A spacecraft pursuit-escape anti-game control method for non-motorized targets

Abstract

The application discloses a spacecraft pursuit-escape anti-game control method for non-motorized targets and belongs to the technical field of spacecrafts, which comprises the following steps: acquiring spacecraft pursuit-escape game parameters of a pursuit star and a defense star; using the spacecraft pursuit-escape game parameters of the pursuit star and the defense star to establish a spacecraft pursuit-escape game control problem model for non-motorized targets, and obtaining optimal control laws of the pursuit star and the defense star; converting the spacecraft pursuit-escape anti-game problem for non-motorized targets into Riccati equations through the optimal control laws of the pursuit star and the defense star; solving the Riccati equations by using backward integration, integrating the trajectories, and obtaining state trajectories of both sides of pursuit and escape according to the integrated results, so as to control the motion trajectory of the defense star. The solving process of the pursuit-escape anti-game problem is simplified, the solving time is shortened, and the solving accuracy is improved. The operation path of the defense star can be controlled in time, so that the defense star located around the main star cannot prevent the pursuit star from approaching the main star in time.

Description

Technical Field

[0001] This invention belongs to the field of spacecraft technology, specifically relating to a spacecraft pursuit and escape anti-game control method for non-maneuvering targets. Background Technology

[0002] Spacecraft in orbit are vulnerable to attacks and close approaches from non-cooperative targets. Due to their high value and the need to use their fuel for on-orbit operations, they cannot autonomously evade threats and require the deployment of companion satellites as defensive satellites for coordinated countermeasures. The competition for position between the companion satellite and the target satellite over the host satellite is known as a game-theoretic control problem of spacecraft pursuit, escape, and defense against non-maneuverable targets. The target satellite aims to approach the host satellite while ensuring it is not intercepted or destroyed by the companion satellite; the defensive satellite aims to prevent the target satellite from approaching the host satellite as much as possible.

[0003] Currently, most modeling and solutions for spacecraft pursuit-escape-defense game-theoretic control problems use non-zero-sum differential game methods. However, since it is a non-zero-sum game in the game type, the cost functions of the pursuing star and the defending star are not the same. Therefore, solving for its optimal control strategy is often very complex, and it is ultimately transformed into a two-point boundary value problem for numerical solution. This is time-consuming and has low accuracy, resulting in the inability to control the running path of the defending star in time. Consequently, the defending star located around the host star cannot prevent the pursuing star from approaching the host star in time. Summary of the Invention

[0004] To overcome the challenges of existing game-theoretic problems where solving for optimal control strategies is complex and ultimately reduces to two-point boundary value problems requiring numerical solutions—which are time-consuming and inaccurate—and prevents defensive satellites around the primary satellite from promptly preventing pursuing satellites from approaching, this invention aims to provide a game-theoretic control method for spacecraft pursuing and escaping non-maneuvering targets. This method describes the game objectives of the pursuing and defensive satellites using a pair of numerically opposite cost functions, significantly simplifying the solution process. It reduces time consumption while rapidly controlling the trajectory of the defensive satellite, giving it greater maneuverability and enabling more precise interception of the pursuing satellite.

[0005] To achieve the above objectives, the present invention adopts the following technical solution:

[0006] A spacecraft pursuit and escape anti-game control method for non-maneuvering targets includes:

[0007] Obtain the spacecraft pursuit and escape game parameters of the pursuing and defending satellites;

[0008] By utilizing the spacecraft pursuit and escape game parameters of the pursuing and defending satellites, a spacecraft pursuit and escape game control problem model is established for non-maneuvering targets, and the optimal control laws of the pursuing and defending satellites are obtained.

[0009] By obtaining the optimal control laws for the pursuing and defending satellites, the spacecraft pursuit-escape-defense game problem against non-maneuvering targets is transformed into the Riccati equation.

[0010] The Riccati equation is solved by using backward integration, and the trajectory is integrated. The state trajectories of the pursuer and the pursuer are obtained based on the integrated result, thereby controlling the trajectory of the defense star.

[0011] As a further improvement of the present invention, the spacecraft pursuit-escape game parameters for obtaining the pursuing and defending stars are wherein:

[0012] The spacecraft pursuit and escape game parameters for the pursuing and defending stars include the initial position of the pursuing star, the initial position of the defending star, the velocity of the pursuing star, and the velocity of the defending star.

