CN113722958A - Efficient modeling method for irregular-shaped small celestial body gravitational field - Google Patents

Efficient modeling method for irregular-shaped small celestial body gravitational field Download PDF

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CN113722958A
CN113722958A CN202111002679.4A CN202111002679A CN113722958A CN 113722958 A CN113722958 A CN 113722958A CN 202111002679 A CN202111002679 A CN 202111002679A CN 113722958 A CN113722958 A CN 113722958A
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CN113722958B (en
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刘继中
尚海滨
赵梓辰
韦炳威
徐瑞
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Beijing Institute of Technology BIT
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Abstract

The invention discloses a high-efficiency modeling method for an irregular-shaped small celestial body gravitational field, and belongs to the technical field of aerospace. The implementation method of the invention comprises the following steps: selecting a modeling task area, and discretizing the whole space in the task area into a series of tiny triangular pyramid spaces; acquiring gravitational field residual data at the vertex by calculating the two-body gravitational acceleration at the vertex and making a difference with the nominal gravitational acceleration; obtaining a gravitational field residual error linear expression form of the inner area of the triangular pyramid through gravity center interpolation; traversing and calculating all triangular pyramids in the task area, and converting the gravitational field residual data into a grid judgment constraint and linear interpolation form; summing the gravitational field parameters and the two-body gravitational acceleration to obtain a local linear gravitational field model based on grid judgment; and obtaining a local gravitational field model near the asteroid meeting the precision requirement by iteratively giving the spatial discrete scale of the linear gravitational field. The method can obviously improve the efficiency of gravitational field modeling under the condition of meeting the task requirements.

Description

Efficient modeling method for irregular-shaped small celestial body gravitational field
Technical Field
The invention relates to a local rapid modeling method for a gravitational field near a deep space celestial body, in particular to a local rapid modeling method for a gravitational field near a asteroid, and belongs to the technical field of aerospace.
Background
The gravitational field modeling is a key technology for realizing asteroid exploration, and has important application value in the asteroid exploration task. Due to the irregular shape and uneven density of the asteroid, the gravitational field near the asteroid is usually strongly nonlinear. On one hand, how to model a strong nonlinear gravitational field near the asteroid to obtain a gravitational field mathematical model with higher precision suitable for an actual detection task is a difficult point to solve; on the other hand, in order to improve the safety and reliability of the actual asteroid exploration task, how to realize the rapid modeling of the gravitational field near the asteroid so as to deal with the emergency in the exploration task is also a key problem to be solved. Gravitational field modeling near the asteroid is one of the hot issues that are currently of interest to technologists.
In The developed method for modeling The Gravitational field near The minor planet, in The prior art [1] (Werner, r.a., "The graphical Potential of a Homogeneous Polyhedron or Don't cuts Corners," Celestial Mechanics and dynamic advancement, 1993), a polyhedral modeling method is proposed for modeling The Gravitational field near The minor planet, which uses a series of spatial polyhedral grids to disperse The minor planet, thereby obtaining The polyhedral Gravitational field of The minor planet in a numerical summation form. The method has the advantages that the calculation precision is high, and the obtained gravitational field near the asteroid is verified in many practical tasks. Meanwhile, the method needs a large amount of calculation, is low in calculation efficiency, and is difficult to be applied to an actual asteroid detection task with a function of rapidly calculating the gravitational field near the asteroid.
In the prior art [2] (Herrera, S.E., Palmer, P.L., and Roberts, R.M., "Modeling the visual Potential of a nonlinear analog, and" Celestial Mechanics and dynamic analog, 2013), a spherical harmonic and Bessel Gravitational field Modeling method is provided for the problem of Modeling the Gravitational field near the minor planet, a reference sphere is selected, a spherical harmonic Modeling method is adopted for the spherical external Gravitational field, and a Bessel function Modeling method is adopted for the spherical internal Gravitational field. Compared with a polyhedral model, the method has the advantages that the gravitational field calculation efficiency can be improved, and the method has the defects that the modeling precision is low, and the method is difficult to be suitable for a small celestial body detection task with high track precision requirement.
