A method for evaluating multiple reachability coverage of constellation orbit maneuver

Abstract

The application discloses a kind of evaluation constellation orbit transfer maneuver multiple reachable coverage method, it is applicable to the quick evaluation of satellite constellation multiple satellite reachable area, belongs to aerospace field.The application implementation method is: in the fixed sphere coordinate system under the earth center, target area is carried out equal-volume dispersion in two dimensions of distance and zenith angle, and with the reachable azimuth interval of discrete unit center line as evaluation index;By calculating the intersection of center line and reachable domain envelope and carrying out detailed judgment, obtain single satellite reachable azimuth interval;Using a node sorting and one-time sequential reading method, obtain the multiple satellite reachable azimuth interval of corresponding target center line, while combining target azimuth angle area for multiple satellite reachable azimuth interval is intercepted;Integrate the multiple satellite reachable interval of all target center lines to obtain the multiple satellite reachable area of constellation to target area, i.e. 2 realize the multiple reachable coverage evaluation of constellation orbit transfer maneuver.The time complexity of the application is compressed to O (n 2 +n).

Description

Technical Field

[0001] This invention relates to a method for evaluating the multi-reachability coverage of satellite constellation orbit change maneuvers, applicable to the rapid assessment of the reachability area of ​​multiple satellites in a satellite constellation, and belongs to the aerospace field. Background Technology

[0002] The reachability domain of a satellite's orbital maneuver characterizes the spatial range it may reach within a future period, which is crucial for maintaining spacecraft on-orbit safety and improving space situational awareness. Reachability domain calculations for individual satellites are relatively comprehensive, including pulse reachability and continuous thrust reachability. These reachability domains are represented using a three-dimensional mesh, making reachability determination for point targets relatively simple, equivalent to determining whether a point lies inside a polyhedron. For three-dimensional space targets, it is necessary to discretize the surface and then determine the reachability of each point. Research on multi-satellite reachability calculations is limited; existing methods mostly involve statistical analysis of the reachability characteristics of each satellite to the target. For M satellites, each with a reachability envelope described by n faces, the multi-satellite reachability of a target region represented by m discrete points requires Mmn calculations, resulting in an unacceptable computational burden. Furthermore, simply discretizing the surface points cannot quantify the volume of the multi-satellite reachable target region, while the number of m discrete points required for the entire target region is extremely large, further increasing the computational load. Therefore, the quantitative calculation and geometric visualization of multi-satellite reachability domains typically rely on Boolean operation algorithms based on existing three-dimensional mesh models.

[0003] First, the spatial Boolean operation of the three-dimensional mesh model is performed using the technique [1] (see Spatial Boolean operation of three-dimensional mesh model [J] Journal of Huazhong University of Science and Technology (Natural Science Edition), Bi Lin, Wang Liguan, Chen Jianhong, Feng Xinglong, 2008, 294(5):82-85). This method finds the intersection line between two intersecting triangular facets through the intersection test, then obtains the polygon from the intersection line of the intersecting triangle, and re-triangulates the polygon. Finally, the selection of other meshes is judged based on the recombined model. However, if the intersection test is simply performed, a Boolean operation is performed on two three-dimensional regions described by n mesh triangular facets, which requires n(n-1) / 2 intersection judgments. The time consumption of this approach is unacceptable.

[0004] Therefore, specific techniques are usually used to quickly detect intersecting parts. The first technique [2] (see the method of octree-based voxelization of three-dimensional mesh model [J] Journal of Engineering Graphics, Wu Xiaojun, Liu Weijun, Wang Tianran, 2005(4):1-7) recursively divides the spatial cube into 8 small spatial cubes. As long as the triangular facets located in the same subspace are detected for intersection, the intersecting triangles can be quickly located, thus improving the overall running efficiency of the Boolean operation algorithm.

[0005] On the other hand, the OBB bounding box algorithm was proposed by the earlier technology [3] (see OBBBree: A Hierarchical Structure for Rapid Interference Detection [C] Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques, Gottschalk S, Lin MC, Manocha D, 1996). It preprocesses and filters out some triangular faces that cannot intersect, reduces the number of face pairs that need to be intersected, and improves the running efficiency of Boolean operations.

[0006] However, these algorithms extract the final model mesh based on intersecting loops, thus requiring consideration of various special and complex situations, and their robustness needs improvement. Furthermore, these algorithms can only operate on pairs of models sequentially; calculating the N-fold reachability domain for M satellites requires M! / (N-1)! (MN)! Boolean operations, which remains computationally prohibitive. Therefore, this invention proposes a multi-satellite reachability domain calculation method based on line coverage. Summary of the Invention

[0007] The main objective of this invention is to provide a method for evaluating the multiple reachability coverage of constellation orbit change maneuvers. In a geocentric fixed spherical coordinate system, the target region is discretized into equal volumes along two dimensions: distance and zenith angle. The reachability azimuth interval of the centerline of the discrete unit is used as the evaluation index. By calculating and judging the intersection of the centerline and the reachability domain envelope, the reachability azimuth interval of a single star is obtained. A node sorting and one-time sequential reading method is used to obtain the multi-star reachability azimuth intervals corresponding to the target centerline. Simultaneously, the multi-star reachability azimuth intervals are truncated based on the target azimuth region. Finally, the multi-star reachability intervals of all target centerlines are integrated to obtain the multi-star reachability region of the constellation relative to the target region, thus achieving the evaluation of the multiple reachability coverage of constellation orbit change maneuvers.

[0008] The objective of this invention is achieved through the following technical solution.

[0009] This invention discloses a method for evaluating the multiple reachability coverage of constellation orbit change maneuvers, comprising the following steps:

[0010] Step 1: Fix the coordinate system S at the Earth's center I or geocentric inertial coordinate system S J Below, the target region S is represented by a partial three-dimensional spherical layer. This indicates that part of the three-dimensional spherical layer will be... Perform equal-volume discretization in the dimensions of distance r and zenith angle θ, and then set the azimuth angle... Analyzing the dimensions, we obtain I×J rings of equal volume R. ij and with the center line C of the ring of equal volume ij The reachability is taken as an iso-volume ring R ij The accessibility status.

[0011] First, define the coordinate system to be used, considering the Earth inertial coordinate system, the Earth fixed coordinate system, and the satellite orbital coordinate system:

[0012] Geocentric inertial coordinate system S J Its coordinate origin is located at the Earth's gravitational center E, and the reference plane is defined as the Earth's mean equatorial plane. The X-axis points to the vernal equinox, the Z-axis is perpendicular to the equatorial plane and points to the North Pole, and the Y-axis is determined by the right-hand rule: Y = Z × X.

[0013] Geocentric fixed coordinate system S I The origin is located at the Earth's gravitational center E, and the reference plane is defined as the Earth's mean equatorial plane. I The axis runs along the intersection of the Greenwich Meridian plane and the Earth's equatorial plane, Z I The axis is perpendicular to the equatorial plane and points to the North Pole, Y I The axis is determined by the right-hand rule: Y I =Z I ×X I .

