CN111695239B - Dynamic polyhedron hybrid model-based asteroid landing segment gravitation calculation method - Google Patents

Dynamic polyhedron hybrid model-based asteroid landing segment gravitation calculation method Download PDF

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CN111695239B
CN111695239B CN202010405291.8A CN202010405291A CN111695239B CN 111695239 B CN111695239 B CN 111695239B CN 202010405291 A CN202010405291 A CN 202010405291A CN 111695239 B CN111695239 B CN 111695239B
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杨洪伟
邵远斌
李爽
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Nanjing University of Aeronautics and Astronautics
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Abstract

The invention discloses a dynamic polyhedron hybrid model-based asteroid landing zone gravitation calculation method, which comprises the following steps: setting the gravitational force calculation problem parameters of the asteroid landing section; establishing a landing zone space target point position equation; establishing the number of polyhedral model surface unitsN p Is a model of the initial mixture of (a); dividing the space of the landing zone into areas and determining the most suitable polyhedral model surface unit number in each sub-area; and calling a mixed model of the corresponding polyhedral model surface unit number according to the subarea where the target point is located, and calculating the gravitation of the mixed model. The invention is suitable for calculating the gravitation in the cylindrical flight space of the asteroid landing section; when each sub-region is sampled, the number of suitable polyhedron faces is determined according to the allowable maximum relative error parameter set in advance, so that the effect of error control on the whole hybrid model can be realized; when the attraction force is calculated, the calculation efficiency is high and the precision is high.

Description

Dynamic polyhedron hybrid model-based asteroid landing segment gravitation calculation method
Technical Field
The invention belongs to the technical field of aerospace, and particularly relates to an asteroid landing segment gravitation calculation method based on a dynamic polyhedral hybrid model.
Background
In recent years, the world space development has generated a great deal of interest in the exploration of the celestial bodies (asteroid, comet) and has performed a series of related tasks. The main reasons are as follows: first, the near-earth asteroid always presents potential collision hazards to the earth, and each large spaceflight mechanism starts to enlarge the work of identifying, classifying and positioning the celestial body. Second, the celestial bodies have abundant mineral resources, which is of great importance for in situ synthesis of spacecraft propellants and for providing raw materials for various manufacturing components. Thirdly, the formation of the solar system is carried out by the celestial body, so far, 46 hundred million years old, and the celestial body is just as important as the activated fossil of the evolution of the solar system, and the celestial body exploration provides rich contents for the research of the origin and the development of the solar system by human beings. Unlike planets, the mass of the celestial body is often insufficient to lend itself to sphericity, and the shape is largely irregular, which can cause considerable difficulties in the calculation of the gravitational field and thus affect a series of probing tasks including wrapping and landing.
A polyhedral model for calculating irregular asteroid attraction is disclosed in the prior literature Werner R A, scheeres D J.Exterior gravitation of a polyhedron derived and compared with harmonic and mascon gravitation representations of asteroid 4769castalia [ J ] Celestial Mechanics and Dynamical Astronomy,1996,65 (3): 313-344, which is accurate and efficient and well suited for studying kinetic problems near the surface of a small celestial body. However, the model is low in calculation efficiency, and is not suitable for on-orbit calculation of a spacecraft, particularly for the situation that the resolution of an asteroid geometric model is high.
The hybrid model in the prior documents Pearl J M, louisos W F, hitt D L.hybrid Gravity Model from Asteroid Surface Topology [ C ]//2018Space Flight Mechanics Meeting.2018:0955 divides cells of the surface of the celestial body divided by a mesh into two types, one type is a face cell which is positioned near a landing point and is processed by a polyhedral model, the other type is a face cell which is processed by a computationally efficient approximation model, by this division, the computation time is kept in the order of magnitude of the approximation model, and the computation precision is kept in the order of magnitude of the polyhedral model, thereby realizing the double benefits of computation time and precision. However, the hybrid model is a static hybrid model, and once the hybrid scheme is determined, the hybrid model cannot be changed, so that the calculation accuracy of the gravitation of the landing zone at different height positions cannot be uniformly controlled, and the calculation time of other target positions is wasted in order to ensure the accuracy of the maximum error position.