Disclosure of Invention
In order to solve the problem that the calculation efficiency and the precision are difficult to be considered when the existing method carries out the modeling of the gravitational field near the asteroid, the invention discloses a method for efficiently modeling the gravitational field of the small celestial body with an irregular shape, which aims to: the method has the advantages that local rapid modeling of the gravitational field near the asteroid is realized, the modeling calculation efficiency of the gravitational field near the asteroid is improved on the premise of ensuring the modeling precision, and the technical problems related to application of the gravitational field near the asteroid are solved. The technical problem includes improving the efficiency of trajectory planning near the asteroid.
The purpose of the invention is realized by the following technical scheme:
the invention discloses an efficient modeling method for an irregular-shaped celestial body gravitational field. And in each triangular pyramid, gravity field residual data at the vertex is obtained by calculating the two-body gravity acceleration at the vertex and making a difference with the nominal gravity acceleration. On the basis, a gravitational field residual error linear expression form of the inner area of the triangular pyramid is obtained through gravity center interpolation. And traversing all triangular pyramids in the task area, and converting gravitational field residual data in the whole task area into a grid judgment constraint and linear interpolation form. And summing the obtained gravitational field parameters and the acceleration of the two-body gravitational force to obtain a local linear gravitational field based on grid judgment. And finally, calculating the space grid discrete scale meeting the precision requirement by iteratively giving the space discrete scale of the linear gravitational field, and obtaining a local gravitational field model near the asteroid meeting the precision requirement. The gravitational field can avoid a large amount of numerical calculation in the traditional modeling method, greatly improves the efficiency of the gravitational field modeling under the condition of meeting task requirements, and further solves the technical problem related to the application of the gravitational field near the asteroid. The technical problem includes improving the efficiency of trajectory planning near the asteroid.
The invention discloses a high-efficiency modeling method for an irregular-shaped small celestial body gravitational field, which comprises the following steps of:
step one, discretizing the area near the asteroid to obtain a series of spatially discrete triangular pyramid grids.
Defining the task area near the asteroid as a regular hexahedron, wherein the hexahedron boundary is represented by six characteristic parameters and is divided into a maximum X-axis coordinate rxmaxMinimum X-axis coordinate rxminMaximum Y-axis coordinate rymaxMinimum Y-axis coordinate ryminMaximum Z-axis coordinate rzmaxMinimum Z-axis coordinate rzmin。
Respectively giving the number of grid discrete points in the three-axis direction and the number of X-axis discrete points nxNumber n of Y-axis discrete pointsyZ-axis discrete point number nzThe region omega is divided into a series of small regular hexahedral meshes, denoted omegaj
Ωj：＝[rx，i，rx，i+1]×[ry，i，ry，i+1]×[rz，i，rz，i+1]，j＝1，2，...，(nx-1)·(ny-1)·(nz-1)
Wherein: qjRepresenting the jth regular hexahedral mesh.
For each small regular hexahedron, it is divided into a number of triangular pyramids.
In order to reduce the number of discrete grids and further improve the calculation efficiency, it is preferable that each of the small regular hexahedrons is divided into five triangular pyramids with different volumes.
For each small regular hexahedron, the small regular hexahedron is divided into five triangular pyramids with unequal volumes, and then N is obtainedS＝5·(nx-1)·(ny-1)·(nz-1) triangular pyramid grids, so that the task area Ω can be represented as
Figure BDA0003236082370000022
And step two, calculating the gravity acceleration of the center of the two bodies at the vertex of each discrete triangular pyramid grid, and subtracting the gravity acceleration from the nominal gravity field to obtain a linear residual form of the gravity field.
Defining the position vector of any point in the triangular pyramid as rkFirst, the gravitational acceleration at that point is calculated. Calculating the acceleration of the two bodies of gravity at the point and making a difference with the nominal acceleration of gravity to obtain a linear residual expression form of the gravitational field
Figure BDA0003236082370000021
Wherein: sigma (r)k) Representing a triangular pyramid thetakPoint r in the regionkThe residual error with respect to the central two-body gravity,
Figure BDA0003236082370000031
represents a point rkAcceleration of two bodies of gravity of the type
Figure BDA0003236082370000032
Wherein: mu.sastThe central two-body attraction constant of the asteroid is shown by observation.
Obtaining the gravitational field residual error form of each triangular pyramid vertex
Figure BDA0003236082370000033
And thirdly, performing linear gravity center interpolation on residual data of each triangular pyramid vertex to obtain a gravity field linear expression form in the whole task area.
