Robust adaptive fusion filtering celestial positioning method and system

Abstract

This invention discloses a robust adaptive fusion filtering astronomical attitude determination method and system, belonging to the field of radio navigation technology. The method includes: constructing a combined navigation model of a starlight and inertial navigation system based on a Lie group description; obtaining the observation residuals and covariance of the corresponding modes; performing robust and adaptive parallel sub-filtering of modes, and updating the posterior states and covariances of the two modes; updating the mode probabilities; fusing the updated posterior states and covariances to obtain the fused output of the parallel sub-filters, and using the fused output of the parallel sub-filters as the final estimate of the starlight and inertial navigation system based on a Lie group description, outputting an astronomical attitude determination result including azimuth, pitch, and roll angles. This invention exhibits strong adaptability, robustness, and accuracy in dealing with complex noise and measurement field values, and can achieve high-precision astronomical attitude determination for starlight and inertial navigation systems in complex dynamic environments.

Description

Technical Field

[0001] This invention relates to the field of radio navigation technology, and in particular to a robust adaptive fusion filtering astronomical attitude determination method and system. Background Technology

[0002] As a high-precision, fully autonomous navigation method, the combined star-based and inertial navigation system occupies a core position in the field of astronomical attitude determination in aerospace. It leverages the complementary advantages of the high-frequency dynamic response characteristics of the inertial measurement unit (IMU) and the absolute attitude observation information from star sensors, making it a key technology for achieving high-precision, fully autonomous navigation of aircraft. At the data fusion and state estimation level, current engineering applications are mainly based on the Bayesian estimation theory framework, with the standard Kalman filter (KF) and its nonlinear variant—the extended Kalman filter (EKF)—forming the existing core algorithm system.

[0003] Existing EKF-based inertial integrated navigation methods typically include the following key steps: First, a state-space model of the system is established, selecting attitude, velocity, position error, and inertial device bias as state vectors; second, the nonlinear system equations are linearized at the current state estimate through Taylor series expansion, usually ignoring second-order and higher-order terms to obtain an approximate linearized state transition matrix, i.e., the Jacobian matrix; subsequently, the algorithm enters a cyclic iterative process of "prediction-update": in the time update stage, the inertial navigation solution results are used to further predict the prior state and covariance of the system; in the measurement update stage, the difference between the sensor observations and the predicted values ​​is calculated, and the prior state is corrected by combining Kalman gain, thereby obtaining the posterior optimal estimate of the system.

[0004] As modern aerospace missions increasingly demand higher requirements for navigation accuracy, long-term stability, and environmental adaptability, the shortcomings of traditional algorithms are mainly reflected in two dimensions: the lack of consistency in geometric modeling and the insufficient ability of the filtering framework to cope with complex noise environments.

[0005] In terms of geometric modeling and error description, existing technologies suffer from a fundamental "geometric inconsistency." The three-dimensional rotational attitude of a rigid body mathematically constitutes a special orthogonal group, essentially belonging to a Riemannian manifold structure in non-Euclidean space. However, the traditional EKF algorithm is based on Euclidean vector space, and its additive error definition forcibly simplifies rotation operations on the manifold to addition operations in vector space. This approach violates the orthogonality constraints of the rotation matrix or the unit modulus constraints of the quaternions, causing the state variables during the filtering iteration process to gradually deviate from the true manifold surface. Although forced normalization is often used in engineering to remedy this, this operation, lacking rigorous mathematical basis, leads to the state covariance matrix failing to accurately reflect the true distribution of errors in the tangent space, resulting in inconsistent variance estimation. Especially under high dynamic conditions such as large maneuvers and large misalignment angles, the truncation error introduced by linearization and the constraint violation effect can couple, severely weakening the system's error convergence capability and even causing filter divergence in long-endurance missions.

