Ultra-low-orbit electric propulsion satellite formation flight control method, device, equipment and medium

Abstract

The present application discloses a method, device, equipment and medium for controlling formation flight of ultra-low-orbit electric propulsion satellites, which relates to the field of satellite flight control technology. The method comprises: establishing a relative orbital dynamics model under the action of perturbation force based on the absolute orbital dynamics model of the ultra-low-orbit electric propulsion satellite under the action of perturbation force, and combining the relative motion mathematical model under the action of control force to establish a relative motion mathematical model under the joint action of the two forces; then combining the Hamiltonian function, co-state equation and canonical equation to establish a mathematical model of the orbit optimization control problem of the ultra-low-orbit electric propulsion satellite formation flight; converting the mathematical model of the above-mentioned optimization control problem into a two-point boundary value problem; based on special points, analytically calculating the initial values ​​of the co-state variables and the initial values ​​of the mass co-state; and determining the optimal control scheme according to the initial values ​​of the co-state variables, the initial values ​​of the mass co-state and the mathematical model of the two-point boundary value problem. The present application can improve the solution success rate, convergence speed and control accuracy of the optimization control.

Description

Technical Field

[0001] The present application relates to the field of satellite flight control technology, and in particular to a method, device, equipment and medium for controlling formation flight of ultra-low-orbit electric propulsion satellites. Background Art

[0002] Ultra-low-orbit satellite formations not only play a key role in scientific exploration and sustainable space development, but also demonstrate enormous potential in areas such as global communications and internet coverage. Furthermore, electric propulsion technology, as an emerging field, offers key advantages in reducing spaceflight fuel consumption and mission costs. Therefore, orbit control methods for ultra-low-orbit electric propulsion satellite formation flying hold significant strategic significance in modern aerospace technology. Through formation flying, satellites can achieve high-precision Earth observation, space environment monitoring, and astronomical observation.

[0003] Although the control technology for formation flight of ultra-low-orbit electric propulsion satellites has made significant progress in recent years, it still faces a series of challenges and difficulties. First, due to the large perturbations such as atmospheric drag in the ultra-low-orbit environment, satellites need to frequently maintain their orbital position, which places higher precision and efficiency requirements on the electric propulsion system. Secondly, formation flight requires that satellites maintain precise relative positions, which requires not only high-precision orbit maintenance, but also autonomous and efficient on-orbit decision-making and solving capabilities to cope with various emergencies and respond quickly. In addition, fast and high-quality response in formation flight is also a key issue. Satellites need to be able to quickly respond to ground commands or autonomous decisions to adjust their orbits and control their attitude. Summary of the Invention

[0004] The purpose of this application is to provide a method, device, equipment and medium for controlling the formation flight of ultra-low-orbit electric propulsion satellites, which can improve the solution success rate, convergence speed and control accuracy of optimization control.

[0005] To achieve the above objectives, this application provides the following solutions:

[0006] In a first aspect, the present application provides a method for controlling formation flight of ultra-low-orbit electric propulsion satellites, comprising:

[0007] Establish the absolute orbital dynamics model of ultra-low-orbit electric propulsion satellite under the action of perturbation force;

[0008] Based on the absolute orbital dynamics model of ultra-low orbit electric propulsion satellites under the action of perturbation force, the relative orbital dynamics model of ultra-low orbit electric propulsion satellite formation under the action of perturbation force is established; the ultra-low orbit electric propulsion satellite formation is composed of several ultra-low orbit electric propulsion satellites;

[0009] Based on the relative motion mathematical model of the ultra-low-orbit electric propulsion satellite formation under the action of control force and the relative orbit dynamics model under the action of perturbation force, a relative motion mathematical model of the ultra-low-orbit electric propulsion satellite formation flying under the combined action of environmental perturbation force and control force is established;

[0010] Taking minimum flight time or minimum fuel consumption as the optimization goal, a mathematical model for the orbit optimization control problem of ultra-low-orbit electric propulsion satellite formation flight is established based on the relative motion mathematical model, Hamiltonian function, costate equation, and canonical equation of ultra-low-orbit electric propulsion satellite formation flight under the combined action of environmental perturbation and control force. The Hamiltonian function, costate equation, and canonical equation are composed of costate variables and mass costates. The costate variables correspond to the orbital elements of the ultra-low-orbit electric propulsion satellite; and the mass costate corresponds to the mass of the ultra-low-orbit electric propulsion satellite.

[0011] Transform the mathematical model of the orbit optimization control problem of ultra-low-orbit electric propulsion satellite formation flying into a two-point boundary value problem; calculate the analytical Jacobian matrix of the two-point boundary value problem;

[0012] Based on the special point, the initial values ​​of the co-state variables and the mass co-state are calculated analytically; the special point is a point on the nominal trajectory of the virtual center of the ultra-low-orbit electric propulsion satellite formation flight;

[0013] According to the initial values ​​of co-state variables and mass co-states, two-point boundary value problems and canonical equations, the optimal thrust direction and optimal throttling coefficient at all times during the flight are calculated; the optimal thrust direction and optimal throttling coefficient at all times during the flight together constitute the optimal control scheme for the ultra-low-orbit electric propulsion satellite formation flight orbit.

