Real-time solving method of spacecraft orbit pursuit game

Abstract

The invention discloses a real-time solving method of a spacecraft orbit pursuit game, and relates to the field of real-time solving of the spacecraft orbit pursuit game. The problems that in an existing spacecraft pursuit game, a boundary value needs to be solved on the basis of a saddle point solution of the maximum value principle, a heuristic method or a nonlinear programming method is difficult to meet the requirements of on-orbit application, and the guarantee of calculation time and convergence becomes the maximum barrier to hinder the on-orbit application of the game theory are solved. The method comprises the following steps: solving an equivalent single-side minimum time optimal control equation by using a Hamiltonian function and a separation property of a linear common-state equation; based on a single-side minimum time optimal control equation, a least square method is adopted to fit a polynomial to obtain a transformation matrix, a dynamic ellipsoid is utilized to determine an analysis boundary, the terminal time of the analysis boundary is a boundary reaching a set, and based on a terminal geometric condition, a zero point of f (tf) is solved to obtain optimal game time and an optimal game strategy; and realizing on-line solving of a spacecraft pursuit game.

Description

Technical Field

[0001] The present invention relates to the technical field of real-time solution of spacecraft orbit pursuit and escape game, and in particular to a real-time solution method of spacecraft orbit pursuit and escape game. Background Art

[0002] The current development and use of space is leading to increased congestion, orbital debris, and the risk of collision and misjudgment. More and more space activities are causing the emergence of "gray areas". In this ever-changing orbital environment, it is necessary to re-evaluate the survivability and response level of spacecraft.

[0003] The calculation of equilibrium solutions is the core issue of game problems, which involves the decision-making in the strategy space of both parties. Nash equilibrium is the basic theory of game problems. It studies how to calculate the optimal strategies of both parties in multi-person dynamic games, that is, the optimal benefits and optimal action strategies of both parties. Generally speaking, if the equilibrium solution of a game problem can be obtained, it means that the game problem has been solved, that is, we know all the knowledge about the game problem. The properties of the equilibrium solution ensure that the equilibrium strategy is the optimal strategy of the player when facing the smartest and most rational opponent. However, from the perspective of computational complexity, the game problem is an NP-hard problem, that is, it cannot be solved in polynomial time. Therefore, there is currently no general method to completely solve the calculation of equilibrium solutions. The calculation and search for equilibrium solutions are the core and key to the study of game problems and the basis of all strategy analysis.

[0004] The spacecraft pursuit game is usually modeled as a two-person zero-sum game with terminal free time. However, the saddle point solution based on the maximum principle requires solving the boundary value problem. Currently, methods based on heuristic methods or nonlinear programming are difficult to meet the needs of on-orbit applications. Both the computing time and the guarantee of convergence have become the biggest obstacles to the on-orbit application of game theory. Summary of the invention

[0005] In order to solve the problem that the saddle point solution based on the maximum principle in the spacecraft pursuit and escape game in the prior art needs to solve the boundary value problem, the current methods based on heuristic methods or nonlinear programming are difficult to meet the needs of on-orbit applications, both in terms of calculation time and guarantee of convergence, which has become the biggest obstacle to the on-orbit application of game theory.

[0006] To solve the above technical problems, the present invention is achieved through the following technical solutions:

[0007] Solution 1: The present invention proposes a real-time solution method for a spacecraft orbit pursuit and escape game, the method comprising the following steps:

[0008] Step 1: Using the separation property of Hamiltonian function and linear co-state equation, an equivalent one-sided minimum time optimal control equation is solved;

[0009] Step 2: Based on the one-sided minimum time optimal control equation described in step 1, the least squares method is used to fit the polynomial to obtain the transformation matrix The dynamic ellipsoid is used to determine the analytical boundary, and its terminal time is the boundary of the arrival set. Based on the terminal geometric conditions, f(t f ) to obtain the optimal game time and optimal game strategy, that is, to achieve online solution of the spacecraft pursuit and escape game.

