A long-term controlled maintenance method for Halo orbits based on attraction domains

Abstract

This invention discloses a long-term control and maintenance method for Halo orbits based on an attraction domain, belonging to the aerospace field. The implementation method is as follows: Based on three-body dynamics, a nominal Halo orbit is obtained through differential correction techniques. Based on the control Lyapunov function, attraction domain discrimination conditions are constructed. The attraction domain corresponding to the nominal Halo orbit is solved using a sampling method, constraining the spacecraft within this attraction domain, ensuring the stability and controllability of the controller. Furthermore, a dead zone within the attraction domain is selected, with the outer boundary of the dead zone as the outer loop and the inner boundary as the inner loop. When the spacecraft touches the outer boundary, the thrusters are activated; when the spacecraft is controlled to the inner boundary, the thrusters cease operation, constructing this dual-loop control strategy. Based on the dual-loop control strategy, with the control Lyapunov function decreasing and the control amplitude as constraints, and fuel consumption as the performance index, the corresponding feedback control law is obtained by solving this optimization problem, enabling control and maintenance of the spacecraft in the Halo orbit.

Description

Technical Field

[0001] This invention relates to a method for long-term control and maintenance of a Halo orbit near the equilibrium point of a Sun-Earth three-body system, belonging to the field of aerospace. Background Technology

[0002] Translation points are not only excellent locations for scientific exploration of the space environment but also important transit points for deep space exploration, holding significant research value and becoming a hot target for deep space exploration missions of major spacefaring nations. Halo orbits near the Sun-Earth L1 / L2 point are a special type of three-dimensional periodic orbit existing near collinear translation points in a three-body dynamic system. These orbits can serve as deployment orbits for missions such as near-Earth asteroid observation and cosmological observation, possessing significant application value. However, due to the instability of collinear translation point orbits and the chaotic nature of three-body systems, spacecraft operating on them will quickly deviate from their original orbits. Therefore, orbit maintenance control is essential in practical missions. For Halo orbit maintenance, low-thrust control offers higher precision and specific impulse compared to pulse control. However, low-thrust control typically requires continuous thrust operation, which, limited by thruster lifespan, makes it unsuitable for long-term Halo orbit maintenance missions.

[0003] Among the developed Halo orbit control and maintenance methods, the prior art [1] (Nazari M, Anthony W, and Butcher EA Continuous Thrust Station-keeping in Earth-Moon L1 Halo Orbits Based on LQR control and Floquet Theory[c]. AIAA / AAS Astrodynamics Specialist Conference, 2014, 4140) proposed an orbit maintenance strategy based on time-varying LQR for the Halo orbit maintenance problem at the Earth-Moon L1 point, and combined it with dead zone control to reduce the thruster's operating time. The advantage of this method is that compared with a continuous controller with constant control gain, time-varying LQR requires lower fuel consumption, and dead zone control takes into account the thruster's operating life limit. The disadvantage is that the dead zone range is not designed, so the spacecraft may deviate too much from the nominal orbit and fly away from the nominal Halo orbit under the limitation of thrust constraints.

[0004] Prior art [2] (Yi Q, Ruiter AdStation-keeping strategy for real translunar libration point orbits using continuous thrust[J].AerospaceScience and Technology,2019,94:105376) proposed an orbit keeping strategy based on backstepping and dead zone control for periodic orbits near the Earth-Moon libration point, and discussed the influence of different dead zone range settings on orbit control and keeping. The advantage of this method is that it uses dead zone control to consider the working life limit of the thruster in the small thrust keeping control strategy, and analyzes the influence of the region range setting in the dead zone scheme on fuel consumption. The disadvantage is that this method cannot consider control constraints when designing the controller and dead zone range, so it cannot guarantee the stability and controllability of the controller. Summary of the Invention

[0005] To address the problem that existing long-term Halo orbit maintenance methods based on dead-zone control fail to consider control constraints when designing the dead-zone range, potentially leading to spacecraft instability due to uncontrollable conditions, this invention aims to provide a long-term Halo orbit control and maintenance method based on the attraction domain. This method calculates the attraction domain of the Halo orbit using a sampling method, ensuring the stability and controllability of the spacecraft's maintenance control. A dual-loop control strategy is designed based on the Halo orbit attraction domain, reducing thruster operating time while maintaining stability and controllability, thus solving the problem of limited thruster lifespan.

