A spacecraft escort problem solving method based on LQR and differential game

Abstract

The application discloses a spacecraft escort problem solving method based on LQR and differential game, constructs a game model of the escort problem based on a CW equation, and comprises a cost function, optimization variables, dynamic constraints, acceleration capacity constraints and the like; the cost function adopts a linear quadratic index, the optimization variables are continuous control force accelerations of both attack and defense sides, the dynamic constraints are described by using the CW equation, and the acceleration capacity constraints are modeled by using maximum acceleration limitation; the method has the advantages of simple model, clear physical meaning and convenient solving; in addition, the application converts a complex non-zero-sum game problem into a phased zero-sum game problem and an optimal control problem, can effectively solve the spacecraft escort problem under the action of continuous thrust, is an effective expansion of a continuous thrust escort problem model and method in the prior art, and significantly improves the solving efficiency of the problem. Through the method, an effective cooperative defense threat solution can be provided for on-orbit spacecraft.

Description

Technical Field

[0001] This invention relates to the field of aerospace technology, and particularly to a method and system for identifying cooperative target control strategies, especially a method for solving spacecraft escort problems based on LQR and differential games. Background Technology

[0002] Spacecraft orbital pursuit and escape is a classic spacecraft game problem. Unlike the classic orbital pursuit and escape problem, the spacecraft escort problem generally involves three participants: the host star, the attacking star, and the defending star. The host star, due to its high fuel value, generally cannot maneuver freely and therefore requires a defending star for coordinated protection. The game between the defending star and the attacking star regarding the host star's position is called the spacecraft escort problem. The attacking star's goal is to make every effort to get as close to the host star as possible while ensuring it is not intercepted or destroyed by the defending star; the defending star's goal is to intercept and destroy the attacking star before it gets too close to the host star.

[0003] Currently, most modeling and solutions for spacecraft escort problems use differential game theory. However, since it is a non-zero-sum game in the game theory category, the cost functions of the attacking and defending stars are not the same, so it is often difficult to solve for the optimal control strategy quickly and effectively. Summary of the Invention

[0004] To address the aforementioned technical problems, this invention provides a solution method for spacecraft escort problems based on LQR and differential games, applicable to master-slave satellite escort missions in space. This method transforms the non-zero-sum game problem into a phased zero-sum game problem and an optimal control problem, greatly simplifying the solution process and achieving good results in simulations.

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

[0006] A method for solving the spacecraft escort problem based on LQR and differential games includes the following steps:

[0007] Obtain the parameters of the spacecraft escort problem and construct a model of the spacecraft escort problem using LQR and differential games;

[0008] By inputting the parameters of the spacecraft escort problem into the spacecraft escort problem model, the control strategies of the attacking and defending stars in phases α and β are obtained, respectively.

[0009] Based on the real-time distance and safe distance between the attacking and defending satellites, determine the current stage of the attacking and defending satellites, and switch the corresponding control strategy according to the current stage.

[0010] The steps for constructing the spacecraft escort problem model are as follows:

[0011] S1: Define the three satellites involved in the spacecraft protection problem as the primary satellite, the attacking satellite, and the defending satellite;

[0012] S2: Initialize the safe distance between the attacking and defending stars, define the attacking star's behavior of leaving the defending star and approaching the host star, and divide the spacecraft escort problem into phase α and phase β based on the attacking star's behavior of leaving the defending star and approaching the host star.

[0013] S3: Based on the different objectives of the attacking and defending stars in phases α and β, the cost function is solved using LQR and differential strategies respectively to obtain the spacecraft escort problem model.

[0014] Furthermore, in S2, when the distance between the attacking star and the defending star is less than the safe distance, the attacking star's act of breaking away from the defending star's interception is defined as breaking away from the defending star; when the distance between the attacking star and the defending star is greater than or equal to the safe distance, the attacking star's act of approaching the main star is defined as approaching the main star.

