Synchronous orbit electric propulsion satellite remote drifting game control method

Abstract

The invention discloses a remote drifting game control method for synchronous orbit electric propulsion satellites. The method comprises the steps that the synchronous orbit electric propulsion satellites are divided into cooperative targets and non-cooperative targets; carrying out remote floating approaching control on the cooperative target by adopting a transverse thrust control method; for a non-cooperative target, establishing an absolute dynamic model considering J2 perturbation, constructing a time-optimal pursuit game saddle point strategy problem of introducing a Hamilton function based on the absolute dynamic model, and converting the time-optimal pursuit game saddle point strategy problem into a two-point boundary value problem; and converting a two-point boundary value problem into an unconstrained parameter optimization problem, performing global search and local optimization solution on the unconstrained parameter optimization problem, and performing remote drifting game control on the non-cooperative target according to an optimal solution. According to the invention, the low-cost floating game control of the geosynchronous orbit electric propulsion satellite is realized by designing a game control strategy with high success rate approaching.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of satellite drift proximity control, and particularly relates to a synchronous orbit electric propulsion satellite remote drift game control method. BACKGROUND

[0002] The geosynchronous orbit, especially the geostationary orbit (GEO), is an important orbital resource. Due to its unique attribute that the orbital period is the same as the rotation period of the earth, more and more high-value satellites are on the orbit, which play a key role in communication, navigation, mapping and meteorology. In this case, countries compete to launch geosynchronous orbit satellites to occupy orbital resources, which gradually leads to the increasingly crowded orbit and the increasing risk of space safety. Due to the high cost of launching high-orbit satellites, the design life of the satellites is usually longer than that of ordinary satellites, and the design cost is higher. Under the double influence of high launch cost and high design cost, the economic loss caused by the damage of the satellites is very great. Therefore, the demand for low-cost on-orbit services and game control in the geosynchronous orbit scenario is increasing, which is of great significance to protect space assets and maintain space safety.

[0003] Drift is a remote phase modulation method, which can also be called as fixed point position adjustment, that is, through thrust control, a spacecraft is transferred from a phase of an orbit to another phase of the same orbit. This technology is a basic technical means to meet the needs of on-orbit service and high-orbit game. As an important part of satellite repositioning and reducing the phase angle of the “pursuit-escape” spacecraft, it is usually guided and controlled by a ground control station, which has many limiting factors and high task cost. In order to enhance the autonomy and real-time performance of the satellite, the importance of the on-board autonomous rendezvous and phase modulation guidance strategy is increasing in the scenarios of on-orbit service and high-orbit game.

[0004] Electric propulsion, as a continuous small thrust propulsion, has the advantages of light weight, large specific impulse and long working time, and can provide stable small thrust for a long time, which is suitable for orbital adjustment tasks that require high precision and low fuel consumption.

[0005] In the prior art, the following problems exist in the remote drift star game control of the electric propulsion satellite: the traditional method mainly uses Hill equation or C-W equation to describe the relative dynamics model, which assumes that the relative distance between the pursuer and the evader is small, and the linearization processing can be performed near the reference orbit. However, for the remote drift star game scenario in the geosynchronous orbit, the initial relative distance between the pursuer and the evader is usually thousands or even tens of thousands of kilometers, the linearization assumption is no longer valid, and the J2 perturbation and other non-spherical gravitational field effects cannot be accurately described, which leads to a significant increase in modeling error and a decrease in orbit prediction accuracy; some technologies use Hohmann transfer or double-pulse orbit transfer to realize drift star phase modulation, which is relatively simple in calculation, but the pulse thrust needs a large instantaneous thrust amplitude, which is not suitable for the continuous small thrust characteristics of the electric propulsion system. Forced pulse approximation will significantly increase fuel consumption, and in the game confrontation scenario, it lacks real-time optimization capability and cannot dynamically adjust the control parameters according to the evader's maneuvering strategy; some existing researches use nonlinear programming, genetic algorithm and other direct numerical optimization methods to solve the electric propulsion orbit transfer problem. However, for the confrontation optimization problem such as the pursuit game, the simple numerical optimization method cannot guarantee to find the real game equilibrium solution (saddle point solution), and often only the unilateral optimal strategy can be obtained. In addition, the continuous small thrust orbit optimization problem has the characteristics of strong nonlinearity, multiple constraints and high dimension, and the traditional algorithm is easy to fall into local optimum, with low success rate and poor convergence.

[0006] However, due to the complexity of the continuous small thrust orbit optimization problem, the convergence of the time-optimal or fuel-optimal solution is poor, and the traditional method is difficult to quickly obtain an efficient solution. Therefore, for the drift star phase modulation control problem of the electric propulsion satellite, it is of great theoretical value and engineering significance to study efficient optimization algorithms and control strategies. SUMMARY

[0007] In view of the above problems in the prior art, the remote drift star game control method for the electric propulsion satellite in the geosynchronous orbit provided by the present application solves the problem of low control efficiency and accuracy in the remote drift star game control of the electric propulsion satellite.

