Lunar satellite formation relative navigation method based on factor graph optimization

Abstract

The invention relates to a lunar satellite formation relative navigation method based on factor graph optimization, which comprises the following steps: in a sliding window with a set length, constructing a factor graph by using relative orbit states of a series of satellite formations as state nodes and constraint factors connected with the state nodes; and optimizing the factor graph to obtain the optimal estimation of the relative orbit state of the satellite formation at the current moment. According to the relative navigation method, historical data are utilized, the historical orbit state is optimized and corrected again in the process of optimizing the factor graph, error accumulation is effectively restrained, and navigation precision and robustness are remarkably improved.

Description

Technical Field

[0001] This invention relates to the field of satellite formation technology, and in particular to a relative navigation method for lunar satellite formations based on factor graph optimization. Background Technology

[0002] Very long-wave radio astronomy observations hold unique scientific value in astronomical research, with applications encompassing the revelation of early universe evolution, the study of extreme celestial phenomena, and the detection of large-scale magnetic field structures in the universe. Due to interference from the Earth's ionosphere and the limitations of strong electromagnetic noise, ground-based platforms struggle to conduct such precise observations. Utilizing the Moon as a natural electromagnetic shield, deploying satellite formations in lunar orbit allows for high-resolution observations and precise measurements in a quiet electromagnetic environment. The inter-satellite distances within the formation are continuously compressed and expanded according to mission requirements, exhibiting a "breathing" characteristic to create spatial interferometric apertures ranging from tens to hundreds of kilometers.

[0003] Unlike traditional lunar satellite formations, the "breathing formation" possesses dynamic characteristics. This flexible formation allows satellites to dynamically adjust their relative positions according to mission requirements, thereby optimizing the efficiency of collaborative work between satellites at different stages. This dynamic formation configuration places higher demands on the relative navigation system, requiring not only real-time acquisition of relative position and velocity information between the satellites in the formation, but also maintaining the continuity and stability of navigation calculations in the event of rapid changes in formation configuration, anomalies in measurement data, or interruptions. Therefore, accurate and efficient autonomous navigation technology is crucial for the normal operation of the lunar "breathing formation," further becoming a key factor in ensuring the success of ultra-long-wave exploration missions.

[0004] Currently, satellite formation relative navigation primarily employs real-time estimation schemes based on filtering algorithms such as Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF). These filtering methods process the current observation data recursively, offering advantages such as high computational efficiency and ease of real-time implementation. However, filtering algorithms are essentially forward-recursive real-time processing methods, updating only using the current observation values. Once the current observation data is processed, historical observation information is discarded, making it impossible to re-optimize and correct the processed historical data. This real-time filtering scheme has significant limitations when facing the complex dynamic scenario of lunar "breathing formations": when the formation configuration changes rapidly, the filtering algorithm is prone to accumulating state estimation biases; when measurement data is abnormal or sensor malfunctions occur, the filtering algorithm lacks the ability to backtrack and optimize historical data, leading to erroneous observations having a continuous impact on subsequent state estimations; furthermore, the complex perturbations in the lunar orbital environment, communication delays, and the intermittent nature of inter-satellite measurements all pose challenges to the robustness and accuracy of real-time filtering. Therefore, traditional filtering schemes are insufficient to meet the high-precision and highly robust relative navigation requirements of lunar "breathing formations." Summary of the Invention

[0005] To address at least some of the problems mentioned above in the prior art, this invention provides a relative navigation method for lunar satellite formations based on factor graph optimization, comprising the following steps: Within a sliding window of a set length, a factor graph is constructed using the relative orbital states of a series of satellite formations as state nodes and constraint factors connecting the state nodes. The factor graph is optimized to obtain the optimal estimate of the relative orbital state of the satellite formation at the current moment.

[0006] Furthermore, the relative orbital state of each epoch within the sliding window serves as the state node for constructing the factor graph, where the state sequence formed by consecutive state nodes within the sliding window is as follows: ; Indicates the first k The relative orbital states of each epoch, among which Indicates the first k The relative position vector of each epochal minor star relative to the primary star This represents the relative velocity vector of the k-th epoch child star relative to the primary star. express The transpose of , express The transpose of .

