Spacecraft robust orbit game method considering uncertainty

Abstract

The invention discloses a spacecraft robust orbit gaming method considering uncertainty, and relates to a spacecraft robust orbit gaming method considering uncertainty. The objective of the invention is to solve the problem of low interception accuracy caused by navigation errors and control errors inevitably existing in practical application due to the fact that most of the traditional research of the pursuit game based on differential game aims at the determinacy problem. The method comprises the following steps: obtaining a state transition matrix of a game dynamics model from an initial moment to a game ending moment; obtaining a differential game pursuit game problem model considering uncertainty; obtaining a nonlinear equation set about the initial value of the co-state variable; the open-loop game optimal control of the pursuit two parties in the current state is obtained; the open-loop game optimal control of two pursuing parties in the current state is substituted into a two-point boundary value problem, and an energy index and a terminal uncertainty index are obtained; and obtaining an optimal quasi-terminal uncertainty index by taking the energy index as a criterion. The method is applied to the field of spacecraft pursuit game.

Description

Technical Field

[0001] The invention relates to a spacecraft robust orbit game method considering uncertainty. Background Art

[0002] With the development of aerospace technology in various countries, the number of spacecrafts competing for space resources has increased dramatically. With the changing international situation, the offensive and defensive confrontation of spacecraft in space has gradually become a research hotspot. In traditional interception problems, most research results are aimed at non-maneuverable targets. However, most modern spacecraft have certain maneuverability and can maneuver and escape when the threat approaches. For such situations, traditional interception methods will no longer be applicable. Correspondingly, the pursuit and escape game of non-cooperative spacecraft considering the maneuverability of both parties, as an important part of the space offensive and defensive confrontation system, has great potential application value and has received extensive attention and research. Traditional research on pursuit and escape game based on differential countermeasures is mostly aimed at deterministic problems. However, in practical applications, there are inevitably navigation errors and control errors, both of which will affect the final interception results. On the basis of the traditional deterministic pursuit and escape game problem, the research on the pursuit and escape game problem considering uncertainty aims to incorporate uncertainty into the solution process of the optimal control of both parties to optimize the final interception result. This method is closer to engineering practice and has important strategic significance. Summary of the invention

[0003] The purpose of this invention is to solve the problem that traditional research on pursuit-escape games based on differential strategies is mostly aimed at deterministic problems. However, in practical applications, navigation errors and control errors inevitably exist, resulting in low interception accuracy. A robust orbit game method for spacecraft considering uncertainty is proposed.

[0004] A spacecraft robust orbit game method considering uncertainty. The specific process is as follows:

[0005] Step 1: Establish a Hill coordinate system and establish a game dynamics model based on the Hill coordinate system; obtain the remaining flight time T′ of the tracker at the initial moment, use time T′ as the time for the pursuit and escape game from the initial moment, and obtain the state transfer matrix of the game dynamics model from the initial moment to the end of the game;

[0006] Based on the state transfer matrix of the game dynamics model from the initial moment to the end of the game, the terminal uncertainty index and the game index are set, and the game index and the terminal uncertainty index are weighted to obtain the differential countermeasure pursuit and escape game problem model considering uncertainty.

[0007] Step 2: Use the indirect method to derive the necessary conditions for the saddle point solution of the differential game pursuit game model considering uncertainty. Based on the necessary conditions, the two-point boundary value problem that must be satisfied by the optimal control of both parties is obtained. Then, the two-point boundary value problem is transformed into a nonlinear system of equations about the initial values ​​of the co-state variables using the optimal necessary conditions.

[0008] Step 3: Solve the nonlinear equations about the initial values ​​of the co-state variables to obtain the initial values ​​of the co-state variables Based on initial values ​​of covariates Get the optimal control of the open-loop game between the chasing and fleeing parties in the current state

[0009] Step 4: Optimize the open-loop game between the two parties in the current state Substitute the two-point boundary value problem and obtain the energy index J U and terminal uncertainty index J P ;

[0010] Energy index J U To obtain the optimal quasi-terminal uncertainty index J P .

[0011] The beneficial effects of the present invention are:

[0012] The present invention takes the close-range pursuit and escape game of low-orbit circular orbit spacecraft as the task background. The thrusters of both spacecraft in the game are continuous thrust. The navigation error and control error of the tracker are considered, wherein the control error model is a linear control error model. A bilateral optimal interception model with thrust amplitude constraint is constructed based on the Clohessey-Whiltshire linear relative motion equation in the Hill coordinate system. Combined with the linear error propagation theory, the two-point boundary value problem of open-loop optimal control of both parties under current conditions is derived, and a homology method for solving the initial value of the cooperative variable is proposed, thereby realizing the robust maneuver of the spacecraft pursuit and escape game considering uncertainty.

[0013] The present invention proposes a robust orbit game method for spacecraft considering uncertainty. In the present invention, firstly, through the variational principle and differential game theory, the target tracking thrust amplitude constraint, terminal position condition, deterministic energy index and position uncertainty index are transformed into a nonlinear equation group containing Lagrangian coefficients, and the importance of energy index and uncertainty index are allocated through weight coefficients. On this basis, the necessary conditions for the optimal solution of the one-to-one game maneuver problem considering the uncertainty of the tracker are derived, and the transformation of the bilateral optimal control problem to the two-point boundary value problem is realized. In addition, in order to solve the problem of the difficulty of guessing the initial value in the indirect method, a homology method based on the semi-analytical optimal solution of the deterministic linear quadratic game is proposed to obtain the initial value of the co-state. The research results of the present invention provide an effective solution to the orbit pursuit game problem considering the uncertainty of the tracker. By introducing uncertainty indicators, the covariance of the terminal interception position under the influence of navigation error and control error of the tracker can be effectively improved, thereby improving the interception accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] Figure 1 It is a flow chart of the present invention;

[0015] Figure 2 for Figure 1 Hill coordinate system and linear control error model diagram;

