Spacecraft elliptical orbit real-time closed-loop pursuit-evasion game method and device

Abstract

The spacecraft elliptical orbit real-time closed-loop pursuit-evasion game method and device belong to the cross field of spacecraft orbit dynamics and intelligent game control, and particularly relate to a spacecraft elliptical orbit real-time closed-loop pursuit-evasion game method based on convex optimization theory. The method solves the problems of traditional orbit game control methods, such as difficulty in solving equilibrium, poor strategy convergence, and difficulty in on-orbit application. The method comprises the following steps: solving an equivalent discounted one-sided minimum time transfer problem to obtain an open-loop optimal strategy pair; recursively updating the state of the spacecraft according to the optimal control strategy, recursively updating the state of the evading spacecraft under unknown control, and updating the target point. The spacecraft elliptical orbit real-time closed-loop pursuit-evasion game method and device are suitable for real-time pursuit-evasion game of spacecrafts in elliptical orbits.

Description

Technical Field

[0001] This invention relates to the interdisciplinary field of spacecraft orbital dynamics and intelligent game control, and in particular to a real-time closed-loop pursuit-escape game method for spacecraft elliptical orbits based on convex optimization theory. Background Technology

[0002] In existing technologies, the spacecraft pursuit-escape game problem often employs a saddle point equilibrium solution method based on the maximum principle. The core of this method is to construct a Hamiltonian function and solve the boundary value problem of the accompanying differential equation, ultimately deriving the optimal control strategy for both sides. This type of method relies on a high-precision mathematical expression of the orbital dynamics model and requires joint iterative solutions to the state equations and co-state equations in the continuous time domain. However, this technical approach has the following inherent drawbacks:

[0003] (1) In the nonlinear dynamics scenario of elliptical orbits, the time-varying characteristics of the state equation and costate equation lead to increased difficulty in solving boundary value problems, and the computational complexity exceeds the real-time processing capability of the spacecraft's onboard computer.

[0004] (2) The numerical optimization methods such as heuristic search used in the existing improvement scheme can find the solution to the boundary value problem, but they are prone to problems such as poor policy convergence and sensitivity to initial value guessing. In particular, they are prone to getting trapped in local optima under multiple constraints of terminal time freedom.

[0005] (3) Existing methods lack dynamic environment perception and closed-loop strategy adjustment mechanisms. The solution process relies on offline pre-computation mode, which makes it difficult to cope with the real-time requirements of target maneuver strategy changes and orbital perturbation interference in space confrontation scenarios.

[0006] The aforementioned shortcomings result in computational delays of more than minutes when traditional game theory solutions are applied in orbit, and the stability of the strategies cannot meet the rigid requirement of millisecond-level response of spacecraft attitude and orbit control systems. Summary of the Invention

[0007] This invention proposes a real-time closed-loop pursuit-escape game method and device for spacecraft elliptical orbits, which solves the problems of difficulty in equilibrium solution, poor strategy convergence, and difficulty in on-orbit application of traditional orbit game control methods.

[0008] The real-time closed-loop pursuit and escape game method for spacecraft in elliptical orbits described in this invention includes the following steps:

[0009] Step S1: Given initial conditions x p (θ0),x e (θ0) and control constraints

[0010] Where: θ0 represents the spacecraft's initial true anomaly angle; x p(θ0) represents the initial state of the tracking spacecraft; x e (θ0) represents the initial state of the escaped spacecraft; This represents the set of feasible control inputs for tracking spacecraft; This represents the set of feasible control inputs for an escape spacecraft;

[0011] Step S2: Iterate to obtain the closed-loop control strategy; wherein, the iterative process in stage s includes the following steps:

[0012] Step S2.1: At θ s The complete states (x) of the two spacecraft in the pursuit-escape game are input at the corresponding time. p (s),x e (s)), where θ s The true anterior angle corresponding to phase s;

[0013] Where: x p (s) indicates the state of the tracked spacecraft during phase s; x e (s) represents the state of the escaped spacecraft in phase s;

[0014] Step S2.2: Solve the equivalent discount one-sided minimum time transition problem to obtain the open-loop optimal strategy pair. and

[0015] in: This indicates the optimal control strategy for tracking spacecraft; This represents the optimal control strategy for an escape spacecraft. The terminal true nearest angle corresponds to the solution of the game problem in stage s.

[0016] Step S2.3: According to the optimal control strategy recursion Recursive unknown control u e The escape spacecraft X e (θ s Update target point

[0017] in: This indicates that the spacecraft's state is tracked according to the optimal control strategy during phase s; x e (θ s () represents the state of the spacecraft escaping according to the optimal control strategy during phase s; This represents the relative state target point corresponding to the terminal true nearest angle in solving a game problem;

[0018] Step S2.3: Determine if ||φ||≤ε is satisfied:

[0019] If so, the iteration ends, the final closed-loop control game strategy is obtained, the target is intercepted, and the game ends.

[0020] Otherwise, obtain θ s+1 Continue the iteration in phase s+1:

[0021] θ s+1 =θ s +Δθ;

[0022] Where: φ represents the terminal constraint condition; ε represents the error threshold termination condition; Δθ represents the difference in true perimeter angle between two adjacent stages.

[0023] Furthermore, a preferred embodiment is provided, wherein the equivalent discount one-sided minimum time transition problem is:

[0024] Based on the control inputs and current states of the tracking and escaping spacecraft, solve the minimum time transition problem:

[0025]

[0026] x pe (θ s )=x p (θ s )-x e (θ s )

[0027] x pe (θ f ) = 0

[0028] Where, x pe The relative state difference between the tracking spacecraft and the escaping spacecraft; u pe Input for equivalent discount control; The maximum value of the equivalent control input set; θ f The optimal true anterior angle; x p (θ s () represents the initial state of the spacecraft during phase s; x e (θ s ) represents the initial state of the escape spacecraft during phase s; x pe (θ f ) represents the terminal state constraint due to the difference in relative motion states; x pe (θ s ) represents the initial state constraint for the difference in relative motion state.

[0029] Furthermore, in a preferred embodiment, step S2.2 is as follows:

[0030] Repeatedly solve for different given terminal times θ f Solving the feasibility problem:

[0031]

[0032] x pe (θ s )=x p (θ s )-x e (θ s )

[0033] Wherein, φ(θ) f Given the terminal error at the time corresponding to the true anterior angle; u pe Equivalent discount control input; for The boundary of.

