A spacecraft escort control method based on zero-sum differential game

Abstract

This invention provides a spacecraft escort control method based on zero-sum differential games, belonging to the field of aerospace technology. The control method includes the following steps: obtaining spacecraft escort problem parameters for both the pursuing and defending satellites; establishing a spacecraft escort model based on zero-sum differential games and inputting initial parameters into the model; transforming the spacecraft escort model into a two-point boundary value problem model; using a genetic algorithm to solve the two-point boundary value problem model to obtain the state trajectory within the game time, thus completing the spacecraft escort control. This spacecraft escort control method can effectively solve the spacecraft escort problem under continuous thrust, and is an effective extension of existing continuous thrust escort problem models and methods. The model employs a differential game method described by a zero-sum cost function, thus simplifying the solution of the difficult-to-solve weighted performance index escort problem and improving the solution efficiency of the spacecraft escort game problem.

Description

Technical Field

[0001] This invention belongs to the field of aerospace technology, specifically relating to a spacecraft escort control method based on zero-sum differential games. Background Technology

[0002] Spacecraft orbital pursuit is a widely studied engineering problem. In classic orbital pursuit problems, the participants aim to either move as far away from or as close as possible to each other. However, some spacecraft in orbit, due to their high-value fuel requirements for on-orbit operations, cannot autonomously evade when threatened by a pursuing satellite and must deploy companion satellites as defensive satellites to coordinate countermeasures. The struggle for position between the defensive and pursuing satellites over the host satellite is known as the spacecraft escort problem. The pursuing satellite aims to approach the host satellite while ensuring it is not intercepted or destroyed by the defensive satellite; the defensive satellite aims to prevent the pursuing satellite from approaching the host satellite as much as possible.

[0003] Currently, most modeling and solutions for spacecraft escort problems use non-zero-sum differential game methods. However, since it is a non-zero-sum game in the game theory category, the cost functions of the pursuing and defending satellites are not the same, making the solution for its optimal control strategy very complex. Summary of the Invention

[0004] To overcome the shortcomings of the prior art, the present invention aims to provide a spacecraft escort control method based on zero-sum differential games. A spacecraft escort problem model based on zero-sum differential games is established, and the game objectives of the pursuing satellite and the defending satellite are described by a pair of cost functions with opposite values. The spacecraft escort problem is transformed into a zero-sum differential game problem by the above method, which greatly simplifies the solution process and has achieved good results in simulation.

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

[0006] This invention provides a spacecraft escort control method based on zero-sum differential games, comprising the following steps:

[0007] S1: Obtain spacecraft escort parameters for the pursuing and defending satellites;

[0008] S2: Establish a spacecraft escort model based on zero-sum differential games, and input the spacecraft escort parameters of the pursuing star and the defending star into the spacecraft escort model to obtain the optimal control law of the pursuing star and the defending star.

[0009] S3: Based on the optimal control laws of the pursuing and defending satellites and the differential equations of the dynamic constraints that the pursuing and defending satellites need to satisfy in space orbit, the spacecraft escort model is transformed into a two-point boundary value problem model.

[0010] S4: Use a genetic algorithm to solve the two-point boundary value problem model, obtain the state trajectory during the game time, and complete the spacecraft escort control.

[0011] In the specific implementation process, the process of establishing the spacecraft protection model based on zero-sum differential games in S2 is as follows:

[0012] S11: Establish an LVLH coordinate system with the position of the primary star as the origin, and establish a cost function based on the LVLH coordinate system and the objectives of the pursuing and defensive stars.

[0013] S12: Based on the cost function and the CW equation, determine the differential equations of the dynamic constraints that the pursuing and defending stars need to satisfy in space orbit, and establish a spacecraft escort model based on zero-sum differential games.

[0014] In the specific implementation process, the cost function in S11 is:

[0015] ;

[0016] Among them, the symmetric positive semi-definite matrix

[0017] , , , , , satisfy:

[0018] , , , , , ;

[0019] In the above formula:

[0020] , These are the weighting coefficients;

[0021] , , , , , ;

[0022] Where 'a' is the subscript of the pursuing star; 'd' is the subscript of the defensive star; The state vector of the tracking star. ; The state vector of the defense star. ; The radial position component of the orbit in the LVLH system; The position component of the flight direction in the LVLH system; The radial velocity component of the orbit in the LVLH system; For the velocity components in the orbital flight direction under the LVLH system; The continuous control amount applied to the tracking star. ; The continuous control amount applied to the defensive star. ; For the cost functions of both the pursuing and defensive stars; The time when the game ends; This is the start time of the game.