[0006] In terms of environmental adaptability and robustness, traditional filtering frameworks struggle to cope with the complex and ever-changing real-world flight environments. Classical Kalman filtering theory relies heavily on two assumptions: "the statistical characteristics of noise are known and fixed" and "the noise follows a Gaussian distribution." However, in practical starlight and inertial navigation applications, star sensors are highly susceptible to stray light interference, thermal noise, or high-energy particles in space, leading to outliers in the observation data that significantly deviate from the Gaussian distribution. Because traditional algorithms are based on… Norm minimization criteria are extremely sensitive to outliers; a single anomalous observation can significantly skew the estimation results, lacking robustness and resilience. Furthermore, the accuracy of an aircraft's dynamic model varies significantly with environmental noise levels during different flight phases, such as trajectory changes and maneuvers. Existing technologies typically employ fixed process noise covariance matrices and measurement noise covariance matrices, failing to sense and follow changes in system noise characteristics online. When the actual noise level does not match the preset parameters, the filter cannot allocate appropriate weighted gains, leading to decreased system accuracy in non-stationary noise environments and failing to meet the requirements of high-reliability navigation. Summary of the Invention

[0007] The technical problem to be solved by this invention is to provide a robust adaptive fusion filtering astronomical attitude determination method and system. It solves the problem of long-term error accumulation caused by EKF linearization from the modeling level, significantly improves the stability and accuracy of the system, and can adjust the noise covariance in real time in dynamic environments. It has strong anti-interference and adaptive capabilities, and effectively overcomes the dependence of traditional Kalman filters on noise statistical assumptions.

[0008] This invention is achieved through the following technical solution:

[0009] A robust adaptive fusion filtering astronomical attitude determination method includes the following steps:

[0010] S1: Construct a combined navigation model that includes the state equation of the inertial navigation error of the starlight and inertial navigation system based on the description of Lie groups and the observation equation of the starlight and inertial navigation system based on the description of Lie groups;

[0011] S2: Based on the pre-set mode transition probability and initial mode probability of the integrated navigation model, calculate the initial mixed state value and initial mixed covariance of the robust mode and adaptive mode. Then, make predictions based on the initial mixed state value and initial mixed covariance of the robust mode and adaptive mode to obtain the observation residual and observation residual covariance of the corresponding mode.

[0012] S3: Based on the observation residuals and observation residual covariance of the corresponding modes, the adaptive mode combines the Sage-Husa adaptive strategy, and the robust mode combines the Huber robust estimation strategy to perform parallel sub-filtering of the two modes. When the adaptive mode sub-filters, the Kalman gain is calculated based on the measurement noise covariance matrix estimated at the previous time step, and the posterior state and covariance of the adaptive mode are updated. When the robust mode sub-filters, the normalized residuals of each observation channel are calculated, the weights of each observation channel are calculated according to the Huber cost function and a weight matrix is ​​constructed, the equivalent weighted observation residual covariance is calculated using the weight matrix, and then the robust Kalman gain is calculated to update the posterior state and covariance of the robust mode.

[0013] S4: Update the mode probabilities based on the updated posterior states and covariances under the two modes;

[0014] S5: Based on the updated mode probabilities, the updated posterior states and covariances of the two modes are fused to obtain the fused output of the parallel sub-filter. The fused output of the parallel sub-filter is used as the final estimate of the starlight and inertial integrated navigation system based on the Lie group description. The output includes astronomical attitude determination results including azimuth, pitch and roll angles.

[0015] The optimized state equation for the inertial navigation error of the starlight and inertial integrated navigation system described by the Lie group in step S1 is Equation (1):

[0016] (1);

[0017] in: The derivative of the inertial navigation error state vector. Represents the system state transition matrix. This represents the inertial navigation error state vector. Represents the noise transition matrix. This represents the process noise vector.

[0018] The optimized observation equation for the starlight and inertial combined navigation system described by the Lie group in step S1 is Equation (2):

[0019] (2);

[0020] in: Represents the observation vector. Represents the observation matrix. This represents the attitude error angle transformation matrix. This indicates observation noise.

[0021] Furthermore, the method for calculating the mixed initial value and mixed initial covariance under the robust mode and adaptive mode in step S2 is as follows:

[0022] S211: Calculation mode according to formula (3) From pattern Mixed weights:

[0023] (3);

[0024] in: Representation pattern From pattern Mixed weights, Indicates from pattern Currently in mode The transition probability, express The filter is in mode at any given time. The posterior pattern probability, This represents all possible source modes of the filter from the previous time step. Indicates from pattern Currently in mode The transition probability, express The filter is in mode at any given time. posterior pattern probability;

[0025] S212: Pattern-based From pattern The mixed weights are constructed according to equation (4). exist Initial values ​​of mixed states at time: (4);

[0026] in: Representation pattern exist The mixed initial values ​​of the state at time 10:00. Representation pattern exist The posterior state estimate vector at time 1;

[0027] S213: Pattern-based exist The initial mixed state value at time t is calculated according to equation (5). exist Mixed initial covariance at time: (5);

[0028] in: Representation pattern exist The mixed initial covariance at time 1, Representation pattern exist The posterior covariance at time 1, This indicates the matrix transpose.

