CN116662733A - Variational filter maneuvering target tracking method based on Gaussian process regression - Google Patents
Variational filter maneuvering target tracking method based on Gaussian process regression Download PDFInfo
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Abstract
The application discloses a variational filter maneuvering target tracking method based on Gaussian process regression, and belongs to the field of target tracking. Firstly, obtaining a predicted posterior distribution of a target state, a predicted posterior distribution of a process noise covariance and a predicted measurement value by adopting a time prediction method, and then updating a combined posterior distribution of the target state and the process noise covariance by using a variational filtering method; the application uses the variation inference technology, solves the problem of unknown process noise in the filtering method by iteratively estimating the joint posterior of the target state and the process noise covariance, and achieves better effect on root mean square error than Kalman filtering using standard noise parameters; by using a Gaussian process regression technology, abnormal measurement is removed through a Gaussian model prediction measurement and threshold method, the problem of abnormal measurement in maneuvering target tracking is solved, and compared with the prior art, a better tracking effect is achieved.
Description
Technical Field
The application relates to a variational filter maneuvering target tracking method based on Gaussian process regression, and belongs to the field of target tracking.
Background
In motorized target tracking, the target may be various moving objects, such as an aircraft, a vehicle, a ship, a pedestrian, and the like. Maneuvering target tracking technology is always a difficulty and hotspot in the field of target tracking due to the unpredictability of target maneuvering. In a real engineering scene of maneuvering target tracking, it is quite possible that enough priori information cannot be obtained, and therefore the performance of the tracking method is greatly reduced.
Kalman filtering is a method for processing the whole signal process in the Gaussian white noise background, and the state transfer equation is used for representing the input-output relationship of a linear system and estimating required information parameters from measurement data. Unlike other filtering methods, kalman filtering does not impose strict requirements on the stationarity of the overall system, nor does it require the storage of historical data. Only the current instantaneous observed value and the estimated value of the previous instantaneous state need to be stored, so that the memory requirement and the calculation complexity are reduced. Thus, kalman filtering is widely used in many applications, including navigation control and target tracking.
Among the prior information required for the kalman filtering, the system process noise in the state space model followed by the target is important prior information that determines the performance of the tracking method, and the system process noise is used to describe possible system interference and time-varying information in the dynamic system, etc. If the process noise information of the tracking method is inaccurate, an estimation error of the tracking method may be increased. In addition, the Kalman filtering assumes that the information of the measurement values is accurate, however, in a real engineering scene, due to the change of the environment, the situation of abnormal measurement values cannot be avoided, the abnormal measurement values greatly influence the performance of the Kalman filtering method, the root mean square error of the algorithm estimation result can be improved, and the tracking accuracy is poor.
Disclosure of Invention
In order to solve the problems of unknown noise and reduced tracking effect of a Kalman filtering method caused by abnormal measurement in the system process and improve tracking precision, the application provides a variational filtering maneuvering target tracking method based on Gaussian process regression, which comprises the following steps:
a first object of the present application is to provide a maneuvering target tracking method, comprising: firstly, obtaining a predicted posterior distribution of a target state, a predicted posterior distribution of a process noise covariance and a predicted measurement value by adopting a time prediction method, and then updating a combined posterior distribution of the target state and the process noise covariance by using a variational filtering method;
the time prediction method comprises the following steps: carrying out one-step time prediction on the target state and the process noise covariance by adopting a Kalman filtering method to obtain a prediction posterior distribution of the target state and the process noise covariance; modeling by using a Gaussian process regression technology and using historical measurement, and predicting the current moment to obtain predicted measurement;
the variational filtering method comprises the following steps: firstly, detecting current measurement by using a measurement threshold processing method, removing abnormal measurement and replacing the abnormal measurement by using prediction measurement; and then, iteratively estimating the joint posterior distribution of the target state and the process noise covariance by using a variation approximation method, ending the variation approximation method when the termination condition is met or the iteration number is reached, and taking the result at the moment as the final joint posterior distribution.
Optionally, the process of obtaining the posterior distribution of the target state and the posterior distribution of the process noise covariance by the time prediction method includes:
for the current time k, the target state mean value x of the time k-1 is used k-1|k-1 Target state covariance P k-1|k-1 And state transition matrix F k Calculating a prediction state mean value x through a prediction step of Kalman filtering k|k-1 And prediction state covariance Σ k ：
x k|k-1 ＝F k x k-1|k-1
Calculating the predictive degree of freedom at time k by inverse Wishare distribution expressionAnd pre-heatingMeasuring matrix phi k|k-1 ：
Φ k|k-1 ＝ρΦ k-1|k-1
wherein , and Φk|k The degree of freedom and scale matrix of the process noise covariance posterior distribution at the k-1 moment.
