Spacecraft limited energy reachable set real-time control method and device, computer and storage medium

Abstract

The invention discloses a real-time control method and device for a limited energy reachable set of a spacecraft, a computer and a storage medium, belongs to the technical field of aerospace orbit control, and solves the problems of low real-time performance and result precision of the limited energy reachable set caused by insufficient high-dimensional and nonlinear characteristics of a spacecraft dynamic model in the prior art. And an analytical solution is difficult to directly obtain. The control method comprises the following steps: establishing a relative orbit dynamics system model; defining a finite energy reachable set; constructing a state uncertainty and control driving range model; establishing a gain set based on the model, and describing an initial shape of a finite energy reachable set; the Minkowski sum is used for carrying out equivalent conversion on the gain set, and a finite energy reachable set shape ellipsoid is obtained; and defining an ellipsoid shape function, and obtaining a finite energy reachable set optimal approximate solution through an ellipsoid trace optimal internal and external approximation method. The method is suitable for the technical field of spacecraft orbit dynamics and autonomous control, and can be applied to spacecraft orbit control research and development.

Description

Technical Field

[0001] The present invention belongs to the technical field of space orbit control, and particularly relates to the technical fields of spacecraft orbit dynamics and autonomous control. Background Art

[0002] Reachability analysis plays a core role in spacecraft mission planning and safety control, which can quantify the maximum reachable maneuvering range of a spacecraft under specific energy, thrust or pulse constraints. This method is widely applied to scenarios such as formation flying, pursuit-evasion games, on-orbit servicing, and life extension. In the on-orbit capture mission of a target spacecraft, accurate reachability calculation is crucial for avoiding collision risks and optimizing service paths. However, due to the high-dimensional and non-linear characteristics of the spacecraft dynamics model, the analytical solution of the reachable set is difficult to obtain directly, and it is usually necessary to rely on numerical approximation or geometric modeling methods to seek a balance between real-time performance and accuracy.

[0003] As a typical problem, the finite-energy reachable set focuses on the maximum maneuvering range of a spacecraft under the constraint of total energy, and is particularly suitable for safety analysis under abnormal engine conditions. Its advantage lies in relying only on the remaining energy constraint, avoiding errors introduced by abnormal thrust amplitude or attitude instability. However, the mathematical description of this set involves the Minkowski sum operation of high-dimensional convex bodies, that is, the superposition of the initial state transfer set and the controllable set. The exact solution of such problems requires the help of optimal control theory or functional analysis, with high computational complexity and difficult to meet the requirements of real-time tasks. Therefore, constructing efficient and high-precision inner / outer approximation methods has become a key challenge in engineering applications.

[0004] Early reachability research mainly adopted numerical solution strategies. For example, the set boundary was approximated by combining the optimal control framework with the grid discretization method, or the series approximation of the system evolution function was carried out using polynomial expansion. Although such methods can obtain high accuracy, the computational load increases exponentially with the dimension and it is difficult to meet the requirements of real-time planning. Another type of research is based on convex optimization theory, where the non-convex constraints are relaxed and then the approximate reachable set is solved, but significant conservatism is easily introduced in strongly non-linear scenarios dominated by control. In addition, geometric approximation of the reachable set using polyhedra or ellipsoidal envelopes is also widely used. Among them, ellipsoidal approximation has attracted much attention due to its simple mathematical form and convenience in integrating quadratic constraints, etc. However, its accuracy is limited by the selection strategy of the ellipsoidal shape matrix. Traditional methods usually solve it by quadratic programming, with low computational efficiency and without fully considering the influence of dynamic characteristics on the approximation error.

[0005] Regarding the approximation problem of the finite-energy reachable set, existing methods have limitations in specific scenarios. Early work deduced boundary conditions through verbal description and empirical assumptions, but overestimation would occur when the control action was dominant; subsequent research introduced a strict L2 space regularization method, but did not solve the weight optimization problem of the coupling between the initial state set and the system dynamics.

[0006] In summary, in the existing technology for calculating the reachable set of the spacecraft dynamics model, due to the lack of high-dimensionality and non-linear characteristics of the dynamics model, it is difficult to directly obtain the analytical solution, which affects the real-time performance and result accuracy of the reachable set. Summary of the Invention

[0007] The present invention solves the technical problem in the existing technology that due to the lack of high-dimensionality and non-linear characteristics of the spacecraft dynamics model, the real-time performance and result accuracy of the finite energy reachable set are low, making it difficult to directly obtain the analytical solution.

[0008] The present invention proposes a real-time control method for the finite energy reachable set of a spacecraft, and the method includes:

[0009] S1. Establish a relative orbit dynamics system model and define the finite energy reachable set using this model;

[0010] S2. Use the relative orbit dynamics system model to construct a state uncertainty and control drive range model;

[0011] S3. Based on the state uncertainty and control drive range model, establish a gain set to describe the initial shape of the finite energy reachable set;

[0012] S4. Use the Minkowski sum to perform an equivalent transformation on the gain set to obtain an ellipsoid representing the shape of the finite energy reachable set;

[0013] S5. Define the shape function of the ellipsoid and obtain the optimal approximate solution of the finite energy reachable set through the optimal inner and outer approximation method of the ellipsoid trace.

