume that the radiation source position in the satellite body coordinate system is x T,b =[x t,b ,y t,b ,z t,b ] T , the arrival angles of the radiation source signal measured at a certain moment are azimuth angle α and elevation angle β respectively, then the angle of the radiation source signal satisfies the following equation:

[0109]

[0110]

[0111] Among them, α and β are the azimuth and elevation angles of the radiation source signal arrival angle, respectively, and x t,b ,y t,b and z t,b is the coordinate of the radiation source position of satellite 1 in the stellar coordinate system

[0112] Step 2: Establish the TDOA (Time Difference of Arrival)-FDOA (Frequency Difference of Arrival)-DOA (Direction of Arrival) positioning model of the SCT (Sparrow Cascading Taylor) algorithm, select two satellites to measure TDOA (Time Difference of Arrival) and FDOA (Frequency Difference of Arrival) respectively, measure the azimuth and elevation angles through the main satellite, and according to the measured time difference and frequency difference of the signals reaching the two satellites, as well as the azimuth and elevation angles, jointly establish the earth surface equations to establish the positioning equation group;

[0113] Assume that the true position of the target point to be estimated is u o =(x o ,y o ,z o ) T , the true positions and velocities of the two satellites are There are errors in the satellite position and speed. For the convenience of expression, in Therefore, the satellite random error at this moment Δβ=β-β 0 ,in Define Δβ to have a mean of zero and a covariance matrix of Q β Gaussian random vector of;

[0114] Satellite 1 is selected as the reference satellite, and the arrival time difference between satellite 2 and satellite 1 is recorded as t 21 , the expression is as follows:

[0115]

[0116] Where c is the signal propagation speed, so r 21 =ct 21 , is the Range Difference of Arrival (RDOA), r i o (i=1,2) is the distance between the actual position of the i-th satellite and the actual position of the desired positioning target point. The calculation formula is as follows:

[0117]

[0118] is the TDOA measurement noise,

[0119] The arrival frequency difference between satellite 2 and satellite 1 is f 21 , the calculation formula is as follows:

[0120]

[0121] Where f is the signal frequency sent by the radiation source, This is the radial velocity difference between the main and auxiliary satellites and the target point at this time. is the actual radial velocity of the ith satellite and the radiation point without error, and its expression is as follows:

[0122]

[0123] The FDOA (Frequency Difference of Arrival) measurement noise is

[0124] In the satellite direction finding model, the DOA (Direction of Arrival) observation of the target source is obtained in the satellite body coordinate system. Assume that the radiation source position in the satellite body coordinate system is x T,b =[x t,b ,y t,b ,z t,b ] T There will be errors in the satellite position. At a certain moment, the arrival angles of the target source signal measured by the satellite array antenna are the azimuth angle α and the elevation angle θ respectively. Here, the measurement noise of the azimuth angle and the elevation angle are η6 and η7, α o =α+η6,θ o =θ+η7, then the angular relationship between the satellite and the target source signal satisfies the following equation:

[0125]

[0126]

[0127] Among them, α o and θ o are the true values ​​of the azimuth and elevation angles of the radiation source signal arrival angle, and is the coordinate of the real position of the radiation source of satellite 1 in the stellar coordinate system.

[0128] At this time, the vector formed by the observation quantity is δ = [r 21 ,f 21 ,α,θ]T , the measurement error is expressed as Δδ, which has a mean of zero and a covariance matrix of Q α A Gaussian random variable, so δ = δ o +Δδ, where

[0129] According to the vector δ formed by the observed quantity, 21 ,f 21 ,α,β] T , and the WGS-84 earth ellipsoid model to construct the positioning equation group

[0130]

[0131] where r 21 is the arrival distance difference observation, f 21 is the arrival frequency difference observation, α is the azimuth observation, β is the elevation observation, A=6378137, is the major semi-axis of the WGS-84 ellipsoid, e 2 =0.00669437999013, which is the square of the first eccentricity.

[0132] Step 3: Derive the fitness function of the sparrow swarm algorithm as a basis for measuring the current performance.

