Method and device for orbit deviation evolution and correction of a spacecraft

Abstract

The application provides a spacecraft orbit deviation evolution and correction method and device, and belongs to the technical field of aerospace. The method quantifies the nonlinear order of the dynamic model of the spacecraft first, and then determines the evolution deviation of the spacecraft and the correction pulse margin based on the quantified nonlinear order in combination with the initial value problem and the boundary value problem of the ordinary differential equation, reduces the complexity of the orbit deviation evolution and deviation correction process, improves the accuracy of the results of the orbit deviation evolution and deviation correction, and enables the spacecraft to better complete the task.

Description

Technical Field

[0001] This invention relates to the field of aerospace technology, and in particular to a method and apparatus for the evolution and correction of orbital deviations of spacecraft. Background Technology

[0002] Because spacecraft (such as satellites and space shuttles) experience various unavoidable errors during mission execution (such as orbit insertion error, orbit determination error, and orbital maneuver execution error), their actual motion can deviate from their nominal orbit, preventing them from completing their mission. Therefore, it is necessary to understand the evolution of these deviations during mission execution and apply corrective pulses to bring the spacecraft back to its nominal orbit.

[0003] However, due to the highly nonlinear changes in the motion state of spacecraft, the existing orbital deviation evolution and correction process is highly complex, and the results of orbital deviation evolution and correction have large errors with the actual situation, which makes it difficult for spacecraft to complete their missions well. Summary of the Invention

[0004] This invention proposes a method and apparatus for orbital deviation evolution and correction of spacecraft. First, the nonlinear order of the spacecraft's dynamic model is quantified. Then, based on the quantized nonlinear order, the initial value problem and boundary value problem of the ordinary differential equation are combined to determine the evolutionary deviation and correction pulse margin of the spacecraft. This reduces the complexity of the orbital deviation evolution and correction process, improves the accuracy of the results, and enables the spacecraft to complete its mission better.

[0005] To achieve the above objectives, the present invention adopts the following technical solution: Firstly, this invention provides a method for the evolution and correction of orbital deviations of a spacecraft. The spacecraft needs to move within its nominal orbit to perform a mission. The method includes: quantifying the nonlinear order of the spacecraft's dynamic model based on its nominal orbit, a preset motion state deviation, and a conjugate unscented transformation method; wherein the preset motion state deviation describes the deviation between the spacecraft's preset motion state and its target motion state, the target motion state referring to the spacecraft's motion state on its nominal orbit; the nonlinear order describes the computational complexity of the spacecraft's dynamic model. Then, based on the nonlinear order of the spacecraft's dynamic model, the motion state formula of the spacecraft's orbit, and the generalized chaotic polynomial, the evolutionary deviation of the spacecraft is determined; the evolutionary deviation demonstrates the evolution of the deviation between the spacecraft's actual motion state and its target motion state over time; the motion state formula of the spacecraft's orbit indicates the change law of the spacecraft's motion state over time and is constructed based on the initial value problem of ordinary differential equations. Based on the nonlinear order of the spacecraft's dynamic model, the correction pulse formula for the spacecraft's orbit, and the generalized chaotic polynomial, the correction pulse margin of the spacecraft is determined. The correction pulse margin is used to adjust the actual motion state of the spacecraft to the target motion state. The correction pulse formula for the spacecraft's orbit describes the velocity increment required for the spacecraft to reach the target motion state from the current motion state and is constructed based on the boundary value problem of ordinary differential equations.

[0006] This invention provides a method for the evolution and correction of spacecraft orbital deviations. It employs a conjugate unscented transformation method to transform the nominal orbit and preset motion state deviations of the spacecraft to determine the nonlinear order of the spacecraft's dynamic model. Then, by combining the nonlinear order and generalized chaotic polynomials, the motion state formula constructed based on the initial value problem of ordinary differential equations is analyzed to obtain the spacecraft's evolutionary deviation. Finally, by combining the nonlinear order and generalized chaotic polynomials, the correction pulse formula constructed based on the boundary value problem of ordinary differential equations is analyzed to obtain the spacecraft's correction pulse margin. This process does not require complex analysis of the space environment or the actual dynamic model of the spacecraft, making it a non-invasive method. This reduces the complexity of the orbital deviation evolution and correction process, improves the accuracy of the results, and enables the spacecraft to better complete its mission.

[0007] In one implementation of the first aspect, the nonlinear order of the spacecraft's dynamic model is quantified, including steps 1 to 5.

[0008] Step 1: Obtain the spacecraft's nominal orbit and preset motion state deviation; the nominal orbit includes the spacecraft's initial motion state; the preset motion state deviation... satisfy , N () represents a normal distribution. The variance matrix of the preset motion state deviation, The mean value of the preset motion state deviation; Step 2: Construct a dynamic model of the spacecraft based on the nominal orbit and the preset motion state deviation; the dynamic model satisfies: ,in, Indicates the actual motion state of the spacecraft. A mapping relationship indicating the state of motion. Indicates the initial motion state of the spacecraft. This indicates the deviation from the preset motion state. This represents the i-th moment within a preset time period; Step 3: Set the order of the conjugate unscented transformation method h The initial value is 1; Step 4: For each of the multiple moments contained within the preset time period, the conjugate unscented transform method is used to obtain multiple integration nodes for that moment. The order distance between the moments is then calculated using these integration nodes and the dynamic model. Each integration node satisfies... ,in, Indicates the first q The motion state deviation of each integration node q Integer and q ∈[1,2( h +1)+2 h+1 ]; Indicates the first q Preset motion state deviation of each integration node; express The decomposition matrix, and T = ; Among them, when the order of the conjugate unscented transformation method h When the order distance is 1, the following conditions are met: When the order of the conjugate unscented transformation method h When the order distance is greater than 1, the following conditions are met: in, express h The order distance at time i when = 1, express h =1 hour q The integral weights of each integral node. Represents the norm; express k The order distance at time i when >1, Indicateh >1 hour q The integral weight of each integral node; Step 5: When the order distance between all times within the preset time period is less than the convergence criterion... ε At that time, 2 h -1 or 2 h -2 is determined as the nonlinear order of the spacecraft's dynamic model; otherwise, the order of the conjugate unscented transformation method is used. h Updated to h +1 and return to step 4.

[0009] Secondly, this invention provides a spacecraft orbit deviation evolution and correction device. The spacecraft needs to move within its nominal orbit to perform its mission. The device includes an order quantization module, an evolution deviation determination module, and a correction pulse margin determination module. The order quantization module is used to quantify the nonlinear order of the spacecraft's dynamic model based on the spacecraft's nominal orbit, a preset motion state deviation, and a conjugate unscented transformation method. The preset motion state deviation describes the deviation between the spacecraft's preset motion state and its target motion state, where the target motion state refers to the spacecraft's motion state on its nominal orbit. The nonlinear order describes the computational complexity of the spacecraft's dynamic model. The evolution deviation determination module is used to determine the spacecraft's evolution deviation based on the nonlinear order of the spacecraft's dynamic model, the spacecraft orbit motion state formula, and a generalized chaotic polynomial. The evolution deviation demonstrates the evolution of the deviation between the spacecraft's actual motion state and its target motion state over time. The spacecraft orbit motion state formula indicates the change in the spacecraft's motion state over time and is constructed based on the initial value problem of ordinary differential equations. The corrected pulse margin determination module is used to determine the corrected pulse margin of the spacecraft based on the nonlinear order of the spacecraft's dynamic model, the corrected pulse formula of the spacecraft orbit, and the generalized chaotic polynomial. The corrected pulse margin is used to adjust the actual motion state of the spacecraft to the target motion state of the spacecraft. The corrected pulse formula of the spacecraft orbit describes the velocity increment required for the spacecraft to reach the target motion state from the current motion state and is constructed based on the boundary value problem of ordinary differential equations.

