A track pursuit and evasion game strategy design method based on optimization theory

Abstract

The present application belongs to the field of aerospace space orbit game and space safety, and relates to a kind of orbit pursuit and evasion interception game strategy design method based on optimization theory, specifically comprising the following processes: determining orbit game scene parameters;Based on the relative motion equation of spacecraft, a round mode pursuit and evasion interception orbit game model is established;The pursuit strategy of the pursuer is designed, the interception strategy of the interceptor is designed, and the escape strategy of the evader is designed;Based on the orbit game scene parameters and the round mode pursuit and evasion interception orbit game model, the game indexes of the three are optimized, and the optimal values of the game indexes of the three are obtained respectively. The application of the method can provide an effective solution for the pursuit and evasion interception cooperation problem of spacecraft using impulse thrust, thereby laying a theoretical foundation for the realization of space failed spacecraft takeover, space failed spacecraft removal and other engineering applications.

Description

Technical Field

[0001] This invention belongs to the field of aerospace space orbital game theory and space security, and in particular relates to a design method for orbital pursuit, escape, and interception game strategy based on optimization theory. Background Technology

[0002] With the continuous development of space technology in various countries, the number of spacecraft is increasing, mission capabilities are rapidly improving, and the space environment is becoming increasingly complex. Under this trend, space security has become a "high frontier" involving national security and interests, and the space domain has become an arena for great power competition and confrontation. As major spacefaring nations continuously enrich their space confrontation methods, rapidly improve their capabilities, and initially establish space combat systems, my country's space security faces significant threats. Therefore, China's spacecraft need to possess commensurate space confrontation capabilities to ensure better mission completion.

[0003] In space orbital maneuvering, multiple pursuers may engage multiple targets for interception. To improve the overall combat performance of multiple pursuers, target allocation systems may assign multiple pursuers to the same target. Compared to a single pursuer against a single escaping spacecraft, the multiple-to-single pursuit problem involves the coupling of relative motions among multiple pursuers. How to coordinate and control each pursuer to achieve optimal pursuit and interception results is a crucial issue in multiple-to-one pursuit-escape-interception research. Summary of the Invention

[0004] The purpose of this invention is to provide a strategy design method for orbital pursuit-escape-interception game based on optimization theory, which solves the problem of mutual coupling of relative motions of multiple pursuers in multi-to-single pursuit.

[0005] This invention is achieved through the following technical solution:

[0006] This invention discloses a strategy design method for orbital pursuit, interception, and interception game based on optimization theory. The round-based system is that the orbital maneuvers of the two sides are carried out alternately in a sequential manner. Only when one side carries out an orbital maneuver and delays its reaction time can the other side carry out an orbital maneuver. The spacecraft are divided into pursuers, interceptors, and escapees.

[0007] Specifically, the process includes the following:

[0008] Determine the parameters for the orbital game scenario;

[0009] Based on the relative motion equations of spacecraft, a round-based game model of pursuit, escape, and interception orbits is established.

[0010] Design the pursuer's pursuit strategy, and optimize the pursuer's game index based on the parameters of the orbital game scenario and the turn-based pursuit-escape-interception orbital game model to obtain the optimal value of the pursuer's game index.

[0011] Design an interceptor's interception strategy, optimize the interceptor's game indicators based on the parameters of the orbital game scenario and the turn-based pursuit-escape-interception orbital game model, and obtain the optimal value of the interceptor's game indicators;

[0012] The escape strategy of the escapee is designed. Based on the parameters of the orbital game scenario and the turn-based pursuit-escape-interception orbital game model, the escapee's game index is optimized to obtain the optimal value of the escapee's game index.

[0013] Furthermore, the parameters of the orbital game scenario include the semi-major axis of the reference spacecraft orbit in the LVLH coordinate system, the pulse control interval time, the distance judgment condition parameters, the initial position and velocity of both parties, and the pulse constraint conditions.

[0014] Furthermore, in S1, the turn-based pursuit-escape-interception trajectory game model is as follows:

[0015]

[0016]

[0017] In formula (1), k represents the number of rounds in the game between the two parties; J represents the bilateral optimization index of the pursuit-escape game.

[0018] P1 represents the pursuer; P2 represents the interceptor; E1 represents the escapee.

