A pure bearing target tracking method considering signal delay

Abstract

The present invention discloses a bearing-only target tracking method considering signal delay; this method estimates the position of a target based on the received delayed bearing signals from the target. First, the probability density of the target at time k is approximated by N Gaussian components. Then, the variable period T corresponding to each Gaussian component i at time k is calculated respectively. k i Then each Gaussian component calculates the corresponding posterior probability density according to the corresponding variable period T. k i Next, the corresponding weights are calculated for each Gaussian component i based on the observation values respectively. Finally, based on the weights of each Gaussian component, the posterior probability density at time k is obtained by weighting. Compared with the traditional method, the present invention can effectively improve the accuracy of bearing-only target tracking that needs to consider delay on the basis of relatively low computational complexity.

Description

Technical Field

[0001] The present invention belongs to the technical field of target detection and tracking, and relates to a bearing-only target tracking method considering signal delay. Background Art

[0002] Bearing-only target tracking (BOT), also known as passive target tracking. Generally, since the time required for the signal to propagate in the medium has little effect on the result, it is usually considered that the signal immediately propagates from the target to the receiver. However, when the signal used is a signal with a slow propagation speed such as an acoustic signal, the signal emitted by the target cannot reach the receiver immediately and requires a certain amount of time. When this time is ignored, it will affect the tracking performance of the tracker and cause a large error in the estimation of the target position. Therefore, when there is a signal delay, it is necessary to estimate the delay time and determine the size of the variable period to improve the accuracy of position estimation.

[0003] Different from general bearing-only target tracking algorithms considering signal delay, the present invention proposes to apply a shifted Rayleigh filter based on Gaussian sum in the prediction and estimation stages of the target state to improve the tracking accuracy. The shifted Rayleigh filter (SRF) is one of the non-linear filters with high accuracy and good computational efficiency for solving the BOT problem. It uses an augmented measurement framework to accurately calculate its posterior density and utilizes the basic structure of the non-linearity existing in bearing-only tracking in a new way. The invention uses the shifted Rayleigh filter to calculate the probability density of each Gaussian component, and finally calculates the final probability density of the target at time k by the method of Gaussian sum at each estimation moment. Summary of the Invention

[0004] The object of the present invention is to propose a target tracking method of a shifted Rayleigh filter based on Gaussian sum to solve the problem that the existing technology for estimating signal delay is difficult to balance the running time and the estimation accuracy, and the method includes the following steps:

[0005] Step (1): Approximate the prior probability density and posterior probability density of the probability density of the target at time k with N Gaussian components, and each Gaussian component is represented by i, where i = 1,..., N;

[0006] Step (2): Calculate the variable period corresponding to each Gaussian component i at time k

[0007] Use linear search to estimate the variable period Suppose a model of a bearing-only target tracking system with signal delay is established in the Cartesian coordinate system, and the state vectors of the target and the receiver at time kT are expressed as and where the position of the receiver is assumed to be known, and the relative state variable is defined as

[0008]

[0009] where X kT denotes the state vector at time kT, and (x kT , y kT ) represents the relative position of the target, denotes the velocity in the x-y coordinate system;

[0010] Assume that the target is moving in an approximately uniform straight line, so the motion equation of the target is

[0011]

[0012] where

[0013]

[0014]

[0015] v kT is Gaussian noise with a mean of zero and a covariance matrix of Q kT , and f kT is the motion function of the target, and the value of Q kT is a coefficient;

[0016] Represent the measurement sequence as Z kT = [z T , z 2T ,..., z kT ', and the measurement z kT satisfies the following equation with the positive direction of the Y-axis as the reference, where the clockwise direction is set as the positive direction;

[0017]

[0018] where w kT is Gaussian noise with a mean of zero and a covariance matrix of R kT ;

[0019] where the true azimuth angle is

[0020]

[0021] For the requirements of real-time tracking in practical applications, a real-time tracking method based on real-time tracking technology is adopted. A given threshold δ is set to search for an appropriate variable period satisfying the following formula:

[0022]

[0023] Step (3), each Gaussian component is based on the corresponding variable period Calculations are performed to obtain each corresponding posterior probability density; each Gaussian component is brought into the SRF filter according to the estimated variable period.

[0024] The predicted mean and variance are calculated as follows:

[0025]

[0026]

[0027] where is the system input signal, Q is the system noise covariance matrix, and F k-1 is the state transition matrix at time k - 1 with the variable period at this time brought in;

[0028] The posterior mean and variance are defined as follows:

[0029]

[0030]

[0031] where the Kalman gain and

[0032] In the above formula

[0033]

[0034]

[0035] and

[0036]

[0037] where the variable σ θ is the variance of the observation noise, b k is the observation value, and var[] represents the variance; in the equation is the mean of the shifted Rayleigh variable;

[0038] Step (4): Calculate the corresponding weights for each Gaussian component i according to the observation values.

