CN110059285B - 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 PDF

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CN110059285B
CN110059285B CN201910324168.0A CN201910324168A CN110059285B CN 110059285 B CN110059285 B CN 110059285B CN 201910324168 A CN201910324168 A CN 201910324168A CN 110059285 B CN110059285 B CN 110059285B
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王磊
郑伟
张洪波
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National University of Defense Technology
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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 J2Vector of term gravityQuantity decomposition and analysis and forecast model derivation of free-range trajectory deviation, 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

Consider J2Item-influenced missile free-section trajectory deviation analysis and prediction method
Technical Field
The invention relates to the technical field of flight dynamics, in particular to a method for considering earth J2The method is an analysis forecasting method for trajectory deviation of a missile free flight segment influenced by terms.
Background
The free flight segment is the segment with the longest flight time in the whole flight segment of the ballistic missile, and accounts for more than 90% of the total flight time. Due to the high flying height, the trajectory of the ballistic missile is similar to a part of an elliptical orbit under the action of earth central gravity mainly in a free flying section, but due to perturbation factors (such as earth non-spherical gravity, thin atmospheric resistance and the like), the real trajectory of the ballistic missile deviates from a standard elliptical orbit. In order to ensure the hit precision of the ballistic missile, the ballistic deviation of the free flight section under perturbation conditions needs to be quickly forecasted when the engine is controlled. In fact, extraterrestrial aircraft orbit prediction considering perturbation factors is one of the classic problems in the field of orbit dynamics, namely the initial value problem. The classical theory for this problem is mainly: a flat root method, an fg series decomposition method, a non-orthogonal decomposition method, an adaptive variable step numerical integration method, and the like. The method has a good effect when being used for predicting the orbits of the near-earth satellites, but the accuracy of the method is obviously reduced when the method is used for predicting the sub-orbits such as the free-section trajectory of the ballistic missile;the fg series method is to perform Taylor expansion on a standard elliptical trajectory by taking flight time as an independent variable, and deducing and considering J on the basis of certain reasonable approximation2The method is only suitable for short-time extrapolation and cannot be used for calculating the free-section trajectory of the ballistic missile; the non-orthogonal decomposition method is proposed in 1982 by Lizhuan of scholars in China2The missile free-section trajectory analysis calculation method based on item perturbation is applied to consider J2In the closed-circuit guidance on-line compensation method of the influence, the method can not ensure that the precision of the prediction of any downward trajectory is less than 100 meters; the self-adaptive variable-step numerical integration method is the most common method for accurate prediction of the current orbit, the efficiency of the traditional fixed-step orbit numerical integration can be greatly improved through self-adaptive condition step length, but under the condition that the guidance period on a ballistic missile is usually less than 20ms, an analytic algorithm with higher precision still needs to be researched.
Aiming at the defects of the current common orbit/trajectory prediction algorithm, a new method is designed by considering the earth J2The analysis and forecast method of the missile free section trajectory deviation under the influence of terms has important significance.
Disclosure of Invention
The object of the present invention is to provide a consideration of 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. The solution of this method is time consuming at 10 compared to other methods of the prior art-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. The specific technical scheme is as follows:
consider J2Term-influenced missile selfThe method for analyzing and forecasting deviation of section trajectory comprises J2Decomposing the item gravitation vector and deducing a free-segment trajectory deviation analysis forecasting model;
J2decomposing the term gravity vector to obtain the perturbation force in different coordinate axis directions as expression 8):
Figure BDA0002035754040000021
in the formula: delta ar、δaβAnd δ azRespectively represents J2The components of the term gravity vector in the directions of an r axis, an β axis and a z axis in an orbit column coordinate system, wherein r is the radius of the earth;
sr＝-3K,sβ＝K,sz＝K；
Figure BDA0002035754040000022
Figure BDA0002035754040000023
Figure BDA0002035754040000024
Figure BDA0002035754040000025
mu is the gravitational constant of the earth, aeIs the average radius of the equator of the earth;
pifor constant coefficients, i is 0,1,2,3,4, as follows:
Figure BDA0002035754040000026
Figure BDA0002035754040000027
latitude of point P, σ is the lateral angle, αAIs the longitude of point A in the polar coordinate system, f0Representing true paraxial angles at point A, i.e.An initial true proximal angle;
q1and q is2The following were used:
Figure BDA0002035754040000028
Figure BDA0002035754040000029
the derivation of the free-range ballistic deviation analysis forecasting model is specifically as follows:
according to the state space perturbation theory, the integral solving expression of the trajectory deviation of the missile free flight section is expressed as expression 9):
Figure BDA0002035754040000031
in the formula, △ vr(f)、△vβ(f) And △ vz(f) △ r (f) and △ z (f) are components of a deviation position vector of the ballistic state along the r axis and the z direction in the orbit cylindrical coordinate system, △ t (f) is the difference between the actual flight time and the standard ballistic flight time of the two bodies, h is the modulus of a momentum moment vector corresponding to the ballistic plane of the two bodies;
Figure BDA0002035754040000032
indicating the corresponding standard two-body ballistic centroid at true anomaly angle ξ, i.e.
