CN105550498B - A kind of ballistic curve approximating method based on Moving Least - Google Patents
A kind of ballistic curve approximating method based on Moving Least Download PDFInfo
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Abstract
The present invention relates to a kind of ballistic curve approximating method based on Moving Least, belongs to ballistic test technical field of data processing.The present invention is first depending on the ballistic function that traditional least square method is fitted ballistic test data, calculates the minimal error, mean error and experiment point tolerance of the fitting ballistic function and experimental data；And according to obtaining minimal error, mean error and the experimental point error calculation weighting function radius of influence；Then weighting function is constructed according to the weighting function radius of influence；Finally ballistic data is fitted using Moving Least according to the weighting function constructed, to obtain corresponding ballistic curve.Compared to the mode of the original piecewise fitting of traditional least square method, the present invention does not need progress piecewise fitting, multinomial order does not need tentative calculation, and the fitting to ballistic data can be realized using Moving Least by the weighting function of building.
Description
Technical field
The present invention relates to a kind of ballistic curve approximating method based on Moving Least, belongs at ballistic test data
Manage technical field.
Background technique
Trajectory processing is the process of a kind of quantity of state and ballistic movement amount dependence when obtaining transmitting from experimental data,
Quantity of state is generally the emission time of carrier aircraft, firing altitude, launch angle etc. when the transmitting that trajectory is handled in engineering practice, and locates
The ballistic movement amount of reason is generally range, time etc., and dependence needs to meet slickness and the variation of curvature small range etc.
Assuming that.Currently, the processing main method of trajectory includes traditional least square method.Traditional least square method is in trajectory treatment process
In, polynomial order needs tentative calculation, when multinomial order is excessively high, is easy to produce high rate of change first derivative, order is too low
When, raw experimental data approximation accuracy is insufficient, and domain is often when data volume is larger, initial data non-linear nature is stronger
It needs to carry out piecewise fitting, to limit the versatility and convenience of its engineer application.
Summary of the invention
It is an object of the invention to overcome existing traditional least square method ballistics-fit method method multinomial order need tentative calculation,
The disadvantage that domain need to be segmented and initial data approximation accuracy is not high is influenced for the weighting function in Moving Least
Domain proposes a kind of algorithm for estimating based on De Giorgi iteration, estimates the radius of influence of the weighting function in local neighborhood, construction
The concrete form of weighting function obtains a kind of new method for relying on fitting experimental data ballistic function.
The present invention provides a kind of ballistic curve fitting side based on Moving Least to solve above-mentioned technical problem
The step of method, this method, is as follows：
1) ballistic function that ballistic test data are fitted according to traditional least square method；
2) minimal error, mean error and experimental point error calculation are taken according to the ballistic function that traditional least square method obtains
The weighting function radius of influence；
3) weighting function is constructed according to the weighting function radius of influence；
4) ballistic data is fitted using Moving Least according to the weighting function constructed, it is corresponding to obtain
Ballistic curve.
The weighting function radius of influence is：
Wherein diIndicate the weighting function radius of influence corresponding to experimental point,Indicate mean error, ΔiIndicate experimental point
Error, ΔminIndicate minimal error, i=1,2 ... n, n are experiment points.
The weighting function constructed in the step 3)：
Wherein ri=| x-xi|/diIndicate that the compactly support radius of influence, θ indicate normalizing factor, CM=ki, kiAt the beginning of indicating experimental point
Beginning weight.
The weighting function meets regression nature, i.e.,：
θ value can be calculated according to above formula.
The minimal error, mean error and experiment point tolerance meet following relationship：
Wherein n indicates experiment points, ΔLIndicate minimal error, ΔR1Indicate mean error, ΔR2Indicate experiment point tolerance,Indicate that the weighting function for being greater than setting normal value, Δ indicate expectation Moving Least fitting function.
The positive effect of the present invention is：The present invention is based on the trajectory processing methods of De Giorgi iterative estimate, pass through selection
First fitting function acquisition weighting function error folder forces estimation, calculates the weighting function radius of influence, constructs weighting function, according to shifting
The processing of dynamic four complete ballistic datas of step of least square method digital simulation function.Compared to traditional least square method by original segmentation
The mode of fitting changes into the calculating of the compactly support radius of neighbourhood, and the present invention does not need to carry out piecewise fitting.By weighting function compactly support
The boundedness of the estimation of the radius of neighbourhood and the regression nature of weight function, can highly precise approach initial data, can fitting data acutely become
The case where change.By the continuous differentiability of cost functional, obtained ballistic function has smooth enough.By the weight point of weighting function
With property, it is not required to the tentative calculation of multinomial order.