[0010] Further, a preferred implementation is provided, in which the method for confirming the minimum time on one side in step 1 is:

[0011]

[0012] Among them, t f is the end time of the game, R is a set of positive real numbers, r p , r e are the tracking spacecraft position vector and the escaping spacecraft position vector respectively.

[0013] Further, a preferred implementation is provided, the one-sided minimum time optimal control equation in step 1 is:

[0014]

[0015] in, H is the bilateral Hamiltonian function, u p To track the optimal control of the spacecraft, U p To track the feasible control set of the spacecraft, u e For optimal control of the escape spacecraft, U e is the feasible control set of the escape spacecraft, λ p,v To track the spacecraft velocity covariate, λ e,v is the escape spacecraft velocity co-state variable.

[0016] Furthermore, a preferred implementation is provided, wherein step 2 also includes the step of decomposing the boundary of the terminal time arrival set to obtain a control input response.

[0017] Further, a preferred implementation is provided, in which the method of decomposing the terminal time as the boundary of the arrival set to obtain the control input response is:

[0018]

[0019] r(t,τ)=r zir (t)+r zsr (t,τ)

[0020] Among them, Φ 1 , Φ 2are the state transfer matrices of the CW equation, τ is the Lagrange multiplier corresponding to the terminal constraint, r zir 、r zsr are the zero input position vector and the zero state position vector respectively.

[0021] Further, a preferred embodiment is provided, wherein the zero-state response r zsr Symmetric about the origin, that is, r zsr (t,τ)+r zsr (t,-τ)=0,

[0022] Where θ = n·t f , n is the angular velocity of the reference circular orbit.

[0023]

[0024] a=(26θ-32sinθ+3sin2θ) / 4n 3

[0025] b=-3(θ-sinθ) 2 / n 3

[0026] c=(14θ+3θ 3 +24θcosθ-32sinθ-3sin2θ) / n 3

[0027] d=(2θ-sin(2θ)) / 4n 3

[0028] Further, a preferred implementation is provided, in which the method for determining the analytical boundary using the dynamic ellipsoid in step 2 is:

[0029] The main axis of the dynamic ellipsoid is obtained by calculating the scaling factor, which is calculated as follows: r(t) = p 1 t n +p 2 t n-1 +...+p n t+p n+1

[0030] Where n is the polynomial order, p 1 are the polynomial coefficients to be determined.

[0031] Further, a preferred implementation is provided, in step 2, based on the one-sided minimum time optimal control equation described in step 1, the least squares method is used to fit the polynomial to obtain the transformation matrix The method is:

[0032]

[0033] in, V is the eigenvector corresponding to the orthogonal decomposition;

[0034]

[0035] Solution 2: An electronic device comprises: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor implements the steps of any method described in Solution 1 when executing the computer program.

[0036] Solution three: A computer-readable storage medium storing a computer program, wherein the computer program, when executed by a processor, implements the steps of the method described in any one of Solution one.

[0037] The present invention is beneficial in that:

[0038] The real-time solution method for the spacecraft orbit pursuit and escape game described in the present invention adopts a quantitative and reachable set method to solve the two-person zero-sum pursuit and escape differential game with linearized relative motion dynamics constraints in real time. First, the separation properties of the Hamiltonian function and the linear co-state equation are used to derive the equivalent one-sided minimum time optimal control problem. Then, the analytical boundary of the reachable set is approximated by a dynamic ellipsoid using a least squares-based polynomial fitting. Even if different control inputs are used, the coefficients only need to be calculated offline once in advance. The optimal terminal time is when the boundary of the reachable set first passes through the origin, and with the help of the proposed terminal geometry, the approximate optimal strategy can be obtained simultaneously within a few milliseconds. The proposed geometry provides insights into the decision-making process of the pursuer and the evader, which helps to promote further research. The proposed method is suitable for on-orbit implementation because it has a very low computational burden and reliable and predictable output, and is not affected by the convergence uncertainty present in numerical methods. The proposed method also has the potential for closed-loop applications.