[0015] Furthermore, the cost function in S3 is:

[0016] when hour,

[0017]

[0018] when hour,

[0019]

[0020]

[0021] Where: symmetric positive semi-definite matrix satisfy:

[0022] Q = k1I 6×6 R d =k2I 3×3 R a =k3I 3×3

[0023] In the formula:

[0024] k1, k2, k3 are weighting coefficients

[0025] Where matrix A represents the spacecraft's relative motion dynamics constraint matrix, its specific form is constructed using the CW equations in orbital dynamics, namely:

[0026]

[0027] in Let represent the orbital angular velocity of the reference satellite, μ be the Earth's gravitational field coefficient, and a0 be the semi-major axis of the reference satellite's orbit; where,

[0028] a — the subscript of the attacking star;

[0029] d — the subscript of the defense star;

[0030] m — the subscript of the primary star;

[0031] J a —Performance indicators of the attacking satellite when the distance between the attacking and defending satellites is greater than the safe distance;

[0032] J d —Performance indicators of the defensive satellite when the distance between the attacking and defensive satellites is greater than the safe distance;

[0033] J—Performance index of both the attacking and defending stars when the distance between them is less than the safe distance;

[0034] u amax —The maximum acceleration exerted by the attacking star;

[0035] u dmax —The maximum acceleration that the defensive star can apply;

[0036] X a —The state vector of the attacking star, specifically X a =[x a ,y a ,z a ,v ax ,v ay ,v az ] T ;

[0037] X d —The state vector of the defense star, specifically X d =[x d ,y d ,z d ,v dx ,v dy ,v dz ] T ;

[0038] u a —The continuous control quantity applied by the attacking star, specifically in the form of u a =[u ax ,u ay ,u az ] T ;

[0039] u d —The continuous control quantity applied by the defensive star, specifically in the form of ud =[u dx ,u dy ,u dz ] T ;

[0040] B—The control matrix for the attacking and defending stars, specifically in the form B = [0 3×3 ,I 3×3 ] T ;

[0041] d safe —The safe distance between the attacking and defensive stars.

[0042] Furthermore, the parameters of the spacecraft escort problem include: the semi-major axis a0 of the reference satellite orbit; and the initial state X of the attacking satellite. a (t0); Initial state X of the defense star d (t0); the initial position of the primary star [x m ,y m ,z m ] T The maximum acceleration u of the attacking star amax The maximum acceleration u of the defense star dmax The safe distance d between the attacking and defending stars safe .

[0043] Furthermore, dynamic programming is used to solve the control strategies of the attacking and defending stars in stage α. The specific steps are as follows:

[0044] Step 1, define the Hamiltonian function as:

[0045]

[0046] Where X is the relative motion state quantity between the defensive star and the attacking star, denoted as X = X a -X d λ is a costate variable;

[0047] Step 2: List the optimality conditions for the bilateral optimization problem:

[0048]

[0049] Step 3: Substitute the Hamiltonian function into the above optimality conditions to obtain the formal solution of the optimal control strategy for the defensive and offensive stars:

[0050]

[0051] Step four: Substitute the optimal control strategy solutions of both parties into the Hamiltonian function to obtain the HJ equation:

[0052]

[0053] Step 5: Under optimal conditions, the costate variables have the following form:

[0054]

[0055] Step 6: Substitute the above equations into the formal solution to obtain the optimal feedback control strategies for the defensive and offensive stars:

[0056]

[0057] Step 7: Rewrite the HJ equation accordingly, as follows:

[0058]

[0059] Summarized as follows:

[0060]

[0061] Step 8: Solve for the unknown matrix P using the Lyapunov iteration method. The iterative algorithm is as follows:

[0062]

[0063] The initial value is selected as:

[0064]

[0065] Step nine, finally obtaining the control strategies for the attacking and defending stars in phase α:

[0066]

[0067] Furthermore, the solution method for the control strategy of the attacking star and the defending star in stage β is as follows:

[0068] Step 1: The control strategy for the defensive star is the same in phases α and β.

[0069] Stage β

[0070] Step two: Use the LQR control method to solve for the control strategy of the attacking star. The optimal strategy is as follows:

[0071]

[0072] Wherein, the matrix to be solved The solution to the following Riccati equation is:

[0073]

[0074] Step 3: Obtain the control strategies for the attacking and defending stars in stage β.

[0075]

[0076] Furthermore, utilizing the control strategies of the attacking and defending stars in phases α and β, the accelerations of the attacking and defending stars at a given moment are calculated. If ||u a ||>u amax or ||u d ||>u dmax Then, the truncation method is adopted, such that ||u a ||=u amax ,||u d ||=u dmax .

[0077] Furthermore, in S1, the primary star does not maneuver, while the attacking star and the defending star can apply continuous thrust to maneuver.