[0019] The column vector representing the incremental control of the pursuer's pulse velocity;

[0020] The column vector representing the incremental control of the interceptor pulse velocity;

[0021] The column vector representing the incremental control of the escapee's pulse velocity;

[0022] The constraint function representing the incremental control of the pursuer's pulse velocity;

[0023] The constraint function representing the incremental control of the interceptor pulse velocity;

[0024] The constraint function representing the incremental control of the escapee's pulse velocity;

[0025] The state vector representing the tracker is specifically:

[0026] The state vector representing the interceptor is specifically:

[0027] The state vector representing the escapee is specifically:

[0028] x represents the radial position component of the orbit under the LVLH system;

[0029] y represents the position component of the flight direction in the LVLH system;

[0030] z represents the position component of the orbital angular momentum direction in the LVLH system;

[0031] v x Represents the radial velocity component of the orbit in the LVLH system;

[0032] v y The velocity component representing the orbital flight direction in the LVLH system;

[0033] v z The velocity component representing the direction of the orbital angular momentum in the LVLH system;

[0034] ΔT is the pulse control interval time;

[0035] in: The matrix satisfies:

[0036] B = [0 3×3 ,I 3×3 ] T (2)

[0037] In formula (2):

[0038]

[0039] The pulse control interval is ΔT, and the state transition matrix φ(k+1,k) is used to realize the state transition from time k·ΔT to time (k+1)·ΔT. Its specific form is constructed through the CW equation in orbital dynamics, as follows:

[0040]

[0041] in The velocity of the origin of the LVLH coordinate system around the Earth is represented by μ, the Earth's gravitational field coefficient is μ, and a is the semi-major axis of the reference spacecraft's orbit in the LVLH coordinate system.

[0042] Furthermore, the specific process of designing the pursuer's pursuit strategy is as follows:

[0043] The strategy index for the k-th control application time of the pursuer P1 is designed as follows:

[0044] J P1 (k)=(X P1 (k+1)-X E1(k+1)) T ·Q P1E1 ·(X P1 (k+1)-X E1 (k+1)) (5)

[0045] In the formula, Q P1E1 Assign a weight matrix, with the following constraints:

[0046]

[0047] Pick

[0048] but

[0049]

[0050] Then the policy index of the pursuer P1 at the k-th control application time is:

[0051]

[0052] Furthermore, the interception strategy of the interceptor P2 is specifically as follows:

[0053] Design an interception index function, which includes an interceptor pursuit term, a relative position-velocity coordination term, and an azimuth angle coordination term, expressed as follows:

[0054] J P2 =η1·J P2E1 +η2·J P1P2 +η3·J a ;

[0055] Where η1, η2, and η3 are the weights of each item; J P2E1 For interceptor pursuit; J P1P2 J represents the relative position and velocity coordination term; a This is the azimuth coordination term.

[0056] Furthermore, the interceptor pursued item J. P2E1 The design is as follows:

[0057] The pursuit term index at the k-th control application moment of the interceptor is:

[0058]

[0059] In the formula, Q P2E1 Assign a weight matrix and take...

[0060]

[0061] in,

[0062]

[0063] Formula (8) is transformed into:

[0064]

[0065] Furthermore, the interceptor's relative position velocity cooperative term J P1P2 The specific design is as follows:

[0066] A directed graph describing the communication topology of the pursuer has a spanning tree and defines the relevant parameter a. ij If the i-th pursuer can obtain information about the j-th pursuer, then a ij =1, otherwise 0; Based on the above assumptions, the local tracking error of the i-th pursuer is:

[0067]

[0068] The design of the relative position velocity coordination term is as follows:

[0069]

[0070] For the pursuit-escape-interception game, let i and j be 1 and 2 respectively, and take:

[0071]

[0072]

[0073] then

[0074] Furthermore, the interceptor's azimuth coordination term J a The design process is as follows:

[0075] Define the angle between the positions of P1, P2 and E1 as θ. To ensure that the pursuing spacecraft can intercept the escaping spacecraft from different azimuth angles, we take θ→π, that is, θ * =π, because

[0076]

[0077] Therefore, the azimuth co-term is taken as:

[0078] Ja = cosθ;

[0079] in,

[0080]

[0081]

[0082] Furthermore, the specific process of designing the escape strategy for the escapee is as follows:

[0083] The target for the k-th control application time of the escapee E1 is:

[0084]

[0085] Among them, Q E1P1 Q E1P2 Assign a weight matrix, with the following constraints:

[0086]

[0087] The optimization metrics for the Escaper E1 are:

[0088] J E1 (k)=J E1P1 (k)+J E1P2 (k)

[0089] =-X E1P1 (k+1) T ·Q E1P1 ·X E1P1 (k+1)-X E1P2 (k+1) T ·Q E1P2 ·X E1P2 (k+1);

[0090] in,

[0091]

[0092]

[0093] Furthermore, depending on the value of k, there are two cases: when k is odd, it means that the pursuer and the interceptor are in pulse control, while the escapee is not; when k is even, it means that the escapee is in pulse control, while the pursuer and the interceptor are not.

