CN102338870B - Method for tracking three-dimensional target by adopting forward scattering radar - Google Patents

Abstract

The invention relates to a method for tracking a three-dimensional target by adopting a forward scattering radar, which belongs to the technical field of target tracking, and comprises the following three steps: carrying out system model definitions including a definition for the geometric configuration of the forward scattering radar, a definition for a target model and a definition for an observation model; carrying out estimation on the initial state of the target; and by taking the initial state of the target as a filter initial value, carrying out recurrence filter by adopting an extended Kalman filter algorithm to realize the tracking on the three-dimensional motion target by adopting the forward scattering radar. In the method for tracking the three-dimensional target by adopting the forward scattering radar, the problems that a matrix singular calculation is easy to generate inaccuracy by Gauss-Newton iteration under a common observation noise background, and a local extremum is easy to involve because the evaluated error of the initial value is too big are solved; the filter initial value which is comparatively accurate can be obtained under the condition that the measurement accuracy is not high; the filter convergence rate is high; the accuracy is high; the data rate is high; and the calculated amount is low.

Description

A kind of objective tracking that adopts the forward scattering radar

Technical field

The present invention relates to a kind of objective tracking that adopts the forward scattering radar, belong to the target following technical field.

Background technology

The forward scattering radar utilizes forward scattering district (135 °～180 ° in the double-basis ditch) detection of a target of bistatic radar cross section long-pending (RCS).In this zone, bistatic RCS is generally than the big 20～40dB of single base RCS, thereby can realize the effective detection and tracking to low speed, little RCS target (comprising stealthy target) in the forward scattering zone.Because its anti-stealthy and anti-low-altitude penetration capability that has is utilized the forward scattering radar to carry out target following and has been received more concern in recent years.

Studies show that the application of continuous accurate harmonic signal in the forward scattering target detection has prospect most.Existing tracking and experimental system all adopt continuous accurate harmonic signal to survey.In continuous quasi-harmonic system, the coordinate estimation of objective is that Doppler shift, position angle and the elevation angle by measurement target obtains, relation between the kinematic parameter (position, speed) of they and target is non-linear, therefore the target following problem of forward scattering radar can be summed up as the nonlinear filtering problem, comprises that the original state of target is estimated and follow the tracks of to keep algorithm.The accurate original state of wanting to utilize existing tracking (classical way) to obtain target is estimated, just require radar system that very high angle-measurement accuracy and Doppler measurement precision must be arranged, yet in the system of reality, be subject to antenna size and system complexity, the measuring accuracy of orientation and luffing angle is also very limited, in this case the iterative initial value that obtains of classical way from actual value away from, especially very inaccurate in the velocity estimation of base direction, error can reach 2～3 times of magnitudes of base direction speed, causes following the tracks of the very slow even filtering divergence of maintenance stage filtering speed of convergence easily.Because classical way adopts Gauss-Newton iterative to carry out filtering, need utilize some groups of measurement vectors following the tracks of the maintenance stage, just can obtain the parameters of target motion in a moment through iteration repeatedly and estimate that calculated amount is very big, data transfer rate is lower.

Summary of the invention

The objective of the invention is to improve data transfer rate simultaneously, reduce calculated amount in order to utilize the forward scattering radar to realize the original state of three-dimensional motion target is estimated and convergence filtering fast, proposed a kind of objective tracking that adopts the forward scattering radar.

The objective of the invention is to be achieved through the following technical solutions.

A kind of objective tracking that adopts the forward scattering radar of the present invention, its step is as follows:

1) carry out the system model definition, comprise the definition of forward scattering radar geometric configuration, the definition of object module and the definition of observation model are respectively:

1.1 forward scattering radar geometric configuration is defined, set x, y, z are the rectangular coordinate of target, R _eFor receiving base, T _rBe firing base, the line segment between firing base and the reception base is called baseline, T _gPosition for any k moment target, ψ is that the angle between targetpath and the baseline also is flight-path angle, θ, β is respectively k and receives azimuth of target and the elevation angle that the base records constantly, h is object height, b is base length, and the real motion track of target setting is straight line AB, and target trajectory is gone up at surface level (xoy plane) and is projected as line segment CD within the radar coverage;

1.2 the objective definition model, target setting at the uniform velocity passes baseline along straight line AB with flight-path angle ψ, and the three-dimensional system state equation of transfer of following the tracks of is

X(k+1)＝ΦX(k)+Gv(k) (1)

Target setting is highly being made linear uniform motion in the surface level of h, and state vector can be expressed as X (k)=[x _ky _kH V _xV _y], x _k, y _k, h is the k three-dimensional rectangular coordinate of target constantly, V _x, V _yBe the x direction of this moment target and the speed component of y direction; Setting T is sampling interval, and v (k) is process noise and is the zero-mean white Gaussian noise that then state-transition matrix Φ and noise profile matrix G are respectively

$\mathrm{\&Phi;}=\left[\begin{array}{ccccc}1& 0& 0& \mathrm{<figure-callout\; id="0"\; label="T"\; filenames="HDA0000086028470000011.png,HDA0000086028470000012.png"\; state="\{\{state\}\}">T</figure-callout>}& 0\\ 0& 1& 0& 0& \mathrm{<figure-callout\; id="0"\; label="T"\; filenames="HDA0000086028470000011.png,HDA0000086028470000012.png"\; state="\{\{state\}\}">T</figure-callout>}\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]$ $G=\left[\begin{array}{ccc}\frac{{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="HDA0000086028470000012.png,HDA0000086028470000021.png"\; state="\{\{state\}\}">T</figure-callout>}}^2}{2}& 0& 0\\ 0& \frac{{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="HDA0000086028470000012.png,HDA0000086028470000021.png"\; state="\{\{state\}\}">T</figure-callout>}}^2}{2}& 0\\ 0& 0& \frac{{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="HDA0000086028470000012.png,HDA0000086028470000021.png"\; state="\{\{state\}\}">T</figure-callout>}}^2}{2}\\ \mathrm{<figure-callout\; id="0"\; label="T"\; filenames="HDA0000086028470000011.png,HDA0000086028470000012.png"\; state="\{\{state\}\}">T</figure-callout>}& 0& 0\\ 0& T& 0\end{array}\right]---\left(2\right)$

1.3 the definition observation model supposes that 1～k moment measurement vector can be designated as:

