CN113722958B - Efficient modeling method for irregular-shape celestial body gravitational field - Google Patents
Efficient modeling method for irregular-shape celestial body gravitational field Download PDFInfo
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Abstract
The invention discloses a high-efficiency modeling method for an irregular-shape 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, 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 linear expression form of a gravitational field residual error of an internal region of the triangular pyramid through gravity center interpolation; traversing all triangular pyramids in the task area, and converting the residual data of the gravitational field into grid judgment constraint and linear interpolation modes; summing the gravitational field parameters and the gravitational acceleration to obtain a local linear gravitational field model based on grid judgment; and (3) obtaining the local gravitational field model near the asteroid meeting the precision requirement by iterating and giving the space discrete scale of the linear gravitational field. The method can remarkably improve the efficiency of gravitational field modeling under the condition of meeting task requirements.
Description
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 suitable for a gravitational field near an asteroid, and belongs to the technical field of aerospace.
Background
The gravitational field modeling is a key technology for realizing the detection of the asteroid, and has important application value in the detection task of the asteroid. The gravitational field around the asteroid is typically strongly nonlinear due to its irregularly shaped and non-uniform density characteristics. On the 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 difficulty to be solved; on the other hand, in order to improve the safety and reliability of the actual asteroid detection task, how to implement rapid modeling of the gravitational field near the asteroid to cope with the emergency in the detection task is also a key problem to be solved. The modeling of the gravitational field near the asteroid is one of the hot spot problems of current technological staff.
Among the developed approaches to modeling of the gravitational field around the asteroid, prior art [1] (Werner, r.a., "The Gravitational Potential of a Homogeneous Polyhedron or Don't Cut cores," Celestial Mechanics and Dynamical Astronomy, 1993), a polyhedral modeling approach was proposed for the problem of modeling of the gravitational field around the asteroid, which uses a series of spatial polyhedral grids to discretize the asteroid, thereby obtaining the asteroid polyhedral gravitational field in the form of numerical summation. The method has the advantages of higher calculation accuracy, and the obtained gravitational field near the asteroid is verified in a plurality of practical tasks. Meanwhile, the method needs larger calculation amount, has lower calculation efficiency, and is difficult to be applied to an actual asteroid detection task with rapid calculation of a gravitational field near the asteroid.
In the prior art [2] (Herrera, s.e., palmer, p.l., and Roberts, r.m., "Modeling the Gravitational Potential of a Nonspherical Asteroid," Celestial Mechanics and Dynamical Astronomy, 2013), a sphere harmonic and Bessel gravitational field modeling method is proposed for the problem of gravitational field modeling near the asteroid, a reference sphere is selected, a spherical harmonic function modeling method is adopted for the sphere external gravitational field, and a Bessel function modeling method is adopted for the sphere internal gravitational field. Compared with a polyhedral model, the method has the advantages that the calculation efficiency of the gravitational field can be improved, and the method has the defects that the modeling accuracy is low, and the method is difficult to be suitable for small celestial body detection tasks with high requirements on track accuracy.
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 is used for modeling the gravitational field near the asteroid, the invention discloses a high-efficiency modeling method for the gravitational field of the small celestial body with an irregular shape, which aims to: the method realizes local rapid modeling of the gravitational field near the asteroid, improves the modeling calculation efficiency of the gravitational field near the asteroid on the premise of ensuring the modeling precision, and solves the application related technical problems of the gravitational field near the asteroid. The technical problem includes improving the efficiency of track planning around the asteroid.