[0014] In a second aspect, the present application provides an orbit control device for an ultra-low-orbit electric propulsion satellite formation, comprising:

[0015] The absolute orbit dynamics model establishment module is used to establish the absolute orbit dynamics model of the ultra-low orbit electric propulsion satellite under the action of perturbation force;

[0016] The module for establishing the absolute orbital dynamics model of ultra-low-orbit electric propulsion satellites is used to: establish the relative orbital dynamics model of an ultra-low-orbit electric propulsion satellite formation under the action of perturbation force based on the absolute orbital dynamics model of the ultra-low-orbit electric propulsion satellites under the action of perturbation force; an ultra-low-orbit electric propulsion satellite formation is composed of several ultra-low-orbit electric propulsion satellites;

[0017] The relative motion mathematical model establishment module is used to: establish the relative motion mathematical model of the ultra-low-orbit electric propulsion satellite formation flying under the combined action of environmental perturbation force and control force based on the relative motion mathematical model of the ultra-low-orbit electric propulsion satellite formation under the action of control force and the relative orbit dynamics model under the action of perturbation force;

[0018] The mathematical model building module for the flight orbit optimization control problem is used to: establish a mathematical model for the orbit optimization control problem of ultra-low-orbit electric propulsion satellite formation flight based on the relative motion mathematical model, Hamiltonian function, costate equation and canonical equation of the ultra-low-orbit electric propulsion satellite formation flight under the combined action of environmental perturbation and control force, with minimum flight time or minimum fuel consumption as the optimization goal; the Hamiltonian function, costate equation and canonical equation are composed of costate variables and mass costates; the costate variables correspond to the orbital elements of the ultra-low-orbit electric propulsion satellite; and the mass costate corresponds to the mass of the ultra-low-orbit electric propulsion satellite;

[0019] The analytic Jacobian matrix calculation module is used to: transform the mathematical model of the orbit optimization control problem of ultra-low-orbit electric propulsion satellite formation flying into a two-point boundary value problem; and calculate the analytic Jacobian matrix of the two-point boundary value problem;

[0020] The co-state initial value analysis module is used to: analytically calculate the initial values ​​of co-state variables and mass co-states based on a special point; the special point is a point on the nominal trajectory of the virtual center of the ultra-low-orbit electric propulsion satellite formation flight;

[0021] The optimal control scheme calculation module is used to calculate the optimal thrust direction and optimal throttling coefficient at all times during the flight according to the initial values ​​of the co-state variables and the mass co-state, the two-point boundary value problem and the canonical equation; the optimal thrust direction and the optimal throttling coefficient at all times during the flight together constitute the optimal control scheme for the ultra-low-orbit electric propulsion satellite formation flight orbit.

[0022] In a third aspect, the present application provides a computer device comprising: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the above-mentioned ultra-low-orbit electric propulsion satellite formation flight control method.

[0023] In a fourth aspect, the present application provides a computer-readable storage medium on which a computer program is stored. When the computer program is executed by a processor, it implements the above-mentioned ultra-low-orbit electric propulsion satellite formation flight control method.

[0024] According to the specific embodiments provided in this application, this application discloses the following technical effects:

[0025] In view of the problems existing in the ultra-low orbit electric propulsion satellite formation flight control technology in terms of orbit maintenance accuracy, autonomous decision-making and solution, and rapid response capability, the present application provides an ultra-low orbit electric propulsion satellite formation flight control method, device, equipment and medium. By converting the mathematical model of the ultra-low orbit electric propulsion satellite formation flight orbit optimization control problem into a two-point boundary value problem, and calculating the analytical Jacobian matrix of the two-point boundary value problem, the convergence speed is improved; based on special points, the initial values ​​of the co-state variables and the initial values ​​of the mass co-state are analytically calculated, and then the optimal thrust direction and the optimal throttling coefficient at all times during the flight are calculated through the initial values ​​of the co-state variables and the initial values ​​of the mass co-state, thereby improving the optimization success rate and control accuracy. Based on the above content, the present application can improve the solution success rate, convergence speed and control accuracy of the optimization control through analytical means when dealing with complex high-dimensional nonlinear controlled systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0027] Figure 1 A flowchart of a method for controlling formation flight of ultra-low-orbit electric propulsion satellites provided in one embodiment of the present application;

[0028] Figure 2 A schematic diagram showing a comparison of the calculation time of the ultra-low-orbit electric propulsion satellite formation flight control method proposed in this application and the Gaussian pseudospectral method, provided in one embodiment of this application;

[0029] Figure 3 Schematic diagram of the control accuracy comparison results of the ultra-low-orbit electric propulsion satellite formation flight control method proposed in this application and the Gaussian pseudospectral method provided in one embodiment of this application

[0030] Figure 4 A schematic diagram of the structure of a computer device provided in one embodiment of the present application. DETAILED DESCRIPTION

[0031] The following will be combined with the drawings in the embodiments of this application to clearly and completely describe the technical solutions in the embodiments of this application. Obviously, the embodiments described are only part of the embodiments of this application, not all of the embodiments. Based on the embodiments in this application, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of this application.

[0032] In order to make the above-mentioned purposes, features and advantages of the present application more obvious and easy to understand, the present application is further described in detail below with reference to the accompanying drawings and specific implementation methods.

[0033] In an exemplary embodiment, Figure 1 As shown, a method for controlling formation flight of ultra-low-orbit electric propulsion satellites is provided, which includes the following steps 101 to 107.

[0034] Step 101: Establish an absolute orbital dynamics model of an ultra-low-orbit electric propulsion satellite under the action of perturbation force.

[0035] Step 102: Based on the absolute orbital dynamics model of the ultra-low orbit electric propulsion satellite under the action of the perturbation force, a relative orbital dynamics model of the ultra-low orbit electric propulsion satellite formation under the action of the perturbation force is established; the ultra-low orbit electric propulsion satellite formation is composed of a number of ultra-low orbit electric propulsion satellites.

[0036] Step 103: Based on the relative motion mathematical model of the ultra-low-orbit electric propulsion satellite formation under the action of control force and the relative orbital dynamics model under the action of perturbation force, a relative motion mathematical model of the ultra-low-orbit electric propulsion satellite formation flying under the combined action of environmental perturbation force and control force is established;

[0037] Step 104: With minimum flight time or minimum fuel consumption as the optimization goal, a mathematical model for the orbit optimization control problem of the ultra-low-orbit electric propulsion satellite formation flight is established based on the relative motion mathematical model, Hamiltonian function, costate equation, and canonical equation of the ultra-low-orbit electric propulsion satellite formation flight under the combined effects of environmental perturbations and control forces. The Hamiltonian function, costate equation, and canonical equation are composed of costate variables and mass costates. The costate variables correspond to the orbital elements of the ultra-low-orbit electric propulsion satellites, and the mass costate corresponds to the mass of the ultra-low-orbit electric propulsion satellites.