[0007] Furthermore, the constraint factors include prior factors, dynamic factors, and joint observation factors; The prior factor is subject to Gaussian constraints and is expressed as follows: , Where X0 is the initial orbital state estimate or the first orbital state after marginalization in the previous window. It is the a priori covariance; Based on the third-order relative motion model, the orbital state at time t is constructed. X i t orbital state at time t+1 X i t+1 kinetic factors: , in, This is the discretized propagation function for the third-order relative motion model; According to the inter-satellite distance observation model in the collaborative measurement model, the inter-satellite distance observation factor is expressed as: , in, d i This represents the distance between the i-th child star and the primary star. The modulus represents the relative orbital position between the i-th child star and the primary star; According to the inter-satellite angle observation model in the collaborative measurement model, the inter-satellite angle observation factor is expressed as: , in, These represent the inter-satellite azimuth and elevation angles estimated from the relative position vectors, respectively. Representing the measurement of the first i The azimuth and elevation angles between the minor star and the primary star; After merging the inter-satellite distance observation factor and the inter-satellite angle measurement observation factor, since the distance measurement noise and the angle measurement noise are independent and follow a zero-mean Gaussian distribution, they are cascaded to construct a joint observation factor. : ; Based on the aforementioned constraint factors, the objective function for factor graph optimization within the sliding window is: , in, For process noise covariance, R i This represents the measurement noise covariance.

[0008] Furthermore, the DogLeg trust-region algorithm is used to iteratively optimize the objective function of the factor graph to optimize the factor graph, including the following steps: At the current estimation point At this point, perform a first-order Taylor expansion on all factor residuals to construct the normal equation: , in, J The joint Jacobian matrix of the residuals of all factors. For joint noise covariance, For state increments; In the trust region radius Inside, the fastest descent direction is: , in, g For gradient, ; The Gauss-Newton direction is: ; Use the following directional search strategy to determine the search direction: , in, Determined by the trust region boundary conditions; After determining the search direction, iteratively optimize according to the following rules to solve the objective function of factor graph optimization: , in This represents the relative orbital state at iteration t. This represents the updated state at iteration t+1.

[0009] Furthermore, when constructing the factor graph, it is first determined whether the length of the sliding window is greater than the set length W. If so, the earliest state node is moved out of the sliding window and the prior factors are updated. The factor graph is constructed with W state nodes, the updated prior factors, the dynamic factors, and the joint observation factors.

[0010] Furthermore, updating the prior factors using the marginalization method includes the following steps: Suppose that the sliding window at the current time contains two types of state vectors: states to be marginalized. and reserved status ; Based on the two types of state vectors, linearizing all factors yields the standard quadratic form of the joint optimization problem: , in, It is in a state of being marginalized. The relevant Hessian submatrix, Is it in the state of preservation? The relevant Hessian submatrix, It is a cross term that describes the coupling relationship between the state to be marginalized and the state to be retained; These are the corresponding linear terms, calculated from the residuals and the Jacobian matrix; According to least squares theory, by treating marginalized states Solve for the minimization condition and substitute back into the standard quadratic form to fold the influence of the state to be marginalized into the influence on the state to be retained. Constraints; The standard quadratic form regarding Taking the partial derivative and setting it to zero, we get: ; The formula is further transformed into ; Substituting the transformed formula into the standard quadratic form, we obtain the result regarding the preserved state. The new quadratic form: , remember , The residual form of the prior factors obtained after marginalization is: , in, A From Factors obtained through matrix factorization b Source ; Finally, the covariance matrix of the prior factors is .

[0011] Furthermore, it also includes: Establish a relative motion model of the lunar satellite formation in the coordinate system of the main star's orbit; and Establish a collaborative measurement model for lunar satellite formations.

[0012] Furthermore, establishing a relative motion model for the lunar satellite formation in the main star's orbital coordinate system includes: Assuming that in the primary star's orbital coordinate system, the relative position vector of the secondary star relative to the primary star is: Combining the gravitational model of the two-body problem, the nonlinear relative motion equations of the sub-star are: , in, This is the distance from the satellite to the center of the moon. For the gravitational term Perform a first-order Taylor expansion: ; Substituting the first-order Taylor expansion into the nonlinear relative motion equations of the sub-star, and neglecting second-order and higher minima, the fundamental relative motion equations are derived: , Gravity term Performing a third-order Taylor expansion down to the third-order minimum yields the third-order nonlinear correction term: , ; By combining the basic equations of relative motion with third-order nonlinear correction terms and rearranging, we obtain the third-order relative motion model: .