[0008] In order to achieve the above-mentioned purposes, the technical scheme adopted by the present application is as follows: a remote drift star game control method for the electric propulsion satellite in the geosynchronous orbit, comprising: The electric propulsion satellite in the geosynchronous orbit is divided into cooperative targets and non-cooperative targets; For cooperative targets, the lateral thrust control method is used for long-range drifting satellite approach control; For non-cooperative objectives, an absolute dynamics model considering J2 perturbation is established. Based on the absolute dynamics model, a time-optimal saddle point strategy problem of pursuit and escape game is constructed by introducing Hamiltonian functions, and it is transformed into a two-point boundary value problem. The two-point boundary value problem is transformed into an unconstrained parameter optimization problem, and a global search and local optimization are performed to solve it. Based on the optimal solution, a remote floating star game is used to control the non-cooperative objective.

[0009] Furthermore, for non-cooperative targets, an absolute dynamic model considering J2 perturbation is established, including: An orbital dynamics model for a geostationary orbit electric propulsion satellite drifting is established using non-singular orbital elements; wherein, the non-singular orbital elements include the satellite's geocentric distance, satellite's geographical longitude, satellite's geographical latitude, satellite's orbital path angle, satellite's velocity, and satellite's orbital azimuth angle; the satellite includes a pursuer and a fleeing satellite; Based on the orbital dynamics model, the satellite's control variables are set, including its velocity in three directions: geocentric distance, geographic longitude, and geographic latitude, as well as its acceleration vector. Considering the J2 perturbation, determine the components of gravitational acceleration in the geodetic coordinate system; Based on the satellite's control variables, the velocity, acceleration vector, and gravitational acceleration components in the geocentric distance, longitude, and latitude directions are used to construct a satellite long-distance game dynamics equation considering J2 perturbation. Then, the absolute dynamics models of the satellites corresponding to the pursuer and the escaper are constructed, and the terminal conditions of the pursuit process are determined.

[0010] Furthermore, the satellite long-distance game dynamics equation is expressed as: In the formula, the subscript , and These represent the satellites corresponding to the pursuing and fleeing sides, respectively. Indicates satellite The rate of change of geocentric distance, Indicates satellite speed, Indicates satellite The running path angle; Indicates satellite The rate of change of geographical longitude, Indicates satellite The azimuth angle of operation, Indicates satellite The distance from the Earth's center, denotes the geographic latitude of the satellite denotes the geographic latitude of the satellite denotes the geographic latitude of the satellite denotes the rate of change of the geographic latitude of the satellite denotes the rate of change of the geographic latitude of the satellite denotes the rate of change of the velocity of the satellite denotes the rate of change of the velocity of the satellite denotes the control variable of the satellite denotes the control variable of the satellite denotes the control variable of the satellite , denotes the angle between the projection of the thrust vector of the satellite's thruster onto the plane spanned by the velocity vector and the angular momentum vector and the velocity vector denotes the angle between the projection of the unit mass thrust vector of the satellite's thruster onto the plane spanned by the velocity vector and the angular momentum vector and the plane denotes the gravitational constant of the Earth denotes the J2 perturbation denotes the velocity of the satellite denotes the rate of change of the flight path angle of the satellite denotes the flight path angle of the satellite denotes the flight path angle of the satellite denotes the flight path angle of the satellite wherein the J2 perturbation is expressed as wherein , and denote the components of the J2 perturbation on the velocity of the satellite, the flight path angle of the satellite and the flight path angle of the satellite, respectively denotes the J2 perturbation coefficient denotes the equatorial radius of the Earth

[0011] Further, the absolute dynamics model of the chaser and the evader corresponding to the satellite are expressed as wherein and denote the rate of change of the state variable vector of the chaser and the evader corresponding to the satellite, respectively and denote the state equation function of the chaser and the evader corresponding to the satellite, respectively and denote the state variable of the chaser and the evader corresponding to the satellite, respectively and denote the control variable of the chaser and the evader corresponding to the satellite, respectively denotes the time The terminal condition of the game process is expressed as: wherein, , and respectively represent the geocentric distance of the corresponding satellite of the pursuer, the satellite geographical longitude and the satellite geographical latitude, , and respectively represent the geocentric distance of the corresponding satellite of the evader, the satellite geographical longitude and the satellite geographical latitude, represents time, respectively represent terminal time.