[0039] The present invention is also applied to the field of attack and defense confrontation of spacecraft and solving the problem of autonomous decision-making on orbit. BRIEF DESCRIPTION OF THE DRAWINGS

[0040] Figure 1 This is a schematic diagram of the unilaterally reachable set and bilateral results described in Implementation Method 11.

[0041] Among them, (a) is a schematic diagram of unilateral terminal results, (b) is a schematic diagram of bilateral result comparison, and (c) is a schematic diagram of bilateral collective perspective.

[0042] Figure 2 This is a schematic diagram for comparing the low-orbit game results described in Implementation Method 11.

[0043] Among them, (a) is a schematic diagram of the perspective of the end moment of the unilaterally reachable game, (b) is a schematic diagram of the bilateral optimal trajectory, and (c) is a schematic diagram of the perspective of the bilaterally reachable set. DETAILED DESCRIPTION

[0044] In order to make the purpose, technical solutions and advantages of the implementation methods of the present application clearer, the technical solutions in the implementation methods of the present application will be clearly and completely described below in conjunction with the drawings in the implementation methods of the present application. Obviously, the described implementation methods are only part of the implementation methods of the present application, not all of the implementation methods.

[0045] Implementation method 1: This implementation method proposes a real-time solution method for a spacecraft orbit pursuit and escape game, and the method includes the following steps:

[0046] Step 1: Using the separation property of Hamiltonian function and linear co-state equation, an equivalent one-sided minimum time optimal control equation is solved;

[0047] Step 2: Based on the one-sided minimum time optimal control equation described in step 1, the least squares method is used to fit the polynomial to obtain the transformation matrix The dynamic ellipsoid is used to determine the analytical boundary, and its terminal time is the boundary of the arrival set. Based on the terminal geometric conditions, f(t f ) to obtain the optimal game time and optimal game strategy, that is, to achieve online solution of the spacecraft pursuit and escape game.

[0048] Implementation method 2: This implementation method further limits the real-time solution method of the spacecraft orbit pursuit and escape game described in implementation method 1. The method for confirming the unilateral minimum time described in step 1 is:

[0049]

[0050] Among them, t f is the end time of the game, R is a set of positive real numbers, rp and re are the position vectors of the tracking spacecraft and the escaping spacecraft respectively.

[0051] Implementation method 3: This implementation method further limits the real-time solution method of the spacecraft orbit pursuit and escape game described in implementation method 1. The one-sided minimum time optimal control equation described in step 1 is:

[0052]

[0053] in, H is the bilateral Hamiltonian function, up is the optimal control of the tracking spacecraft, Up is the feasible control set of the tracking spacecraft, ue is the optimal control of the escaping spacecraft, Ue is the feasible control set of the escaping spacecraft, λ p,v To track the spacecraft velocity covariate, λ e,vis the escape spacecraft velocity co-state variable.

[0054] Implementation method 4: This implementation method further limits the real-time solution method of the spacecraft orbit pursuit and escape game described in implementation method 1. Step 2 also includes the step of decomposing the terminal time as the boundary of the arrival set to obtain the control input response.

[0055] Implementation mode 5: This implementation mode further limits the real-time solution method of the spacecraft orbit pursuit and escape game described in implementation mode 4. The method of decomposing the terminal time as the boundary of the arrival set to obtain the control input response is:

[0056]

[0057] r(t,τ)=r zir (t)+r zsr (t,τ)

[0058] Among them, Φ 1 , Φ 2 are the state transfer matrices of the CW equation, τ is the Lagrange multiplier corresponding to the terminal constraint, r zir 、r zsr are the zero input position vector and the zero state position vector respectively.

[0059] Implementation 6: This implementation is a further limitation on the real-time solution method of the spacecraft orbit pursuit and escape game described in Implementation 5. The zero-state response r zsr Symmetric about the origin, that is, r zsr (t,τ)+r zsr (t,-τ)=0,

[0060] where θ = n·t f .

