On-orbit spacecraft rendezvous trajectory optimization method based on Koopman operator
The Koopman operator is used to perform dynamic constraint convexification and trust region optimization, which solves the problem of insufficient accuracy in the existing technology of on-orbit spacecraft rendezvous trajectory optimization and achieves higher fitting accuracy and quality.
Patent Information
- Application Number
- CN202510770117.6
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-06-10
- Publication Date
- 2025-09-19
AI Technical Summary
Existing convex optimization methods suffer from pseudo-feasibility or infeasibility problems under non-convex dynamic models and nonlinear process constraints. The accuracy of the initial guess is sensitive to the convexification process, resulting in insufficient accuracy in the optimization of on-orbit spacecraft rendezvous trajectories.
The Koopman operator is used to convexify the dynamic constraints. By establishing a kinematic model and using the Koopman operator to linearize the state observation function, the nonlinear constraints are converted into linear forms by combining the trust region and slack variable optimization, which reduces the sensitivity to the initial guess and improves the fitting accuracy.
The accuracy and quality of on-orbit spacecraft rendezvous trajectory optimization are improved, the sensitivity to initial guesses is reduced, and the fitting accuracy of nonlinear systems is improved.
Smart Images
Figure CN120670709A_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the technical field of spacecraft trajectory optimization, and in particular to an on-orbit spacecraft rendezvous trajectory optimization method based on a Koopman operator. Background Art
[0002] In recent years, with the continuous development of on-orbit servicing missions, high-precision, safe and reliable rendezvous trajectory optimization for spacecraft has attracted increasing attention. For spacecraft trajectory optimization problems, there are currently analytical methods, indirect methods, direct methods, and intelligent optimization methods. Among them, convex optimization has attracted widespread attention in research due to its polynomial time solvability. However, convex optimization has certain limitations when facing most spacecraft trajectory optimization problems. Constrained convexification is a necessary process for solving non-convex optimization problems, and sequential convexification has been effectively applied to many problems. However, its convexification process relies on the accuracy of the initial guess, which makes it pseudo-feasible or even infeasible when facing non-convex dynamic models and nonlinear process constraints. Summary of the Invention
[0003] The purpose of the present invention is to address the shortcomings of the above-mentioned background technology and provide an on-orbit spacecraft rendezvous trajectory optimization method by using convex optimization and Koopman operator technology, taking into account non-convex dynamics and safety constraints, so as to achieve the purpose of reducing the sensitivity of the convexification process to the initial guess and improving the fitting accuracy of the original nonlinear system.
[0004] In order to achieve the above object, the present invention provides an on-orbit spacecraft rendezvous trajectory optimization method based on the Koopman operator, comprising the following steps:
[0005] S1, establishes a kinematic model for the on-orbit spacecraft rendezvous mission;
[0006] The thrust acceleration of the spacecraft meets the amplitude constraint; the rendezvous trajectory is ensured to be outside the no-fly zone, and a spherical no-fly zone is established. The optimization goal is to minimize fuel consumption, and the objective function is the time integral of the thrust acceleration modulus.
[0007] S2, performs dynamic constraint convexification based on Koopman operator;
[0008] The Koopman operator gives a discrete evolution form of the observation function of the state. When there is a control force input, the Koopman operator realizes linear expression through high-dimensional states. A set of state observation functions is selected. By using a set of random control inputs and a set of initial states distributed around the initial constraints, the evolving on-orbit spacecraft trajectory dataset is obtained. The trajectory point dataset after the dimension increase of the observation function is defined. Based on the extended dynamic mode decomposition of the Koopman operator, an analytical solution is obtained, and the linear dynamic model after the dimension increase is obtained. The original nonlinear dynamic constraints are converted into a high-dimensional discrete linear form.
[0009] S3, perform convexification of the no-fly zone process constraints;
[0010] The spherical constraint is converted into a hyperplane constraint, where the hyperplane is composed of the tangent planes of the trajectory points where the mass of the no-fly zone center violates the no-fly zone; slack variables are introduced to convert the objective function into a linear form;
[0011] S4, construct the trust region and perform iterative calculations;
[0012] Trust region constraints and relaxation term constraints are added to participate in the optimization of the objective function. The trust region constraints constrain the thrust acceleration term, and the relaxation term constraints serve as virtual compensation for the control quantity to compensate for the error when the dynamic constraints are insufficient.
