CN114911167A - An analytical solution method and system for the finite-time pursuit and escape game control of spacecraft - Google Patents

Abstract

The invention discloses an analytical solution method and system for the limited-time pursuit and escape game control of a spacecraft. The method includes: in the LVLH coordinate system, using the zero-control miss amount as a state variable, establishing a fixed-time differential game model for the spacecraft orbital pursuit and escape; The optimal control law of the form and its corresponding optimal game path. The invention can quickly solve the saddle point of the fixed-time differential game of the spacecraft pursuit and escape game, the bilateral optimal control law of pursuit and escape corresponding to the saddle point can plan the game path for the spacecraft, avoid space debris and other unknown goals, and has the advantages of correct and reasonable design method, The calculation process is fast and effective, and the applicability to practical tasks is good.

Description

An analytical solution method and system for the finite-time pursuit and escape game control of spacecraft

technical field

The invention belongs to the technical field of spacecraft pursuit and escape game path planning, and in particular relates to an analytical solution method and system for spacecraft limited-time pursuit and escape game control.

Background technique

With the continuous development of space rendezvous and docking technology and on-orbit service technology, the control method for cooperative targets has become mature, and the rendezvous or interception control method (chasing and fleeing game control technology) considering that both parties have maneuverability needs to be developed urgently. It is of great significance for avoiding unknown targets such as space debris and for national space security defense. Correspondingly, the control problem of the pursuit-flight game between two spacecraft has also become a hot issue in the current cutting-edge technology theory. The problem is essentially a continuous-time dynamic programming problem under bilateral active control, in which the two sides of the game have conflicting goals: the tracking spacecraft aims to approach the escape spacecraft, and the escape spacecraft needs to work hard to escape the approach. Therefore, compared with the rendezvous control problem of cooperative targets, the spacecraft pursuit-flight game control problem has more confrontation and uncertainty, and its problem dimension and solution difficulty are also doubled.

Differential game is a theory that uses the differential method to study infinite dynamic games in continuous time, which is very suitable for modeling and analyzing such problems. Differential games can be divided into qualitative differential games and quantitative differential games according to whether the payoff function exists or not. Fixed-time differential games are a kind of quantitative differential games, which mainly study the solution of the differential game saddle point under the premise of a given game time. The saddle point of the differential game refers to the Nash equilibrium point of both sides of the pursuit and escape. The solution of the differential game at the saddle point corresponds to the optimal game path and control law of the pursuit and escape, which can provide an effective reference for the design of the game strategy of the spacecraft of the pursuit and escape. It is of great significance to solve my country's space security issues. For a detailed introduction to the theory of differential games, please refer to the book "Differential Games and Their Applications" by Li Dengfeng.

When solving the saddle point of the spacecraft fixed-time differential game, the differential game model of the spacecraft orbital pursuit and escape is usually established as a linear quadratic zero-sum differential game. The solution method of the linear quadratic zero-sum differential game can be divided into numerical method and analytical method. At present, the numerical method is basically used. The relative motion state of the spacecraft and the corresponding fixed-time differential game model are established, and the Hamilton-Jacobi-Bellman (HJB) partial differential equation (or according to Pontryagin) of the game is constructed according to the principle of dynamic programming. The maximum value principle constructs the Hamiltonian function of the game) and converts it into a matrix Riccati differential equation, and the closed-loop optimal linear feedback control law of both sides is obtained by solving the matrix Riccati differential equation. Considering that the solution of numerical methods usually requires the integration of matrix Riccati differential equations, the solution time is long, and it is difficult to apply to engineering practice. Realize the fast and real-time solution of the spacecraft path planning problem and its similar derivative problems under the premise of the chase-flight game time.

SUMMARY OF THE INVENTION

The present invention proposes an analytical solution method and system for the game control of a spacecraft's limited-time chase and escape game, so as to overcome the defects of the prior art such as complicated solution process and long solution time, reduce the solution complexity, greatly shorten the solution time, and satisfy the Fast and real-time solution requirements in engineering practice.

In order to achieve the above purpose, the present invention provides an analytical solution method for the limited-time pursuit and escape game control of a spacecraft, comprising the following steps:

S1: In the LVLH coordinate system, the zero-control miss amount is used as the state variable to establish a fixed-time differential game model for spacecraft orbital pursuit and escape;

S2: Based on the fixed-time differential game model established in S1, solve the optimal control law and its corresponding optimal game path in analytical form for the spacecraft of the pursuit and escape, respectively.

In order to achieve the above object, the present invention also provides an analytical solution system for spacecraft limited-time pursuit-escape game control, including a memory and a processor, wherein the memory stores an analytical solution program for spacecraft limited-time pursuit-escape game control, the The processor performs the following steps when running the analytical solution program for the limited-time pursuit and escape game control of the spacecraft:

S0: Obtain the initial parameters of the orbital pursuit and escape game of the two spacecraft under the fixed-time differential game;

S1: In the LVLH coordinate system, the zero-control miss amount is used as the state variable to establish a fixed-time differential game model for spacecraft orbital pursuit and escape;

S2: Based on the fixed-time differential game model established in S1, solve the optimal control law and its corresponding optimal game path in analytical form for the spacecraft of the pursuit and escape, respectively.

