A geometric representation and fast calculation method for the reachable area of ​​a spacecraft with single pulse maneuverability

Abstract

The present invention belongs to the field of aerospace technology, and relates to a method for geometric representation and rapid calculation of a single-pulse maneuverable reachable domain of a spacecraft, comprising the following steps: constructing a reachable domain envelope geometric feature representation model; randomly generating a fixed-time reachable domain geometric shape feature sample and a long-axis rotation angle sample database; respectively constructing deep neural networks that fit shape features and rotation angle features, and using the acquired samples to complete network training; combining the well-trained deep neural network to respectively map the shape features and rotation angle features of the reachable domain at a reference position, so as to achieve efficient and accurate calculation of the geometric features of the single-pulse fixed-time reachable domain of a spacecraft; the present invention makes up for the shortcomings of the existing fixed-time reachable domain optimization solution method, which has low computational efficiency, and the uncertainty propagation solution method, which is difficult to accurately characterize the reachable domain envelope shape.

Description

Technical Field

[0001] The present invention belongs to the field of new generation information technology, and specifically relates to a method for geometric representation and rapid calculation of a single-pulse maneuverable reachable domain of a spacecraft. Background Art

[0002] The reachable domain of a spacecraft single pulse maneuver is highly flexible and practical in position space, and is widely used in orbital game decision-making and resource allocation. As an important means to grasp and predict the situation of non-cooperative targets, the reachable domain can effectively quantify the threat of the opponent's spacecraft. Geometric representation and rapid calculation of the reachable domain of a spacecraft single pulse maneuver is an important basis for ensuring that we are in a game advantage.

[0003] The reachable domain refers to the set of spatial positions that a spacecraft can reach after maneuvering under the constraints of given maneuverability and mission time. Based on the relationship between the reachable domain and time, the reachable domain is divided into a time-free reachable domain and a fixed-time reachable domain. The fixed-time reachable domain emphasizes time constraints and provides a key space and time framework, enabling game participants to make dynamic decisions and manage resources more accurately. Due to its time sensitivity and strong irregularity, the fixed-time reachable domain mostly uses numerical solution methods that rely on dynamic models, which has an inherent conflict between computational efficiency and accuracy. How to accurately and efficiently calculate the reachable domain of a spacecraft's single-pulse maneuver is a key issue that needs to be solved at this stage.

[0004] At present, the methods for calculating the reachable domain at fixed time are mainly divided into two categories: extreme value optimization solution method and uncertainty propagation method. The existing literature (Holzinger MJ, Scheeres D J. Reachability results for nonlinear systems with ellipsoidal initial sets[J]. IEEE Transactions on Aerospace and Electronic Systems, 2012, 48(2): 1583-1600.) adopts the extreme value optimization solution method, uses the relative motion model to propagate the state of the spacecraft after maneuvering, and solves the extreme value of the reachable distance relative to the nominal position at fixed time to obtain the relative reachable domain. When solving the reachable domain at fixed time by the optimization method, a large number of numerical integrations are still required to solve the envelope surface, resulting in low computational efficiency. Existing literature (Shihido N, Tomlin C J. Ellipsoidal approximations of reachable sets for linear games[C]. Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, 2000.) propagates the uncertainty of the initial orbit of the spacecraft based on uncertainty theory and describes the shape of the envelope outside the reachable domain by an ellipsoid. However, as the propagation time increases, the irregularity of the envelope outside the reachable domain makes it difficult for the covariance ellipsoid to describe the actual shape of the envelope of the reachable domain.

[0005] The existing fixed-time reachable domain calculation methods, whether extreme value optimization solution methods or uncertainty propagation methods, cannot simultaneously guarantee high accuracy of the reachable domain solution and high computational efficiency. Therefore, based on the high dynamics of orbital games, it is necessary to establish a geometric representation model that can solve the inherent contradiction between computational efficiency and accuracy of existing numerical methods, so as to achieve fast and high-precision calculation of the fixed-time reachable domain of a spacecraft single pulse. Summary of the invention

[0006] In view of the above-mentioned deficiencies of the prior art, the purpose of the present invention is to provide a geometric representation and fast calculation method of the reachable domain of a spacecraft single-pulse maneuver, so as to solve the problem that the fixed-time reachable domain solution method in the prior art cannot take into account both the calculation efficiency and calculation accuracy of the reachable domain envelope.

[0007] To achieve the above object, the technical solution adopted by the present invention is as follows:

[0008] A method for geometric representation and rapid calculation of a single-pulse maneuverable reachable region of a spacecraft comprises the following steps:

[0009] S1. Aiming at the problem that the envelope of the outer reachable domain of a single pulse fixed time of a spacecraft is highly time sensitive and has a highly irregular spatial shape, the envelope of the outer reachable domain is decoupled into geometric shape features composed of the major axis rotation angle and the sampling point position. The geometric feature representation model of the envelope of the outer reachable domain is constructed according to the major axis rotation angle. Its expression is:

[0010] , (1)

[0011] In the formula, RD is the reachable domain; f is the major axis rotation angle; [ x j , y j , z j ] is the first j The location coordinates of the sampling points; C() Represents the functional relationship of combining the major axis rotation angle with the geometric shape features composed of all sampling point positions; where:

[0012] The characterization model of the long axis rotation angle is:

[0013] , (2)

[0014] In the formula, OE are the spacecraft orbit parameters; Δv is the pulse maneuver speed amplitude; t is the propagation time; W φ () It represents the nonlinear functional relationship between maneuvering parameters and the long axis rotation angle;

[0015] When the propagation time t Less than switching time t s When the propagation time is t Greater than or equal to the switching time t s When the reachable outer envelope is a long-term fixed time reachable outer envelope; wherein the switching time t s Defined as:

[0016] , (3)

[0017] In the formula, oh max and oh min Respectively represent the maximum and minimum orbital angular velocities among the sampling points after the maneuver; ts Indicates the switching time;

[0018] The geometric shape features are divided into short-term fixed time reachable domain outer envelope geometric shape features and long-term fixed time reachable domain outer envelope geometric shape features, the short-term fixed time reachable domain outer envelope geometric shape features are composed of short-term independent variable range and short-term farthest reachable distance in the plane and farthest reachable distance in different planes, and the long-term fixed time reachable domain outer envelope geometric shape features are composed of long-term independent variable range and long-term farthest reachable distance in the plane and farthest reachable distance in different planes; wherein:

[0019] The characterization model of the short-term independent variable range is:

[0020] , (4)

[0021] In the formula, x max and x min are the upper and lower bounds of the independent variable range; F S () Represents the nonlinear functional relationship between the characteristic parameters and the range of the independent variable after rotation;

[0022] The characterization model of the farthest reachable distance in the short-term plane is:

[0023] , (5)

[0024] In the formula, y The sampling point is within the surface S Coordinates maximum reachable distance; subscript in and out Represent the inner and outer curves respectively; G () It represents the functional relationship between the characteristic parameter and the farthest reachable distance; δ is the relative position of the sampling point within the range of the independent variable;

[0025] The characterization model of the farthest reachable distance of the different surfaces is:

[0026] , (6)

[0027] In the formula, s It is the distance feature of the sampling points in the two-dimensional reachable domain; H() It represents the nonlinear functional relationship between the characteristic parameters and the farthest reachable distance of the different surfaces; z is the maximum distance that can be reached on different surfaces of the sampling points;

[0028] The characterization model of the long-term independent variable range is:

[0029] , (7)