[0012] Further, the saddle point strategy problem of the pursuit-evasion game is described as: wherein, represents the game performance index function, and respectively represent the thrust amplitude of the corresponding satellite of the pursuer and the optimal thrust control strategy of the corresponding satellite of the pursuer, and respectively represent the thrust amplitude of the corresponding satellite of the evader and the optimal thrust control strategy of the corresponding satellite of the evader, represents the Hamilton function, and respectively represent the co-state variable related to the state variable of the corresponding satellite of the pursuer and the evader, and respectively represent the state equation function of the corresponding satellite of the pursuer and the evader, and respectively represent the Hamilton function introduced by the corresponding satellite of the pursuer and the evader.

[0013] Further, the two-point boundary value problem obtained by converting the saddle point strategy problem of the pursuit-evasion game includes the state equation, the co-state equation, the state terminal condition, the co-state terminal condition and the Hamilton function terminal condition.

[0014] Further, in the two-point boundary value problem: the state equation is a satellite long-distance game dynamics equation, and the state terminal condition is a terminal condition of the game process; the co-state equation is: wherein, denotes the satellite co-state, denotes the satellite introduced Hamilton function, denotes the satellite state variable; the co-state terminal condition is: wherein, ~ denotes the chaser corresponding satellite co-state variable, ~ denotes the escape corresponding satellite co-state variable, denotes the terminal time; the Hamilton function terminal condition is: wherein, and denote time and terminal time respectively, denotes the satellite geographic latitude, denotes the satellite four, denotes the terminal constraint condition vector.

[0015] Further, based on the terminal condition in the saddle point strategy problem of pursuit-evasion game, the optimal control output of the corresponding satellite of the chaser and the escape is respectively: wherein, denotes the optimal yaw angle of the chaser corresponding satellite thruster, denotes the optimal pitch angle of the chaser corresponding satellite thruster, ~ denotes the chaser corresponding satellite co-state variable, denotes the chaser corresponding satellite speed, denotes the chaser corresponding satellite running path angle; denotes the optimal yaw angle of the escape corresponding satellite thruster, denotes the optimal pitch angle of the escape corresponding satellite thruster, ~ denotes the escape corresponding satellite co-state variable, denotes the escape corresponding satellite speed, denotes the escape corresponding satellite running path angle.

[0016] Further, the two-point boundary value problem is converted into an unconstrained parameter optimization problem, and global search and local optimization are carried out, including: The 6 initial values of the co-state of the satellite corresponding to the pursuit and escape sides in the two-point boundary value problem and the terminal time are taken as the optimization parameters, and a parameter optimization problem without constraints is constructed by combining the state terminal condition, the co-state terminal condition and the Hamilton function terminal condition; The parameter optimization problem is globally searched by simulating the natural evolution mechanism through a heuristic or evolutionary algorithm to obtain a suboptimal solution in the convergence domain of the optimal solution; The optimal solution is obtained by locally optimizing the suboptimal solution through a sequential quadratic programming algorithm, and then the non-cooperative target is controlled by remote drift star game.

[0017] The beneficial effects of the present application are: (1) The absolute dynamics modeling method considering J2 perturbation is introduced in the present application, and the satellite remote game dynamics equation is established in the geodetic coordinate system, which effectively overcomes the problem of insufficient precision of the traditional relative dynamics model in high-orbit long-distance rendezvous. The model can simultaneously describe the independent dynamic evolution process of the pursuit and escape sides in a wide range of space, providing a high-precision state prediction basis for remote game control.

[0018] (2) In the present application, the time-optimal pursuit and escape game problem is converted into a saddle point optimization problem by introducing the Hamilton function, and a two-point boundary value system composed of state equations, co-state equations and terminal conditions is established. This scheme can systematically balance the strategies of the pursuit and escape sides, so that the control process converges to the saddle point solution, thereby obtaining the time-optimal pursuit and escape result and improving the game stability and convergence speed of the control strategy.

[0019] (3) The present application proposes a hybrid parameter optimization algorithm of "global search + local optimization", which determines the initial convergence domain of the co-state initial value and the terminal time by using a heuristic algorithm, and then realizes local precise solution by a sequential quadratic programming algorithm. This hybrid optimization mechanism has both globality and accuracy, effectively overcomes the common multi-peak non-convergence phenomenon in continuous small thrust control problems, and improves the robustness and computational efficiency of the optimal solution.

[0020] (4) The present application is aimed at the continuous small thrust characteristics of electric propulsion satellites, and realizes remote drift star phase modulation and approach by controlling the thrust direction angle. This method can realize large-scale phase adjustment under low thrust conditions, reduce propellant consumption, prolong on-orbit mission life, and provide a high-performance control means for high-orbit on-orbit service and safety protection. BRIEF DESCRIPTION OF DRAWINGS

[0021] Figure 1 The present application provides a flowchart of the remote drift star game control method for synchronous orbit electric propulsion satellites.

[0022] Figure 2 The present application provides the state change trend of the pursuit and escape game process.

[0023] Figure 3 The pursuit and evasion control angle change trend is provided for the application.