[0016] Figure 3 is the nominal relative motion trajectory of the tracker in the game coordinate system;

[0017] Figure 4 is the nominal motion trajectory diagram of the chasing and escaping parties in the Hill coordinate system;

[0018] Figure 5 Schematic diagram of uncertainty index and energy index of optimal control under different index weight coefficients;

[0019] Figure 6 It is the optimal control curve diagram of the pursuit and escape parties under different indicator weight coefficients, a is the X-axis component of the control amount of the pursuit spacecraft, b is the X-axis component of the control amount of the escape spacecraft, c is the Y-axis component of the control amount of the pursuit spacecraft, d is the Y-axis component of the control amount of the escape spacecraft, e is the Z-axis component of the control amount of the pursuit spacecraft, f is the Z-axis component of the control amount of the escape spacecraft, g is the control amplitude of the pursuit spacecraft, and h is the control amplitude of the escape spacecraft;

[0020] Figure 7 The relative positions of MC simulation terminals and their error ellipses for different indicator weight coefficients, a is the XY plane of the game coordinate system, and b is the YZ plane of the game coordinate system. DETAILED DESCRIPTION

[0021] Specific implementation method 1: In this implementation method, a spacecraft robust orbit game method considering uncertainty has the following specific process:

[0022] Step 1: Establish a Hill coordinate system, and establish a game dynamics model based on the Hill coordinate system (5); obtain the remaining flight time T′ of the tracker at the initial moment, and use time T′ as the time for the pursuit and escape game from the initial moment to obtain the state transfer matrix of the game dynamics model from the initial moment to the end of the game;

[0023] Based on the state transfer matrix of the game dynamics model from the initial moment to the end of the game, the terminal uncertainty index and the game index are set, and the game index and the terminal uncertainty index are weighted to obtain the differential countermeasure pursuit and escape game problem model considering uncertainty.

[0024] Step 2: Use the indirect method to derive the necessary conditions (17) and (23) for the saddle point solution of the differential game pursuit game model considering uncertainty. Based on the necessary conditions, the two-point boundary value problem (24) that the optimal control of the pursuit and escape parties must satisfy is obtained. Then, the optimal necessary conditions are used to transform the two-point boundary value problem into a set of nonlinear equations about the initial values ​​of the co-state variables.

[0025] Step 3: Solve the nonlinear equations about the initial values ​​of the co-state variables to obtain the initial values ​​of the co-state variables Based on initial values ​​of covariates Get the optimal control of the open-loop game between the chasing and fleeing parties in the current state

[0026] Step 4: Optimize the open-loop game between the two parties in the current state Substitute the two-point boundary value problem (25) and obtain the energy index J U and terminal uncertainty index J P ;

[0027] Energy index J U Based on formula (13), the optimal quasi-terminal uncertainty index J is obtained. P .

[0028] The optimal control of the open-loop game between the chasing and escaping parties in the current state Substitute the two-point boundary value problem (25) to obtain the optimal control at all times Optimal control at all times Substituting into formula (9), we get J P represents the terminal uncertainty index;

[0029] Optimal control at all times Substitute into formula (11) and get the energy index J U ;

[0030] Energy index J U Based on formula (13), the optimal quasi-terminal uncertainty index J is obtained. P .

[0031] Specific implementation method 2: This implementation method is different from specific implementation method 1 in that in step 1, a Hill coordinate system is established, and a game dynamics model is established based on the Hill coordinate system; the remaining flight time T′ of the tracker at the initial moment is obtained, and the time T′ is used as the time for the pursuit and escape game from the initial moment, and the state transfer matrix (7) of the game dynamics model (5) from the initial moment to the end moment of the game is obtained;

[0032] Based on the state transfer matrix of the game dynamics model from the initial moment to the end of the game, the terminal uncertainty index and the game index are set, and the game index and the terminal uncertainty index are weighted to obtain the differential countermeasure pursuit and escape game problem model considering uncertainty.

[0033] The specific process is:

[0034] Step 1: Establish the Hill coordinate system; the specific process is:

[0035] Given the nominal orbital parameters of the tracking spacecraft and the escaping spacecraft at the initial time t0 (the spacecraft orbit requires 6 parameters, namely, the semi-major axis, eccentricity, orbital inclination, right ascension of the ascending node, perigee time, and perigee argument);

[0036] Given the position vector and velocity vector of the escaping spacecraft in the geocentric inertial coordinate system at the initial moment;

[0037] The escaping spacecraft at the initial moment is taken as the reference satellite, and the center of mass of the reference satellite is taken as the origin to establish a Hill coordinate system orbiting the earth. The x-axis of the Hill coordinate system is outward along the position vector of the escaping spacecraft in the geocentric inertial coordinate system, the z-axis is the direction of the orbital angular momentum of the escaping spacecraft in the geocentric inertial coordinate system, and the y-axis is determined by the right-hand rule.

[0038] Step 1 and 2: Establish the game dynamics model (5) based on the Hill coordinate system; the specific process is as follows:

[0039] Assume that the nominal states (position and velocity) of the pursuit spacecraft and the evasion spacecraft in the Hill coordinate system are x p 、x e , then the mechanical model of the pursuit spacecraft and the escaping spacecraft in the Hill coordinate system is described by the Clohessey-Whiltshire equation as follows

[0040]

[0041] in, To track the time derivative of the spacecraft's position velocity vector in the Hill coordinate system, The time derivative of the position velocity vector of the escape spacecraft in the Hill coordinate system;

[0042] u p =[u px u py u pz ] T To track the control acceleration vector of the spacecraft, u px is the control acceleration component of the tracking spacecraft on the x-axis of the Hill coordinate system, u py is the control acceleration component of the tracking spacecraft on the y-axis of the Hill coordinate system, u pz is the control acceleration component of the tracking spacecraft in the Hill coordinate system z-axis, and the superscript T indicates the transpose;