Figure BDA0002035754040000033
p represents the radius of the two-body trajectory, e represents the eccentricity of the two-body trajectory;
Figure BDA0002035754040000034
Figure BDA0002035754040000035
Figure BDA0002035754040000036
Figure BDA0002035754040000037
Figure BDA0002035754040000038
Figure BDA0002035754040000039
Figure BDA0002035754040000041
Figure BDA0002035754040000042
mu is an earth gravity constant;
substituting expression 8) into expression 9), and integrating to obtain a complete analytical expression 10) -15 of each deviation):
Figure BDA0002035754040000043
Figure BDA0002035754040000044
Figure BDA0002035754040000045
Figure BDA0002035754040000046
Figure BDA0002035754040000047
Figure BDA0002035754040000048
in the formula:
Figure BDA0002035754040000049
Figure BDA00020357540400000410
Figure BDA00020357540400000411
Figure BDA00020357540400000412
Figure BDA00020357540400000413
Figure BDA0002035754040000051
Figure BDA0002035754040000052
Figure BDA0002035754040000053
Figure BDA0002035754040000054
Figure BDA0002035754040000055
Figure BDA0002035754040000056
Λi,jas shown in table 1:
TABLE 1 function Λi,jExpression statistical table of
Figure BDA0002035754040000057
Figure BDA0002035754040000058
Figure BDA0002035754040000059
Figure BDA00020357540400000510
Figure BDA00020357540400000511
Figure BDA0002035754040000061
Figure BDA0002035754040000062
Figure BDA0002035754040000063
Figure BDA0002035754040000064
Figure BDA0002035754040000065
Figure BDA0002035754040000066
Preferred in the above technical solution, J2The decomposition of the term gravity vector specifically comprises the following steps:
let U2Represents the earth J2Term gravitational potential, J in the earth's inertial system2The term gravitational potential is expression 1):
Figure BDA0002035754040000067
wherein: mu is the gravitational constant of the earth, aeIs the average radius of the equator of the earth, J is a constant and
Figure BDA0002035754040000068
r is the radius of the earth and r is the radius of the earth,
Figure BDA0002035754040000069
the latitude of the earth;
converting the independent variable of the expression 1) into a true anomaly angle f from the geocentric latitude according to spherical trigonometry; taking N as the north pole and the great arc of the celestial coordinate system
Figure BDA00020357540400000610
The projection of a standard two-body trajectory determined by the parameters of the shutdown point of the missile on the spherical surface, a curve AB represents the projection of the missile shooting motion track on the spherical surface, and a straight line OP is vertical to a plane OAB*，αA△ f are dihedral angle of POC and POA and POB, respectively*The angle of the two-sided angle of (c),
Figure BDA00020357540400000611
respectively, the latitude of the point A and the latitude, lambda, of the point PAAnd λPThe longitudes of point a and point P, respectively, are:
in spherical triangle ANP, expression 2) -4) is obtained:
Figure BDA00020357540400000612
Figure BDA00020357540400000613
Figure BDA00020357540400000614
wherein gamma is the azimuth angle corresponding to the shutdown point A, αAThe longitude of the point A in the new pole coordinate system;
in the spherical triangular BPN, expression 5) is obtained:
Figure BDA0002035754040000071
in the formula: σ is a lateral angle; f. of0Representing the true proximal angle at point a, i.e. the initial true proximal angle; f represents a true near point angle corresponding to any moment on the missile, wherein P represents any point;
substituting expression 5) into expression 1), i.e. J is obtained2The function of the term gravitational potential with respect to the true perigee angle is expression 6):
Figure BDA0002035754040000072
separately calculate U2(f) Partial derivatives of r, f, sigma, i.e. J2The expression of the term gravity vector in the rail cylindrical coordinate system is expression 7):
Figure BDA0002035754040000073
calculation is performed based on the two-body standard trajectory when σ is 0, the following coefficient in expression 7) is reduced to zero, that is
Figure BDA0002035754040000074
The perturbation forces in different coordinate axis directions are uniformly expressed by expression 8):
Figure BDA0002035754040000075
in addition to the objects, features and advantages described above, other objects, features and advantages of the present invention are also provided. The present invention will be described in further detail below with reference to the accompanying drawings.