[0034] This invention also proposes a real-time closed-loop pursuit and escape game device for spacecraft in elliptical orbits, the device comprising the following modules:

[0035] Module S1: Given initial conditions x p (θ0), x e (θ0) and control constraints

[0036] Where: θ0 represents the spacecraft's initial true anomaly angle; x p (θ0) represents the initial state of the tracking spacecraft; x e (θ0) represents the initial state of the escaped spacecraft; This represents the set of feasible control inputs for tracking spacecraft; This represents the set of feasible control inputs for an escape spacecraft;

[0037] Module S2: Iteratively obtains the closed-loop control strategy; the iterative process in stage s includes the following sub-modules:

[0038] Submodule S2.1: at θ s The complete states (x) of the two spacecraft in the pursuit-escape game are input at the corresponding time. p (s),x e (s)), where θ s The true anterior angle corresponding to phase s;

[0039] Where: x p (s) indicates the state of the tracked spacecraft during phase s; x e (s) represents the state of the escaped spacecraft in phase s;

[0040] Submodule S2.2: Solve the equivalent discounted one-sided minimum time transition problem, and obtain the open-loop optimal strategy pair. and

[0041] in: This indicates the optimal control strategy for tracking spacecraft; This represents the optimal control strategy for an escape spacecraft. The terminal true nearest angle corresponds to the solution of the game problem in stage s.

[0042] Submodule S2.3: According to the optimal control strategy recursion Recursive unknown control u e The escape spacecraft X e (θ s Update target point

[0043] in: This indicates that the spacecraft's state is tracked according to the optimal control strategy during phase s; x e (θ s () represents the state of the spacecraft escaping according to the optimal control strategy during phase s; This represents the relative state target point corresponding to the terminal true nearest angle in solving a game problem;

[0044] Submodule S2.3: Determine if ||φ||≤ε is satisfied:

[0045] If so, the iteration ends, the final closed-loop control game strategy is obtained, the target is intercepted, and the game ends.

[0046] Otherwise, obtain θ s+1 Continue the iteration in phase s+1:

[0047] θ s+1 =θ s +Δθ;

[0048] Where: φ represents the terminal constraint condition; ε represents the error threshold termination condition; Δθ represents the difference in true perimeter angle between two adjacent stages.

[0049] The present invention also proposes a computer device comprising: a processor and a memory, the memory being used to store executable instructions of the processor, the processor being configured to perform the real-time closed-loop pursuit and escape game of spacecraft elliptical orbit as described above by executing the executable instructions.

[0050] The present invention also proposes a computer storage medium storing a computer program, wherein when the computer program is executed, it performs the real-time closed-loop pursuit and escape game of the spacecraft elliptical orbit described in any of the above-mentioned embodiments.

[0051] The present invention also proposes a computer program product, including a computer program / instruction, which, when executed by a processor, implements the steps of the real-time closed-loop pursuit and escape game in the elliptical orbit of a spacecraft as described in any of the above-mentioned embodiments.

[0052] The present invention has the following beneficial effects:

[0053] 1. The real-time closed-loop pursuit-escape game method for spacecraft elliptical orbits described in this invention does not require manual parameter setting and can solve for the optimal closed-loop strategy, making the closed-loop strategy based on game theory-guided optimization feasible. This is crucial for the actual on-orbit application of spacecraft.

[0054] 2. The spacecraft elliptical orbit real-time closed-loop pursuit-escape game method described in this invention does not require hyperparameter adjustment, its output is deterministic, and convergence is guaranteed, making it suitable for airborne applications, especially on resource-constrained spaceborne embedded platforms.

[0055] 3. The real-time closed-loop pursuit-escape game theory method for spacecraft in elliptical orbits described in this invention is directly applicable to autonomous pursuit-escape confrontation missions of spacecraft in elliptical orbit environments, including scenarios such as space target interception, non-cooperative target avoidance, and multi-spacecraft coordinated encirclement. Its application areas cover aerospace offensive and defensive confrontation, on-orbit service safety control, and active space debris avoidance. Through high-speed strategy solving and real-time iteration of game strategies, it solves the problems of traditional orbital game control methods, such as difficulty in equilibrium solution, poor strategy convergence, and difficulty in on-orbit application, providing millisecond-level autonomous decision-making capabilities for complex space confrontation missions.

[0056] The real-time closed-loop pursuit and escape game method and device for spacecraft elliptical orbits described in this invention are applicable to real-time pursuit and escape games in spacecraft elliptical orbits. Attached Figure Description

[0057] To more clearly illustrate the technical solutions in the embodiments of the present invention, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0058] Figure 1 This is a rolling time-domain schematic diagram of a closed-loop pursuit-escape game in one embodiment of the present invention.

[0059] Figure 2 This is a schematic diagram illustrating the equivalence transformation condition between bilateral and unilateral problems in one embodiment of the present invention.

[0060] Figure 3 This is a schematic diagram illustrating the feasibility determination of a convex optimization subproblem in one embodiment of the present invention;

[0061] Figure 4 This is a schematic diagram comparing the closed-loop terminal time using a trivial strategy and a expected strategy in one embodiment of the present invention.

[0062] Figure 5 This is a schematic diagram of time consistency analysis in one embodiment of the present invention;

[0063] Figure 6 This is a schematic diagram of unilateral and bilateral trajectories for a short-term interception problem in one embodiment of the present invention; wherein, (a) is a unilateral trajectory; and (b) is a bilateral trajectory.

[0064] Figure 7 This is a schematic diagram of the optimal control input for the short-term interception problem in one embodiment of the present invention; wherein, (a) is trapezoidal discretization; and (b) is pseudospectral discretization.

[0065] Figure 8 This is a schematic diagram of the optimal trajectory for a short-run intersection problem in one embodiment of the present invention; wherein, (a) is a one-sided transition result; and (b) is a two-sided intersection result.

[0066] Figure 9 This is a schematic diagram of the optimality verification of a closed-loop game in one embodiment of the present invention; wherein, (a) is a closed-loop interception game; and (b) is a closed-loop intersection game. Detailed Implementation

[0067] To make the technical solutions and advantages of the present invention clearer, the specific embodiments of the present invention will be described in further detail and completely below with reference to the accompanying drawings. The various embodiments described below are only some preferred embodiments of the present invention, and not all of them; the various embodiments described below are intended to explain the present invention and should not be construed as limiting the present invention; reasonable combinations of the technical features defined in the various embodiments of the present invention, as well as all other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort, are all within the scope of protection of the present invention.

