Fuel-optimal side-by-side four-star configuration periodic maintenance control method

Abstract

The invention discloses a fuel-optimal side-by-side four-star configuration periodical maintenance control method. The method comprises the following steps: establishing a complete kinetic model containing J2 perturbation and atmospheric perturbation effects in an LVLH coordinate system; based on the model, the change of the phase along with time under the influence of atmospheric resistance is obtained, and the monotonous change rule of the center deviation of different satellite pairs in the formation satellites along with time is obtained; the baseline length of the satellite pair and the center deviation distance between the first and fourth tracks of the satellite and the second and third tracks of the satellite are used as core constraint conditions. In the optimization framework, weighted fuel cost is used as a target function, and differentiated fuel weight distribution is carried out on different satellite pairs. A hybrid optimization strategy of global search and local refining is carried out by adopting a genetic algorithm, an optimal control variable is efficiently solved, and finally, the fuel consumption is minimized on the premise of meeting all constraints.

Description

Technical Field

[0001] This invention relates to the field of distributed satellite formation flight control technology, and more specifically to a fuel-optimal distributed satellite formation relative configuration maintenance control strategy. Background Technology

[0002] Distributed satellite swarm flight technology is an important development direction in modern aerospace technology, with broad application prospects in fields such as Earth observation, deep space exploration, and communication and navigation. In the design of distributed satellite Earth observation swarm missions, it is required that the swarm satellites maintain a relatively stable formation configuration to obtain a relatively stable spatial baseline, ensuring that multiple satellites work together to complete the observation mission.

[0003] In low Earth orbit (LEO) environments, distributed satellite formations are subject to complex perturbations such as J2 perturbation and atmospheric drag, leading to significant formation drift. This necessitates periodic configuration maintenance control. The TanDEM-X binary satellite formation employs a co-orbiting approach, using relative orbit measurements to determine the relative distance between the two satellites, which is then used as a relative orbit control strategy to maintain the formation configuration. For coplanar multi-satellite formations used for three-dimensional Earth observation, the different areal-to-mass ratios of the satellites within the formation cause phase drift, resulting in an approximately linear change in the formation baseline length. This configuration drift characteristic provides crucial insights for optimizing configuration maintenance strategies.

[0004] Traditional formation maintenance methods primarily employ continuous control or instantaneous pulse control, which negatively impacts satellite mission execution and consumes significant fuel. Therefore, there is an urgent need for a relative maintenance control strategy that fully considers the characteristics of distributed, side-by-side multi-satellite configuration changes. This strategy would perform continuous thrust configuration maintenance during satellite mission intervals, achieving fuel-optimal periodic formation maintenance control and improving the overall efficiency of the formation system. Summary of the Invention

[0005] The technical problem solved by this invention is to address the shortcomings of traditional formation configuration maintenance control methods, fully consider the characteristics of distributed side-by-side multi-star configuration changes, and propose a fuel-optimal periodic configuration relative maintenance control strategy to solve the maintenance efficiency and energy consumption problems of side-by-side four-star configurations.

[0006] The technical solution of this invention is: a fuel-optimal side-by-side four-star configuration periodic maintenance control method, comprising:

[0007] A mathematical model of the relative motion of the formation satellites is established, and the mathematical model is simplified to obtain a multi-satellite relative motion model of low Earth orbit circular orbit formation. Based on the obtained multi-satellite relative motion model of low Earth orbit circular orbit formation, a mathematical expression for the inter-satellite baseline of the formation system in the LVLH coordinate system is established, thereby obtaining the direct correspondence between the three-dimensional baseline of the formation satellites and the orbital parameters of the formation satellites.

[0008] Based on the dynamic equations of formation satellites in the LVLH coordinate system, a complete perturbation dynamic model is established to include the configuration drift evolution of J2 perturbation and atmospheric perturbation. Based on the complete perturbation dynamic model, the variation law of single-satellite orbital phase drift of formation satellites under the influence of atmospheric drag is established. According to the direct correspondence between the three-dimensional baseline of formation satellites and the orbital parameters of formation satellites, an approximate analytical relationship is obtained for the elliptical center drift of each pair of satellites in a four-satellite formation as a function of time t.

[0009] Based on the obtained approximate analytical relation, the constraint of formation satellite configuration maintenance control is established with the relative drift threshold of the elliptical centers of each pair of independent satellites. Under this constraint, the minimum value tmin that satisfies the constraint expression is obtained with the drift difference of the elliptical centers formed by each pair of satellites not exceeding the relative drift threshold as the boundary. Any value T that satisfies T≤tmin is used as the control period for the periodic maintenance control of formation configuration.

[0010] Based on the control period of the obtained formation configuration periodic maintenance control, an optimization constraint framework for formation configuration periodic maintenance control is established with the baseline maintenance accuracy along the track and the maintenance accuracy at the center of the ellipse as control constraints.

[0011] Based on the obtained optimization constraint framework for the periodic maintenance control of the formation configuration, a weighted fuel cost objective function is established, and differentiated fuel weight allocation is implemented for the formation satellite pairs. The weighted fuel cost objective function is combined with the baseline maintenance accuracy along the flight path to construct a composite objective function. The thrust vector is used as the optimization variable, and the goal is to minimize the composite objective function. The optimal thrust vector is obtained by solving the problem using a genetic algorithm and a sequential quadratic programming algorithm.

[0012] The mathematical model for the relative motion of the formation satellites is established, and the model is simplified to obtain a multi-satellite relative motion model for low-Earth orbit circular orbit formation, including:

[0013] An LVLH coordinate system is constructed using the virtual center of a satellite formation operating in a circular orbit as the reference star. In the LVLH coordinate system, the positive x-axis points from the Earth's center to the reference star, the positive y-axis is the track direction of the orbit where the reference star is located, and the z-axis is the normal direction on the orbital plane. The x, y, and z axes of the LVLH coordinate system conform to the right-hand rule.

[0014] Establish the LVLH coordinates of the secondary star relative to the primary star, then the expression for the relative motion of the secondary star relative to the primary star is:

[0015] Δx=a a -a c +a c [-(e a cosω a -e c cosω c cosu c -(e a sinω a -e c sinω c )sinu c ]

[0016]

[0017] Δz=a c [-(Ω a -Ω c )sini c cosu c +(i a -i c )sinu c ]

[0018] Where Δx represents the radial relative motion of the formation satellite system in the x-direction, Δy represents the relative motion of the formation satellite system along the track in the y-direction, and Δz represents the normal relative motion of the formation satellite system in the z-direction; u c It is the mean latitude argument of the primary star, u c =ω c +M c ω c M is the perigee argument. c It is the angle of approach, Ω c and r c These are the right ascension of the ascending node and the geocentric distance of the primary star, i c It is the inclination of the primary star, e c It is the eccentricity of the primary star, a c It is the semi-major axis of the primary star; a a e a u a M a Ω a i a r a For the orbital variables of the secondary star, its physical meaning is the same as that of the primary star mentioned above;

[0019] For a satellite formation system with a primary star in a circular orbit, i.e. e c =0, let a a -a c=0, simplifying the three mathematical expressions for the relative motion Δx in the radial x direction, the relative motion Δy along the trajectory y direction of the formation satellite system, and the relative motion Δz in the normal z direction of the formation satellite system, we obtain the relative motion model of multiple satellites in a low-Earth orbit circular orbit formation:

[0020] Δx=-a c e a cos(u c -ω a )

[0021] Δy=a c (u a -u c )+2a c e a sin(u c -ω a )

[0022]

[0023] in,

[0024] The process involves establishing mathematical expressions for the inter-satellite baselines of the formation system in the LVLH coordinate system based on the obtained low-Earth orbit circular orbit formation multi-satellite relative motion model, thereby obtaining the direct correspondence between the three-dimensional baselines of the formation satellites and the orbital parameters of the formation satellites, including:

[0025] Δx, Δy, and Δz in the LVLH coordinate system are defined as the radial baseline, track baseline, and normal baseline of the formation satellites, respectively, thereby establishing a direct correspondence between the three-dimensional baseline of the formation satellites and the orbital parameters of the formation satellites.