The gravitational field residual data at each vertex of the triangular pyramid is represented as { sigma }(r1)，σ(r2)，σ(r3)，σ(r4)}。
Conversion coefficient { alpha) for defining barycentric interpolation coordinate and three-dimensional rectangular space coordinate system1，α2，α3，α4Rewriting the gravitational field residual error into the following gravity center coordinate form
σ(rk)＝α1σ(r1)+α2σ(r2)+α3σ(r3)+α4σ(r4)
The coefficients of the coordinate system satisfy the following relationship
Figure BDA0003236082370000034
The conversion factor α can then be solved back1(rk)，α2(rk)，α3(rk)，α4(rk) Substituting the residual error into a gravitational field residual error calculation formula under a gravity center coordinate system to obtain an interpolation form residual error of any point in the triangular pyramid
σ(rk)＝α1(rk)σ(r1)+α2(rk)σ(r2)+α3(rk)σ(r3)+α4(rk)σ(r4)
Traversing and calculating all triangular pyramid grids to obtain a gravitational field in the whole task area, namely expressing the gravitational field in an omega area as
Figure BDA0003236082370000035
Wherein r represents the space three-dimensional position coordinate and vector of any point in the whole task area
Figure BDA00032360823700000310
l, and the matrix Φ are calculated as follows.
l＝(rT，1)T
Figure BDA0003236082370000036
Figure BDA00032360823700000311
And step four, gradually increasing the grid discrete scale, calculating the modeling precision of the asteroid gravitational field, obtaining the asteroid gravitational field meeting the precision requirement, and realizing the rapid modeling of the gravitational field near the asteroid.
Defining a set of detection points in a task area as
Figure BDA0003236082370000037
For any one
Figure BDA0003236082370000038
Corresponding nominal gravitational field data of
Figure BDA0003236082370000039
For a given residual gravitational field model, the modeling accuracy of the model is defined as
Figure BDA0003236082370000041
Wherein: the function Λ (r) represents the gravitational acceleration at the detection point r output by the residual gravitational field model, and the function max {. cndot.) represents taking the maximum value for all elements in the set.
Given the discrete scale of the spatial grid, denoted as ρ ═ nx×ny×nzCalculating the asteroid gravitational field through the first step to the third step to further obtain the space grid modeling precision, and defining the relationship between the space grid modeling precision and the grid discrete scale described by the following mapping function
Figure BDA0003236082370000042
Wherein, ΛρRepresenting a model of the gravitational field near the asteroid at a particular grid discrete scale.
Given an allowable modeling error epsilon, an iteration step lambda of a modeling precision improvement process, and an initial grid discrete density rho0Then, in the ith iteration, define the grid discrete density as
ρi＝ρ0+λ·i
Obtaining the modeling precision P (rho) in the ith iteration through a mapping functioni)。
If the accuracy does not meet the requirement, i.e. P (ρ)i) And if the dispersion density is larger than epsilon, increasing the dispersion density and carrying out next iteration.
If the accuracy meets the requirement, i.e., P (ρ)i) Less than or equal to epsilon, and outputting a asteroid gravitational field model Lambda meeting the precision requirementρAnd corresponding discretized grid { ΘkAnd acquiring the asteroid gravitational field meeting the precision requirement, and realizing rapid modeling of the gravitational field near the asteroid.
Further comprises the following steps: and acquiring the asteroid gravitational field meeting the precision requirement according to the step four, and solving the technical problem related to the application of the gravitational field near the asteroid. The technical problem includes improving the efficiency of trajectory planning near the asteroid.
Has the advantages that:
1. the invention discloses an efficient modeling method for an irregular-shaped small celestial body gravitational field. And in each triangular pyramid, gravity field residual data at the vertex is obtained by calculating the two-body gravity acceleration at the vertex and making a difference with the nominal gravity acceleration. On the basis, a gravitational field residual error linear expression form of the inner area of the triangular pyramid is obtained through gravity center interpolation. And traversing all triangular pyramids in the task area, and converting gravitational field residual data in the whole task area into a grid judgment constraint and linear interpolation form. And summing the obtained gravitational field parameters and the acceleration of the two-body gravitational force to obtain a local linear gravitational field model based on grid judgment. And finally, calculating the space grid discrete scale meeting the precision requirement by iteratively giving the space discrete scale of the linear gravitational field, and obtaining a local gravitational field model near the asteroid meeting the precision requirement. The method can avoid a large amount of numerical calculation in the traditional modeling method, and greatly improves the efficiency of gravitational field modeling under the condition of meeting task requirements.