[0038] Step 105: Convert the mathematical model of the orbit optimization control problem of the ultra-low-orbit electric propulsion satellite formation flying into a two-point boundary value problem; and calculate the analytical Jacobian matrix of the two-point boundary value problem.

[0039] Step 106: Based on the special point, the initial values ​​of the co-state variables and the initial values ​​of the mass co-state are analytically calculated; the special point is a point on the nominal trajectory of the virtual center of the ultra-low-orbit electric propulsion satellite formation flight;

[0040] Step 107: Based on the initial values ​​of the co-state variables and the mass co-state, the two-point boundary value problem and the canonical equation, the optimal thrust direction and the optimal throttling coefficient at all times during the flight are calculated; the optimal thrust direction and the optimal throttling coefficient at all times during the flight together constitute the optimal control scheme for the ultra-low-orbit electric propulsion satellite formation flight orbit.

[0041] The main advantages of implementing steps 101 to 108 above over existing methods (Gaussian pseudospectral methods) are a high optimization success rate, rapid convergence, and high control accuracy. The analytical Jacobian matrix obtained in step 105 can significantly increase the speed of high-dimensional and high-order vector operations; the co-state initial values ​​obtained in step 106 are the best performing initial values, ensuring both a high optimization success rate and high control accuracy.

[0042] In another exemplary embodiment of the present application, the above step 101 specifically includes the following steps 201 to 202:

[0043] Step 201: describing the orbital motion of the ultra-low-orbit electric propulsion satellite based on the improved vernal equinox orbital elements, and establishing a perturbation force model affecting the orbital motion of the ultra-low-orbit electric propulsion satellite; the perturbation force model includes a mechanical model of the Earth's non-spherical gravity and a mechanical model of atmospheric drag;

[0044] Step 202: Substitute the mechanical model of the Earth's non-spherical gravity into the Lagrangian equation of planetary motion, and substitute the mechanical model of atmospheric drag into the Gaussian perturbation equation to establish the absolute orbital dynamics model of the ultra-low orbit electric propulsion satellite under the perturbation force.

[0045] Using the improved vernal equinox orbit elements [a,e x ,e x ,i x ,i y ,L] describes the orbital motion of ultra-low-orbit electric propulsion satellite, [a,e x ,e x ,i x ,i y , L] refer to the orbit semi-major axis, the x-component of the eccentricity vector, the y-component of the eccentricity vector, the x-component of the orbital inclination, the y-component of the orbital inclination, and the mean longitude, respectively. The improved equinox orbit elements can better handle the singularity problems caused by small inclinations and small eccentricities. The mapping relationship between the improved equinox orbit elements and the classic orbit elements [a, e, i, ω, Ω, M] is:

[0046]

[0047] Where [a, e, i, ω, Ω, M] refer to the orbital semi-major axis, eccentricity, orbital inclination, argument of perigee, right ascension of ascending node, and mean anomaly, respectively. g Represents the sidereal hour angle relative to Greenwich.

[0048] The main perturbations that affect orbital motion are the Earth's non-spherical gravity and atmospheric drag.

[0049] The non-spherical gravity of the earth is a conservative force, using the potential function U ENP Describing the mechanical model, the mechanical model of the earth's non-spherical gravity is in the form of:

[0050]

[0051] Among them, r is the distance from the center of the earth to the satellite, μ is the gravitational coefficient of the earth, λ is the longitude of the satellite, R e is the radius of the Earth, J2 is the harmonic coefficient of the non-spherical perturbation, J 22 ,λ 22 is the coefficient of the Tian harmonic term of the non-spherical perturbation.

[0052] Atmospheric drag is a non-conservative force, and the mechanical model of atmospheric drag is:

[0053]

[0054] Where K1=(C D S1 / m)Q, Q=(1-r p ω e cosi / v p ) 2 , S1, S2 are the cross-sectional areas in the vertical velocity direction and the normal direction of the track surface, respectively, C D is the drag coefficient, m is the satellite mass, ρ is the atmospheric density, n is the average angular velocity of the satellite, u is the latitude argument, r p is the distance from the center of the earth to the perigee, ω e is the angular velocity of the Earth's rotation, v p is the perigee satellite velocity, F T , F N , F W are the track, normal and lateral atmospheric drag, respectively. The subscripts T, N and W represent the track, normal and lateral directions, respectively.

[0055] Affected by the environmental perturbation force, the changes in the orbital elements of the improved vernal equinox of the ultra-low-orbit electric propulsion satellite in the inertial coordinate system are the absolute orbital dynamics model.

[0056] The non-spherical gravity of the Earth is a conservative force. Substituting the mechanical model of the non-spherical gravity of the Earth shown in formula (2) into the Lagrangian planetary motion equation, the changes of orbital elements under the action of the non-spherical gravity of the Earth are described in the following form:

[0057]

[0058] Atmospheric drag is a non-conservative force. The mechanical model of atmospheric drag shown in formula (3) is substituted into the Gaussian perturbation equation to describe the changes of orbital elements under the action of atmospheric drag. The form is as follows:

[0059]

[0060] Among them, a s represents the nominal orbital semi-major axis of the satellite, ωe represents the angular velocity of the Earth's rotation, V s is the satellite velocity, and l = Ω + ω + M is the equatorial longitude of the satellite.

[0061] By combining formula (4) and formula (5), the absolute orbital dynamics model of the ultra-low-orbit electric propulsion satellite under the action of perturbation force is obtained.