[0023] In practical implementation, the differential equations for the dynamic constraints that the pursuing and defending stars need to satisfy in space orbit are as follows:

[0024] ;

[0025] Among them, matrix This represents the dynamic constraint matrix for the relative motion of the spacecraft. The control matrix for the pursuing and defensive stars, wherein, ;

[0026] The matrix The matrix is ​​constructed using the CW equation. as follows:

[0027]

[0028] in, This represents the angular velocity of the Earth's orbit around the primary star. The coefficient of Earth's gravitational field. , The semi-major axis of the main star's orbit.

[0029] In the specific implementation process, the process of obtaining the optimal control law for the pursuing star and the defensive star is as follows:

[0030] Introducing costate variables based on the Lagrange multiplier method and The bilateral optimization problem with differential equations and equality constraints is transformed into an unconstrained bilateral optimization problem, and the auxiliary cost function is obtained by processing the cost function.

[0031] By processing the auxiliary cost function and solving it under set conditions, the optimal control law for the pursuing and defensive stars is obtained.

[0032] In the specific implementation process, the auxiliary cost function is:

[0033] ;

[0034] The setting conditions are:

[0035] ; ; ; ; ; ;

[0036] The optimal control laws for the pursuing and defensive satellites are as follows:

[0037] ;

[0038] .

[0039] In practical implementation, the two-point boundary value problem model is as follows:

[0040] .

[0041] In practical implementation, the steps for solving the two-point boundary value problem model using a genetic algorithm are as follows:

[0042] S21: Set the population as the initial value of the costate variable. and ;

[0043] S22: Initial values and Perform integration to obtain the terminal values ​​of the costate variables. ;

[0044] S23: Combine the population's fitness function to obtain the initial values ​​of the costate variables that minimize the fitness function. ;

[0045] The fitness function of the population is: .

[0046] In the specific implementation process, the process of obtaining the state trajectory within the game time is as follows:

[0047] by The initial values ​​of the variables are used to solve the differential equations in the two-point boundary value problem model, so as to obtain the trajectory of the pursuing star and the defending star during the game time, the velocity changes of the pursuing star and the defending star during the game time, the distance changes between the pursuing star and the defending star and the distance changes between the pursuing star and the main star during the game time, and thus obtain the state trajectory during the game time.

[0048] Compared with the prior art, the present invention has the following beneficial effects:

[0049] This invention provides a spacecraft escort control method based on zero-sum differential games. The solution method transforms a complex non-zero-sum game problem into a zero-sum game problem, and further into a two-point boundary value problem, significantly improving the solution efficiency. The spacecraft escort problem model is a differential game model based on linear CW equations, which has the advantages of model simplicity and small linearization error. The above model and method can provide an effective threat solution for spacecraft employing master-slave satellite cooperation. Attached Figure Description

[0050] Figure 1 This is a game-theoretic scenario diagram for the spacecraft protection problem to which this invention applies;

[0051] Figure 2 This is a flowchart illustrating the specific implementation of the spacecraft protection problem of the present invention;

[0052] Figure 3 This is a simulation trajectory diagram of the present invention;

[0053] Figure 4 This is a simulation speed change graph of the present invention;

[0054] Figure 5 This is a diagram showing the distances between the pursuing star and the defensive star, as well as the distances between the pursuing star and the primary star, according to the present invention. Detailed Implementation

[0055] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0056] It should be noted that the terms "first," "second," etc., in the specification, claims, and accompanying drawings of this invention are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate so that the embodiments of the invention described herein can be implemented in orders other than those illustrated or described herein. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover a non-exclusive inclusion; for example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0057] The present invention will now be described in further detail with reference to the accompanying drawings:

[0058] This invention discloses a spacecraft escort control method based on zero-sum differential games. The specific contents include: firstly, establishing a spacecraft escort model based on differential games, including optimization indices, optimization variables, and dynamic constraints; wherein the optimization indices are described by a weighted quadratic function that includes inter-satellite distance and fuel consumption, the optimization variables are the continuous control force accelerations of the pursuing and defending sides, and the dynamic constraints are described by the CW equation under the relative motion model; then, establishing a solution method (escort control method) for the above optimization problem.