[0026] The formation satellite dynamics equations based on the LVLH coordinate system are used to establish a complete perturbation dynamics model that includes J2 perturbation and atmospheric perturbation in the configuration drift evolution.

[0027] Define a satellite formation consisting of four satellites, arranged in the order of satellite 1, satellite 2, satellite 3, and satellite 4. For one of the satellites, k, k∈{1,2,3,4}, define [x k ,y k ,z k Let x be the position vector of satellite k in the LVLH coordinate system, where x k These are the x-axis position coordinates of satellite k in the LVLH coordinate system, and y-axis position coordinates. k These are the y-axis position coordinates of satellite k in the LVLH coordinate system, and the z-axis position coordinates are... k These are the z-axis position coordinates of satellite k in the LVLH coordinate system;

[0028] definition Let $k$ be the velocity vector of satellite $k$ in the LVLH coordinate system. It is the x-axis velocity coordinate of satellite k in the LVLH coordinate system. It is the y-axis velocity coordinate of satellite k in the LVLH coordinate system. It is the z-axis velocity coordinate of satellite k in the LVLH coordinate system;

[0029] definition

[0030] Let $\mathbf{k}$ be the acceleration vector of satellite $k$ in the LVLH coordinate system, where $\mathbf{k}$ is the acceleration vector of satellite $k$. These are the x-axis acceleration coordinates of satellite k in the LVLH coordinate system. These are the y-axis acceleration coordinates of satellite k in the LVLH coordinate system. These are the z-axis acceleration coordinates of satellite k in the LVLH coordinate system;

[0031] The complete perturbation dynamics model, including J2 perturbation and atmospheric perturbation, is as follows:

[0032]

[0033] in, Let a be the average angular velocity of the reference orbit, a be the orbital radius, and μ = 3.986 × 10⁻⁶. 5 km 3 / s 2 The gravitational constant, The three-axis acceleration vector provided for the satellite's k-thrust. Let be the three-axis acceleration vector of the perturbation experienced by satellite k in the LVLH coordinate system.

[0034] The method establishes the variation law of single-satellite orbital phase drift under atmospheric drag in formation satellites based on a complete perturbation dynamics model; and obtains approximate analytical relationships of the elliptical center drift of each pair of satellites in a four-satellite formation with time t based on the direct correspondence between the three-dimensional baseline of the formation satellites and the orbital parameters of the formation satellites, including:

[0035] Based on the standard atmospheric drag model and the Lagrange planetary equations, the variation of the single-star orbital phase drift with time, du(t), under atmospheric drag perturbation, is as follows:

[0036]

[0037] Where μ is the Earth's standard gravitational constant, and a c The semi-major axis of the main star The atmospheric density at the satellite's altitude is given by m, the total mass of the satellite is given by S, and the windward area of ​​the satellite is given by C. d This is the drag coefficient;

[0038] Then we obtain an approximate analytical relationship for the elliptical center drift of any pair of satellites in the formation as a function of time t. Let any satellite k and satellite j form a satellite pair, k, j∈{1,2,3,4}, and k≠j. The approximate analytical relationship for the elliptical center drift of satellite k and satellite j as a function of time t is:

[0039]

[0040] Among them, u k u j The orbital phases of satellites k and j are respectively, m k m j S represents the total mass of satellite k and satellite j, respectively. k S j Let be the windward area of ​​satellites k and j.

[0041] The constraint for maintaining the formation satellite configuration based on the obtained approximate analytical relationship and the relative drift thresholds of each pair of independent satellites with respect to the elliptical center includes:

[0042] If the drift difference between the centers of the ellipses formed by two independent pairs of satellites does not exceed the relative drift threshold, then constraints for maintaining formation configuration control are established.

[0043] Let satellites k and j be a pair, and satellites p and q be another pair, where k, j, p, q ∈ {1, 2, 3, 4}, and k ≠ j ≠ p ≠ q. Then the relative drift threshold of the center of the ellipse formed by satellite pair k / j and satellite pair p / q is ΔL. Tmax The constraints for maintaining the formation satellite configuration control are:

[0044]

[0045] The minimum value tmin, satisfying the constraint expression, is obtained by taking the drift difference between the centers of the ellipses formed by each pair of satellites not exceeding the relative drift threshold as the boundary. Any value T satisfying T≤tmin is used as the control period for the periodic maintenance control of the formation configuration, including:

[0046] Let the relative drift threshold of the ellipse centers formed by satellite pairs k, j and p, q be ΔL. Tmax Solve the following constraint expression:

[0047]

[0048] Find the minimum value t that satisfies the constraint expression. min ;

[0049] Any value T satisfying T≤tmin is used as the control period for the periodic maintenance control of the formation configuration; that is, the formation configuration maintenance frequency is once every T days, and the formation configuration is maintained once every T days of free drift.

[0050] The control period based on the obtained formation configuration periodic maintenance control, using the baseline maintenance accuracy along the track and the ellipse center maintenance accuracy as control constraints, establishes an optimization constraint framework for formation configuration periodic maintenance control, including:

[0051] Let L be the length of the baseline along the track direction of satellites k and j after configuration maintenance. kj The expected length of the baseline along the track direction for satellites k and j Permissible baseline deviation δL between satellite k and satellite j along the track direction kj Then, the baseline maintenance accuracy constraint along the flight path is established as follows:

[0052] Let χ be the deviation distance of relatively independent satellites from the center of their ellipse after configuration maintenance, and let δ be the allowable deviation distance between the centers of the ellipse of satellite k and satellite j. χ The accuracy constraint for maintaining the elliptical center of the satellite formation is χ≤δ χ ;

[0053] The optimization constraint framework for maintaining the periodicity of formation configuration is as follows:

[0054] Baseline maintenance accuracy constraints for satellites k and j along their paths:

[0055] Precision constraint for maintaining the elliptical center of relatively independent satellite pairs: χ≤δ χ .

