Low fuel consumption low earth orbit satellite orbit recovery method

Abstract

The application discloses a low-fuel-consumption low-orbit satellite orbit recovery method, and belongs to the technical field of satellite orbit recovery.The application aims at the problem that the existing orbit maneuver method needs frequent orbit adjustment, which is not conducive to the long-term operation of the satellite in orbit with low fuel consumption.The application comprises the following steps: establishing an orbit dynamics equation considering the earth oblateness perturbation; using a semi-analytical method to solve the initial orbit elements of a low-fuel-consumption transition orbit according to the current orbit and the target orbit of the satellite; simultaneously, combining the orbit dynamics equation, calculating the drift time required for the satellite to drift on the transition orbit by using the earth oblateness perturbation without fuel consumption and the final orbit elements of the transition orbit at the end of the transition; and based on the Pontryagin maximum principle, calculating the optimal control rate sequence of the thruster fuel of the satellite from the current orbit to the transition orbit and from the transition orbit to the target orbit, and controlling the thruster to realize the orbit recovery of the satellite.The application is used for satellite orbit recovery.

Description

Technical Field

[0001] This invention relates to a low-fuel-consumption method for restoring the orbit of low-Earth orbit satellites, belonging to the field of satellite orbit restoration technology. Background Technology

[0002] In the aerospace field, orbit restoration refers to adjusting, correcting, or rectifying the orbits of satellites, spacecraft, or other space vehicles through the action of actuators (usually thrusters). With technological advancements and increased satellite lifespans, the presence of various disturbances makes orbital deviations more pronounced over their lifespan. Simultaneously, the increasing amount of low-Earth orbit debris increases the risk of orbital collisions between low-Earth orbit satellites and probes, making satellite orbit restoration technology increasingly important.

[0003] In recent years, the development of low-Earth orbit (LEO) satellites has shown trends towards clustering, miniaturization, longer lifespan, and lower cost. Electric thrusters, due to their high specific impulse and low fuel consumption, have gradually replaced solid and liquid rocket engines in LEO satellites, becoming the preferred choice for orbital maneuvering engines. Existing orbital maneuvering methods for continuously thrust spacecraft include direct, indirect, and hybrid methods. To facilitate convergence of the iterative process, simplified spacecraft dynamics models are typically chosen, treating the effects of Earth's oblateness and solar radiation pressure as external disturbances. This necessitates frequent orbital adjustments for LEO spacecraft throughout their orbital lifespan, which is detrimental to long-term, low-fuel-consumption operation. Summary of the Invention

[0004] To address the problem that existing orbital maneuvering methods require frequent orbital adjustments, which are detrimental to the long-term on-orbit operation of satellites with low fuel consumption, this invention provides a low-fuel-consumption low-Earth orbit satellite orbit recovery method.

[0005] The present invention provides a low-fuel-consumption method for low-Earth orbit satellite orbit recovery, comprising:

[0006] Establish orbital dynamic equations that take into account the perturbation of Earth's oblateness;

[0007] The initial orbital elements of the low-fuel-consumption transition orbit are solved using a semi-analytical method based on the satellite's current orbit and target orbit. Simultaneously, combined with the orbital dynamics equations, the drift time required for the satellite to drift without fuel consumption using Earth's oblateness perturbation in the transition orbit and the final orbital elements of the transition orbit at the end of the transition are calculated. The satellite's current orbit is the perturbed orbit.

[0008] Based on the Pontryagin maximum principle, the optimal thruster fuel control rate sequence for the satellite from its current orbit to the transition orbit and from the transition orbit to the target orbit are calculated by combining the performance index function, Hamiltonian function and switching control function. The thrusters are then controlled to achieve orbit recovery for the satellite.

[0009] According to the low-fuel-consumption low-Earth orbit satellite orbit recovery method of the present invention, the orbital dynamic equation considering the perturbation of Earth's oblateness is as follows:

[0010]

[0011] In the formula, V is the satellite velocity, f is the thrust acceleration, β is the thrust direction angle, representing the angle between the satellite thrust vector and the velocity vector, I is the satellite orbital inclination, Ω is the right ascension of the satellite's ascending node, and k is the precession constant of the right ascension of the ascending node, k = 1.0425 × 10⁻⁶. -33 s 6 / m 7 .