[0014] The satellite orbital coordinate system S0 has the satellite's center of mass o as its origin and the current position vector r of the spacecraft as its reference point. Sat Let x0 be the x-axis, z0 be the z-axis perpendicular to the plane of the orbit and pointing in the direction of the positive normal, and y0 be determined by the right-hand rule, i.e., y0 = z0 × x0.

[0015] For single-satellite reachability calculations, the reachability envelope is represented in the orbital coordinate system S0. For multi-satellite reachability calculations, the reachability envelope needs to be uniformly represented in the inertial coordinate system S0. J Below. For a specific satellite represented by the six orbital roots [a,e,i,Ω,ω,f], the reachable domain envelope needs to be uniformly represented in the inertial coordinate system S by the transformation matrix shown in formula (1). J Down:

[0016]

[0017] Where u = ω + f, and M is the rotation matrix about the corresponding axis.

[0018] The transformation between the geocentric inertial frame and the geocentric fixed frame only considers the Earth's uniform rotation, neglecting the effects of precession, nutation, and polar motion. The coordinate system rotates only around the Z-axis, and the corresponding transformation matrix is:

[0019]

[0020] GAST stands for Greenwich Mean Sidereal Time, which is the Earth's rotation angle from the mean vernal equinox to the Greenwich meridian when calculating the satellite's position.

[0021] Based on the different characteristics of the target, in the geocentric fixed coordinate system S I or geocentric inertial coordinate system S J Below, the target region S is given by latitude, longitude, and distance, or by the envelope of a three-dimensional mesh. The distance interval [r] is set according to S and task requirements. L ,r U ] and the zenith angle interval [θ L ,θ U Define a three-dimensional spherical layer Represented as:

[0022]

[0023] Then part of the three-dimensional spherical layer volume for:

[0024]

[0025] For a three-dimensional sphere Discretize in the dimensions of distance r and zenith angle θ, and in the azimuth angle Analyzing the dimensions, we obtain I×J discrete circular rings R of equal volume. ij Represented as:

[0026]

[0027] Where r i and θ j For the nodes to be discrete, the following conditions must be met:

[0028] r0 = r L ,r I =r U ,θ0=θ L ,θ J =θ U

[0029] Given the discrete ring volume V of three-dimensional information ij for:

[0030]

[0031] As the zenith angle and distance increase, the equal spacing (same as Δθ=θ) j -θ j-1 and Δr=r i -r i-1 The increasing volume differences between discrete elements represented by the mesh method are detrimental to the reachability analysis of constellations. Therefore, the equal-volume discretization method is used to determine the node r.i and θ j The volume of a portion of the three-dimensional sphere is given by equation (3), then the volume V of each discrete ring is... ij for

[0032]

[0033] Combining equations (4) and (5), node r i and θ j Represented as:

[0034]

[0035]

[0036] Discrete unit R ij The centerline is defined as C ij , represented as:

[0037]

[0038] Where C ij It is caused by node r i and θ j The reachability of the centerline of the determined circle is determined by the azimuth angle interval corresponding to the reachability domain of the satellites in the constellation. express.

[0039] When the parameters I and J of the discrete element are sufficiently large, the volume V of the resulting discrete ring is... ij The reachability R for a single discrete unit is relatively small. RD(ij) Accessibility using its centerline Φ RD(ij) The approximate performance improves, as expressed by:

[0040]

[0041] The method achieves volumetric discretization of three-dimensional spatial targets using a two-dimensional discrete computation of distance and zenith angle I×J. The spatial targets are represented by the reachable interval of the analytical azimuth center ring, and it is applicable to any target region given by latitude and longitude or grid envelope.

[0042] Step 2: For the center line C of the equal-volume ring obtained by equal-volume discretization in Step 1 ij and transformation to inertial coordinate system S J The reachable envelope of a single star is calculated by decomposing the three-dimensional reachable envelope of a single star into multiple triangular patches, and then calculating the centerline C in spherical coordinates. ij The in-plane intersections of the triangle and the triangular facet yield the corresponding single-star reachable azimuth matrix. and target azimuth matrix Φ TAR(ij) .

[0043] The reachable envelope of a single star is represented as r(x,y,z)0 in the orbital coordinate system S0, where the positions of the three axes are given by an m×n grid matrix. This is then transformed to the inertial coordinate system S... J The following is represented as r(x,y,z). J Or in the fixed coordinate system S I The following is represented as r(x,y,z). I .

[0044] Calculating the azimuth interval corresponding to the satellite's reachable domain is equivalent to finding the center circle C. ij Intersection points with all grid surfaces constituting the envelope. To simplify the calculation, the envelope is represented in spherical coordinates, also using an m×n grid matrix. The problem is equivalent to finding r = (r i-1 +r i ) / 2 and θ=(θ j-1 +θ j The line ) / 2 intersects with all the grid surfaces that make up the envelope.

[0045] Consider adjacent 2×2 matrices in an m×n grid matrix as a single non-planar grid surface element, with a total of k = (n-1)(m-1) such elements. Calculate the extreme values ​​of distance and zenith angle, and determine C. ij Whether it is within the range of r and θ of the grid surface, the selected grid surfaces are represented as set K.

[0046] For the initially selected set of mesh faces K, determine the number of unique points. A set of 3 points forms one triangular facet, and a set of 4 points forms two triangular facets. For all triangular facets, calculate the sum of C. ij Find the intersection point and determine whether the intersection point is inside the triangle.

[0047] Calculate the azimuth angles of all intersection points P in set K that satisfy the requirements. Since the envelope is a closed surface, the number of intersection points must be an even number, 2K. m The reachability region of the m-th satellite is defined with respect to C. ij The intersection points are arranged in ascending order of azimuth angle to obtain the following sequence:

[0048]

[0049] Where +1 represents the start point of the reachable interval, -1 represents the end point of the reachable interval, and there are a total of K. m each interval

[0050] Since the azimuth angle ranges from [0, 2π] when all grid points are transformed to spherical coordinates. When There are two possibilities: 1) the Z-axis intersects the envelope; 2) the envelope intersects the plane formed by the positive X-axis and the positive Z-axis, but the Z-axis does not intersect the envelope. Changing the azimuth range to [-π, π], if it still... If the condition is not met, it is determined to be case 1; otherwise, it is determined to be case 2.

[0051] For case 1, when C ij When there is an intersection with the envelope in spherical coordinates, it is further determined whether the intersection corresponds to a reachable or inaccessible region. In Cartesian fixed coordinates, the midpoint of the two azimuth nodes is taken. Draw a ray along the positive Z-axis and find its intersection with the reachable region envelope, using the same method as in spherical coordinates. If the number of intersection points is odd, the corresponding azimuth interval is a reachable region; otherwise, it is an inaccessible region. The change corresponds to the sign of the two nodes.

[0052] If C ij When there is no intersection with the envelope in spherical coordinates, further judgment is needed for C. ij Whether it lies inside the envelope. Also in the Cartesian inertial coordinate system, take any C. ij Draw a ray along the positive Z-axis from the previous point and find its intersection with the envelope of the reachable region. If the number of intersection points is odd, then C ij Located inside the envelope, the reachable azimuth angle range is [0, 2π].