[0013] As a further improvement of the present invention, the establishment of a spacecraft pursuit and escape game control problem model for non-maneuvering targets includes:

[0014] An LVLH coordinate system is established with the position of the primary star t as the origin. Based on the LVLH coordinate system and the spacecraft pursuit and escape game parameters of the pursuing and defensive stars, a cost function is designed.

[0015] Based on the CW equations, we determine the differential equations of the dynamic constraints that need to be satisfied by the pursuing and defending stars in space orbits.

[0016] Based on the above process, a spacecraft pursuit and escape defense game model for non-maneuvering targets was completed, based on differential game theory.

[0017] As a further improvement of the present invention, the step of establishing an LVLH coordinate system with the position of the primary star t as the origin, and designing a cost function based on the spacecraft pursuit-escape game parameters of the LVLH coordinate system and the pursuing and defending stars, includes:

[0018]

[0019] The cost function consists of two parts: a terminal term and an integral term.

[0020] Terminal Item The meaning is the relative state between the pursuing satellite and the primary satellite at the terminal moment, and the relative state between the defensive satellite and the primary satellite; integral term The meaning is the fuel consumption of both sides throughout the entire game;

[0021] Among them, J a The cost function for the tracking star; J d The cost function for the defense star; x a x is the state vector of the tracking star. d The state vector of the defense star; t fQ1 represents the terminal moment of the game; Q2 is the weight matrix of the terminal distance term between the pursuing star and the main star; Q2 is the weight matrix of the terminal distance term between the defensive and pursuing stars; R a R is the weight matrix for the fuel consumption term of the tracking star; d The weight matrix for the fuel consumption term of the defense star; u a The continuous control quantity applied to the tracking star; u d The continuous control amount applied to the defensive star.

[0022] As a further improvement of the present invention, the method for obtaining the optimal control law for the pursuing and defending satellites includes:

[0023] By introducing costate variables using the Lagrange multiplier method, the bilateral optimization problem with differential equation equality constraints is transformed into an unconstrained bilateral optimization problem. The auxiliary cost function is then obtained by processing the cost function.

[0024]

[0025] Where: x a The state vector of the tracking star; t f R represents the final moment of the game; y represents the state difference between the defensive star and the pursuing star; Q1 is the weight matrix of the final distance term between the pursuing star and the main star; Q2 is the weight matrix of the final distance term between the defensive star and the pursuing star; R a R is the weight matrix for the fuel consumption term of the tracking star; d The weight matrix for the fuel consumption term of the defense star; u a The continuous control quantity applied to the tracking star; u d The continuous control quantity applied to the defensive star; λ is the introduced Lagrange multiplier; v is the introduced second Lagrange multiplier.

[0026] As a further improvement of the present invention, the cost function is processed to obtain an auxiliary cost function, and then the variable integral of the auxiliary cost function is calculated:

[0027]

[0028] in:

[0029]

[0030]

[0031] Where: λ(t) is the introduced Lagrange multiplier; ν(t) is the introduced second Lagrange multiplier; x a x is the state vector of the tracking star. d The state vector of the defense star; u a The continuous control quantity applied to the tracking star; u dThe continuous control quantity applied to the defensive star; H is the terminal constraint function; H is the Hamiltonian function.

[0032] The optimal control laws for the pursuing and defensive stars are obtained as follows:

[0033]

[0034]

[0035] Where: R a R is the weight matrix for the fuel consumption term of the tracking star; d The weight matrix for the fuel consumption term of the defense star; u a The continuous control quantity applied to the tracking star; u d The continuous control quantity applied to the defense star; λ(t) is the introduced Lagrange multiplier; v(t) is the introduced second Lagrange multiplier.