[0062] The absolute orbital dynamics model of the ultra-low-orbit electric propulsion satellite under the perturbation force obtained in the above steps is written in the following matrix form:

[0063]

[0064] Where x=[a,e x ,e y ,i x ,i y ,L] T , D is the environmental perturbation matrix, which is the earth's non-spherical perturbation matrix D ENP and the atmospheric drag perturbation matrix D drag The Earth's non-spherical perturbation matrix D ENP is the matrix form of the right side of the equation (4), the atmospheric drag perturbation matrix D drag is the matrix form of the right side of the equation (5). The Earth non-spherical perturbation matrix D ENP and the atmospheric drag perturbation matrix D drag They are all matrices with 6 rows and 1 column, and each row represents the rate of change of the orbital elements caused by the corresponding perturbation force.

[0065] The number of satellites in the ultra-low orbit electric propulsion satellite formation is recorded as N, and the center point of the ultra-low orbit electric propulsion satellite formation is The calculation method is as follows:

[0066]

[0067] Where m k represents the mass of the kth satellite in the ultra-low-orbit electric propulsion satellite formation, is the orbital element change rate of the kth satellite.

[0068] The core of the control task of the ultra-low-orbit electric propulsion satellite formation is to make the orbital element deviation of each satellite in the ultra-low-orbit electric propulsion satellite formation and the center point of the ultra-low-orbit electric propulsion satellite formation present an expected distribution. That is, the relative orbital dynamics model of the ultra-low orbit electric propulsion satellite formation is as follows:

[0069]

[0070] Where D' represents the environmental perturbation matrix deviation between the kth satellite and the center point of the ultra-low-orbit electric propulsion satellite formation.

[0071] In another exemplary embodiment of the present application, the above step 103 includes the following steps 301 and 302:

[0072] Step 301: Establish a mathematical model of an ultra-low-orbit electric propulsion satellite formation under the action of control force.

[0073]

[0074] Among them, T max is the maximum thrust amplitude; I sp is the thruster specific impulse; g0 is the standard gravitational acceleration at sea level; u∈[0,1] is the throttling coefficient of the thruster; α is the unit vector of the thrust direction; M is the thrust control coefficient matrix.

[0075] Step 302: Establish a relative motion mathematical model of ultra-low-orbit electric propulsion satellite formation flying under the combined effects of environmental perturbation forces and control forces.

[0076] By combining formula (8) and formula (9), we can obtain the relative motion mathematical model of ultra-low-orbit electric propulsion satellite formation flying under the combined action of environmental perturbation force and control force, which is as follows:

[0077]

[0078] Establish the optimization indicators, Hamiltonian functions, co-state equations and canonical equations of the controlled system (ultra-low orbit electric propulsion satellite formation).

[0079] The goal of optimizing control for ultra-low-orbit electric propulsion satellite formation flight is usually to optimize time or fuel, so the performance index J can be uniformly expressed as shown in the following formula:

[0080]

[0081] Among them, t0 and t f are the initial moment and the terminal moment respectively; is the Lagrangian function, for the time optimization problem (Minimum Time, MT) For the minimum fuel problem (MF), we have

[0082] Therefore, the Hamiltonian function H of the controlled system is expressed as shown in the following formula:

[0083]

[0084] in, is the covariate variable corresponding to x; m is the mass costate corresponding to the satellite mass m.

[0085] The equation of motion for the covariate variables is:

[0086]

[0087] Let y = [δx T ,m,λ x T ,λ m ] T , the canonical equation of the controlled system is expressed as follows:

[0088]

[0089] Where, represents the rate of change of the covariate variable corresponding to x; when i=1, 2, ..6, it refers to λ a , λ L .

[0090] Based on the above steps, a mathematical model for the orbit optimization control problem of ultra-low-orbit electric propulsion satellite formation flight is established.

[0091] The mathematical model of the orbit optimization control problem of the ultra-low-orbit electric propulsion satellite formation flight in step 104 can be expressed as follows: Design the optimal throttling coefficient u * and the optimal thrust direction α * This ensures that the regular equation shown in formula (14) always holds true and the performance index J shown in formula (11) is minimized.

[0092] In another exemplary embodiment of the present application, in the above step 105, the mathematical model of the ultra-low-orbit electric propulsion satellite formation flight orbit optimization control problem is converted into a two-point boundary value problem, specifically including steps 401 to 402:

[0093] Step 401: Use the Pontryagin maximum principle to describe the optimal thrust direction and the optimal throttling coefficient.

[0094] Step 402: Convert the mathematical model of the orbit optimization control problem of ultra-low-orbit electric propulsion satellite formation flight into a two-point boundary value problem.

[0095] By using the Pontryagin maximum principle, the orbit optimization control problem of ultra-low-orbit electric propulsion satellite formation flying described in step 105 is transformed into a two-point boundary value problem.

[0096] According to the Pontryagin maximum principle, the optimal thrust direction α * and the optimal throttling coefficient u *Has the following form:

[0097]

[0098] Among them, α * is the optimal thrust direction, M is the thrust control coefficient matrix, λ x is the co-state variable corresponding to the orbital element of the ultra-low orbit electric propulsion satellite, u * is the optimal throttling coefficient, ρ is the switching function, and for the time optimal problem, ρ=-I sp g0||M T λ x || / m-λ m , for the fuel optimization problem, ρ=1-I sp g0||M T λ x || / m-λ m .

[0099] At this point, the track optimization control problem described in step 104 can be transformed into a two-point boundary value problem. The mathematical model of the two-point boundary value problem is as follows:

[0100]

[0101] Among them, δx0 and m0 are the relative orbital elements and mass of the satellite at time t0 respectively; δx f is the spacecraft at the terminal time t f The relative orbital elements of f free” and H(t f )=0 matches and corresponds to the flight time free case, while “t f fixed” and “H(t f )free” matches and corresponds to the case where the flight time is fixed.