[(f _d1，θ ₁，β ₁)，L，(f _dk，θ _k，β _k)] ^T

Wherein, f _Dk, θ _k, β _kBe respectively the k observed reading at Doppler shift, echo bearing angle and the elevation angle constantly, then observation equation is:

Wherein

Be k moment measurement vector (f _Dk, θ _k, β _k), Be k moment state vector [x _ky _kH V _xV _y], observation noise

Be the zero-mean white Gaussian noise, noise variance is respectively

2) original state of target is estimated that concrete steps are:

2.1 the observation data that radar is obtained target adopts least square method to carry out match, sets in the original state estimation and adopts [(f _D1, θ ₁, β ₁), L, (f _DN, θ _N, β _N)] ^TN (N 〉=24) group observations calculates altogether, and this N group observations is divided into three groups, is respectively (f _D1, f _D2, L, f _DN), (θ ₁, θ ₂, L θ _N) and (β ₁, β ₂, L, β _N), and respectively with least square fitting to the single order polynomial expression, namely measured value is carried out smoothly observed reading (the f after obtaining smoothly _D1LS, f _D2LS, L, f _DNLS), (θ _1LS, θ _2LS, L, θ _NLS), (β _1LS, β _2LS, L, β _NLS);

2.2 get the observed reading of step 2.1 after level and smooth, even

f _d1，f _d2，L f _dN＝f _d1LS，f _d2LS，L，f _dNLS，θ ₁，θ ₂，L，θ _N＝θ _1LS，θ _2LS，L，θ _NLS

β ₁，β ₂，L β _N＝β _1LS，β _2LS，L，β _NLS

Carry out original state and estimate that concrete steps are:

(1＜n≤N) state vector of target is expressed as [x constantly to set n _n, y _n, h, V _x, V _y], provide x _n, V _x, V _yRelational expression between the three

$\left[\begin{array}{cc}{a}_1& a_3\\ b_1& b_3\end{array}\right]\left[\begin{array}{c}{x}_k\\ V_x\end{array}\right]=\left[\begin{array}{c}{-a}_2\\ {-b}_2\end{array}\right]V_y---\left(5\right)$

Coefficient a wherein ₁, a ₂, a ₃, b ₁, b ₂, b ₃Definite method be

$\left\{\begin{array}{cc}{a}_1=({\tan \mathrm{\&theta;}}_n-{\tan \mathrm{\&theta;}}_1)& b_1=({\tan \mathrm{\&theta;}}_n-{\tan \mathrm{\&theta;}}_2)\\ a_2=-(n-1)T& b_2=-(n-2)T\\ a_3={\tan \mathrm{\&theta;}}_1(n-1)T& b_3={\tan \mathrm{\&theta;}}_2(n-2)T\end{array}\right.---\left(6\right)$

The group of solving an equation (5) obtains x _n, V _x, y _nWith V _yBetween relational expression be respectively

$x_n=\frac{a_3{b}_2-a_2{b}_3}{a_1{b}_3-a_3{b}_1}\&CenterDot;V_y=d_{n1}{V}_y---\left(7\right)$

$V_x=\frac{a_2{b}_1-a_1{b}_2}{a_1{b}_3-a_3{b}_1}\&CenterDot;V_y=d_{n2}{V}_y---\left(8\right)$

y _n＝tanθ _kgd _n1V _y＝d _n3V _y (9)

Provide n dbjective state component h constantly, x _n, y _nBetween relational expression

$h={\tan \mathrm{\&beta;}}_n\sqrt{{x_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA0000086028470000012.png,HDA0000086028470000021.png"\; state="\{\{state\}\}">n</figure-callout>}}}^2+{y_n}^2}---\left(10\right)$

Formula (7) and (9) substitution (10) can be got h and V _yBetween relational expression

$h={\tan \mathrm{\&beta;}}_{\mathrm{<figure-callout\; id="1"\; label="n\; d\; n"\; filenames="HDA0000086028470000021.png,HDA0000086028470000031.png"\; state="\{\{state\}\}">n</figure-callout>}}\sqrt{d_n^{}}\sqrt{1_2^{}+{\mathrm{<figure-callout\; id="3"\; label="d\; n"\; filenames="HDA0000086028470000021.png,HDA0000086028470000022.png"\; state="\{\{state\}\}">d</figure-callout>}}_n^3}\left|V_y\right|={\mathrm{<figure-callout\; id="4"\; label="d\; n"\; filenames="HDA0000086028470000012.png,HDA0000086028470000022.png"\; state="\{\{state\}\}">d</figure-callout>}}_n\left|V_y\right|---\left(11\right)$

Provide f _DnWith V _yBetween relational expression

$f_{\mathrm{dn}}+\frac{1}{\mathrm{\&lambda;}}[\frac{(d_{n1}{\mathrm{<figure-callout\; id="2"\; label="d\; n"\; filenames="HDA0000086028470000012.png,HDA0000086028470000021.png"\; state="\{\{state\}\}">d</figure-callout>}}_n+d_{n3})V_y^2}{\sqrt{({\mathrm{<figure-callout\; id="1"\; label="d\; n"\; filenames="HDA0000086028470000021.png,HDA0000086028470000031.png"\; state="\{\{state\}\}">d</figure-callout>}}_n^1+{\mathrm{<figure-callout\; id="3"\; label="d\; n"\; filenames="HDA0000086028470000021.png,HDA0000086028470000022.png"\; state="\{\{state\}\}">d</figure-callout>}}_n^3+{\mathrm{<figure-callout\; id="4"\; label="d\; n"\; filenames="HDA0000086028470000012.png,HDA0000086028470000022.png"\; state="\{\{state\}\}">d</figure-callout>}}_n^4)V_{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HDA0000086028470000012.png,HDA0000086028470000021.png"\; state="\{\{state\}\}">y</figure-callout>}}^2}}+\frac{d_{n3}{V}_y^2-(b-d_{n1}{V}_y)d_{n2}{V}_y}{\sqrt{{(b-d_{n1}{V}_y)}^2+{\mathrm{<figure-callout\; id="3"\; label="d\; n"\; filenames="HDA0000086028470000021.png,HDA0000086028470000022.png"\; state="\{\{state\}\}">d</figure-callout>}}_n^3{\mathrm{<figure-callout\; id="2"\; label="V\; y"\; filenames="HDA0000086028470000012.png,HDA0000086028470000021.png"\; state="\{\{state\}\}">V</figure-callout>}}_y^{}+{\mathrm{<figure-callout\; id="4"\; label="d\; n"\; filenames="HDA0000086028470000012.png,HDA0000086028470000022.png"\; state="\{\{state\}\}">d</figure-callout>}}_n^4{V}_y^2}}]=0---\left(12\right)$