[0013] Furthermore, establishing a collaborative measurement model for lunar satellite formations includes: The inter-satellite angle observation model is defined as follows: , in, Representing the first i Information on the azimuth and elevation angles between the minor star and the primary star. ; Indicates the first i The relative position vector of a minor star in the coordinate system of the primary star's orbit. This represents noise in inter-satellite angle measurements. The inter-satellite distance observation model is defined as follows: , in, d i Representing the i The distance between the minor star and the primary star. This indicates noise in inter-satellite distance measurements; Based on inter-satellite angle and inter-satellite distance observation information, a collaborative measurement model for satellite formations M Established as: , in, Representing the i Information on the interstellar distances and angles between the minor stars and the primary star. .

[0014] The present invention has at least the following beneficial effects: The relative navigation method of the present invention first constructs a factor graph within a sliding window of a set length by using the relative orbital states of a series of satellite formations as state nodes and constraint factors connecting the state nodes. Then, the factor graph is optimized to transform the relative navigation problem into a nonlinear least squares optimization problem for solution. By utilizing historical data, the historical orbital states are re-optimized and corrected during the optimization of the factor graph, which effectively suppresses error accumulation and significantly improves navigation accuracy and robustness. This invention employs a sliding window strategy, which jointly optimizes historical orbit states and observation information within a fixed-length sliding window. By using an edge-mapping mechanism, the historical orbit state information outside the sliding window is compressed into prior factors. The historical information contained in the historical orbit states outside the sliding window is retained through prior factors and passed on to subsequent optimizations, thus avoiding the infinite growth of computational load and improving navigation accuracy while reducing computational complexity.

[0015] This invention constructs a third-order relative motion model, which more accurately describes the complex dynamic characteristics of the "breathing formation" configuration change process and provides precise dynamic constraints for factor graph optimization. Attached Figure Description

[0016] To further illustrate the above and other advantages and features of the various embodiments of the present invention, a more specific description of the embodiments of the invention will be presented with reference to the accompanying drawings. It is to be understood that these drawings depict only typical embodiments of the invention and are therefore not intended to limit its scope. In the drawings, identical or corresponding parts will be indicated by identical or similar reference numerals for clarity.

[0017] Figure 1 The flowchart of a relative navigation method for lunar satellite formation based on factor graph optimization according to an embodiment of the present invention is shown.

[0018] Figure 2 The simulation verification results according to an embodiment of the present invention are shown. Detailed Implementation

[0019] It should be noted that the components in the accompanying drawings may be shown exaggerated for illustrative purposes and may not be to scale.

[0020] In this invention, the various embodiments are merely intended to illustrate the solutions of the invention and should not be construed as limiting.

[0021] In this invention, unless otherwise specified, the quantifiers “a” and “one” do not exclude scenarios involving multiple elements.

[0022] It should also be noted that, in the embodiments of the present invention, only a portion of the parts or components may be shown for clarity and simplicity. However, those skilled in the art will understand that, under the teachings of the present invention, the required parts or components can be added as needed for specific scenarios.

[0023] It should also be noted that within the scope of this invention, the terms "same", "equal", and "equal to" do not mean that the two values ​​are absolutely equal, but allow for a certain reasonable error. In other words, the terms also cover "substantially the same", "substantially equal", and "substantially equal to".

[0024] It should also be noted that in the description of this invention, the terms "center," "longitudinal," "lateral," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," and "outer," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are used only for the convenience of describing the invention and for simplifying the description, and do not explicitly or implicitly suggest that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on the invention. Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance.

[0025] Furthermore, the embodiments of the present invention describe the process steps in a specific order. However, this is only for the convenience of distinguishing each step, and is not a limitation on the order of each step. In different embodiments of the present invention, the order of each step can be adjusted according to the process.