[0036] As a further improvement of the present invention, the transformation of the spacecraft pursuit and escape game problem against non-maneuvering targets into a Riccati equation solving problem includes:

[0037] Substituting the optimal control laws of the pursuing and defending satellites into the dynamic constraint differential equations of the pursuing and defending satellites:

[0038]

[0039] The joint state vector is obtained based on the above dynamic constraint equations. With joint costate variables The relationship between them is as follows:

[0040]

[0041] The terminal relationship between the joint state vector and the joint costate variables is obtained based on the transverse condition:

[0042]

[0043] There is a linear relationship between the joint state vector and the joint costate variables:

[0044] Λ(t)=P(t)K(t)

[0045] Differentiating both sides simultaneously, we get:

[0046]

[0047] simultaneous formula and have to:

[0048]

[0049] Where: Q1 is the weight matrix for the terminal distance term of the tracking star to the primary star; Q2 is the weight matrix for the terminal distance term of the tracking star to the primary star; X a R is the state vector of the tracking star. a R is the weight matrix for the fuel consumption term of the tracking star; d t is the weight matrix for the fuel consumption term of the defense star; λ(t) is the introduced Lagrange multiplier; v(t) is the introduced second Lagrange multiplier; P(t) is the relation matrix between the joint state vector and the joint costate vector; t f This is the final moment of the game.

[0050] As a further improvement of the present invention, the spacecraft pursuit and escape game problem against non-maneuvering targets is transformed into a problem of solving the Riccati equation, the solution of which is:

[0051]

[0052]

[0053] Where: u d The continuous control quantity applied to the defensive star; u d The continuous control amount applied to the defensive star; R a R is the weight matrix for the fuel consumption term of the tracking star; d S is the weight matrix for the fuel consumption term of the defense star; a For the control transformation matrix of the tracking star; S d K is the control transformation matrix of the defense star; K(t) is the joint costate variable.

[0054] As a further improvement of the present invention, the method of solving the Riccati equation using backward integrals includes:

[0055] Starting with the final value of the Riccati equation, we take the negative of the differential equation:

[0056]

[0057] As a further improvement of the present invention, the method of solving the Riccati equation using backward integration and integrating the trajectory is described, wherein the integration of the trajectory is performed using an integration algorithm to obtain the matrix P(t).

[0058] Compared with the prior art, the present invention has the following beneficial effects:

[0059] This invention provides a spacecraft pursuit-escape game-theoretic control method for non-maneuvering targets. By acquiring the spacecraft pursuit-escape game parameters of the pursuing and defending satellites, a model of the spacecraft pursuit-escape game-theoretic control problem for non-maneuvering targets is established. By obtaining the optimal control laws of the pursuing and defending satellites, the spacecraft pursuit-escape game-theoretic problem for non-maneuvering targets is transformed into a Riccati equation problem, which is solved using backward integration. Simultaneously, this invention provides an effective threat solution for spacecraft employing master-slave satellite cooperation. The spacecraft pursuit-escape game-theoretic control model for non-maneuvering targets established in this invention is a differential game-theoretic model based on linear CW equations, possessing advantages such as model simplicity and small linearization error. According to the proposed method for solving the pursuit-escape game-theoretic problem, a complex non-zero-sum game problem can be transformed into a zero-sum game problem, and further into a Riccati equation problem, significantly improving solution efficiency, shortening solution time, and quickly obtaining the state trajectories of both the pursuing and fleeing satellites. When the primary satellite, which is performing its original mission in orbit, is approached by a pursuing satellite, the orbits of the defensive satellites around the primary satellite can be adjusted in a timely manner based on the obtained trajectories of both the pursuer and the pursuer. This allows the defensive satellites to intercept the pursuing satellite in a timely manner, preventing it from getting close to the primary satellite. This gives the defensive satellites greater maneuverability and makes the interception of the pursuing satellite more precise. Attached Figure Description

[0060] Figure 1 This is a flowchart illustrating the specific implementation of the game-theoretic control problem for spacecraft pursuit and escape against non-maneuvering targets according to the present invention.

[0061] Figure 2 This is a game scenario diagram of the spacecraft pursuit and escape defense game control problem against non-maneuverable targets to which this invention applies;

[0062] Figure 3 This is a simulation trajectory diagram from an embodiment of the present invention;

[0063] Figure 4 This is a simulation speed change graph from an embodiment of the present invention;

[0064] Figure 5 This is a simulated thrust acceleration variation diagram in an embodiment of the present invention;

[0065] Figure 6 This is a diagram showing the distances between the pursuing satellite and the defensive satellite, as well as the distances between the pursuing satellite and the primary satellite, in an embodiment of the present invention. Detailed Implementation

[0066] The present invention will now be described in detail with reference to the accompanying drawings and embodiments. It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other.

