A method and product for constructing a spacecraft pursuit-evasion game payoff function based on reachable sets

Abstract

The present application relates to a kind of based on reachable domain spacecraft pursuit and evasion game payoff function construction method and product, the method includes: establishing dynamic model and relative reference system, for describing the position, velocity of tracking party and escape spacecraft position, speed change with time;Introduce the initial orbit of escape as reference orbit and establish VNC relative reference system;Calculating reachable domain: the position point set of both sides is obtained, reflect the space confrontation relationship in pursuit and evasion process;Pursuit and evasion game payoff function is constructed: according to whether there is overlap area in the relative reference system of the reachable domain of tracking party and escape, different payoff functions are constructed, the optimal solution of pursuit and evasion game is obtained;Optimization initial guess estimate: using the distribution characteristics of reachable domain, provide initial guess for actual calculation.The method proposed in the present application introduces the qualitative analysis link of feasibility before orbit optimization calculation for spacecraft pursuit and evasion problem, to construct reasonable payoff function, reduce repeated calculation, reduce overall calculation cost.

Description

Technical Field

[0001] This application relates to the aerospace field, and in particular to a method and product for constructing a payoff function for a spacecraft pursuit and escape game based on reachability domains. Background Technology

[0002] With the increasing demands for deep space exploration, on-orbit servicing missions, and space security, the spacecraft chase-escape game problem has gradually attracted widespread attention. In mission scenarios such as space target surveillance, on-orbit interception, defense and evasion, and spacecraft formation control, spacecraft often engage in adversarial relationships. One party (the tracking party) attempts to approach or intercept the target, while the other party (the escaping party) maneuvers to evade its approach. This type of problem can generally be viewed as a spacecraft chase-escape game problem. How to construct a reasonable game model and design effective decision-making strategies under complex dynamic environments is one of the important issues in space mission planning and space adversarial research.

[0003] In traditional spacecraft pursuit-escape game theory research, performance index functions (i.e., payoff functions) are typically constructed based on optimal control theory or differential game theory. The optimal strategies for both sides are then obtained by solving the Hamilton-Jacobi-Isaacs equations or related optimal control problems. The specific form of the payoff function varies depending on the maneuverability of the pursuer and the escapee, and it is usually designed around indicators such as relative distance, approach time, or fuel consumption. Therefore, the construction of the payoff function directly affects the rationality and effectiveness of the game strategy. Improper design may lead to biased strategy evaluation; for example, in some situations, the pursuer can approach the target regardless of circumstances, but the payoff function still sets the relative distance as the optimization objective.

[0004] Furthermore, solving the pursuit-escape game often requires extensive numerical propagation and iterative calculations of orbital dynamics. When the payoff function is poorly designed, it often needs to be repeatedly adjusted and verified through multiple calculations. This process consumes significant computational resources, reduces computational efficiency, and consequently affects the effectiveness of the method in complex task planning or real-time decision-making scenarios.

[0005] Accessibility theory primarily studies the range of states a system can reach within a given time period under finite control input conditions. This theory is commonly used for system analysis, testing, and verification, and has been extensively studied in fields such as vehicle control and hybrid systems. In recent years, the reachability domain method has been gradually introduced into spacecraft orbital maneuver analysis and mission design. The reachability domain describes the set of all states a spacecraft can reach from a given initial state, given a dynamic model, time constraints, and control capability limitations. By analyzing the reachability domain of a spacecraft within a certain time range, its maneuverability and spatial coverage can be intuitively assessed, providing important basis for orbital transfer design, intercept window analysis, and mission feasibility assessment. In a pursuit-escape game scenario, the reachability domains of both the pursuer and the escapee reflect their potential positional distribution over a future time period. By analyzing the spatial relationship between their reachability domains, the competitive situation of the pursuer and escapee in future maneuver space can be more intuitively depicted, thus providing important reference for constructing the payoff function.

[0006] Therefore, how to utilize spacecraft reachability domain information to construct a payoff function, thereby improving the rationality and computational efficiency of the pursuit-escape game model, is an important technical issue in spacecraft orbital game research. Summary of the Invention

[0007] The purpose of this invention is to overcome the problems of insufficient prior information in the main solution of the spacecraft orbital pursuit and escape problem and inaccurate construction of the payoff function, thereby providing a method and product for constructing the payoff function of the spacecraft pursuit and escape game based on the reachability domain.

[0008] In view of this, the present invention provides a method for constructing a payoff function for a spacecraft pursuit and escape game based on reachability domain, comprising: Step 1: Establish dynamic model and relative reference frame: Establish a dynamic model to describe the changes in position and velocity of the tracking and escaping spacecraft over time; introduce the initial orbit of the escaping spacecraft as a reference orbit and establish a VNC relative reference frame; Step 2: Calculate the reachable domain: Under the constraints of maximum maneuverability and maximum pursuit time, calculate the trajectory propagation in the reference frame to obtain the position point set of both the pursuer and the escaper; construct the two-dimensional or three-dimensional reachable domain envelope of the pursuer and the escaper to reflect the spatial adversarial relationship during the pursuit process; Step 3: Constructing the payoff function for the pursuit-escape game: Under given time constraints, based on whether the reachable domains of the pursuer and the escapee overlap in the relative reference frame, different payoff functions are constructed to obtain the optimal solution for the pursuit-escape game; Step 4: Optimize the initial guess estimation: Utilize the reachability domain distribution characteristics to provide an initial guess for single-pulse or multi-pulse scenarios in actual calculations through single-pulse reachability domain analysis.