Drawings
The accompanying drawings, which are incorporated in and constitute a part of this application, illustrate embodiments of the invention and, together with the description, serve to explain the invention and not to limit the invention. In the drawings:
FIG. 1 is a schematic diagram of a rail column coordinate system in an embodiment;
FIG. 2 is a diagram showing the relationship between spherical angles in the embodiment;
FIG. 3 is example J2Influence characteristics of the gravitational force on different tracks;
FIG. 4 is a view of the invention2The term-influenced missile free-section trajectory deviation analysis forecasting method and the flat root method are relative to a calculation residual error comparison diagram of a numerical integration result.
Detailed Description
Embodiments of the invention will be described in detail below with reference to the drawings, but the invention can be implemented in many different ways, which are defined and covered by the claims.
Example (b):
consider earth J2The method for analyzing and forecasting the deviation of the missile free section trajectory under the influence of terms comprises J2The method comprises the following steps of (1) decomposing an item gravitation vector and deducing a free-segment ballistic deviation analysis forecasting model, wherein the details are as follows:
1、J2the decomposition of the term gravity vector specifically comprises the following steps:
let U2Represents the earth J2Term gravitational potential, J in the earth's inertial system2The term gravitational potential is expression 1):
Figure BDA0002035754040000081
wherein: mu is the gravitational constant of the earth, aeIs the average radius of the equator of the earth, J is a constant and
Figure BDA0002035754040000082
r is the radius of the earth and r is the radius of the earth,
Figure BDA0002035754040000083
the latitude of the earth.
Converting the independent variable of the expression 1) into a true anomaly angle f from the geocentric latitude according to spherical trigonometry. As shown in FIG. 2, N is the north pole, the great arc, of the celestial coordinate system
Figure BDA0002035754040000084
The projection of a standard two-body trajectory determined by the parameters of the shutdown point of the missile on the spherical surface, a curve AB represents the projection of the missile shooting motion track on the spherical surface, and a straight line OP is vertical to a plane OAB*。αA△ f are dihedral angle of POC and POA and POB, respectively*The dihedral angle of (1).
Figure BDA0002035754040000085
Respectively, the latitude of the point A and the latitude, lambda, of the point PAAnd λPThe longitude of points a and P, respectively.
In the spherical triangle ANP, expressions 2) to 4) can be obtained:
Figure BDA0002035754040000086
Figure BDA0002035754040000087
Figure BDA0002035754040000088
wherein gamma is the azimuth angle corresponding to the shutdown point A, αAThe longitude of point a in the new polar coordinate system can be considered, see fig. 2 in particular.
In the spherical triangular BPN, expression 5) can be derived:
Figure BDA0002035754040000091
in the formula: σ is a lateral angle; f. of0Representing the true proximal angle at point a, i.e. the initial true proximal angle; f represents the true paraxial point angle corresponding to any time on the missile, wherein P represents any point.
Substituting expression 5) into expression 1), J can be obtained2The function of the term gravitational potential with respect to the true perigee angle is expression 6):
Figure BDA0002035754040000092
wherein:
Figure BDA0002035754040000093
mu is the gravitational constant of the earth, aeIs the average radius of the equator of the earth;
pifor constant coefficients, i is 0,1,2,3,4, specifically:
Figure BDA0002035754040000094
σ is the lateral angle.