Optionally, the process adopting the gaussian process regression technology includes:
historical measurement using k time { z } 1 ,z 2 ,…,z k-1 And adopting a zero-mean function and a square index covariance function as training data, calculating super parameters through a maximum likelihood estimation method, and carrying out Gaussian process modeling. Then, using the Gaussian process model to predict the measurement at k time to obtain a predicted measurement wherein ,z1:k-1 ＝{z 1 ,z 2 ,…,z k-1 And all measurement data from the starting time to the k-1 time.
Optionally, the measurement thresholding method includes:
calculation of the mean value of the information using the historical information at time kCalculation of the New Standard deviation Using historical New and New means +.>Determination of measurement threshold T using innovation standard deviation D ；
If the current measurement is z k Greater than or equal to threshold T D Consider the measurement z at the current time k Is an anomaly measure, which is rejected and a robust predictive measure using gaussian process regressionContinuing the method as the measurement value of the current moment; when the innovation is smaller than the threshold, i.e. r k <T D Consider the measurement z at the current time k Is normal measurement, and does not process the continuing method.
Optionally, the measurement threshold T D The calculation method of (1) is as follows:
wherein ,{r1 ,r 2 ,…,r k-1 And is a historical information set at the moment k.
Optionally, the variation approximation method includes:
first, iterative initialization is performed, using the prediction state mean x k|k-1 Initializing iteration hidden variable mean valueUse of predictive degrees of freedom->Prediction scale matrix Φ k|k-1 Initializing the iterative degree of freedom->And an iteration scale matrix->
Then, carrying out iterative updating, and using the hidden variable mean value of the last iterationAnd current measurement z k Updating target states in current iterationsx k Posterior distribution q (x) k ) The method comprises the steps of carrying out a first treatment on the surface of the Update state mean using current iteration +.>Sum prediction covariance Σ k Updating theta in current iteration k Posterior distribution q (θ) k ) The method comprises the steps of carrying out a first treatment on the surface of the Use of predictive degrees of freedom->Prediction scale matrix Φ k|k-1 And the updated state mean of the current iteration +.>Updating hidden variable mean +.>Covariance Q of process noise in current iteration k Posterior distribution Q (Q) k ) Updating;
finally, checking the iteration condition, when the difference value of the iteration target states is smaller than the threshold delta x Or stopping iteration when the iteration times reach N, otherwise, continuing iteration.
Optionally, the specific steps of the iterative updating include:
using the mean of the inverse Wishart distribution of the last iterationSum of covariance->Calculating an estimate A of the process noise covariance matrix in the current iteration k ：
Using current measurements z k And the mean value of hidden variables of the last iterationCalculating innovation:
from calculated innovation r k Abnormality measurement threshold T D And predictive measurement of GPRProcessing the anomaly measurement value:
based on the updating step of Kalman filtering, the Kalman filtering gain K is calculated x,k And using the mean value of hidden variables of the last iterationAnd current measurement z k Updating x in current iteration k Posterior distribution q (x) k ) Mean and covariance of (c):
P k|k ＝A k -K x,k H k A k
based on the updating step of Kalman filtering, the Kalman filtering gain K is calculated θ,k And uses the updated state mean of the current iterationSum prediction covariance Σ k Updating theta in current iteration k Posterior distribution q (θ) k ) Mean and covariance of (c):
K θ,k ＝Σ k (Σ k +A k ) -1
using a predictive mean value of a predictive inverse Wishare distribution based on an inverse Wishare distribution expression and an update step of Kalman filteringPrediction covariance Φ k|k-1 And update state mean +.>Updating hidden variable meansFor Q in the current iteration k Posterior distribution Q (Q) k ) Updating the mean and covariance of (c):
optionally, the measurement update iteration ends prematurely when the following condition is satisfied:
wherein ,δx Is the threshold for the termination of the iteration.
Optionally, when the iteration is terminated, a filtered mean value x of the target state at the k moment is obtained k|k Filtering covariance P of target state k|k Filtered mean of process noise covarianceAnd filtering covariance Φ k|k 。
A second object of the present application is to provide a computer-readable storage medium storing computer-executable instructions which, when executed by a processor, implement the maneuvering target tracking method according to any one of the above.
The application has the beneficial effects that:
(1) The application uses the variation inference technology, solves the problem of unknown process noise in the filtering method by the combined posterior of the iteration estimation target state and the process noise covariance, reduces the error of target tracking, achieves better effect on root mean square error than Kalman filtering using standard noise parameters, and improves tracking precision;
(2) The application uses Gaussian process regression technology, removes abnormal measurement by Gaussian model prediction measurement and a threshold method, solves the problem of abnormal measurement in maneuvering target tracking, and further improves tracking precision, so that the application has better tracking effect compared with the prior art.