[0030] In the formula, y min1 It is the lower limit of the independent variable range of curves 1 and 2; y min2 It is the upper limit of the independent variable range of curves 3 and 4; F L () Represents the nonlinear computational relationship between characteristic parameters and the range of the independent variable after rotation; y max1 It is the upper limit of the independent variable range of curves 1 and 3; y max2 It is the upper limit of the independent variable range of curves 2 and 4;

[0031] The characterization model of the longest reachable distance in the long term is:

[0032] , (8)

[0033] In the formula, x The sampling point is within the surface R The maximum reachable distance of the axis coordinates; G() Represents the nonlinear functional relationship between feature parameters and the farthest reachable distance; i Number the curves; d 2 is the relative position of the sampling point within the range of the independent variable;

[0034] S2. Randomly generate an envelope outside the fixed-time reachable domain within a given maneuvering parameter range based on Monte Carlo simulation, extract geometric shape feature samples and long-axis rotation angle samples of the fixed-time reachable domain through the generated envelope outside the reachable domain, establish a fixed-time reachable domain envelope database, and use the fixed-time reachable domain envelope database to complete the training of the geometric shape feature mapping network and the long-axis rotation angle mapping network;

[0035] Among them, the geometric shape feature mapping network and the long axis rotation angle mapping network net W They are respectively fitted by the deep neural network of geometric shape features and long axis rotation angle; wherein the geometric shape feature mapping network includes the long-term and short-term reachable domain outer envelope independent variable range mapping network net F 、 Long-term and short-term maximum reachable distance mapping network net G The farthest reachable distance mapping network of sampling points in different planes and reachable domain net H ;in:

[0036] Long axis rotation angle mapping network netW The mapping relationship is:

[0037] , (9)

[0038] In the formula, OE are the spacecraft orbit parameters; Δv is the pulse maneuver speed amplitude; t is the normalized propagation time; net W It is a deep neural network for the long axis rotation angle. Its input is the spacecraft orbit parameters, pulse maneuver velocity amplitude, and normalized propagation time. Its output is the long axis rotation angle.

[0039] The mapping relationship of the short-term reachable outer envelope independent variable range mapping network is:

[0040] , (10)

[0041] In the formula, OE are the spacecraft orbit parameters; Δv is the pulse maneuver speed amplitude; t is the normalized propagation time; x max and x min are the upper and lower bounds of the independent variable range, respectively; net FS is a deep neural network with an envelope independent variable range outside the reachable domain. The input is the spacecraft orbit parameters, pulse maneuver velocity amplitude, and normalized propagation time. The output is the envelope curve of the reachable domain. R The upper and lower bounds of the axis independent variable;

[0042] The mapping relationship of the long-term reachable domain envelope independent variable mapping network is:

[0043] ; (11)

[0044] In the formula, OE are the spacecraft orbit parameters; Δv is the pulse maneuver speed amplitude; t is the normalized propagation time; y min1 For curves 1 and 2 S The lower bound of the axis independent variable range; y min2 For curves 3 and 4 S The upper bound of the axis independent variable range; y max1 For curves 1 and 3 S The upper bound of the axis independent variable range; y max2 For curves 2 and 4 SThe upper bound of the axis independent variable range; net FL is a deep neural network with a range of independent variables in the long-term reachable domain envelope. The input is the spacecraft orbit parameters, pulse maneuver velocity amplitude, and normalized propagation time. The output is the reachable domain envelope curve in S The upper and lower bounds of the axis independent variable;

[0045] The mapping relationship of the short-term maximum reachable distance mapping network is:

[0046] , (12)

[0047] Where OE is the spacecraft orbit parameter; Δv is the pulse maneuver velocity amplitude; τ is the normalized propagation time; δ is the relative position of the sampling point within the range of the independent variable; y is the farthest reachable distance of the sampling point on the S axis; net Gin / out It is a deep neural network for the inner and outer farthest reachable distances within the short-term sampling point plane. The input is the spacecraft orbit parameters, the pulse maneuver velocity amplitude, the normalized propagation time, and the position of the sampling point within the R-axis independent variable range. The output is the farthest reachable distance of the sampling point on the S-axis.

[0048] The long-term in-plane maximum reachable distance mapping network The mapping relationship is:

[0049] , (13)

[0050] In the formula, OE are the spacecraft orbit parameters; Δv is the pulse maneuver speed amplitude; t is the normalized propagation time; d 2 is the relative position of the sampling point within the range of the independent variable; x The sampling point is R The maximum reachable distance of the axis; is the longest reachable distance deep neural network on the curve number i within the long-term sampling point surface, and the input is the spacecraft orbit parameters, pulse maneuver velocity amplitude, normalized propagation time, sampling point S Position within the axis independent variable range, output sampling point R The maximum reachable distance of the axis;

[0051] The farthest reachable distance mapping network of the reachable domain sampling points net H The mapping relationship is:

[0052] , (14)

[0053] In the formula, OE are the spacecraft orbit parameters; Δv is the pulse maneuver speed amplitude; t is the normalized propagation time; s It is the distance feature of the sampling points in the two-dimensional reachable domain; z The sampling point is W The maximum reachable distance of the axis; net H The sampling point is inside the W The deep neural network of the farthest reachable distance of the coordinate axis takes as input the orbital parameters of the spacecraft, the amplitude of the pulse maneuvering velocity, the normalized propagation time, and the distance characteristics of the sampling points in the two-dimensional reachable domain, and outputs the farthest reachable distance of the sampling points in different planes;

[0054] S3. Based on the geometric shape feature mapping network and the long-axis rotation angle mapping network, a method for quickly calculating the geometric shape features of the envelope outside the fixed-time reachable domain is proposed to realize the calculation of the geometric shape features of the envelope outside the fixed-time reachable domain of a single pulse of a spacecraft.

[0055] As a preferred solution of the present invention, the construction process of the geometric representation model of the reachable outer envelope is as follows:

[0056] Considering the high time sensitivity of the reachable domain, in order to reduce the complexity of characterizing irregular shapes in the reachable domain, the two types of features, long axis rotation angle and geometric shape, are decoupled, and characterization methods are designed for the two types of features respectively. f Calculated by formula (2);

[0057] The geometric shape features are composed of the range of independent variables, the farthest reachable distance in the plane, and the farthest reachable distance in different planes. As the propagation time increases, the degree of bending and distortion of the reachable domain becomes severe. Short-term and long-term fixed-time reachable domain geometric shape characterization models are designed respectively: to meet the propagation time t < t s When , the reachable domain is a short-term fixed-time reachable domain, and the outer envelope of the short-term fixed-time reachable domain is composed of the inner curve close to the earth and the outer curve far from the earth. On the contrary, it is a long-term fixed-time reachable domain, and the outer envelope of the long-term fixed-time reachable domain is composed of the end-to-end splicing of curves 1 to 4. Switching time t s The definition of is shown in formula (3);

[0058] The outer envelope of the short-term fixed-time reachable domain is a closed figure in the orbital plane, which is difficult to be directly represented by a single-valued function. Therefore, the closed reachable domain is split into an inner curve and an outer curve by the two points with the largest phase difference on both sides of the outer envelope. The range of the curve independent variable is calculated by formula (4);

[0059] xis the horizontal coordinate of any sampling point within the range of the envelope independent variable that can be reached outside the domain in a short-term fixed time. The relative position of the sampling point within the range of the independent variable is d It is expressed as:

[0060] , (15)

[0061] In the formula, x is the horizontal coordinate of the sampling point; x max and x min The upper and lower bounds of the range of the independent variable of the reachable domain envelope curve;

[0062] The farthest reachable distance on the inner and outer curves within the surface of any sampling point within the independent variable is calculated by formula (5);