With d _N1, d _N2, d _N3, d _N4Substitution (12) obtains about V _yThe monobasic nonlinear equation, variable n in this monobasic nonlinear equation is got n=(N/2+1): N successively, calculate the d of each n correspondence _N1, d _N2, d _N3, d _N4, obtain following Nonlinear System of Equations

$\left\{\begin{array}{c}{\mathrm{<figure-callout\; id="2"\; label="f\; d"\; filenames="HDA0000086028470000012.png,HDA0000086028470000021.png"\; state="\{\{state\}\}">f</figure-callout>}}_d+\frac{1}{\mathrm{\&lambda;}}[\frac{(d_{21}{\mathrm{<figure-callout\; id="22"\; label="d"\; filenames="HDA0000086028470000022.png"\; state="\{\{state\}\}">d</figure-callout>}}_{22}+d_{23})V_y^2}{\sqrt{(d_{21}^2+d_{23}^2+{\mathrm{<figure-callout\; id="24"\; label="d"\; filenames="HDA0000086028470000022.png"\; state="\{\{state\}\}">d</figure-callout>}}_{24}^2)V_y^2}}+\frac{d_{23}{V}_y^2-(b-d_{21}{V}_y)d_{22}{V}_y}{\sqrt{{(b-d_{21}{V}_y)}^2+d_{23}^2{V}_y^2+{\mathrm{<figure-callout\; id="24"\; label="d"\; filenames="HDA0000086028470000022.png"\; state="\{\{state\}\}">d</figure-callout>}}_{24}^2{V}_y^2}}]=0\\ {\mathrm{<figure-callout\; id="3"\; label="f\; d"\; filenames="HDA0000086028470000021.png,HDA0000086028470000022.png"\; state="\{\{state\}\}">f</figure-callout>}}_d+\frac{1}{\mathrm{\&lambda;}}[\frac{(d_{31}{d}_{32}+d_{33})V_y^2}{\sqrt{(d_{31}^2+d_{33}^2+d_{34}^2)V_y^2}}+\frac{d_{33}{V}_y^2-(b-d_{31}{V}_y)d_{32}{V}_y}{\sqrt{{(b-d_{31}{V}_y)}^2+d_{33}^2{V}_y^2+d_{34}^2{V}_y^2}}]=0\\ M\\ f_{\mathrm{dN}}+\frac{1}{\mathrm{\&lambda;}}[\frac{({\mathrm{<figure-callout\; id="1"\; label="d\; N"\; filenames="HDA0000086028470000021.png,HDA0000086028470000031.png"\; state="\{\{state\}\}">d</figure-callout>}}_N+d_{N2}+d_{N3})V_y^2}{\sqrt{({\mathrm{<figure-callout\; id="1"\; label="d\; N"\; filenames="HDA0000086028470000021.png,HDA0000086028470000031.png"\; state="\{\{state\}\}">d</figure-callout>}}_N^1+d_{N3}^2+{\mathrm{<figure-callout\; id="4"\; label="d\; N"\; filenames="HDA0000086028470000012.png,HDA0000086028470000022.png"\; state="\{\{state\}\}">d</figure-callout>}}_N^4)V_y^2}}+\frac{d_{N3}{V}_y^2-(b-d_{N1}{V}_y)d_{N3}{V}_y}{\sqrt{{(b-d_{N1}{V}_y)}^2+d_{N3}^2{V}_y^2+d_{N4}^2{V}_y^2}}]=0\end{array}\right.---\left(13\right)$

Adopt Levenberg-Marquardt algorithm solution Nonlinear System of Equations (13), obtain V _yOptimum solution, according to this V _yOptimum solution, based on formula (7), (8), (9), (11), find the solution and obtain x _N, V _x, y _N, the state value of h also namely obtains n-hour dbjective state (x _N, y _N, h, V _x, V _y) numerical solution;

3) with step 2) the target original state that obtains also is n-hour dbjective state (x _N, y _N, h, V _x, V _y) numerical solution as the filtering initial value, adopt EKF (EKF) algorithm to carry out Recursive Filtering, obtain after N and the N each state estimation of target constantly, namely realized the tracking of forward scattering radar to the three-dimensional motion target.

Beneficial effect

The present invention compared with prior art, it is advantageous that: at first, in the target original state is estimated, obtain Nonlinear System of Equations based on analytic derivation, and adopt the Levenberg-Marquardt algorithm group of solving an equation, overcome Gauss-Newton iteration under the general observation noise background and occurred the problem that the inaccurate and initial value evaluated error of the unusual calculating of matrix is absorbed in local extremum too greatly easily easily, can under the not high situation of measuring accuracy, obtain more accurate filtering initial value; Secondly, utilize the accurate original state estimated value that obtains, combine with EKF (EKF) algorithm and carry out Recursive Filtering, filtering fast convergence rate and precision height, data transfer rate height, calculated amount are little.

Description of drawings

Fig. 1 is the geometry synoptic diagram of three-dimensional forward scattering radar among the present invention;

The location estimation square error that the initial value that Fig. 2 estimates for the target original state method of estimation that adopts in the embodiment of the invention and classical way carries out EKF filtering under the same conditions contrasts synoptic diagram;

The velocity estimation square error that the initial value that Fig. 3 estimates for the target original state method of estimation that adopts in the embodiment of the invention and classical way carries out EKF filtering under the same conditions contrasts synoptic diagram;

The emulation contrast effect figure that Fig. 4 carries out target following for the objective tracking that adopts in the embodiment of the invention and classical way, unwise Kalman filtering (UKF) algorithm;

Fig. 5 contrasts synoptic diagram for the base direction location estimation square error of the objective tracking that adopts in the embodiment of the invention and classical way;

Fig. 6 is the objective tracking that adopts in the embodiment of the invention and the base direction speed V of classical way _xEstimate square error contrast synoptic diagram;

Fig. 7 is the objective tracking that adopts in the embodiment of the invention and the vertical parallax direction speed V of classical way _yEstimate square error contrast synoptic diagram.

