Satellite orbit game approaching control rapid calculation method

Abstract

The invention discloses a satellite orbit game approaching control rapid calculation method. The method comprises the following steps: constructing a satellite orbit game model; based on the remaining game time, the game remaining radian, the zero-control off-target vector, the game remaining radian matrix, the amplitude of the tracking satellite control acceleration and the amplitude of the escape satellite control acceleration, determining suboptimal approaching control of the tracking satellite and the escape satellite; calculating an integral term g (theta) when theta is used as game remaining time and a proportional term h (theta) of a maximum singular value and a minimum singular value of a game remaining radian matrix at theta based on a polynomial fitting method; performing calculation optimization on tracking satellite approaching control; according to the method, the polynomial coefficient of the integral item is calculated and fitted off line through a polynomial fitting method, only the integral upper bound needs to be input for calculation to obtain the integral value during on-orbit calculation, multiple times of calculation of matrix singular values are avoided, and the solving time consumption of the game remaining time and the calculation time of suboptimal approach control are effectively shortened.

Description

Technical Field

[0001] The present invention relates to the technical field of satellite approaching, and in particular to a fast calculation method for satellite orbit game approaching control. Background Art

[0002] Due to the increase in space debris and the gradual increase in the importance of national space security, the strategic significance of space attack and defense technologies has gradually emerged. As one of the key technologies of space attack and defense, satellite orbit game control has become the forefront and hot spot of current space technology research, and can be widely applied to tasks such as non-cooperative space target observation, reconnaissance, and towing. As a type of zero-sum game, in the orbit pursuit-evasion game, the two players have conflicting goals. The tracking satellite hopes to reach the vicinity of the escaping satellite in the shortest time by applying control, while the latter tries its best to apply control to avoid or delay the approach. Due to the unpredictable maneuver of the escaping satellite, the calculation complexity of the optimal approaching control of the tracking satellite is high, and the traditional optimal control method considering only one side is difficult to ensure the completion of the approach.

[0003] For the above satellite orbit game, the approaching control calculation methods of the tracking satellite can be roughly divided into (1) exact control calculation based on the saddle point of differential game, (2) approximate fitting of approaching control based on neural network, and (3) suboptimal control calculation based on zero-control miss distance. Among them, method (1) can give the exact approaching control that satisfies the game saddle point condition. However, affected by the complexity of satellite orbit dynamics, there is no analytical expression for the game saddle point solution. The existing method (1) needs to solve at least a 4-dimensional two-point boundary value problem, and can only numerically give the saddle point solution of the game. The calculation involves at least thousands of integrations of nonlinear differential equations, which takes a long time and is difficult to apply on the satellite. Method (2) needs to be offline trained to obtain a neural network with a small fitting error for the game saddle point solution. The scale of the neural network required for fitting is large, and the requirements for control calculation and weight storage are high when used online on the satellite. Under the premise that the on-board computing power cannot be significantly improved, it is difficult to execute on orbit. Method (3) can give an approaching control with similar performance to the game saddle point solution, but the current calculation method needs to repeatedly calculate the integral of a certain type of time-varying matrix or its singular value to solve the remaining game time. Limited by the current on-board computing speed and processing power, the on-board solution efficiency of the approaching control still needs to be further improved. Especially under the limited on-board computing power, quickly calculating the approaching control is of great significance for reducing the response time and approaching time of the tracking satellite and enhancing its adaptability to the rapidly changing game situation. Summary of the Invention

[0004] To overcome the deficiencies of the prior art, the present invention provides a fast calculation method for satellite orbit game approaching control, which realizes offline calculation and fitting of the polynomial coefficients of the integral term through polynomial fitting method, saves them in the on-board computer, and only needs to input the integral upper bound during on-orbit calculation to obtain the integral value through simple addition and multiplication operations, avoiding multiple calculations of matrix singular values, and effectively reducing the solution time of the game remaining time and the calculation time of sub-optimal approaching control.

[0005] To achieve the above invention purpose, the present invention adopts the following technical solutions:

[0006] The present application provides a fast calculation method for satellite orbit game approaching control, including the following steps:

[0007] S101. Construct a satellite orbit game model;

[0008] S102. Based on the remaining game time, the remaining game arc, the zero-control miss vector, the remaining game arc matrix, the magnitude of the tracking satellite control acceleration, and the magnitude of the escaping satellite control acceleration, determine the sub-optimal approaching control of the tracking satellite and the escaping satellite;

[0009] S103. Based on the polynomial fitting method, calculate the integral term g(θ) when θ is the remaining game time and the proportional term h(θ) of the maximum singular value to the minimum singular value of the remaining game arc matrix at θ;

[0010] S104. Calculate and optimize the approaching control of the tracking satellite.

[0011] Further, constructing the satellite orbit game model includes the following steps:

[0012] Based on the Oxyz reference coordinate system, construct the orbital dynamics equations of the tracking satellite and the escaping satellite, and their expressions are as follows:

[0013]

[0014] Among them, A is the system matrix, B is the input matrix of the control acceleration, is the state of satellite i = P, E, x i is the projection of the position vector of satellite i relative to point O on the Ox axis, y i is the projection of the position vector of satellite i relative to point O on the Oy axis, z i is the projection of the position vector of satellite i relative to point O on the Oz axis, is the projection of the velocity vector of satellite i relative to point O on the Ox axis, is the projection of the velocity vector of satellite i relative to point O on the Oy axis, is the projection of the velocity vector of satellite i relative to point O on the Oz axis, X iis the time derivative of the satellite \(i = P, E\) state, \(u\) i is the control acceleration vector of the satellite \(i = P, E\); is the angular velocity of the virtual satellite orbit, \(\mu\) is the gravitational constant of the earth, \(r\) O is the orbital radius of the virtual satellite, \(I\) 3 and \(0\) 3×3 are the 3D identity matrix and the \(3\times3\) zero matrix respectively;