[0012] According to the low-fuel-consumption low-Earth orbit satellite orbit recovery method of the present invention, the current orbital elements and the target orbital elements of the satellite are set as follows:

[0013] Assume the satellite's maneuvering process takes place over a period of time from t0 to t1. f Where t0 is the initial orbit recovery time of the satellite in its current orbit, t f The moment the satellite returns to its target orbit; then the corresponding change in orbital elements is V0,I0,Ω0→V f ,I f ,Ω f Where V0 is the satellite velocity in the current orbit, I0 is the satellite inclination in the current orbit, and Ω0 is the right ascension of the satellite's ascending node in the current orbit; V f For the satellite velocity in the target orbit, I f Ω represents the orbital inclination of the satellite in the target orbit. f The right ascension of the ascending node of the satellite in the target orbit.

[0014] The satellite's maneuvering process during the time period t0→t f A satellite orbits in a given orbital period. The process of transferring the satellite from its current orbit to its target orbit within each orbital period is divided into three phases: a primary propulsion phase, a drift phase, and a secondary propulsion phase. The primary propulsion phase's time is denoted as t1, the drift phase's time as t2, and the secondary propulsion phase's time as t3. Then:

[0015] t f = t1 + t2 + t3.

[0016] According to the low-fuel-consumption low-Earth orbit satellite orbit recovery method of the present invention, the drift time t2 is calculated as follows:

[0017] Calculate the precession velocity of the right ascension of the ascending node of the initial satellite in the transition orbit. Precession velocity of the right ascension of the ascending node of the satellite in the target orbit Precession speed difference

[0018]

[0019] In the formula V d For the initial satellite velocity in the transition orbit, I d The initial satellite orbital inclination for the transition orbit;

[0020] The drift time t2 is:

[0021]

[0022] In the formula Ω d Ω is the right ascension of the ascending node of the initial satellite in the transition orbit. d =Ω0.

[0023] According to the low-fuel-consumption low-Earth orbit satellite orbit recovery method of the present invention, the calculation method for the initial orbital elements of the transition orbit is as follows:

[0024] Establish an optimization model for maintaining the optimal fuel trajectory:

[0025]

[0026] stΩ(t f )=Ω f

[0027] In the formula m f This represents the total propellant consumption for both propulsion phases.

[0028] Let λ V,1 Let λ be the satellite velocity co-state variable during the first propulsion phase. I,1 Let λ be the co-state variable of the satellite's orbital inclination during a single propulsion phase. V,3 λ is the satellite velocity co-state variable during the second propulsion phase. I,3 Let the satellite orbital inclination be the co-state variable during the second propulsion phase. Then, the gradient of the total propellant consumption function is expressed as:

[0029]

[0030] Select the initial satellite orbit inclination I for the transition orbit d As a free variable, combined with formula (2), formula (4) can be transformed into:

[0031]

[0032] In the formula λ I,2 Let λ be the co-state variable of the satellite's orbital inclination during the drift segment. V,2 For the satellite velocity co-state variable during the drift segment;

[0033] Solve for λ using the Pontryagin minimum principle in equation (5). V,1 , λ I,1 , λ V,2 and λ I,2 Then, make the gradient of formula (5) zero, and solve for the initial satellite orbit inclination I of the transition orbit. d .

[0034] According to the low-fuel-consumption low-Earth orbit satellite orbit recovery method of the present invention, the method for determining the final orbital elements of the transition orbit at the end of the transition is as follows:

[0035] At time t0, the satellite orbital elements V0, I0, Ω0, which begin at the start of time period t1, are used for one advance phase. At the end of time period t1, the initial orbital elements V0, I0, Ω0 of the transition orbit are obtained. d ,I d ,Ω d After the rest period caused by the J2 term, the right ascension of the ascending node precesses, and the final orbital element V of the transition orbit is obtained at the end of time interval t2. d ,I d ,Ω f ;

[0036] Then the drift time t2 is calculated.

[0037] According to the low-fuel-consumption low-Earth orbit satellite orbit recovery method of the present invention, the performance index function is selected as follows:

[0038]

[0039] T max Where c is the maximum thrust of the thruster, and c is the thrust constant, c = I sp ·g0,I sp denoted by thrust specific impulse, g0 is the gravitational acceleration at the Earth's surface, u is the thrust fuel control rate sequence, and t is time.