Embodiment

The present invention will be further described below in conjunction with drawings and Examples.

Embodiment

Adopt parameter in the bistatic forward scattering radar experimental system of people such as Russian A.B.Blyakhman development to carry out objective and follow the tracks of emulation.Getting operation wavelength is 0.77m, base length is 40km, and data transfer rate is 1Hz, and the Doppler shift standard deviation is 1, the measurement of azimuth standard deviation is 0.5 °, the measurement of elevation standard deviation is 0.5 °, and the process noise standard deviation is 1, and target velocity is 150m/s, flight-path angle is 15 °, the three-dimensional rectangular coordinate of target initial position is (20 ,-6,2) km.

A kind of objective tracking that adopts the forward scattering radar, its step is as follows:

A kind of objective tracking that adopts the forward scattering radar of the present invention, its step is as follows:

1) carry out the system model definition, comprise the definition of forward scattering radar geometric configuration, the definition of object module and the definition of observation model are respectively:

1.1 forward scattering radar geometric configuration is defined, as shown in Figure 1, set x, y, z are the rectangular coordinate of target, R _eFor receiving base, T _rBe firing base, the line segment between firing base and the reception base is called baseline, T _gPosition for any k moment target, ψ is the angle between targetpath and the baseline, it also is flight-path angle, θ, β is respectively k and receives azimuth of target and the elevation angle that the base records constantly, and h is object height, and b is base length, the real motion track of target setting is straight line AB, and target trajectory is gone up at surface level (xoy plane) and is projected as line segment CD within the radar coverage;

1.2 the objective definition model, target setting at the uniform velocity passes baseline along straight line AB with flight-path angle ψ, and the three-dimensional system state equation of transfer of following the tracks of is

X(k+1)＝ΦX(k)+Gv(k) (14)

Target setting is highly being made linear uniform motion in the surface level of h, and state vector can be expressed as X (k)=[x _ky _kH V _xV _y], x _k, y _k, h is the k three-dimensional rectangular coordinate of target constantly, V _x, V _yBe the x direction of this moment target and the speed component of y direction; Setting T is sampling interval, and v (k) is process noise, is the zero-mean white Gaussian noise, and then state-transition matrix Φ and noise profile matrix G are respectively

$\mathrm{\&Phi;}=\left[\begin{array}{ccccc}1& 0& 0& \mathrm{<figure-callout\; id="0"\; label="T"\; filenames="HDA0000086028470000011.png,HDA0000086028470000012.png"\; state="\{\{state\}\}">T</figure-callout>}& 0\\ 0& 1& 0& 0& \mathrm{<figure-callout\; id="0"\; label="T"\; filenames="HDA0000086028470000011.png,HDA0000086028470000012.png"\; state="\{\{state\}\}">T</figure-callout>}\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]$ $G=\left[\begin{array}{ccc}\frac{{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="HDA0000086028470000012.png,HDA0000086028470000021.png"\; state="\{\{state\}\}">T</figure-callout>}}^2}{2}& 0& 0\\ 0& \frac{{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="HDA0000086028470000012.png,HDA0000086028470000021.png"\; state="\{\{state\}\}">T</figure-callout>}}^2}{2}& 0\\ 0& 0& \frac{{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="HDA0000086028470000012.png,HDA0000086028470000021.png"\; state="\{\{state\}\}">T</figure-callout>}}^2}{2}\\ \mathrm{<figure-callout\; id="0"\; label="T"\; filenames="HDA0000086028470000011.png,HDA0000086028470000012.png"\; state="\{\{state\}\}">T</figure-callout>}& 0& 0\\ 0& T& 0\end{array}\right]---\left(15\right)$

1.3 the definition observation model supposes that 1:k moment measurement vector can be designated as:

[(f _d1，θ ₁，β ₁)，L，(f _dk，θ _k，β _k)] ^T

Wherein, f _Dk, θ _k, β _kBe respectively the k observed reading at Doppler shift, echo bearing angle and the elevation angle constantly, then observation equation is:

Wherein

Be k moment measurement vector (f _Dk, θ _k, β _k),

Be k moment state vector [x _ky _kH V _xV _y], observation noise

Be the zero-mean white Gaussian noise, noise variance is respectively

2) original state of target is estimated that concrete steps are:

2.1 the observation data that radar is obtained target adopts least square method to carry out match, sets in the original state estimation and adopts [(f _D1, θ ₁, β ₁), L, (f _DN, θ _N, β _N)] ^TN (N 〉=24) group observations calculates altogether, and this N group observations is divided into three groups, is respectively (f _D1, f _D2, L, f _DN), (θ ₁, θ ₂, L θ _N) and (β ₁, β ₂, L, β _N), and respectively with least square fitting to the single order polynomial expression, namely measured value is carried out smoothly observed reading (the f after obtaining smoothly _D1LS, f _D2LS, L, f _DNLS), (θ _1LS, θ _2LS, L, θ _NLS), (β _1LS, β _2LS, L, β _NLS);

2.2 get the observed reading of step 2.1 after level and smooth, even

f _d1，f _d2，L f _dN＝f _d1LS，f _d2LS，L，f _dNLS，θ ₁，θ ₂，L，θ _N＝θ _1LS，θ _2LS，L，θ _NLS

β ₁，β ₂，L β _N＝β _1LS，β _2LS，L，β _NLS

Carry out original state and estimate that concrete steps are:

(1＜n≤N) state vector of target is expressed as [x constantly to set n _n, y _n, h, V _x, V _y], because x _n, V _x, V _yRelation below satisfying

$\left[\begin{array}{cc}{a}_1& a_3\\ b_1& b_3\end{array}\right]\left[\begin{array}{c}{x}_k\\ V_x\end{array}\right]=\left[\begin{array}{c}{-a}_2\\ {-b}_2\end{array}\right]V_y---\left(18\right)$

Coefficient a wherein ₁, a ₂, a ₃, b ₁, b ₂, b ₃Definite method be the preceding n group observations after will be level and smooth