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In order to more clearly illustrate the technical solutions of the embodiments of the present application, the drawings required for the description of the embodiments will be briefly described below, and it is apparent that the drawings in the following description are only some embodiments of the present application, and other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is an overall flow chart of a variational filter maneuver target tracking method of the present application based on Gaussian process regression.
Fig. 2 is a diagram of the true motion trajectory of the object in experiment one of the present application.
Fig. 3 is a graph of the root mean square error versus the target position for experiment one of the present application.
Fig. 4 is a graph of the root mean square error versus target velocity for experiment one of the present application.
Fig. 5 is a diagram of the true motion trajectory of the target of experiment two of the present application.
Fig. 6 is a graph of the rms error of the target location versus the experiment two of the present application.
FIG. 7 is a graph comparing the root mean square error of the target velocity for experiment two of the present application.
FIG. 8 is a graph of average normalized error squares comparison for experiment two of the present application.
Detailed Description
For the purpose of making the objects, technical solutions and advantages of the present application more apparent, the embodiments of the present application will be described in further detail with reference to the accompanying drawings.
The following description will be first made of the related content related to the present application:
1. regression principle of Gaussian process
The Gaussian Process (GP) is a non-parametric probabilistic model for modeling random functions of arbitrary dimensions. It can be seen as a distribution of a set of random variables, where the joint distribution of any finite number of variables obeys a multidimensional gaussian distribution. The gaussian process can be used for prediction and uncertainty quantization through the derivation of a priori and posterior distributions.
Gaussian process regression (Gaussian Process Regression, GPR) is a method of modeling regression problems using gaussian processes. In gaussian process regression, the input and output are considered random variables and it is assumed that the relationship between them can be described by an unknown random function. By using a gaussian process, this unknown random function can be modeled as a gaussian distribution, so that for new inputs, the corresponding outputs and their uncertainties can be derived by bayesian inference.
(1) Gaussian process prediction
A gaussian process is defined entirely by its mean function m (u) and covariance function k (u, u'). Gaussian processes are generalizations of gaussian distributions, which are typically defined on vectors, while gaussian processes are defined on functions. The gaussian process is generally defined as:
wherein u, u' e R d Is a random variable. Consider the following model:
y＝f(u)+∈
where u is the input vector, f is a function, y is a measurement value containing additive noise, noiseThe a priori distribution of the measurement y is therefore:
measured value y and predicted value f * Is:
wherein K (U, U) =k n ＝(k ij ) Is a covariance matrix of size n×n, matrix element k ij ＝k(u i ,u j ) For measuring u i and uj Correlation between; k (U, U) * )＝K(u * ,U) T Is test point u * An n x 1 size covariance matrix with the training set U; k (u) * ,u * ) Is test point u * Is a covariance of (2); i n Is an n-dimensional identity matrix. From this, a predicted value f can be obtained * Posterior distribution of (c):
wherein ,is test point u * Corresponding to the predicted value f * Mean and variance of (c).
(2) Gaussian process training
If there is sufficient prior information about the dataset, the prior mean and covariance functions can be accurately specified. However, in machine learning applications, this scenario with detailed a priori information is not typical. In order for the gaussian process technique to be of value in practice, it must be possible in practice to choose between different mean and covariance functions depending on the data. This process is referred to as training a gaussian process model.
For gaussian process regression, various covariance functions can be chosen, with the most common covariance function being the square-index covariance, i.e.:
wherein ,representing the variance of the metrology data, m=diag (l 2 ) L represents the scale, parameter set +.>Commonly referred to as superparameters, may be derived by maximum likelihood estimation. First, a negative log likelihood function L (θ) = -logp (y|x, θ) of training data is determined, and the super parameter θ is biased. Then, the partial derivative is minimized by an optimization method such as a conjugate gradient method, etc., so as to obtain an optimal solution of the super parameter. The negative log likelihood function L (θ) is in the form of:
wherein ,the partial derivative of L (θ) with respect to θ is:
wherein ,
2. principle of variable decibel leaf-Sitting filtering
The variational Bayesian inference is one of applications of a variational method in statistical inference, and can perform local optimal estimation on hidden variable posterior distribution of a probability model in an iterative mode. And (3) carrying out variational Bayesian inference, expanding the posterior of the hidden variable according to dimensions by using an average field theory to obtain a calculation frame, and iteratively updating an estimation result according to dimensions until the method converges. The variational Bayesian inference is a probability-based method that can be used to deal with noise and uncertainty. Complex integrals can be effectively approximated by variational Bayesian inference, but a suitable hidden variable conjugate prior model must be determined. Assume that the system state obeys a gaussian distribution:
wherein ,xk|k-1 ＝F k x k-1|k-1 And is also provided with
The conjugate prior distribution of covariance of the multivariate gaussian distribution is an inverse Wishart distribution, so the inverse Wishart distribution can be used to represent the unknown process noise covariance Q k The satisfied a priori distribution. But unknown process noise covariance Q k Only system state x k A portion of the covariance of the gaussian distribution obeyed. To decompose covariance matrix P k|k-1 Introducing an intermediate hidden variable theta k The above formula is rewritten as:
wherein ,
assuming unknown process noise covariance Q k Meeting GaussianThe distribution, using the inverse Wishart distribution as its prior distribution, can be expressed as the distribution that the noise covariance of the unknown process satisfies:
wherein ,representing Q compliance degrees of freedom +.>The inverse scale matrix is the inverse Wishart distribution of Φ.