[0024] Figure 4 The pursuit and evasion game trajectory is provided for the application. DETAILED DESCRIPTION

[0025] The specific embodiments of the application are described below to facilitate the understanding of the application for those skilled in the art, but it should be clear that the application is not limited to the scope of the specific embodiments, and for those skilled in the art, it is obvious that various changes are within the spirit and scope of the application defined and determined by the appended claims, and all the application and creation utilizing the concept of the application are within the scope of protection.

[0026] The embodiment of the application provides a remote drift star game control method for a synchronous orbit electric propulsion satellite, as shown in the figure, which comprises the following steps: Figure 1 The synchronous orbit electric propulsion satellite is divided into cooperative targets and non-cooperative targets; For the cooperative targets, the transverse thrust control method is used for remote drift star approach control; For the non-cooperative targets, the absolute dynamics model considering J2 perturbation is established, the time-optimal pursuit and evasion game saddle point strategy problem is constructed based on the absolute dynamics model, and the problem is converted into a two-point boundary value problem; The two-point boundary value problem is converted into an unconstrained parameter optimization problem, and the problem is solved by global search and local optimization, and the remote drift star game control is performed on the non-cooperative targets according to the optimal solution.

[0027] In the embodiment, the target spacecraft located in the geosynchronous orbit is divided into non-maneuvering targets (or cooperative targets) and maneuvering targets (or non-cooperative targets); wherein for the cooperative targets, the transverse thrust control method is used for remote drift star approach control, and the trade-off surface is provided for the drift star mission design by analyzing the drift star characteristics in the geostationary orbit. For the non-cooperative targets, since the relative distance is far, the relative dynamics modeling is not suitable for common use, the absolute dynamics model is established in the application, and the global optimization and precise solution are combined to control the remote game process, and the remote drift star game control is realized.

[0028] The embodiment mainly describes the remote drift star game control process of the non-cooperative targets: specifically, for the non-cooperative targets, the absolute dynamics model considering J2 perturbation is established, comprising: ​An orbital dynamics model for a geostationary orbit electric propulsion satellite drifting is established using non-singular orbital elements; wherein, the non-singular orbital elements include the satellite's geocentric distance, satellite's geographical longitude, satellite's geographical latitude, satellite's orbital path angle, satellite's velocity, and satellite's orbital azimuth angle; the satellite includes a pursuer and a fleeing satellite; Based on the orbital dynamics model, the satellite's control variables are set, including its velocity in three directions: geocentric distance, geographic longitude, and geographic latitude, as well as its acceleration vector. Considering the J2 perturbation, determine the components of gravitational acceleration in the geodetic coordinate system; Based on the satellite's control variables, the velocity, acceleration vector, and gravitational acceleration components in the geocentric distance, longitude, and latitude directions are used to construct a satellite long-distance game dynamics equation considering J2 perturbation. Then, the absolute dynamics models of the satellites corresponding to the pursuer and the escaper are constructed, and the terminal conditions of the pursuit process are determined.

[0029] For non-cooperative targets, their maneuverability means they may employ escape maneuvers during the approach, making conventional phasing control or trajectory-changing rendezvous difficult to achieve. Therefore, a game theory strategy analysis is necessary for this pursuit and escape problem.

[0030] For games in geosynchronous orbits, due to perturbation effects and high orbital altitude, the relative dynamic equations commonly used in studying game problems are no longer applicable. Therefore, it is necessary to establish an absolute dynamic model to deal with this long-distance game problem.

[0031] Establish an orbital dynamics model in the geodetic coordinate system ,in, Satellites are represented in sequence. The distance from the center of the earth, geographical longitude, geographical latitude, flight path angle, speed, and flight orientation. and Let represent the satellites corresponding to the pursuing and escaping sides, respectively. Further, the satellite's path angle is defined as the angle between the satellite's velocity vector and the local horizontal plane, and the satellite's azimuth angle is defined as the angle between the projection of the satellite's velocity vector onto the local horizontal plane and the due east direction of the local horizontal plane. Thus, we obtain the complete state variables of the satellite.

[0032] Regarding the control parameters of the satellite, let the thrust amplitudes of the pursuing and escaping forces be respectively... and In order to ensure that the pursuing party can successfully intercept the escaping party, it is necessary to make The satellite adjusts the thruster angle... and The game is completed by adjusting the trajectory. The angle between the projection of the thrust vector onto the plane spanned by the velocity vector and angular momentum and the velocity vector; Defined as the angle between the thrust vector per unit mass and the plane.