[0067] The following detailed description is exemplary and intended to provide further detailed explanation of the invention. Unless otherwise specified, all technical terms used in this invention have the same meaning as commonly understood by one of ordinary skill in the art to which this application pertains. The terminology used in this invention is for the purpose of describing particular embodiments only and is not intended to limit the scope of exemplary embodiments according to the invention.

[0068] To address the problem that most current modeling and solution methods for spacecraft pursuit and escape anti-game theory problems rely on non-zero-sum differential game methods, resulting in long processing times and low accuracy, this invention provides a spacecraft pursuit and escape anti-game theory control method for non-maneuvering targets, such as... Figure 1 As shown, the method includes:

[0069] Obtain the spacecraft pursuit and escape game parameters of the pursuing and defending satellites;

[0070] By utilizing the spacecraft pursuit and escape game parameters of the pursuing and defending satellites, a spacecraft pursuit and escape game control problem model is established for non-maneuvering targets, and the optimal control laws of the pursuing and defending satellites are obtained.

[0071] By obtaining the optimal control laws for the pursuing and defending satellites, the spacecraft pursuit-escape-defense game problem against non-maneuvering targets is transformed into the Riccati equation.

[0072] The Riccati equation is solved by using backward integration, and the trajectory is integrated. The state trajectories of the pursuer and the pursuer are obtained based on the integrated result, thereby controlling the trajectory of the defense star.

[0073] This invention simplifies the solution process for the escape-defense game problem, shortens the solution time, and improves the accuracy of the solution. When the primary satellite performing its original mission in orbit is approached by a pursuing satellite, the defensive satellites around the primary satellite can be controlled in time to intercept the pursuing satellite and prevent it from getting close to the primary satellite.

[0074] The present invention will now be described in detail:

[0075] A spacecraft pursuit and escape anti-game control method for non-maneuvering targets includes:

[0076] S1: Obtain the spacecraft pursuit, escape, and defense game parameters of the pursuing and defending stars, including the initial positions and velocities of the pursuing and defending stars;

[0077] The spacecraft pursuit-escape-defense game parameters for the pursuing and defending satellites include the initial position of the pursuing satellite, the initial position of the defending satellite, the velocity of the pursuing satellite, and the velocity of the defending satellite.

[0078] S2: Establish a game-theoretic model of spacecraft pursuit, escape, and defense against non-maneuverable targets, and solve it using the Lagrange multiplier method and variational method to obtain the optimal control laws for the pursuing and defending satellites.

[0079] S21: Establish an LVLH coordinate system with the position of the primary star t as the origin, and design the following cost function based on the LVLH coordinate system and the game objectives of the pursuing star and the defensive star.

[0080]

[0081] The cost function consists of two parts: a terminal term and an integral term.

[0082] Among them, terminal item The meaning is the relative state between the pursuing satellite and the primary satellite at the terminal moment, and the relative state between the defensive satellite and the primary satellite; integral term The meaning is the fuel consumption of both sides throughout the entire game; J a The cost function for the tracking star; J b The cost function for the defense star; x a Let X be the state vector of the tracking star. a =[x a ,y a ,v ax ,v ay ] T ;x d X is the state vector of the defense star. d =[x d ,y d ,v dx ,v dy ] T ;t f R represents the final moment of the game; Q1 is the weight matrix of the final distance term between the pursuing star and the primary star; Q2 is the weight matrix of the final distance term between the pursuing star and the primary star; a R is the weight matrix for the fuel consumption term of the tracking star; d The weight matrix for the fuel consumption term of the defense star; u a The continuous control quantity applied to the tracking star, u a =[u ax ,u ay ] T ;u d The continuous control quantity applied to the defensive star, u d =[u dx ,u dy ] T .

[0083] S22: Based on the CW equations, determine the differential equations of the dynamic constraints that need to be satisfied by the pursuing and defending stars in space orbits;

[0084]

[0085] Where: matrix A represents the spacecraft's relative motion dynamics constraint matrix; B is the control matrix for the pursuing and defending satellites, B = [0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 12; 13; 18; 19; 12; 19; 2×2 ,I 2×2 ] T .

[0086] Matrix A is constructed using the CW equation, as shown below:

[0087]

[0088] in, The value represents the angular velocity of the primary star's orbit around Earth, and μ is the Earth's gravitational field coefficient, μ = 3.986 × 10⁻⁶. 14 , where a0 is the semi-major axis of the main star's orbit.