[0006] The objective of this invention is achieved through the following technical solution:

[0007] This invention discloses a long-term control and maintenance method for Halo orbits based on an attraction domain. Based on three-body dynamics, the nominal Halo orbit is obtained through differential correction techniques. An attraction domain discrimination condition is constructed based on the control Lyapunov function. The attraction domain corresponding to the nominal Halo orbit is solved using a sampling method, constraining the spacecraft within this domain and ensuring the stability and controllability of the controller. Furthermore, a dead zone within the attraction domain is selected, with the outer boundary of the dead zone as the outer loop and the inner boundary as the inner loop. When the spacecraft touches the outer boundary, the thrusters are activated; when the spacecraft is controlled to the inner boundary, the thrusters deactivate, constructing this dual-loop control strategy. Based on this dual-loop control strategy, with the control Lyapunov function decreasing and the control amplitude as constraints, and fuel consumption as the performance index, the corresponding feedback control law is obtained by solving this optimization problem, enabling control and maintenance of the spacecraft in the Halo orbit. This control method ensures the stability and controllability of the spacecraft, reduces thruster operating time, and achieves long-term stable maintenance of the spacecraft in the Halo orbit.

[0008] This invention discloses a long-term controlled maintenance method for Halo orbits based on an attraction field, comprising the following steps:

[0009] Step one involves constructing the spacecraft's natural equations of motion. Based on the circular restricted three-body model, a third-order analytical solution for the periodic orbit near the translation point in the Sun-Earth circular restricted three-body system is calculated. Differential correction techniques are then used to refine the third-order analytical solution to obtain a high-precision nominal Halo orbit. For the Halo orbit maintenance problem, a controlled equations of motion for the spacecraft are constructed. Combining the controlled and natural equations of motion, the tracking error dynamics are derived.

[0010] The specific implementation method for step one is as follows:

[0011] The natural equations of motion for a spacecraft are expressed as follows:

[0012]

[0013] Where x, y, and z represent the spacecraft's position, μ is the three-body gravitational coefficient, and r1 and r2 represent the spacecraft's distances from the Sun and Earth, respectively.

[0014]

[0015] Based on the circular restricted three-body model, the third-order analytical solution of the periodic orbit near the translation point in the Sun-Earth circular restricted three-body system is calculated. The third-order analytical solution is corrected using differential correction techniques to obtain a high-precision nominal Halo orbit.

[0016] The natural motion equation (1) of the spacecraft can be simplified as follows:

[0017]

[0018] Where the subscript r represents the dynamics of the nominal Halo orbit,

[0019] For the Halo orbit maintenance problem, the equations of controlled motion for the spacecraft are constructed as follows:

[0020]

[0021] in, Let B be the position and velocity state vector of the controlled spacecraft, where B = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19 ... 3×3 I 3×3 ] T u = [u1, u2, u3] T To control the input, and satisfy u∈U={u|u T Pu≤U max}, where P is a symmetric positive definite matrix.

[0022] Combining the controlled motion equation (3) and the natural motion equation (1) of the spacecraft, the tracking error dynamics are obtained as follows:

[0023]

[0024] Where, Δx=xx r ,

[0025] Step two involves discretizing the Halo orbit to obtain discrete points, each of which can be considered an equilibrium point. Solving the attraction domain problem of the Halo orbit is transformed into solving the attraction domain problem of the error dynamic equilibrium point. Based on the control Lyapunov function, attraction domain discrimination conditions and an optimization problem maximizing the attraction domain are constructed. Using a sampling method, the set of all sampling points satisfying these discrimination conditions is solved, obtaining the attraction domain of the nominal Halo orbit from step one. The stability and controllability of the controller are ensured based on this attraction domain.

[0026] The specific implementation method for step two is as follows:

[0027] For error dynamics, the uncontrolled equilibrium point is Δx = 0, that is, when u(t) ≡ 0. The Halo orbit is discretized to obtain discrete points, each of which can be considered as an equilibrium point. Therefore, solving the attraction domain problem of the Halo orbit is transformed into solving the attraction domain of the equilibrium point Δx = 0 in the error dynamics (4). Based on the method of controlling the Lyapunov function, under the control constraint u T Pu≤U max Next, solve for the attraction region corresponding to the nominal Halo orbit obtained in step one. If the spacecraft's state is within this attraction region at the initial moment, then it is determined that there exists a control law that satisfies the control constraints, enabling the spacecraft to converge to the nominal trajectory.