[0061]

[0062] a=(26θ-32sinθ+3sin2θ) / 4n 3

[0063] b=-3(θ-sinθ) 2 / n 3

[0064] c=(14θ+3θ 3 +24θcosθ-32sinθ-3sin2θ) / n 3

[0065] d=(2θ-sin(2θ)) / 4n 3

[0066] Implementation method 7: This implementation method further limits the real-time solution method of the spacecraft orbit pursuit and escape game described in implementation method 1. The method of determining the analytical boundary using the dynamic ellipsoid in step 2 is:

[0067] The main axis of the dynamic ellipsoid is obtained by calculating the scaling factor, which is calculated as follows:

[0068] r(t)=p 1 t n +p 2 t n-1 +...+p n t+p n+1

[0069] Where n is the polynomial order, p 1 are the coefficients of the unknown polynomial.

[0070] Implementation 8. This implementation is a further limitation of the real-time solution method for the spacecraft orbit pursuit game described in Implementation 1. In step 2, based on the one-sided minimum time optimal control equation described in step 1, the least squares method is used to fit the polynomial to obtain the transformation matrix The method is:

[0071]

[0072] in, V is the eigenvector corresponding to the orthogonal decomposition;

[0073]

[0074] Embodiment 9. This embodiment proposes an electronic device, comprising: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor implements the steps of any one of the methods described in Scheme 1 when executing the computer program.

[0075] Embodiment 10: This embodiment proposes a computer-readable storage medium, which stores a computer program. When the computer program is executed by a processor, the steps of the method described in any one of the schemes 1 are implemented.

[0076] Implementation eleven: This implementation provides an example, which is used to explain the above implementations one to eight. The specific example is as follows:

[0077] See also Figure 1 to Figure 2 To illustrate this implementation, for the two-spacecraft pursuit-escape game problem, the terminal time-free pursuit-escape game is modeled as a two-person zero-sum problem, that is, the pursuit spacecraft p and the escape spacecraft e. The relative motion under the circular reference orbit is described by the relative motion equation CW in the LVLH coordinate system:

[0078]

[0079] It is usually assumed that the gaming spacecraft / player maneuvers according to the maximum maneuverability, that is, fixed thrust: ||u p ||=u p,max >||u e ||=u e,max

[0080] In order to describe the impact of mobility differences on the game results, the mobility ratio β is defined as p,max / u e,max >1. It is generally assumed that the maneuverability of the pursuit spacecraft is greater than that of the escape spacecraft to ensure the end of the game, that is, the pursuit spacecraft must be able to capture the escape spacecraft. The result of the pursuit-escape game is only related to the position. Generally, only the position is concerned. The end condition of the pursuit-escape game is:

[0081] r p (t f )=r e (t f ),φ(x p ,x e )= p -r e =r pe =0

[0082] The goal of the spacecraft pursuit and escape game is to minimize the capture time of the tracking spacecraft and maximize the capture time of the escaping spacecraft. Zero-sum means that the sum of the goals or utilities of the tracking spacecraft and the escaping spacecraft is zero:

[0083] J p (u p ,u e ) = t f

[0084] J e (u p ,u e )=-t f

[0085] J p (u p ,u e )+J e (u p ,u e )=0

[0086] The optimal capture time is:

[0087] Similar to the unilateral optimal control problem, the optimality condition of the differential game problem is guaranteed by the minimum principle, which defines the sum of the bilateral Hamiltonian functions of the optimal control:

[0088]

[0089] in

[0090] The tracking spacecraft maximizes the Hamiltonian function, and the escaping spacecraft minimizes the Hamiltonian function. The optimal control is:

[0091]

[0092] Terminal values ​​of comorphic variables:

[0093] λ p,r (t f )=τ,λ e,r (t f )=-τ

[0094] λ p,v (t f )=0,λ e,v (t f )=0

[0095] It can be proved that in the case of a two-person zero-sum pursuit problem, it is equivalent to the origin transfer problem of minimizing time:

[0096] P 2 :mint f

[0097]

[0098] x(t 0 )=x pe (t 0 )

[0099] r(t f )=0

[0100] u max =u p,max -u e,max

[0101] Decompose the points on the terminal reachable set to obtain the control input response. The time optimal reachable boundary r(t f ) is equal to zero input response r zir and zero-state response r zsr The sum of:

[0102]

[0103] r(t,τ)=r zir (t)+r zsr (t,τ)

[0104] It can be found that the zero state response r zsr Symmetric about the origin, that is, r zsr (t,τ)+r zsr (t,-τ)=0.

