Satellite group collaborative prediction method based on Lie group differential manifold and covariant optimization

Abstract

The invention discloses a satellite group collaborative prediction method based on Lie group differential manifold and covariant optimization, and mainly solves the problems of inaccurate disturbance modeling, poor physical interpretation and large prediction error in the prior art. According to the scheme, the method comprises the following steps: mapping the position and attitude of a satellite to an SE (3) manifold, and describing the movement speed of the satellite by using Lie algebra; establishing a multi-level Lie algebra disturbance decomposition mechanism including absolute perturbation, relative coupling and observation noise; constructing a Riemannian loss function by using the orbit state after Lie group manifold embedding and the orbit state deviation obtained by disturbance decomposition; defining an orbital state updating rule by using a covariant derivative of the function, and iteratively solving the orbital state updating rule on the SE (3) manifold to output an orbital state prediction result of the satellite; for an abnormal state of inter-satellite link interruption, a differential geometric connection form is defined, and virtual relative position observation of an integral generation orbit state is carried out. According to the method, the physical interpretability and long-term prediction stability of orbit prediction can be improved, prediction errors are reduced, and the method can be used for low orbit communication constellation deployment of frequency band inter-satellite links.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of spacecraft orbit dynamics and differential geometry application intersection, and specifically relates to a satellite constellation autonomous orbit prediction method, which can be used for a low-orbit communication constellation such as Hongyan and Hongyun system deploying Ka-band inter-satellite links. BACKGROUND

[0002] With the deepening of human exploration of the universe, spacecraft as the main tool for exploring the universe plays an increasingly important role in scientific research, earth observation, communication navigation and other fields. The development of spacecraft orbit prediction technology is crucial to ensure the success of space missions. In the field of space, accurately determining and predicting the orbit of a spacecraft is not only the basis for effective mission planning, but also one of the key factors to ensure the safe operation of spacecraft.

[0003] After launch, spacecraft enter the predetermined orbit and are affected by various factors in the subsequent operation process, including but not limited to the Earth's non-spherical gravitational field, atmospheric resistance, solar radiation pressure, and the gravitational perturbation of other celestial bodies. These factors will cause small but continuous changes in the spacecraft's orbit, so accurate orbit prediction methods are needed to update the spacecraft's orbit parameters in a timely manner to ensure that it can perform tasks according to the predetermined plan.

[0004] Traditional dynamic models rely on single-satellite orbit mechanics equations such as J2 perturbation models, which describe the influence of the Earth's non-spherical gravitational field on the spacecraft's orbit through the second-order term J2 of the Earth's gravitational field, but do not consider the synergistic effect of the star cluster and cannot adapt to regional perturbations such as sudden changes in atmospheric density, significantly accumulating long-term prediction errors.

[0005] The patent document with publication number CN118227973A discloses a spacecraft in-orbit operation state evaluation method based on an algorithm model, which uses a neural network model constructed by CNN and RNN to perform nonlinear dynamics prediction on the fused multi-source data, and attempts to model satellite correlation. However, due to its black box model, it lacks physical mechanism support, and training relies on a large amount of data, so there is a deficiency of high on-board deployment computing load.

[0006] The patent document with publication number CN111399023A discloses an inertial base combined navigation filtering method based on Lie group nonlinear state error, which proposes a solution defined by nonlinear state error in the strict Lie group to solve the problem of poor robustness in complex environments caused by linear state error in existing cooperative filtering methods such as distributed EKF. However, this method only handles double-satellite relative measurement, ignores multi-satellite joint dynamics constraints, and relies on empirical thresholds for anomaly handling, so the prediction result has a certain accuracy error. SUMMARY

[0007] The purpose of the present application is to overcome the shortcomings of the prior art, and provide a satellite swarm cooperative prediction method based on Lie group differential manifold and covariant optimization, so as to realize analytical modeling and high-precision autonomous prediction of the dynamics coupling of the star swarm without relying on a neural network, improve the physical interpretability, reduce the prediction error, and make the method have stronger theoretical basis and mechanism transparency in actual application scenarios.

[0008] The technical idea for achieving the above-mentioned purpose is that by establishing a satellite swarm cooperative prediction framework of'manifold embedding-covariant optimization', the satellite swarm motion is modeled on an SE(3) Lie group manifold, and by combining multi-level Lie algebra disturbance decomposition and covariant derivative optimization, high-precision inter-satellite dynamics coupling modeling and prediction are realized.

