Low-dimensional control method for pursuit-evasion game of underactuated spacecraft

Abstract

The application discloses a kind of low-dimensional control methods for radial underactuated spacecraft pursuit-evasion game, comprising the following steps: step S1, based on differential game theory, construct the 24-dimensional two-point boundary value problem of radial underactuated spacecraft optimal time game and high-dimensional game control strategy;Step S2, deduce the state condition of reducing underactuated high-dimensional two-point boundary value to 12-dimensional two-point boundary value problem, and design low-dimensional game optimal control strategy.The application demonstrates the feasibility of underactuated pursuit-evasion game when radial thrust is missing, deduces the low-dimensional optimal time game control strategy for realizing three-dimensional pursuit and evasion only through trace and normal thrust, expands the theoretical analysis and engineering application of underactuated spacecraft, and the method can be used for underactuated spacecraft interception, on-orbit service, approach of out-of-control spacecraft and other applications.

Description

Technical Field

[0001] This invention relates to the field of spacecraft pursuit and escape game, and more particularly to a low-dimensional control method for radially underactuated spacecraft pursuit and escape game. Background Technology

[0002] Since the 1960s, with the increasing frequency of space launches, the number of spacecraft and space debris in space has been rapidly increasing. Spacecraft chase-escape game theory has received increasing attention in recent years due to its promising applications in clearing space debris and evading or approaching runaway spacecraft. Unlike the classic one-sided optimization problem of intercepting or rendezvous and docking non-maneuvering target spacecraft, the chase-escape game problem involves a conflict of interest: the tracking spacecraft attempts to approach the escape spacecraft, while the escape spacecraft tries to escape the tracking spacecraft. Therefore, this problem is usually viewed as a two-sided optimization problem and studied using differential game theory. Differential game theory, proposed by Isaacs, is a branch of game theory that can be used to analyze various dynamic game problems such as missile interception, aircraft combat, and drone swarm flight.

[0003] It is worth noting that almost all current game strategies are based on fully driven relative orbital dynamics design. Within the local-vertical-local-horizontal coordinate system framework, in the underactuated case where radial or directional thrust is lost, the degrees of freedom of the control input are less than the degrees of freedom of the controlled variables. Therefore, fully driven game strategies are not applicable to underactuated situations. According to linear system theory, the system is controllable when radial thrust is missing. Therefore, the following problems need to be solved in the underactuated spacecraft pursuit-escape game: First, obtaining the rendezvous or formation flight of underactuated spacecraft through optimal control methods is a one-sided optimal feedback strategy, while the underactuated pursuit-escape game is a two-sided optimization problem. Second, unlike the fully driven pursuit-escape game, the lack of radial thrust may impose dynamic constraints on the player's flight trajectory. Next, how to construct a high-dimensional two-point boundary value problem using differential game theory and give a high-dimensional underactuated saddle point strategy is also a key research point. Finally, deriving the state of reducing the high-dimensional game problem to a low-dimensional problem and giving a low-dimensional game strategy to drive the tracker and escaper to achieve a three-dimensional pursuit-escape game is also a major challenge. Summary of the Invention

[0004] To address the technical problems existing in the prior art, the present invention aims to provide a low-dimensional control method for radially underactuated spacecraft pursuit-escape game. This method derives the underactuated game strategy under optimal time using differential game theory, constructing a 24-dimensional two-point boundary value problem. Then, it derives a method to reduce the high-dimensional two-point boundary value problem to a 12-dimensional low-dimensional two-point boundary value problem and obtains the low-dimensional game strategy, enabling the tracker and escaper to complete a three-dimensional zero-sum pursuit-escape game under the action of trajectory and normal thrust.

[0005] To achieve the above-mentioned objectives, this invention provides a low-dimensional control method for radially underactuated spacecraft in a pursuit-escape game, comprising the following steps:

[0006] Step S1: Based on differential game theory, construct a 24-dimensional two-point boundary value problem and a high-dimensional game optimal control strategy for the radial underactuated spacecraft optimal time game.