[0043] u e =[u ex u ey u ez ] T is the control acceleration vector of the escaping spacecraft, u ex is the control acceleration component of the escape spacecraft on the x-axis of the Hill coordinate system, u ey is the control acceleration component of the escaping spacecraft on the y-axis of the Hill coordinate system, u ez is the control acceleration component of the escaping spacecraft in the z-axis of the Hill coordinate system;

[0044] A and B are matrices;

[0045] The matrices A and B are

[0046]

[0047] Where n is the orbital angular velocity of the reference satellite in the Hill coordinate system;

[0048] The game state is defined as the difference between the nominal states of the pursuer and the evader;

[0049] x=x p -x e (4)

[0050] Among them, x is the game state quantity;

[0051] Then the game dynamics model is constructed as:

[0052]

[0053] in, is the derivative of the game state with respect to time;

[0054] Step 13: Get the remaining flight time T′ of the tracker at the initial moment, expressed as:

[0055]

[0056] Where r0 and v0 are the position and velocity vector of the pursuer relative to the evader at the initial moment, respectively; r0′ is the modulus of r0; · represents the dot product;

[0057] Step 14: Take the remaining flight time T′ of the tracker at the initial moment as the time for the pursuit-escape game from the initial moment, then the state transfer matrix Φ(T′) of the game dynamics model (5) from the initial moment to the end of the game;

[0058] The expression is:

[0059]

[0060] Step 15: Assume that the initial navigation error covariance matrix of the tracker at the initial time t0 is P0. According to the derivation of covariance analysis of linear systems (see Jenson EL, Scheeres D J. Multi-Objective Optimization of Covariance and Energy for Asteroid Transfers [J / OL]. Journal of Guidance, Control, and Dynamics, 2021, 44 (7): 1253-1265.), the tracker covariance propagated from the initial time t0 to any time t is expressed as

[0061]

[0062] Where P(t) is the tracker covariance propagated from the initial time t0 to any time t, Φ(t-t0) is the state transfer matrix from the initial time t0 to time t, σ is the linear control error proportional coefficient, Φ(t-τ) is the state transfer matrix from time τ to time t, τ is the integral variable, and u p (τ) is the tracking spacecraft control acceleration vector at time τ;

[0063] Assume that the thrust configuration of both the pursuer and the fugitive is three orthogonal thrusters along the xyz axis of the Hill coordinate system; consider the initial navigation error and control error of the pursuer, and adopt the linear control error model (see Figure 2 , in the figure u x ,u y ,u z are the nominal control amplitudes of the spacecraft on the xyz axis of the Hill coordinate system, u x,r ,uy,r ,u z,r are the control errors of the spacecraft on the xyz axis of the Hill coordinate system, that is, the standard deviations of the control errors in the xyz three-axis directions are proportional to the nominal control amplitudes on their respective axes, and the proportionality coefficient is σ = 0.01;

[0064] Step 16: Based on the tracker covariance P(t), considering the uncertainty of the tracker, set the terminal uncertainty index to the trace of the terminal position covariance matrix;

[0065] The specific process is:

[0066]

[0067] Among them, trace() represents the matrix trace operation, P(t f ) represents the tracker state covariance matrix at the terminal time, Φ(t f -τ) represents the time from τ to the terminal time t f The state transfer matrix of J P represents the terminal uncertainty index; W is a matrix;

[0068] Terminal time t f =t0+T′; the matrix W represents the pair P(t f ) are weighted, and only the terminal position uncertainty is considered in this study;

[0069] J P The dispersion of the game terminals around the nominal position of the tracker at all times is described under the influence of the initial navigation error and control error of the tracker.

[0070] Step 17: Set the terminal certainty index;

[0071] Step 18. The present invention considers the uncertainty of the tracker in the pursuit-escape game, weights the terminal certainty index (11) and the terminal uncertainty index (9), and obtains a differential countermeasure pursuit-escape game problem model (13) considering uncertainty.

[0072] The other steps and parameters are the same as those in the first or second embodiment.

[0073] Specific implementation method three: This implementation method is different from specific implementation methods one or two in that: in step one six, the matrix W is set to

[0074]

[0075] Among them, I 3×3 represents the identity matrix with 3 rows and 3 columns, 0 3×3 Represents a 3-row, 3-column zero matrix.

[0076] The other steps and parameters are the same as those in the first or second embodiment.

[0077] Specific implementation method 4: This implementation method is different from one of the specific implementation methods 1 to 3 in that: in step 17, a terminal certainty index is set; the specific process is:

[0078] For the deterministic linear quadratic pursuit-escape game problem with a given initial remaining flight time T of the tracker, many traditional studies have used classical quadratic indicators including the nominal terminal position deviation and the energy of both the pursuer and the escapee.

[0079]

[0080] Among them, J G is the terminal certainty index, J X is the nominal terminal distance indicator, J U is the energy index; x(t f ) is the game state quantity at the terminal moment, S is the terminal state weight matrix, R p is the control weight matrix, R e is the control weight matrix, u p (t) is the control acceleration vector of the tracker at time t, u e (t) is the control acceleration vector of the escaping device at time t;

[0081] R p =r p I 3×3 , R e =r e I 3×3 , r p represents the control weight coefficient of the tracker, r e represents the control weight coefficient of the evader;

[0082] Take r p =1,r e =1.0001, r e >r p Indicates that the evading party is unacceptable to fuel consumption, and r e With r p Close enough to make both parties try their best to avoid one party making almost no maneuvers due to too large r value;

[0083] Terminal state weight matrix

[0084]

[0085] Among them, s r Represents the terminal position weight coefficient, s v represents the terminal velocity weight coefficient, I 3×3 represents a unit matrix with 3 rows and 3 columns, 03×3 Represents a 0 matrix with 3 rows and 3 columns;

[0086] In the interception problem studied in this paper, only the relative positions of the terminals of the pursuer and the evader are required, and there is no requirement for the relative speed of the terminals. Therefore, the parameter in equation (12) is set to s r =1,s v =0.

