CN110059285A - Consider J2Item-influenced missile free-section trajectory deviation analysis and prediction method - Google Patents
Consider J2Item-influenced missile free-section trajectory deviation analysis and prediction method Download PDFInfo
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- CN110059285A CN110059285A CN201910324168.0A CN201910324168A CN110059285A CN 110059285 A CN110059285 A CN 110059285A CN 201910324168 A CN201910324168 A CN 201910324168A CN 110059285 A CN110059285 A CN 110059285A
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Abstract
The invention provides a method for considering J2The method for analyzing and forecasting the deviation of the missile free section trajectory under the influence of terms comprises J2The decomposition of the term gravity vector and the derivation of a free-segment trajectory deviation analysis forecasting model are carried out through J2Decomposing the term gravity vector to obtain expressions of the perturbation force in different coordinate axis directions; according to the state space perturbation theory, decomposing J2And (4) substituting the term gravitation vector expression into an integral solving expression of the missile free flight section trajectory deviation, and obtaining a complete analytic expression of each term deviation through integration. Compared with other methods in the prior art, the method of the invention has the advantage that the resolving time of the method is 10‑5And in the magnitude of s, the calculation error of any downward position is less than 50 meters, and the calculation result is expressed in an inertial system and can directly participate in the missile-borne guidance calculation without additional coordinate conversion.
Description
Technical field
The present invention relates to flight dynamics technical fields, particularly, are related to a kind of consideration earth J2The guided missile that item influences is certainly
Forecasting procedure is parsed by flight trajectory deviation.
Background technique
Free flight phasel is longest one section of flight time in the entire inflight phase of ballistic missile, accounts for about total flight time
90% or more.Since flying height is higher, ballistic missile is in free flight phasel mainly by the effect of earth center gravitation, flight
Track is approximately a part of elliptic orbit, but due to perturbation factors (such as earth non-sphere gravitation, rarefied atmosphere resistance)
In the presence of again real trace can deviate standard ellipse track.For the accuracy at target for guaranteeing ballistic missile, engine cutoff is being carried out
It needs to carry out Fast Prediction to the Study of Free Flight Phase Trajectory deviation under perturbation conditions when control.In fact, considering perturbation factors
Extra-atmospheric flight device orbit prediction is one of the classical problem in dynamics of orbits field, i.e. initial-value problem.For this problem
Classical theory mainly have: mean element method, fg series decomposition method, non-orthogonal decomposition method and Step-varied back propagation numerical integration side
Method etc..Mean element method effect when being used for near-earth satellite orbit prediction is preferable, but for the sub- rail such as Ballistic Missile in Its Free trajectory
When road forecasting problem, the precision of this method can be then decreased obviously；Fg Series Method is using the flight time as independent variable, in standard ellipse
Taylor expansion is carried out on trajectory, on the basis of certain rationally approximate, derived and is considered J2The analytical expression of item, but this method
It is only applicable to short time extrapolation, it is impossible to be used in the resolving of Ballistic Missile in Its Free trajectory；Non-orthogonal decomposition method is Chinese scholar Lee
Lian Zhong proposes a kind of consideration earth J in nineteen eighty-two2The guided missile trajectory of free flight phase Analytic Calculation Method of item perturbation, has been applied to
Consider J2In the Closed Loop Guidance online compensation method that item influences, but this method cannot be guaranteed the precision of trajectory prediction under any directive
Respectively less than 100 meters；Step-varied back propagation numerical integrating is the most popular method of current orbit Exact Forecast, passes through adaptive item
Part step-length, can significant increase tradition fixed step size orbit numeral integral efficiency, but it is usually small in the ballistic missile missie-borne guidance period
Under conditions of 20ms, it is still necessary to the research higher analytical algorithm of precision.
For the deficiency of current each common track/trajectory prediction algorithm, design it is a kind of it is new, consider earth J2What item influenced
Guided missile trajectory of free flight phase deviation parsing forecasting procedure is of great significance.