[0053] For cases 1 and 2, the azimuth of the intersection point may lie in the interval [-π, 0]. We transform it to the interval [0, 2π] for the next calculation. Nodes with azimuths in the interval [-π, 0] are equivalently transformed to the interval [0, 2π], and then arranged again in ascending order of azimuth. If the second row of the first column is -1, the second row of the last column must be +1; therefore, we add a starting node. and a termination node

[0054] By decomposing the three-dimensional envelope of a single star into multiple triangular facets, the centerline C is calculated in spherical coordinates. ij The in-plane intersection of the triangle and the target centerline C under any condition is obtained by formula (5). ij Azimuth sequence Target azimuth sequence Φ TAR(ij) Obtained using the same method.

[0055] Step 3: For the reachable region of the m-th satellite obtained in Step 2, represented by start and end nodes, for C ij Reachable azimuth sequence The constellation's alignment with centerline C is obtained through node sorting and a one-time sequential read. ij Multi-star reachable matrix According to Φ TAR(ij) The target azimuth interval By performing a cut-off, a multi-star reachability matrix for a given target region is obtained.

[0056] The reachable azimuth angles of all M satellites obtained in step two The result of merging is:

[0057]

[0058] At this time, the matrix The number of nodes in the data is:

[0059]

[0060] Next, the matrix Arranged in ascending order according to the first row:

[0061]

[0062] Define the reachable stars as N and set its initial value N0 = 0. Then, read the matrix sequentially. The second row of elements. When the s-th element is +1 (s∈{1,2,...,S}), the corresponding angle in the first row represents the starting point of a new reachable interval, thus increasing the number of reachable stars by N. s =N s-1 +1. Conversely, when the s-th element is -1 (s∈{1,2,...,S}), the corresponding first row angle represents the termination point of an existing reachable interval, thus reducing the number of reachable stars by N. s =N s-1 -1. After reading all elements, set the corresponding N... s This means that a completely new matrix is ​​obtained in the third row of the matrix. for:

[0063]

[0064] Where N s For interval The number of stars that can be reached.

[0065] Subsequently, it is necessary to determine the target azimuth interval Φ TAR(ij) For multi-star reachable matrices Perform the transformation. Transform Φ TAR(ij) The 2p azimuth nodes in the diagram are represented as follows: and The nodes are arranged in ascending order of azimuth to obtain a new matrix. for:

[0066]

[0067] in The third row of the column For the number of reachable stars N in the third row of the previous column, ... Add a column of all zeros to the first column of the table, so that... hour When multiple points have the same When the zenith angle is the same, it is necessary to ensure Located at the end of the sort, It is at the top of the sort.

[0068] position and The column and the pair By performing truncation, we obtain p corresponding target multi-star reachability matrices. Represented as:

[0069]

[0070] The constellation's alignment with centerline C is obtained through node sorting and a one-time sequential read. ij Given a multi-star reachability matrix for a target region

[0071] Step 4: Based on the centerline C obtained in Step 3 ij Given a multi-star reachability matrix for a target region Calculate a single discrete unit R ij The N-star can reach the azimuth angle Γ RDNX(ijp) (N) integrates the multi-star reachable ranges of all target centerlines to obtain the multi-star reachable range of the constellation for the target area, that is, to realize the multi-reachable coverage assessment of constellation orbit change maneuvers.

[0072] The maximum number of reachable stars is defined as N. max =max(N) ps The initial reachable angle of multiple stars is defined as Γ. RDNX(ijp) (N) = 0, where N ∈ {1,...,N} max Then, iterate through... N in the third line s =The azimuth node corresponding to N, then the reachable azimuth angle of star N is:

[0073]

[0074] After obtaining the centerline C of each discrete unit ij Corresponding target range After the reachable angle of star N, the reachable angle Γ CovNX(ijp) (N) and target angle The proportion is approximated as the reachability of the target region within the entire discrete unit. Then the centerline C... ij The multi-star reachable volume V of all targets RDNX(ij) (N) satisfies:

[0075]

[0076] Once the reachability interval of a single discrete element is given, its reachable volume can be directly obtained. The sum of the reachable volumes of all discrete elements is the reachable volume of the entire spatial target region. For I×J discrete annexes of equal volume, their reachable volume is expressed as:

[0077]

[0078] Where V RDNX (N) is the reachable volume of N stars in the entire target region of space.

[0079] The process also includes step five: based on the multi-reachability coverage of the constellation orbit-changing maneuvers assessed in step four, a visual representation of the multi-satellite reachable region is achieved, improving the accuracy and efficiency of multi-satellite reachable region assessment in multi-satellite collaboration, thereby improving the accuracy and efficiency of multi-satellite collaborative mission execution. The multi-satellite collaborative mission includes observation, guidance, and interception.

[0080] Beneficial effects:

[0081] 1. This invention discloses a method for evaluating the multi-reachability coverage of constellation orbital maneuvers. In a geocentric fixed spherical coordinate system, the reachable region and target region are represented by I×J discrete circular ring centerline azimuth intervals. The reachable azimuth interval of the discrete unit centerline is used as the evaluation index. By decomposing the single-star reachable three-dimensional envelope into multiple triangular facets and analytically calculating the in-plane intersections, the reachable azimuth accuracy and computational complexity are similar to those of a three-dimensional mesh envelope. Furthermore, the n azimuth intervals represented by start and end nodes can be used for rapid calculation of multi-star reachable intervals based on node sorting and sequential reading, with a time complexity of O(n^2). 2 +n). The time complexity of calculating the m-fold coverage area corresponding to n satellites using three-dimensional Boolean operations is O(n! / (m-1)!(nm)!).

[0082] 2. The present invention discloses a method for evaluating the multiple reachability coverage of satellite constellation maneuvers. By dividing the conventional effective area into two special cases, namely the intersection of the envelope and the Z-axis, and the intersection of the plane formed by the positive X-axis and the positive Z-axis without the Z-axis intersecting the envelope, the azimuth angle interval of the intersection of the three-dimensional envelope and the center line of the discrete circular ring at any position and shape in space is obtained. It is applicable to any satellite reachable area and target area given by latitude and longitude or grid envelope.

[0083] 3. This invention discloses a method for evaluating the multi-reachability coverage of constellation maneuvers. Building upon the aforementioned beneficial effects 1 and 2, it achieves quantitative evaluation and geometric visualization of the multi-satellite reachability domain. Based on the visualized representation results of the multi-satellite reachability domain, it improves the accuracy and efficiency of multi-satellite reachability domain evaluation in multi-satellite collaboration, thereby enhancing the accuracy and efficiency of multi-satellite collaborative mission execution. The multi-satellite collaborative mission includes observation, guidance, and interception. Attached Figure Description

[0084] Figure 1 This is a flowchart of a method for evaluating the multiple reachability coverage of constellation orbit change maneuvers disclosed in this invention.

[0085] Figure 2 The geocentric inertial coordinate system S in step one of this invention J Geocentric fixed coordinate system S I and satellite orbital coordinate system S o Positional relationship.

[0086] Figure 3 The reachable domain envelope and centerline C in the Cartesian coordinate system in step two of this invention. ij The positional relationship of the satellite is shown in the red grid, where the initial orbital root number of the satellite is [6778.137km, 0, 70°, 50°, 0°, 20°]. The satellite flies for 20 minutes with a 15g overload and a pulse increment of 3km / s. The center line is shown in the blue solid line, where r = 6617.081km and θ = 9.755°.