[0094] When k is odd, the specific process of optimizing the game metric for pursuer P1 is as follows:

[0095] Formula (1) simplifies to:

[0096]

[0097]

[0098] The constraint function for the incremental control of the pursuer's pulse velocity is:

[0099]

[0100] but

[0101]

[0102] The game strategy is solved using the Lagrange multiplier method by introducing auxiliary variables, as detailed below:

[0103] By introducing auxiliary variables, the optimization problem under inequality constraints is transformed into one under equality constraints; the Lagrangian function L(ΔV) is taken. P ,λ)=J k -λ T ·h(ΔV P );

[0104] Among them, take λ=[λ1,λ2,λ3,λ4,λ5,λ6,λ7,λ8,λ9,λ 10 ,λ 11 ,λ 12 ] T ;

[0105] Using the exterior point method, a penalty term is introduced to transform the optimization problem under equality constraints into an unconstrained optimization problem. The augmented objective function with the penalty term is:

[0106]

[0107] get:

[0108]

[0109] For interceptor P2, the specific process of optimizing the game metric for interceptor P2 is as follows:

[0110] Formula (1) simplifies to:

[0111]

[0112]

[0113] The constraint function for the interceptor pulse velocity increment control is:

[0114]

[0115] but

[0116]

[0117] The game strategy is solved using the Lagrange multiplier method by introducing auxiliary variables, as detailed below:

[0118] By introducing auxiliary variables, the optimization problem under inequality constraints is transformed into one under equality constraints; the Lagrangian function L(ΔV) is taken. P ,λ)=J k -λ T ·h(ΔV P );

[0119] Among them, take λ=[λ1,λ2,λ3,λ4,λ5,λ6,λ7,λ8,λ9,λ 10 ,λ 11 ,λ 12 ] T ;

[0120] Using the exterior point method, a penalty term is introduced to transform the optimization problem under equality constraints into an unconstrained optimization problem. The augmented objective function with the penalty term is:

[0121]

[0122] get:

[0123]

[0124] When k is even, the specific process of optimizing the game metric for escapee E1 is as follows:

[0125] Formula (1) simplifies to:

[0126]

[0127]

[0128] The constraint function for the incremental control of the pursuer's pulse velocity is:

[0129]

[0130] but

[0131]

[0132] The game strategy is solved using the Lagrange multiplier method by introducing auxiliary variables, as detailed below:

[0133] By introducing auxiliary variables, the optimization problem under inequality constraints is transformed into one under equality constraints; the Lagrange function is then used.

[0134] Among them, take λ=[λ1,λ2,λ3,λ4,λ5,λ6,λ7,λ8,λ9,λ 10 ,λ 11 ,λ 12 ] T ;

[0135] Using the exterior point method, a penalty term is introduced to transform the optimization problem under equality constraints into an unconstrained optimization problem. The augmented objective function with the penalty term is:

[0136]

[0137] get:

[0138]

[0139] Finally, equations (20), (21), and (22) are processed as follows:

[0140] An approximate solution is obtained by iteratively solving using a quasi-Newton method.

[0141] The quasi-Newton algorithm is used to process the unconstrained subproblems of the optimization model. Then, it is determined whether the optimization termination condition is met. If not, σ and λ are updated, and the quasi-Newton algorithm is used again until the termination condition is met. At this point, the optimal value of the game strategy is obtained.

[0142] Compared with the prior art, the present invention has the following beneficial technical effects:

[0143] This invention discloses a strategy design method for orbital pursuit-escape-interception game based on optimization theory. It is a collaborative pursuit-interception mechanism in orbital game scenarios, characterized by its simple model, clear physical meaning, and easy strategy solution. Unlike traditional pursuit-escape game theory, which only considers distance, this method considers factors such as positional and directional collaboration in the pursuit-escape-interception game scenario, enabling the pursuit and interception of non-cooperative targets. The application of this model and method in the pursuit and interception of the same target by multiple pursuers can provide an effective solution to the collaborative pursuit-escape-interception problem of spacecraft using pulse thrust, thus laying a theoretical foundation for engineering applications such as space orbital games and the removal of spacecraft that have failed in space. Attached Figure Description

[0144] Figure 1 The trajectory curves of the three-way game of pursuit, interception, and escape are used to illustrate the relative positions of the pursuer, interceptor, and escapee during the actual game process.

[0145] Figure 2 The trajectory and maneuvering points of the pursuer in the pursuit-escape-interception game are illustrated in the diagram.

[0146] Figure 3 The interceptor's trajectory and maneuver points are shown in the diagram of the interceptor's position curve and maneuver points in the chase-and-intercept game.

