Real-time solution method for spacecraft orbit pursuit-evasion game

Abstract

The application relates to a real-time solution method for spacecraft orbit pursuit-evasion game, and relates to the field of real-time solution of spacecraft orbit pursuit-evasion game. The existing spacecraft pursuit-evasion game based on the maximum principle needs to solve the boundary value problem of the saddle point solution, and the method based on the heuristic method or the nonlinear programming method cannot meet the demand of on-orbit application in terms of calculation time and convergence guarantee, which becomes the biggest obstacle for the on-orbit application of the game theory. The method comprises the following steps: solving equivalent one-sided minimum time optimal control equations by using the separation property of the Hamilton function and the linear co-state equation; adopting the least square method to fit a transformation matrix based on the one-sided minimum time optimal control equation, determining an analytical boundary by using a dynamic ellipsoid, and taking the terminal time as the boundary of the set; based on the terminal geometric condition, solving the zero point of f(t f ) to obtain the optimal game time and the optimal game strategy, and realizing the online solution of the spacecraft pursuit-evasion game.

Description

Technical Field

[0001] This invention relates to the field of real-time solution technology for spacecraft orbital pursuit and escape games, and specifically to a real-time solution method for spacecraft orbital pursuit and escape games. Background Technology

[0002] Current space development and use are leading to increased congestion, orbital debris, and risks of collisions and misjudgments, with increasing space activity creating a "grey area." In this ever-changing orbital environment, it is necessary to reassess the survivability and response capabilities of spacecraft.

[0003] Equilibrium solution calculation is a core issue in game theory, involving decisions within the strategy space of both players. Nash equilibrium is a fundamental theory of game theory, studying how to calculate the optimal strategies for both sides in dynamic multiplayer games, i.e., the optimal payoffs and optimal action strategies. Generally, if an equilibrium solution to a game problem can be obtained, it means the game problem is solved, meaning we know all the information about it. The properties of equilibrium solutions guarantee that the equilibrium strategy is the optimal strategy for players facing the smartest and most rational opponents. However, from a computational complexity perspective, game theory problems are NP-hard, meaning they cannot be solved in polynomial time. Therefore, there is currently no universally applicable method to completely solve equilibrium solution calculation. The calculation and search for equilibrium solutions are the core and key to game theory research and the foundation of all strategy analysis.

[0004] Spacecraft chase-escape game is usually modeled as a two-person zero-sum game with terminal free time. However, saddle point solutions based on the maximum principle require solving boundary value problems. Currently, heuristic or nonlinear programming methods are difficult to meet the requirements of on-orbit applications. Both computation time and convergence guarantees are the biggest obstacles to the on-orbit application of game theory. Summary of the Invention

[0005] This invention addresses the problem that existing spacecraft pursuit-escape game theory solutions based on the maximum principle require solving boundary value problems. Currently, heuristic or nonlinear programming methods are insufficient for on-orbit applications, hindering both computation time and convergence guarantees. These issues are the biggest obstacles to the on-orbit application of game theory.

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

[0007] Option 1: This invention proposes a real-time solution method for a spacecraft orbital pursuit-escape game, the method comprising the following steps:

[0008] Step 1: Solve the equivalent one-sided minimum time optimal control equation using the separation properties of the Hamiltonian function and the linear common-state equation;

[0009] Step 2: Based on the one-sided minimum time optimal control equation described in Step 1, the transformation matrix is ​​obtained by fitting the polynomial using the least squares method. The analytic boundary is determined using a dynamic ellipsoid, with the terminal time being the boundary of the arrival set. Based on the terminal geometry, f(t) is solved. f The optimal game time and optimal game strategy are obtained at the zero point of the spacecraft pursuit and escape game, thus realizing the online solution of the spacecraft pursuit and escape game.

[0010] Furthermore, a preferred embodiment is provided, wherein the method for confirming the minimum unilateral time in step 1 is as follows:

[0011]

[0012] Among them, t f Let R be the set of positive real numbers, and r be the time when the game ends. p r e These are the position vectors for the tracking spacecraft and the escape spacecraft, respectively.

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

[0014]

[0015] in, H is a two-sided Hamiltonian function, u p To track the optimal control of the spacecraft, U p To track the feasible control set of spacecraft ， u e For optimal control of the escape spacecraft, U e For the feasible control set of the escape spacecraft, λ p,v To track the spacecraft's velocity costate variable, λ e,v The velocity covariance of the escaping spacecraft.

