CN108037663A

CN108037663A - A kind of maneuvering target tracking method based on change dimension filtering algorithm

Info

Publication number: CN108037663A
Application number: CN201711284472.4A
Authority: CN
Inventors: 于雪莲; 欧能杰; 唐永昊; 赵林森; 周云
Original assignee: University of Electronic Science and Technology of China
Current assignee: University of Electronic Science and Technology of China
Priority date: 2017-12-07
Filing date: 2017-12-07
Publication date: 2018-05-15

Abstract

The disclosure of the invention is a kind of based on the maneuvering target tracking method for becoming dimension filtering algorithm, belongs to signal processing technology, and in particular to becomes dimension filtering algorithm.The present invention is compared with becoming and tieing up filtering algorithm, when the k moment detect it is motor-driven, the state estimation for thinking the k s moment under non-maneuver model is rational, and the acceleration at state estimation and the measuring value at k moment the calculating k s moment using the k s moment, so as to improve the accuracy of acceleration estimation, error during models switching is reduced.The present invention is compared with becoming and tieing up filtering algorithm, continuous item in the state covariance matrix of maneuver modeling is adjusted using k s moment acceleration estimation, acceleration variance is reduced, so that wave filter can rapidly and accurately adjust filter state, accelerates the convergent speed of filtering.

Description

Maneuvering target tracking method based on variable-dimension filtering algorithm

Technical Field

The invention belongs to the signal processing technology, and particularly relates to a variable-dimension filtering algorithm.

Background

Maneuvering target tracking belongs to a technology in the field of signal processing, plays an important role in national defense and economic construction, and a plurality of scholars are dedicated to researching a target tracking algorithm all the time. However, as the maneuverability of modern aircraft continues to increase, reliable and accurate tracking of targets has become a major challenge.

The current maneuvering target tracking algorithm can be roughly divided into two types, one is an adaptive tracking algorithm without maneuvering detection, and the other is a tracking algorithm with maneuvering detection. The maneuver detection is essentially to determine and adjust the state of an object using the residual error of the object. The conventional maneuvering detection algorithm comprises an adjustable white noise model, a variable-dimension filtering algorithm, an output estimation algorithm and the like. Among them, the Variable dimension filtering algorithm was proposed in 1982 by Bar-Shalom and Birmiwal in Variable dimension filter for varying target tracking. The variable dimension filtering algorithm adopts 'switch' type switching, determines the maneuvering condition of the target according to the detection means and switches between a non-maneuvering model and a maneuvering model. The original variable-dimension filtering algorithm has the advantages that the prior assumption of target maneuver is not relied on, but the switching of the filter between the non-maneuvering state and the maneuvering state often causes a large error in target state estimation, so that the tracking error is increased.

Disclosure of Invention

The invention aims to provide a maneuvering target tracking method based on a variable-dimension filtering algorithm, and the maneuvering target tracking performance is improved.

The basic idea for realizing the invention is as follows: and detecting the occurrence of the maneuver by using the attenuation memory average value, and initializing the state estimation of the maneuver moment by using the measurement value of the current moment and the state estimation of the maneuver occurrence moment once the maneuver hypothesis is accepted, and simultaneously updating the state covariance matrix of the moment so as to enable the filtering under the maneuver model to be rapidly converged.

The technical scheme of the invention is a maneuvering target tracking method based on a variable-dimension filtering algorithm, which comprises the following steps:

step 1: establishing a target motion model

1a) Respectively establishing a motion state equation of a non-maneuvering model and a maneuvering model:

X_k＝FX_k-1+W_k-1

wherein, X_kAnd X_k-1Representing the state vectors of the non-motorized model at times k and k-1 respectively,x_kandindicating the position and velocity of the target k at time,andrespectively representing the state vectors of the maneuvering model at the time k and the time k-1; andrespectively representing the position, velocity and acceleration of the target at time k, F and F^mRespectively as a state transition matrix in non-maneuvering and maneuvering; w_k-1Andis a white noise sequence of discrete time at the moment of k-1, the mean value is zero, and the variances are respectively Q delta_kjAnd Q^mδ_kjQ and Q^mNoise covariance matrix, δ, representing non-motorized and motorized models_kjRepresenting a two-dimensional impulse function, wherein when k is j, the impulse function is 1, otherwise, the impulse function is 0, k and j are randomly generated, and the superscript m represents that the mobile mode is in;

1b) the measurement equations of the target non-maneuvering model and the maneuvering model are established using the following equations:

Z_k＝HX_k+V_k

wherein Z is_kIndicating the k time radarThe measured value of (a); h and H^mMeasuring matrixes of a non-maneuvering model and a maneuvering model respectively; v_kFor measuring noise, the mean is 0 and the variance is R_kAnd independently of process noise, R_kA covariance matrix for the measured noise;

step 2: state prediction for non-motorized models

2a) And (3) completing one-step prediction of the target state by the non-maneuvering model established in the step 1 and the state estimation at the last moment:

wherein,representing the state of the prediction target at the k moment at the k-1 moment;representing the state update value at the target k-1 moment;

2b) the one-step prediction of the target state covariance matrix is determined by:

P_k|k-1＝FP_k-1|k-1F^T+Q

wherein, P_k|k-1Representing the state covariance at the predicted k time at k-1; p_k-1|k-1Representing the state covariance update value at the k-1 moment; f^TIs the transpose of the non-motorized state transition matrix; q represents a noise covariance matrix;

2c) and determining the prediction of the measured value of the target at the k-1 moment according to the state prediction value as follows:

wherein,the measured value of the predicted target at the k-1 moment is shown;representing the state of the prediction target at the k moment at the k-1 moment;

2d) the filtering innovation (residual) is determined using the following equation:

wherein v is_kFor filtering innovation at time k, Z_kIs a measured value at the moment k;

2e) determining a covariance matrix of the prediction error according to:

S_k＝HP_k|k-1H^T+R_k

wherein S is_kA covariance matrix that is a prediction error; p_k|k-1Representing the state covariance of the predicted target at the k-1 moment at the k moment; matrix H^TTransposing the measurement matrix; r_kA covariance matrix for the measured noise;

and step 3: detecting whether target maneuver occurs

3a) The average value of the attenuation memory of the filtering innovation is calculated according to the following formula:

μ＝1-1/s

where ρ is_kAnd ρ_k-1Filtering for respectively representing k and k-1 timeThe attenuation of the wave event memorizes the average value,is the square of the normalized filtering innovation and follows a chi-square distribution with a degree of freedom of 1/(1-mu); mu is a discount factor, s is a sliding window length, v'_kDenotes v_kTransposing;

3b) detecting the occurrence of maneuver according to the following formula:

P{ρ_k≤ρ_max}＝1-α

where ρ is_maxα is the significance level set for the threshold value of maneuver hypothesis, if p_kExceeds a threshold value rho set by the following formula_maxIf yes, the assumption of maneuver occurrence is accepted, the estimator is converted from the non-maneuver model to the maneuver model at the threshold point, and the step 5 is entered; otherwise, refusing to accept the hypothesis of maneuver occurrence, and entering step 4;

and 4, step 4: state update for non-motorized models

4a) The filter gain is determined from the covariance matrix of the prediction error according to:

wherein, K_kFilter gain at time k; p_k|k-1Representing the state covariance of the predicted target at the k-1 moment at the k moment; s_kA covariance matrix that is a prediction error; h^TTransposing the measurement matrix;

4b) the update of the target state is accomplished as follows:

wherein,representing the state update value at the target k moment;representing the state of the prediction target at the k moment at the k-1 moment; z_kRepresenting a target measurement value;the measured value of the predicted target at the k-1 moment is shown;

4c) the update of the target state covariance is done as follows:

wherein, P_k|kRepresenting the state covariance update value at the target k moment;is a transpose of the filter gain matrix;

4d) repeating the steps 2-3;

and 5: initialization of maneuver modes

5a) When the maneuver is detected at the moment k, the maneuver is considered to be started at the moment k-s, therefore, the acceleration dimension is increased, the state estimation of the maneuver model at the moment k-s is initialized, s is a set fixed value, and the expression is as follows:

wherein,representing the state update value of the mobile model at the time k-s,representing the position estimate of the non-motorized model at time k-s,representing the velocity estimate of the non-motorized model at time k-s, z_kRepresenting the measurement value at the k moment in a one-dimensional state, and T represents the radar sampling interval;

5b) initialization of the state covariance matrix at time k-s is accomplished as follows:

wherein,representing the state covariance matrix of the motor model at time k-s,to representElement of row i and column j, P_k-s|k-sState covariance matrix, P, representing the non-motorized model at time k-s_k-s|k-s(i, j) represents P_k-s|k-sRow i and column j;

step 6: state prediction for a mobility model

6a) And completing one-step prediction of the target state by the established maneuvering model and the state update value at the last moment:

wherein,representing the state of the prediction target at the k moment at the k-1 moment;representing the state update value at the target k-1 moment; f^mRepresenting a state transition matrix when maneuvering;