[0068] A brief description of the existing technology:

[0069] Existing research on spacecraft pursuit and escape problems typically involves enormous computational demands, often relying on heuristic search methods or nonlinear programming techniques to derive open-loop strategies. These methods hinder their on-orbit implementation, especially on platforms with limited computational resources. This invention proposes a real-time closed-loop method to solve the spacecraft pursuit and escape problem in two game-theoretic scenarios with time-varying thrust profiles in elliptical orbits: interception and rendezvous / docking. This method is based on convex optimization, and the specific steps are summarized as follows:

[0070] 1. Problem Transformation: The bilateral optimal control problem is transformed into an equivalent discounted minimum time transition problem.

[0071] 2. Determine the optimal terminal time: Determine the optimal terminal time by iteratively solving a series of convex subproblems using the bisection method.

[0072] 3. Extend to a rolling time-domain pursuit scheme: Extend the proposed open-loop strategy to a rolling time-domain pursuit scheme and ensure its success.

[0073] Numerical results demonstrate that the proposed method can solve the chase-escape game problem in real time within approximately 100 milliseconds, significantly reducing computation time compared to existing methods and eliminating the need for manual parameter tuning. Furthermore, Monte Carlo simulations show that the proposed closed-loop strategy is robust against escapees with unknown actions, providing empirical evidence to guarantee performance.

[0074] When spacecraft perform close-range maneuvers (such as interception missions) around non-cooperative targets, it is reasonable to assume that the target will take action to mitigate the potential threat or attempt to escape (if the target is rational and well-informed). However, even if players employ an optimal strategy derived from static trajectory optimization, there is still a risk of misjudging the opponent's actions. This is because the agent's responses can invalidate the initial assumptions made before the game began. Therefore, the traditional one-sided trajectory optimization problem evolves into a more complex problem involving two agents making dynamic optimal decisions.

[0075] Game theory is used to model and analyze the dynamic interactions between multiple players and the process of making optimal decisions. When the evolution of player states is controlled by differential equations, the game is called a differential game.

[0076] Optimal interception or close-range guidance of a spacecraft against a potentially maneuvering target can be formulated as a chase-escape game, a variant of dynamic non-cooperative differential games. In this game, the pursuer aims to capture the escapee, while the escapee attempts to avoid capture. A fundamental assumption is that the pursuer's maneuverability exceeds that of the escapee, ensuring the game ends within an unknown but finite timeframe. Therefore, the pursuer's goal is to minimize the terminal time, while the escapee attempts to maximize it. This situation constitutes a zero-sum game, which is of significant importance in the aerospace field.

[0077] The main challenge of pursuit-escape game theory lies in developing an efficient and reliable method to find saddle point strategies, especially in real-time applications in practical scenarios. Currently, methods for solving pursuit-escape game theory problems are mainly divided into indirect methods, semi-direct nonlinear programming methods, and reachability-based methods.

[0078] 1. Existing methods and their limitations:

[0079] Indirect methods, based on the Pontryagin Minimum Principle (PMP), are the main approach in existing literature for solving spacecraft chase-escape games. However, they face significant difficulties in solving the resulting boundary value problems, especially when reliable convergence on onboard equipment is required. The utility of open-loop PMP strategies is limited due to disturbances, uncertainties, and unpredictable opponent responses. These factors necessitate closed-loop strategies, which require efficient repetitive computation guidance. Chase-escape games lead to high-dimensional boundary value problems, which are particularly challenging even for problems involving linear dynamics under limited resources.

[0080] Reachability-based methods: Terminal times are determined based on the forward reachability set inclusion criterion, providing intuitive insights. Those skilled in the art have used reachability to study pursuit-escape games with impulsive maneuvers, and have also used the Hamilton-Jacobi-Isax (HJI) reachability method to solve continuous games with multiple pursuers and escapees. While the HJI method provides a comprehensive framework for obtaining closed-loop solutions, it is far from real-time, especially in time-varying scenarios. It is primarily used for simple motions; typically, only planar kinematics are used. Analytical solutions and closed-loop strategies can be obtained. However, for spacecraft pursuit-escape games, even for linear relative dynamics, explicit analytical solutions are difficult to obtain. Furthermore, reachability-based methods are computationally expensive due to the need to traverse approximate reachable surfaces. Additionally, due to the difficulty in handling exact reachability sets, terminal time errors are large, and optimal control strategies are often suboptimal because they are typically derived from grids.

[0081] 2. Advantages of the method of the present invention:

[0082] Real-time guidance and trajectory optimization are crucial for online pursuit-escape game theory. To the best of the art, there is currently no method for real-time computation of solutions to pursuit-escape differential games. However, with advancements in hardware computing power, this invention addresses the problem of real-time computation of pursuit-escape differential games based on convex optimization. The main advantage of convex optimization lies in its ability to provide a globally optimal solution and its fast convergence speed, which is essential for autonomous decision-making. Using state-of-the-art interior-point methods, a customized convex optimization algorithm can be implemented on embedded systems.

[0083] Spacecraft pursuit and escape is a typical minimum-time problem, while the timeless minimum problem is a typical nonlinear programming problem. The optimal time is usually determined by the Probabilistic Minimum Problem (PMP) and depends on the initial costate. Approximate estimations are typically made to guess the optimal time, which is then used to solve for the exact optimal value. However, generating reasonable initial costate guesses and obtaining convergent solutions is difficult. Sequential convex optimization is also a well-established method for solving free-time problems, treating free time as an optimization variable to handle unknown time. However, for linear free-time problems, the optimal time can be obtained losslessly by solving a series of convex (optimization) subproblems; this invention employs this method.

[0084] 3. The main contributions of the method of this invention can be summarized as follows:

[0085] (1) Propose an equivalent discount minimum time (transition) problem: A completely equivalent discount minimum time problem is proposed, whose optimal solution is consistent with the original bilateral chase game in almost all places. This is applicable to interception and rendezvous docking games with time-varying thrust.

[0086] (2) Bisection Method for Determining Minimum Feasible Time: A bisection search method is adopted, using the minimum transition convex subproblem as the feasibility criterion for game termination, to determine the minimum feasible time within a finite number of iterations. The proposed rolling time-domain closed-loop strategy has been proven to handle the control of unknown escapees. Numerical empirical evidence shows that the proposed method guarantees performance compared to open-loop equilibrium solutions.

[0087] (3) Comprehensive numerical verification and comparison: The effectiveness and reliability of the proposed method are demonstrated. This method avoids obtaining initial guesses and solves the unreliable convergence problem encountered in long-term games. It utilizes an efficient convex optimization algorithm, making it very suitable for real-time applications in chase-and-escape games.