The invention aims at realizing the following technical scheme:
according to the efficient modeling method of the irregular-shape celestial body gravitational field, a modeling task area is selected according to the asteroid characteristics and an expected modeling area, discretization processing is carried out on the whole space in the task area, and continuous three-dimensional space is discretized into a series of tiny triangular pyramid spaces. In each triangular pyramid, the gravitational field residual data at the vertex is obtained by calculating the two-body gravitational acceleration at the vertex and making a difference from the nominal gravitational acceleration. On the basis, the linear expression form of the residual error of the gravitational field of the triangular pyramid internal area is obtained through gravity center interpolation. And performing traversal calculation on all triangular pyramids in the task area, and converting the residual data of the gravitational field in the whole task area into grid judgment constraint and linear interpolation modes. And summing the obtained gravitational field parameters with the gravitational acceleration of the two bodies to obtain the local linear gravitational field based on grid judgment. Finally, the space discrete scale of the linear gravitational field is given through iteration, the space grid discrete scale meeting the precision requirement is calculated, and the local gravitational field model near the asteroid meeting the precision requirement is obtained. The gravitational field can avoid a large amount of numerical calculation in the traditional modeling method, greatly improve the efficiency of gravitational field modeling under the condition of meeting task requirements, and further solve the related technical problems of application of gravitational fields near the asteroid. The technical problem includes improving the efficiency of track planning around the asteroid.
The invention discloses a high-efficiency modeling method for an irregular-shape celestial body gravitational field, which comprises the following steps:
step one, discretizing the area near the asteroid to obtain a series of triangular pyramid grids with space dispersion.
Defining the task area near the asteroid as a regular hexahedron denoted as Ω, the hexahedron boundary being represented by six characteristic parameters, divided into maximum X-axis coordinates r xmax Minimum X-axis coordinate r xmin Maximum Y-axis coordinate r ymax Minimum Y-axis coordinate r ymin Maximum Z-axis coordinate r zmax Minimum Z-axis coordinate r zmin 。
Respectively giving grid discrete points in three-axis directions and X-axis discrete points n x Discrete number n of Y-axis points y Z-axis discrete point number n z The region Ω is divided into a series of small regular hexahedral meshes, denoted Ω j
Ω j ：＝[r x，i ，r x，i+1 ]×[r y，i ，r y，i+1 ]×[r z，i ，r z，i+1 ]，j＝1，2，...，(n x -1)·(n y -1)·(n z -1)
Wherein: q (Q) j Representing the j-th regular hexahedral mesh.
For each small regular hexahedron, it is divided into several triangular pyramids.
In order to reduce the number of discrete meshes and further to improve the calculation efficiency, it is preferable to divide each small regular hexahedron into five triangular pyramids of unequal volumes.
For each small regular hexahedron, it is divided into five triangular pyramids of unequal volumes, thus obtaining N S ＝5·(n x -1)·(n y -1)·(n z -1) triangular pyramid meshes, so that the task area Ω can be represented as
And step two, calculating the gravitational acceleration of the two body centers at the vertex of each discrete triangular pyramid grid, and differencing with the nominal gravitational field to obtain the linear residual form of the gravitational field.
Defining the position vector of any point in triangular pyramid as r k First, the gravitational acceleration at this point is calculated. Calculating the gravitational acceleration of the two bodies at the point and differencing the gravitational acceleration with the nominal gravitational acceleration to obtain the linear residual expression form of the gravitational field
Wherein: sigma (r) k ) Representing triangular pyramid Θ k In-region point r k A residual error with respect to the central disomic force,representation point r k Two-body gravitational acceleration at the point, in the form of
Wherein: mu (mu) ast Represents the center-to-body gravitational constant of the asteroid obtained by observation.
Obtaining residual form of gravitational field at vertex of each triangular pyramid
And thirdly, carrying out linear gravity center interpolation on residual data of each triangular pyramid vertex to obtain a linear expression form of the gravitational field in the whole task area.
