CN106227697B - A kind of non-linear space target state based on manifold determines method - Google Patents

A kind of non-linear space target state based on manifold determines method Download PDF

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CN106227697B
CN106227697B CN201610541665.2A CN201610541665A CN106227697B CN 106227697 B CN106227697 B CN 106227697B CN 201610541665 A CN201610541665 A CN 201610541665A CN 106227697 B CN106227697 B CN 106227697B
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刘吉英
段晓君
朱炬波
晏良
彭利军
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National University of Defense Technology
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Abstract

The present invention provides a kind of non-linear space target states based on manifold to determine method, the described method includes: establishing the observational equation of extraterrestrial target track, it is embedded in the physical characteristic of low dimensional manifold in space based on target trajectory, using manifold as fundamental space, using the multiscale analysis method in manifold, dbjective state parameter is subjected to multi-scale Representation in manifold, obtains multi-scale Representation coefficient；Based on observational equation and multi-scale Representation formula, manifold multi-scale Representation coefficient is obtained using nonlinear iteration method；To obtain the estimated value of dbjective state parameter.Method of the invention can simplify the solution process of dbjective state parameter, improve calculation accuracy；And method of the invention has versatility, is suitable for space industry, under a variety of Instrumentation systems, the high-precision estimation problem of the motion state of a variety of extraterrestrial targets.

Description

A kind of non-linear space target state based on manifold determines method
Technical field
The invention belongs to the high-precision estimation problems of extraterrestrial target state, are related to a kind of non-linear space mesh based on manifold Mark motion state determines method, that is, utilizes the distance to extraterrestrial target, azimuth, pitch angle or distance (and/poor) change rate Deng measurement element information, by the track solution process in manifold, the high-precision estimation of target position, speed is obtained, is suitable for The technical fields such as high-precision post-processing of measurement data in spaceflight TT&C system.
Background technique
In traditional aircraft measurement data high accuracy data processing, using spline restraint calculation method, i.e., by flight rail Mark is indicated with spline function, converts the estimation to spline coefficients for the direct estimation of position and speed, to utilize batten letter Several slickness improves calculation accuracy.But in this method in use, the selection of spline function node always exists difficulty, need Take considerable time the optimization for carrying out node.
Manifold indicates that the hypothesis based on " local linear, global non-linear " carries out mathematical modeling, it is intended to from observation space Inherent regularity and overall structure are found in data set distribution.How the geometrical property priori of trajectory range curve is made full use of Information, and the feature being embedded in low dimensional manifold are realized in low dimensional manifold in conjunction with metamessage and Model representation approach is surveyed Track is estimated, further saves required parameter when resolving, challenge existing for traditional method is avoided, such as batten section Point selection etc..
According to the literature, the research in terms of manifold both at home and abroad is mainly used for extensive, high dimensional data dimension and about subtracts, And the object studied in extraterrestrial target data processing problem is the estimation of the position and speed of target, research for this problem is still It is not reported.
Summary of the invention
It is an object of the invention to overcome spline function section present in current extraterrestrial target state high-precision estimation method The problems such as selection of point is difficult, computationally intensive；The characteristics of being substantially the one-dimensional manifold in three-dimensional space using aerial vehicle trajectory, Non-linear estimations theory and method in Differential Manifold is introduced, a kind of nonlinear motion state determination side based on manifold is provided Method, this method utilizes the curve manifold characteristic of space tracking in space, using the track representation method based on manifold, to tradition Batten representation method improve, while improving the calculation accuracy of track, it is complicated to avoid Spline Node optimization etc. Process；Traditional batten representation method can be improved to the track representation method in manifold, so that further saving indicates ginseng The complexity that number improves precision of state estimation, and avoids node optimization problem, reduces solution process.
To achieve the goals above, the present invention provides a kind of non-linear space target states based on manifold to determine Method, which comprises
The observational equation for establishing extraterrestrial target track, the physics for being embedded in low dimensional manifold in space based on target trajectory are special Property, using manifold as fundamental space, using the multiscale analysis method in manifold, dbjective state parameter is carried out in manifold more Scale indicates, obtains multi-scale Representation coefficient；Based on observational equation and multi-scale Representation formula, obtained using nonlinear iteration method Manifold multi-scale Representation coefficient；To obtain the estimated value of dbjective state parameter.