Separately calculate U2(f) Partial derivatives of r, f, sigma, i.e. J2The expression of the term gravity vector in the rail cylindrical coordinate system (the definition of the rail cylindrical coordinate system is shown in fig. 1) is expression 7):
Figure BDA0002035754040000095
wherein, δ ar、δaβAnd δ azRespectively represents J2The components of the term gravity vector in the r-axis, β -axis and z-direction in the orbital cylindrical coordinate system.
In the trajectory error propagation analytic solution derivation process based on the state space perturbation method, the perturbation force is not calculated based on the current missile real position, but is calculated based on the two-body standard trajectory, at the moment, sigma is 0, and the following coefficient in the expression 7) is reduced to zero, namely, the coefficient is reduced to zero
Figure BDA0002035754040000096
The perturbation forces in different coordinate axis directions are uniformly expressed by expression 8):
Figure BDA0002035754040000097
in the formula:
sr＝-3K,sβ＝K,sz＝K；
Figure BDA0002035754040000101
Figure BDA0002035754040000102
Figure BDA0002035754040000103
wherein q is1And q is2The following were used:
Figure BDA0002035754040000104
Figure BDA0002035754040000105
2. derivation of a free-segment ballistic deviation analysis forecasting model is specifically as follows:
according to the state space perturbation theory, the integral solving expression of the trajectory deviation of the missile free flight section is expressed as expression 9):
Figure BDA0002035754040000106
in the formula, △ vr(f)、△vβ(f) And △ vz(f) Components of deviation velocity vector of ballistic state along r-axis, β -axis and z-direction in orbit cylindrical coordinate system, △ r (f) and △ z (f) components of deviation position vector of ballistic state along r-axis and z-direction in orbit cylindrical coordinate system, △ t (r and 3978:)f) Is made ofThe difference between the inter-flight time and the standard two-body ballistic flight time; h is the mode of the momentum moment vector corresponding to the two-body ballistic plane;
Figure BDA0002035754040000107
indicating the corresponding standard two-body ballistic centroid at true anomaly angle ξ, i.e.
Figure BDA0002035754040000108
And p represents the radius of the two-body trajectory, e represents the eccentricity of the two-body trajectory;
Figure BDA0002035754040000109
Figure BDA00020357540400001010
Figure BDA00020357540400001011
Figure BDA0002035754040000111
Figure BDA0002035754040000112
Figure BDA0002035754040000113
Figure BDA0002035754040000114
Figure BDA0002035754040000115
substituting expression 8) into expression 9), and integrating to obtain the complete analytical expression 10) -15 of each deviation):
Figure BDA0002035754040000116
Figure BDA0002035754040000117
Figure BDA0002035754040000118
Figure BDA0002035754040000119
Figure BDA00020357540400001110
Figure BDA00020357540400001111
in the formula:
Figure BDA00020357540400001112
all coefficients are constant, k is 1,2,3,4,5 and 6; taking 1,2,3,4,5,6 and 7; and has the following components:
Figure BDA00020357540400001113
Figure BDA00020357540400001114
Figure BDA0002035754040000121
Figure BDA0002035754040000122
Figure BDA0002035754040000123
Figure BDA0002035754040000124
Figure BDA0002035754040000125
Figure BDA0002035754040000126
Figure BDA0002035754040000127
Figure BDA0002035754040000128
Figure BDA0002035754040000129
Λi,jas shown in table 1:
TABLE 1 function Λi,jExpression statistical table of
Figure BDA00020357540400001210
Figure BDA0002035754040000131
Figure BDA0002035754040000132
Figure BDA0002035754040000133
Figure BDA0002035754040000134
Figure BDA0002035754040000135
Figure BDA0002035754040000136
Figure BDA0002035754040000137
Figure BDA0002035754040000138
Figure BDA0002035754040000139
Figure BDA00020357540400001310
Suppose the location vector of the shutdown point in the geocentric inertial system is x0＝[0，6578140，0]TThe initial velocity in the direction of the geodesic vector is 3300m/s and the initial velocity in the plane of the missile and perpendicular to the direction of the geodesic vector is 6680 m/s. While traversing the azimuth angle from-90 to 90. The trajectory deviation is analyzed and forecasted by adopting a numerical integration method, the analytic solution derived in the embodiment and the flat root method (assuming that the missile flies for 2700 seconds under different azimuth angles), and the calculation result of the numerical integration method is used as a reference for evaluating the accuracy of the analytic solution and the flat root method. The average number method adopts a first-order solution to calculate, namely only a first-order/second-order long-term, a first-order long-period term and a first-order short-period term of each orbit number are considered. In addition, in order to ensure the calculation precision of the first-order long period term of the mean-near point angle, the second-order long period term and the short period term of the semi-long axis are considered at the same time.