Disclosure of Invention
Aiming at the defects of the prior art, the invention aims to provide a dynamic polyhedron hybrid model-based asteroid landing zone gravitation calculation method, so as to solve the problem that the gravitation of an asteroid with irregular shape is difficult to calculate rapidly and accurately in the prior art.
In order to achieve the above purpose, the invention adopts the following technical scheme:
the invention discloses a dynamic polyhedron hybrid model-based asteroid landing zone gravitation calculation method, which comprises the following steps:
1) Setting the gravitational force calculation problem parameters of the asteroid landing section;
2) Establishing a landing zone space target point position equation;
3) Establishing the number N of polyhedral model surface units p Is a model of the initial mixture of (a);
4) Dividing the space of the landing zone into areas and determining the most suitable polyhedral model surface unit number in each sub-area;
5) And calling a mixed model corresponding to the polyhedral model surface unit number according to the subarea where the target point is located, and calculating the gravitation of the mixed model.
Further, the step 1) specifically includes: setting the density rho and surface grid parameters of the target asteroid; selecting landing point position, setting maximum height h of cylindrical flight space constraint condition parameters of landing section max And a maximum radius d max Setting the maximum relative error delta allowed by the mixed model tol 。
Further, the step 2) specifically includes: sequentially establishing an asteroid body coordinate system A and a landing point local cylindrical coordinate system B for describing a cylindrical flight space, wherein in the coordinate system A, an origin is positioned at an asteroid centroid, a coordinate axis x axis and a coordinate axis z axis are respectively coincident with a minimum inertia main shaft and a maximum inertia main shaft of the asteroid body, and a coordinate axis y axis points to meet a right hand system; in the coordinate system B, the origin is positioned at the landing point, the coordinate axis i is a unit vector arbitrarily selected in the local horizontal plane of the landing point, the coordinate axis k is vertical to the local horizontal plane of the landing point, and the coordinate axis j points to meet the right-hand system; recording the position of a landing point under a coordinate system A c And if the target point position is r, the equation of the target point position is as follows:
r＝(hk+dcosθi+dsinθj)+r c (1)
wherein h is E [0, h max ]D E [0, d is the distance between the target point and the landing point along the k-axis direction max ]For the distance from the target point to the k-axis, θ ε [0,2π]Is the angle between the projection of the target position vector in the coordinate system B in the ij plane and the i axis.
Further, the step 3) specifically includes: for a pair ofN around landing sites p The surface units adopt a polyhedral model, the rest surface units adopt an approximate model to calculate gravitational acceleration g and Laplacian thereof, and the specific calculation formula is as follows:
wherein A is f For the area of each surface unit of the polyhedral model, r f R is the vector from the sampling point to the center of the face unit e To obtain a vector of positions of the sampling points relative to any point on the edge, the variables areA unit normal vector representing the outward facing of the face unit, is->Representing the unit normal vector in the plane, facing outwards at the plane edge of the face unit, +.>The unit vectors respectively representing three vertexes from the sampling point to the grid surface, a and b are respectively distances from the sampling point to the end points of the edge, e is the length of the edge, omega f Representing the signed solid angle L with the side facing the sampling point e Representing the line integral term associated with the edge of the side.
Further, the step 4) specifically includes: dividing the landing leg space into a plurality of sub-regions:
h 0 ＝0,h i ＝h i-1 +Δh i ，i∈[1,10] (6)
Δh 1 ＝0.01h max ,Δh i ＝Δh 1 +0.02(i-1)h max (7)
d 0 ＝0,d j ＝d j-1 +Δd j ，j∈[1,10] (8)
Δd 1 ＝0.01d max ,Δd j ＝Δd 1 +0.02(j-1)d max (9)
θ 0 ＝0,θ k ＝θ k-1 +Δθ k ，k∈[1,10] (10)
then selecting sampling points in each subarea, and calculating by adopting a polyhedral model method and a mixed model method to obtain gravitational acceleration g p And g h Subspace is { [ h ] i-1 ,h i ],[d j-1 ,d j ],[θ k-1 ,θ k ]}：
Calculating the position vector of the sampling point in the A coordinate system according to the formula (1), and respectively g p And g h The relative error of each positional hybrid model is thus calculated as:
if the maximum relative error delta among the sampling points max ≤δ tol The optimal polyhedral model surface unit number of the dynamic mixed model in the region is kept to be the original N p Unchanged; otherwise N p ＝N p +dN p The relative error of the position is again calculated until the condition is met.