2. The invention discloses an efficient modeling method for an irregular-shaped celestial body gravitational field. Due to the linear gravity interpolation characteristic of the model, under the condition of any small tolerance error, a proper space grid discretization scale can be found, namely, on the premise of ensuring the modeling precision of the gravitational field, the modeling efficiency of the gravitational field near the asteroid is remarkably improved.
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FIG. 1 is a flow chart of an efficient modeling method for a small celestial body gravitational field with an irregular shape, which is disclosed by the invention;
FIG. 2 is a schematic diagram of a spatial triangular pyramid grid discretization method;
FIG. 3 is a schematic diagram of a force field gravity center interpolation method in a space triangular pyramid grid;
FIG. 4 is a simulation diagram of the relationship between the modeling precision of the gravitational field near the irregular asteroid and the grid discrete scale.
FIG. 5 is a comparison of the calculation efficiency of the gravitational field model and the polyhedral gravitational field model
Detailed Description
For a better understanding of the objects and advantages of the present invention, reference should be made to the following detailed description taken in conjunction with the accompanying drawings and examples.
Example 1: asteroid 1999KW4 local gravitational field rapid modeling
As shown in fig. 1, in the local rapid modeling method for the gravitational field near the asteroid disclosed in this embodiment, a asteroid 1999KW4 is selected for performing gravitational field modeling, and the specific steps are as follows:
step one, giving a discretization scale of a space grid, and discretizing an area near a asteroid to obtain a series of spatially discrete small regular hexahedron grids.
Defining the task area near the asteroid as a regular hexahedron, wherein the hexahedron boundary is represented by six characteristic parameters and is divided into a maximum X-axis coordinate rxmax1.27km, minimum X-axis coordinate rxminMaximum Y-axis coordinate r of 0.37kmymax1.24km, minimum Y-axis coordinate rymin0.35km, maximum Z-axis coordinate rzmax0.83km, minimum Z-axis coordinate rzmin＝0.33km。
Respectively giving the number of grid discrete points in the three-axis direction and the number of X-axis discrete points nxNumber of discrete points n on Y-axis of 5yNumber of discrete points n on Z-axiszThe region Ω is divided into a series of small regular hexahedral meshes denoted as Ω, 5j
Ωj：＝[rx，i，rx，i+1]×[ry，i，ry，i+1]×[rz，i，rz，i+1]，j＝1，2，...，(nx-1)·(ny-1)·(nz-1)
Wherein: omegajRepresenting the jth regular hexahedral mesh.
And step two, giving a small regular hexahedral mesh in the task area, and performing further mesh discretization on the small regular hexahedral mesh to obtain a series of small triangular pyramid meshes which are spatially discretized.
For each small regular hexahedron, the small regular hexahedron is divided into five triangular pyramids with unequal volumes, and the specific division rule is shown in fig. 1, so that N can be obtainedS＝5·(nx-1)·(ny-1)·(nz-1) 80 triangular pyramid grids, the task area Ω can then be represented as
Figure BDA0003236082370000069
And step three, calculating the gravity acceleration of the two bodies at the vertex of each small triangular pyramid grid.
Defining the position vector of any point in the triangular pyramid as rkFirst, the gravitational acceleration at that point is calculated. Calculating the acceleration of gravity of the two bodies at the point and subtracting from the nominal acceleration of gravity to obtain
Figure BDA0003236082370000061
Wherein: sigma (r)k) Representing a triangular pyramid thetakPoint r in the regionkThe residual error with respect to the central two-body gravity,
Figure BDA0003236082370000062
represents a point rkAcceleration of two bodies of gravity of the type
Figure BDA0003236082370000063
Wherein: mu.sastThe central two-body attraction constant of the asteroid is shown by observation.
And step four, calculating the residual error at the vertex of each triangular pyramid grid according to the nominal gravitational acceleration data.
Selecting a triangular pyramid grid area, and defining the triangular pyramid grid area as thetakThe method described in the first step and the second step can obtain four vertexes of the triangular pyramid region, and the four vertexes are expressed as { r1，r2，r3，r4Acquiring gravitational acceleration at the vertex according to the nominal gravitational field data, respectively defining as
Figure BDA0003236082370000064
Accordingly, a residual error of the nominal gravitational acceleration at each vertex of the triangular pyramid with respect to the two-body gravitational acceleration can be calculated and expressed as
Figure BDA0003236082370000065
And step five, calculating barycentric interpolation coordinates in the small triangular pyramid region in each small triangular pyramid grid according to residual error data.