[0010] In one implementation of the first and second aspects, the evolutionary deviation of the spacecraft is described by the mean, variance, and covariance of the spacecraft's motion states. The mean of the motion state satisfies: The variance of the motion state satisfies: The covariance of the motion state satisfies: in, Let i represent the i-th component of the spacecraft's motion state. The spacecraft's motion state satisfies the formula for the motion state of its orbit, and , Let be the orthogonal basis functions of the motion state. The coefficients of the orthogonal basis functions of the motion state are . , , , The first term represents the result obtained after transformation by the conjugate unscented transformation method. q The integral weights of each integral node. The formula representing the motion state of the spacecraft orbit in the direction of the i-th component is as follows: Let k be the orthogonal basis function numbered k in the generalized chaotic polynomial that constructs the deviation evolution process. The first term represents the result obtained after transformation by the conjugate unscented transformation method. q Each integration node q Integer and q ∈[1,2( l +1)+2 l+1 ], l The nonlinear order of the spacecraft's dynamic model. The mean of the i-th component representing the motion state of the spacecraft; This represents the constant term in the generalized chaotic polynomial constructed for the i-th component of the spacecraft's motion state; Let represent the variance of the i-th component of the spacecraft's motion state. The first [unclear] refers to the motion state of the spacecraft. i The coefficients of the orthogonal basis function numbered m in the generalized chaotic polynomial constructed from the components. Let m denote the orthogonal basis function numbered m in the generalized chaotic polynomial; The j-th component represents the motion state of the spacecraft. This represents the covariance between the i-th and j-th components of the spacecraft's motion state. Let m be the coefficient of the orthogonal basis function numbered m in the generalized chaotic polynomial constructed for the j-th component of the spacecraft's motion state.

[0011] In one implementation of the first and second aspects, the spacecraft's correction pulse margin satisfies: in, The magnitude of the correction pulse margin of the spacecraft. Indicates reliability index, This represents the magnitude of the correction pulse for the spacecraft. Indicates the correction pulse of the spacecraft. It satisfies the corrected pulse formula for the spacecraft orbit, and , express The i-th component, Describes the orthogonal basis functions of the modified pulse. Let represent the coefficients of the orthogonal basis functions of the modified pulse, and , , , The first term represents the result obtained after transformation by the conjugate unscented transformation method. q The integral weights of each integral node. The formula for the corrected pulse of the spacecraft orbit in the direction of the i-th component is given. Let j be the orthogonal basis function numbered j in the generalized chaotic polynomial used to construct the modified impulse quantity. The first term represents the result obtained after transformation by the conjugate unscented transformation method. q Each integration node q Integer and q ∈[1,2( l +1)+2 l+1 ], l This represents the nonlinear order of the spacecraft's dynamics model. The probability density distribution function represents the corrected pulse quantity that meets the reliability index. The probability density distribution function is obtained based on the Monte Carlo numerical simulation method and the corrected pulse formula for the spacecraft orbit.

[0012] In one implementation of the first and second aspects, the formula for the motion state of the spacecraft orbit satisfies: in, Indicates the end time. Indicates the spacecraft's motion state at the final moment. Indicates the initial motion state of the spacecraft. The mapping relationship indicating the state of motion, at the same time Formulas representing the motion state of a spacecraft's orbit.

[0013] In one implementation of the first and second aspects, the formula for the correction pulse of the spacecraft orbit satisfies: in, Indicates the initial time. Indicates the end time. Indicates the spacecraft's position at the initial moment. Indicates the spacecraft's position at the time of termination. This represents the correction pulse required to move the spacecraft from its initial position to its final position. The formula for the corrected pulse of a spacecraft orbit.

[0014] Thirdly, the present invention provides an electronic device including a processor and a memory coupled to the processor; the memory is used to store computer instructions, and when the electronic device is running, the processor executes the computer instructions stored in the memory to cause the electronic device to perform the method described in the first aspect above or any implementation thereof.

[0015] Fourthly, the present invention provides a computer-readable storage medium including computer program instructions that, when executed by a computer, cause the computer to perform the method described in the first aspect above or any implementation thereof.

[0016] Fifthly, the present invention provides a computer program product, including computer program instructions, which, when executed on a computer, cause the computer to perform the method described in the first aspect above or any implementation thereof.

[0017] The technical effects corresponding to the second to fifth aspects and their possible implementations can be referred to the above description of the technical effects of the first aspect and its possible implementations, and will not be repeated here. Attached Figure Description

[0018] Figure 1 This is a schematic diagram of a method for the evolution and correction of orbital deviations of a spacecraft, provided in an embodiment of this application. Figure 2 This is a schematic diagram of the integral node positions of the two-dimensional unscented transform provided in the embodiments of this application; Figure 3 This is a schematic diagram of the integral node positions of the two-dimensional conjugate unscented transform provided in the embodiments of this application; Figure 4 This is a schematic diagram of the integral node positions of the three-dimensional conjugate unscented transform provided in the embodiments of this application; Figure 5 This is a schematic diagram showing the quantitative analysis results of the nonlinearity of the Earth-Moon transfer system provided in the embodiments of this application; Figure 6 This is a schematic diagram of the relative values ​​of the coefficients of a position-dependent third-order chaotic polynomial provided in an embodiment of this application; Figure 7 This is a schematic diagram of the relative values ​​of the coefficients of a velocity-related third-order chaotic polynomial provided in an embodiment of this application; Figure 8This is a comparison chart showing the distribution of z-axis position errors obtained from 2000 simulations using a surrogate model and the Monte Carlo method, as provided in an embodiment of this application. Figure 9 This is a comparison chart showing the distribution of z-axis velocity errors obtained from 2000 simulations using a surrogate model and the Monte Carlo method, as provided in an embodiment of this application. Figure 10 This is a comparison chart of the statistical characteristics of z-axis velocity using the surrogate model and the Monte Carlo method provided in the embodiments of this application; Figure 11 This is a comparison chart of the statistical characteristics of the z-axis position using the surrogate model and the Monte Carlo method provided in the embodiments of this application; Figure 12 This is a schematic diagram of the PCE coefficients related to the x-axis velocity, y-axis velocity, and z-axis velocity in the corrected pulse quantity provided in the embodiments of this application; Figure 13 The corrected pulse quantity provided in the embodiments of this application is related to v 2 A diagram illustrating the relevant PCE coefficients; Figure 14 This is a comparison diagram of the distribution of the surrogate model provided in the embodiments of this application and the corrected pulse obtained from 500 Monte Carlo simulations in the x-direction velocity; Figure 15 This is a comparison diagram of the distribution of the surrogate model provided in the embodiments of this application and the corrected pulse obtained from 500 Monte Carlo simulations in the y-direction velocity; Figure 16 This is a comparison diagram of the distribution of the surrogate model provided in the embodiments of this application and the corrected pulse obtained from 500 Monte Carlo simulations in the z-direction velocity; Figure 17 This is a comparison diagram of the pulse distribution obtained by the surrogate model provided in the embodiments of this application and the pulse distribution obtained by the Monte Carlo method; Figure 18 This is a comparison chart of the modified impulse statistical characteristics of the surrogate model provided in the embodiments of this application and the Monte Carlo method; Figure 19 This is a schematic diagram of the structure of a spacecraft orbital deviation evolution and correction device provided in an embodiment of this application. Detailed Implementation

[0019] In the specification and claims of this invention, the terms "first" and "second," etc., are used to distinguish different objects, rather than to describe a specific order of objects.

[0020] In the embodiments of this application, "and / or" indicates a relationship between objects. For example, A and / or B can represent the following three situations: A exists alone, B exists alone, and A and B exist simultaneously.

[0021] In the embodiments of this application, the terms "exemplary" or "for example" are used to indicate that something is an example, illustration, or description. Any embodiment or design that is described as "exemplary" or "for example" in the embodiments of this application should not be construed as being more preferred or advantageous than other embodiments or design. Specifically, the use of the terms "exemplary" or "for example" is intended to present the relevant concepts in a specific manner.

[0022] In the description of this invention, unless otherwise stated, "a plurality of" means two or more. For example, a plurality of motion states means two or more motion states.

[0023] The methods and apparatus provided in this application relate to the control of spacecraft motion state and can be used for the evolution and correction of spacecraft orbital deviations.

[0024] The following example, using a spacecraft that is a lunar probe and is carrying out a lunar exploration mission, will explain the necessity of the aforementioned orbital deviations and deviation corrections.