[0013] Furthermore, the kinematic model established in S1 is:
[0014]
[0015] Among them, r c =[x c y c z c ] T is the vector from the spacecraft to the target point, u c is the thrust acceleration of the spacecraft, R is the relative position vector from the center of the Earth to the target point, ω is the orbital velocity of the rotating coordinate system, μ is the Earth's gravitational constant, R=|R|,
[0016] Furthermore, the thrust acceleration of the spacecraft in S1 satisfies the amplitude constraint:
[0017] ||u c (t)||2≤u max
[0018] Among them, u max is the maximum thrust acceleration allowed, t is the time;
[0019] The spherical no-fly zone is represented by:
[0020] ||r c(t)-r out (t)||2≥r min
[0021] Among them, r out (t) is the position of the center of the no-fly zone at time t in the LVLH coordinate system, r min is the radius of the no-fly zone.
[0022] Furthermore, the objective function in S1 is:
[0023]
[0024] Among them, t0 is the initial time, t f is the target time, and J is the fuel consumption.
[0025] Furthermore, when there is a control force u input in S2, the Koopman operator realizes the linear expression through the high-dimensional state z:
[0026] z + =A z +Bu
[0027] x=Cz
[0028] dim(z)>>dim(x)
[0029] Among them, A is the state transfer matrix, B is the control input matrix, C is the output matrix, z + Represents the new state, and x represents the original state space;
[0030] The selected state observation function table is
[0031]
[0032] Among them, N ψ is the number of observation functions, and g(·) is the observation function.
[0033] Furthermore, the trajectory dataset of the spacecraft in orbit that is evolved in S2 is:
[0034]
[0035] in, All of them are trajectory point datasets, satisfying Represents a trajectory point; set is the trajectory point dataset after dimension increase based on the observation function ψ(x);
[0036] definition Based on the trajectory point dataset after the dimension increase of the observation function ψ(x), the computational problem of the extended dynamic modulus decomposition Koopman operator is described as follows:
[0037]
[0038] in,‖·‖ F is the Frobenius norm;
[0039] The analytical solution to the least squares optimization problem is:
[0040]
[0041] in, is the Moore-Penrose pseudoinverse;
[0042] Thus, the linear dynamic model after dimensionality increase is obtained, and the original nonlinear dynamic constraints have been converted into a high-dimensional discrete linear form:
[0043] X k+1 =A koop X k +B koop u k ,k=1,2,...,N
[0044] Among them, X k 、X k+1 is the dimensionality-increasing state value of the dynamic model, u k represents the thrust acceleration modulus, A koop 、B koop Represents the state transfer matrix and control input matrix after the convexification of the Koopman operator dynamic constraints.
[0045] Furthermore, the no-fly zone constraint in S3 is converted to:
[0046]
[0047] in, Represents X p The first three dimensions of the dimensional state are equivalent to r c , is the set of discrete points p that violate the no-fly zone constraint in the reference trajectory:
[0048]
[0049] Furthermore, the objective function in S3 in discrete form is expressed as the sum of thrust acceleration moduli at discrete points:
[0050]
[0051] Where m is the number of discrete points;
[0052] Introducing the slack variable η i , let ||ui ||2≤η i , the objective function and the original thrust acceleration constraint are converted to:
[0053]
[0054] 0≤η k ≤u max
[0055] Among them, u max is the maximum thrust acceleration of the spacecraft.
[0056] Furthermore, trust region constraints are added in S4 and slack constraints
[0057] The trust region constraint on the thrust acceleration term is:
[0058]
[0059] The form of the relaxation constraint to compensate for the error when the dynamic constraint is insufficient is:
[0060]
[0061] Where i represents the iteration number;
[0062] Incorporate the trust region constraint and the relaxation term constraint into the objective function to participate in the optimization, combined with the preset penalty coefficient ω u and ω f , the objective function is expanded to:
[0063]
[0064] 10. The on-orbit spacecraft rendezvous trajectory optimization method based on Koopman operator according to claim 9, characterized in that the convergence condition of the optimization iteration in S4 is determined by the optimization variable and Decision, subject to the following conditions:
[0065]
[0066] Among them, ε u , ε f These are all preset convergence values.
[0067] The above solution of the present invention has the following beneficial effects:
[0068] The Koopman operator-based on-orbit spacecraft rendezvous trajectory optimization method provided by this invention uses the Koopman operator for global linearization and linearized dynamic constraints, resulting in higher approximation accuracy than traditional linearization methods. Furthermore, by selecting an appropriate observation function, the sensitivity of the optimization solution to the initial guess can be further reduced. This improves the fitting accuracy of the original nonlinear system, thereby enhancing the accuracy and quality of on-orbit spacecraft rendezvous trajectory optimization.