Detailed description of the invention
Fig. 1 is quadravalence substrate, Δ in the embodiment of the present inventionL=Δmin,ΔR=ΔiMoving Least fitting data
Figure.
Specific embodiment
A specific embodiment of the invention is described further with reference to the accompanying drawing.
The present invention is according to least square method above problem present in trajectory treatment process, and the present invention provides a kind of bases
In the ballistic curve approximating method of Moving Least, this method is first depending on traditional least square method to ballistic test data
The ballistic function being fitted calculates minimal error, mean error and the experiment of the fitting ballistic function and experimental data
Point tolerance；And according to obtaining minimal error, mean error and the experimental point error calculation weighting function radius of influence；Then according to power
Weight function influences radius constructs weighting function；Finally use Moving Least to Ballistic Number according to the weighting function constructed
According to being fitted, to obtain corresponding ballistic curve.
Wherein Moving Least (Moving least square method, MLSM) is as a kind of fitting algorithm
It is proposed, and initially applied in Solid Mechanics field in nineteen sixty-eight by Shepard, Nayroles in 1992 et al. will be moved most earliest
Small square law is gradually widely applied in the field for solving partial differential equation in gridless routing.The base of MLSM
Present principles are, first to complete characteristic curve in entire domain subregion, then use least square method on different regions
Fitting uses subregion local fit, this will have larger improvement to fitting precision, ties with theoretical value and least square method fitting
Fruit comparison, the result shows that Moving Least is versatile, initial data approximation accuracy is high, is applicable to ballistics-fit method.
It is right below using the practical flight height of the practical flight sampling of certain model bomb, time as raw experimental data
Specific implementation process of the invention is described in detail.
1. the ballistic function being fitted according to traditional least square method to ballistic test data
Choosing first fitting function acquisition weighting function error folder forces estimation to refer to：The substrate for choosing certain order, according to biography
System least square method is fitted the ballistic function obtained, average by the minimal error of digital simulation ballistic function and experimental data
Error tests point tolerance, and experiment point tolerance is by raw experimental data and classical square law according to L2Mould acquires, and specific formula is
Wherein u*Indicating experimental data, u indicates the data obtained by classical square law,
The substrate forms for the first fitting function chosen when being fitted in the present embodiment using traditional least square method are such as
Under：
R=a0+a1H+a2H2
According to least square method acquisition fit time, the minimal error of digital simulation time and initial data time are averaged
Error and experiment point tolerance.Minimal error, mean error and experiment point tolerance meet following relationship：
Wherein n indicates experiment points, ΔLIndicate minimal error, ΔR1Indicate mean error, ΔR2Indicate experiment point tolerance,Indicate that the weighting function for being greater than setting normal value, Δ indicate expectation Moving Least fitting function.
2. calculating weighting function according to minimal error, mean error, experiment point tolerance and experimental point initial weight influences half
Diameter di。
The calculating process of the weighting function radius of influence is as follows：
The first step introduces De Giorgi iteration lemma, ifIt is defined in [k0, ∞) on non-negative dullness do not increase
Function, and meet
Wherein α > 0, β > 1.Then
Wherein
Second step provides definition
By one gather estimate note into | E |, define simultaneously
E (k)=x ∈ D | ω (x) > k }
Di=x ∈ D | 0≤| | x-xi| | < Ri}
If Γ (x) indicates the function being fitted by observation data by Classical Least-Squares, definition
Δi=| Γ (xi)-u(xi)|
Wherein n indicates the number of observation data, u (xi) indicate observation data.
Assume that weighting function meets in the overall situation：
And meet
Wherein(ωi-k)+=max { (ωi- k), 0 }, 0≤k≤1.
To inequality (1), it is noted that as h > k,And on E (h)We have
To (1) right end, byInequality, Wo Menyou
Wherein p > 1, q > 1, andConsider that weight function is not more than 1, joint above formula can obtain
I.e.