[0040] According to the low-fuel-consumption low-Earth orbit satellite orbit recovery method of the present invention, the Hamiltonian function H is selected as follows:

[0041]

[0042] In the formula λ V Let λ represent the satellite velocity costate variable. I λ represents the co-state variable of satellite orbital inclination. Ω Let λ be the right ascension covariance of the satellite's ascending node. V ,λ I ,λ Ω The equation satisfies the following:

[0043]

[0044] According to the low-fuel-consumption low-Earth orbit satellite orbit recovery method of the present invention, the switch control function S is:

[0045]

[0046] The thruster fuel control rate sequence u is made subject to the switching control function S, and is selected as follows:

[0047]

[0048] In the formula u * (t) represents the optimal control rate sequence of the thruster fuel control rate sequence u under optimal fuel conditions;

[0049] In the optimal control rate sequence u * Under the condition of (t), the optimal thrust direction angle sequence β is obtained. * (t) is:

[0050]

[0051] According to the low-fuel-consumption low-Earth orbit satellite orbit recovery method of the present invention, the optimal control law sequence u * (t) and the optimal thrust direction angle sequence β * The solution method for (t) is as follows:

[0052] Set the target shooting variable z = [λ] V (t0),λ I (t0),λ Ω (t0),λ m (t0),t f ] T In the formula λ m Let m be the costate variable corresponding to the satellite mass m; then combine it with the initial state variables [V(t0), I(t0), Ω(t0), m(t0)]. T Solve for the following boundary conditions:

[0053]

[0054] In the formula t 10 Let t be the end time of time interval t1. 20 This is the end time of time interval t2;

[0055] The boundary conditions are solved using a nonlinear solver to obtain the optimal control rate sequence u. * (t) and the optimal thrust direction angle sequence β * (t); Calculate the Jacobian matrix at each step during the solution process to accelerate the convergence of the solution process.

[0056] The beneficial effects of this invention are as follows: The method of this invention can improve the efficiency of satellite orbit adjustment. It achieves low-fuel-consumption orbital maneuvers by selecting a suitable transition orbit and combining a semi-analytical method with the Pontryagin maximum principle, utilizing Earth's oblateness perturbation. This method can reduce the fuel costs of long-life, low-Earth orbit satellites and can converge quickly through simulation analysis. It is expected to bring significant improvements to the sustainability and economy of satellite operation and space missions. Attached Figure Description

[0057] Figure 1 This is a flowchart of the low-fuel-consumption low-Earth orbit satellite orbit recovery method described in this invention;

[0058] Figure 2 This is a schematic diagram of the satellite's orbital plane; in the diagram... Using the J2000 geocentric fixed coordinate system, The satellite's center of mass vector;

[0059] Figure 3 This is a schematic diagram of the thrust action sequence; f in the diagram max This represents the maximum thrust acceleration.

[0060] Figure 4 This is a schematic diagram of the reorganization of the thrust action sequence;

[0061] Figure 5 This is a schematic diagram showing the change of satellite velocity over time under a thrust sequence.

[0062] Figure 6 This is a schematic diagram showing the change of satellite velocity over time after the recombination of the thrust action sequence.

[0063] Figure 7 This is a schematic diagram showing the change of satellite orbital inclination over time under a thrust sequence.

[0064] Figure 8 This is a schematic diagram showing the change of satellite orbital inclination over time after the thrust sequence is recombined;

[0065] Figure 9 This is a schematic diagram showing the change of the right ascension of the ascending node of a satellite over time under a thrust sequence.

[0066] Figure 10 This is a schematic diagram showing the change of the right ascension of the satellite's ascending node over time after the thrust sequence is recombined. Detailed Implementation

[0067] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0068] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other.

[0069] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, but this is not intended to limit the scope of the invention.