Assuming an initial process noise covariance Q 0 Obeying the inverse Wishart distributionThe initial process noise covariance Q can be corrected according to the expression of the inverse Wishart distribution 0 The mean value of (c) is expressed as:
wherein epsilon is the tuning parameter. The transition posterior probability distribution p (Q) of process noise k |Q k-1 ) Can be defined as:
Φ k|k-1 ＝ρΦ k-1
wherein ρ ε [0,1] is a forgetting factor.
To estimate the system state x k Unknown process noise covariance Q k Hidden variable theta k It is necessary to calculate their joint posterior probability density function p (x k ,θ k ,Q k |z 1:k ). This joint posterior typically does not yield an analytical solution, but can be approximated by a variational Bayesian inference technique. By mean field approximation [63] The method can obtain:
p(x k ,θ k ,Q k |z 1:k )≈q(x k ,θ k ,Q k )≈q(x k )q(θ k )q(Q k )
the inference problem can be translated into a minimization of p (x) by a variational Bayesian inference technique k ,θ k ,Q k |z 1:k) and q(xk ,θ k ,Q k ) And the problem of optimizing the KL divergence between the two. Due to the non-negative nature of KL divergence, minimizing the KL divergence problem can be equivalent to finding a solution that allows for lovp (z k ) Lower bound of evidence (Evidence Lower Bound, ELBO)Maximized joint posterior q (x k ,θ k ,Q k ) I.e. +.> wherein />The method comprises the following steps:
the coordinate lifting method is a non-gradient optimization method, which searches along the direction of one coordinate in each iteration, and the local maximum value of the objective function is achieved by using different coordinate methods in a circulating way. The coordinate lifting method may be used to solve some specific optimization problems, such as variance inference. By using the coordinate lifting method:
where c is a constant, the joint probability density function lovp (x) k ,θ k ,Q k ,z k |z 1:k-1 ) The method is divided into the following forms:
to update state x k Solving for the mean value x of the posterior distribution k|k And covariance matrix P k|k X can be k The logarithmic expression of the posterior distribution of (c) is expressed as:
order theThe following formula can be obtained by performing an exponential operation on both sides of the above formula:
from the above, the system state x can be seen k The posterior probability density function of (2) is also subject to a gaussian distribution, and the mean and covariance matrices thereof are x respectively k|k and Pk|k All can be calculated by the update step of the kalman filter. First through iterative process noise matrix A k And measuring noise matrix R k Calculation of Kalman gain K x,k Then calculating updated filter state x by Kalman gain k|k And filtering covariance P k|k ：
P k|k ＝A k -K x,k H k A k
To update the hidden variable theta k Solving for the mean value θ of the posterior distribution k|k And covariance matrix P θ,k|k θ can be set k The logarithmic expression of the posterior distribution of (c) is expressed as:
the following formula can be obtained by performing an exponential operation on both sides of the above formula:
the hidden variable theta can be seen through the above k The posterior probability density function of (2) is also subject to a gaussian distribution, and the mean and covariance matrices thereof are respectively θ k|k and Pθ,k|k All can be calculated by Kalman filtering, firstly by iterative process noise matrix A k Sum-of-hidden-variable prediction covariance matrix Σ k Calculation of Kalman gain K θ,k Then calculating updated hidden variable mean value theta through Kalman gain k|k Sum covariance P θ,k|k ：
K θ,k ＝Σ k (Σ k +A k ) -1
P θ,k|k ＝Σ k -K θ,k Σ k
After respectively solving the joint posterior q (x k ,θ k ,Q k ) X in the middle k and θk After posterior distribution of (1), in order to update Q k Next, solving for the mean of the posterior distributionAnd covariance matrix phi k|k Q can be set k The logarithmic expression of the posterior distribution of (c) is expressed as:
wherein ：
the form of the inverse Wishart distribution is:
wherein Q and Φ are both positive definite matrices of n x n. The logarithm of the two sides of the upper part is obtained:
thus, the inverse matrixObeying the Wishart distribution +.>The average value is as follows:
3. kalman filtering principle
Kalman filtering is summarized in three points:
A. calculating new prior state estimation and prior error estimation by using the optimal estimation state and the optimal estimation error of the previous frame;
B. calculating a Kalman gain by using the prior error estimation and the measurement noise of the current frame;
C. then using gain and prior state estimation to calculate the optimal state estimation of the current frame;
the process is as follows:
1) Prediction
x k|k-1 ＝F k x k-1|k-1 +B k α
wherein ,Fk As a state transfer function, B k Alpha can be regarded as a control variable, P k|k-1 For posterior error estimation, Q k Is the process noise covariance.