[0029] Furthermore, in step S2, the method for predicting the observed residuals and observed residual covariance of the corresponding modes based on the mixed initial values ​​and mixed initial covariance of the states under both robust and adaptive modes is as follows:

[0030] S214: Change the mode exist The initial mixed state value and initial mixed covariance at time step are advanced by one sampling time, and the pattern is obtained according to equation (6). exist Prior state vector and prior covariance at each time step: (6);

[0031] in: Representation pattern exist Prior state vector at time step This indicates that a starlight and inertial navigation system based on the Lie group description is in... The state transition matrix at time t, Representation pattern exist Prior covariance at time, Representation pattern exist The system process noise covariance matrix at time t;

[0032] S215: Based on current observation vectors and patterns exist Given the prior state vector and prior covariance at time step, calculate the observation residuals and observation residual covariance according to equation (7): (7);

[0033] in: express Time Mode The observation residuals express The observation vector at time t, express The observation matrix at time, express Time Mode The observed residual covariance express The transpose of the observation matrix at time t. express Time Mode The observation noise covariance matrix.

[0034] Furthermore, the methods for adaptive mode sub-filtering and updating the posterior state and covariance in step S3 are as follows:

[0035] S311: The adaptive Kalman gain is calculated based on the estimated value of the observation noise covariance matrix according to equation (8):

[0036] (8);

[0037] in: Indicates in Adaptive Kalman gain at time step Indicates adaptive mode in Prior covariance at time, Indicates adaptive mode in The estimated value of the observation noise covariance matrix at time t;

[0038] S312: Mode based on adaptive Kalman gain according to equation (9) exist Posterior state and covariance update at time step:

[0039] (9);

[0040] in: Indicates adaptive mode in The posterior state vector at time t. Indicates adaptive mode in Prior state vector at time step express Observation residuals in time-adaptive mode Indicates adaptive mode in The posterior covariance at time 1, Represents the identity matrix;

[0041] S313: Introduce an adaptive forgetting factor and update it according to Equation (10) using an exponentially weighted average. Estimates of the observation noise covariance matrix at time step:

[0042] (10); where: Indicates adaptive mode in The estimated value of the observation noise covariance matrix at time t. Represents the adaptive forgetting factor. express The transpose matrix of the observation residuals in the time-adaptive mode.

[0043] Furthermore, the methods for robust mode sub-filtering and updating the posterior state and covariance in step S3 are as follows:

[0044] S321: Implement robust mode in The first moment The residual components of each observation channel are normalized according to their prediction variance using equation (11) to obtain the robust model in... Normalized residuals at time:

[0045] (11);

[0046] in: Indicates robust mode in The first moment Normalized residuals of each observation channel, Indicates robust mode in Normalized residuals at time, express The observation residuals of the time-robust model at the 1st moment The components of each observation channel, express The diagonal component of the covariance of the observation residuals of the time-robust model. This represents diagonal matrix operations. Indicates the number of observation channels;

[0047] S322: Calculate the weight of each channel based on the Huber cost function according to equation (12): (12);

[0048] in: Indicates robust mode in The first moment The weights of each observation channel, The threshold representing the Huber cost function;

[0049] S323: Introduce a robust smoothing factor. According to formula (13), the weights of each channel are smoothed by a small exponential smoothing to obtain the smoothing weights of each observation channel in the robust mode.