[0009] According to the above idea, the technical scheme of the satellite swarm cooperative prediction method based on Lie group differential manifold and covariant optimization of the present application comprises the following:

[0010] The position and attitude of the satellite are mapped to the SE(3) manifold, and the motion velocity of the satellite is described by using Lie algebra, so as to realize Lie group manifold embedding of the orbit state;

[0011] A multi-level Lie algebra disturbance decomposition mechanism including absolute perturbation, relative coupling and observation noise is established, so as to realize hierarchical modeling of the orbit state deviation;

[0012] The orbit state after Lie group manifold embedding and the orbit state deviation information obtained by multi-level Lie algebra disturbance decomposition are used to construct a Riemann loss function that fuses the single-satellite kinematic error and the three-element hyperedge structure in the multi-satellite geometric constraint;

[0013] The covariant gradient of the Riemann loss function is used to define the orbit state update rule, and the orbit state update rule is iteratively solved on the SE(3) manifold to output the orbit state prediction result of the satellite;

[0014] For the abnormal state of the interruption of the inter-satellite link, the connection form of differential geometry is defined, and the virtual relative position observation of the orbit state is generated by integrating along the historical path;

[0015] Further, the multi-level Lie algebra disturbance decomposition mechanism including absolute perturbation, relative coupling and observation noise is established, and the implementation thereof comprises the following:

[0016] The absolute orbit of the satellite is dynamically modeled under the consideration of the non-spherical gravity of the earth, so as to analytically solve the absolute perturbation such as the non-spherical gravity of the earth;

[0017] The relative motion coupling term between the satellite swarm is constructed based on the Hamiltonian;

[0018] The observation noise model is established by using a Gaussian process;

[0019] Further, the orbit state after embedding the Lie group manifold and the orbit state deviation information obtained by multi-level Lie algebra perturbation decomposition are used to construct a Riemann loss function that fuses single satellite kinematic error and three-element hyperedge structure in multi-satellite geometric constraint, and the implementation includes:

[0020] Map the satellite kinematic state to the SE(3) Lie group manifold to obtain the corresponding Lie group element T i (t), and use Lie algebra to describe the measured satellite speed as Obtain the orbit state after embedding the Lie group manifold;

[0021] Multi-level Lie algebra perturbation decomposition is performed on the above orbit state to obtain orbit state deviation information, including absolute perturbation of earth's non-spherical gravity Relative motion state change rate And observation noise model n ij ;

[0022] Combine absolute perturbation And relative motion state change rate And add observation noise n ij , to obtain the predicted value of satellite speed

[0023] Combine satellite speed prediction value And measured satellite speed Calculate the single satellite kinematic error in the satellite group under the metric tensor G;

[0024] Select any three satellites from the satellite group to form a three-element group (i, j, k), obtain the kinematic state of each satellite at the current time, map these states to the SE(3) Lie group manifold to obtain the corresponding Lie group elements T i (t), T j (t), T k (t), and define a three-element hyperedge constraint term based on this

[0025] Sum all single satellite kinematic errors and three-element hyperedge constraint terms, and multiply the latter by a weight coefficient λ to construct a comprehensive Riemann loss function

[0026] Further, the orbit state update rule is defined using the covariant gradient of the Riemann loss function, and the implementation includes:

[0027] Calculate the covariant gradient of the Riemann loss function

[0028] Starting from the current orbit state ξ i k , define the Lie algebra update amount using the covariant gradient of the Riemann loss function

[0029] The Lie algebra update is converted into the corresponding Lie group transformation increment δT by exponential mapping i k ;

[0030] The Lie algebra update is converted into the corresponding Lie group transformation increment δT by exponential mapping The Lie algebra update is converted into the corresponding Lie group transformation increment δT by exponential mapping i k , the satellite speed prediction value ξ on the Lie algebra is obtained i k+1 , and the satellite attitude prediction value T on the Lie group is obtained i k+1 ;

[0031] Further, the virtual relative position observation of the orbit state generated by the integral of the defined connection form along the historical path is the defined inter-satellite connection form ω ij First, the integral form virtual orbit relative position is generated by integrating along the historical path Then, the discrete form virtual orbit relative position is obtained by discretization

[0032] Compared with the prior art, the present application has the following advantages:

[0033] Firstly, in the present application, the Lie group differential geometry SE(3) manifold is introduced into satellite orbit prediction, and the geometric dynamics equation of group motion is established, which breaks through the limitation that the traditional state space modeling is difficult to effectively describe the complex relative motion relationship between satellites, and at the same time, the invariance structure of the SE(3) manifold is used to unify the geometric space, avoiding the error accumulation problem caused by coordinate transformation in the traditional method, and significantly improving the long-term prediction stability;

[0034] Secondly, in the present application, the covariant derivative optimization process is adopted to adapt to the multi-layer disturbance changes in interstellar space, so as to ensure the stable update of the orbit state, and the virtual observation prolongs the tolerance time of single-satellite link interruption, realizes the breakthrough of anti-disturbance ability, and reduces the prediction error;

[0035] Thirdly, in the present application, all modules of the manifold optimization process are based on physical equation derivation, so that the analytical modeling and high-precision autonomous prediction of star group dynamics coupling can be realized without relying on neural network, the calculation amount is much lower than that of CNN and RNN neural network, the black box operation is avoided, the physical interpretability is improved, the stronger theoretical basis and mechanism transparency are provided in the actual application scene, the high reliability requirement of aerospace is met, and good engineering value is obtained. BRIEF DESCRIPTION OF DRAWINGS

[0036] Figure 1 is the implementation flowchart of the present application;

[0037] Figure 2 The Riemann loss function of the three-element hyper-edge structure in the fusion of single-star kinematic error and multi-star geometric constraint is constructed in the present application to realize a sub-flowchart;

[0038] Figure 3 The sub-flowchart for predicting the orbit state by using the Riemann loss function is realized in the present application;

[0039] Figure 4 The sub-flowchart for constructing the virtual relative position observation is realized in the present application. DETAILED DESCRIPTION

[0040] The embodiments of the present application are further described in detail below with reference to the accompanying drawings.

[0041] The satellite group motion environment in the low-orbit space is selected as the embodiment background in the present application, wherein the low-orbit satellite attitude is approximately kept unchanged, and the angular velocity vector is approximately zero.

[0042] Referring to Figure 1 The implementation steps of the present application include the following:

[0043] Step 1, Lie group manifold embedding of the orbit state.

[0044] The Lie group manifold refers to a special Euclidean group SE(3), which is a mathematical structure of rigid body motion in a three-dimensional space, including rotation and movement, and the tangent space thereof near the unit element can be expressed as a Lie algebra The latter is often used to describe the instantaneous motion of a rigid body.

[0045] The orbit state refers to the position, attitude and velocity information of the satellite on the orbit.

[0046] This step is realized as follows:

[0047] 1.1) Map the position and attitude of the satellite to the satellite kinematic state group of the SE(3) manifold to obtain the SE(3) group element of the i-th satellite at time t:

[0048]

[0049] Wherein, T i (t)∈SE(3) is a 4×4 matrix representing the SE(3) group element of the i-th satellite at time t,

[0050] I 3×3 is a unit matrix, which remains constant in a short time according to the property of the low-orbit satellite attitude being approximately unchanged,

[0051] is the position vector in the ECI coordinate system, with a unit of meters,

[0052] is the angular velocity vector, which is approximated to zero in low earth orbit satellite dynamics modeling;

[0053] 1.2) Map the satellite's motion velocity to Lie algebra:

[0054]

[0055] where, is the 6x1 Lie algebra vector,

[0056] v i (t) = [v x , v y , v z ] - is the velocity vector in ECI frame, in m / s.

[0057] Step 2, build a multi-layer perturbation decomposition model for the orbit state in Lie algebra space.

[0058] This example assumes, but is not limited to, a three-layer perturbation decomposition, in which:

[0059] 2.1) Analyze the Earth's non-spherical gravity and other absolute perturbations at the first level, i.e., consider the Earth's non-spherical gravity to model the satellite's absolute orbit dynamics:

[0060]

[0061] where, is the 6x1 vector, representing the state rate of change caused by absolute perturbations, in m / s 2 ,

[0062] J2 = 1.0826 x 10 -3 is the Earth's flattening coefficient, dimensionless,

[0063] μ = 3.986 x 10 6 is the Earth's gravitational constant, in m 3 / s 2 ,

[0064] R e = 6.378 x 10 6 is the Earth's radius, in meters,

[0065] represents the distance from the satellite to the Earth's center, in meters,

[0066] is the position component in the ECI coordinate system, in meters,

[0067] represents the control input, the first three items are thrust acceleration, in m / s2 ; the last three correspond to the angular acceleration terms, which are approximately zero in low earth orbit satellite dynamics modeling;

[0068] 2.2) At the second level, the inter-satellite relative motion coupling term is constructed based on the Hamiltonian:

[0069] The coupling term refers to a mathematical expression used to describe the interaction between different parts in a multi-body system.

