A filter-based satellite orbit extrapolation method

Abstract

The application provides a filter-based satellite orbit extrapolation method, comprising: taking the motion state of a satellite as a wave to establish a mathematical model of a wave function, integrating the mathematical model of the wave function into a wave function filtering module, a physical quantity calculation module and a wave function evolution module; taking the wave function filtering module, the physical quantity calculation module and the wave function evolution module as a basis to construct a satellite trajectory prediction model; inputting the satellite orbit extrapolation data of the last moment and real-time observation data into the satellite trajectory prediction model to realize wave function evolution, wave function filtering and physical quantity calculation, and outputting the satellite orbit extrapolation data of the current moment. The method of the application realizes a signal analysis method with clear physical meaning through the use of a cycle module, wave function filtering and evolution, realizes observation-trajectory integrated processing and short-arc orbit extrapolation performance improvement.

Description

Technical Field

[0001] This invention belongs to the field of satellite orbit research, specifically relating to a filtering-based satellite orbit extrapolation method. Background Technology

[0002] Over the past decade or so, the number of satellites in orbit has grown rapidly, and the space environment has become increasingly complex, making accurate prediction and correlation of satellite trajectories a core issue in space situational awareness. For maneuvering targets, due to their variable trajectories and complex behavior, traditional prediction methods based on classical orbital mechanics are ill-suited.

[0003] Current research on satellite orbits is characterized by multidisciplinary integration, specifically:

[0004] 1) Statistical / information theory methods (Gaussian filtering, Kalman filtering, KL divergence, mutual information, Bach distance) are used to construct trajectory association criteria;

[0005] 2) Combining control theory and optimal control with Monte Carlo simulation for maneuver detection and track-orbit correlation;

[0006] 3) Emerging technologies such as quantum computing and quantum neural networks attempt to use quantum parallelism to alleviate the computational pressure of large-scale constellation orbit prediction.

[0007] While the above methods have made some progress, they still have the following drawbacks in scenarios involving maneuvering targets:

[0008] 1) The lack of an effective description of the evolution of uncertainty caused by maneuvers leads to a rapid amplification of long-term extrapolation errors;

[0009] 2) The model has a complex structure and high computational cost, making it difficult to maintain low computing power consumption while meeting timeliness requirements;

[0010] 3) Most methods focus on “observation-trajectory” matching or pure dynamic extrapolation, failing to unify the two within a physically meaningful framework, which limits the simultaneous improvement of forecast timeliness and correlation accuracy. Summary of the Invention

[0011] The purpose of this invention is to provide a filtering-based satellite orbit extrapolation method to achieve uncertainty evolution description of maneuvering targets, low computational consumption, and integrated observation-trajectory processing based on quantum computing.

[0012] To achieve the above objectives, the present invention provides a satellite orbit extrapolation method based on filtering, comprising:

[0013] Step S1: Using the satellite's motion state as a wave, establish a mathematical model of the wave function, and integrate the mathematical model of the wave function into a wave function filtering module, a physical quantity calculation module, and a wave function evolution module;

[0014] Step S2: Based on the wave function filtering module, physical quantity calculation module, and wave function evolution module, construct a satellite trajectory prediction model;

[0015] Step S3: Input the satellite orbit extrapolation data from the previous moment and the real-time observation data into the satellite trajectory prediction model to realize wave function evolution, wave function filtering and physical quantity calculation, and output the satellite orbit extrapolation data at the current moment.

[0016] Step S1 specifically includes:

[0017] Step S11: Establish the definitions of wave function and uncertainty, the conditions that the uncertainty modification operator must satisfy, the definition of the degree of uncertainty change, and the state of the wave function after observation. Uncertainty Modification Operator The definition formulas of the wave function and the definition formulas of the observed state of the position representation are integrated into the wave function filtering module.

[0018] Step S12: Based on the definition of the wave function, the corresponding calculation formula for the position distribution is obtained. The calculation formulas for the Gaussian wave of the wave function, the position distribution, the position expectation, and the momentum expectation are integrated into a physical quantity calculation module.

[0019] Step S13: Establish the formula for the time evolution operator and integrate it to obtain the wave function evolution module; the formula for the time evolution operator includes the definition and decomposition formula of the time evolution operator, the operation formula for representation transformation, and the calculation formula for kinetic energy evolution and potential energy evolution.

[0020] Step S11 specifically includes:

[0021] Step S111: Define the wave function in position space and measurement uncertainty In the wave function, the satellite's position distribution is analogous to the time domain information of the wave, the velocity distribution is analogous to the frequency domain information of the wave, and the measurement uncertainty is defined as the product of the position uncertainty and the momentum uncertainty.

[0022] Step S112: Establish the uncertainty modification operator The formula for the conditions that need to be met;

[0023] Uncertainty Modification Operator The formula for the required conditions is:

[0024] ,

[0025] ,

[0026] In the formula, It is the wave function in position space. Current position It is a positional operator. It is a momentum operator. It is an uncertainty modification operator. It refers to the degree of change in measurement uncertainty. The magnitude is determined by the ratio of the change in the inner product of the wavefunction before and after the observation. The + sign is the dagger symbol, which can be interpreted as transpose and conjugate from the perspective of linear algebra.

[0027] Step S113: Based on the observed state of the position representation and physical state The degree of change in measurement uncertainty during the observation process is derived from the direct product. ;

[0028] The degree of overall probability change during the observation process Defined as:

[0029] ,

[0030] in, It is a wave function. It is the spatial dimension of the domain of the wave function, and the operator ⊙ denotes the pointwise multiplication of the functional expressions corresponding to two states in space. The observed state representing the positional representation. It is a multidimensional position coordinate. It is the differential of the multidimensional position coordinates; the degree of change in the overall probability is used to determine the degree of change in measurement uncertainty;

[0031] Step S114: Establish the state of the wave function after observation The calculation formula is used as the observation state operator. Definition;

[0032] The state of the wave function after it is observed The calculation formula is:

[0033] ,

[0034] in, Represents the observation state operator; It is the wave function in position space; It is the uncertainty correction operator; the operator ⊙ represents pointwise multiplication in space; It is a normalization operator;

[0035] Step S115: Establish a Gaussian wave for the wave function as a scalar wave function model applicable to conservative forces and define the wave function;

[0036] Gaussian wave of wave function for:

[0037] ,

[0038] in, This is the current position. A is the second-order coefficient matrix of the wave function, which is a symmetric matrix. B is the first-order coefficient vector of the wave function. C is the normalization constant of the wave function.

[0039] Step S116: Modify the operator based on the measurement uncertainty The required formulas and the degree of uncertainty variation during the observation process. Give the uncertainty modification operator. The definition formula is then used; subsequently, the Gaussian wave of the wave function is substituted to determine the uncertainty modification operator. Does it meet the requirements?

[0040] Uncertainty Modification Operator The defining formula is:

[0041] ,

[0042] in, It refers to the degree of change in measurement uncertainty. To measure uncertainty, It is the uncertainty corrected by modifying the operator based on the uncertainty. It is the wave function in position space. This is the current location.

[0043] Step S117: Provide the observed state of the location representation Definition formula;

[0044] The observed state of the position representation is:

[0045] ,

[0046] Where Σ is the covariance matrix corresponding to a single observation, and T denotes the transpose. Current position The location center of the observed data.

[0047] Step S13 includes:

[0048] Step S131: For the wave function in the location space Adding a time dimension to obtain the wave function that evolves over time The definition of the time evolution operator is obtained by using the solution of the equation of motion;

[0049] The definition of the time evolution operator is ;

[0050] Step S132: Based on the split-step Fourier method, obtain the decomposition formula of the time evolution operator;

[0051] Time evolution operator of wave function The decomposition formula is:

[0052] ,

[0053] in, , , These are the Hamiltonian operator, the kinetic energy operator, and the potential energy operator, respectively. It is a wave function that evolves over time;

[0054] Step S133: Establish the operational formula for representation transformation in order to facilitate the calculation of the decomposition formula of the time evolution operator;

[0055] Step S134: Establish calculation formulas for potential energy evolution and kinetic energy evolution to facilitate the calculation of the decomposition formula of the time evolution operator.

[0056] The satellite trajectory prediction model includes a loop module, which comprises a wavefunction evolution module, a wavefunction filtering module, and a physical quantity calculation module; the wavefunction evolution module receives the wavefunction from the previous time step. The superscript S represents the Schrödinger picture and utilizes the time evolution operator. The evolution yields the evolved wave function. The wavefunction filtering module receives the observation state at the current moment. And the evolved wave function, through the observed state at the current moment Weighting is achieved jointly with the evolved wave function, and the operator is modified using uncertainty after weighting. Reduce the measurement uncertainty of the wave function to output the wave function at the current moment. The physical quantity calculation module receives the wave function at the current moment. Combined with physical quantity operators, it outputs the expected physical quantity at the current moment.

[0057] Step S1 further includes:

[0058] Step S14': Establish the damped wave equation to give the calculation formulas for generalized momentum, generalized position and extended Hamiltonian; establish the operator evolution module using the calculation formulas for generalized momentum and generalized position;

[0059] The generalized position X and the generalized momentum P satisfy the following transformation relationship:

[0060] ,

[0061] Where x is the current position, p is the current momentum, and M is the satellite mass. It is the damping coefficient, and t is the current time;

[0062] The Hamiltonian H is:

[0063] ,

[0064] Where M is the satellite mass, Let X be the generalized momentum, X be the generalized position, and K be the second derivative of the potential energy. The damping coefficient is... For Hamiltonian;

[0065] The damped wave equation is:

[0066] ,

[0067] in, It is a damped wave function.

[0068] Step S15': Establish the formula for the damping parameter β, and use the generalized position, generalized momentum, and damping parameter to establish the model of the damping wave function. The definition of the wave function in the wave function filtering module is replaced; the formulas for position expectation and momentum expectation in the physical quantity calculation module are corrected based on the model of the damped wave function.

[0069] The formula for the damping parameter β is:

[0070] ,

[0071] in, It was the first moment Damping parameters, It is the second moment The damping parameter, e is the natural base. Damping coefficient, M is the satellite mass.

[0072] Location expectation and momentum expectation The formula is:

[0073] ,

[0074] ,

[0075] in, It is a positional operator. It is a momentum operator. It is a damped wave function model. It refers to generalized momentum, uncertainty, and damping parameters. t is the damping coefficient, M is the satellite mass, and t is the current time.

[0076] The satellite trajectory prediction model of this invention includes a loop module, which comprises a wavefunction evolution module, an operator evolution module, a representation transformation module, a wavefunction filtering module, and a physical quantity calculation module; the wavefunction evolution module receives the damped wavefunction in the interaction scene at the previous moment. And utilize time evolution operators The evolution yields the evolved damped wave function; the operator evolution module is configured to receive physical quantity operators from the previous time step in the interaction scene. And process to obtain the physical quantity operators in the interaction scene at the current moment. The representation conversion module is configured to receive the current observation state in the Schrödinger picture. and the physical quantity operators in the interaction scene at the current moment. The process yields the observed state in the interaction picture at the current moment. The wavefunction filtering module receives the observation state in the interaction scene at the current moment. and the evolved wave function The damped wave function in the interaction scene at the current moment is obtained through processing. The physical quantity calculation module receives the physical quantity operators in the interaction scene at the current moment. and the damped wave function in the interaction scene at the current moment. Processing yields the expected physical quantity at the current moment. ;

[0077] Or, momentum expectation Approximately: ,in, It is a momentum operator. It is a damped wave function model. It refers to generalized momentum, uncertainty, and damping parameters. Here, M is the damping coefficient, M is the satellite mass, and t is the current time. The satellite trajectory prediction model includes a loop module, which comprises a wave function evolution module, an operator evolution module, a representation transformation module, a wave function filtering module, and a physical quantity calculation module. The wave function evolution module receives the damping wave function in the Schrödinger picture from the previous time step. The evolved damped wave function is obtained by using a time evolution operator; the operator evolution module is configured to receive physical quantity operators in the Schrödinger picture from the previous time step. The module processes and obtains the physical quantity operators in the interaction picture at the current moment; the representation transformation module is configured to transform the evolved damped wave function and the physical quantity operators in the interaction picture at the current moment into the Schrödinger picture respectively; the wave function filtering module is configured to filter the observed state in the Schrödinger picture at the current moment based on the evolved damped wave function and the observed state in the Schrödinger picture at the current moment. The damped wave function in the Schrödinger picture at the current moment is obtained through processing. The physical quantity calculation module is configured to calculate the damped wave function in the Schrödinger picture at the current moment. And the physical quantity operators in the Schrödinger picture at the current moment. Processing yields the expected physical quantity at the current moment. ;

[0078] Alternatively, it may include step S17': replacing the formulas for generalized position X and generalized momentum P with the following formulas to correct the operator evolution module: ; , It is a generalized positional operator, representing the current position. , It is the momentum operator for generalized momentum and current momentum. Here, M is the damping coefficient, M is the satellite mass, and t is the current time.