[0078] Furthermore, after switching the current stage to the corresponding control strategy, a simulation experiment was conducted to obtain the experimental results.

[0079] Furthermore, the experimental results were plotted as trajectory diagrams and acceleration change curves.

[0080] Compared with the prior art, the present invention has the following beneficial effects:

[0081] This invention provides a solution method for the spacecraft escort problem based on LQR and differential game theory. It constructs a game model of the escort problem based on the CW equation, including a cost function, optimization variables, dynamic constraints, and acceleration capability constraints. The cost function uses a linear quadratic form index, the optimization variables are the continuous control force accelerations of both attackers and defenders, the dynamic constraints are described by the CW equation, and the acceleration capability constraints are modeled using maximum acceleration limits. This method has the advantages of simple model, clear physical meaning, and convenient solution. Furthermore, this invention transforms the complex non-zero-sum game problem into a staged zero-sum game problem and an optimal control problem, effectively solving the spacecraft escort problem under continuous thrust. It is an effective extension of existing continuous thrust escort problem models and methods, significantly improving the solution efficiency. The application of this model and method can provide an effective collaborative threat defense solution for spacecraft in orbit. Attached Figure Description

[0082] Figure 1 This is a game-theoretic scenario diagram of the spacecraft escort problem provided in an embodiment of the present invention;

[0083] Figure 2 A detailed flowchart of a method for solving the spacecraft escort problem based on LQR and differential games is provided for an embodiment of the present invention;

[0084] Figure 3 The simulation trajectory diagram provided for the embodiments of the present invention;

[0085] Figure 4 The simulation acceleration variation diagram provided for the embodiments of the present invention;

[0086] Figure 5 The flowchart shows a method for solving the spacecraft escort problem based on LQR and differential games, which is provided by the present invention. Detailed Implementation

[0087] This invention provides a method for solving the spacecraft escort problem based on LQR and differential games, comprising the following steps:

[0088] The first step is to obtain the parameters of the spacecraft escort problem and construct a model of the spacecraft escort problem using LQR and differential games;

[0089] The steps for constructing the spacecraft escort problem model are as follows:

[0090] S1: Define the three satellites involved in the spacecraft escort problem as the primary satellite, the attacking satellite, and the defending satellite. The primary satellite does not maneuver, while the attacking and defending satellites can apply continuous thrust to maneuver.

[0091] S2: Initialize the safe distance between the attacking and defending stars, define the attacking star's detachment from the defending star and its approach to the host star, and divide the spacecraft escort problem into phase α and phase β based on the attacking star's detachment from the defending star and its approach to the host star; where, when the distance between the attacking and defending stars is less than the safe distance, the attacking star's detachment from the defending star is defined as the detachment from the defending star; when the distance between the attacking and defending stars is greater than or equal to the safe distance, the attacking star's approach to the host star is defined as the approach to the host star.

[0092] S3: Based on the different objectives of the attacking and defending stars in phases α and β, the cost function is solved using LQR and differential strategies respectively to obtain the spacecraft escort problem model.

[0093] The cost function to be solved is:

[0094] when hour,

[0095]

[0096] when hour,

[0097]

[0098]

[0099] Where: symmetric positive semi-definite matrix satisfy:

[0100] Q = k1I 6×6 R d =k2I 3×3 R a =k3I 3×3

[0101] In the formula:

[0102] k1, k2, k3 are weighting coefficients

[0103] Where matrix A represents the spacecraft's relative motion dynamics constraint matrix, its specific form is constructed using the CW equations in orbital dynamics, namely:

[0104]

[0105] in Let represent the orbital angular velocity of the reference satellite, μ be the Earth's gravitational field coefficient, and a0 be the semi-major axis of the reference satellite's orbit; where,

[0106] a — the subscript of the attacking star;

[0107] d — the subscript of the defense star;

[0108] m — the subscript of the primary star;

[0109] J a —Performance indicators of the attacking satellite when the distance between the attacking and defending satellites is greater than the safe distance;

[0110] J d —Performance indicators of the defensive satellite when the distance between the attacking and defensive satellites is greater than the safe distance;

[0111] J—Performance index of both the attacking and defending stars when the distance between them is less than the safe distance;

[0112] u amax —The maximum acceleration exerted by the attacking star;