The residual data of the gravitational field at each vertex of the triangular pyramid is expressed as { sigma (r) 1 )，σ(r 2 )，σ(r 3 )，σ(r 4 )}。
Conversion coefficient { alpha } defining barycentric interpolation coordinate and three-dimensional space rectangular coordinate system 1 ，α 2 ，α 3 ，α 4 The gravitational field residual error is rewritten into the following barycentric coordinate form
σ(r k )＝α 1 σ(r 1 )+α 2 σ(r 2 )+α 3 σ(r 3 )+α 4 σ(r 4 )
The coefficients of the coordinate system satisfy the following relationship
The conversion coefficient { alpha }, can then be inversely solved 1 (r k )，α 2 (r k )，α 3 (r k )，α 4 (r k ) Substituting the residual calculation formula of the gravitational field under the gravity center coordinate system to obtain the residual of any point in the triangular pyramid in an interpolation mode
σ(r k )＝α 1 (r k )σ(r 1 )+α 2 (r k )σ(r 2 )+α 3 (r k )σ(r 3 )+α 4 (r k )σ(r 4 )
Traversing and calculating all triangular pyramid grids to obtain a gravitational field in the whole task area, namely representing the gravitational field in the omega area as
Wherein r represents the spatial three-dimensional position coordinate and vector of any point in the whole task areal, and matrix Φ are calculated from the following formulas.
l＝(r T ，1) T
And step four, gradually improving the discrete scale of the grid, 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 the detection point set in the task area asFor arbitrary->The corresponding nominal gravitational field data is +.>For a given residual gravitational field model, the modeling accuracy of the model is defined as
Wherein: the function Λ (r) represents gravitational acceleration at the detection point r output by the residual gravitational field model, and the function max {. Cndot } represents maximizing all elements in the set.
Given the discrete scale of the spatial grid, expressed as ρ=n x ×n y ×n z Calculating the asteroid gravitational field through the steps one to three, further obtaining 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
Wherein, lambda ρ Representing a model of the gravitational field around the asteroid at a particular grid discrete scale.
Given the allowable modeling error epsilon, the iteration step lambda of the modeling accuracy improvement process and the initial grid discrete density rho 0 Then in the ith iteration, the grid discrete density is defined as
ρ i ＝ρ 0 +λ·i
Obtaining modeling accuracy P (ρ) in the ith iteration by the mapping function i )。
If the accuracy does not meet the requirement, P (ρ) i ) And (3) increasing the discrete density to enable the next iteration to be performed.
If the accuracy meets the requirement, i.e. P (ρ) i ) Less than or equal to epsilon, outputting an asteroid gravitational field model lambda meeting the precision requirement ρ Corresponding discretized grid { Θ k And acquiring the asteroid gravitational field meeting the precision requirement, and realizing rapid modeling of the gravitational field near the asteroid.
The method also comprises the following steps: and (3) acquiring an asteroid gravitational field meeting the precision requirement according to the step four, and solving the application related technical problems of the gravitational field near the asteroid. The technical problem includes improving the efficiency of track planning around the asteroid.
The beneficial effects are that:
1. the invention discloses an efficient modeling method for an irregular-shape celestial body gravitational field. In each triangular pyramid, the gravitational field residual data at the vertex is obtained by calculating the two-body gravitational acceleration at the vertex and making a difference from the nominal gravitational acceleration. On the basis, the linear expression form of the residual error of the gravitational field of the triangular pyramid internal area is obtained through gravity center interpolation. And performing traversal calculation on all triangular pyramids in the task area, and converting the residual data of the gravitational field in the whole task area into grid judgment constraint and linear interpolation modes. And summing the obtained gravitational field parameters with the gravitational acceleration of the two bodies to obtain a local linear gravitational field model based on grid judgment. Finally, the space discrete scale of the linear gravitational field is given through iteration, the space grid discrete scale meeting the precision requirement is calculated, and the local gravitational field model near the asteroid meeting the precision requirement is obtained. The invention can avoid a large amount of numerical calculation in the traditional modeling method, and greatly improve the efficiency of gravitational field modeling under the condition of meeting task demands.