Shown in FIG. 3 as J2The influence characteristics of the attractive force can be seen as follows: under current simulation conditions, J2The influence of the attractive force on the ballistic position is close to 18km at most, and the influence of the attractive force on the ballistic position is not lower than 4km at least; j. the design is a square2The influence of gravity on the track varies with the track parameters,but the general trend is: when the azimuth angle approaches-90 °, 0 ° and 90 °, J2The influence of the gravitational force is most significant; when the azimuth angle approaches-50 ° or 50 °, J2The influence of gravity is the weakest.
FIG. 4 shows J derived from this embodiment under different azimuth conditions2The comparison between the term-influenced ballistic deviation analysis forecasting model and the calculated residual error of the flat root method relative to the numerical integration result is shown in table 2:
table 2 statistical analysis results of the flat root number calculation residuals in this embodiment and the prior art
Method of producing a composite material Maximum value (m) Mean value (m) Mean square error (m)
Method of the present embodiment 29.6629 12.4930 7.3667
Root number balancing method in prior art 366.7923 148.3389 112.0958
As can be seen from table 2: under the current simulation condition, the mean value of residual errors calculated by the root averaging method is 148.3389m, and the calculated relative error is about 1%; under the current simulation condition, the analytic solution provided by the embodiment is higher than the flat number by one magnitude in precision, and the calculated relative error is better than 2 per thousand.
The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention, and various modifications and changes may be made by those skilled in the art. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (2)

1. Consider J2The method for analyzing and forecasting the deviation of the missile free section trajectory under the influence of terms is characterized by comprising the following steps: comprising J2Decomposing the item gravitation vector and deducing a free-segment trajectory deviation analysis forecasting model;
J2decomposing the term gravity vector to obtain the perturbation force in different coordinate axis directions as expression 8):
Figure FDA0002402708060000011
in the formula: delta ar、δaβAnd δ azRespectively represents J2The components of the term gravity vector in the directions of an r axis, an β axis and a z axis in an orbit column coordinate system, wherein r is the radius of the earth;
sr＝-3K,sβ＝K,sz＝K；
Figure FDA0002402708060000012
Figure FDA0002402708060000013
Figure FDA0002402708060000014
Figure FDA0002402708060000015
mu is the gravitational constant of the earth, aeIs the average radius of the equator of the earth;
pifor constant coefficients, i is 0,1,2,3,4, as follows:
Figure FDA0002402708060000016
Figure FDA0002402708060000017
latitude of point P, σ is the lateral angle, αAIs the longitude of point A in the polar coordinate system, f0Representing the true proximal angle at point a, i.e. the initial true proximal angle;
q1and q is2The following were used:
Figure FDA0002402708060000018
Figure FDA0002402708060000019
the derivation of the free-range ballistic deviation analysis forecasting model is specifically as follows:
according to the state space perturbation theory, the integral solving expression of the trajectory deviation of the missile free flight section is expressed as expression 9):
Figure FDA0002402708060000021
in the formula: Δ vr(f)、Δvβ(f) And Δ vz(f) The components of the deviation speed vector of the ballistic state in the orbit cylindrical coordinate system along the r axis, β axis and z direction, respectively, [ delta ] r (f) and [ delta ] z (f) are the components of the deviation position vector of the ballistic state in the orbit cylindrical coordinate system along the r axis and the z direction, respectively, [ delta ] t (f) is the difference between the actual flight time and the flight time of the standard two-body ballistic trajectory, and h is the mode of the momentum moment vector corresponding to the two-body ballistic plane;
Figure FDA0002402708060000022
indicating the corresponding standard two-body ballistic centroid at true anomaly angle ξ, i.e.