6b) the one-step prediction of the target state covariance matrix is determined by:

wherein,representing the state covariance at the predicted k time at k-1;representing the state covariance update value at the k-1 moment; q^mRepresenting a noise covariance matrix;

6c) and determining the prediction of the measured value of the target at the k-1 moment according to the state prediction value as follows:

wherein,the measured value of the predicted target at the k-1 moment is shown;representing the state of the prediction target at the k moment at the k-1 moment; h^mRepresenting a measurement matrix when the target is maneuvering;

6d) the filtering innovation (residual) is determined using the following equation:

wherein,for filtering innovation at time k, Z_kIs a measured value at the moment k;

6e) determining a covariance matrix of the prediction error according to:

wherein,a covariance matrix that is a prediction error;representing the state covariance of the predicted target at the k-1 moment at the k moment; (H)^m)^TRepresenting the transpose of the measurement matrix; r_kA covariance matrix for the measured noise;

and 7: detecting if a maneuver has ended

7a) When the acceleration suddenly drops to 0, namely the maneuver is suddenly ended, the maneuver model is stoppedWhen the confidence area exceeds 95%, switching to a non-maneuvering model, and entering step 2;

7b) the statistics of non-maneuver detections are calculated as follows:

wherein, delta_kA statistic representing the significance test of the acceleration estimate at time k,represents an estimate of the acceleration component at time k,a block in which the covariance matrix representing the maneuvering model corresponds to the acceleration component,representing the sum of the acceleration estimation significance test statistics over a sliding window of length p;

7c) when in useFalls in the set threshold valueWhen the acceleration is not significant, switching to a non-maneuvering model, and entering step 2; otherwise, the target is considered to be in a maneuvering state, and the step 8 is entered;

and 8: state update for a maneuver mode

8a) The filter gain is determined from the covariance matrix of the prediction error according to:

wherein,filter gain at time k;representing the state covariance of the predicted target at the k-1 moment at the k moment; (H)^m)^TRepresenting the transpose of the measurement matrix;a covariance matrix that is a prediction error;

8b) the update of the target state is accomplished as follows:

8c) the update of the target state covariance is done as follows:

wherein,representing the state covariance update value at the target k moment;is a transpose of the filter gain matrix;

8d) and 6-7 repeating the steps.

Compared with the prior art, the invention has the following advantages:

firstly, compared with a variable-dimension filtering algorithm, when maneuvering is detected at the moment k, the state estimation of the moment k-s under a non-maneuvering model is considered to be reasonable, and the acceleration at the moment k-s is calculated by using the state estimation of the moment k-s and a measurement value of the moment k, so that the accuracy of the acceleration estimation is improved, and the error during model switching is reduced.

Secondly, compared with a variable-dimension filtering algorithm, the method utilizes the acceleration estimation at the k-s moment to adjust the related items in the state covariance matrix of the maneuvering model, so that the acceleration variance is reduced, the filter can rapidly and accurately adjust the filtering state, and the filtering convergence speed is accelerated.

Drawings

FIG. 1 is a schematic flow diagram of the present invention;

FIG. 2 is a comparison graph of simulation errors of the present invention and a variable-dimension filtering algorithm.

Detailed Description

With reference to fig. 1, the present invention has the following steps:

step 1, establishing a target motion model

Sampling in a period T, and respectively establishing a motion state equation of a non-maneuvering model and a maneuvering model for a target:

X_k＝FX_k-1+W_k-1

wherein, X_kAnd X_k-1Representing the state vectors of the non-motorized model at times k and k-1 respectively,x_kandindicating the position and velocity of the target k at time,andrepresenting the state vectors of the maneuver model at times k and k-1 respectively, andrespectively representing the position, velocity and acceleration of the target at time k, F and F^mThe state transition matrix when the vehicle is not maneuvering and the state transition matrix when the vehicle is maneuvering respectively has the following expression:

W_k-1andis a discrete-time white noise sequence, the mean value is zero, and the variance is E [ W ] respectively_kW_j]＝Q_kδ_kjAnd

in the examples of the present invention, givenThe process noise variance matrix is then:

the measurement equations of the non-maneuvering model and the maneuvering model are as follows:

Z_k＝HX_k+V_k

wherein Z is_kRepresenting the measured value of the radar at the k moment; h and H^mThe measurement matrixes of the non-maneuvering model and the maneuvering model respectively have the following expressions:

H＝[1 0]

H^m＝[1 0 0]

V_kfor measuring noise, the mean is 0 and the variance isR_kAnd is independent of process noise.