[0014] Furthermore, the process of S1 includes:

[0015] S101. Based on the linearized equation C-W, establish the relative orbit dynamics system model as follows:

[0016]

[0017] where A is the system matrix, n represents the average orbital angular velocity of the spacecraft, 0 and I represent the zero matrix and the identity matrix respectively; x(t) is the state of the spacecraft, indicating that the state vector includes relative position and velocity components; B is the input matrix, u(t) is the spacecraft control input, indicating the thrust acceleration;

[0018] S102. Based on the relative orbit dynamics system model, define the finite energy reachable set as follows:

[0019]

[0020] Among them, R represents the reachable set (mathematical script); t f is the end time, representing the time for calculating the reachable set; U represents the control feasible region; x(t f ) is the end state, representing the elements in the reachable set; t0 represents the initial time; t represents the time variable; d represents the differential symbol; x(t0) represents the initial state.

[0021] Furthermore, the state uncertainty and control drive range model described in S2 are as follows:

[0022]

[0023] Among them, τ represents the time variable.

[0024] Furthermore, the process of S3 includes:

[0025] S301. Decompose the state uncertainty and control drive range model into state gain and control gain;

[0026] S302. Establish the gain set; the gain set includes the state gain set and the control gain set;

[0027] S303. Describe the initial shape of the finite energy reachable set.

[0028] Furthermore, the process of S301 includes:

[0029] When the spacecraft control input u(t) is independent of the state x(t) for energy-optimal control, decompose the state uncertainty and control drive range model into state gain and control gain;

[0030] The state gain is: Φ(t f , t0)x(t0),

[0031] The control gain is:

[0032] Furthermore, the equivalent transformation of the gain set using the Minkowski sum in S4 results in the following finite energy reachable set shape ellipsoid:

[0033]

[0034] Among them, ε is the ellipsoid notation, is the Minkowski sum.

[0035] Furthermore, the ellipsoid trace optimal inner and outer approximation method described in S5 includes ellipsoid trace optimal inner approximation and ellipsoid trace optimal outer approximation. The process of obtaining the optimal inner and outer approximation solutions of the finite energy reachable set using this method includes:

[0036] S501. Define an ellipsoid using a convex geometric auxiliary function as follows:

[0037]

[0038] where q is the center of the ellipsoid, Q is the ellipsoid shape function, and l is an arbitrary direction vector; n n - dimensional Euclidean space;

[0039] S502. Solve the Minkowski sum of the ellipsoids;

[0040] Based on the convex geometric auxiliary function, the Minkowski sum of the ellipsoids ε1 = ε(q1, Q1) and ε2 = ε(q2, Q2) is:

[0041]

[0042] The relationship of the approximate solution of the ellipsoid is:

[0043] q = q1 + q2

[0044] (l, Ql) 1 / 2 ≈(l, Q1l) 1 / 2 +(l, Q2l) 1 / 2

[0045] The shape function of the inner - approximate ellipsoid is Q (p), and the shape function of the outer - approximate ellipsoid is The condition to be satisfied is

[0046] S503. Optimize based on the optimal inner - approximation and optimal outer - approximation of the ellipsoid trace;

[0047] Derive the outer - approximate solution of the finite - energy reachable set using inequalities as follows:

[0048]

[0049] Take the trace of the matrix as the optimization variable Obtain the unique optimal parameter: Furthermore, obtain the optimal outer - approximation of the finite - energy reachable set as:

[0050]

[0051] Derive the inner - approximate solution of the finite - energy reachable set using inequalities as follows:

[0052]

[0053] Take the trace of the matrix as the optimization variable f(p)=tr Q (p)], and obtain the unique optimal parameter: p *, and then the optimal inner approximation of the finite energy reachable set is obtained as:

[0054]

[0055] S504. Based on the optimal inner approximation solution and the optimal outer approximation solution, the optimal inner and outer approximation solution of the finite energy reachable set is:

[0056]

[0057] The present invention also provides a real-time control device for the finite energy reachable set of a spacecraft, and the device includes:

[0058] Orbit modeling module: used to establish a relative orbit dynamics system model and define the finite energy reachable set by using this model;

[0059] Model construction module: used to construct a state uncertainty and control drive range model by using the relative orbit dynamics system model;

[0060] Shape modeling module: used to establish a gain set based on the state uncertainty and control drive range model to describe the initial shape of the finite energy reachable set;

[0061] Reachable set transformation module: used to perform equivalent transformation on the gain set by using the Minkowski sum to obtain the finite energy reachable set shape ellipsoid;

[0062] Reachable set approximation optimization module: used to define the shape function of the ellipsoid and obtain the optimal approximation solution of the finite energy reachable set through the ellipsoid trace optimal inner and outer approximation method.

[0063] The present invention provides a computer, including a processor and a memory, and a computer program is stored in the memory. When the processor runs, it executes the computer program to implement the method described in any one of claims 1 to 7.

[0064] The present invention provides a storage medium, and a computer program is stored in the storage medium. When the computer program runs, it implements the method described in any one of claims 1 to 7.