[0105] where θ = n·t f

[0106]

[0107] a=(26θ-32sinθ+3sin2θ) / 4n 3

[0108] b=-3(θ-sinθ) 2 / n 3

[0109] c=(14θ+3θ 3 +24θcosθ-32sinθ-3sin2θ) / n 3

[0110] d=(2θ-sin(2θ)) / 4n 3

[0111] W(t) is a symmetric positive definite matrix. Except for the terminal moment, W(t f )=0. So r zsr The reachable boundary of is similar to the ellipsoid E(0,W), and the ellipsoid is proportional to the control input size. In order to obtain the semi-axis length and main axis direction of the ellipsoid, SVD / eigenvalue decomposition is performed on W:

[0112] W=V -1 ΣV

[0113] The eigenvector is the principal axis direction, and the three eigenvectors correspond to the directions of the principal axes of the ellipsoid.

[0114]

[0115] The Boundary of Reachable Sets for Unilateral Time Optimal Control Problem The ellipsoid E(r zsr ,P(t f )) approximation, but it is difficult to calculate the explicit analytical expression due to the integral in, and it is difficult to obtain the true analytical ellipsoid. For this reason, a polynomial fitting ellipsoid approximation is proposed. Since the actual reachable set boundary and the approximate reachable set boundary E(0,W) only have a scaling relationship, the scaling factor is recalculated, that is, the major axis approximation of the ellipsoid. The eigenvalue, that is, the corresponding semi-major axis of the ellipsoid, is yet to be determined.

[0116] E={r T Pr = 1}

[0117] P=V -1 DV

[0118] r=1 / D 2

[0119] Where V is still the eigenvector of P, which is the same as the eigenvector of W. The actual game problem will not last too long, and the present invention only focuses on one period t f ∈[0,T]. Sample K samples in the time range, where the eigenvalue r(k) can be calculated offline in advance, and the time-varying principal axis based on polynomial fitting is the W eigenvalue:

[0120] r(t)=p 1 t n +p 2 t n-1 +...+p n t+p n+1

[0121] Where n is the polynomial order, the corresponding transformation matrix is ​​approximated

[0122]

[0123] in

[0124]

[0125] From the perspective of unilateral time optimal control, the solution to the unilateral time optimal control problem corresponding to the pursuit-escape game is converted into the zero-crossing problem of the reachable set. The solution to the unilateral optimal control problem is to transfer to the origin of the coordinates, so from the perspective of the reachable set, the game ends when the boundary of the optimal time reachable set passes through the origin for the first time:

[0126]

[0127] The judgment condition is r zsr (t f ) is at the boundary of the reachable set. Using the approximate ellipsoid E(0,P(t f ) The estimated end time of the game is:

[0128]

[0129] The equation equivalent to the judgment is:

[0130]

[0131] If f(t f )<0, it means the game time has not ended, otherwise the game is over. The calculation of is equivalent to calculating f(t f ) zero point, since f(t f ) is analytic, so the bisection method can be used to calculate an approximation with a given error.

[0132] Once the end time of the game is determined, the covariate variables are determined:

[0133] τ=W -1 r zsr

[0134] Where W -1 The computation can be explicitly parsed.

[0135]

[0136] Then the corresponding original game problem P 1 Terminal comorphic variables in:

[0137]

[0138] In the geometric sense of the terminal co-state variables, the three points are in a straight line at the terminal moment. p,zsr (t f )r e,zsr (t f ) terminals intersect and are parallel so r p,zir (t f )r e,zir (t f )r p (t f )=r e (t f ) are collinear.