[0087] The other steps and parameters are the same as those in Specific Embodiments 1 to 3.

[0088] Specific implementation method 5: This implementation method is different from one of the specific implementation methods 1 to 4 in that: in the step 18, the present invention considers the uncertainty of the tracker in the pursuit and escape game, and weights the terminal certainty index (11) and the terminal uncertainty index (9) to obtain the differential countermeasure pursuit and escape game problem model (13) considering uncertainty; the expression is:

[0089] J=ηJ U +(1-η)J P +J X (13)

[0090] Among them, η represents the weight coefficient, η∈(0,1); J represents the weighted indicator function;

[0091] The weight coefficient η∈(0,1) indicates the importance that both parties attach to different indicators in the indicator function J. The larger the η, the more attention both parties pay to the energy indicator J. U , on the contrary, more attention is paid to the terminal uncertainty index J P ;

[0092] Note that the nominal terminal distance index J X There is no weighting involved because the relative distance between the terminals is the criterion that directly determines whether the interception mission can be successful, and it is also the core of the game between the two parties in the pursuit.

[0093] The other steps and parameters are the same as those in Specific Embodiments 1 to 4.

[0094] Specific implementation method six: This implementation method is different from any one of specific implementation methods one to five in that: in step two, the necessary conditions (17) and (23) of the saddle point solution of the differential game pursuit game problem model considering uncertainty are derived by indirect method, and based on the necessary conditions, the two-point boundary value problem (24) that the optimal control of the pursuit and escape parties must satisfy is obtained, and then the two-point boundary value problem is transformed into a nonlinear equation group about the initial value of the co-state variable by using the optimal necessary conditions;

[0095] The specific process is:

[0096] Step 21: Use the indirect method to derive the necessary conditions (17) and (23) for the saddle point solution of the differential game pursuit game model considering uncertainty;

[0097] Step 22: Based on necessary conditions (17) and (23), we obtain the two-point boundary value problem (25) that must be satisfied for the optimal control of both parties.

[0098] Step 23: Use necessary conditions (17) and (23) to transform the two-point boundary value problem (25) into a system of nonlinear equations with respect to the initial values ​​of the covariates.

[0099] The other steps and parameters are the same as those in Specific Implementation Methods 1 to 5.

[0100] Specific implementation method 7: This implementation method is different from specific implementation methods 1 to 6 in that: in step 21, the necessary conditions (17) and (23) for the saddle point solution of the differential game pursuit and escape game problem model considering uncertainty are derived by indirect method;

[0101] The specific process is:

[0102] Transform the weighted index function J into Bolza type:

[0103]

[0104] In the differential game pursuit-escape game problem, for the pursuer, the goal is to find the optimal control to make the index (14) as small as possible, while the evader requires to make the index (14) as large as possible;

[0105] To facilitate subsequent expressions, we use R as the symbol to represent the content within the integral sign in equation (14), and let

[0106] R=(1-η)σ 2 B T Φ(t f -τ) T WΦ(t f -τ)B+ηR p (15)

[0107] Among them, R represents the intermediate variable;

[0108] Based on formula (14), the Hamilton function is constructed as

[0109]

[0110] Where H represents the value of Hamilton function, λ represents the covariate vector;

[0111] When solving the Nash equilibrium saddle point solution for the pursuit-escape game problem with uncertainty as an indicator, the solution is only from the perspective of the optimality of the control quantity of both parties. The obtained control quantity may far exceed the maximum thrust amplitude that the spacecraft thruster can provide, which has no practical significance in engineering. Therefore, the maximum thrust amplitude that the pursuit spacecraft can provide on the xyz axis of the Hill coordinate system is used as the control amplitude constraint. On the basis of the Nash equilibrium saddle point solution, the thrust direction of the control quantity of both parties is kept unchanged, and the control amplitude is reduced to meet the control amplitude constraint of the spacecraft.

[0112] Without considering the control amplitude constraint, the Nash equilibrium saddle point solution of the game problem should satisfy the first-order necessary condition

[0113]

[0114] The control amplitude constraint is the maximum thrust amplitude that the pursuer or the escaper can provide on the xyz axis of the Hill coordinate system respectively;

[0115] The optimal control of the pursuer and evader is solved as

[0116]

[0117] in, represents the optimal control acceleration vector of the tracker, represents the optimal control acceleration vector of the evader; Substitute equation (18) into the Hamilton function (16) and derive the game state x to obtain the canonical equation of the co-state variable:

[0118]

[0119] in, represents the derivative of the covariate vector with respect to time;

[0120]

[0121] Among them, Φ(t f -t) means time t to terminal time t f The state transition matrix of

[0122] Since the dynamics in the present invention is based on the CW equation, it is a linear steady-state system,

[0123]

[0124] Therefore, formula (19) is simplified to

[0125]

[0126] The terminal transversal condition corresponding to the covariate variable is

[0127] λ(t f )=Sx(t f ) (twenty three)

[0128] Among them, λ(t f ) represents the co-state variable vector at the terminal time t f The value of x(t f ) represents the game state at the terminal time t f The value of

[0129] The other steps and parameters are the same as those in Specific Implementation 16 to 1.