2) Updating
x k|k ＝x k|k-1 +B k α
P k|k ＝(I-K k H k )P k|k-1
wherein ,Hk S is a measurement function k For the covariance matrix between the actual value and the estimated value, K k Is the calculated kalman gain.
Embodiment one:
the embodiment provides a maneuvering target tracking method, which comprises the following steps: firstly, obtaining a predicted posterior distribution of a target state, a predicted posterior distribution of a process noise covariance and a predicted measurement value by adopting a time prediction method, and then updating a combined posterior distribution of the target state and the process noise covariance by using a variational filtering method;
the time prediction method comprises the following steps: carrying out one-step time prediction on the target state and the process noise covariance by adopting a Kalman filtering method to obtain a prediction posterior distribution of the target state and the process noise covariance; modeling by using a Gaussian process regression technology and using historical measurement, and predicting the current moment to obtain predicted measurement;
the variational filtering method comprises the following steps: firstly, detecting current measurement by using a measurement threshold processing method, removing abnormal measurement and replacing the abnormal measurement by using prediction measurement; and then, iteratively estimating the joint posterior distribution of the target state and the process noise covariance by using a variation approximation method, ending the variation approximation method when the termination condition is met or the iteration number is reached, and taking the result at the moment as the final joint posterior distribution.
Embodiment two:
the embodiment provides a variational filter maneuvering target tracking method based on Gaussian process regression, and a flow chart is shown in fig. 1, wherein the method comprises the following steps:
step one: parameter initialization
For an initial time k=1, a measurement z of time k=1 is used 1 Initializing target state mean x k|k And target state covariance P k|k The initialization parameter tau is set to 10, and the noise mean value of the process is initialized by tauSum covariance phi k|k The attenuation factor was set to 0.9999, the upper limit of the number of iterations of the variational filter was set to 10, and the iteration termination threshold was set to 10 -6 。
Step two: temporal prediction
For the current time k, the target state mean value x of the time k-1 is used k-1|k-1 Target state covariance P k-1|k-1 And state transition matrix F k Calculating a prediction state mean value x through a prediction step of Kalman filtering k|k-1 And prediction state covariance Σ k ：
x k|k-1 ＝F k x k-1|k-1
Degree of freedom using k-1 moment process noise covariance posterior distributionAnd a scale matrix phi k|k Calculating the prediction degree of freedom +.>Prediction scale matrix Φ k|k-1 ：
Φ k|k-1 ＝ρΦ k-1|k-1
Historical measurement using k time { z } 1 ,z 2 ,…,z k-1 And adopting a zero-mean function and a square index covariance function as training data, calculating super parameters through a maximum likelihood estimation method, and carrying out Gaussian process modeling. Then, using the Gaussian process model to predict the measurement at k time to obtain a predicted measurement
Using historical information at time k { r ] 1 ,r 2 ,…,r k-1 Calculating an anomaly measurement threshold T D ：
Step three: measurement update
3.1 iterative initialization
Counting iterationsi is set to 1, and hidden variable iteration mean is initialized by using prediction state meanInitializing iteration parameters using predictive degrees of freedom>Initializing iteration parameters using a prediction scale matrix
3.2 iterative updating
Using the mean of the inverse Wishart distribution of the last iterationSum of covariance->Calculating an estimate A of the process noise covariance matrix in the current iteration k ：
Using current measurements z k And the mean value of hidden variables of the last iterationCalculating innovation:
from calculated innovation r k Abnormality measurement threshold T D And predictive measurement of GPRProcessing the anomaly measurement value:
/>
kalman filtering-based update stepStep, calculating Kalman filtering gain K x,k And using the mean value of hidden variables of the last iterationAnd current measurement z k Updating x in current iteration k Posterior distribution q (x) k ) Mean and covariance of (c):
P k|k ＝A k -K x,k H k A k
based on the updating step of Kalman filtering, the Kalman filtering gain K is calculated θ,k And uses the updated state mean of the current iterationSum prediction covariance Σ k Updating theta in current iteration k Posterior distribution q (θ) k ) Mean and covariance of (c):
K θ,k ＝Σ k (Σ k +A k ) -1
using a predictive mean value of a predictive inverse Wishare distribution based on an inverse Wishare distribution expression and an update step of Kalman filteringPrediction covariance Φ k|k-1 And update state mean +.>Updating hidden variable meansFor Q in the current iteration k Posterior distribution Q (Q) k ) Updating the mean and covariance of (c):
the measurement update iteration ends prematurely when the following conditions are satisfied:
wherein ,δx Is the threshold for the termination of the iteration.