Figure FDA0002402708060000023
p represents the radius of the two-body trajectory, e represents the eccentricity of the two-body trajectory;
Figure FDA0002402708060000024
Figure FDA0002402708060000025
Figure FDA0002402708060000026
Figure FDA0002402708060000027
Figure FDA0002402708060000028
Figure FDA0002402708060000029
Figure FDA00024027080600000210
Figure FDA0002402708060000031
mu is an earth gravity constant;
substituting expression 8) into expression 9), and integrating to obtain a complete analytical expression 10) -15 of each deviation):
Figure FDA0002402708060000032
Figure FDA0002402708060000033
Figure FDA0002402708060000034
Figure FDA0002402708060000035
Figure FDA0002402708060000036
Figure FDA0002402708060000037
in the formula:
Figure FDA0002402708060000038
Figure FDA0002402708060000039
Figure FDA00024027080600000310
Figure FDA00024027080600000311
Figure FDA00024027080600000312
Figure FDA00024027080600000313
Figure FDA0002402708060000041
Figure FDA0002402708060000042
Figure FDA0002402708060000043
Figure FDA0002402708060000044
Figure FDA0002402708060000045
Figure FDA0002402708060000046
representing the standard two-body ballistic centroid distance corresponding to a true anomaly f, i.e.
Figure FDA0002402708060000047
p represents the radius of the two-body trajectory, e represents the eccentricity of the two-body trajectory;
Λi,jas shown in table 1:
TABLE 1 function Λi,jExpression statistical table of
Figure FDA0002402708060000048
Figure FDA0002402708060000049
Figure FDA00024027080600000410
Figure FDA00024027080600000411
Figure FDA0002402708060000051
Figure FDA0002402708060000052
Figure FDA0002402708060000053
Figure FDA0002402708060000054
Figure FDA0002402708060000055
Figure FDA0002402708060000056
Figure FDA0002402708060000057
2. Consideration J according to claim 12The method for analyzing and forecasting the deviation of the missile free section trajectory under the influence of terms is characterized by comprising the following steps: j. the design is a square2The decomposition of the term gravity vector specifically comprises the following steps:
let U2Represents the earth J2Term gravitational potential, J in the earth's inertial system2The term gravitational potential is expression 1):
Figure FDA0002402708060000058
wherein: mu is the gravitational constant of the earth, aeIs the average radius of the equator of the earth, J is a constant and
Figure FDA0002402708060000059
r is the radius of the earth and r is the radius of the earth,
Figure FDA00024027080600000510
the latitude of the earth;
converting the independent variable of the expression 1) into a true anomaly angle f from the geocentric latitude according to spherical trigonometry; taking N as north pole of celestial coordinate system and big arc AB*The projection of a standard two-body trajectory determined by the parameters of the shutdown point of the missile on the spherical surface, a curve AB represents the projection of the missile shooting motion track on the spherical surface, and a straight line OP is vertical to a plane OAB*，αAΔ f are dihedral angle of the plane POC and POA and plane POA and POB, respectively*The angle of the two-sided angle of (c),
Figure FDA00024027080600000511
respectively, the latitude of the point A and the latitude, lambda, of the point PAAnd λPThe longitudes of point a and point P, respectively, are:
in spherical triangle ANP, expression 2) -4) is obtained:
Figure FDA00024027080600000512
Figure FDA00024027080600000513
Figure FDA00024027080600000514
wherein gamma is the azimuth angle corresponding to the shutdown point A, αAThe longitude of the point A in the new pole coordinate system;
in the spherical triangular BPN, expression 5) is obtained:
Figure FDA0002402708060000061
in the formula: σ is a lateral angle; f. of0Representing the true proximal angle at point a, i.e. the initial true proximal angle; f represents a true near point angle corresponding to any moment on the missile, wherein P represents any point;
substituting expression 5) into expression 1), i.e. J is obtained2The function of the term gravitational potential with respect to the true perigee angle is expression 6):
Figure FDA0002402708060000062
separately calculate U2(f) Partial derivatives of r, f, sigma, i.e. J2The expression of the term gravity vector in the rail cylindrical coordinate system is expression 7):
Figure FDA0002402708060000063
Figure FDA0002402708060000064
Figure FDA0002402708060000065
calculation is performed based on the two-body standard trajectory when σ is 0, the following coefficient in expression 7) is reduced to zero, that is
Figure FDA0002402708060000066
The perturbation forces in different coordinate axis directions are uniformly expressed by expression 8):
Figure FDA0002402708060000067
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