[0130] Specific implementation eight: This implementation differs from specific implementations one to seven in that: in step 22, based on necessary conditions (17) and (23), a two-point boundary value problem (25) that must be satisfied for optimal control of both parties in pursuit and escape is obtained;

[0131] The specific process is:

[0132] Substituting the optimal control quantities (18) of the pursuer and the evader into the game dynamics model (5) and combining the co-state variable equation (22) and the terminal transversality condition (23) to obtain the two-point boundary value problem that the state quantity and the co-state variable need to satisfy when there is no control amplitude constraint;

[0133]

[0134] Among them, x(t0) represents the value of the game state at the initial time t0, and x0 represents the given initial value of the game state;

[0135] Taking into account the control amplitude constraints of real spacecraft, the two-point boundary value problem (24) is modified as follows:

[0136]

[0137] Among them, u pmax Indicates the maximum thrust acceleration of the tracker on each coordinate axis of the Hill coordinate system (Hill coordinate system xyz axis) (Hill coordinate system xyz axis each has a maximum thrust acceleration, and the three maximum thrust accelerations are equal), u emax Indicates the maximum thrust acceleration of the evader on each coordinate axis of the Hill coordinate system (Hill coordinate system xyz axis) (the Hill coordinate system xyz axis each has a maximum thrust acceleration, and the three maximum thrust accelerations are equal);

[0138] It means taking the maximum value on the three axes;

[0139] Formula (25) imposes an amplitude constraint on the control quantity of the pursuer and the evader, namely the thrust acceleration. When the optimal thrust in a certain axis exceeds the maximum allowable thrust amplitude, the thrust amplitudes in the three axes are reduced at the same time so that the thrust amplitude in each axis meets the constraint condition.

[0140] Note that the control amount in equation (25) is not consistent with the saddle point solution expressed by equation (18), but the control amount after amplitude reduction is the maximum acceptable amplitude of the control amount of both parties in the pursuit-escape game in the saddle point direction, which still conforms to the Pontryagin extreme value theorem, that is, satisfy

[0141]

[0142] Therefore, according to formula (25), and It is the optimal solution under thrust limitation conditions.

[0143] The other steps and parameters are the same as those in Specific Implementation 1 to 7-1.

[0144] Specific implementation method 9: This implementation method is different from the specific implementation methods 1 to 8 in that: in the steps 2 and 3, the necessary conditions (17) and (23) are used to transform the two-point boundary value problem (25) into a nonlinear equation system about the initial values ​​of the co-state variables; the specific process is:

[0145] Considering the boundary condition λ(t f )=Sx(t f ), the two-point boundary value problem of equation (25) is equivalent to the problem about the terminal time t f And the nonlinear equations with initial value λ0 of the covariate state:

[0146]

[0147] in, express With Sx F (t f ); F represents the consideration of the uncertainty pursuit game problem, represents the initial value of the covariate variable considering the uncertainty pursuit and escape game problem, Indicates is the initial value of the covariate variable, and the terminal value of the covariate variable after integration according to formula (25), x F (t f ) represents the game state quantity of the uncertain pursuit-escape game problem at the terminal time t f The value of .

[0148] The other steps and parameters are the same as those in Specific Embodiments 1 to 8.

[0149] Specific implementation method 10: This implementation method is different from any one of specific implementation methods 1 to 9 in that: in step 3, the nonlinear equation group about the initial value of the co-state variable is solved to obtain the initial value of the co-state variable Based on initial values ​​of covariates Get the optimal control of the open-loop game between the chasing and fleeing parties in the current state

[0150] The specific process is:

[0151] If the shooting method is used directly to solve equation (27), there is a problem that the random initial value guess is difficult to make the Newton method converge. To avoid the above difficulties, the present invention uses a homotopy method combined with a semi-analytical solution of a linear quadratic game to obtain the initial value of the co-state variable.

[0152] Step 31: For the deterministic linear quadratic pursuit game problem, the co-state variable is expressed as the feedback form of the game state quantity

[0153] λ G (t) = C (Tt) x (t) (28)

[0154] Among them, λ G (t) represents the value of the co-state variable at time t, G represents the deterministic linear quadratic pursuit and escape game problem, C(Tt) represents the coefficient matrix at time t, x(t) represents the game state vector at time t, and T is the remaining game time;

[0155] The coefficient matrix C(Tt) is obtained by a semi-analytical method (see Liao Tian. Research on the control and solution method of spacecraft pursuit and escape game [D]. Harbin Institute of Technology, 2021);

[0156] Similar to formula (27), the initial value of the covariate variable in the deterministic linear quadratic pursuit game problem is The nonlinear equations should be satisfied

[0157]

[0158] in, express With Sx G (t f ) difference, represents the initial value of the covariate variable considering the deterministic linear quadratic pursuit and escape game problem, Indicates is the initial value of the co-state variable, and the terminal value of the co-state variable after integration according to the deterministic linear quadratic pursuit-escape game dynamics model (5); x G (t f ) represents the game state quantity of the deterministic linear quadratic pursuit game problem at the terminal time t f The value of

[0159] Step 32: Based on equation (27) and equation (29), construct the homotopy function:

[0160]

[0161] in, represents the homotopy function, represents the independent variable of the homotopy function;

[0162] ε is the homotopy parameter;

[0163] When ε=0, It is a deterministic linear quadratic pursuit and escape game problem with a known semi-analytical solution;

[0164] When ε=1, It is a game problem of uncertainty pursuit and escape.

[0165] Step 33: Obtain the independent variable of the homotopy function through the homotopy algorithm and homotopy parameter ε;

[0166] Step 34: Independent variables based on homology functions get Initialize the covariate variables As the initial value of λ in equation (25), we integrate equation (25) and obtain the optimal control of the open-loop game between the chasing and escaping parties in the current state:

[0167] The homotopy algorithm used in the present invention (see Eugene L. Allgower, Kurt Georg. Numerical Continuation Methods [M]. Springer, Berlin, Heidelberg) converts the homotopy parameter ε and the homotopy function independent variable Merge into an augmented column vector, namely

[0168]

[0169] Wherein, z represents the iterative solution vector in the homotopy algorithm;

[0170] The homotopic path tracking algorithm uses a prediction-correction method to start from the initial point Start by tracking a space curve that satisfies equation (30) and gradually reach the target point By changing ε from 0 to 1, we can find the terminal boundary condition (27) That is, the initial value of the co-state variable of the two-point boundary value problem (25); based on the initial value of the co-state variable, the open-loop optimal control quantity of each of the chasing and fleeing parties at the current moment can be obtained by integrating equation (25):

[0171] The other steps and parameters are the same as those in Specific Embodiments 1 to 9.