[0015] Constraints are imposed on the control accelerations of the chaser satellite and the escape satellite, and the magnitude of the control acceleration of the chaser satellite is greater than that of the escape satellite to ensure that the chaser satellite can successfully approach the escape satellite. The expression is as follows:

[0016] \(\left\lVert u\right\rVert\) P \(\leq\rho\) P , \(\left\lVert u\right\rVert\) E \(\leq\rho\) E

[0017] where \(\left\lVert\cdot\right\rVert\) is the 2-norm of the vector, \(\left\lVert u\right\rVert\) P is the 2-norm of the control acceleration of the chaser satellite, \(\left\lVert u\right\rVert\) E is the 2-norm of the control acceleration of the escape satellite, \(\rho\) P is the magnitude of the control acceleration of the chaser satellite, \(\rho\) E is the magnitude of the control acceleration of the escape satellite;

[0018] Based on the position vectors of the chaser satellite and the escape satellite relative to the virtual satellite, a game end time objective function is constructed to ensure that the chaser satellite reaches the escape satellite as soon as possible. The expression is as follows:

[0019] \(t\) f \(:=\arg\min\{t\in[0,\pi / 2\omega]\mid\left\lVert r\right\rVert\) P (t)-r E (t)\left\lVert = 0\right\}\)

[0020] where \(t\) f is the game end time, \(r\) i =\left[x\right. i \ y i \ z i \) T is the position vector of the satellite \(i = P, E\) relative to the virtual satellite. During the game, the chaser satellite \(P\) applies control to reduce \(t\) f , while the escape satellite \(E\) applies control to increase \(t\) f .

[0021] Further, determining the sub-optimal approaching control of the tracking satellite and the escaping satellite based on the remaining game time, the remaining game arc, the zero-effort miss vector, the remaining game arc matrix, the magnitude of the control acceleration of the tracking satellite, and the magnitude of the control acceleration of the escaping satellite includes the following steps:

[0022] Based on the relative state between the tracking satellite and the escaping satellite, the control acceleration vector of the tracking satellite, and the control acceleration vector of the escaping satellite, construct the relative state differential equation between the tracking satellite and the escaping satellite, and its expression is as follows:

[0023]

[0024] Wherein, is the relative state derivative between the tracking satellite and the escaping satellite, X PE is the relative state between the tracking satellite P and the escaping satellite E, X PE =X P -X E , A is the system matrix, B is the input matrix of the control acceleration, u P is the control acceleration vector of the tracking satellite, u E is the control acceleration vector of the escaping satellite;

[0025] The zero-effort miss vector differential equation can be obtained through the relative state differential equation between the tracking satellite and the escaping satellite

[0026] Define the zero-effort miss distance as J(t f , t)=||Y(t f , t)||, and take the derivative with respect to time t to obtain

[0027] Based on the remaining game time and angular velocity, determine the remaining game arc, and its expression is as follows:

[0028] θ go =ωt go

[0029] Wherein, θ go is the remaining game arc, ω is the angular velocity, t go is the remaining game time, t go =t f -t, t f is the game end time;

[0030] Based on the zero-effort miss vector, the remaining game arc matrix, the magnitude of the control acceleration of the tracking satellite, and the magnitude of the control acceleration of the escaping satellite, determine the sub-optimal approaching control of the tracking satellite and the escaping satellite, and its expression is as follows:

[0031]

[0032] Among them, \(u\) P is the sub-optimal approaching control of the tracking satellite, and \(u\) E is the sub-optimal approaching control of the escaping satellite, \(\rho\) P is the magnitude of the control acceleration of the tracking satellite, and \(\rho\) E is the magnitude of the control acceleration of the escaping satellite, \(B(\theta\) go ) is the game remaining radian matrix, \(Y(t\) f ,t)\) is the zero-control miss vector, \(\xi(t\) f ,t)=Y(t\) f ,t) / ||Y(t\) f ,t)||, is the zero-control miss vector coefficient matrix, \(C = [I 3 0 3×3 .

[0033] Furthermore, calculating the game remaining time includes the following steps:

[0034] Construct a derivative equation of the zero-control miss distance with respect to time \(t\) according to the game remaining radian matrix, the magnitude of the control acceleration of the tracking satellite, and the magnitude of the control acceleration of the escaping satellite And perform scaling to obtain the zero-control miss distance scaling objective function Among them, \(\lambda\) m (\theta\) go ) is the minimum singular value of the matrix \(B(\theta\) go ), and \(\lambda\) M (\theta\) go ) is the maximum singular value of the matrix \(B(\theta\) go );

[0035] Integrate the time from \(t\) to \(t\) f The integration can obtain Among them, \(\eta\) is the integration variable, and \(\omega\eta\) represents the corresponding game remaining radian;

[0036] Based on the game end condition of the game end time objective function, combined with the zero-control miss distance scaling objective function, perform Solve;

[0037] Based on the relationship between the game remaining radian and the game remaining time, solve \(J(\theta\) go ) - g(\theta\) go ) = 0 to obtain the game remaining radian, where

[0038] According to \(t\) go =\theta\) go / \omega\) for calculation to obtain the game remaining time, where \(t\)go is the remaining time of the game, θ go is the remaining radian of the game, and ω is the angular velocity of the virtual satellite orbit.

[0039] Furthermore, solving for the remaining radian of the game based on the gradient method and the secant method includes the following steps:

[0040] Set the iteration termination error of the secant method and the gradient method. At the current time t i , calculate the state X of the tracking satellite relative to the escaping satellite PE ;

[0041] If the current time t i is the initial time of the game, then calculate the initial iteration left and right endpoints of the secant method

[0042] With as the initial iteration left and right endpoints, use the secant method to solve the equation J(θ go ) - g(θ go ) = 0 to obtain θ 0 corresponding to time t go (t 0 ), and save θ go (t 0 );

[0043] If the current time t i is not the initial time of the game, then set the saved θ go (t i-1 ) as the initial value Use the gradient method to solve the equation J(θ go ) - g(θ go ) = 0 to obtain θ i corresponding to time t go (t i ), and replace the saved θ go (t i-1 ) with θ go (t i ).