[0089] Symmetric positive semidefinite matrix Q1∈R 4×4 Q2∈R 4×4 R d ∈R 2×2 R a ∈R 2×2 satisfy:

[0090] Q1=k1I 4×4 Q2=k2I 4×4 R d =k d I 2×2 R a =k a I 2×2 (3)

[0091] in:

[0092] k1,k2,k a ,k d These are the weighting coefficients;

[0093] x represents the radial position component of the orbit in the LVLH frame; y represents the directional position component of the flight in the LVLH frame; v x v represents the radial velocity component of the orbit in the LVLH system. y denoted as the velocity component of the orbital flight direction in the LVLH system; J represents the cost function for both the pursuing and defending stars.

[0094] Based on the above process, a spacecraft pursuit and escape defense game model for non-maneuverable targets based on differential game theory is established.

[0095] S3: The process of obtaining the optimal control laws for both sides based on the established game-theoretic model of spacecraft pursuit and escape against non-maneuvering targets is as follows:

[0096] Define the relative state difference between the defensive star and the pursuing star as: y = xd -x a The constraint differential equation is changed to:

[0097]

[0098] By introducing costate variables based on the Lagrange multiplier method, the bilateral optimization problem with differential equation equality constraints is transformed into an unconstrained bilateral optimization problem, and the auxiliary cost function is obtained by processing the cost function.

[0099]

[0100] Where λ and v are costate variables.

[0101] To facilitate the variational calculation of the auxiliary cost function, we first perform integration by parts on the auxiliary cost function:

[0102]

[0103] in:

[0104]

[0105]

[0106] Variational analysis of the auxiliary cost function:

[0107]

[0108] Since a necessary condition for solving this problem using the variational method is that the variation of the auxiliary cost function is 0, i.e., , the following necessary condition can be obtained:

[0109]

[0110]

[0111]

[0112]

[0113]

[0114]

[0115] The optimal control laws for both parties are obtained from formulas (15) and (16), namely:

[0116]

[0117]

[0118] Where λ(t) is the introduced Lagrange multiplier; ν(t) is the second introduced Lagrange multiplier.

[0119] S3: Input the optimal control laws of the pursuing and defending satellites into the spacecraft pursuit-escape-defense game problem model for non-maneuvering targets, and transform the spacecraft pursuit-escape-defense game problem for non-maneuvering targets into a problem of solving the Riccati equation.

[0120] The steps to transform the spacecraft pursuit and escape defense game problem against non-maneuvering targets into a problem of solving the Riccati equation are as follows:

[0121] S31: Substitute the optimal control laws of both parties into the dynamic constraint differential equations of both parties:

[0122]

[0123] S32: Define the joint state vector With joint costate variables Based on the state equation and the costate equation, the relationship between the joint state vector and the joint costate variables can be obtained as follows:

[0124] Let the matrices be:

[0125]

[0126]

[0127]

[0128]

[0129] The above relation can then be written as:

[0130]

[0131] S33: The terminal relationship between the joint state vector and the joint costate variables can be obtained based on the transverse condition:

[0132]

[0133] S34: There is a linear relationship between the joint state vector and the joint costate variables.

[0134] Λ(t)=P(t)K(t) (22) Differentiating both sides simultaneously, we get:

[0135]

[0136] Combining equations (21) and (22), we get:

[0137]

[0138] S35: The final solution to this problem is:

[0139]

[0140] in:

[0141] S a =[I4,-I4],S d =[04,I4] (26)

[0142] Wherein: S a For the control transformation matrix of the tracking star; S d This is the control transformation matrix for the defense star.

[0143] The matrix P(t) is obtained by solving the matrix Riccati equation shown in equation (24):

[0144] According to equation (21), the final value condition can be obtained as follows:

[0145]

[0146] S4: Solve the Riccati equation using backward integration and integrate the trajectory to finally solve the spacecraft pursuit and escape game problem against non-maneuvering targets.

[0147] S41: Starting with the final value of the Riccati equation, take the negative of the differential equation:

[0148]

[0149] S42: Use the ode45 integration algorithm in MATLAB to integrate and obtain P(t).

[0150] Substituting P(t) into equation (26) yields the control law, and substituting the control law into the state equation yields the state trajectories of both parties.