[0033] Therefore, the control quantities of the satellites corresponding to the pursuing and escaping sides are obtained as follows: From the definition of a state variable, its velocity in the three directions of distance from the Earth's center, geographical longitude, and geographical latitude can be obtained as follows: In the formula, The time derivative of the distance from the Earth's center; The acceleration vector is: In the formula, , and Let represent the acceleration components in the directions of geocentric distance, geographic latitude, and geographic longitude, respectively. The second time derivative of the distance from the Earth's center. The time derivative representing geographical latitude, The second time derivative representing geographical latitude. The time derivative of geographic longitude, This represents the second time derivative of geographical longitude.

[0034] In long-range game theory, considering the J2 perturbation, the acceleration component in the geodetic coordinate system is introduced as follows: In the formula, The gravitational constant representing the Earth. This represents the J2 perturbation coefficient.

[0035] Based on the above parameters, the satellite long-distance game dynamics equation is constructed as follows: In the formula, the subscript , and These represent the satellites corresponding to the pursuing and fleeing sides, respectively. Indicates satellite The rate of change of geocentric distance, Indicates satellite speed, Indicates satellite The running path angle; Indicates satellite The rate of change of geographical longitude, Indicates satellite The azimuth angle of operation, Indicates satellite The distance from the Earth's center, Indicates satellite Geographical latitude; Indicates satellite The rate of change of geographical latitude; Indicates satellite The rate of change of velocity, Indicates satellite The control quantity, , Indicates satellite The angle between the projection of the thrust vector of the thruster onto the plane spanned by the velocity vector and angular momentum and the velocity vector. Indicates satellite The angle between the thrust vector per unit mass of the thruster and the plane. The gravitational constant representing the Earth. Indicates J2 perturbation, Indicates satellite speed; Indicates satellite The rate of change of the running path angle, Indicates the satellite's orbital path angle; Indicates the satellite's azimuth angle; The J2 perturbation is represented as: In the formula, , and These represent the components of the J2 perturbation in satellite velocity, satellite path angle, and satellite azimuth angle, respectively. Indicates the J2 perturbation coefficient. It represents the radius of the Earth's equator.

[0036] The absolute dynamic models of the satellites corresponding to the pursuing and escaping sides are expressed as follows: In the formula, and These represent the rate of change vectors of the state variables of the satellites corresponding to the pursuing and escaping sides, respectively. and Let these represent the state equation functions of the satellites corresponding to the pursuing and escaping sides, respectively. and These represent the state variables of the satellites corresponding to the pursuing and escaping sides, respectively. and These represent the control quantities of the satellites corresponding to the pursuing and escaping sides, respectively. Indicates time.

[0037] Since the pursuer needs to successfully catch up with the fleeing player at the end, the terminal condition of the game process is set as follows: In the formula, , and These represent the geocentric distance, longitude, and latitude of the satellite corresponding to the pursuing party, respectively. , and These represent the geocentric distance, longitude, and latitude of the satellite corresponding to the escaping party, respectively. Indicates time, These represent the terminal time.

[0038] In this embodiment, to ensure the optimality of the game, a time-optimal saddle point strategy problem for the chase-escape game is constructed based on the absolute dynamics model, incorporating the Hamiltonian function. The solution to this problem is the saddle point solution; when the saddle point solution is satisfied, the time-optimal game is achieved. Furthermore, if either the chaser or the escaper fails to choose the optimal strategy, both will lose. Therefore, the Hamiltonian function is introduced.

[0039] Therefore, in this embodiment, the saddle point strategy problem in the pursuit-escape game is described as follows: In the formula, This represents a function that measures the performance of a game. and These represent the thrust amplitude of the pursuing satellite and the optimal thrust control strategy of the pursuing satellite, respectively. and These represent the thrust amplitude of the satellite corresponding to the escaping party and the optimal thrust control strategy of the satellite corresponding to the escaping party, respectively. Represents the Hamiltonian function. and These represent the co-state variables related to the state variables of the satellites corresponding to the pursuing and escaping sides, respectively. and Let these represent the state equation functions of the satellites corresponding to the pursuing and escaping sides, respectively. and These represent the Hamiltonian functions introduced by the satellites for the pursuing and escaping sides, respectively.

[0040] in, and It is used to reflect the immediate impact of changes in state variables at a certain moment on performance indicators.

[0041] To address the above issues, the costate equation is constructed as follows: In the formula, Indicates satellite co-state, Indicates satellite The introduced Hamiltonian function, Indicates satellite State variables; For both sides involved in the pursuit and capture: This allows us to obtain the detailed form of the costate equations for both the pursuer and the fleeing party. Since the costate equations are quite complex, we will use the pursuing party as an example here. The overall equation structure for the fleeing party's costate equation is the same as that for the pursuing party; simply replace the corresponding costates with the values ​​corresponding to the fleeing party.

[0042] In the above equation, to These are the six state variables corresponding to the satellite being pursued. to For the six state variables of the satellite corresponding to the pursuing party, the co-state variables are... to This represents the rate of change of each costate variable of the satellite corresponding to the pursuing party.