[0044] Furthermore, calculating the initial iteration left and right endpoints of the secant method includes the following steps:

[0045] According to the magnitude of the control acceleration ρ P of the tracking satellite and the magnitude of the control acceleration ρ E of the escaping satellite, calculate ρ P / ρ E ;

[0046] Define the function h(η) = λ M (η) / λ m(η) η ∈ [0, π / 2], and introduce the turning point θ c , the turning point gives the initial iteration right endpoint in the secant method for the vast majority of game scenarios When ρ P / ρ E is greater than h(π / 2), the turning point θ c = π / 2; otherwise, use the gradient method to solve the following equation h(θ c ) = ρ P / ρ E to obtain the turning point θ c ;

[0047] Calculate and judge whether J(θ c ) - g(θ c ) ≤ 0 holds. If it holds, let the initial iteration left and right endpoints of the secant method be

[0048] If J(θ c ) - g(θ c ) ≤ 0 does not hold, select N 1 +1 points at equal intervals in the interval [0, π / 2] Let

[0049] Furthermore, the calculation of the integral terms g(θ) and h(θ) based on the polynomial fitting method includes the following steps:

[0050] Discretize the feasible interval [0, π / 2] of the game remaining radian θ go into N = 1000 subintervals at equal intervals, and obtain N + 1 grid points θ j = jπ / 2N, j = 0, 1,..., N;

[0051] According to the virtual satellite orbital angular velocity ω, control the acceleration amplitudes ρ P and ρ E , for each θ j , based on the formula calculate the matrix B(θ j ), and calculate the maximum singular value λ M (θ j ) and the minimum singular value λ m (θ j ), for each grid point θ j , calculate h(θ j ), and use Simpson's numerical integration for g(θ j );

[0052] Let the fitting polynomial of g(θ) be a n θ n + an-1 θ n-1 +…+a 1 θ + a 0 , where a i , i = 0, 1,..., n are the coefficients of θ i , n is the highest order of the polynomial, which needs to be selected according to the fitting accuracy. To calculate all coefficients, let The coefficients a i , i = 0, 1,..., n of the fitting polynomial are calculated based on the least squares method, and its linear equations are as follows:

[0053]

[0054] Let the fitting polynomial of h(θ) be b m θ m + b m-1 θ m-1 +…+ b 1 θ + b 0 , where b i , i = 0, 1,..., m are the coefficients of θ i , m is the highest order of the polynomial, which needs to be selected according to the fitting accuracy. To calculate all coefficients b i , i = 0, 1,..., m;

[0055] After offline calculation of the coefficients a i , i = 0, 1,..., n and b i , i = 0, 1,..., m, save them on the on-board computer;

[0056] During on-orbit calculation, for any θ ∈ [0, π / 2], the on-board computer calculates the values of g(θ) and h(θ) through the formula

[0057] Furthermore, the calculation optimization of the approaching control of the tracking satellite includes the following steps:

[0058] According to the virtual satellite orbital angular velocity ω, the tracking satellite control acceleration amplitude ρ P and the escape satellite control acceleration amplitude ρ E , calculate the coefficients a i , i = 0, 1,..., n, b i , i = 0, 1,..., m of the fitting polynomial and save them on the on-board computer;

[0059] According to the state vectors X P , X E ​, the system matrix A and the input matrix B for controlling acceleration are used to calculate the relative state X PE ;

[0060] Based on the remaining radian θ of the game at the current moment go , the remaining time t of the game is calculated go , the zero-effort miss vector Y(t f ,t) and the remaining radian matrix B(t go );

[0061] Based on the zero-effort miss vector, the remaining radian matrix and the amplitude of the tracking satellite's control acceleration, the sub-optimal approaching control u of the tracking satellite is calculated P , and its expression is as follows:

[0062]

[0063] where u P is the sub-optimal approaching control of the tracking satellite, ρ P is the amplitude of the tracking satellite's control acceleration, B(θ go ) is the remaining radian matrix, and Y(t f ,t) is the zero-effort miss vector.

[0064] Advantages of this application: By using polynomial fitting to perform offline fitting on the key and complex integral involved in solving the remaining time of the game, calculating and storing the polynomial coefficients, when used on the satellite, the corresponding integral can be calculated only based on the polynomial coefficients. To avoid the repeated calculation of the time-varying matrix or its singular value in the existing calculation method, the complexity and calculation time of solving the remaining time of the game and the approaching control of the tracking satellite can be effectively reduced.

[0065] The existing calculation method for sub-optimal approaching control based on the zero-effort miss vector needs to solve a non-linear equation about the remaining time of the game through an iterative method. The complex integral term in this equation has no analytical expression and can only rely on numerical calculation, involving multiple calculations of the singular value of the time-varying matrix. The polynomial fitting method proposed in this application calculates and fits the polynomial coefficients of this integral term offline and stores them in the on-board computer. When calculating on orbit, only the upper bound of the integral needs to be input to obtain the integral value through simple addition and multiplication operations, avoiding multiple calculations of the matrix singular value, and effectively reducing the solving time of the remaining time of the game and the calculation time of the sub-optimal approaching control. Description of the Drawings

[0066] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on these drawings.