[0016] Furthermore, in a preferred embodiment, step 2 further includes the step of decomposing the terminal time as the boundary of the arrival set to obtain the control input response.

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

[0018]

[0019] r(t,τ)=r zir (t)+r zsr (t,τ)

[0020] Where Φ1 and Φ2 are the state transition matrices of the CW equation, τ is the Lagrange multiplier corresponding to the terminal constraint, and r zir r zsr These are the zero-input position vector and the zero-state position vector, respectively.

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

[0022] Where θ=n·t f , where 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] Furthermore, a preferred embodiment is provided, wherein the method for determining the analytical boundary using a dynamic ellipsoid in step 2 is as follows:

[0029] The principal axes of the dynamic ellipsoid are obtained by calculating the scaling factor. The calculation method is: r(t) = p1t n +p2t n-1 +...+p n t+p n+1

[0030] Where n is the order of the polynomial, and p1 is the polynomial coefficient to be determined.

[0031] Furthermore, a preferred embodiment is provided, in step 2, the transformation matrix is ​​obtained by fitting the polynomial using the least squares method based on the one-sided minimum time optimal control equation described in step 1. The method is as follows:

[0032]

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

[0034]

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

[0036] Option 3: A computer-readable storage medium storing a computer program that, when executed by a processor, implements the steps of the method as described in any of Option 1.

[0037] The advantages of this invention are:

[0038] The real-time solution method for spacecraft orbital pursuit-escape game described in this invention employs a quantitative and reachable set approach to solve a two-player zero-sum pursuit-escape differential game with linearized relative motion dynamics constraints in real time. First, an equivalent one-sided minimum-time optimal control problem is derived using the separation properties of Hamiltonian functions and linear common-state equations. Then, a polynomial fitting based on least squares is used to approximate the analytical boundary of the reachable set with a dynamic ellipsoid. Even with different control inputs, only one offline calculation of coefficients is required beforehand. 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 milliseconds. The proposed geometry provides insights into the decision-making processes of both the pursuer and the escapee, which helps to facilitate further research. The proposed method is suitable for in-orbit implementation because it has extremely low computational burden and reliable and predictable output, and is not affected by the convergence uncertainties present in numerical methods. The method proposed in this invention also has the potential for closed-loop applications.

[0039] This invention is also applied to spacecraft space offensive and defensive confrontation, and to solving the problem of autonomous decision-making in orbit. Attached Figure Description

[0040] Figure 1 This is a schematic diagram of the one-sided reachable set and the two-sided result as described in Implementation Method Eleven.

[0041] Among them, (a) is a schematic diagram of the results of a single-sided terminal, (b) is a schematic diagram of the comparison of results of two sides, and (c) is a schematic diagram of the perspective of two-sided set.

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

[0043] Among them, (a) is a schematic diagram from the perspective of the end time of a one-sided reachable game, (b) is a schematic diagram from the perspective of the optimal trajectory of a two-sided game, and (c) is a schematic diagram from the perspective of the two-sided reachable set. Detailed Implementation

[0044] To make the objectives, technical solutions, and advantages of the embodiments of this application clearer, the technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of this application, and not all of them.

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

[0046] Step 1: Solve the equivalent one-sided minimum time optimal control equation using the separation properties of the Hamiltonian function and the linear common-state equation;

[0047] Step 2: Based on the one-sided minimum time optimal control equation described in Step 1, the transformation matrix is ​​obtained by fitting the polynomial using the least squares method. The analytic boundary is determined using a dynamic ellipsoid, with the terminal time being the boundary of the arrival set. Based on the terminal geometry, f(t) is solved. f The optimal game time and optimal game strategy are obtained at the zero point of the spacecraft pursuit and escape game, thus realizing the online solution of the spacecraft pursuit and escape game.

[0048] Implementation Method Two: This implementation method further defines the real-time solution method for the spacecraft orbital pursuit and escape game described in Implementation Method One. The method for confirming the minimum time on one side in step 1 is as follows:

[0049]

[0050] Among them, t f Let R be the set of positive real numbers, and let rp and re be the position vectors of the tracking spacecraft and the escaping spacecraft, respectively.