To achieve the above object, the present invention also provides a computer-readable storage medium on which a computer program is stored, and when the computer program is executed by a processor, the following steps are implemented:

S0: Obtain the initial parameters of the orbital pursuit and escape game of the two spacecraft under the fixed-time differential game;

S1: In the LVLH coordinate system, the zero-control miss amount is used as the state variable to establish a fixed-time differential game model for spacecraft orbital pursuit and escape;

S2: Based on the fixed-time differential game model established in S1, solve the optimal control law and its corresponding optimal game path in analytical form for the spacecraft of the pursuit and escape, respectively.

The analytical solution method and system for a spacecraft limited-time pursuit and escape game control provided by the present invention have the following advantages:

1. The optimal game control law obtained by the analytical method in the present invention is completely consistent with the numerical method, but the solution is simpler and faster. Considering that the pursuit and escape game of the spacecraft is more likely to be carried out in orbit in real time, and the computing power of the on-board computer is relatively weak; therefore, the analytical solution method of the spacecraft's fixed-time differential game is more conducive to the on-board computer to solve the game quickly and accurately. The maneuvering strategy has better applicability in practical engineering tasks.

2. Compared with the solution process of the optimal game control law in the numerical method, the derivation and application of the optimal game control law in the analytical form of the present invention is more intuitive and concrete, and it is also easier to reflect the pursuit and escape game of the two spacecraft under the premise of a given game time. general rule.

Description of drawings

Fig. 1 is the technical scheme diagram of the analytical solution method for the limited-time pursuit and escape game control of the spacecraft according to the present invention;

2 is a schematic flowchart of the processor executing the solving program in the memory in the limited-time pursuit and escape game control analysis and solving system of the spacecraft according to the present invention;

Fig. 3 is the optimal path diagram of the game of pursuit and escape of the two spacecraft in the preferred embodiment 1;

FIG. 4 is a graph showing the variation law of the position and distance of the spacecraft of both sides relative to the reference point in the preferred embodiment 1, wherein: (a) is a graph of the variation law of the relative position component x with time, and (b) is a relative The change law of the position component y with time, (c) is the change law of the relative position component z with time, (d) is the change law of the relative distance s with time;

Fig. 5 is a graph of the variation law of the optimal thrust acceleration with time of the spacecraft of the pursuit and escape in the preferred embodiment 1, wherein: (a) is a graph of the variation of the optimal thrust acceleration component a _x with time, and (b) is the optimal thrust acceleration component a x. The variation law of the optimal thrust acceleration component a _y with time, (c) is the variation law of the optimal thrust acceleration component a _z with time, (d) is the variation law of the optimal thrust acceleration magnitude a with time;

Fig. 6 is the optimal path diagram of the game of pursuit and escape of the two spacecraft in the preferred embodiment 2;

FIG. 7 is a graph showing the variation law of the position and distance of the spacecraft of both parties relative to the reference point relative to the reference point in the preferred embodiment 2, wherein: (a) is a graph of the variation law of the relative position component x with time, and (b) is the relative position component x. The change law of the position component y with time, (c) is the change law of the relative position component z with time, (d) is the change law of the relative distance s with time;

Fig. 8 is a graph showing the variation law of the speed of the two spacecraft relative to the reference point in the preferred embodiment 2, wherein: (a) is the variation law of the relative velocity component v _x with time, and (b) is the relative velocity The graph of the variation rule of the component v _y with time, (c) is the variation rule diagram of the relative velocity component v _z with time, (d) is the variation rule diagram of the relative velocity magnitude v with time;

Fig. 9 is the variation law diagram of the optimal thrust acceleration with time of the spacecraft of the pursuit and escape in the preferred embodiment 2, wherein: (a) is the variation rule diagram of the optimal thrust acceleration component a _x with time, (b) is the optimal thrust acceleration component a x with time variation diagram The variation law of the optimal thrust acceleration component a _y with time, (c) is the variation law of the optimal thrust acceleration component a _z with time, (d) is the variation law of the optimal thrust acceleration magnitude a with time;

FIG. 10 is an internal structure diagram of the computer equipment used in the analytical solution system for the limited-time pursuit and escape game control of the spacecraft according to the present invention.

Detailed ways

In order to make the purpose, technical solutions and advantages of the present application more clearly understood, the present application will be described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present application, but not to limit the present application.

The specific embodiments of the present invention are as follows:

In one embodiment, as shown in FIG. 1 , an analytical solution method for limited-time pursuit and escape game control of a spacecraft is provided, including the following steps:

S1: In the LVLH coordinate system, a fixed-time differential game model for spacecraft orbital pursuit and escape is established by using the zero-control miss amount as the state variable.