[0028] Based on the method of controlling Lyapunov functions, under control constraint u T Pu≤U max Next, the attraction region corresponding to the nominal Halo orbital obtained in step one is solved. The specific implementation method is as follows:

[0029] In a finite state space Upper selection of candidate control Lyapunov functions V is chosen as a star function, and in Shangzhengding, that is For each point within the attraction domain The following criteria must be met:

[0030]

[0031] in, Let V be the gradient, and we have:

[0032]

[0033] The discrimination condition (5) is used as a necessary but not sufficient condition for determining whether point Δx belongs to the attraction domain. Based on this necessary but not sufficient condition, the attraction domain is estimated by sampling method.

[0034] For finite state spaces Uniform sampling is performed to obtain N sampling points, denoted as Δx. i Let i = 1, 2, ..., N. Verify whether the discrimination condition (5) holds for each sampling point. The set of all sampling points that satisfy the discrimination condition (5) is X.

[0035] The estimation equation for the attraction region is:

[0036]

[0037] Wherein, γ is a parameter related to the size of the signature attraction domain, which is obtained by solving the following attraction domain optimization problem:

[0038]

[0039] Based on the sampling method, the attraction domain of the Halo orbit near the L2 point of the Sun-Earth three-body system is obtained by simultaneously solving the discrimination condition (5), the estimation equation of the attraction domain (6), and the characteristic attraction domain optimization problem (7). The spacecraft is constrained within this attraction domain to ensure the stability and controllability of the controller.

[0040] Step three involves dividing the control strategy design into two parts: dead zone design and control law design. A dead zone within the attraction domain obtained in step two is selected. Based on this, to account for thruster lifespan limitations and avoid continuous thruster operation, a dual-loop control strategy is constructed: the outer boundary of the dead zone is the outer loop, and the inner boundary is the inner loop. The thrusters are activated when the spacecraft touches the outer boundary and deactivated when the spacecraft reaches the inner boundary. Combining this dual-loop control strategy, with constraints on the decrease of the Lyapunov function and the control amplitude, and fuel consumption as the performance index, a long-term control optimization problem for the Halo orbit is constructed. Solving this long-term control optimization problem yields the corresponding feedback control law. This feedback control law is used to maintain the spacecraft's position, ensuring the stability and controllability of the controller and achieving long-term stable maintenance of the Halo orbit.

[0041] The specific implementation method for step three is as follows:

[0042] Control strategy design consists of two parts: dead zone design and control law design. The dead zone is selected to be contained within the attraction region obtained in step two; that is, the dead zone is... The correlation coefficients between the outer and inner boundaries of the dead zones α1 and α2. The dual-loop control strategy based on the attraction domain is as follows: the outer boundary is the outer loop, and the inner boundary is the inner loop. When the spacecraft touches the outer boundary, the thruster is activated, and when the spacecraft is controlled to the inner boundary, the thruster stops working. Based on this dual-loop control strategy, the lifespan limit of the thruster is considered to avoid continuous operation of the thruster. When the spacecraft is within the attraction domain, according to the discrimination condition (5) in step two, there exists a control law. u∈U This ensures that the derivative of the control Lyapunov function is less than zero, thus guaranteeing the spacecraft's controllability. The control logic is as follows:

[0043]

[0044] The spacecraft state x is measured at each moment to obtain the error state. Δx With the control of the decreasing Lyapunov function and the control amplitude as constraints, and fuel consumption as the performance index, the long-term control optimization problem of the Halo orbit is constructed as shown in equation (9):

[0045]

[0046] By solving the long-term control optimization problem of the Halo orbit (9), the corresponding feedback control law is obtained. The spacecraft is maintained according to the feedback control law to ensure the stability and controllability of the controller, reduce the working time of the thruster, and achieve long-term stable maintenance of the Halo orbit.

[0047] Beneficial effects:

[0048] 1. This invention discloses a long-term control and maintenance method for Halo orbits based on an attraction domain. Based on the Halo orbit attraction domain, a dead zone is selected within the attraction domain. The outer boundary of the dead zone is defined as the outer ring, and the inner boundary as the inner ring. When the spacecraft touches the outer boundary, the thrusters are activated; when the spacecraft is controlled to the inner boundary, the thrusters cease operation. Utilizing this dual-ring control strategy, long-term control and maintenance of the Halo orbit can solve the problem of continuous thruster operation and increase the spacecraft's on-orbit operating time.