[0009] As an improvement to the above method, the establishment of the relative reference system in step 1 specifically includes: by With the origin at the x-axis along the tangential direction of the orbit, the z-axis along the direction of the orbit's angular momentum, and the y-axis determined by the right-hand rule as the lateral direction, a right-hand coordinate system is formed. The coordinate transformation relationship at any time t is as follows: ; in, This indicates the positional deviation of the tracking device relative to the reference orbit. This indicates the positional deviation of the escaping party relative to the reference orbit. This represents the rotation matrix from the inertial frame to the VNC coordinate system. for Reference position at any moment To track the relative position of the parties, The relative position of the fleeing party.

[0010] As an improvement to the above method, step 2 includes: Step 2.1: Uniformly discretize the amplitude and direction angle of the velocity pulses to construct a finite set of maneuvers; Step 2.2: Under the given dynamic model conditions, perform numerical integration on the updated state to obtain the trajectories of the tracking and escaping parties within a certain time interval. By traversing the maneuver parameter grid one by one, obtain the set of trajectories of both parties. Step 2.3: Extract the position vectors of the pursuer and the escaper from the trajectory obtained by numerical integration through interpolation, and construct the position vectors of the pursuer and the escaper relative to the reference orbit. This forms the position point set of the pursuer and the escaper relative to the reference orbit at a certain moment, reflecting the distribution of all possible relative states of the pursuer and the escaper at that moment. Step 2.4: For the obtained set of relative position points, the convex hull algorithm is used to construct the two-dimensional or three-dimensional reachable domain envelopes of the pursuer and the escaper. The two-dimensional reachable domain envelope is used to describe the planar reachable region at a fixed time, and the three-dimensional reachable domain envelope is used to describe the overall reachable region that evolves over time. This obtains the reachable domains of the pursuer and the escaper, which intuitively reflects the spatial adversarial relationship in the pursuit and escape process.

[0011] As an improvement to the above method, the discretization in step 2.1 specifically includes: If both the pursuer and the pursuer apply an instantaneous velocity pulse maneuver at the initial moment of the mission, the maneuver form is represented as follows: ; in, The velocity before the pulse is applied. The velocity after the pulse is applied. The pulse velocity increment is constrained by thrust capability and its magnitude does not exceed the respective maximum permissible maneuverability; Discretize the velocity pulse amplitude at equal intervals; ; Among them, by adjusting the number of discrete points N, where N>0, This represents the maximum speed increment that can be applied during a single maneuver. The pursuing spacecraft moves within the same orbital plane, with no out-of-plane maneuvers, pitch angle β=0, azimuth angle α (0, 2π), and maneuver speed. Represented as: ; By combining velocity amplitude and direction angle, a finite set of maneuvers is constructed. ; Where M represents the number of maneuvers, which is equal to N+1.

[0012] As an improvement to the above method, step 2.2 specifically includes: From the tracking party's maneuver assembly Combined with the escape party's maneuver Select the corresponding velocity pulse , The initial velocities of both sides are superimposed to update the initial state; under the given two-body dynamics model, the updated state is numerically integrated to obtain the trajectories of the tracking and escaping sides in the time interval; by traversing the maneuver parameter grid one by one, a trajectory set covering all discrete maneuver combinations is obtained.

[0013] As an improvement to the above method, step 2.3 specifically includes: at a specified terminal time The trajectory is processed; the position vectors of the tracking and escaping forces are extracted from the integral trajectory using interpolation, and their relative position vectors with respect to the reference trajectory are calculated to form the specified terminal time. The set of relative positions of the two sides reflects the distribution of all possible relative states of the spacecraft at that moment under all maneuver combinations.

[0014] As an improvement to the above method, the reachable regions of the tracking and escaping parties in a relative reference frame are constructed. and Step 3 specifically includes the following situations: Case 1: When the reachable regions have no intersection, i.e. This indicates that regardless of the maneuvers employed by the tracking and fleeing parties within permissible limits, neither can reach the same relative position area, meaning the pursuit mission is infeasible; a payoff function constrained by a fixed time is constructed. : ; in, Represents a given time t The location of the tracking party Indicates the location of the fleeing party. Indicates the relative distance between the two parties. , Indicates the control inputs of both parties. This represents a max-min function where the tracking party aims to minimize the relative distance between the two parties, while the escaping party aims to maximize the relative distance between the two parties. Case 2: When the reachable regions partially overlap, i.e. There are two cases: When the reachable domain of the pursuer does not intersect with the z-axis, even if the escaper does not perform any maneuvers, the pursuer cannot achieve interception under its maneuverability constraints. Therefore, the no-maneuver strategy is the optimal solution. When the reachable domain of the pursuer intersects with the z-axis, it indicates that the pursuer has the potential to approach the escaper, and the escaper will evade by choosing an appropriate maneuvering strategy. Case 3: When the reachable region of the tracking side includes the reachable region of the escape side, i.e. This indicates that, under a given time constraint, the tracking party can cover all possible relative locations reachable by the escapee within its maneuverability range; at this point, regardless of the escapee's maneuver strategy, the tracking party has a corresponding maneuver plan to approach or intercept it; a payoff function constrained by distance is constructed. The expression is as follows: ; in, Indicates a fixed distance. Indicates the time of interception. The function represents the maximum lower bound of the set. The tracking party aims to intercept the target in the shortest time, while the escape party aims to extend the interception time. Case 4: The escape reachable region includes the tracking reachable region, i.e. This indicates that, given the escapee's maneuverability, its reachable space always covers the pursuer's reachable space. Therefore, regardless of the pursuer's maneuvering strategy, it cannot guarantee approaching the escapee within that timeframe. A payoff function constrained by distance is constructed, expressed as follows: ; in, Indicates a fixed distance. This indicates the interception time. The tracking party aims to achieve escape in the longest possible time, while the escaping party aims to shorten the escape time.