[0088] In one embodiment, the basic concepts of the chase differential game and the equivalent formula for the discount minimum time transition problem are explained:

[0089] 1. On the bilateral game of fugitive repatriation:

[0090] For a chase-escape game played on an elliptical reference orbit, the relative dynamics are described by the following Tschauner-Hempel(TH) equations, where the true anomaly angle θ is the independent variable:

[0091]

[0092] In the formula:

[0093]

[0094] in:

[0095] This represents the position (r) of player (or participant, spacecraft) i in the LVLH (Local Vertical and Local Horizontal) coordinate system. i ) and velocity (v) i ), where the subscripts i∈{p, e} represent the pursuer (p, or the pursuing spacecraft) and the escapee (e, or the escape spacecraft).

[0096] Formula (1) represents a linear time-varying system, which is also a control affine (dynamic) system.

[0097] u i ρ is the unit control vector; ρ is an introduced intermediate variable, ρ = 1 / (e + cosθ), where e is the reference orbital eccentricity; denoted as ω, μ is the Earth's gravitational constant, and h is the orbital angular momentum.

[0098] The player's initial state at θ0 is given by the following formula:

[0099] x i (θ0)=x i，0 (2)

[0100] Without loss of generality, we assume that the mass of the spacecraft changes during the maneuver.

[0101] Without loss of generality, assume that the spacecraft has a changing mass during maneuvering. Introduce a scalar function. This function represents the time-varying thrust profile of the game player. For a specific fixed thrust amplitude, this function degenerates into a constant function. Specifically, it is defined as follows:

[0102]

[0103] in:

[0104] u i，max c represents the initial maximum control speed of participant i; i Its effective exhaust velocity; t represents time;

[0105] The alternative control quantity for participant i consists of all feasible control quantities, denoted as:

[0106]

[0107] in, The optimal terminal true nearest angle at the end of the game.

[0108] Generally, the pursuit and escape problem is modeled as a zero-sum game, where the pursuer's goal is to minimize the terminal time, while the escapee's goal is to maximize the terminal time. The corresponding payoff function is the time θ that satisfies the following terminal condition. f :

[0109] φ(x p ,x e )=0 (5)

[0110] Specifically, it is given by the following formula:

[0111] J(u p ,u e )=θ f (6)

[0112] Consider two types of pursuit games: interception game and rendezvous and docking game. The terminal conditions are expressed as follows:

[0113] Interception Game Theory:

[0114] Meeting and docking game:

[0115] The optimal strategy pair in a game Described by Nash equilibrium, also known as a saddle point in zero-sum games, it satisfies the following inequality:

[0116]

[0117] The core advantage of Nash equilibrium lies in the fact that no participant can improve their own payoff by unilaterally changing their strategy. The information structure of the chase game is assumed to be complete information, meaning that both the pursuer and the fleeing player are fully aware of all relevant information, including the strategy set.

[0118] Therefore, the zero-sum pursuit game problem can be formulated as an optimization problem.

[0119]

[0120] Where: st(1), (2), (4), (5), (6), (7) represent the constraints of the above formulas (1), (2), (4), (5), (6), (7).

[0121] The necessary condition for a bilateral optimal control solution is derived from Pontryagin's minimum principle. The bilateral Hamiltonian function is constructed as follows:

[0122] H = H p +H e =λ p ·(Ax p +Bu p g p )+λ e ·(Ax e +Bu e g e ) (9)

[0124] in, It is the costate variable of player i.

[0125] Note that the Hamiltonian function is separable, so the optimal control for both the pursuer and the escapee is given by the following equation:

[0126]

[0127] The specific expression is:

[0128]

[0129] The costate variables are solved using the following equations:

[0130]

[0131] The terminal values ​​of the costate variables are determined by the Lagrange multipliers τ:

[0132]

[0133] Intersection Game With full state constraints Interception Game Only position constraints And the terminal costate variable λ i The zero component needs to be extended, i.e., λ. v,i =0. The transverse condition of the terminal Hamiltonian function is:

[0134] H(θ f )+1=0 (13)

[0135] Obviously λ p (θ f )+λ e (θ f Since ) = 0, and considering the linear costate equation, the optimal control for the pursuer and the escapee is almost the same everywhere, except that the interception game exhibits control singularity at the terminal time.

[0136] Therefore, the derived boundary value problem Formulated as:

[0137]

[0138] That is, to find the parameters τ and θ. f This makes formulas (12) and (13) valid.

[0139] Clearly, the rendezvous and docking game is more challenging than the interception problem because the terminal constraints are more complex.

[0140] 2. Regarding the issue of minimum time transfer for discounts:

[0141] To reduce complexity, the control affine dynamics of equation (1) are expressed in a simplified state space:

[0142]

[0143] Where x pe =xp -x e It is the state difference. Based on the necessary condition, the optimal control direction of the two players is the same. It is assumed that the escapee's control input is always the same as the pursuer's, thus effectively making the escapee a mimicking agent. This assumption is a relaxation of the original problem, and it will be equivalent when the optimal solution is taken. Therefore, formula (15) is reformulated as:

[0144]

[0145] Here g pe =g p -g e It is the difference in control amplitude, u pe Indicates the direction of thrust, and is related to u. p and u e Same. Discount feasibility control. for:

[0146]

[0147] Then, the initial state for order reduction is:

[0148] x pe (θ0)=x pe,0 (18)

[0149] The corresponding terminal conditions for the interception game and the rendezvous and docking game are as follows:

[0150]

[0151] Therefore, an equivalent (equivalent) discount minimum time (transition) problem is derived, denoted as . This problem involves changing the system from its initial state x. pe (θ0) is transferred to the origin 0, that is:

[0152]

[0153] Similar to the bilateral Pontryagin minimum principle (PMP), The Hamiltonian function is defined as:

[0154]

[0155] And minimized:

[0156]

[0157] The costate variables satisfy:

[0158]

[0159] and

[0160]

[0161] Reduce the terminal Hamiltonian constraint, and the cross-sectional condition must be satisfied:

[0162] H pe (θ f )+1=0 (24)

[0163] Where: τ pe For the unknown Lagrange multipliers in the original transition problem; for intercept games, we have For rendezvous games, there are

[0164] Will The discount boundary value problem is expressed as

[0165]

[0166] That is, finding the parameter τ pe ,θ f This makes formulas (23) and (24) valid.

[0167] The original bilateral optimal control problem has the same optimal solution as the discounted one-sided minimum time transition problem, and the two are equivalent in almost all places.

[0168] Analyze the sufficient conditions. Since the costate equations and optimal control of both bilateral and discounted unilateral problems have the same construction, the corresponding optimal control can be determined by assigning terminal values ​​to the costate.