[0043] To determine the terminal conditions of costate variables, we define... To match terminal conditions The corresponding costate variables are derived as follows: The co-mode terminal condition is: In the formula, ~ This represents the co-state variable of the satellite corresponding to the pursuing side. ~ This represents the co-state variable of the satellite corresponding to the escaping party. Indicates terminal time; The terminal condition for the Hamilton function is: In the formula, and These represent the time and the terminal time, respectively. Indicates the satellite's geographical latitude. Indicates the satellite's fourth degree. This represents the terminal constraint vector.

[0044] To achieve the saddle point solution, the optimal control outputs of both the pursuer and the pursuer should satisfy: Therefore, the optimal control outputs for the satellites of the pursuing and escaping sides can be obtained as follows: In the formula, This indicates the optimal yaw angle for the pursuing satellite's thrusters. This indicates the optimal pitch angle of the satellite thruster corresponding to the pursuing target. ~ This represents the co-state variable of the satellite corresponding to the pursuing side. This indicates the speed of the satellite corresponding to the pursuing party. This indicates the orbital angle of the satellite corresponding to the pursuing party; This represents the optimal yaw angle for the satellite thruster corresponding to the escaping party. This represents the optimal pitch angle of the satellite thruster corresponding to the escaping side. ~ This represents the co-state variable of the satellite corresponding to the escaping party. This represents the satellite velocity corresponding to the escaping party. This indicates the satellite's orbital path angle corresponding to the escaping party.

[0045] Based on the above process, the two-point boundary value problem obtained by transforming the saddle point strategy problem of the pursuit and escape game includes the state equation, costate equation, state terminal condition, costate terminal condition and Hamilton function terminal condition in the above formula; wherein, the state equation is the satellite long-distance game dynamics equation, and the state terminal condition is the terminal condition of the game process. In this embodiment, the two-point boundary value problem is transformed into an unconstrained parameter optimization problem, and a global search and local optimization are performed on it, including: The initial values ​​of the six co-state variables of each satellite corresponding to the pursuer and the escaper in the two-point boundary value problem, as well as the terminal time, are used as optimization parameters. Combined with the state terminal condition, the co-state terminal condition, and the Hamiltonian function terminal condition, an unconstrained parametric optimization problem is constructed. By simulating the natural evolution mechanism through heuristic or evolutionary algorithms, a global search is performed on the parameter optimization problem to obtain the suboptimal solution within the convergence region of the optimal solution; Based on the suboptimal solution, a sequential quadratic programming algorithm is used for local optimization to obtain the optimal solution, and then a remote floating star game control is performed on the non-cooperative objective.

[0046] Specifically, the initial values ​​of six co-state variables for each side in the pursuit and pursuit process, and the terminal time. A total of 13 quantities are used as optimization parameters. Since the right-hand sides of the equations for the state termination condition, the co-state termination condition, and the Hamiltonian function termination condition all need to be equal to zero at the terminal time, the optimization objective is set as the square of the distance between the function set formed by these three and the zero point, which is: In the formula, This indicates whether the equation is true or false. When it is 0, it means that the objective function value is less than the minimum value. When the objective function approaches 0, it means that the equation condition is met.

[0047] To achieve parameter optimization, this invention employs a "global optimization + exact solution" approach. First, a heuristic or evolutionary algorithm is used to simulate the natural evolutionary mechanism, exploring possible solution regions through a population-based approach. After a global search, a suboptimal solution converges to the optimal solution. Then, a sequential quadratic programming (SQP) algorithm is used for local optimization to obtain the optimal solution. This hybrid solution strategy effectively improves global applicability while simultaneously enhancing the stability and accuracy of numerical convergence.

[0048] This embodiment provides a simulation example of the above-mentioned remote drifting star approach control for non-cooperative targets.

[0049] Since geostationary orbit is a high orbit with a long orbital period, this section normalizes the problem to simplify the calculations. We choose orbital altitude as the normalized unit for distance, i.e., DU = 38540053 m, and time as the normalized unit, TU = 4877.5 s. This normalization ensures the simplified Earth's gravitational constant. This simplifies the calculation. In this case, the normalized unit of velocity can be obtained from VU = DU / TU, and the normalized unit of acceleration can be obtained from AU = VU / TU.

[0050] The initial states of the game between the two spacecraft during the pursuit and escape process are given in Table 1. A hybrid parameter optimization algorithm combining global search and local optimization is adopted. The GA and fmincon functions in MATLAB are called to obtain the initial costate values ​​and terminal times of the pursuer and the escaper. Substituting these values ​​into the state equation and the costate equation and integrating them, the parameter changes of the long-range game can be obtained.