[0063] Sampling is performed uniformly within the range of the independent variable of the outer envelope of the reachable domain in a short-term fixed time, and the farthest reachable distance of the sampling points on the inner and outer curves that constitute the outer envelope of the reachable domain are calculated respectively. y , connect all sampling points in sequence to form a short-term fixed-time reachable domain plane envelope connected end to end;

[0064] After the long-term fixed time reachable domain envelope is rotated, the inner and outer curves are separated into their respective y The axis minimum point is split into two curves, and curves 1 to 4 are obtained, all of which satisfy a S Axis ordinate y Value and a R Axis coordinates x The single-valued function relationship of the value;

[0065] The range of independent variables of the envelope curve outside the domain that can be reached in the long-term fixed time is calculated by formula (7);

[0066] y is the ordinate of any sampling point within the reachable domain envelope independent variable, then the relative position of the sampling point within the independent variable range d 2 It is expressed as:

[0067] , (16)

[0068] The farthest reachable distance of any sampling point in the independent variable on curves 1 to 4 is calculated by formula (8);

[0069] The distance characteristic of any sampling point in the plane outside the envelope of the two-dimensional reachable domain is s ,The farthest reachable distance of sampling points on different planes is calculated by formula (6);

[0070] The long axis rotation angle of the spacecraft single pulse fixed time reachable domain is calculated f With all sampling point locations [ x j , y j , z j ] is composed of geometric shape features, and a geometric representation model of the envelope outside the reachable domain is obtained based on the combination of formulas.

[0071] As a preferred solution of the present invention, the storage form of the fixed time reachable external envelope database is as follows:

[0072] , (17)

[0073] In the formula, SU S and SU L They are the short-term and long-term geometric feature storage units of the reachable domain; j is the sampling point number; x j , y j , z j They are the reachable domain sampling points j exist R、S、W Axis coordinate values.

[0074] As a preferred embodiment of the present invention, the normalized propagation time τ is as follows:

[0075] ,

[0076] In the formula, t is the propagation time; T n is the orbital period of the reference orbit.

[0077] As a preferred solution of the present invention, the process of obtaining the envelope outside the short-term fixed time reachable domain includes the following steps:

[0078] S51. Input initial orbit parameters OE , propagation time t The upper limit of the spacecraft single pulse maneuver speed increment Δv , combined with the long axis rotation angle deep neural network net W The major axis rotation angle is calculated by formula (9): f ;

[0079] S52, by calculating the switching time t s Determine whether the input propagation time corresponds to the long-term or short-term fixed-time reachable domain envelope, and fit the deep neural network with the trained short-term envelope shape independent variable range net FS The envelope outside the reachable domain is calculated by formula (10) in the R-axis independent variable range;

[0080] S53, in the range of independent variables [ x max , x min ] uniformly sampled, record the k Sampling points R Axis coordinates x k , calculate the relative position of the sampling point within the range of the independent variable d k , and combined with deep neural networks net Gin / out Calculate the reachable domain envelope orbital surface S Axis maximum reach distance y k As shown in formula (19), we get the envelope point of the two-dimensional reachable domain [ x k , y k ], all the two-dimensional reachable domain envelope points form the outer envelope of the plane reachable domain:

[0081] , (19)

[0082] In the formula, OE are the spacecraft orbit parameters; Δv is the pulse maneuver speed amplitude; t is the normalized propagation time; d k It is the sampling point k relative position within the range of the independent variable; y k It is the sampling point k exist S The maximum reachable distance of the axis; net Gin / out Fitting the neural network to the longest reachable distances inside and outside the short-term sampling point surface for training;

[0083] S54, uniformly sample within the two-dimensional envelope of the reachable domain, record the j The sampling points have coordinates in the orbital plane [ x j , y j ], calculate the position characteristics of the sampling points in the surface s j , and combined with deep neural networks net H Calculate the reachable domain envelope orbital out of plane W Axis maximum reach distance zj As shown in formula (20), the envelope point of the three-dimensional reachable domain is obtained [ x j , y j , z j ], all three-dimensional reachable domain envelope points constitute the three-dimensional reachable domain outer envelope:

[0084] , (20)

[0085] In the formula, OE are the spacecraft orbit parameters; Δv is the maneuvering speed amplitude; t is the normalized propagation time; s j It is the distance feature of the sampling points in the two-dimensional reachable domain; z j is the farthest reachable distance of the sampling point on the W axis; net H Fitting the neural network to the farthest reachable distance of the sampling point surface on the W coordinate axis to complete the training;

[0086] S55, rotate the major axis by an angle φ and all points on the three-dimensional envelope surface [ x j , y j , z j ] By combining the following formulas, we can obtain the three-dimensional reachable domain outer envelope RD as shown in formula (1);

[0087] S56, the initial orbital parameters OE Transition to orbital state [ r 0 ,v 0 ], recursively to the propagation time t Time position speed[ r t ,v t ],Will RSW The reachable domain in the coordinate system is transformed into the inertial system to obtain the fixed time reachable domain.

[0088] As a preferred embodiment of the present invention, the process of obtaining the long-term fixed time reachable outer envelope comprises the following steps:

[0089] S61. Input initial orbit parameters OE , propagation time t The upper limit of the spacecraft single pulse maneuver speed increment Δv , combined with the long axis rotation angle deep neural network net WThe major axis rotation angle is calculated by formula (9): f ,

[0090] S62, by calculating the switching time t s Determine whether the input propagation time corresponds to the long-term or short-term fixed-time reachable domain envelope, and fit the neural network with the trained short-term envelope shape independent variable range net FS The envelope outside the reachable domain is calculated in the S-axis independent variable range as shown in formula (11);

[0091] S63, respectively, in the independent variable range of curves 1 to 4 [ y min1 , y max1 ]、[ y min1 , y max2 ]、[ y min2 , y max1 ]、[ y min2 , y max2 ] uniformly sampled, record the k Sampling points S Axis coordinates y k , calculate the relative position of the sampling point within the range of the independent variable d 2k , and combined with deep neural networks net Gi Calculate the reachable domain envelope orbital surface R Axis maximum reach distance x k As shown in formula (13), we get the envelope point of the two-dimensional reachable domain [ x k , y k ], all the envelope points of the two-dimensional reachable domain form the outer envelope of the plane reachable domain;

[0092] S64, uniformly sample within the two-dimensional envelope of the reachable domain, record the j The sampling points have coordinates in the orbital plane [ x j , y j ], calculate the position characteristics of the sampling points in the surface s j , and combined with deep neural networks net H Calculate the reachable domain envelope orbital out of plane W Axis maximum reach distance zj As shown in formula (20), the envelope point of the three-dimensional reachable domain is obtained [ x j , y j , z j ], all three-dimensional reachable domain envelope points constitute the three-dimensional reachable domain outer envelope:

[0093] , (20)

[0094] In the formula, OE are the spacecraft orbit parameters; Δv is the pulse maneuver speed amplitude; t is the normalized propagation time; s j It is the distance feature of the sampling points in the two-dimensional reachable domain; z j is the farthest reachable distance of the sampling point on the W axis; net H Fitting the neural network to the farthest reachable distance of the sampling point surface on the W coordinate axis to complete the training;

[0095] S65, rotate the major axis by an angle φ and all points on the three-dimensional envelope surface [ x j , y j , z j ] By combining the following formulas, we can get the outer envelope RD of the three-dimensional reachable domain;

[0096] ; (1)

[0097] In the formula, RD is the reachable domain; f is the major axis rotation angle; [ x j , y j , z j ] is the first j The location coordinates of the sampling points; C() Represents the functional relationship of combining the rotation angle feature with the geometric shape feature composed of all sampling point positions;

[0098] S66, the initial orbital parameters OE Transition to orbital state [ r 0 ,v 0 ], recursively to the propagation time t Time position speed[ r t,v t ],Will RSW The reachable domain in the coordinate system is transformed into the inertial system to obtain the fixed time reachable domain.