[0151] In summary, the spacecraft pursuit-escape game control model established in this invention for non-maneuvering targets is a differential game theory model based on linear CW equations. It has the advantages of model simplicity and small linearization error. According to the proposed method for solving the pursuit-escape game problem, the complex non-zero-sum game problem can be transformed into a zero-sum game problem, and further into a problem solving the Riccati equation, significantly improving solution efficiency, shortening solution time, and quickly obtaining the state trajectories of both the pursuer and the pursuer. When the primary satellite performing its original mission in orbit is approached by a pursuing satellite, the state trajectories of the surrounding defensive satellites can be adjusted in a timely manner based on the obtained state trajectories of both sides. This allows the defensive satellites around the primary satellite to intercept the pursuing satellite in a timely manner, preventing it from approaching the primary satellite and avoiding attacks on the primary satellite.

[0152] When this method is applied to real-world spacecraft game scenarios, it enhances the maneuverability of accompanying satellites and enables more precise interception of threats. When a spacecraft is threatened, the accompanying satellite is controlled to mitigate the threat, thereby ensuring the spacecraft's safe operation in orbit, saving fuel, and extending its operational time.

[0153] Example

[0154] This embodiment simulates a scenario where a primary satellite performing its original mission in orbit is approached by a pursuing satellite, and the primary satellite is equipped with a defensive satellite for protection. See also Figure 2 Suppose that at an initial time t0 = 0, there are 3 satellites near a circular orbit with an orbital radius of 500 km, namely the primary satellite m, the pursuing satellite a, and the defensive satellite d. The primary satellite cannot maneuver, while the pursuing and defensive satellites can maneuver in the plane. An LVLH coordinate system is established with the primary satellite as the origin. Its initial state is shown in Table 1, which shows the initial position and velocity (m, m / s) of the pursuing satellite a and the defensive satellite d.

[0155] Table 1

[0156]

[0157] Obtain the spacecraft pursuit-escape-defense game parameters for the pursuing and defending satellites, including the initial positions and velocities of the pursuing and defending satellites. The initial position of the pursuing satellite is [0, 20000m], and the initial velocity is 0. The initial position of the defending satellite is [-500m, 20000m], and the initial velocity is 0.

[0158] A game-theoretic model of spacecraft pursuit, escape, and defense against non-maneuverable targets is established, and the optimal control laws for the pursuing and defending satellites are obtained by using the Lagrange multiplier method and variational method.

[0159] The optimal control laws of the pursuing and defending satellites are input into the spacecraft pursuit-escape-defense game problem model for non-maneuverable targets, and the spacecraft pursuit-escape-defense game problem for non-maneuverable targets is transformed into a problem of solving the Riccati equation.

[0160] The Riccati equation is solved using backward integrals, and the trajectory is integrated to ultimately solve the game-theoretic control problem of spacecraft pursuit and escape against non-maneuvering targets. Based on the results, the trajectory of the defense satellite is controlled to intercept the pursuing satellite.

[0161] Figures 3 to 6 This is the state trajectory diagram obtained in this embodiment, wherein... Figure 3 To track the movement of the pursuing and defensive stars; Figure 4 The velocity components of the pursuing and defensive stars on the x and y axes; Figure 5 The changes in the x-axis and y-axis components of the acceleration of the pursuing and defending stars; Figure 6This represents the changes in the distance between the pursuing star and the primary star, and the distance between the defensive star and the pursuing star throughout the entire process.

[0162] The second objective of this invention is to propose a spacecraft pursuit and escape anti-game control method system for non-maneuvering targets, comprising:

[0163] Parameter Acquisition Module: Used to acquire spacecraft pursuit and escape game parameters for the pursuing and defending satellites;

[0164] Model building module: Used to establish a spacecraft pursuit and escape game control problem model for non-maneuverable targets using the spacecraft pursuit and escape game parameters of the pursuing and defending satellites, and obtain the optimal control law for the pursuing and defending satellites.

[0165] Transformation Equation Module: This module is used to transform the spacecraft pursuit-escape-defense game problem against non-maneuverable targets into the Riccati equation by obtaining the optimal control laws of the pursuing and defending satellites.

[0166] The solution module is used to solve the Riccati equation using backward integration and to integrate the trajectory. Based on the integrated result, the state trajectories of the pursuing and fleeing parties are obtained, thereby controlling the motion trajectory of the defense star.