[0056] The optimization constraint framework based on the obtained periodic maintenance control of formation configuration establishes a weighted fuel cost objective function and implements differentiated fuel weight allocation for formation satellite pairs, including:

[0057] The weighted fuel cost objective function J:

[0058]

[0059] Where, λ kj Let λ be the fuel weighting coefficient for satellites k and j. pq Let Δ be the fuel weighting coefficient for satellites p and q. k Let Δ be the thrust vector of satellite k. j Let Δ be the thrust vector of satellite j. p Let Δ be the thrust vector of satellite p. q Let t0 be the thrust vector of satellite q, and t0 be the configuration maintenance start time. fThe configuration is maintained until the end time;

[0060] By setting λ kj >λ pq This makes the fuel weight of the k and j satellite pairs higher, and tends to reduce the fuel consumption of the k and j satellite pairs during the optimization process; or it sets λ kj <λ pq This makes the fuel weight of the p and q satellite pairs higher, and tends to reduce the fuel consumption of the p and q satellite pairs during the optimization process.

[0061] The process involves combining the weighted fuel cost objective function with the baseline maintenance accuracy along the flight path to construct a composite objective function. Using the thrust vector as the optimization variable and minimizing the composite objective function as the objective, the optimal thrust vector is obtained by solving the problem using a genetic algorithm and a sequential quadratic programming algorithm. This includes:

[0062] Introducing weighting coefficients and The composite objective function cost is:

[0063]

[0064] By adjusting the coefficient and Make J and J are able to remain on the same order of magnitude;

[0065] The configuration maintenance process is discretized into n time periods. During configuration maintenance using thrust, each thrust vector is a 3-dimensional vector. The four-star formation system has a total of 4n 3-dimensional thrust vectors. These thrust vectors are combined into a 12n-dimensional thrust vector Δ. total Δ total These are the optimization variables;

[0066] During configuration maintenance, the thrust vector is used as the optimization variable, and the composite objective function cost is used as the optimization objective. When the composite objective function cost reaches its minimum value, the solution obtained is... This is the optimal thrust vector;

[0067] The method uses the thrust vector as the optimization variable and the composite objective function cost as the optimization objective. When the composite objective function cost reaches its minimum value, the solution obtained is... That is, the optimal thrust vector, including:

[0068] The optimal thrust vector is obtained by combining global search and local refinement, using both genetic algorithm and sequential quadratic programming algorithm. The specific process is as follows:

[0069] Global search phase: In view of the high-dimensional non-convexity characteristics of maintaining the thrust vector in the satellite formation configuration, as well as the multi-constraint characteristics of the periodic maintenance control of the formation configuration and the nonlinear characteristics of the baseline length change, a genetic algorithm is used to search for the global optimal neighborhood of the thrust vector in the 12n-dimensional space.

[0070] Local refinement stage: Based on the global optimal neighborhood of the thrust vector in 12n-dimensional space, the sequential quadratic programming algorithm is further used to construct the Lagrangian function and solve it using the KKT conditions under the constraints of baseline maintenance accuracy and ellipse center maintenance accuracy along the track direction.

[0071] Introducing Lagrange multipliers υ1,υ2,υ3>0, we construct the Lagrange function:

[0072]

[0073] Where, υ1 is the Lagrange multiplier constrained by satellite j and satellite k along the track baseline; υ2 is the Lagrange multiplier constrained by satellite p and satellite q along the track baseline; and υ3 is the Lagrange multiplier constrained by the ellipse center deviation.

[0074] The constructed Lagrange function can be simplified as follows:

[0075] Where Υ is the set of all Lagrange multipliers, h(Δ total () is the set of all constraint functions;

[0076] The simplified Lagrangian function is solved using the KKT optimality condition:

[0077] Condition 1: The gradient of the objective function is expressed as a linear combination of the gradients of the constraint functions. Under the optimal thrust vector That is, under the premise of satisfying all constraints, even a small change in any direction cannot reduce fuel consumption;

[0078] Condition 2: That is, it satisfies the baseline maintenance accuracy constraint along the track and the ellipse center maintenance accuracy constraint;

[0079] The optimization variable Δ that simultaneously satisfies the above KKT conditions total The set of Lagrange multipliers Υ is the optimal thrust vector. and the corresponding Lagrange multipliers Thus, the optimal thrust vector is obtained, satisfying the baseline maintenance accuracy constraints along the track direction and the ellipse center maintenance accuracy constraints.

[0080] The advantages of this invention compared to the prior art are:

[0081] (1) A mathematical model of the relative motion of the formation satellites was established based on the theory of orbital kinematics, and a direct correspondence between the inter-satellite baseline and orbital parameters of the formation satellites was established;

[0082] (2) A complete dynamic model including J2 perturbation and atmospheric perturbation effects was established in the LVLH coordinate system, which accurately described the long-term drift evolution process of the formation configuration. The drift characteristics of the formation configuration were fully utilized, and a calculation method for the formation configuration maintenance control period was designed and determined.

[0083] (3) This invention designs a baseline constraint along the flight path and a configuration center deviation constraint, and designs a fuel-optimal control strategy for periodically maintaining the formation configuration. At the same time, it introduces the baseline control accuracy along the flight path as a constraint, and designs a configuration maintenance control algorithm of global genetic algorithm + local sequence quadratic programming, thus realizing fuel-optimal control for periodically maintaining the configuration.

[0084] (4) The method constructed by the present invention can solve the energy consumption problem in the process of maintaining the on-orbit configuration of the formation satellites, which helps the mission to respond flexibly and has practical engineering significance. Attached Figure Description

[0085] Figure 1 This is a flowchart of the method of the present invention; Figure 2 A three-dimensional plot of the satellite trajectories in the LVLH coordinate system; Figure 3 The tracking error diagram for each satellite; Figure 4 This is a schematic diagram of the motion trajectories of each satellite in the LVLH coordinate system; Figure 5 A schematic diagram illustrating the terminal error maintained by a side-by-side four-star configuration; Figure 6 This is a schematic diagram illustrating the actual mass consumption during the first configuration maintenance. Figure 7 A schematic diagram of the three-axis control input for four satellites; Figure 8 This is a schematic diagram showing the change in total thrust of each satellite during configuration maintenance. Detailed Implementation

[0086] like Figure 1 As shown, this invention provides a fuel-optimal side-by-side four-star configuration periodic maintenance control strategy, comprising the following steps:

[0087] S1. Based on orbital kinematics theory, establish a mathematical model of the relative motion of the formation satellites, simplify it to obtain a multi-satellite relative motion model of low Earth orbit circular orbit formation, establish a mathematical expression for the inter-satellite baseline of the formation system in the LVLH coordinate system, and thus obtain the direct correspondence between the three-dimensional baseline of the formation satellites and the orbital parameters of the formation satellites.

[0088] S2. Based on the dynamic equations of the formation satellites in the LVLH coordinate system, a complete perturbation dynamic model including the configuration drift evolution of J2 perturbation and atmospheric perturbation is established. Based on this model, the variation law of the orbital phase drift of a single satellite under the influence of atmospheric drag is established. Furthermore, based on the correspondence between the baseline along the track direction and the orbital phase in S1, an approximate analytical relationship is obtained for the elliptical center drift of each pair of satellites in a four-satellite formation as a function of time t.