[0049] 2. The Halo orbit long-term control and maintenance method disclosed in this invention utilizes a controlled Lyapunov function to construct attraction domain discrimination conditions, and solves for the attraction domain corresponding to the nominal Halo orbit that satisfies the thrust constraint using a sampling method. Constraining the spacecraft within this attraction domain can solve the problems of uncontrollability or controller instability during Halo orbit control and maintenance, ensuring the stability and controllability of the spacecraft while operating in Halo orbit. Attached Figure Description

[0050] Figure 1 This is a flowchart of the long-term control and maintenance method for Halo orbits based on the attraction domain of the present invention.

[0051] Figure 2 In a specific implementation plan, the amplitude A near the L2 point of the Sun-Earth circular restricted three-body system... z =200,000 km Halo simulation image.

[0052] Figure 3 The image shows a simulation of the Halo attraction domain near the L2 point of the Sun-Earth circular restricted three-body system, obtained under thrust constraints, in a specific implementation plan.

[0053] Figure 4 This is a schematic diagram of a retention control strategy based on the attraction domain in a specific implementation plan.

[0054] Figure 5 The following is a simulation diagram of the long-term controlled thrust curve near the L2 point of the attraction domain in the specific implementation plan.

[0055] Figure 6 The following is a simulation diagram of the long-term Halo control Lyapunov function variation curve near the L2 point of the attraction domain in a specific implementation plan. Detailed Implementation

[0056] To better illustrate the purpose and advantages of the present invention, the invention will be further described below in conjunction with the accompanying drawings and examples.

[0057] To verify the feasibility of the method, a simulation of the control and maintenance of the basic attraction domain was conducted, taking the Halo orbit mission at the L2 point of the Sun-Earth three-body system as an example. Figure 1 As shown in the figure, the specific implementation steps of the Halo orbit long-term controlled maintenance method based on the attraction field disclosed in this example are as follows:

[0058] Step one involves constructing the spacecraft's natural equations of motion. Based on the circular restricted three-body model, a third-order analytical solution for the periodic orbit near the translation point in the Sun-Earth circular restricted three-body system is calculated. Differential correction techniques are then used to refine the third-order analytical solution to obtain a high-precision nominal Halo orbit. For the Halo orbit maintenance problem, a controlled equations of motion for the spacecraft are constructed. Combining the controlled and natural equations of motion, the tracking error dynamics are derived.

[0059] The specific implementation method for step one is as follows:

[0060] In this example, the Sun-Earth three-body system is chosen, and the natural motion equations of the spacecraft are expressed as follows:

[0061]

[0062] Where x, y, and z represent the spacecraft's positions, and μ = 3.0035 × 10⁻⁶. -6 Here, r1 and r2 represent the gravitational coefficients of the Sun-Earth system, and r1 and r2 represent the distances of the spacecraft from the Sun and Earth, respectively.

[0063]

[0064] Based on the circularly restricted three-body model, the third-order analytical solution of the periodic orbit near the translational point in the Sun-Earth circularly restricted three-body system is calculated. Differential correction techniques are used to correct the third-order analytical solution to obtain a high-precision nominal Halo orbit. This example selects the amplitude A near the Sun-Earth L2 point. z =200,000 km Halo orbit, the obtained Halo orbit is as follows Figure 2 As shown.

[0065] The natural motion equation (1) of the spacecraft can be simplified as follows:

[0066]

[0067] Where the subscript r represents the dynamics of the nominal Halo orbit,

[0068] For the Halo orbit maintenance problem, the equations of controlled motion for the spacecraft are constructed as follows:

[0069]

[0070] in, Let B be the position and velocity state vector of the controlled spacecraft, where B = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19 ... 3×3 I 3×3 ] T u = [u1, u2, u3] T To control the input, and satisfy u∈U={u|u T Pu≤U max In this case, U max =250mN, where P is a symmetric positive definite matrix, and in this case, it is a 3×3 identity matrix.