[0065] Compared with the prior art, the beneficial effects of the present invention are:

[0066] (1) The approximate method based on the optimal ellipsoidal trace proposed in this invention solves the problem of real-time calculation of the reachable set of spacecraft under finite energy constraints and initial state errors by establishing an equivalent Minkowski sum mapping and a matrix trace optimization strategy, and realizes the rapid solution of high-precision analytical solutions. This method breaks through the contradiction between computational efficiency and approximation accuracy in traditional numerical methods, providing theoretical support for real-time path planning under the limited computing power of on-board processors. This method can significantly improve computational efficiency while ensuring high computational accuracy. By analytically approximating the shape of the reachable set, this method significantly reduces the computational complexity and avoids complex calculations such as numerical calculations and traversal operations. Compared with the most cited literature "Reachable set computation for spacecraft relative motion with energy-limited low-thrust" in the top journal "Aerospace Science and Technology" in the aerospace discipline, although their computational efficiencies are similar, the approximate method based on the optimal ellipsoidal trace proposed in this invention greatly improves the accuracy of reachable set estimation. Especially in the general case dominated by control gain, this method solves the overestimation problem in classical methods. It can be seen that this method can better approximate the exact solution compared with classical literature, and has comparable computational efficiency, overcoming the overestimation of the outer approximation solution in classical literature methods for the case dominated by control gain.

[0067] (2) The equivalent transformation method based on Minkowski sum in this invention realizes the efficient approximate calculation of the finite energy reachable set and improves the result accuracy of the finite energy reachable set real-time control method. First, through Minkowski sum operation, the state gain set and the control gain set are equivalently transformed to establish an analytical description of the reachable set. This avoids the computational burden caused by the direct calculation of high-dimensional convex bodies in traditional numerical methods, reduces the computational complexity, can more accurately represent the boundary of the reachable set, and at the same time ensures the feasibility of analytical calculation, making the calculation results more accurate and reliable. Second, this method uses the ellipsoidal approximation strategy to transform the calculation problem of Minkowski sum into a shape function optimization problem, and combines the matrix trace optimization strategy to solve the optimal inner and outer approximation solutions. This strategy avoids the overestimation problem in traditional methods in the case dominated by control gain while ensuring computational efficiency.

[0068] (3) The state uncertainty and control drive range model of the present invention accurately describes the on-orbit state evolution process of spacecraft, improving the accuracy and reliability of the calculation of the finite energy reachable set. First, by constructing a state uncertainty model and combining the dynamic characteristics of the spacecraft, the error range of the system state is accurately quantified; under the action of gravitational influence and time variation, the navigation error forms a state gain set, providing an accurate initial boundary for the calculation of the reachable set and avoiding the calculation deviation caused by state estimation errors in traditional methods. Second, the control drive range model reasonably quantifies the energy consumption of the spacecraft by setting a weight function, enabling the control gain set to accurately represent the maximum maneuvering range; this reduces the limitations of traditional fixed control assumptions, making the reachable set calculation more flexible, applicable to different energy constraints and thrust conditions, and improving the calculation adaptability and engineering application value. In addition, decomposing the state uncertainty and control drive range into state gain and control gain not only optimizes the calculation structure but also enhances the interpretability of the model, and the Minkowski sum operation provides a more reasonable technical basis for subsequent real-time control of spacecraft.

[0069] Generally, the state uncertainty and control drive range model of the present invention avoids the overestimation problem of the outer approximation solution in the case of dominant control gain in the classical method, improves the accuracy of the reachable set shape description, and makes the calculation result more in line with the actual physical characteristics.

[0070] The method described in the present invention is applicable to the field of aerospace orbit control technology. BRIEF DESCRIPTION OF THE DRAWINGS

[0071] The drawings constituting a part of the present invention are used to provide a further understanding of the present invention. The schematic embodiments and descriptions thereof of the present invention are used to explain the present invention and do not constitute an improper limitation of the present invention. In the drawings:

[0072] Figure 1 is a flowchart of a real-time control method for the finite energy reachable set of a spacecraft according to the present invention;

[0073] Figure 2 is a flowchart of the optimal inner and outer approximation solutions in the ninth specific embodiment;

[0074] Figure 3 is a schematic diagram of the approximate solution of the finite energy reachable set in the eleventh specific embodiment;

[0075] Figure 4 is a schematic diagram of the optimal approximation and calculation of the finite energy reachable set in the eleventh specific embodiment;

[0076] Figure 5 is a schematic diagram of the variation of the exact solution of the finite energy reachable set with time in the eleventh specific embodiment;

[0077] Figure 6Schematic diagram for comparing the exact solution and the inner approximation solution of the finite energy reachable set in Embodiment XI;

[0078] Figure 7 Schematic diagram for comparing the exact solution and the outer approximation solution of the finite energy reachable set in Embodiment XI;

[0079] Figure 8 Schematic diagram for comparing the outer approximation solutions of the method of the present invention and the traditional method in Embodiment XI;

[0080] Figure 9 Schematic diagram of the results of the method of the present invention and the traditional method under the comparison of long period and large energy in Embodiment XI. Specific Embodiment

[0081] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. It should be noted that, without conflict, the embodiments in the present invention and the features in the embodiments can be combined with each other. The described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments.