[0089] S3. Based on the approximate analytical relationship between the elliptical center drift of each pair of satellites in the formation and time t, establish the constraint of formation satellite configuration maintenance control with the relative drift threshold of the elliptical centers of each pair of independent satellites; under this constraint, with the drift difference of the elliptical centers formed by each pair of satellites not exceeding the relative drift threshold as the boundary, obtain the minimum value tmin that satisfies the constraint expression, and take any value T that satisfies T≤tmin as the control period of the periodic maintenance control of the formation configuration.

[0090] S4. Based on the control period of the formation configuration periodic maintenance control obtained in S3, the baseline maintenance accuracy along the track and the maintenance accuracy of the ellipse center are used as control constraints to establish an optimization constraint framework for the formation configuration periodic maintenance control.

[0091] S5. Based on the optimization constraint framework of the periodic maintenance control of the formation configuration obtained in S4, a weighted fuel cost objective function is established, and differentiated fuel weight allocation is implemented for the formation satellite pairs. The weighted fuel cost objective function is combined with the baseline maintenance accuracy along the track direction to construct a composite objective function. The thrust vector is used as the optimization variable, and the goal is to minimize the composite objective function. The optimal thrust vector is obtained by solving the problem using a genetic algorithm and a sequential quadratic programming algorithm.

[0092] S1 involves establishing a mathematical model of the relative motion of the formation satellites based on orbital kinematics theory, simplifying it to obtain a multi-satellite relative motion model in a low-Earth orbit circular orbit, and establishing a mathematical expression for the inter-satellite baseline of the formation system in the LVLH coordinate system, thereby obtaining a direct correspondence between the three-dimensional baseline of the formation satellites and the orbital parameters of the formation satellites. The specific process is as follows:

[0093] S1.1 Based on orbital kinematics theory, a mathematical model of the relative motion of the formation satellites is established and simplified to obtain a multi-satellite relative motion model in a low-Earth orbit circular orbit formation. The specific process is as follows:

[0094] The LVLH coordinate system is constructed using the virtual center of a satellite formation in a circular orbit as the reference star. In the LVLH coordinate system, the positive x-axis points from the Earth's center to the reference star, the positive y-axis is the track direction of the orbit of the reference star, and the z-axis is the normal direction on the orbital plane. The x, y, and z axes of the LVLH coordinate system conform to the right-hand rule.

[0095] Establish the LVLH coordinates of the satellite relative to the host star. Then, the expression for the relative motion of the satellite relative to the host star is as follows: Δx is the radial relative motion of the satellite system in the x-direction, Δy is the relative motion of the satellite system along the track in the y-direction, and Δz is the normal relative motion of the satellite system in the z-direction.

[0096] Δx=a a -a c +a c [-(e a cosω a -e c cosω c cosu c -(e a sinω a -e c sinω c )sinu c ]

[0097]

[0098] Δz=a c [-(Ω a -Ω c )sini c cosu c +(i a -i c )sinu c ]

[0099] Where u c It is the mean latitude argument of the primary star, u c =ω c +M c ω c M is the perigee argument. c It is the angle of approach, Ω c and r c These are the right ascension of the ascending node and the geocentric distance of the primary star, i c It is the inclination of the primary star, e c It is the eccentricity of the primary star, a c It is the semi-major axis of the primary star; a a e a u a M a Ω a i a r a For the orbital variables of the secondary star, its physical meaning is the same as that of the primary star mentioned above;

[0100] For a satellite formation system with a primary star in a circular orbit, i.e. e c =0, let aa -a c =0. Simplifying the three mathematical expressions for the relative motion Δx in the radial x-direction, the relative motion Δy along the trajectory y-direction of the formation satellite system, and the relative motion Δz in the normal z-direction of the formation satellite system, we get:

[0101] Δx=-a c e a cos(u c -ω a )

[0102] Δy=a c (u a -u c )+2a c e a sin(u c -ω a )

[0103]

[0104] in,

[0105] S1.2 Defines Δx, Δy, and Δz in the LVLH coordinate system as the radial baseline, track baseline, and normal baseline of the formation satellites, respectively, thereby establishing a direct correspondence between the three-dimensional baseline of the formation satellites and the orbital parameters of the formation satellites.

[0106] S2, based on the dynamic equations of the formation satellites in the LVLH coordinate system, establishes a complete perturbation dynamic model of configuration drift evolution, including J2 perturbation and atmospheric perturbation. Based on this model, the variation law of the orbital phase drift of a single satellite under the influence of atmospheric drag is established over time. Furthermore, based on the correspondence between the baseline along the track direction and the orbital phase in S1, an approximate analytical relationship is obtained for the elliptical center drift of each pair of satellites in a four-satellite formation as a function of time t. The specific process is as follows:

[0107] S2.1 Based on the formation satellite dynamics equations in the LVLH coordinate system, a complete perturbation dynamics model is established that includes the configuration drift evolution of J2 perturbation and atmospheric perturbation.

[0108] For a satellite formation consisting of four satellites arranged in the order of satellite 1, satellite 2, satellite 3, and satellite 4, for one of the satellites k (k∈{1,2,3,4}), define [x k ,y k ,z k ] represents the position vector of satellite k in the LVLH coordinate system, where x k These are the x-axis position coordinates of satellite k in the LVLH coordinate system, and y-axis position coordinates. kThese are the y-axis position coordinates of satellite k in the LVLH coordinate system, and the z-axis position coordinates are... k These are the z-axis position coordinates of satellite k in the LVLH coordinate system;

[0109] definition Let $k$ be the velocity vector of satellite $k$ in the LVLH coordinate system. It is the x-axis velocity coordinate of satellite k in the LVLH coordinate system. It is the y-axis velocity coordinate of satellite k in the LVLH coordinate system. It is the z-axis velocity coordinate of satellite k in the LVLH coordinate system;

[0110] definition Let $\mathbf{k}$ be the acceleration vector of satellite $k$ in the LVLH coordinate system, where $\mathbf{k}$ is the acceleration vector of satellite $k$. These are the x-axis acceleration coordinates of satellite k in the LVLH coordinate system. These are the y-axis acceleration coordinates of satellite k in the LVLH coordinate system. These are the z-axis acceleration coordinates of satellite k in the LVLH coordinate system;

[0111] The dynamic equations for the LVLH coordinate system, which include J2 perturbations and atmospheric perturbations, are established as follows:

[0112]

[0113] in, Let a be the average angular velocity of the reference orbit, a be the orbital radius, and μ = 3.986 × 10⁻⁶. 5 km 3 / s 2 The gravitational constant, The three-axis acceleration vector provided for the satellite's k-thrust. Let be the three-axis acceleration vector of the perturbation experienced by satellite k in the LVLH coordinate system.

[0114] S2.2 is based on the complete perturbation dynamics model of configuration drift evolution that includes J2 perturbation and atmospheric perturbation in S2.1. It establishes the variation law of single-satellite orbital phase drift with time under the influence of atmospheric drag. Furthermore, based on the correspondence between the baseline along the track direction and the orbital phase in S1, it obtains the approximate analytical relationship of the elliptical center drift of each pair of satellites in a four-satellite formation with time t.