The substitution following formula

$\left\{\begin{array}{cc}{a}_1=({\tan \mathrm{\&theta;}}_n-{\tan \mathrm{\&theta;}}_1)& {\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="HDA0000086028470000021.png,HDA0000086028470000031.png"\; state="\{\{state\}\}">b</figure-callout>}}_1=({\tan \mathrm{\&theta;}}_n-{\tan \mathrm{\&theta;}}_2)\\ a_2=-(n-1)\mathrm{<figure-callout\; id="2"\; label="T\; b"\; filenames="HDA0000086028470000012.png,HDA0000086028470000021.png"\; state="\{\{state\}\}">T</figure-callout>}& b_{}{}_2=-(n-2)T\\ a_3={\tan \mathrm{\&theta;}}_1(n-1)\mathrm{<figure-callout\; id="3"\; label="T\; b"\; filenames="HDA0000086028470000021.png,HDA0000086028470000022.png"\; state="\{\{state\}\}">T</figure-callout>}& b_{}{}_3={\tan \mathrm{\&theta;}}_2(n-2)T\end{array}\right.---\left(19\right)$

The group of solving an equation (5) obtains x _n, V _x, y _nWith V _yBetween relational expression be

$x_n=\frac{a_3{b}_2-a_2{b}_3}{a_1{b}_3-a_3{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="HDA0000086028470000021.png,HDA0000086028470000031.png"\; state="\{\{state\}\}">b</figure-callout>}}_1}\&CenterDot;V_y=d_{n1}{V}_y---\left(20\right)$

$V_x=\frac{a_2{b}_1-a_1{b}_2}{a_1{b}_3-a_3{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="HDA0000086028470000021.png,HDA0000086028470000031.png"\; state="\{\{state\}\}">b</figure-callout>}}_1}\&CenterDot;V_y=d_{n2}{V}_y---\left(21\right)$

y _n＝tanθ _kgd _n1V _y＝d _n3V _y (22)

According to n moment dbjective state component h, x _n, y _nRelation below satisfying

$h={\tan \mathrm{\&beta;}}_{\mathrm{<figure-callout\; id="2"\; label="n\; x\; n"\; filenames="HDA0000086028470000012.png,HDA0000086028470000021.png"\; state="\{\{state\}\}">n</figure-callout>}}\sqrt{{x_n}^{}}\sqrt{{{}_{}}^2+{y_n}^2}---\left(23\right)$

Formula (7) and (9) substitution (10) can be got h and V _yBetween relational expression

$h={\tan \mathrm{\&beta;}}_{\mathrm{<figure-callout\; id="1"\; label="n\; d\; n"\; filenames="HDA0000086028470000021.png,HDA0000086028470000031.png"\; state="\{\{state\}\}">n</figure-callout>}}\sqrt{d_n^{}}\sqrt{1_2^{}+{\mathrm{<figure-callout\; id="3"\; label="d\; n"\; filenames="HDA0000086028470000021.png,HDA0000086028470000022.png"\; state="\{\{state\}\}">d</figure-callout>}}_n^3}\left|V_y\right|=d_{\mathrm{<figure-callout\; id="4"\; label="n"\; filenames="HDA0000086028470000012.png,HDA0000086028470000022.png"\; state="\{\{state\}\}">n</figure-callout>}4}\left|V_y\right|---\left(24\right)$

According to f _DnWith V _yRelation below satisfying

$f_{\mathrm{dn}}+\frac{1}{\mathrm{\&lambda;}}[\frac{(d_{n1}{\mathrm{<figure-callout\; id="2"\; label="d\; n"\; filenames="HDA0000086028470000012.png,HDA0000086028470000021.png"\; state="\{\{state\}\}">d</figure-callout>}}_n+d_{n3})V_y^2}{\sqrt{({\mathrm{<figure-callout\; id="1"\; label="d\; n"\; filenames="HDA0000086028470000021.png,HDA0000086028470000031.png"\; state="\{\{state\}\}">d</figure-callout>}}_n^1+{\mathrm{<figure-callout\; id="3"\; label="d\; n"\; filenames="HDA0000086028470000021.png,HDA0000086028470000022.png"\; state="\{\{state\}\}">d</figure-callout>}}_n^3+{\mathrm{<figure-callout\; id="4"\; label="d\; n"\; filenames="HDA0000086028470000012.png,HDA0000086028470000022.png"\; state="\{\{state\}\}">d</figure-callout>}}_n^4){\mathrm{<figure-callout\; id="2"\; label="V\; y"\; filenames="HDA0000086028470000012.png,HDA0000086028470000021.png"\; state="\{\{state\}\}">V</figure-callout>}}_y^{}}}+\frac{d_{n3}{V}_y^2-(b-d_{n1}{V}_y)d_{n2}{V}_y}{\sqrt{{(b-d_{n1}{V}_y)}^2+{\mathrm{<figure-callout\; id="3"\; label="d\; n"\; filenames="HDA0000086028470000021.png,HDA0000086028470000022.png"\; state="\{\{state\}\}">d</figure-callout>}}_n^3{\mathrm{<figure-callout\; id="2"\; label="V\; y"\; filenames="HDA0000086028470000012.png,HDA0000086028470000021.png"\; state="\{\{state\}\}">V</figure-callout>}}_y^{}+{\mathrm{<figure-callout\; id="4"\; label="d\; n"\; filenames="HDA0000086028470000012.png,HDA0000086028470000022.png"\; state="\{\{state\}\}">d</figure-callout>}}_n^4{V}_y^2}}]=0---\left(25\right)$

With d _N1, d _N2, d _N3, d _N4Substitution (12) obtains about V _yThe monobasic nonlinear equation, variable n in this monobasic nonlinear equation is got n=(N/2+1): N successively, calculate the d of each n correspondence _N1, d _N2, d _N3, d _N4, obtain following Nonlinear System of Equations