[0050] (13);

[0051] in: Indicates robust mode in The first moment Smoothing weights for each observation channel, Indicates robust mode in The first moment The weights of each observation channel, Indicates the robust smoothing factor;

[0052] S324: Assemble the smoothing weights of all robust mode observation channels into a weight diagonal matrix according to equation (14):

[0053] (14);

[0054] in: Represents the robust mode weight diagonal matrix;

[0055] S325: Based on the robust mode weight diagonal matrix, calculate the equivalent weighted observation residual covariance according to equation (15), and based on the equivalent weighted observation residual covariance, calculate the robust Kalman gain according to equation (16):

[0056] (15);

[0057] (16);

[0058] in: Indicates robust mode in Equivalent weighted observation residual covariance at time points express Observation residual covariance of time-robust mode Indicates robust Kalman gain. Indicates robust mode in Prior covariance at time, Indicates robust mode in The equivalent weighted observation residual covariance inverse matrix at time points;

[0059] S326: Based on robust Kalman gain, the robust mode posterior state and covariance are updated according to equation (17):

[0060] (17);

[0061] in: Indicates robust mode in The posterior state vector at time t. Indicates robust mode in Prior state vector at time step Indicates robust mode in The posterior covariance at time 1, express The observation noise covariance matrix of the time-robust model. This represents the robust Kalman gain transpose matrix.

[0062] Furthermore, the method for updating the pattern probability in step S4 is as follows:

[0063] S411: According to equation (18), map the posterior mode probability of the previous time step to the prior mode probability of the current time step using the preset transition probability:

[0064] (18);

[0065] in: express The filter is in mode at any given time. The prior pattern probability, express The filter is in mode at any given time. posterior pattern probability;

[0066] S412: Calculate the pattern according to equation (19) Likelihood of the next observation vector:

[0067] (19);

[0068] in: express Always in mode The likelihood of the observed vector. Represents pi (π). This represents the operation of the exponential function. Indicates the dimension of the observation vector;

[0069] S413: According to Bayes' theorem, based on equation (20), the current pattern is... The likelihood of the observed vector is multiplied by the prior mode probability and normalized to obtain the current filter mode. The posterior pattern probability is used to update the pattern probability:

[0070] (20);

[0071] in: express The filter is in mode at any given time. The posterior pattern probability.

[0072] Furthermore, in step S5, the updated posterior states and covariances under the two modes are fused according to equation (21) to obtain the fused output of the integrated navigation model:

[0073] (twenty one);

[0074] in: This represents the fused posterior state vector output of the integrated navigation model. Representation pattern exist The posterior state vector at time t. This represents the fusion posterior covariance output of the integrated navigation model. Representation pattern exist Posterior covariance at time step.

[0075] A robust adaptive fusion filtering astronomical attitude determination system is provided for executing a robust adaptive fusion filtering astronomical attitude determination method as described in any of the above, comprising an integrated navigation model construction module, a robust adaptive parallel sub-filter module, a mode probability update module, and a fusion output module.

[0076] The integrated navigation model construction module is used to construct an integrated navigation model that includes the inertial navigation error state equation of the starlight and inertial integrated navigation system described by Lie group and the observation equation of the starlight and inertial integrated navigation system described by Lie group.

[0077] The robust adaptive parallel sub-filter module is used to calculate the observation residuals and observation residual covariance of the corresponding mode; based on the observation residuals and observation residual covariance of the corresponding mode, robust mode and adaptive mode parallel sub-filters are performed, and the posterior state and covariance of the two modes are updated.

[0078] The mode probability update module is used to update the mode probability based on the updated posterior state and covariance under the two modes;

[0079] The fusion output module is used to fuse the updated posterior states and covariances under the two modes to obtain the fusion output of the parallel sub-filter. The fusion output of the parallel sub-filter is used as the final estimate of the starlight and inertial integrated navigation system based on the Lie group description, and the output includes astronomical attitude determination results including azimuth, pitch and roll angles.

[0080] Beneficial effects of the invention:

[0081] The robust adaptive fusion filtering astronomical attitude determination method and system provided by this invention have the following advantages:

[0082] By introducing Lie group theory, the geometric consistency of state estimation on rotating manifolds is fundamentally guaranteed, making up for the theoretical defects of traditional EKF and effectively solving the problem of inconsistent estimation of unobservable error state variance. On this basis, by designing a robust adaptive fusion filtering algorithm, the classical optimal estimation theory is extended to more challenging non-Gaussian, time-varying noise environments, effectively solving the problems of complex noise characteristics and measurement outliers in complex dynamic environments. This significantly improves the robustness and accuracy of starlight and inertial navigation in complex dynamic environments, and achieves high-precision astronomical attitude determination. Attached Figure Description

[0083] Figure 1 This is a schematic diagram of the process of this invention.