[0051] After 12105.955 seconds, the pursuing spacecraft successfully intercepted the escaping spacecraft. The trends of the state variables during the game between the pursuing and escaping spacecraft are as follows: Figure 2 As shown in the diagram, the red dashed line represents the changes in the parameters of the escaping spacecraft, and the blue solid line represents the changes in the parameters of the pursuing spacecraft. It can be seen that the changes in the Earth-center distance between the two sides were relatively small, and the pursuing spacecraft intercepted the escaping spacecraft by increasing its orbital altitude. Both the geographical longitude and latitude of the two sides increased, ultimately concluding the game around 66.62° longitude and 40.51° latitude.

[0052] The trend of the change in the control angle between the pursuing and fugitive sides is as follows: Figure 3 As shown in the figure, the pitch angle of the pursuing spacecraft's thrusters generally decreases, while the pitch angle of the escaping spacecraft first increases and then decreases. Although it made adjustments, it was eventually captured by the pursuing spacecraft. The yaw angle of the pursuing spacecraft's thrusters gradually decreases, while the yaw angle of the escaping spacecraft gradually increases.

[0053] The trajectory of the game between the pursuers and the fugitives is as follows Figure 4 As shown, after 12105.955 seconds, the pursuing spacecraft intercepted the escaping spacecraft at approximately 66.62° longitude and 40.51° latitude.

[0054] Specific embodiments have been used to illustrate the principles and implementation methods of this invention. The descriptions of the embodiments above are only for the purpose of helping to understand the method and core ideas of this invention. At the same time, for those skilled in the art, there will be changes in the specific implementation methods and application scope based on the ideas of this invention. Therefore, the content of this specification should not be construed as a limitation of this invention.

[0055] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the principles of the invention, and should be understood that the scope of protection of the invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of this invention.

Claims

1. A method for remote drifting satellite control in geosynchronous orbit with electric propulsion, characterized in that, include: Electric propulsion satellites in geosynchronous orbit are divided into cooperative and non-cooperative targets; For cooperative targets, the lateral thrust control method is used for long-range drifting satellite approach control; For non-cooperative objectives, an absolute dynamics model considering J2 perturbation is established. Based on the absolute dynamics model, a time-optimal saddle point strategy problem of pursuit and escape game is constructed by introducing Hamiltonian functions, and it is transformed into a two-point boundary value problem. The two-point boundary value problem is transformed into an unconstrained parameter optimization problem, and a global search and local optimization are performed to solve it. Based on the optimal solution, a remote floating star game is used to control the non-cooperative objective.

2. The remote drifting star game control method for geosynchronous orbit electric propulsion satellites according to claim 1, characterized in that, For non-cooperative targets, an absolute dynamic model considering J2 perturbation is established, including: An orbital dynamics model for a geostationary orbit electric propulsion satellite drifting is established using non-singular orbital elements; wherein, the non-singular orbital elements include the satellite's geocentric distance, satellite's geographical longitude, satellite's geographical latitude, satellite's orbital path angle, satellite's velocity, and satellite's orbital azimuth angle; the satellite includes a pursuer and a fleeing satellite; Based on the orbital dynamics model, the satellite's control variables are set, including its velocity in three directions: geocentric distance, geographic longitude, and geographic latitude, as well as its acceleration vector. Considering the J2 perturbation, determine the components of gravitational acceleration in the geodetic coordinate system; Based on the satellite's control variables, the velocity, acceleration vector, and gravitational acceleration components in the geocentric distance, longitude, and latitude directions are used to construct a satellite long-distance game dynamics equation considering J2 perturbation. Then, the absolute dynamics models of the satellites corresponding to the pursuer and the escaper are constructed, and the terminal conditions of the pursuit process are determined.

3. The remote drifting star game control method for geosynchronous orbit electric propulsion satellites according to claim 2, characterized in that, The satellite long-distance game dynamics equation is expressed as: In the formula, the subscript , and These represent the satellites corresponding to the pursuing and fleeing sides, respectively. Indicates satellite The rate of change of geocentric distance, Indicates satellite speed, Indicates satellite The running path angle; Indicates satellite The rate of change of geographical longitude, Indicates satellite The azimuth angle of operation, Indicates satellite The distance from the Earth's center, Indicates satellite Geographical latitude; Indicates satellite The rate of change of geographical latitude; Indicates satellite The rate of change of velocity, Indicates satellite The control quantity, , Indicates satellite The angle between the projection of the thrust vector of the thruster onto the plane spanned by the velocity vector and angular momentum and the velocity vector. Indicates satellite The angle between the thrust vector per unit mass of the thruster and the plane. The gravitational constant of the Earth, Indicates J2 perturbation, Indicates satellite speed; Indicates satellite The rate of change of the running path angle, Indicates the satellite's orbital path angle; Indicates the satellite's azimuth angle; The J2 perturbation is represented as: In the formula, , and These represent the components of the J2 perturbation in satellite velocity, satellite path angle, and satellite azimuth angle, respectively. Indicates the J2 perturbation coefficient. It represents the radius of the Earth's equator.