[0007] Step S2: Derive the state conditions for reducing the underactuated high-dimensional two-point boundary value problem to a 12-dimensional two-point boundary value problem and design the optimal control strategy for low-dimensional game theory.

[0008] According to a technical solution of the present invention, step S1 specifically includes:

[0009] Step S11: Given a chase-escape game scenario: Assume a virtual navigator spacecraft is flying in a near-Earth circular orbit, with a radially underactuated tracker and a radially underactuated escape device nearby. Construct a local-vertical-local-horizontal coordinate system with the navigator's center of mass as the origin. Then, the player... or The relative distance and relative speed with the navigator are respectively expressed as: and ; where subscript or These represent the tracker and the escape device, respectively.

[0010] The underradial driving dynamics equation for the pursuit-escape game is expressed as follows:

[0011] (1)

[0012] in, Indicates underactuated control acceleration and has , and They represent players respectively Acceleration is achieved through control of the trajectory and normal, while This represents nonlinear relative orbital dynamics, specifically expressed as...

[0013]

[0014] in, For the dimensional perspective of the navigator, and These are angular velocity and angular acceleration, respectively. Indicates the orbital radius of the navigator. Indicates player or orbital radius, And there are , and Representing the navigator and the player respectively. orbital angular velocity, This is Earth's gravitational constant;

[0015] The above equation (1) is linearized as follows:

[0016] (2)

[0017] In the formula, This represents the player's initial state, which includes relative position and relative velocity. , For the system matrix, , , For control matrix;

[0018] Step S12: Based on differential game theory, the differential game between the radial underactuated tracker and the escaper is constructed as follows:

[0019] (3)

[0020] In the formula, This represents the acceleration controlled by player P. This represents the acceleration controlled by player E, and has... ;

[0021] In the optimal time differential game, the tracker attempts to minimize the interception time. The escape vehicle is captured, while the escape vehicle tries to maximize the interception time. The quadratic cost function of the underdriven pursuit game is constructed as follows:

[0022] (4)

[0023] in, This represents a performance index for quadratic differential games.

[0024] Step S13: Solve for saddle point policy pairs, introducing the underactuated Hamiltonian function. and underactuated terminal conditions underdriven Hamiltonian function and underactuated terminal conditions They are respectively represented as

[0025] (5)

[0026] (6)

[0027] in, Represents the Lagrange multipliers. For the tracker's costate variables, ; For the costate variables of the escaper, ; and These are the position and velocity components of the tracker's costate variables, respectively. and Let be the position and velocity components of the escapement costate variables, respectively, and satisfy the following relationship:

[0028] (7)

[0029] Saddle point strategies satisfy the adjoint equation and terminal boundary conditions ,in, or Then the accompanying equation and terminal boundary conditions Represented as

[0030] (8)

[0031] (9)

[0032] According to the transverse condition The following equation is obtained:

[0033] (10)

[0034] The optimal time game strategies for the under-radial driven tracker and the escaper are as follows:

[0035] (11)

[0036] (12)

[0037] In the formula, and These represent the maximum accelerations of the tracker and the escape vehicle, respectively. This represents the high-dimensional optimal time game strategy for player P. This represents the optimal time game strategy for player E in a higher dimension;

[0038] Equations (6) to (10) constitute the 24-dimensional two-point boundary value problem for two players in the radial underactuated case. The underactuated saddle point strategy pairs (11) and (12) are the control laws for the underactuated game tracker and escaper. Under the action of the control laws, the tracker and escaper can complete the three-dimensional chase-escape game.

[0039] According to one technical solution of the present invention, step S2 specifically includes:

[0040] Step S21, Definition This is an error state. For new costate variables and and These represent the position and velocity components of the costate variable, respectively.

[0041] Then the differential game model of equation (3) and the adjoint equation of equation (8) are simplified to:

[0042] (13)

[0043] According to the definition of error state, the initial condition is:

[0044] (14)

[0045] The location conditions for interception are expressed as follows:

[0046] (15)

[0047] According to the transverse condition of equation (9), the new costate variable satisfies the following equation.