[0015] As an improvement to the above method, step 4 is specifically as follows: Define the minimum distance function from point r to the boundary of the reachable domain as: ; in, Indicates the reachable domain for tracking The boundary set; Represents a set The point closest to r in the middle distance; From the escape route to the reachable domain Select the state point that is furthest from the boundary of the reachable domain of the tracking target. Defined as: ; This corresponds to the extreme state point where the fleeing party has the greatest escape advantage relative to the pursuing party under given time and maneuverability constraints; when the pursuing party can construct a feasible pursuit and escape trajectory for this state point, it also has the ability to cover other state points within the reachable domain of the fleeing party. by For the target position, construct the initial velocity direction estimation vectors for both the tracking and escaping sides. and : ; in, and The initial spatial positions of the pursuer and the escapee are given, respectively. Further, by combining the maximum maneuverability constraints of both parties, the initial velocity pulse estimate is obtained, which serves as the initial input for accurate calculation of the pursuit and escape trajectory or nonlinear optimization algorithm.

[0016] On the other hand, the present invention also provides a computer program product, including a computer program, characterized in that the computer program, when executed by a processor, implements the steps of the above-described method. Compared with the prior art, the advantages of the present invention are: 1. A spacecraft pursuit reachability modeling method based on VNC relative reference frame is proposed, which maps the three-dimensional pursuit motion from inertial space to the orbital, normal and radial directions, realizes the decoupled expression of relative orbital offset and maneuver direction, and improves the physical interpretability and structural clarity of reachability analysis.

[0017] 2. A method for constructing the payoff function in a pursuit-escape game based on the relationship between the reachable domain set is proposed. By analyzing the intersection and inclusion relationship between the reachable domains of the pursuer and the escapee, the feasibility of the pursuit task can be determined, and the payoff function can be constructed on this basis. This avoids the problem of relying on experience to select the payoff function in traditional methods, and improves the rationality of game modeling and the stability of solution.

[0018] 3. An initial value generation method for multi-pulse orbit optimization is proposed. By utilizing the distribution characteristics of the reachable domain, an initial guess is provided for multi-pulse optimization through single-pulse reachable domain analysis, thereby improving the convergence efficiency and stability of numerical solutions. Attached Figure Description

[0019] Figure 1 This is a schematic diagram of the task concept; Figure 2 A schematic diagram showing that the reachable areas of both parties in the pursuit and escape process have no intersection; Figure 3 A schematic diagram showing that the reachable domains of both the pursuer and the pursuer intersect (but do not intersect with the z-axis); Figure 4 A schematic diagram showing that the reachable domains of both the pursuer and the pursuer intersect (intersect with the z-axis); Figure 5 A schematic diagram showing the reachable domain of the tracking party including the escape party; Figure 6 A schematic diagram showing the reachable domain of the escaped party including the tracking party; Figure 7 The tracks of the two parties in the pursuit scene 1; Figure 8 The tracks for both sides in the pursuit and escape in Scene 2; Figure 9 The tracks for both sides in scenario three of the pursuit; Figure 10 The tracks for both sides in scenario four of the pursuit; Figure 11 This is a flowchart of the calculation process. Detailed Implementation

[0020] The technical solutions provided in this application are further illustrated below with reference to the embodiments.

[0021] Example 1 Embodiment 1 of the present invention proposes a method for constructing a payoff function for a spacecraft pursuit and escape game based on reachability domain.

[0022] This invention addresses the spacecraft pursuit and escape problem by proposing a method for constructing a game-theoretic payoff function based on reachability domains. This method introduces a qualitative feasibility analysis step before orbit optimization calculations to determine whether the pursuer has the possibility of achieving escape under given dynamic models and control constraints. This allows for the construction of a reasonable payoff function, reducing redundant calculations and lowering the overall computational cost. The specific technical solution is as follows: 1. Task Concept In spacecraft pursuit and escape problems, the participants typically include the pursuer and the escaping target spacecraft. Under constraints of a limited dynamics model and control capabilities, the pursuer's primary objective is to effectively approach, capture, or drive the target spacecraft to a designated space region within a finite timeframe using appropriate maneuvering strategies. The escaping spacecraft, on the other hand, attempts to avoid being approached or intercepted by the pursuer, or to delay capture as much as possible, by implementing evasive maneuvers. A conceptual mission diagram is shown below. Figure 1 As shown.

[0023] This type of pursuit problem is essentially an adversarial game under complex dynamic environments and multiple constraints. Its outcome depends not only on the initial orbital states of both sides but also on available control resources, maneuver opportunities, and strategy choices. To effectively guide subsequent orbit optimization and decision-making processes, it is necessary to construct a payoff function that reflects the relative states of both sides, thereby improving solution efficiency and enhancing the rationality of the strategy.

[0024] 2 Design Methodology 2.1 Dynamic Model To address the spacecraft pursuit and escape problem, a dynamic model of the pursuer and the escaper is established. Assuming both the pursuer and the escaper move in a geocentric inertial coordinate system, and considering only the gravitational pull of the Earth's center, their equations of motion are expressed using a classical two-body dynamic model: (1) in, r , v Let these represent the spacecraft's position vector and velocity vector in the geocentric inertial coordinate system, respectively. μ is the Earth's gravitational constant.

[0025] The state vectors of the tracking and escaping parties are represented as follows: (2) At the start of the pursuit mission, the initial states of both the pursuer and the fleeing party are given. Consider that both parties can perform a momentary velocity pulse maneuver at the beginning of the mission, the form of which is expressed as: (3) Where, Δ v The pulse velocity increment is constrained by thrust capability and its magnitude does not exceed the respective maximum permissible maneuverability.