Summary of the invention
The purpose of the present invention is to provide a kind of consideration J2The guided missile trajectory of free flight phase deviation that item influences parses forecasting procedure,
Including J2The decomposition of item gravitation vector and trajectory of free flight phase deviation parsing forecasting model derive, and pass through J2Point of item gravitation vector
Solution obtains the expression formula of the perturbative force on different change in coordinate axis direction；According to state space perturbation theory, by the J after decomposition2Xiang Yin
Force vector expression formula brings the Integration Solving expression formula of guided missile Study of Free Flight Phase Trajectory deviation into, and integral obtains each single item deviation
Complete analytical expression.Other methods in compared with the existing technology, the resolving time-consuming of this method is 10-5S magnitude, any directive
Under position calculate error and be respectively less than 50 meters, and calculated result indicates in inertial system, can directly participate in missie-borne guidance calculate and
It is converted without additional coordinate.Specific technical solution is as follows:
A kind of consideration J2The guided missile trajectory of free flight phase deviation that item influences parses forecasting procedure, including J2Point of item gravitation vector
Solution and trajectory of free flight phase deviation parsing forecasting model derive；
J2The decomposition of item gravitation vector obtains the perturbative force such as expression formula 8 on different change in coordinate axis direction):
In formula: δ ar、δaβWith δ azRespectively indicate J2Item gravitation vector is in track cylindrical coordinate on r axis, β axis and the direction z
Component；R is earth radius；
sr=-3K, sβ=K, sz=K；
μ is Gravitational coefficient of the Earth, aeFor terrestrial equator mean radius；
piFor constant coefficient, i takes 0,1,2,3,4, specific as follows:
For the latitude of P point, σ is lateral angle, αAFor longitude of the A point in pole coordinate system, f0Indicate very close at A point
Point angle, i.e., initial true anomaly；
q1And q2It is as follows:
The parsing forecasting model derivation of trajectory of free flight phase deviation is specifically:
According to state space perturbation theory, the Integration Solving expression of guided missile Study of Free Flight Phase Trajectory deviation is expression formula
9):
In formula: △ vr(f)、△vβ(f) and △ vzIt (f) is respectively trajectory state deviation velocity vector in track cylindrical coordinate
It is middle along r axis, the component of β axis and the direction z；△ r (f) and △ z (f) is respectively trajectory state deviation position vector in track cylindrical coordinates
Along the component of r axis and the direction z in system；△ t (f) is the difference of actual flying time and standard two-body ballistic flight time；H is two-body
The mould of the corresponding angular momentum vector of trajectory plane；Indicate that corresponding standard two-body trajectory the earth's core is away from that is, when true anomaly is ξP indicates the semi-latus rectum of two-body trajectory, and e indicates the eccentricity of two-body trajectory；
μ is Gravitational coefficient of the Earth；
By expression formula 8) substitute into expression formula 9), and the complete analytical expression 10 integrated up to each single item deviation) -15):
In formula:
Λi,jIt is as shown in table 1:
1 function Λ of tablei,jExpression formula statistical form
It is preferred in above technical scheme, J2The decomposition of item gravitation vector, specifically:
Enable U2Indicate earth J2Item gravitational potential, the J in Earth centered inertial system2Item gravitational potential is expression formula 1):
Wherein: μ is Gravitational coefficient of the Earth, aeFor terrestrial equator mean radius, J be constant andR is the earth half
Diameter,For terrestrial latitude；
According to spherical trigonometry by expression formula 1) independent variable true anomaly f is converted to by reduced latitude；Take N for celestial sphere seat
Mark the north pole of system, orthodromeThe projection for being the standard two-body trajectory that is determined by guided missile Burnout parameter on spherical surface, it is bent
Line AB indicates projection of the guided missile disturbed motion track on spherical surface, and straight line OP is perpendicular to plane OAB*, αA, △ f be respectively plane
The dihedral angle and plane POA and POB of POC and POA*Dihedral angle,The latitude of respectively A point and the latitude of P point, λAWith
λPThe longitude of respectively A point and P point, then have:
In spherical triangle ANP, expression formula 2 is obtained) -4):
Wherein: γ is the corresponding azimuth Burnout A, αAFor longitude of the A point in new pole coordinate system；
In spherical triangle BPN, expression formula 5 is obtained):
In formula: σ is lateral angle；f0Indicate the true anomaly at A point, i.e., initial true anomaly；F indicates any time on bullet
Corresponding true anomaly, P indicates arbitrary point here；
By expression formula 5) bring expression formula 1 into) to get to J2Item gravitational potential is expression formula 6 about the function of true anomaly):
U is sought respectively2(f) about r, f, σ partial derivative to get arrive J2Expression of the item gravitation vector in track cylindrical coordinate
Formula is expression formula 7):
Calculated based on two-body normal trajectory, at this time σ=0, expression formula 7) in following coefficient move back cancellation, i.e.,
Perturbative force on different change in coordinate axis direction is uniformly used into expression formula 8) it indicates:
Other than objects, features and advantages described above, there are also other objects, features and advantages by the present invention.