[0087] Figure 4 The reachable domain envelope and centerline C in the spherical coordinate system of step two of this invention are... ij The positional relationship. The satellite maneuverable reachable envelope is represented by a red grid, the centerline by a blue solid line, and the intersection by a black circle. The satellite reachable envelope and target centerline data are related to... Figure 3 same.

[0088] Figure 5 The center line C in the Cartesian coordinate system in step two of this invention. ij The reachable azimuth range is defined. The satellite maneuverable envelope is represented by a red grid, the reachable area by a green solid line, the unreachable area by a blue solid line, the satellite orbit by a black solid line, and the initial satellite position by a red pentagram. The satellite's initial orbital root numbers are [6778.137km, 0, 70°, 310°, 0°, 0°], flying for 15 minutes with a 15g overload and a 3km / s pulse increment. The target area is set at a distance of 6778.137km, with a zenith angle range of [0°, 60°] and an azimuth angle range of [0°, 360°]. The discrete model parameters are set to 10×1.

[0089] Figure 6 In step two of this invention, under Cartesian coordinates, case 1: when the Z-axis intersects the envelope, the centerline C... ijThe reachable azimuth range. The satellite's maneuverable reachable envelope is represented by a red grid, the reachable area by a green solid line, the unreachable area by a blue solid line, the satellite orbit by a black solid line, and the satellite's initial position by a red pentagram. The satellite's initial orbital root numbers are [6778.137km, 0, 70°, 310°, 0°, 0°], and its maneuverability is... Figure 5 Same. Target area data and Figure 5 same.

[0090] Figure 7 In step two of this invention, under Cartesian coordinates, case 2 occurs when the envelope intersects the plane formed by the positive X-axis and the positive Z-axis, but the Z-axis does not intersect the envelope. The centerline C... ij The reachable azimuth range is indicated by a yellow circle at the start and a yellow cross at the end. The satellite's maneuverable reachable envelope is represented by a red grid, reachable areas by a green solid line, unreachable areas by a blue solid line, the satellite orbit by a black solid line, and the satellite's initial position by a red pentagram. The satellite's initial orbital root numbers are [6778.137km, 0, 85°, 50°, 0°, 30°], and its maneuverability is... Figure 5 Same. Target area data and Figure 5 same.

[0091] Figure 8 This is a schematic diagram of the calculation of the reachable interval of the centerline multi-star in step three of this invention.

[0092] Figure 9 This relates to step three of the present invention, which describes the multi-star reachability of the Walker constellation to a target area given latitude, longitude, and distance. The Walker constellation has an orbital semi-major axis of 6778.137 km, a configuration code of 70°:30 / 5 / 2, and maneuverability similar to... Figure 5 Same. The maneuverable reachable area envelope is represented by a blue grid, the reachable area of ​​1 satellite is represented by a blue solid line, the reachable area of ​​2 satellites is represented by a green solid line, the reachable area of ​​3 satellites is represented by a red solid line, the satellite orbit is represented by a black solid line, and the starting point of the reachable azimuth interval is represented by a yellow circle and the ending point is represented by a yellow cross.

[0093] Figure 10 This is a schematic diagram of the multi-star reachability interval calculation for a given target region in step three of this invention.

[0094] Figure 11 This refers to the multi-star reachability of the Walker constellation with respect to a given maneuverable reachable envelope representing the target region in step three of this invention. The Walker constellation and... Figure 9 The same, mobility and Figure 5 The target satellite's initial orbital root numbers are [6778.137km, 0, 60°, 310°, 0°, 30°], and its maneuverability is the same as... Figure 5 Same. The target reachability envelope is represented by an orange grid, the maneuver reachability envelope by a blue grid, the reachable area of ​​1 satellite by a blue solid line, the reachable area of ​​2 satellites by a green solid line, the reachable area of ​​3 satellites by a red solid line, the satellite orbit by a black solid line, and the starting point of the reachable azimuth interval is represented by a yellow circle and the ending point by a yellow cross.

[0095] Figure 12 This refers to the reachability of a single satellite under different numbers of satellites representing the target area, where latitude, longitude, and distance are used in an example of this invention.

[0096] Figure 13 This refers to the 2-satellite reachability in this invention example, considering different numbers of satellites representing the target area using latitude, longitude, and distance.

[0097] Figure 14 This refers to the 1-satellite reachability in this invention example, for different numbers of satellites representing the target area by the maneuverable reachability domain.

[0098] Figure 15 This refers to the 2-satellite reachability in this invention example, considering different numbers of satellites representing the target area by the maneuverable reachability domain. Detailed Implementation

[0099] To better illustrate the purpose and advantages of the present invention, the invention will be further described below in conjunction with the accompanying drawings and examples.

[0100] Example 1:

[0101] To verify the feasibility of the method, a target region represented by latitude, longitude, and distance, and a target region represented by a maneuverable reachability domain, are given respectively, and a discrete cell partitioning method using equal volume is employed. Then, multiple Walker circular orbit constellations are used to calculate the multi-star reachability region along the centerline of the discrete cell, finally obtaining the percentage of multi-star reachability of the target region. The constellation parameters used are shown in the table below:

[0102] Table 1 Zodiac Parameters

[0103]

[0104] The target regions represented by latitude, longitude, and distance, as well as the target regions represented by the maneuverability domain, are both discretized in the three-dimensional sphere shown in the table below.

[0105] Table 2 Target Area Parameters

[0106]

[0107] like Figure 1 As shown in the figure, this embodiment discloses a method for evaluating the multiple reachability coverage of constellation orbit change maneuvers, and the specific implementation steps are as follows:

[0108] Step 1: In the geocentric inertial coordinate system S J Below, the target region S is represented by a partial three-dimensional spherical layer. This indicates that part of the three-dimensional spherical layer will be... Perform equal-volume discretization in the dimensions of distance r and zenith angle θ, and then set the azimuth angle... Analyzing the dimensions, we obtain I×J rings of equal volume R. ij and its centerline C ij The reachability of R ij The accessibility status.

[0109] This example represents the target area in the Earth's inertial coordinate system, while defining the satellite orbital coordinate system as follows:

[0110] Geocentric inertial coordinate system S J Its coordinate system originates at the Earth's gravitational center E, and the reference plane is defined as the Earth's mean equatorial plane. The X-axis points to the vernal equinox, the Z-axis is perpendicular to the equatorial plane and points to the North Pole, and the Y-axis is determined by the right-hand rule: Y = Z × X.

[0111] Geocentric fixed coordinate system S I The origin is located at the Earth's gravitational center E, and the reference plane is defined as the Earth's mean equatorial plane. I The axis runs along the intersection of the Greenwich Meridian plane and the Earth's equatorial plane, Z I The axis is perpendicular to the equatorial plane and points to the North Pole, Y I The axis is determined by the right-hand rule: Y I =Z I ×X I .