[0082] Specific Embodiment I, see Figure 1 This embodiment describes a real-time control method for the finite energy reachable set of a spacecraft. The method includes:

[0083] S1. Establish a relative orbit dynamics system model and define the finite energy reachable set using this model;

[0084] S2. Use the relative orbit dynamics system model to construct a state uncertainty and control drive range model;

[0085] S3. Based on the state uncertainty and control drive range model, establish a gain set to describe the initial shape of the finite energy reachable set;

[0086] S4. Use the Minkowski sum to perform an equivalent transformation on the gain set to obtain an ellipsoid of the finite energy reachable set shape;

[0087] S5. Define the shape function of the ellipsoid and obtain the optimal approximation solution of the finite energy reachable set through the ellipsoid trace optimal inner and outer approximation method.

[0088] In this embodiment, first, the present invention constructs a relative orbit dynamics system model to precisely define the finite energy reachable set, so as to improve the calculation accuracy and optimize the calculation efficiency. Secondly, based on the relative orbit dynamics system model, a state uncertainty and control drive range model is established, thereby quantifying the operable space of the spacecraft under different energy constraints, accurately describing the system state, and serving as the basis for the real-time control method of the finite energy reachable set. The gain set method introduced by the present invention, as a key tool for describing the initial shape of the finite energy reachable set, fully considers the system dynamics characteristics in the establishment of the gain set, making the approximate calculation of the reachable set more accurate. Especially in the case dominated by control gain, it can effectively reduce the overestimation problem in traditional methods and improve the credibility of the calculation results.

[0089] Furthermore, in order to reduce the calculation complexity and ensure the approximate accuracy at the same time, the present invention uses the Minkowski sum theory to equivalently transform the gain set and constructs an ellipsoidal approximation model of the finite energy reachable set. This method not only improves the calculation efficiency but also avoids the numerical instability problem caused by the dimension growth in the traditional polyhedron approximation method, making it more applicable to the high-dimensional orbit dynamics environment. The present invention defines an ellipsoidal shape function and combines the inner approximation optimization and outer approximation optimization strategies to obtain the optimal approximate solution of the finite energy reachable set; this optimization strategy can significantly improve the result accuracy while maintaining the calculation efficiency. Compared with traditional methods, it can better approximate the exact solution under the strong nonlinear conditions of spacecraft dynamics.

[0090] Generally speaking, a real-time control method for the finite energy reachable set of a spacecraft proposed by the present invention shows extensive engineering value in application scenarios such as spacecraft formation flight, orbit capture, obstacle avoidance control, and on-orbit maintenance, providing strong technical support for the autonomous planning and safety control of space missions.

[0091] Specific Embodiment 2, this embodiment further limits the S1 process described in Embodiment 1. The process of S1 includes:

[0092] S101. Construct a 6-dimensional state space model using the Clohessy-Wiltshire (C-W) linearization equation C-W, that is, the relative orbit dynamics system model is as follows:

[0093]

[0094] Among them, A is the system matrix, n represents the average orbital angular velocity of the spacecraft, 0 and I respectively represent the zero matrix and the identity matrix; x(t) is the spacecraft state, indicating that the state vector includes relative position and velocity components; B is the input matrix, u(t) is the spacecraft control input, indicating the thrust acceleration;

[0095] S102. Define the finite energy reachable set based on the relative orbit dynamics system model as follows:

[0096]

[0097] where \(R\) represents the set of reachable sets (mathematical script); \(t\) f is the terminal time, representing the time for calculating the reachable set; \(U\) represents the control feasible region; \(x(t\) f ) is the terminal state, representing the elements in the reachable set; \(t_0\) represents the initial time; \(t\) represents the time variable; \(d\) represents the differential symbol; \(x(t_0)\) represents the initial state.

[0098] This embodiment further defines the process of S1 in Embodiment 2. In this embodiment, based on the Clohessy-Wiltshire (C-W) linearization equation, a 6D state space model, i.e., the relative orbit dynamics system model, is constructed to accurately describe the relative orbit dynamics of the spacecraft, and the finite energy reachable set is defined to improve the calculation accuracy and optimization efficiency. Among them, the system matrix \(A\) describes the state evolution of the spacecraft, the state vector includes the relative position and velocity, the input matrix \(B\) depicts the influence of the thrust on the spacecraft, and the control input represents the thrust acceleration, so that the model can accurately quantify the operable space of the spacecraft under different energy constraints and provide a technical basis for the control method of the reachable set. Based on this, this embodiment defines the finite energy reachable set to determine the maximum maneuvering range of the spacecraft under energy constraints; compared with the traditional method, this method reduces the calculation complexity, improves the real-time calculation ability, and effectively overcomes the overestimation problem under the condition of dominant control gain, making the result of the real-time control method of the finite energy reachable set more accurate and reliable.

[0099] Specific Embodiment 3 is a further limitation on S2 described in Embodiment 2. The state uncertainty and control drive range model in S2 is as follows:

[0100]

[0101] where \(\tau\) represents the time variable.

[0102] This embodiment further defines the process of S2 in Embodiment 2. In this embodiment, based on the state uncertainty and control drive range model, the uncertainty of the spacecraft state and the influence of the control input on the orbit maneuver are quantified. By accurately modeling the state evolution and thrust constraints of the spacecraft, it is ensured that the dynamic characteristics are fully considered during the calculation process, thereby improving the accuracy of the real-time control method of the finite energy reachable set, achieving an accurate description of the operable space of the spacecraft under different energy constraints, and optimizing the accuracy and efficiency of the real-time control method of the finite energy reachable set.