[0071]

[0072] Combining the controlled motion equation (3) and the natural motion equation (1) of the spacecraft, the tracking error dynamics are obtained as follows:

[0073]

[0074] Where, Δx=xx r ,

[0075] Step two involves discretizing the Halo orbit to obtain discrete points, each of which can be considered an equilibrium point. Solving for the attraction domain of the Halo orbit is then transformed into solving for the attraction domain of the error dynamic equilibrium point. Based on the control Lyapunov function, attraction domain discrimination conditions and an optimization problem maximizing the attraction domain are constructed. Using a sampling method, the set of all sampled points satisfying these conditions is solved, yielding the nominal Halo orbit's attraction domain from Step one. This attraction domain ensures the stability and controllability of the controller.

[0076] The specific implementation method for step two is as follows:

[0077] For error dynamics, the uncontrolled equilibrium point is Δx = 0, that is, when u(t) ≡ 0. The Halo orbit is discretized to obtain discrete points, each of which can be considered as an equilibrium point. Therefore, solving the attraction domain problem of the Halo orbit is transformed into solving the attraction domain of the equilibrium point Δx = 0 in the error dynamics (4). Based on the method of controlling the Lyapunov function, under the control constraint u T Pu≤U max Next, solve for the attraction region corresponding to the nominal Halo orbit obtained in step one. If the spacecraft's state is within this attraction region at the initial moment, then it is determined that there exists a control law that satisfies the control constraints, enabling the spacecraft to converge to the nominal trajectory.

[0078] Based on the method of controlling Lyapunov functions, under control constraint u T Pu≤U max Next, the attraction region corresponding to the nominal Halo orbital obtained in step one is solved. The specific implementation method is as follows:

[0079] In a finite state space Upper selection of candidate control Lyapunov functions V is chosen as a star function, and in Shangzhengding, that is For each point within the attraction domain The following criteria must be met:

[0080]

[0081] in, Let V be the gradient of V, and we have:

[0082]

[0083] The discrimination condition (5) is used as a necessary but not sufficient condition for determining whether point Δx belongs to the attraction domain. Based on this necessary but not sufficient condition, the attraction domain is estimated by sampling method.

[0084] For finite state spaces Uniform sampling is performed to obtain N sampling points, denoted as Δx. i Let i = 1, 2, ..., N. Verify whether the discrimination condition (5) holds for each sampling point. The set of all sampling points that satisfy the discrimination condition (5) is X.

[0085] The estimation equation for the attraction region is:

[0086]

[0087] Wherein, γ is a parameter related to the size of the signature attraction domain, which is obtained by solving the following attraction domain optimization problem:

[0088]

[0089] Based on the sampling method, by simultaneously solving the discrimination condition (5), the estimation equation of the attraction domain (6), and the characteristic attraction domain optimization problem (7), the attraction domain of the Halo orbit near the L2 point of the Sun-Earth three-body system is obtained. Constraining the spacecraft within this attraction domain ensures the stability and controllability of the controller. In this case, for the amplitude A near the Sun-Earth L2 point... z For the Halo orbit at 200,000 km, the corresponding attraction region was calculated using the algorithm described above, ultimately yielding γ = 0.5. The attraction region of the Halo orbit is shown below. Figure 3 As shown.

[0090] Step three involves dividing the control strategy design into two parts: dead zone design and control law design. A dead zone within the attraction domain obtained in step two is selected. Based on this, to account for thruster lifespan limitations and avoid continuous thruster operation, a dual-loop control strategy is constructed: the outer boundary of the dead zone is the outer loop, and the inner boundary is the inner loop. The thrusters are activated when the spacecraft touches the outer boundary and deactivated when the spacecraft reaches the inner boundary. Combining this dual-loop control strategy, with constraints on the decrease of the Lyapunov function and the control amplitude, and fuel consumption as the performance index, a long-term control optimization problem for the Halo orbit is constructed. Solving this long-term control optimization problem yields the corresponding feedback control law. This feedback control law is used to maintain the spacecraft's position, ensuring the stability and controllability of the controller and achieving long-term stable maintenance of the Halo orbit.