[0112] The satellite orbital coordinate system S0 has the satellite's center of mass o as its origin and the current position vector r of the spacecraft as its reference point. Sat Let x0 be the x-axis, z0 be the z-axis perpendicular to the plane of the orbit and pointing in the direction of the positive normal, and y0 be determined by the right-hand rule, i.e., y0 = z0 × x0.

[0113] Figure 2 For the geocentric inertial coordinate system S J Geocentric fixed coordinate system S I and satellite orbital coordinate system S o Positional relationship.

[0114] For single-satellite reachability calculations, the reachability envelope is represented in the orbital coordinate system S0. For multi-satellite reachability calculations, the reachability envelope needs to be uniformly represented in the inertial coordinate system S0. J Below. For a specific satellite represented by the six orbital roots [a, e, i, Ω, ω, f], the reachable domain envelope needs to be uniformly represented in the inertial coordinate system S using the following transformation matrix. J Down:

[0115]

[0116] Where u = ω + f, and M is the rotation matrix about the corresponding axis.

[0117] The transformation between the geocentric inertial frame and the geocentric fixed frame only considers the Earth's uniform rotation, neglecting the effects of precession, nutation, and polar motion. The coordinate system rotates only around the Z-axis, and the corresponding transformation matrix is:

[0118]

[0119] GAST stands for Greenwich Mean Sidereal Time, which is the Earth's rotation angle from the mean vernal equinox to the Greenwich meridian when calculating the satellite's position.

[0120] This example is in the geocentric inertial coordinate system S. J Below, a three-dimensional sphere is set according to the target region S and Table 2. express:

[0121]

[0122] The distance range is [6578.137, 6778.137] km, the zenith angle range is [0, 60°], and the azimuth angle range is [0, 360°].

[0123] but volume for:

[0124]

[0125] In the example

[0126] For a three-dimensional sphere Discretize in the dimensions of distance r and zenith angle θ, and in the azimuth angle Analyzing the dimensions, we obtain I×J discrete circular rings R of equal volume. ij Represented as:

[0127]

[0128] Where r i and θ j For the nodes to be discrete, the following conditions must be met:

[0129] r0 = r L ,r I =r U ,θ0=θ L ,θ J =θ U

[0130] Given the discrete ring volume V of three-dimensional information ij for:

[0131]

[0132] As the zenith angle and distance increase, the traditional equal spacing (same as Δθ=θ) becomes less effective. j -θ j-1 and Δr=r i -r i-1 The increasing volume differences between discrete elements represented by the mesh method are detrimental to the reachability analysis of constellations. Therefore, the equal-volume discretization method is used to determine the node r. i and θ j The volume V of each discrete ring ij for

[0133]

[0134] Node r i and θ j Represented as:

[0135]

[0136]

[0137] Discrete unit R ij The centerline is defined as C ij , represented as:

[0138]

[0139] Where C ij It is caused by node r i and θ j The reachability of the centerline of the determined circle is determined by the azimuth angle interval corresponding to the reachability domain of the satellites in the constellation. express.

[0140] When the parameters I and J of the discrete element are sufficiently large, the volume V of the resulting discrete ring is... ij The reachability R for a single discrete unit is relatively small. RD(ij) Accessibility using its centerline Φ RD(ij) The approximate performance improves, as expressed by:

[0141]

[0142] The method achieves volumetric discretization of three-dimensional spatial targets using a two-dimensional discrete computation of distance and zenith angle I×J. The spatial targets are represented by the reachable interval of the analytical azimuth center ring, and it is applicable to any target region given by latitude and longitude or grid envelope.

[0143] Step 2: For the center line C of the equal-volume ring obtained by equal-volume discretization in Step 1 ij and transformation to inertial coordinate system S J The reachable envelope of a single star is calculated by decomposing the three-dimensional reachable envelope of a single star into multiple triangular patches, and then calculating the centerline C in spherical coordinates. ij The in-plane intersections of the triangle and the triangular facet yield the corresponding single-star reachable azimuth matrix. and target azimuth matrix Φ TAR(ij) .

[0144] The reachable envelope of a single star is represented as r(x,y,z)0 in the orbital coordinate system S0, where the positions of the three axes are given by an m×n grid matrix. This is then transformed to the inertial coordinate system S... J The following is represented as r(x,y,z). J .

[0145] Calculating the azimuth interval corresponding to the satellite's reachable domain is equivalent to finding the center circle C. ij Intersection with all grid surfaces that make up the envelope. Figure 3 Let C be the envelope of the reachable region in Cartesian coordinates and the centerline. ij The positional relationship of the satellite is shown in the red grid, where the initial orbital root number of the satellite is [6778.137km, 0, 70°, 50°, 0°, 20°]. The satellite flies for 20 minutes with a 15g overload and a pulse increment of 3km / s. The center line is shown in the blue solid line, where r = 6617.081km and θ = 9.755°.

[0146] To simplify the calculation, the envelope is represented in spherical coordinates, also using an m×n grid matrix. The problem is equivalent to finding r = (r i-1 +r i ) / 2 and θ=(θ j-1 +θ j The line ) / 2 intersects with all the grid surfaces that make up the envelope. Figure 4 Let C be the envelope of the reachable domain in spherical coordinates and the centerline. ij The positional relationship is shown, with the intersection point represented by a black circle.

[0147] Consider adjacent 2×2 matrices in an m×n grid matrix as a single non-planar grid surface element, with a total of k = (n-1)(m-1) such elements. Calculate the extreme values ​​of distance and zenith angle, and determine C. ij Whether it is within the range of r and θ of the grid surface, the selected grid surfaces are represented as set K.

[0148] For the initially selected set of mesh faces K, determine the number of unique points. A set of 3 points forms one triangular facet, and a set of 4 points forms two triangular facets. For all triangular facets, calculate the sum of C. ijFind the intersection point and determine whether the intersection point is inside the triangle.

[0149] Calculate the azimuth angles of all intersection points P in set K that satisfy the requirements. Since the envelope is a closed surface, the number of intersection points must be an even number, 2K. m The reachability region of the m-th satellite is defined with respect to C. ij The intersection points are arranged in ascending order of azimuth angle to obtain the following sequence:

[0150]

[0151] Where +1 represents the start point of the reachable interval, -1 represents the end point of the reachable interval, and there are a total of K. m each interval

[0152] Figure 5 Centerline C in Cartesian coordinate system ij The reachable azimuth range is defined. The satellite maneuverable envelope is represented by a red grid, the reachable area by a green solid line, the unreachable area by a blue solid line, the satellite orbit by a black solid line, and the initial satellite position by a red pentagram. The satellite's initial orbital root numbers are [6778.137km, 0, 70°, 310°, 0°, 0°], flying for 15 minutes with a 15g overload and a 3km / s pulse increment. The target area is set at a distance of 6778.137km, with a zenith angle range of [0°, 60°] and an azimuth angle range of [0°, 360°]. The discrete model parameters are set to 10×1.

[0153] Since the azimuth angle ranges from [0, 2π] when all grid points are transformed to spherical coordinates. When There are two possibilities: 1) the Z-axis intersects the envelope; 2) the envelope intersects the plane formed by the positive X-axis and the positive Z-axis, but the Z-axis does not intersect the envelope. Changing the azimuth range to [-π, π], if it still... If the condition is not met, it is determined to be case 1; otherwise, it is determined to be case 2.