[0051] Implementation Method 3: This implementation method further defines the real-time solution method for the spacecraft orbital pursuit-escape game described in Implementation Method 1. The single-sided minimum time optimal control equation in step 1 is:

[0052]

[0053] in, H is the bilateral Hamiltonian function, up is the optimal control for the tracking spacecraft, Up is the set of feasible control options for the tracking spacecraft, ue is the optimal control for the escaping spacecraft, Ue is the set of feasible control options for the escaping spacecraft, and λ p,v To track the spacecraft's velocity costate variable, λ e,vThe velocity covariance of the escaping spacecraft.

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

[0055] Implementation Method Five: This implementation method further defines the real-time solution method for the spacecraft orbital pursuit and escape game described in Implementation Method Four. The method for obtaining the control input response by decomposing the boundary of the arrival set at the terminal time is as follows:

[0056]

[0057] r(t,τ)=r zir (t)+r zsr (t,τ)

[0058] Where Φ1 and Φ2 are the state transition matrices of the CW equation, τ is the Lagrange multiplier corresponding to the terminal constraint, and r zir r zsr These are the zero-input position vector and the zero-state position vector, respectively.

[0059] Implementation Method Six: This implementation method further defines the real-time solution method for the spacecraft orbital pursuit-escape game described in Implementation Method Five. The zero-state response r... zsr Symmetric about the origin, i.e., 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 Seven: This implementation method further defines the real-time solution method for the spacecraft orbital pursuit and escape game described in Implementation Method One. The method for determining the analytical boundary using a dynamic ellipsoid in step 2 is as follows:

[0067] The principal axes of the dynamic ellipsoid are obtained by calculating the scaling factor. The calculation method is as follows:

[0068] r(t) = p1t n +p2t n-1 +...+p n t+p n+1

[0069] Where n is the order of the polynomial and p1 is the coefficient of the undetermined polynomial.

[0070] Implementation Method Eight: This implementation method further defines the real-time solution method for the spacecraft orbital pursuit and escape game described in Implementation Method One. In step 2, the transformation matrix is ​​obtained by fitting the polynomial using the least squares method based on the one-sided minimum time optimal control equation described in step 1. The method is as follows:

[0071]

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

[0073]

[0074] Implementation Method Nine: This implementation method provides an electronic device, including: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the steps of the method as described in any of Scheme One.

[0075] Implementation Method 10: This implementation method provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the steps of the method as described in any of Scheme 1.

[0076] Implementation Method Eleven: This implementation method provides an example, which is used to explain the above-described implementation methods one through eight. The specific example is as follows:

[0077] See Figures 1 to 2 This implementation describes a two-spacecraft chase-escape game problem, modeling the time-free chase-escape game as a zero-sum problem between two spacecraft, i.e., the pursuing spacecraft p and the escaping spacecraft e. The relative motion in the circular reference orbit is described using the relative motion equation CW in the LVLH coordinate system:

[0078]

[0079] It is typically assumed that the spacecraft / player in the game maneuvers at its maximum maneuverability, i.e., with a fixed thrust: ||u p ||=u p,max >||u e ||=u e,max

[0080] To describe the impact of differences in maneuverability on the game outcome, we define the maneuverability ratio β = u p,max / u e,max >1. It is generally assumed that the maneuverability of the tracking spacecraft is greater than that of the escape spacecraft to ensure the end of the game; that is, the tracking spacecraft will always be able to capture the escape spacecraft. The outcome of the chase-escape game depends only on position; generally, only position is of concern. The end condition of the chase-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-escape game is for the tracking spacecraft to minimize its capture time and the escape spacecraft to maximize its capture time. A zero-sum game means that the sum of the goals or utilities of the tracking and escape 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 one-sided optimal control problem, the optimality condition of the differential game problem is guaranteed by the minimum principle. The sum of the two-sided Hamiltonian functions of optimal control is defined as follows:

[0088]

[0089] in

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

[0091]

[0092] Terminal values ​​of costate 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 proven that, under the two-person zero-sum pursuit problem, it is equivalent to minimizing the origin transition problem in time:

[0096] P2:mint f

[0097]

[0098] x(t0)=x pe (t0)

[0099] r(t f ) = 0

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

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

[0102]

[0103] r(t,τ)=r zir (t)+r zsr (t,τ)

[0104] The zero-state response r can be observed. zsr Symmetric about the origin, i.e., 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 that it is singular at the terminal time. f Therefore, r = 0. zsr The reachable boundary is similar to the ellipsoid E(0,W), and the ellipsoid is proportional to the size of the control input. To obtain the ellipsoid's semi-axis length and principal axis direction, SVD / eigenvalue decomposition is performed on W:

[0112] W = V -1 ΣV

[0113] The eigenvectors are the directions of the principal axes, and the three eigenvectors correspond to the directions of the principal axes of the ellipsoid.