[0079] Replace the Hamiltonian with the following formula The formula:

[0080] ,

[0081] Where P is generalized momentum, X is generalized position, and K is the second derivative of potential energy;

[0082] Furthermore, the calculation formula for the time evolution operator is replaced with the following formula to correct the calculation formula for the extended Hamiltonian:

[0083] ,

[0084] in, Let X be the generalized momentum and X be the generalized position. For momentum operator, For positional operators, The damping coefficient is... For Hamiltonian operators, For measurement uncertainty, t is the current time, M is the satellite mass, and K is the second derivative of the potential energy;

[0085] Each loop module of the satellite trajectory prediction model includes a wavefunction evolution module, an operator evolution module, a wavefunction filtering module, and a physical quantity calculation module; the wavefunction evolution module receives the damped wavefunction in the Schrödinger picture from the previous moment. The evolved damped wave function is obtained by using a time evolution operator; the operator evolution module is configured to receive the position operator in the Schrödinger picture. The momentum operator in the interaction scene at the previous moment The position operator in the Schrödinger painting is obtained through processing. and the momentum operator in the interaction scene at the current moment. The wavefunction filtering module is configured to filter based on the evolved damped wavefunction and the current observation state in the Schrödinger picture. The damped wave function in the Schrödinger picture at the current moment is obtained through processing. The physical quantity calculation module is configured to calculate the damped wave function in the Schrödinger picture at the current moment. And the physical quantity operators in the Schrödinger picture at the current moment. , Processing yields the expected physical quantity at the current moment. , .

[0086] Step S1 includes:

[0087] Step S11”: Establish the definition of measurement uncertainty, introduce phase to establish the spinor wave function model for each spin component, and establish the weight transfer function of the spinor space and the calculation formula of each component of the weighted spinor wave function model.

[0088] Step S12”: Establish the Gaussian wave of the screw wave function model and measurement uncertainty Formulas for the intensity of each component of the wave function, and the location distribution of the spinor wave function. The formula is derived, and the motion formulas for each spin component of the spinor wave function are established.

[0089] Step S13”: Establish the definition of the degree of uncertainty change and the state of the wave function after observation. Uncertainty Modification Operator Definition formula; Establishment of time evolution operator The decomposition formula;

[0090] Step S14”: Based on the results of steps S11”-S13”, the weight transfer module, wave function evolution module, wave function filtering module, and physical quantity calculation module are integrated to obtain the weight transfer module, wave function evolution module, wave function filtering module, and physical quantity calculation module.

[0091] In step S1, the weight transfer module is configured as follows:

[0092] Calculate the expected position of the current spinor wavefunction as a whole. Covariance In order to Centered on The spin space component of the spinor wavefunction in a cube space of length denoted by . Uniform sampling; calculate the discrete values ​​of the weight transfer function, and calculate the expected weight-transferred component of the i-th spin space of the spinor wavefunction after weight transfer. ; Calculate the expected component of the spinor wavefunction after weighted shift in the i-th spin space. Expected position Covariance ; Construct the expected component of the spinor wavefunction after weight transfer in the i-th spin space. The positional distribution; an expression for estimating the phase using the degree of phase change at the expected position; integration to obtain the wave function after weight transfer;

[0093] The wavefunction evolution module is set up to: construct evolution operators respectively. and For components and Perform the operation. The potential energy required for radial maneuvering along the track, It is the potential energy required for maneuvering along the normal to the orbital plane. and Let S be the spin-space components of the orbital tangential maneuver S and the orbital normal maneuver W of the spinor wavefunction; according to the time evolution operator The decomposition formula performs batch operations on each component of the spinor wavefunction; for the components Perform the operation. Let T be the spin-space component of the spinor wavefunction along the orbital radial maneuver T.

[0094] The wavefunction filtering module is set to: filter the observed state Perform batch point-to-point multiplication with each component of the spinor wave function; calculate the degree of uncertainty variation according to the definition of uncertainty variation; modify the operator according to uncertainty. The defined formula is used to batch calculate the results of adjusting the measurement inaccuracy for each component;

[0095] The physical quantity calculation module is set to calculate the expected value of the position of the spinor wave function;

[0096] In step S2, the satellite trajectory prediction model includes multiple loop modules, each loop module including a weight transfer module, a wavefunction evolution module, a wavefunction filtering module, and a physical quantity calculation module; the weight transfer module receives the wavefunction from the previous time step. The superscript S represents the Schrödinger picture, which is processed to obtain the wavefunction after weight transfer; the wavefunction evolution module receives the wavefunction after weight transfer and processes it to obtain the evolved wavefunction; the wavefunction filtering module receives the observation state at the current time. And the evolved wave function, outputting the wave function at the current moment. The physical quantity calculation module receives the wave function at the current moment. and physical quantity operators , Output the expected physical quantity at the current moment. , .

[0097] In step S1, step S14 further includes: simplifying the spinor wavefunction model; the simplified spinor wavefunction model is:

[0098] ,

[0099] in, Let be the i-th spin-space component of the spinor wavefunction. It is the second-order coefficient matrix of the i-th spin space component. It is the first-order coefficient vector of the i-th spin space component. It is the intensity constant of the i-th spin space component; and / or

[0100] Step S1 further includes: simplifying the weight transfer module by simplifying the calculation formulas for momentum, angular momentum, and energy;

[0101] momentum angular momentum ,energy for:

[0102]

[0103]

[0104] ,

[0105] in, , , These are momentum, angular momentum, and energy. For momentum operator, For positional operators, Let be the potential energy function. For satellite quality.

[0106] The method of this invention utilizes cyclic modules, wavefunction filtering, and evolution to achieve a signal analysis method with clear physical meaning, realizing integrated observation-trajectory processing and improving the performance of short-arc orbit extrapolation. Furthermore, this invention can also draw upon the structure of spin-particle wavefunctions and introduce a trajectory prediction model with non-spatiotemporal degrees of freedom to improve the timeliness of maneuvering target prediction and realize the evolutionary description of maneuvering uncertainty. Through these two technical approaches, a new framework distinct from classical orbital mechanics is explored, thereby achieving accurate prediction and reliable correlation of maneuvering target trajectories with low computational power consumption. Attached Figure Description

[0107] Figure 1 This is a structural block diagram of a mathematical model of a wave function established by a filtering-based satellite orbit extrapolation method according to a first embodiment of the present invention.

[0108] Figure 2 This is a structural block diagram of a satellite trajectory prediction model established by a filtering-based satellite orbit extrapolation method according to a first embodiment of the present invention.

[0109] Figure 3 This is a structural block diagram of a satellite trajectory prediction model established by a filtering-based satellite orbit extrapolation method according to a second embodiment of the present invention.

[0110] Figure 4 This is a structural block diagram of a satellite trajectory prediction model established by a filtering-based satellite orbit extrapolation method according to a third embodiment of the present invention.

[0111] Figure 5 This is a structural block diagram of a satellite trajectory prediction model established by a filtering-based satellite orbit extrapolation method according to the fourth embodiment of the present invention.

[0112] Figure 6 This is a structural block diagram of a satellite trajectory prediction model established by a filtering-based satellite orbit extrapolation method according to the fifth embodiment of the present invention. Detailed Implementation

[0113] The preferred embodiments of the present invention are given below with reference to the accompanying drawings, and described in detail to enable a better understanding of the functions and features of the ultraviolet light monitoring device and method of the present invention.

[0114] The basic principle of the filtering-based satellite orbit extrapolation method of this invention is as follows: first, the theoretical derivation of the wave function is performed; then, a mathematical model of the wave function is constructed; and finally, a satellite trajectory prediction model is constructed. The mathematical model of the wave function includes all formulas used in wave function evolution, wave function filtering, and physical quantity calculations, such as formulas 2.1 to 2.16, which will be detailed below. The relationships between these formulas are as follows: Figure 1As shown in the figure, the numbers in parentheses are formula numbers. The mathematical model of the wave function is integrated into a physical quantity calculation module, a wave function filtering module, and a wave function evolution module. The satellite trajectory prediction model is established using these modules. The relationships between the various modules of the satellite trajectory prediction model are as follows: Figure 2 As shown.

[0115] First embodiment: Satellite orbit extrapolation method based on filtering idea, considering only conservative forces.

[0116] like Figure 1 and Figure 2 As shown, according to a first embodiment of the present invention, the satellite orbit extrapolation method based on the filtering concept of the present invention includes the following steps:

[0117] Step S1: Using the satellite's motion state as a wave, establish a mathematical model of the wave function, and integrate the mathematical model of the wave function into a wave function filtering module, a physical quantity calculation module, and a wave function evolution module;

[0118] Step S2: Construct a satellite trajectory prediction model based on the wave function filtering module, physical quantity calculation module, and wave function evolution module.

[0119] Step S3: Input the satellite orbit extrapolation data from the previous moment and the real-time observation data into the satellite trajectory prediction model to realize wave function evolution, wave function filtering and physical quantity calculation, and output the satellite orbit extrapolation data at the current moment.

[0120] This invention processes satellite observation data from a signal analysis perspective, representing the satellite's motion state as waves and using the observation data for wave function filtering. The satellite's position distribution is represented as the time-domain information of the wave, and its velocity distribution is represented as the frequency-domain information of the wave.

[0121] The mathematical model of the wave function includes all the formulas used in wave function evolution, wave function filtering, and physical quantity calculation.

[0122] Step S1 specifically includes:

[0123] Step S11: Establish the definitions of wave function and uncertainty, the conditions that the uncertainty modification operator must satisfy, the definition of the degree of uncertainty change, and the state of the wave function after observation. Uncertainty Modification Operator The definition formulas of the wave function and the definition formulas of the observed state of the position representation are integrated into the wave function filtering module.

[0124] This invention borrows the representational transformation formula from quantum mechanics from the time-frequency conversion formula for satellites, introducing the definition of uncertainty ℏ to participate in the calculation of wave function representational transformation. Furthermore, it considers the degree of variation in satellite uncertainty. The present invention also constructs an operator for modifying measurement uncertainty. ,in It refers to the degree of change in measurement uncertainty.

[0125] Therefore, the process of filtering out wavefunctions with lower uncertainty from wavefunctions using observational data can be viewed as a classification of wavefunctions, involving wave construction, wave evolution, and wave observation-related calculations. The wavefunctions in this invention are observed in a macroscopic manner, and thus differ from those in quantum mechanics.

[0126] Step S11 specifically includes:

[0127] Step S111: Define the wave function in position space and measurement uncertainty In the wave function, the satellite's position distribution is analogous to the time domain information of the wave, the velocity distribution is analogous to the frequency domain information of the wave, and the measurement uncertainty is defined as the product of the position uncertainty and the momentum uncertainty.

[0128] Referring to quantum mechanics, the motion of a satellite can be described in different representations (position representation and momentum representation), and the descriptions in the two representations are equivalent. Therefore, the wave function can describe either the satellite's position or momentum. Specifically, the wave function in position space... The description in the position representation can be equivalently derived from the momentum representation through Fourier transform. The wave function in position space... The square of the modulus represents the probability density of where the satellite's center of mass might be located in space.