[0084] Figure 2a This is a schematic diagram of the actual attitude angle of a spacecraft during motion, as shown in this invention.

[0085] Figure 2b This invention provides a schematic diagram of the measurement values ​​of a star sensor simulating the motion of a spacecraft.

[0086] Figure 3 This is a schematic diagram showing the actual value, observed value, and filtered estimate of the azimuth angle of this invention.

[0087] Figure 4 This is a schematic diagram showing the actual value, observed value, and filtered estimate of the pitch angle of this invention.

[0088] Figure 5 This is a schematic diagram showing the actual value, observed value, and filtered estimate of the roll angle of this invention.

[0089] Figure 6 This is a schematic diagram illustrating the estimation errors of the traditional EKF method, the robust KF method, and the method presented in this paper. Detailed Implementation

[0090] A robust adaptive fusion filtering astronomical attitude determination method, the flowchart of which is shown below. Figure 1 As shown, the specific steps include the following:

[0091] S1: Construct a combined navigation model that includes the state equation of the inertial navigation error of the starlight and inertial navigation system based on the description of Lie groups and the observation equation of the starlight and inertial navigation system based on the description of Lie groups;

[0092] Specifically, based on the state equation of the inertial navigation error of the starlight and inertial combined navigation system described by Lie group, it is equation (1):

[0093] (1);

[0094] in: The derivative of the inertial navigation error state vector. Represents the system state transition matrix. This represents the inertial navigation error state vector. Represents the noise transition matrix. This represents the process noise vector.

[0095] Inertial navigation error state vector have , Indicates the angle of inaccuracy under Lie group, Represents the left Jacobian matrix. Indicates the velocity error under the Lie group. This represents the positional error under the Lie group. This indicates the constant drift error of the gyroscope. This indicates the accelerometer constant drift error. This represents the transpose of the matrix.

[0096] The system state transition matrix is ;

[0097] in: express System relative to Tied in The angular velocity vector represented in the system. Indicates an antisymmetric matrix, The carrier is represented by Tie The attitude transformation matrix of the system. Indicates gravitational acceleration in The projection under the system, express System relative to Tied to The carrier velocity vector represented in the system. express System relative to Tied in The estimated value of the carrier position vector represented in the system. Represents the identity matrix. The system represents the geocentric coordinate system. The system represents the geocentric inertial coordinate system. The system represents the coordinate system of the carrier.

[0098] For local navigation on a typical vehicle, the change in gravitational acceleration is negligible. It can be approximated as a constant value. In this case, in the system state transition matrix, the relatively stable gravity term replaces the specific force term, which is easily affected by quantization noise, thus improving the calculation accuracy and enabling the unobservable state to have better variance consistency.

[0099] The noise transfer matrix is ;

[0100] The process noise vector is in: express White noise of an axial gyroscope express White noise of an axial gyroscope express White noise of an axial gyroscope express White noise from the axial accelerometer express White noise from the axial accelerometer express White noise from the axial accelerometer.

[0101] Because the strapdown matrix output by the inertial navigation system (INS) includes the platform misalignment angle. Therefore, the attitude angles obtained from the inertial navigation attitude calculation Includes attitude error angle That is:

[0102] ;

[0103] in: Indicates the actual pitch angle of the carrier. Indicates the actual azimuth angle of the carrier. Indicates the actual roll angle of the carrier. This represents the pitch angle obtained from the inertial navigation attitude calculation. This represents the azimuth angle obtained from the inertial navigation attitude calculation. This represents the roll angle obtained from the inertial navigation attitude calculation. This indicates pitch angle attitude error. This indicates the azimuth attitude error. This indicates the roll angle attitude error.

[0104] Celestial navigation systems (CNS) achieve high accuracy in star-based attitude determination using star sensors, and the calculated attitude angles... This can be expressed as the superposition of the actual attitude value and the corresponding observation noise, i.e.:

[0105] ;in: This represents the pitch angle calculated by the astronomical navigation system. This represents the azimuth angle calculated by the astronomical navigation system. This represents the roll angle calculated by the astronomical navigation system. The observation noise representing the pitch angle, The observation noise representing the azimuth angle, This represents the observation noise of the roll angle.