[0154] For case 1, when C ij When there is an intersection with the envelope in spherical coordinates, it is further determined whether the intersection corresponds to a reachable or inaccessible region. In Cartesian fixed coordinates, the midpoint of the two azimuth nodes is taken. Draw a ray along the positive Z-axis and find its intersection with the reachable region envelope, using the same method as in spherical coordinates. If the number of intersection points is odd, the corresponding azimuth interval is a reachable region; otherwise, it is an inaccessible region. The change corresponds to the sign of the two nodes.

[0155] If C ij When there is no intersection with the envelope in spherical coordinates, further judgment is needed for C. ijWhether it lies inside the envelope. Also in the Cartesian inertial coordinate system, take any C. ij Draw a ray along the positive Z-axis from the previous point and find its intersection with the envelope of the reachable region. If the number of intersection points is odd, then C ij Located inside the envelope, the reachable azimuth angle range is [0, 2π].

[0156] Figure 6 For case 1 in Cartesian coordinate system, the centerline C ij The reachable azimuth range. The initial orbital root numbers of the satellite are [6778.137km, 0, 70°, 310°, 0°, 0°].

[0157] For cases 1 and 2, the azimuth of the intersection point may lie in the interval [-π, 0]. We transform it to the interval [0, 2π] for the next calculation. Nodes with azimuths in the interval [-π, 0] are equivalently transformed to the interval [0, 2π], and then arranged again in ascending order of azimuth. If the second row of the first column is -1, the second row of the last column must be +1; therefore, we add a starting node. and a termination node

[0158] Figure 6 For case 2 in Cartesian coordinate system, the centerline C ij The reachable azimuth range. The initial orbital root numbers of the satellite are [6778.137km, 0, 85°, 50°, 0°, 30°].

[0159] In summary, the reachability region of a single satellite and the target centerline C under any conditions are obtained. ij Azimuth sequence Target azimuth sequence Φ TAR(ij) Obtained using the same method.

[0160] Step 3: For the reachable region of the m-th satellite obtained in Step 2, represented by start and end nodes, for C ij Reachable azimuth sequence The constellation's alignment with centerline C is obtained through node sorting and a one-time sequential read. ij Multi-star reachable matrix According to Φ TAR(ij) The target azimuth interval By performing a cut-off, a multi-star reachability matrix for a given target region is obtained.

[0161] The reachable azimuth angles of all M satellites obtained in step two The result of merging is:

[0162]

[0163] At this time, the matrix The number of nodes in the data is:

[0164]

[0165] Next, the matrix Arranged in ascending order according to the first row:

[0166]

[0167] Define the reachable stars as N and set its initial value N0 = 0. Then, read the matrix sequentially. The second row of elements. When the s-th element is +1 (s∈{1,2,...,S}), the corresponding angle in the first row represents the starting point of a new reachable interval, thus increasing the number of reachable stars by N. s =N s-1 +1. Conversely, when the s-th element is -1 (s∈{1,2,...,S}), the corresponding first row angle represents the termination point of an existing reachable interval, thus reducing the number of reachable stars by N. s =N s-1 -1. After reading all elements, set the corresponding N... s This means that a completely new matrix is ​​obtained in the third row of the matrix. for:

[0168]

[0169] Where N s For interval The number of stars that can be reached. Figure 8 This is a schematic diagram for calculating the reachable range of multiple stars along the centerline.

[0170] Figure 9 This represents the multi-satellite reachability of the Walker constellation for a given latitude, longitude, and distance to a target area. The Walker constellation has an orbital semi-major axis of 6778.137 km and a configuration code of 70°:30 / 5 / 2. The maneuverable reachability envelope is represented by a blue grid, the reachable area with 1 satellite is represented by a solid blue line, the reachable area with 2 satellites is represented by a solid green line, and the reachable area with 3 satellites is represented by a solid red line.

[0171] Subsequently, it is necessary to determine the target azimuth interval Φ TAR(ij) For multi-star reachable matrices Perform the transformation. Transform Φ TAR(ij) The 2p azimuth nodes in the diagram are represented as follows: and The nodes are arranged in ascending order of azimuth to obtain a new matrix. for:

[0172]

[0173] in The third row of the column For the number of reachable stars N in the third row of the previous column, ... Add a column of all zeros to the first column of the table, so that... hour It should be noted that when multiple points have the same characteristics as... When the zenith angle is the same, it is necessary to ensure Located at the end of the sort, It is at the top of the sort.

[0174] position and The column and the pair By performing truncation, we obtain p corresponding target multi-star reachability matrices. Represented as:

[0175]

[0176] Therefore, we can obtain the constellation relative to the center line C. ij Given a multi-star reachability matrix for a target region Figure 10 A schematic diagram illustrating the calculation of multi-star reachability intervals for a given target region. Figure 11 To and Figure 9 The Walker constellation represents the reachability of multiple satellites in a given maneuverable reachable envelope for a target region. The initial orbital root numbers of the target satellites are [6778.137 km, 0, 60°, 310°, 0°, 30°]. The target reachable region envelope is represented by an orange grid, the maneuverable reachable region envelope by a blue grid, the 1-satellite reachable region by a blue solid line, the 2-satellite reachable region by a green solid line, and the 3-satellite reachable region by a red solid line.

[0177] Step 4: Based on the centerline C obtained in Step 3 ij Given a multi-star reachability matrix for a target region Calculate a single discrete unit R ij The N-star can reach the azimuth angle Γ RDNX(ijp) (N) integrates the multi-star reachable ranges of all target centerlines to obtain the multi-star reachable range of the constellation for the target area, that is, to realize the multi-reachable coverage assessment of constellation orbit change maneuvers.

[0178] The maximum number of reachable stars is defined as N. max =max(N) ps The initial reachable angle of multiple stars is defined as Γ. RDNX(ijp) (N) = 0, where N ∈ {1,...,N} max Then, iterate through... N in the third line s =The azimuth node corresponding to N, then the reachable azimuth angle of star N is:

[0179]

[0180] After obtaining the centerline C of each discrete unit ij Corresponding target range After the reachable angle of star N, the reachable angle Γ CovNX(ijp) (N) and target angle The proportion is approximated as the reachability of the target region within the entire discrete unit. Then the centerline C... ij The multi-star reachable volume V of all targets RDNX(ij) (N) satisfies:

[0181]

[0182] Once the reachability interval of a single discrete element is given, its reachable volume can be directly obtained. The sum of the reachable volumes of all discrete elements is the reachable volume of the entire spatial target region. For I×J discrete annexes of equal volume, their reachable volume is expressed as:

[0183]

[0184] Where V RDNX (N) is the reachable volume of N stars in the entire target region of space.

[0185] The 996 Walker constellations in Table 1 are used to perform multi-star reachability calculations for the target areas represented by the given latitude, longitude, and distance in Table 2. Figure 12 To achieve a target reachability rate of over 90% with a minimum of 36 satellites, a maximum reachability rate of 94.2585% is required. As the number of satellites increases, the reachability rate increases more gradually, and the remaining approximately 10% of unreachable areas require a large number of satellites. Figure 13 The table below shows the reachability rate of a target with two satellites for different numbers of satellites. As the number of satellites increases, the reachability rate increases significantly, but the target has not yet reached saturation. Considering the optimal inclination angle, the constellation configurations with a reachability rate of over 90% for a single satellite are shown in the table below:

[0186] Table 3. Configurations with a 90% or higher reachability rate for a single satellite in the three-dimensional spherical target region.