Finally, when iteration is terminated, a filtering mean value x of a k-moment target state is obtained k|k Filtering covariance P of target state k|k Filtered mean of process noise covarianceAnd filtering covariance Φ k|k 。
In order to verify the variational filtering method based on Gaussian process regression, the application provides the following specific experiment:
1. experimental conditions and evaluation index
The method is realized by adopting MATLAB R2022a, and runs on a personal desktop computer with a processor of Intel Core i5-12400F 2.5GHz and a memory of 32 GB. The experiment uses the root mean square error (the root mean square error, RMSE) and the average normalized error square (Average Normalized Estimation Error Squared, ANEES) of the position and the velocity targeted by the performance evaluation index, and is defined as follows:
1) Root Mean Square Error (RMSE)
/>
wherein ,Ns Is the number of Monte Carlo simulations, x k 、x k,t 、 and />The estimated position, the true position, the estimated velocity and the true velocity in the x-direction, y, respectively k 、y k,t 、/> and />The estimated position, the true position, the estimated velocity and the true velocity in the y-direction, respectively.
The root mean square error reflects the deviation between the predicted value and the true value, with lower root mean square error indicating that the predicted value is closer to the true value.
2) Average normalized error square (ANEES)
wherein ,Ns Is the total number of Monte Carlo simulations, x k Is a true state of the device and is,is the estimated state and P is the estimated covariance matrix. ANEES describes the stability and consistency of the filtering method, the closer the value of ANEES is to the dimension of the estimated state, the better the consistency of the filtering method and the stronger the stability of the method.
2. Experiment and result analysis
The method of the present application is compared with VBAKF method proposed in published paper A Novel Adaptive Kalman Filter With Inaccurate Process and Measurement Noise Covariance Matrices by Huang Y et al 2018, KF (KF-NC) method using standard process noise covariance, and KF (KF-TC) method using real process noise covariance.
Specific experiments the performance of the method of the application was evaluated in two ways, namely: the process noise is unknown and the anomaly is measured, and the experimental result is as follows:
experiment one: unknown process noise
The first experiment aims to test the performance of the method in the unknown process noise covariance scene. In experiment one, the initial position of the target was [234.92km,85.50km ]] T The target initial speed is [ -141.4m/s, -141.4m/s] T The sampling period t=3s, the sampling duration is 600s, 200 measurement values are obtained by total sampling, and 200 Monte Carlo simulation experiments are performed.
Measuring noise covariance matrix wherein σr =50m and σ θ =1°. The process noise covariance matrix Q is given in the form of time-varying noise:
wherein ,σv ＝1m/s。
The maneuvering trajectory of the target is divided into the following four phases:
(1) performing uniform linear motion between 1s and 200 s;
(2) a right turning movement with a turning angular velocity ω= -0.05rad/s between 200s and 218 s;
(3) then, uniform linear motion is carried out between 218s and 480 s;
(4) a left turning movement with a turning angular velocity ω=0.022 rad/s is performed between 480s and 600 s.
The real motion trail of the target is shown in fig. 2, wherein the circle marks the initial position of the target, and the curve is the motion trail of the target.
Table 1, fig. 3 and fig. 4 show the position root mean square error and velocity root mean square error pairs for the various methods of experiment one.
As can be seen from table 1, fig. 3 and fig. 4, the influence of the unknown process noise covariance on the kalman filter is large, and when the uncertainty contained in the unknown process noise covariance is large, the performance of the kalman filter may be reduced rapidly. The VBAKF method can estimate the process noise covariance to a certain extent, the performance of the VBAKF method is obvious in target speed estimation, the VBAKF method is better than the KF-NC method using standard process noise in estimation performance, and the VBAKF method is slightly better than the KF-NC method in estimation performance of the target position.
TABLE 1 root mean square error comparison of target position and velocity
The performance of the method provided by the application is close to that of a KF-TC method using real process noise covariance on the position root mean square error, is obviously lower than that of a VBAKF method, and has almost the same estimation precision as that of the KF-TC method in the position aspect. This result benefits to a great extent from the modeling and updating of the unknown process noise covariance in the method of the present application, which makes the method of the present application robust against unknown uncertainties.
Even if the system process noise of the target is complex unknown time-varying noise, the method can have good tracking performance, and the usability of the method is improved. The inventive method performs somewhat worse than the KF-TC method using real noise, but also far better than the KF-NC method using standard process noise, in terms of speed root mean square error.