[0039] The weight corresponding to the i-th Gaussian component at time k is

[0040]

[0041] where where is the predicted measurement of the i-th Gaussian component, and b​k is the measurement value at time k, is the innovation variance of filter i given by the following formula,

[0042]

[0043] wherein and are respectively the linearized measurement matrix and the predicted covariance of filter i, is the variance of the predicted angle;

[0044] Step (5), according to the weights of each Gaussian component, weighted to obtain the posterior probability density at time k;

[0045] Step (6), according to the posterior probability mean, obtain the target position at time k;

[0046]

[0047] Preferably, the prior probability density in step (1) is expressed as:

[0048]

[0049] where N is the number of Gaussian components, w i is the weight of the i-th Gaussian component, and i = 1, ···, N; represents a Gaussian distribution, x k is the state vector of the target at time k, are respectively the mean and variance predicted by the i-th Gaussian component according to the state vector at time k - 1; the posterior probability density is written as

[0050]

[0051] Preferably, said weighted to obtain the posterior probability density at time k according to the weights of each Gaussian component;

[0052]

[0053] wherein represents a Gaussian distribution, x k is the state vector of the target at time k, are respectively the mean and variance estimated by the i-th Gaussian component according to the predicted value and the actual observed value obtained from the state vector at time k - 1.

[0054] Through the above steps, the position of the target can be estimated considering signal delay. And using the Gaussian sum and shifted Rayleigh filter algorithm in the present invention can improve the estimation accuracy without significantly increasing the calculation time. Description of the Drawings

[0055] Figure 1 is the algorithm flow chart of the present invention;

[0056] Figure 2 is the root mean square error (RMSE) effect diagram of the filtering result of the present invention;

[0057] Figure 3 is the BOT schematic diagram of the present invention with time-delay signals; Detailed implementation manners

[0058] The following combines the attached Figure 1 to elaborate in detail on the principle of the method of the present invention.

[0059] Step (1):

[0060] Approximate the prior and posterior probability densities of the target by means of the weighted sum of N Gaussian components. The prior probability density can be expressed as

[0061]

[0062] where N is the number of Gaussian components, w i is the weight of the i-th Gaussian component, and i = 1, ···, N; represents the Gaussian distribution, and x k is the state vector of the target at time k, are respectively the mean and variance predicted by the i-th Gaussian component according to the state vector at time k - 1; the posterior probability density is written as

[0063]

[0064] Step (2): Calculate the variable period corresponding to each Gaussian component at time k In the present invention, linear search is used to estimate the variable period Assume that the model of a bearing-only target tracking system with signal delay is established in the Cartesian coordinate system as Figure 3 shown. At time kT, express the state vectors of the target and the receiver as and (the position of the receiver is assumed to be known), and define the relative state variable as

[0065]

[0066] where X kT represents the state vector at time kT, and (x kT , y kT ) represents the relative position of the target, represents the velocity in the x - y coordinate system.

[0067] We assume that the target is approximately in uniform rectilinear motion, so the motion equation of the target is

[0068]

[0069] where

[0070]

[0071]

[0072] v kT is Gaussian noise with a mean of zero and a covariance matrix of Q kT and f kT is the motion function of the target. The value of Q kT in the present invention is set according to the actual situation.

[0073] The measurement sequence is represented as Z kT =[z T ,z 2T ,...,z kT ', and the measurement z kT satisfies the following equation with the positive direction of the Y-axis as the reference (clockwise is the positive direction):

[0074]

[0075] where w kT is Gaussian noise with a mean of zero and a covariance matrix of R kT The value of R kT in the present invention is set according to the actual situation.

[0076] where the true azimuth angle is

[0077]

[0078] For the requirements of real-time tracking in practical applications, a real-time tracking method based on real-time tracking technology is adopted. Only a given threshold δ needs to be approximately set. The size of the threshold δ in the present invention can be selected according to the actual situation to search for an appropriate variable period satisfying the following formula:

[0079]

[0080] Step (3): Each Gaussian component calculates the corresponding posterior probability density according to the corresponding variable period. Each Gaussian component is substituted into the SRF filter according to the estimated variable period.

[0081] The specific steps are as follows:

[0082] The predicted mean and variance The calculation of

[0083]

[0084]

[0085] wherein, is the system input signal, and Q is the system noise covariance matrix.