Below with reference to accompanying drawings, the present invention is described in further detail.
Detailed description of the invention
The attached drawing constituted part of this application is used to provide further understanding of the present invention, schematic reality of the invention
It applies example and its explanation is used to explain the present invention, do not constitute improper limitations of the present invention.In the accompanying drawings:
Fig. 1 is embodiment middle orbit cylindrical coordinate schematic diagram；
Fig. 2 is spherical angle relational graph in embodiment；
Fig. 3 is J in embodiment2Item gravitation is to not co-orbital influencing characterisitic；
Fig. 4 is consideration J of the invention2The guided missile trajectory of free flight phase deviation parsing forecasting procedure and mean element method phase that item influences
For the calculating residual error comparison diagram of numerical integration result.
Specific embodiment
The embodiment of the present invention is described in detail below in conjunction with attached drawing, but the present invention can be limited according to claim
Fixed and covering multitude of different ways is implemented.
Embodiment:
A kind of consideration earth J2The guided missile trajectory of free flight phase deviation that item influences parses forecasting procedure, including J2Item gravitation vector
Decomposition and trajectory of free flight phase deviation parsing forecasting model derive, Details as Follows:
1、J2The decomposition of item gravitation vector, specifically:
Enable U2Indicate earth J2Item gravitational potential, the J in Earth centered inertial system2Item gravitational potential is expression formula 1):
Wherein: μ is Gravitational coefficient of the Earth, aeFor terrestrial equator mean radius, J be constant andR is the earth half
Diameter,For terrestrial latitude.
According to spherical trigonometry by expression formula 1) independent variable true anomaly f is converted to by reduced latitude.As shown in Fig. 2, N
For the north pole of celestial coordinate system, orthodromeIt is the standard two-body trajectory that is determined by guided missile Burnout parameter on spherical surface
Projection, curve AB indicate projection of the guided missile disturbed motion track on spherical surface, and straight line OP is perpendicular to plane OAB*。αA, △ f difference
For the dihedral angle and plane POA and POB of plane POC and POA*Dihedral angle.The latitude of respectively A point and the latitude of P point
Degree, λAAnd λPThe longitude of respectively A point and P point.
In spherical triangle ANP, expression formula 2 can be obtained) -4):
Wherein: γ is the corresponding azimuth Burnout A, αALongitude of the A point in new pole coordinate system can be considered as, specifically
It can be seen that Fig. 2.
In spherical triangle BPN, expression formula 5 can be obtained):
In formula: σ is lateral angle；f0Indicate the true anomaly at A point, i.e., initial true anomaly；F indicates any time on bullet
Corresponding true anomaly, P indicates arbitrary point here.
By expression formula 5) bring expression formula 1 into), J can be obtained2Item gravitational potential is expression formula about the function of true anomaly
6):
Wherein:μ is Gravitational coefficient of the Earth, aeFor terrestrial equator mean radius；
piFor constant coefficient, i takes 0,1,2,3,4, specifically:
σ is lateral angle.
U is sought respectively2(f) about r, f, σ partial derivative to get arrive J2Item gravitation vector in track cylindrical coordinate, (sit by track column
Mark system definition it is as shown in Figure 1) in expression formula be expression formula 7):
Wherein, δ ar、δaβWith δ azRespectively indicate J2Item gravitation vector is in track cylindrical coordinate on r axis, β axis and the direction z
Component.