[0114]

[0115] Boundary of the reachable set of the one-sided time optimal control problem It can be obtained through the ellipsoid E(r) zsr ,P(t f While the approximation is approximate, the integral in the equation is difficult to calculate with an explicit analytical expression, making it hard to obtain a true analytical ellipsoid. Therefore, this section proposes a polynomial-fitted ellipsoid approximation. 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, which is the approximation of the ellipsoid's principal axis. The eigenvalues, which correspond to the semi-major axis of the ellipsoid, are undetermined.

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

[0117] P = V -1 DV

[0118] r = 1 / D 2

[0119] Here, V remains the eigenvector of P, and is the same as the eigenvector of W. The actual game problem will not last very long; this invention focuses only on one period t. f The approximate state within [0,T]. K samples are sampled over a time period, where the feature value r(k) can be pre-calculated offline. The time-varying principal axis based on polynomial fitting is the W feature value:

[0120] r(t) = p1t n +p2t n-1 +...+p n t+p n+1

[0121] Where n is the order of the polynomial, the corresponding transformation matrix is ​​obtained approximately.

[0122]

[0123] in

[0124]

[0125] From the perspective of one-sided time-optimal control, solving the one-sided time-optimal control problem in a chase-escape game is transformed into a zero-crossing problem of the reachable set. The solution to the one-sided optimal control problem is to move to the origin. Therefore, from the perspective of the reachable set, the game ends when the boundary of the optimal time reachable set first sweeps across the origin.

[0126]

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

[0128]

[0129] The equation that is further equivalent to the judgment is:

[0130]

[0131] If f(t) f If the value is less than 0, it means the game has not ended; otherwise, the game ends. In fact... The calculation is equivalent to the calculation of f(t) f The zero of f(t) f Since it is analytical, the bisection method can be used to calculate the approximate value under a given error.

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

[0133] τ=W -1 r zsr

[0134] Among them W -1 It can be calculated explicitly using analytical methods.

[0135]

[0136] Then, the terminal costate variable in the corresponding original game problem P1 is:

[0137]

[0138] From a geometric perspective, the terminal co-state variables are aligned at any given time using a straight line. p,zsr (t f )r e,zsr (t f The terminals intersect and are parallel, so r p,zir (t f )r e,zir (t f )r p (t f ) = r e (t f 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 The optimal game time and optimal game strategy are obtained at the zero point of the game, thereby realizing the online solution of the spacecraft game.

[0144] To verify the algorithm's stability under different orbital periods (orbital altitudes), the orbital period range is T = [6000, 86400] seconds, and the spacecraft's absolute maneuverability and relative maneuverability ranges are: u max ∈[0.0001,1]m / s 2β∈[1.2,4]. All simulation results are based on Matlab and an Intel Core 13700K processor.

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

[0146] Table 3-1 Initial Parameters for Example 1

[0147]

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

[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 for Example 2

[0151]

[0152]

[0153] To verify the effectiveness and correctness of the proposed method, it is compared with current heuristic search-based (IHM) and nonlinear programming (NLP)-based methods. A comparison of the three methods is presented, among which the proposed Approximate Analytical Reachability Set (ARS) method has the highest computational efficiency, requiring only 2 milliseconds to obtain the optimal terminal time and optimal costate variables.

[0154] Table 3-3 Numerical Solution Results of Example 1

[0155]

[0156] The angles of the unilaterally reachable set and the bilateral optimal results are as follows: Figure 1 As shown in (a), (b), and (c), it can be observed that the terminal error of the approximate reachable set method is greater than that of the heuristic search-based method. This is mainly due to the approximation error of the terminal set.

[0157] To further analyze the impact of the results under a low reference orbit, the initial parameters of Example 2 were selected. The same procedure was followed to solve the above problem, and the results are shown below:

[0158] Table 3-4 Initial Parameters for Example 2

[0159]

[0160]

[0161] See Figure 2 Numerical simulations (a), (b), and (c) show that the proposed method improves solution efficiency by two orders of magnitude compared to traditional methods, meeting the requirements for real-time on-orbit solution. In the low-orbit scenario, due to the very short game termination time, the set approximation error is small, resulting in more accurate results.