[0070] The orbit perturbation term refers to various factors that affect the actual orbit of a satellite or celestial body, causing it to deviate from the ideal orbit, such as the non-spherical gravity of the Earth, atmospheric drag, and solar radiation pressure.

[0071] Considering the coupling effect of the relative motion relationship between multiple satellites and the orbit perturbation term, the Hamiltonian is introduced to construct the inter-satellite relative motion coupling term:

[0072]

[0073] where, is a 6 × 1 vector representing the rate of change of the relative motion state of satellite i, j, with units of m / s 2 ,

[0074] is a 6 × 6 symplectic matrix,

[0075] is the relative motion Hamiltonian,

[0076] δr ij = r i - r j represents the relative position vector with units of meters,

[0077] δp ij = m i (v i - v j ) represents the relative momentum vector with units of kg·m / s, and M is the mass matrix.

[0078] 2.3) At the third level, the observation noise model is established using a Gaussian process:

[0079]

[0080] where, ij is a 3 × 1 vector representing the inter-satellite link measurement noise with units of meters,

[0081] represents the Gaussian process,

[0082] σ = 0.05 is the noise intensity with units of meters,

[0083] for spatial correlation scale, unit: meter,

[0084] δr ij denotes the relative position vector.

[0085] Step 3, construct the Riemannian loss function by using the model of multi-layer perturbation decomposition.

[0086] Referring to Figure 2 , the implementation of this step includes the following:

[0087] 3.1) Using the above multi-layer perturbation decomposition model, the SE(3) group element T i (t) obtained in step 1 and the satellite orbit velocity in the Lie algebra space Solve the orbit state deviation information, including the absolute perturbation of the earth's non-spherical gravity Relative motion state rate And the observation noise model n ij ;

[0088] 3.2) Combine the absolute perturbation And the relative motion state rate And add the observation noise n ij , get the predicted value of the satellite velocity

[0089]

[0090] Where a ij is the weight coefficient, which is adjusted according to the specific application scenario;

[0091] 3.3) Combine the satellite velocity prediction value And the measured satellite velocity Calculate the square norm ‖ Under the metric tensor G, that is, the single-satellite kinematic error in the satellite constellation;

[0092] 3.4) Select any three satellites from the satellite constellation to form a triplet (i, j, k), obtain the kinematic state of each satellite at the current time, map these states to the SE(3) Lie group manifold to obtain the corresponding Lie group elements T i (t), T j (t), T k (t), based on which define a three-element hyper-edge constraint term:

[0093]

[0094] Where i, j, k ∈ {1, 2,..., N} and are different, representing the i, j, k th satellite.

[0095] 3.5) Summing up all the single-star kinematic errors and the three-element hyper-edge constraint term respectively, and multiplying the latter by the weight coefficient λ to obtain the comprehensive Riemann loss function

[0096]

[0097] where G is a 6x6 positive definite matrix representing the manifold metric tensor,

[0098] λ = 0.1 is the hyper-edge constraint weight coefficient,

[0099] log(·) represents the logarithmic mapping of the SE(3) group.

[0100] Step 4, detecting the inter-satellite link of the satellite.

[0101] Judging that the satellite is in an abnormal state of inter-satellite link;

[0102] If the ground station detects that the satellite signal meets the working parameters and technical standards, it is judged that the link is normal, and step 5 is executed;

[0103] If the ground station detects that the satellite signal is lost, the bit error rate is too high, the connection is interrupted, etc., it is judged that the link is abnormal, and step 7 is executed again.

[0104] Step 5, predicting the orbit state using the Riemann loss function.