[0103] Embodiment 4: This embodiment further defines the process of S3 described in Embodiment 3. The process of S3 includes:

[0104] S301. Decompose the state uncertainty and control drive range model into state gain and control gain;

[0105] S302. Establish the gain set; the gain set includes a state gain set and a control gain set;

[0106] S303. Describe the initial shape of the finite energy reachable set.

[0107] This embodiment further defines the process of S3 in Embodiment 3. In this embodiment, by decomposing the state uncertainty and control drive range model, the accuracy and efficiency of the finite energy reachable set calculation are improved. First, the state uncertainty and control drive range are decomposed into state gain and control gain to accurately describe the impact of spacecraft state changes and control inputs on the maneuverability; then, based on this decomposition, a gain set is established, including a state gain set and a control gain set, to realize the quantification of the reachable range of the system under different energy constraints; finally, the gain set is used to describe the initial shape of the finite energy reachable set, improving the accuracy of the real-time control method of the finite energy reachable set.

[0108] Embodiment 5: This embodiment further defines S301 described in Embodiment 4. The process of S301 includes:

[0109] When the spacecraft control input u(t) is energy-optimal control and is independent of the state x(t), decompose the state uncertainty and control drive range model into state gain and control gain;

[0110] The state gain is: Φ(t f , t0)x(t0),

[0111] The control gain is:

[0112] This embodiment further defines the process of S301 in Embodiment 4. In this embodiment, by decomposing the state uncertainty and control drive range model into state gain and control gain, the accuracy and efficiency of the finite energy reachable set calculation are improved.

[0113] Embodiment 6: This embodiment further defines the process of S302 described in Embodiment 5. The process of S302 is as follows:

[0114] Use a Kalman filter to calculate the navigation error of the spacecraft state as follows:

[0115]

[0116] Among them, x0 is the state estimation value; X0 is the covariance matrix; X0 is the coupling variable, representing the orbit determination error and the velocity measurement error;

[0117] The navigation error χ0 forms a state gain set ε(x0(t), X0(t)) that describes the range of spacecraft state uncertainty with the influence of gravity and time variation:

[0118] Set the weight function R to quantify the energy consumption of the spacecraft as:

[0119]

[0120] where L2 represents the second-order integrable space, and u(τ) T represents the transpose of the control;

[0121] The maximum maneuvering range of the spacecraft control drive is the control gain set: ε(0, 2W(t f , t0)).

[0122] This embodiment further defines the process of S302 in Embodiment 5. In this embodiment, the Kalman filter is used to estimate the state of the spacecraft. Among them, the state estimation value is the actual position and velocity of the spacecraft, the covariance matrix describes the uncertainty of the estimation error, and the orbit determination error and the velocity measurement error respectively quantify the orbit and velocity measurement errors of the spacecraft. With the influence of gravity and time variation, the navigation error will affect the range of spacecraft state uncertainty, and this error is described by the state gain set, thereby quantifying the dynamic changes of the spacecraft. The weight function is used to quantify the energy consumption of the spacecraft, and the parameters in the weight function respectively represent different influencing factors of the spacecraft energy consumption.

[0123] Specific Embodiment 7, this embodiment further defines the process of S303 in Embodiment 6. The process of S303 includes:

[0124] The shape function 2W(t f , t0) of the control gain set describes the outer shape of the finite energy reachable set;

[0125] Based on the state estimation value and the covariance matrix of the state gain set, the center of the sphere of the finite energy reachable set is described as: Φ(t f , t0)x(t0);

[0126] Traversing the irregular convex body of the state gain ellipsoid represents the initial shape of the finite energy reachable set as:

[0127]

[0128] This embodiment further defines the process of S301 described in Embodiment 1. In this embodiment, the effective combination of the control gain set and the state gain set describes the initial shape of the finite energy reachable set. Among them, the shape function of the control gain set is used to describe the outer shape of the finite energy reachable set, so as to quantify the maximum maneuvering range of the spacecraft under energy constraints; the state estimate value and covariance matrix of the state gain set can describe the center position of the finite energy reachable set, ensuring the reasonable distribution of the uncertainty of the spacecraft state within the reachable set; the traversal of the irregular convex body of the state gain ellipsoid represents the initial shape of the finite energy reachable set, effectively overcoming the problem of accuracy loss in traditional methods.

[0129] Specific Embodiment 8. This embodiment further defines the S4 process described in Embodiment 7. In the S4, the Minkowski sum is used to equivalently transform the gain set to obtain the shape ellipsoid of the finite energy reachable set as follows:

[0130]

[0131] Among them, ε is the ellipsoid notation, is the Minkowski sum.

[0132] This embodiment further defines the S4 process described in Embodiment 7. In this embodiment, the Minkowski sum is used to equivalently transform the gain set, further improving the calculation accuracy of the finite energy reachable set. The Minkowski sum describes the maximum operable space of the spacecraft under energy constraints by precisely combining the state gain and the control gain, thus effectively simplifying the calculation process and improving the accuracy.