[0147] Figure 4 The escapee's trajectory and maneuver points are illustrated in the pursuit-escape-interception game.

[0148] Figure 5 This is a schematic diagram of a coordinated pursuit and interception scenario based on orbital game theory.

[0149] Figure 6This is a flowchart illustrating a design method for a trajectory pursuit-escape-interception game strategy based on optimization theory, according to the present invention. Detailed Implementation

[0150] To make the objectives, technical solutions, and advantages of the present invention clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention; that is, the described embodiments are only a part of the embodiments of the present invention, and not all of them.

[0151] The components described and illustrated in the accompanying drawings and embodiments of this invention can be arranged and designed in various different configurations. Therefore, the detailed description of the embodiments of the invention provided in the following drawings is not intended to limit the scope of the claimed invention, but merely to illustrate one selected embodiment of the invention. All other embodiments obtained by those skilled in the art based on the accompanying drawings and embodiments of this invention without inventive effort are within the scope of protection of this invention.

[0152] It should be noted that the terms “comprising,” “including,” or any other variations are intended to cover non-exclusive inclusion, such that a process, element, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to the process, element, method, article, or apparatus.

[0153] In the process of multiple pursuers intercepting the same target, the coordination strategy adopted by the pursuers to intercept the target has a significant impact on the actual combat effectiveness.

[0154] like Figure 6 As shown, this invention provides a strategy design method for a trajectory pursuit-escape-interception game based on optimization theory, comprising the following steps:

[0155] Determine the parameters for the orbital game scenario;

[0156] Based on the relative motion equations of spacecraft, a round-based game model of pursuit, escape, and interception orbits is established.

[0157] Design the pursuer's pursuit strategy, and optimize the pursuer's game index based on the parameters of the orbital game scenario and the turn-based pursuit-escape-interception orbital game model to obtain the optimal value of the pursuer's game index.

[0158] Design an interceptor's interception strategy, optimize the interceptor's game indicators based on the parameters of the orbital game scenario and the turn-based pursuit-escape-interception orbital game model, and obtain the optimal value of the interceptor's game indicators;

[0159] The escape strategy of the escapee is designed. Based on the parameters of the orbital game scenario and the turn-based pursuit-escape-interception orbital game model, the escapee's game index is optimized to obtain the optimal value of the escapee's game index.

[0160] The parameters of the orbital game scenario include the semi-major axis of the reference spacecraft orbit in the LVLH coordinate system, the pulse control interval time, the distance judgment condition parameters, the initial position and velocity of both parties, and the pulse constraint conditions.

[0161] The application of this model and method can provide an effective solution to the coordinated pursuit, escape, and interception problem of spacecraft using pulse thrust, thereby laying a theoretical foundation for engineering applications such as takeover of spacecraft with space failures and removal of spacecraft that have failed.

[0162] The features and performance of the present invention will be further described in detail below with reference to embodiments.

[0163] This invention discloses a strategy design method for orbital pursuit-escape-interception game based on optimization theory. The spacecraft consists of a pursuer (P1), an interceptor (P2), and an escapee (E1). For a "two-on-one" pursuit-escape-interception game scenario, based on the motion states of pursuer P1, escapee E1, and interceptor P2, the invention studies the design and optimization of orbital game strategies in a round-based scenario where both sides use pulsed orbital maneuvers as their response. The round-based nature of this method is that the orbital maneuvers of both sides are implemented sequentially and alternately; that is, only after one side has implemented an orbital maneuver and there is a certain delay (called the reaction time) can the other side implement an orbital maneuver. The method includes the following steps:

[0164] S1. Based on the spacecraft relative motion (CW) equations, a round-based pursuit-escape-interception orbital game model is established as follows:

[0165]

[0166]

[0167] The symbols in equation (1) have the following meanings:

[0168] k — the number of rounds in the game between the two sides;

[0169] J—a two-sided optimization index in the pursuit-escape game;

[0170] P1—The Pursuer;

[0171] P2 – Interceptor;

[0172] E1 – The escapee;

[0173] —A column vector formed by the incremental control of the pursuer's pulse velocity;

[0174] —A column vector formed by the incremental control of the interceptor pulse velocity;

[0175] —A column vector formed by the incremental control of the escapee's pulse velocity;

[0176] —The constraint function for incremental control of the pursuer's pulse velocity;

[0177] —The constraint function for the incremental control of the interceptor pulse velocity;

[0178] —The constraint function for the incremental control of the escapee's pulse velocity;