[0139]

[0140] And || r p,zsr (t f ,τ)|| / ||r e,zsr (t f ,τ)||=u p,max / u e,max

[0141] so,

[0142]

[0143] Finally, by solving f(t f ) to obtain the optimal game time and the optimal game strategy, thereby realizing the online solution of the spacecraft game.

[0144] In order to verify the algorithm stability under different orbital periods (orbital altitudes), the orbital period range is T = [6000, 86400] seconds, the absolute maneuverability range and relative maneuverability range of the spacecraft are: u max ∈[0.0001,1]m / s 2 β∈[1.2,4]. All simulation results are based on Matlab and Intel Core13700K processor.

[0145] Example 1: The initial parameters of the GEO reference orbit are as follows: the escape spacecraft is located at the origin of the coordinate system. T = 86400s, u p,max =0.0686m / s 2 ,u e,max =0.0343m / s 2

[0146] Table 3-1 Initial parameters of Example 1

[0147]

[0148] Example 2: Initial parameters of low-orbit reference orbit,

[0149] T = 1 × 10 4 s,u p,max =0.02979m / s 2 ,u e,max =0.02482m / s 2 ,

[0150] Table 3-2 Initial parameters of Example 2

[0151]

[0152]

[0153] In order to verify the effectiveness and correctness of the method, it is compared with the current heuristic search-based method (IHM) and the nonlinear programming (NLP)-based method. A comparison of the three methods is given, among which the approximate analytically reachable set method (ARS) proposed in this paper has the highest computational efficiency, and only needs 2 milliseconds to obtain the optimal terminal time and the optimal co-state variable.

[0154] Table 3-3 Numerical solution results of Example 1

[0155]

[0156] The angle of the unilaterally reachable set and the bilateral optimal result are as follows Figure 1 (a), (b), and (c) show that the terminal error of the method based on approximate reachable sets is greater than that of the method based on heuristic search, which is mainly due to the approximate error of the terminal set.

[0157] To further analyze the impact of the results under the low reference orbit, the initial parameters of Example 2 are selected. The above problem is solved according to the same procedure, and the results are as follows:

[0158] Table 3-4 Initial parameters of Example 2

[0159]

[0160]

[0161] See also Figure 2 The numerical simulation results in (a), (b), and (c) show that the proposed method is two orders of magnitude more efficient than the traditional method, meeting the requirements of on-orbit real-time solution. In the case of low orbit, since the game ends in a short time, the set approximation error is very small, resulting in more accurate results.

[0162] The proposed method solves a two-player zero-sum pursuit-evasion differential game with linearized relative motion dynamics constraints in real time using a quantitative and reachable set approach. First, the separation properties of the Hamiltonian and the linear co-state equations are used to derive an equivalent one-sided minimum-time optimal control problem. Then, the analytical boundary of the reachable set is approximated by a dynamic ellipsoid using least-squares-based polynomial fitting. The coefficients only need to be calculated offline once in advance, even when different control inputs are used. The optimal terminal time is when the boundary of the reachable set first crosses the origin, and with the help of the proposed terminal geometry, an approximate optimal strategy can be obtained simultaneously within a few milliseconds. The proposed geometry provides insights into the decision-making process of the pursuer and the evader, which can help promote further research. The proposed method is suitable for on-orbit implementation because it has a very low computational burden and reliable and predictable output, and it does not suffer from the convergence uncertainty present in numerical methods. The proposed method also has the potential for closed-loop applications.

[0163] Those skilled in the art will appreciate that the above are only preferred embodiments of the present invention, and the various embodiments of the present disclosure and / or the features described in the claims may be combined or combined in various ways, even if such combinations or combinations are not explicitly described in the present disclosure. It is not intended to limit the present invention. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art may still modify the technical solutions described in the aforementioned embodiments, or perform equivalent substitutions on some of the technical features therein. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention shall be included in the protection scope of the present invention.