2. According to the high-efficiency modeling method for the irregular-shape celestial body gravitational field, the space discrete scale of the linear gravitational field modeling method is given through iteration, the modeling precision in a detection point calculation task area is randomly selected, and the space discrete scale is gradually improved to obtain higher modeling precision until a local gravitational field model near the asteroid meeting the precision requirement is obtained. Due to the linear gravity center interpolation characteristic of the model, under the condition of given any small allowable error, a proper space grid discretization scale can be found, namely, the modeling efficiency of the gravitational field near the asteroid is obviously improved on the premise of ensuring the modeling precision of the gravitational field.
Drawings
FIG. 1 is a flow chart of a method for efficiently modeling an irregular-shape celestial body gravitational field;
FIG. 2 is a schematic diagram of a method of discretizing a spatial triangular pyramid grid;
FIG. 3 is a schematic diagram of a gravity field gravity center interpolation method in a triangular spatial pyramid grid;
FIG. 4 is a simulation graph of the modeling accuracy of the gravitational field around an irregular asteroid versus the discrete scale of the grid.
FIG. 5 is a graph showing the comparison of the calculation efficiency of the gravitational field model and the polyhedral gravitational field model
Detailed Description
For a better description of the objects and advantages of the present invention, the following description will be given with reference to the accompanying drawings and examples.
Example 1: fast modeling of local gravitational field of asteroid 1999KW4
As shown in fig. 1, the method for locally and rapidly modeling the gravitational field near the asteroid disclosed in the embodiment selects the asteroid 1999KW4 to perform gravitational field modeling, and specifically includes the following steps:
step one, discretizing the area nearby the asteroid to obtain a series of small regular hexahedral meshes with space discretization, wherein the discretization scale of the space mesh is given.
Defining the task area around the asteroid as a regular hexahedron, denoted omega, the hexahedron boundary being represented by six characteristic parametersIs divided into a maximum X-axis coordinate r xmax =1.27 km, minimum X-axis coordinate r xmin =0.37 km, maximum Y-axis coordinate r ymax =1.24 km, minimum Y-axis coordinate r ymin =0.35 km, maximum Z-axis coordinate r zmax =0.83 km, minimum Z-axis coordinate r zmin ＝0.33km。
Respectively giving grid discrete points in three-axis directions and X-axis discrete points n x =5, Y-axis discrete points n y =5, Z-axis discrete points n z =5, dividing the region Ω into a series of small regular hexahedral meshes, denoted Ω j
Ω j ：＝[r x，i ，r x，i+1 ]×[r y，i ，r y，i+1 ]×[r z，i ，r z，i+1 ]，j＝1，2，...，(n x -1)·(n y -1)·(n z -1)
Wherein: omega shape j Representing the j-th regular hexahedral mesh.
Step two, giving a small regular hexahedral grid in the task area, and further performing grid dispersion on the small regular hexahedral grid to obtain a series of small triangular pyramid grids with space dispersion.
For each small regular hexahedron, it is divided into five triangular pyramids of unequal volumes, the specific division rule is shown in FIG. 1, and N can be obtained S ＝5·(n x -1)·(n y -1)·(n z -1) =80 triangular pyramid meshes, so that the task area Ω can be represented as
And thirdly, calculating the two-body gravitational acceleration at the vertex of each small triangular pyramid grid.
Defining the position vector of any point in triangular pyramid as r k First, the gravitational acceleration at this point is calculated. Calculating the gravitational acceleration at the point and making a difference from the nominal gravitational acceleration to obtain
Wherein: sigma (r) k ) Representing triangular pyramid Θ k In-region point r k A residual error with respect to the central disomic force,representation point r k Two-body gravitational acceleration at the point, in the form of
Wherein: mu (mu) ast Represents the center-to-body gravitational constant of the asteroid obtained by observation.
And step four, calculating residual errors at the vertexes of each triangular pyramid grid according to the nominal gravitational acceleration data.