[0102] The shooting function corresponding to the two-point boundary value problem is expressed as shown in the following formula (18):

[0103]

[0104] Among them, z MT =[λ(t0) T ,t f ] T , z MF =λ(t0) T , the superscripts MT and MF represent the time-optimal and fuel-optimal situations, respectively, MT and z MF The independent variable of the shooting function representing the time-optimal and fuel-optimal problems; and Targeting function representing time-optimal and fuel-optimal problems; δx f represents the expected terminal time relative to the orbital element, δx(t f ) represents the real terminal time relative orbit element.

[0105] Based on the two-point boundary value problem in step 105, a technique for calculating the analytical Jacobian matrix of the two-point boundary value problem is proposed below.

[0106] Let the shooting function corresponding to the two-point boundary value problem shown in formula (18) be equal to zero to solve z MT =[λ(t0) T ,t f ] T or z MF =λ(t0) T When the multidimensional partial differential equation is calculated, it involves multiple iterative integration processes, which is computationally intensive and time-consuming. Therefore, it is necessary to calculate the analytical Jacobian matrix to speed up the iterative integration efficiency and improve the accuracy and robustness of the target function calculation process. Introducing the homotopy parameter ε, the analytical Jacobian matrix J corresponding to formula (18) is as follows:

[0107]

[0108] in, Target shooting function The Jacobian matrix of the target variable z can be derived directly; Target shooting function The Jacobian matrix of the homology parameter ε cannot be directly derived, so the state transfer matrix method needs to be used.

[0109] Considering the homotopy ε as an additional state variable, the extended regular variable is The corresponding dynamic equation is The following formula:

[0110]

[0111] The corresponding extended state transfer matrix can be expressed as The following formula:

[0112]

[0113] Where t is a certain moment; Φ is the state transition matrix of y; for The state transition matrix of ; I represents the identity matrix.

[0114] because The initial information of contains (z; ε), so yes function, thereby realizing The solution.

[0115] In summary, through and By performing calculations, the analytical Jacobian matrix J can be directly obtained.

[0116] Because the range of possible co-state initial values ​​is large, but the effective co-state initial value radius that makes the optimization problem converge is very small, how to effectively guess the co-state initial value has always been a difficulty and challenge in solving two-point boundary value problems. The analytical guessing technique for co-state initial values ​​proposed in this application consists of two parts: first, determining special points, and second, solving the analytical co-state initial value based on the special points.

[0117] A special point is defined as a point on the nominal trajectory of the virtual center of the satellite formation flying. The definition of the nominal trajectory of the virtual center of the satellite formation flying is as follows:

[0118] (1) Satellite formation flying has a virtual center nominal trajectory that is approximately circular, that is, each satellite in the satellite formation always flies within a fixed value range of the virtual nominal center;

[0119] (2) The orbital position deviation of each satellite in the satellite formation from the virtual center can be maintained through no more than three orbital maneuvers.

[0120] The above step 106 specifically includes: step 501: determining the special point; step 502: analyzing the initial values ​​of the co-state variables and the initial values ​​of the mass co-state according to the orbital elements corresponding to the special point.

[0121] Step 501 includes the following steps 601 to 603:

[0122] Step 601: The points closest to and farthest from the center of the Earth on the approximately circular virtual center nominal trajectory are recorded as perigee and apogee respectively;

[0123] Step 602: Record the point on the virtual center nominal trajectory where the ultra-low-orbit electric propulsion satellite formation performs the most orbit-maintaining maneuvers as the maneuvering point;

[0124] Step 603: The virtual center nominal trajectory of the approximate circle is divided into three segments by perigee, apogee and maneuvering point. The orbit element change rate corresponding to the midpoints of the three segments is calculated respectively. The point with the largest orbit element change rate is determined as a special point. The orbit element corresponding to the special point is recorded as

[0125] By solving formula (22), we can obtain the guessed initial value of the co-state variable λ x (t0).

[0126]

[0127] Then the initial value of the co-state variable λ x Substituting (t0) into equation (23), we can solve the guessed initial value of mass co-state λ m (t0).

[0128]

[0129] In another exemplary embodiment of the present application, the above step 107 specifically includes the following steps 701 to 703:

[0130] Step 701: Calculate the optimal thrust direction and the optimal throttling coefficient at the initial moment according to the initial values ​​of the co-state variables and the initial values ​​of the mass co-state;

[0131] Step 702: Substitute the initial values ​​of the co-state variables and the initial values ​​of the mass co-state into the canonical equation to obtain the co-state variables and mass co-state at all times during the flight process;

[0132] Step 703: Calculate the optimal thrust direction and the optimal throttling coefficient at all times during the flight based on the co-state variables and mass co-state at all times during the flight.

[0133] Initialize the covariate variable λ x (t0) and the initial value of mass co-state λ m Substitute (t0) into formula (15) and formula (16) to calculate the optimal thrust direction α* and optimal throttling coefficient u* at the initial moment. x (t0) and the initial value of mass co-state λ m Substitute (t0) into formula (14) to obtain the co-state variables and mass co-states at all times during the flight process, and then substitute into formulas (15) and (16) to calculate the optimal thrust direction α at all times during the flight process. * and the optimal throttling coefficient u * The optimal thrust direction α at all times during flight * and the optimal throttling coefficient u * Together they form the optimal control scheme for the ultra-low-orbit electric propulsion satellite formation flight orbit.

[0134] Steps 105 to 107 together constitute an optimization control algorithm assisted by a new analytical method for solving the orbit optimization control problem of the ultra-low-orbit electric propulsion satellite formation flight described in step 104.