[0113] u dmax —The maximum acceleration that the defensive star can apply;

[0114] X a —The state vector of the attacking star, specifically X a =[x a ,y a ,z a ,v ax ,v ay ,v az ] T ;

[0115] X d —The state vector of the defense star, specifically X d =[x d ,y d ,z d ,v dx ,v dy ,v dz ] T ;

[0116] u a —The continuous control quantity applied by the attacking star, specifically in the form of u a =[u ax ,u ay ,u az ] T ;

[0117] u d —The continuous control quantity applied by the defensive star, specifically in the form of u d =[u dx ,u dy ,u dz ] T ;

[0118] B—The control matrix for the attacking and defending stars, specifically in the form B = [0 3×3 ,I 3×3 ] T ;

[0119] d safe —The safe distance between the attacking and defensive stars.

[0120] The second step involves inputting the spacecraft escort problem parameters into the spacecraft escort problem model to obtain the control strategies for the attacking and defending satellites in phases α and β, respectively. The spacecraft escort problem parameters include: the semi-major axis a0 of the reference satellite orbit; and the initial state X of the attacking satellite. a (t0); Initial state X of the defense star d (t0); the initial position of the primary star [x m ,y m ,z m ] T The maximum acceleration u of the attacking star amax The maximum acceleration u of the defense star dmax The safe distance d between the attacking and defending stars safe .

[0121] The dynamic programming method is used to solve the control strategies of the attacking and defending stars in stage α. The specific steps are as follows:

[0122] Step 1, define the Hamiltonian function as:

[0123]

[0124] Where X is the relative motion state quantity between the defensive star and the attacking star, denoted as X = X a -X d λ is a costate variable;

[0125] Step 2: List the optimality conditions for the bilateral optimization problem:

[0126]

[0127] Step 3: Substitute the Hamiltonian function into the above optimality conditions to obtain the formal solution of the optimal control strategy for the defensive and offensive stars:

[0128]

[0129] Step four: Substitute the optimal control strategy solutions of both parties into the Hamiltonian function to obtain the HJ equation:

[0130]

[0131] Step 5: Under optimal conditions, the costate variables have the following form:

[0132]

[0133] Step 6: Substitute the above equations into the formal solution to obtain the optimal feedback control strategies for the defensive and offensive stars:

[0134]

[0135] Step 7: Rewrite the HJ equation accordingly, as follows:

[0136]

[0137] Summarized as follows:

[0138]

[0139] Step 8: Solve for the unknown matrix P using the Lyapunov iteration method. The iterative algorithm is as follows:

[0140]

[0141] The initial value is selected as:

[0142]

[0143] Step nine, finally obtaining the control strategies for the attacking and defending stars in phase α:

[0144]

[0145] The solution method for the control strategy of the attacking and defensive stars in stage β is as follows:

[0146] Step 1: The control strategy for the defensive star is the same in phases α and β.

[0147] Stage β

[0148] Step two: Use the LQR control method to solve for the control strategy of the attacking star. The optimal strategy is as follows:

[0149]

[0150] Wherein, the matrix to be solved The solution to the following Riccati equation is:

[0151]

[0152] Step 3: Obtain the control strategies for the attacking and defending stars in stage β.

[0153]

[0154] The third step is to determine the current stage of the attacking and defending satellites based on the real-time distance and safe distance between them, and then switch the corresponding control strategy according to the current stage.

[0155] Furthermore, by utilizing the control strategies of the attacking and defending stars in phases α and β, the accelerations of the attacking and defending stars at a certain moment are calculated. If the calculation results yield ||u a ||>u amax or ||u d ||>u dmax Then, the truncation method is adopted, such that ||u a ||=u amax ,||u d ||=u dmax .

[0156] After switching to the corresponding control strategy at the current stage, simulation experiments can be conducted to obtain experimental results. These results can then be plotted as trajectory graphs and acceleration change curves for observation and analysis.

[0157] Example

[0158] This embodiment provides a method for solving the spacecraft escort problem based on LQR and differential games, such as Figure 5 As shown, it includes the following steps:

[0159] Obtain the parameters of the spacecraft escort problem and construct a model of the spacecraft escort problem using LQR and differential games;

[0160] By inputting the parameters of the spacecraft escort problem into the spacecraft escort problem model, the control strategies of the attacking and defending stars in phases α and β are obtained, respectively.