Ballistic error based on state space perturbation method is propagated in analytic solutions derivation process, and perturbative force is not based on when leading
Play actual position to be calculated, but calculated based on two-body normal trajectory, at this time σ=0, expression formula 7) in following coefficient
Cancellation is moved back, i.e.,
Perturbative force on different change in coordinate axis direction is uniformly used into expression formula 8) it indicates:
In formula:
sr=-3K, sβ=K, sz=K；
Wherein, q1And q2It is as follows:
2, trajectory of free flight phase deviation parsing forecasting model derives, specifically:
According to state space perturbation theory, the Integration Solving expression of guided missile Study of Free Flight Phase Trajectory deviation is expression formula
9):
In formula: △ vr(f)、△vβ(f) and △ vzIt (f) is respectively trajectory state deviation velocity vector in track cylindrical coordinate
It is middle along r axis, the component of β axis and the direction z；△ r (f) and △ z (f) is respectively trajectory state deviation position vector in track cylindrical coordinates
Along the component of r axis and the direction z in system；△t(f) be actual flying time and standard two-body ballistic flight time difference；H is two-body
The mould of the corresponding angular momentum vector of trajectory plane；Indicate that corresponding standard two-body trajectory the earth's core is away from that is, when true anomaly is ξAnd p indicates the semi-latus rectum of two-body trajectory, e indicates the eccentricity of two-body trajectory；
By expression formula 8) substitute into expression formula 9), and integrate and obtain the complete analytical expression 10 of each single item deviation) -15):
In formula:It is constant coefficient, k takes 1,2,3,4,5,6；L takes 1,2,3,4,5,6,7；And have:
Λi,jIt is as shown in table 1:
1 function Λ of tablei,jExpression formula statistical form
Assuming that Burnout position vector is x in the inertial system of the earth's core0=[0,6578140,0]T, along the initial of the earth's core arrow direction
Speed is 3300m/s, is 6680m/s in missile flight plane and perpendicular to the initial velocity on the arrow direction of the earth's core.It is right simultaneously
Azimuth ° is traversed from -90 ° to 90.Numerical integrating is respectively adopted, the analytic solutions and mean element method that the present embodiment derives
Ballistic deflection is carried out parsing forecast (assuming that guided missile flies 2700 seconds in different orientations), and by numerical integrating
Calculated result as evaluation analytic solutions and mean element method precision benchmark.Mean element method is calculated using single order solution, i.e.,
Only consider single order/second order secular term, the single order long period term, single order short-period term of each orbital tracking.In addition, flat close in order to guarantee
The computational accuracy of point angle single order long period term, while considering the second order long and short period item of semi-major axis.
Fig. 3 show J2The influencing characterisitic of item gravitation, it will thus be seen that under current simulated conditions, J2Item gravitation is to trajectory position
Maximum is influenced close to 18km, and minimum is not less than 4km；J2Influence of the item gravitation to track changes with the variation of orbit parameter, but
Total trend is: when azimuth is -90 °, 0 ° and 90 ° close, J2The influence of item gravitation is most significant；When azimuth close to -50 ° or
At 50 °, J2The influence of item gravitation is most weak.
Fig. 4 show the J of the present embodiment derivation under the conditions of different orientations2Influence ballistic deflection parsing forecasting model with
Calculating residual error comparative situation of the mean element method relative to numerical integration result, see Table 2 for details:
2 the present embodiment of table and the in the prior art statistic analysis result of mean element hair calculating residual error
| Method | Maximum value (m) | Mean value (m) | Mean square deviation (m) |
| The present embodiment method | 29.6629 | 12.4930 | 7.3667 |
| Mean element method in the prior art | 366.7923 | 148.3389 | 112.0958 |
As can be seen from Table 2: under current emulation simulated conditions, the mean value that mean element method calculates residual error is 148.3389m,
Relative error is calculated 1% or so；Under current simulated conditions, the analytic solutions that the present embodiment proposes are higher than mean element by one in precision
A magnitude calculates relative error and is better than 2 ‰.
The foregoing is only a preferred embodiment of the present invention, is not intended to restrict the invention, for the skill of this field
For art personnel, the invention may be variously modified and varied.All within the spirits and principles of the present invention, made any to repair
Change, equivalent replacement, improvement etc., should all be included in the protection scope of the present invention.