[0099] Beneficial effects: The present invention designs a geometric feature characterization model for the reachable domain of a spacecraft single-pulse maneuver, and uses a deep neural network to fit the geometric features of the reachable domain. By combining geometric features, the outer envelope of the reachable domain of a spacecraft single-pulse maneuver is quickly calculated, which greatly improves the calculation efficiency of the reachable domain compared to other reachable domain calculation methods. By constructing deep neural networks that fit shape features and rotation angle features respectively, the complexity of shape feature mapping is reduced, and the convergence stability of the network is improved, so that accurate and efficient calculation of irregular shapes in the reachable domain can be achieved. The overall idea is novel, with strong innovation and engineering application prospects. BRIEF DESCRIPTION OF THE DRAWINGS

[0100] Figure 1 Figure a in the figure is a schematic diagram of the geometric characterization method of the two-dimensional envelope of the short-term reachable domain of the present invention; Figure b is a schematic diagram of the geometric characterization method of the two-dimensional envelope of the long-term reachable domain of the present invention;

[0101] Figure 2 It is a schematic diagram of the characteristics of the sampling point positions within the two-dimensional envelope of the reachable domain of the present invention;

[0102] Figure 3 It is the mean square error performance effect of the deep neural network of the present invention;

[0103] Figure 4 is a reachable domain calculation flow chart of the present invention;

[0104] Figure 5 It is a schematic diagram of the envelope of the two-dimensional reachable domain calculated by the present invention;

[0105] Figure 6 It is a schematic diagram of the envelope of the three-dimensional reachable domain calculated by the present invention. DETAILED DESCRIPTION

[0106] In order to facilitate the understanding of those skilled in the art, the present invention is further described below in conjunction with embodiments and drawings. The contents mentioned in the implementation modes are not intended to limit the present invention.

[0107] As an embodiment of the present invention, Figure 1 to Figure 6 As shown, a method for geometric representation and rapid calculation of a single pulse reachable domain of a spacecraft includes the following steps:

[0108] S1. Aiming at the problem that the envelope of the reachable domain outside the fixed time of a single pulse of a spacecraft is time-sensitive and has a strong irregular shape, a geometric feature characterization model of the reachable domain is constructed. The reachable domain envelope is accurately characterized by combining the major axis rotation angle with the geometric shape features as follows:

[0109] ,

[0110] Where RD is the reachable domain; [x j ,y j ,z j ] is the position coordinate of sampling point j; φ is the major axis rotation angle; C() represents the functional relationship combining the rotation angle feature with the geometric shape feature composed of all sampling point positions.

[0111] The angular characteristics of the single pulse reachable domain of the spacecraft are composed of the major axis rotation angle φ, and the geometric shape characteristics are composed of the independent variable range of the sampling point, the farthest reachable distance in the plane, and the farthest reachable distance in different planes. The specific calculation process of the two types of geometric characteristics is as follows:

[0112] S11, calculation of the major axis rotation angle. Figure 1 As shown in Figure a, the reachable domain rotates with time in the RS plane of the spacecraft reference position RSW coordinate system. In order to make the shape representation of the reachable domain not affected by the rotation angle, the relative reachable domain angle is rotated so that the reachable domain has the largest projection on the R axis, and the major axis rotation angle is f Calculated by the following formula,

[0113] ,

[0114] Where φ is the major axis rotation angle; OE is the spacecraft orbit parameter; Δv is the velocity amplitude of the spacecraft pulse maneuver; t is the propagation time; W φ () represents the nonlinear functional relationship between maneuvering parameters and the long axis rotation angle;

[0115] S12, long / short-term reachable domain judgment. As the propagation time increases, the reachable domain is bent and distorted severely. We design geometric shape characterization models for short-term and long-term fixed-time reachable domains respectively; the propagation time t< switching time t s When , the reachable domain is a short-term reachable domain, and the reachable domain envelope is composed of the two curves of the inner side close to the earth and the outer side far away from the earth; on the contrary, it is a long-term reachable domain, and the reachable domain envelope is composed of the curves 1 to 4 spliced ​​together; where: the switching time t s The definition of is as follows:

[0116] ,

[0117] In the formula, ω max and ω min Respectively represent the maximum and minimum orbital angular velocities at the sampling point after the maneuver; t s Indicates the switching time;

[0118] S13. Calculation of the range of independent variables in the short-term reachable domain. The envelope of the short-term relative reachable domain is a closed figure in the orbital plane. The rotated figure is a multi-valued function of the R-axis coordinate of the spacecraft reference position relative to the RSW coordinate system, which is difficult to solve. Therefore, the inner curve and the outer curve are calculated separately and then spliced ​​to form the outer envelope.

[0119] The closed reachable domain is divided into an inner curve and an outer curve by the farthest points on both sides of the envelope from the center. After rotation, the projection of the line connecting the farthest points on both sides on the R coordinate axis is the largest; the range of the independent variables of the inner and outer curves that constitute the envelope of the short-term reachable domain is calculated by the following formula:

[0120] ,

[0121] In the formula, x max With x min are the upper and lower bounds of the independent variable range; OE is the spacecraft orbit parameter; Δv is the velocity amplitude of the spacecraft pulse maneuver; t is the propagation time; F S () represents the nonlinear functional relationship between the characteristic parameter and the range of the independent variable after rotation;

[0122] S14, short-term two-dimensional reachable domain envelope calculation; x is the horizontal coordinate of any sampling point within the reachable domain envelope independent variable range, then the relative position δ of the sampling point within the independent variable range is expressed as:

[0123] ,

[0124] Where x is the horizontal coordinate of the sampling point; max With x min The upper and lower bounds of the reachable domain envelope independent variable range;

[0125] The maximum reachable distance within the plane of any sampling point in the independent variable is calculated by the following formula:

[0126] ,

[0127] Where y is the farthest reachable distance of the S coordinate within the sampling point plane; the subscripts in and out represent the inner and outer curves respectively; G() represents the functional relationship between the feature parameter and the farthest reachable distance;

[0128] Uniformly sample within the range of the reachable domain envelope independent variable, calculate the farthest reachable distance y of the sampling points on the inner and outer curves that constitute the reachable domain envelope, and connect all the sampling points in sequence to form a short-term reachable domain plane envelope that is connected end to end;

[0129] S15, long-term reachable domain independent variable range calculation. After the long-term relative reachable domain envelope graph is rotated, as shown in Figure 1As shown in Figure b, based on the short-term closed figure splitting method, the inner and outer curves are split into two curves at their respective y-axis minimum points, and curves 1 to 4 are obtained, all of which satisfy a single-valued function relationship between an S-axis ordinate y value and an R-axis coordinate x value;

[0130] The range of the long-term reachable domain shape independent variable is calculated by the following formula:

[0131] ,

[0132] In the formula, y max1 is the upper limit of the independent variable range of curves 1 and 3; y max2 is the upper limit of the independent variable range of curves 2 and 4; y min1 is the lower limit of the independent variable range of curves 1 and 2; y min2 is the upper limit of the independent variable range of curves 3 and 4; F L () represents the nonlinear computational relationship between the characteristic parameter and the range of the independent variable after rotation;

[0133] y is the ordinate of any sampling point within the reachable domain envelope independent variable, then the relative position δ2 of the sampling point within the independent variable range is expressed as:

[0134] ,

[0135] S16, long-term two-dimensional reachable domain envelope calculation; the farthest reachable distance within the surface of any sampling point in the independent variable is calculated by the following formula:

[0136] ,

[0137] Where x is the farthest reachable distance of the R coordinate within the sampling point surface; i is the curve number; G() represents the nonlinear functional relationship between the characteristic parameter and the farthest reachable distance;

[0138] Take samples evenly within the range of the independent variable, calculate the maximum distance x of the S coordinate of each sampling point on the four curves, connect the two curves end to end to form a closed figure, that is, the outer envelope of the domain plane that can be reached in the short term after rotation;

[0139] S17, calculation of the position characteristics of the sampling points within the surface. Figure 2 As shown, a sampling point A is taken in the envelope of the plane reachable domain. The point closest to the major axis is point AM. The modulus length of AAM is calculated as L0; the modulus length of the intersection of point A and the inner envelope curve along the AAM direction is L1; the modulus length of the intersection of point A and the inner envelope curve along the AMA direction is L2; ​​denote L S =L1+L2; ML and MR are the intersection points of the major axis and the outer envelope respectively; L Mis the length of the major axis MLMR; L3 is the length of the curve AMML; NM is the point on the major axis closest to the reference track point N, and L4 is the length of the curve MLNM;

[0140] Furthermore, the plane relative position of point A is normalized as shown in the following formula:

[0141] ,

[0142] Where: s0 represents the position of point A on the cross section of the rope-shaped reachable area relative to the major axis; s1 and s2 represent the position of point A on the cross section relative to the inner and outer curve boundaries; s3 and s4 represent the position of the projection AM on the midline of point A on the major axis MLMR relative to the left and right endpoints respectively;

[0143] Let s = [s0, s1, s2, s3, s4] be the position feature of the two-dimensional point A in the plane envelope;

[0144] S18, calculation of the farthest reachable distance of sampling points in different planes; based on the position feature s of the sampling points in the two-dimensional reachable domain envelope, the three-dimensional farthest reachable distance of the sampling points is calculated by the following formula:

[0145] ,

[0146] Where z is the farthest distance that can be reached on the different planes of the sampling points; s is the distance feature of the sampling points in the two-dimensional reachable domain; H() represents the nonlinear functional relationship between the feature parameters and the farthest distance that can be reached on the different planes;

[0147] In summary, the two types of geometric features of the reachable domain are expressed as follows, which can be combined with equation (1) to obtain the reachable domain of a spacecraft with a single pulse at a fixed time;

[0148] ,

[0149] In the formula, f is the major axis rotation angle; [x j ,y j ,z j ] is the position coordinate of the outer envelope sampling point j.

[0150] S2. Using Monte Carlo simulation, randomly generate a fixed-time reachable domain envelope within a given maneuvering parameter range, extract the reachable domain envelope geometric feature samples and establish a fixed-time reachable domain outer envelope geometric feature database, further, respectively construct a deep neural network fitting shape features and rotation angle features, and use the reachable domain outer envelope geometric feature database to complete the deep neural network training. The specific steps are as follows:

[0151] S21, reachable domain envelope generation; In geosynchronous orbit, the range of single pulse maneuver parameters is shown in Table 1;

[0152] Table 1 Single-pulse maneuver parameter range

[0153]

[0154] Generate 10,000 sampling points in each given parameter combination, evenly diverge in space, and recursively calculate the propagation time t through the following orbit to obtain 2,400 fixed-time reachable domain envelopes composed of the final sampling point set;

[0155] ,

[0156] Where: r, v are the spacecraft position and velocity; , is the first-order derivative of the spacecraft position and velocity with respect to time; μ is the earth's gravitational constant; f p It is the power of perturbation;

[0157] S22, extraction and storage of geometric features of reachable domain; For the reachable domain envelope in the RSW coordinate system, first, the PSO algorithm is used to search for the rotation angle φ in the range of [0,360°], so that the envelope curve formed after the reachable domain is rotated φ in the RS plane has the largest range of independent variables on the R axis, and the bending direction of the long axis is the positive direction of the S axis. Then, by sampling on the reachable domain envelope, the three-axis coordinate values ​​of the sampling points, the in-plane features and the reachable domain independent variable range are extracted;

[0158] The storage format of short-term and long-term geometric features of the reachable domain is as follows:

[0159] ,

[0160] In the formula, x j ,y j 、z j are the coordinates of the sampling point j in the reachable domain on the R, S, and W axes after rotation; SU S with SU L They are the short-term and long-term geometric feature storage units of the reachable domain;

[0161] S23, geometric feature fitting deep neural network construction; as shown in the following formula, the normalized propagation time τ is defined as follows:

[0162] ,

[0163] Where t is the propagation time; T n is the orbital period of the reference orbit;

[0164] The long axis rotation angle deep neural network mapping relationship is as follows:

[0165] ,

[0166] The neural network input is the spacecraft orbital parameters, the spacecraft single pulse maneuvering velocity amplitude, and the normalized propagation time, and the output is the long axis rotation angle; where OE is the spacecraft orbital parameters; Δv is the spacecraft single pulse maneuvering velocity amplitude; τ is the normalized propagation time; φ is the long axis rotation angle; net W Fit a neural network for the rotation angle;

[0167] The mapping relationship of the short-term reachable domain envelope shape independent variable deep neural network is as follows:

[0168] ,

[0169] The neural network input is the spacecraft orbital parameters, the spacecraft single pulse maneuvering velocity amplitude, and the normalized propagation time, and the output is the upper and lower bounds of the reachable domain envelope curve on the R axis independent variable; where OE is the spacecraft orbital parameters; Δv is the spacecraft single pulse maneuvering velocity amplitude; τ is the normalized propagation time; x max and x min are the upper and lower bounds of the independent variable respectively; net FS Fitting a neural network for a range of short-term envelope shape independent variables;

[0170] The long-term envelope shape independent variable deep neural network mapping relationship is as follows:

[0171] ,

[0172] The neural network input is the spacecraft orbital parameters, the spacecraft single pulse maneuvering velocity amplitude, and the normalized propagation time, and the output is the upper and lower bounds of the reachable domain envelope curve on the S-axis independent variable; where OE is the spacecraft orbital parameters; Δv is the spacecraft single pulse maneuvering velocity amplitude; τ is the normalized propagation time; y min1 is the lower limit of the independent variable range of curves 1 and 2 on the S axis; y min2 is the upper limit of the independent variable range of curves 3 and 4 on the S axis; y max1 is the upper limit of the independent variable range of curves 1 and 3 on the S axis; y max2 is the upper limit of the independent variable range of curves 2 and 4 on the S axis; net FL Fitting a neural network for a range of long-term envelope shape independent variables;

[0173] The deep neural network mapping relationship of the farthest reachable distance of the sampling points in the short-term reachable domain on the track surface on the envelope curve inside and outside the reachable domain is as follows:

[0174] ,

[0175] The neural network inputs are the spacecraft orbit parameters, pulse maneuver velocity amplitude, normalized propagation time, and the position of the sampling point within the R-axis independent variable range, and outputs the farthest reachable distance of the sampling point on the S-axis; where OE is the spacecraft orbit parameter; Δv is the single pulse maneuver velocity amplitude; τ is the normalized propagation time; δ is the relative position of the sampling point within the independent variable range; y is the farthest reachable distance of the sampling point on the S-axis; net Gin / out Fitting a neural network for the furthest reachable distances inside and outside the short-term sampling point surface;

[0176] The deep neural network mapping relationship of the farthest reachable distance of the sampling points in the long-term reachable domain in the orbital plane on curves 1 to 4 on the R axis is as follows:

[0177] ,

[0178] The neural network input is the spacecraft orbital parameters, the spacecraft single pulse maneuver velocity amplitude, the normalized propagation time, and the position of the sampling point within the S-axis independent variable range, and the output is the farthest reachable distance of the sampling point on the R-axis. In the formula, OE is the spacecraft orbital parameters; Δv is the spacecraft single pulse maneuver velocity amplitude; τ is the normalized propagation time; δ2 is the relative position of the sampling point within the independent variable range; x is the farthest reachable distance of the sampling point on the R-axis; net Gi Fitting a neural network to the longest reachable distance on the curve number i within the long-term sampling point surface;

[0179] The mapping relationship between the farthest reachable distance of sampling points in the reachable domain and the deep neural network is as follows:

[0180] ,

[0181] The neural network input is the spacecraft orbit parameter, the spacecraft single pulse maneuver velocity amplitude, the normalized propagation time, and the distance characteristics of the sampling point in the two-dimensional reachable domain, and outputs the farthest reachable distance of the sampling point. In the formula, OE is the spacecraft orbit parameter; Δv is the spacecraft single pulse maneuver velocity amplitude; τ is the normalized propagation time; s is the distance characteristics of the sampling point in the two-dimensional reachable domain; z is the farthest reachable distance of the sampling point on the W axis; net H Fitting a neural network to the farthest reachable distance on the W coordinate axis within the sampling point surface;

[0182] S24, geometric feature fitting deep neural network training; in order to improve the fitting effect of the deep neural network on the geometric features of the reachable domain, before training, it is necessary to use the minimum-maximum normalization method to normalize the input and output parameters of the samples, and the maximum and minimum values ​​generated are shown in Table 2;

[0183] Table 2 Normalized maximum values ​​of input and output parameters

[0184]

[0185] The specific settings of various deep neural network structures are shown in Table 3;

[0186] Table 3 Deep neural network structure settings

[0187]

[0188] The above deep neural network is trained using the trainlm method using the Levenberg-Marquardt algorithm. The training set, validation set, and test set are divided into 8:1:1 ratios. After training, the square error between the true value of the sample and the fitting output of the training set and test set is calculated as follows: Figure 3 As shown, the network has high accuracy;

[0189] S3. According to the upper limit of the single pulse maneuvering velocity increment, propagation time and initial position vector of the mission spacecraft, the shape characteristics and rotation angle characteristics of the reachable domain at the reference position are fitted using the neural network trained in step 2, and combined into the reachable domain envelope. The specific calculation process is as follows: Figure 4 As shown;

[0190] S31. Parameter input: Input the initial orbital parameters OE of the spacecraft, as shown in Table 4.

[0191] Table 4 Spacecraft initial orbit parameters

[0192]

[0193] The input spacecraft single pulse maneuver parameters are shown in Table 5;

[0194] Table 5 Spacecraft maneuvering parameters

[0195]

[0196] S32, terminal reference position calculation. Combined with the propagation time t=18h, the terminal reference position and velocity of the spacecraft are calculated as follows:

[0197] ,

[0198] S33, rotation angle calculation. Combined with the spacecraft single pulse maneuver velocity amplitude Δv max =100m / s and propagation time t=18h, by net W The calculated major axis rotation angle is 95.25°;

[0199] S34, independent variable range calculation. Combined with the spacecraft single pulse maneuver velocity amplitude Δv max =100m / s and propagation time t=18h, the switching time is calculated to be 61h, and the corresponding reachable domain is determined to be a short-term reachable domain;FS The envelope curve of the calculated reachable domain has an independent variable range of [-25084, 22677] km on the R coordinate axis;

[0200] S35, 2D envelope calculation. Uniform sampling within the independent variable range of R axis [-25084,22677] km, by net G Calculate the farthest reachable distance of all sampling points on the inner and outer curves, connect them end to end to form a two-dimensional reachable domain envelope, and rotate the major axis rotation angle counterclockwise to obtain the two-dimensional reachable domain envelope in the RSW coordinate system, as shown in Figure 5 As shown;

[0201] S36, 3D envelope calculation. Uniform sampling is performed within the 2D envelope, and the position characteristics of the sampling points are calculated, combined with the spacecraft single pulse maneuver velocity amplitude Δv max =100m / s and propagation time t=18h, by net H The farthest reachable distance of all sampling points is calculated to obtain the envelope of the three-dimensional reachable domain, and the single pulse reachable domain of the spacecraft in the inertial coordinate system is obtained by the coordinate transformation relationship, such as Figure 6 shown.

[0202] The technical solution of the present invention is described in detail above in conjunction with the embodiments / drawings, but the present invention is not limited to the above technical solution. For ordinary technicians in this technical field, after knowing the contents recorded in the present invention, they can make several equivalent changes and substitutions without departing from the principle of the present invention. These equivalent changes and substitutions should also be regarded as belonging to the protection scope of the present invention.