[0167] A third objective of this invention is to provide an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the steps of the aforementioned spacecraft pursuit and escape anti-game control method for non-maneuvering targets.

[0168] The aforementioned spacecraft pursuit and escape anti-game control method for non-maneuvering targets includes the following steps:

[0169] Obtain the spacecraft pursuit and escape game parameters of the pursuing and defending satellites;

[0170] By utilizing the spacecraft pursuit and escape game parameters of the pursuing and defending satellites, a spacecraft pursuit and escape game control problem model is established for non-maneuvering targets, and the optimal control laws of the pursuing and defending satellites are obtained.

[0171] By obtaining the optimal control laws for the pursuing and defending satellites, the spacecraft pursuit-escape-defense game problem against non-maneuvering targets is transformed into the Riccati equation.

[0172] The Riccati equation is solved by using backward integration, and the trajectory is integrated. The state trajectories of the pursuer and the pursuer are obtained based on the integrated result, thereby controlling the trajectory of the defense star.

[0173] A fourth objective of this invention is to provide a computer-readable storage medium storing a computer program that, when executed by a processor, implements the steps of the aforementioned spacecraft pursuit and escape anti-game control method for non-maneuvering targets.

[0174] The aforementioned spacecraft pursuit and escape anti-game control method for non-maneuvering targets includes the following steps:

[0175] Obtain the spacecraft pursuit and escape game parameters of the pursuing and defending satellites;

[0176] By utilizing the spacecraft pursuit and escape game parameters of the pursuing and defending satellites, a spacecraft pursuit and escape game control problem model is established for non-maneuvering targets, and the optimal control laws of the pursuing and defending satellites are obtained.

[0177] By obtaining the optimal control laws for the pursuing and defending satellites, the spacecraft pursuit-escape-defense game problem against non-maneuvering targets is transformed into the Riccati equation.

[0178] The Riccati equation is solved by using backward integration, and the trajectory is integrated. The state trajectories of the pursuer and the pursuer are obtained based on the integrated result, thereby controlling the trajectory of the defense star.

[0179] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0180] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0181] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0182] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0183] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the specific implementation of the present invention. Any modifications or equivalent substitutions that do not depart from the spirit and scope of the present invention should be covered within the scope of protection of the claims of the present invention.

Claims

1. A spacecraft pursuit and escape anti-game control method for non-maneuvering targets, characterized in that, include: Obtain the spacecraft pursuit and escape game parameters of the pursuing and defending satellites; By utilizing the spacecraft pursuit and escape game parameters of the pursuing and defending satellites, a spacecraft pursuit and escape game control problem model is established for non-maneuvering targets, and the optimal control laws of the pursuing and defending satellites are obtained. By obtaining the optimal control laws for the pursuing and defending satellites, the spacecraft pursuit-escape-defense game problem against non-maneuvering targets is transformed into the Riccati equation. The Riccati equation is solved by using backward integration, and the trajectory is integrated. The state trajectories of the pursuer and the pursuer are obtained based on the integrated result, thereby controlling the movement trajectory of the defense star. The transformation of the spacecraft pursuit and escape game problem against non-maneuvering targets into a Riccati equation solution problem includes: Substituting the optimal control laws of the pursuing and defending satellites into the dynamic constraint differential equations of the pursuing and defending satellites: The joint state vector is obtained based on the above dynamic constraint equations. With joint costate variables The relationship between them is as follows: The terminal relationship between the joint state vector and the joint costate variables is obtained based on the transverse condition: There is a linear relationship between the joint state vector and the joint costate variables: Differentiating both sides simultaneously, we get: simultaneous formula and have to: in: The weight matrix for the terminal distance term between the tracking star and the primary star; The weight matrix for the terminal distance term between the tracking star and the primary star; The state vector of the tracking star; The weight matrix for the fuel consumption of the tracking star; The weight matrix for the fuel consumption of the defense star; For the introduction of Lagrange multipliers; This is the second Lagrange multiplier introduced; This is the relationship matrix between the joint state vector and the joint costate vector; The final moment of the game; The game-theoretic problem of spacecraft pursuit and escape against non-maneuvering targets is transformed into a problem of solving the Riccati equation. The solution to this problem is: in: The continuous control amount applied to the tracking star; The continuous control quantity applied to the defensive star; The weight matrix for the fuel consumption of the tracking star; The weight matrix for the fuel consumption of the defense star; For controlling the conversion matrix of the tracking star; For the defense star control conversion matrix; For joint costate variables; The method of solving the Riccati equation using backward integrals includes: Starting with the final value of the Riccati equation, we take the negative of the differential equation: 。 2. The spacecraft pursuit and escape anti-game control method for non-maneuvering targets according to claim 1, characterized in that, The spacecraft pursuit-escape game parameters for obtaining the pursuing and defending satellites are as follows: The spacecraft pursuit and escape game parameters for the pursuing and defending stars include the initial position of the pursuing star, the initial position of the defending star, the velocity of the pursuing star, and the velocity of the defending star.

