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

Abstract

The invention discloses an ultra-low orbit electric propulsion satellite formation flight control method, device and equipment and a medium, and relates to the technical field of satellite flight control, and the method comprises the steps: building a relative orbit dynamics model under the action of perturbative force based on an absolute orbit dynamics model of an ultra-low orbit electric propulsion satellite under the action of perturbative force, establishing a relative motion mathematical model under the combined action of the two forces by combining the relative motion mathematical model under the action of the control force; then, a Hamiltonian function, a co-state equation and a regular equation are combined, and a mathematical model of the ultra-low orbit electric propulsion satellite formation flight orbit optimization control problem is established; converting the mathematical model of the optimization control problem into a two-point boundary value problem; based on the special points, analyzing and calculating a co-state variable initial value and a quality co-state initial value; according to the co-state variable initial value, the quality co-state initial value and the two-point boundary value problem mathematical model, the optimal control scheme is determined, and the solution success rate, the convergence speed and the control precision of optimization control can be improved.

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 development of space, but also show great potential in areas such as global communications and Internet coverage. At the same time, electric propulsion technology, as an emerging field technology, has core advantages in saving space flight fuel consumption and reducing flight mission costs. Therefore, the orbit control method of ultra-low-orbit electric propulsion satellite formation flight has important strategic significance in modern aerospace technology. Through formation flying, satellites can achieve high-precision earth observation, space environment monitoring, astronomical observation and other tasks.

[0003] Although the control technology of ultra-low-orbit electric propulsion satellite formation flight has made significant progress in recent years, it still faces a series of challenges and difficulties. First, due to the large perturbation forces such as atmospheric drag in the ultra-low-orbit environment, satellites need to frequently maintain their orbits, which places higher precision and efficiency requirements on the electric propulsion system. Secondly, formation flight requires satellites to maintain precise relative positions, which requires not only high-precision orbit maintenance, but also autonomous and efficient decision-making and solving capabilities on orbit 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 respond quickly to ground commands or autonomous decisions to adjust orbits and attitude control. Summary of the invention

[0004] The purpose of this application is to provide a method, device, equipment and medium for ultra-low-orbit electric propulsion satellite formation flight control, 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 an ultra-low-orbit electric propulsion satellite formation flight, 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 orbital dynamics model under the action of perturbation force, a relative motion mathematical model of the ultra-low-orbit electric propulsion satellite formation flight under the combined action of environmental perturbation force and control force is established;

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

[0011] The mathematical model of the orbit optimization control problem of ultra-low-orbit electric propulsion satellite formation flight is transformed into a two-point boundary value problem; the analytical Jacobian matrix of the two-point boundary value problem is calculated;

[0012] Based on the special point, the initial values ​​of the co-state variables and the initial values ​​of 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 costate variables and mass costates, two-point boundary value problems and canonical equations, 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.

[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 building module is used to: build 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 the 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; the 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 flight 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: take the minimum flight time or the minimum fuel consumption as the optimization target, and establish the mathematical model of the orbit optimization control problem of the ultra-low orbit electric propulsion satellite formation flight based on the relative motion mathematical model, Hamiltonian function, co-state equation and canonical equation of the ultra-low orbit electric propulsion satellite formation flight under the joint action of environmental perturbation force and control force; the Hamiltonian function, co-state equation and canonical equation are composed of co-state variables and mass co-states; the co-state variables correspond to the orbital elements of the ultra-low orbit electric propulsion satellite; the mass co-state corresponds to the mass of the ultra-low orbit electric propulsion satellite;

[0019] The analytical 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 flight into a two-point boundary value problem; calculate the analytical Jacobian matrix of the two-point boundary value problem;

[0020] The co-state initial value analysis module is used to: analyze and calculate the initial values ​​of co-state variables and mass co-state initial values ​​based on special points; 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 the optimal throttling coefficient at all times during the flight according to the initial values ​​of the co-state variables and the initial values ​​of 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, including: 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 having a computer program stored thereon, which, when executed by a processor, 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 costate variables and the initial values ​​of the mass costate 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 costate variables and the initial values ​​of the mass costate, 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 for 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 drawings required for use in the embodiments will be briefly introduced below. 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 paying creative work.