[0133] For r 21 , f 21 , by moving the terms on both sides of the equal sign, we can get the following expression:

[0134]

[0135]

[0136] Assuming that the target point to be measured is on the surface of the earth, the WGS-84 equation expressing the earth's surface is introduced and transformed to obtain the following expression:

[0137]

[0138] Will Formula 4 and Substitute formula 6 into Formula 12 and Formula 13 can be expressed as follows:

[0139]

[0140]

[0141] Since the true position and speed of the satellite are unknown, we use s, Instead of the satellite's true position and speed so , Replace the above Formula 15 and The true position and velocity of the satellite in formula 16 are expressed as follows:

[0142] r 21 -||u o -s2||+||u o -s1||=η4 Formula 17

[0143]

[0144] For α and θ, we have:

[0145] α-α o =η6 Formula 19

[0146] θ-θ o =η7 Formula 20

[0147] Lian Li Formula 14, r 21 -||u o -s2||+||u o -s1||=η4 Formula 17, Formula 18, α-α o =η6 Formula 19, θ-θ o =η7 Formula 20, construct the fitness value calculation function required for SCT (Sparrow Cascading Taylor algorithm), let ε = [η3, η4, η5, η6, η7] T , and the 2-norm operation can be performed to obtain the fitness function

[0148] fit(u)=||ε|| Formula 21

[0149] Step 4: Improve the sparrow algorithm. First, in the sparrow position update strategy, the sine-cosine SCA algorithm idea, Levy flight strategy, and random walk strategy are introduced to improve it. The skylark search algorithm is used to iteratively update the sparrow position and obtain the solution corresponding to the optimal fitness value.

[0150] The sparrow algorithm is improved to avoid falling into local extreme values ​​and to improve positioning accuracy and convergence speed.

[0151] The position update strategy of the skylark finder adopts the sine-cosine SCA algorithm idea and introduces a nonlinear sine learning factor; the learning factor formula and the improved finder position formula are expressed as follows:

[0152] ω=ω min +(ωmax -ω min )·sin(tπ / iter max ) Formula 22

[0153]

[0154] Where r1 is a random number in [0, 2π], r2 is a random number in [0, 2], ω is the learning factor, It indicates the position of the i-th individual in the j-th dimension when the number of iterations is t, R2∈[0,1] and ST∈[0.5,1] represent the warning value and safety value respectively;

[0155] Combine the Levy flight strategy and the random walk strategy to fully explore the search space while maintaining the search for the local optimal solution;

[0156] The Levy flight strategy is introduced into the Skylark joiner update formula, and the improved formula is expressed as follows:

[0157]

[0158] Where d is the vector dimension,

[0159] The calculation formula of the Levi flight strategy is as follows:

[0160]

[0161]

[0162] Where Γ(x)=(x-1)!, Q is a random number that obeys the normal distribution, r3,r4∈[0,1], and ξ is 1.5. The position update formula of the individual joiner after introducing the random walk strategy is expressed as follows:

[0163]

[0164] in, and They represent two random solutions when the number of iterations is t, τ is the scaling factor, and they obey the uniform distribution of (0,1);

[0165] The present invention obtains the current optimal value through the position of the skylark obtained in this round, records it, and then compares the optimal value of this round with the global optimal value. If the current optimal value is better than the global optimal value, it is replaced. Otherwise, the global optimal value is retained and the next round is entered until the maximum number of iterations is reached to output the optimal value.

[0166] Step 5: Use Skylark and Taylor series method in cascade to further improve positioning accuracy.

[0167] like Figure 4 As shown, the Taylor series needs an accurate initial value to iterate out the accurate target point. When the initial value is selected reasonably, a more accurate result can be obtained. The result of the Skylark algorithm is used as its initial value.

[0168] Let the solution of the above sparrow algorithm be expressed as the initial position Let G be the Jacobian matrix, according to the best linear unbiased estimate, we can get:

[0169]

[0170]

[0171] Where: W is the covariance matrix of the observation error, ρ = [Δx, Δy, Δz].

[0172] Finally, the initial value is continuously iterated according to the set threshold value until it is smaller than the threshold value to obtain the radiation source position.

[0173] Step 6: Under the same conditions, four different positioning technologies are used together with the SCT (Sparrow Cascading Taylor) positioning technology of the dual-satellite time-frequency difference joint direction-finding positioning system to locate the radiation source respectively, and the simulation results are compared.

[0174] The present invention conducts simulation on a dedicated experimental platform and verifies the performance of the proposed positioning technology by changing experimental parameters, which specifically includes the following:

[0175] 1. Simulation parameters

[0176] The positions and velocities of the two satellites are s1 = [30.957, 115.401, 1253799.098], s2 = [33.347, 115.546, 1254600.266], v1 = [-5624.52, 113.895, -3892.451], v2 = [-5821.051, -147.535, 3632.683]. The positioning algorithm parameters are set as follows: population size N = 200, SCT (Sparrow Cascading Taylor) maximum iteration number t max =100.