[0179] —The tracker's state vector, specifically

[0180] —The interceptor's state vector, specifically

[0181] —The escapee's state vector, specifically

[0182] The six components of the state vector are explained as follows:

[0183] x — the radial position component of the orbit relative to the coordinate system (LVLH system);

[0184] y — the position component of the flight direction in a relative coordinate system (LVLH system);

[0185] z — Position component of the orbital angular momentum direction relative to the coordinate system (LVLH system);

[0186] v x —The radial velocity component of the orbit in the relative coordinate system (LVLH system);

[0187] v y —The velocity components in the orbital flight direction under the relative coordinate system (LVLH system);

[0188] v z —The velocity component in the direction of the orbital angular momentum in a relative coordinate system (LVLH system);

[0189] ΔT is the pulse control interval time;

[0190] in: satisfy:

[0191] B = [0 3×3 ,I 3×3 ]T (2)

[0192] In formula (2):

[0193]

[0194] The pulse control interval is ΔT, and the state transition matrix φ(k+1,k) is used to realize the state transition from time k·ΔT to time (k+1)·ΔT. Its specific form is constructed using the CW equation in orbital dynamics, namely:

[0195]

[0196] in denoted by ω, μ represents the orbital angular velocity at the origin of the LVLH coordinate system, μ is the Earth's gravitational field coefficient, and a is the semi-major axis of the reference satellite's orbit.

[0197] S2, the pursuit strategy design for pursuer P1, the specific steps are as follows:

[0198] The policy index at the k-th moment when the pursuer applies control is:

[0199] J P1 (k)=(X P1 (k+1)-X E1 (k+1)) T ·Q P1E1 ·(X P1 (k+1)-X E1 (k+1)) (5)

[0200] In the formula, Q P1E1 Assign a weight matrix, with constraints.

[0201]

[0202] Pick

[0203] but

[0204]

[0205] Therefore, the policy index of the pursuer P1 at the k-th control application time is:

[0206]

[0207] S3. Design the interception strategy for interceptor P2. The interception index function of interceptor P2 is specifically composed of the "pursuit term", "position-velocity coordination term", and "azimuth coordination term". The specific steps are as follows:

[0208] S3.1 Interceptor Pursuit Design

[0209] The pursuit term index of interceptor P2 at the k-th control application moment is:

[0210]

[0211] In the formula, Q P2E1 Assign a weight matrix, with constraints.

[0212]

[0213] Pick

[0214]

[0215] in,

[0216]

[0217] Therefore, the pursuit term index at the kth control application moment of the interceptor is:

[0218]

[0219] S3.2 Interceptor Position-Velocity Coordination Term Design

[0220] A directed graph describing the communication topology of the pursuing spacecraft (pursuer and interceptor) has a spanning tree and defines the relevant parameter a. ij If the i-th pursuer can obtain information about the j-th pursuer, then a ij =1, otherwise 0. Based on the above assumptions, the local tracking error of the i-th pursuer is:

[0221]

[0222] The design of the relative position velocity coordination term is as follows:

[0223]

[0224] For a 2V1 chase-escape-block game, let i and j be 1 and 2 respectively, and take...

[0225]

[0226]

[0227] Therefore, we can conclude that:

[0228]

[0229] In equation (12):

[0230]

[0231] S3.3 Interceptor Azimuth Coordination Term Design

[0232] Angle constraints are added to the optimization indicators, and an azimuth constraint term is designed. To ensure that interceptors can "pincer" the escaped spacecraft from different azimuth angles, we take θ→π, i.e., θ * =π, let

[0233]

[0234] according to Figure 5 Angle constraint illustration:

[0235]

[0236] so:

[0237]

[0238] because Therefore, the azimuth co-term can be taken as: Ja = cosθ; that is...

[0239]

[0240] In summary, the interceptor's index function consists of an interception term, a position-velocity term, and an azimuth-cooperative term, which is:

[0241] J P2 =η1·J P2E1 +η2·J P1P2 +η3·J a (15)

[0242] Where η1, η2, and η3 are the weight values ​​of each item.

[0243] S4. Design the escape strategy for escapee E:

[0244] The index for the k-th control application time of the escapee is:

[0245]

[0246] Among them, Q E1P1 Q E1P2 Assign a weight matrix, with constraints.

[0247]

[0248] The optimization metrics for escapees are:

[0249] J E1 (k)=J E1P1 (k)+J E1P2 (k)

[0250] =-X E1P1(k+1) T ·Q E1P1 ·X E1P1 (k+1)-X E1P2 (k+1) T ·Q E1P2 ·X E1P2 (k+1) (17)

[0251] in,

[0252]

[0253]

[0254] S5. Determine the reference spacecraft orbital elements and the pulse control interval time ΔT, set the distance judgment condition parameters, input the initial position and velocity of both parties, set the pulse constraint conditions, solve the game strategy, and obtain the optimal values ​​of the game indicators of the pursuer, interceptor, and escapee.