[0129] Measurement uncertainty The definition is the product of position uncertainty and momentum uncertainty, used in representation transformations. There is a term in the Fourier transform formula... The formula shows the role of uncertainty in representation transformation. From a mathematical perspective, it can be considered that the representation transformation can be achieved by normalizing the uncertainty and then performing a Fourier transform.

[0130] Step S112: Establish the uncertainty modification operator The formula for the conditions that need to be met;

[0131] After multiple observations, the position and momentum of a macroscopically moving object become more certain. However, in quantum mechanics, the momentum distribution of a state becomes more uncertain after its position representation is observed. Therefore, this invention modifies the measurement uncertainty of the wave function after observation to prevent the momentum distribution from expanding, thus ensuring that the observation behavior conforms to macroscopic principles. While changing the measurement uncertainty, it is necessary to maintain the expected values ​​of position and momentum unchanged; therefore, an uncertainty modification operator is used. The formula for the required conditions is:

[0132] (2.1)

[0133] (2.2)

[0134] In the formula, It is the wave function in position space. Current position It is a positional operator. It is a momentum operator. It is an uncertainty modification operator. It refers to the degree of change in measurement uncertainty. The magnitude is determined by the ratio of the change in the inner product of the wavefunction before and after the observation. The + symbol is the dagger symbol, which can be interpreted as transpose and conjugate from the perspective of linear algebra.

[0135] It should be noted that the degree of variation in measurement uncertainty The degree of change in overall probability The meanings are different; N is generally taken as 3, representing the dimension of the location space.

[0136] Formulas (2.1) and (2.2) constrain the newly constructed mathematical model to still satisfy the invariance of the center of position and momentum after the addition of the concept of "adjustable uncertainty".

[0137] Step S113: Based on the observed state of the position representation and physical state The degree of change in measurement uncertainty during the observation process is derived from the direct product. ;

[0138] The existence of the observed state is independent in different representations, so there is no change in representation, and therefore no parameter of uncertainty.

[0139] The direct product of the observed state and the physical state is the pointwise multiplication of the observed state and the physical state in space.

[0140] Therefore, the degree of overall probability change during the observation process Defined as:

[0141] (2.3)

[0142] in, It is a wave function. It is the spatial dimension of the domain of the wave function, and the operator ⊙ denotes the pointwise multiplication of the functional expressions corresponding to two states in space. The observed state representing the positional representation. It is a multidimensional position coordinate. It is the differential of the multidimensional position coordinates.

[0143] The degree of variation in measurement uncertainty It is the degree of change in overall probability. The Nth root of .

[0144] Formula (2.3) is based on the principle that the total probability of the wave function decreases after it is observed (essentially a screening process), thus requiring normalization. The normalization coefficients are used to calculate the degree of change in the overall probability. This allows us to view the observation of macroscopic objects as a combination of wave function collapse and uncertainty adjustment.

[0145] Step S114: Establish the state of the wave function after observation The calculation formula is used as the observation state operator. Definition;

[0146] The state of the wave function after it is observed The calculation formula is:

[0147] (2.4)

[0148] in, The observation state operator is defined by formula (2.4); It is the wave function in position space; It is the uncertainty modification operator, whose parameter γ is calculated by formula (2.3); the operator ⊙ represents pointwise multiplication in space, that is, multiplication of the probability densities of two state distributions; It is a normalization operator used to make the integral of the probability density over the entire space equal to 1.

[0149] In formula (2.4), multiplying the two states can be considered as multiplying the probability densities of the distributions. Therefore, the integral over the entire space will certainly not reach 1. However, the integral of the probability distribution over the entire space should be 1 to be reasonable. Thus, the function of this operator is normalization. In other words, the left side of the equals sign in formula (2.3) represents the observation state operator defined by the observed states. The right side of the equals sign represents the relationship between the observed state and the physical state (i.e., Normalization of the direct product of ).

[0150] Step S115: Establish a Gaussian wave for the wave function as a scalar wave function model applicable to conservative forces and define the wave function;

[0151] In this embodiment, the wave function adopts a scalar wave function model suitable for conservative forces. The wave function is defined as a Gaussian function in the complex domain, i.e., a Gaussian wave. In physics, this mathematical model is generally a harmonic oscillator, which is characterized by conservative forces. That is to say, for high-orbit short arcs, it is sufficient to consider only the Earth's gravitational field, thus making the model relatively simple.

[0152] In this embodiment, the wave function is defined as the Gaussian wave of the wave function. Gaussian wave of wave function It is derived based on the normal distribution assumption and constructed using Gaussian functions in the complex field.

[0153] In the Gaussian wave of the wave function, the distribution of physical states in each representation conforms to a normal distribution. This requires appropriate approximations of the operators in the model to form a closed mathematical structure.

[0154] The wave function is represented by a normalized Gaussian function defined in three-dimensional real space with a range in one-dimensional complex space; that is, the Gaussian wave function. for:

[0155] (2.5)

[0156] in, This is the current position. A is the second-order coefficient matrix of the wave function, which is a symmetric matrix. B is the first-order coefficient vector of the wave function. C is the normalization constant of the wave function.

[0157] It should be noted that the wave function in location space mentioned above... This is a practical physical concept; the wave function here refers to a Gaussian wave. wave function in location space The mathematical model under the assumption of normal distribution can also be called a Gaussian wave of the wave function.

[0158] Since the covariance matrix of the normal distribution is positive definite, it is necessary to constrain the real part Re(A) of the second-order coefficient matrix A of the wave function to be positive definite. Thus, the Gaussian wave function is now clearly defined. The definition of the corresponding function space is that all vectors in the function space are functions.

[0159] Step S116: Modify the operator based on the measurement uncertainty The required formulas and the degree of uncertainty variation during the observation process. Give the uncertainty modification operator. The specific definition formula is then used; subsequently, the Gaussian wave of the wave function is substituted to determine the uncertainty modification operator. Does it meet the requirements?

[0160] Uncertainty Modification Operator The defining formula is:

[0161] ,

[0162] in, It refers to the degree of change in measurement uncertainty. To measure uncertainty, It is the uncertainty corrected by modifying the operator based on the uncertainty. It is the wave function in position space. This is the current location.

[0163] Gaussian wave of wave function Substitute the uncertainty modification operator After defining the specific formula, the uncertainty modification operator is then used. The defining formula is:

[0164] , (2.13)

[0165] in, It refers to the degree of change in measurement uncertainty. To measure uncertainty, It is the uncertainty corrected by modifying the operator based on the uncertainty. It is the wave function in position space. Current position For the real part, Let A be the imaginary part, and let B be the second-order coefficient matrix of the wave function, which is a symmetric matrix. Let B be the first-order coefficient vector of the wave function, and let C be the normalization constant of the wave function.

[0166] Clearly, adjusting the uncertainty does not change the real part of the second-order coefficients, therefore the uncertainty modification operator... It meets the requirement of closure.

[0167] Step S117: Provide the observed state of the location representation The definition formula.

[0168] The location distribution of the observation data is obtained as follows: The location center of the observation data is located based on the observation results. Then, based on the observation progress of the equipment (such as the completeness and time span of data acquisition), the standard deviation is defined to obtain a normal distribution of the locations, which serves as the location distribution of the observation data. The location distribution of the observation data can be represented by the square of the modulus of the observation state; therefore, the observation state of the location representation is defined based on the location distribution of the observation data. , is used to represent the location distribution of observed data.

[0169] In this embodiment, since the observation state is defined as a Gaussian function in the real number field, the observation state of the position representation is:

[0170] (2.14)

[0171] Where Σ is the covariance matrix corresponding to a single observation, and T denotes the transpose. Current position The location center of the observed data.

[0172] Therefore, through the observed state and the physical state (i.e., the wave function in position space) The point-to-point multiplication of the observed state enables the observation of the physical state in a quantum sense, i.e., the collapse of the wave function. Obviously, the second-order term on the exponent in the expression of the observed state is a positive definite matrix, and the collapsed state obtained after multiplication is still in the state space corresponding to the Gaussian wave of the wave function. Therefore, the state space is closed under the condition of collapse.

[0173] Step S12: Based on the definition of the wave function, obtain the corresponding calculation formula for the position distribution. Integrate the Gaussian wave of the wave function, the calculation formula for the position distribution, and the calculation formulas for the expected values ​​of position and momentum into a physical quantity calculation module.

[0174] Among them, the Gaussian wave of the wave function The squared modulus of a satellite is used as a formula to calculate its position distribution, representing the probability distribution of satellite positions, i.e., the normal distribution in position space. Therefore, the formula for calculating the position distribution is:

[0175] (2.6)

[0176] in, This is the current position. A is the second-order coefficient matrix of the wave function, which is a symmetric matrix. B is the first-order coefficient vector of the wave function. C is the normalization constant of the wave function.

[0177] Among them, position expectation By using the formula for calculating the spatial position coordinates It is obtained by weighted averaging. That is, the formula for calculating the expected location is: = =∫ d , This is a positional operator.

[0178] In this embodiment, the present invention constructs a scalar wave function model applicable to conservative forces as a Gaussian wave based on the concept of uncertainty, providing a theoretical basis for subsequent trajectory prediction.

[0179] Step S13: Establish the formula for the time evolution operator and integrate it to obtain the wave function evolution module;

[0180] The formulas for the time evolution operator include the definition and decomposition formulas of the time evolution operator, the operational formulas for representation transformation, and the calculation formulas for kinetic energy evolution and potential energy evolution.

[0181] Step S13 specifically includes:

[0182] Step S131: For the wave function in the location space Adding a time dimension to obtain the wave function that evolves over time The definition of the time evolution operator is obtained by solving the equation of motion; thus, based on the time evolution operator of the wave function, the operation between the wave function and the observation data at different times is realized.

[0183] The time evolution operator of the wave function is given by the equation of motion, which has the same form as the Schrödinger equation.

[16] The equation of motion is expressed as:

[0184] (2.7)

[0185] in, It is a wave function that evolves over time. It is uncertainty. and These are spatial location and time. It is the Hamiltonian operator, V It represents potential energy, and t represents the current time.

[0186] Therefore, the solution to the equations of motion, expressed using the initial state and time evolution operators, is as follows:

[0187] (2.8)

[0188] The exponential part is the time evolution operator. T is the time sorting operator. It is the Hamiltonian operator. and These are the initial state time and the current time, respectively. It is the time to be integrated. It represents the measurement uncertainty for each time interval being integrated, where i is the imaginary unit.

[0189] Therefore, the definition of the time evolution operator is Its evolution over multiple moments between the initial state and the current time can be represented as .

[0190] Step S132: Based on the split-step Fourier method, obtain the decomposition formula of the time evolution operator;

[0191] Time evolution operator of wave function The decomposition formula is:

[0192] (2.9)

[0193] in, , , These are the Hamiltonian operator, the kinetic energy operator, and the potential energy operator, respectively. It is a wave function that evolves over time.

[0194] Step S133: Establish the operational formula for representation transformation in order to facilitate the calculation of the decomposition formula of the time evolution operator;

[0195] According to the split-step Fourier method, the time evolution operator of the wave function needs to perform one operation each in the position representation and the momentum representation (i.e., potential energy evolution operation and kinetic energy evolution operation) for each evolution, which requires representation transformation.

[0196] The formula for representation transformation is based on Fourier transform. The formula for representation transformation is:

[0197] (2.10)

[0198] in, It is the Gaussian wave corresponding to the wave function in the momentum representation. The Gaussian wave of the wave function is the Gaussian wave in the position representation; It is the current momentum. This is the current position. A is the second-order coefficient matrix of the wave function, B is the first-order coefficient vector of the wave function, and C is the normalization constant of the wave function. It is measurement uncertainty.

[0199] because It is a positive real number. It is a matrix whose eigenvalues ​​all have positive real parts, and its inverse matrix is... This property is also maintained, so its real part is also positive definite. That is, the real part of the matrix corresponding to the coefficients of the second-order terms on the exponent in the wave function in the momentum representation is also positive definite. This implies that the Gaussian wave function... The parameter space is closed for representation transformations.

[0200] Step S134: Establish calculation formulas for potential energy evolution and kinetic energy evolution to facilitate the calculation of the decomposition formula of the time evolution operator.