[0026] To address the high-precision relative navigation requirements of lunar orbital "breathing formations," this invention proposes a lunar satellite formation relative navigation method based on factor graph optimization. This method, grounded in the sliding window principle, utilizes observational data over a period of time to jointly optimize the orbital state and obtain the optimal estimate, effectively suppressing error accumulation and significantly improving navigation accuracy and robustness. Simultaneously, by constructing a third-order relative motion model, it accurately describes the dynamic changes in the formation configuration, providing precise dynamic constraints for the navigation algorithm and meeting the stringent requirements of high-precision autonomous navigation for ultra-long-wave radio astronomy observation missions.

[0027] Figure 1 The flowchart of a relative navigation method for lunar satellite formation based on factor graph optimization according to an embodiment of the present invention is shown.

[0028] like Figure 1 As shown, a relative navigation method for lunar satellite formations based on factor graph optimization includes the following steps: Step 1: Establish a relative motion model of the lunar satellite formation in the coordinate system of the main star's orbit.

[0029] To accurately describe the relative motion relationships between satellites in the lunar "breathing formation," a relative motion model is established in the primary star orbital coordinate system (LocalVertical Local Horizontal, LVLH). The primary star orbital coordinate system is defined as follows: the origin is located at the primary star's center of mass, the x-axis points away from the lunar center along the primary star's radial direction, the y-axis is along the primary star's orbital velocity direction, and the z-axis is along the primary star's orbital angular momentum direction. These three axes form a right-handed coordinate system.

[0030] Considering that the primary star is orbiting in a near-circular orbit, its orbital angular velocity can be approximated as: Where μ is the lunar gravitational constant, rc Let be the orbital radius. Assume that, in the primary star's orbital coordinate system, the relative position vector of the secondary star relative to the primary star is... Combining the gravitational model of the two-body problem, the nonlinear relative motion equations of the sub-star are: , in, This is the distance from the satellite to the center of the moon.

[0031] Because the relative distance between the satellites in the formation is much smaller than the orbital radius, it is possible to measure the gravitational term. Perform a first-order Taylor expansion: , Substituting the first-order Taylor expansion into the nonlinear relative motion equations of the sub-star, and neglecting second-order and higher minima, the fundamental relative motion equations are derived: , To improve the accuracy of relative motion modeling, the gravitational term is... Performing a third-order Taylor expansion down to the third-order minimum yields the third-order nonlinear correction term: , .

[0032] By combining the basic equations of relative motion with third-order nonlinear correction terms and rearranging, we obtain the third-order relative motion model: .

[0033] Compared with the classical linear CW equation, the motion model of this invention can more accurately describe the dynamic characteristics of "breathing formation" when the inter-satellite distance is large, providing accurate dynamic constraints for factor graph optimization.

[0034] Step 2: Establish a collaborative measurement model for the lunar satellite formation.

[0035] In the formation measurement system, the main satellite is equipped with lidar and Ku-band communication equipment. The lidar measures the line-of-sight angle between the main satellite and each satellite, and the Ku-band inter-satellite communication function is used to obtain the distance information between the main satellite and each satellite.

[0036] The inter-satellite angle observation model is defined as follows: , in, Representing the first i Information on the azimuth and elevation angles between the minor star and the primary star. ; Indicates the first i The relative position vector of a minor star in the coordinate system of the primary star's orbit. This represents noise in inter-satellite angle measurements.

[0037] The inter-satellite distance observation model is defined as follows: , in, d i Representing the i The distance between the minor star and the primary star. This indicates noise in inter-satellite distance measurements.

[0038] Based on the above inter-satellite angle and inter-satellite distance observation information, the collaborative measurement model of the satellite formation... M Established as: , in, Representing the i Information on the interstellar distances and angles between the minor stars and the primary star. .

[0039] A collaborative measurement model based on lidar and Ku-band communication was established. The primary satellite measures the azimuth and elevation angles of each satellite using lidar and acquires inter-satellite distance information via Ku-band communication. In the primary satellite's LVLH coordinate system, the relative position vectors of the satellites are mapped to the measurement space, establishing an observation model for inter-satellite angles and distances, while considering the impact of measurement noise. The collaborative measurement model of the satellite formation provides observational constraints for subsequent factor map optimization, ensuring that the estimated state is consistent with the actual measurement data.