[0026] 2.2 Establishment of a relative reference frame To intuitively describe the relative motion between the pursuer and the fleeing party, this invention introduces a reference trajectory and establishes a relative reference frame. The reference trajectory is selected as the fleeing party's initial trajectory, which can intuitively reflect the fleeing party's deviation relative to the initial trajectory, while clearly defining the relative positional relationship between the pursuer and the fleeing party on the initial trajectory. The initial state of this reference trajectory can be represented as follows: (4) The reference position at any given time can be obtained by integrating over the reference orbit. r ref ( t Directly calculating the relative positions of the pursuer and the escaper at any given moment makes it difficult to intuitively reflect their velocity changes relative to the orbital direction. Therefore, the following reference coordinate system is introduced: r ref ( t With the origin at (x, y), the x-axis is along the tangential direction of the orbit (velocity direction), the z-axis is along the direction of the orbital angular momentum (cross product of position and velocity vectors), and the y-axis is determined by the right-hand rule as the lateral direction, thus forming a right-handed coordinate system. This coordinate system can intuitively describe the relative motion of the spacecraft along the orbit, out-of-plane, and lateral directions, providing convenience for relative maneuver analysis, reachability domain construction, and pursuit / escape strategy design.

[0027] The following is an example of any time. t Coordinate transformation relationship: (5) Where RVNC represents the rotation matrix from the inertial frame to the VNC coordinate system. Through the above definition of relative position, the pursuit problem is transformed from an absolute motion description to a relative motion description with respect to a reference orbit, facilitating the construction and analysis of the reachable domain.

[0028] 2.3 Reachable Domain Calculation Method Under the constraints of maximum maneuverability and maximum pursuit time, this invention discretizes the maneuver parameter space of the pursuer and the escaper, transforming the originally continuous, high-dimensional, and difficult-to-solve-analytical pursuit dynamics problem into a finite-dimensional, controllable-precision parameter grid search problem, thereby generating the reachable domain of both the pursuer and the escaper within an acceptable computational complexity.

[0029] Unlike methods based on random sampling or single-step optimal control, this method employs a grid search strategy, enabling overall coverage of the reachable space with a given discrete precision. The specific implementation steps are as follows.

[0030] (1) Discrete maneuver parameters Under the constraint of maximum maneuverability, the velocity pulse parameters of the pursuer and the escaper are discretized respectively. Assume that the single velocity pulses of the pursuer and the escaper satisfy... (6) Where Δ v p,max With Δ v e,max These represent the maximum speed increments that the pursuing and escaping parties can apply in a single maneuver, respectively.

[0031] Based on this, the velocity pulse amplitude is discretized at equal intervals. (7) Simultaneously, the maneuver direction is discretized. In three-dimensional space, the maneuver direction can be represented by the azimuth angle α and the pitch angle β, where α takes the value [0, 2π) and β takes the value [0, π]. However, since the pursuing and fleeing parties are often located on the same orbital plane in the two-body problem, the z-direction is always 0, i.e., β = 0. Therefore, the maneuver velocity can be expressed as: (8) By combining velocity amplitude and direction angle, a finite set of maneuvers is constructed. (9) The above process completes the construction of the kinematic parameter mesh. This is achieved by adjusting the number of discrete points. N The angle division accuracy can be adjusted, and the grid size can be adjusted.

[0032] (2) State update and trajectory propagation After constructing the maneuver parameter mesh, it is necessary to update the states of both the pursuer and the fleeing party after they have performed maneuvers, and then calculate their trajectory propagation through numerical integration. Let the pursuer and the fleeing party be at the initial moment... t The states of 0 are respectively (10) Apply a corresponding velocity pulse Δ to the initial velocities of both parties. v p With Δ v e ,in v p ∈ V P Δ v e ∈ V e , V P , V e Representing the set of maneuvers for the tracking and escaping parties, we obtain the updated initial state: (11) Numerical integration is performed on the updated state under the given dynamic model conditions to obtain the time interval [t0, t] for the tracking and escaping parties. max The trajectory within the [ ] . By iterating through the maneuver parameter grid one by one, a set of trajectories covering all discrete maneuver combinations can be obtained.

[0033] (3) End state extraction To describe the relative spatial relationship between the pursuer and the pursuer under given time conditions, at a specified terminal time... t k The trajectory results are processed. Position vectors of the tracking and escaping parties are extracted from the trajectory obtained through numerical integration using interpolation methods. (12) And construct the position vectors of the pursuing and fleeing parties relative to the reference orbit: (13) This is thus formed at time t k The set of position points of the pursuing and fleeing parties relative to the reference orbit is given below. This set of points comprehensively reflects the distribution of all possible relative states of the pursuing and fleeing parties at that moment within the maneuver parameter grid space.

[0034] (4) Construction of reachable domains For the obtained set of relative positions (in the two-body model, only two-dimensional positions are considered within the same orbital plane, i.e., the pursuer and the fleeing party only have xy coordinates), a convex hull algorithm is used to construct the two-dimensional or three-dimensional reachable domain envelopes of the pursuer and the fleeing party. The two-dimensional reachable domain describes the planar reachable region at a fixed time, while the three-dimensional reachable domain describes the overall reachable region evolving over time. Using this method, the reachable domains of both the pursuer and the fleeing party can be obtained simultaneously, thus intuitively reflecting the spatial adversarial relationship during the pursuit process.

[0035] 2.4 Method for Constructing the Payoff Function in the Pursuit Game Based on the aforementioned method for constructing the relative reachability domains of the tracking party and the escape party, this invention further proposes a method for constructing a pursuit and escape payment function based on reachability domain spatial relationship analysis.

[0036] 2.4.1 Reachability Domain Relationship Analysis Given time constraints t ∈[0, t f [or specify terminal time] t k Next, we construct the reachable regions of the tracking and escaping parties in the relative reference frame, denoted as . U p ( t k )and U e ( t k Furthermore, since the initial trajectory of the escapee is selected as the reference trajectory, the initial trajectory of the escapee coincides with the z-axis in the three-dimensional coordinate system. By analyzing the relationship between the pursuer and the z-axis and the set relationship between the pursuer and the escapee in space, the feasibility of the pursuit problem can be divided into the following typical cases.