[0067] Figure 1 are the virtual satellite, the reference orbit Oxyz, and the chasing and escaping satellite;

[0068] Figure 2 is the position change curve of the chasing and escaping satellite obtained by using the calculation method of the present invention;

[0069] Figure 3 is the control change curve of the chasing and escaping satellite obtained by using the calculation method of the present invention;

[0070] Figure 4 is the flight trajectory of the chasing and escaping satellite in the game process obtained by using the calculation method of the present invention;

[0071] Figure 5 is the step schematic diagram of a fast calculation method for satellite orbit game approaching control of the present invention. Specific Embodiments

[0072] The following will describe the embodiments of the present invention in detail with reference to the drawings.

[0073] The following illustrates the embodiments of the present invention through specific examples. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of them. The present invention can also be implemented or applied through other different specific embodiments. Various details in this specification can also be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that, without conflict, the following embodiments and the features in the embodiments can be combined with each other. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts belong to the scope of protection of the present invention.

[0074] Embodiment 1:

[0075] A fast calculation method for satellite orbit game approaching control includes the following steps:

[0076] S101. Construct a satellite orbit game model;

[0077] As Figure 1As shown in the figure, since the distance between the tracking satellite P and the escaping satellite E is much smaller than the distance to the earth's center, a moving point near both and operating in a circular orbit around the earth is selected as the virtual satellite. With the position O of this virtual satellite as the coordinate origin, a local reference coordinate system Oxyz is established, where the Ox axis points along the direction from the earth's center to the position O of the virtual satellite, the Oy axis points along the direction of the current flight speed of the virtual satellite, and the Oz axis forms a right-handed system with the Ox and Oy axes and points along the orbital angular momentum direction.

[0078] Constructing the satellite orbit game model includes the following steps:

[0079] Based on the Oxyz reference coordinate system, construct the orbital dynamics equations of the tracking satellite and the escaping satellite, and their expressions are as follows:

[0080]

[0081] Among them, A is the system matrix, B is the input matrix of the control acceleration, is the state of satellite i = P, E, x i is the projection of the position vector of satellite i relative to point O on the Ox axis, y i is the projection of the position vector of satellite i relative to point O on the Oy axis, z i is the projection of the position vector of satellite i relative to point O on the Oz axis, is the projection of the velocity vector of satellite i relative to point O on the Ox axis, is the projection of the velocity vector of satellite i relative to point O on the Oy axis, is the projection of the velocity vector of satellite i relative to point O on the Oz axis, X i is the derivative of the state of satellite i = P, E with respect to time, u i is the control acceleration vector of satellite i = P, E; is the orbital angular velocity of the virtual satellite, μ is the gravitational constant of the earth, r O is the orbital radius of the virtual satellite, I 3 and 0 3×3 are the 3D identity matrix and the 3×3 zero matrix respectively;

[0082] Constrain the control accelerations of the tracking satellite and the escaping satellite, and the magnitude of the control acceleration of the tracking satellite is greater than that of the escaping satellite to ensure that the tracking satellite can successfully approach the escaping satellite. Its expression is as follows:

[0083] ||u P || ≤ ρ P , ||u E || ≤ ρ E

[0084] Among them, ||·|| is the 2-norm of the vector, ||uP || is the 2-norm of the control acceleration of the chaser satellite, ||u E || is the 2-norm of the control acceleration of the escaping satellite, ρ P is the magnitude of the control acceleration of the chaser satellite, ρ E is the magnitude of the control acceleration of the escaping satellite;

[0085] Based on the position vectors of the chaser satellite and the escaping satellite relative to the virtual satellite, a game end time objective function is constructed to ensure that the chaser satellite reaches the escaping satellite as soon as possible, and its expression is as follows:

[0086] t f :=arg min{t∈[0,π / 2ω]|||r P (t)-r E (t)||=0}

[0087] where, t f is the game end time, r i =[x i y i z i T is the position vector of satellite i = P, E relative to the virtual satellite. During the game process, the chaser satellite P applies control to reduce t f , while the escaping satellite E applies control to increase t f .

[0088] It should be noted that the satellite orbit game considered in this application is a short-range orbit pursuit-evasion game. In this scenario, the distance between the two satellites in the pursuit-evasion game is less than the detection range of their on-board radar / camera and much less than their distance to the earth's center. Therefore, both satellites in the game can obtain the position and velocity vector information of the other party.

[0089] S102. Determine the sub-optimal approaching control of the chaser satellite and the escaping satellite based on the remaining game time, the remaining game arc, the zero-control miss vector, the remaining game arc matrix, the magnitude of the control acceleration of the chaser satellite, and the magnitude of the control acceleration of the escaping satellite;

[0090] The remaining game arc θ go can be solved by combining the gradient method and the secant method. Since the gradient method requires initial value information while the secant method only needs to give the initial iteration interval Therefore, at the initial moment t 0 of the game, the secant method is used to calculate the θ 0 corresponding to the moment t go (t 0 ). At the subsequent sampling decision moments t i , i = 1, 2,..., t 1 <t​2 <t 3 <…, take the game remaining arc θ i-1 calculated at the previous moment t go (t i-1 ) as the initial value Use the gradient method to calculate the game remaining arc θ i at the current moment t go (t i ).