[0069] Other beneficial effects of the present invention will be described in detail in the subsequent specific implementation section. BRIEF DESCRIPTION OF THE DRAWINGS
[0070] Figure 1 is a flow chart of the steps of the present invention;
[0071] Figure 2 This is the trajectory result diagram at 1240s in an embodiment of the present invention;
[0072] Figure 3 This is the trajectory result diagram at 2340s in an embodiment of the present invention;
[0073] Figure 4 This is a comparison diagram of the relative errors between the present invention and the traditional linearization method. DETAILED DESCRIPTION
[0074] The following describes the embodiments of the present disclosure through specific examples, and those skilled in the art can easily understand other advantages and effects of the present disclosure from the contents disclosed in this specification. Obviously, the described embodiments are only a part of the embodiments of the present disclosure, rather than all of the embodiments. The present disclosure can also be implemented or applied through other different specific embodiments, and the details in this specification can also be modified or changed in various ways based on different viewpoints and applications without departing from the spirit of the present disclosure. It should be noted that, in the absence of conflict, the following embodiments and features in the embodiments can be combined with each other. Based on the embodiments in the present disclosure, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present disclosure.
[0075] It should be noted that various aspects of the embodiments within the scope of the appended claims are described below. It should be apparent that the aspects described herein can be embodied in a wide variety of forms, and any specific structure and / or function described herein is merely illustrative. Based on this disclosure, it should be understood by those skilled in the art that an aspect described herein can be implemented independently of any other aspect, and two or more of these aspects can be combined in various ways. For example, any number of aspects described herein can be used to implement an apparatus and / or practice a method. In addition, other structures and / or functionalities other than one or more of the aspects described herein can be used to implement this apparatus and / or practice this method.
[0076] It should also be noted that the diagrams provided in the following embodiments are merely schematic illustrations of the basic concepts of the present disclosure. The diagrams only show components relevant to the present disclosure and are not drawn according to the number, shape, and size of components in actual implementation. In actual implementation, the configuration, quantity, and proportion of each component may be varied at will, and the component layout may be more complex. Furthermore, in the following description, specific details are provided to facilitate a thorough understanding of the examples. However, those skilled in the art will appreciate that the described aspects may be practiced without these specific details.
[0077] like Figure 1 As shown, an embodiment of the present invention provides an on-orbit spacecraft rendezvous trajectory optimization method based on the Koopman operator, comprising the following steps:
[0078] S1, establishes a kinematic model for the on-orbit spacecraft rendezvous mission.
[0079] In this embodiment, the kinematic model established is specifically:
[0080]
[0081] Among them, r c =[x c y c z c ] T is the vector from the spacecraft to the target point, u c is the thrust acceleration of the spacecraft, R is the relative position vector from the center of the Earth to the target point, ω is the orbital velocity of the rotating coordinate system, μ is the Earth's gravitational constant, R=|R|, The thrust acceleration of the spacecraft satisfies the amplitude constraint:
[0082] ||u c (t)‖2≤u max (2)
[0083] Among them, u maxis the maximum thrust acceleration allowed, and t is the time.
[0084] Considering the safety of the rendezvous process, it is necessary to ensure that the rendezvous trajectory is outside the no-fly zone. Therefore, a spherical no-fly zone is considered in this embodiment:
[0085] ||r c (t)-r out (t)||2≥r min (3)
[0086] Among them, r out (t) is the position of the center of the no-fly zone at time t in the LVLH coordinate system (Local Vertical Local Horizontal, a local reference system commonly used to describe the relative motion of spacecraft, especially in rendezvous and docking, formation flying, or operations near the space station), r min is the radius of the no-fly zone.
[0087] Under the goal of minimizing fuel consumption, the optimization problem corresponding to this embodiment is modeled as:
[0088]
[0089] Among them, t0 is the initial time, t f is the target time, J is the fuel consumption, and we define:
[0090] r c (t0) = r c,0 ,v c (t0) = v c,0 ,r c (t f )=r c,f ,v c (t f )=v c,f (5)
[0091] S2, performs dynamic constraint convexification based on Koopman operator.
[0092] In this embodiment, for a general nonlinear system ds(t) / dt=f(s(t)), the Koopman operator The discrete evolution form of the state observation function g(·) is given as:
[0093]
[0094] Among them, s(t) is the independent variable of the nonlinear system, s k+1 =F(s k ) is the discrete form of the original system.