[0133] Specific Embodiment 9. This embodiment further defines the S5 process described in Embodiment 8. As Figure 2 shown, the ellipsoid trace optimal inner and outer approximation method described in S5 includes the ellipsoid trace optimal inner approximation and the ellipsoid trace optimal outer approximation. The process of obtaining the optimal inner and outer approximation solutions of the finite energy reachable set by using this method includes:

[0134] S501. Define an ellipsoid by using a convex geometry auxiliary function as follows:

[0135]

[0136] Among them, q is the ellipsoid center, Q is the ellipsoid shape function, and l is an arbitrary direction vector; n represents the n-dimensional Euclidean space.

[0137] S502. Solve the Minkowski sum of the ellipsoids;

[0138] Based on the convex geometry auxiliary function, the Minkowski sum of the ellipsoid ε1 = ε(q1, Q1) and the ellipsoid ε2 = ε(q2, Q2) is:

[0139]

[0140] The relationship of the ellipsoidal approximate solution is:

[0141] q = q1 + q2

[0142] (l, Ql) 1 / 2 ≈(l, Q1l) 1 / 2 +(l, Q2l) 1 / 2

[0143] The shape function of the inner approximate ellipsoid is Q (p), and the shape function of the outer approximate ellipsoid is The satisfaction condition is

[0144] S503. Based on the optimal inner approximation optimization and optimal outer approximation optimization of the ellipsoidal trace;

[0145] The outer approximate solution of the finite energy reachable set is derived by using the indeterminate form as follows:

[0146]

[0147] Take the trace of the matrix as the optimization variable Get the unique optimal parameter: Furthermore, the optimal outer approximation of the finite energy reachable set is:

[0148]

[0149] The inner approximate solution of the finite energy reachable set is derived by using the inequality as follows:

[0150]

[0151] Take the trace of the matrix as the optimization variable f(p) = tr Q (p)], and get the unique optimal parameter: p * , and furthermore, the optimal inner approximation of the finite energy reachable set is:

[0152]

[0153] S504. Based on the optimal inner approximate solution and optimal outer approximate solution, the optimal inner and outer approximate solution of the finite energy reachable set is:

[0154]

[0155] This embodiment further defines the process of S5 described in Embodiment 8. In this embodiment, an ellipsoid is defined using a convex geometric auxiliary function to accurately describe the geometric shape of the finite energy reachable set. Among them, the center of the ellipsoid represents the reference position of the reachable set, and the ellipsoid shape function determines its expansion range, thereby accurately describing the maximum maneuverability of the spacecraft under energy constraints. Based on the state gain set and the control gain set, the shape of the reachable set is calculated and the Minkowski sum of the ellipsoids is solved. This method integrates the effects of different gains through geometric superposition, making the shape description of the reachable set more accurate; this solution introduces the inner and outer approximation methods, and approximates the reachable set with the optimal inner and outer ellipsoids. During the optimal approximation solution process, the inner approximation solution and the outer approximation solution are optimized respectively through inequality derivation, and the trace of the matrix is used as the optimization variable to ensure the uniqueness and accuracy of the calculation; furthermore, by combining the optimal inner and outer approximation solutions, a high-precision description of the finite energy reachable set is constructed, making the calculation results not only conform to the dynamic characteristics of the spacecraft, but also effectively avoid the overestimation problem in traditional methods. This embodiment further improves the calculation efficiency, improves the real-time performance while ensuring the calculation accuracy, and is more adaptable to the requirements of complex space missions.

[0156] Specific Embodiment 10. This embodiment further defines a real-time control device for the finite energy reachable set of a spacecraft. The device includes:

[0157] Orbit modeling module: used to establish a relative orbit dynamics system model and define the finite energy reachable set using this model;

[0158] Model construction module: used to construct a state uncertainty and control drive range model using the relative orbit dynamics system model;

[0159] Shape modeling module: used to establish a gain set based on the state uncertainty and control drive range model to describe the initial shape of the finite energy reachable set;

[0160] Reachable set transformation module: used to perform equivalent transformation on the gain set using the Minkowski sum to obtain the ellipsoid of the finite energy reachable set shape;

[0161] Reachable set approximation optimization module: used to define the shape function of the ellipsoid and obtain the optimal approximation solution of the finite energy reachable set through the optimal inner and outer approximation method of the ellipsoid trace.

[0162] This embodiment further defines a real-time control device for the finite energy reachable set of a spacecraft. In this embodiment, this solution proposes an efficient calculation device for the finite energy reachable set, which improves the calculation accuracy and real-time performance through modular design.

[0163] First, the orbit modeling module is responsible for establishing a relative orbit dynamics system model and precisely defining the finite energy reachable set based on this model, providing a dynamics foundation for subsequent calculations. Then, the model construction module constructs a state uncertainty and control drive range model through this orbit dynamics system model, thereby quantifying the operable space of the spacecraft under different energy constraints and making the calculation of the reachable set more accurate. Then, the shape modeling module establishes a gain set based on the state uncertainty and control drive range model to describe the initial shape of the finite energy reachable set, ensuring that the geometric representation of the reachable set conforms to the actual dynamics characteristics. On this basis, the reachable set transformation module uses the Minkowski sum to equivalently transform the gain set to obtain the ellipsoidal shape of the finite energy reachable set, making the calculation more concise and efficient while reducing the computational complexity. Finally, the reachable set approximation optimization module obtains the optimal approximate solution of the finite energy reachable set by defining the shape function of the ellipsoid and combining inner approximation optimization and outer approximation optimization. This optimization strategy effectively avoids the overestimation problem of traditional methods and improves the calculation accuracy.