[0115] Based on the standard atmospheric drag model and the Lagrange planetary equations, the variation of the orbital phase drift of a single star over time under atmospheric drag perturbation is as follows:

[0116]

[0117] Where μ is the Earth's standard gravitational constant (m 3 / s2 ), a c The semi-major axis of the main star (m), Atmospheric density at the satellite's altitude (kg / m³) 3 ), m is the total mass of the satellite, S is the windward area of ​​the satellite, m and S may vary depending on the satellite, C d This is the drag coefficient.

[0118] Then, we obtain an approximate analytical relationship between the elliptical center drift of each pair of satellites in the formation and time t. Taking satellite k and satellite j (k, j∈{1,2,3,4}(k≠j)) as an example, the approximate analytical relationship between the elliptical center drift of satellite k and satellite j and time t is:

[0119]

[0120] Among them, u k u j The orbital phases of satellites k and j are respectively, m k m j S represents the total mass of satellite k and satellite j, respectively. k S j Let be the windward area of ​​satellites k and j.

[0121] S3, based on the approximate analytical relationship between the elliptical center drift of each pair of satellites in S2 and time t, establishes a constraint for maintaining the formation satellite configuration control using the relative drift threshold of the elliptical centers of mutually independent pairs of satellites. Under this constraint, with the drift difference between the elliptical centers of each pair of satellites not exceeding the relative drift threshold as the boundary, the minimum value t satisfying the constraint expression is obtained. min Any value T satisfying T≤tmin is used as the control period for the periodic maintenance control of the formation configuration. The specific process is as follows:

[0122] S3.1 Based on the approximate analytical relationship of the elliptical center drift of each pair of satellites in the formation as a function of time t in S2, constraints for maintaining the formation satellite configuration are established using the relative drift thresholds of the elliptical centers of each pair of satellites that are independent of each other.

[0123] If the drift difference between the centers of the ellipses formed by two independent pairs of satellites does not exceed the relative drift threshold, then constraints for maintaining formation configuration control are established.

[0124] Taking satellite pairs of 1 / 4 and 2 / 3 as examples, the relative drift threshold of the resulting ellipse center is ΔL. Tmax Then the constraints for maintaining formation configuration control are established as follows:

[0125]

[0126] S3.2 Under the constraint of formation configuration maintenance control, with the drift difference between the centers of the ellipses formed by any two satellite pairs not exceeding the relative drift threshold as the boundary, the minimum value t that satisfies the constraint expression is obtained. min , T(T≤t min As a formation configuration, the control cycle needs to be maintained.

[0127] Taking satellite pair 1 / 4 and satellite pair 2 / 3 as examples again, the relative drift threshold of the center of the formed ellipse is ΔL. Tmax Solve the following constraint expression:

[0128]

[0129] Find the minimum value t that satisfies the constraint expression. min Any value T satisfying T≤tmin is used as the control period for the periodic maintenance control of the formation configuration. That is, the formation configuration maintenance frequency is T, and the formation configuration is maintained once every T days of free drift.

[0130] The control period for the periodic maintenance control of formation configuration obtained in S4 and S3 is used as control constraints, with the baseline maintenance accuracy along the track and the ellipse center maintenance accuracy as control constraints, to establish an optimization constraint framework for the periodic maintenance control of formation configuration. The specific process is as follows:

[0131] Assume the length of the baseline along the track direction of satellite k and satellite j after configuration maintenance is L. kj The expected length of the baseline along the track direction for satellites k and j Permissible baseline deviation δL between satellite k and satellite j along the track direction kj Then, the baseline maintenance accuracy constraint along the flight path is established as follows:

[0132] Assuming that the deviation χ of the elliptical center of relatively independent satellites is maintained by configuration, the allowable deviation δ between the elliptical centers of satellite k and satellite j is... χ Then the accuracy constraint for maintaining the elliptical center of the satellite formation is established as χ≤δ χ .

[0133] Therefore, the optimization constraint framework for the periodic maintenance control of formation configuration is established as follows:

[0134] Satellite baseline maintenance accuracy constraints along the track direction for (satellite k and satellite j):

[0135] Precision constraint for maintaining the elliptical center of relatively independent satellite pairs: χ≤δ χ .

[0136] Taking satellite pairs of 1 / 4 and 2 / 3 as examples again, the optimization constraint framework for the periodic maintenance control of the formation configuration is as follows:

[0137] Baseline length constraints for Satellite 1 and Satellite 4:

[0138] Baseline length constraints for Satellite 2 and Satellite 3:

[0139] Center deviation constraint between the trajectories of satellites 1 and 4 and those of satellites 2 and 3: χ≤δ χ ;

[0140] Under the optimization constraint framework of the periodic maintenance control of formation configuration obtained from S4 (S5), a weighted fuel cost objective function is established, and differentiated fuel weight allocation is implemented for the formation satellite pairs. The weighted fuel cost objective function is combined with the baseline maintenance accuracy along the flight path to construct a composite objective function. With the thrust vector as the optimization variable and minimizing the composite objective function as the objective, the optimal thrust vector is obtained by solving the problem using a genetic algorithm and a sequential quadratic programming algorithm. The specific process is as follows:

[0141] S5.1 In the constrained optimization framework obtained in S4, the following weighted fuel cost objective function is established:

[0142]

[0143] Where, λ 14 λ represents the fuel weighting coefficients for satellites 1 and 4. 23 Let λ be the fuel weighting coefficient for satellites 2 and 3, and let λ be the fuel weighting coefficient for satellite 2 and satellite 3. 14 >λ 23 >0, Δ1 is the thrust vector of satellite 1, Δ2 is the thrust vector of satellite 2, Δ3 is the thrust vector of satellite 3, Δ4 is ​​the thrust vector of satellite 4, t0 is the configuration maintenance start time, t f End time for configuration maintenance.

[0144] By setting λ 14 >λ 23 This increases the fuel weight of satellite pairs 1 and 4, tending to reduce their fuel consumption during optimization; or it sets λ... 14 <λ 23 This results in a higher fuel weight for satellite pairs 2 and 3, and tends to reduce the fuel consumption of satellite pairs 2 and 3 during the optimization process.

[0145] S5.2 Combine the weighted fuel cost objective function with the baseline maintenance accuracy along the flight path to construct a composite objective function. With the thrust vector as the optimization variable and minimizing the composite objective function as the objective, use a genetic algorithm and a sequential quadratic programming algorithm to solve for the optimal thrust vector.

[0146] Introducing weighting coefficients and The composite objective function cost is:

[0147]

[0148] By adjusting the coefficient and Make J and J are able to remain on the same order of magnitude.

[0149] The configuration maintenance process is discretized into n time periods. During configuration maintenance using thrust, each thrust Δ is a 3-dimensional vector. The four-star formation system has a total of 4n 3-dimensional thrust vectors. These thrust vectors are merged into a 12n-dimensional thrust vector Δ. total This refers to the optimization variable.

[0150] Therefore, during configuration maintenance, the thrust vector is used as the optimization variable, and the composite objective function cost is used as the optimization objective. When the composite objective function cost reaches its minimum value, the solution obtained is... This is the optimal thrust vector.