[0025] The lunar probe's mission mainly involves designing a parking orbit around Earth, a lunar transfer orbit, and a mission orbit around the Moon. However, during the lunar transfer, unavoidable uncertainties exist, such as orbital insertion errors during separation from the launch vehicle, orbit determination errors during navigation and orbital positioning, and errors in orbital maneuvers during the transfer. This results in a difference between the lunar probe's actual orbit and its nominal orbit. Furthermore, due to the highly nonlinear and sensitive characteristics of the Earth-Moon system, this difference increases significantly over time, eventually leading to unintended orbits at the end of the trajectory, increasing the burden of terminal guidance, or even preventing the probe from being captured by the Moon. Mid-course corrections must be performed at certain points. Therefore, it is necessary to study the evolution of orbital deviations during the lunar transfer and formulate appropriate mid-course correction strategies to ensure the successful execution of the lunar exploration mission.

[0026] To address the problems in the prior art where the existing orbital deviation evolution and correction processes are highly complex and the results deviate significantly from reality, thus hindering spacecraft from fulfilling their missions effectively, this application provides a method and apparatus for orbital deviation evolution and correction. First, the nonlinear order of the spacecraft's dynamic model is quantified. Then, based on the quantized nonlinear order, the initial and boundary value problems of the ordinary differential equations are combined to determine the spacecraft's evolutionary deviation and correction pulse margin. This reduces the complexity of the orbital deviation evolution and correction process, improves the accuracy of the results, and enables the spacecraft to fulfill its mission more effectively.

[0027] For example, the spacecraft orbital deviation evolution and correction method provided in this embodiment of the invention can be executed by an electronic device with processing capabilities, such as a computer or server. Taking a computer as an example, the hardware components of the computer may include: a processor, memory, a network interface, a user interface, a communication bus, etc.

[0028] The processor controls the electronic equipment to perform related processing and computational tasks, such as quantifying the nonlinear order of the spacecraft's dynamic model, determining the spacecraft's evolutionary deviations, and determining the spacecraft's correction pulse margin. The processor may include a central processing unit (CPU) or other processors, and may be single-core or multi-core; for example, the processor may include multiple CPUs.

[0029] The memory is used to store computer instructions and related data, such as the nonlinear order of a spacecraft's dynamic model, evolutionary deviations of the spacecraft, and correction pulse margins. The memory can be random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM), flash memory, optical storage, magnetic disk storage media, or other magnetic storage devices, or any other medium capable of storing program code or data accessible by a computer. Optionally, the memory can be integrated into the processor, or it can be independent of the processor.

[0030] A network interface is used for communication between a computer and other devices or communication networks. A network interface can be a transceiver with transmit and receive capabilities. Optionally, a network interface may include standard wired interfaces or wireless interfaces (such as Wi-Fi interfaces, Bluetooth interfaces, and 5G interfaces).

[0031] The communication bus is used to enable communication between different components. For example, the processor, memory, network interface and user interface mentioned above can be interconnected through the communication bus.

[0032] The user interface may include a display screen and an input unit (such as a keyboard). Optionally, the user interface may also include a standard wired interface or a wireless interface.

[0033] Those skilled in the art will understand that the computer described above may include more or fewer components, or combine certain components, or have different component arrangements; the embodiments of this application do not limit this.

[0034] In this embodiment, the spacecraft needs to move within its nominal orbit to perform the mission. For example, such as... Figure 1 As shown in the embodiment of this application, a method for the evolution and correction of orbital deviation of a spacecraft includes S101-S103.

[0035] S101. Based on the nominal orbit of the spacecraft, the preset motion state deviation, and the conjugate unscented transformation method, the nonlinear order of the spacecraft's dynamic model is quantified.

[0036] Here, the nominal orbit of the spacecraft refers to the operational orbit required for the spacecraft to achieve its mission; the preset motion state deviation describes the deviation between the preset motion state and the target motion state, whereby the target motion state refers to the spacecraft's motion state on the nominal orbit; and the nonlinear order describes the computational complexity of the spacecraft's dynamic model.

[0037] It should be understood that the Conjugate Unscented Transformation (CUT) method described above, as an improvement on the Unscented Transformation (UT), follows deterministic rules, selecting a minimal but information-rich set of "sample points" and assigning weights to them to calculate the mean, variance (i.e., "moments"), and other statistical properties of a random variable after a nonlinear function transformation. Using the CUT method to quantize the nonlinear order of a spacecraft's dynamic model significantly reduces the computational load required for the aforementioned process compared to Monte Carlo quantization.

[0038] In one implementation, S101 includes steps 1 to 5.

[0039] Step 1: Obtain the nominal orbit and preset motion state deviation of the spacecraft.

[0040] The aforementioned nominal orbit includes the spacecraft's initial motion state. This initial motion state refers to the spacecraft's motion state at the initial moment. The aforementioned preset motion state deviation... satisfy , N () represents a normal distribution. The variance matrix of the preset motion state deviation, The mean value of the preset motion state deviation (generally set to 0).

[0041] Step 2: Construct a dynamic model of the spacecraft based on the nominal orbit and the preset motion state deviation.

[0042] Step 2.1: Construct an initial dynamic model based on the nominal orbit.

[0043] Since the change in the motion state of a spacecraft is a nonlinear system, and a nonlinear system in fact establishes a nonlinear mapping relationship between input variables and output variables, the initial dynamic model based on the nominal orbit satisfies the following formula.

[0044] in, Indicates input variables, Indicates the output variable. It represents the non-linear mapping relationship between input variables and output variables.

[0045] Step 2.2: Based on the initial dynamic model, apply preset motion state deviations to construct the spacecraft's dynamic model.

[0046] The aforementioned preset motion state deviation is applied to the input variables of the initial dynamic model. Thus, the dynamic model of the spacecraft is obtained. The above dynamic model satisfies: ,in, Indicates the actual motion state of the spacecraft. A mapping relationship indicating the state of motion. Indicates the initial motion state of the spacecraft. This indicates the deviation from the preset motion state. This represents the i-th moment within a preset time period.

[0047] Step 3: Set the order of the conjugate unscented transformation method h The initial value is 1.

[0048] Step 4: For each of the multiple moments contained in the preset time period, the conjugate unscented transformation method is used to transform the moment to obtain multiple integration nodes, and the order distance of the moment is calculated through the multiple integration nodes and the dynamic model.

[0049] Each of the above multiple integration nodes satisfies ,in, Indicates the first qThe motion state deviation of each integration node q Integer and q ∈[1,2( h +1)+2 h+1 ]; Indicates the first q Preset motion state deviation of each integration node; express The decomposition matrix, and T = .

[0050] For the above conjugate unscented transformation method, when the order of the conjugate unscented transformation method... h When the order distance is 1, the following conditions are met: When the order of the conjugate unscented transformation method h When the order distance is greater than 1, the following conditions are met: in, express h The order distance at time i when = 1, express h =1 hour q The integral weights of each integral node. Represents the norm; express k The order distance at time i when >1, Indicate h >1 hour q The integral weights of each integral node.

[0051] The transformation process of the above-mentioned conjugate unscented transformation method is described in detail below.

[0052] Assuming that the nonlinear mapping relationship of the spacecraft's motion state change is smooth, and considering the isochronous variational relationship, the above dynamic model can be transformed into the following form.

[0053] Where, δ y δ represents the perturbation value of the mapped output. x This represents the perturbation value of the mapped input. x = x x . x The nominal point represents the initial state of motion of a spacecraft in its nominal orbit.

[0054] Using Einstein's summation convention (when the same index appears once in the upper right corner and once in the lower left corner of a monomial, it is assumed that the index is summed over all possible values), the summation sign is omitted and written in the form of tensor component indices. The above formula can be expanded into the following form.

[0055] In the two formulas above and All represent mapped outputs y AND mapping f The i-th component represents time. t At the nominal point The value at the location, component i , j , k , l This represents the tensor component index. d This represents the dimension of the input variable x, when the initial motion state of the spacecraft is... At that time, the above dimensions d It is 6.

[0056] After obtaining the formula in the above form, the i-th component of the above mapping output is applied using the moment matching idea. Find the expected value. As can be understood, the moment matching concept described above refers to adjusting the parameters of a probability distribution so that several of its moments are equal to the corresponding moments of the target distribution or sample. These moments can include the first-order raw moment (mean, describing the central location of the distribution), the second-order central moment (variance, describing the dispersion of the distribution), and the third-order central moment (skewness, describing the asymmetry of the distribution), etc.