[0095] When there is a control force u input, the Koopman operator realizes linear expression through the high-dimensional state z:
[0096] z + =A z +Bu
[0097] x=C z (7)
[0098] dim(z)>>dim(x)
[0099] Among them, A is the state transfer matrix, B is the control input matrix, C is the output matrix, z + represents the new state, and x represents the original state space.
[0100] However, in theory, the Koopman operator is an infinite-dimensional linear operator. In order to be able to apply it to practical problems, it must be approximated in finite dimensions. Therefore, in this embodiment, a set of state observation functions is selected:
[0101]
[0102] Among them, N ψ is the number of observation functions.
[0103] In order to ensure the quantity of data while expecting them to capture the nonlinearity of the original system, by using a set of random control inputs and a set of initial states distributed around the initial constraints, the evolved on-orbit spacecraft trajectory dataset is obtained as follows:
[0104]
[0105] in, All are trajectory point data sets (matrix), satisfying Represents a trajectory point.
[0106] definition is the trajectory point dataset after dimension increase based on the observation function ψ(x). Therefore, the computational problem based on the extended dynamic modulus decomposition Koopman operator can be described as:
[0107]
[0108] Among them, ||·|| F is the Frobenius norm.
[0109] Therefore, the analytical solution to this least squares optimization problem is:
[0110]
[0111] in, is the Moore-Penrose pseudoinverse.
[0112] Substituting the calculation result of Equation (11) into Equation (7) yields the linear dynamic model after dimensionality increase. Therefore, the original nonlinear dynamic constraint has been converted into a high-dimensional discrete linear form:
[0113] X k+1 =A koop X k +B koop u k ,k=1,2,...,N(12)
[0114] Among them, X k 、X k+1 is the dimensionality-increasing state value of the dynamic model, u k represents the thrust acceleration modulus, A koop 、B koop Represents the state transfer matrix and control input matrix after the convexification of the Koopman operator dynamic constraints.
[0115] S3, perform convexification of the no-fly zone process constraints.
[0116] In this embodiment, the no-fly zone constraint is processed by converting the original spherical constraint into a hyperplane constraint. These hyperplanes are formed by the tangent planes of the no-fly zone center mass violating the no-fly zone trajectory point. Therefore, in this case, the no-fly zone constraint becomes:
[0117]
[0118] in, Represents X p The first three dimensions of the dimensional state are equivalent to r because the first six dimensions of the selected observation function retain the original system state. c
[0119]
[0120] It is the set of discrete points p that violate the no-fly zone constraint in the reference trajectory.
[0121] The original objective function is the integral form of the thrust acceleration modulus over time, while in discrete form, it is expressed as the sum of the thrust acceleration modulus at discrete points:
[0122]
[0123] Where m is the number of discrete points.
[0124] In order to transform the objective function into a linear form, a set of slack variables η are introduced.i , let ||u i ||2≤η i , so the objective function and the original thrust acceleration constraint are converted to:
[0125]
[0126] 0≤η k ≤u max
[0127] Among them, u max is the maximum thrust acceleration of the spacecraft.
[0128] S4, construct the trust region and perform iterative calculations.
[0129] In this embodiment, in order to prevent the linear convexification process from causing the problem to be unbounded or pseudo-feasible solution, a trust region constraint is added during optimization. and slack constraints The trust region constraint on the thrust acceleration term is:
[0130]
[0131] The relaxation term constraint is used as a virtual compensation for the control quantity to compensate for the error when the dynamic constraint is insufficient. The form is:
[0132]
[0133] Here, i represents the iteration number.
[0134] Therefore, the above-mentioned trust region constraints and slack term constraints will be incorporated into the objective function to participate in the optimization, and the preset penalty coefficient ω will be set. u and ω f , the objective function is expanded to:
[0135]
[0136] It should be noted that the no-fly zone constraint processing process introduces a reference trajectory. When the current iteration number is greater than 1, the previous iteration result can be used as the reference trajectory. However, the first iteration calculation can only use the initial guess as the reference trajectory. To solve this problem, this embodiment decomposes the original problem into subproblems. In these subproblems, only the relaxed constraints of dynamics, initial terminal, and thrust acceleration are considered. The solution of the subproblem is used as the accurate reference trajectory for the collision avoidance constraint.
[0137] In this embodiment, the convergence condition of the optimization iteration is determined by the optimization variable and Decided, the conditions are as follows:
[0138]
[0139] Among them, ε u , ε f These are all preset convergence values.