Experiment II: abnormality measurement
The second objective of experiment was to test the performance of the method of the application when there was an abnormal amount in the measurements. In experiment two, some normal measurements in the measurement set are replaced by abnormal measurements to test the processing capacity of the proposed method for the abnormal measurements.
In experiment two, the initial position of the target is [234.92km,85.50km ]] T The target initial speed is [ -141.4m/s, -141.4m/s] T The sampling period t=3s, the sampling duration is 600s, 200 measurement values are obtained by total sampling, and 200 Monte Carlo simulation experiments are performed. Measuring noise covariance matrix wherein σr =50m and σ θ =1°. The process noise covariance matrix Q is given in the form of time-varying noise, the expression of which is the same as that of experiment one.
The maneuvering trajectory of the target is divided into the following four phases:
(1) performing uniform linear motion between 1s and 150 s;
(2) a right turning movement with a turning angular velocity ω= -0.033rad/s between 150s and 218 s;
(3) performing uniform linear motion between 218s and 480 s;
(4) finally, a left-hand turning movement is performed with a turning angular velocity ω=0.02 rad/s between 480s and 600 s.
After the measurements are generated, normal measurements of 60s,150s,300s and 420s are replaced with abnormal measurements.
The real motion trail of the target is shown in fig. 5, wherein the circle marks the initial position of the target, and the curve is the motion trail of the target.
The position root mean square error and the velocity root mean square error for the various methods in experiment two are compared in fig. 6 and 7. As can be seen from fig. 6, the root mean square error of the VBAKF method significantly fluctuates because experiment two adds an anomaly measure to the measure, while the VBAKF method does not process the anomaly measure. The anomaly measurement also causes the overall performance of the VBAKF method to be reduced, and the performance of the VBAKF method at part of the time is similar to that of the KF-NC method, and the performance of the VBAKF method at part of the time is even far less than that of the KF-NC method.
It can be found from fig. 6 and fig. 7 that the overall root mean square error curve of the method of the present application is smooth, has small fluctuation, and the method performance is significantly better than KF-NC and VBAKF methods, approaching KF-TC methods using real process noise. The method adopts the GPR technology and threshold value processing abnormality measurement, improves the robustness of the method, and simultaneously ensures that the proposed method has adaptability to unknown time-varying process noise through a variation technology, and has stable performance.
In fig. 8, the ANEES of the methods in the second experimental scenario are compared, and it can be seen that the stability and consistency of the method of the present application are better than those of the other methods, which benefits from the GPR-based anomaly measure processing portion of the method of the present application, making the method of the present application insensitive to anomaly measures.
Therefore, according to the first experiment and the second experiment, the variational filter maneuvering target tracking method based on the Gaussian process regression provided by the application is used for modeling and predicting through the Gaussian process regression, the threshold processing method is used for processing abnormal measurement, the combined posterior of the target state and the process noise is iteratively estimated through the variational inference technology, the uncertainty and the abnormal measurement value contained in the covariance of the unknown process noise can be processed, and the tracking precision, the robustness and the universality of the method are improved.
Some steps in the embodiments of the present application may be implemented by using software, and the corresponding software program may be stored in a readable storage medium, such as an optical disc or a hard disk.
The foregoing description of the preferred embodiments of the application is not intended to limit the application to the precise form disclosed, and any such modifications, equivalents, and alternatives falling within the spirit and scope of the application are intended to be included within the scope of the application.
Claims (10)
1. A motorized target tracking method, the method comprising: firstly, obtaining a predicted posterior distribution of a target state, a predicted posterior distribution of a process noise covariance and a predicted measurement value by adopting a time prediction method, and then updating a combined posterior distribution of the target state and the process noise covariance by using a variational filtering method;
the time prediction method comprises the following steps: carrying out one-step time prediction on the target state and the process noise covariance by adopting a Kalman filtering method to obtain a prediction posterior distribution of the target state and the process noise covariance; modeling by using a Gaussian process regression technology and using historical measurement, and predicting the current moment to obtain predicted measurement;
the variational filtering method comprises the following steps: firstly, detecting current measurement by using a measurement threshold processing method, removing abnormal measurement and replacing the abnormal measurement by using prediction measurement; and then, iteratively estimating the joint posterior distribution of the target state and the process noise covariance by using a variation approximation method, ending the variation approximation method when the termination condition is met or the iteration number is reached, and taking the result at the moment as the final joint posterior distribution.
2. The maneuvering target tracking method according to claim 1, wherein the process of obtaining a posterior distribution of target states and a posterior distribution of process noise covariance by the time prediction method includes:
for the current time k, the target state mean value x of the time k-1 is used k-1|k-1 Target state covariance P k-1|k-1 And state transition matrix F k Calculating a prediction state mean value x through a prediction step of Kalman filtering k|k-1 And prediction state covariance Σ k ：
x k|k-1 ＝F k x k-1|k-1
Calculating the predictive degree of freedom at time k by inverse Wishare distribution expressionPrediction scale matrix Φ k|k-1 ：
Φ k|k-1 ＝ρΦ k-1|k-1
wherein , and Φk|k The degree of freedom and scale matrix of the process noise covariance posterior distribution at the k-1 moment.