[0037] (1) The reachable regions have no intersection. Figure 2 The pursuit mission is not feasible. When the reachable region of the pursuit side and the reachable region of the escape side do not intersect under given time conditions, i.e., satisfying: (14) This means that within this timeframe, regardless of the permitted maneuvers employed by either the pursuer or the fleeing party, neither can reach the same relative location area. Therefore, it can be directly determined that the pursuit mission is infeasible under these time conditions.

[0038] (2) The reachable domains partially overlap: the pursuit problem has a feasible solution. When the reachable domains of the tracking and escaping parties overlap in a relative reference frame, i.e.: (15) Meanwhile, there is no strict inclusion relationship between the reachable domains of both parties, indicating that under a given time constraint, there exists a spatial region that both parties can reach within their respective maneuverability constraints. This overlapping region corresponds to the set of potential feasible solutions for pursuit and escape. For the escaping party, when the reachable domain of the pursuing party does not intersect with the z-axis (e.g., Figure 3 As shown), even if the fleeing party does not exert any maneuver, the tracking party cannot intercept it due to its maneuverability constraints. Therefore, the no-maneuver strategy is the optimal solution in this case. When the reachable domain of the tracking party intersects with the z-axis (e.g., Figure 4 As shown in the figure, this indicates that the tracking party has the potential to approach the escaping party, at which point the escaping party needs to choose an appropriate maneuvering strategy to evade.

[0039] In the above scenario, whether the system can escape or intercept depends on the outcome of the game between the strategies of both parties, thus requiring further refinement of trajectory design and optimization of control strategies.

[0040] (3) The reachable domain of the tracking side includes the reachable domain of the escape side. Figure 5 It is definitely traceable. When the tracing reachable region completely contains the escape reachable region, i.e., satisfying (16) This demonstrates that, under a given time constraint, the tracking party can cover all possible relative locations the fleeing party can reach within its maneuverability range. At this point, regardless of the fleeing party's maneuver strategy, the tracking party always has a corresponding maneuver plan to approach or intercept it.

[0041] (4) The escape reachable domain includes the tracking reachable domain. Figure 6 ): Will inevitably escape When the escape reachable region completely contains the pursuit reachable region, i.e.: (17) This means that, given the escapee's maneuverability, its reachable space always overlaps with the pursuer's reachable space. Therefore, regardless of the pursuer's maneuvering strategy, it cannot guarantee that it will close in on the escapee within that timeframe.

[0042] 2.4.2 Construction of Payment Function For the four scenarios described above, different payoff functions can be constructed to achieve the optimal solution to the pursuit-escape game. ① When the reachable regions have no intersection, the tracking party has no possibility of interception. Therefore, the game problem at this point focuses on the extreme value of the distance between the two parties. A payoff function with a fixed time constraint can be constructed, as shown below. (18) in r p ( t ) represents a given time. t The location of the tracking party r e ( t ) indicates the position of the fleeing party, d r ( t () indicates the relative distance between the two parties. v p , v e This represents the control inputs of both parties. The tracking party aims to minimize the relative distance between the two parties, while the escaping party aims to maximize the relative distance between them.

[0043] ② When the reachable region of the pursuer contains the reachable region of the escaper, or vice versa, these correspond to the scenarios of inevitable interception or inevitable escape, respectively. In this case, the focus of the game problem shifts to the time required to achieve interception or escape. Therefore, a payoff function constrained by distance can be constructed, expressed as follows: (19) in d c Indicates a fixed distance. t c The interception time is indicated by the tracking party's goal of achieving interception in the shortest time (or escaping in the longest time), while the escaping party's goal is to extend the interception time (or shorten the escape time).

[0044] ③ When the reachable domains partially overlap, if the reachable domains of the pursuer and the initial trajectory of the escaper do not intersect, the escaper can escape even without maneuvering. In this case, J2 is selected as the payoff function. When the reachable domains of the pursuer and the initial trajectory of the escaper intersect, the ability to intercept or escape is significantly affected by time. In this case, the solutions under fixed time or fixed distance conditions are uncertain. Therefore, it is necessary to construct two types of payoff functions (i.e., J1 and J2): one optimizes the distance between the two parties under fixed time conditions, and the other optimizes the maneuver time under fixed distance conditions.

[0045] 2.4.3 Optimize the initial guess estimation By utilizing the geometric structure characteristics of the reachable region, initial guesses can be provided for subsequent pursuit trajectories, thereby effectively narrowing the search solution space and improving the convergence speed and computational efficiency of numerical solutions.

[0046] To characterize the spatial relationship between any state point in the escape reachable region and the tracking reachable region, the minimum distance function from point r to the boundary of the tracking reachable region is defined as: (20) in U p ( t k ) indicates the reachable domain for tracking. U p ( t k The boundary set of ).

[0047] To construct the initial guessed state, from the escape reachable domain U e ( t k The state point furthest from the boundary of the reachable domain is selected in the ) and defined as: (twenty one) This point corresponds to the extreme state point where the escaping party has the greatest escape advantage relative to the pursuer, given time and maneuverability constraints. When the pursuer can construct a feasible pursuit trajectory for this state point, it also has the ability to cover other state points within the reachable domain of the escaping party.

[0048] by r * For the target position, the initial velocity direction estimation vectors of the tracking and escaping sides can be constructed respectively. (twenty two) Where rp(t0) and re(t0) are the spatial positions of the pursuer and the escaper at the initial moment, respectively. By further combining the maximum maneuverability constraints of both parties, the initial velocity pulse estimate can be obtained, which serves as the initial input for accurate calculation of the pursuit and escape trajectory or nonlinear optimization algorithm.