[0169] Let τ = τ pe Then λ p =λ pe , λ e =-λ pe =-λ p Therefore, formula (24) is the same as the bilateral Hamiltonian function (13). Then, bilateral optimal control can be constructed from the discounted optimal solution:

[0170]

[0171] This is the same as bilateral optimal control (10). And from The solution satisfies the problem All conditions. Through the same operation, it is also possible to... Replace τ with To obtain the necessary conditions.

[0172] For a chase-escape game problem with a complete information structure, there is no difference between the original two-sided optimal control problem and the discounted one-sided minimum time transition problem. The transformation roadmap is as follows: Figure 2 As shown. This equivalence can be explained from two perspectives:

[0173] First, the game (or strategy) depends only on the differences in the initial state and control inputs. Therefore, due to linear terminal conditions and linear common states, the player's optimal control is almost identical everywhere when adopting a saddle point strategy.

[0174] Secondly, with a complete information structure, the hunter can eliminate the existence of control over the escaping spacecraft (or escapee) in advance. The escapee's control effect acts as a discount factor applied to the hunter's control amplitude. This property benefits from separable Hamiltonian functions, linear dynamics, and linear terminal conditions (applicable to interception and rendezvous games).

[0175] In one implementation, a convex optimization-based method for solving the minimum time problem in interception and rendezvous (docking) games is proposed, and its convergence is analyzed:

[0176] 1. Regarding the feasibility subproblem of minimum-time game based on convex optimization:

[0177] Minimum time transition problem A lossless solution can be achieved through a series of feasible subproblems, employing a direct method rather than the indirect method of BMP. For optimization problems with free terminal time and linear dynamics, assuming (A, B) is controllable, linear search methods (such as the golden section method or bisection method) can be used to determine the optimal terminal time.

[0178] Minimum Discount Time Transfer Problem A series of fixed-time feasible convex subproblems can be solved losslessly, considering the minimum final error problem. And given θ f The terminal constraint φ is then formalized as a cost formula:

[0179]

[0180] Regarding the question If ||φ||>0, then the subproblem is infeasible, unless Furthermore, for infeasible subproblems, the control amplitude lies within the feasible set. The boundary. The entire interval is divided into two parts: the capture region and the escape region, corresponding to feasible subproblems and infeasible solutions, respectively, such as... Figure 3As shown. In fact, the feasibility of the subproblems provides a qualitative criterion for determining the outcome of the game, win or lose. Given θ... f If a subproblem is feasible, the pursuer wins; otherwise, from the pursuer's perspective, the game is lost. If a subproblem is infeasible, the escapee wins. This is based on the maximum maneuverability assumption g. e ≤g p The game guarantees that within a limited time θ f The problem ends within ≤∞. Therefore, the problem... The optimal value is equivalent to the minimum feasible solution.

[0181] We discuss numerical solutions to the convex subproblem. For simplicity, the state variable x is omitted without loss of generality. pe Control quantity u pe and constraints g pe The subscript symbols are used. Two discretization methods are proposed: the trapezoidal discretization method and the pseudospectral discretization method, for solving subproblems caused by time-varying dynamics.

[0182] Given a fixed terminal time θ f , for the interval [θ0, θ f The control and state variables within the [node name] are discretized into N nodes:

[0183] [u0;…;u N-1 ] and [x0; ...; x N-1 ].

[0184] For each discrete time θ k ∈[θ0, θ f ], k = 0, 1, ..., N-1, the trapezoidal discretization of formula (16) constitutes a local approximate expression:

[0185]

[0186] The step size is defined as follows:

[0187] Δθ=(θ f -θ0) / (N-1).

[0188] Pseudospectral discretization is a global approximation method that exhibits exponential convergence when the solution is smooth. A time-affine transformation is required to adapt to the Chebyshev-Gauss-Lobart (CGL) collocation interval.

[0189]

[0190] Where: parameter l∈[-1,1].

[0191]

[0192] Where: the differential matrix Dki Defined by CGL methods.

[0193] Finally, the feasible subproblems are stacked into a second-order cone programming problem (SOCP), with the following formula:

[0194]

[0195] Among them: decision variables Matrix M is composed of a sparse coefficient matrix.

[0196] The binary search method is used to find the minimum feasible time, which is reliable and has the advantage of output prediction. Assume that within a given search interval [θ]... f，min θ f，max There exists an optimal equilibrium solution in the equation, and given the tolerance ∈, the maximum number of iterations can be predetermined, and is determined by... Definition: If an optimal terminal time exists within a given interval, the method guarantees convergence. If the optimal final time output is θ... f，max If the problem is not feasible, then the problem is not feasible; in other words, the pursuers cannot capture the escapee within the given search area.

[0197] because This is a convex optimization problem, which guarantees a lossless solution at arbitrary precision without considering numerical errors. Assuming the maneuverability g... p >g e When the value is greater than 0, it can be assumed that there is always a saddle point. Furthermore, by appropriately selecting the search interval for the bisection method, the algorithm guarantees termination within a finite number of iterations, thus ensuring the attainment of the optimal equilibrium solution.

[0198] 2. Regarding the rolling time-domain closed-loop minimum time escape strategy:

[0199] The saddle point strategy obtained based on the Minimax Principle (PMP) is an open-loop control, with limited practical applicability. In reality, the escapee may take any feasible but irrational action, even if it's not the optimal strategy. If the pursuer continues to follow the initially obtained "optimal strategy," the desired outcome cannot be achieved. Furthermore, considering the uncertainty of irrational escapees, a closed-loop strategy must be implemented. However, a globally closed-loop pursuit strategy u describing global closed-loop equilibrium is not feasible. * (x pe Described by HJI, it provides closed-loop optimal equilibrium solutions for all states, but it faces implementation obstacles: the grid-based partial differential equation solution is only applicable to problems below four dimensions, which is both time-consuming and impractical for on-orbit applications.

[0200] To the knowledge of those skilled in the art, there is currently no existing method that can analytically obtain the closed-loop equilibrium of a spacecraft pursuit-escape game. Constructing a closed-loop strategy... This is unavoidable, especially in terms of real-time applicability. This is the first time the game's requirements for a closed-loop pursuit strategy have been proposed. First, the closed-loop strategy should ensure the capture of the escapee under a given assumption of maneuverability. This condition is satisfied because the state space is unconstrained in the spacecraft pursuit game. Second, the closed-loop capture time... It should approximate the global closed-loop equilibrium u derived from HJI theory. * (x pe The time when θ) is generated And less than u from the open-loop equilibrium strategy * (θ) Optimal time like Figure 4 As shown. Conversely, even a poor, trivial strategy can achieve capture, but the returns will be worse than an ideal equilibrium strategy.