Step 2, predicting the state of the non-maneuvering model

And completing one-step prediction of the target state by the established non-maneuvering model and the state update value at the last moment:

wherein,representing the state of the prediction target at the k moment at the k-1 moment;representing the state update value at the target k-1 moment; f denotes a state transition matrix.

The one-step prediction of the covariance matrix of the target state is determined by

P_k|k-1＝FP_k-1|k-1F^T+Q

Wherein, P_k|k-1Representing the state covariance at the predicted k time at k-1; p_k-1|k-1Represents the state covariance update at time k-1, F^TRepresenting the transpose of the state transition matrix and Q the noise variance matrix.

And determining the prediction of the measured value of the target at the k-1 moment according to the state prediction value as follows:

wherein,the measured value of the predicted target at the k-1 moment is shown;the state of the prediction target at the time k-1 is shown, and H is a measurement matrix.

The filtering innovation (residual) is determined using the following equation:

wherein v is_kFor filtering innovation at time k, Z_kMeasured values at time k.

Determining a covariance matrix of the prediction error according to:

S_k＝HP_k|k-1H^T+R_k

wherein S is_kA covariance matrix that is a prediction error; p_k|k-1Representing the state covariance of the predicted target at the k-1 moment at the k moment; h^TTransposing the measurement matrix; r_kIs a covariance matrix of the measured noise.

Step 3, detecting whether the target maneuver occurs

The average value of the attenuation memory of the filtering innovation is calculated according to the following formula:

μ＝1-1/s

where ρ is_kAnd ρ_k-1Representing filtering information at time k and k-1, respectivelyThe mean value of the attenuation memory of (1) subject to the degree of freedom of 1/(1-. mu.m)) Chi fang distribution; mu is the discount factor and s is the sliding window length, s being 5 in the present example.

Detecting the occurrence of maneuver according to the following formula:

P{ρ_k≤ρ_max}＝1-α

where ρ is_maxα is set significance level for the threshold value of maneuver hypothesis, α in the embodiment of the invention is 0.005, and the corresponding threshold value rho is obtained when the degree of freedom is 5 according to the chi-square distribution table_max＝16.75。

If ρ_kExceeds a threshold value rho set by the following formula_maxIf yes, the assumption of maneuver occurrence is accepted, the estimator is converted from the non-maneuver model to the maneuver model at the threshold point, and the step 5 is entered; otherwise, refusing to accept the hypothesis of maneuver occurrence, and entering step 4;

step 4, updating the state of the non-maneuvering model

The filter gain is determined from the covariance matrix of the prediction error according to:

wherein, K_kFilter gain at time k; p_k|k-1Representing the state covariance of the predicted target at the k-1 moment at the k moment; h^TTransposing the measurement matrix; [. the]^-1Representing the inverse of the matrix.

The update of the target state is accomplished as follows:

wherein,representing the state update value at the target k moment;representing the state of the prediction target at the k moment at the k-1 moment;the measured value of the predicted target at the k-1 moment is shown; z_kRepresenting the target measurement value, K_kA filter gain matrix.

The update of the target state covariance is done as follows:

wherein, P_k|kRepresenting the state covariance update value at the target k moment; s_kA covariance matrix that is a prediction error;is a transpose of the filter gain matrix.

And (4) repeating the steps 2-3.

Step 5, initialization of the maneuver State

When the maneuver is detected at the moment k, the target at the moment k-s is considered to start the maneuver, so that the acceleration dimension is increased, and the state estimation of the maneuver model at the moment k-s is initialized, and the expression is as follows:

wherein,representing the state update value of the mobile model at the time k-s,representing a non-motorized model inThe estimation of the position at the time k-s,representing the velocity estimation of the non-motorized model at time k-s.

Initialization of the state covariance matrix at time k-s is accomplished as follows:

wherein,representing the state covariance matrix of the motor model at time k-s,to representElement of row i and column j, P_k-s|k-sState covariance matrix, P, representing the non-motorized model at time k-s_k-s|k-s(i, j) represents P_k-s|k-sRow i and column j.