Conversion coefficient { alpha) for defining barycentric interpolation coordinate and three-dimensional rectangular space coordinate system1，α2，α3，α4Rewriting the gravitational field residual error into the following gravity center coordinate form
σ(rk)＝α1σ(rk)+α2σ(rk)+α3σ(rk)+α4σ(rk)
The coefficients of the coordinate system satisfy the following relationship
Figure BDA0003236082370000066
The conversion factor α can then be solved back1(rk)，α2(rk)，α3(rk)，α4(rk) Substituting the obtained result into a gravitational field residual error calculation formula under a gravity center coordinate system, wherein a specific coordinate system conversion schematic diagram is shown in fig. 2.
And step six, obtaining a local gravitational field model near the asteroid in a two-body residual error mode.
Expressed as nominal gravitational field data
Figure BDA0003236082370000067
And the following calculation formula
Figure BDA0003236082370000068
Residual data at each vertex of the triangular pyramid can be obtained, denoted as { σ (r) }1)，σ(r2)，σ(r3)，σ(r4)}。
Thus, the gravity acceleration in the form of linear interpolation of any point in space can be obtained
Figure BDA0003236082370000071
Wherein: sigma (r)k) Representing a triangular pyramid thetakPoint r in the regionkThe residual error with respect to acceleration of the two-body gravity.
Further, a linear residual form gravitational field inside the triangular pyramid can be obtained
Figure BDA0003236082370000072
And seventhly, performing gravity center interpolation on the residual error data of the vertex in each small triangular pyramid grid to obtain a linear gravitational field in the region.
Conversion coefficient { alpha) for defining barycentric interpolation coordinate and three-dimensional rectangular space coordinate system1，α2，α3，α4Rewriting the gravitational field into the following gravity coordinate form
σ(rk)＝{α1σ(r1)，α2σ(r2)，α3σ(r3)，α4σ(r4)}
The coefficients of the coordinate system satisfy the following relationship
Figure BDA0003236082370000073
The conversion factor α can then be solved back1(rk)，α2(rk)，α3(rk)，α4(rk) Substituting the obtained result into a gravitational field calculation formula under a gravity center coordinate system, and obtaining a specific coordinate system conversion schematic diagram as shown in fig. 2.
σ(rk)＝{α1(rk)σ(r1)，α2(rk)σ(r2)，α3(rk)σ(r3)，α4(rk)σ(r4)}
And step eight, integrating all the small triangular pyramid grids to obtain a gravitational field model in the whole task area.
According to the calculation method in the seventh step, all triangular pyramid grids are calculated in a traversing manner, and the gravitational field in the whole task area, namely the gravitational field in the omega area can be represented as
Figure BDA0003236082370000074
Wherein r represents the space three-dimensional position coordinate and vector of any point in the whole task area
Figure BDA0003236082370000076
l, and the matrix Φ are calculated as follows.
l＝(rT，1)T
Figure BDA0003236082370000075
Figure BDA0003236082370000077
And step nine, randomly selecting inspection points in the task area, calculating gravitational acceleration at the inspection points, and obtaining the modeling precision of the gravitational field near the asteroid under the grid discrete scale.
Defining a set of detection points in a task area as
Figure BDA0003236082370000078
The set contains 1000 random detection points for any
Figure BDA0003236082370000079
Corresponding nominal gravitational field data of
Figure BDA00032360823700000710
For a given residual gravitational field model, the modeling accuracy of the model is defined as
Figure BDA0003236082370000081
Wherein: the function Λ (r) represents the gravitational acceleration at the detection point r output by the residual gravitational field model, and the function max {. cndot.) represents taking the maximum value for all elements in the set.
And step ten, judging whether the modeling precision meets the task requirement.
Regarding the calculation process from the first step to the ninth step as a mapping function, the input of the function is the discrete scale of the space grid, and is expressed as rho ═ nx×ny×nzThe output is the modeling precision of the space grid, and the following mapping function is obtained by combining the modeling precision definition formula
Figure BDA0003236082370000082
Wherein, ΛρRepresenting a model of the gravitational field near the asteroid at a particular grid discrete scale.