[0201] Given initial parameters or game settings, repeatedly solve quantitative problems. Constructing a closed-loop update decision is a reasonable and feasible solution. Assume the player's exact state is known. When the game starts, the optimal terminal time is determined by the initial state and the control profile: Corresponding to open-loop policy pairs:

[0202]

[0203] However, considering that player i makes a reaction decision and observes the states of other players, the minimum observation time delay ΔT is taken into account. Following the rolling time-domain strategy:

[0204]

[0205] in:

[0206] It is the value of the game; u i (s) is the strategy of player i in phase s;

[0207] The values ​​of the game sequence {V(0),...,V(s-1),V(s),V(s+1),...,V} cl} corresponds to

[0208] use This indicates the terminal time of the closed-loop strategy.

[0209] Next, we analyze the properties of the closed-loop strategy. Generally, if it exists, the global closed-loop equilibrium terminal time... It represents the best performance that can be expected.

[0210] like Figure 5 As shown, where x p (s), x e(s) is from x p (s-1), x e (s-1) and equilibrium The equilibrium state achieved.

[0211] x′ e (s) represents the updated state of player e, where u′ e (s-1) is unknown. If the open-loop policy... If strong time consistency is satisfied, then the following inequality holds:

[0212] V(s-1)=V(x p (s-1),x e (s-1))≥V(x p (s),x e (s))≥V(x p (s),x′ e (s))=V(s)

[0213] Then V(s-1)≥V(s) satisfies the condition that the sequence is monotonically decreasing, and:

[0214]

[0215] Unfortunately, the open-loop equilibrium strategy is not strongly time-consistent and does not satisfy the subgame equilibrium property. Therefore, the decreasing property of the V(s) sequence cannot be guaranteed. Since the open-loop equilibrium strategy is only weakly time-consistent, the results show that it only maintains equilibrium for states along the equilibrium path. Note that the rolling time-domain method is not an equilibrium strategy, but a normal solution. However, subsequent numerical simulations demonstrate that the proposed rolling time-domain method behaves like a time-consistent strategy and guarantees performance in both short-time and long-time games.

[0216] Figure 1 This shows a rolling time-domain diagram of a closed-loop chase-escape game. Since the opponent's strategy is unknown, the only available information includes the observation time t. s state x -i and control profile g -i θ, where -i represents player i's opponent. θ is the value of player i at any stage s. s At this point, player i poses a one-sided discount problem. Solving the game of fugitive pursuit At the same time, a pair of optimal strategies are derived. And one-sided The process is propagated. After a minimum ΔT, a refresh decision is made using the updated state; this process is repeated until the terminal tolerance ||φ|| is satisfied.

[0217] In one embodiment, a numerical simulation comparison experiment is provided. The method described in this invention is compared with existing indirect methods. Based on the comparison results, weak time consistency is verified, and Monte Carlo simulations demonstrate that the proposed rolling time-domain closed-loop strategy can guarantee victory.

[0218] The existing indirect methods used for comparison are described in the following literature: "Study on the optimal tracking / escape trajectory of SpaceX spacecraft in the Hill reference frame", "Study on differential game theory of close-range operation in elliptical orbits", and "Solving the tracking-escape game in elliptical orbits by precise gradient".

[0219] To verify the effectiveness of the proposed method, it will be compared with existing numerical methods (Indirect Heuristic Method IHM and Direct Pseudospectral Method PSM). The Indirect Heuristic Method (IHM) is used to solve the two-sided boundary value problem. The Direct Pseudospectral Method (PSM) is used to solve the one-sided optimal problem.

[0220] Both interception games and intersection games are considered using two discretization methods.

[0221] The initial parameters of the GEO reference orbit are as follows:

[0222] The escape spacecraft is located at the origin of the coordinate system;

[0223] The orbital period is T = 86400s, u p,max =0.0686m / s 2 ,u e,max =0.0343m / s 2 .

[0224] Table 1 Initial state parameters of the spacecraft

[0225] <![CDATA[r x (t0)m]]> <![CDATA[r y (t0)m]]> <![CDATA[r z (t0)m]]> <![CDATA[v x (t0)m / s]]> <![CDATA[v y (t0)m / s]]> <![CDATA[v z (t0)m / s]]> <![CDATA[x p (t0)]]> <![CDATA[-3.89328×10 4 ]]> <![CDATA[-1×10 5 ]]> <![CDATA[1×10 4 ]]> 0 0 0 <![CDATA[x e (t0)]]> 0 0 0 0 0 0

[0226] To discretize the minimum time transition problem, pseudospectral discretization and the trapezoidal method are used to discretize it at 50 collocation points.

[0227] The indirect heuristic is implemented using the particle swarm optimization method, with a population size of 80 to ensure comparable computation time.

[0228] The minimum-time nonlinear programming problem is solved by GPOPS-II.

[0229] The average results shown in Table 2 were obtained through 100 simulations. It can be observed that the three methods yield consistent results, but the convex optimization-based method has the highest computational efficiency.

[0230] Optimal trajectory and control are respectively in Figure 6 and Figure 7The proposed LCVX method is significantly faster than both indirect heuristics and nonlinear programming methods. The results obtained using pseudospectral discretization and trapezoidal discretization are consistent.

[0231] Table 2 Comparison of Short-Term Interception Results

[0232]

[0233] Compared to interception game problems, intersection game problems are computationally more difficult due to the increased terminal constraints. To verify the effectiveness of the proposed method on intersection game problems, validation was performed on both short-run and long-run intersection game problems.

[0234] The intersection game problem contains seven free parameters, almost twice the four parameters of the interception game. This increase in parameters makes it more difficult for indirect heuristics to find the optimal solution, significantly increasing runtime. In contrast, methods based on (lossless) convex optimization only add three additional equality constraints compared to the interception game problem. Therefore, the computational complexity of this problem remains nearly the same and the performance is consistent regardless of the endpoint.

[0235] The results for the same short-run intersection problem are shown in Table 3. It can be seen that in the intersection game, the IHM method using PSO failed to find the optimal solution even with manual parameter adjustment.

[0236] Figure 8 The paper depicts the one-sided trajectories (one-sided transition results) and two-sided optimal trajectories (two-sided intersection results) for short-term and long-term games. For long-term games, the relative motion is more complex compared to short-term games. Both trapezoidal and pseudospectral discretization produce the same solution, but trapezoidal discretization is more efficient. Compared to the interception problem, the game-solving method based on convex optimization does not significantly increase the time required.