Further, the step 5) specifically includes: giving h, d and theta of a target point, and importing the optimal polyhedral model surface unit number N of the sub-region where the target point is located p Substituting formula (1) to convert the coordinates under the A coordinate system, and substituting formula (2) to obtain the gravitation of the target point.
The gravitation calculated by the method is the gravitation of the asteroid landing section; the landing space target point position described in said step 2) belongs to a cylindrical space.
The invention has the beneficial effects that:
the calculation method is suitable for calculating the gravitation in the cylindrical flight space of the asteroid landing section; the method is characterized in that the number of the optimal polyhedral model surface units suitable for each region is dynamically adjusted according to the calculated relative error by sampling the points of the subareas in the space, so that a dynamic polyhedral hybrid model is determined. When the gravitation is calculated, the calculation accuracy of the whole model is controllable and can be changed according to the requirement, so that unnecessary calculation accuracy or waste of calculation time is reduced.
Drawings
FIG. 1 is a schematic flow chart of a calculation method of the present invention;
FIG. 2 is a schematic diagram of a coordinate system definition and hybrid model according to the present invention.
Detailed Description
The invention will be further described with reference to examples and drawings, to which reference is made, but which are not intended to limit the scope of the invention.
Referring to fig. 1, the method for calculating the attraction force of the asteroid landing segment based on the dynamic polyhedral hybrid model comprises the following steps:
1) Setting the gravitational force calculation problem parameters of the asteroid landing section;
2) Establishing a landing zone space target point position equation;
3) Establishing the number N of polyhedral model surface units p Is a model of the initial mixture of (a);
4) Dividing the space of the landing zone into areas and determining the most suitable polyhedral model surface unit number in each sub-area;
5) And calling a mixed model corresponding to the polyhedral model surface unit number according to the subarea where the target point is located, and calculating the gravitation of the mixed model.
The asteroid Eros is taken as an example:
1. setting the density ρ=2.67 g/cm of the target asteroid Eros 3 Surface mesh parameters; the asteroid surface mesh parameter has 25350 vertices and 49152 faces; select landing site location as [13.0281, -4.7835,3.6562]km, setting a cylindrical flight space constraint condition parameter h of a landing section max ＝5km,d max =2.5 km; setting the allowable maximum relative error as delta tol ＝0.05％。
2. An asteroid body coordinate system a and a landing site local cylindrical coordinate system B describing a cylindrical flight space are sequentially established as shown in fig. 2. In the coordinate system A, the origin is positioned at the center of mass of the asteroid, the x-axis and the z-axis of the coordinate axes are respectively coincident with the minimum inertia main shaft and the maximum inertia main shaft of the asteroid body, and the y-axis of the coordinate axis points to satisfy the right-hand system. In the coordinate system B, the origin is located at the landing point, the coordinate axis i is selected as a unit vector [0,0,0.4205] in the local horizontal plane of the landing point (namely, the triangle plane of the polyhedral surface where the landing point is located), the coordinate axis k is a unit vector [0.5769, -0.0586,0.8147] perpendicular to the local horizontal plane of the landing point, and the direction of the coordinate axis j meets the right-hand system.
Recording the position of a landing point under a coordinate system A c And if the target point position is r, the equation of the target point position is as follows:
r＝(hk+dcosθi+dsinθj)+r c (1)
wherein h is E0,h max ]D E [0, d is the distance between the target point and the landing point along the k-axis direction max ]For the distance from the target point to the k-axis, θ ε [0,2π]Is the angle between the projection of the target position vector in the coordinate system B in the ij plane and the i axis.