Claims (2)
1. a kind of consideration J2The guided missile trajectory of free flight phase deviation that item influences parses forecasting procedure, it is characterised in that: including J2Item gravitation
The decomposition of vector and trajectory of free flight phase deviation parsing forecasting model derive；
J2The decomposition of item gravitation vector obtains the perturbative force such as expression formula 8 on different change in coordinate axis direction):
In formula: δ ar、δaβWith δ azRespectively indicate J2Point of the item gravitation vector in track cylindrical coordinate on r axis, β axis and z-axis direction
Amount；R is earth radius；
sr=-3K, sβ=K, sz=K；
μ is Gravitational coefficient of the Earth, aeFor terrestrial equator mean radius；
piFor constant coefficient, i takes 0,1,2,3,4, specific as follows:
For the latitude of P point, σ is lateral angle, αAFor longitude of the A point in pole coordinate system, f0Indicate the true anomaly at A point,
I.e. initial true anomaly；
q1And q2It is as follows:
The parsing forecasting model derivation of trajectory of free flight phase deviation is specifically:
According to state space perturbation theory, the Integration Solving expression of guided missile Study of Free Flight Phase Trajectory deviation is expression formula 9):
In formula: △ vr(f)、△vβ(f) and △ vz(f) be respectively trajectory state deviation velocity vector in track cylindrical coordinate along r
The component of axis, β axis and z-axis direction；△ r (f) and △ z (f) is respectively trajectory state deviation position vector in track cylindrical coordinate
The middle component along r axis and z-axis direction；△ t (f) is the difference of actual flying time and standard two-body ballistic flight time；H is two-body
The mould of the corresponding angular momentum vector of trajectory plane；Indicate that corresponding standard two-body trajectory the earth's core is away from that is, when true anomaly is ξP indicates the semi-latus rectum of two-body trajectory, and e is the eccentricity of two-body trajectory；
λ1,1(f, ξ)=cos (f- ξ)；
λ3,1(f, ξ)=- sin (f- ξ)；
μ is Gravitational coefficient of the Earth；
By expression formula 8) substitute into expression formula 9), and the complete analytical expression 10 integrated up to each single item deviation) -15):
In formula:
Λi,jIt is as shown in table 1:
1 function Λ of tablei,jExpression formula statistical form
2. consideration J according to claim 12The guided missile trajectory of free flight phase deviation that item influences parses forecasting procedure, and feature exists
In: J2The decomposition of item gravitation vector, specifically:
Enable U2Indicate earth J2Item gravitational potential, the J in Earth centered inertial system2Item gravitational potential is expression formula 1):
Wherein: μ is Gravitational coefficient of the Earth, aeFor terrestrial equator mean radius, J be constant andR is earth radius,
For terrestrial latitude；
According to spherical trigonometry by expression formula 1) independent variable true anomaly f is converted to by reduced latitude；Taking N is celestial coordinate system
North pole, orthodromeThe projection for being the standard two-body trajectory that is determined by guided missile Burnout parameter on spherical surface, curve AB
Indicate projection of the guided missile disturbed motion track on spherical surface, straight line OP is perpendicular to plane OAB*, αA, △ f be respectively plane POC with
The dihedral angle and plane POA and POB of POA*Dihedral angle,The latitude of respectively A point and the latitude of P point, λAAnd λPPoint
Not Wei A point and P point longitude, then have:
In spherical triangle ANP, expression formula 2 is obtained) -4):
Wherein: γ is the corresponding azimuth Burnout A, αAFor longitude of the A point in new pole coordinate system；
In spherical triangle BPN, expression formula 5 is obtained):
In formula: σ is lateral angle；f0Indicate the true anomaly at A point, i.e., initial true anomaly；F indicates that any time is corresponding on bullet
True anomaly, here P i.e. indicate arbitrary point；
By expression formula 5) bring expression formula 1 into) to get to J2Item gravitational potential is expression formula 6 about the function of true anomaly):
U is sought respectively2(f) about r, f, σ partial derivative to get arrive J2The expression formula of gravitation vector in track cylindrical coordinate is
Expression formula 7):
Calculated based on two-body normal trajectory, at this time σ=0, expression formula 7) in following coefficient move back cancellation, i.e.,
Perturbative force on different change in coordinate axis direction is uniformly used into expression formula 8) it indicates:
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| CN110609972A (en) * | 2019-09-30 | 2019-12-24 | 中国科学院紫金山天文台 | Free trajectory construction method for appointed launching elevation angle |
| CN111475767A (en) * | 2020-03-18 | 2020-07-31 | 中国科学院紫金山天文台 | Minimum energy trajectory strict construction method considering earth rotation influence |
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| CN110489879A (en) * | 2019-08-22 | 2019-11-22 | 中国人民解放军32035部队 | A kind of extraterrestrial target in the case of the disturbance suitable for space environment passes away forecasting procedure |
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| CN110609972A (en) * | 2019-09-30 | 2019-12-24 | 中国科学院紫金山天文台 | Free trajectory construction method for appointed launching elevation angle |
| CN111475767A (en) * | 2020-03-18 | 2020-07-31 | 中国科学院紫金山天文台 | Minimum energy trajectory strict construction method considering earth rotation influence |
| CN111475767B (en) * | 2020-03-18 | 2021-03-16 | 中国科学院紫金山天文台 | Minimum energy trajectory strict construction method considering earth rotation influence |
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