[0237] Table 3 Results of the Short-Term Intersection Problem

[0238]

[0239] Closed-loop game verification:

[0240] In a chase game, both sides adopt optimal control strategies, and then the game progresses in a rolling fashion until the end, with the following result: Figure 9 As shown, the results of closed-loop game theory indicate that even if the opponent adopts the optimal strategy, the outcome is consistent with the prediction of the open-loop strategy. In other words, the outcome of the closed-loop strategy is no worse than that of the open-loop strategy, making it a risk-free strategy. Regardless of the opponent's strategy, they cannot escape being intercepted or captured.

[0241] In summary, this invention proposes a novel method for solving the chase-escape differential game. Under the assumptions of complete information and three conventional conditions, the bilateral optimal control problem corresponding to the chase-escape game is transformed into an equivalent discounted minimum time transformation problem. This transformation effectively offsets the influence of the opponent, treating the control input of the other player (or game participant) as a completely known disturbance that can be eliminated in advance. Leveraging the computational advantages of convex optimization, a binary search method for optimal capture time is proposed. This method is implemented in real-time using a state-of-the-art interior-point method, enabling the game to be solved efficiently and reliably within approximately 100 milliseconds.

[0242] In one embodiment, based on the above embodiments, the real-time closed-loop pursuit and escape game method for spacecraft elliptical orbits is summarized as follows:

[0243] A real-time closed-loop pursuit and escape game theory method for spacecraft in elliptical orbits, the method comprising the following steps:

[0244] Step S1: Given initial conditions x p (θ0), x e (θ0) and control constraints

[0245] Where: θ0 represents the spacecraft's initial true anomaly angle; x p (θ0) represents the initial state of the tracking spacecraft; x e (θ0) represents the initial state of the escaped spacecraft; This represents the set of feasible control inputs for tracking spacecraft; This represents the set of feasible control inputs for an escape spacecraft;

[0246] Step S2: Iterate to obtain the closed-loop control strategy; wherein, the iterative process in stage s includes the following steps:

[0247] Step S2.1: At θ s The complete states (x) of the two spacecraft in the pursuit-escape game are input at the corresponding time. p (s),x e (s)), where θ s The true anterior angle corresponding to phase s;

[0248] Where: x p (s) indicates the state of the tracked spacecraft during phase s; x e (s) represents the state of the escaped spacecraft in phase s;

[0249] Step S2.2: Solve the equivalent discount one-sided minimum time transition problem to obtain the open-loop optimal strategy pair. and

[0250] in: This indicates the optimal control strategy for tracking spacecraft; This represents the optimal control strategy for an escape spacecraft. The terminal true nearest angle corresponds to the solution of the game problem in stage s.

[0251] Step S2.3: According to the optimal control strategy recursion Escape spacecraft x under recursive unknown control ue e (θ s Update target point

[0252] in: This indicates that the spacecraft's state is tracked according to the optimal control strategy during phase s; x e (θ s () represents the state of the spacecraft escaping according to the optimal control strategy during phase s; This represents the relative state target point corresponding to the terminal true nearest angle in solving a game problem;

[0253] Step S2.3: Determine if ||φ||≤ε is satisfied:

[0254] If so, the iteration ends, the final closed-loop control game strategy is obtained, the target is intercepted, and the game ends.

[0255] Otherwise, obtain θ s+1 Continue the iteration in phase s+1:

[0256] θ s+1 =θ s +Δθ;

[0257] Where: φ represents the terminal constraint condition; ε represents the error threshold termination condition; Δθ represents the difference in true perimeter angle between two adjacent stages.

[0258] In this embodiment, the equivalent discount one-sided minimum time transition problem is:

[0259] Based on the control inputs and current states of the tracking and escaping spacecraft, solve the minimum time transition problem:

[0260]

[0261] x pe (θ s )=x p (θ s )-x e (θ s )

[0262] x pe (θ f ) = 0

[0263] Where, x pe The relative state difference between the tracking spacecraft and the escaping spacecraft; u pe Input for equivalent discount control; The maximum value of the equivalent control input set; θ f The optimal true anterior angle; x p (θ s () represents the initial state of the spacecraft during phase s; x e (θ s ) represents the initial state of the escape spacecraft during phase s; x pe (θ f ) represents the terminal state constraint due to the difference in relative motion states; x pe (θ s ) represents the initial state constraint for the difference in relative motion state.

[0264] In this embodiment, step S2.2 is:

[0265] Repeatedly solve for different given terminal times θ f Solving the feasibility problem:

[0266]

[0267] x pe (θ s )=x p (θ s )-x e (θ s )

[0268] Wherein, φ(θ) f Given the terminal error at the time corresponding to the true anterior angle; u pe Equivalent discount control input; for The boundary of.

[0269] In this embodiment, the true perimeter angle at each moment and at each stage is one-to-one and can be converted to each other.

[0270] In this embodiment, by constructing a one-sided equivalent model of orbital game dynamics and combining it with a model prediction closed-loop feedback mechanism, real-time strategy generation and dynamic trajectory optimization in high-speed orbital game scenarios are achieved.

[0271] In this embodiment, the autonomous pursuit and escape confrontation missions directly used by spacecraft in elliptical orbit environments include scenarios such as space target interception, non-cooperative target avoidance, and multi-spacecraft cooperative encirclement. Its application areas cover aerospace offensive and defensive confrontation, on-orbit service safety control, and active avoidance of space debris. Through high-speed strategy solving and real-time iteration of game strategy, it solves the problems of traditional orbital game control methods, such as difficulty in equilibrium solving, poor strategy convergence, and difficulty in on-orbit application, and provides millisecond-level response autonomous decision-making capabilities for complex space confrontation missions.

[0272] It should be noted that traditional open-loop game theory strategies for spacecraft are essentially static decision-making processes. Their control commands, once determined at the beginning of the mission, cannot be updated online, making them unable to handle sudden maneuvers by escaping spacecraft. The optimal strategy derived under the assumption of complete information is unilateral, leading to potential mismatches with unknown adversaries in practical applications. In contrast, the method described in this embodiment employs a closed-loop game theory strategy, which can offset the impact of adversary strategy uncertainty, thus meeting the requirements of practical applications.

[0273] In this embodiment, the method does not require manual parameter setting and can solve the closed-loop optimal strategy, making the closed-loop strategy based on game theory-guided optimization feasible, which is crucial for the actual on-orbit application of spacecraft.