[0057] Specifically, based on practical engineering experience, it is believed that the spacecraft's orbital insertion error and initial navigation and orbit determination error can be quantified using a probability distribution, i.e., the input variable deviation. δx Given the probability distribution that satisfies the condition, and considering that the values ​​of each order sensitivity matrix (i.e., the partial derivatives below) are only functions of the nominal point and time, the i-th component of the above mapping output... The expectation satisfies the following formula (1).

[0058] Formula (1) The left side of the above formula (1) is a high-dimensional definite integral problem, which can be solved by numerical integration, resulting in the following formula (2).

[0059] Formula (2) Solving the above formulas (1) and (2) yields the following formula (3).

[0060] Formula (3) Specifically, if we let the deviation amount δx Each dimension is independent and follows a standard normal distribution. The expansion order is 2, and a nominal point is chosen (with integral weights set to...). w The distances from the nominal point on the coordinate axes of the Cartesian coordinate system are 0) and 1. r The point (with integral weight set as) w 1) As an integration node, for d In Euclidean space, the number of such integration nodes is q =2 d +1, with x Taking the axis as an example, the nodes are respectively: ,in e i Indicates along the first i The basis vectors of the coordinate axes (0 (1) ,0 (2) ,……,1 (i) ,……,0 (n) Only counted in x For the case on the axis, the moment matching equation it satisfies is given by substituting the specific values ​​into the following equation.

[0061] Based on the symmetry of the selected nodes and the symmetry of the deviation distribution, it can be found that the second and third equations obviously satisfy (both equal to 0), therefore there are only two valid equations: ,Replenish Alternatively, the constraint that minimizes the fourth-order center distance error yields a closed system of equations (explanatory description). This is the idea behind the Unscented Transform (UT) method. The integration nodes of the two-dimensional unscented transform are as follows: Figure 2 As shown.

[0062] As discussed above, the UT method restricts sample points to the coordinate axes, thus failing to match cross moments. To match higher-order moments, it's not sufficient to limit integration nodes to the principal axes. Furthermore, to ensure the favorable condition that odd-order moments are naturally zero, integration nodes should be selected at symmetrical positions. The CUT method utilizes this approach to selecting integration nodes, defining a first principal axis (based on the basis vector direction) and a second principal axis (based on the direction vector). The integration nodes are obtained by using the first or second principal axis as the axis of symmetry to find the points symmetrical to the sample points. In the two-dimensional case, the integration node selection scheme of the CUT method is as follows: Figure 3 As shown, the integration node selection scheme in the three-dimensional case is as follows: Figure 4 As shown.

[0063] To further illustrate the CUT method described above, the expected expansion is truncated to the fourth order, resulting in the corresponding moment matching equation (odd-order moments are omitted) as shown below.

[0064] The four formulas above correspond to the 0th-order moment, the 2nd-order central moment, the 4th-order cross moment, and the 4th-order central moment, respectively. Based on these four formulas, supplementary... Alternatively, the constraint that minimizes the sixth-order center distance error can be used to obtain a closed system of equations, which, when solved, yields the integration nodes and weights. This method, because it can precisely match the fourth-order moments, is called the CUT4 method. It can be observed that this method has fifth-order algebraic accuracy. For a 6-dimensional random variable, the number of integration nodes required by this method is 76 (…). ).

[0065] Step 5: When the order distance between all times within the preset time period is less than the convergence criterion... ε (Right now When ), 2 h -1 or 2 h -2 is determined as the nonlinear order of the spacecraft's dynamic model; otherwise, the order of the conjugate unscented transformation method is used. h Updated to h +1 and return to step 4.

[0066] It can be seen that the above-mentioned method of using CUT to quantify the nonlinearity of spacecraft motion models is non-invasive and does not require the system to provide explicit mapping relationships, which has significant advantages for the analysis of black box dynamic models.

[0067] The following is a verification process for the feasibility of the nonlinear order of the dynamic model of the quantized spacecraft in S101.

[0068] The above quantization scheme is based on three assumptions (as shown below), which hold true for general nonlinear dynamic systems that are quite sensitive to various factors of initial value perturbation.

[0069] Assumption 1: The magnitude of a higher-order sensitivity partial derivative is always a smaller higher-order quantity of the magnitude of a lower-order sensitivity partial derivative.

[0070] Assumption 2: The nonlinearity of the system is stable over a certain time interval, and if a certain sensitivity is not negligible, then this sensitivity should be time-varying.

[0071] Assumption 3: The mapping relationship is smooth enough.

[0072] Proof: According to Assumption 3, expand the mapping at the nominal point into a Taylor polynomial of arbitrary order; then, according to Assumption 1, we can illustrate this using a first-order distance as an example.k The same applies to the next distance.

[0073] Therefore: Furthermore, according to assumption two, the linear distance can be viewed as the inner product of vectors in a finite-dimensional Euclidean space, with respect to a fixed vector. δx (q)j δ x (q)k orthogonal vectors Since the distance is always finite, if discrete sampling and distance calculation converges within a stable nonlinear time interval, it indicates that this convergence is not due to an "orthogonality" relationship. In fact, the probability of such an "orthogonality" relationship occurring is extremely low. Sampling 1-2 time nodes near the point of interest is sufficient, thus acknowledging that: If the first-order distance converges, it indicates that the second-order sensitivity matrix has a very small influence on the system and can be ignored. This means that the nonlinear system can be expanded using a first-order polynomial, which is algebraic first-order (not considering linear systems).

[0074] If the first distance does not converge, it indicates that the second-order sensitivity matrix has a significant impact on the system, and it is necessary to improve the accuracy of moment matching and calculate the second distance.

[0075] This approach does not require solving for the Jacobi, Hessian, and higher-order sensitivity matrices of the nonlinear system, nor does it require explicitly defining the specific nonlinear system model, which is essential for black-box, modularly designed system models. Furthermore, this method uses the computationally inexpensive conjugate unscented transform (CUT) to measure the error caused by higher-order terms, thus quantifying the nonlinearity of the nonlinear system with minimal computational cost.

[0076] S102. Determine the evolutionary deviation of the spacecraft based on the nonlinear order of the spacecraft's dynamic model, the motion state formula of the spacecraft's orbit, and the generalized chaotic polynomial.

[0077] The above-mentioned formula for the motion state of a spacecraft orbit indicates the change of the spacecraft's motion state over time and is constructed based on the initial value problem of ordinary differential equations. Optionally, the formula for the motion state of a spacecraft orbit satisfies the following formula (4).

[0078] Formula (4) in, Indicates the end time. Indicates the spacecraft's motion state at the final moment. Indicates the initial motion state of the spacecraft. The mapping relationship indicating the state of motion, at the same time Formulas representing the motion state of a spacecraft's orbit.

[0079] The aforementioned evolutionary deviation illustrates the evolution of the discrepancy between the spacecraft's actual motion state and its target motion state over time. This evolutionary deviation is described by the mean, variance, and covariance of the spacecraft's motion state.

[0080] For example, the mean of the above motion states satisfies the following formula (5).

[0081] Formula (5) The variance of the above motion states satisfies the following formula (6).

[0082] Formula (6) The covariance of the above motion states satisfies the following formula (7).

[0083] Formula (7) in, The first character representing the motion state of the spacecraft i Each component ( i = 1,2,…,6), the motion state of the spacecraft satisfies the motion state formula of the spacecraft orbit, and is obtained through the generalized chaotic polynomial. ( k = 0,1,2,… P -1), P The number of monomials contained in a polynomial is determined by formula (10). The orthogonal basis functions for the motion states are determined using the Askey scheme shown in Table 1. The coefficients of the orthogonal basis functions of the motion state are . , , , The first term represents the result obtained after transformation by the conjugate unscented transformation method. q The integral weights of each integral node. The formula representing the motion state of the spacecraft orbit in the direction of the i-th component is as follows: Let k be the orthogonal basis function numbered k in the generalized chaotic polynomial that constructs the deviation evolution process. The first term represents the result obtained after transformation by the conjugate unscented transformation method. q Each integration node q Integer and q ∈[1,2( l+1)+2 l +1 ], l The nonlinear order of the spacecraft's dynamic model. The mean of the i-th component representing the motion state of the spacecraft; This represents the constant term in the generalized chaotic polynomial constructed for the i-th component of the spacecraft's motion state; Let represent the variance of the i-th component of the spacecraft's motion state. Let m be the coefficient of the orthogonal basis function numbered m in the generalized chaotic polynomial constructed for the i-th component of the spacecraft's motion state. Let m denote the orthogonal basis function numbered m in the generalized chaotic polynomial; The j-th component represents the motion state of the spacecraft. This represents the covariance between the i-th and j-th components of the spacecraft's motion state. Let m be the coefficient of the orthogonal basis function numbered m in the generalized chaotic polynomial constructed for the j-th component of the spacecraft's motion state.