[0105] Referring to Figure 3 The implementation steps of this step include the following:

[0106] 5.1) Calculate the covariant derivative of the Riemann loss function:

[0107]

[0108] where, is the covariant derivative operator, is the Riemann loss function,

[0109] G is a 6x6 positive definite matrix representing the manifold metric tensor,

[0110] is a set of generalized coordinates representing the current orbit state of satellite i at the kth iteration,

[0111] ξ j and ξ h are the jth and hth components of the orbit state ξ i k

[0112] is the Christoffel symbol, which is derived from the Earth's gravitational field metric g hl ​Export, g hl =diag(1,1,1,-Φ(r)) represents the gravitational field metric tensor, Φ(r) is the earth's gravitational potential function, g mj ,g mh ,g jh is the metric tensor g hl The mth, hth, and jth components of x h ,x j ,x m Represents different components of space-time coordinates;

[0113] 5.2) From the current orbital state ξ i k Starting from, we use the covariant gradient of the Riemann loss function to define the Lie algebra update:

[0114]

[0115] Where η = 0.01 is the adaptively adjustable learning rate, and k is the number of iterations;

[0116] 5.3) Through exponential mapping, the Lie algebra update is converted into the corresponding Lie group transformation increment:

[0117] δT i k =exp(δξ i k ),

[0118] Here, exp(·) represents the exponential mapping from Lie algebra to Lie group;

[0119] 5.4) Using Lie algebra to update the quantity and Lie group transformation increment δT i k , and obtain the predicted value of satellite velocity ξ on Lie algebra i k+1 And the satellite attitude prediction value T on the Lie group i k+1 :

[0120]

[0121] Among them, v i k+1 is the predicted value of satellite velocity, r i k+1 is the predicted value of satellite position.

[0122] Step 6: Iterative optimization of orbital state.

[0123] 6.1) Set the initial state of the satellite ξ i k And the control parameters include learning rate η, number of iterations k;

[0124] 6.2) Define the loss function according to the difference between the orbit state and the reference orbit

[0125]

[0126] where, is the predicted position, r ref is the position on the precise ephemeris provided by the ground station;

[0127] 6.3) Calculate the gradient of the loss function with respect to the correlation coefficient hyperedge constraint weight λ and the metric tensor G respectively according to the definition of the Riemann loss function

[0128]

[0129] 6.4) Update the results of 6.3) using the gradient descent method to obtain the updated hyperedge constraint weight λ k+1 and the metric tensor G k+1 :

[0130]

[0131] where, λ k represents the hyperedge constraint weight coefficient after the kth iteration, G k is the metric tensor after the kth iteration;

[0132] 6.5) Repeat steps 6.2)-6.4) until the loss function converges, and output the optimized satellite orbit state velocity and position prediction value.

[0133] Step 7, fiber bundle reconstruction of virtual orbit relative position observations.

[0134] Referring Figure 4 , the implementation steps of this step include the following:

[0135] 7.1) When it is detected that the satellite is in an abnormal state of inter-satellite link interruption, define the connection form of differential geometry, whose formula is as follows:

[0136]

[0137] where, ω ij is the connection form representing the satellite i, j, and represents the differential geometric object,

[0138] represents the pth component of the relative position vector,

[0139] ​Christoffel symbols, which are derived from the Earth gravitational field metric g gl g gl = diag(1,1,1,-Φ(r)) represents the gravitational field metric tensor, Φ(r) is the Earth gravitational potential function, nh g nl g hl are the nth, the lth, the hth component of the metric tensor g gl x l x h x n represent different components of the spacetime coordinates;

[0140] 7.2) The integral of the connection form generates the integral form of the virtual relative position of the orbits

[0141]

[0142] where γ is the historical path, and T is the historical time;

[0143] 7.3) Discretize the above integral to obtain the virtual relative position of the orbit state

[0144]

[0145] where Δt = 0.1 is the integral step, the unit is second,

[0146] N = T / Δt = 10 is the integral step number.

[0147] The above only describes the preferred embodiments of the present application and is not used to limit the present application. Obviously, for those skilled in the art, after understanding the content and principles of the present application, various modifications and changes in form and details can be made without departing from the principles and structures of the present application, but these modifications and changes based on the idea of the present application are still within the protection scope of the claims of the present application.

[0148] It should be noted that the step numbers in the specification and claims of the present application are only for clearly and completely describing the embodiments of the present application, facilitating understanding, and the sequence of the numbers is not limited. The part not described in detail in the present application belongs to the common knowledge of those skilled in the art.