In above-mentioned technical proposal, the method is specifically included:
Step 1) establishes the observational equation of extraterrestrial target track:
Y (t)=H (X (ti)), i=1 ..., N (1)
Wherein, Y (t) be extraterrestrial target track；X(ti) it is dbjective state parameter to be estimated；H is observational equation；Its InDbjective state parameter i.e. to be estimated is the position in three-dimensional space It sets and speed；
Step 2) indicates the multi-scale transform operator F () in dbjective state parameter manifold to be estimated:
Wherein X (t1),…,X(tN) be N number of moment target position and speed parameter, C1,…,CMFor multi-scale Representation Coefficient, M < < N, that is, complete the rarefaction representation of dbjective state parameter；
Step 3) establishes partial derivative matrix of the metrical information relative to dbjective state parameter, while establishing dbjective state parameter Relative to the partial derivative matrix of multi-scale Representation coefficient, two matrix multiples are finally obtained into metrical information about multi-scale Representation system Several compound partial derivative matrixs；Using nonlinear iteration method, the estimated value C of multi-scale Representation coefficient is obtained1,…,CM；
Step 3) is calculated multi-scale Representation coefficient C by step 4)1,…,CMIt substitutes into formula (2) and calculates estimating for dbjective state Evaluation: X(k)(t1),…,X(k)(tN)。
In above-mentioned technical proposal, the construction process of the multi-scale transform operator F () in the step 2) is specifically included:
Step 201) sets initial gauges as j0A scale, j0Selection pass through to ti, i=1 ..., N are down-sampled to be obtained, and is enabled tj,k=k/2j,j≥0,k∈Z+,βj,k=F (tj,k) it is value of the F on each sampled point of current scale；
Step 202) refines sampled point by dichotomy, calculates the thick scale sampling of low-densityAnd small echo Coefficient ((αj,k)k∈Z)j≥0；
It enables:
It has thus obtained 2 in next scalej+1On even number point value；
For the numerical value on odd point, interpolation is carried out with D rank multinomial, for point 2k+1, D+1 are chosen around the point Point, specifically chooses process are as follows: if D+1 is even number, successively chooses to the left on the left side point 2k+1A, the right is to the right Successively chooseIt is a, if D+1 is odd number, successively chosen to the left on the left side point 2k+1A, the right is to the right Successively chooseA point；Multinomial π is obtained with polynomial interopolation to D+1 point of selectionj,k(t), multinomial meets:
And then it obtains
In βj,k=F (k/2j), the interpolation of odd pointOn the basis of, define " small echo " system Number are as follows: the difference between " predicted value " on the odd point that interpolation obtains and the true value on odd point:
I.e.
Step 203) is sampled by the thick scale of low-densityWith wavelet coefficient ((αj,k)k∈Z)j≥0Reconstruct arbitrarily connects Continuous function F, i.e.,
Wavelet coefficient ((αj,k)k∈Z)j≥0Numerical values recited have the characteristics that exponential damping, that is, there is constant C and R so that
In above-mentioned technical proposal, the value of the β in the step 202) is replaced with the average value on section；I.e.
βj,k=Ave F | Ij,k, j, k ∈ Z,
Here Ij,k=[k/2j,(k+1)/2j), and have βj,k=(βj+1,2k+βj+1,2k+1)/2。
In above-mentioned technical proposal, the step 3) is specifically included:
Step 301) calculates the difference of target trajectory observation and calculated value according to the initial value of dbjective state parameter；
Assuming that dbjective state initial parameter values are X(0)(t1),…,X(0)(tN), multi-scale Representation coefficient isThen The calculated value of target trajectory is obtained according to (1) formula are as follows:
The observation of target trajectory is Y(0)(t1),...,Y(0)(tN)；
The observation of target trajectory and the difference of calculated value are as follows:
Step 302) calculates the correction value of multi-scale Representation coefficient according to the observation of target trajectory and the difference of calculated value； Enable k=1；
It can be obtained according to (1) and (2):
Take single order Taylor approximate in above formula, then the correction value of multi-scale Representation coefficient are as follows:
Wherein,Indicate pseudoinverse operator, i.e.,
The updated value of multi-scale Representation coefficient are as follows:
Step 303) basisCalculate X(k)(t1),...,X(k)(tN), to calculate the calculated value of target trajectoryTo calculate
Step 304) is calculated according to the process of step 302)
Step 305) judge whether be less than threshold value ε, if a determination be made that certainly, be transferred to step 4)；Otherwise, k is enabled =k+1, is transferred to step 303).