Given an allowable modeling error ε of 1 × 10-9The iteration step length lambda of the modeling accuracy improvement process is 5, and the discrete density rho of the initial grid05. Through the modeling precision calculation formula, the modeling precision P (rho) in the ith iteration can be obtainedi)。
If the accuracy meets the requirement, i.e., P (ρ)i) And E, executing the step eleven.
If the accuracy does not meet the requirement, i.e. P (ρ)i) Greater than epsilon, increasing the discrete density, order
ρi＝ρ0+λ·i
And repeating the step one to the step ten.
And step eleven, outputting the gravitational field model near the asteroid obtained in the step eleven.
According to the eleventh step, obtaining the grid discretization scale meeting the precision requirement, and defining the grid discretization scale as rhom＝nxm×nym×nzm. According to the method described in the first to the ninth steps,obtaining a gravitational field model ΛρAnd corresponding discretized grid { Θk}。
And step twelve, establishing an orbit dynamics model near the asteroid according to the gravitational field model output in the step eleven.
Kinetic parameters were defined from the 1999KW4 asteroid parameters, where spin angular velocity ω ═ 0, 0, 6.31 × 10-4)Trad/s, the following kinetic model was constructed.
if r∈Θk，then
Figure BDA0003236082370000083
Figure BDA0003236082370000084
The modeling accuracy of the 1999KW4 asteroid-nearby gravitational field calculated by the example is related to the grid discretization scale as shown in FIG. 3, and the calculation efficiency ratio of the polyhedron model is shown in FIG. 4.
And thirteen, according to the orbital dynamics model near the asteroid established in the step twelve, the planning efficiency of the orbit near the asteroid is improved.
The above detailed description is intended to illustrate the objects, aspects and advantages of the present invention, and it should be understood that the above detailed description is only exemplary of the present invention and is not intended to limit the scope of the present invention, and any modifications, equivalents, improvements and the like made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (7)

1. An efficient modeling method for a small celestial body gravitational field with an irregular shape is characterized by comprising the following steps: comprises the following steps of (a) carrying out,
carrying out discretization treatment on an area near a minor planet to obtain a series of spatially discrete triangular pyramid grids;
calculating the gravity acceleration of the center of the two bodies at the vertex of each discrete triangular pyramid grid, and subtracting the gravity acceleration from the nominal gravity field to obtain a linear residual form of the gravity field;
performing linear gravity center interpolation on residual data of each triangular pyramid vertex to obtain a gravity field linear expression form in the whole task area;
and step four, gradually increasing the grid discrete scale, calculating the modeling precision of the asteroid gravitational field, obtaining the asteroid gravitational field meeting the precision requirement, and realizing the rapid modeling of the gravitational field near the asteroid.
2. The method for efficiently modeling the gravitational field of small irregular-shaped celestial bodies of claim 1, wherein: the method also comprises a fifth step of obtaining the asteroid gravitational field meeting the precision requirement according to the fourth step, and solving the technical problem related to the application of the gravitational field near the asteroid; the technical problem includes improving the efficiency of trajectory planning near the asteroid.
3. The method for efficiently modeling the gravitational field of small irregular-shaped celestial bodies as claimed in claim 1 or 2, wherein: the first implementation method comprises the following steps of,
defining the task area near the asteroid as a regular hexahedron, wherein the hexahedron boundary is represented by six characteristic parameters and is divided into a maximum X-axis coordinate rxmaxMinimum X-axis coordinate rxminMaximum Y-axis coordinate rymaxMinimum Y-axis coordinate ryminMaximum Z-axis coordinate rzmaxMinimum Z-axis coordinate rzmin；
Respectively giving the number of grid discrete points in the three-axis direction and the number of X-axis discrete points nxNumber n of Y-axis discrete pointsyZ-axis discrete point number nzThe region omega is divided into a series of small regular hexahedral meshes, denoted omegaj
Ωj:＝[rx,i,rx,i+1]×[ry,i,ry,i+1]×[rz,i,rz,i+1],j＝1,2,...,(nx-1)·(ny-1)·(nz-1)
Wherein: omegajIs shown asj regular hexahedral meshes;
for each small regular hexahedron, it is divided into a number of triangular pyramids.