Further, the S1 includes the following steps:

S101: In the LVLH coordinate system, establish a fixed-time differential game initial model based on the C-W equation of relative motion of the spacecraft;

1) Establish the LVLH coordinate system as the reference coordinate system, and use the C-W equation to describe the relative motion of the spacecraft on both sides;

Taking the escape spacecraft as the reference spacecraft and the virtual circular orbit corresponding to the position of the escape spacecraft as the reference spacecraft orbit, the LVLH coordinate system is established. At this time, the LVLH coordinate system is established as follows: the origin of the coordinate system is located at the center of mass O of the reference spacecraft, the Ox axis is along the radial vector direction of the reference spacecraft, the Oz axis is along the normal direction of the virtual circular orbit, and the Oy axis moves along the reference spacecraft. The tangent direction of the trajectory and the Ox axis and the Oz axis form a right-hand coordinate system.

In the LVLH coordinate system, the C-W equation shown in equation (1) is used to describe the relative motion of the pursuit and escape spacecraft:

分别为航天器相对于参考坐标系沿径向、迹向和法向的位置分量和速度分量，ω为航天器在虚拟圆轨道上运动的角速度，a_x、a_y、a_z分别为航天器沿径向、迹向和法向的推力加速度分量。Among them, x, y, z,

are the position component and velocity component of the spacecraft relative to the reference coordinate system along the radial, track and normal directions, respectively, ω is the angular velocity of the spacecraft moving on the virtual circular orbit, a _x , a _y , and a _z are the spacecraft Thrust acceleration components in radial, trace, and normal directions.

推力加速度矢量为u＝[a_x,a_y,a_z]^T，则C-W方程可写成状态空间形式为Let the relative motion state of the spacecraft be

The thrust acceleration vector is u=[a _x , a _y , a _z ] ^T , then the CW equation can be written in the state space form as

in,

Equation (2) is a linear inhomogeneous constant coefficient system equation; under the premise of given initial condition x(t ₀ )=x ₀ , the solution of the equation is

In formula (3), the state transition matrix Φ(t,t ₀ ) is

Wherein, s=sinωΔt, c=cosωΔt, Δt=tt ₀ .

2) Establish the differential game state equation of the spacecraft pursuit and escape game;

且二者均满足C-W方程的使用条件；令

为航天器追逃微分对策的状态变量，a_P,x、a_P,y、a_P,z、a_E,x、a_E,y、a_E,z分别为追逃双方航天器沿径向、迹向和法向的推力加速度分量，则式(1)可进一步写为Suppose the relative motion states of the chasing and fleeing spacecraft are as follows:

And both meet the conditions of use of the CW equation; let

are the state variables of the differential countermeasures for the pursuit and escape of the spacecraft, a _P,x , a _P,y , a _P,z , a _E,x , a _E,y , and a _E,z are the radial direction of the pursuit and escape spacecraft, respectively , the thrust acceleration components in the trace and normal directions, then equation (1) can be further written as

则推力加速度沿径向、迹向和法向的各分量为Assuming that the spacecraft on both sides of the pursuit and escape use continuous thrust control, the maximum thrust acceleration is _TP and TE respectively, and the direction of the thrust acceleration is the yaw angle α _{(the angle between the projection of T i on} the xOy plane and the Ox axis) and the pitch. The angle β (the included angle between T _i and the xOy plane) represents, α∈[0,2π],

Then the components of thrust acceleration along the radial, track and normal directions are

where i=P,E. Substituting Equation (6) into Equation (5), the differential game state equation of the spacecraft pursuit and escape game can be obtained as

3) Establish the initial model of the fixed-time differential game of the spacecraft pursuit and escape game;

和

且满足|u_P|≤T_P、|u_E|≤T_E，则由式(2)、式(6)和式(7)可得航天器追逃博弈的微分对策状态方程为Set the given game time of pursuit and escape as T=t _f -t ₀ , and the control quantities of the spacecraft of both parties are respectively the thrust acceleration vector

and

And satisfy |u _P |≤T _P , |u _E |≤T _E , then the differential game state equation of the spacecraft pursuit game can be obtained from equations (2), (6) and (7) as:

Considering that the pursuit spacecraft hopes to reach the end time t _f of the pursuit and escape game, the closer the distance between the two spacecrafts is, the better, and the less fuel is consumed during the pursuit and escape game, the better, and the purpose of the escape spacecraft is On the contrary; the payoff function of the pursuit and escape of the two spacecraft can be defined as

为半正定对称矩阵，表征了追逃博弈结束时刻支付函数中双方航天器之间的距离所占比重；R_P＝r_PI₃和R_E＝r_EI₃为正定对称矩阵，分别表征了支付函数中追踪航天器和逃逸航天器消耗的燃料所占比重。特别地，当固定时间微分对策中追踪航天器的目的是交会而非拦截时，Q_f＝q_fI₆。in,

is a semi-positive definite symmetric matrix, which represents the proportion of the distance between the two spacecraft in the payoff function at the end of the chase-and-flight game; R _P =r _P I ₃ and R _E =r _E I ₃ are positive definite symmetric matrices, which respectively represent The fraction of fuel consumed by the tracking and escape spacecraft in the payout function. In particular, when the objective of tracking the spacecraft in the fixed-time differential game is rendezvous rather than interception, Q _f = q _f I ₆ .