3. N around the land point p The surface units adopt a polyhedral model, N p Set to 0, and calculate gravitational acceleration g and its laplace operator by using an approximation model for the rest of the surface units, the specific calculation formula is as follows:
wherein A is f For the area of each surface unit of the polyhedral model, r f R is the vector from the sampling point to the center of the face unit e To obtain a vector of positions of the sampling points relative to any point on the edge, the variables areA unit normal vector representing the outward facing of the face unit, is->Representing the unit normal vector in the plane, facing outwards at the plane edge of the face unit, +.>Respectively represent from sampling point to netThe unit vectors of three vertexes of the grid, a and b are the distances from the sampling point to the end point of the edge, and e is the length of the edge, omega f Representing the signed solid angle L with the side facing the sampling point e Representing the line integral term associated with the edge of the side.
4. The landing leg space is divided into a number of sub-regions (here into 1000):
h 0 ＝0,h i ＝h i-1 +Δh i ，i∈[1,10] (6)
Δh 1 ＝0.01h max ,Δh i ＝Δh 1 +0.02(i-1)h max (7)
d 0 ＝0,d j ＝d j-1 +Δd j ，j∈[1,10] (8)
Δd 1 ＝0.01d max ,Δd j ＝Δd 1 +0.02(j-1)d max (9)
θ 0 ＝0,θ k ＝θ k-1 +Δθ k ，k∈[1,10] (10)
then selecting 32 sampling points in each subarea, and respectively adopting a polyhedral model method and a mixed model method to calculate and obtain gravitational acceleration g p And g h In subspace { [ h ] i-1 ,h i ],[d j-1 ,d j ],[θ k-1 ,θ k ]Examples are:
calculating the position vector of the sampling point in the A coordinate system according to the formula (1), and respectively g p And g h The relative error of each positional hybrid model is thus calculated as:
if the maximum relative error delta among 32 sampling points max ≤δ tol The optimal polyhedral model surface unit number of the dynamic mixed model in the region is kept to be the original N p Unchanged; otherwise N p ＝N p +dN p (dN p The specific values of (a) can be based on the resolution, delta of the asteroid geometry model max And delta tol Appropriately adjusted) and then calculates the relative error for that position until the condition is met.
5. 1000 sampling points are selected at equal intervals in space along the k-axis 0 to 5km above the landing site. When calculation is performed using the polyhedral model, the calculation time for each point is 0.0282s on average. When the dynamic mixing model is adopted for calculation, the calculation time of each point is 0.00385s, the average relative error of 1000 sampling points is 0.0078%, and the maximum relative error is 0.0518%. Of the 1000 points calculated, the maximum N called by the dynamic hybrid model p A value of 36, and N p When the static mixed model with the value of 36 is calculated, the average relative error is 0.0119 percent, the maximum relative error is 0.0521 percent, and the calculation time of each point is 0.004s. In summary, the calculation efficiency of the hybrid model is 7 times higher than that of the polyhedral model, and compared with the static hybrid model, the dynamic hybrid model not only achieves the unified error control effect, but also reduces unnecessary calculation time waste. The present invention has been described in terms of the preferred embodiments thereof, and it should be understood by those skilled in the art that various modifications can be made without departing from the principles of the invention, and such modifications should also be considered as being within the scope of the invention.