[0161] Based on the real-time distance and safe distance between the attacking and defending satellites, determine the current stage of the attacking and defending satellites, and switch the corresponding control strategy according to the current stage.

[0162] The steps for constructing the spacecraft escort problem model are as follows:

[0163] S1: Define the three satellites involved in the spacecraft protection problem as the primary satellite, the attacking satellite, and the defending satellite;

[0164] S2: Initialize the safe distance between the attacking and defending stars, define the attacking star's behavior of leaving the defending star and approaching the host star, and divide the spacecraft escort problem into phase α and phase β based on the attacking star's behavior of leaving the defending star and approaching the host star.

[0165] S3: Based on the different objectives of the attacking and defending stars in phases α and β, the cost function is solved using LQR and differential strategies respectively to obtain the spacecraft escort problem model.

[0166] To enable those skilled in the art to better understand the technical solutions of this invention, the technical solutions of this invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of this invention. Obviously, the described embodiments are only some embodiments of this invention, and not all embodiments.

[0167] See Figure 1 Assume there are three satellites near a circular orbit with an altitude of 2000 km (corresponding to a semi-major axis a0 = 8731 km): the primary satellite m, the attacking satellite a, and the defending satellite d. Establish an LVLH coordinate system with the primary satellite as the origin, and its initial state is shown in Table 1. Assume there are upper limits on the acceleration control values ​​of the attacking and defending satellites, with the attacking satellite's acceleration not exceeding 6 m / s². 2 The speed of the defensive star does not exceed 5 m / s 2 .

[0168] Table 1. Initial relative position and velocity (m, m / s) between attacking star a and defending star d

[0169] j <![CDATA[x j (t0)]]> <![CDATA[y j (t0)]]> <![CDATA[z j (t0)]]> <![CDATA[v xj (t0)]]> <![CDATA[v yj (t0)]]> <![CDATA[v zj (t0)]]> a 2700 1300 0 -150 50 0 d 2400 1200 0 -90 20 0

[0170] Based on the above conditions, specific embodiments of the present invention are given below, see [link to specific embodiments]. Figure 2 .

[0171] The first step in constructing the spacecraft escort problem model is as follows:

[0172] 1.1. First, the participants in the escort problem are identified as three satellites, referred to as the primary satellite, the attacking satellite, and the defending satellite, denoted by m, a, and d, respectively. The primary satellite does not maneuver, while the attacking and defending satellites can apply continuous thrust maneuvers.

[0173] 1.2. Design the safe distance d between the attacking and defending stars. safe When the distance between the attacking star and the defending star is less than the safe distance d safe At that time, the attacking star's goal is to escape the defense star's interception; when the distance between the attacking star and the defense star is greater than or equal to the safe distance d. safe At that time, the attacking star's objective is to get close to the host star. Based on the different objectives of the attacking star, the guardian problem is divided into two phases: phase α and phase β.

[0174] 1.3. Design cost functions for the attacking and defending stars in phases α and β respectively:

[0175] when hour,

[0176]

[0177] when hour,

[0178]

[0179] However, due to limitations in aerospace dynamics and acceleration, there are also corresponding constraints.

[0180] 1.4. Its overall model is as follows:

[0181] when hour,

[0182]

[0183] when hour,

[0184]

[0185]

[0186] Where: symmetric positive semi-definite matrix satisfy:

[0187] Q = k1I 6×6 R d =k2I 3×3 R a =k3I 3×3 (6)

[0188] In the formula:

[0189] k1 = 10 -5 k2 = 1, k3 = 0.8 are the weighting coefficients.

[0190] Where matrix A represents the spacecraft's relative motion dynamics constraint matrix, its specific form is constructed using the CW equations in orbital dynamics, namely:

[0191]

[0192] in The reference satellite's orbital angular velocity is represented by μ, the Earth's gravitational field coefficient is μ, and a0 = 8731 km is the semi-major axis of the reference satellite's orbit.