[0086] The mean and variance of the posterior are defined as follows:

[0087]

[0088]

[0089] where the Kalman gain and

[0090] In the above formula

[0091]

[0092]

[0093] and

[0094]

[0095] where the variable σ θ is the variance of the observation noise, and var[] represents the variance. In the equation is the mean of the shifted Rayleigh variable, and the expression is:

[0096]

[0097] where is the cumulative distribution function of the standard normal variable.

[0098] Step (4): Each Gaussian component calculates the corresponding weight according to the observation value

[0099] The weight corresponding to the i-th Gaussian component at time k is

[0100]

[0101] where where is the predicted measurement of the i-th Gaussian component, and b k is the measurement value at time k, is the innovation variance of filter i given by the following formula,

[0102]

[0103] Among them, and are respectively the linearized measurement matrix and the predicted covariance of filter i, is the variance of the predicted angle;

[0104] Step (5): According to the weights of each Gaussian component, the posterior probability density at time k is obtained by weighting

[0105]

[0106] Where represents a Gaussian distribution, and x k is the state vector of the target at time k, are respectively the mean and variance estimated from the predicted value and the actual observed value obtained by the i-th Gaussian component according to the state vector at time k - 1.

[0107] Step (6): According to the posterior probability mean, the target position at time k is obtained.

[0108]

[0109] As Figure 2 shown, it is the tracking effect of the present invention. Compared with the traditional method, it can effectively increase the estimation accuracy and improve the estimation accuracy on the basis of lower computational complexity.

Claims

1. A bearing-only target tracking method considering signal delay, characterized in that, the method specifically includes the following steps: Step (1): Approximate the prior probability density and the posterior probability density of the target at time k with N Gaussian components, and each Gaussian component is represented by i, where i = 1,..., N; The prior probability density is expressed as: where N is the number of Gaussian components, and w i is the weight of the i-th Gaussian component, and i = 1, …, N; represents a Gaussian distribution, and x k is the state vector at the target time k, are respectively the mean and variance predicted by the i-th Gaussian component based on the state vector at time k-1; Step (2), calculate the variable period corresponding to each Gaussian component i at time k respectively Using linear search to estimate the variable period Suppose a bearing-only target tracking system model with signal delay is established in the Cartesian coordinate system. At time kT, the state vectors of the target and the receiver are expressed as and where the position of the receiver is assumed to be known, and the relative state variable is defined as where X kT represents the state vector at time kT, (x kT , y kT ) represents the relative position of the target, represents the velocity in the x-y coordinate system; Assume that the target is approximately moving in a uniform straight line, so the motion equation of the target is where v kT is Gaussian noise with a mean of zero and a covariance matrix of Q kT , and f kT is the motion function of the target, and the value of Q kT is the coefficient; Represent the measurement sequence as Z kT = [z T , z 2T ,..., z kT ', where the measurement z kT satisfies the following equation with the positive direction of the Y-axis as the reference, where the clockwise direction is set as the positive direction; where w kT is Gaussian noise with zero mean and covariance matrix R kT ; where the true azimuth angle is In response to the requirements for real-time tracking in practical applications, a real-time tracking method based on real-time tracking technology is adopted, and a given threshold δ is set to search for an appropriate variable period Satisfy the following formula: Step (3), each Gaussian component calculates each corresponding posterior probability density according to the corresponding variable period ; each Gaussian component substitutes the estimated variable period into the F of the SRF filter k-1 ; Predicted mean and variance are calculated as follows: Among them, the system input signal, Q is the system noise covariance matrix, F k-1 is the variable period brought to this moment at the (k - 1)th moment, and it is the state transition matrix; The posterior mean variance is defined as follows: where the Kalman gain and In the above formula and where the variable σ θ is the variance of the observation noise, b k is the observed value, and var[] represents the variance; in the equation is the mean of the shifted Rayleigh variable; Step (4): Calculate the corresponding weights for each Gaussian component i according to the observation values; The weight corresponding to the i-th Gaussian component at time k is wherein wherein is the predicted measurement of the i-th Gaussian component, and b k is the measurement value at time k, is the innovation variance of filter i given by the following formula, Among them, and are the linearized measurement matrix and the predicted covariance of filter i respectively, is the variance of the predicted angle; Step (5): According to the weights of each Gaussian component, weighted to obtain the posterior probability density at time k: where represents a Gaussian distribution, and x k is the state vector at the target time k, are respectively the mean and variance estimated from the predicted value and the actual observed value obtained by the i-th Gaussian component according to the state vector at time k-1; Step (6): Obtain the target position at time k according to the posterior probability mean.

Images