[0091] The specific implementation method for step three is as follows:

[0092] The control strategy design consists of two parts: dead zone design and control law design. The dead zone is selected to be contained within the attraction region obtained in step two. In this case, a region with a size of 0.1 to 0.5 times the attraction region is chosen as the dead zone. The dual-loop control strategy based on the attraction domain is as follows: the outer boundary is the outer loop, and the inner boundary is the inner loop. When the spacecraft touches the outer boundary, the thrusters are activated; when the spacecraft is controlled to the inner boundary, the thrusters are deactivated. Figure 4 As shown. Based on this dual-loop control strategy, considering the thruster's lifespan limitation, continuous thruster operation is avoided. When the spacecraft is within the attraction domain, according to the judgment condition (5) in step two, a control law exists. u∈U This ensures that the derivative of the control Lyapunov function is less than zero, thus guaranteeing the spacecraft's controllability. The control logic is as follows:

[0093]

[0094] The spacecraft state x is measured at each moment to obtain the error state. Δx With the control of the decreasing Lyapunov function and the control amplitude as constraints, and fuel consumption as the performance index, the long-term control optimization problem of the Halo orbit is constructed as shown in equation (9):

[0095]

[0096] By solving the long-term control optimization problem of the Halo orbit (9), the corresponding feedback control law is obtained. The spacecraft is maintained according to the feedback control law, ensuring the stability and controllability of the controller, reducing the working time of the thruster, and realizing the long-term stable maintenance of the Halo orbit.

[0097] A dual-loop control strategy based on the attraction domain is used to control and maintain a spacecraft in Halo orbit. Figure 5 The control variation curves are presented, showing that the dual-loop control strategy proposed in this invention can effectively reduce the thruster's operating time, thereby increasing the spacecraft's on-orbit operating time. Furthermore, the Halo long-term hold control Lyapunov function variation curve near the L2 point of the attraction domain is shown below. Figure 6 As shown, the Lyapunov function exhibits a decreasing trend, indicating that the corresponding controller is stable.

[0098] The above detailed description further illustrates the purpose, technical solution, and beneficial effects of the invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for long-term controlled maintenance of Halo orbits based on an attraction field, characterized in that: Includes the following steps: Step 1: Construct the natural equations of motion for the spacecraft. Based on the circular restricted three-body model, calculate the third-order analytical solution of the periodic orbit near the translation point in the Sun-Earth circular restricted three-body system. Use differential correction techniques to correct the third-order analytical solution to obtain a high-precision nominal Halo orbit. For the Halo orbit maintenance problem, construct the controlled equations of motion for the spacecraft. Combine the controlled equations of motion and the natural equations of motion to obtain the tracking error dynamics. Step 2: Discretize the Halo orbit to obtain discrete points. Each discrete point can be regarded as an equilibrium point. Solving the attraction domain problem of the Halo orbit is transformed into solving the attraction domain problem of the error dynamic equilibrium point. Based on the control of the Lyapunov function, the attraction domain discrimination condition and the optimization problem of maximizing the attraction domain are constructed. The set of all sampling points that satisfy the discrimination condition is solved by the sampling method to obtain the attraction domain of the nominal Halo orbit in step one. Step 3: Divide the control strategy design into two parts: dead zone design and control law design. A dead zone within the attraction domain obtained in step two is selected. Based on this, to account for the thruster's lifespan limitations and avoid continuous thruster operation, a dual-loop control strategy is constructed: the outer boundary of the dead zone is the outer loop, and the inner boundary of the dead zone is the inner loop. When the spacecraft touches the outer boundary, the thruster is activated; when the spacecraft is controlled to the inner boundary, the thruster stops operating. Combining this dual-loop control strategy, with the control Lyapunov function decreasing and the control amplitude as constraints, and fuel consumption as the performance index, a long-term control optimization problem for the Halo orbit is constructed. By solving this long-term control optimization problem for the Halo orbit, the corresponding feedback control law is obtained. The feedback control law is used to maintain the spacecraft, ensuring the stability and controllability of the controller, and achieving long-term stable maintenance of the Halo orbit.