[0106] In inertial navigation attitude calculation, the platform misalignment angle With attitude error angle The following conversion relationship exists between them:

[0107] ;in: This indicates the misalignment angle of the eastward-facing platform. This indicates the misalignment angle of the northbound platform. This indicates that the astronautical platform is out of alignment. This represents the attitude error angle transformation matrix.

[0108] Using the difference between the two sets of attitude angles calculated by INS and CNS respectively as the observation vector, we can obtain the following formula:

[0109] ;

[0110] Therefore, the observation equation of the starlight and inertial combined navigation system based on the Lie group description is obtained as equation (2):

[0111] (2);

[0112] in: Represents the observation vector. Represents the observation matrix. This indicates observation noise.

[0113] S2: Calculate the initial mixed state value and initial mixed covariance of the robust mode and adaptive mode based on the pre-set mode transition probability and initial mode probability of the integrated navigation model. Then, make predictions based on the initial mixed state value and initial mixed covariance of the robust mode and adaptive mode to obtain the observation residual and observation residual covariance of the corresponding mode.

[0114] Specifically, the method for calculating the mixed initial values ​​and mixed initial covariance under the robust mode and adaptive mode is as follows:

[0115] S211: Calculation mode according to formula (3) From pattern Mixed weights:

[0116] (3);

[0117] in: Representation pattern From pattern Mixed weights, Indicates from pattern Currently in mode The transition probability, express The filter is in mode at any given time. The posterior pattern probability, This represents all possible source modes of the filter from the previous time step. Indicates from pattern Currently in mode The transition probability, express The filter is in mode at any given time. posterior pattern probability;

[0118] When the pattern When this occurs, it indicates that the filter is in adaptive mode, performing adaptive filtering based on the Sage-Husa strategy; mode When the value is 0, it indicates that the filter is in robust mode and performs robust filtering based on the Huber function.

[0119] S212: Pattern-based From pattern The mixed weights are constructed according to equation (4). exist Initial values ​​of mixed states at time: (4);

[0120] in: Representation pattern exist The mixed initial values ​​of the state at time 10:00. Representation pattern exist The posterior state estimate vector at time 1;

[0121] S213: Pattern-based exist The initial mixed state value at time t is calculated according to equation (5). exist Mixed initial covariance at time: (5);

[0122] in: Representation pattern exist The mixed initial covariance at time 1, Representation pattern exist The posterior covariance at time 1, This indicates the matrix transpose.

[0123] This step takes into account both the inherent uncertainties of each model and the additional uncertainties arising from inconsistencies in estimation between models.

[0124] Overall, the mixed initial covariance It can be divided into two items before and after the plus sign, representing the uncertainty of the pattern itself and the additional uncertainty generated by the conversion between patterns, respectively. Representation pattern exist The posterior covariance at time; Representation pattern exist The posterior state estimate vector at time 1; The mean squared deviation compensation term between modes is represented as the outer product of the difference vectors.

[0125] Specifically, the method for predicting the observation residuals and observation residual covariance of the corresponding modes based on the mixed initial values ​​and mixed initial covariance of the states under both robust and adaptive modes is as follows:

[0126] S214: Change the mode exist The initial mixed state value and initial mixed covariance at time step are advanced by one sampling time, and the pattern is obtained according to equation (6). exist Prior state vector and prior covariance at each time step: (6);

[0127] in: Representation pattern exist Prior state vector at time step This indicates that a starlight and inertial navigation system based on the Lie group description is in... The state transition matrix at time t, Representation pattern exist Prior covariance at time, Representation pattern exist The system process noise covariance matrix at time t;

[0128] S215: Based on current observation vectors and patterns exist Given the prior state vector and prior covariance at time step, calculate the observation residuals and observation residual covariance according to equation (7): (7);

[0129] in: express Time Mode The observation residuals express The observation vector at time t, express The observation matrix at time, express Time Mode The observed residual covariance express The transpose of the observation matrix at time t. express Time Mode The observation noise covariance matrix.

[0130] Time Mode Observation residuals Used to directly reflect the deviation between observations and the model. This can be obtained from the previously constructed integrated navigation model.

[0131] Time Mode Observational residual covariance It combines the components of the predicted state uncertainty transmitted through the observation matrix with the magnitude of the observation noise itself.

[0132] The larger, The smaller the value, the greater the correction the fusion filter should provide; conversely, the larger the value, the more conservative the correction should be.