[0193] The meanings of the other symbols in equations (1) to (3) are as follows:

[0194] a — the subscript of the attacking star;

[0195] d — the subscript of the defense star;

[0196] m — the subscript of the primary star;

[0197] J a —Performance indicators of the attacking satellite when the distance between the attacking and defending satellites is greater than the safe distance;

[0198] J d —Performance indicators of the defensive satellite when the distance between the attacking and defensive satellites is greater than the safe distance;

[0199] J—Performance index of both the attacking and defending stars when the distance between them is less than the safe distance;

[0200] u amax —The maximum acceleration exerted by the attacking star;

[0201] u dmax —The maximum acceleration that the defensive star can apply;

[0202] X a —The state vector of the attacking star, specifically X a =[x a ,y a ,z a ,v ax ,v ay ,v az ] T ;

[0203] X d —The state vector of the defense star, specifically X d =[x d ,y d ,z d ,vdx ,v dy ,v dz ] T ;

[0204] u a —The continuous control quantity applied by the attacking star, specifically in the form of u a =[u ax ,u ay ,u az ] T ;

[0205] u d —The continuous control quantity applied by the defensive star, specifically in the form of u d =[u dx ,u dy ,u dz ] T ;

[0206] B—The control matrix for the attacking and defending stars, specifically in the form B = [0 3×3 ,I 3×3 ] T ;

[0207] d safe —The safe distance between the attacking star and the defending star, d safe =200m;

[0208] The second step involves inputting the spacecraft escort problem parameters into the spacecraft escort problem model to obtain the control strategies for the attacking and defending satellites in phases α and β, respectively. Figure 2 As shown, the specific steps are as follows: S1, specific parameters related to the spacecraft escort problem, including: the semi-major axis of the reference satellite orbit a0 = 8731 km; the initial state of the attacking satellite X. a (t0)=[2700m,1300m,0,-150m / s,50m / s,0] T The initial state of the defensive star X d (t0)=[2400m,1200m,0,-90m / s,20m / s,0] T The initial position of the primary star [x] m ,y m ,z m ] T The maximum acceleration of the attacking star [x] m ,y m ,z m ] T =[0,0,0] T The maximum acceleration u of the attacking star amax =6m / s; the maximum acceleration u of the defense star dmax= 5m / s; the safe distance d between the attacking and defending satellites safe =200m;

[0209] S2, the dynamic programming method is used to solve the bilateral optimization problem of stage α above. The specific steps are as follows:

[0210] S21, Define the Hamiltonian function as:

[0211]

[0212] Where X is the relative motion state quantity between the defensive star and the attacking star, denoted as X = X a -X d λ is a costate variable.

[0213] S22, List the optimality conditions for the bilateral optimization problem:

[0214]

[0215] S23, Substituting the Hamiltonian function into the above optimality conditions yields the formal solution of the optimal control strategy for both parties:

[0216]

[0217] S24, Substituting the optimal control strategy solutions of both parties into the Hamiltonian function yields the Hamilton-Jacobi equation (HJ equation):

[0218]

[0219] S25. To obtain the feedback control strategy, assume that the costate variables have the following form under optimal conditions:

[0220]

[0221] S26, Substituting the above equation into the formal solution yields the optimal feedback control strategy for both the attacker and defender:

[0222]

[0223] S27, The HJ equation is rewritten accordingly, as follows:

[0224]

[0225] Summarized as follows:

[0226]

[0227] S28. The unknown matrix P is solved using the Lyapunov iterative method, that is, the rewritten and simplified HJ equation, also known as the Riccati equation. The iterative algorithm is as follows:

[0228]

[0229] The initial value is selected as:

[0230]

[0231] S29, ultimately yielding the control strategies for the attacking and defensive stars in phase α:

[0232]

[0233] S3, design control strategies for the attacking and defensive stars in phase β, the specific steps are as follows:

[0234] S31, the cost function of the defensive star remains the same in phases α and β, so its control strategy can be considered to be the same in phases α and β:

[0235]

[0236] S32. The following section designs the control strategy for the attacking satellite in phase β. Since its performance index is a weighted average of the distance to the host satellite and fuel consumption, and the host satellite is a non-maneuvering spacecraft, the LQR control method can be directly used to design the control strategy for the attacking satellite. The optimal strategy is as follows:

[0237]

[0238] Wherein, the matrix to be solved The solution to the following Riccati equation is:

[0239]

[0240] S4 yields the control strategies for the attacking and defending stars in phases α and β, respectively, with the following expressions:

[0241]

[0242]

[0243] When calculating the accelerations of the attacking and defending stars using the above formulas, if the calculated acceleration of the attacking star at a certain moment exceeds the corresponding maximum acceleration, a truncation method is adopted, assuming that its acceleration at that moment should be the maximum applicable acceleration, such that ||u a ||>u amax =6m / s 2 ,||u d ||>u dmax =5m / s 2 .