By above formula and De Giorgi iteration lemma is combined, then
|E(li+di) |=0
Wherein li=ωi(xi), one Small initial data can be set the several numbers of data point according to the observation, consider each observation herein
The weight of point is equal, is set asHave simultaneously
It is defined by E (k), it is known that the radius of influence of weighting function is di。
Wherein diIndicate the weighting function radius of influence corresponding to experimental point,Indicate mean error, ΔiIndicate experimental point
Error, ΔminIndicate minimal error, i=1,2 ... n, n are experiment points.
3. constructing weighting function according to the weighting function radius of influence.
Constructed weighting function：
Wherein ri=| x-xi|/diIndicate that the compactly support radius of influence, θ indicate normalizing factor, CM=ki, kiAt the beginning of indicating experimental point
Beginning weight, value takeWeighting function meets regression nature, i.e.,：
θ value can be calculated according to above formula.
4. ballistic data is fitted using Moving Least according to the weighting function constructed, it is corresponding to obtain
Ballistic curve.Ballistic data, initial data are calculated to patented method by taking the practical flight sampled data of certain model bomb as an example
The Weapon Range under the conditions of height when for certain type bomb practical flight.
If Γ (x) indicates the function being fitted by observation data by Classical Least-Squares, definition
Δi=| Γ (xi)-u(xi)|
Wherein n indicates the number of observation data, u (xi) indicate observation data.Choose least square method and mobile minimum two
Multiply and is using polynomial basis bottom
B=a0+a1x+a2x2+a3x3+a4x4
Choose ΔL=Δmin,ΔR=Δi, weighting function calculated result is as follows：
10.0*exp(-0.752487*abs(x))-7.1751e-65
10.0*exp(-0.552226*abs(x-10.0))-7.1751e-65
10.0*exp(-0.9306*abs(x-20.0))-7.1751e-65
10.0*exp(-2.50413*abs(x-30.0))-7.1751e-65
10.0*exp(-0.880331*abs(x-40.0))-7.1751e-65
10.0*exp(-0.984*abs(x-50.0))-7.1751e-65
10.0*exp(-2.56958*abs(x-60.0))-7.1751e-65
10.0*exp(-2.95324*abs(x-70.0))-7.1751e-65
10.0*exp(-1.19171*abs(x-80.0))-7.1751e-65
10.0*exp(-1.15845*abs(x-90.0))-7.1751e-65
10.0*exp(-2.24491*abs(x-100.0))-7.1751e-65
10.0*exp(-3.75*abs(x-110.0))-7.1751e-65
10.0*exp(-1.1415*abs(x-120.0))-7.1751e-65
10.0*exp(-1.29846*abs(x-130.0))-7.1751e-65
10.0*exp(-1.2336*abs(x-140.0))-7.1751e-65
Its fitting result is as shown in Figure 1.
Claims (4)
1. a kind of ballistic curve approximating method based on Moving Least, which is characterized in that the step of the approximating method such as
Under：
1) ballistic function that ballistic test data are fitted according to least square method；
2) minimal error, mean error and experimental point error calculation weighting function are taken according to the ballistic function that least square method obtains
The radius of influence；The weighting function radius of influence is：
Wherein diIndicate the weighting function radius of influence corresponding to experimental point,Indicate mean error, ΔiIndicate experiment point tolerance,
ΔminIndicate minimal error, i=1,2 ... n, n are experiment points；
3) weighting function is constructed according to the weighting function radius of influence；
4) ballistic data is fitted using Moving Least according to the weighting function constructed, to obtain corresponding bullet
Road curve.
2. the ballistic curve approximating method according to claim 1 based on Moving Least, which is characterized in that described
Step 3) in construct weighting function：
Wherein ri=| x-xi|/diIndicate that the compactly support radius of influence, θ indicate normalizing factor, CM=ki, kiIndicate that experimental point is initially weighed
Weight.
3. the ballistic curve approximating method according to claim 2 based on Moving Least, which is characterized in that described
Weighting function meets regression nature, i.e.,：
θ value can be calculated according to above formula.
4. the ballistic curve approximating method according to claim 3 based on Moving Least, which is characterized in that described
Minimal error, mean error and experiment point tolerance meet following relationship：
Wherein n indicates experiment points, ΔLIndicate minimal error, ΔR1Indicate mean error, ΔR2Indicate experiment point tolerance,Table
Show that the weighting function for being greater than setting normal value, Δ indicate expectation Moving Least fitting function.
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