2. The method for long-term controlled maintenance of Halo orbits based on an attraction field as described in claim 1, characterized in that: The specific implementation method for step one is as follows: The natural equations of motion for a spacecraft are expressed as follows: Where x, y, and z represent the spacecraft's position, μ is the three-body gravitational coefficient, and r1 and r2 represent the spacecraft's distances from the Sun and Earth, respectively. Based on the circular restricted three-body model, the third-order analytical solution of the periodic orbit near the translation point in the Sun-Earth circular restricted three-body system is calculated. The third-order analytical solution is corrected by differential correction technique to obtain a high-precision nominal Halo orbit. The natural motion equation (1) of the spacecraft can be simplified as follows: Where the subscript r represents the dynamics of the nominal Halo orbit, For the Halo orbit maintenance problem, the equations of controlled motion for the spacecraft are constructed as follows: in, Let B be the position and velocity state vector of the controlled spacecraft, where B = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19 ... 3×3 I 3×3 ] T u = [u1, u2, u3] T To control the input, and satisfy u∈U={u|u T Pu≤U max }, where P is a symmetric positive definite matrix; Combining the controlled motion equation (3) and the natural motion equation (1) of the spacecraft, the tracking error dynamics are obtained as follows: Where, Δx=xx r , 3. The method for long-term controlled maintenance of Halo orbits based on an attraction field as described in claim 2, characterized in that: The specific implementation method for step two is as follows: For error dynamics, the uncontrolled equilibrium point is Δx = 0, that is, when u(t) ≡ 0. Discretize the Halo orbit to obtain discrete points, each of which can be regarded as an equilibrium point; therefore, the problem of solving the attraction domain of the Halo orbit is transformed into solving the attraction domain of the equilibrium point Δx=0 of the error dynamics (4); Based on the method of controlling Lyapunov functions, under control constraint u T Pu≤U max Next, solve for the attraction domain corresponding to the nominal Halo orbit obtained in step one; if the spacecraft's state is in the attraction domain at the initial moment, it is determined that there exists a control law that satisfies the control constraints, so that the spacecraft can converge to the nominal trajectory; Based on the method of controlling Lyapunov functions, under control constraint u T Pu≤U max Next, the attraction region corresponding to the nominal Halo orbital obtained in step one is solved. The specific implementation method is as follows: In a finite state space Upper selection of candidate control Lyapunov functions V is chosen as a star function, and in Shangzhengding, that is For each point within the attraction domain The following criteria must be met: in, Let V be the gradient of V, and we have: The discrimination condition (5) is used as a necessary but not sufficient condition for determining whether point Δx belongs to the attraction domain; based on this necessary but not sufficient condition, the attraction domain is estimated by sampling method; For finite state spaces Uniform sampling is performed to obtain N sampling points, denoted as Δx. i , i = 1, 2, ... N; verify whether the discrimination condition (5) holds for each sampling point; the set of all sampling points that satisfy the discrimination condition (5) is X; The estimation equation for the attraction region is: Wherein, γ is a parameter related to the size of the signature attraction domain, which is obtained by solving the following attraction domain optimization problem: Based on the sampling method, the attraction domain of the Halo orbit near the L2 point of the Sun-Earth three-body system is obtained by simultaneously solving the discrimination condition (5), the estimation equation of the attraction domain (6), and the characteristic attraction domain optimization problem (7). The spacecraft is constrained within this attraction domain to ensure the stability and controllability of the controller.

4. The method for long-term controlled maintenance of Halo orbits based on an attraction field as described in claim 3, characterized in that: The specific implementation method for step three is as follows: Control strategy design consists of two parts: dead zone design and control law design. The dead zone is selected to be contained within the attraction region obtained in step two, i.e., the dead zone is... The correlation coefficients between the outer and inner boundaries of the dead zones α1 and α2; the dual-loop control strategy based on the attraction domain is as follows: the outer boundary is the outer loop, and the inner boundary is the inner loop. When the spacecraft touches the outer boundary, the thruster is activated, and when the spacecraft is controlled to the inner boundary, the thruster stops working; based on this dual-loop control strategy, the life limit of the thruster is considered to avoid the thruster from working continuously; When the spacecraft is within the attraction domain, according to the discrimination condition (5) in step two, a control law exists. u∈U This ensures that the derivative of the control Lyapunov function is less than zero, thus guaranteeing the spacecraft's controllability. The control logic is as follows: The spacecraft state x is measured at each moment to obtain the error state. Δx With the control of the decrease of the Lyapunov function and the control amplitude as constraints, and fuel consumption as the performance index, the long-term control optimization problem of the Halo orbit is constructed as shown in equation (9): By solving the long-term control optimization problem of the Halo orbit (9), the corresponding feedback control law is obtained. The spacecraft is maintained according to the feedback control law to ensure the stability and controllability of the controller, reduce the working time of the thruster, and achieve long-term stable maintenance of the Halo orbit.

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