[0027] Figure 1 A schematic diagram of a flow chart 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 the present application and the Gaussian pseudospectral method provided in an embodiment of the present 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 pseudo-spectral method provided in an 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 the present application to clearly and completely describe the technical solutions in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, not all of the embodiments. Based on the embodiments in the present application, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of this application.

[0032] In order to make the above-mentioned objects, 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 ultra-low-orbit electric propulsion satellite formation flight is provided, and the method includes the following steps 101 to 107. Among them:

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

[0037] Step 104: Taking the minimum flight time or the minimum fuel consumption as the optimization target, a mathematical model of 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 effect of environmental perturbation force 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; the mass costate corresponds to the mass of the ultra-low orbit electric propulsion satellite;

[0038] Step 105: transforming the mathematical model of the orbit optimization control problem of the ultra-low-orbit electric propulsion satellite formation flight into a two-point boundary value problem; calculating the analytical Jacobian matrix of the two-point boundary value problem;

[0039] Step 106: Based on the special point, the initial value of the co-state variable and the initial value 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: According to the initial values ​​of the co-state variables and the initial values ​​of 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 implementation of the above steps 101 to 108 has the main advantages of high optimization success rate, fast convergence speed and high control accuracy compared to the existing method (Gaussian pseudospectral method). The analytical Jacobian matrix obtained in step 105 can greatly improve the speed of high-dimensional and high-order vector operations; the initial value of the co-state obtained in step 106 is the initial value with the best performance, which ensures the 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 resistance;

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

[0045] The improved vernal equinox orbit element [a,e x ,e x ,i x ,i y ,L] describes the orbital motion of the 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 orbit inclination, the y component of the orbit inclination, and the mean longitude. The improved vernal equinox orbit element can better handle the singularity problem caused by small inclination and small eccentricity. The mapping relationship between the improved vernal equinox orbit element and the classical orbit element [a,e,i,ω,Ω,M] is:

[0046]

[0047] Where [a, e, i, ω, Ω, M] refers 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 in 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 of the satellite from the center of the earth, μ is the gravitational coefficient of the earth, λ is the satellite's longitude, and 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 field 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 direction perpendicular to the velocity and in the normal direction to the track surface, respectively, C D is the drag coefficient, m is the satellite mass, ρ is the atmospheric density, n is the satellite average angular velocity, u is the latitude angle, 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, and the subscripts T, N and W represent the track, normal and lateral directions, respectively.

[0055] Affected by the environmental perturbation force, the changes of 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. Substituting the mechanical model of atmospheric drag shown in formula (3) 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 is 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 non-spherical perturbation matrix D of the earth 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, then the center point of the ultra-low-orbit electric propulsion satellite formation is The calculation method is as follows:

[0066]

[0067] In the formula, 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 to 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 effect 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 ultra-low-orbit electric propulsion satellite formation flight optimization control is usually time optimization or fuel optimization, so the performance index J can be uniformly expressed as shown in the following form:

[0080]

[0081] Among them, t0 and t f are the initial time and the terminal time respectively; is the Lagrangian function, for the minimum time problem (MT), we have 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 variable is:

[0086]

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

[0088]

[0089] In the formula, represents the rate of change of the covariate variable corresponding to x; when i in the subscript is 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: 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 transformed 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 the ultra-low-orbit electric propulsion satellite formation flight 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 covariate 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 free flight time situation, while “t f fixed” and “H(t f )free” matches and corresponds to the fixed flight time situation.

[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 representing the shooting function for 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 to the orbital element.

[0105] Based on the two-point boundary value problem in step 105, an analytical Jacobian matrix calculation technique for 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 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 analytic Jacobian matrix to speed up the iterative integration efficiency and improve the accuracy and robustness of the calculation process of the shooting function. By introducing the homology parameter ε, the analytic Jacobian matrix J corresponding to formula (18) is as follows:

[0107]

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

[0109] Considering the homology 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 transfer matrix of y; for The state transfer matrix of ; I represents the identity matrix.