[0177] 2. Error judgment criteria

[0178] The performance of the positioning algorithm is judged by the positioning error. The smaller the value, the better the positioning algorithm performance. The positioning error is the distance between the result coordinates estimated by the positioning algorithm and the actual coordinates. Let the estimated result be The actual position is u o =(xo ,y o ,z o ) T , then the calculation formula is:

[0179]

[0180] 3. Comparison of simulation results

[0181] Simulation 1SCT (Sparrow Cascading Taylor) algorithm verifies the improved positioning performance compared with PSO (Particle Swarm Optimization) algorithm.

[0182] Figure 5 The positioning error change curve of the SCT (Sparrow Cascading Taylor) algorithm and the PSO (Particle Swarm Optimization) algorithm after 20 Monte Carlo experiments when the DOA (Direction of Arrival) error is [0.01°, 0.1°] and the radiation source position is (B, L, H) = (32.1, 118.78, 0).

[0183] By observation Figure 5 The red curve is the SCT (Sparrow Cascading Taylor) algorithm. It can be seen that the positioning error is smaller than that of the PSO (Particle Swarm Optimization) algorithm after the direction finding error is 0.06°. However, the PSO (Particle Swarm Optimization) algorithm has a greater impact on the positioning error as the direction finding error changes.

[0184] The positioning performance curves of the simulated 2SCT (Sparrow Cascading Taylor) and PSO (Particle Swarm Optimization) algorithms are compared with the changes in TDOA (Time Difference of Arrival) measurement error.

[0185] Figure 6The positioning error change curve of the SCT (Sparrow Cascading Taylor) algorithm and the PSO (Particle Swarm Optimization) algorithm after 20 Monte Carlo experiments when the TDOA (Time Difference of Arrival) error is [-90, -60] dB and the radiation source position is (B, L, H) = (32.1, 118.78, 0).

[0186] By observation Figure 6 The red curve is the SCT (Sparrow Cascading Taylor) algorithm. The error of the PSO (Particle Swarm Optimization) algorithm is much larger than that of the SCT (SparrowCascading Taylor) algorithm. However, with the change of the time difference error, the positioning error of the SCT (SparrowCascading Taylor) algorithm is less affected.

[0187] The positioning performance curves of the simulated 3SCT (Sparrow Cascading Taylor) and PSO (Particle Swarm Optimization) algorithms are compared with the changes in TDOA (Time Difference of Arrival) measurement error.

[0188] Figure 7 The positioning error change curve of the SCT (Sparrow Cascading Taylor) algorithm and the PSO (Particle Swarm Optimization) algorithm after 20 Monte Carlo experiments when the FDOA (Frequency Difference of Arrival) error is [-90, -60] dB and the radiation source position is (B, L, H) = (32.1, 118.78, 0).

[0189] By observation Figure 7The red curve is the SCT (Sparrow Cascading Taylor) algorithm. The error of the single PSO (Particle Swarm Optimization) algorithm is much larger than that of the SCT (Sparrow Cascading Taylor) algorithm. However, with the change of frequency difference, the positioning error of the SCT (Sparrow Cascading Taylor) algorithm is less affected. However, due to the instability of the PSO (Particle Swarm Optimization) algorithm, the change of frequency difference has a greater impact on the positioning error of the PSO (Particle Swarm Optimization) algorithm.

[0190] The simulation verifies the performance of the 4SCT (Sparrow Cascading Taylor) algorithm in improving the convergence speed compared to the SSA (Sparrow Search Algorithm) algorithm.

[0191] Figure 8 The fitness value convergence curves of the SCT (Sparrow Cascading Taylor) algorithm and the SSA (Sparrow Search Algorithm) algorithm are obtained by performing 100 Monte Carlo tests when the TDOA (Time Difference of Arrival) / FDOA (Frequency Difference of Arrival) / DOA (Direction of Arrival) errors are 200ns, 5, and 1° respectively, and the target original position is (B, L, H) = (32.1, 118.78, 0). By observing Figure 8It can be seen that the convergence speed of SCT (Sparrow Cascading Taylor) is faster than that of SSA (Sparrow Search Algorithm). After about 6 iterations, the fitness value of SCT (Sparrow Cascading Taylor) algorithm reaches the minimum, while SSA (Sparrow Search Algorithm) still has a large error at this time. Therefore, SCT (Sparrow Cascading Taylor) converges faster than SSA (Sparrow Search Algorithm) algorithm and achieves the purpose of accelerating the convergence speed.

[0192] The above description is only a specific implementation of the present invention, but the protection scope of the present invention is not limited thereto. Any changes or substitutions that can be easily conceived by a technician familiar with the technical field within the technical scope disclosed by the present invention should be covered by the protection scope of the present invention.