[0070] Specific Implementation Method 1: Combination Figure 1 As shown, this invention provides a low-fuel-consumption method for restoring the orbit of a low-Earth orbit satellite. Due to factors such as the Earth's non-spherical perturbation, solar radiation pressure, Earth's magnetic field gravity, and the drag of a thin atmosphere, the orbit of a low-Earth orbit satellite may deviate from its intended orbit. To restore the satellite to its original orbit, the orbit restoration method includes:

[0071] An orbital dynamics equation considering the perturbation of Earth's oblateness is established; this orbital dynamics equation can accurately describe the motion of a spacecraft in a complex gravitational field.

[0072] Then, the orbital parameters are calculated: the initial orbital elements of the low-fuel-consumption transition orbit are solved using a semi-analytical method based on the satellite's current orbit and target orbit; at the same time, combined with the orbital dynamics equations, the drift time required for the satellite to drift without fuel consumption using the Earth's oblateness perturbation in the transition orbit and the final orbital elements of the transition orbit at the end of the transition are calculated; the satellite's current orbit is the orbit after the perturbation.

[0073] Optimal control rate determination: Based on the Pontryagin maximum principle, and combining the performance index function, Hamiltonian function, and switching control function, the optimal control rate sequence for thruster fuel is calculated for the satellite from its current orbit to the transition orbit, and the optimal control rate sequence for thruster fuel is calculated for the satellite from the transition orbit to the target orbit. The thrusters are then controlled to achieve orbit restoration of the satellite. This ensures that orbit correction is completed using the minimum amount of fuel.

[0074] This implementation combines the propulsion process of the thruster with the fuel-free drift process on the transition track calculated by the semi-analytical method, together forming a complete track recovery process.

[0075] The orbital dynamics equations of this embodiment are derived from the Gaussian dynamics equations, considering the following four assumptions:

[0076] (1) Due to the use of a small thrust engine, the evolution of the orbital elements is taken as the average value over one period;

[0077] (2) During the orbital maneuver, the transfer track is always circular;

[0078] (3) During the orbital transfer process, the engine thrust is constant;

[0079] (4) The engine thrust vector is perpendicular to the satellite position vector, always forming an angle β with the orbital plane throughout one orbital period, and its sign changes at the nodes. This ensures that the right ascension of the ascending node does not change within one orbital period, while the rate of change of the orbital inclination is maximized.

[0080] Combination Figure 2 As shown, the orbital dynamics equation considering the Earth's oblateness perturbation is:

[0081]

[0082] In the formula, V is the satellite velocity, f is the thrust acceleration, β is the thrust direction angle, representing the angle between the satellite thrust vector and the velocity vector, I is the satellite orbital inclination, Ω is the right ascension of the satellite's ascending node, and k is the precession constant of the right ascension of the ascending node, k = 1.0425 × 10⁻⁶. -33 s 6 / m 7 .

[0083] Figure 2 In the diagram, the pericentric angular distance φ represents the angle between the position vector and the vernal equinox radius vector, and the "node" is the point corresponding to φ = ±90°. To ensure that the orbital inclination changes consistently within one period, the β angle is selected according to the following strategy: when φ∈[0,90°)∪(270°,360°], the "+β" angle is selected; when φ∈[90°,270°], the "-β" angle is selected.

[0084] This implementation uses the concept of orbital averaging, that is, under the assumption of small thrust, it is assumed that the semi-major axis a, eccentricity e, right ascension of the ascending node Ω, orbital inclination I, and pericentric angular distance φ remain constant within one orbital period, while the true pericentric angle w changes in different periods.

[0085] After considering the influence of Earth's oblateness on the gravitational field model, the orbital dynamics equations need to be corrected by introducing a first-order band harmonic term (J2 term). The J2 term does not produce long-period oscillations of the semi-major axis, eccentricity, and orbital inclination, but it will produce a precession velocity of the right ascension of the ascending node that depends on the semi-major axis and orbital inclination.

[0086] According to the orbital dynamics equations, the control variables of the system are the thrust acceleration f and the thrust direction angle β.

[0087] The change in the right ascension Ω of the ascending node is passively accomplished due to the thrust of the J2 term, and the precession rate... Influenced by the semi-major axis and orbital inclination, the orbital dynamics equations reflect the impact of Earth's oblateness perturbation on spacecraft orbital dynamics. The core idea of ​​low-energy transition orbit design is to utilize dynamic effects to achieve energy-free drift and minimum total energy transfer of the spacecraft in the transition orbit.