Selecting a triangular pyramid grid area, which is defined as Θ k The method of the first and second steps can obtain four vertexes of the triangular pyramid area, which is expressed as { r } 1 ，r 2 ，r 3 ，r 4 Acquiring gravitational acceleration at vertices based on nominal gravitational field data, defined as
Thus, the residual error of the nominal gravitational acceleration at each vertex of the triangular pyramid relative to the gravitational acceleration of the two bodies can be calculated and expressed as
And fifthly, in each small triangular pyramid grid, calculating the barycenter interpolation coordinates in the small triangular pyramid area according to residual data.
Conversion coefficient { alpha } defining barycentric interpolation coordinate and three-dimensional space rectangular coordinate system 1 ，α 2 ，α 3 ，α 4 Residual gravitational fieldThe difference is rewritten as the following barycentric coordinate form
σ(r k )＝α 1 σ(r k )+α 2 σ(r k )+α 3 σ(r k )+α 4 σ(r k )
The coefficients of the coordinate system satisfy the following relationship
The conversion coefficient { alpha }, can then be inversely solved 1 (r k )，α 2 (r k )，α 3 (r k )，α 4 (r k ) And substituting the residual calculation formula of the gravitational field under the barycentric coordinate system to obtain a specific coordinate system conversion schematic diagram shown in figure 2.
And step six, obtaining a local gravitational field model near the asteroid in a two-body residual form.
From the nominal gravitational field data, expressed asThe following calculation formula
Residual data at each vertex of the triangular pyramid, denoted as { sigma (r) 1 )，σ(r 2 )，σ(r 3 )，σ(r 4 )}。
Thus, the linear interpolation type gravitational acceleration of any point in space can be obtained
Wherein: sigma (r) k ) Representing triangular pyramid Θ k In-region point r k A residual error with respect to the acceleration of the gravitational force.
Further, a linear residual form gravitational field inside the triangular pyramid can be obtained
And seventhly, in each small triangular pyramid grid, carrying out gravity center interpolation on residual data of the vertexes to obtain a linear gravitational field in the area.
Conversion coefficient { alpha } defining barycentric interpolation coordinate and three-dimensional space rectangular coordinate system 1 ，α 2 ，α 3 ，α 4 Rewriting the gravitational field into the form of barycentric coordinates
σ(r k )＝{α 1 σ(r 1 )，α 2 σ(r 2 )，α 3 σ(r 3 )，α 4 σ(r 4 )}
The coefficients of the coordinate system satisfy the following relationship
The conversion coefficient { alpha }, can then be inversely solved 1 (r k )，α 2 (r k )，α 3 (r k )，α 4 (r k ) And substituting the data into a gravitational field calculation formula under a gravity center coordinate system to obtain a specific coordinate system conversion schematic diagram shown in figure 2.
σ(r k )＝{α 1 (r k )σ(r 1 )，α 2 (r k )σ(r 2 )，α 3 (r k )σ(r 3 )，α 4 (r k )σ(r 4 )}
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 through traversal, and the gravitational field in the whole task area, namely the gravitational field in the omega area, can be expressed as
Wherein r represents the spatial three-dimensional position coordinate and vector of any point in the whole task areal, and matrix Φ are calculated from the following formulas.
l＝(r T ，1) T
And step nine, randomly selecting a check point in the task area, calculating gravitational acceleration at the check point, and obtaining the modeling precision of the gravitational field near the asteroid under the discrete scale of the grid.
Defining the detection point set in the task area asThe set contains 1000 random detection points, for anyThe corresponding nominal gravitational field data is +.>For a given residual gravitational field model, the modeling accuracy of the model is defined as
Wherein: the function Λ (r) represents gravitational acceleration at the detection point r output by the residual gravitational field model, and the function max {. Cndot } represents maximizing all elements in the set.
And step ten, judging whether the modeling accuracy meets the task requirement.
The above-mentioned step one to step nine calculation processes are regarded as a mapping function whose input is the discrete scale of the spatial grid, expressed as ρ=n x ×n y ×n z The output is the modeling precision of the space grid, and the following mapping function is obtained by combining the modeling precision definition formula
Wherein, lambda ρ Representing a model of the gravitational field around the asteroid at a particular grid discrete scale.