[0048] (16)

[0049] Step S22: According to equation (7), the costate variables of the tracker and the escaper satisfy the following relationship in differential game theory.

[0050] (17)

[0051] The high-dimensional game strategies shown in equations (11) and (12) can be rewritten in the following low-dimensional form:

[0052] (18)

[0053] (19)

[0054] in, Let P represent the low-dimensional optimal time game strategy. Denotes the low-dimensional optimal time game strategy of player E;

[0055] Equations (13) to (19) constitute the low-dimensional two-point boundary value problem of radial underactuated spacecraft pursuit and escape game.

[0056] Compared with the prior art, the present invention has the following beneficial effects:

[0057] The low-dimensional control method for radially underactuated spacecraft pursuit-escape game provided by this invention constructs an underactuated game cost function based on differential game theory and derives the underactuated saddle point strategy pair under complete information as the three-dimensional optimal time game control law, which expands the theoretical analysis and engineering application of underactuated spacecraft. The spacecraft can still conduct three-dimensional pursuit-escape game even if it loses radial thrust. The proposed method can be used for applications such as underactuated spacecraft interception, on-orbit servicing, and approaching runaway spacecraft. Attached Figure Description

[0058] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly described below. Obviously, the drawings described below are merely some embodiments of the present invention, and those skilled in the art can obtain other drawings based on these drawings without creative effort.

[0059] Figure 1 This schematic diagram illustrates a low-dimensional control method for radially underactuated spacecraft pursuit and escape game according to one embodiment of the present invention.

[0060] Figure 2 This illustration illustrates a sub-radial drive pursuit-escape game trajectory according to one embodiment of the present invention.

[0061] Figure 3 A schematic diagram illustrating the time history of the relative distance between a tracker and an escape device according to an embodiment of the present invention;

[0062] Figure 4 A schematic diagram illustrating the time history of the trace and normal control inputs of a tracker according to an embodiment of the present invention;

[0063] Figure 5 This is a schematic diagram illustrating the time history of the trajectory and normal control inputs of an escape device according to an embodiment of the present invention. Detailed Implementation

[0064] The description of the embodiments in this specification should be taken in conjunction with the accompanying drawings, which should form part of the complete specification. In the drawings, the shape or thickness of the embodiments may be exaggerated and may be indicated in a simplified or convenient manner. Furthermore, parts of the various structures in the drawings will be described separately; it is worth noting that elements not shown in the figures or not described in words are in a form known to those skilled in the art.

[0065] The descriptions of the embodiments herein, including any references to directions and orientations, are for ease of description only and should not be construed as limiting the scope of the invention. The following description of preferred embodiments involves combinations of features, which may exist independently or in combination; the invention is not particularly limited to the preferred embodiments. The scope of the invention is defined by the claims.

[0066] like Figure 1 As shown, the present invention provides a low-dimensional control method for radially underactuated spacecraft pursuit-escape game, comprising the following steps:

[0067] Step S1: Based on differential game theory, construct a 24-dimensional two-point boundary value problem and a high-dimensional game optimal control strategy for the radial underactuated spacecraft optimal time game.

[0068] Step S1 specifically includes:

[0069] Step S11: Given a chase-escape game scenario: Assume a virtual navigator spacecraft is flying in a near-Earth circular orbit, with a radially underactuated tracker and a radially underactuated escape device nearby. Construct a local-vertical-local-horizontal coordinate system with the navigator's center of mass as the origin. Then, the player... or The relative distance and relative speed with respect to the navigator are defined as follows: and subscript or These represent the tracker and the escape device, respectively.

[0070] Therefore, the under-radial driving dynamics of the pursuit-escape game can be constructed as follows:

[0071] (1)

[0072] In the formula Indicates underactuated control acceleration and has , and They represent players respectively Acceleration is achieved through control of the trajectory and normal, while This represents nonlinear relative orbital dynamics, specifically expressed as...