[0088] The thrust directly changes the semi-major axis a and the orbital inclination I, and the rate of change of both is not affected by the right ascension of the ascending node.

[0089] Combination Figures 3 to 10 As shown, the current orbital elements and target orbital elements of the satellite are set as follows:

[0090] Assume the satellite's maneuvering process takes place over a period of time from t0 to t1. f Where t0 is the initial orbit recovery time of the satellite in its current orbit, t f The moment the satellite returns to its target orbit; then the corresponding change in orbital elements is V0,I0,Ω0→V f ,I f ,Ω f Where V0 is the satellite velocity in the current orbit, I0 is the satellite inclination in the current orbit, and Ω0 is the right ascension of the satellite's ascending node in the current orbit; V f For the satellite velocity in the target orbit, I f Ω represents the orbital inclination of the satellite in the target orbit. f The right ascension of the ascending node of the satellite in the target orbit.

[0091] The satellite's maneuvering process during the time period t0→t f A satellite orbits in a given orbital period. The process of transferring the satellite from its current orbit to its target orbit within each orbital period is divided into three phases: a primary propulsion phase, a drift phase, and a secondary propulsion phase. The primary propulsion phase's time is denoted as t1, the drift phase's time as t2, and the secondary propulsion phase's time as t3. Then:

[0092] t f = t1 + t2 + t3.

[0093] Pontryagin's maximum principle is an important principle in mathematics. It concerns how to choose control variables in a given dynamic system for a performance index (usually expressed as an integral objective function) that needs to be minimized or maximized, while satisfying certain constraints (such as dynamic equations, initial and termination conditions).

[0094] Pontryagin's maximum principle can reduce the optimization problem of low-thrust transfer trajectory to a two-point boundary value problem that satisfies the first-order necessary condition, and then use a nonlinear solver to obtain the optimal control.

[0095] The control strategy using the Pontryagin maximum principle requires that the thrust direction in any state be selected in accordance with the objective of minimizing the index function, and the thrust magnitude be selected in a way that makes the objective function reach an extreme value.

[0096] Solving the fuel optimization problem using the Pontryagin control strategy typically involves a switch control structure with discontinuous control functions, requiring a significant amount of fuel to maintain the low-Earth orbit satellite's orbit.

[0097] Analyzing the changes in orbital elements over time under optimal control reveals that velocity and orbital inclination only change during the thrust phase. Therefore, it is considered to reorganize the original thrust sequence, connecting the various thrust segments into a longer time period while maintaining the total thrust time constant, during which the thrusters operate continuously.

[0098] The recombined thrust sequence divides the complete orbital transfer process into three stages: thrust (t1) - rest (t2) - thrust (t3). Let:

[0099]

[0100] Here, k∈(1,n) are the stage division points, meaning that the advancement process of the first [1,k] sequence is concentrated in the time interval t1, and the advancement process of the second [k+1,n] sequence is concentrated in the time interval t3. The intermediate time interval t2 is a thrustless rest period. During this time interval, the satellite's velocity and orbital inclination remain unchanged, while the right ascension of the ascending node only undergoes a constant-rate drift under the influence of the Earth's J2 term.

[0101] The terminal-time-free near-circular orbit transfer problem can be transformed into: how to select the appropriate transition orbit parameter V. d ,I d This results in the propellant consumption m during the transfer process. f It is approximately the minimum, and the total transfer time is approximately the shortest.

[0102] Furthermore, the drift time t2 is calculated as follows:

[0103] Calculate the precession velocity of the right ascension of the ascending node of the initial satellite in the transition orbit. Precession velocity of the right ascension of the ascending node of the satellite in the target orbit Precession speed difference

[0104]

[0105] In the formula V d For the initial satellite velocity in the transition orbit, I d The initial satellite orbital inclination for the transition orbit;

[0106] The drift time t2 is:

[0107]

[0108] In the formula Ω d Ω is the right ascension of the ascending node of the initial satellite in the transition orbit. d =Ω0.

[0109] In this embodiment, the initial orbital elements of the transition orbit are calculated as follows:

[0110] Establish an optimization model for maintaining the optimal fuel trajectory:

[0111]

[0112] stΩ(t f )=Ω f

[0113] In the formula m f This represents the total propellant consumption for both propulsion phases.