[0049] Example 2 Embodiment 2 of the present invention provides a method for constructing a payoff function for a spacecraft pursuit-escape game based on reachability domains. The calculation process is as follows: Figure 11 As shown, it includes the following steps: (1) Initialize the initial states of the tracking and escaping parties. The initial state vectors of the tracking party and the escaping party in a unified coordinate system are obtained, and the state vectors include position components and velocity components; at the same time, the maximum maneuver speed increment of the tracking party, the maximum maneuver speed increment of the escaping party, and the maximum pursuit time are set.

[0050] (2) Constructing a search grid for incremental maneuver speed to calculate the reachable domain Within a preset range of maneuver speed amplitudes, the velocity increments of both the pursuer and the escaper are discretely sampled, and the maneuver direction angles are uniformly discretized in the plane, forming a set of amplitude-direction combinations of maneuver speed increments. For each combination of maneuver speed increments, the corresponding velocity increments are superimposed onto the initial velocity components of both the pursuer and the escaper, thereby generating a joint initial state set containing both parties. For each joint initial state, a numerical integration method is used for trajectory propagation to obtain discrete trajectory points from time 0 to the maximum pursuit / escape time. The spatial positions of the pursuer and the escaper at the target time are extracted using time interpolation, thus obtaining the reachability domains of both parties at any given time.

[0051] (3) Constructing a reference orbit Based on the initial state of the preset reference orbit, orbit propagation is performed under the same dynamic model to obtain the reference orbit, which is used for subsequent relative position calculations.

[0052] (4) Generate the relative reachable domain At the target determination moment, the spatial positions of the tracking and escaping parties are obtained respectively, and the difference between these positions and the spatial position of the reference orbit at the corresponding moment is calculated to obtain the set of relative positions of the tracking and escaping parties relative to the reference orbit. Based on this set of relative positions, a three-dimensional coordinate system is constructed with spatial position and time as its dimensions, where the x-axis and y-axis represent relative spatial position, and the z-axis represents time. Furthermore, the envelope of the set of relative positions is calculated in this three-dimensional coordinate system to obtain the position-time three-dimensional reachable domain.

[0053] (5) Determine the feasibility of the fugitive repatriation problem and construct the payment function. Calculate the reachable domain of the tracking U PWith the escape route reachable domain U e The relationship between them.

[0054] If the two have no intersection, or satisfy the following conditions: U P yes U e If the subset of the target indicates that the fleeing party has superior mobility and the tracking party cannot cover the area it may reach, it is determined that the fleeing party will definitely escape successfully. In this case, a payoff function that optimizes the distance over a fixed time is selected. If satisfied U e yes U P If the set is a subset of the escape route, it means that the reachable domain of the pursuer completely covers the reachable domain of the escape route. The pursuer has the advantage in maneuverability and it is determined that it will definitely be able to complete the pursuit. In this case, a payoff function with fixed distance optimization time is selected. If the two have an intersection and do not contain each other, but the reachable domain of the tracker and the reference orbit (z-axis) do not intersect, it means that although the tracker can reach part of the reachable area of ​​the escapee, it cannot reach the reference orbit position that the escapee may pass through, and still cannot achieve effective interception. In this case, the payment function with fixed time optimization distance is still selected.

[0055] If the two have an intersection and do not contain each other, but the reachable domain of the tracking path intersects with the reference trajectory, it indicates that the pursuit result is related to the specific maneuver strategies of both parties. In this case, it is necessary to solve two types of payoff functions simultaneously: fixed-time optimized distance and fixed-distance optimized time.

[0056] Example 3 The feasibility of pursuit is determined using the method described in this invention, wherein the initial states of both parties are shown in Table 1 below: Table 1

[0057] Based on this, simulation analysis was conducted using four different combinations of maneuverability scenarios: (1) Scenario 1: When the maximum pulse velocity of both the tracking party and the escaping party is 0.5 m·s -1 When the maximum pursuit time is 8 hours, the calculated location-time three-dimensional reachability domain envelope is as follows: Figure 2 As shown, there is no intersection between the reachable domain of the pursuer and the reachable domain of the escaper, indicating that the pursuer cannot enter the reachable range of the escaper within a given time.

[0058] Therefore, in this scenario, the fleeing party will inevitably be able to escape successfully, and the pursuit game problem degenerates into a distance optimization problem, corresponding to the selection of a payoff function that optimizes the distance over a fixed time.

[0059] Figure 7The relative trajectories under the corresponding strategies are given, and it can be observed that: the pursuer moves towards the direction closest to the reachable domain of the escaper; while the escaper maneuvers in the direction away from the reachable domain of the pursuer.

[0060] (2) Scenario 2: When the maximum pulse velocity of the tracking side increases to 1 m·s -1 At times, such as Figure 3 As shown, the reachable domains of both parties begin to intersect. However, the reachable domain of the pursuer still does not intersect with the direction (z-axis) of the escapee's initial trajectory, meaning that the pursuer still cannot reach the trajectory area where the escapee might exist. Therefore, in this scenario, it is still determined that the pursuer cannot achieve effective pursuit, and the payoff function still chooses the fixed-time optimized distance form.

[0061] Figure 8 The relative trajectories under the corresponding optimal strategies are shown, and it can be seen that both sides still maneuver along the direction closest to / farthest from each other.

[0062] (3) Scenario 3: When the maximum pulse velocity of the tracking side further increases to 1.5 m·s -1 At times, such as Figure 4 As shown, there is a clear intersection between the reachable domains of the pursuing and escaping parties, but they do not contain each other. In this case, the outcome of the pursuit is no longer solely determined by the geometric relationship of the reachable domains, but depends on the maneuver strategies chosen by both parties. Therefore, it is necessary to construct and solve two types of payoff functions: fixed-time distance optimization and fixed-distance time optimization.