[0073]

[0074] in, For the dimensional perspective of the navigator, and These are angular velocity and angular acceleration, respectively. Indicates the orbital radius of the navigator. Indicates player or orbital radius, And there are , and Representing the navigator and the player respectively. orbital angular velocity, This is Earth's gravitational constant;

[0075] According to linear system theory, the closed-loop system is controllable in the under-radial drive case. Therefore, equation (1) above can be linearized as follows:

[0076] (2)

[0077] In the formula This represents the player's initial state, which includes relative position and relative velocity. , For the system matrix, , , For control matrix;

[0078] Step S12: Based on differential game theory, the differential game between the radial underactuated tracker and the escaper can be constructed as follows:

[0079] (3)

[0080] In the formula, This represents the acceleration controlled by player P. This represents the acceleration controlled by player E, and has... .

[0081] In the optimal time differential game, the tracker attempts to minimize the interception time. The escape vehicle is captured, while the escape vehicle tries to maximize the interception time. The quadratic cost function of the underdriven pursuit game can be constructed as follows:

[0082] (4)

[0083] in, This represents a performance index for quadratic differential games.

[0084] Step S13: Solving for typical saddle point strategy pairs involves introducing the underactuated Hamiltonian function. and underactuated terminal conditions Initially, the underactuated Hamiltonian function and underactuated terminal conditions They can be represented as

[0085] (5)

[0086] (6)

[0087] in, Represents the Lagrange multipliers. For the tracker's costate variables, ; For the costate variables of the escaper, ; and These are the position and velocity components of the tracker's costate variables, respectively. and Let be the position and velocity components of the escapement costate variables, respectively, and satisfy the following relationship:

[0088] (7)

[0089] The saddle point strategy also satisfies the adjoint equation. and terminal boundary conditions ,in, or Then the accompanying equation and terminal boundary conditions Represented as

[0090] (8)

[0091] (9)

[0092] Furthermore, based on the cross section condition The following equation can be obtained:

[0093] (10)

[0094] Therefore, the optimal time game strategies for the under-radial driven tracker and the escaper can be obtained as follows:

[0095] (11)

[0096] (12)

[0097] In the formula, and These represent the maximum accelerations of the tracker and the escape vehicle, respectively. This represents the high-dimensional optimal time game strategy for player P. This represents the optimal time game strategy for player E in a higher dimension.

[0098] Equations (6) to (10) constitute the 24-dimensional two-point boundary value problem for two players in the radial underactuated case, while the underactuated saddle point strategy pairs (11) and (12) are the control laws for the underactuated game tracker and escaper. Under the action of the above control laws, the tracker and escaper can still complete the three-dimensional chase-escape game.

[0099] Step S2: Derive the state conditions for reducing the underactuated high-dimensional two-point boundary value problem to a 12-dimensional two-point boundary value problem and design the optimal control strategy for low-dimensional game theory.

[0100] Step S2 specifically includes:

[0101] Step S21, Definition This is an error state. For new costate variables and and These represent the position and velocity components of the costate variable, respectively.

[0102] Therefore, the differential game model in equation (3) and the adjoint equation in equation (8) can be simplified to the following game dynamics.

[0103] (13)

[0104] According to the definition of error state, the initial condition is:

[0105] (14)

[0106] The location conditions for interception are expressed as follows:

[0107] (15)

[0108] According to the transverse condition of equation (9), the new costate variable satisfies the following equation.

[0109] (16)

[0110] Step S22: According to equation (7), the costate variables of the tracker and the escaper satisfy the following relationship in the differential game:

[0111] (17)

[0112] Therefore, the high-dimensional game strategies shown in equations (11) and (12) can be rewritten in the following low-dimensional form.

[0113] (18)

[0114] (19)

[0115] in, Let P represent the low-dimensional optimal time game strategy. Denotes the low-dimensional optimal time game strategy of player E;

[0116] Thus, equations (13) to (19) constitute the low-dimensional two-point boundary value problem of the radially underactuated spacecraft chase-escape game. Compared with the 24-dimensional two-point boundary value problem constructed by equations (6) to (10), the low-dimensional problem has only 12 dimensions, thereby reducing the difficulty of finding the saddle point solution. The low-dimensional game strategy given by equations (18) and (19) is equivalent to the original high-dimensional game strategies (11) and (12). At the same time, even if the tracker and escaper lose radial thrust, the two players can still complete the three-dimensional chase-escape game by controlling the two channels of trace and normal through equations (18) and (19).