[0151] S5.3 The optimal thrust vector is solved using a combination of global search and local refinement, employing genetic algorithm and sequential quadratic programming algorithm respectively. The specific process is as follows:

[0152] Global search phase: In view of the high-dimensional non-convexity characteristics of maintaining the thrust vector in the satellite formation configuration, as well as the multi-constraint characteristics of the periodic maintenance control of the formation configuration and the nonlinear characteristics of the baseline length change, a genetic algorithm is used to search for the global optimal neighborhood of the thrust vector in the 12n-dimensional space.

[0153] Local refinement stage: Based on the global optimal neighborhood of the thrust vector in 12n-dimensional space, the sequential quadratic programming algorithm is further used to construct the Lagrangian function and solve it using the KKT conditions under the constraints of baseline maintenance accuracy and ellipse center maintenance accuracy along the track.

[0154] Introducing Lagrange multipliers υ1,υ2,υ3>0, we construct the Lagrange function:

[0155]

[0156] Wherein, υ1 is the Lagrange multiplier constrained by satellite j and satellite k along the track baseline; υ2 is the Lagrange multiplier constrained by satellite p and satellite q along the track baseline; and υ3 is the Lagrange multiplier constrained by the ellipse center deviation.

[0157] The constructed Lagrange function can be simplified as follows:

[0158] Where Υ is the set of all Lagrange multipliers, h(Δ total ) is the set of all constraint functions.

[0159] The simplified Lagrangian function is solved using the Karush-Kuhn-Tucker (KKT) optimality condition:

[0160] Condition 1: The gradient of the objective function is expressed as a linear combination of the gradients of the constraint functions. Under the optimal thrust vector That is, under the premise of satisfying all constraints, even a small change in any direction cannot reduce fuel consumption.

[0161] Condition 2: That is, it satisfies the baseline maintenance accuracy constraint along the flight path and the ellipse center maintenance accuracy constraint.

[0162] The optimization variable Δ that simultaneously satisfies the above KKT conditions total The set of Lagrange multipliers Υ is the optimal thrust vector. and the corresponding Lagrange multipliers Thus, the optimal thrust vector is obtained, satisfying the baseline maintenance accuracy constraints along the track direction and the ellipse center maintenance accuracy constraints.

[0163] The effects of the present invention will be briefly introduced and explained below with specific examples.

[0164] The mission requires designing a four-satellite formation with a semi-major axis of 6873.68 km. The four satellites (1, 2, 3, and 4) are approximately side-by-side and evenly distributed. The initial orbital parameters for this side-by-side four-satellite configuration are shown in Table 1. The configuration will be maintained using the periodic control strategy of this invention. Figures 4-8 The simulation results show that the configuration was maintained every 3 days.

[0165] Table 1. Initial orbital parameters of each star in a side-by-side four-star configuration.

[0166]

[0167] In the simulation, the simulation object is a leaderless formation of 4 satellites with a circular orbit of radius 6873.68 km as the reference orbit. The masses of the 4 satellites are satellite 1: 2000 kg, satellite 2: 1800 kg, satellite 3: 1800 kg, and satellite 4: 2000 kg. It is assumed that the configuration will function normally when the center of the trajectory ellipse of satellites 2 and 3 drifts within 200 m from the center of the ellipse of satellites 2 and 3, the baseline of satellites 1 and 4 is within 1000 m ± 5 m, and the baseline of satellites 2 and 3 is within 1000 / 3 m ± 2 m.

[0168] The simulation objective is to achieve the following: all three-dimensional control input channels are limited to [-0.08, 0.08], in N. The allowable deviation δ is set. 14 =5m, δ 23 =2m,δ χ =5m. After 3.11 days, when the elliptical center drift difference between satellites 1 and 4 and satellites 2 and 3 is 200m, the formation configuration maintenance period is set to 3 days, and the configuration maintenance duration is set to 2 hours. Optimization strategy parameters are set: Genetic algorithm global search phase: population size 50-100, crossover probability 0.8, mutation probability 0.1; Sequence quadratic programming local refinement phase: interior-point algorithm is used for accurate solution.

[0169] The trajectories of each satellite in the LVLH coordinate system are as follows Figure 4 As shown. The terminal error maintained by the side-by-side four-star configuration is as follows. Figure 5 The baseline length termination error for satellites 1 and 4 is 0.01m, and the baseline length termination error for satellites 2 and 3 is 0.15m. The center deviation between the trajectories of satellites 1 and 4 and those of satellites 2 and 3 decreased from 189.75m before maintenance to 4.19m, both within the allowable deviation range. Figure 6 It can be seen that the mass consumption of satellite 1 is 0.30g, satellite 2 is 4.46g, satellite 3 is 5.55g, and satellite 4 is 3.17g. The mass consumption of satellites 1 and 4 is significantly less than that of satellites 2 and 3, which meets the fuel consumption requirements. The total fuel consumption is 13.49g, achieving optimal fuel configuration maintenance control that meets the fuel consumption requirements; according to Figure 7 The 3-axis control inputs of all four satellites meet the limit range [-0.08, 0.08], in N. Figure 8 The diagram shows the total thrust variation of each satellite during configuration maintenance. Therefore, this simulation uses a fuel-optimal side-by-side four-satellite configuration periodic maintenance control strategy proposed in this invention to achieve periodic fuel-optimal configuration maintenance control where the fuel consumption of satellites 1 and 4 is less than that of satellites 2 and 3.

[0170] Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make possible changes and modifications to the technical solutions of the present invention based on the above-disclosed technical content without departing from the spirit and scope of the present invention. Therefore, any simple modifications, equivalent changes and modifications made to the above embodiments based on the technical essence of the present invention without departing from the content of the technical solutions of the present invention shall fall within the protection scope of the technical solutions of the present invention.

Claims

1. A fuel-optimal side-by-side four-star configuration periodic maintenance control method, characterized by, The application relates to a method for periodic maintenance control of a formation satellite configuration. A mathematical model of relative motion of formation satellites is established, and a low-orbit circular-orbit formation multi-satellite relative motion model is obtained by simplifying the mathematical model; According to the obtained low-orbit circular-orbit formation multi-satellite relative motion model, a mathematical expression of an inter-satellite baseline of the formation system in an LVLH coordinate system is established, so that a direct corresponding relationship between three-dimensional baselines of the formation satellites and orbit parameters of the formation satellites is obtained; Based on the formation satellite dynamics equation in the LVLH coordinate system, a complete perturbation dynamics model of configuration drift evolution is established, which contains J2 perturbation and atmospheric perturbation; Based on the complete perturbation dynamics model, a change rule of a single-satellite orbit phase drift of the formation satellites under the influence of atmospheric resistance with time is established; According to the direct corresponding relationship between the three-dimensional baselines of the formation satellites and the orbit parameters of the formation satellites, an approximate analytical relationship formula of the elliptical center drift of two-satellite pairs in a side-by-side four-satellite formation satellite with time t is obtained; According to the obtained approximate analytical relationship formula, a constraint of formation satellite configuration maintenance control is established by taking the relative drift threshold of the elliptical centers of the two-satellite pairs as the constraint; under the constraint, a minimum value tmin is obtained by taking the difference between the elliptical center drifts of the two-satellite pairs as the boundary and taking the difference not exceeding the relative drift threshold as the control period of the formation configuration periodic maintenance control; Based on the obtained control period of the formation configuration periodic maintenance control, an optimization constraint framework of the formation configuration periodic maintenance control is established by taking the baseline maintenance accuracy along the track direction and the elliptical center maintenance accuracy as the control constraints; Based on the obtained optimization constraint framework of the formation configuration periodic maintenance control, a weighted fuel cost objective function is established by implementing differential fuel weight distribution to the formation satellite pairs; a composite objective function is constructed by combining the weighted fuel cost objective function and the baseline maintenance accuracy along the track direction, the thrust vector is taken as the optimization variable, minimization of the composite objective function is taken as the target, and the genetic algorithm and the sequential quadratic programming algorithm are used for solving to obtain the optimal thrust vector.