Claims

1. A satellite swarm collaborative prediction method based on Lie group differential manifolds and covariant optimization, characterized by: include: The satellite's position and attitude are mapped to the SE(3) manifold, and the satellite's motion velocity is described by Lie algebra to achieve the Lie group manifold embedding of the orbital state. A multi-level Lie algebraic perturbation decomposition mechanism including absolute perturbation, relative coupling, and observation noise is established to achieve hierarchical modeling of orbital state deviations. By using the orbital state after embedding into the Lie group manifold and the orbital state deviation information obtained by multi-level Lie algebraic perturbation decomposition, a Riemann loss function is constructed that integrates the single-star kinematic error and the ternary hyperedge structure in the multi-star geometric constraint. The orbit state update rule is defined using the covariant derivative of the Riemann loss function, and it is iteratively solved on the SE(3) manifold to output the satellite's orbit state prediction result. For the abnormal state of intersatellite link interruption, a differential geometry contact form is defined, and the virtual relative position observation of the orbital state is generated by integrating it along the historical path.

2. The method according to claim 1, characterized in that The satellite position and attitude are mapped to the SE(3) manifold, and the formula is as follows: Among them, T i (t)∈SE(3) is a 4×4 matrix, representing the SE(3) group element of the i-th satellite at time t, I 3×3 is the unit matrix, which remains constant in a short time according to the property that the attitude of low-orbit satellites is approximately constant. is the position vector in the ECI coordinate system, in meters, is the angular velocity vector, which is approximately zero in the dynamic modeling of low-orbit satellites.

3. The method according to claim 1, characterized in that The Lie algebra is used to describe the satellite's motion speed, and its formula is as follows: in, represents a 6×1 Lie algebra vector, is the velocity vector in the ECI coordinate system, in m / s, is the angular velocity vector, which is approximately zero in the dynamic modeling of low-orbit satellites.

4. The method according to claim 1, wherein A multi-level Lie algebraic perturbation decomposition mechanism including absolute perturbation, relative coupling and observation noise is established. Its implementation includes the following: The absolute perturbation is to perform dynamic modeling of the satellite absolute orbit under the consideration of the non-spherical gravity of the Earth: in, Vector, representing the rate of change of state caused by absolute perturbation, in m / s 2 , J2=1.0826×10 -3 is the Earth's oblateness coefficient, dimensionless, μ=3.986×10 6 is the Earth's gravitational constant, in m 3 / s 2 , R e =6.378×10 6 is the radius of the Earth in meters, Indicates the distance from the satellite to the center of the earth in meters. is the position component in the ECI coordinate system, in meters, Represents control input, the first three items are thrust acceleration, unit is m / s 2 The last three terms correspond to angular acceleration terms, which are approximately zero in the dynamic modeling of low-orbit satellites. The relative coupling is based on the Hamiltonian to construct the relative motion coupling term between satellite groups: in, is a 6×1 vector, representing the rate of change of the relative motion state of satellites i and j, in m / s 2 , is a 6×6 symplectic matrix, is the relative motion Hamiltonian, δr ij =r i -r j Represents the relative position vector in meters. δp ij =m i (v i -v j ) represents the relative momentum vector in kg·m / s, and M is the mass matrix; The observation noise is modeled using a Gaussian process: Among them, n ij is a 3×1 vector representing the intersatellite link measurement noise in meters. represents a Gaussian process, σ=0.05 is the noise intensity, in meters. =100×10 3 is the spatial correlation scale in meters, δr ij Represents the relative position vector.

5. The method according to claim 1, wherein The orbital state after embedding the Lie group manifold and the orbital state deviation information obtained by multi-level Lie algebraic perturbation decomposition are used to construct a Riemann loss function that integrates the single-star kinematic error and the ternary hyperedge structure in the multi-star geometric constraint. The implementation includes: 5a) Map the satellite kinematic state onto the SE(3) Lie group manifold to obtain the corresponding Lie group element T i (t), and the measured satellite velocity is described by Lie algebra as Get the orbital state after embedding the Lie group manifold; 5b) Perform multi-level Lie algebraic perturbation decomposition on the above orbital states to obtain orbital state deviation information, including the absolute perturbation of the Earth's non-spherical gravity Relative motion state change rate and observation noise model n ij ; 5c) Combined with absolute perturbation and the rate of change of relative motion state And add observation noise n ij , get the predicted value of satellite speed Among them, a ij is the weight coefficient, which is adjusted according to the specific application scenario; 5d) Combined with satellite velocity prediction and the measured satellite speed Compute the squared norm under the metric tensor G represents the kinematic error of a single satellite in the satellite constellation; 5e) Randomly select three satellites from the satellite constellation to form a triplet (i, j, k), obtain the kinematic state of each satellite at the current moment, and map these states onto the SE(3) Lie group manifold to obtain the corresponding Lie group element T i (t), T j (t), T k (t), based on which the ternary hyperedge constraint term is defined: Among them, i,j,k∈{1,2,...,N} and are different from each other, representing the i,j,kth satellite; 5f) Sum all single star kinematic errors and the three-dimensional hyperedge constraint terms respectively, and multiply the latter by the weight coefficient λ to construct a comprehensive Riemann loss function Where G is a 6×6 positive definite matrix, representing the manifold metric tensor, λ=0.1 is the hyperedge constraint weight coefficient, log(·) represents the logarithmic mapping of the SE(3) group.