[0059] like Figure 2 As shown, it includes the following steps:

[0060] Step 1: Obtain the spacecraft escort parameters for the pursuing and defending satellites;

[0061] Step 2: Establish a spacecraft protection model based on zero-sum differential games, and input the initial parameters into the spacecraft protection model;

[0062] Step 3: Transform the spacecraft escort model into a 16-dimensional two-point boundary value problem model;

[0063] Step 4: Use a genetic algorithm to solve the 16-dimensional two-point boundary value problem model, obtain the state trajectory within the game time, and complete the spacecraft escort control.

[0064] This invention provides a spacecraft escort control method based on zero-sum differential games, the specific steps of which are as follows:

[0065] S1: Obtain the spacecraft escort parameters of the pursuing and defending satellites; including 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;

[0066] S2: Establish a spacecraft escort model based on zero-sum differential game theory. This involves establishing an LVLH coordinate system with the host satellite's position as the origin, and then establishing a cost function based on the LVLH coordinate system and the objectives of the pursuing and defending satellites. Based on the cost function and the CW equations, determine the differential equations of the dynamic constraints that the pursuing and defending satellites must satisfy in their space orbits, thus completing the model establishment. Input the spacecraft escort parameters of the pursuing and defending satellites into the spacecraft escort model to obtain the optimal control laws for the pursuing and defending satellites. The spacecraft escort model is a differential game theory model based on linear CW equations, possessing the advantages of model simplicity and very small linearization error.

[0067] The cost function is as follows:

[0068] (1)

[0069] The cost function is divided into two parts: a terminal term and an integral term. The terminal term represents the relative states between the pursuing star and the primary star at the final moment, and between the defending star and the primary star. The integral term represents the relative states between the pursuing star and the primary star, and between the defending star and the primary star, as well as the fuel consumption of both sides throughout the entire game.

[0070] The differential equations for the dynamic constraints that the aforementioned tracking and defense stars must satisfy in space orbits are as follows: (2)

[0071] Among them, the symmetric positive semi-definite matrix

[0072] , , , , , satisfy:

[0073] , , , , , (3)

[0074] In the above formula:

[0075] , These are the weighting coefficients;

[0076] , , , , , ;

[0077] The meanings of the other symbols in equations (1) to (4) are as follows:

[0078] a — the subscript of the chasing star;

[0079] d —The subscript of the defense star;

[0080] —The state vector of the tracking star, specifically ;

[0081] —The state vector of the defense star, specifically ;

[0082] —The radial position component of the orbit in the relative coordinate system (LVLH system);

[0083] —Position component of the flight direction in a relative coordinate system (LVLH system);

[0084] —The radial velocity component of the orbit in the relative coordinate system (LVLH system);

[0085] —The velocity components in the orbital flight direction under a relative coordinate system (LVLH system);

[0086] —The continuous control quantity applied by the tracking star, specifically in the form of ;

[0087] —The continuous control applied by the defensive star, specifically in the form of ;

[0088] —The control matrix for the pursuing and defensive stars, specifically in the form of ;

[0089] —The cost functions for both the pursuing and defensive stars;

[0090] —The end time of the game;

[0091] —The start time of the game;

[0092] Among them, matrix This represents the dynamic constraint matrix for the relative motion of the spacecraft. The control matrix for the pursuing and defensive stars, wherein, ;

[0093] The matrix The matrix is ​​constructed using the CW equation. as follows:

[0094] (4)

[0095] in, This represents the angular velocity of the Earth's orbit around the primary star. The coefficient of Earth's gravitational field. , The semi-major axis of the main star's orbit.

[0096] The process of obtaining the optimal control law for the pursuit star and the defense star is as follows:

[0097] S11: Introducing Costate Variables Based on the Lagrange Multiplier Method and The bilateral optimization problem with differential equations and equality constraints is transformed into an unconstrained bilateral optimization problem, and the auxiliary cost function is obtained by processing the cost function.

[0098] (5)

[0099] in, and These are costate variables.

[0100] S12: To facilitate the variational calculation of the auxiliary cost function, we first perform integration by parts on the auxiliary cost function:

[0101] (6)

[0102] in:

[0103] (7)

[0104] (8)

[0105] S13: Taking the variation of the auxiliary cost function:

[0106] (9)

[0107] S14: 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:

[0108] (10)

[0109] (11)

[0110] (12)

[0111] (13)

[0112] (14)

[0113] (15)

[0114] The optimal control laws for both parties are derived from the latter two equations, namely:

[0115] (16)

[0116] (17).