[0187]

[0188]

[0189] Similarly, the 996 Walker constellations in Table 1 are used to perform multi-satellite reachability calculations for the target areas represented by the given maneuverable reachability in Table 2. The initial orbital root numbers of the target satellites are [6778.137km, 0, 60°, 310°, 0°, 30°], and they fly for 15 minutes with a 15g overload and a 3km / s pulse increment. Figure 12 The reachability of a target with one satellite for different numbers of satellites. Figure 2 This represents the reachability rate for two stars. Due to the reduced target area, most constellations with the same configuration have better reachability. The maximum reachability for one star is 99.9956% when the configuration is 70°:39 / 3 / 1. Considering the optimal tilt angle, the constellation configurations with over 99% reachability for one star are shown in the table below:

[0190] Table 4. Target Area Reachability with 1-Star Reach of Over 99% in Reachable Envelope Target Region

[0191]

[0192] The above detailed description further illustrates the purpose, technical solution, and beneficial effects of the invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for evaluating the multiple reachability coverage of constellation orbital maneuvers, characterized in that: Includes the following steps, Step 1: Fix the coordinate system S at the Earth's center I or geocentric inertial coordinate system S J Below, the target region S is represented by a partial three-dimensional spherical layer. This indicates that part of the three-dimensional spherical layer will be... Perform equal-volume discretization in the dimensions of distance r and zenith angle θ, and then set the azimuth angle... Analyzing the dimensions, we obtain I×J rings of equal volume R. ij and with the center line C of the ring of equal volume ij The reachability is taken as an iso-volume ring R ij Accessibility status; Step 2: For the center line C of the equal-volume ring obtained by equal-volume discretization in Step 1 ij and transformation to inertial coordinate system S J The three-dimensional envelope of a single star is obtained by decomposing it into multiple triangular facets and calculating the centerline C in spherical coordinates. ij The in-plane intersections of the triangle and the triangular facet yield the corresponding single-star reachable azimuth matrix. and target azimuth matrix Φ TAR(ij) ; Step 3: For the reachable region of the m-th satellite obtained in Step 2, represented by start and end nodes, for C ij Reachable azimuth sequence The constellation's alignment with centerline C is obtained through node sorting and a one-time sequential read. ij Multi-star reachable matrix According to Φ TAR(ij) The target azimuth interval By performing a cut-off, a multi-star reachability matrix for a given target region is obtained. Step 4: Based on the centerline C obtained in Step 3 ij Given a multi-star reachability matrix for a target region Calculate a single discrete unit R ij The N-star can reach the azimuth angle Γ RDNX(ijp) (N) integrates the multi-star reachable ranges of all target centerlines to obtain the multi-star reachable range of the constellation for the target area, that is, to realize the multi-reachable coverage assessment of constellation orbit change maneuvers.

2. The method for evaluating the multiple reachability coverage of constellation orbit change maneuvers as described in claim 1, characterized in that: It also includes step five: based on the multi-reachability coverage of the constellation orbit change maneuver obtained from step four, realize the visualization representation of the multi-star reachable area, improve the accuracy and efficiency of the multi-star reachable area assessment in multi-star collaboration, and thus improve the accuracy and efficiency of multi-star collaboration mission execution.

3. The method for evaluating the multiple reachability coverage of constellation orbit change maneuvers as described in claim 2, characterized in that: The multi-satellite collaborative mission includes observation, guidance, and interception.

4. A method for evaluating the multiple reachability coverage of constellation orbital maneuvers as described in claim 1, 2, or 3, characterized in that: The implementation method for step one is as follows: First, define the coordinate system to be used, considering the Earth inertial coordinate system, the Earth fixed coordinate system, and the satellite orbital coordinate system: Geocentric inertial coordinate system S J Its coordinate origin is located at the Earth's gravitational center E, and the reference plane is defined as the Earth's mean equatorial plane; the X-axis points to the vernal equinox, the Z-axis is perpendicular to the equatorial plane and points to the North Pole, and the Y-axis is determined by the right-hand rule: Y = Z × X; Geocentric fixed coordinate system S I The origin is located at the Earth's gravitational center E, and the reference plane is defined as the Earth's mean equatorial plane. I The axis runs along the intersection of the Greenwich Meridian plane and the Earth's equatorial plane, Z I The axis is perpendicular to the equatorial plane and points to the North Pole, Y I The axis is determined by the right-hand rule: Y I =Z I ×X I ; The satellite orbital coordinate system S0 has the satellite's center of mass o as its origin and the current position vector r of the spacecraft as its reference point. Sat Let x0 be the x-axis, z0 be the z-axis perpendicular to the positive normal of the orbital plane, and y0 be determined by the right-hand rule, i.e., y0 = z0 × x0. For single-satellite reachability calculations, the reachability envelope is represented in the orbital coordinate system S0; for multi-satellite reachability calculations, the reachability envelope needs to be uniformly represented in the inertial coordinate system S0. J Below; for a specific satellite represented by the six orbital roots [a,e,i,Ω,ω,f], the reachable domain envelope needs to be uniformly represented in the inertial coordinate system S by the transformation matrix shown in formula (1). J Down: Where u = ω + f, and M is the rotation matrix about the corresponding axis; In the transformation between the geocentric inertial frame and the geocentric fixed frame, only the Earth's uniform rotation is considered, while the effects of precession, nutation, and polar motion are ignored; the coordinate system rotates only around the Z-axis, and the corresponding transformation matrix is: GAST is Greenwich Mean Sidereal Time, which is the Earth's rotation angle from the mean vernal equinox to the Greenwich meridian when calculating the satellite's position. Based on the different characteristics of the target, in the geocentric fixed coordinate system S I or geocentric inertial coordinate system S J Below, the target region S is given by latitude, longitude, and distance, or by the envelope of a three-dimensional mesh; the distance interval [r] is set according to S and task requirements. L ,r U ] and the zenith angle interval [θ L ,θ U Define a three-dimensional spherical layer Represented as: Then part of the three-dimensional spherical layer volume for: For a three-dimensional sphere Discretize in the dimensions of distance r and zenith angle θ, and in the azimuth angle Analyzing the dimensions, we obtain I×J discrete circular rings R of equal volume. ij Represented as: Where r i and θ j For the nodes to be discrete, the following conditions must be met: r0=r L ,r I =r U ,θ0=θ L ,i J =θ U Given the discrete ring volume V of three-dimensional information ij for: The node r is determined using the equal volume discretization method. i and θ j The volume of a portion of the three-dimensional sphere is given by equation (3), then the volume V of each discrete ring is... ij for Combining equations (4) and (5), node r i and θ j Represented as: Discrete unit R ij The centerline is defined as C ij , represented as: Where C ij It is caused by node r i and θ j The reachability of the centerline of the determined circle is determined by the azimuth angle interval corresponding to the reachability domain of the satellites in the constellation. express; When the parameters I and J of the discrete element are sufficiently large, the volume V of the resulting discrete ring is... ij The reachability R for a single discrete unit is relatively small. RD(ij) Accessibility using its centerline Φ RD(ij) The approximate performance improves, as expressed by: The method achieves volumetric discretization of three-dimensional spatial targets using a two-dimensional discrete computation of distance and zenith angle I×J. The spatial targets are represented by the reachable interval of the analytical azimuth center ring, and it is applicable to any target region given by latitude and longitude or grid envelope.