3. The maneuvering target tracking method according to claim 1, wherein the process employing gaussian process regression technique includes:
historical measurement using k time { z } 1 ,z 2 ,…,z k-1 And adopting a zero-mean function and a square index covariance function as training data, calculating super parameters through a maximum likelihood estimation method, and carrying out Gaussian process modeling. Then, using the Gaussian process model to predict the measurement at k time to obtain a predicted measurement wherein ,z1:k-1 ＝{z 1 ,z 2 ,…,z k-1 And all measurement data from the starting time to the k-1 time.
4. The maneuvering target tracking method according to claim 1, wherein the measurement thresholding method includes:
calculation of the mean value of the information using the historical information at time kCalculation of the New Standard deviation Using historical New and New means +.>Determination of measurement threshold T using innovation standard deviation D ；
If the current measurement is z k Greater than or equal to threshold T D Consider the measurement z at the current time k Is an anomaly measure, which is rejected and a robust predictive measure using gaussian process regressionContinuing the method as the measurement value of the current moment; when the innovation is smaller than the threshold, i.e. r k <T D Consider the measurement z at the current time k Is normal measurement, and does not process the continuing method.
5. The maneuvering target tracking method according to claim 4, wherein the measurement threshold T D The calculation method of (1) is as follows:
wherein ,{r1 ,r 2 ,…,r k-1 And is a historical information set at the moment k.
6. The maneuvering target tracking method according to claim 4, wherein the variance approximation method includes:
first, iterative initialization is performed, using the prediction state mean x k|k-1 Initializing iteration hidden variable mean valueUse of predictive degrees of freedom->Prediction scale matrix Φ k|k-1 Initializing the iterative degree of freedom->And an iteration scale matrix->
Then, carrying out iterative updating, and using the hidden variable mean value of the last iterationAnd current measurement z k Updating target state x in current iteration k Posterior distribution q (x) k ) The method comprises the steps of carrying out a first treatment on the surface of the Update state mean using current iteration +.>Sum prediction covariance Σ k Updating theta in current iteration k Posterior distribution q (θ) k ) The method comprises the steps of carrying out a first treatment on the surface of the Use of predictive degrees of freedom->Prediction scale matrix Φ k|k-1 And the updated state mean of the current iteration +.>Updating hidden variable mean +.>Covariance Q of process noise in current iteration k Posterior distribution Q (Q) k ) Updating;
finally, checking the iteration condition, when the difference value of the iteration target states is smaller than the threshold delta x Or stopping iteration when the iteration times reach N, otherwise, continuing iteration.
7. The maneuvering target tracking method according to claim 6, wherein the iterative updating includes the specific steps of:
using the mean of the inverse Wishart distribution of the last iterationSum of covariance->Calculating an estimate A of the process noise covariance matrix in the current iteration k ：
Using current measurements z k And the mean value of hidden variables of the last iterationCalculating innovation:
from calculated innovation r k Abnormality measurement threshold T D And predictive measurement of GPRProcessing the anomaly measurement value:
based on the updating step of Kalman filtering, the Kalman filtering gain K is calculated x,k And using the mean value of hidden variables of the last iterationAnd current measurement z k Updating x in current iteration k Posterior distribution q (x) k ) Mean and covariance of (c):
P k|k ＝A k -K x,k H k A k
based on the updating step of Kalman filtering, the Kalman filtering gain K is calculated θ,k And uses the updated state mean of the current iterationSum prediction covariance Σ k Updating theta in current iteration k Posterior distribution q (θ) k ) Mean and covariance of (c):
K θ,k ＝Σ k (Σ k +A k ) -1
using a predictive mean value of a predictive inverse Wishare distribution based on an inverse Wishare distribution expression and an update step of Kalman filteringPrediction covariance Φ k|k-1 And update state mean +.>Updating hidden variable mean +.>For the currentQ in iteration k Posterior distribution Q (Q) k ) Updating the mean and covariance of (c):
8. the maneuvering target tracking method according to claim 7, wherein the measurement update iteration ends prematurely when the following conditions are met:
wherein ,δx Is the threshold for the termination of the iteration.
9. The maneuvering target tracking method according to claim 7, wherein when iteration is terminated, a filtered mean value x of the target state at the k moment is obtained k|k Filtering covariance P of target state k|k Filtered mean of process noise covarianceAnd filtering covariance Φ k|k 。
10. A computer readable storage medium storing computer executable instructions which when executed by a processor implement the maneuvering target tracking method according to any one of claims 1 to 9.
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