$\left\{\begin{array}{c}{f}_{d2}+\frac{1}{\mathrm{\&lambda;}}[\frac{(d_{21}{d}_{22}+d_{23})V_y^2}{\sqrt{(d_{21}^2+d_{23}^2+d_{24}^2)V_y^2}}+\frac{d_{23}{V}_y^2-(b-d_{21}{V}_y)d_{22}{V}_y}{\sqrt{{(b-d_{21}{V}_y)}^2+d_{23}^2{V}_y^2+d_{24}^2{V}_y^2}}]=0\\ f_{d3}+\frac{1}{\mathrm{\&lambda;}}[\frac{(d_{31}{d}_{32}+d_{33})V_y^2}{\sqrt{(d_{31}^2+d_{33}^2+d_{34}^2)V_y^2}}+\frac{d_{33}{V}_y^2-(b-d_{31}{V}_y)d_{32}{V}_y}{\sqrt{{(b-d_{31}{V}_y)}^2+d_{33}^2{V}_y^2+d_{34}^2{V}_y^2}}]=0\\ M\\ f_{\mathrm{dN}}+\frac{1}{\mathrm{\&lambda;}}[\frac{(d_{N1}+d_{N2}+d_{N3})V_y^2}{\sqrt{(d_{N1}^2+d_{N3}^2+d_{N4}^2)V_y^2}}+\frac{d_{N3}{V}_y^2-(b-d_{N1}{V}_y)d_{N3}{V}_y}{\sqrt{{(b-d_{N1}{V}_y)}^2+d_{N3}^2{V}_y^2+d_{N4}^2{V}_y^2}}]=0\end{array}\right.---\left(26\right)$

Adopt Levenberg-Marquardt algorithm solution Nonlinear System of Equations (13), obtain V _yOptimum solution;

According to above-mentioned V _yOptimum solution, based on formula (7), (8), (9), (11), find the solution and obtain x _N, V _x, V _y, the state value of h also namely obtains n-hour dbjective state (x _N, y _N, h, V _x, V _y) numerical solution;

X in the step 2.2 _n, V _x, V _yThe relation that satisfies, h, x _n, y _nThe relation that satisfies, and f _DnWith V _yThe relation derivation process that satisfies is as follows:

According to the systematic observation equation, for n observed reading (f constantly _Dn, θ _n, β _n), have

$f_{\mathrm{dn}}=-\frac{1}{\mathrm{\&lambda;}}[\frac{x_n{V}_x+y_n{V}_{\mathrm{<figure-callout\; id="2"\; label="y\; x\; n"\; filenames="HDA0000086028470000012.png,HDA0000086028470000021.png"\; state="\{\{state\}\}">y</figure-callout>}}}{\sqrt{x_n^{}}}+\frac{y_n{V}_y-(b-x_n)V_x}{\sqrt{{(b-x_n)}^2+{\mathrm{<figure-callout\; id="2"\; label="y\; n"\; filenames="HDA0000086028470000012.png,HDA0000086028470000021.png"\; state="\{\{state\}\}">y</figure-callout>}}_n^{}+h^2}}]---\left(27\right)$

${\mathrm{\&theta;}}_n=\arctan \frac{y_n}{x_n}---\left(28\right)$

${\mathrm{\&beta;}}_n=\arctan \frac{h}{\sqrt{{{\mathrm{<figure-callout\; id="2"\; label="x\; n"\; filenames="HDA0000086028470000012.png,HDA0000086028470000021.png"\; state="\{\{state\}\}">x</figure-callout>}}_n}^2+{y_n}^2}}---\left(29\right)$

Can be got by equation (28)

$\frac{y_1}{x_1}={\tan \mathrm{\&theta;}}_{1}L\frac{y_n}{x_n}={\tan \mathrm{\&theta;}}_n---\left(30\right)$

By target movement model y as can be known ₁With y _nBetween have relation

$\left\{\begin{array}{c}{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="HDA0000086028470000021.png,HDA0000086028470000031.png"\; state="\{\{state\}\}">y</figure-callout>}}_1=y_n-(n-1){\mathrm{TV}}_y\\ x_1=x_n-(n-1){\mathrm{TV}}_x\end{array}\right.---\left(31\right)$

Formula (31) substitution (30) can be obtained

$\frac{y_n-(k-1){\mathrm{TV}}_y}{x_n-(k-1){\mathrm{TV}}_x}={\tan \mathrm{\&theta;}}_1---\left(32\right)$

In conjunction with (30) and (32), have simultaneously

(tanθ _n-tanθ ₁)x _n-(n-1)TV _y+tanθ ₁(n-1)TV _x＝0(33)

In like manner can get

(tanθ _n-tanθ ₂)x _n-(n-2)TV _y+tanθ ₂(n-2)TV _x＝0(34)

Following system of equations can be listed in simultaneous (33) and (34)

Make x in the system of equations (35) _n, V _y, V _xCoefficient be respectively (a ₁, a ₂, a ₃), (b ₁, b ₂, b ₃), namely

$\left\{\begin{array}{cc}{a}_1=({\tan \mathrm{\&theta;}}_n-{\tan \mathrm{\&theta;}}_1)& {\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="HDA0000086028470000021.png,HDA0000086028470000031.png"\; state="\{\{state\}\}">b</figure-callout>}}_1=({\tan \mathrm{\&theta;}}_n-{\tan \mathrm{\&theta;}}_2)\\ a_2=-(n-1)\mathrm{<figure-callout\; id="2"\; label="T\; b"\; filenames="HDA0000086028470000012.png,HDA0000086028470000021.png"\; state="\{\{state\}\}">T</figure-callout>}& b_{}{}_2=-(n-2)T\\ a_3={\tan \mathrm{\&theta;}}_1(n-1)\mathrm{<figure-callout\; id="3"\; label="T\; b"\; filenames="HDA0000086028470000021.png,HDA0000086028470000022.png"\; state="\{\{state\}\}">T</figure-callout>}& b_{}{}_3={\tan \mathrm{\&theta;}}_2(n-2)T\end{array}\right.---\left(36\right)$

Then system of equations (35) can be write as following form

$\left[\begin{array}{cc}{a}_1& a_3\\ b_1& b_3\end{array}\right]\left[\begin{array}{c}{x}_k\\ V_x\end{array}\right]=\left[\begin{array}{c}{-a}_2\\ {-b}_2\end{array}\right]V_y---\left(37\right)$

Be x _n, V _x, V _yThe relation that satisfies;

Solve an equation (37) can get

$x_n=\frac{a_3{b}_2-a_2{b}_3}{a_1{b}_3-a_3{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="HDA0000086028470000021.png,HDA0000086028470000031.png"\; state="\{\{state\}\}">b</figure-callout>}}_1}\&CenterDot;V_y=d_{n1}{V}_y---\left(38\right)$