[0084] The process of determining the evolutionary deviations of the aforementioned spacecraft is described in detail below.

[0085] Step 1: Construct the above formula (4).

[0086] Since nonlinear dynamic systems often use ordinary differential equations to describe the nonlinear mapping relationship of the spacecraft's motion state changes, and since the initial value problem of ordinary differential equations has the ability to describe the evolution process of the dynamic system after giving the initial values, the above formula (4) is constructed based on the initial value problem of ordinary differential equations.

[0087] In one application scenario, taking the Earth-Moon transfer of a probe as an example, the dynamic model of the spacecraft is adopted, considering the gravitational perturbation terms of the Sun and Moon and the perturbation term of Earth's oblateness J2 during the Earth-Moon transfer process: in, It is the gravitational constant of celestial bodies. These are the probe's position vector and velocity vector relative to Earth. It is the detector relative to the celestial body i position vector, It is the position vector of celestial body i relative to the Earth. The initial motion state is obtained by reading the DE440 ephemeris. Let the initial deviation satisfying the normal distribution be . . This represents the position component of the detector on the x-axis in the ECI coordinate system (i.e., the Earth-centered Earth-fixed coordinate system) at the initial moment. This represents the position component of the detector on the y-axis of the ECI coordinate system at the initial moment. This represents the position component of the detector on the z-axis of the ECI coordinate system at the initial moment. This represents the velocity component of the detector on the y-axis of the ECI coordinate system at the initial moment. This represents the velocity component of the detector on the y-axis of the ECI coordinate system at the initial moment. This represents the velocity component of the detector on the y-axis of the ECI coordinate system at the initial moment; This represents the position deviation component of the detector on the x-axis of the ECI coordinate system at the initial moment. This represents the initial position deviation component of the detector on the y-axis of the ECI coordinate system. This represents the initial position deviation component of the detector on the z-axis of the ECI coordinate system. This represents the position deviation component of the detector on the x-axis of the ECI coordinate system at the initial moment. This represents the initial position deviation component of the detector on the y-axis of the ECI coordinate system. This represents the position deviation component of the detector on the z-axis of the ECI coordinate system at the initial moment.

[0088] Step 2: Transform the nonlinear mapping relationship embodied in the above formula (4) into the form of tensor component index, as shown below.

[0089] The elements of the sensitivity matrix in the above form ( )use b The form of the index is used to obtain the following formula (8).

[0090] Formula (8) Step 3: Solve based on generalized chaotic polynomials and the CUT method .

[0091] Perform the above formula (8) p Rank ( p The value of the order is equal to the nonlinear order of the dynamic model obtained in S101 above. h Truncate and replace the deviation in each term with an unknown polynomial. Combine like terms to obtain new combination coefficients. c ,get y i The simplified formula is shown in formula (9) below.

[0092] Formula (9) in, The number corresponding to the i-th component of the deviation is The order of the polynomial should satisfy the constraints. , This indicates the total order of the polynomial. Let be the orthogonal basis functions of the motion state. These are the coefficients of the orthogonal basis functions of the motion state. Therefore, it can be considered that... .

[0093] It is easy to prove that for d Dimensional input variables, p The total number of terms in the truncated proxy model. P It satisfies the following formula (10).

[0094] Formula (10) The above The selection adopts the one-dimensional orthogonal basis function in the Askey scheme. When the component satisfies a certain probability distribution (e.g., Gaussian distribution), the unknown expression can be quickly obtained in the function space of its corresponding standard probability distribution (i.e., standard normal distribution) using the Gram-Schmidt orthogonalization method, i.e., a set of orthogonal polynomials. The weight function has the same form as the standard probability density function corresponding to the deviation, and the convergence speed can reach exponential. Therefore, in the embodiments of this application, the motion state deviations corresponding to different probability distributions are... The above orthogonal basis functions The selection rules are shown in Table 1 below.

[0095] Table 1. Optional orthogonal basis functions corresponding to the probability deviation distribution of motion states.

[0096] therefore, It is the product of one-dimensional orthogonal basis functions. If the components of the bias are independent of each other, the weight function of the high-dimensional function space can also be directly written as the product of the weight functions of the corresponding function spaces of the components.

[0097] To better understand the following method, the weight function and orthogonal basis function mentioned above will be introduced below.

[0098] For a one-dimensional continuous function over the real number field, the following mapping relationship is defined: If selected reasonably If this mapping satisfies the fundamental property of the inner product in Euclidean space, then this mapping relation is defined as the inner product in the function space. It is called the weight function. Then it is called a function. f and g Weighted orthogonal under the action of the weight function, real number Called a function f The length in the sense of the weight function. The basis vectors in the function space. It can be transformed into orthogonal basis functions that depend on the weight function, just like an array in Euclidean space, through Gram-Schmidt orthogonalization.

[0099] Under the premise that the deviation between the true value and the predicted value should be minimized as required by the above formula (4), the orthogonal basis function should be selected reasonably. The length of the vector has a minimum value, i.e., the length of the vector is a minimum value. The length of the vector is the smallest. The length of satisfies the following formula (11).

[0100] Formula (11) For the above formula (11) Taking the derivative, we obtain the equation satisfied by the coefficients of the orthogonal basis functions. Solving the equation yields the following formula (12).

[0101] Formula (12) Then for the deviations that are independent of each component, and the corresponding weight function All can be written as the product of one-dimensional orthogonal basis functions and weight functions to separate variables. Therefore, as long as the value of one one-dimensional inner product is 0, the value of this multi-dimensional inner product is 0. The resulting variable separation formula is shown below.

[0102] in, This is Kronecker notation, a mathematical notation that encapsulates the logical principle of "equal to 1, unequal to 0". It should be noted that the above formula does not use the Einstein summation convention for summation.

[0103] Based on the above variable separation formula, the coefficients of the orthogonal basis functions of the motion state can be quickly obtained. The coefficients of the orthogonal basis functions of the above motion states satisfy .

[0104] For numerical integrals of the numerator without analytical expression in the above formula The calculation process can be found in the relevant content about the CUT method mentioned above. , In the calculation After that, you can pass (i.e., a surrogate model of the spacecraft orbital motion state formula) is solved to obtain .

[0105] Step 4: Analyze the spacecraft's deviation evolution process using a surrogate model based on the motion state formula.

[0106] For the deviation evolution process of spacecraft, the surrogate model of the motion state formula can be statistically analyzed using Monte Carlo numerical simulation, or the polynomial coefficients of the surrogate model of the motion state formula can be directly used for statistical analysis.

[0107] Based on the above formulas (1), (3) and the above proxy formula, the first moment (the mean of the motion state), the second center distance (the variance of the motion state), and the covariance (the covariance of the motion state) of the spacecraft motion state can be calculated.

[0108] The above first moment ( k Indicates from 0 to P The summation satisfies the following formula.

[0109] The above second-order center distance ( m Indicates from 1 to P Summation, P The above formula (10) satisfies the following formula.

[0110] The above covariance ( m Indicates from 1 to P The summation satisfies the following formula.

[0111] S103. Determine the correction pulse margin of the spacecraft based on the nonlinear order of the spacecraft's dynamic model, the correction pulse formula for the spacecraft's orbit, and the generalized chaotic polynomial.

[0112] The correction pulse margin is used to adjust the actual motion state of the spacecraft to the target motion state of the spacecraft; the correction pulse formula for the spacecraft orbit describes the velocity increment required for the spacecraft to reach the target motion state from the current motion state and is constructed based on the boundary value problem of ordinary differential equations.