[0133] S3: Based on the observation residuals and observation residual covariance of the corresponding modes, the adaptive mode combines the Sage-Husa adaptive strategy, and the robust mode combines the Huber robust estimation strategy to perform parallel sub-filtering of the two modes. When the adaptive mode sub-filters, the Kalman gain is calculated based on the measurement noise covariance matrix estimated at the previous time step, and the posterior state and covariance of the adaptive mode are updated. When the robust mode sub-filters, the normalized residuals of each observation channel are calculated, the weights of each observation channel are calculated according to the Huber cost function and a weight matrix is ​​constructed, the equivalent weighted observation residual covariance is calculated using the weight matrix, and then the robust Kalman gain is calculated to update the posterior state and covariance of the robust mode.

[0134] Specifically, the methods for adaptive mode sub-filtering and updating the posterior state and covariance are as follows:

[0135] S311: The adaptive Kalman gain is calculated based on the estimated value of the observation noise covariance matrix according to equation (8):

[0136] (8);

[0137] in: Indicates in Adaptive Kalman gain at time step Indicates adaptive mode in Prior covariance at time, Indicates adaptive mode in The estimated value of the observation noise covariance matrix at time t;

[0138] Here Unlike the observation noise covariance used by the standard KF in calculating the Kalman gain, this substitution allows the gain to automatically reflect the intensity of recent observation noise.

[0139] S312: Mode based on adaptive Kalman gain according to equation (9) exist Posterior state and covariance update at time step:

[0140] (9);

[0141] in: Indicates adaptive mode in The posterior state vector at time t. Indicates adaptive mode in Prior state vector at time step express Observation residuals in time-adaptive mode Indicates adaptive mode in The posterior covariance at time 1, Represents the identity matrix;

[0142] S313: Introduce an adaptive forgetting factor and update it according to Equation (10) using an exponentially weighted average. Estimates of the observation noise covariance matrix at time step:

[0143] (10); where: Indicates adaptive mode in The estimated value of the observation noise covariance matrix at time t. Represents the adaptive forgetting factor. express The transpose matrix of the observation residuals in the time-adaptive mode.

[0144] Adaptive forgetting factor Commonly used Generally, the larger this value is, the faster new data is adopted.

[0145] The square of the current residual is used as the total residual statistic. Then, the influence caused by state uncertainty is subtracted from it to estimate the pure observation noise component. An adaptive forgetting factor is introduced and the data is updated using an exponentially weighted average. Estimating the observation noise covariance at any given time can yield a more accurate observation noise covariance, which is beneficial for achieving high-precision astronomical attitude determination.

[0146] Specifically, the methods for robust mode sub-filtering and updating the posterior state and covariance are as follows:

[0147] S321: Implement robust mode in The first moment The residual components of each observation channel are normalized according to their prediction variance using equation (11) to obtain the robust model in... Normalized residuals at time:

[0148] (11);

[0149] in: Indicates robust mode in The first moment Normalized residuals of each observation channel, Indicates robust mode in Normalized residuals at time, express The observation residuals of the time-robust model at the 1st moment The components of each observation channel, express The diagonal component of the covariance of the observation residuals of the time-robust model. This represents diagonal matrix operations. Indicates the number of observation channels;

[0150] Normalized residuals are an important reference for robust filtering in determining the field value of the measurement.

[0151] S322: Calculate the weight of each channel based on the Huber cost function according to equation (12): (12);

[0152] in: Indicates robust mode in The first moment The weights of each observation channel, The threshold representing the Huber cost function; generally satisfies Experience values ​​are often used. .

[0153] S323: Introduce a robust smoothing factor. According to formula (13), the weights of each channel are smoothed by a small exponential smoothing to obtain the smoothing weights of each observation channel in the robust mode.

[0154] (13);

[0155] in: Indicates robust mode in The first moment Smoothing weights for each observation channel, Indicates robust mode in The first moment The weights of each observation channel, Indicates the robust smoothing factor;

[0156] Introducing a robust smoothing factor and using small-amplitude exponential smoothing can avoid sudden fluctuations in weights with measurement field values, which could affect robustness and filtering performance.

[0157] Robust smoothing factor satisfies .