Given the allowable modeling error epsilon=1×10 -9 Iterative step lambda=5 of modeling accuracy improvement process, initial grid discrete density ρ 0 =5. The modeling accuracy P (ρ) in the ith iteration can be obtained by the modeling accuracy calculation formula i )。
If the accuracy meets the requirement, i.e. P (ρ) i ) And E, executing the step eleventh.
If the accuracy does not meet the requirement, P (ρ) i ) Increasing the discrete density to make-
ρ i ＝ρ 0 +λ·i
Repeating the steps one to ten.
And step eleven, outputting the gravitational field model near the asteroid obtained in the step ten.
According to step eleven, a grid discretization scale meeting the precision requirement is obtained, which is defined as ρ m ＝n xm ×n ym ×n zm . Obtaining a gravitational field model Λ according to the method of steps one to nine ρ Corresponding discretized grid { Θ k }。
And step twelve, establishing an orbit dynamics model nearby the asteroid according to the gravitational field model output in the step eleven.
According to 1999KW4 asteroid parameters, kinetic parameters are defined, wherein the spin angular velocity ω= (0,0,6.31)×10 -4 ) T rad/s, the following kinetic model was constructed.
if r∈Θ k ，then
The relationship between the modeling accuracy of the gravity field near 1999KW4 asteroid obtained by calculation of the example and the discretization scale of the grid is shown in fig. 3, and the calculation efficiency pair of the gravity field near 1999KW4 asteroid and the calculation efficiency pair of the polyhedron model are shown in fig. 4.
And step thirteen, according to the orbit dynamics model near the asteroid established in the step twelve, improving the planning efficiency of the orbit near the asteroid.
While the foregoing is directed to embodiments of the present invention, other and further embodiments of the invention may be devised without departing from the basic scope thereof, and the scope thereof is determined by the claims that follow.
Claims (3)
1. A high-efficiency modeling method for an irregular-shape celestial body gravitational field is characterized by comprising the following steps of: comprises the following steps of the method,
step one, discretizing the area near the asteroid to obtain a series of triangular pyramid grids with space dispersion;
the first implementation method of the step is that,
defining the task area near the asteroid as a regular hexahedron, denoted by Ω, the hexahedron boundaries being represented by six characteristic parameters, respectively the maximum X-axis coordinates r xmax Minimum X-axis coordinate r xmin Maximum Y-axis coordinate r ymax Minimum Y-axis coordinate r ymin Maximum Z-axis coordinate r zmax Minimum Z-axis coordinate r zmin ；
Respectively giving grid discrete points in three-axis directions and X-axis discrete points n x Discrete number n of Y-axis points y Z-axis discrete point number n z The region Ω is divided into a series of small regular hexahedral meshes, denoted Ω j
Ω j :＝[r x,i ,r x,i+1 ]×[r y,i ,r y,i+1 ]×[r z,i ,r z,i+1 ],j＝1,2,...,(n x -1)·(n y -1)·(n z -1)
Wherein: omega shape j Representing a j-th regular hexahedral mesh;
for each small regular hexahedron, dividing the small regular hexahedron into a plurality of triangular pyramids;
calculating the gravitational acceleration of the two centers of gravity at the vertex of each discrete triangular pyramid grid, and differencing with the nominal gravitational field to obtain a linear residual form of the gravitational field;
the implementation method of the second step is that,
defining the position vector of any point in triangular pyramid as r k First, calculating gravitational acceleration at the point; calculating the gravitational acceleration of the two bodies at the point and differencing the gravitational acceleration with the nominal gravitational acceleration to obtain the linear residual expression form of the gravitational field
Wherein: sigma (r) k ) Representing triangular pyramid Θ k In-region point r k A residual error with respect to the central disomic force,representation point r k Two-body gravitational acceleration at the point, in the form of
Wherein: mu (mu) ast Representation ofThe center two-body gravitational constant of the asteroid is obtained through observation;
obtaining residual form of gravitational field at vertex of each triangular pyramid