[0113] For example, the formula for the corrected pulse of the above spacecraft orbit satisfies the following formula (13).

[0114] Formula (13) in, Indicates the initial time. Indicates the end time. Indicates the spacecraft's position at the initial moment. Indicates the spacecraft's position at the time of termination. This represents the correction pulse required to move the spacecraft from its initial position to its final position. The formula for the corrected pulse of a spacecraft orbit.

[0115] The correction pulse margin of the above spacecraft satisfies the following formula (14).

[0116] Formula (14) in, The magnitude of the correction pulse margin of the spacecraft. Indicates reliability index, This represents the magnitude of the correction pulse for the spacecraft. Indicates the correction pulse of the spacecraft. It satisfies the corrected pulse formula for the spacecraft orbit, and , express The i-th component, Describes the orthogonal basis functions of the modified pulse. Let represent the coefficients of the orthogonal basis functions of the modified pulse, and , , , The first term represents the result obtained after transformation by the conjugate unscented transformation method. q The integral weights of each integral node. The formula for the corrected pulse of the spacecraft orbit in the direction of the i-th component is given. Let j be the orthogonal basis function numbered j in the generalized chaotic polynomial used to construct the modified impulse quantity. The first term represents the result obtained after transformation by the conjugate unscented transformation method. q Each integration node q Integer and q ∈[1,2( l +1)+2 l+1 ], l This represents the nonlinear order of the spacecraft's dynamics model. The probability density distribution function represents the corrected pulse quantity that meets the reliability index. The probability density distribution function is obtained based on the Monte Carlo numerical simulation method and the corrected pulse formula for the spacecraft orbit.

[0117] The process of determining the correction pulse margin of the aforementioned spacecraft is described in detail below.

[0118] Step 1: Construct the dynamic equations of the spacecraft and solve for the differential equations satisfied by the state transition matrix.

[0119] The dynamic equations of the above spacecraft satisfy Applying isochronous variational methods to the dynamic equations of the aforementioned spacecraft, we obtain the isochronous variational formulas for the dynamic equations. Let the solution of the isochronous variational formula of the above dynamic equations satisfy the following formula (15).

[0120] Formula (15) in, Here is the state transition matrix. t 0 represents the initial moment of the orbital transfer.

[0121] Differentiating both sides of the above formula (15) with respect to time yields the following formula (16).

[0122] Formula (16) The differential equation satisfied by the state transition matrix can be obtained by the above formula (16), as shown in the following formula (17).

[0123] Formula (17) Taking a lunar probe as an example, whose mission is to transfer between Earth and the Moon, the differential equation satisfied by the state transition matrix of the lunar probe during the Earth-Moon transfer process is as follows.

[0124] The variable coefficient part of the above differential equation is shown below.

[0125] As can be seen from the above formula, the state transition matrix can be obtained through the spacecraft's position vector. r Determined, that is, whenever the spacecraft's position vector r If changes occur, the spacecraft's state transition matrix can be obtained by solving the above formula.

[0126] Step 2: Construct the orbital transfer formula and perform multiple differential corrections on the orbital transfer formula to obtain the correction pulse.

[0127] According to the above Numerical integration of a transfer trajectory with initial deviation and numerical integration of the state transition matrix yield the trajectory transfer formula as shown in formula (18) below.

[0128] Formula (18) in, t f Indicates the end time. t TCMIndicates the pulse correction time. Indicates the pulse velocity increment at the correction time and , This represents the deviation between the actual orbit and the nominal orbit at the end of the integration (i.e., the position of the spacecraft at the end of the integration). Target position on nominal orbit (the deviation between them), and .

[0129] Inverting the above formula (18) is actually a boundary value problem. After multiple differential corrections to this boundary value problem, the correction pulse of the spacecraft can be obtained. In the process of multiple differential corrections, after each differential correction, the orbital numerical integration is required to recalculate the deviation and state transition matrix. Note that the first-order sensitivity cannot completely describe the state transition relationship, and multiple target tests are generally required for convergence.

[0130] In each of the above differential correction processes, the embodiments of this application employ a generalized chaotic polynomial as a non-intrusive nonlinear quantization method, which can be applied to the analysis of pulse correction quantities.

[0131] Specifically, take an initial deviation that satisfies a normal distribution. And establish the above formula (13).

[0132] Based on the nonlinear order of the above spacecraft motion model, the generalized chaotic polynomial corresponding to the above formula (13) is obtained, and the coefficients are solved using the CUT method. (i.e., a surrogate model for the corrected pulse formula of the spacecraft orbit), where, express The i-th component, Describes the orthogonal basis functions of the modified pulse. Let represent the coefficients of the orthogonal basis functions of the modified pulse, and , , The process of solving the generalized chaotic polynomial corresponding to the nonlinear order of the above-mentioned spacecraft motion model and the above formula (13) using the CUT method can be referred to the relevant content in S102 above. The embodiments of this application will not be repeated here.

[0133] Step 3: Statistical analysis of the surrogate model of the correction pulse formula using Monte Carlo numerical simulation is used to obtain the correction pulse margin of the spacecraft.

[0134] After obtaining the surrogate model of the modified pulse formula, a statistical analysis of the surrogate model was performed using Monte Carlo numerical simulation to obtain the probability density distribution function of the modified pulse quantity. Then based on reliability indicators (Value range is 0~1) Estimate the corresponding correction pulse margin, i.e. By transforming the formula, we can obtain the above formula (14). It can be understood that the above correction pulse quantity refers to the correction pulse that satisfies a certain probability distribution due to the uncertainty of the deviation, which is a random variable. The above correction pulse margin refers to the pulse correction size that satisfies the reliability index by taking the corresponding quantile of the correction pulse quantity according to the reliability index.

[0135] It should be noted that the execution order of S102 and S103 can be S102 first and then S103, or S103 first and then S102, or S102 and S103 can be executed simultaneously.

[0136] The following example, using a spacecraft as a probe performing a lunar transfer mission, verifies the effectiveness of the nonlinear order of the quantified spacecraft dynamics model in S101.

[0137] Set the initial state of the probe's nominal orbit. In the ECI coordinate system, it is represented in the following format.

[0138] Let the initial deviation disturbance (1σ) be: .

[0139] Seeking k The initial deviation disturbance is treated as dimensionless for the second distance, and the convergence index ε = 0.01 is set.

[0140] The evolution of the first-order, second-order, third-order, and fourth-order distances is calculated respectively, as follows: Figure 5 As shown. By Figure 5 It can be seen that, under the condition of satisfying this convergence rate, the following characteristics of the Earth-Moon transfer system can be observed: 1. During most of the Earth-Moon transfer time, the algebraic order of the system is 2-3, and linearization methods will introduce certain errors. 2. The algebraic order of the system in the near-lunar segment of the Earth-Moon transfer is at least 8, and the nonlinearity increases sharply in the near-lunar segment, necessitating the use of higher-order nonlinear uncertainty analysis methods. 3. Impulse correction should be avoided in the strongly nonlinear near-lunar segment; impulse correction should be performed as early as possible. Common timing for correction pulses is... Figure 4 It is noted that the algebraic order of the system is 2-3 when performing mid-course corrections. When studying the deviation evolution and impulse correction amount at the correction time, the order of the surrogate model can be taken as 3.

[0141] Continuing with the example of a spacecraft being a probe performing an Earth-Moon transfer mission, we will verify the effectiveness of determining the evolutionary deviation of the spacecraft in S102 above.

[0142] Take the initial deviation disturbance (1σ) as A third-order chaotic polynomial surrogate model was constructed using the construction process and analysis results of the generalized chaotic polynomial in S102, with a total number of terms. P There are 84 items.

[0143] The coefficients of the orthogonal basis functions have an algebraic order of 6 in both the denominator and numerator. Since the denominator has an analytical expression, it can be solved directly using the Gauss-Hermite orthogonality rule by directly applying the tensor product. The numerator is solved using the CUT-6 method with 7th-order algebraic precision.

[0144] Figure 6 and Figure 7 The convergence of the third-order chaotic polynomial was verified, showing that the convergence accuracy for position reaches 10. -7 For speed convergence accuracy to reach 10 -5 .