Claims

1. A method for geometric representation and rapid calculation of the reachable area of ​​a spacecraft single pulse maneuver, characterized in that: The steps include: S1. Aiming at the problem that the envelope of the outer reachable domain of a single pulse fixed time of a spacecraft is highly time sensitive and has a highly irregular spatial shape, the envelope of the outer reachable domain is decoupled into geometric shape features composed of the long-axis rotation angle feature and the sampling point position. The geometric feature representation model of the envelope of the outer reachable domain is constructed based on the long-axis rotation angle feature, and its expression is: ， （1） In the formula, RD is the reachable domain; φ is the major axis rotation angle; [ x j , y j , z j ] is the first j The location coordinates of the sampling points; C() Represents the functional relationship of combining the major axis rotation angle with the geometric shape features composed of all sampling point positions; where: The major axis rotation angle φ The representation model is: ， （2） In the formula, OE are the spacecraft orbit parameters; Δv is the pulse maneuver speed amplitude; t is the propagation time; W φ () It represents the nonlinear functional relationship between maneuvering parameters and the long axis rotation angle; When the propagation time t Less than switching time t s When the propagation time is t Greater than or equal to the switching time t s When the reachable outer envelope is a long-term fixed time reachable outer envelope; wherein the switching time t s Defined as: ， （3） In the formula, ω max and ω min Respectively represent the maximum and minimum orbital angular velocities among the sampling points after the maneuver; t s Indicates the switching time; The geometric shape features are divided into short-term fixed time reachable domain outer envelope geometric shape features and long-term fixed time reachable domain outer envelope geometric shape features, the short-term fixed time reachable domain outer envelope geometric shape features are composed of short-term independent variable range and short-term farthest reachable distance in the plane and farthest reachable distance in different planes, and the long-term fixed time reachable domain outer envelope geometric shape features are composed of long-term independent variable range and long-term farthest reachable distance in the plane and farthest reachable distance in different planes; wherein: The characterization model of the short-term independent variable range is: ， （4） In the formula, x max and x min are the upper and lower bounds of the independent variable range; F S () Represents the nonlinear functional relationship between the characteristic parameters and the range of the independent variable after rotation; The characterization model of the farthest reachable distance in the short-term plane is: ， （5） In the formula, y The sampling point is within the surface S Coordinates maximum reachable distance; subscript in and out Represent the inner and outer curves respectively; G() It represents the functional relationship between the characteristic parameter and the farthest reachable distance; δ is the relative position of the sampling point within the range of the independent variable; The characterization model of the farthest reachable distance of the different surfaces is: ， （6） In the formula, s It is the distance feature of the sampling points in the two-dimensional reachable domain; H() It represents the nonlinear functional relationship between the characteristic parameters and the farthest reachable distance of the different surfaces; z is the maximum distance that can be reached on different surfaces of the sampling points; The characterization model of the long-term independent variable range is: ， （7） In the formula, y min1 It is the lower limit of the independent variable range of curves 1 and 2; y min2 It is the upper limit of the independent variable range of curves 3 and 4; F L () Represents the nonlinear computational relationship between characteristic parameters and the range of the independent variable after rotation; y max1 It is the upper limit of the independent variable range of curves 1 and 3; y max2 It is the upper limit of the independent variable range of curves 2 and 4; The characterization model of the longest reachable distance in the long term is: ， （8） In the formula, x The sampling point is within the surface R The maximum reachable distance of the axis coordinates; G() Represents the nonlinear functional relationship between feature parameters and the farthest reachable distance; i Number the curves; δ 2 is the relative position of the sampling point within the range of the independent variable; S2. Randomly generate an envelope outside the fixed-time reachable domain within a given maneuvering parameter range based on Monte Carlo simulation, extract geometric shape feature samples and long-axis rotation angle feature samples of the fixed-time reachable domain through the generated envelope outside the reachable domain, establish a fixed-time reachable domain envelope database, and use the fixed-time reachable domain envelope database to complete the training of the geometric shape feature mapping network and the long-axis rotation angle mapping network; Among them, the geometric shape feature mapping network and the long axis rotation angle mapping network net W They are respectively fitted by the deep neural network of geometric shape features and long axis rotation angle; wherein the geometric shape feature mapping network includes the long-term and short-term reachable domain outer envelope independent variable range mapping network net F 、 Long-term and short-term maximum reachable distance mapping network net G The farthest reachable distance mapping network of sampling points in different planes and reachable domain net H ;in: Long axis rotation angle mapping network net W The mapping relationship is: ， （9） In the formula, OE are the spacecraft orbit parameters; Δv is the pulse maneuver speed amplitude; τ is the normalized propagation time; net W It is a deep neural network for the long axis rotation angle. Its input is the spacecraft orbit parameters, pulse maneuver velocity amplitude, and normalized propagation time. Its output is the long axis rotation angle. The mapping relationship of the short-term reachable outer envelope independent variable range mapping network is: ， （10） In the formula, OE are the spacecraft orbit parameters; Δv is the pulse maneuver speed amplitude; τ is the normalized propagation time; x max and x min are the upper and lower bounds of the independent variable range, respectively; net FS is a deep neural network with an envelope independent variable range outside the reachable domain. The input is the spacecraft orbit parameters, pulse maneuver velocity amplitude, and normalized propagation time. The output is the envelope curve of the reachable domain. R The upper and lower bounds of the axis independent variable; The mapping relationship of the long-term reachable domain envelope independent variable mapping network is: ； （11） In the formula, OE are the spacecraft orbit parameters; Δv is the pulse maneuver speed amplitude; τ is the normalized propagation time; y min1 For curves 1 and 2 S The lower bound of the axis independent variable range; y min2 For curves 3 and 4 S The upper bound of the axis independent variable range; y max1 For curves 1 and 3 S The upper bound of the axis independent variable range; y max2 For curves 2 and 4, S The upper bound of the axis independent variable range; net FL is a deep neural network with a range of independent variables in the long-term reachable domain envelope. The input is the spacecraft orbit parameters, pulse maneuver velocity amplitude, and normalized propagation time. The output is the reachable domain envelope curve in S The upper and lower bounds of the axis independent variable; The mapping relationship of the short-term maximum reachable distance mapping network is: ， （12） Where OE is the spacecraft orbit parameter; Δv is the pulse maneuver velocity amplitude; τ is the normalized propagation time; δ is the relative position of the sampling point within the range of the independent variable; y is the farthest reachable distance of the sampling point on the S axis; net Gin / out It is a deep neural network for the inner and outer farthest reachable distances within the short-term sampling point plane. The input is the spacecraft orbit parameters, the pulse maneuver velocity amplitude, the normalized propagation time, and the position of the sampling point within the R-axis independent variable range. The output is the farthest reachable distance of the sampling point on the S-axis. The long-term in-plane maximum reachable distance mapping network The mapping relationship is: ， （13） In the formula, OE are the spacecraft orbit parameters; Δv is the pulse maneuver speed amplitude; τ is the normalized propagation time; δ 2 is the relative position of the sampling point within the range of the independent variable; x The sampling point is R The maximum reachable distance of the axis; is the longest reachable distance deep neural network on the curve number i within the long-term sampling point surface, and the input is the spacecraft orbit parameters, pulse maneuver velocity amplitude, normalized propagation time, sampling point S Position within the axis independent variable range, output sampling point R The maximum reachable distance of the axis; The farthest reachable distance mapping network of the reachable domain sampling points net H The mapping relationship is: ， （14） In the formula, OE are the spacecraft orbit parameters; Δv is the pulse maneuver speed amplitude; τ is the normalized propagation time; s It is the distance feature of the sampling points in the two-dimensional reachable domain; z The sampling point is W The maximum reachable distance of the axis; net H The sampling point is inside the W The deep neural network of the farthest reachable distance of the coordinate axis takes as input the orbital parameters of the spacecraft, the amplitude of the pulse maneuvering velocity, the normalized propagation time, and the distance characteristics of the sampling points in the two-dimensional reachable domain, and outputs the farthest reachable distance of the sampling points in different planes; S3. Based on the geometric shape feature mapping network and the long-axis rotation angle mapping network, a method for quickly calculating the geometric shape features of the envelope outside the fixed-time reachable domain is proposed to realize the calculation of the geometric shape features of the envelope outside the fixed-time reachable domain of a single pulse of a spacecraft.

2. The method for geometric representation and rapid calculation of the reachable region of a spacecraft single pulse maneuver according to claim 1 is characterized in that: The construction process of the geometric representation model of the reachable outer envelope is as follows: Considering the high time sensitivity of the reachable domain, in order to reduce the complexity of characterizing irregular shapes in the reachable domain, the two types of features, long axis rotation angle and geometric shape, are decoupled, and characterization methods are designed for the two types of features respectively. φ Calculated by formula (2); The geometric shape features are composed of the range of independent variables, the farthest reachable distance in the plane, and the farthest reachable distance in different planes. As the propagation time increases, the degree of bending and distortion of the reachable domain becomes severe. Short-term and long-term fixed-time reachable domain geometric shape characterization models are designed respectively: to meet the propagation time t < t s When , the reachable domain is a short-term fixed-time reachable domain, and the outer envelope of the short-term fixed-time reachable domain is composed of the inner curve close to the earth and the outer curve far from the earth. On the contrary, it is a long-term fixed-time reachable domain, and the outer envelope of the long-term fixed-time reachable domain is composed of the end-to-end splicing of curves 1 to 4. Switching time t s The definition of is shown in formula (3); The outer envelope of the short-term fixed-time reachable domain is a closed figure in the orbital plane, which is difficult to be directly represented by a single-valued function. Therefore, the closed reachable domain is split into an inner curve and an outer curve by the two points with the largest phase difference on both sides of the outer envelope. The range of the curve independent variable is calculated by formula (4); x is the horizontal coordinate of any sampling point within the range of the envelope independent variable that can be reached outside the domain in a short-term fixed time. The relative position of the sampling point within the range of the independent variable is δ It is expressed as: ， （15） In the formula, x is the horizontal coordinate of the sampling point; x max and x min The upper and lower bounds of the range of the independent variable of the reachable domain envelope curve; The farthest reachable distance on the inner and outer curves within the surface of any sampling point within the independent variable is calculated by formula (5); Sampling is performed uniformly within the range of the independent variable of the outer envelope of the reachable domain in a short-term fixed time, and the farthest reachable distance of the sampling points on the inner and outer curves that constitute the outer envelope of the reachable domain are calculated respectively. y , connect all sampling points in sequence to form a short-term fixed-time reachable domain plane envelope connected end to end; After the long-term fixed time reachable domain envelope is rotated, the inner and outer curves are separated into their respective y The axis minimum point is split into two curves, and curves 1 to 4 are obtained, all of which satisfy a S Axis ordinate y Value and a R Axis coordinates x The single-valued function relationship of the value; The range of independent variables of the envelope curve outside the domain that can be reached in the long-term fixed time is calculated by formula (7); y is the ordinate of any sampling point within the reachable domain envelope independent variable, then the relative position of the sampling point within the independent variable range δ 2 It is expressed as: ， （16） The farthest reachable distance of any sampling point in the independent variable on curves 1 to 4 is calculated by formula (8); The distance characteristic of any sampling point in the plane outside the envelope of the two-dimensional reachable domain is s ,The farthest reachable distance of sampling points on different planes is calculated by formula (6); The long axis rotation angle of the spacecraft single pulse fixed time reachable domain is calculated φ With all sampling point locations [ x j , y j , z j ] is composed of geometric shape features, and a geometric representation model of the envelope outside the reachable domain is obtained based on the combination of formulas.