[0172] Implementation of the case study

[0173] In the simulation example, the parameters of the task scenario are set as shown in Table 1.

[0174] Table 1 Spacecraft pursuit and escape game mission scenario parameters

[0175]

[0176] Assume that the nominal position and velocity vectors of the evader and the pursuer in the Hill coordinate system at the initial moment are:

[0177]

[0178] Among them, r E0 represents the position vector of the evader at the initial moment, v E0 represents the velocity vector of the evader at the initial moment, r P0 Represents the position vector of the tracker at the initial moment, v P0 represents the velocity vector of the tracker at the initial moment;

[0179] From formula (6), we can get the game duration T′=500s. Let the initial time t0=0s, then the terminal time t f = 500s. Assume that the initial navigation error of the tracker (the first three units are m 2 , the last three items are in m 2 / s 2 )for

[0180] P0=diag([10 2 10 2 10 2 0.01 2 0.01 2 0.01 2 ]) (32)

[0181] Taking the indicator weight coefficient η = 0.5 as an example, the initial value of the covariate variable corresponding to formula (25) can be calculated by the homology method as follows:

[0182]

[0183] The origin of the Hill coordinate system is translated to the evader, and the new coordinate system is defined as the game coordinate system. The game state represents the nominal relative motion of the tracker to the evader in this coordinate system.

[0184] Substitute the initial value of the co-state variable in equation (33) into equation (25) and integrate it. Draw the nominal relative motion trajectory of the tracker and the nominal maneuver trajectory of the two game players in the game coordinate system and the Hill coordinate system respectively. Figure 3 , Figure 4 shown.

[0185] from Figure 3 , Figure 4 It can be seen from the figure that the nominal trajectory of the tracker reaches the vicinity of the evader at the end of the game, which means that the open-loop optimal control obtained by the present invention can achieve the interception of the evader by the nominal trajectory of the tracker when the maneuverability of the tracker is greater than that of the evader.

[0186] By changing the value of the index weight coefficient η, the optimal control corresponding to 20 different η values ​​is calculated from η = 0.02 to η = 0.98, and the uncertainty index J is obtained. P and energy index J U As η changes, Figure 5 shown.

[0187] Assume that the maximum energy index threshold allowed by the tracker in this embodiment is J Umax =20, then according to Figure 5 The analysis results in show that, under the division of the indicator weight coefficients in this embodiment, selecting the indicator weight coefficient as η = 0.7779 can achieve the minimum uncertainty index, that is, the uncertainty of the terminal interception position is optimal. Selecting two indicator weight coefficients of η = 0.7779 and η = 0.98 for comparison, the optimal control of the tracker and escaper under the two weight coefficients is as follows: Figure 6 shown.

[0188] Taking the initial nominal position and velocity of the tracker as the mean, the initial navigation error covariance matrix is ​​sampled 1000 times according to the given formula. Monte Carlo simulation is performed on the optimal control obtained with the index weight coefficients η = 0.7779 and η = 0.98 at the sampling points, and the uncertainty index at the game terminal moment is shown in Table 2. The error ellipse of the relative position of the tracker in the game coordinate system is shown in Figure 7 shown.

[0189] Table 2 Terminal uncertainty index and MC simulation results under different index weights

[0190]

[0191] The above results show that when the tracker has initial navigation error and linear control error, the present invention can select the optimal index weight coefficient η, and use the solved open-loop game optimal control to intercept the evader with the tracker's nominal trajectory while achieving the optimal uncertainty of the terminal's relative position within the energy index threshold allowed by the tracker. This can be used as a robust maneuvering strategy for spacecraft space attack, defense, pursuit and escape games.

[0192] The present invention may also have many other embodiments. Without departing from the spirit and essence of the present invention, those skilled in the art may make various corresponding changes and modifications based on the present invention, but these corresponding changes and modifications should all fall within the scope of protection of the claims attached to the present invention.

Claims

1. A robust orbit game method for spacecraft considering uncertainty, Features: The specific process of the method is: Step 1: Establish a Hill coordinate system and establish a game dynamics model based on the Hill coordinate system; obtain the remaining flight time T′ of the tracker at the initial moment, use time T′ as the time for the pursuit and escape game from the initial moment, and obtain the state transfer matrix of the game dynamics model from the initial moment to the end of the game; Based on the state transfer matrix of the game dynamics model from the initial moment to the end of the game, the terminal uncertainty index and the game index are set, and the game index and the terminal uncertainty index are weighted to obtain the differential countermeasure pursuit and escape game problem model considering uncertainty. Step 2: Use the indirect method to derive the necessary conditions for the saddle point solution of the differential game pursuit game model considering uncertainty. Based on the necessary conditions, the two-point boundary value problem that must be satisfied by the optimal control of both parties is obtained. Then, the two-point boundary value problem is transformed into a nonlinear system of equations about the initial values ​​of the co-state variables using the optimal necessary conditions. Step 3: Solve the nonlinear equations about the initial values ​​of the co-state variables to obtain the initial values ​​of the co-state variables Based on initial values ​​of covariates Get the optimal control of the open-loop game between the chasing and fleeing parties in the current state Step 4: Optimize the open-loop game between the two parties in the current state Substitute the two-point boundary value problem and obtain the energy index J U and terminal uncertainty index J P ; Energy index J U To obtain the optimal quasi-terminal uncertainty index J P .