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

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

[0116] Because the possible range of the initial value of the co-state is large, but the effective initial value radius of the co-state that makes the optimization problem converge is very small, how to effectively guess the initial value of the co-state has always been a difficulty and challenge in solving the two-point boundary value problem. The analytical guessing technology of the initial value of the co-state proposed in this application includes two parts: one is to determine the special points, and the other is to solve the analytical initial value of the co-state based on the special points.

[0117] The special point is defined as a point on the nominal trajectory of the virtual center of the satellite formation flight. The definition of the nominal trajectory of the virtual center of the satellite formation flight 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] Then the above step 106 specifically includes: step 501: determining the special point; step 502: analyzing the initial value of the co-state variable and the initial value 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: Record the points closest to and farthest from the center of the earth on the virtual center nominal trajectory of the approximate circle 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 rates corresponding to the midpoints of the three segments are 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 according to the co-state variables and mass co-states 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, and then substitute them into formulas (15) and (16) to calculate the optimal thrust direction α at all times during the flight. * 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 plan 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 as follows:

[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 pseudo-spectral 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 the 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 pseudo-spectral method, and the average calculation time is one-fifth of the Gaussian pseudo-spectral method, that is, the algorithm converges quickly. Figure 3 The horizontal and vertical axes are respectively the example number and the control accuracy. 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 plan 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 mathematical model of the orbit optimization control problem of ultra-low orbit electric propulsion satellite formation flight is established by combining the Hamiltonian function, the comorphic equation and the canonical equation; 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 comorphic variables and the mass comorphic initial values ​​are analytically calculated; according to the initial values ​​of the comorphic variables and the mass comorphic initial values, the two-point boundary value problem and the canonical equation, the optimal control scheme is determined.

[0139] Based on the same inventive concept, the embodiment of the present application also provides an ultra-low-orbit electric propulsion satellite formation orbit control device for implementing the ultra-low-orbit electric propulsion satellite formation flight control method involved above. The implementation scheme for solving the problem provided by the device is similar to the implementation scheme recorded in the above method, so the specific limitations in one or more ultra-low-orbit electric propulsion satellite formation orbit control device embodiments provided below can refer to the limitations of the ultra-low-orbit electric propulsion satellite formation flight control method above, and will not be repeated here.

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

[0141] The absolute orbit dynamics model building module is used to: build 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 the 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; the 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 flight 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: take the minimum flight time or the minimum fuel consumption as the optimization target, and establish the mathematical model of the orbit optimization control problem of the ultra-low orbit electric propulsion satellite formation flight based on the relative motion mathematical model, Hamiltonian function, co-state equation and canonical equation of the ultra-low orbit electric propulsion satellite formation flight under the joint action of environmental perturbation force and control force; the Hamiltonian function, co-state equation and canonical equation are composed of co-state variables and mass co-states; the co-state variables correspond to the orbital elements of the ultra-low orbit electric propulsion satellite; the mass co-state corresponds to the mass of the ultra-low orbit electric propulsion satellite;

[0145] The analytical 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 flight into a two-point boundary value problem; calculate the analytical Jacobian matrix of the two-point boundary value problem;

[0146] The co-state initial value analysis module is used to: analyze and calculate the initial values ​​of co-state variables and mass co-state initial values ​​based on special points; 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 the optimal throttling coefficient at all times during the flight according to the initial values ​​of the co-state variables and the initial values ​​of 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, referred to as I / O) and a communication interface. Among them, the processor, the memory and the input / output interface are connected through a system bus, and the communication interface is connected to the system bus through the input / output interface. Among them, 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 the 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 those shown in the figure, or combine certain components, or have a different arrangement of components.