[0040] Step 3: Within a sliding window of a set length, construct a factor graph using the relative orbital states of a series of satellite formations as state nodes and constraint factors connecting the state nodes; optimize the factor graph to obtain the optimal estimate of the relative orbital states of the satellite formations at the current moment.

[0041] The steps for constructing a factor graph using relative orbital states as state nodes and constraint factors connecting the state nodes are as follows: With a sliding window length of W Within the time interval, Indicates the first k The relative orbital states of each epoch, among which Indicates the first k The relative position vector of each epochal minor star relative to the primary star This represents the relative velocity vector of the k-th epoch child star relative to the primary star. express The transpose of , express The transpose of .

[0042] The relative orbital state of each epoch within the sliding window serves as the state node for constructing the factor graph. W The state sequence formed by consecutive state nodes within a sliding window of length is as follows: ; Among them, X k The preliminary relative orbital state at the current moment is calculated using a third-order relative motion model. To X k-1 It belongs to the historical trajectory state.

[0043] The constraint factors connecting these state nodes mainly include three categories: prior factors, dynamic factors, and joint observation factors.

[0044] The initial state at the start of the sliding window, or the prior information obtained from marginalizing the previous window, needs to be introduced into the optimization problem through prior factors. These prior factors are Gaussian constraints, and are expressed as follows: , Where X0 represents the initial orbital state estimate or the first orbital state after marginalization in the previous window; that is, X0 represents the earliest state node within the moving window. It is the a priori covariance.

[0045] Based on the third-order relative motion model, the orbital state at time t can be constructed. X i t orbital state at time t+1 X i t+1 kinetic factors: , in, This is the discretized propagation function for the third-order relative motion model.

[0046] The joint observation factors include inter-satellite distance observation factors and inter-satellite angle observation factors.

[0047] According to the inter-satellite distance observation model in the collaborative measurement model, the inter-satellite distance observation factor is expressed as: , in, d i Representing the i The distance measurement between the minor star and the primary star. The magnitude of the relative orbital position between the i-th sub-star and the main star, i.e., the theoretical distance; According to the inter-satellite angle observation model in the collaborative measurement model, the inter-satellite angle observation factor is expressed as: , in, These represent the inter-satellite azimuth and elevation angles estimated from the relative position vectors, respectively. Representing the measurement of the first i The azimuth and elevation angles between the minor star and the primary star.

[0048] After merging the above inter-satellite distance observation factors and inter-satellite angle measurement observation factors, since the distance measurement and angle measurement noises are independent and follow a zero-mean Gaussian distribution, they are cascaded to construct a joint observation factor: ; Based on the above constraints, the objective function for factor graph optimization within the sliding window is: , in For process noise covariance, To measure the noise covariance.

[0049] The trust-region Dog-Leg algorithm is then used to optimize the factor graph and solve for the optimal value of the state sequence. The goal of factor graph optimization is to find a set of states that minimizes the weighted sum of squares of all factor residuals. The factor residuals are specific values ​​calculated from the factors, representing the degree of constraint violation. Factor graph optimization is essentially a nonlinear least squares problem.

[0050] This invention employs the trust-region DogLeg algorithm to optimize the factor graph, outputting the optimal estimate of the state sequence to obtain the optimal estimate of the relative orbital state of the satellite formation at the current moment. As a trust-region nonlinear optimization scheme, the DogLeg algorithm combines the fast convergence of the Gauss-Newton method with the stability of gradient descent.

[0051] The DogLeg algorithm with a trust region is used to iteratively optimize the objective function of the above factor graph optimization. In each iteration step, the algorithm cleverly combines the advantages of the steepest descent method and the Gauss-Newton method within a trust region to find the optimal update direction and step size until it converges to the optimal solution.

[0052] The specific process is as follows: At the current estimation point At this point, perform a first-order Taylor expansion on all factor residuals to construct the normal equation: , in, J The joint Jacobian matrix of the residuals of all factors. For joint noise covariance, This represents the state increment.

[0053] The current estimated point refers to the current value assigned by the optimization algorithm to all state variables (relative to the orbital state) to be optimized in the current iteration step.