2. A spacecraft robust orbit game method considering uncertainty according to claim 1, Features: In the step 1, a Hill coordinate system is established, and a game dynamics model is established based on the Hill coordinate system; the remaining flight time T′ of the tracker at the initial moment is obtained, and the time T′ is used as the time for the pursuit and escape game from the initial moment to obtain the state transfer matrix of the game dynamics model from the initial moment to the end of the game; Based on the state transfer matrix of the game dynamics model from the initial moment to the end of the game, the terminal uncertainty index and the game index are set, and the game index and the terminal uncertainty index are weighted to obtain the differential countermeasure pursuit and escape game problem model considering uncertainty. The specific process is: Step 1: Establish the Hill coordinate system; the specific process is: Given an initial time t 0 Nominal orbital parameters of the pursuit and evasion spacecraft; Given the position vector and velocity vector of the escaping spacecraft in the geocentric inertial coordinate system at the initial moment; The escaping spacecraft at the initial moment is taken as the reference satellite, and the center of mass of the reference satellite is taken as the origin to establish a Hill coordinate system orbiting the earth. The x-axis of the Hill coordinate system is outward along the position vector of the escaping spacecraft in the geocentric inertial coordinate system, the z-axis is the direction of the orbital angular momentum of the escaping spacecraft in the geocentric inertial coordinate system, and the y-axis is determined by the right-hand rule. Step 1 and 2: Establish a game dynamics model based on the Hill coordinate system; the specific process is: Assume that the nominal states of the pursuit spacecraft and the evasion spacecraft in the Hill coordinate system are x p 、x e , then the mechanical model of the pursuit spacecraft and the escaping spacecraft in the Hill coordinate system is described by the Clohessey-Whiltshire equation as follows in, To track the time derivative of the spacecraft's position velocity vector in the Hill coordinate system, The time derivative of the position velocity vector of the escape spacecraft in the Hill coordinate system; u p =[u px u py u pz ] T To track the control acceleration vector of the spacecraft, u px is the control acceleration component of the tracking spacecraft on the x-axis of the Hill coordinate system, u py is the control acceleration component of the tracking spacecraft on the y-axis of the Hill coordinate system, u pz is the control acceleration component of the tracking spacecraft in the Hill coordinate system z-axis, and the superscript T indicates the transpose; u e =[u ex u ey u ez ] T is the control acceleration vector of the escaping spacecraft, u ex is the control acceleration component of the escape spacecraft on the x-axis of the Hill coordinate system, u ey is the control acceleration component of the escaping spacecraft on the y-axis of the Hill coordinate system, u ez is the control acceleration component of the escaping spacecraft in the z-axis of the Hill coordinate system; A and B are matrices; The matrices A and B are Where n is the orbital angular velocity of the reference satellite in the Hill coordinate system; The game state is defined as the difference between the nominal states of the pursuer and the evader; x=x p -x e (4) Among them, x is the game state quantity; Then the game dynamics model is constructed as: in, is the derivative of the game state with respect to time; Step 13: Get the remaining flight time T′ of the tracker at the initial moment, expressed as: Among them, r 0 and v 0 are the position and velocity vector of the pursuer relative to the evader at the initial moment; r 0 ′ is r 0 The modulus of; represents the dot product; Step 14: Take the remaining flight time T′ of the tracker at the initial moment as the time for the pursuit-escape game from the initial moment, then the state transfer matrix Φ(T′) of the game dynamics model (5) from the initial moment to the end of the game; The expression is: Step 15: Set the initial time t 0 The initial navigation error covariance matrix of the tracker is P 0 According to the derivation of linear system covariance analysis, the initial time t 0 The tracker covariance propagated to any time t is expressed as Among them, P(t) is the initial time t 0 The tracker covariance propagated to any time t, Φ(tt 0 ) is the initial time t 0 The state transfer matrix from time τ to time t, σ is the linear control error proportional coefficient, Φ(t-τ) is the state transfer matrix from time τ to time t, u p (τ) is the tracking spacecraft control acceleration vector at time τ; Step 16: Based on the tracker covariance P(t), considering the uncertainty of the tracker, set the terminal uncertainty index to the trace of the terminal position covariance matrix; The specific process is: Among them, trace() represents the matrix trace operation, P(t f ) represents the tracker state covariance matrix at the terminal time, Φ(t f -τ) represents the time from τ to the terminal time t f The state transfer matrix of J P represents the terminal uncertainty index; W is a matrix; Terminal time t f =t 0 +T′; Step 17: Set the terminal certainty indicator; Step 18: Considering the uncertainty of the tracker, weight the terminal certainty index (11) and the terminal uncertainty index (9) to obtain the differential countermeasure pursuit and escape game problem model considering uncertainty.

3. A spacecraft robust orbit game method considering uncertainty according to claim 2, Features: In step 16, the matrix W is set to Among them, I 3×3 represents the identity matrix with 3 rows and 3 columns, 0 3×3 Represents a 3-row, 3-column zero matrix.

4. A spacecraft robust orbit game method considering uncertainty according to claim 3, Features: In the step 17, the terminal certainty index is set; the specific process is: Among them, J G is the terminal certainty index, J X is the nominal terminal distance indicator, J U is the energy index; x(t f ) is the game state quantity at the terminal moment, S is the terminal state weight matrix, R p is the control weight matrix, R e is the control weight matrix, u p (t) is the control acceleration vector of the tracker at time t, u e (t) is the control acceleration vector of the escaping device at time t; R p =r p I 3×3 , R e =r e I 3×3 , r p represents the control weight coefficient of the tracker, r e represents the control weight coefficient of the evader; Take r p =1,r e =1.0001; Terminal state weight matrix Among them, s r Represents the terminal position weight coefficient, s v represents the terminal velocity weight coefficient, I 3×3 represents a unit matrix with 3 rows and 3 columns, 0 3×3 Represents a 0 matrix with 3 rows and 3 columns; s r =1,s v =0。 5. A spacecraft robust orbit game method considering uncertainty according to claim 4, Features: In the step 18, the uncertainty of the tracker is taken into account, and the terminal certainty index (11) and the terminal uncertainty index (9) are weighted to obtain a differential countermeasure pursuit and escape game problem model considering uncertainty; the expression is: J=ηJ U +(1-η)J P +J X (13) Among them, η represents the weight coefficient, η∈(0,1); J represents the weighted indicator function.