4. The remote drifting star game control method for geosynchronous orbit electric propulsion satellites according to claim 3, characterized in that, The absolute dynamic models of the satellites corresponding to the pursuing and escaping sides are expressed as follows: In the formula, and These represent the rate of change vectors of the state variables of the satellites corresponding to the pursuing and escaping sides, respectively. and Let these represent the state equation functions of the satellites corresponding to the pursuing and escaping sides, respectively. and These represent the state variables of the satellites corresponding to the pursuing and escaping sides, respectively. and These represent the control quantities of the satellites corresponding to the pursuing and escaping sides, respectively. Indicates time; The terminal condition of the game process is expressed as: In the formula, , and These represent the geocentric distance, longitude, and latitude of the satellite corresponding to the pursuing party, respectively. , and These represent the geocentric distance, longitude, and latitude of the satellite corresponding to the escaping party, respectively. Indicates time, These represent the terminal time.

5. The remote drifting star game control method for geosynchronous orbit electric propulsion satellites according to claim 3, characterized in that, The saddle point strategy problem in the pursuit-escape game is described as follows: In the formula, This represents a function that measures the performance of a game. and These represent the thrust amplitude of the pursuing satellite and the optimal thrust control strategy of the pursuing satellite, respectively. and These represent the thrust amplitude of the satellite corresponding to the escaping party and the optimal thrust control strategy of the satellite corresponding to the escaping party, respectively. Represents the Hamiltonian function. and These represent the co-state variables related to the state variables of the satellites corresponding to the pursuing and escaping sides, respectively. and Let these represent the state equation functions of the satellites corresponding to the pursuing and escaping sides, respectively. and These represent the Hamiltonian functions introduced by the satellites for the pursuing and escaping sides, respectively.

6. The remote drifting star game control method for geosynchronous orbit electric propulsion satellites according to claim 3, characterized in that, The two-point boundary value problem derived from the saddle point strategy problem in the pursuit-escape game includes the state equation, costate equation, state terminal condition, costate terminal condition, and Hamiltonian function terminal condition.

7. The remote drifting star game control method for geosynchronous orbit electric propulsion satellites according to claim 6, characterized in that, In the aforementioned two-point boundary value problem: The state equation is the satellite long-distance game dynamics equation, and the state terminal condition is the terminal condition of the game process. The costate equation is: In the formula, Indicates satellite co-state, Indicates satellite The introduced Hamiltonian function, Indicates satellite State variables; The co-mode terminal condition is: In the formula, ~ This represents the co-state variable of the satellite corresponding to the pursuing side. ~ This represents the co-state variable of the satellite corresponding to the escaping party. Indicates terminal time; The Hamilton function termination condition for: In the formula, and These represent the time and the terminal time, respectively. Indicates the satellite's geographical latitude. Indicates the satellite's fourth degree. This represents the terminal constraint vector.

8. The remote drifting star game control method for geosynchronous orbit electric propulsion satellites according to claim 6, characterized in that, Based on the terminal conditions in the saddle point strategy problem of the pursuit-escape game, the optimal control outputs of the satellites corresponding to the pursuer and the escaper are obtained as follows: In the formula, This indicates the optimal yaw angle for the pursuing satellite's thrusters. This indicates the optimal pitch angle of the satellite thruster corresponding to the pursuing target. ~ This represents the co-state variable of the satellite corresponding to the pursuing side. This indicates the speed of the satellite corresponding to the pursuing party. This indicates the orbital angle of the satellite corresponding to the pursuing party; This represents the optimal yaw angle for the satellite thruster corresponding to the escaping party. This represents the optimal pitch angle of the satellite thruster corresponding to the escaping side. ~ This represents the co-state variable of the satellite corresponding to the escaping party. This represents the satellite velocity corresponding to the escaping party. This indicates the satellite's orbital path angle corresponding to the escaping party.

9. The remote drifting star game control method for geosynchronous orbit electric propulsion satellites according to claim 6, characterized in that, The two-point boundary value problem is transformed into an unconstrained parametric optimization problem, and then subjected to global search and local optimization, including: Using the initial costate values ​​of the satellites corresponding to the pursuer and the escaper in the two-point boundary value problem as optimization parameters, and combining the state terminal condition, the costate terminal condition, and the Hamiltonian function terminal condition, we construct an unconstrained parameter optimization problem. By simulating the natural evolution mechanism through heuristic or evolutionary algorithms, a global search is performed on the parameter optimization problem to obtain the suboptimal solution within the convergence region of the optimal solution; Based on the suboptimal solution, a sequential quadratic programming algorithm is used for local optimization to obtain the optimal solution, and then a remote floating star game control is performed on the non-cooperative objective.

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