[0114] The essence of the transfer trajectory problem using the "push-pause-push" control strategy is that it contains two optimization variables (V d ,I d A single objective (minm) with an equality constraint f For optimization problems, the values ​​of optimization variables can be determined by combining gradient analysis and nonlinear solutions.

[0115] For each propulsion stage, the total propellant consumption m f The partial derivatives are equal to the values ​​of the costate variables at the terminal time. The relevant parameters of the transition trajectory can be determined using a combination of gradient analysis and nonlinear solutions.

[0116] Let λ V,1 Let λ be the satellite velocity co-state variable during the first propulsion phase. I,1 Let λ be the co-state variable of the satellite's orbital inclination during a single propulsion phase. V,3 λ is the satellite velocity co-state variable during the second propulsion phase. I,3 Let the satellite orbital inclination be the co-state variable during the second propulsion phase. Then, the gradient of the total propellant consumption function is expressed as:

[0117]

[0118] V that makes the partial derivative of the total propellant consumption function zero d and Id These are the optimal transition trajectory parameters.

[0119] Select the initial satellite orbit inclination I for the transition orbit d As a free variable, combined with formula (2), formula (4) can be transformed into:

[0120]

[0121] In the formula λ I,2 Let λ be the co-state variable of the satellite's orbital inclination during the drift segment. V,2 For the satellite velocity co-state variable during the drift segment;

[0122] Solve for λ using the Pontryagin minimum principle in equation (5). V,1 , λ I,1 , λ V,2 and λ I,2 Then, make the gradient of formula (5) zero, and solve for the initial satellite orbit inclination I of the transition orbit. d .

[0123] The transfer strategy for the low-energy transition trajectory can be summarized as follows: First, the velocity V of the transition trajectory is given. d The initial conjecture value is then used, followed by the Pontryagin minimum principle to solve for the costate variable value λ at the terminal point. V,1 ,λ I,1 ,λ V,2 ,λ I,2 Calculate I that makes the gradient of the mass function zero. d Through the above operations, the original bi-objective optimization problem with two optimization variables and one equality constraint is simplified into a single-objective optimization problem with one optimization variable, which can be easily solved by a nonlinear solver (such as Grubi).

[0124] The method for determining the final orbital elements of the transition trajectory at the end of the transition is as follows:

[0125] At time t0, the satellite orbital elements V0, I0, Ω0, which begin at the start of time period t1, are used for one advance phase. At the end of time period t1, the initial orbital elements V0, I0, Ω0 of the transition orbit are obtained. d ,I d ,Ω d After the rest period caused by the J2 term, the right ascension of the ascending node precesses, and the final orbital element V of the transition orbit is obtained at the end of time interval t2. d ,I d ,Ω f ;

[0126] Then the drift time t2 is calculated.

[0127] Furthermore, to solve the fuel minimum optimization problem during the rendezvous process, the homotopy method is used to gradually transition from energy optimality to fuel optimality, overcoming the problems of discontinuity and convergence difficulties in the fuel optimal function. To solve the thruster fuel optimal control rate sequence, a performance index function is selected:

[0128]

[0129] T max Where c is the maximum thrust of the thruster, and c is the thrust constant, c = I sp ·g0,I sp denoted by thrust specific impulse, g0 is the gravitational acceleration at the Earth's surface, u is the thrust fuel control rate sequence, and t is time.

[0130] Choose the Hamiltonian function H:

[0131]

[0132] In the formula λ V Let λ represent the satellite velocity costate variable. I λ represents the co-state variable of satellite orbital inclination. Ω Let λ be the right ascension covariance of the satellite's ascending node. V ,λ I ,λ Ω The equation satisfies the following:

[0133]

[0134] According to Pontryagin's minimum principle, optimal control should minimize the Hamiltonian function. Therefore, the switching control function S is introduced as follows:

[0135]

[0136] The thruster fuel control rate sequence u is made subject to the switching control function S, and is selected as follows:

[0137]

[0138] In the formula u * (t) represents the optimal control rate sequence of the thruster fuel control rate sequence u under optimal fuel conditions;

[0139] In the optimal control rate sequence u * Under the condition of (t), the optimal thrust direction angle sequence β is obtained. * (t) is:

[0140]

[0141] Because within one orbital period, λ V,λ I There will be no significant change, so there is no ambiguity regarding the sign of β.