[0255] The turn-based game model is based on analysis and modeling of relative orbital dynamics. Therefore, when constructing the orbital game scenario, a reference spacecraft needs to be defined. Assume the orbital elements of the reference spacecraft are as shown in Table 1:

[0256] Table 1 Reference Spacecraft Orbital Elements

[0257]

[0258] In a turn-based game, the pursuer and the escapee take turns controlling the game, with a fixed time interval. Assuming the orbital recursion integration time unit dt = 42.8306 s, the control pulse interval is ΔT = k. T ·dt, initially set k T =60, meaning that the two sides take turns controlling the device approximately every 42 minutes. The upper limit of the pulse control for both sides is V. P max =20m / s and V E max =20m / s, the pursuer takes control first, followed by the escapee. Assume the initial states of the pursuer and escapee are as shown in Table 2:

[0259] Table 2 Initial Relative Position and Velocity in the Pursuit and Escape Game

[0260]

[0261] The simulation begins with the active spacecraft initiating a close maneuver and lasts approximately 24 hours. It is assumed that the pursuit mission is completed when the relative distance is less than 5.0 km. That is, let L be the distance between the two spacecraft. min =1.5km, T max=12h. The pursuer and the escapee alternately apply control. Therefore, depending on the value of k, there are two cases. When k is odd, the pursuer applies impulse control, and the escapee has no control. In this case, the original game problem simplifies to:

[0262]

[0263]

[0264] And determined by initial conditions

[0265]

[0266] Take the impulse constraint function as

[0267]

[0268] but

[0269]

[0270] S51. By introducing auxiliary variables, the optimization problem under inequality constraints is transformed into an equality constraint problem; the Lagrangian function is used.

[0271] L(ΔV P ,λ)=J k -λ T ·h(ΔV P )

[0272] Among them, take λ=[λ1,λ2,λ3,λ4,λ5,λ6,λ7,λ8,λ9,λ 10 ,λ 11 ,λ 12 ] T .

[0273] S52. Using the exterior point method, a penalty term is introduced to transform the optimization problem under equality constraints into an unconstrained optimization problem; the augmented objective function introduced by the penalty term is:

[0274]

[0275] achievable

[0276]

[0277] S53. Use the quasi-Newton method for iterative solution to obtain an approximate solution. The quasi-Newton algorithm is used to process the unconstrained subproblems of the optimization model. Then, it is determined whether the optimization termination condition is met. If not, σ and λ are updated, and the quasi-Newton algorithm is used again until the termination condition is met, for example, reaching the maximum number of iterations of 2000.

[0278] The index function for interceptors and escapees adopts the same as J. P1 (k) Using the same approach, the problem is transformed into an unconstrained optimization problem. Newton's iteration method is then used to solve an approximate solution for the model. The pulse control quantities of each spacecraft in each round of the game are shown in Table 3, and their spatial motion trajectories are as follows: Figure 1 As shown, the trajectories of the pursuer, interceptor, and escapee are as follows: Figure 2 , Figure 3 and Figure 4 As shown.

[0279] Table 3. Pulse sequences (m / s) of the pursuer, interceptor, and escapee.

[0280]

[0281]

[0282] For interceptor P2, the specific process of optimizing the game metric for interceptor P2 is as follows: Formula (1) simplifies to:

[0283]

[0284]

[0285] The constraint function for the interceptor pulse velocity increment control is:

[0286]

[0287] but

[0288]

[0289] The game strategy is solved using the Lagrange multiplier method by introducing auxiliary variables, as detailed below:

[0290] By introducing auxiliary variables, the optimization problem under inequality constraints is transformed into one under equality constraints; the Lagrangian function L(ΔV) is taken. P ,λ)=J k -λ T ·h(ΔV P );

[0291] Among them, take λ=[λ1,λ2,λ3,λ4,λ5,λ6,λ7,λ8,λ9,λ 10 ,λ 11 ,λ 12 ] T ;

[0292] Using the exterior point method, a penalty term is introduced to transform the optimization problem under equality constraints into an unconstrained optimization problem. The augmented objective function with the penalty term is:

[0293]

[0294] get:

[0295]

[0296] When k is even, the specific process of optimizing the game metric for escapee E1 is as follows:

[0297] Formula (1) simplifies to:

[0298]

[0299]

[0300] The constraint function for the incremental control of the pursuer's pulse velocity is:

[0301]

[0302] but

[0303]

[0304] Similarly, the Lagrange multiplier method is used to solve the game strategy by introducing auxiliary variables, as follows:

[0305] By introducing auxiliary variables, the optimization problem under inequality constraints is transformed into one under equality constraints; the Lagrange function is then used.