[0135] The performance of the ultra-low-orbit electric propulsion satellite formation flight control method proposed in this application is verified below:

[0136] In ten randomly selected formation flight examples, the ultra-low-orbit electric propulsion satellite formation flight control method proposed in this application was compared with the Gaussian pseudospectral method. The simulation results are as follows: Figure 2 and Figure 3As shown:

[0137] Figure 2 The horizontal and vertical axes are the example number and calculation time, respectively. It can be seen that the calculation time of the ultra-low-orbit electric propulsion satellite formation flight control method proposed in this application is less than that of the Gaussian pseudospectral method, and the average calculation time is one-fifth of the Gaussian pseudospectral method, that is, the algorithm converges quickly. Figure 3 The horizontal and vertical axes are the example number and control accuracy, respectively. It can be seen that the control accuracy of the ultra-low orbit electric propulsion satellite formation flight control method proposed in this application is better than that of the Gaussian pseudospectral method, and the average control accuracy is improved by one order of magnitude compared with the Gaussian pseudospectral method. That is, compared with the Gaussian pseudospectral method, the control accuracy of the ultra-low orbit electric propulsion satellite formation flight control method proposed in this application is higher.

[0138] The present application also provides an application scenario, which applies the above-mentioned ultra-low orbit electric propulsion satellite formation flight control method. Specifically: the ultra-low orbit electric propulsion satellite formation flight control method provided in this embodiment can be applied in the satellite flight control scenario. The satellite flight control scenario includes a parameter acquisition link and a satellite formation flight control link; the basic parameters of the ultra-low orbit electric propulsion satellite formation enter the satellite formation flight control link from the parameter acquisition link to obtain the optimal control solution for the corresponding flight orbit. The ultra-low orbit electric propulsion satellite formation flight control method provided in this embodiment belongs to the satellite formation flight control link. Specifically, in the process of satellite formation flight control link for satellite formations, a relative orbital dynamics model under the action of perturbation force can be established based on the absolute orbital dynamics model of the ultra-low orbit electric propulsion satellite under the action of perturbation force, and combined with the mathematical model of relative motion under the action of control force, a mathematical model of relative motion under the joint action of the two forces is established; then, the Hamiltonian function, costate equation and canonical equation are combined to establish a mathematical model of the orbit optimization control problem of the ultra-low orbit electric propulsion satellite formation flight; the mathematical model of the above-mentioned optimization control problem is converted into a two-point boundary value problem; based on special points, the initial values ​​of the costate variables and the initial values ​​of the mass costate are analytically calculated; according to the initial values ​​of the costate variables and the initial values ​​of the mass costate, the two-point boundary value problem and the canonical equation, the optimal control scheme is determined.

[0139] Based on the same inventive concept, the embodiments of the present application also provide an ultra-low-orbit electric propulsion satellite formation flight control device for implementing the ultra-low-orbit electric propulsion satellite formation flight control method involved above. The implementation solution provided by this device is similar to the implementation solution described in the above method. Therefore, the specific limitations of one or more ultra-low-orbit electric propulsion satellite formation flight control device embodiments provided below can be found in the above-mentioned limitations of the ultra-low-orbit electric propulsion satellite formation flight control method, and will not be repeated here.

[0140] In an exemplary embodiment, the present application further provides an ultra-low-orbit electric propulsion satellite formation orbit control device, comprising the following modules:

[0141] The absolute orbit dynamics model establishment module is used to establish the absolute orbit dynamics model of the ultra-low orbit electric propulsion satellite under the action of perturbation force;

[0142] The module for establishing the absolute orbital dynamics model of ultra-low-orbit electric propulsion satellites is used to: establish the relative orbital dynamics model of an ultra-low-orbit electric propulsion satellite formation under the action of perturbation force based on the absolute orbital dynamics model of the ultra-low-orbit electric propulsion satellites under the action of perturbation force; an ultra-low-orbit electric propulsion satellite formation is composed of several ultra-low-orbit electric propulsion satellites;

[0143] The relative motion mathematical model establishment module is used to: establish the relative motion mathematical model of the ultra-low-orbit electric propulsion satellite formation flying under the combined action of environmental perturbation force and control force based on the relative motion mathematical model of the ultra-low-orbit electric propulsion satellite formation under the action of control force and the relative orbit dynamics model under the action of perturbation force;

[0144] The mathematical model building module for the flight orbit optimization control problem is used to: establish a mathematical model for the orbit optimization control problem of ultra-low-orbit electric propulsion satellite formation flight based on the relative motion mathematical model, Hamiltonian function, costate equation and canonical equation of the ultra-low-orbit electric propulsion satellite formation flight under the combined action of environmental perturbation and control force, with minimum flight time or minimum fuel consumption as the optimization goal; the Hamiltonian function, costate equation and canonical equation are composed of costate variables and mass costates; the costate variables correspond to the orbital elements of the ultra-low-orbit electric propulsion satellite; and the mass costate corresponds to the mass of the ultra-low-orbit electric propulsion satellite;

[0145] The analytic Jacobian matrix calculation module is used to: transform the mathematical model of the orbit optimization control problem of ultra-low-orbit electric propulsion satellite formation flying into a two-point boundary value problem; and calculate the analytic Jacobian matrix of the two-point boundary value problem;

[0146] The co-state initial value analysis module is used to: analytically calculate the initial values ​​of co-state variables and mass co-states based on a special point; the special point is a point on the nominal trajectory of the virtual center of the ultra-low-orbit electric propulsion satellite formation flight;

[0147] The optimal control scheme calculation module is used to calculate the optimal thrust direction and optimal throttling coefficient at all times during the flight according to the initial values ​​of the co-state variables and the mass co-state, the two-point boundary value problem and the canonical equation; the optimal thrust direction and the optimal throttling coefficient at all times during the flight together constitute the optimal control scheme for the ultra-low-orbit electric propulsion satellite formation flight orbit.

[0148] In an exemplary embodiment, a computer device is provided. The computer device may be a server or a terminal. The internal structure diagram thereof may be as follows: Figure 4 As shown. The computer device includes a processor, a memory, an input / output interface (Input / Output, abbreviated as I / O) and a communication interface. The processor, memory and input / output interface are connected through a system bus, and the communication interface is connected to the system bus through the input / output interface. The processor of the computer device is used to provide computing and control capabilities. The memory of the computer device includes a non-volatile storage medium and an internal memory. The non-volatile storage medium stores an operating system, a computer program and a database. The internal memory provides an environment for the operation of the operating system and computer program in the non-volatile storage medium. The database of the computer device is used to store satellite formation flight control data. The input / output interface of the computer device is used to exchange information between the processor and an external device. The communication interface of the computer device is used to communicate with an external terminal through a network connection. When the computer program is executed by the processor, a method for controlling the formation flight of ultra-low-orbit electric propulsion satellites is implemented.