[0201] This invention refers to the two steps of computation required for evolution as kinetic energy evolution and potential energy evolution. Since a Gaussian distribution remains a Gaussian distribution after Gaussian integration, approximating the computation process according to the Gaussian integral formula can satisfy closure, thereby approximating the potential energy as a second-order expansion at each point in space.

[0202] Based on the potential energy approximation method, the formula for calculating potential energy is:

[0203] (2.11)

[0204] in, For the prior instantaneous position, Current position Let be the potential energy function. Let be the gradient of the potential energy function. The Hessian matrix is ​​the potential energy parameter.

[0205] The formula for calculating the evolution of potential energy is:

[0206] (2.12)

[0207] in, It is a potential energy evolution term. For the prior instantaneous position, Current position Let be the potential energy function. Let be the gradient of the potential energy function. The Hessian matrix is ​​the potential energy parameter. Gaussian wave of wave function ∠A represents the current position, A is the second-order coefficient matrix of the wave function, B is the first-order coefficient vector of the wave function, C is the normalization constant of the wave function, and ℏ is the uncertainty.

[0208] Since the potential energy function is a real function, the wave function after the potential energy evolves... Gaussian waves that still belong to the wave function The parameter space. From a mathematical perspective, kinetic energy evolution can be viewed as a special case of potential energy evolution, therefore, this invention will not repeat the details.

[0209] As mentioned above, step S2 includes: constructing a satellite trajectory prediction model based on the wave function filtering module, the physical quantity calculation module, and the wave function evolution module.

[0210] In this embodiment, the constructed satellite trajectory prediction model is a satellite trajectory prediction model that only considers conservative forces. This model, which only considers conservative forces, is the simplest model in this invention. Its construction process does not require the introduction of probabilistic methods such as least squares; instead, the idea of ​​path integrals is hidden in the data fitting process. It also does not include data-driven machine learning methods; its iterative process is solely to reduce the influence of initial assumptions on the final result. Although the satellite trajectory prediction model only uses physical concepts, its model structure can be analogized to a recurrent neural network for easy understanding.

[0211] like Figure 2 As shown, the satellite trajectory prediction model includes a loop module, which comprises a wavefunction filtering module, a physical quantity calculation module, and a wavefunction evolution module; the wavefunction evolution module receives the wavefunction from the previous time step. The superscript S represents the Schrödinger picture and utilizes the time evolution operator. The evolution yields the evolved wave function. The wavefunction filtering module receives the observation state at the current moment. And the evolved wave function, through the observed state at the current moment Weighting is achieved jointly with the evolved wave function, and the operator is modified using uncertainty after weighting. Reduce the measurement uncertainty of the wave function to output the wave function at the current moment. The physical quantity calculation module receives the wave function at the current moment. and physical quantity operators , Output the expected physical quantity at the current moment. , . For positional operators, For momentum operator, For position expectation, For the expected value of momentum.

[0212] Therefore, the model structure can be compared with that of a recurrent neural network, using the observation state... As input, with wave function As a neural network, it takes the expectation of a physical quantity as its output, starting with an initial hypothesis wavefunction with extremely low confidence; and utilizes time evolution operators. Evolving the wave function to the nearest observation time, from the observation state Filtering is achieved by weighting the wave function, and the data fitting process is achieved through multiple filtering of the observed data.

[0213] The physical quantity operators are initialized using the physical quantity operators in the Schrödinger picture, while the initialization of the wave function requires the introduction of some assumptions for construction. Therefore, step S2 may also include: establishing initial value assumptions for the uncertainty and the wave function.

[0214] In satellite trajectory prediction models, the physical state specifically refers to the satellite's motion state. Observational data directly provides the distribution of the wave function in the position representation, while the distribution in the momentum representation is derived through assumptions. Assuming the satellite's orbit is a perfect circle passing through the observation position at a certain moment, the expected velocity of the satellite at that moment can be calculated, and the standard deviation of the velocity is assumed to be 0.5% of the velocity. In the prediction model, the satellite's mass has no practical effect; for computational convenience, it is assumed to be 1 kg, thus the momentum and velocity have the same numerical value.

[0215] In this embodiment, based on the initial value assumption, the initial value of the measurement uncertainty is... initial values ​​of wave function for:

[0216] (2.15)

[0217] (2.16)

[0218] In the formula, Σ is the observation variance. It is the expected location. This is the expected value of momentum. The normalization coefficient is omitted in the formula.

[0219] It should be noted that, in this embodiment, although the satellite trajectory prediction model can be structurally analogous to a recurrent neural network, it is not a machine learning method and therefore does not have a loss function. In other words, the wave model's learning of wave function parameters does not require reducing the loss function. This is because the evolution-observation related operations are, to some extent, consistent with the idea of ​​path integrals. Therefore, during the iteration process, the parts of the satellite trajectory prediction model that do not conform to the true trajectory are directly suppressed, without the need for a loss function. In fact, the model most similar to this invention is various dynamic methods, such as the generalized Laplace method, which has complete physical meaning.

[0220] In other embodiments of the present invention, the satellite trajectory prediction model can be implemented using physical machine learning methods, such as the restricted Boltzmann machine, which can be considered a physics-inspired machine learning method because its corresponding loss function is the fitting error, and the main components of the fitting error are similar to the propagator in quantum mechanics.

[0221] Second embodiment: Satellite orbit extrapolation method based on filtering concept considering conservative forces and damping terms.

[0222] Satellite motion is also influenced by non-conservative forces such as atmospheric damping and solar radiation pressure. In the first embodiment, this invention proposes a satellite trajectory prediction model based on a scalar wave function, which assumes that the effects of conservative forces are the primary basis. However, in actual low-Earth orbit satellite trajectory prediction, the influence of non-conservative forces such as atmospheric damping cannot be ignored (the positional shift is approximately 100 meters per day). Therefore, in this embodiment, this invention introduces a damping term into the equation of motion to construct a damped wave function model, thereby expanding the applicability of the wave function model and aiming to verify its improved fitting ability. If the prediction accuracy is significantly improved, it will not only demonstrate the general applicability of the theory but also, to some extent, show that the equations of motion studied in the quantum realm can be transferred to the theory of this invention.

[0223] According to a second embodiment of the present invention, the satellite orbit extrapolation method based on filtering principles includes the following steps:

[0224] Step S1': Using the satellite's motion state as a wave, establish a mathematical model of the wave function, and integrate the mathematical model of the wave function into a wave function filtering module, a physical quantity calculation module, and a wave function evolution module; wherein, the mathematical model of the wave function is a mathematical model of the wave function that takes into account atmospheric damping (or other non-conservative forces).

[0225] Step S2': Based on the wave function filtering module, physical quantity calculation module and wave function evolution module, construct a satellite trajectory prediction model; correspondingly, the satellite trajectory prediction model established in step S2' is a satellite damped trajectory prediction model.

[0226] Step S3': Input the satellite orbit extrapolation data from the previous moment and the real-time observation data into the satellite trajectory prediction model to realize wave function evolution, wave function filtering and physical quantity calculation, and output the satellite orbit extrapolation data at the current moment.

[0227] In this embodiment, steps S11'-S13' of step S1' are exactly the same as steps S11-S13 of step S1 in the first embodiment. In addition, step S1' further includes:

[0228] Step S14': Establish the damped wave equation to give the calculation formulas for generalized momentum, generalized position and extended Hamiltonian; establish the operator evolution module using the calculation formulas for generalized momentum and generalized position;

[0229] The formula for calculating atmospheric damping perturbation is:

[0230] (3.1)

[0231] In the formula It is resistance. It is atmospheric density. It is the atmospheric damping coefficient. It is the windward area. It's speed. Atmospheric density. Referring to "Space Debris - Models and Risk Analysis", the table is obtained as follows: Theoretically, a more accurate value can be obtained by using gradient descent with the fitting error as the loss function.

[0232] In the symbolic model, the satellite mass is always taken. The above formula is changed to:

[0233] (3.2)

[0234] In the formula, This represents the acceleration of the satellite due to atmospheric drag. This indicates the satellite's area-to-mass ratio. It is the current momentum. It is the atmospheric damping coefficient.

[0235] In this invention, some parameters in the simulation environment are selected as follows:

[0236] (3.3)

[0237] In the reference equations of motion, atmospheric drag is proportional to momentum. Since it is a first-order term, the following approximation is made:

[0238] (3.4)

[0239] In the formula, This is the expected momentum of the satellite at the current calculation step. It is the damping coefficient.

[0240] The damped wave equation was established using the method proposed by Peng Huanwu in 1980.

[0241] The damped wave equation is established, specifically including:

[0242] The first step is to use analytical mechanics to derive the equations of motion for the satellite with the damping term introduced:

[0243] (3.5)

[0244] Where x is position, p is momentum, and M is satellite mass. is the damping coefficient, and K is the coefficient related to the local potential energy. For the sake of convenience, the potential energy is directly approximated as the potential energy of the harmonic oscillator.

[0245] The second step is to find variables that satisfy the regular commutation relation, which are the generalized position X and the generalized momentum P.

[0246] Wherein, the generalized position X and the generalized momentum P satisfy:

[0247] (3.6)

[0248] Treat x, representing position, and p, representing momentum, as non-commuting operators, and find the time derivative of their commutator.

[0249] (3.7)

[0250] Where x is the current position, p is the current momentum, and M is the satellite mass. It is the damping coefficient. , The meaning is the derivative of the current position and current momentum.

[0251] The first equality in equation (3.7) is obtained by differentiation, while the second equality needs to be obtained by substituting into the equation of motion.

[0252] Integrating the above equation, we obtain the damping commutation relation, which is:

[0253] (3.8)

[0254] The generalized position X and the generalized momentum P satisfy the following transformation relationship:

[0255] (3.9)

[0256] Where x is the current position, p is the current momentum, and M is the satellite mass. t is the damping coefficient, and t is the current time.

[0257] The third step is to apply the Hamiltonian canonical equation.

[0258] (3.10)

[0259] Substituting the equation of motion into the expression on the left side of the equation, and then integrating, we obtain the Hamiltonian H:

[0260] (3.11)

[0261] Where M is the satellite mass, Let X be the generalized momentum, X be the generalized position, and K be the second derivative of the potential energy. The damping coefficient is... For Hamiltonian.

[0262] The damped wave equation is constructed as follows:

[0263] (3.12)

[0264] in, It is a damped wave function.

[0265] This invention extends the wavefunction model by introducing a damping term, aiming to enable the model to handle the influence of atmospheric damping on satellite trajectories. Furthermore, Peng Huanwu's method also calculates time-dependent forces, thus enabling the calculation of light pressure perturbations and further extending the wavefunction model.

[0266] Step S15': Establish the formula for the damping parameter β, and use the generalized position, generalized momentum, and damping parameter to establish the model of the damping wave function. The definition of the wave function in the wave function filtering module is replaced; based on the model of the damped wave function, the formulas for position expectation and momentum expectation in the physical quantity calculation module are corrected.

[0267] In other words, the damped wave function model Compared to the scalar wave function model applicable to conservative forces, a damping parameter β is added to establish a scalar wave function model considering atmospheric damping, and the generalized position X and generalized momentum P are used to describe the state.

[0268] Observe the transformation relationship between x, p and X, P, using damping parameters. It can calculate the position x and momentum p. Therefore, the data description structure of the damped wave function is as follows: When calculating the evolution of the state over time, the damping parameter β needs to be considered. The formula for the damping parameter β is:

[0269] (3.13)

[0270] in, It was the first moment Damping parameters, It is the second moment The damping parameter, e is the natural base. Damping coefficient, M is the satellite mass.

[0271] Location expectation and momentum expectation The formula is:

[0272] (3.14)

[0273] (3.15)

[0274] in, It is a positional operator. It is a momentum operator. It is a damped wave function model. It refers to generalized momentum, uncertainty, and damping parameters. Positional operators is the damping coefficient, and M is the satellite mass.

[0275] Uncertainty Modification Operator The following conditions must be met:

[0276] (3.16)

[0277] (3.17)

[0278] in, It is a positional operator. It is a momentum operator. It is a damped wave function model. It refers to generalized momentum, uncertainty, and damping parameters. Positional operators is the damping coefficient, and M is the satellite mass.