3. The method for geometric representation and rapid calculation of the reachable region of a spacecraft single-pulse maneuver according to claim 1 is characterized in that: The storage form of the fixed time reachable external envelope database is as follows: ， （17） In the formula, SU S and SU L They are the short-term and long-term geometric feature storage units of the reachable domain; j is the sampling point number; x j , y j , z j They are the reachable domain sampling points j exist R, S, W Axis coordinate values.

4. The method for geometric representation and rapid calculation of the reachable region of a spacecraft single pulse maneuver according to claim 1 is characterized in that: The normalized propagation time τ is as follows: ， （18） In the formula, t is the propagation time; T n is the orbital period of the reference orbit.

5. The method for geometric representation and rapid calculation of the reachable region of a spacecraft single pulse maneuver according to claim 1 is characterized in that: The process of obtaining the envelope of the short-term fixed time reachable outer domain comprises the following steps: S51. Input initial orbit parameters OE , propagation time t The upper limit of the spacecraft single pulse maneuver speed increment Δv , combined with the long axis rotation angle deep neural network net W The major axis rotation angle is calculated by formula (9): φ ; S52, by calculating the switching time t s Determine whether the input propagation time corresponds to the long-term or short-term fixed-time reachable domain envelope, and fit the deep neural network with the trained short-term envelope shape independent variable range net FS The envelope outside the reachable domain is calculated by formula (10) in the R-axis independent variable range; S53, in the range of independent variables [ x max , x min ] uniformly sampled, record the k Sampling points R Axis coordinates x k , calculate the relative position of the sampling point within the range of the independent variable δ k , and combined with deep neural networks net Gin / out Calculate the reachable domain envelope orbital surface S Axis maximum reach distance y k As shown in formula (19), the envelope point of the two-dimensional reachable domain is obtained [ x k , y k ], all the two-dimensional reachable domain envelope points form the outer envelope of the plane reachable domain: ， （19） In the formula, OE are the spacecraft orbit parameters; Δv is the pulse maneuver speed amplitude; τ is the normalized propagation time; δ k It is the sampling point k relative position within the range of the independent variable; y k It is the sampling point k exist S The maximum reachable distance of the axis; net Gin / out Fitting the neural network to the longest reachable distances inside and outside the short-term sampling point surface for training; S54, uniformly sample within the two-dimensional envelope of the reachable domain, record the j The sampling points have coordinates in the orbital plane [ x j , y j ], calculate the position characteristics of the sampling points in the surface s j , and combined with deep neural networks net H Calculate the reachable domain envelope orbital out of plane W Axis maximum reach distance z j As shown in formula (20), the envelope point of the three-dimensional reachable domain is obtained [ x j , y j , z j ], all three-dimensional reachable domain envelope points constitute the three-dimensional reachable domain outer envelope: ， （20） In the formula, OE are the spacecraft orbit parameters; Δv is the maneuvering speed amplitude; τ is the normalized propagation time; s j It is the distance feature of the sampling points in the two-dimensional reachable domain; z j is the maximum reachable distance of the sampling point on the W axis; net H Fitting the neural network to the farthest reachable distance of the sampling point surface on the W coordinate axis to complete the training; S55, rotate the major axis by an angle φ and all points on the three-dimensional envelope surface [ x j , y j , z j ] By combining the following formulas, we can obtain the three-dimensional reachable domain outer envelope RD as shown in formula (1); S56, the initial orbital parameters OE Transition to orbital state [ r 0 ,v 0 ], recursively to the propagation time t Time position speed[ r t , v t ],Will RSW The reachable domain in the coordinate system is transformed into the inertial system to obtain the fixed time reachable domain.

6. The method for geometric representation and rapid calculation of the reachable region of a spacecraft single pulse maneuver according to claim 1 is characterized in that: The process of obtaining the long-term fixed time reachable outer envelope comprises the following steps: S61. Input initial orbit parameters OE , propagation time t The upper limit of the spacecraft single pulse maneuver speed increment Δv , combined with the long axis rotation angle deep neural network net W The major axis rotation angle is calculated by formula (9): φ , S62, by calculating the switching time t s Determine whether the input propagation time corresponds to the long-term or short-term fixed-time reachable domain envelope, and fit the neural network with the trained short-term envelope shape independent variable range net FS The envelope outside the reachable domain is calculated in the S-axis independent variable range as shown in formula (11); S63, respectively, in the independent variable range of curves 1 to 4 [ y min1 , y max1 ]、[ y min1 , y max2 ]、[ y min2 , y max1 ]、[ y min2 , y max2 ] uniformly sampled, record the k Sampling points S Axis coordinates y k , calculate the relative position of the sampling point within the range of the independent variable δ 2k , and combined with deep neural networks net Gi Calculate the reachable domain envelope orbital surface R Axis maximum reach distance x k As shown in formula (13), we get the envelope point of the two-dimensional reachable domain [ x k , y k ], all the envelope points of the two-dimensional reachable domain form the outer envelope of the plane reachable domain; S64, uniformly sample within the two-dimensional envelope of the reachable domain, record the j The sampling points have coordinates in the orbital plane [ x j , y j ], calculate the position characteristics of the sampling points in the surface s j , and combined with deep neural networks net H Calculate the reachable domain envelope orbital out of plane W Axis maximum reach distance z j As shown in formula (20), the envelope point of the three-dimensional reachable domain is obtained [ x j , y j , z j ], all three-dimensional reachable domain envelope points constitute the three-dimensional reachable domain outer envelope: ， （20） In the formula, OE are the spacecraft orbit parameters; Δv is the pulse maneuver speed amplitude; τ is the normalized propagation time; s j It is the distance feature of the sampling points in the two-dimensional reachable domain; z j is the maximum reachable distance of the sampling point on the W axis; net H Fitting the neural network to the farthest reachable distance of the sampling point surface on the W coordinate axis to complete the training; S65, rotate the long axis by an angle φ and all points on the three-dimensional envelope surface [ x j , y j , z j ] By combining the following formulas, we can get the outer envelope RD of the three-dimensional reachable domain; ； （1） In the formula, RD is the reachable domain; x j , y j , z j ] is the first j The location coordinates of the sampling points; C() Represents the functional relationship of combining the rotation angle feature with the geometric shape feature composed of all sampling point positions; S66, the initial orbital parameters OE Transition to orbital state [ r 0 ,v 0 ], recursively to the propagation time t Time position speed[ r t , v t ],Will RSW The reachable domain in the coordinate system is transformed into the inertial system to obtain the fixed time reachable domain.

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