By combining Equation (8) and Equation (9), the initial model of the fixed-time differential game of the spacecraft pursuit and escape game can be established.

S102: Define the zero-control miss-target amount as a new state variable, and on the basis of the original initial model, re-derive and establish a fixed-time differential game model for spacecraft orbital pursuit and escape.

Define the zero-control off-target amount y(t) as a new state variable, namely

y(t)=DΦ(t _f ,t)x(t) (10)

特别地，当固定时间微分对策中追踪航天器的目的是交会而非拦截时，D＝I₆。where, D=[I ₃ O ₃ ],

In particular, when the objective of tracking the spacecraft in the fixed-time differential game is to rendezvous rather than to intercept, D=I ₆ .

make

G(t _f ,t)=DΦ(t _f ,t)B (11)

According to formula (10), the new state variable y(t) is derived and substituted into formula (8) and formula (11), the new differential game state equation can be obtained as:

Correspondingly, the payment function of the pursuit and escape of the two spacecraft can be defined as

为半正定对称矩阵，表征了追逃博弈结束时刻支付函数中追踪航天器相对于逃逸航天器的零控脱靶量所占比重；特别地，当固定时间微分对策中追踪航天器的目的是交会而非拦截时，

in,

It is a semi-positive definite symmetric matrix, which represents the proportion of the tracking spacecraft relative to the zero-control misses of the escape spacecraft in the payoff function at the end of the chase-and-flight game; When not intercepted,

Combining equations (12) and (13), the fixed-time differential game model of the spacecraft pursuit and escape game can be established with the zero-control miss amount as the state variable.

So far, on the basis of the original initial model, the fixed-time differential game model of spacecraft orbital pursuit and escape has been re-derived and established.

S2: Based on the fixed-time differential game model established in S1, solve the optimal control law and its corresponding optimal game path in analytical form for the spacecraft of the pursuit and escape, respectively.

Further, the S2 includes the following steps:

S201: Based on the fixed-time differential game model established in S1, the variational method is used to solve the optimal game control law for the pursuit and escape of the two spacecraft;

Based on the fixed-time differential game model of spacecraft orbital pursuit and escape composed of equations (12) and (13), the Hamiltonian function of the game is first constructed according to the Pontryagin maximum principle, and then the variational method is used to solve the pursuit and escape. The optimal game control law for both spacecraft. For specific methods and steps, please refer to the literature [Y.C.Ho, A.E.Bryson Jr., S.Baron.Differential games and optimal pursuit-evasion strategies[J].IEEETransactions on Automatic Control,AC-10(4),385-389,1965. ].

Using the variational method to solve:

in,

Solving Equation (11), Equation (16), Equation (15) and Equation (14) in turn, the optimal game control law for the pursuit and escape of the two spacecraft can be obtained.

S202: Analytically solve the optimal game control law of the pursuit and escape spacecraft and obtain the corresponding optimal game path.

From equation (4), it can be obtained that for the fixed-time differential game model of spacecraft orbital pursuit and escape newly established in S1, we have

Wherein, s=sinωτ, c=cosωτ, τ=t _f -t.

Substituting equation (17) into equation (11), we can get

Substituting Equation (18) into Equation (16) and Equation (15) in turn, we can get

in,

Substituting Equation (19) and Equation (10) into Equation (14), we can obtain the optimal game control law of the analytical form of the spacecraft of the pursuit and escape, respectively:

Among them, K ^-1 (t _f ,t) can be obtained by LU decomposition and inverse matrix method.

It is not difficult to see from equation (20): when the end time t _f of the pursuit and escape game is given, the optimal game control laws u _P and u _E of the two spacecrafts in pursuit and escape are only related to the current time t and the relative motion of the two spacecrafts at the current time. State x is related.

In particular, when the objective of tracking the spacecraft in the fixed-time differential game is to rendezvous rather than intercept,

where τ=t _f -t,

So far, the analytical solution of the spacecraft's fixed-time differential game has been completed.

Considering that the spacecraft on both sides of the pursuit and escape use continuous thrust control, the maximum thrust accelerations are _{T P} and TE respectively; for the current moment t∈[t ₀ ,t _f ], when the optimal thrust acceleration obtained by the analytical method satisfies When |u _P |>T _P or |u _E |>T _E , the thrust saturation limit is adopted as

where i=P,E. When |u _P |≤T _P and |u _E | _≤TE , the obtained optimal thrust accelerations u _P and u _E remain unchanged.

Convert the optimal thrust accelerations u _P and u _E of the chasing and fleeing spacecraft into the geocentric inertial coordinate system, and then substitute them into the orbital dynamics equation of the spacecraft and solve it to obtain the orbital dynamic equations of the two spacecraft in the geocentric inertial coordinate system. the optimal game path.