Claims

1. A cascade positioning method in a dual-satellite joint direction finding and positioning system, characterized in that: The method comprises the following steps: Step 1: Measure the TDOA, FDOA and DOA parameters of the signal sent by the ground radiation source to the two observation satellites; Step 2: Establish the TDOA-FDOA-DOA positioning model of the Skylark Cascade Taylor SCT algorithm positioning technology, select two satellites to perform TDOA and FDOA measurements respectively, and a main satellite to measure the DOA information of the target source, and according to the measured time difference and frequency difference of the signal reaching the two satellites, as well as the receiving angle information, jointly establish the earth surface equations to establish the positioning equation group, including the establishment of the positioning equation group of the Skylark Cascade Taylor SCT algorithm positioning technology of the dual-satellite time-frequency difference joint direction-finding positioning system; Assume that the true position of the target point to be estimated is u o =(x o ,y o ,z o ) T , the true positions and velocities of the two satellites are There are errors in the actual position and speed of the satellite. For the convenience of expression, in Therefore, the satellite random error at this moment is Δβ=β-β 0 ,in Define Δβ to have a mean of zero and a covariance matrix of Q β Gaussian random vector of; Satellite 1 is selected as the reference satellite, t 21 The expression is as follows: Where c is the speed of light, t 21 r is the time difference between the signal sent by the radiation source and the arrival of the two satellites, measured by the satellite. 21 is the distance difference between the radiation source and the two satellites, so r 21 =ct 21 , r i o is the distance between the actual position of the i-th satellite and the actual position of the required positioning target point. The calculation formula is as follows: is the TDOA measurement noise, The arrival frequency difference between satellite 2 and satellite 1 is f 21 , the calculation formula is as follows: Where f is the signal frequency sent by the radiation source, This is the radial velocity difference between the main satellite and the target point at this time, is the actual radial velocity of the ith satellite and the radiation point without error, and its expression is as follows: Measure noise for FDOA; In the satellite direction finding model, the DOA observation of the target source is obtained by observing in the satellite body coordinate system. Assume that the position of the radiation source in the satellite body coordinate system is x T,b =[x t,b ,y t,b ,z t,b ] T There will be errors in the satellite position. At a certain moment, the arrival angles of the target source signal measured by the satellite array antenna are respectively and pitch angle Here, the measurement noise of azimuth and elevation angles is η6 and η7, The angular relationship between the satellite and the target source signal satisfies the following equation: Among them, α o and θ o are the true values ​​of the azimuth and elevation angles of the radiation source signal arrival angle, and is the coordinate of the real position of the radiation source of satellite 1 in the stellar coordinate system. At this time, the vector formed by the observed quantity is Its measurement error is represented as Δδ, which has a mean of zero and a covariance matrix of Q α A Gaussian random variable, so δ = δ o +Δδ, where The vector formed by the observed quantity The positioning equations constructed with the WGS-84 earth ellipsoid model are expressed as follows: where r′ 21 is the arrival distance difference observation, f′ 21 is the arrival frequency difference observation, ε is the azimuth observation, is the elevation angle observation, s1 and s2 are the position coordinates of satellite 1 and satellite 2 respectively, and are the speeds of satellite 1 and satellite 2, respectively, A = 6378137, is the major semi-axis of the WGS-84 ellipsoid, e 2 =0.00669437999013, which is the square of the first eccentricity; Step 3: Derive the fitness function of the Skylark algorithm as a basis for measuring the current particle performance; Step 4: Improve the sparrow algorithm to the skylark algorithm. First, in the sparrow's position update strategy, the sine-cosine SCA algorithm idea, Levy flight strategy, and random walk strategy are introduced to improve it. The skylark search algorithm is calculated to continuously iterate and update the individual position and obtain the solution corresponding to the optimal fitness value. Step 5: Use Skylark and Taylor iteration method in cascade to further improve positioning accuracy; Step 6: Under the same conditions, the PSO particle swarm algorithm and the SCT positioning technology of the dual-satellite time-frequency difference joint direction finding positioning system are used to locate the radiation source respectively, and the simulation results are compared.