[0142] Optimal control law sequence u * (t) and the optimal thrust direction angle sequence β * The solution method for (t) is as follows:

[0143] Set the target shooting variable z = [λ] V (t0),λ I (t0),λ Ω (t0),λ m (t0),t f ] T In the formula λ m Let m be the costate variable corresponding to the satellite mass m; then combine it with the initial state variables [V(t0), I(t0), Ω(t0), m(t0)]. T Solve for the following boundary conditions:

[0144]

[0145] In the formula t 10 Let t be the end time of time interval t1. 20 This is the end time of time interval t2;

[0146] The boundary conditions are solved using a nonlinear solver to obtain the optimal control law sequence u. * (t) and the optimal thrust direction angle sequence β * (t); Calculate the Jacobian matrix at each step during the solution process to accelerate the convergence of the solution process.

[0147] While the invention has been described herein with reference to specific embodiments, it should be understood that these embodiments are merely examples of the principles and applications of the invention. Therefore, it should be understood that many modifications can be made to the exemplary embodiments, and other arrangements can be designed without departing from the spirit and scope of the invention as defined by the appended claims. It should be understood that different dependent claims and features described herein can be combined in ways different from those described in the original claims. It is also understood that features described in conjunction with individual embodiments can be used in other described embodiments.

Claims

1. A method for restoring the orbit of a low-fuel-consumption low-Earth orbit satellite, characterized in that... include, Establish orbital dynamic equations that take into account the perturbation of Earth's oblateness; The initial orbital elements of the low-fuel-consumption transition orbit are solved using a semi-analytical method based on the satellite's current orbit and target orbit. Simultaneously, combined with the orbital dynamics equations, the drift time required for the satellite to drift without fuel consumption using Earth's oblateness perturbation in the transition orbit and the final orbital elements of the transition orbit at the end of the transition are calculated. The satellite's current orbit is the perturbed orbit. Based on the Pontryagin maximum principle, the optimal thruster fuel control rate sequence for the satellite from its current orbit to the transition orbit and from the transition orbit to the target orbit are calculated by combining the performance index function, Hamiltonian function and switching control function. The thrusters are then controlled to achieve orbit recovery for the satellite. The orbital dynamics equation considering the Earth's oblateness perturbation is as follows: In the formula, V is the satellite velocity, f is the thrust acceleration, β is the thrust direction angle, representing the angle between the satellite thrust vector and the velocity vector, I is the satellite orbital inclination, Ω is the right ascension of the satellite's ascending node, and k is the precession constant of the right ascension of the ascending node, k = 1.0425 × 10⁻⁶. - 33 s 6 / m 7 ; The satellite's current orbital elements and target orbital elements are set as follows: Assume the satellite's maneuvering process takes place over a period of time from t0 to t1. f Where t0 is the initial orbit recovery time of the satellite in its current orbit, t f The moment the satellite returns to its target orbit; then the corresponding change in orbital elements is V0,I0,Ω0→V f ,I f ,Ω f Where V0 is the satellite velocity in the current orbit, I0 is the satellite inclination in the current orbit, and Ω0 is the right ascension of the satellite's ascending node in the current orbit; V f For the satellite velocity in the target orbit, I f Ω represents the orbital inclination of the satellite in the target orbit. f The right ascension of the ascending node of the satellite in the target orbit; The satellite's maneuvering process during the time period t0→t f A satellite orbits in a given orbital period. The process of transferring the satellite from its current orbit to its target orbit within each orbital period is divided into three phases: a primary propulsion phase, a drift phase, and a secondary propulsion phase. The primary propulsion phase's time is denoted as t1, the drift phase's time as t2, and the secondary propulsion phase's time as t3. Then: t f =t1+t2+t3; The drift time t2 is calculated as follows: Calculate the precession velocity of the right ascension of the ascending node of the initial satellite in the transition orbit. Precession velocity of the right ascension of the ascending node of the satellite in the target orbit Precession speed difference In the formula V d For the initial satellite velocity in the transition orbit, I d The initial satellite orbital inclination for the transition orbit; The drift time t2 is: In the formula Ω d Ω is the right ascension of the ascending node of the initial satellite in the transition orbit. d =Ω0.