Step 6, predicting the state of the maneuvering model

And completing one-step prediction of the target state by the established maneuvering model and the state update value at the last moment:

wherein,representing the state of the prediction target at the k moment at the k-1 moment;representing the state update value at the target k-1 moment; f^mRepresenting the target state transition matrix.

The one-step prediction of the target state covariance matrix is determined by:

wherein,representing the state covariance at the predicted k time at k-1;representing the state covariance update value at the k-1 moment; (F)^m)^TRepresenting a transpose of a target state transition matrix; q^mRepresenting the noise covariance matrix.

wherein,the measured value of the predicted target at the k-1 moment is shown;representing the state of the prediction target at the k moment at the k-1 moment; h^mRepresenting the target metrology matrix.

The filtering innovation (residual) is determined using the following equation:

wherein,for filtering innovation at time k, Z_kMeasured values at time k.

Determining a covariance matrix of the prediction error according to:

wherein,a covariance matrix that is a prediction error;representing the state covariance of the predicted target at the k-1 moment at the k moment; (H)^m)^TRepresenting the transpose of the measurement matrix.

Step 7, detecting whether the maneuver is finished

When the acceleration suddenly drops to 0 (namely the maneuver is suddenly ended), namely the innovation of the maneuver modelAnd when the confidence area exceeds 95%, switching to a non-maneuvering model and entering step 2.

The statistics of non-maneuver detections are calculated as follows:

wherein, delta_kA statistic representing the significance test of the acceleration estimate at time k,represents an estimate of the acceleration component at time k,a block in which the covariance matrix representing the maneuvering model corresponds to the acceleration component,represents the sum of the acceleration estimation significance test statistics over a sliding window of length p, which in the present example is taken to be 2.

When in useFalls in the set threshold valueWhen the acceleration is not significant, switching to a non-maneuvering model, and entering step 2; otherwise, the target is considered to be in a maneuvering state, and the step 8 is entered;

in the embodiment of the invention, when the significance level is 0.005, the degree of freedom is 2 corresponding to the threshold value according to the chi-square distribution table

Step 8, updating the state of the maneuvering mode

wherein,filter gain at time k;representing the state covariance of the predicted target at the k-1 moment at the k moment; (H)^m)^TRepresenting the transpose of the measurement matrix;is a covariance matrix of the prediction error.

The update of the target state is accomplished as follows:

wherein,representing the state update value at the target k moment;representing the state of the prediction target at the k moment at the k-1 moment; z_kRepresenting a target measurement value;the measured value of the predicted target at the time k-1 is shown.

The update of the target state covariance is done as follows:

wherein,representing the state covariance update value at the target k moment;is a transpose of the filter gain matrix.

And 6-7 repeating the steps.

The effect of the present invention will be further explained with reference to fig. 2.

1. Simulation conditions

Setting the initial state of the real track of the target as [10000m,100m/s,0m/s²]And carrying out 180-second sampling observation on the target, wherein the specific motion of the target is as follows:

within 1-30 s, the target does uniform linear motion, and the acceleration within 31-60 s is 30m/s²Reducing the acceleration to 0, wherein when the acceleration is 61-90 s, the target performs uniform acceleration linear motion, and the acceleration is increased from 0 to 30m/s in 91-120 s²And performing uniform linear motion within 121-180 s.

In a Cartesian coordinate system, 1000 Monte Carlo experiments are adopted, a radar sampling interval T is set to be 1s, and a measurement position variance R of a radar is set to be 2500m²。

The evaluation index of the simulation is root mean square error, namely RMSE, and the calculation formula is as follows:

wherein,andrespectively representing the true value and the estimated value of the j component of the k-th operation time state.

2. Emulated content

The method and the variable-dimension filtering algorithm are adopted to respectively track and estimate the position and the speed of the target, and the tracking effect is compared.

3. Simulation analysis

Fig. 2(a) shows the root mean square error of the position when the present invention and the variable dimension filter algorithm track a one-dimensional target, the solid line shows the tracking error curve of the present invention, and the dotted line shows the tracking error curve of the variable dimension filter algorithm. As can be seen from the figure, when the mobility of the target is weak, the position error level of the variable dimension filtering algorithm is similar to that of the variable dimension filtering algorithm; when the target mobility is strong, the invention can effectively reduce the position error (by about 40-50 percent relatively), and the filtering convergence speed is faster than that of the variable-dimension filtering algorithm.