The preferred embodiment 1 is an orbital pursuit and escape game between two spacecrafts under a fixed-time differential game. In the preferred embodiment 1, the purpose of tracking the spacecraft is to intercept the escaping spacecraft at the end of the game (that is, the relative distance between the chasing and fleeing spacecraft at the end of the game is 0), and the purpose of the escaping spacecraft is the same as that of the escaping spacecraft. on the contrary.

Preferred Embodiment 1 provides an analytical solution system for the limited-time pursuit and escape game control of a spacecraft, including a memory and a processor, wherein the memory stores an analytical solution program for the limited-time pursuit and escape game control of the spacecraft, and the processor is in The following steps are performed when running the analytical solution program for the limited-time pursuit and escape game control of the spacecraft:

S0: Obtain the initial parameters of the orbital pursuit and escape game of the two spacecraft under the fixed-time differential game;

Table 1 Example 1 Initial parameter configuration

According to the initial parameters shown in Table 1, the relevant physical quantities of the differential game state equation of the orbital pursuit and escape game of the two spacecraft can be obtained as:

为微分对策的状态变量初值，x_P，0、x_E，0为初始时刻追逃双方航天器相对于虚拟圆轨道参考点的运动状态，ω₀为初始时刻参考虚拟圆轨道的角速度，μ为地心引力常数，r_ref,0＝[x_ref,0,y_ref,0,z_ref,0]^T为参考点处航天器在地心惯性坐标系下的位置矢量。in,

is the initial value of the state variables of the differential game, x _{P, 0} , x _{E, 0} are the motion states of the chasing and fleeing spacecraft relative to the reference point of the virtual circular orbit at the initial moment, ω ₀ is the angular velocity of the reference virtual circular orbit at the initial moment, μ is the gravitational constant, r _ref,0 =[x _ref,0 ,y _ref,0 ,z _ref,0 ] ^T is the position vector of the spacecraft at the reference point in the earth-centered inertial coordinate system.

In addition, set the relevant parameters of the payoff function of the spacecraft of both parties as follows: q _f =1×10 ¹⁰ , r _P =8×10 ⁷ , r _E =1×10 ⁸ , t ₀ =0, t _f =3600s; , the acquisition of the initial parameters of the orbital pursuit and escape game of the two spacecraft under the fixed-time differential game has been completed.

S1: In the LVLH coordinate system, the zero-control miss amount is used as the state variable to establish a fixed-time differential game model for spacecraft orbital pursuit and escape;

Calculate r _ref,0 according to the parameters given in Table 1 and substitute it into Equation (24) to obtain ω ₀ ; Substitute ω ₀ and t _f into Equation (18) to obtain the analytical expression of G(t _f ,t); The analytical expression of t _f , t) is substituted into equation (12), and the relevant parameters of the payment function are substituted into equation (13), the fixed-time differential game model of spacecraft orbital pursuit and escape with y(t) as the state variable can be obtained.

S2: Based on the fixed-time differential game model established in S1, the optimal control law and the corresponding optimal game path in analytical form of the spacecraft of the pursuit and escape are solved respectively.

Substitute ω ₀ and t _f into Equation (17) to obtain the analytical expression of Φ(t _f ,t); Substitute q _f , r _P , r _E , ω ₀ and t _f into Equation (19) and obtain it by LU decomposition Inverse matrix, get the analytical expression of K ^-1 (t _f ,t); combine the analytical expressions of G(t _f ,t), K ^-1 (t _f ,t), Φ(t _f ,t) and r Substitute _P , r _E into Equation (20), and the analytical expression of the optimal control law of the pursuit and escape spacecraft can be obtained. Calculate x _P,0 , x _E,0 according to the parameters given in Table 1 and substitute them into Equation (24) to obtain x ₀ ; substitute x ₀ into the analytical expression of the optimal control law of the pursuit and escape spacecraft, and then t ₀ can be obtained The optimal control laws u _P,0 and u _E,0 of the tracking spacecraft and escape spacecraft corresponding to the time.

Determine whether u _P,0 and u _E,0 meet the thrust saturation limit processing conditions? If satisfied, substitute u _P,0 and u _E,0 into equations (23) and (6) in turn to update the current optimal control law of the tracking spacecraft and escape spacecraft; otherwise, the tracking and escape spacecraft The current optimal control law remains unchanged.

Convert u _P,0 and u _E,0 into the geocentric inertial coordinate system, substitute the two-body orbital motion equation of the spacecraft and solve it to obtain the motion of the two spacecraft in the geocentric inertial coordinate system at the next moment. state. According to the motion state of the escaping spacecraft at the next moment, calculate the motion state at the reference point corresponding to the virtual circular orbit; repeat steps S1 and S2 until the end time t _f of the chasing and fleeing game, and then the spacecraft of both parties chasing and escaping within a given game time can be obtained the optimal game path.