[0244] The third step is to determine the timing of the attacking and defending stars. With d safe The magnitude of the values ​​is used to determine whether the current stage is α or β, and to continuously update the control strategy.

[0245] The fourth step is to conduct numerical simulation experiments using the above solution method, such as... Figure 3 and Figure 4 As shown, the trajectory diagrams and acceleration variation curves of both sides in the spacecraft escort problem are obtained.

[0246] This embodiment provides a solution method for the spacecraft protection problem based on LQR and differential game theory. First, a spacecraft protection problem model is established, including a cost function, optimization variables, dynamic constraints, and acceleration capability constraints. The cost function uses a linear quadratic index, the optimization variables are the continuous control force accelerations of both attackers and defenders, the dynamic constraints are described by the CW equation, and the acceleration capability constraints are modeled using maximum acceleration limits. Then, a solution method for the above optimization problem is established, specifically including the following steps: S1, the optimization problem is divided into a bilateral optimization problem and optimal control based on the stages; S2, for the bilateral optimization problem, dynamic programming is used to solve it, transforming the bilateral optimization problem into a two-point boundary value problem, and finally obtaining the optimal feedback control solution by solving the Riccati equation; S3, for the optimal control, dynamic programming is also used to solve it, and finally obtaining the optimal feedback control solution by solving the Riccati equation; S4, the control laws of both attackers and defenders in the two different stages are obtained; S5, the respective control strategies are switched simultaneously according to the stage transition. This proposed method for solving the spacecraft escort problem effectively addresses the issue of spacecraft escort under continuous thrust, representing a significant extension of existing models and methods for continuous thrust escort problems. The model employs a stage-switching approach, making the solution of non-zero-sum game problems simpler and faster.

[0247] The above embodiments are merely one of the implementation methods for achieving the technical solution of the present invention. The scope of protection claimed by the present invention is not limited to this embodiment, but also includes any variations, substitutions and other implementation methods that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention.

Claims

1. A method for solving the spacecraft escort problem based on LQR and differential games, characterized in that, Includes the following steps: Obtain the parameters of the spacecraft escort problem and construct a model of the spacecraft escort problem using LQR and differential games; By inputting the parameters of the spacecraft escort problem into the spacecraft escort problem model, the control strategies of the attacking and defending stars in phases α and β are obtained, respectively. Based on the real-time distance and safe distance between the attacking and defending satellites, determine the current stage of the attacking and defending satellites, and switch the corresponding control strategy according to the current stage. The steps for constructing the spacecraft escort problem model are as follows: S1: Define the three satellites involved in the spacecraft protection problem as the primary satellite, the attacking satellite, and the defending satellite; S2: Initialize the safe distance between the attacking and defending stars, define the attacking star's behavior of leaving the defending star and approaching the host star, and divide the spacecraft escort problem into phase α and phase β based on the attacking star's behavior of leaving the defending star and approaching the host star. S3: Based on the different objectives of the attacking and defending stars in phases α and β, the cost function is solved using LQR and differential strategies respectively to obtain the spacecraft escort problem model.

2. The method for solving the spacecraft escort problem based on LQR and differential games according to claim 1, characterized in that, In S2, when the distance between the attacking star and the defending star is less than the safe distance, the attacking star's act of breaking away from the defending star's interception is defined as breaking away from the defending star; when the distance between the attacking star and the defending star is greater than or equal to the safe distance, the attacking star's act of approaching the main star is defined as approaching the main star.