[0140] As described above, the on-orbit spacecraft rendezvous trajectory optimization method provided in this embodiment uses global linearization based on the Koopman operator. By linearizing the dynamic constraints, it achieves higher approximation accuracy than traditional linearization methods. Furthermore, by selecting an appropriate observation function, it can further reduce the sensitivity of the optimization solution to the initial guess. This improves the fitting accuracy of the original nonlinear system, thereby enhancing the precision and quality of on-orbit spacecraft rendezvous trajectory optimization.
[0141] The following is a specific case to further illustrate the effect of the present invention. For the optimization of the specific on-orbit spacecraft rendezvous trajectory under minimum fuel consumption, the above-mentioned optimization method is used to solve and optimize. Figure 2 The following is the trajectory result at 1240s. Figure 3 The figure shows the trajectory result at 2340s. It can be seen that the on-orbit spacecraft avoided the no-fly zone and approached the desired position, achieving the purpose of trajectory optimization.
[0142] Figure 4 The figure shows the relative error comparison results between this method and the traditional linearization method (KL corresponds to the solution result based on the Koopman operator, and TL corresponds to the solution result based on the first-order Taylor linearization). It can be seen that the error of this method is significantly reduced compared with the traditional linearization method, thus proving the effectiveness of this method.
[0143] Based on the same inventive concept, this embodiment further provides a device comprising at least one processor and a memory communicatively connected to the at least one processor. The memory stores instructions executable by the at least one processor, and the instructions are executed by the at least one processor to enable the at least one processor to perform the aforementioned method for optimizing on-orbit spacecraft rendezvous trajectories based on a Koopman operator.
[0144] Based on the same inventive concept, this embodiment also provides a computer-readable storage medium on which a computer program is stored. When the program is executed by a processor, it implements the aforementioned on-orbit spacecraft rendezvous trajectory optimization method based on the Koopman operator.
[0145] The computer-readable medium includes, but is not limited to, any type of disk (including floppy disks, hard disks, optical disks, CD-ROMs, and magneto-optical disks), ROM, RAM, EPROM (Erasable Programmable Read-Only Memory), EEPROM, flash memory, magnetic cards, or optical cards. In other words, the computer-readable medium includes any medium that can store or transmit information in a form that can be read by a device (such as a printer).
[0146] The device, computer-readable storage medium, etc. provided in this embodiment have the same inventive concept and the same beneficial effects as the aforementioned method, and will not be described in detail here.
[0147] The technical features of the above embodiments can be combined arbitrarily. In order to make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.
[0148] The above embodiments merely illustrate several implementation methods of the present application. While the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the present application. It should be noted that a person of ordinary skill in the art may make various modifications and improvements without departing from the spirit of the present application, all of which fall within the scope of protection of the present application. Therefore, the scope of protection of the present application shall be determined by the appended claims.
Claims
1. A method for optimizing on-orbit spacecraft rendezvous trajectory based on Koopman operator, characterized in that: The steps include: S1, establishes a kinematic model for the on-orbit spacecraft rendezvous mission; The thrust acceleration of the spacecraft meets the amplitude constraint; the rendezvous trajectory is ensured to be outside the no-fly zone, and a spherical no-fly zone is established. The optimization goal is to minimize fuel consumption, and the objective function is the time integral of the thrust acceleration modulus. S2, performs dynamic constraint convexification based on Koopman operator; The Koopman operator gives a discrete evolution form of the observation function of the state. When there is a control force input, the Koopman operator realizes linear expression through high-dimensional states. A set of state observation functions is selected. By using a set of random control inputs and a set of initial states distributed around the initial constraints, the evolving on-orbit spacecraft trajectory dataset is obtained. The trajectory point dataset after the dimension increase of the observation function is defined. Based on the extended dynamic mode decomposition of the Koopman operator, an analytical solution is obtained, and the linear dynamic model after the dimension increase is obtained. The original nonlinear dynamic constraints are converted into a high-dimensional discrete linear form. S3, perform convexification of the no-fly zone process constraints; The spherical constraint is converted into a hyperplane constraint, where the hyperplane is composed of the tangent planes of the trajectory points where the mass of the no-fly zone center violates the no-fly zone; slack variables are introduced to convert the objective function into a linear form; S4, construct the trust region and perform iterative calculations; Trust region constraints and relaxation term constraints are added to participate in the optimization of the objective function. The trust region constraints constrain the thrust acceleration term, and the relaxation term constraints serve as virtual compensation for the control quantity to compensate for the error when the dynamic constraints are insufficient.