4. The method for efficiently modeling the gravitational field of small irregular-shaped celestial bodies of claim 3, wherein: the second step is realized by the method that,
defining the position vector of any point in the triangular pyramid as rkFirstly, calculating the gravitational acceleration at the point; calculating the acceleration of the two bodies of gravity at the point and making a difference with the nominal acceleration of gravity to obtain a linear residual expression form of the gravitational field
Figure FDA0003236082360000011
Wherein: sigma (r)k) Representing a triangular pyramid thetakPoint r in the regionkThe residual error with respect to the central two-body gravity,
Figure FDA0003236082360000012
represents a point rkAcceleration of two bodies of gravity of the type
Figure FDA0003236082360000013
Wherein: mu.sastRepresents the observed gravitational constant of the central two bodies of the asteroid;
obtaining the gravitational field residual error form of each triangular pyramid vertex
Figure FDA0003236082360000021
5. The method for efficiently modeling the gravitational field of small irregular-shaped celestial bodies of claim 4, wherein: the third step is to realize the method as follows,
the gravitational field residual data at each vertex of the triangular pyramid is denoted as { σ (r) }1),σ(r2),σ(r3),σ(r4)}；
Conversion coefficient { alpha) for defining barycentric interpolation coordinate and three-dimensional rectangular space coordinate system1,α2,α3,α4Rewriting the gravitational field residual error into the following gravity center coordinate form
σ(rk)＝α1σ(r1)+α2σ(r2)+α3σ(r3)+α4σ(r4)
The coefficients of the coordinate system satisfy the following relationship
Figure FDA0003236082360000022
Then the conversion coefficient [ alpha ] is solved reversely1(rk),α2(rk),α3(rk),α4(rk) Substituting the residual error into a gravitational field residual error calculation formula under a gravity center coordinate system to obtain an interpolation form residual error of any point in the triangular pyramid
σ(rk)＝α1(rk)σ(r1)+α2(rk)σ(r2)+α3(rk)σ(r3)+α4(rk)σ(r4)
Traversing and calculating all triangular pyramid grids to obtain a gravitational field in the whole task area, namely expressing the gravitational field in an omega area as
Figure FDA0003236082360000023
Wherein r represents the space three-dimensional position coordinate and vector of any point in the whole task area
Figure FDA00032360823600000210
l, and the matrix phi are calculated by the following formula;
l＝(rT,1)T
Figure FDA0003236082360000024
Figure FDA0003236082360000025
6. the method for efficiently modeling the gravitational field of small irregular-shaped celestial bodies of claim 5, wherein: the implementation method of the fourth step is that,
defining a set of detection points in a task area as
Figure FDA0003236082360000026
For any one
Figure FDA0003236082360000027
Corresponding nominal gravitational field data of
Figure FDA0003236082360000028
For a given residual gravitational field model, the modeling accuracy of the model is defined as
Figure FDA0003236082360000029
Wherein: the function Λ (r) represents the gravitational acceleration at the detection point r output by the residual gravitational field model, and the function max {. is the maximum value of all elements in the set;
given the discrete scale of the spatial grid, denoted as ρ ═ nx×ny×nzCalculating the asteroid gravitational field through the first step to the third step to further obtain the space grid modeling precision, and defining the relationship between the space grid modeling precision and the grid discrete scale described by the following mapping function
Figure FDA0003236082360000031
Wherein, ΛρRepresenting a asteroid nearby gravitational field model at a particular grid discrete scale;
given an allowable modeling error epsilon, an iteration step lambda of a modeling precision improvement process, and an initial grid discrete density rho0Then, in the ith iteration, define the grid discrete density as
ρi＝ρ0+λ·i
Obtaining the modeling precision P (rho) in the ith iteration through a mapping functioni)；
If the accuracy does not meet the requirement, i.e. P (ρ)i) If the dispersion density is larger than epsilon, increasing the dispersion density and carrying out next iteration;
if the accuracy meets the requirement, i.e., P (ρ)i) Less than or equal to epsilon, and outputting a asteroid gravitational field model Lambda meeting the precision requirementρAnd corresponding discretized grid { ΘkAnd acquiring the asteroid gravitational field meeting the precision requirement, and realizing rapid modeling of the gravitational field near the asteroid.
7. The method for efficiently modeling the gravitational field of small irregular-shaped celestial bodies of claim 3, wherein: for each small regular hexahedron, dividing the small regular hexahedron into five triangular pyramids with unequal volumes, and obtaining
Figure FDA0003236082360000033
A triangular pyramid grid, so that the task area Ω is represented as
Figure FDA0003236082360000032
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