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

[0151] In an exemplary embodiment, a computer-readable storage medium is provided, storing a computer program, and when the computer program is executed by a processor, the steps in the above 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 of ordinary skill in the art can understand that all or part of the processes in the above-mentioned embodiment methods can be completed 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 the memory, database or other medium used in the embodiments provided in the present application can include at least one of non-volatile and volatile memory. Non-volatile memory can 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 can 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 database involved in each embodiment provided in this application may include at least one of a relational database and a non-relational database. The non-relational database may include a distributed database based on blockchain, etc., but is not limited thereto. The processor involved in each embodiment provided in this application may be a general-purpose processor, a central processing unit, a graphics processor, a digital signal processor, a programmable logic device, a data processing logic device based on quantum computing, etc., but is not limited thereto.

[0155] The technical features of the above embodiments may 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 article uses specific examples to illustrate the principles and implementation methods of this application. The description of the above embodiments is only used to help understand the method and core ideas of this application. At the same time, for those skilled in the art, according to the ideas of this application, there will 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 comprises: 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 orbital dynamics model under the action of perturbation force, a relative motion mathematical model of the ultra-low-orbit electric propulsion satellite formation flight under the combined action of environmental perturbation force and control force is established; Taking the minimum flight time or the minimum fuel consumption as the optimization goal, based on the relative motion mathematical model of ultra-low orbit electric propulsion satellite formation flight under the joint action of environmental perturbation and control force, Hamiltonian function, comorphic equation and canonical equation, a mathematical model of the orbit optimization control problem of ultra-low orbit electric propulsion satellite formation flight is established; Hamiltonian function, comorphic equation and canonical equation are composed of comorphic variables and mass comorphic; comorphic variables correspond to the orbital elements of ultra-low orbit electric propulsion satellite; mass comorphic corresponds to the mass of ultra-low orbit electric propulsion satellite; The mathematical model of the orbit optimization control problem of ultra-low-orbit electric propulsion satellite formation flight is transformed into a two-point boundary value problem; the analytical Jacobian matrix of the two-point boundary value problem is calculated; Based on the special point, the initial values ​​of the co-state variables and the initial values ​​of 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 costate variables and mass costates, two-point boundary value problems and canonical equations, 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.

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 the ultra-low-orbit electric propulsion satellite is described, and a perturbation force model affecting the orbital motion of the ultra-low-orbit electric propulsion satellite is established; the perturbation force model includes the mechanical model of the earth's non-spherical gravity and the mechanical model of atmospheric resistance; 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 flight is transformed into a two-point boundary value problem, including: The optimal thrust direction and optimal throttling coefficient are described by using the Pontryagin maximum principle. The mathematical model of the orbit optimization control problem of ultra-low-orbit electric propulsion satellite formation flight 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 covariate 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, and 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 a special point; According to the orbital elements corresponding to the special points, the initial values ​​of the costate variables and the initial values ​​of the mass costate are analyzed.

6. The ultra-low-orbit electric propulsion satellite formation flight control method according to claim 1, characterized in that: According to the initial values ​​of the co-state variables and the mass co-state initial values, 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, 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 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; According to the co-state variables and mass co-states at all times during the flight, the optimal thrust direction and the 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 orbit inclination, the y-component of the orbit 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 comprises: The absolute orbit dynamics model building module is used to: build 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 the 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; the 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 flight 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: take the minimum flight time or the minimum fuel consumption as the optimization target, and establish the mathematical model of the orbit optimization control problem of the ultra-low orbit electric propulsion satellite formation flight based on the relative motion mathematical model, Hamiltonian function, co-state equation and canonical equation of the ultra-low orbit electric propulsion satellite formation flight under the joint action of environmental perturbation force and control force; the Hamiltonian function, co-state equation and canonical equation are composed of co-state variables and mass co-states; the co-state variables correspond to the orbital elements of the ultra-low orbit electric propulsion satellite; the mass co-state corresponds to the mass of the ultra-low orbit electric propulsion satellite; The analytical 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 flight into a two-point boundary value problem; calculate the analytical Jacobian matrix of the two-point boundary value problem; The co-state initial value analysis module is used to: analyze and calculate the initial values ​​of co-state variables and mass co-state initial values ​​based on special points; 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 the optimal throttling coefficient at all times during the flight according to the initial values ​​of the co-state variables and the initial values ​​of 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 described in 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 1-7 is implemented.

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