[0054] In the trust region radius Inside, the fastest descent direction is: , in, g For gradient, .

[0055] The Gauss-Newton direction is: .

[0056] Finally, the search direction is determined using the following directional search strategy: , in, Determined by the trust region boundary conditions.

[0057] After determining the search direction, iteratively optimize according to the following rules to solve the objective function of factor graph optimization: , in This represents the relative orbital state at iteration t. This represents the updated state at iteration t+1.

[0058] After the factor graph is constructed, the trust-region Dog-Leg algorithm is used to optimize the factor graph and obtain the optimal estimate of the relative orbital state of the satellite formation at the current moment. When the next epoch arrives, the new state node (calculated using the third-order relative motion model) is... k The initial relative orbital state of each epoch enters the sliding window. Since the sliding window has a fixed length, the earliest state node needs to be moved out of the window, but its historical information still needs to be retained through prior factors and passed to subsequent optimizations. To achieve this, a Schur complement-based edge-mapping mechanism is used to process the state nodes moved out of the sliding window to update the prior factors. In the next epoch (the initial relative orbital state of each epoch), the state nodes are processed. k+1 (epoch), based on updated prior factors, dynamic factors, observation factors, and the epoch of the first epoch. k+1 From the first epoch to the ( k+1 ) -W+1 A new factor graph is constructed from a series of relative orbital states of each epoch, and then the factor graph is optimized. The entire process can be summarized as a loop of factor graph construction - factor graph optimization - sliding update.

[0059] In other words, when constructing the factor graph, first determine if the length of the sliding window is greater than the set length W. If so, move the earliest state node out of the sliding window and update the prior factors. Construct the factor graph using W state nodes, the updated prior factors, the dynamic factors, and the observed factors.

[0060] The following describes the process of updating the prior factors.

[0061] Suppose that the sliding window at the current time contains two types of state vectors: states to be marginalized. and reserved status Two types of state vectors, linearizing all factors, yield the standard quadratic form of the joint optimization problem: , in, It is in a state of being marginalized. The relevant Hessian submatrix, Is it in the state of preservation? The relevant Hessian submatrix, It is a cross term that describes the coupling relationship between the state to be marginalized and the state to be retained; It corresponds to the linear term, which is calculated from the residual and the Jacobian matrix.

[0062] because These variables are no longer retained and need to be completely eliminated from the optimization variables. According to least squares theory, by... Solving for the minimization condition and substituting back into the standard quadratic form above, we can fold its effects into a state that preserves the state. Constraints.

[0063] The above standard quadratic form (cost function) is related to Taking the partial derivative and setting it to zero, we get: ; The formula is further transformed into ; Substituting the transformed formula into the above standard quadratic form, we obtain the information regarding the preserved state. The new quadratic form: , remember: , The residual form of the prior factors obtained after marginalization is: , in, A From Factors obtained through matrix factorization b Source Finally, the covariance matrix (prior covariance) of the prior factors is: .

[0064] The prior factors obtained after marginalization are added as new factors to the next optimization, so that the constraints of the removed states are preserved.

[0065] A factor graph optimization algorithm for relative navigation based on a sliding window is designed. A factor graph is constructed, using the relative orbital states at different times within the sliding window as state nodes, and dynamic factors and observation factors as constraint factors, transforming the relative navigation problem into a nonlinear least squares optimization problem. Dynamic factors describe the dynamic constraints between states, and observation factors describe the constraint relationships between orbital states and measured values ​​(inter-satellite distances and inter-satellite angles). The DogLeg optimization algorithm is used to solve the problem. This algorithm adaptively adjusts the step size through a trust region mechanism to ensure convergence and efficiency. A sliding window strategy is employed to jointly optimize historical states and observation data within a fixed time window. A marginalization mechanism compresses historical information outside the window into prior constraints, preventing the computational load from growing indefinitely. This algorithm can utilize historical observation data to jointly optimize orbital states, effectively suppressing error accumulation and significantly improving navigation accuracy and robustness.

[0066] Figure 2 The simulation verification results according to an embodiment of the present invention are shown.

[0067] The effectiveness of the lunar satellite formation relative navigation method based on factor graph optimization proposed in this invention is verified through simulation.