In the preferred embodiment 1, a schematic flowchart of the processor executing the limited-time pursuit and escape game control analysis program of the spacecraft in the memory is shown in FIG. 2 . Specifically, the optimal game path diagram of the pursuit and escape of the two spacecraft of the preferred embodiment 1 is shown in Figure 3, and the relative position components x, y, z and the relative distance s of the pursuit and escape two spacecraft are shown in Figure 4. , the variation of the optimal thrust acceleration components a _x , a _y , _az and the optimal thrust acceleration magnitude a of the spacecraft on both sides of the pursuit and escape are shown in Fig. 5. By comparison, the traditional numerical method is used to solve the preferred embodiment 1. The variation laws of the corresponding physical quantities obtained with time are consistent with Fig. 4 and Fig. 5.

The steps and initial parameters of the preferred embodiment 2 are exactly the same as those of the preferred embodiment 1, and the main difference is: the purpose of tracking the spacecraft is to meet the escape spacecraft at the end of the game (that is, at the end of the game, chasing and escaping between the two spacecrafts) The relative distance and relative velocity are both 0), and the purpose of the escape spacecraft is the opposite.

The second preferred embodiment provides an analytical solution system for the limited-time pursuit and escape game control of a spacecraft, including a memory and a processor, wherein the memory stores an analytical solution program for the limited-time pursuit and escape game control of the spacecraft, the processor is in The following steps are performed when running the analytical solution program for the limited-time pursuit and escape game control of the spacecraft:

S0: Obtain the initial parameters of the orbital pursuit and escape game of the two spacecraft under the fixed-time differential game;

The obtained initial parameters are the same as the initial parameters in the first preferred embodiment.

S1: In the LVLH coordinate system, the zero-control miss amount is used as the state variable to establish a fixed-time differential game model for spacecraft orbital pursuit and escape;

Calculate r _ref,0 according to the parameters given in Table 1 and substitute it into Equation (24) to obtain ω ₀ ; Substitute ω ₀ and t _f into Equation (21) to obtain the analytical expression of G(t _f ,t); The analytical expression of t _f , t) is substituted into equation (12), and the relevant parameters of the payment function are substituted into equation (13), the fixed-time differential game model of spacecraft orbital pursuit and escape with y(t) as the state variable can be obtained.

S2: Based on the fixed-time differential game model established in S1, the optimal control law and the corresponding optimal game path in analytical form of the spacecraft of the pursuit and escape are solved respectively.

Substitute ω ₀ and t _f into Equation (17) to obtain the analytical expression of Φ(t _f ,t); Substitute q _f , r _P , r _E , ω ₀ and t _f into Equation (22) and obtain it by LU decomposition Inverse matrix, get the analytical expression of K ^-1 (t _f ,t); combine the analytical expressions of G(t _f ,t), K ^-1 (t _f ,t), Φ(t _f ,t) and r Substitute _P , r _E into Equation (20), and the analytical expression of the optimal control law of the pursuit and escape spacecraft can be obtained. Calculate x _P,0 , x _E,0 according to the parameters given in Table 1 and substitute them into Equation (24) to obtain x ₀ ; substitute x ₀ into the analytical expression of the optimal control law of the pursuit and escape spacecraft, and then t ₀ can be obtained The optimal control laws u _P,0 and u _E,0 of the tracking spacecraft and escape spacecraft corresponding to the time.

Determine whether u _P,0 and u _E,0 meet the thrust saturation limit processing conditions? If satisfied, substitute u _P,0 and u _E,0 into equations (23) and (6) in turn to update the current optimal control law of the tracking spacecraft and escape spacecraft; otherwise, the tracking and escape spacecraft The current optimal control law remains unchanged.

Convert u _P,0 and u _E,0 into the geocentric inertial coordinate system, substitute the two-body orbital motion equation of the spacecraft and solve it to obtain the motion of the two spacecraft in the geocentric inertial coordinate system at the next moment. state. According to the motion state of the escaping spacecraft at the next moment, calculate the motion state at the reference point corresponding to the virtual circular orbit; repeat steps S1 and S2 until the end time t _f of the chasing and fleeing game, and then the spacecraft of both parties chasing and escaping within a given game time can be obtained the optimal game path.

In the preferred embodiment 2, a schematic flowchart of the processor executing the limited-time pursuit and escape game control analysis program of the spacecraft in the memory is shown in FIG. 2 . Specifically, in the preferred embodiment 2, the optimal game path diagram of the spacecraft of both parties to pursue and escape is shown in Figure 6, and the relative position components x, y, z and relative distance s of the spacecraft of both parties to chase and escape are shown in Figure 7. , the relative velocity components v _x , v _y , v _z and relative velocity v of the chasing and fleeing spacecraft are shown in Fig. 8. The optimal thrust acceleration components a _x , a _y , _az and The variation of the optimal thrust acceleration magnitude a with time is shown in Figure 9. By comparison, the traditional numerical method is used to solve the preferred embodiment 2, and the variation laws of the corresponding physical quantities obtained with time are consistent with those in Fig. 7, Fig. 8 and Fig. 9, and the obtained optimal game path of the spacecraft of both parties is consistent with that in Fig. 6.