[0117] It is worth noting that the control acceleration output by the low-dimensional game strategy is exactly the same as that of the high-dimensional game strategy. The difference lies in whether the constructed game model is a high-dimensional or low-dimensional two-point boundary value problem. Compared with high-dimensional problems, low-dimensional two-point boundary value problems are easier to solve for the optimal solution.

[0118] In one embodiment of the present invention, the low-dimensional control method for radial underactuated spacecraft pursuit-escape game provided by the present invention is adopted, and the following experimental results are obtained:

[0119] The orbital parameters of the virtual navigator spacecraft are shown in Table 1. The maximum accelerations of the tracker and escape vehicle are set as follows: and The initial states of the tracker and the escape device are set to... and .

[0120] Table 1: Track Elements of the Virtual Navigator

[0121]

[0122] The pursuit-escape game trajectory under the condition of insufficient radial drive is as follows Figure 2 As shown, where the symbol Indicates the initial position of the tracker. Indicates the initial position of the tracker. For the interception location, This indicates the interception trajectory of the tracker, and The diagram shows the escape trajectory of the escape vehicle, indicating that the tracker captured it after a certain period of time. Figure 3 This shows the time history of the relative positions of the tracker and the escape vehicle. , , and These represent the three-dimensional position error, radial position error, trace position error, and normal position error, respectively.

[0123] This demonstrates that even if the two players lack radial thrust, they can still engage in three-dimensional orbital game, and because the tracker's maximum acceleration is greater than the escaper's maximum acceleration, the tracker ultimately achieves its capture objective.

[0124] Figure 4 For the control acceleration of the tracker, where and These are the control accelerations of the tracker in the track and normal directions, respectively. To shorten the relative distance with the escape vehicle, a larger control input than that of the escape vehicle is required.

[0125] Similarly, Figure 5 The time history of the escape device's controlled acceleration, in which and These represent the control accelerations of the escaper in the track and normal directions, respectively. It is worth noting that in the low-dimensional game strategy, the co-state variables of the tracker and the escaper are the same, which corresponds to the fact that in the simulation results, the positive and negative switching times of the two players are the same regardless of whether it is the track or normal thrust.

[0126] The pursuit-escape game strategy for fully driven spacecraft is not applicable to the underactuated case. The low-dimensional control method for radially underactuated spacecraft pursuit-escape game provided by this invention only requires thrust in the path and normal directions to achieve a three-dimensional pursuit-escape game. In this invention, the underactuated game strategy derived from differential game theory corresponds to the underactuated rather than the saddle point solution of the bilateral pursuit-escape optimization problem; at the same time, this invention constructs a low-dimensional two-point boundary value problem and an optimal time game strategy in the radially underactuated case, thereby reducing the difficulty of finding the saddle point solution.

[0127] Finally, it should be noted that the above description represents a preferred embodiment of the present invention. It should be pointed out that although preferred embodiments have been described, those skilled in the art, once they understand the basic inventive concept of the present invention, can make various improvements and modifications without departing from the principles described herein. These improvements and modifications should also be considered within the scope of protection of the present invention. Therefore, the appended claims are intended to be interpreted as including both the preferred embodiments and all changes and modifications falling within the scope of the embodiments of the present invention.