FIG. 2(b) is the root mean square error of the speed when the invention and the variable dimension filtering algorithm track a one-dimensional target, and it can be seen from the figure that when the mobility of the target is weak, the speed error level of the invention and the variable dimension filtering algorithm is almost the same; when the target mobility is strong, the method keeps good speed tracking performance, and the variable-dimension filtering algorithm almost loses the speed tracking capability. Therefore, the invention greatly improves the tracking performance of the variable dimension filtering algorithm on the speed when the target maneuvers.

Claims

1. A maneuvering target tracking method based on a variable dimension filtering algorithm comprises the following steps:

step 1: establishing a target motion model

X_k＝FX_k-1+W_k-1

Z_k＝HX_k+V_k

wherein Z is_kRepresenting the measured value of the radar at the k moment; h and H^mMeasuring matrixes of a non-maneuvering model and a maneuvering model respectively; v_kFor measuring noise, the mean is 0 and the variance is R_kAnd independently of process noise, R_kA covariance matrix for the measured noise;

step 2: state prediction for non-motorized models

P_k|k-1＝FP_k-1|k-1F^T+Q

2d) the filtering innovation is determined using the following equation:

2e) determining a covariance matrix of the prediction error according to:

S_k＝HP_k|k-1H^T+R_k

and step 3: detecting whether target maneuver occurs

ρ_k＝μρ_k-1+ε_vk

<mrow> <msub> <mi>&epsiv;</mi> <msub> <mi>v</mi> <mi>k</mi> </msub> </msub> <mo>=</mo> <msubsup> <mi>v</mi> <mi>k</mi> <mo>&prime;</mo> </msubsup> <msubsup> <mi>S</mi> <mi>k</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>v</mi> <mi>k</mi> </msub> </mrow>

μ＝1-1/s

where ρ is_kAnd ρ_k-1Representing the attenuation memory average of the filtered innovation at time k and k-1 respectively,is made ofNormalizing the square of the filter innovation and obeying a chi-square distribution with a degree of freedom of 1/(1- μ); mu is a discount factor, s is a sliding window length, v'_kDenotes v_kTransposing;

3b) detecting the occurrence of maneuver according to the following formula:

P{ρ_k≤ρ_max}＝1-α

and 4, step 4: state update for non-motorized models

4b) the update of the target state is accomplished as follows:

4c) the update of the target state covariance is done as follows:

4d) repeating the steps 2-3;

and 5: initialization of maneuver modes

<mrow> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mi>s</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mi>s</mi> </mrow> <mi>m</mi> </msubsup> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mi>s</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mi>s</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mi>s</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mi>s</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mn>2</mn> <mrow> <msup> <mi>s</mi> <mn>2</mn> </msup> <msup> <mi>T</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>&lsqb;</mo> <msub> <mi>z</mi> <mi>k</mi> </msub> <mo>-</mo> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mi>s</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mi>s</mi> </mrow> </msub> <mo>-</mo> <mi>s</mi> <mi>T</mi> <msub> <mover> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mi>s</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mi>s</mi> </mrow> </msub> <mo>&rsqb;</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow>

step 6: state prediction for a mobility model

6d) the filtering innovation is determined using the following equation:

6e) determining a covariance matrix of the prediction error according to:

wherein,a covariance matrix that is a prediction error;state covariance representing predicted target at time k-1 at time kA difference; (H)^m)^TRepresenting the transpose of the measurement matrix; r_kA covariance matrix for the measured noise;

and 7: detecting if a maneuver has ended

7b) the statistics of non-maneuver detections are calculated as follows:

<mrow> <msub> <mi>&delta;</mi> <mi>k</mi> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msubsup> <mover> <mover> <mi>x</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> </mrow> <mi>m</mi> </msubsup> <mo>)</mo> </mrow> <mo>&prime;</mo> </msup> <msup> <mrow> <mo>&lsqb;</mo> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> </mrow> <mi>m</mi> </msubsup> <mrow> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo>)</mo> </mrow> <mo>&rsqb;</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msubsup> <mover> <mover> <mi>x</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> </mrow> <mi>m</mi> </msubsup> </mrow>

<mrow> <msubsup> <mi>&rho;</mi> <mi>k</mi> <mi>m</mi> </msubsup> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mi>k</mi> <mo>-</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>k</mi> </munderover> <msub> <mi>&delta;</mi> <mi>j</mi> </msub> </mrow>

and 8: state update for a maneuver mode

8b) the update of the target state is accomplished as follows:

8c) the update of the target state covariance is done as follows:

8d) and 6-7 repeating the steps.