2. The cascade positioning method in a dual-satellite joint direction finding and positioning system according to claim 1, characterized in that: The step 1 includes measuring the TDOA and FDOA of the signals sent by the ground radiation source to the two observation satellites and the azimuth and elevation angles measured by the primary satellite; The dual-satellite joint direction finding positioning uses two satellites, one of which is a main satellite with direction finding function, to calculate the position of the target based on the TDOA / FDOA generated by the movement of the radiation source and the corresponding relationship between the azimuth and pitch angle measured by the main satellite and the position and speed of the radiation source. The dual-satellite joint direction-finding system consists of a main satellite with direction-finding function, an auxiliary satellite, a receiving station and a radiation source. For a radiation source stationary on the ground or in the air, the time difference and frequency difference generated by the two satellites are combined with the azimuth and elevation angles measured by the main satellite to obtain a positioning solution through four parameters. Therefore, t 21 The expression is as follows: The frequency f of the signal sent by the radiation source measured by the satellite 21 The relationship between the actual frequency f of the ground radiation source signal is: where v 21 is the radial velocity difference between the radiation source and the two satellites, The arrival angles of the radiation source signal measured at a certain moment are azimuth angle α and elevation angle θ respectively, then the angle of the radiation source signal satisfies the following equation: Where α and θ are the azimuth and elevation angles of the radiation source signal arrival angle, respectively, and x t,b ,y t,b and z t,b is the coordinate of the radiation source position of satellite 1 in the stellar coordinate system.

3. The cascade positioning method in a dual-satellite joint direction finding and positioning system according to claim 1, characterized in that: The step 3 includes deriving a formula for the fitness function of the Skylark algorithm of the technology based on the SCT algorithm cascade positioning model of the dual-satellite time-frequency difference joint direction finding positioning system as a basis for measuring the current particle performance. For r 21 , f 21 , and move the terms on both sides of the equal sign to obtain the following expression: Assuming that the target point to be measured is on the surface of the earth, the WGS-84 equation expressing the earth's surface is introduced and transformed to obtain the following expression: The following expression is obtained: Since the true position and speed of the satellite are unknown, we use s, Instead of the satellite's true position and speed s o , Replace the above with Formula 15 and The true position and velocity of the satellite in formula 16 are expressed as follows: for and have: Combined with the above Equation 19 and Formula 20 and the earth surface equation are used to construct the fitness value calculation function required for SCT. j=[eta3, eta4, eta5, eta6, eta7] T , and the 2-norm operation is performed to obtain the fitness function, which is expressed as follows: fit(u)=||j|| Formula 21.

4. The cascade positioning method in a dual-satellite joint direction finding and positioning system according to claim 1, characterized in that: The step 4 includes that the Skylark algorithm is an improvement on the ISSA algorithm, thereby avoiding falling into local extreme values ​​and improving positioning accuracy and convergence speed; The position update strategy of the skylark finder adopts the sine-cosine SCA algorithm idea and introduces a nonlinear sine learning factor; the learning factor formula and the improved finder position formula are expressed as follows: ω = ω min +(ω max -ω min )·sin(tπ / iter max ) Formula 22 Where r1 is a random number in [0, 2π], r2 is a random number in [0, 2], ω is the learning factor, It represents the position of the i-th skylark individual in the j-th dimension when the number of iterations is t, R2∈[0,1] and ST∈[0.5,1] represent the warning value and safety value respectively; Combine the Levy flight strategy and the random walk strategy to fully explore the search space while maintaining the search for the local optimal solution; The Levy flight strategy is introduced into the Skylark joiner update formula, and the improved formula is expressed as follows: Where d is the vector dimension, The calculation formula of the Levi flight strategy is as follows: Where Γ(x)=(x-1)!, Q is a random number that obeys the normal distribution, r3,r4∈[0,1], and ξ is 1.

5. The position update formula of the skylark joiner after introducing the random walk strategy is expressed as follows: in, and They represent two random solutions when the number of iterations is t, τ is the scaling factor, and they obey the uniform distribution of (0,1); The current optimal value is obtained through the position of the skylark obtained in this round, and it is recorded. Then the optimal value of this round is compared with the global optimal value. If the current optimal value is better than the global optimal value, it is replaced. Otherwise, the global optimal value is retained and the next round is entered until the maximum number of iterations is reached to output the optimal value.

5. The cascade positioning method in a dual-satellite joint direction finding and positioning system according to claim 1, characterized in that: The step 5 includes further improving the positioning accuracy by cascading the Skylark algorithm and the Taylor algorithm; The Taylor series method iterates the precise target point through precise initial values. When the initial values ​​are selected reasonably, accurate results are obtained, and the results of the Skylark algorithm are used as its initial values. Let the solution of the above Skylark algorithm be expressed as the initial position Let G be the Jacobian matrix, and according to the best linear unbiased estimate, we get: Where: W is the covariance matrix of the observation error, ρ = [Δx, Δy, Δz], Finally, the initial value is continuously iterated according to the set threshold value until it is smaller than the threshold value to obtain the radiation source position.

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