6. The method according to claim 1, characterized in that The covariant gradient of the Riemann loss function is used to define the track state update rule, and its implementation includes: 6a) Compute the covariant derivative of the Riemann loss function: in, is the covariant derivative operator, is the Riemann loss function, G is a 6×6 positive definite matrix, representing the manifold metric tensor, ξ i k is a set of generalized coordinates representing the current orbital state of satellite i at the kth iteration, ξ j and ξ h They are orbital states ξ i k The jth and hth components of is the Christoffel symbol, which is derived from the Earth's gravitational field metric g hl Export, g hl =diag(1,1,1,-Φ(r)) represents the gravitational field metric tensor, Φ(r) is the earth's gravitational potential function, g mj ,g mh ,g jh is the metric tensor g hl The weight, x h ,x j ,x m Represents different components of space-time coordinates; 6b) From the current orbital state ξ i k Starting from, we use the covariant derivative of the Riemann loss function to define the Lie algebra update: Where η = 0.01 is the adaptively adjustable learning rate, and k is the number of iterations; 6c) Through exponential mapping, the Lie algebra update is converted into the corresponding Lie group transformation increment: δT i k =exp(δξ i k ), Here, exp(·) represents the exponential mapping from Lie algebra to Lie group; 6d) Update the quantity δξ using Lie algebra i k and Lie group transformation increment δT i k , and obtain the predicted value of satellite velocity ξ on Lie algebra i k+1 And the satellite attitude prediction value T on the Lie group i k+1 : Among them, v i k+1 is the predicted value of satellite velocity, r i k+1 is the predicted value of satellite position.

7. The method according to claim 1, characterized in that The orbit state update rule is iteratively solved on the SE (3) manifold to output the satellite's orbit state prediction result, which is implemented by: 7a) Set the initial state of the satellite ξ i k And the control parameters include learning rate η, number of iterations k; 7b) Define the loss function based on the difference between the orbital state and the reference orbit in, is the predicted position, r ref Position on the precise ephemeris provided to the ground station; 7c) According to the definition of the Riemann loss function, calculate the gradient of the correlation coefficient hyperedge constraint weight λ in the loss function respectively and the gradient of the metric tensor G 7d) Use the gradient descent method to update the result of 7c) to obtain the updated hyperedge constraint weight λ k+1 and the metric tensor G k+1 : Among them, λ k represents the hyperedge constraint weight coefficient after the kth iteration, G k is the metric tensor after the kth iteration; 7e) Repeat steps 7b)-7d) until the iterations converge, and output the final velocity and position prediction values ​​of the satellite orbit state.

8. The method according to claim 1, characterized in that For the abnormal state of intersatellite link interruption, the differential geometry connection form is defined, and its formula is as follows: Among them, ω ij To express the contact form between satellites i and j, to express differential geometry objects, represents the pth component of the relative position vector, is the Christoffel symbol, which is derived from the Earth's gravitational field metric g gl Export, g gl =diag(1,1,1,-Φ(r)) represents the gravitational field metric tensor, Φ(r) is the earth's gravitational potential function, g nh , g nl , g hl is the metric tensor g gl The nth, lth, and hth components of x l , x h , x n Represents the different components of space-time coordinates.

9. The method according to claim 1, characterized in that The integration along the historical path of the connection form defining the differential geometry generates virtual relative position observations of the orbital state, which is realized by: 9a) Integrating the connection form to generate the relative positions of the virtual orbitals in the integral form Among them, γ is the historical path, T is the historical time; 9b) Discretize the above integral to obtain the relative position of the virtual orbit in discrete form Among them, Δt=0.1 is the integration step, the unit is second, N=T / Δt=10 is the number of integration steps.

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