3. The method for solving the spacecraft escort problem based on LQR and differential games according to claim 1, characterized in that, The cost function in S3 is: when hour, when hour, Where: symmetric positive semi-definite matrix satisfy: Q=k1I 6×6 ,R d =k2I 3×3 ,R a =k3I 3×3 In the formula: k1, k2, k3 are weighting coefficients Where matrix A represents the spacecraft's relative motion dynamics constraint matrix, its specific form is constructed using the CW equations in orbital dynamics, namely: in Let represent the orbital angular velocity of the reference satellite, μ be the Earth's gravitational field coefficient, and a0 be the semi-major axis of the reference satellite's orbit; where, a — the subscript of the attacking star; d — the subscript of the defense star; m — the subscript of the primary star; J a —Performance indicators of the attacking satellite when the distance between the attacking and defending satellites is greater than the safe distance; J d —Performance indicators of the defensive satellite when the distance between the attacking and defensive satellites is greater than the safe distance; J—Performance index of both the attacking and defending stars when the distance between them is less than the safe distance; u amax —The maximum acceleration exerted by the attacking star; u dmax —The maximum acceleration that the defensive star can apply; X a —The state vector of the attacking star, specifically X a =[x a ,y a ,z a ,v ax ,v ay ,v az ] T ; X d —The state vector of the defense star, specifically X d =[x d ,y d ,z d ,v dx ,v dy ,v dz ] T ; u a —The continuous control quantity applied by the attacking star, specifically in the form of u a =[u ax ,u ay ,u az ] T ; u d —The continuous control quantity applied by the defensive star, specifically in the form of u d =[u dx ,u dy ,u dz ] T ; B—The control matrix for the attacking and defending stars, specifically in the form B = [0 3×3 ,I 3×3 ] T ; d safe —The safe distance between the attacking and defensive stars.

4. The method for solving the spacecraft escort problem based on LQR and differential games according to claim 3, characterized in that, The parameters of the spacecraft escort problem include: the semi-major axis a0 of the reference satellite orbit; and the initial state X of the attacking satellite. a (t0); Initial state X of the defense star d (t0); the initial position of the primary star [x m ,y m ,z m ] T The maximum acceleration u of the attacking star amax The maximum acceleration u of the defense star dmax The safe distance d between the attacking and defending stars safe .

5. The method for solving the spacecraft escort problem based on LQR and differential games according to claim 4, characterized in that, The dynamic programming method is used to solve the control strategies of the attacking and defending stars in stage α. The specific steps are as follows: Step 1, define the Hamiltonian function as: Where X is the relative motion state quantity between the defensive star and the attacking star, denoted as X = X a -X d λ is a costate variable; Step 2: List the optimality conditions for the bilateral optimization problem: Step 3: Substitute the Hamiltonian function into the above optimality conditions to obtain the formal solution of the optimal control strategy for the defensive and offensive stars: Step four: Substitute the optimal control strategy solutions of both parties into the Hamiltonian function to obtain the HJ equation: Step 5: Under optimal conditions, the costate variables have the following form: Step 6: Substitute the above equations into the formal solution to obtain the optimal feedback control strategies for the defensive and offensive stars: Step 7: Rewrite the HJ equation accordingly, as follows: Summarized as follows: Step 8: Solve for the unknown matrix P using the Lyapunov iteration method. The iterative algorithm is as follows: The initial value is selected as: Step nine, finally obtaining the control strategies for the attacking and defending stars in phase α:

6. The method for solving the spacecraft escort problem based on LQR and differential games according to claim 5, characterized in that, The solution method for the control strategy of the attacking and defensive stars in stage β is as follows: Step 1: The control strategy for the defensive star is the same in phases α and β. Step two: Use the LQR control method to solve for the control strategy of the attacking star. The optimal strategy is as follows: Wherein, the matrix to be solved The solution to the following Riccati equation is: Step 3: Obtain the control strategies for the attacking and defending stars in stage β.

7. The method for solving the spacecraft escort problem based on LQR and differential games according to claim 6, characterized in that, Using the control strategies of the attacking and defending stars in phases α and β, calculate the accelerations of the attacking and defending stars at a given moment. If ||u a ||>u amax or ||u d ||>u dmax Then, the truncation method is adopted, such that ||u a ||=u amax ,||u d ||=u dmax .

8. The method for solving the spacecraft escort problem based on LQR and differential games according to claim 7, characterized in that, In S1, the primary star does not maneuver, while the attacking star and the defending star can apply continuous thrust to maneuver.

9. The method for solving the spacecraft escort problem based on LQR and differential games according to claim 1, characterized in that, After switching the current stage to the corresponding control strategy, a simulation experiment was conducted to obtain the experimental results.

10. A method for solving the spacecraft escort problem based on LQR and differential games according to claim 9, characterized in that, The experimental results were plotted as trajectory graphs and acceleration change curves.

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