5. The method for evaluating the multiple reachability coverage of constellation orbit change maneuvers as described in claim 4, characterized in that: The second step is implemented as follows: The reachable envelope of a single star is represented as r(x,y,z)0 in the orbital coordinate system S0, where the positions of the three axes are given by an m×n grid matrix; this is then transformed into the inertial coordinate system S... J The following is represented as r(x,y,z). J Or in the fixed coordinate system S I The following is represented as r(x,y,z). I ; Calculating the azimuth interval corresponding to the satellite's reachable domain is equivalent to finding the center circle C. ij Intersection with all grid surfaces that constitute the envelope; To simplify the calculation, the envelope is represented in spherical coordinates, also using an m×n grid matrix. The problem is equivalent to finding r = (r i-1 +r i ) / 2 and θ=(θ j-1 +θ j The line ) / 2 intersects with all the grid surfaces that make up the envelope; Consider adjacent 2×2 matrices in an m×n grid matrix as a single non-planar grid surface element, with a total of k = (n-1)(m-1) such elements; calculate the extreme values ​​of distance and zenith angle respectively, and determine C. ij Whether a grid surface is within the range of r and θ, the selected grid surfaces are represented as set K; For the initially selected set of mesh faces K, determine the number of unique points. A set of 3 points forms one triangular facet, and a set of 4 points forms two triangular facets. For all triangular facets, calculate the sum C. ij Find the intersection point and determine whether the intersection point is inside the triangle; Calculate the azimuth angles of all intersection points P in set K that satisfy the requirements; since the envelope is a closed surface, the number of intersection points must be an even number, 2K. m The reachability region of the m-th satellite is defined for C. ij The intersection points are arranged in ascending order of azimuth angle to obtain the following sequence: Where +1 represents the start point of the reachable interval, -1 represents the end point of the reachable interval, and there are a total of K. m each interval Since the azimuth angle ranges from [0, 2π] when all grid points are transformed to spherical coordinates; when At this time, there are two situations: 1) The Z-axis intersects with the envelope; 2) The envelope intersects the plane formed by the positive X-axis and the positive Z-axis, but the Z-axis does not intersect the envelope; change the azimuth range to [-π, π], if it still... This is identified as case 1; Conversely, it is determined to be case 2; For case 1, when C ij When there is an intersection with the envelope in spherical coordinates, further determine whether the intersection corresponds to an accessible or inaccessible region; in Cartesian fixed coordinates, take the midpoint of the two azimuth nodes. Draw a ray along the positive Z-axis and find its intersection with the reachable region envelope, using the same method as in spherical coordinates. If the number of intersection points is odd, the corresponding azimuth interval is a reachable region; otherwise, it is an inaccessible region. The change corresponds to the sign of the two nodes. If C ij When there is no intersection with the envelope in spherical coordinates, further judgment is needed for C. ij Whether it lies inside the envelope; also in the Cartesian inertial coordinate system, take any C ij Find the intersection point of a ray drawn from the previous point along the positive Z-axis and the envelope of the reachable region; If the number of intersections is odd, then C ij Located inside the envelope, the reachable azimuth angle range is [0, 2π]. For cases 1 and 2, the azimuth of the intersection point may be in the interval [-π, 0]. We transform it to the interval [0, 2π] for the next calculation. We then equivalently transform the nodes whose azimuths are in the interval [-π, 0] to the interval [0, 2π] and arrange them again in ascending order of azimuth. If the second row of the first column is -1, the second row of the last column must be +1. We add a starting node. and a termination node By decomposing the three-dimensional envelope of a single star into multiple triangular facets, the centerline C is calculated in spherical coordinates. ij The in-plane intersection of the triangle and the target centerline C under any condition is obtained by formula (5). ij Azimuth sequence Target azimuth sequence Φ TAR(ij) Obtained using the same method.

6. The method for evaluating the multiple reachability coverage of constellation orbit change maneuvers as described in claim 5, characterized in that: Step three is implemented as follows: The reachable azimuth angles of all M satellites obtained in step two The result of merging is: At this time, the matrix The number of nodes in the data is: Next, the matrix Arranged in ascending order according to the first row: Define the reachable stars as N and set its initial value N0 = 0; then, read the matrix sequentially. The second row of elements; when the s-th element is +1 (s∈{1,2,…,S}), the corresponding angle in the first row represents the starting point of a new reachable interval, thus increasing the number of reachable stars by N. s =N s-1 +1; Conversely, when the s-th element is -1 (s∈{1,2,…,S}), the corresponding first row angle represents the termination point of an existing reachable interval, thus reducing the number of reachable stars by N. s =N s-1 -1; after reading all elements, set the corresponding N... s This means that a completely new matrix is ​​obtained in the third row of the matrix. for: Where N s For interval The number of stars that can be reached; Subsequently, it is necessary to determine the target azimuth interval Φ TAR(ij) For multi-star reachable matrices Perform the transformation; change Φ TAR(ij) The 2p azimuth nodes in the diagram are represented as follows: and The nodes are arranged in ascending order of azimuth to obtain a new matrix. for: in The third row of the column For the number of reachable stars N in the third row of the previous column, ... Add a column of all zeros to the first column of the table, so that... hour When multiple points have the same When the zenith angle is the same, it is necessary to ensure Located at the end of the sort, Located at the top of the sort; position and The column and the pair By performing truncation, we obtain p corresponding target multi-star reachability matrices. Represented as: The constellation's alignment with centerline C is obtained through node sorting and a one-time sequential read. ij Given a multi-star reachability matrix for a target region 7. The method for evaluating the multiple reachability coverage of constellation orbit change maneuvers as described in claim 6, characterized in that: Step four is implemented as follows: The maximum number of reachable stars is defined as N. max =max(N) ps The initial reachable angle of multiple stars is defined as Γ. RDNX(ijp) (N) = 0, where N ∈ {1, ..., N} max Then, iterate through... N in the third line s =The azimuth node corresponding to N, then the reachable azimuth angle of star N is: After obtaining the centerline C of each discrete unit ij Corresponding target range After the reachable angle of star N, the reachable angle Γ CovNX(ijp) (N) and target angle The proportion is approximated as the reachability of the target region in the entire discrete unit; then the centerline C ij The multi-star reachable volume V of all targets RDNX(ij) (N) satisfies: Once the reachability interval of a single discrete unit is given, its reachable volume can be obtained directly; the sum of the reachable volumes of all discrete units is the reachable volume of the entire spatial target region. For I×J discrete circular annexes of equal volume, the reachable volume can be expressed as: Where V RDNX (N) is the reachable volume of N stars in the entire target region of space.

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