[0149] Those skilled in the art will understand that Figure 4 The structure shown in the figure is only a block diagram of a part of the structure related to the solution of the present application, and does not constitute a limitation on the computer device to which the solution of the present application is applied. The specific computer device may include more or fewer components than shown in the figure, or combine certain components, or have a different component arrangement.

[0150] In an exemplary embodiment, a computer device is further provided, including a memory and a processor. The memory stores a computer program, and the processor implements the steps in the above method embodiments when executing the computer program.

[0151] In an exemplary embodiment, a computer-readable storage medium is provided, storing a computer program. When the computer program is executed by a processor, the steps in the above-mentioned method embodiments are implemented.

[0152] It should be noted that the user information (including but not limited to user device information, user personal information, etc.) and data (including but not limited to data used for analysis, stored data, displayed data, etc.) involved in this application are all information and data authorized by the user or fully authorized by all parties, and the collection, use and processing of relevant data must comply with relevant regulations.

[0153] Those skilled in the art will understand that all or part of the processes in the above-mentioned embodiment methods can be implemented by instructing the relevant hardware through a computer program, and the computer program can be stored in a non-volatile computer-readable storage medium. When the computer program is executed, it can include the processes of the embodiments of the above-mentioned methods. Among them, any reference to memory, database or other media used in the embodiments provided in this application may include at least one of non-volatile and volatile memory. Non-volatile memory may include read-only memory (ROM), magnetic tape, floppy disk, flash memory, optical memory, high-density embedded non-volatile memory, resistive random access memory (ReRAM), magnetic random access memory (MRAM), ferroelectric random access memory (FRAM), phase change memory (PCM), graphene memory, etc. Volatile memory may include random access memory (RAM) or external cache memory, etc. By way of illustration and not limitation, RAM may be in various forms, such as static random access memory (SRAM) or dynamic random access memory (DRAM).

[0154] The databases involved in the various embodiments provided herein may include at least one of a relational database and a non-relational database. Non-relational databases may include, but are not limited to, distributed databases based on blockchains. The processors involved in the various embodiments provided herein may include, but are not limited to, general-purpose processors, central processing units, graphics processing units, digital signal processors, programmable logic units, data processing logic units based on quantum computing, and the like.

[0155] The technical features of the above embodiments can be combined arbitrarily. To make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0156] This document uses specific examples to illustrate the principles and implementation methods of this application. The description of the above examples is only intended to help understand the method and core concept of this application. At the same time, for those skilled in the art, based on the concept of this application, there may be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as limiting this application.

Claims

1. A method for controlling formation flight of ultra-low-orbit electric propulsion satellites, characterized in that: The ultra-low-orbit electric propulsion satellite formation flight control method includes: Establish the absolute orbital dynamics model of ultra-low-orbit electric propulsion satellite under the action of perturbation force; Based on the absolute orbital dynamics model of ultra-low orbit electric propulsion satellites under the action of perturbation force, the relative orbital dynamics model of ultra-low orbit electric propulsion satellite formation under the action of perturbation force is established; the ultra-low orbit electric propulsion satellite formation is composed of several ultra-low orbit electric propulsion satellites; Based on the relative motion mathematical model of the ultra-low-orbit electric propulsion satellite formation under the action of control force and the relative orbit dynamics model under the action of perturbation force, a relative motion mathematical model of the ultra-low-orbit electric propulsion satellite formation flying under the combined action of environmental perturbation force and control force is established; Taking minimum flight time or minimum fuel consumption as the optimization goal, a mathematical model for the orbit optimization control problem of ultra-low-orbit electric propulsion satellite formation flight is established based on the relative motion mathematical model, Hamiltonian function, costate equation, and canonical equation of ultra-low-orbit electric propulsion satellite formation flight under the combined action of environmental perturbation and control force. The Hamiltonian function, costate equation, and canonical equation are composed of costate variables and mass costates. The costate variables correspond to the orbital elements of the ultra-low-orbit electric propulsion satellite; and the mass costate corresponds to the mass of the ultra-low-orbit electric propulsion satellite. Transform the mathematical model of the orbit optimization control problem of ultra-low-orbit electric propulsion satellite formation flying into a two-point boundary value problem; calculate the analytical Jacobian matrix of the two-point boundary value problem; Based on the special point, the initial values ​​of the co-state variables and the mass co-state are calculated analytically; the special point is a point on the nominal trajectory of the virtual center of the ultra-low-orbit electric propulsion satellite formation flight; According to the initial values ​​of co-state variables and mass co-states, two-point boundary value problems and canonical equations, the optimal thrust direction and optimal throttling coefficient at all times during the flight are calculated; the optimal thrust direction and optimal throttling coefficient at all times during the flight together constitute the optimal control scheme for the ultra-low-orbit electric propulsion satellite formation flight orbit.

2. The ultra-low-orbit electric propulsion satellite formation flight control method according to claim 1, characterized in that: The absolute orbital dynamics model of the ultra-low-orbit electric propulsion satellite under the action of perturbation force is established, including: Based on the improved vernal equinox orbital elements, the orbital motion of ultra-low-orbit electric propulsion satellites is described, and a perturbation force model affecting the orbital motion of ultra-low-orbit electric propulsion satellites is established; the perturbation force model includes the mechanical model of the Earth's non-spherical gravity and the mechanical model of atmospheric drag; The mechanical model of the Earth's non-spherical gravity is substituted into the Lagrangian equation of planetary motion, and the mechanical model of atmospheric drag is substituted into the Gaussian perturbation equation to establish the absolute orbital dynamics model of the ultra-low orbit electric propulsion satellite under the action of perturbation force.