2. A method for optimal fuel and periodic maintenance control of a side-by-side four-star configuration as claimed in claim 1, characterized in that, The method for establishing the mathematical model of relative motion of formation satellites and simplifying the mathematical model to obtain the low-orbit circular-orbit formation multi-satellite relative motion model comprises the following steps: An LVLH coordinate system is constructed by taking a virtual center of a circular-orbit satellite formation configuration as a reference star; in the LVLH coordinate system, the positive direction of the x-axis is directed from the center of the Earth to the reference star, the positive direction of the y-axis is the track direction of the orbit of the reference star, and the z-axis is in the normal direction of the orbit plane; the x, y and z axes of the LVLH coordinate system comply with the right-hand rule; An LVLH coordinate of a slave star relative to a master star is established, and the relative motion expression of the slave star relative to the master star is as follows: Δx = a a - a c + a c [ - (e a cos ω a - e c cos ω c ) cos u c - (e a sin ω a - e c sin ω c ) sin u c ] Δz = a c [-(Ω a -Ω c )sini c cosu c +(i a -i c )sinu c ] wherein Δx is the relative motion of the formation satellite system in the radial x direction, Δy is the relative motion of the formation satellite system along the track y direction, and Δz is the relative motion of the formation satellite system in the normal z direction; u c is the argument of latitude of the primary satellite, u c = ω c + M c , ω c is the argument of perigee, M c is the mean anomaly, Ω c and r c are the right ascension of the ascending node and the geocentric distance of the primary satellite, i c is the inclination of the primary satellite, e c is the eccentricity of the primary satellite, a c is the semi-major axis of the primary satellite; a a , e a , u a , M a , Ω a , i a , r a are the orbital variables of the secondary satellite, and have the same physical meaning as the above-mentioned primary satellite; For a satellite formation system with a primary star in a circular orbit, i.e. e c =0, let a a -a c =0, simplifying the three mathematical expressions for the relative motion Δx in the radial x direction, the relative motion Δy along the trajectory y direction of the formation satellite system, and the relative motion Δz in the normal z direction of the formation satellite system, we obtain the relative motion model of multiple satellites in a low-Earth orbit circular orbit formation: Δx = -a c e a cos(u c -ω a ) Δy = a c (u a -u c )+2a c e a sin(u c -ω a ) wherein 3. A method of optimal fuel and periodic maintenance control of a side-by-side four-star configuration according to claim 2, characterized in that, The method for establishing the mathematical expression of the inter-satellite baseline of the formation system in the LVLH coordinate system according to the obtained low-orbit circular-orbit formation multi-satellite relative motion model, so as to obtain the direct corresponding relationship between the three-dimensional baselines of the formation satellites and the orbit parameters of the formation satellites, comprises the following steps: The Delta x, Delta y and Delta z in the LVLH coordinate system are defined as the radial baseline, the along-track baseline and the normal baseline of the formation satellites respectively, so that the direct corresponding relationship between the three-dimensional baselines of the formation satellites and the orbit parameters of the formation satellites is established.

4. A method of optimal fuel and periodic maintenance control of a side-by-side four-star configuration according to claim 3, characterized in that, The formation satellite dynamics equation based on the LVLH coordinate system is used to establish a complete perturbation dynamics model of the configuration drift evolution containing J2 perturbation and atmospheric perturbation. Define a satellite formation consisting of four satellites, arranged in the order of satellite 1, satellite 2, satellite 3, and satellite 4. For one of the satellites, k, k∈{1,2,3,4}, define [x k ,y k ,z k Let x be the position vector of satellite k in the LVLH coordinate system, where x k These are the x-axis position coordinates of satellite k in the LVLH coordinate system, and y-axis position coordinates. k These are the y-axis position coordinates of satellite k in the LVLH coordinate system, and the z-axis position coordinates are... k These are the z-axis position coordinates of satellite k in the LVLH coordinate system; Definitions denotes the velocity vector of satellite k in the LVLH coordinate system, where is the x-axis velocity coordinate of satellite k in the LVLH coordinate system, is the y-axis velocity coordinate of satellite k in the LVLH coordinate system, is the z-axis velocity coordinate of satellite k in the LVLH coordinate system; Definitions denotes the acceleration vector of satellite k in the LVLH coordinate system, where, is the x-axis acceleration coordinate of satellite k in the LVLH coordinate system, is the y-axis acceleration coordinate of satellite k in the LVLH coordinate system, is the z-axis acceleration coordinate of satellite k in the LVLH coordinate system; The complete perturbation dynamics model of the configuration drift evolution containing J2 perturbation and atmospheric perturbation is as follows: where is the reference orbit mean angular velocity, a is the orbit radius, μ = 3.986 x 10 5 km 3 / s 2 is the gravitational constant, is the three-axis acceleration vector provided by the satellite k thrusters, is the three-axis acceleration vector in the LVLH frame experienced by the satellite k due to perturbations.

5. A method of optimal fuel and periodic maintenance control of a side-by-side four-star configuration according to claim 4, characterized in that, The single-satellite orbit phase drift of the formation satellite under the influence of atmospheric resistance is obtained based on the complete perturbation dynamics model, and the approximate analytical relationship of the elliptical center drift of the two-satellite pair in the side-by-side four-satellite formation satellite with respect to time t is obtained according to the direct correspondence between the three-dimensional baseline of the formation satellite and the orbit parameters of the formation satellite, including: The change rule of the single-satellite orbit phase drift with respect to time t under the influence of atmospheric resistance is as follows based on the standard atmospheric resistance model and the Lagrange planetary equation: where μ is the standard gravitational constant of the Earth, a c is the semi-major axis of the primary star, is the atmospheric density at the satellite's altitude, m is the total mass of the satellite, S is the frontal area of the satellite, C d is the drag coefficient; The approximate analytical relationship of the elliptical center drift of the two-satellite pair in the formation satellite with respect to time t is obtained, and the approximate analytical relationship of the elliptical center drift of satellite k and satellite j with respect to time t is as follows: k, j ∈ {1, 2, 3, 4}, and k ≠ j. where u k , u j are the orbital phase of satellite k and satellite j, respectively, m k , m j are the total mass of satellite k and satellite j, respectively, S k , S j are the windward area of satellite k and satellite j, respectively.