[0164] Overall, through modular design, the device realizes the efficient modeling, optimization calculation, and real-time solution of the finite energy reachable set, and is applicable to application scenarios such as spacecraft orbit maneuvering, obstacle avoidance planning, and mission optimization.

[0165] Specific Embodiment XI, the following combines the attached Figure 2 to limit the solution of the present invention. In this embodiment,

[0166] The verification orbit is a GEO orbit, and the initial navigation error is:

[0167] x0(t0) = [10, 10, 10, 1, 1, 1] T m, m / s

[0168] X0(t0) = diag([8000, 4000, 2000, 0.01, 0.01, 0.0324]) m 2 , m 2 / s 2

[0169] The control feasible region is:

[0170]

[0171] Figure 3 The exact solution of the finite energy reachable set obtained by directly traversing the union of ellipsoids as shown.

[0172] Figure 4 、 Figure 5 、 Figure 6The main content is to plot the exact solutions of the finite energy reachable set and the corresponding optimal approximate solutions at different times at intervals of 1000s within an orbital period (86400s), and distinguish them by the colors of the examples in the right figure. Figure 4 The solution in is the exact solution obtained by solving the Minkowski sum, and the enlarged part describes the exact solution of the reachable set at 43200s and its corresponding Monte Carlo experimental results. The results show that the Minkowski sum equivalent transformation used in the present invention can completely represent the finite energy reachable set. Figure 5 The outer side of the main body is the reachable set exact solution, and the inner side is the optimal approximate solution; the outer side of the corresponding enlarged part is a comparative experiment of the reachable set exact solution and the optimal inner approximate solution of 43200s. Figure 6 The outside of the main body is the optimal outer approximate solution of the reachable set, and the inside is the exact solution of the reachable set; the outside of the corresponding enlarged part is a 43200s comparison experiment of the optimal outer approximate solution of the reachable set and the exact solution of the reachable set.

[0173] Figure 7 It is a comparison between the method of the present invention and the inner approximate solution of the classical literature; Figure 8 It is a comparison between the method of the present invention and the outer approximate solution of the classical literature; Figure 9 This is the result of the comparison between the method of the present invention and classical literature under long period and high energy.

[0174] Figure 7 , Figure 8 and Figure 9 For comparison with classic literature, the initial value of simulation navigation error is:

[0175] x0(t0)=0,X0(t0)=diag([500 2 ,300 2 ,200 2 ,0.05 2 ,0.03 2 ,0.02 2 ])m 2 ,m 2 / s 2

[0176] Figure 7 and Figure 8 The corresponding energy weight matrix is ​​R = c -1 I,c=[6,60,600,6000]×10 -7 m 2 / s 3 , the corresponding times are 1 / 4T, 1 / 2T, 3 / 4T, 1T respectively; Figure 9 Specific simulation results of comparative experiments with large energy R=I and long period 1T, where the method of the present invention is drawn in red, the exact solution is drawn in blue, and the method of the classical literature is drawn in green.Figure 7 In the drawn inner approximation comparison experiment, the red result of this paper is closer to the exact solution and has a higher approximation degree than the green result of the classical literature; Figure 8 also shows that the outer approximation result proposed by the present invention is superior to the classical literature, and the error between the classical literature method and the exact solution increases with time and energy, but the approximation degree of the method of the present invention to the exact solution is very high. Figure 9 The simulation results of large energy and long period are verified. The upper part describes the inner approximation of the method of the present invention and the classical method to the exact solution, and both can approximate the exact solution well; the method of the present invention in the lower part maintains a high outer approximation to the exact solution, but the approximation error of the classical method is large.

[0177] As shown by Figure 7 , 8, and 9, combined with the comparison of the calculation efficiency and accuracy of different methods in the calculation of the finite energy reachable set shown in Table 1, the method of the present invention can approximate the exact solution better than the classical literature, and the calculation efficiency is comparable, and it overcomes the overestimation of the outer approximation solution of the classical literature method in the case of control gain dominance.

[0178] Let R = c -1 ·I c = 6×10 -6 m 2 / s 3 , the simulation time and accuracy at 1 / 2T moment are summarized in Table 1 as follows:

[0179]

[0180]

[0181] The specific embodiments of the present invention disclosed above are only used to help explain the present invention. The specific embodiments do not describe all the details in detail, nor limit the invention to the specific embodiments described. According to the content of this specification, many modifications and changes can be made. These embodiments are selected and specifically described in this specification to better explain the principle and practical application of the present invention, so that those skilled in the art in the relevant technical field can understand and utilize the present invention well.

Claims

1. A real-time control method for the reachable set of a spacecraft with limited energy, characterized in that, The method includes: S1. Establish a relative orbit dynamics system model and define a finite energy reachable set using this model; S2. Construct a state uncertainty and control drive range model using the relative orbit dynamics system model; S3. Establish a gain set based on the state uncertainty and control drive range model to describe the initial shape of the finite energy reachable set; S4. Use the Minkowski sum to perform an equivalent transformation on the gain set to obtain an ellipsoid of the finite energy reachable set shape; S5. Define a shape function of the ellipsoid and obtain an optimal approximation solution of the finite energy reachable set through the optimal inner and outer approximation method of the ellipsoid trace.