[0117] S3: Based on the optimal control laws of the pursuing and defending stars and the differential equations of the dynamic constraints that the pursuing and defending stars need to satisfy in the space orbit, the spacecraft escort model is transformed into a 16-dimensional two-point boundary value problem model; that is, the optimal control laws of both sides are substituted into the dynamic constraint differential equations of both sides, and combined with the necessary conditions, namely equations (10)-(17), to obtain the following two-point boundary value problem, which transforms the complex non-zero-sum game problem into a zero-sum game problem, and further into a two-point boundary value problem, significantly improving the solution efficiency of the problem.

[0118] Specifically, it includes the following steps: Step 1: Use the variational principle to obtain the necessary conditions for the saddle point solution of the differential game problem; Step 2: Combine the necessary conditions for the saddle point solution with the initial conditions of the game to form a 16-dimensional two-point boundary value problem.

[0119] The model for the two-point boundary value problem is as follows:

[0120] (18)

[0121] S4: Using a genetic algorithm, the population is set as the initial value of the costate variable, and the fitness function is the modulus of the difference between the final integral value and the theoretical final value of the costate variable. The 16-dimensional two-point boundary value problem model is solved to obtain the change curves of various state variables of the pursuing and defending sides during the game time, thus obtaining the solution to the spacecraft escort problem and completing the spacecraft escort control.

[0122] The specific steps are as follows:

[0123] S41: Set the population as the initial value of the costate variable. and ;

[0124] S42: Initial values and We perform integration by integrating the differential equations in the two-point boundary value problem to obtain the integrated terminal values ​​of the costate variables. .

[0125] S43: Set the fitness function of the population to... ,in, This represents the final value of a costate variable obtained by integrating the initial value of the costate variable through the differential equation in a two-point boundary value problem. Physically, it represents the error between the final value obtained by integrating the initial value of the costate variable through the differential equation in a two-point boundary value problem and the condition that the final value of the costate variable should satisfy. The smaller this error, the higher the fitness of the individuals in the population.

[0126] S44: Obtain the fitness function Minimum initial value of costate variable .

[0127] S45: with The initial values ​​of the variables are used to calculate the differential equations in the two-point boundary value problem, obtaining the trajectories of the pursuing and defending stars during the game time, the velocity changes of the pursuing and defending stars during the game time, the distance changes between the pursuing and defending stars and the distance changes between the pursuing and defending stars during the game time, and the distance changes between the pursuing and defending stars and the main star. The change curves of various state quantities of the pursuing and defending sides during the game time are obtained, and the solution to the spacecraft escort problem is obtained, thus completing the spacecraft escort control.

[0128] Example

[0129] See Figure 1 Suppose at a certain initial moment Near a circular orbit with a radius of 7000km, there are three satellites: 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.

[0130] Table 1

[0131]

[0132] The genetic algorithm is used to solve the above two-point boundary value problem. The specific steps are as follows:

[0133] Input the relevant parameters for the spacecraft escort problem, including: the semi-major axis of the reference satellite orbit. Initial state of the chasing star Initial state of the defensive star Coefficients of each weight matrix , , , , , Game termination time ;

[0134] Set the population as a costate variable. and initial value and The fitness function is ,in The meaning represented is the final value obtained by integrating the initial value of the costate variable through the differential equation in the two-point boundary value problem.

[0135] Find the initial values ​​of the costate variables that minimize the fitness function. and .

[0136] by Integrating the differential equation in the two-point boundary value problem with the initial values ​​of the variables, we obtain the following: Figure 1 , Figure 3 , Figure 4 and Figure 5 The motion trajectories of the pursuing and defending stars during the game time, the velocity changes of the pursuing and defending stars during the game time, the distance changes between the pursuing and defending stars and the distance changes between the pursuing and the primary star during the game time are shown. The state trajectories are obtained, and the spacecraft escort problem is solved.

[0137] The proposed spacecraft escort control method effectively solves the spacecraft escort problem under continuous thrust, representing an effective extension of existing models and methods for continuous thrust escort problems. This model employs a differential game method described by a zero-sum cost function, thus simplifying the solution of the difficult-to-solve weighted performance index escort problem and improving the efficiency of solving the spacecraft escort game problem.

[0138] The above content is only for illustrating the technical concept of the present invention and should not be construed as limiting the scope of protection of the present invention. Any modifications made to the technical solution based on the technical concept proposed in this invention shall fall within the scope of protection of the claims of this invention.