[0091] Based on the remaining game time, game remaining arc, zero-control miss vector, game remaining arc matrix, magnitude of the tracking satellite control acceleration, and magnitude of the escaping satellite control acceleration, determining the sub-optimal approaching control of the tracking satellite and the escaping satellite includes the following steps:

[0092] Based on the relative state between the tracking satellite and the escaping satellite, the control acceleration vector of the tracking satellite, and the control acceleration vector of the escaping satellite, construct the relative state differential equation between the tracking satellite and the escaping satellite, and its expression is as follows:

[0093]

[0094] where is the relative state derivative between the tracking satellite and the escaping satellite, X PE is the relative state between the tracking satellite P and the escaping satellite E, X PE = X P - X E , A is the system matrix, B is the input matrix of the control acceleration, u P is the control acceleration vector of the tracking satellite, u E is the control acceleration vector of the escaping satellite;

[0095] The zero-control miss vector differential equation can be obtained through the relative state differential equation between the tracking satellite and the escaping satellite

[0096] Define the zero-control miss distance as J(t f , t) = ||Y(t f , t)||, and take the derivative with respect to time t to obtain

[0097] Based on the remaining game time and angular velocity, determine the game remaining arc, and its expression is as follows:

[0098] θ go = ωt go

[0099] where θ go is the game remaining arc, ω is the angular velocity, t gois the remaining time of the game, t go = t f - t, t f is the end time of the game;

[0100] Based on the zero-effort miss vector, the remaining game radian matrix, the magnitude of the tracking satellite control acceleration, and the magnitude of the escaping satellite control acceleration, determine the sub-optimal approaching control of the tracking satellite and the escaping satellite, and its expression is as follows:

[0101]

[0102] where, u P is the sub-optimal approaching control of the tracking satellite, u E is the sub-optimal approaching control of the escaping satellite, ρ P is the magnitude of the tracking satellite control acceleration, ρ E is the magnitude of the escaping satellite control acceleration, B(θ go ) is the remaining game radian matrix, Y(t f , t) is the zero-effort miss vector, ξ(t f , t) = Y(t f , t) / ||Y(t f , t)||, is the zero-effort miss vector coefficient matrix, C = [I 3 0 3×3 .

[0103] Calculating the remaining time of the game includes the following steps:

[0104] According to the remaining game radian matrix, the magnitude of the tracking satellite control acceleration, and the magnitude of the escaping satellite control acceleration, construct the derivative equation of the zero-effort miss distance with respect to time t and perform scaling to obtain the scaled objective function of the zero-effort miss distance where, λ m (θ go ) is the minimum singular value of the matrix B(θ go ), λ M (θ go ) is the maximum singular value of the matrix B(θ go );

[0105] Integrate the time from t to t f to obtain where, η is the integration variable, and ωη represents the corresponding remaining game radian;

[0106] Based on the game end condition of the game end time objective function, combined with the scaled objective function of the zero-effort miss distance, solve ;

[0107] Based on the relationship between the remaining game arc and the remaining game time, solve J(θ go ) - g(θ go ) = 0 to obtain the remaining game arc, where

[0108] According to t go = θ go / ω for calculation to obtain the remaining game time, where t go is the remaining game time, θ go is the remaining game arc, and ω is the angular velocity of the virtual satellite orbit.

[0109] Solving for the remaining game arc based on the gradient method and the secant method includes the following steps:

[0110] Set the iteration termination error of the secant method and the gradient method. At the current time t i , calculate the state X of the tracking satellite relative to the escaping satellite PE ;

[0111] If the current time t i is the initial game time, then calculate the initial iteration left and right endpoints of the secant method

[0112] Taking as the initial iteration left and right endpoints, use the secant method to solve the equation J(θ go ) - g(θ go ) = 0 to obtain θ 0 corresponding to the time t go (t 0 ), and save θ go (t 0 );

[0113] If the current time t i is not the initial game time, then set the saved θ go (t i-1 ) as the initial value Use the gradient method to solve the equation J(θ go ) - g(θ go ) = 0 to obtain θ i corresponding to the time t go (t i ), and replace the saved θ go (t i-1 ) with θ go (t i ).

[0114] Calculating the initial iteration left and right endpoints of the secant method includes the following steps:

[0115] According to the control acceleration amplitude ρ of the tracking satellite P and the control acceleration amplitude ρ of the escaping satellite E , calculate ρ P / ρ E ;

[0116] Define the function h(η) = λ M (η) / λ m (η) where η ∈ [0, π / 2], and introduce the turning point θ c , and the turning point gives the initial iteration right endpoint in the secant method for the vast majority of game scenarios When ρ P / ρ E is greater than h(π / 2), the turning point θ c = π / 2; otherwise, use the gradient method to solve the following equation in the interval [0, π / 2] h(θ c ) = ρ P / ρ E to obtain the turning point θ c ;

[0117] Calculate and judge whether J(θ c ) - g(θ c ) ≤ 0 holds. If it holds, let the initial iteration left and right endpoints of the secant method be

[0118] If J(θ c ) - g(θ c ) ≤ 0 does not hold, equally spaced select N 1 +1 points in the interval [0, π / 2] Let

[0119] S103. Perform calculations based on the polynomial fitting method to obtain the integral term g(θ) when θ is the remaining game time and the ratio term h(θ) of the largest singular value to the smallest singular value of the game remaining radian matrix at θ;

[0120] In the process of solving J(θ go ) - g(θ go ) = 0, it is necessary to calculate the integral term g(θ) in many times, and this term involves the repeated calculation of the singular values of the matrix B(η). In addition, when solving h(η) = λ M (η) / λ m (η) where η ∈ [0, π / 2] to calculate the turning point θ cIt is also necessary to repeatedly calculate the singular values of B(η). However, the on-board computer takes a long time to process this type of operation. To avoid repeatedly calculating the singular values, the polynomial fitting method is used to calculate g(θ) and h(θ), calculating the coefficients of the fitting polynomial offline and calculating the values of the fitting polynomial on orbit.