6. A method of optimal fuel and periodic maintenance control of a side-by-side four-star configuration as claimed in claim 5, wherein, The constraint of the formation satellite configuration maintenance control is established according to the obtained approximate analytical relationship, including: The constraint of the formation configuration maintenance control is established with the drift difference of the elliptical center formed by the two-satellite pair as the boundary. Let satellite k, satellite j be a pair of satellite pairs, satellite p, satellite q be a pair of satellite pairs, k, j, p, q ∈ {1, 2, 3, 4}, and k≠j≠p≠q, then the relative drift threshold of the center of the ellipse formed by the satellite pair k / j, the satellite pair p / q is ΔL Tmax Then the constraint of formation satellite configuration maintenance control is:

7. A method of optimal fuel and periodic maintenance control of a side-by-side four-star configuration according to claim 6, characterized in that, The minimum value tmin that satisfies the constraint expression is obtained with the drift difference of the elliptical center formed by the two-satellite pair as the boundary, and any value T that satisfies T ≤ tmin is taken as the control period of the periodic maintenance control of the formation configuration, including: Let the threshold of relative drift of the center of the ellipse formed by satellite pair k, j, satellite pair p, q be ΔL Tmax , the following constraint expression is solved, obtaining a minimum value t under the satisfied constraint expression min ; The control period of the periodic maintenance control of the formation configuration is taken as any value T that satisfies T ≤ tmin, that is, the formation configuration maintenance frequency is T days once, and the formation configuration is maintained once every T days of free drift.

8. A method of optimal fuel and periodic maintenance control of a side-by-side four-star configuration according to claim 7, characterized in that, The optimization constraint framework of the periodic maintenance control of the formation configuration is established based on the obtained control period of the periodic maintenance control of the formation configuration, including: Let L be the length of the along-track baseline between satellite k and satellite j after the along-track baseline has been maintained kj L is the desired length of the along-track baseline between satellite k and satellite j δL is the allowable deviation of the along-track baseline between satellite k and satellite j kj The along-track baseline maintenance accuracy constraint is then established as Let the relative independent satellite pair elliptic center after the configuration to maintain the deviation distance χ, the allowable deviation distance of the elliptic center of satellite k and satellite j is δ χ Then the formation satellite elliptic center maintenance accuracy constraint is χ≤δ χ ; The optimization constraint framework of the periodic maintenance control of the formation configuration is as follows: Satellite k and satellite j along track baseline maintenance accuracy constraint: The relative independence of the satellite pairs maintains an elliptic center accuracy constraint: χ < δ χ .

9. A method of optimal fuel and periodic maintenance control of a side-by-side four-star configuration according to claim 8, characterized in that, The weighted fuel cost objective function is established based on the obtained optimization constraint framework of the periodic maintenance control of the formation configuration, and the fuel weight is allocated differently to the formation satellite pairs, including: The weighted fuel cost objective function J is as follows: where λ kj is a fuel weight coefficient of satellite k and satellite j, λ pq is a fuel weight coefficient of satellite p and satellite q, Δ k is a thrust vector of satellite k, Δ j is a thrust vector of satellite j, Δ p is a thrust vector of satellite p, Δ q is a thrust vector of satellite q, t0is a configuration maintenance start time, t f is a configuration maintenance end time; By setting λ kj > λ pq , the fuel weight of the k, j satellite pair is made higher, and in the optimization process, the fuel consumption of the k, j satellite pair tends to be reduced; or setting λ kj < λ pq , the fuel weight of the p, q satellite pair is made higher, and in the optimization process, the fuel consumption of the p, q satellite pair tends to be reduced.

10. A method of optimal fuel and periodic maintenance control of a side-by-side four-star configuration according to claim 9, characterized in that, The composite objective function is constructed by combining the weighted fuel cost objective function with the along-track baseline maintenance accuracy, the thrust vector is taken as the optimization variable, the minimum value of the composite objective function is taken as the target, and the genetic algorithm and the sequential quadratic programming algorithm are used for solving to obtain the optimal thrust vector, including: Introducing a weight coefficient and The composite objective function cost is: By adjusting the coefficients and so that and J remain of the same order of magnitude; The configuration maintenance process is discretized into n time periods, in the configuration maintenance process using thrust, each thrust vector is a 3-dimensional vector, and the four-star formation system has a total of 4n 3-dimensional thrust vectors. These thrust vectors are combined into a 12n-dimensional thrust vector Δ total , Δ total is the optimization variable; In the configuration maintenance process, the thrust vector is taken as the optimization variable, and the composite objective function is taken as the optimization objective. When the composite objective function reaches the minimum value, the solution obtained is That is, the optimal thrust vector.

11. A method of optimal fuel and periodic maintenance control of a side-by-side four-star configuration according to claim 10, characterized in that, The thrust vector is taken as an optimization variable, a composite objective function is taken as an optimization objective, and when the composite objective function reaches a minimum value, the obtained That is, the optimal thrust vector, comprising: The genetic algorithm and the sequential quadratic programming algorithm are used for solving to obtain the optimal thrust vector in a way of combining global search and local refinement, and the specific process is as follows: Global search stage: In view of the high-dimensional non-convex characteristics of the satellite formation configuration maintaining thrust vector, the multi-constraint characteristics of the periodic formation configuration maintaining control and the nonlinear characteristics of the baseline length variation, a genetic algorithm is used to search the global optimal neighborhood of the thrust vector in 12n-dimensional space; Local refining stage: Based on the global optimal neighborhood of the thrust vector in 12n-dimensional space, a sequential quadratic programming algorithm is further used to construct a Lagrange function and solve it by using KKT conditions under the constraints of maintaining the accuracy of the baseline along the track and the maintenance accuracy of the elliptical center, so as to obtain the optimal thrust vector.

12. A method of optimal fuel and periodic maintenance control of a side-by-side four-star configuration according to claim 11, characterized in that, The optimal thrust vector is obtained by constructing a Lagrange function and solving it by using KKT conditions, including: The Lagrange multipliers υ1, υ2, υ3>0 are introduced to construct a Lagrange function: Wherein, υ1 is the Lagrange multiplier of the satellite j and the satellite k along the track baseline constraint; υ2 is the Lagrange multiplier of the satellite p and the satellite q along the track baseline constraint; and υ3 is the Lagrange multiplier of the elliptical center deviation constraint; The constructed Lagrangian function is simplified as: where Y is the set of all Lagrange multipliers, h(A total ) is the set of all constraint functions. The KKT optimality condition is used to solve the simplified Lagrange function: Condition 1: Express the gradient of the objective function as a linear combination of the gradients of the constraint functions, At the optimal thrust vector, That is, no small change in any direction can reduce fuel consumption while satisfying all constraints; Condition 2: i.e., the along track baseline maintenance accuracy constraint and the maintenance accuracy constraint of the center of the ellipse are satisfied; the optimization variables Δ satisfying the above KKT conditions total and the set of Lagrange multipliers Υ, is the optimal thrust vector and the corresponding Lagrange multipliers Thus, the optimal thrust vector satisfying the along-track baseline accuracy constraint and the elliptical center accuracy constraint is obtained

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