$V_x=\frac{a_2{b}_1-a_1{b}_2}{a_1{b}_3-a_3{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="HDA0000086028470000021.png,HDA0000086028470000031.png"\; state="\{\{state\}\}">b</figure-callout>}}_1}\&CenterDot;V_y=d_{n2}{V}_y---\left(39\right)$

y _n＝tanθ _kgd _n1V _y＝d _n3V _y (40)

According to equation (29)

$h={\tan \mathrm{\&beta;}}_n\mathrm{<figure-callout\; id="2"\; label="g\; x\; n"\; filenames="HDA0000086028470000012.png,HDA0000086028470000021.png"\; state="\{\{state\}\}">g</figure-callout>}\sqrt{{x_n}^{}}\sqrt{{{}_{}}^2+{y_n}^2}---\left(41\right)$

Be h, x _n, y _nThe relation that satisfies;

With (38), (40) substitution (41) can get

$h={\tan \mathrm{\&beta;}}_{\mathrm{<figure-callout\; id="1"\; label="n\; d\; n"\; filenames="HDA0000086028470000021.png,HDA0000086028470000031.png"\; state="\{\{state\}\}">n</figure-callout>}}\sqrt{d_n^{}}\sqrt{1_2^{}+{\mathrm{<figure-callout\; id="3"\; label="d\; n"\; filenames="HDA0000086028470000021.png,HDA0000086028470000022.png"\; state="\{\{state\}\}">d</figure-callout>}}_n^3}\left|V_y\right|=d_{\mathrm{<figure-callout\; id="4"\; label="n"\; filenames="HDA0000086028470000012.png,HDA0000086028470000022.png"\; state="\{\{state\}\}">n</figure-callout>}4}\left|V_y\right|---\left(42\right)$

With formula (38), (39), (40), (42) substitution equation (27) obtains about V _yThe monobasic nonlinear equation

$f_{\mathrm{dn}}+\frac{1}{\mathrm{\&lambda;}}[\frac{(d_{n1}{\mathrm{<figure-callout\; id="2"\; label="d\; n"\; filenames="HDA0000086028470000012.png,HDA0000086028470000021.png"\; state="\{\{state\}\}">d</figure-callout>}}_n+d_{n3})V_y^2}{\sqrt{({\mathrm{<figure-callout\; id="1"\; label="d\; n"\; filenames="HDA0000086028470000021.png,HDA0000086028470000031.png"\; state="\{\{state\}\}">d</figure-callout>}}_n^1+{\mathrm{<figure-callout\; id="3"\; label="d\; n"\; filenames="HDA0000086028470000021.png,HDA0000086028470000022.png"\; state="\{\{state\}\}">d</figure-callout>}}_n^3+{\mathrm{<figure-callout\; id="4"\; label="d\; n"\; filenames="HDA0000086028470000012.png,HDA0000086028470000022.png"\; state="\{\{state\}\}">d</figure-callout>}}_n^4){\mathrm{<figure-callout\; id="2"\; label="V\; y"\; filenames="HDA0000086028470000012.png,HDA0000086028470000021.png"\; state="\{\{state\}\}">V</figure-callout>}}_y^{}}}+\frac{d_{n3}{V}_y^2-(b-d_{n1}{V}_y)d_{n2}{V}_y}{\sqrt{{(b-d_{n1}{V}_y)}^2+{\mathrm{<figure-callout\; id="3"\; label="d\; n"\; filenames="HDA0000086028470000021.png,HDA0000086028470000022.png"\; state="\{\{state\}\}">d</figure-callout>}}_n^3{\mathrm{<figure-callout\; id="2"\; label="V\; y"\; filenames="HDA0000086028470000012.png,HDA0000086028470000021.png"\; state="\{\{state\}\}">V</figure-callout>}}_y^{}+{\mathrm{<figure-callout\; id="4"\; label="d\; n"\; filenames="HDA0000086028470000012.png,HDA0000086028470000022.png"\; state="\{\{state\}\}">d</figure-callout>}}_n^4{V}_y^2}}]=0---\left(43\right)$

Be f _DnWith V _yRelation;

3) with step 2) the target original state that obtains also is n-hour dbjective state (x _N, y _N, h, V _x, V _y) numerical solution adopt EKF (EKF) algorithm to carry out Recursive Filtering as the filtering initial value, obtain 〉=each state estimation of target constantly of N, namely realized the tracking of forward scattering radar to the three-dimensional motion target.

Simulation comparison experiment target original state estimated accuracy is to influence such as Fig. 2, shown in Figure 3 of filtering speed of convergence and precision, the target original state that adopts classical way and method of the present invention to estimate is respectively carried out EKF filtering as initial value, finishes 100 Monte Carlo simulations and is averaged; Among Fig. 2, solid line is to adopt original state estimated value that classical way obtains as the filtering initial value, carries out X (baseline) the direction location estimation square error curve of EKF filtering then; Dotted line is to adopt original state estimated value that new method obtains as the filtering initial value, carries out X (baseline) the direction location estimation square error curve of EKF filtering; By two curve contrasts as can be known, initial value method of estimation in this paper can obtain more accurate filtering initial value under identical measuring accuracy, thereby the EKF algorithm is restrained fast, and tracking accuracy is also than higher; And the initial value that classical way is estimated makes that the speed of convergence of EKF filtering is very slow, passes through baseline in target and does not constantly also converge in the final tracking accuracy scope; The speed V that the initial value that " new method base direction speed " in Fig. 3 legend and " new method vertical parallax direction speed " are represented respectively to adopt method of the present invention to obtain carries out EKF filtering _x, V _yThe speed V that the initial value that evaluated error curve, " classical way base direction speed " and " classical way vertical parallax direction speed " are represented respectively to adopt classical way to obtain carries out EKF filtering _x, V _yThe evaluated error curve, four curves can be found in the comparison diagram 3, when adopting original state that classical way estimates as EKF filtering initial value, the speed estimation error of directions X is very big, causes that the filtering convergence is slow, precision is low; During initial value substitution EKF filtering that new method in this paper is estimated, V _xAnd V _ySpeed estimation error is all very little, so the filtering convergence is fast, and position estimation accuracy is than higher; The performance of the different trackings of simulation comparison experiment more as shown in Figure 4 to 7, be (0.5Hz in observation noise, 0.5 °, 0.5 °) time, the original state evaluated error that classical way obtains is bigger, and in the filtering of back, significantly do not reduce the initial estimation error, and the filtering performance instability, filtering accuracy is relatively poor; The original state evaluated error less (the especially position of base direction and speed) that the tracking of this paper invention obtains, in the filtering of back, can effectively reduce error and convergence, UKF algorithm filtering accuracy is also than higher, but performance is not as good as method of the present invention, and filtering error has the trend of increase; The calculated amount that emulation experiment contrasts different trackings is as follows, carry out 100 Monte Carlo simulations and be averaged, the average operating time that three kinds of methods are finished a target following is respectively: classical way needs 8.564927s, method of the present invention needs 0.14912884s, and the UKF algorithm needs 0.32414323s.