[0121] Calculating the integral term g(θ) when θ is the remaining game time and the ratio term h(θ) of the maximum singular value to the minimum singular value of the game remaining radian matrix based on the polynomial fitting method includes the following steps:

[0122] Let the remaining game radian be θ go The feasible interval [0, π / 2] is discretized into N = 1000 subintervals at equal intervals, obtaining N + 1 grid points θ j = jπ / 2N, j = 0, 1,..., N;

[0123] According to the angular velocity ω of the virtual satellite orbit, control the acceleration amplitude ρ P and ρ E , for each θ j , based on the formula calculate the matrix B(θ j ), and calculate the maximum singular value λ M (θ j ) and the minimum singular value λ m (θ j ), for each grid point θ j , calculate h(θ j ), and use Simpson's numerical integration for g(θ j );

[0124] Let the fitting polynomial of g(θ) be a n θ n + a n-1 θ n-1 + … + a 1 θ + a 0 , where a i , i = 0, 1,..., n are the coefficients of θ i , n is the highest order of the polynomial, which needs to be selected according to the fitting accuracy. To calculate all the coefficients, let Calculate the coefficients a i , i = 0, 1,..., n of the fitting polynomial based on the least squares method, and its linear equation is shown as follows:

[0125]

[0126] Let the fitting polynomial of h(θ) be b m θ m + b m-1 θ m-1+…+b 1 θ + b 0 where b i , i = 0, 1, ..., m are the coefficients of θ i , m is the highest order of the polynomial, which needs to be selected according to the fitting accuracy, and all coefficients b i , i = 0, 1, ..., m;

[0127] During offline calculation of the coefficients a i , i = 0, 1, ..., n and b i , i = 0, 1, ..., m, save them on the on-board computer;

[0128] During on-orbit calculation, for any θ ∈ [0, π / 2], the on-board computer calculates the values of g(θ) and h(θ) through the formula .

[0129] S104. Optimize the calculation of the approaching control of the tracking satellite;

[0130] Optimizing the calculation of the approaching control of the tracking satellite includes the following steps:

[0131] Based on the angular velocity ω of the virtual satellite orbit, the amplitude ρ of the tracking satellite control acceleration P and the amplitude ρ of the escaping satellite control acceleration E , calculate the coefficients a i , i = 0, 1, ..., n, b i , i = 0, 1, ..., m of the fitting polynomial and save them on the on-board computer;

[0132] Based on the state vectors X P , X E of the tracking satellite and the escaping satellite measured at the current moment in the Oxyz coordinate system, the system matrix A, and the input matrix B of the control acceleration, calculate the relative state X PE ;

[0133] Based on the remaining radian θ of the game at the current moment go , calculate the remaining time t go of the game, the zero-control miss vector Y(t f , t) and the remaining radian matrix B(t go );

[0134] Based on the zero-control miss vector, the remaining radian matrix, and the amplitude of the tracking satellite control acceleration, calculate the sub-optimal approaching control u P of the tracking satellite, and its expression is as follows:

[0135]

[0136] Among them, u P is the sub-optimal approaching control of the tracking satellite, ρ P is the amplitude of the control acceleration of the tracking satellite, B(θ go ) is the remaining arc matrix of the game, and Y(t f , t) is the zero-control miss vector.

[0137] For example, the orbital angular velocity ω of the virtual satellite is 9.962×10 -4 rad / s. The states of the tracking satellite and the escaping satellite in the Oxyz coordinate system are shown in Table 1 below. ρ P = 0.2 m / s 2 , ρ E = 0.1 m / s 2 .

[0138] Table 1 States of the tracking satellite and the escaping satellite in the Oxyz coordinate system

[0139]

[0140] Use the approaching control calculation method proposed in this application to calculate u P , and consider the escaping control acceleration of the escaping satellite as follows

[0141]

[0142] Simulate this game scenario in Matlab 2021b, and the computing hardware is I7-12700KF 3.61 GHZ and 64GB RAM. In the simulation, the chasing and escaping satellites update their controls every 0.02 seconds. When the distance between the tracking satellite and the escaping satellite is less than 1 m, it is considered that the tracking satellite has successfully approached the escaping satellite and the game ends.

[0143] In the simulation, the average value T P of the single calculation time consumption of the approaching control u ave , the maximum value T max , the minimum value T min , the total game duration t f and the distance ||r(t f )|| between the chasing and escaping satellites at the end of the game are shown in Table 2 below. Among them, the polynomial fitting integral is used for the approaching control calculation method proposed in this patent, and the traditional direct numerical integral g(θ) is not used for the approaching control calculation method. It can be seen from this table that the approaching control calculation method proposed in this patent can achieve the approach to the escaping satellite, and the total game duration is not much different from the traditional method. However, this method improves the approaching control speed by 1-2 orders of magnitude compared with the traditional method, which shows the effectiveness of the method proposed in this patent.

[0144] Table 2 Time-consuming of single approach control calculation, total duration of the game, and distance between two satellites at the end of the game in the example

[0145]

[0146] For this example, Figure 2 the position change curve of the chasing and escaping satellites when the method of this patent is adopted is given, Figure 3 the control change curve of the chasing and escaping satellites when the method of this patent is adopted is given, Figure 4 the flight trajectory of the chasing and escaping satellites during the game process when the method of this patent is adopted is given. It can be found that when the method proposed in this patent is adopted, the distance between the tracking satellite and the escaping satellite gradually decreases on each direction axis until they meet. When the distance between the two satellites is far, the approach control of the tracking satellite changes little with the control of the escaping satellite, and the main purpose is to reduce the distance between the two satellites. When the distance between the two satellites is close, the periodic change and escape control of the escaping satellite have a greater impact on the change of the distance between the two satellites. Therefore, the approach control of the tracking satellite has a similar change trend to the control of the escaping satellite.

[0147] Those skilled in the art can clearly understand that for the convenience and simplicity of description, the specific working processes of the above-described systems and units can refer to the corresponding processes in the foregoing method embodiments and will not be repeated here.

[0148] The terms "including" and "having" in the description of this application and any variations thereof are intended to cover non-exclusive inclusion. For example, a process, method, system, product, or device that includes a series of steps or units does not necessarily have to be limited to those steps or units clearly listed, but may include other steps or units that are not clearly listed or are inherent to these processes, methods, products, or devices.