The invention has the following advantages over the prior art:
1, model is established in the track that method of the invention moves target；Using manifold as fundamental space, by using stream Multiscale analysis method in shape establishes whole multiscale analysis method, obtains multiple dimensioned table of the target trajectory in manifold Show；It converts the estimation problem of trajectory parameters to the estimation of model coefficient, carries out surveying the high-precision resolving of member to track；
2, method of the invention has the advantage for significantly improving calculation accuracy relative to traditional point-by-point least square method； Compared to traditional batten calculation method, calculation accuracy is identical, but the problem that be that of avoiding node optimization etc. complicated, can significant letter Change solution process；
3, multi-scale transform operator F () used in method of the invention, with the multiple dimensioned calculation such as traditional wavelet transformation The maximum difference of son is: conventional transformation is to be indicated curve on fixed basic function, therefore the property of basic function will Influence the precision indicated, if the motion profile of extraterrestrial target and selected basic function property mismatch, there may be compared with Big expression error.And the multi-scale transform in the present invention is then the representation method of data-driven, by down-sampled (thick scale) Data are set out, and are the direct applications to " slickness " prior information by interpolation and Difference Calculation wavelet coefficient, therefore adaptability It greatly enhances；
4, method of the invention has versatility, be suitable for space industry, under a variety of Instrumentation systems, a variety of extraterrestrial targets The high-precision estimation problem of motion state.
Detailed description of the invention
Fig. 1 is that the local tangential plane projection of target trajectory shows that (and inverse transformation) is intended to；
Fig. 2 is multi-scale transform operator schematic diagram of the invention.
Specific embodiment
In general, the motion profile of target is space curve, it is whole to estimate not in a plane, it is therefore desirable at certain A part is projected to tangent plane, the plane curve after being projected.As shown in Figure 1, specifically: it will be on some part of space curve If do to be mapped to cut and spatially obtainAnd one group of base (e of oriented quantity spacej), then exist (τ1,…,τd), so thatIt is average using polynomial interopolation in multiscale analysis method in manifold or multiple spot value The interpolation at each coordinate midpoint of acquisition of (see below statement), obtains cutting the expression in spaceIt is reflected by inverse It penetratesObtain the expression in manifold.
The characteristics of carrying out multi-scale Representation to the curve in plane, be exactly using its slickness, first carries out low-density to it Sampling, then during gradually refinement (refining in the present invention using dichotomy), calculate " small echo " coefficient.The wavelet coefficient has The characteristics of index decreased, therefore after refinement to a certain extent, error is negligible, after abandoning negligible coefficient, i.e., The sub-reduced parameter model of curve can be realized under conditions of meeting precision.
A kind of non-linear space target state based on manifold determines method, and this method is by target trajectory position and speed The estimation problem of degree parameter is converted into the estimation of multi-scale Representation coefficient, which comprises
Step 1) establishes the observational equation of extraterrestrial target track:
Y (t)=H (X (ti)), i=1 ..., N (1)
Wherein, in the present embodiment, Y (t) is distance (and/poor) change rate for being target；X(ti) it is target to be estimated State parameter；H is observational equation；
In the present embodiment, the expression formula of the distance between target and observation station change rate H are as follows:
WhereinDbjective state parameter i.e. to be estimated is three-dimensional space Between in position and speed, (xk,yk,zk)TFor the position coordinates of observation station.
In other embodiments, observational equation can indicate are as follows:
Distance:
Azimuth:
Pitch angle:
Step 2) indicates the multi-scale transform operator F () in dbjective state parameter manifold to be estimated:
Wherein X (t1),...,X(tN) be N number of moment track position and speed parameter, C1,...,CMFor multiple dimensioned table Show coefficient, M < < N complete the rarefaction representation (sub-reduced parameter model) of trajectory parameters.