[0306] Among them, take λ=[λ1,λ2,λ3,λ4,λ5,λ6,λ7,λ8,λ9,λ 10 ,λ 11 ,λ 12 ] T ;

[0307] Using the exterior point method, a penalty term is introduced to transform the optimization problem under equality constraints into an unconstrained optimization problem. The augmented objective function with the penalty term is:

[0308]

[0309] get:

[0310]

[0311] For the above three cases, the final processing method is the same, that is, the following processing is performed on equations (20), (21) and (22):

[0312] An approximate solution is obtained by iteratively solving using a quasi-Newton method.

[0313] The quasi-Newton algorithm is used to process the unconstrained subproblems of the optimization model. Then, it is determined whether the optimization termination condition is met. If not, σ and λ are updated, and the quasi-Newton algorithm is used again until the termination condition is met. At this point, the optimal value of the game strategy is obtained.

[0314] 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. An optimal theory-based track pursuit-evasion game strategy design method, characterized in that, The turn-based system shown in the diagram involves the alternating sequential orbital maneuvers of the two sides. Only after one side has performed an orbital maneuver and delayed its reaction time can the other side perform its own orbital maneuver. The spacecraft are divided into pursuers, interceptors, and escapees. Specifically, the process includes the following: Determine the parameters for the orbital game scenario; Based on the relative motion equations of spacecraft, a round-based game model of pursuit, escape, and interception orbits is established. Design the pursuer's pursuit strategy, and optimize the pursuer's game index based on the parameters of the orbital game scenario and the turn-based pursuit-escape-interception orbital game model to obtain the optimal value of the pursuer's game index. Design an interceptor's interception strategy, optimize the interceptor's game indicators based on the parameters of the orbital game scenario and the turn-based pursuit-escape-interception orbital game model, and obtain the optimal value of the interceptor's game indicators; The escape strategy of the escapee is designed. Based on the parameters of the orbital game scenario and the turn-based pursuit and interception orbital game model, the escapee's game index is optimized to obtain the optimal value of the escapee's game index. The parameters of the orbital game scenario include the semi-major axis of the reference spacecraft orbit in the LVLH coordinate system, the pulse control interval time, the distance judgment condition parameters, the initial position and velocity of both parties, and the pulse constraint conditions. The turn-based pursuit and interception orbital game model is as follows: （1） In equation (1), k J represents the bilateral optimization index of the pursuit-evasion game; P 1 represents the pursuer; P 2 represents the interceptor; E 1 represents the escapee; The column vector representing the incremental control of the pursuer's pulse velocity; The column vector representing the incremental control of the interceptor pulse velocity; The column vector representing the incremental control of the escapee's pulse velocity; The constraint function representing the incremental control of the pursuer's pulse velocity; The constraint function representing the incremental control of the interceptor pulse velocity; The constraint function representing the incremental control of the escapee's pulse velocity; The state vector representing the tracker is specifically: ; The state vector representing the interceptor is specifically: ; The state vector representing the escapee is specifically: ; Represents the radial position component of the orbit in the LVLH coordinate system; The position component representing the flight direction in the LVLH coordinate system; The position component representing the direction of the orbital angular momentum in the LVLH coordinate system; Represents the radial velocity component of the orbit in the LVLH coordinate system; The velocity component representing the direction of orbital flight in the LVLH coordinate system; The velocity component representing the direction of the orbital angular momentum in the LVLH coordinate system; This refers to the pulse control interval time. in: The matrix satisfies: （2） In formula (2): （3） The pulse control interval time is State transition matrix Used to implement from Time's up The state transition at time t is specifically constructed using the CW equations in orbital dynamics, as follows: （4） in This represents the speed at which the origin of the LVLH coordinate system rotates around the Earth. The coefficient of Earth's gravitational field. Let be the semi-major axis of the reference spacecraft orbit in the LVLH coordinate system.

2. The method for designing a trajectory pursuit-escape-interception game strategy based on optimization theory according to claim 1, characterized in that, The specific process of designing the pursuit strategy for the pursuer is as follows: The design of the Pursuer P1 k The policy metrics at each control application time are: （5） In the formula, Assign a weight matrix, with the following constraints: ； Pick ; but （6） Then the Pursuer P1's k The policy metrics at each control application time are: （7）。 3. The method for designing a trajectory pursuit-escape-interception game strategy based on optimization theory according to claim 1, characterized in that, The design interceptor P The specific interception strategy for 2 is as follows: Design an interception index function, which includes an interceptor pursuit term, a relative position-velocity coordination term, and an azimuth angle coordination term, expressed as follows: ； in, , , These are the weights of each item; For the interceptor to pursue the item; For relative position and velocity coordination terms; This is the azimuth coordination term.