[0279] After extracting the damping parameter from formula (3.16), it was found that it is equivalent to the condition that the symbolic model considering only conservative forces needs to satisfy. Therefore, it is not necessary to modify the operator for uncertainty. Make corrections.

[0280] Step S16': Replace the decomposition formula of the time evolution operator with the extended Hamiltonian calculation formula to obtain the improved wave function evolution module;

[0281] Therefore, the evolution process in the improved wavefunction evolution module is modified according to the extended Hamiltonian calculation formula (3.11) to realize the calculation of the extended Hamiltonian.

[0282] Among them, the improved wave function evolution module incorporates formulas for damping parameters, potential energy, and the damping term of the Hamiltonian to facilitate the calculation of the extended Hamiltonian.

[0283] In the extended formula for the Hamiltonian (3.11), the second differential K of the potential energy represents the local expression of the potential energy, requiring consideration of the instantaneous center position of the harmonic oscillator. To calculate the second differential K of the potential energy, the formula (2.11) is modified as follows:

[0284] (3.18)

[0285] in, This is the current location. It is the a priori instantaneous position. It is the potential energy function at the a priori instantaneous position The Hessian matrix, It is the potential energy function at the a priori instantaneous position The gradient.

[0286] Therefore, in the extended formula for calculating the Hamiltonian (3.11), the expression for the damping term of the Hamiltonian in the Schrödinger representation is:

[0287] (3.19)

[0288] As mentioned above, step S2': Based on the wave function filtering module, physical quantity calculation module and wave function evolution module, a satellite trajectory prediction model is constructed; correspondingly, the satellite trajectory prediction model established in step S2' is a satellite damped trajectory prediction model.

[0289] This invention refers to the satellite trajectory prediction model considering atmospheric damping perturbations as the satellite damped trajectory prediction model. Considering the time-dependent Hamiltonian of the damped wavefunction model, and drawing inspiration from quantum mechanics, a Heisenberg model with time-dependent operators is required. Since the description of the observation state is most intuitive in the Schrödinger model, the model is constructed by combining the interaction models of both. The difference between the satellite trajectory prediction model in this embodiment and the satellite trajectory prediction model in the first embodiment that only considers conservative forces is that it uses an interaction representation. It is completely equivalent to the satellite trajectory prediction model in the first embodiment that only considers conservative forces at the initial moment.

[0290] In this embodiment, as Figure 3 As shown, the model structure of the satellite trajectory prediction model of the present invention can be similar to that of a long short-term memory network. It takes the observation state as input, physical quantity operators as hidden layers, physical state as memory network, and physical quantity expectation as output. The physical state guided by observation filters out the forgetting of analogous memory.

[0291] The satellite trajectory prediction model of this invention includes a loop module, which comprises a wavefunction evolution module, an operator evolution module, a representation transformation module, a wavefunction filtering module, and a physical quantity calculation module. The wavefunction evolution module receives the damped wavefunction from the interaction scene at the previous moment. And utilize time evolution operators The evolution yields the evolved damped wave function; the operator evolution module is configured to receive physical quantity operators from the previous time step in the interaction scene. (i.e., position operator, momentum operator) and process them to obtain the physical quantity operators in the interaction scene at the current moment. The representation conversion module is configured to receive the current observation state in the Schrödinger picture. and the physical quantity operators in the interaction scene at the current moment. The process yields the observed state in the interaction picture at the current moment. The wavefunction filtering module receives the observation state in the interaction scene at the current moment. and the evolved wave function The damped wave function in the interaction scene at the current moment is obtained through processing. The physical quantity calculation module receives the physical quantity operators in the interaction scene at the current moment. and the damped wave function in the interaction scene at the current moment. Processing yields the expected physical quantity at the current moment. The physical quantity operator represents all possible operators. Since this embodiment only uses the position and momentum operators, the result includes both position and momentum operators. Furthermore, It can also be understood as two types of wave functions: position representation and momentum representation, which can be obtained by conversion between each other.

[0292] The wave function evolution module corresponds to formula (2.8) above, but the difference from the first embodiment is that the Hamiltonian is calculated according to formula (3.11), while the damping parameter needs to be calculated according to formula (3.13). The wave function filtering module corresponds to formulas (2.3) and (2.4) above, that is, the representation transformation module is the observation state in the Schrödinger picture at the current moment. Physical quantity operators in the interaction scene at the current moment The product of the inverses. The operator evolution module is implemented based on formula (3.9). The physical quantity calculation module is based on formulas (3.14) and (3.15) above.

[0293] In this embodiment, as Figure 3 The model shown can be directly used in the equation of motion using the central potential energy of the harmonic oscillator, but in the calculation process of this invention, the center of the instantaneous potential energy is actually moving.

[0294] Third embodiment: Satellite orbit extrapolation method based on filtering concept considering conservative forces and damping terms.

[0295] For a target with a semi-major axis of 7000 km, due to the damping coefficient The order of magnitude is The geocentric distance from the center of the instantaneous potential energy (a quadratic function in three-dimensional space) is... From equation (3.15), we can see the damping coefficient. The order of magnitude of the effect on speed is Therefore, its effect on momentum is small and negligible. Thus, the momentum operator and the position operator can be approximated as having the same form, i.e., the momentum expectation can be approximated as:

[0296] (3.20)

[0297] In the third embodiment, step S1' of the filtered satellite orbit extrapolation method of the present invention is basically the same as step S1' in the second embodiment, except that it also includes: replacing formula (3.15) with formula (3.20) to simplify the calculation of momentum expectation. This modification allows the momentum operator and the position operator in formula (3.14) to be transformed into the Schrödinger representation through coordinate scaling.

[0298] Thus, the advantage of this approximation is that after evolution via the wave function evolution module, the damped wave function (i.e., the physical state) can always maintain its position representation in the Schrödinger picture S when switching back from the interaction picture I to the Schrödinger picture S.

[0299] like Figure 4 As shown, in step S2', the established satellite trajectory prediction model includes a loop module, which comprises a wave function evolution module, an operator evolution module, a representation transformation module, a wave function filtering module, and a physical quantity calculation module. The wave function evolution module receives the damped wave function in the Schrödinger picture from the previous moment. The evolved damped wave function is obtained by using a time evolution operator; the operator evolution module is configured to receive physical quantity operators in the Schrödinger picture from the previous time step. The module processes and obtains the physical quantity operators in the interaction picture at the current moment; the representation transformation module is configured to transform the evolved damped wave function and the physical quantity operators in the interaction picture at the current moment to the Schrödinger picture respectively; the wave function filtering module is configured to filter the observed state in the Schrödinger picture at the current moment based on the evolved damped wave function and the observed state in the Schrödinger picture at the current moment. The damped wave function in the Schrödinger picture at the current moment is obtained through processing. The physical quantity calculation module is configured to calculate the damped wave function in the Schrödinger picture at the current moment. And the physical quantity operators in the Schrödinger picture at the current moment. Processing yields the expected physical quantity at the current moment. The physical quantity operators include position and momentum operators. Therefore, the model structure is simplified from a long short-term memory network to a recurrent neural network.

[0300] Fourth Implementation: A Satellite Orbit Extrapolation Method Based on Filtering Considering Conservative Forces and Damping Terms

[0301] Following the approach of the fourth embodiment, and further considering formula (3.8), it is found that formula (3.9) can be replaced by another interaction representation that may not conform to physics. In this interaction representation, the position operator is the same as that in the Schrödinger representation, while the momentum operator is time-dependent.

[0302] (3.21)

[0303] , It is a generalized positional operator, representing the current position. , It is the momentum operator for generalized momentum and current momentum. Here, M is the damping coefficient, M is the satellite mass, and t is the current time.

[0304] It does not conform to physics because the evolution operator is derived from the understanding of wave functions in this invention, rather than being derived.

[0305] Therefore, in the fourth embodiment, step S1' of the filter-based satellite orbit extrapolation method of the present invention is basically the same as step S1' in the second embodiment, except that it also includes step S17': replacing formula (3.9) for generalized position X and generalized momentum P with formula (3.21) to correct the operator evolution module; replacing Hamiltonian with formula (3.22). Formula (3.11) is used, and formula (3.23) is used to replace the calculation formula (2.9) of the time evolution operator to correct the calculation formula of the extended Hamiltonian.

[0306] Hamiltonian Represented as:

[0307] (3.22)

[0308] However, the time evolution operator of the corresponding wavefunction is:

[0309] (3.23)

[0310] in, Let X be the generalized momentum and X be the generalized position. For momentum operator, For positional operators, The damping coefficient is... For Hamiltonian operators, For measurement uncertainty, t is the current time, M is the satellite mass, and K is the second derivative of the potential energy.

[0311] The correction in formula (3.23) is to treat the evolution operators involving momentum as equivalent to adjustments to position, and the evolution operators involving position as equivalent to adjustments to momentum. Therefore, the corrected result can be understood as shifting the object of momentum adjustment from... Transferred to superior.

[0312] In step S2', the structure of the established satellite trajectory prediction model is as follows: Figure 5 As shown. Each loop module of the satellite trajectory prediction model includes a wavefunction evolution module, an operator evolution module, a wavefunction filtering module, and a physical quantity calculation module. The wavefunction evolution module receives the damped wavefunction from the Schrödinger picture at the previous moment. The evolved damped wave function is obtained by using a time evolution operator; the operator evolution module is configured to receive the position operator in the Schrödinger picture. The momentum operator in the interaction scene at the previous moment The position operator in the Schrödinger painting is obtained through processing. and the momentum operator in the interaction scene at the current moment. The wavefunction filtering module is configured to filter based on the evolved damped wavefunction and the current observation state in the Schrödinger picture. The damped wave function in the Schrödinger picture at the current moment is obtained through processing. The physical quantity calculation module is configured to calculate the damped wave function in the Schrödinger picture at the current moment. And the physical quantity operators in the Schrödinger picture at the current moment. , Processing yields the expected physical quantity at the current moment. , Among them, physical quantity operators include position operators. and momentum operator .

[0313] Fifth Implementation: A Satellite Orbit Extrapolation Method Based on Filtering Concept Introducing Non-Spatiotemporal Degrees of Freedom

[0314] The previous first to fourth embodiments all established satellite physical trajectory prediction models based on wave function theory, describing in detail the motion laws of satellites under ideal physical conditions and laying the theoretical foundation for trajectory prediction. However, actual orbital satellite trajectories often undergo complex changes due to non-physical factors such as human maneuvering, making it difficult for physical trajectory models to fully characterize these maneuvered trajectory characteristics. Therefore, in the fifth embodiment of this invention, from the perspective of trajectory function expansion, non-spatiotemporal degrees of freedom are introduced to construct a screw wave function model, aiming to better describe the satellite's maneuvering behavior and its impact on the trajectory. This invention will achieve modeling of maneuvering trajectories, promoting the development of trajectory prediction from purely physical trajectories to comprehensive trajectory prediction that includes human factors. It should be noted that in the fifth embodiment, the research focus is on expanding the model's function, which is fundamentally different from the purpose of improving the model's fitting ability by introducing damping terms in the second to fourth embodiments.

[0315] The motivations and methods for satellite orbit changes exhibit a certain behavioral logic, and this invention assumes that the entirety of this behavioral logic can be described in spin space. Each component in spin space calculates its time evolution according to different equations of motion, and the specific behavioral logic of each target can be orthogonally decomposed into several known behavioral logics. It should be noted that this invention directly borrows the term "spin space," but its specific physical meaning is different.

[0316] To allocate the weights of the wavefunction in spin space, this invention considers the energy-momentum tensor of the wavefunction at each point in space. This approach obviously violates the non-commutation relation, but a corresponding concept exists in quantum mechanics: the Wigner distribution. As a quasi-probability distribution, the Wigner distribution can simultaneously describe the position and momentum information of a quantum state in phase space. Although it does not strictly conform to classical probability rules, it effectively captures the spatiotemporal locality of quantum systems. This invention adopts a similar idea, conceiving an approximate calculation method that aims to calculate the energy-momentum tensor at each point to achieve spin space allocation without affecting the evolution of the wavefunction in shape-position space. This method is suitable for the application scenarios of this invention.