3. The ultra-low-orbit electric propulsion satellite formation flight control method according to claim 1, characterized in that: The mathematical model of the orbit optimization control problem of ultra-low-orbit electric propulsion satellite formation flying is transformed into a two-point boundary value problem, specifically including: The optimal thrust direction and optimal throttling coefficient are described using the Pontryagin maximum principle. The mathematical model of the orbit optimization control problem of ultra-low-orbit electric propulsion satellite formation flying is transformed into a two-point boundary value problem.

4. The ultra-low-orbit electric propulsion satellite formation flight control method according to claim 3, characterized in that: The optimal thrust direction and optimal throttling coefficient are expressed as follows: Among them, α * is the optimal thrust direction, M is the thrust control coefficient matrix, λ x is the co-state variable corresponding to the orbital element of the ultra-low orbit electric propulsion satellite, u * is the optimal throttling coefficient, and ρ is the switching function.

5. The ultra-low-orbit electric propulsion satellite formation flight control method according to claim 1, characterized in that: Based on the special points, the initial values ​​of the co-state variables and the initial values ​​of the mass co-state are calculated analytically, including: The points closest to and farthest from the center of the earth on the nominal trajectory of the virtual center of the approximate circle are recorded as perigee and apogee respectively; The point on the virtual center nominal trajectory where the ultra-low-orbit electric propulsion satellite formation performs the most orbit-maintaining maneuvers is recorded as the maneuvering point; The virtual center nominal trajectory of the approximate circle is divided into three segments by perigee, apogee and maneuvering point. The orbital element change rates corresponding to the midpoints of the three segments are calculated respectively, and the point with the largest orbital element change rate is determined as the special point. According to the orbital elements corresponding to the special points, the initial values ​​of the co-state variables and the initial values ​​of the mass co-state are analyzed.

6. The ultra-low-orbit electric propulsion satellite formation flight control method according to claim 1, characterized in that: Based on the initial values ​​of co-state variables and mass co-states, two-point boundary value problems, and canonical equations, the optimal thrust direction and optimal throttling coefficient at all times during flight are calculated, including: According to the initial values ​​of the co-state variables and the initial values ​​of the mass co-state, the optimal thrust direction and the optimal throttling coefficient at the initial moment are calculated; Substitute the initial values ​​of the co-state variables and the mass co-state into the canonical equation to obtain the co-state variables and mass co-state at all times during the flight process; According to the co-state variables and mass co-state at all times during the flight, the optimal thrust direction and optimal throttling coefficient at all times during the flight are calculated.

7. The ultra-low-orbit electric propulsion satellite formation flight control method according to claim 2, characterized in that: The improved vernal equinox orbit elements include the orbit semi-major axis, the x-component of the eccentricity vector, the y-component of the eccentricity vector, the x-component of the orbital inclination, the y-component of the orbital inclination and the mean longitude.

8. An orbit control device for an ultra-low-orbit electric propulsion satellite formation, characterized in that: The ultra-low-orbit electric propulsion satellite formation orbit control device includes: The absolute orbit dynamics model establishment module is used to establish the absolute orbit dynamics model of the ultra-low orbit electric propulsion satellite under the action of perturbation force; The module for establishing the absolute orbital dynamics model of ultra-low-orbit electric propulsion satellites is used to: establish the relative orbital dynamics model of an ultra-low-orbit electric propulsion satellite formation under the action of perturbation force based on the absolute orbital dynamics model of the ultra-low-orbit electric propulsion satellites under the action of perturbation force; an ultra-low-orbit electric propulsion satellite formation is composed of several ultra-low-orbit electric propulsion satellites; The relative motion mathematical model establishment module is used to: establish the relative motion mathematical model of the ultra-low-orbit electric propulsion satellite formation flying under the combined action of environmental perturbation force and control force based on the relative motion mathematical model of the ultra-low-orbit electric propulsion satellite formation under the action of control force and the relative orbit dynamics model under the action of perturbation force; The mathematical model building module for the flight orbit optimization control problem is used to: establish a mathematical model for the orbit optimization control problem of ultra-low-orbit electric propulsion satellite formation flight based on the relative motion mathematical model, Hamiltonian function, costate equation and canonical equation of the ultra-low-orbit electric propulsion satellite formation flight under the combined action of environmental perturbation and control force, with minimum flight time or minimum fuel consumption as the optimization goal; the Hamiltonian function, costate equation and canonical equation are composed of costate variables and mass costates; the costate variables correspond to the orbital elements of the ultra-low-orbit electric propulsion satellite; and the mass costate corresponds to the mass of the ultra-low-orbit electric propulsion satellite; The analytic Jacobian matrix calculation module is used to: transform the mathematical model of the orbit optimization control problem of ultra-low-orbit electric propulsion satellite formation flying into a two-point boundary value problem; and calculate the analytic Jacobian matrix of the two-point boundary value problem; The co-state initial value analysis module is used to: analytically calculate the initial values ​​of co-state variables and mass co-states based on a special point; the special point is a point on the nominal trajectory of the virtual center of the ultra-low-orbit electric propulsion satellite formation flight; The optimal control scheme calculation module is used to calculate the optimal thrust direction and optimal throttling coefficient at all times during the flight according to the initial values ​​of the co-state variables and the mass co-state, the two-point boundary value problem and the canonical equation; the optimal thrust direction and the optimal throttling coefficient at all times during the flight together constitute the optimal control scheme for the ultra-low-orbit electric propulsion satellite formation flight orbit.

9. A computer device comprising: A memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the ultra-low-orbit electric propulsion satellite formation flight control method according to any one of claims 1 to 7.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the ultra-low-orbit electric propulsion satellite formation flight control method described in any one of claims 1 to 7 is implemented.

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