[0068] The lunar "breathing formation" operates in a near-circular orbit at an altitude of 300 km, consisting of one primary satellite and nine secondary satellites. In the inter-satellite measurement system, the lidar angle measurement accuracy is 1' (3σ), and the Ku-band link ranging accuracy is 3m (3σ). Assuming the primary satellite's absolute orbit is known with an orbit determination error of 1.8 km (3σ), the initial relative position error of each secondary satellite is set to 100m (3σ). Navigation simulations are conducted based on a 100 km formation baseline, where the maximum relative distance between the secondary satellites and the primary satellite can reach 110 km.

[0069] After 5 hours of navigation simulation verification, the system can achieve rapid convergence using the relative navigation method proposed in this invention. Figure 2 As shown, the maximum relative position error of the nine satellites is 35.4 m (3σ), and the maximum relative velocity error is 0.03 m / s (3σ). Simulation results demonstrate that this method can achieve high-precision relative state estimation, meeting the accuracy requirements of formation control for orbital state.

[0070] In summary, the relative navigation method proposed in this invention effectively suppresses error accumulation and significantly improves navigation accuracy and robustness by constructing a third-order nonlinear relative motion model and a sliding window optimization strategy, providing a reliable technical solution for autonomous navigation of lunar exploration missions such as ultra-long-wave radio astronomy observations.

[0071] While some embodiments of the present invention have been described in this application, those skilled in the art will understand that these embodiments are merely illustrative. Numerous variations, alternatives, and improvements will arise in those skilled in the art under the teachings of this invention without departing from its scope. The appended claims are intended to define the scope of the invention and thereby cover methods and structures within the scope of the claims themselves and their equivalents.

Claims

1. A relative navigation method for lunar satellite formations based on factor graph optimization, characterized in that, Includes the following steps: Within a sliding window of a set length, a factor graph is constructed using the relative orbital states of a series of satellite formations as state nodes and constraint factors connecting the state nodes. The factor graph is optimized to obtain the optimal estimate of the relative orbital state of the satellite formation at the current moment.

2. The lunar satellite formation relative navigation method based on factor graph optimization according to claim 1, characterized in that, The relative orbital state of each epoch within the sliding window serves as the state node for constructing the factor graph. The state sequence formed by consecutive state nodes within the sliding window is as follows: ； Indicates the first k The relative orbital states of each epoch, among which Indicates the first k The relative position vector of each epochal minor star relative to the primary star This represents the relative velocity vector of the k-th epoch child star relative to the primary star. express The transpose of , express The transpose of .

3. The lunar satellite formation relative navigation method based on factor graph optimization according to claim 2, characterized in that, The constraint factors include prior factors, dynamic factors, and joint observation factors; The prior factor is subject to Gaussian constraints and is expressed as follows: ， Where X0 is the initial orbital state estimate or the first orbital state after marginalization in the previous window. It is the a priori covariance; Based on the third-order relative motion model, the orbital state at time t is constructed. X i t orbital state at time t+1 X i t+1 kinetic factors: ， in, This is the discretized propagation function for the third-order relative motion model; According to the inter-satellite distance observation model in the collaborative measurement model, the inter-satellite distance observation factor is expressed as: ， in, d i This represents the distance between the i-th child star and the primary star. The modulus represents the relative orbital position between the i-th child star and the primary star; According to the inter-satellite angle observation model in the collaborative measurement model, the inter-satellite angle observation factor is expressed as: ， in, These represent the inter-satellite azimuth and elevation angles estimated from the relative position vectors, respectively. Representing the measurement of the first i The azimuth and elevation angles between the minor star and the primary star; After merging the inter-satellite distance observation factor and the inter-satellite angle measurement observation factor, since the distance measurement noise and the angle measurement noise are independent and follow a zero-mean Gaussian distribution, they are cascaded to construct a joint observation factor. : ； Based on the aforementioned constraint factors, the objective function for factor graph optimization within the sliding window is: ， in, For process noise covariance, R i This represents the measurement noise covariance.