3. The spacecraft pursuit and escape anti-game control method for non-maneuvering targets according to claim 1, characterized in that, The establishment of a game-theoretic control model for spacecraft pursuit and escape against non-maneuvering targets includes: An LVLH coordinate system is established with the position of the primary star t as the origin. Based on the LVLH coordinate system and the spacecraft pursuit and escape game parameters of the pursuing and defensive stars, a cost function is designed. Based on the CW equations, we determine the differential equations of the dynamic constraints that need to be satisfied by the pursuing and defending stars in space orbits. Based on the above process, a spacecraft pursuit and escape defense game model for non-maneuvering targets was completed, based on differential game theory.

4. The spacecraft pursuit and escape anti-game control method for non-maneuvering targets according to claim 3, characterized in that, The establishment of an LVLH coordinate system with the position of the primary star t as the origin, and the design of a cost function based on the spacecraft pursuit-escape game parameters of the pursuing and defending stars in the LVLH coordinate system, including: The cost function consists of two parts: a terminal term and an integral term. Terminal Item The meaning is the relative state between the pursuing satellite and the primary satellite at the terminal moment, and the relative state between the defensive satellite and the primary satellite; integral term The meaning is the fuel consumption of both sides throughout the entire game; in, The cost function for the tracking star; The cost function for the defensive star; The state vector of the tracking star; The state vector of the defense star; The final moment of the game; The weight matrix for the terminal distance term between the tracking star and the primary star; The weight matrix for the defensive-targeting satellite terminal distance term; The weight matrix for the fuel consumption of the tracking star; The weight matrix for the fuel consumption of the defense star; The continuous control amount applied to the tracking star; The continuous control amount applied to the defensive star.

5. The spacecraft pursuit and escape anti-game control method for non-maneuvering targets according to claim 1, characterized in that, The optimal control laws for obtaining the pursuing and defensive satellites include: By introducing costate variables using the Lagrange multiplier method, the bilateral optimization problem with differential equation equality constraints is transformed into an unconstrained bilateral optimization problem. The auxiliary cost function is then obtained by processing the cost function. in: The state vector of the tracking star; The final moment of the game; The difference in status between defensive and pursuing stars; The weight matrix for the terminal distance term between the tracking star and the primary star; The weight matrix for the defensive-targeting satellite terminal distance term; The weight matrix for the fuel consumption of the tracking star; The weight matrix for the fuel consumption of the defense star; The continuous control amount applied to the tracking star; The continuous control quantity applied to the defensive star; For the introduction of Lagrange multipliers; This is the second Lagrange multiplier introduced.

6. The spacecraft pursuit and escape anti-game control method for non-maneuvering targets according to claim 5, characterized in that, The cost function is processed to obtain an auxiliary cost function, and then the variable integral of the auxiliary cost function is calculated: in: in: For the introduction of Lagrange multipliers; (t) represents the second Lagrange multiplier introduced; The state vector of the tracking star; The state vector of the defense star; The continuous control amount applied to the tracking star; The continuous control quantity applied to the defensive star; For terminal constraint functions; It is a Hamiltonian function; The optimal control laws for the pursuing and defensive stars are obtained as follows: in: The weight matrix for the fuel consumption of the tracking star; The weight matrix for the fuel consumption of the defense star; The continuous control amount applied to the tracking star; The continuous control quantity applied to the defensive star; For the introduction of Lagrange multipliers; This is the second Lagrange multiplier introduced.

7. The spacecraft pursuit and escape anti-game control method for non-maneuvering targets according to claim 1, characterized in that, The method involves solving the Riccati equation using backward integration and integrating over the trajectory. The integration over the trajectory is performed using an integration algorithm to obtain a matrix. .

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