Claims

1. A low-dimensional control method for a radially underactuated spacecraft in a pursuit-escape game, characterized in that, Includes the following steps: Step S1: Based on differential game theory, construct a 24-dimensional two-point boundary value problem and a high-dimensional game optimal control strategy for the radial underactuated spacecraft optimal time game. Step S2: Derive the state conditions for reducing the dimensionality of the underactuated high-dimensional two-point boundary value problem to a 12-dimensional two-point boundary value problem and design the optimal control strategy for the low-dimensional game. Step S1 specifically includes: Step S11: Given a chase-escape game scenario: Assume a virtual navigator spacecraft is flying in a near-Earth circular orbit, with a radially underactuated tracker and a radially underactuated escape device nearby. Construct a local-vertical-local-horizontal coordinate system with the navigator's center of mass as the origin. The player... or The relative distance and relative speed with the navigator are respectively expressed as: and ; where subscript or These represent the tracker and the escape device, respectively. The underradial driving dynamics equation for the pursuit-escape game is expressed as follows: (1) in, Indicates underactuated control acceleration and has , and They represent players respectively Acceleration is achieved through control of the trajectory and normal, while This represents nonlinear relative orbital dynamics, specifically expressed as... in, For the dimensional perspective of the navigator, and These are angular velocity and angular acceleration, respectively. Indicates the orbital radius of the navigator. Indicates player or orbital radius, And there are , and Representing the navigator and the player respectively. orbital angular velocity, This is Earth's gravitational constant; According to linear system theory, the closed-loop system is controllable in the under-radial drive case, and the above equation (1) can be linearized as follows: (2) In the formula, This represents the player's initial state, which includes relative position and relative velocity. , For the system matrix, , , For control matrix; Step S12: Based on differential game theory, the differential game between the radial underactuated tracker and the escaper is constructed as follows: (3) in, This represents the acceleration controlled by player P. This represents the acceleration controlled by player E, and has... ; In the optimal time differential game, the tracker attempts to minimize the interception time. The escape vehicle is captured, while the escape vehicle tries to maximize the interception time. The quadratic cost function of the underdriven pursuit game is constructed as follows: (4) in, This represents a performance index for quadratic differential games. Step S13: Solve for saddle point policy pairs, introducing the underactuated Hamiltonian function. and underactuated terminal conditions underdriven Hamiltonian function and underactuated terminal conditions They are respectively represented as (5) (6) in, Represents the Lagrange multipliers. For the tracker's costate variables, ; For the costate variables of the escaper, ; and These are the position and velocity components of the tracker's costate variables, respectively. and Let be the position and velocity components of the escapement costate variables, respectively, and satisfy the following relationship: (7) Saddle point strategies satisfy the adjoint equation and terminal boundary conditions ,in, or Then the accompanying equation and terminal boundary conditions Represented as (8) (9) According to the transverse condition The following equation is obtained: (10) in, This indicates the relative speed between the tracker and the navigator. This indicates the relative speed between the escape vehicle and the navigator; The optimal time game strategies for the under-radial driven tracker and the escaper are as follows: (11) (12) In the formula, and These represent the maximum accelerations of the tracker and the escape vehicle, respectively. This represents the high-dimensional optimal time game strategy for player P. This represents the optimal time game strategy for player E in a higher dimension; Equations (6) to (10) constitute the 24-dimensional two-point boundary value problem for two players in the radial underactuated case. Equations (11) and (12) are the underactuated saddle point strategy pairs, which are the control laws for the underactuated game tracker and escaper. Under the action of the control laws, the tracker and escaper can complete the three-dimensional chase-escape game.

2. The low-dimensional control method for radially underactuated spacecraft pursuit-escape game as described in claim 1, characterized in that, Step S2 specifically includes: Step S21, Definition This is an error state. For new costate variables and and These represent the position and velocity components of the costate variable, respectively. Then the differential game model of equation (3) and the adjoint equation of equation (8) are simplified to: (13) According to the definition of error state, the initial condition is: (14) The location conditions for interception are expressed as follows: (15) in, Indicates the relative distance between the tracker and the navigator. Indicates the relative distance between the escape vehicle and the navigator; According to the transverse condition of equation (9), the new costate variable satisfies the following equation. (16) Step S22: According to equation (7), the costate variables of the tracker and the escaper satisfy the following relationship in the differential game. (17) The high-dimensional game strategies shown in equations (11) and (12) can be rewritten in the following low-dimensional form. (18) (19) in, This represents the low-dimensional optimal time game strategy of player P. This represents player E's low-dimensional optimal time game strategy; Equations (13) to (19) constitute the low-dimensional two-point boundary value problem of radial underactuated spacecraft pursuit and escape game.

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