By comparing the specific execution steps when the processor runs the program in the above-mentioned preferred embodiment with the schematic flowchart (as shown in Figure 2), it can be found that: when the processor runs the program, the actual process flow is not in the order described in the specific execution steps. To be implemented sequentially. It should be understood that, unless explicitly stated herein, the execution of specific steps when the processor runs the program is not strictly limited to the sequence, and these steps may be executed in other sequences. Moreover, at least a part of the steps includes multiple sub-steps or multiple stages, these sub-steps or stages are not necessarily executed at the same time, but can be executed at different times, and the execution order of these sub-steps or stages is not necessarily is performed sequentially, but may be performed in turn or alternately with other steps or at least a portion of sub-steps or stages of other steps.

In another embodiment, a computer device used in an analytical solution system for a spacecraft limited-time chase-and-flight game control is provided. The computer device may be a terminal, and its internal structure diagram may be as shown in FIG. 10 . The computer equipment includes a processor, memory, a network interface, a display screen, and an input device connected by a system bus. The processor of the computer device is used to provide computing and control capabilities, and the memory of the computer device includes a non-volatile storage medium and an internal memory. The nonvolatile storage medium stores an operating system and a computer program. The internal memory provides an environment for the execution of the operating system and computer programs in the non-volatile storage medium. The network interface of the computer device is used to communicate with an external terminal through a network connection. When the computer program is executed by the processor, an analytical solution method for the limited-time pursuit and escape game control of the spacecraft is realized. The display screen of the computer equipment may be a liquid crystal display screen or an electronic ink display screen, and the input device of the computer equipment may be a touch layer covered on the display screen, or a button, a trackball or a touchpad set on the shell of the computer equipment , or an external keyboard, trackpad, or mouse.

Those skilled in the art can understand that the structure shown in FIG. 10 is only a block diagram of a part of the structure related to the solution of the present application, and does not constitute a limitation on the computer equipment to which the solution of the present application is applied. Include more or fewer components than shown in the figures, or combine certain components, or have a different arrangement of components.

In another embodiment, a computer-readable storage medium is provided, on which a computer program is stored, and when the computer program is executed by a processor, implements the steps in the above-described preferred embodiments.

Those of ordinary skill in the art can understand that all or part of the processes in the above preferred embodiments can be implemented by instructing relevant hardware through a computer program, and the computer program can be stored in a non-volatile computer-readable storage In the medium, when the computer program is executed, it may include the processes of the above-mentioned method embodiments. Wherein, any reference to memory, storage, database or other medium used in the various embodiments provided in this application may include non-volatile and/or volatile memory. Nonvolatile memory may include read only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM), or flash memory. Volatile memory may include random access memory (RAM) or external cache memory. By way of illustration and not limitation, RAM is available in various forms such as static RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rate SDRAM (DDRSDRAM), enhanced SDRAM (ESDRAM), synchronous chain Road (Synchlink) DRAM (SLDRAM), memory bus (Rambus) direct RAM (RDRAM), direct memory bus dynamic RAM (DRDRAM), and memory bus dynamic RAM (RDRAM), etc.

The technical features of the above embodiments can be combined arbitrarily. For the sake of brevity, all possible combinations of the technical features in the above embodiments are not described. However, as long as there is no contradiction in the combination of these technical features, it should be considered that is the range described in this manual.

The above-mentioned embodiments only express several embodiments of the present invention, and their descriptions are more specific and detailed, but the protection scope of the present invention is not limited to the above-mentioned embodiments, and all technical solutions under the idea of the present invention belong to the present invention scope of protection. It should be pointed out that for those of ordinary skill in the art, some modifications and improvements without departing from the concept of the present invention should also be regarded as the protection scope of the present invention. Therefore, the scope of protection of the present invention should be determined by the appended claims.

Claims (10)

1. the analytical solution method of a spacecraft limited time chasing and escape game control, is characterized in that, comprises the steps: S1: In the LVLH coordinate system, the zero-control miss amount is used as the state variable to establish a fixed-time differential game model for spacecraft orbital pursuit and escape; The S1 includes the following steps: S101: In the LVLH coordinate system, establish a fixed-time differential game initial model based on the C-W equation of relative motion of the spacecraft; S102: Defining the zero-control miss-target amount as a new state variable, and re-deriving and establishing a fixed-time differential game model for spacecraft orbital pursuit and escape based on the initial fixed-time differential game model. S2: Based on the fixed-time differential game model established in S1, respectively solve the optimal control law and its corresponding optimal game path in analytical form of the spacecraft of the pursuit and escape parties. The S2 includes the following steps: S201: Based on the fixed-time differential game model established in S1, a variational method is used to solve the optimal game control law for the pursuit and escape of the two spacecraft; S202: Analytically solve the optimal game control law of the chasing and fleeing spacecrafts and obtain a corresponding optimal game path. 2. the method for solving the game control analysis of the limited-time pursuit and escape game of spacecraft according to claim 1, is characterized in that, in the described S101, the virtual circular orbit corresponding to the position of the escape spacecraft is used as the reference spacecraft and the escape spacecraft. With reference to the orbit of the spacecraft, establish the LVLH coordinate system. 3. spacecraft limited time pursuit and escape game control analytical solution method according to claim 1, is characterized in that, in described S101, establishes the fixed time differential strategy initial model of spacecraft pursuit and escape game as follows: The differential game state equation for establishing the spacecraft pursuit game is as follows