2. The low-fuel-consumption low-Earth orbit satellite orbit recovery method according to claim 1, characterized in that, The method for calculating the initial orbital elements of the transition orbit is as follows: Establish an optimization model for maintaining the optimal fuel trajectory: In the formula m f This represents the total propellant consumption for both propulsion phases. Let λ V,1 Let λ be the satellite velocity co-state variable during the first propulsion phase. I,1 Let λ be the co-state variable of the satellite's orbital inclination during a single propulsion phase. V,3 λ is the satellite velocity co-state variable during the second propulsion phase. I,3 Let the satellite orbital inclination be the co-state variable during the second propulsion phase. Then, the gradient of the total propellant consumption function is expressed as: Select the initial satellite orbit inclination I for the transition orbit d As a free variable, combined with formula (2), formula (4) can be transformed into: In the formula λ I,2 Let λ be the co-state variable of the satellite's orbital inclination during the drift segment. V,2 For the satellite velocity co-state variable during the drift segment; Solve for λ using the Pontryagin minimum principle in equation (5). V,1 , λ I,1 , λ V,2 and λ I,2 Then, make the gradient of formula (5) zero, and solve for the initial satellite orbit inclination I of the transition orbit. d .

3. The low-fuel-consumption low-Earth orbit satellite orbit recovery method according to claim 2, characterized in that, The method for determining the final orbital elements of the transition trajectory at the end of the transition is as follows: At time t0, the satellite orbital elements V0, I0, Ω0, which begin at the start of time period t1, are used for one advance phase. At the end of time period t1, the initial orbital elements V0, I0, Ω0 of the transition orbit are obtained. d ,I d ,Ω d After the rest period caused by the J2 term, the right ascension of the ascending node precesses, and the final orbital element V of the transition orbit is obtained at the end of time interval t2. d ,I d ,Ω f ; Then the drift time t2 is calculated.

4. The low-fuel-consumption low-Earth orbit satellite orbit recovery method according to claim 3, characterized in that, Selecting a performance metric function: T max Where c is the maximum thrust of the thruster, and c is the thrust constant, c = I sp ·g0,I sp denoted by thrust specific impulse, g0 is the gravitational acceleration at the Earth's surface, u is the thrust fuel control rate sequence, and t is time.

5. The low-fuel-consumption low-Earth orbit satellite orbit recovery method according to claim 4, characterized in that, Choose the Hamiltonian function H: In the formula λ V Let λ represent the satellite velocity costate variable. I λ represents the co-state variable of satellite orbital inclination. Ω Let λ be the right ascension covariance of the satellite's ascending node. V ,λ I ,λ Ω The equation satisfies the following:

6. The low-fuel-consumption low-Earth orbit satellite orbit recovery method according to claim 5, characterized in that, The switch control function S is: The thruster fuel control rate sequence u is made subject to the switching control function S, and is selected as follows: In the formula u * (t) represents the optimal control rate sequence of the thruster fuel control rate sequence u under optimal fuel conditions; In the optimal control rate sequence u * Under the condition of (t), the optimal thrust direction angle sequence β is obtained. * (t) is:

7. The low-fuel-consumption low-Earth orbit satellite orbit recovery method according to claim 6, characterized in that, Optimal control law sequence u * (t) and the optimal thrust direction angle sequence β * The solution method for (t) is as follows: Set the target shooting variable z = [λ] V (t0),λ I (t0),λ Ω (t0),λ m (t0),t f ] T In the formula λ m Let m be the costate variable corresponding to the satellite mass m; then combine it with the initial state variables [V(t0), I(t0), Ω(t0), m(t0)]. T Solve for the following boundary conditions: In the formula t 10 Let t be the end time of time interval t1. 20 This is the end time of time interval t2; The boundary conditions are solved using a nonlinear solver to obtain the optimal control rate sequence u. * (t) and the optimal thrust direction angle sequence β * (t); Calculate the Jacobian matrix at each step during the solution process to accelerate the convergence of the solution process.

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