[0063] Figure 9 The relative trajectory results under the optimal distance payment function at a fixed time are presented.

[0064] (4) Scenario 4: When the maximum pulse velocity of the tracking side increases to 2 m / s -1 At times, such as Figure 5 As shown, the reachable domain of the pursuer completely covers the reachable domain of the escaper, indicating that the pursuer has an absolute advantage in maneuverability. In this case, no matter what maneuvering strategy the escaper adopts, it cannot escape the pursuer's interception. Therefore, the shortest time for a fixed distance should be selected as the payoff function. Figure 10 The corresponding results are shown, demonstrating that the tracking party successfully intercepted the escapee.

[0065] Depend on Figures 7-10 It can be seen that, based on the relative positional relationship of the reachable domains of both parties, an effective initial value estimate can be provided for maneuver strategies in a single-pulse chase-escape game. However, the applicability of this initial value method to the multi-pulse maneuver scenario still needs further verification.

[0066] Therefore, the initial maneuver estimation results for single-pulse and four-pulse scenarios in Scenario 1 and Scenario 2 were compared and analyzed, as shown in Table 2. The results show that the optimized solution for the first maneuver in the multi-pulse scenario is similar to that in the single-pulse scenario. Therefore, it can be concluded that the initial value estimation method based on the reachability domain is also applicable to the multi-pulse pursuit problem and can provide a reasonable initial guess for its numerical optimization, thereby improving the solution efficiency and convergence stability.

[0067] Table 2

[0068] As can be seen from the above detailed description of this application, in order to address the problem that the principal solution of the existing spacecraft orbital escape problem lacks prior information and the payment function is not accurately constructed, this invention proposes a method for constructing a spacecraft escape game payment function based on the reachability domain.

[0069] This invention constructs the reachability regions of both the pursuer and the escaper under given dynamic models and control input constraints, and analyzes the relative relationship between their reachability regions to quickly determine the feasibility of spacecraft pursuit and escape problems, thereby constructing a reasonable payoff function form. By introducing reachability region information as a priori basis, the repeated adjustments to the payoff function form during the actual solution process can be reduced, and invalid calculations caused by unreasonable payoff function design can be avoided, thus improving the solution efficiency of spacecraft pursuit and escape problems.

[0070] Furthermore, for scenarios where there is a possibility of pursuit and escape, this invention can also provide an estimate of the initial pursuit and escape direction based on the spatial distribution relationship between the relative reachable domains of the pursuer and the escaper. This provides reasonable initial conditions for subsequent trajectory optimization or search algorithms, reduces the global search space, lowers computational complexity, and further improves the overall efficiency of solving the trajectory pursuit and escape problem.

[0071] Example 4 Embodiments of the present invention may also provide a computer program product, including a computer program. When the computer program is executed by a processor, it can implement the various steps in the above method embodiments.

[0072] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to the embodiments, those skilled in the art should understand that modifications or equivalent substitutions to the technical solutions of the present invention do not depart from the spirit and scope of the technical solutions of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A method for constructing a payoff function in a spacecraft pursuit-escape game based on reachability domains, comprising: Step 1: Establish dynamic model and relative reference frame: Establish a dynamic model to describe the changes in position and velocity of the tracking and escaping spacecraft over time; introduce the initial orbit of the escaping spacecraft as a reference orbit and establish a VNC relative reference frame; Step 2: Calculate the reachable domain: Under the constraints of maximum maneuverability and maximum pursuit time, calculate the trajectory propagation in the reference frame to obtain the position point set of both the pursuer and the escaper; construct the two-dimensional or three-dimensional reachable domain envelope of the pursuer and the escaper to reflect the spatial adversarial relationship during the pursuit process; Step 3: Constructing the payoff function for the pursuit-escape game: Under given time constraints, based on whether the reachable domains of the pursuer and the escapee overlap in the relative reference frame, different payoff functions are constructed to obtain the optimal solution for the pursuit-escape game; Step 4: Optimize the initial guess estimation: Utilize the reachability domain distribution characteristics to provide initial guesses for single-pulse or multi-pulse scenarios in actual calculations through single-pulse reachability domain analysis; For the reachable regions of the constructed tracking and escaping sides in the relative reference frame and Step 3 specifically includes the following situations: Case 1: When the reachable regions have no intersection, i.e. This indicates that regardless of the maneuvers employed by the tracking and fleeing parties within permissible limits, neither can reach the same relative position area, meaning the pursuit mission is infeasible; a payoff function constrained by a fixed time is constructed. : ； in, Represents a given time t The location of the tracking party Indicates the location of the fleeing party. Indicates the relative distance between the two parties. , Indicates the control inputs of both parties. This represents a max-min function where the tracking party aims to minimize the relative distance between the two parties, while the escaping party aims to maximize the relative distance between the two parties. Case 2: When the reachable regions partially overlap, i.e. There are two cases: When the reachable domain of the pursuer does not intersect with the z-axis, even if the escaper does not perform any maneuvers, the pursuer cannot achieve interception under its maneuverability constraints. Therefore, the no-maneuver strategy is the optimal solution. When the reachable domain of the pursuer intersects with the z-axis, it indicates that the pursuer has the potential to approach the escaper, and the escaper will evade by choosing an appropriate maneuvering strategy. Case 3: When the reachable region of the tracking side includes the reachable region of the escape side, i.e. This indicates that, under a given time constraint, the tracking party can cover all possible relative locations reachable by the escapee within its maneuverability range; at this point, regardless of the escapee's maneuver strategy, the tracking party has a corresponding maneuver plan to approach or intercept it; a payoff function constrained by distance is constructed. The expression is as follows: ； in, Indicates a fixed distance. Indicates the time of interception. The function represents the maximum lower bound of the set. The tracking party aims to intercept the target in the shortest time, while the escape party aims to extend the interception time. Case 4: The escape reachable region contains the tracking reachable region, i.e. This indicates that, given the escapee's maneuverability, its reachable space always covers the pursuer's reachable space. Therefore, regardless of the pursuer's maneuvering strategy, it cannot guarantee approaching the escapee within the interception time. A payoff function constrained by distance is constructed, expressed as follows: ； in, Indicates a fixed distance. This indicates the interception time. The tracking party aims to achieve escape in the longest possible time, while the escaping party aims to shorten the escape time.