[0317] According to a fifth embodiment of the present invention, the satellite orbit extrapolation method based on filtering principles includes the following steps:

[0318] Step S1”: Using the motion state of the satellite as a wave, establish a mathematical model of the wave function, and integrate the mathematical model of the wave function into a weight transfer module, a wave function evolution module, a wave function filtering module, and a physical quantity calculation module; among them, the mathematical model of the wave function is a spinor wave function model that introduces non-spatiotemporal degrees of freedom.

[0319] Step S2”: Based on the weight transfer module, wave function evolution module, wave function filtering module, and physical quantity calculation module, construct a satellite trajectory prediction model; correspondingly, the satellite trajectory prediction model established in step S2” is a satellite maneuver trajectory prediction model.

[0320] Step S3”: Input the satellite orbit extrapolation data from the previous moment and the real-time observation data into the satellite trajectory prediction model to realize wave function evolution, wave function filtering and physical quantity calculation, and output the satellite orbit extrapolation data at the current moment.

[0321] Therefore, step S1” includes:

[0322] Step S11”: Establish the definition of measurement uncertainty, introduce phase to establish the spinor wave function model for each spin component, and establish the weight transfer function of the spinor space and the calculation formula of each component of the weighted spinor wave function model.

[0323] The momentum information of the wave function is contained in the phase of the position representation. This invention assumes that the degree of phase change is proportional to the magnitude of momentum. Therefore, in order to calculate the a priori instantaneous position... The momentum at a given point is first expressed as the product of amplitude and phase in the spinor wave function model. For each spin component, the spinor wave function model... for:

[0324] (4.1)

[0325] In the formula, and These are functions of amplitude and phase, respectively. This is the current location.

[0326] Then replace the amplitude part with the a priori instantaneous position. A normal spherical distribution centered at the center is obtained, with prior instantaneous location. spinor wave function model :

[0327] (4.2)

[0328] In the formula, It is uncertainty. It is the identity matrix. It is the normalization coefficient. For the prior instantaneous position, Current position It is a function of phase.

[0329] Subsequently, the a priori instantaneous position was... The momentum expectation of the spinor wave function model is taken as the prior instantaneous position. Momentum at the point Then the a priori instantaneous position Momentum at the point for:

[0330] (4.3)

[0331] In the formula, It is a momentum operator. For having a priori instantaneous position The spinor wave function model.

[0332] The angular momentum and energy at each point can be calculated from the momentum at each point:

[0333] (4.4)

[0334] (4.5)

[0335] in, For the prior instantaneous position Momentum at that point For the prior instantaneous position Angular momentum at that point For the prior instantaneous position The energy at the location, For the prior instantaneous position Potential energy at the location, For satellite quality.

[0336] The spin space consists of four components, representing no maneuvering, tangential maneuvering along the orbit, radial maneuvering along the orbit, and normal maneuvering along the orbit. This invention constructs a position-dependent weight transfer function based on the physical quantities of the wave function in position space for each component, to calculate the weight allocation in the spin space.

[0337] The formulas for calculating each component of the spinor wavefunction model after weight transfer are as follows:

[0338] (4.6)

[0339] In the formula, , , , This represents the components of the spinor wavefunction model after weight transfer. The subscripts N, S, T, and W correspond to no maneuver, tangential maneuver along the track, radial maneuver along the track, and normal maneuver along the track, respectively. The elements in the matrix... It is the weight transfer function, i,j=N,S,T,W.

[0340] Weight transfer function (i,j=N,S,T,W) need to satisfy the normalization relation, that is:

[0341] (4.7)

[0342] This invention focuses on orbit-keeping maneuvers. Satellites, due to perturbation forces, gradually deviate from their designed orbits and typically adjust periodically to maintain their orbits. In this invention, orbit-keeping maneuvers are considered continuous maneuvers, and the angular momentum and energy of the system can be approximated as conserved quantities. A weighted transfer function in spinor space is designed based on the angular momentum and energy at each point. Therefore, the weighted transfer function... (i,j=N,S,T,W) is:

[0343] (4.8)

[0344] In the formula, , These are the energy and angular momentum of the satellite's designated working orbit. , , These represent the maximum differences in angular momentum, energy, and inclination between a satellite at the same altitude and its operational orbit within one month. It is a constant with a value greater than 1, used to adjust the maneuver probability. The purpose of this design is to achieve the following: the greater the energy difference, the greater the probability of maneuvering radially along the track; the greater the difference in angular momentum, the greater the probability of maneuvering tangentially along the track; and the greater the angle between the angular momentum, the greater the probability of maneuvering along the normal phase of the track plane.

[0345] Step S12”: Establish the Gaussian wave of the screw wave function model and measurement uncertainty Formulas for the intensity of each component of the wave function, and the location distribution of the spinor wave function. The formula is derived, and the motion formulas for each spin component of the spinor wave function are established.

[0346] in, The potential energy required for maneuvering along the normal direction of the orbital plane, It is the angular momentum of the working trajectory.

[0347] The observation formula for the spinor wave function is the same as that in equations (2.3) and (2.4), but the calculation method for the spatial distribution of the position in the formula is different.

[0348] Gaussian wave of spinor wave function yes:

[0349] (4.13)

[0350] Where x represents the current position, , The components of the Gaussian wave function in spin space are represented by the following: For measurement uncertainty, t represents the current time.

[0351] Components of the Gaussian wave function in spin space Expressed using a Gaussian mixture function defined in three-dimensional real space with a range in one-dimensional complex space, it is:

[0352] (4.14)

[0353] In the formula, The current position is indicated by the subscript i, which represents the index of the spin space, and the subscript j represents the component index of the Gaussian mixture function. It is the second-order coefficient matrix of the components of the j-th mixture Gaussian function in the i-th spin space. It is the first-order coefficient vector of the component of the j-th mixture Gaussian function in the i-th spin space. It is the intensity constant of the component of the j-th mixed Gaussian function in the i-th spin space, and the superscript T is the transpose.

[0354] It is a symmetric matrix and its real part is positive definite. It is a vector. It is a constant. Here... Instead of normalization coefficients, it expresses the intensity of each component, which is different from the concept of proportion in a Gaussian mixture distribution, because different components are not necessarily in phase at the same point.

[0355] The intensity of the i-th spin-space j-th mixture Gaussian function component of the wave function The Gaussian integral equal to this component is:

[0356] (4.15)

[0357] Location distribution of spinor wave function The formula is:

[0358] (4.16)

[0359] The momentum at each point in space is defined as the degree of abrupt phase change at that point, i.e., the derivative of the imaginary part of the exponential term of the Gaussian function. The momentum of the i-th spin space... for:

[0360] (4.17)

[0361] Angular momentum of the i-th spin space and energy for:

[0362] (4.18)

[0363] (4.19)

[0364] In the formula, Let be the momentum of the i-th spin space. Current position It is the potential energy function of two-body gravitational force. The reason why the above formula uses the ideal two-body gravitational potential energy is that an ideal ellipse is generally used as the working orbit in satellite design.

[0365] The motion formulas for each spin component of the spinor wave function are established by directly modifying the wave function or the Hamiltonian. The construction idea is as follows: by modifying the imaginary part of the wave function, a motion formula for tangential maneuvering along the track is established; by adding potential energy centered on the Earth's center, a motion formula for radial maneuvering along the track is established; and by adding potential energy with the working track plane as the plane of symmetry, a motion formula for normal motion along the track plane is established.

[0366] The method of tangential maneuvering along the track involves contracting the coordinates of the wave function in the momentum representation around the momentum corresponding to the working track. To suppress the tendency for the phase change of the wave function to increase with coordinate contraction, the phase needs to be adjusted to maintain the probability density in the position representation.

[0367] The motion formula for tangential maneuvering along the track is:

[0368] (4.9)

[0369] (4.10)

[0370] In the formula, These are the modified momentum coordinates. It is the momentum at the current position calculated using the angular momentum of the working trajectory. This is the coordinate shrinkage coefficient, whose value is between 0 and 1. This represents the current momentum.

[0371] The method for radial maneuvering along the track involves constructing a spherically symmetric potential energy (only for circular tracks), with the working track located at the minimum of this potential energy. The motion formula for radial maneuvering along the track is:

[0372] (4.11)

[0373] In the formula, The potential energy required for radial maneuvering along the track, It is the radius of the working track. This is the current location.

[0374] The method for maneuvering along the normal direction of the orbital plane involves constructing a planar symmetric potential energy, with the working orbital plane located at the minimum of this potential energy. The equation for the motion of maneuvering along the normal direction of the orbital plane is:

[0375] (4.12)

[0376] in, It is the potential energy required for maneuvering along the normal to the orbital plane. It is the angular momentum of the working trajectory.

[0377] The method for tangential maneuvering along the track involves contracting the coordinates of the wave function in the momentum representation around the momentum corresponding to the working track, and then transforming it according to the motion formula (4.9) for tangential maneuvering along the track.

[0378] (4.20)

[0379] In the formula, This is the coefficient of coordinate contraction. However, since such a change would affect the positional distribution of the wave function, further modifications are made to obtain the motion formula for the tangential maneuver along the orbit:

[0380] (4.21)

[0381] in, These are the modified momentum coordinates. It is the momentum at the current position calculated using the angular momentum of the working trajectory. This is the coordinate shrinkage coefficient, whose value is between 0 and 1. For the current momentum, It is the result of adjusting the tangential motion component of the spinor wavefunction, increasing in each step of evolution. This adjustment simulates satellite maneuvers. It is the second-order coefficient matrix of the components of the j-th mixture Gaussian function in the i-th spin space. It is the first-order coefficient vector of the component of the j-th mixture Gaussian function in the i-th spin space. It is the intensity constant of the component of the j-th mixed Gaussian function in the i-th spin space, and T is the transpose.

[0382] Potential energy required for radial maneuver along the track The potential energy can be approximated locally by a plane symmetrical about the perpendicular tangent to the track and the normal to the track surface. For a circular track, this plane is the one whose normal vector is parallel to the track's radial direction and tangent to the working track. After this approximation, the motion formula for maneuvering along the track's radial direction is obtained:

[0383] (4.22)

[0384] (4.23)

[0385] In the formula, It is the radius of the working track.

[0386] After transformation, the motion formula for maneuvering along the normal direction of the orbital plane is obtained:

[0387] , (4.24)

[0388] In the formula, It is the potential energy required for maneuvering along the normal to the orbital plane. It is the angular momentum of the working trajectory.

[0389] Step S13”: Establish the definition of the degree of uncertainty change and the state of the wave function after observation. Uncertainty Modification Operator Definition formula; Establishment of time evolution operator The decomposition formula;

[0390] In the wave function, the satellite's position distribution is analogous to the time domain information of the wave, the velocity distribution is analogous to the frequency domain information of the wave, and the measurement uncertainty is defined as the product of the position uncertainty and the momentum uncertainty.

[0391] The evolution and observation process of the wave function needs to be clearly distinguished into weight transfer, evolution, collapse, and uncertainty adjustment. The evolution and collapse processes involve the separate evolution of each sub-component in the mixture Gaussian distribution, and the formulas are the same as the corresponding parts in the first embodiment of this invention (i.e., formulas (2.3) and (2.4)). Establish the time evolution operator. The decomposition formula is consistent with the corresponding part in the first embodiment of the present invention (i.e., formula (2.9)).

[0392] The uncertainty adjustment targets the entire wave function; the uncertainty modification operator... The defining formula is:

[0393] (4.25)

[0394] (4.26)

[0395] In the formula, The subscript indicates the index of the spin space, and "*" indicates the Gaussian mixture function corresponding to the calculated wave function component.

[0396] Step S14”: Based on the results of steps S11”-S13”, the weight transfer module, wave function evolution module, wave function filtering module, and physical quantity calculation module are integrated to obtain the weight transfer module, wave function evolution module, wave function filtering module, and physical quantity calculation module.