The expression formula that the multi-scale transform operator F () does not show, as shown in Fig. 2, multi-scale transform operator F (): construction process specifically include:
Step 201) sets initial gauges as j0A scale (j0Selection can be by ti, i=1 ..., N is down-sampled to be obtained Arrive, such as 8 times are down-sampled), enable tj,k=k/2j,j≥0,k∈Z+,βj,k=F (tj,k) it is that F in current scale respectively samples (moment) point On value；
Step 202) refines sampled point by dichotomy:
It has thus obtained 2 in next scalej+1On even number point value；
For the numerical value on odd point, interpolation is carried out with D rank multinomial, for point 2k+1, D+1 are chosen around the point Point, specifically chooses process are as follows: if D+1 is even number, successively chooses to the left on the left side point 2k+1A, the right is to the right Successively chooseIt is a, if D+1 is odd number, successively chosen to the left on the left side point 2k+1It is a, the right to It successively chooses on the right sideA point；Multinomial π is obtained with polynomial interopolation to D+1 point of selectionj,k(t), common multinomial Constant value method, such as Lagrange Polynomial interpolating can be met the requirements.
And then it obtains
In βj,k=F (k/2j), the interpolation of odd pointOn the basis of, define " small echo " system Number are as follows: the difference between " predicted value " on the odd point that interpolation obtains and the true value on odd point:
I.e.
In above-mentioned representation method, the F functional value of odd point is obtained by polynomial interopolation, is calculated again to further decrease The value of miscellaneous degree, β can be replaced with the average value on section；I.e.
βj,k=Ave F | Ij,k, j, k ∈ Z,
Here Ij,k=[k/2j,(k+1)/2j), and have βj,k=(βj+1,2k+βj+1,2k+1)/2。
Step 203) is sampled by the thick scale of low-densityWith wavelet coefficient ((αj,k)k∈Z)j≥0It can reconstruct any Continuous function F, i.e.,
Wavelet coefficient ((αj,k)k∈Z)j≥0Numerical values recited have the characteristics that exponential damping, that is, there is constant C and R so that
When j is sufficiently large, give up the wavelet coefficient of subsequent scale, the error introduced in reconstruct can be ignored, thus Under conditions of meeting precision, the sub-reduced parameter model of curve F may be implemented.Therefore, the effect of multi-scale Representation operator F () is just It is the parameter X (t that target is moved1),...,X(tN), it is expressed as C1,...,CM, they are some thick scale j0On sampled value, And the wavelet coefficient on subsequent scale (wavelet coefficient is zero when j is sufficiently large).
Step 3) calculates multi-scale Representation coefficient C1,...,CM；It specifically includes:
Step 301) calculates the initial value of target trajectory, calculates the observation of distance and the difference of calculated value；
Assuming that target trajectory initial value is X(0)(t1),...,X(0)(tN), the multi-scale Representation coefficient after being indicated by model isThe calculated value of distance is then obtained according to (1) formula are as follows:
The observation of distance is Y(0)(t1),...,Y(0)(tN)；
Then, the difference of the observation of distance and calculated value are as follows:
Step 302) calculates the correction value of multi-scale Representation coefficient according to the observation of distance and the difference of calculated value；Enable k= 1；
It can be obtained according to (1) and (2):
Take single order Taylor approximate in above formula, then the correction value of multi-scale Representation coefficient are as follows:
Wherein,Indicate pseudoinverse operator, i.e.,
The updated value of multi-scale Representation coefficient are as follows:
Step 303) basisCalculate X(k)(t1),...,X(k)(tN), to calculate the calculated value of distanceTo calculate
Step 304) is calculated according to the process of step 302)
Step 305) judge whether be less than threshold value ε, if a determination be made that certainly, be transferred to step 4)；Otherwise, k is enabled =k+1, is transferred to step 303)；
The estimated value of step 4) output dbjective state: X(k)(t1),...,X(k)(tN)。

Claims (4)

1. a kind of non-linear space target state based on manifold determines method, which comprises
The observational equation for establishing extraterrestrial target track is embedded in the physical characteristic of low dimensional manifold based on target trajectory in space, Using manifold as fundamental space, using the multiscale analysis method in manifold, dbjective state parameter is subjected to more rulers in manifold Degree indicates, obtains multi-scale Representation coefficient；Based on observational equation and multi-scale Representation formula, flowed using nonlinear iteration method Shape multi-scale Representation coefficient；To obtain the estimated value of dbjective state parameter；
The method specifically includes:
Step 1) establishes the observational equation of extraterrestrial target track:
Y (t)=H (X (ti)), i=1 ..., N (1)
Wherein, Y (t) be extraterrestrial target track；X(ti) it is dbjective state parameter to be estimated；H is observational equation；WhereinDbjective state parameter i.e. to be estimated is the position in three-dimensional space And speed；
Step 2) indicates the multi-scale transform operator F () in dbjective state parameter manifold to be estimated:
Wherein X (t1),...,X(tN) be N number of moment track position and speed parameter, C1,...,CMFor multi-scale Representation system Number, M < < N complete the rarefaction representation of dbjective state parameter；
Step 3) establishes partial derivative matrix of the metrical information relative to dbjective state parameter, while it is opposite to establish dbjective state parameter In the partial derivative matrix of multi-scale Representation coefficient, two matrix multiples are finally obtained into metrical information about multi-scale Representation coefficient Compound partial derivative matrix；Using nonlinear iteration method, the estimated value C of multi-scale Representation coefficient is obtained1,...,CM；
Step 3) is calculated multi-scale Representation coefficient C by step 4)1,...,CMSubstitute into the estimation that formula (2) calculate dbjective state Value: X (t1),...,X(tN)。
2. the non-linear space target state according to claim 1 based on manifold determines method, which is characterized in that The construction process of multi-scale transform operator F () in the step 2) specifically includes:
Step 201) sets initial gauges as j0A scale, j0Selection pass through to ti, i=1 ..., N is down-sampled obtains enables tj,k= k/2j,j≥0,k∈Z+,βj,k=F (tj,k) it is value of the F on each sampled point of current scale；
Step 202) refines sampled point by dichotomy, calculates the thick scale sampling of low-densityAnd wavelet coefficient ((αj,k)k∈Z)j≥0；
It enables:
It has thus obtained 2 in next scalej+1On even number point value；
For the numerical value on odd point, interpolation is carried out with D rank multinomial, for point 2k+1, D+1 point is chosen around the point, It is specific to choose process are as follows: if D+1 is even number, successively to be chosen to the left on the left side point 2k+1A, the right is to the right successively It choosesIt is a, if D+1 is odd number, successively chosen to the left on the left side point 2k+1A, the right is to the right successively It choosesA point；Multinomial π is obtained with polynomial interopolation to D+1 point of selectionj,k(t),
And then it obtains
In βj,k=F (k/2j), the interpolation of odd pointOn the basis of, define " small echo " coefficient Are as follows: the difference between " predicted value " on the odd point that interpolation obtains and the true value on odd point:
I.e.
Step 203) is sampled by the thick scale of low-densityWith wavelet coefficient ((αj,k)k∈Z)j≥0Reconstruct arbitrary continuous function F, i.e.,
Wavelet coefficient ((αj,k)k∈Z)j≥0Numerical values recited have the characteristics that exponential damping, that is, there is constant C and R so that
3. the non-linear space target state according to claim 2 based on manifold determines method, which is characterized in that The value of β in the step 202) is replaced with the average value on section；I.e.
βj,k=Ave F | Ij,k, j, k ∈ Z,
Here Ij,k=[k/2j,(k+1)/2j], and have βj,k=(βj+1,2k+βj+1,2k+1)/2。
4. the non-linear space target state according to claim 2 or 3 based on manifold determines that method, feature exist In the step 3) specifically includes:
Step 301) calculates the difference of target trajectory observation and calculated value according to the initial value of dbjective state parameter；
Assuming that dbjective state initial parameter values are X(0)(t1),...,X(0)(tN), multi-scale Representation coefficient isThen root The calculated value of target trajectory is obtained according to (1) formula are as follows:
The observation of target trajectory is Y(0)(t1),...,Y(0)(tN)；
The observation of target trajectory and the difference of calculated value are as follows:
Step 302) calculates the correction value of multi-scale Representation coefficient according to the observation of target trajectory and the difference of calculated value；Order changes Generation number l=1；
It can be obtained according to (1) and (2):
Take single order Taylor approximate in above formula, then the correction value of multi-scale Representation coefficient are as follows:
Wherein,Indicate pseudoinverse operator, i.e.,
The updated value of multi-scale Representation coefficient are as follows:
Step 303) basisCalculate X(l)(t1),...,X(l)(tN), to calculate the calculated value of target trajectoryTo calculate
Step 304) is calculated according to the process of step 302)
Step 305) judge whether be less than threshold value ε, if a determination be made that certainly, be transferred to step 4)；Otherwise, l=l+ is enabled 1, it is transferred to step 303).
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