2. The method for constructing the payoff function for a spacecraft pursuit and escape game based on reachability domains as described in claim 1, characterized in that, The establishment of the relative reference frame in step 1 specifically includes: by With the origin at the x-axis along the tangential direction of the orbit, the z-axis along the direction of the orbit's angular momentum, and the y-axis determined by the right-hand rule as the lateral direction, a right-hand coordinate system is formed. The coordinate transformation relationship at any time t is as follows: ； in, This indicates the positional deviation of the tracking device relative to the reference orbit. This indicates the positional deviation of the escaping party relative to the reference orbit. This represents the rotation matrix from the inertial frame to the VNC coordinate system. for Reference position at any time To track the relative position of the parties, The relative position of the fleeing party.

3. The method for constructing the payoff function for a spacecraft pursuit-escape game based on reachability domains according to claim 1, characterized in that, Step 2 includes: Step 2.1: Uniformly discretize the amplitude and direction angle of the velocity pulses to construct a finite set of maneuvers; Step 2.2: Under the given dynamic model conditions, perform numerical integration on the updated state to obtain the trajectories of the tracking and escaping parties within a certain time interval. By traversing the maneuver parameter grid one by one, obtain the set of trajectories of both parties. Step 2.3: Extract the position vectors of the pursuer and the escaper from the trajectory obtained by numerical integration through interpolation, and construct the position vectors of the pursuer and the escaper relative to the reference orbit. This forms the position point set of the pursuer and the escaper relative to the reference orbit at a certain moment, reflecting the distribution of all possible relative states of the pursuer and the escaper at that moment. Step 2.4: For the obtained set of relative position points, the convex hull algorithm is used to construct the two-dimensional or three-dimensional reachable domain envelopes of the pursuer and the escaper. The two-dimensional reachable domain envelope is used to describe the planar reachable region at a fixed time, and the three-dimensional reachable domain envelope is used to describe the overall reachable region that evolves over time. This obtains the reachable domains of the pursuer and the escaper, which intuitively reflects the spatial adversarial relationship in the pursuit and escape process.

4. The method for constructing the payoff function for a spacecraft pursuit and escape game based on reachability domains as described in claim 3, characterized in that, The discretization in step 2.1 specifically includes: If both the pursuer and the pursuer apply an instantaneous velocity pulse maneuver at the initial moment of the mission, the maneuver form is represented as follows: ； in, The velocity before the pulse is applied. The velocity after the pulse is applied; The pulse velocity increment is constrained by thrust capability and its magnitude does not exceed the maximum permissible maneuverability of either the pursuing or fleeing party. velocity pulse increment Discretize the amplitude at equal intervals: ； Among them, by adjusting the number of discrete points N, where N>0 This represents the maximum speed increment that can be applied during a single maneuver. The pursuing spacecraft moves within the same orbital plane, with no out-of-plane maneuvers, pitch angle β=0, azimuth angle α (0, 2π), and maneuver speed. Represented as: ； By combining velocity amplitude and direction angle, a finite set of maneuvers is constructed. ; Where M represents the number of maneuvers, which is equal to N+1.

5. The method for constructing the payoff function for a spacecraft pursuit and escape game based on reachability domains as described in claim 3, characterized in that, Step 2.2 specifically includes: From the tracking party's maneuver assembly Combined with the escape party's maneuver Select the corresponding velocity pulse , The initial velocities of both sides are superimposed to update the initial state; under the given two-body dynamics model, the updated state is numerically integrated to obtain the trajectories of the tracking and escaping sides in the time interval; by traversing the maneuver parameter grid one by one, a trajectory set covering all discrete maneuver combinations is obtained.

6. The method for constructing the payoff function for a spacecraft pursuit and escape game based on reachability domains as described in claim 3, characterized in that, Step 2.3 specifically includes: at a specified terminal time The trajectory is processed; the position vectors of the tracking and escaping forces are extracted from the integral trajectory using interpolation, and their relative position vectors with respect to the reference trajectory are calculated to form the specified terminal time. The set of relative positions of the two sides reflects the spacecraft's position under all maneuver combinations. The distribution of all possible relative states at any given time.

7. The method for constructing the payoff function for a spacecraft pursuit and escape game based on reachability domains as described in claim 1, characterized in that, Step 4 specifically involves; Define the minimum distance function from point r to the boundary of the reachable domain as: ； in, Indicates the reachable domain for tracking The boundary set; Represents a set The point closest to r in the middle distance; From the escape route to the reachable domain Select the state point that is furthest from the boundary of the reachable domain of the tracking target. Defined as: ； This corresponds to the extreme state point where the escaping party has the greatest escape advantage relative to the pursuing party, given time and maneuverability constraints; when the pursuing party can target this state point... When constructing a feasible pursuit and escape trajectory, it also has the ability to cover other state points within the reachable domain of the escapee; by For the target position, construct the initial velocity direction estimation vectors for both the tracking and escaping sides. and : ； in, and The initial spatial positions of the pursuer and the escapee are given, respectively. Further, by combining the maximum maneuverability constraints of both parties, the initial velocity pulse estimate is obtained, which serves as the initial input for accurate calculation of the pursuit and escape trajectory or nonlinear optimization algorithm.

8. A computer program product, comprising a computer program, characterized in that, When the computer program is executed by a processor, it implements the steps of the method of claim 1.

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