[0149] As mentioned above, the above embodiments are only used to illustrate the technical solutions of this application and are not intended to limit them. Although this application has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions recorded in the foregoing embodiments, or perform equivalent replacements for some of the technical features. And these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the spirit and scope of the technical solutions of the embodiments of this application.

Claims

1. A fast calculation method for satellite orbit game approach control, characterized in that: The following steps are involved: S101. Construct satellite orbit game model; S102, determining suboptimal approach control of the tracking satellite and the escape satellite based on the remaining game time, the remaining game arc, the zero control miss vector, the remaining game arc matrix, the amplitude of the tracking satellite control acceleration, and the amplitude of the escape satellite control acceleration; S103, calculating the integral term g(θ) when θ is the remaining game time and the proportional term h(θ) of the maximum singular value and the minimum singular value of the remaining arc matrix of the game at θ based on the polynomial fitting method; S104, performing calculation optimization on the tracking satellite approach control.

2. The satellite orbit game approach control fast calculation method according to claim 1 is characterized in that: The satellite orbit game model construction comprises the following steps: The orbital dynamics equations of the tracking satellite and the escape satellite are constructed based on the Oxyz reference coordinate system; The control acceleration of the tracking satellite and the escaping satellite is constrained, and the amplitude of the tracking satellite control acceleration is greater than the amplitude of the escaping satellite control acceleration to ensure that the tracking satellite can successfully approach the escaping satellite. The expression is as follows: ||in P ||ρ P ,||in E ||≤ρ E Among them, ||·|| is the 2-norm of the vector, ||u P || is the 2-norm of the tracking satellite control acceleration, ||u E || is the 2-norm of the control acceleration of the escape satellite, ρ P To track the amplitude of satellite control acceleration, ρ E Control the magnitude of acceleration for the escaping satellite; Based on the position vectors of the tracking satellite and the escape satellite relative to the virtual satellite, the game end time objective function is constructed to ensure that the tracking satellite reaches the escape satellite as soon as possible. Its expression is as follows: t f :=argmin{t∈[0,π / 2ω]|||r P (t)-r E (t)||=0} Among them, t f is the end time of the game, r i =[x i y i z i ] T is the position vector of satellite i=P,E relative to the virtual satellite. During the game, the tracking satellite P exerts control to reduce t f , while the escape satellite E exerts control to increase t f .

3. The satellite orbit game approach control fast calculation method according to claim 2 is characterized in that: The orbital dynamics equations of the tracking satellite and the escape satellite are constructed based on the Oxyz reference coordinate system, and the expressions are as follows: Among them, A is the system matrix, B is the input matrix of control acceleration, is the state of satellite i=P,E, x i is the projection of the position vector of satellite i relative to point O on the Ox axis, y i is the projection of the position vector of satellite i relative to point O on the Oy axis, z i is the projection of the position vector of satellite i relative to point O on the Oz axis, is the projection of the velocity vector of satellite i relative to point O on the Ox axis, is the projection of the velocity vector of satellite i relative to point O on the Oy axis, is the projection of the velocity vector of satellite i relative to point O on the Oz axis, X i is the time derivative of the satellite i=P,E state, u i is the control acceleration vector of satellite i=P,E; is the virtual satellite orbital angular velocity, μ is the earth's gravitational constant, r O is the orbital radius of the virtual satellite, I3 and 0 3×3 They are the 3D identity matrix and the 3×3 zero matrix respectively.

4. The satellite orbit game approach control fast calculation method according to claim 1, characterized in that: Determining the suboptimal approach control of the tracking satellite and the escape satellite based on the remaining game time, the remaining game arc, the zero control miss vector, the remaining game arc matrix, the amplitude of the tracking satellite control acceleration, and the amplitude of the escape satellite control acceleration comprises the following steps: Based on the relative state between the tracking satellite and the escaping satellite, the control acceleration vector of the tracking satellite and the control acceleration vector of the escaping satellite, the relative state differential equation between the tracking satellite and the escaping satellite is constructed, and its expression is as follows: in, is the relative state derivative between the tracking satellite and the escape satellite, X PE To track the relative state between satellite P and escaped satellite E, X PE =X P -X E , A is the system matrix, B is the input matrix of the control acceleration, u P is the control acceleration vector of the tracking satellite, u E is the control acceleration vector of the escaping satellite; The zero-control miss vector differential equation can be obtained by the relative state differential equation between the tracking satellite and the escaping satellite Define the zero-control miss distance as J(t f ,t)=||Y(t f ,t)||, and taking the derivative with respect to time t, we get The remaining arc of the game is determined based on the remaining game time and angular velocity, and its expression is as follows: i go =ωt go Among them, θ go is the remaining arc of the game, ω is the angular velocity, t go is the remaining time of the game, t go =t f -t,t f The end time of the game; Based on the zero control miss vector, the game residual arc matrix, the amplitude of the tracking satellite control acceleration and the amplitude of the escape satellite control acceleration, the suboptimal approach control of the tracking satellite and the escape satellite is determined.

5. The satellite orbit game approach control fast calculation method according to claim 4 is characterized in that: The suboptimal approach control of the tracking satellite and the escape satellite is determined based on the zero control miss vector, the game residual arc matrix, the amplitude of the tracking satellite control acceleration, and the amplitude of the escape satellite control acceleration. The expression is as follows: Among them, u P To track the suboptimal approach control of the satellite, u E is the suboptimal approach control of the escaping satellite, ρ P To track the amplitude of satellite control acceleration, ρ E is the magnitude of the control acceleration of the escape satellite, B(θ go ) is the residual arc matrix of the game, Y(t f ,t) is the zero control miss vector, ξ(t f ,t)=Y(t f ,t) / ||Y(t f ,t)||, is the zero control miss vector coefficient matrix, C = [I3 0 3×3 ].