Claims

1. A spacecraft escort control method based on zero-sum differential games, characterized in that, Includes the following steps: S1: Obtain spacecraft escort parameters for the pursuing and defending satellites; S2: Establish a spacecraft escort model based on zero-sum differential games, and input the spacecraft escort parameters of the pursuing and defending satellites into the spacecraft escort model to obtain the optimal control laws for the pursuing and defending satellites; wherein, the process of establishing the spacecraft escort model based on zero-sum differential games in S2 is as follows: S11: Establish an LVLH coordinate system with the position of the primary star as the origin, and establish a cost function based on the LVLH coordinate system and the objectives of the pursuing and defensive stars. S12: Based on the cost function and the CW equation, determine the differential equations of the dynamic constraints that the pursuing and defending stars need to satisfy in space orbit, and establish a spacecraft escort model based on zero-sum differential games. S3: Based on the optimal control laws of the pursuing and defending satellites and the differential equations of the dynamic constraints that the pursuing and defending satellites need to satisfy in space orbit, the spacecraft escort model is transformed into a two-point boundary value problem model. S4: Use a genetic algorithm to solve the two-point boundary value problem model, obtain the state trajectory during the game time, and complete the spacecraft escort control; the steps of using a genetic algorithm to solve the two-point boundary value problem model are as follows: S21: Set the population as the initial value of the costate variable. and ; S22: Initial values and Perform integration to obtain the terminal values ​​of the costate variables. ; S23: Combining the fitness function of the population, obtain the initial values ​​of the costate variables that minimize the fitness function. ; The fitness function of the population is: ; Among them, the symmetric positive semi-definite matrix , ; satisfy: , ; ; These are the weighting coefficients; , ; The radial position component of the orbit of the tracking star in the LVLH coordinate system; The radial position component of the orbit of the tracking star in the LVLH coordinate system.

2. The spacecraft escort control method based on zero-sum differential games according to claim 1, characterized in that, In S11, the cost function is: ； Among them, the symmetric positive semi-definite matrix , , , , , satisfy: ， ， ， ， ， ； In the above formula: , These are the weighting coefficients; ， ， ， ， ， ； Where 'a' is the subscript of the pursuing star; 'd' is the subscript of the defensive star; The state vector of the tracking star. ; The state vector of the defense star. ; The radial position component of the orbit in the LVLH system; The position component of the flight direction in the LVLH system; The radial velocity component of the orbit in the LVLH system; For the velocity components in the orbital flight direction under the LVLH system; The continuous control amount applied to the tracking star. ; The continuous control amount applied to the defensive star. ; For the cost functions of both the pursuing and defensive stars; The time when the game ends; This is the start time of the game.

3. The spacecraft escort control method based on zero-sum differential games according to claim 2, characterized in that, The differential equations for the dynamic constraints that the pursuing and defending stars must satisfy in space orbit are as follows: ； Among them, matrix This represents the dynamic constraint matrix for the relative motion of the spacecraft. The control matrix for the pursuing and defensive stars, wherein, ; The matrix The matrix is ​​constructed using the CW equation. as follows: in, This represents the angular velocity of Earth's orbit around the primary star. The coefficient of Earth's gravitational field. , The semi-major axis of the main star's orbit.

4. The spacecraft escort control method based on zero-sum differential games according to claim 3, characterized in that, The process of obtaining the optimal control law for the pursuit star and the defense star is as follows: Introducing costate variables based on the Lagrange multiplier method and The bilateral optimization problem with differential equations and equality constraints is transformed into an unconstrained bilateral optimization problem, and the auxiliary cost function is obtained by processing the cost function. By processing the auxiliary cost function and solving it under set conditions, the optimal control law for the pursuing and defensive stars is obtained.

5. The spacecraft escort control method based on zero-sum differential games according to claim 4, characterized in that, The auxiliary cost function is: ； The setting conditions are: ； ； ； ； ； ； The optimal control laws for the pursuing and defensive satellites are as follows: ； 。 6. The spacecraft escort control method based on zero-sum differential games according to claim 5, characterized in that, The model for the two-point boundary value problem is as follows: 。 7. The spacecraft escort control method based on zero-sum differential games according to claim 6, characterized in that, The process of obtaining the state trajectory within the game time is as follows: by The initial values ​​of the variables are used to solve the differential equations in the two-point boundary value problem model, so as to obtain the trajectory of the pursuing star and the defending star during the game time, the velocity changes of the pursuing star and the defending star during the game time, the distance changes between the pursuing star and the defending star and the distance changes between the pursuing star and the main star during the game time, and thus obtain the state trajectory during the game time.

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