[0145] like Figure 8 and Figure 9 As shown, the propagation time of the error to the near end of the moon is about 2 days. The distribution of the error can be obtained by performing 2000 simulations using the surrogate model and the Monte Carlo method. The distribution of the z-axis position and velocity is given below, and the higher-order moments of the state variables are obtained using the surrogate model, as shown in Table 2.

[0146] Table 2. Moments of each order after 2 days of error propagation

[0147] It can be observed that the higher-order moments (skewness and kurtosis) of the position vector are relatively small, exhibiting an approximate normal distribution (see reference). Figure 8 However, the magnitude of the higher-order moments of the velocity vector is not negligible, and the z-axis velocity exhibits a clear left-skewed distribution (see reference). Figure 9 It is insufficient to analyze the state random variable of velocity using only the covariance method or the UT transformation method.

[0148] In addition, the generalized chaotic polynomial method for propagation analysis of Earth-Moon transfer orbit deviation only requires calculating the response values ​​at 137 specific sample points to obtain statistical characteristics, while the Monte Carlo method requires a large number of sample points to obtain statistical characteristics with the same accuracy.

[0149] Depend on Figure 10 and Figure 11It can be observed that the convergence efficiency of the Monte Carlo method is much lower than that of the generalized chaotic polynomial method, and a large number of simulations are required to achieve the same accuracy as the generalized chaotic polynomial method. Therefore, using the generalized chaotic polynomial model greatly reduces the computational load while ensuring computational accuracy and preserving more of the nonlinear characteristics of the system.

[0150] Continuing with the example of a spacecraft being a probe performing an Earth-Moon transfer mission, we will verify the effectiveness of determining the spacecraft's corrected pulse margin in S103 above.

[0151] Take the initial deviation disturbance (1σ) as The coefficients of each component of the chaotic polynomial of the corrected pulse quantity are obtained using the S103 method, as follows: Figure 12 and Figure 13 As shown.

[0152] Figure 14 , Figure 15 as well as Figure 16 The convergence of the third-order chaotic polynomial was verified, by Figures 14 to 16 It can be seen that the convergence accuracy of each component of the corrected pulse reaches 10. -7 The corrected pulse distribution was obtained through a surrogate model and 500 Monte Carlo simulations.

[0153] By using a surrogate model to analyze the statistical characteristics of each component, the moments of each component of the mid-course correction pulse can be obtained, as shown in Table 3 below.

[0154] Table 3. Moments (PCE) of Each Component of the Mid-course Correction Pulse

[0155] pass Figure 17 It can be seen that the mid-course correction pulse distribution predicted by the surrogate model of the generalized chaotic polynomial and the mid-course correction pulse distribution obtained by the 1000 Monte-Carlo shooting method have much better convergence than the surrogate model obtained by the direct method.

[0156] The statistical characteristics obtained by the two methods are shown in Table 4 below.

[0157] Table 4. Comparison of statistical characteristics obtained by PCE method and Monte Carlo method

[0158] in addition Figure 18 The results of statistical characteristics obtained from Monte Carlo numerical simulations with different sample sizes are presented, demonstrating that the surrogate model method can accurately and efficiently estimate the statistical characteristics of modified impulses. The former only requires about 100 samples to obtain very accurate results.

[0159] In summary, the orbital deviation evolution and correction method for a spacecraft provided in this application employs a conjugate unscented transformation method to transform the nominal orbit and preset motion state deviations of the spacecraft to determine the nonlinear order of the spacecraft's dynamic model. Then, by combining the nonlinear order and generalized chaotic polynomials, the motion state formula constructed based on the initial value problem of ordinary differential equations is analyzed to obtain the spacecraft's evolutionary deviation. Finally, by combining the nonlinear order and generalized chaotic polynomials, the correction pulse formula constructed based on the boundary value problem of ordinary differential equations is analyzed to obtain the spacecraft's correction pulse margin. This process does not require complex analysis of the space environment and the actual dynamic model of the spacecraft, making it a non-invasive method. This reduces the complexity of the orbital deviation evolution and correction process, improves the accuracy of the results, and enables the spacecraft to better complete its mission.

[0160] Accordingly, embodiments of this application provide a spacecraft orbit deviation evolution and correction device. The spacecraft needs to move within its nominal orbit to perform its mission, such as... Figure 19 As shown, the above-mentioned device includes an order quantization module 501, an evolution deviation determination module 502, and a correction pulse margin determination module 503.

[0161] The order quantization module 501 is used to quantify the nonlinear order of the spacecraft's dynamic model based on the spacecraft's nominal orbit, preset motion state deviation, and conjugate unscented transformation method. The preset motion state deviation describes the deviation between the spacecraft's preset motion state and the target motion state, where the target motion state refers to the spacecraft's motion state on its nominal orbit. The nonlinear order describes the computational complexity of the spacecraft's dynamic model. For example, the order quantization module 501 is used to implement S101 of the above method.

[0162] The evolution deviation determination module 502 is used to determine the evolution deviation of the spacecraft based on the nonlinear order of the spacecraft's dynamic model, the motion state formula of the spacecraft's orbit, and the generalized chaotic polynomial. The evolution deviation shows the evolution process of the deviation between the actual motion state and the target motion state of the spacecraft over time. The motion state formula of the spacecraft's orbit indicates the change law of the spacecraft's motion state over time and is constructed based on the initial value problem of ordinary differential equations. For example, the evolution deviation determination module 502 is used to implement S102 of the above method.

[0163] The corrected pulse margin determination module 503 is used to determine the corrected pulse margin of the spacecraft based on the nonlinear order of the spacecraft's dynamic model, the corrected pulse formula for the spacecraft's orbit, and the generalized chaotic polynomial. The corrected pulse margin is used to adjust the spacecraft's actual motion state to its target motion state. The corrected pulse formula for the spacecraft's orbit describes the velocity increment required for the spacecraft to reach the target motion state from its current motion state and is constructed based on the boundary value problem of ordinary differential equations. For example, the corrected pulse margin determination module 503 is used to implement S103 of the above method.

[0164] The modules of the above-mentioned spacecraft orbit deviation evolution and correction device can also be used to perform other steps in the above method embodiments. All relevant contents involved in the above method embodiments can be referred to the functional descriptions of the corresponding functional modules, and will not be repeated here.

[0165] This application also provides an electronic device, including: a processor and a memory coupled to the processor; the memory is used to store computer instructions, and when the electronic device is running, the processor executes the computer instructions stored in the memory to cause the electronic device to perform the methods described in the above embodiments. The processor can implement the order quantization module 501, the evolution deviation determination module 502, and the correction pulse margin determination module 503; the memory can also be used to store the nonlinear order of the spacecraft's dynamic model, the spacecraft's evolution deviation, and the correction pulse margin, etc.

[0166] This application also provides a computer-readable storage medium including a computer program that, when run on a computer, performs the methods described in the above embodiments.

[0167] This application also provides a computer program product, which includes computer program instructions that, when run on a computer, execute the methods described in the above embodiments.

[0168] The various embodiments in this specification are described in a progressive manner. The same or similar parts between the various embodiments can be referred to each other. Each embodiment focuses on describing the differences from other embodiments.

[0169] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of this application.

Claims

1. A method for the evolution and correction of orbital deviations of a spacecraft, characterized in that, The spacecraft needs to move in its nominal orbit to perform a mission, the method comprising: Based on the spacecraft's nominal orbit, preset motion state deviation, and conjugate unscented transformation method, the nonlinear order of the spacecraft's dynamic model is quantified; wherein, the preset motion state deviation is used to describe the deviation between the spacecraft's preset motion state and the target motion state, and the target motion state refers to the spacecraft's motion state on the nominal orbit; the nonlinear order is used to describe the computational complexity of the spacecraft's dynamic model. Based on the nonlinear order of the spacecraft's dynamic model, the motion state formula of the spacecraft's orbit, and the generalized chaotic polynomial, the evolutionary deviation of the spacecraft is determined; the evolutionary deviation shows the evolution process of the deviation between the actual motion state of the spacecraft and the target motion state of the spacecraft over time; the motion state formula of the spacecraft's orbit indicates the change law of the spacecraft's motion state over time and is constructed based on the initial value problem of ordinary differential equations; Based on the nonlinear order of the spacecraft's dynamic model, the correction pulse formula for the spacecraft's orbit, and the generalized chaotic polynomial, the correction pulse margin of the spacecraft is determined. The correction pulse margin is used to adjust the actual motion state of the spacecraft to the target motion state of the spacecraft. The correction pulse formula for the spacecraft's orbit describes the velocity increment required for the spacecraft to reach the target motion state from the current motion state and is constructed based on the boundary value problem of ordinary differential equations.