[0274] In this embodiment, the method does not require hyperparameter adjustment, its output is deterministic, and convergence is guaranteed, making it suitable for airborne applications, especially on resource-constrained spaceborne embedded platforms.

[0275] The above description of several specific embodiments further details the technical solution provided by the present invention in order to highlight the advantages and benefits of the technical solution provided by the present invention. However, the above-described specific embodiments are not intended to limit the present invention. Any reasonable modifications and improvements to the present invention, reasonable combinations of embodiments, and equivalent substitutions based on the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A real-time closed-loop pursuit-escape game method for spacecraft in elliptical orbits, characterized in that, The method includes the following steps: Step S1: Given initial conditions x p (θ0), x e (θ0) and control constraints Where: θ0 represents the spacecraft's initial true anomaly angle; x p (θ0) represents the initial state of the tracking spacecraft; x e (θ0) represents the initial state of the escaped spacecraft; This represents the set of feasible control inputs for tracking spacecraft; This represents the set of feasible control inputs for an escape spacecraft; Step S2: Iterate to obtain the closed-loop control strategy; wherein, the iterative process in stage s includes the following steps: Step S2.1: At θ s The complete states (x) of the two spacecraft in the pursuit-escape game are input at the corresponding time. p (s), x e (s)), where θ s The true anterior angle corresponding to phase s; Where: x p (s) indicates the state of the tracked spacecraft during phase s; x e (s) represents the state of the escaped spacecraft in phase s; Step S2.2: Solve the equivalent discount one-sided minimum time transition problem to obtain the open-loop optimal strategy pair. and in: This indicates the optimal control strategy for tracking spacecraft; This represents the optimal control strategy for an escape spacecraft. The terminal true nearest angle corresponds to the solution of the game problem in stage s. Step S2.3: According to the optimal control strategy recursion Recursive unknown control u e The escape spacecraft X e (θ s Update target point in: This indicates that the spacecraft's state is tracked according to the optimal control strategy during phase s; x e (θ s () represents the state of the spacecraft escaping according to the optimal control strategy during phase s; This represents the relative state target point corresponding to the terminal true nearest angle in solving a game problem; Step S2.3: Determine if ||φ||≤ε is satisfied: If so, the iteration ends, the final closed-loop control game strategy is obtained, the target is intercepted, and the game ends. Otherwise, obtain θ s+1 Continue the iteration in phase s+1: i s+1 =θ s +Δθ; Where: φ represents the terminal constraint condition; ε represents the error threshold termination condition; Δθ represents the difference in true perimeter angle between two adjacent stages.

2. The real-time closed-loop pursuit and escape game method for spacecraft in elliptical orbits according to claim 1, characterized in that, The equivalent discount one-sided minimum time transition problem is: Based on the control inputs and current states of the tracking and escaping spacecraft, solve the minimum time transition problem: x pe (i s )=x p (i s )-x e (i s ) x pe (i f )=0 Where, x pe The relative state difference between the tracking spacecraft and the escaping spacecraft; u pe Input for equivalent discount control; The maximum value of the equivalent control input set; θ f The optimal true anterior angle; x p (θ s () represents the initial state of the spacecraft during phase s; x e (θ s ) represents the initial state of the escape spacecraft during phase s; x pe (θ f ) represents the terminal state constraint due to the difference in relative motion states; x pe (θ s ) represents the initial state constraint for the difference in relative motion state.

3. The real-time closed-loop pursuit and escape game method for spacecraft in elliptical orbits according to claim 2, characterized in that, Step S2.2 is as follows: Repeatedly solve for different given terminal times θ f Solving the feasibility problem: x pe (i s )=x p (i s )-x e (i s ) in, Φ (θ f Given the terminal error at the time corresponding to the true anterior angle; u pe Equivalent discount control input; for The boundary of.

4. A real-time closed-loop pursuit and escape game device for spacecraft in elliptical orbits, characterized in that, The device includes the following modules: Module S1: Given initial conditions x p (θ0), x e (θ0) and control constraints Where: θ0 represents the spacecraft's initial true anomaly angle; x p (θ0) represents the initial state of the tracking spacecraft; x e (θ0) represents the initial state of the escaped spacecraft; This represents the set of feasible control inputs for tracking spacecraft; This represents the set of feasible control inputs for an escape spacecraft; Module S2: Iteratively obtains the closed-loop control strategy; the iterative process in stage s includes the following sub-modules: Submodule S2.1: at θ s The complete states (x) of the two spacecraft in the pursuit-escape game are input at the corresponding time. p (s), x e (s)), where θ s The true anterior angle corresponding to phase s; Where: x p (s) indicates the state of the tracked spacecraft during phase s; x e (s) represents the state of the escaped spacecraft in phase s; Submodule S2.2: Solve the equivalent discounted one-sided minimum time transition problem, and obtain the open-loop optimal strategy pair. and in: This indicates the optimal control strategy for tracking spacecraft; This represents the optimal control strategy for an escape spacecraft. The terminal true nearest angle corresponds to the solution of the game problem in stage s. Submodule S2.3: According to the optimal control strategy recursion Recursive unknown control u e The escape spacecraft X e (θ s Update target point in: This indicates that the spacecraft's state is tracked according to the optimal control strategy during phase s; x e (θ s () represents the state of the spacecraft escaping according to the optimal control strategy during phase s; This represents the relative state target point corresponding to the terminal true nearest angle in solving a game problem; Submodule S2.3: Determine if ||φ||≤ε is satisfied: If so, the iteration ends, the final closed-loop control game strategy is obtained, the target is intercepted, and the game ends. Otherwise, obtain θ s+1 Continue the iteration in phase s+1: i s+1 =θ s +Δθ; Where: φ represents the terminal constraint condition; ε represents the error threshold termination condition; Δθ represents the difference in true perimeter angle between two adjacent stages.

5. A computer device, comprising: A processor and a memory, characterized in that the memory is used to store executable instructions of the processor, the processor being configured to execute the spacecraft elliptical orbit real-time closed-loop pursuit and escape game method according to any one of claims 1 to 3 by executing the executable instructions.

6. A computer storage medium, characterized in that, The storage medium stores a computer program, which, when executed, performs the real-time closed-loop pursuit and escape game method for spacecraft elliptical orbits as described in any one of claims 1 to 3.

7. A computer program product comprising a computer program / instructions, characterized in that, When the computer program / instruction is executed by the processor, it implements the real-time closed-loop pursuit and escape game method for spacecraft elliptical orbits as described in any one of claims 1 to 3.

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