The The above results explanation, in the target following of forward scattering radar, adopt technical scheme provided by the invention, tolerance for measuring error is bigger, and under the general measure precision, the original state evaluated error that this method obtains is than the little order of magnitude of evaluated error of classical way; The original state that adopts this method to estimate is carried out EKF (EKF) as the filtering initial value, can effectively improve speed of convergence and the estimated accuracy of filtering algorithm, thereby can before target is passed through baseline, form stable flight path and estimate that calculated amount is little, the data transfer rate height; Can satisfy the application requirements that the forward scattering radar is followed the tracks of three-dimensional uniform motion target.

The above is preferred embodiment of the present invention, and the present invention should not be confined to the disclosed content of this embodiment and accompanying drawing.Everyly do not break away from the equivalence of finishing under the spirit disclosed in this invention or revise, all fall into the scope of protection of the invention.

Claims (1)

1. objective tracking that adopts the forward scattering radar is characterized in that step is as follows:

1) carry out the system model definition, comprise the definition of forward scattering radar geometric configuration, the definition of object module and the definition of observation model are respectively:

1.1 forward scattering radar geometric configuration is defined, set x, y, z are the rectangular coordinate of target, R _eFor receiving base, T _rBe firing base, the line segment between firing base and the reception base is called baseline, T _gPosition for any k moment target, ψ is that the angle between targetpath and the baseline also is flight-path angle, θ, β is respectively k and receives azimuth of target and the elevation angle that the base records constantly, h is object height, b is base length, and the real motion track of target setting is straight line AB, and target trajectory also is to be projected as line segment CD within the radar coverage on the xoy plane at surface level;

1.2 the objective definition model, target setting at the uniform velocity passes baseline along straight line AB with flight-path angle ψ, and the three-dimensional system state equation of transfer of following the tracks of is

X(k+1)＝ΦX(k)+Gv(k) (1)

Target setting is highly being made linear uniform motion in the surface level of h, and state vector can be expressed as X (k)=[x _ky _kH V _xV _y], x _k, y _k, h is the k three-dimensional rectangular coordinate of target constantly, V _x, V _yBe the x direction of this moment target and the speed component of y direction; Setting T is sampling interval, and v (k) is process noise and is the zero-mean white Gaussian noise that then state-transition matrix Ф and noise profile matrix G are respectively

，

1.3 the definition observation model supposes that 1～k moment measurement vector can be designated as:

[(f _d1，θ ₁，β ₁)，...，(f _dk，θ _k，β _k)] ^T

Wherein, f _Dk, θ _k, β _kBe respectively the k observed reading at Doppler shift, echo bearing angle and the elevation angle constantly, then observation equation is:

Wherein

Be k moment measurement vector (f _Dk, θ _k, β _k), Be k moment state vector [x _ky _kH V _xV _y], observation noise

Be the zero-mean white Gaussian noise, noise variance is respectively

2) original state of target is estimated that concrete steps are:

2.1 the observation data that radar is obtained target adopts least square method to carry out match, sets in the original state estimation and adopts [(f _D1, θ ₁, β ₁) ..., (f _DN, θ _N, β _N)] ^TThe N group observations calculates altogether, and this N group observations is divided into three groups, is respectively (f _D1, f _D2..., f _DN), (θ ₁, θ ₂... θ _N) and (β ₁, β ₂..., β _N), and respectively with least square fitting to the single order polynomial expression, namely measured value is carried out smoothly observed reading (the f after obtaining smoothly _D1LS, f _D2LS..., f _DNLS), (θ _1LS, θ _2LS..., θ _NLS), (β _1LS, β _2LS..., β _NLS);

2.2 get the observed reading of step 2.1 after level and smooth, even

f _d1，f _d2，... f _dN＝f _d1LS，f _d2LS，...，f _dNLS，θ ₁，θ ₂，...，θ _N＝θ _1LS，θ _2LS，...，θ _NLS，β ₁，β ₂，... β _N＝β _1LS，β _2LS，...，β _NLS

Carry out original state and estimate that concrete steps are:

(1＜n≤N) state vector of target is expressed as [x constantly to set n _n, y _n, h, V _x, V _y], provide x _n, V _x, V _yTriadic relation's formula

Coefficient a wherein ₁, a ₂, a ₃, b ₁, b ₂, b ₃Definite method be

The group of solving an equation (5) obtains x _n, V _x, y _nWith V _yBetween relational expression be respectively

y _n＝tanθ _k·d _n1V _y＝d _n3V _y (9)

Provide n dbjective state component h constantly, x _n, y _nRelational expression

Formula (7) and (9) substitution (10) can be got h and V _yBetween relational expression

Provide f _DnWith V _yBetween relational expression

With d _N1, d _N2, d _N3, d _N4Substitution (12) obtains about V _yThe monobasic nonlinear equation, variable n in this monobasic nonlinear equation is got n=(N/2+1): N successively, calculate the d of each n correspondence _N1, d _N2, d _N3, d _N4, obtain following Nonlinear System of Equations

Adopt Levenberg-Marquardt algorithm solution Nonlinear System of Equations (13), obtain V _yOptimum solution, according to this V _yOptimum solution, based on formula (7), (8), (9), (11), find the solution and obtain x _N, V _x, y _N, the state value of h also namely obtains n-hour dbjective state [x _N, y _N, h, V _x, V _y] numerical solution;

3) with step 2) the target original state that obtains also is n-hour dbjective state [x _N, y _N, h, V _x, V _y] numerical solution as the filtering initial value, adopt expanded Kalman filtration algorithm to carry out Recursive Filtering, obtain after N and the N each state estimation of target constantly, namely realized the tracking of forward scattering radar to the three-dimensional motion target.

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