2. The real-time control method for the reachable set of limited energy of a spacecraft according to claim 1, wherein The process of S1 includes: S101. Establish the relative orbit dynamics system model based on the linearized equation C-W as follows: where A is the system matrix, n represents the average orbital angular velocity of the spacecraft, 0 and I represent the zero matrix and the identity matrix respectively; x(t) is the spacecraft state, indicating that the state vector contains relative position and velocity components; B is the input matrix, u(t) is the spacecraft control input, representing the thrust acceleration; S102. Define the finite energy reachable set based on the relative orbit dynamics system model as follows: Among them, R represents the set of reachable sets (mathematical script); t f is the terminal time, representing the time for calculating the reachable sets; U represents the control feasible region; x(t f ) is the terminal state, representing the elements in the reachable set; t0 represents the initial time; t represents the time variable; d represents the differential symbol; x(t0) represents the initial state.

3. The real-time control method for the reachable set of limited energy of a spacecraft according to claim 2, wherein The state uncertainty and control drive range model in S2 is as follows: where τ represents the time variable.

4. The real-time control method for the reachable set of limited energy of a spacecraft according to claim 3, characterized in that, The process of S3 includes: S301. Decompose the state uncertainty and control drive range model into state gains and control gains; S302. Establish the gain set; the gain set includes a state gain set and a control gain set; S303. Describe the initial shape of the finite energy reachable set.

5. The real-time control method for the reachable set of limited energy of a spacecraft according to claim 4, characterized in that The process of S301 includes: When the spacecraft control input u(t) is independent of the state x(t) for energy-optimal control, decompose the state uncertainty and control drive range model into state gains and control gains; The state gain is: Φ(t f , t0)x(t0), The control gain is as follows:

6. The real-time control method for the reachable set of limited energy of a spacecraft according to claim 5, characterized in that The equivalent transformation of the gain set using the Minkowski sum in S4 to obtain an ellipsoid of the finite energy reachable set shape is as follows: where ε is the ellipsoid notation, is the Minkowski sum.

7. The real-time control method for the reachable set of limited energy of a spacecraft according to claim 6, characterized in that The optimal inner and outer approximation method of the ellipsoid trace in S5 includes the optimal inner approximation of the ellipsoid trace and the optimal outer approximation of the ellipsoid trace. The process of obtaining the optimal inner and outer approximation solutions of the finite energy reachable set using this method includes: S501. Define an ellipsoid using a convex geometry auxiliary function as follows: where q is the ellipsoid center, Q is the ellipsoid shape function, and l is an arbitrary direction vector; n n-dimensional Euclidean space; S502. Solve the Minkowski sum of the ellipsoids; Based on the convex geometry auxiliary function, the Minkowski sum of the ellipsoid ε1 = ε(q1,Q1) and the ellipsoid ε2 = ε(q2,Q2) is: The relationship of the ellipsoid approximation solution is: q = q1 + q2 (l, Ql) 1 / 2 ≈(l, Q1l) 1 / 2 +(l, Q2l) 1 / 2 The shape function of the inner approximate ellipsoid is Q (p), and the shape function of the outer approximate ellipsoid is The satisfaction condition is S503. Optimize based on the optimal inner approximation of the ellipsoid trace and the optimal outer approximation; Derive the outer approximation solution of the finite energy reachable set using an indefinite form as follows: Take the trace of the matrix as the optimization variable Obtain the unique optimal parameter: Furthermore, the optimal outer approximation of the finite energy reachable set is obtained as: Derive the inner approximation solution of the finite energy reachable set using an inequality as follows: Take the trace of the matrix as the optimization variable \(f(p)=\text{tr} Q (p)]\), and obtain the unique optimal parameter: \(p * \), and further obtain the optimal inner approximation of the finite energy reachable set as: S504. Based on the optimal inner approximation solution and the optimal outer approximation solution, the optimal inner and outer approximation solution of the finite energy reachable set is:

8. A real-time control device for the reachable set of a spacecraft with limited energy, characterized in that, The device includes: Orbit modeling module: used to establish a relative orbit dynamics system model and define a finite energy reachable set using this model; Model construction module: used to construct a state uncertainty and control drive range model using the relative orbit dynamics system model; Shape modeling module: used to establish a gain set based on the state uncertainty and control drive range model to describe the initial shape of the finite energy reachable set; Reachable set transformation module: used to perform an equivalent transformation on the gain set using the Minkowski sum to obtain an ellipsoid of the finite energy reachable set shape; Reachable set approximation optimization module: used to define the shape function of the ellipsoid, and obtain the optimal approximate solution of the finite energy reachable set through the optimal inner and outer approximation method of the ellipsoid trace.

9. A computer, comprising a processor and a memory, characterized in that, A computer program is stored in the memory, and the processor executes the computer program during operation to implement the method according to any one of claims 1 to 7.

10. A storage medium, characterized in that, A computer program is stored in the storage medium, and the computer program implements the method according to any one of claims 1 to 7 during operation.

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