4. The method for designing a trajectory pursuit-escape-interception game strategy based on optimization theory according to claim 3, characterized in that, Interceptor Pursuit The design is as follows: Take the interceptor's first k The tracking indicator at each control application moment is: （8） In the formula, Assign a weight matrix and take... ； in, （9） Formula (8) is transformed into: （10）。 5. The method for designing a trajectory pursuit-escape-interception game strategy based on optimization theory according to claim 3, characterized in that, The relative position velocity coordination term of the interceptor The design is as follows: A directed graph describing the communication topology of the pursuer has a spanning tree and defines relevant parameters. a ij If the first i The pursuer can obtain the first j Information about the pursuers, then a ij =1, otherwise 0; based on the above assumptions, the first i Local tracking error of the individual pursuer: The design of the relative position velocity coordination term is as follows: （11） In the game of pursuing and intercepting fugitives, take... i , j Let 1 and 2 be the values, and take: then (12).

6. The method for designing a trajectory pursuit-escape-interception game strategy based on optimization theory according to claim 3, characterized in that, The interceptor's azimuth coordination term The design process is as follows: Define the angle between the positions of P1, P2 and E1 as . In order to ensure that the pursuing spacecraft can intercept the escapee from different azimuth angles, ,Right now ,because Therefore, the azimuth co-term is taken as: ； in, ； 。 7. The method for designing a trajectory pursuit-escape-interception game strategy based on optimization theory according to claim 1, characterized in that, The escape strategy designed for the escapee is as follows: The design of the Escape E1 k The indicators for each control application time are: in, , Assign a weight matrix, with the following constraints: The optimization metrics for the Escaper E1 are: ； in, ； 。 8. A method for designing a trajectory pursuit-escape-interception game strategy based on optimization theory, as described in claim 2, 3, or 7, characterized in that... according to k Different values ​​result in two cases: when k When the number is odd, it indicates that the pursuer and interceptor are using pulse control, while the escapee has no control; when... k When the number is even, it indicates that the escapee is using pulse control, while the pursuer and interceptor have no control. when k When the number is odd, the specific process of optimizing the game metric for pursuer P1 is as follows: Formula (1) simplifies to: The constraint function for the incremental control of the pursuer's pulse velocity is: ； but ； The game strategy is solved using the Lagrange multiplier method by introducing auxiliary variables, as detailed below: By introducing auxiliary variables, the optimization problem under inequality constraints is transformed into one under equality constraints; the Lagrangian function is then used. ; Among them, take ; Using the exterior point method, a penalty term is introduced to transform the optimization problem under equality constraints into an unconstrained optimization problem. The augmented objective function with the penalty term is: ； get: （20） For interceptor P2, the specific process of optimizing the game metric for interceptor P2 is as follows: Formula (1) simplifies to: The constraint function for the interceptor pulse velocity increment control is: ； but The game strategy is solved using the Lagrange multiplier method by introducing auxiliary variables, as detailed below: By introducing auxiliary variables, the optimization problem under inequality constraints is transformed into one under equality constraints; the Lagrangian function is then used. ; Among them, take ; Using the exterior point method, a penalty term is introduced to transform the optimization problem under equality constraints into an unconstrained optimization problem. The augmented objective function with the penalty term is: ； get: （21） when k When the number is even, the specific process of optimizing the game metric for escapee E1 is as follows: Formula (1) simplifies to: The constraint function for the incremental control of the pursuer's pulse velocity is: but The game strategy is solved using the Lagrange multiplier method by introducing auxiliary variables, as detailed below: By introducing auxiliary variables, the optimization problem under inequality constraints is transformed into one under equality constraints; the Lagrangian function is then used. ; Among them, take ; Using the exterior point method, a penalty term is introduced to transform the optimization problem under equality constraints into an unconstrained optimization problem. The augmented objective function with the penalty term is: get: （22） Finally, equations (20), (21), and (22) are processed as follows: An approximate solution is obtained by iteratively solving using a quasi-Newton method. A quasi-Newton algorithm is used to handle the unconstrained subproblems of the optimization model. Then, it is determined whether the optimization termination condition is met. If not, the algorithm is further updated. and Then, the quasi-Newton algorithm is used again until the termination condition is met, at which point the optimal value of the game strategy is obtained.

Images