6. A spacecraft robust orbit game method considering uncertainty according to claim 5, Features: In the step 2, the necessary conditions for the saddle point solution of the differential game pursuit game model considering uncertainty are derived by an indirect method, and based on the necessary conditions, a two-point boundary value problem that must be satisfied by the optimal control of both parties of pursuit and escape is obtained, and then the two-point boundary value problem is transformed into a nonlinear equation group about the initial value of the co-state variable by using the optimal necessary conditions; The specific process is: Step 21: Use the indirect method to derive the necessary conditions for the saddle point solution of the differential game pursuit and escape game model considering uncertainty; Step 22: Based on the necessary conditions, we obtain the two-point boundary value problem that the optimal control of the pursuit and escape parties must satisfy; Step 23: Use necessary conditions to transform the two-point boundary value problem into a system of nonlinear equations about the initial values ​​of the covariates.

7. A spacecraft robust orbit game method considering uncertainty according to claim 6, Features: In step 21, the necessary conditions for the saddle point solution of the differential game pursuit and escape game model considering uncertainty are derived by indirect method; the specific process is: Transform the weighted index function J into Bolza type: make R=(1-η)σ 2 B T Φ(t f -t) T WΦ(t f -τ)B+ηR p (15) Among them, R represents the intermediate variable; Based on formula (14), the Hamilton function is constructed as Where H represents the value of Hamilton function, λ represents the covariate vector; Without considering the control amplitude constraint, the Nash equilibrium saddle point solution of the game problem should satisfy the first-order necessary condition The control amplitude constraint is the maximum thrust amplitude that the pursuer or the escaper can provide on the xyz axis of the Hill coordinate system respectively; The optimal control of the pursuer and evader is solved as in, represents the optimal control acceleration vector of the tracker, represents the optimal control acceleration vector of the evader; Substituting equation (18) into Hamilton function (16) and taking the derivative of the game state variable x, we can obtain the canonical equation of the co-state variable: in, represents the derivative of the covariate vector with respect to time; Among them, Φ(tf-t) represents the time from time t to the terminal time t f The state transition matrix of have Therefore, formula (19) is simplified to The terminal transversal condition corresponding to the covariate variable is λ(t f )=Sx(t f ) (23) Where λ(tf) represents the covariate variable vector at the terminal time t f The value of x(tf) represents the game state at the terminal time t f The value of .

8. A spacecraft robust orbit game method considering uncertainty according to claim 7, Features: In the step 22, based on the necessary conditions, a two-point boundary value problem that must be satisfied for the optimal control of both parties in pursuit and escape is obtained; The specific process is: Substituting the optimal control quantities (18) of the pursuer and the evader into the game dynamics model (5) and combining the co-state variable equation (22) and the terminal transversality condition (23) to obtain the two-point boundary value problem that the state quantity and the co-state variable need to satisfy when there is no control amplitude constraint; Among them, x(t 0 ) represents the game state at the initial time t 0 The value of x 0 Represents the given initial value of the game state; Taking into account the control amplitude constraints of real spacecraft, the two-point boundary value problem (24) is modified as follows: Among them, u pmax represents the maximum thrust acceleration of the tracker on each axis of the Hill coordinate system, u emax It represents the maximum thrust acceleration of the escaper on each coordinate axis of the Hill coordinate system; satisfy 9. A spacecraft robust orbit game method considering uncertainty according to claim 8, Features: In the steps 2 and 3, the two-point boundary value problem is transformed into a nonlinear equation system about the initial value of the co-state variable by using necessary conditions; the specific process is: Considering the boundary condition λ(t f )=Sx(t f ), the two-point boundary value problem of equation (25) is equivalent to the problem about the terminal time t f and initial value of covariate variable λ 0 The nonlinear equations are: in, express With Sx F (t f ); F represents the consideration of the uncertainty pursuit game problem, represents the initial value of the covariate variable considering the uncertainty pursuit and escape game problem, Indicates is the initial value of the covariate variable, and the terminal value of the covariate variable after integration according to formula (25), x F (t f ) represents the game state quantity of the uncertain pursuit-escape game problem at the terminal time t f The value of .

10. A spacecraft robust orbit game method considering uncertainty according to claim 9, Features: In the step 3, the nonlinear equations about the initial values ​​of the co-state variables are solved to obtain the initial values ​​of the co-state variables. Based on initial values ​​of covariates Get the optimal control of the open-loop game between the chasing and fleeing parties in the current state The specific process is: Step 31: For the deterministic linear quadratic pursuit game problem, the co-state variable is expressed as the feedback form of the game state quantity λ G (t)=C(T-t)x(t) (28) Among them, λ G (t) represents the value of the co-state variable at time t, G represents the deterministic linear quadratic pursuit and escape game problem, C(Tt) represents the coefficient matrix at time t, x(t) represents the game state vector at time t, and T is the remaining game time; The coefficient matrix C(Tt) is obtained by a semi-analytical method; Initial Values ​​of Covariates in Deterministic Linear Quadratic Pursuit and Escape Game Problems The nonlinear equations should be satisfied in, express With Sx G (t f ) difference, represents the initial value of the covariate variable considering the deterministic linear quadratic pursuit and escape game problem, Indicates is the initial value of the covariate variable and the terminal value of the covariate variable after integration according to the kinetic model (5); x G (t f ) represents the game state quantity of the deterministic linear quadratic pursuit game problem at the terminal time t f The value of Step 32: Based on equation (27) and equation (29), construct the homotopy function: in, represents the homotopy function, represents the independent variable of the homotopy function; ε is the homotopy parameter; When ε=0, It is a deterministic linear quadratic pursuit and escape game problem with a known semi-analytical solution; When ε=1, It is a game problem of uncertainty pursuit and escape. Step 33: Obtain the independent variable of the homotopy function through the homotopy algorithm and homotopy parameter ε; Step 34: Independent variables based on homology functions get Initialize the covariate variables As the initial value of λ in equation (25), we integrate equation (25) and obtain the optimal control of the open-loop game between the chasing and escaping parties in the current state:

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