2. The method for optimizing on-orbit spacecraft rendezvous trajectory based on Koopman operator according to claim 1, characterized in that: The kinematic model established in S1 is: Among them, r c =[x c y c z c ] T is the vector from the spacecraft to the target point, u c is the thrust acceleration of the spacecraft, R is the relative position vector from the center of the Earth to the target point, ω is the orbital velocity of the rotating coordinate system, μ is the Earth's gravitational constant, R=|R|, 3. The on-orbit spacecraft rendezvous trajectory optimization method based on Koopman operator according to claim 2, characterized in that: The thrust acceleration of the spacecraft in S1 satisfies the amplitude constraint: ||u c (t)||2≤u max Among them, u max is the maximum thrust acceleration allowed, t is the time; The spherical no-fly zone is represented by: ||r c (t)-r out (t)||2≥r min Among them, r out (t) is the position of the center of the no-fly zone at time t in the LVLH coordinate system, r min is the radius of the no-fly zone.
4. The method for optimizing on-orbit spacecraft rendezvous trajectory based on Koopman operator according to claim 3, characterized in that: The objective function in S1 is: Among them, t0 is the initial time, t f is the target time, and J is the fuel consumption.
5. The method for optimizing on-orbit spacecraft rendezvous trajectory based on Koopman operator according to claim 4, characterized in that: When there is a control force u input in S2, the Koopman operator realizes linear expression through the high-dimensional state z: z + =Az+Bu x=Cz dim(z)>>dim(x) Among them, A is the state transfer matrix, B is the control input matrix, C is the output matrix, z + Represents the new state, and x represents the original state space; The selected state observation function table is Among them, N ψ is the number of observation functions, and g(·) is the observation function.
6. The method for optimizing on-orbit spacecraft rendezvous trajectory based on Koopman operator according to claim 5, characterized in that: The trajectory dataset of the spacecraft on orbit evolved in S2 is: in, All of them are trajectory point datasets, satisfying Represents a trajectory point; set is the trajectory point dataset after dimension increase based on the observation function ψ(x); definition Based on the trajectory point dataset after the dimension increase of the observation function ψ(x), the computational problem of the extended dynamic modulus decomposition Koopman operator is described as follows: Among them, ||·|| F is the Frobenius norm; The analytical solution to the least squares optimization problem is: in, is the Moore-Penrose pseudoinverse; Thus, the linear dynamic model after dimensionality increase is obtained, and the original nonlinear dynamic constraints have been converted into a high-dimensional discrete linear form: X k+1 =A koop X k +B koop u k ,k=1,2,…,Nwhere,X k 、X k+1 is the dimensionality-increasing state value of the dynamic model, u k represents the thrust acceleration modulus, A koop 、B koop Represents the state transfer matrix and control input matrix after the convexification of the Koopman operator dynamic constraints.
7. The method for optimizing on-orbit spacecraft rendezvous trajectory based on Koopman operator according to claim 6, characterized in that: The no-fly zone constraint in S3 is converted to: in, Represents X p The first three dimensions of the dimensional state are equivalent to r c , is the set of discrete points p that violate the no-fly zone constraint in the reference trajectory:
8. The method for optimizing on-orbit spacecraft rendezvous trajectory based on Koopman operator according to claim 7, characterized in that: In S3, the objective function in discrete form is expressed as the sum of the thrust acceleration modulus at discrete points: Where m is the number of discrete points; Introducing the slack variable η i , let ||u i ||2≤η i , the objective function and the original thrust acceleration constraint are converted to: 0≤η k ≤u max Among them, u max is the maximum thrust acceleration of the spacecraft.
9. The method for optimizing on-orbit spacecraft rendezvous trajectory based on Koopman operator according to claim 8, characterized in that: Adding trust region constraints in S4 and slack constraints The trust region constraint on the thrust acceleration term is: The form of the relaxation constraint to compensate for the error when the dynamic constraint is insufficient is: Where i represents the iteration number; Incorporate the trust region constraint and the relaxation term constraint into the objective function to participate in the optimization, combined with the preset penalty coefficient ω u and ω f , the objective function is expanded to:
10. The method for optimizing on-orbit spacecraft rendezvous trajectory based on Koopman operator according to claim 9, characterized in that: The convergence condition of the optimization iteration in S4 is determined by the optimization variable and Decision, subject to the following conditions: Among them, ε u , ε f These are all preset convergence values.