4. The lunar satellite formation relative navigation method based on factor graph optimization according to claim 3, characterized in that, The DogLeg algorithm with trust region is used to iteratively optimize the objective function of factor graph optimization to optimize the factor graph, including the following steps: At the current estimation point At this point, perform a first-order Taylor expansion on all factor residuals to construct the normal equation: ， in, J The joint Jacobian matrix of the residuals of all factors. For joint noise covariance, For state increments; In the trust region radius Inside, the fastest descent direction is: ， in, g For gradient, ; The Gauss-Newton direction is: ; Use the following directional search strategy to determine the search direction: ， in, Determined by the trust region boundary conditions; After determining the search direction, iteratively optimize according to the following rules to solve the objective function of factor graph optimization: ， in This represents the relative orbital state at iteration t. This represents the updated state at iteration t+1.

5. The lunar satellite formation relative navigation method based on factor graph optimization according to claim 3, characterized in that, When constructing the factor graph, first determine whether the length of the sliding window is greater than the set length W. If so, move the earliest state node out of the sliding window and update the prior factors. Construct the factor graph with W state nodes, the updated prior factors, the dynamic factors, and the joint observation factors.

6. The lunar satellite formation relative navigation method based on factor graph optimization according to claim 5, characterized in that, Updating prior factors using the marginalization method involves the following steps: Suppose that the sliding window at the current time contains two types of state vectors: states to be marginalized. and reserved status ; Based on the two types of state vectors, linearizing all factors yields the standard quadratic form of the joint optimization problem: ， in, It is in a state of being marginalized. The relevant Hessian submatrix, Is it in the state of preservation? The relevant Hessian submatrix, It is a cross term that describes the coupling relationship between the state to be marginalized and the state to be retained; These are the corresponding linear terms, calculated from the residuals and the Jacobian matrix; According to least squares theory, by treating marginalized states Solve for the minimization condition and substitute back into the standard quadratic form to fold the influence of the state to be marginalized into the influence on the state to be retained. Constraints; The standard quadratic form regarding Taking the partial derivative and setting it to zero, we get: ； The formula is further transformed into ; Substituting the transformed formula into the standard quadratic form, we obtain the result regarding the preserved state. The new quadratic form: ， remember , The residual form of the prior factors obtained after marginalization is: ， in, A From Factors obtained through matrix factorization b Source ; Finally, the covariance matrix of the prior factors is .

7. The lunar satellite formation relative navigation method based on factor graph optimization according to claim 1, characterized in that, Also includes: Establish a relative motion model of the lunar satellite formation in the coordinate system of the main star's orbit; as well as Establish a collaborative measurement model for lunar satellite formations.

8. The lunar satellite formation relative navigation method based on factor graph optimization according to claim 7, characterized in that, The relative motion model of the lunar satellite formation in the main star orbit coordinate system includes: Assuming that in the primary star's orbital coordinate system, the relative position vector of the secondary star relative to the primary star is: Combining the gravitational model of the two-body problem, the nonlinear relative motion equations of the sub-star are: ， in, This is the distance from the satellite to the center of the moon. For the gravitational term Perform a first-order Taylor expansion: ； Substituting the first-order Taylor expansion into the nonlinear relative motion equations of the sub-star, and neglecting second-order and higher minima, the fundamental relative motion equations are derived: ， Gravity term Performing a third-order Taylor expansion down to the third-order minimum yields the third-order nonlinear correction term: ， ； By combining the basic equations of relative motion with third-order nonlinear correction terms and rearranging, we obtain the third-order relative motion model: 。 9. The lunar satellite formation relative navigation method based on factor graph optimization according to claim 7, characterized in that, Establishing a collaborative measurement model for lunar satellite formations includes: The inter-satellite angle observation model is defined as follows: ， in, Representing the first i Information on the azimuth and elevation angles between the minor star and the primary star. ; Indicates the first i The relative position vector of a minor star in the coordinate system of the primary star's orbit. This represents noise in inter-satellite angle measurements. The inter-satellite distance observation model is defined as follows: ， in, d i Representing the i The distance between the minor star and the primary star. This indicates noise in inter-satellite distance measurements; Based on inter-satellite angle and inter-satellite distance observation information, a collaborative measurement model for satellite formations M Established as: ， in, Representing the i Information on the interstellar distances and angles between the minor stars and the primary star. .

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