[0397] Step S14 further includes: simplifying the screw wave function model. This further simplifies the screw wave function model in the prediction model, reducing each component in the screw space from a mixture of Gaussian distributions to a Gaussian distribution.

[0398] The simplified spinor wavefunction model is as follows:

[0399] (4.27)

[0400] in, Let be the i-th spin-space component of the spinor wavefunction. It is the second-order coefficient matrix of the i-th spin space component. It is the first-order coefficient vector of the i-th spin space component. It is the intensity constant of the i-th spin space component.

[0401] This is because, according to the assumptions of this embodiment, the positional distribution of the track-maintaining maneuver is concentrated and smooth, without obvious multiple peaks. Therefore, such simplification is reasonable and reduces the complexity of data processing.

[0402] The weight transfer module is configured as follows:

[0403] 1. Calculate the expected position of the current global spinor wavefunction. Covariance ;

[0404] , (4.28)

[0405] (4.29)

[0406] (4.30)

[0407] In the formula, It is the intensity of the i-th spin-space component of the wavefunction. It is the expected value of the position of the i-th spin space of the wavefunction. It is the expected value of the position of the entire spinor wavefunction. It is the global covariance of the spinor wavefunction. , Let be the second-order coefficient matrix and first-order coefficient vector of the components in the i-th spin space. , Based on the calculation results in 4.16, these are the components of the simplified Gaussian wave in spin space for each wavefunction. The parameters in the expression.

[0408] Intensity of the i-th spin space component of the wave function The spinor wave function was calculated by combining the components of the spinor wave function using formula (4.16).

[0409] 2. In the context of Centered on The spin space component of the spinor wavefunction in a cube space of length denoted by . Uniform sampling;

[0410] 3. Calculate the discrete values ​​of the weight transfer function according to formulas (4.4), (4.5), and (4.8), where formula (4.8) takes... ;

[0411] 4. Calculate the expected component of the spinor wavefunction after weight transfer in the i-th spin space according to formula (4.6). ;

[0412] Formula (4.6) can be simplified as follows:

[0413] (4.31)

[0414] In the formula, An indicator representing spin space;

[0415] 5. Calculate the expected component of the spinor wavefunction after weighted shift in the i-th spin space. Expected position Covariance ;

[0416] The asterisk (*) indicates the result after adjusting each spinor wave component during the evolution process, which allows for the calculation of satellite maneuvers; otherwise, it is a normal evolution process.

[0417] (4.32)

[0418] (4.33)

[0419] The denominator in the formula is used to normalize the components.

[0420] 6. Construct the expected component of the i-th spin space of the spinor wavefunction after weight transfer. Location distribution:

[0421] (4.34)

[0422] In the formula, It is the intensity adjustment coefficient of the i-th spin space component of the spinor wavefunction.

[0423] The value is set to satisfy the following formula:

[0424] (4.35)

[0425] The summation in the formula is the probability density of discrete samples.

[0426] Therefore, the amplitude of the i-th spin space component of the spinor wavefunction for:

[0427] (4.36)

[0428] In the formula, This corresponds to the amplitude in formula (4.1), and should not be confused with the amplitude in formula (4.14);

[0429] 7. An expression for estimating the phase using the degree of phase change at the expected position; the phase of the i-th spin space component of the spinor wavefunction. The expression is:

[0430] (4.37)

[0431] In the formula, This corresponds to the phase in formula (4.1), and the purpose of the denominator is to normalize;

[0432] 8. Combining formulas (4.36) and (4.37), we obtain the wave function after weight transfer;

[0433] (4.38)

[0434] in, , , , This represents the amplitude of the spinor wavefunction's components in different spin spaces. , , , This represents the phase of the spinor wavefunction's components in different spin spaces.

[0435] The wave function evolution module is set as follows:

[0436] 1. Construct evolution operators according to equations (4.22) and (4.24) respectively. and For components and Perform the operation. The potential energy required for radial maneuvering along the track, It is the potential energy required for maneuvering along the normal to the orbital plane. and Let S be the spin-space components of the spinor wave function in the tangential maneuver S and the normal maneuver W along the orbit.

[0437] 2. According to time evolution operators The decomposition formula (2.9) performs batch operations on each component of the spinor wavefunction;

[0438] 3. Apply equations (4.20) and (4.21) to the components. Perform the operation. Let T be the spin space component of the spinor wave function in the radial maneuver T along the orbit.

[0439] The wave function filtering module is set as follows:

[0440] 1. Observation state Perform batch point-to-point multiplication with each component of the spinor wave function;

[0441] 2. Calculate the degree of change of measurement uncertainty according to the definition of the degree of change of measurement uncertainty (2.3), where the probability density of the whole wave function is calculated according to the formula (4.16) for the numerator part;

[0442] 3. Modify the operator according to the measurement uncertainty. The definition formula (4.26) is used to calculate the results of adjusting the inaccuracy of each component in batches.

[0443] The physical quantity calculation module is implemented by calculating the expected value of the position of the spinor wave function according to formula (4.29).

[0444] Then, as described above, step S2” is executed: based on the weight transfer module, wave function evolution module, wave function filtering module, and physical quantity calculation module, a satellite trajectory prediction model is constructed; correspondingly, the satellite trajectory prediction model established in step S2” is a satellite maneuver trajectory prediction model.

[0445] like Figure 6 As shown, the satellite trajectory prediction model includes multiple loop modules. Each loop module includes a weight transfer module, a wavefunction evolution module, a wavefunction filtering module, and a physical quantity calculation module. The weight transfer module receives the wavefunction from the previous time step. The superscript S represents the Schrödinger picture, which is processed to obtain the wavefunction after weight transfer; the wavefunction evolution module receives the wavefunction after weight transfer and processes it to obtain the evolved wavefunction; the wavefunction filtering module receives the observation state at the current time. And the evolved wave function, outputting the wave function at the current moment. The physical quantity calculation module receives the wave function at the current moment. and physical quantity operators , Output the expected physical quantity at the current moment. , . For positional operators, For momentum operator, For position expectation, This represents the expected value of momentum.

[0446] In other embodiments, step S1 may further include: simplifying momentum. angular momentum ,energy The calculation formula is used to simplify the weight transfer module.

[0447] Specifically, the weight transfer module uses a sampling and fitting method to calculate the distribution of the wavefunction in screw space. Using the analytical method of screw wavefunctions to implement the model's functionality does have some inherent drawbacks, mainly in terms of computational accuracy and program complexity.

[0448] The computational accuracy suffers from two sources of error: sampling error and fitting error. While the weight transfer module uniformly samples each component of the screw wavefunction, theoretically covering a cubic space containing the main data features, the actual number of samples still affects the final computational accuracy. Furthermore, the weight transfer module uses a Gaussian mixture model to approximate the distribution shape, which is complex to implement. Using a simple Gaussian distribution model for fitting can easily overlook the true structure of the data, thus affecting the reliability of the final result.

[0449] The program complexity stems from the fact that while the model primarily employs analytical computation, it relies on sampling and fitting operations in the discrete space. The constant switching between analytical and discrete operations in this hybrid model makes programming and debugging extremely difficult. Therefore, this invention further simplifies the process.

[0450] The simplified part is the matrix in equation (4.6). The position-related weight transfer function in the matrix is ​​modified to a position-independent form. This is done by changing the momentum, angular momentum, and energy in equations (4.3), (4.4), and (4.5) to be expressed using the expected value of the physical quantity of the wave function. That is, the momentum is... angular momentum ,energy Simplified to:

[0451] (4.39)

[0452] (4.40)

[0453] (4.41)

[0454] in, , , These are momentum, angular momentum, and energy. For momentum operator, For positional operators, Let be the potential energy function. For satellite quality.

[0455] Correspondingly, the components on the left side of equation (4.6) This can be represented by the sum of four Gaussian functions, and it can be approximately assumed that the phase change of each Gaussian function is consistent. Therefore, the expected value and covariance of the overall position of the spinor wave function can be directly calculated using formulas (4.28), (4.29), and (4.30). The expected value and covariance of the position are obtained, and then the components of the spinor wave function after weight transfer are directly obtained using formulas (4.36), (4.37), and (4.38). This avoids sampling and fitting in the weight transfer function.

[0456] The above description is merely a preferred embodiment of the present invention and is not intended to limit the scope of the invention. Various variations can be made to the above embodiments of the present invention. That is, all simple and equivalent changes and modifications made based on the claims and description of this invention fall within the protection scope of the claims of this patent. All aspects not described in detail in this invention are conventional technical content.

Claims

1. A satellite orbit extrapolation method based on filtering, characterized in that, include: Step S1: Using the satellite's motion state as a wave, establish a mathematical model of the wave function, and integrate the mathematical model of the wave function into a wave function filtering module, a physical quantity calculation module, and a wave function evolution module; Step S2: Based on the wave function filtering module, physical quantity calculation module, and wave function evolution module, construct a satellite trajectory prediction model; Step S3: Input the satellite orbit extrapolation data from the previous moment and the real-time observation data into the satellite trajectory prediction model to realize wave function evolution, wave function filtering and physical quantity calculation, and output the satellite orbit extrapolation data at the current moment; Step S1 specifically includes: Step S11: Establish the definitions of wave function and uncertainty, the conditions that the uncertainty modification operator must satisfy, the definition of the degree of uncertainty change, and the state of the wave function after observation. Uncertainty Modification Operator The definition formulas of the wave function and the definition formulas of the observed state of the position representation are integrated into the wave function filtering module. Step S12: Based on the definition of the wave function, the corresponding calculation formula for the position distribution is obtained. The calculation formulas for the Gaussian wave of the wave function, the position distribution, the position expectation, and the momentum expectation are integrated into a physical quantity calculation module. Step S13: Establish the formula for the time evolution operator and integrate it to obtain the wave function evolution module; the formula for the time evolution operator includes the definition and decomposition formula of the time evolution operator, the operation formula for representation transformation, and the calculation formula for kinetic energy evolution and potential energy evolution.

2. The satellite orbit extrapolation method based on filtering according to claim 1, characterized in that, Step S11 specifically includes: Step S111: Define the wave function in position space and measurement uncertainty In the wave function, the satellite's position distribution is analogous to the time domain information of the wave, the velocity distribution is analogous to the frequency domain information of the wave, and the measurement uncertainty is defined as the product of the position uncertainty and the momentum uncertainty. Step S112: Establish the uncertainty modification operator The formula for the conditions that need to be met; Uncertainty Modification Operator The formula for the required conditions is: ， ， In the formula, It is the wave function in position space. Current position It is a positional operator. It is a momentum operator. It is an uncertainty modification operator. It refers to the degree of change in measurement uncertainty. The magnitude is determined by the ratio of the change in the inner product of the wavefunction before and after the observation. The + sign is the dagger symbol, which can be interpreted as transpose and conjugate from the perspective of linear algebra. Step S113: Based on the observed state of the position representation and physical state The degree of change in measurement uncertainty during the observation process is derived from the direct product. ; The degree of overall probability change during the observation process Defined as: ， in, It is a wave function. It is the spatial dimension of the domain of the wave function, and the operator ⊙ denotes the pointwise multiplication of the functional expressions corresponding to two states in space. The observed state representing the positional representation. It is a multidimensional position coordinate. It is the differential of the multidimensional position coordinates; the degree of change in the overall probability is used to determine the degree of change in measurement uncertainty; Step S114: Establish the state of the wave function after observation The calculation formula is used as the observation state operator. Definition; The state of the wave function after it is observed The calculation formula is: ， in, Represents the observation state operator; It is the wave function in position space; It is the uncertainty correction operator; the operator ⊙ represents pointwise multiplication in space; It is a normalization operator; Step S115: Establish a Gaussian wave for the wave function as a scalar wave function model applicable to conservative forces and define the wave function; Gaussian wave of wave function for: ， in, This is the current position. A is the second-order coefficient matrix of the wave function, which is a symmetric matrix. B is the first-order coefficient vector of the wave function. C is the normalization constant of the wave function. Step S116: Modify the operator based on the measurement uncertainty The required formulas and the degree of uncertainty variation during the observation process. Give the uncertainty modification operator. The definition formula is then used; subsequently, the Gaussian wave of the wave function is substituted to determine the uncertainty modification operator. Does it meet the requirements? Uncertainty Modification Operator The defining formula is: ， in, It refers to the degree of change in measurement uncertainty. To measure uncertainty, It is the uncertainty corrected by modifying the operator based on the uncertainty. It is the wave function in position space. Current position; Step S117: Provide the observed state of the location representation Definition formula; The observed state of the position representation is: ， Where Σ is the covariance matrix corresponding to a single observation, and T denotes the transpose. Current position The location center of the observed data.