[0162] This method employs a quantitative and reachable set approach to solve a two-player zero-sum pursuit-escape differential game with linearized relative motion dynamics constraints in real time. First, an equivalent one-sided minimum-time optimal control problem is derived using the separation properties of Hamiltonian functions and linear common-state equations. Then, a polynomial fitting based on least squares is used to approximate the analytical boundary of the reachable set with a dynamic ellipsoid. Even with different control inputs, only one offline calculation of coefficients is required beforehand. 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 milliseconds. The proposed geometry provides insights into the decision-making processes of both the pursuer and the escapee, which helps facilitate further research. The proposed method is suitable for in-orbit implementation due to its extremely low computational burden and reliable, predictable output, and it is unaffected by the convergence uncertainties present in numerical methods. The proposed method also has the potential for closed-loop applications.

[0163] Those skilled in the art will understand that the above description is merely a preferred embodiment of the present invention, and the features described in the various embodiments and / or claims of this disclosure can be combined or combined in various ways, even if such combinations or combinations are not explicitly described in this disclosure. This is not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

[0164] Although preferred embodiments of the invention have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including both the preferred embodiments and all changes and modifications falling within the scope of the invention. Clearly, those skilled in the art can make various alterations and modifications to the invention without departing from its spirit and scope. Thus, if these modifications and modifications of the invention fall within the scope of the claims and their equivalents, the invention is also intended to include these modifications and modifications.

Claims

1. A real-time solution method for spacecraft orbital pursuit and escape game, characterized in that, The method includes the following steps: Step 1: Solve the equivalent one-sided minimum time optimal control equation using the separation properties of the Hamiltonian function and the linear common-state equation; Step 2: Based on the one-sided minimum time optimal control equation described in Step 1, the transformation matrix is ​​obtained by fitting the polynomial using the least squares method. The analytical boundary is determined using a dynamic ellipsoid, with the terminal time being the boundary of the arrival set. Based on the terminal geometric conditions, the judgment equation is solved. The optimal game time and optimal game strategy are obtained at the zero point, that is, the online solution of the spacecraft pursuit and escape game is realized. The method for determining the minimum time on one side as described in step 1 is as follows: Where tf is the end time of the game, R is the set of positive real numbers, and rp and re are the position vectors of the tracking spacecraft and the escape spacecraft, respectively; The one-sided minimum time optimal control equation mentioned in step 1 is: in, H is a two-sided Hamiltonian function, u p To track the optimal control of the spacecraft, u e For optimal control of the escape spacecraft, To track the spacecraft's velocity costate variable, For the velocity covariance of the escaping spacecraft, For Lagrange multipliers; The method for determining the analytical boundary using a dynamic ellipsoid in step 2 is as follows: The principal axes of the dynamic ellipsoid are obtained by calculating the scaling factor. The calculation method is as follows: in, Let be the order of the polynomial. The coefficients of the polynomial are to be determined. In step 2, based on the one-sided minimum time optimal control equation described in step 1, the transformation matrix is ​​obtained by fitting the polynomial using the least squares method. The method is as follows: in, V is the eigenvector corresponding to the orthogonal decomposition; , In the formula, rzir and rzsr are the zero-input position vector and the zero-state position vector, respectively; Judgment equation for: 。 2. The real-time solution method for spacecraft orbital pursuit and escape game according to claim 1, characterized in that, Step 2 also includes the step of decomposing the boundary of the arrival set at the terminal time to obtain the control input response.

3. The real-time solution method for spacecraft orbital pursuit and escape game according to claim 2, characterized in that, The method for decomposing the control input response by the boundary of the arrival set at the terminal time is as follows: in, , These are the state transition matrices for the CW equations.

4. The real-time solution method for spacecraft orbital pursuit and escape game according to claim 3, characterized in that, The zero-state response Symmetric about the origin, i.e. , in , where n is the orbital angular velocity.

5. An electronic device, comprising: Memory, processor, and other components stored in the memory and available on the processor. A running computer program, characterized in that, when the processor executes the computer program, it implements the steps of the method as described in any one of claims 1 to 4.

6. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by a processor, it implements the steps of the method as described in any one of claims 1 to 4.

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