Claims (2)

1. The asteroid landing segment gravitation calculation method based on the dynamic polyhedral hybrid model is characterized by comprising the following steps of:
1) Setting the gravitational force calculation problem parameters of the asteroid landing section;
2) Establishing a landing zone space target point position equation;
3) Establishing the number N of polyhedral model surface units p Is a model of the initial mixture of (a);
4) Dividing the space of the landing zone into areas and determining the most suitable polyhedral model surface unit number in each sub-area;
5) Calling a mixed model corresponding to the polyhedral model surface unit number according to the subarea where the target point is located and calculating the gravitation of the mixed model;
the step 1) specifically comprises the following steps: setting the density rho and surface grid parameters of the target asteroid; selecting landing point position, setting maximum height h of cylindrical flight space constraint condition parameters of landing section max And a maximum radius d max Setting the maximum relative error delta allowed by the mixed model tol ；
The step 2) specifically comprises the following steps: sequentially establishing an asteroid body coordinate system A and a landing point local cylindrical coordinate system B for describing a cylindrical flight space, wherein in the coordinate system A, an origin is positioned at an asteroid centroid, a coordinate axis x axis and a coordinate axis z axis are respectively coincident with a minimum inertia main shaft and a maximum inertia main shaft of the asteroid body, and a coordinate axis y axis points to meet a right hand system; in the coordinate system B, the origin is positioned at the landing point, the coordinate axis i is a unit vector arbitrarily selected in the local horizontal plane of the landing point, the coordinate axis k is vertical to the local horizontal plane of the landing point, and the coordinate axis j points to meet the right-hand system; recording the position of a landing point under a coordinate system A c And if the target point position is r, the equation of the target point position is as follows:
r＝(hk+dcosθi+dsinθj)+r c (1)
wherein h is E [0, h max ]D E [0, d is the distance between the target point and the landing point along the k-axis direction max ]For the distance from the target point to the k-axis, θ ε [0,2π]Is the target position in the coordinate system BSetting an included angle between the projection of the vector in the ij plane and the i axis;
the step 3) specifically comprises the following steps: n around the land point p The surface units adopt a polyhedral model, the rest surface units adopt an approximate model to calculate gravitational acceleration g and Laplacian thereof, and the specific calculation formula is as follows:
wherein A is f For the area of each surface unit of the polyhedral model, r f R is the vector from the sampling point to the center of the face unit e To obtain a vector of positions of the sampling points relative to any point on the edge, the variables areA unit normal vector representing the outward facing of the face unit, is->Representing the unit normal vector in the plane, facing outwards at the plane edge of the face unit, +.>The unit vectors respectively representing three vertexes from the sampling point to the grid surface, a and b are respectively distances from the sampling point to the end points of the edge, e is the length of the edge, omega f Representing the band stretched laterally to the sampling pointSolid angle of symbol L e Representing a line integral term associated with the edge of the side;
the step 4) specifically comprises the following steps: dividing the landing leg space into a plurality of sub-regions:
h 0 ＝0,h i ＝h i-1 +Δh i ，i∈[1,10] (6)
Δh 1 ＝0.01h max ,Δh i ＝Δh 1 +0.02(i-1)h max (7)
d 0 ＝0,d j ＝d j-1 +Δd j ，j∈[1,10] (8)
Δd 1 ＝0.01d max ,Δd j ＝Δd 1 +0.02(j-1)d max (9)
θ 0 ＝0,θ k ＝θ k-1 +Δθ k ，k∈[1,10] (10)
then selecting sampling points in each subarea, and calculating by adopting a polyhedral model method and a mixed model method to obtain gravitational acceleration g p And g h Subspace is { [ h ] i-1 ,h i ],[d j-1 ,d j ],[θ k-1 ,θ k ]}：
Calculating the A coordinate of the sampling point according to the formula (1)The position vectors in the system are calculated g respectively p And g h The relative error of each positional hybrid model is thus calculated as:
if the maximum relative error delta among the sampling points max ≤δ tol The optimal polyhedral model surface unit number of the dynamic mixed model in the region is kept to be the original N p Unchanged; otherwise N p ＝N p +dN p The relative error of the position is again calculated until the condition is met.
2. The method for calculating the attraction force of the asteroid landing leg based on the dynamic polyhedral hybrid model according to claim 1, wherein the step 5) specifically comprises: giving h, d and theta of a target point, and importing the optimal polyhedral model surface unit number N of the sub-region where the target point is located p Substituting formula (1) to convert the coordinates under the A coordinate system, and substituting formula (2) to obtain the gravitation of the target point.
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