3. The satellite orbit extrapolation method based on filtering according to claim 2, characterized in that, Step S13 includes: Step S131: For the wave function in the location space Adding a time dimension to obtain the wave function that evolves over time The definition of the time evolution operator is obtained by using the solution of the equation of motion; The definition of the time evolution operator is ; Step S132: Based on the split-step Fourier method, obtain the decomposition formula of the time evolution operator; Time evolution operator of wave function The decomposition formula is: ， in, , , These are the Hamiltonian operator, the kinetic energy operator, and the potential energy operator, respectively. It is a wave function that evolves over time; Step S133: Establish the operational formula for representation transformation in order to facilitate the calculation of the decomposition formula of the time evolution operator; Step S134: Establish calculation formulas for potential energy evolution and kinetic energy evolution to facilitate the calculation of the decomposition formula of the time evolution operator.

4. The satellite orbit extrapolation method based on filtering according to claim 1, characterized in that, The satellite trajectory prediction model includes a loop module, which comprises a wavefunction evolution module, a wavefunction filtering module, and a physical quantity calculation module; the wavefunction evolution module receives the wavefunction from the previous time step. The superscript S represents the Schrödinger picture and utilizes the time evolution operator. The evolution yields the evolved wave function. ; The wave function filtering module receives the observation state at the current time. And the evolved wave function, through the observed state at the current moment Weighting is achieved jointly with the evolved wave function, and the operator is modified using uncertainty after weighting. Reduce the measurement uncertainty of the wave function to output the wave function at the current moment. The physical quantity calculation module receives the wave function at the current moment. Combined with physical quantity operators, it outputs the expected physical quantity at the current moment.

5. The satellite orbit extrapolation method based on filtering according to claim 1, characterized in that, Step S1 further includes: Step S14': Establish the damped wave equation to give the calculation formulas for generalized momentum, generalized position and extended Hamiltonian; establish the operator evolution module using the calculation formulas for generalized momentum and generalized position; The generalized position X and the generalized momentum P satisfy the following transformation relationship: ， Where x is the current position, p is the current momentum, and M is the satellite mass. It is the damping coefficient, and t is the current time; The Hamiltonian H is: ， Where M is the satellite mass, Let X be the generalized momentum, X be the generalized position, and K be the second derivative of the potential energy. The damping coefficient is... For Hamiltonian; The damped wave equation is: ， in, It is a damped wave function; Step S15': Establish the formula for the damping parameter β, and use the generalized position, generalized momentum, and damping parameter to establish the model of the damping wave function. The definition of the wave function in the wave function filtering module is replaced; the formulas for position expectation and momentum expectation in the physical quantity calculation module are corrected based on the model of the damped wave function. The formula for the damping parameter β is: ， in, It was the first moment Damping parameters, It is the second moment The damping parameter, e is the natural base. Damping coefficient, M is the satellite mass; Location expectation and momentum expectation The formula is: ， ， in, It is a positional operator. It is a momentum operator. It is a damped wave function model. It refers to generalized momentum, uncertainty, and damping parameters. t is the damping coefficient, M is the satellite mass, and t is the current time.

6. The satellite orbit extrapolation method based on filtering according to claim 5, characterized in that, The satellite trajectory prediction model includes a loop module, which comprises a wavefunction evolution module, an operator evolution module, a representation transformation module, a wavefunction filtering module, and a physical quantity calculation module; the wavefunction evolution module receives the damped wavefunction in the interaction scene at the previous moment. And utilize time evolution operators The evolution yields the evolved damped wave function; the operator evolution module is configured to receive physical quantity operators from the previous time step in the interaction scene. And process to obtain the physical quantity operators in the interaction scene at the current moment. The representation conversion module is configured to receive the current observation state in the Schrödinger picture. and the physical quantity operators in the interaction scene at the current moment. The process yields the observed state in the interaction picture at the current moment. The wavefunction filtering module receives the observation state in the interaction scene at the current moment. and the evolved wave function The damped wave function in the interaction scene at the current moment is obtained through processing. The physical quantity calculation module receives the physical quantity operators in the interaction scene at the current moment. and the damped wave function in the interaction scene at the current moment. Processing yields the expected physical quantity at the current moment. ; Or, momentum expectation Approximately: ,in, It is a momentum operator. It is a damped wave function model. It refers to generalized momentum, uncertainty, and damping parameters. Here, M is the damping coefficient, M is the satellite mass, and t is the current time. The satellite trajectory prediction model includes a loop module, which comprises a wave function evolution module, an operator evolution module, a representation transformation module, a wave function filtering module, and a physical quantity calculation module. The wave function evolution module receives the damping wave function in the Schrödinger picture from the previous time step. The evolved damped wave function is obtained by using a time evolution operator; the operator evolution module is configured to receive physical quantity operators in the Schrödinger picture from the previous time step. The module processes and obtains the physical quantity operators in the interaction picture at the current moment; the representation transformation module is configured to transform the evolved damped wave function and the physical quantity operators in the interaction picture at the current moment into the Schrödinger picture respectively; the wave function filtering module is configured to filter the observed state in the Schrödinger picture at the current moment based on the evolved damped wave function and the observed state in the Schrödinger picture at the current moment. The damped wave function in the Schrödinger picture at the current moment is obtained through processing. The physical quantity calculation module is configured to calculate the damped wave function in the Schrödinger picture at the current moment. And the physical quantity operators in the Schrödinger picture at the current moment. Processing yields the expected physical quantity at the current moment. ; Alternatively, it may include step S17': replacing the formulas for generalized position X and generalized momentum P with the following formulas to correct the operator evolution module: ; , It is a generalized positional operator, representing the current position. , It is the momentum operator for generalized momentum and current momentum. Here, M is the damping coefficient, M is the satellite mass, and t is the current time. Replace the Hamiltonian with the following formula The formula: ， Where P is generalized momentum, X is generalized position, and K is the second derivative of potential energy; Furthermore, the calculation formula for the time evolution operator is replaced with the following formula to correct the calculation formula for the extended Hamiltonian: ， in, Let X be the generalized momentum and X be the generalized position. For momentum operator, For positional operators, The damping coefficient is... For Hamiltonian operators, For measurement uncertainty, t is the current time, M is the satellite mass, and K is the second derivative of the potential energy; Each loop module of the satellite trajectory prediction model includes a wavefunction evolution module, an operator evolution module, a wavefunction filtering module, and a physical quantity calculation module; the wavefunction evolution module receives the damped wavefunction in the Schrödinger picture from the previous moment. The evolved damped wave function is obtained by using a time evolution operator; the operator evolution module is configured to receive the position operator in the Schrödinger picture. The momentum operator in the interaction scene at the previous moment The position operator in the Schrödinger painting is obtained through processing. and the momentum operator in the interaction scene at the current moment. The wavefunction filtering module is configured to filter based on the evolved damped wavefunction and the current observation state in the Schrödinger picture. The damped wave function in the Schrödinger picture at the current moment is obtained through processing. The physical quantity calculation module is configured to calculate the damped wave function in the Schrödinger picture at the current moment. And the physical quantity operators in the Schrödinger picture at the current moment. , Processing yields the expected physical quantity at the current moment. , .

7. A satellite orbit extrapolation method based on filtering, characterized in that, include: Step S1: Using the satellite's motion state as a wave, establish a mathematical model of the wave function, and integrate the mathematical model of the wave function into a wave function filtering module, a physical quantity calculation module, and a wave function evolution module; Step S2: Based on the wave function filtering module, physical quantity calculation module, and wave function evolution module, construct a satellite trajectory prediction model; Step S3: Input the satellite orbit extrapolation data from the previous moment and the real-time observation data into the satellite trajectory prediction model to realize wave function evolution, wave function filtering and physical quantity calculation, and output the satellite orbit extrapolation data at the current moment; Step S1 includes: Step S11”: Establish the definition of measurement uncertainty, introduce phase to establish the spinor wave function model for each spin component, and establish the weight transfer function of the spinor space and the calculation formula of each component of the weighted spinor wave function model. Step S12”: Establish the Gaussian wave of the screw wave function model and measurement uncertainty Formulas for the intensity of each component of the wave function, and the location distribution of the spinor wave function. The formula is derived, and the motion formulas for each spin component of the spinor wave function are established. Step S13”: Establish the definition of the degree of uncertainty change and the state of the wave function after observation. Uncertainty Modification Operator Definition formula; Establishment of time evolution operator The decomposition formula; Step S14”: Based on the results of steps S11”-S13”, the weight transfer module, wave function evolution module, wave function filtering module, and physical quantity calculation module are integrated to obtain the weight transfer module, wave function evolution module, wave function filtering module, and physical quantity calculation module.

8. The satellite orbit extrapolation method based on filtering according to claim 7, characterized in that, In step S1, the weight transfer module is configured as follows: Calculate the expected position of the current spinor wavefunction as a whole. Covariance In order to Centered on The spin space component of the spinor wavefunction in a cube space of length denoted by . Uniform sampling; calculating the discrete values ​​of the weight transfer function, and calculating the expected weighted component of the spinor wavefunction in the i-th spin space. ; Calculate the expected component of the spinor wavefunction after weight transfer in the i-th spin space. Expected position Covariance ; Construct the expected component of the spinor wavefunction after weight transfer in the i-th spin space. The positional distribution; an expression for estimating the phase using the degree of phase change at the expected position; integration to obtain the wave function after weight transfer; The wavefunction evolution module is set up to: construct evolution operators respectively. and For components and Perform the operation. The potential energy required for radial maneuvering along the track, It is the potential energy required for maneuvering along the normal to the orbital plane. and Let S be the spin-space components of the orbital tangential maneuver S and the orbital normal maneuver W of the spinor wavefunction; according to the time evolution operator The decomposition formula performs batch operations on each component of the spinor wavefunction; for the components Perform the operation. Let T be the spin-space component of the spinor wavefunction along the orbital radial maneuver T. The wavefunction filtering module is set to: filter the observed state Perform batch point-to-point multiplication with each component of the spinor wave function; calculate the degree of uncertainty variation according to the definition of uncertainty variation; modify the operator according to uncertainty. The defined formula is used to batch calculate the results of adjusting the measurement inaccuracy for each component; The physical quantity calculation module is set to calculate the expected value of the position of the spinor wave function; In step S2, the satellite trajectory prediction model includes multiple loop modules, each loop module including a weight transfer module, a wavefunction evolution module, a wavefunction filtering module, and a physical quantity calculation module; the weight transfer module receives the wavefunction from the previous time step. The superscript S represents the Schrödinger picture, which is processed to obtain the wavefunction after weight transfer; the wavefunction evolution module receives the wavefunction after weight transfer and processes it to obtain the evolved wavefunction; the wavefunction filtering module receives the observation state at the current time. And the evolved wave function, outputting the wave function at the current moment. The physical quantity calculation module receives the wave function at the current moment. and physical quantity operators , Output the expected physical quantity at the current moment. , .

9. The satellite orbit extrapolation method based on filtering according to claim 8, characterized in that, In step S1, step S14 further includes: simplifying the spinor wavefunction model; the simplified spinor wavefunction model is: ， in, Let be the i-th spin-space component of the spinor wavefunction. It is the second-order coefficient matrix of the i-th spin space component. It is the first-order coefficient vector of the i-th spin space component. It is the intensity constant of the i-th spin space component; and / or Step S1 further includes: simplifying the weight transfer module by simplifying the calculation formulas for momentum, angular momentum, and energy; momentum angular momentum ,energy for: ， ， ， in, , , These are momentum, angular momentum, and energy. For momentum operator, For positional operators, Let be the potential energy function. For satellite quality.

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