[0164] Although preferred embodiments of the present invention have been described, additional changes and modifications may be made to these embodiments by those skilled in the art once the basic inventive concepts are known. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments and all changes and modifications that fall within the scope of the present invention. Obviously, those skilled in the art may make various changes and modifications to the present invention without departing from the spirit and scope of the present invention. Thus, if these modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is also intended to include these modifications and variations.

Claims

1. A real-time solution method for the spacecraft orbit pursuit and escape game, characterized in that: The method comprises the following steps: Step 1: Using the separation property of Hamiltonian function and linear co-state equation, an equivalent one-sided minimum time optimal control equation is solved; Step 2: Based on the one-sided minimum time optimal control equation described in step 1, the least squares method is used to fit the polynomial to obtain the transformation matrix The dynamic ellipsoid is used to determine the analytical boundary, and its terminal time is the boundary of the arrival set. Based on the terminal geometric conditions, f(t f ) to obtain the optimal game time and optimal game strategy, that is, to achieve online solution of the spacecraft pursuit and escape game.

2. The real-time solution method for the spacecraft orbit pursuit and escape game according to claim 1 is characterized in that: The method for confirming the minimum time on one side described in step 1 is: Among them, t f is the end time of the game, R is a set of positive real numbers, r p , r e are the tracking spacecraft position vector and the escaping spacecraft position vector respectively.

3. The real-time solution method for the spacecraft orbit pursuit and escape game according to claim 1 is characterized in that: The one-sided minimum time optimal control equation described in step 1 is: in, H is the bilateral Hamiltonian function, u p To track the optimal control of the spacecraft, U p To track the feasible control set of the spacecraft, u e For optimal control of the escape spacecraft, U e is the feasible control set of the escape spacecraft, λ p,v To track the spacecraft velocity covariate, λ e,v is the escape spacecraft velocity co-state variable.

4. The real-time solution method for the spacecraft orbit pursuit and escape game according to claim 1 is characterized in that: Step 2 also includes the step of decomposing the terminal time as the boundary of the arrival set to obtain a control input response.

5. The real-time solution method for the spacecraft orbit pursuit and escape game according to claim 4 is characterized in that: The method of decomposing the terminal time as the boundary of the arrival set to obtain the control input response is: r(t,τ)=r zir (t)+r zsr (t,τ) Among them, Φ1 and Φ2 are the state transfer matrices of the CW equation, τ is the Lagrange multiplier, and r zir 、r zsr are the zero input position vector and the zero state position vector respectively.

6. The real-time solution method for the spacecraft orbit pursuit and escape game according to claim 5 is characterized in that: The zero state response r zsr Symmetric about the origin, that is, r zsr (t,τ)+r zsr (t,-τ)=0, a=(26θ-32sinθ+3sin2θ) / 4n 3 b=-3(θ-sinθ) 2 / n 3 c=(14θ+3θ 3 +24θcosθ-32sinθ-3sin2θ) / n 3 <h2 style=";text-align:left;direction:ltr">d = (2θ - sin(2θ)) / 4n<h2 style=";text-align:left;direction:ltr"> 3 where θ = n·t f , n is the orbital angular velocity.

7. The real-time solution method for the spacecraft orbit pursuit and escape game according to claim 1 is characterized in that: The method for determining the analytical boundary using the dynamic ellipsoid in step 2 is: The main axis of the dynamic ellipsoid is obtained by calculating the scaling factor, which is calculated as follows: r(t)=p1t n +p2t n-1 +...+p n t+p n+1 Where n is the polynomial order and p1 is the polynomial coefficient to be determined.

8. The real-time solution method for the spacecraft orbit pursuit and escape game according to claim 1 is characterized in that: In step 2, based on the one-sided minimum time optimal control equation described in step 1, the least squares method is used to fit the polynomial to obtain the transformation matrix The method is: in, V is the eigenvector corresponding to the orthogonal decomposition; 9. An electronic device, comprising: A memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor implements the steps of the method according to any one of claims 1 to 8 when executing the computer program.

10. A computer-readable storage medium storing a computer program, characterized in that: When the computer program is executed by a processor, the steps of the method according to any one of claims 1 to 8 are implemented.

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