6. The satellite orbit game approach control fast calculation method according to claim 1, characterized in that: The calculation of the remaining game time comprises the following steps: According to the residual arc matrix of the game, the amplitude of the control acceleration of the tracking satellite, and the amplitude of the control acceleration of the escape satellite, the derivative equation of the zero-control miss distance with respect to time t is constructed: And scale it up to get the zero control miss distance scaling objective function Among them, λ m (θ go ) is the minimum singular value of the matrix, λ M (θ go ) is the matrix B(θ go )'s maximum singular value; Integrate the time from t to t f Points can be obtained Among them, η is the integral variable, ωη represents the corresponding game residual arc; The game end condition based on the game end time objective function is combined with the zero control miss distance scaling objective function. To solve; Based on the relationship between the remaining arc of the game and the remaining time of the game, J(θ go )-g(θ go )=0 to solve the remaining arc of the game, where According to t go =θ go / ω is calculated to get the remaining time of the game, where t go is the remaining time of the game, θ go is the remaining arc of the game, and ω is the orbital angular velocity of the virtual satellite.

7. The satellite orbit game approach control fast calculation method according to claim 1, characterized in that: The method of solving the remaining arc of the game based on the gradient method and the secant method includes the following steps: Set the iteration termination error of the secant method and gradient method at the current time t i , calculate the state X of the tracking satellite relative to the escaping satellite PE ; If the current time t i is the initial moment of the game, then calculate the initial iteration left and right endpoints of the secant method by The left and right endpoints of the initial iteration are used to solve the equation J(θ go )-g(θ go )=0 to obtain the θ corresponding to time t0 go (t0), and save θ go (t0); If the current time t i If it is not the initial moment of the game, then the saved θ go (t i-1 ) is set to the initial value The gradient method is used to solve the equation J(θ go )-g(θ go )=0 to obtain t i The θ corresponding to the moment go (t i ), and save the θ go (t i-1 ) is replaced by θ go (t i ).

8. The satellite orbit game approach control fast calculation method according to claim 7, characterized in that: The initial iteration left and right endpoints of the secant method are calculated The following steps are involved: According to the control acceleration amplitude ρ of the tracking satellite P and the control acceleration amplitude ρ of the escape satellite E , calculate ρ P / ρ E ; Define the function h(η)=λ M (η) / λ m (η), η∈[0,π / 2], and introduce the turning point θ c The turning point gives the initial iterative right endpoint of the secant method in most game scenarios. When P / ρ E Greater than h(π / 2), turning point θ c =π / 2; otherwise, the gradient method is used to solve the following equation h(θ c )=ρ P / ρ E Get the turning point θ c ; Calculate and judge J(θ c )-g(θ c )≤0 is true, if true, let the left and right endpoints of the initial iteration of the secant method be If J(θ c )-g(θ c )≤0 does not hold, select N1+1 points at equal intervals in the interval [0,π / 2] make 9. The satellite orbit game approach control fast calculation method according to claim 1, characterized in that: The method for calculating the integral terms g(θ) and h(θ) based on the polynomial fitting method comprises the following steps: The remaining arc of the game θ go The feasible interval [0,π / 2] is discretized into N=1000 subintervals with equal intervals, and N+1 grid points θ are obtained. j =jπ / 2N, j = 0, 1, ..., N; According to the virtual satellite orbital angular velocity ω, the acceleration amplitude ρ is controlled P and ρ E , for each θ j , based on the formula Calculate the matrix B(θ j ), and calculate the maximum singular value λ M (θ j ) and the minimum singular value λ m (θ j ), for each grid point θ j , calculate h(θ j ), and use Simpson numerical integration g(θ j ); Let the fitting polynomial of g(θ) be a n θ n +a n-1 θ n-1 +…+a1θ+a0, where a i ,i=0,1,...,n is θ i The coefficients of n are the highest order of the polynomial, which needs to be selected according to the accuracy of the fit. To calculate all the coefficients, let The coefficient a of the fitting polynomial is calculated based on the least squares method i ,i=0,1,...,n, its linear equation is as follows: Let the fitting polynomial of h(θ) be b m θ m +b m-1 θ m-1 +…+b1θ+b0, where b i ,i=0,1,...,m is θ i coefficients, m is the highest order of the polynomial, which needs to be selected according to the accuracy of the fit to calculate all coefficients b i ,i=0,1,...,m; Calculate the coefficients a of the fitted polynomial offline i ,i=0,1,...,n and b i ,i=0,1,...,m, and save it in the onboard computer; During on-orbit calculations, for any θ∈[0,π / 2], the onboard computer uses the formula Calculate the values ​​of g(θ) and h(θ).

10. The satellite orbit game approach control fast calculation method according to claim 1, characterized in that: The calculation optimization of the tracking satellite approach control comprises the following steps: According to the virtual satellite orbital angular velocity ω, the tracking satellite controls the acceleration amplitude ρ P and the escape satellite control acceleration amplitude ρ E , calculate the coefficient a of the fitting polynomial i ,i=0,1,...,n,b i ,i=0,1,...,m, and saved in the onboard computer; According to the state vector X of the tracking satellite and the escape satellite in the Oxyz coordinate system measured at the current moment P ,X E , system matrix A and input matrix B of control acceleration, calculate relative state X PE ; According to the remaining arc θ of the game at the current moment go , calculate the remaining game time t go , zero control miss vector Y(t f ,t) and the game residual arc matrix B(t go ); According to the zero control miss vector, the game residual arc matrix and the amplitude of the tracking satellite control acceleration, the suboptimal approach control u of the tracking satellite is calculated. P , whose expression is as follows: Among them, u P To track the suboptimal approach control of the satellite, ρ P To track the amplitude of satellite control acceleration, B(θ go ) is the residual arc matrix of the game, Y(t f ,t) is the zero control miss-target vector.

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