in,

is the relative motion state of the chasing and fleeing spacecraft in the LVLH coordinate system, u _P = [a _P,x ,a _P,y ,a _P,z ] ^T and u _E =[a _E,x ,a _E,y , a _{E, z} ] ^T are the thrust acceleration vectors of the tracking spacecraft and the escape spacecraft, respectively, u _P and u _E satisfy |u _P |≤T _P , |u _E | _≤TE , and _{T P} and TE are the tracking Maximum thrust acceleration of spacecraft and escape spacecraft in continuous thrust control mode,

and

are the system matrix and control matrix of the CW equation in the state space description form, respectively, and ω is the motion angular velocity of the reference spacecraft in the CW equation. Set the given game time of chasing and fleeing as T=t _f -t ₀ , and the payoff functions of the spacecraft of both parties are respectively

Among them, t is the current time, t ₀ is the start time of the chase and escape game, and t _f is the end time of the chase and escape game;

is a semi-positive definite symmetric matrix, which represents the proportion of the distance between the two spacecraft in the payoff function at the end of the chase-and-flight game; R _P =r _P I ₃ and R _E =r _E I ₃ are positive definite symmetric matrices, which respectively represent The fraction of fuel consumed by the tracking and escape spacecraft in the payout function. By combining Equation (1) and Equation (2), the initial model of the fixed-time differential game of the spacecraft pursuit and escape game can be established. 4. The method for solving game control analysis of spacecraft limited time pursuit and escape according to claim 3, is characterized in that, in described S102, with zero control miss amount as new state variable, establishes the fixed time of spacecraft orbital pursuit and escape The differential game model is as follows: Define the zero-control off-target amount y(t) as a new state variable, namely y(t)=DΦ(t _f ,t)x(t) (3) Among them, D=[I ₃ O ₃ ] is the coefficient matrix,

The state transition matrix corresponding to the system matrix in the equation, where s=sinωτ, c=cosωτ, τ=t _f -t is the remaining time of the chase and escape game corresponding to the current moment. Let the coefficient matrix G(t _f ,t)=DΦ(t _f ,t)B (4) According to formula (3), the new state variable y(t) is derived and substituted into formula (1) and formula (4), the new differential game state equation can be obtained as:

Correspondingly, the payment functions of the spacecrafts of the pursuit and escape are respectively:

in,

is a semi-positive definite symmetric matrix, which represents the proportion of the zero-control misses of the pursuit spacecraft relative to the escape spacecraft in the payoff function at the end of the pursuit-escape game; R _P =r _P I ₃ and R _E =r _E I ₃ are positive definite Symmetric matrix, representing the proportion of fuel consumed by the tracking spacecraft and escape spacecraft in the payoff function, respectively. By combining Equation (5) and Equation (6), a fixed-time differential game model of the spacecraft pursuit and escape game can be established with the zero-control miss amount as the state variable. 5. The method for solving the game control analysis of the limited-time pursuit and escape game of spacecraft according to claim 4, characterized in that, when the purpose of tracing the spacecraft is rendezvous rather than interception, in described S101, Q _f =q _f I ₆ ; in the S102, D=I ₆ ,

6. Spacecraft limited-time pursuit and escape game control analytical solution method according to claim 4, is characterized in that, in described S201, adopt variational method to solve the optimal game control law of pursuit and escape both spacecraft as:

Among them, G(t _f ,t) and K(t _f ,t) are coefficient matrices; Further, there are

Among them, M _P (t _f ,t) and M _E (t _f ,t) are coefficient matrices. Solving equation (4), equation (9), equation (8) and equation (7) in turn, the optimal game control law for the pursuit and escape of the two spacecraft can be obtained. 7. The analytical method for solving the game control of chasing and fleeing in a limited time of the spacecraft according to claim 6, is characterized in that, in the described S202, the method for solving the optimal game control law of the analytical form of the spacecraft of both parties in pursuit and escape is:

in,

Substituting Equation (10), Equation (11), and Equation (3) into Equation (7), we can obtain the optimal game control law of the analytical form of the spacecraft of both parties:

Among them, K ^-1 (t _f , t) can be obtained by LU decomposition and inverse matrix method. 8. The spacecraft limited-time pursuit and escape game control analytical solution method according to claim 7, characterized in that, in the S202, when the purpose of tracing the spacecraft is rendezvous rather than interception,

in,

9. An analytical solution system for spacecraft limited-time pursuit and escape game control, characterized in that it includes a processor and a memory: the memory stores an analytical solution program for spacecraft limited-time pursuit and escape game control, and the processor is in The steps of the method according to any one of the claims 1 to 8 are executed when the analytical solving program for the limited-time pursuit and escape game control of the spacecraft is run. 10. A computer-readable storage medium on which a computer program is stored, characterized in that, when the computer program is executed by a processor, the steps of the method according to any one of claims 1 to 8 are implemented.

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