Thirdly, carrying out linear gravity center interpolation on residual data of each triangular pyramid vertex to obtain a linear expression form of a gravitational field in the whole task area;
the implementation method of the third step is that,
the residual data of the gravitational field at each vertex of the triangular pyramid is expressed as { sigma (r) 1 ),σ(r 2 ),σ(r 3 ),σ(r 4 )}；
Conversion coefficient { alpha } defining barycentric interpolation coordinate and three-dimensional space rectangular coordinate system 1 ,α 2 ,α 3 ,α 4 The gravitational field residual error is rewritten into the following barycentric coordinate form
σ(r k )＝α 1 σ(r 1 )+α 2 σ(r 2 )+α 3 σ(r 3 )+α 4 σ(r 4 )
The coefficients of the coordinate system satisfy the following relationship
Then inverse solution to obtain the conversion coefficient { alpha } 1 (r k ),α 2 (r k ),α 3 (r k ),α 4 (r k ) Substituting the residual calculation formula of the gravitational field under the gravity center coordinate system to obtain the residual of any point in the triangular pyramid in an interpolation mode
σ(r k )＝α 1 (r k )σ(r 1 )+α 2 (r k )σ(r 2 )+α 3 (r k )σ(r 3 )+α 4 (r k )σ(r 4 )
Traversing and calculating all triangular pyramid grids to obtain a gravitational field in the whole task area, namely representing the gravitational field in the omega area as
Wherein r represents the spatial three-dimensional position coordinate and vector of any point in the whole task areal and matrix phi are calculated by the following formula;
step four, gradually lifting the discrete scale of the grid, 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;
the realization method of the fourth step is that,
defining the detection point set in the task area asFor arbitrary->The corresponding nominal gravitational field data is ∈U r The method comprises the steps of carrying out a first treatment on the surface of the For a given residual gravitational field model, the modeling accuracy of the model is defined as
Wherein: the function Λ (r) represents gravitational acceleration at a detection point r output by the residual gravitational field model, and the function max { · } represents that all elements in the set are maximized;
given the discrete scale of the spatial grid, expressed as ρ=n x ×n y ×n z Calculating the asteroid gravitational field through the steps one to three, further obtaining 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
Wherein, lambda ρ Representing a gravitational field model around the asteroid at a particular grid discrete scale;
given the allowable modeling error epsilon, the iteration step lambda of the modeling accuracy improvement process and the initial grid discrete density rho 0 Then in the ith iteration, the grid discrete density is defined as
ρ i ＝ρ 0 +λ·i
Obtaining modeling accuracy P (ρ) in the ith iteration by the mapping function i )；
If the accuracy does not meet the requirement, P (ρ) i ) Increasing the discrete density more than epsilon, and carrying out the next iteration;
if the accuracy meets the requirement, i.e. P (ρ) i ) Less than or equal to epsilon, outputting an asteroid gravitational field model lambda meeting the precision requirement ρ Corresponding discretized grid { Θ k And acquiring the asteroid gravitational field meeting the precision requirement, and realizing rapid modeling of the gravitational field near the asteroid.
2. The method for efficiently modeling the irregular-shaped celestial body gravitational field according to claim 1, wherein the method comprises the following steps of: the fifth step is to obtain the asteroid gravitational field meeting the precision requirement according to the fourth step, and solve the application related technical problems of the gravitational field near the asteroid; the technical problem includes improving the efficiency of track planning around the asteroid.
3. The method for efficiently modeling the irregular-shaped celestial body gravitational field according to claim 1, wherein the method comprises the following steps of: for each small regular hexahedron, it is divided into five triangular pyramids of unequal volumes, thus obtainingN S ＝5·(n x -1)·(n y -1)·(n z -1) triangular pyramid meshes, the task area Ω being then denoted as
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