2. The method as described in claim 1, characterized in that, The nonlinear order of the quantification of the dynamic model of the spacecraft includes the following steps 1 to 5; Step 1: Obtain the nominal orbit and preset motion state deviation of the spacecraft; the nominal orbit includes the initial motion state of the spacecraft; The preset motion state deviation satisfy , N () represents a normal distribution. The variance matrix of the preset motion state deviation. The mean value of the preset motion state deviation; Step 2: Construct a dynamic model of the spacecraft based on the nominal orbit and the preset motion state deviation; the dynamic model satisfies: ,in, This indicates the actual motion state of the spacecraft. A mapping relationship indicating the state of motion. This indicates the initial motion state of the spacecraft. This indicates the deviation of the preset motion state. This represents the i-th moment within a preset time period; Step 3: Set the order of the conjugate unscented transformation method h The initial value is 1; Step 4: For each of the multiple moments contained in the preset time period, the conjugate unscented transform method is used to transform the moment to obtain multiple integration nodes, and the order distance of the moment is calculated using the multiple integration nodes and the dynamic model; each of the multiple integration nodes satisfies ,in, Indicates the first q The motion state deviation of each integration node q Integer and q ∈[1,2( h +1)+2 h+1 ]; Indicates the first q Preset motion state deviation of each integration node; express The decomposition matrix, and T = ; Among them, when the order of the conjugate unscented transformation method h When the order distance is equal to 1, the following condition is met: When the order of the conjugate unscented transformation method h When the order distance is greater than 1, the following condition is met: in, express h The order distance at time i when = 1, express h =1 hour q The integral weights of each integral node. Represents the norm; express k The order distance at time i when >1, Indicate h >1 hour q The integral weight of each integral node; Step 5: When the order distance of each moment within the preset time period is less than the convergence index... ε At that time, 2 h -1 or 2 h -2 is determined as the nonlinear order of the spacecraft's dynamic model; otherwise, the order of the conjugate unscented transformation method is used. h Updated to h +1 and return to step 4.

3. The method as described in claim 1, characterized in that, The evolutionary deviation of the spacecraft is described by the mean, variance, and covariance of the spacecraft's motion states. The mean of the motion state satisfies: The variance of the motion state satisfies: The covariance of the motion state satisfies: in, The first [representation] of the motion state of the spacecraft i Each component represents a component, and the motion state of the spacecraft satisfies the motion state formula of the spacecraft orbit. , Let be the orthogonal basis functions of the motion state. The coefficients of the orthogonal basis functions of the motion state are . , , , The first term represents the result obtained after transformation by the conjugate unscented transformation method. q The integral weights of each integral node. The formula representing the motion state of the spacecraft orbit in the direction of the i-th component is as follows: Let k be the orthogonal basis function numbered k in the generalized chaotic polynomial that constructs the deviation evolution process. The first term represents the result obtained after transformation by the conjugate unscented transformation method. q Each integration node q Integer and q ∈[1,2( l +1)+2 l+1 ], l The nonlinear order of the dynamic model of the spacecraft is given. The mean value of the i-th component representing the motion state of the spacecraft; This represents the constant term in the generalized chaotic polynomial constructed for the i-th component of the spacecraft's motion state; The variance of the i-th component of the spacecraft's motion state is represented by... The generalized chaotic polynomial numbered is constructed for the i-th component of the spacecraft's motion state. m The coefficients of the orthogonal basis functions, Let m denote the orthogonal basis function numbered m in the generalized chaotic polynomial; The j-th component represents the motion state of the spacecraft. This represents the covariance between the i-th and j-th components of the spacecraft's motion state. Let m be the coefficient of the orthogonal basis function numbered m in the generalized chaotic polynomial constructed for the j-th component of the spacecraft's motion state.

4. The method as described in claim 1, characterized in that, The spacecraft's correction pulse margin satisfies: in, This represents the magnitude of the correction pulse margin of the spacecraft. Indicates reliability index, This represents the magnitude of the correction pulse of the spacecraft. This indicates the correction pulse of the spacecraft. Satisfies the corrected pulse formula for the spacecraft orbit, and , express The i-th component, Denotes the orthogonal basis functions of the modified pulse. The coefficients of the orthogonal basis functions of the modified pulse are represented, and , , , The first term represents the result obtained after transformation by the conjugate unscented transformation method. q The integral weights of each integral node. The formula for the corrected pulse of the spacecraft orbit in the direction of the i-th component is given. Let j be the orthogonal basis function numbered j in the generalized chaotic polynomial used to construct the modified impulse quantity. The first term represents the result obtained after transformation by the conjugate unscented transformation method. q Each integration node q Integer and q ∈[1,2( l +1)+2 l+1 ], l represents the nonlinear order of the dynamic model of the spacecraft; The probability density distribution function represents the corrected pulse quantity that meets the reliability index. The probability density distribution function is obtained based on the Monte Carlo numerical simulation method and the corrected pulse formula of the spacecraft orbit.

5. The method as described in claim 1 or 3, characterized in that, The motion state formula of the spacecraft orbit satisfies: in, Indicates the end time. This indicates the motion state of the spacecraft at the time of termination. This indicates the initial motion state of the spacecraft. The mapping relationship indicating the state of motion, at the same time The formula representing the motion state of the spacecraft's orbit.

6. The method as described in claim 1 or 4, characterized in that, The corrected pulse formula for the spacecraft orbit satisfies: in, Indicates the initial time. Indicates the end time. This indicates the initial position of the spacecraft. Indicates the position of the spacecraft at the time of termination. This represents the correction pulse required to change the spacecraft's position from the initial position to the final position. The formula for the corrected pulse of the spacecraft's orbit is given.

7. A spacecraft orbital deviation evolution and correction device, characterized in that, The spacecraft needs to move in the spacecraft's nominal orbit to perform the mission. The device includes an order quantization module, an evolution deviation determination module, and a correction pulse margin determination module. The order quantization module is used to quantify the nonlinear order of the spacecraft's dynamic model based on the spacecraft's nominal orbit, preset motion state deviation, and conjugate unscented transformation method; wherein, the preset motion state deviation is used to describe the deviation between the spacecraft's preset motion state and the target motion state, and the target motion state refers to the spacecraft's motion state on the nominal orbit; the nonlinear order is used to describe the computational complexity of the spacecraft's dynamic model. The evolution deviation determination module is used to determine the evolution deviation of the spacecraft based on the nonlinear order of the spacecraft's dynamic model, the motion state formula of the spacecraft's orbit, and the generalized chaotic polynomial. The evolution deviation shows the evolution process of the deviation between the actual motion state of the spacecraft and the target motion state of the spacecraft over time. The motion state formula of the spacecraft's orbit indicates the change law of the spacecraft's motion state over time and is constructed based on the initial value problem of ordinary differential equations. The correction pulse margin determination module is used to determine the correction pulse margin of the spacecraft based on the nonlinear order of the spacecraft's dynamic model, the correction pulse formula of the spacecraft orbit, and the generalized chaotic polynomial. The correction pulse margin is used to adjust the actual motion state of the spacecraft to the target motion state of the spacecraft. The correction pulse formula of the spacecraft orbit describes the velocity increment required for the spacecraft to reach the target motion state from the current motion state and is constructed based on the boundary value problem of ordinary differential equations.

8. An electronic device, characterized in that, The device includes a processor and a memory coupled to the processor; the memory is used to store computer instructions, which, when the electronic device is running, are executed by the processor to cause the electronic device to perform the method as described in any one of claims 1 to 6.

9. A computer-readable storage medium, characterized in that, It includes computer program instructions that, when executed by a computer, cause the computer to perform the method as described in any one of claims 1 to 6.

10. A computer program product, characterized in that, It includes computer program instructions that, when executed on a computer, cause the computer to perform the method as described in any one of claims 1 to 6.

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