[0158] S324: Assemble the smoothing weights of all robust mode observation channels into a weight diagonal matrix according to equation (14):

[0159] (14);

[0160] in: Represents the robust mode weight diagonal matrix;

[0161] S325: Based on the robust mode weight diagonal matrix, calculate the equivalent weighted observation residual covariance according to equation (15), and based on the equivalent weighted observation residual covariance, calculate the robust Kalman gain according to equation (16):

[0162] (15);

[0163] (16);

[0164] in: Indicates robust mode in Equivalent weighted observation residual covariance at time step express Observation residual covariance of time-robust mode Indicates robust Kalman gain. Indicates robust mode in Prior covariance at time, Indicates robust mode in The equivalent weighted observation residual covariance inverse matrix at time points;

[0165] Using the equivalent weighted observation residual covariance to calculate the robust Kalman gain can effectively mitigate the impact of large residual channels on state updates. Intuitively, robust filtering automatically reduces the gain when faced with outlier measurements, minimizing the possibility of the state being skewed by abnormal observations, while maintaining high efficiency when observations are normal.

[0166] S326: Based on robust Kalman gain, the robust mode posterior state and covariance are updated according to equation (17):

[0167] (17);

[0168] in: Indicates robust mode in The posterior state vector at time t. Indicates robust mode in Prior state vector at time step Indicates robust mode in The posterior covariance at time 1, express The observation noise covariance matrix of the time-robust model. This represents the robust Kalman gain transpose matrix.

[0169] This represents the residual after robust weighted correction. By introducing this robustly weighted residual into the standard Kalman filter, the impact of outlier observations is automatically reduced, making the estimation results more robust. Under the Huber weighting mechanism, the filter effectively suppresses error propagation and maintains stable convergence performance when outlier measurements exist.

[0170] S4: Update the mode probabilities based on the updated posterior states and covariances under the two modes;

[0171] Furthermore, the method for updating the pattern probability in step S4 is as follows:

[0172] S411: According to equation (18), map the posterior mode probability of the previous time step to the prior mode probability of the current time step using the preset transition probability:

[0173] (18);

[0174] in: express The filter is in mode at any given time. The prior pattern probability, express The filter is in mode at any given time. posterior pattern probability;

[0175] Prior mode probability reflects the transition characteristics of the mode itself. Intuitively, it characterizes how the mode probability will evolve due to the filter's internal switching mechanism if there is no observation information.

[0176] S412: Calculate the pattern according to equation (19) Likelihood of the next observation vector:

[0177] (19);

[0178] in: express Always in mode The likelihood of the observed vector. Represents pi (π). This represents the operation of the exponential function. Indicates the dimension of the observation vector;

[0179] If equation (19) is in robust filtering mode, It should be replaced with equivalent weighted covariance. .

[0180] S413: According to Bayes' theorem, based on equation (20), the current pattern is... The likelihood of the observed vector is multiplied by the prior mode probability and normalized to obtain the current filter mode. The posterior pattern probability is used to update the pattern probability:

[0181] (20);

[0182] in: express The filter is in mode at any given time. The posterior pattern probability.

[0183] for Should meet This step combines model priors with observational evidence to obtain normalized posterior model probabilities. Intuitively, it achieves a Bayesian update: the prior reveals the tendency in the absence of observations, while the likelihood expresses the support of observations for each model; multiplying and normalizing the two yields a new probability. In this way, a soft switch balancing "model inertia" and "observation-driven" approaches is achieved through the interactive multi-model (IMM) framework.

[0184] S5: Based on the updated mode probabilities, the updated posterior states and covariances of the two modes are fused to obtain the fused output of the parallel sub-filter. The fused output of the parallel sub-filter is used as the final estimate of the starlight and inertial integrated navigation system based on the Lie group description. The output includes astronomical attitude determination results including azimuth, pitch and roll angles.

[0185] Furthermore, in step S5, the updated posterior states and covariances under the two modes are fused according to equation (21) to obtain the fused output of the parallel sub-filter:

[0186] (twenty one);

[0187] in: This represents the fused posterior state vector output of the integrated navigation model. Representation pattern exist The posterior state vector at time t. This represents the fusion posterior covariance output of the integrated navigation model. Representation pattern exist Posterior covariance at time step.

[0188] This step uses the posterior mode probability to perform a weighted average of the posterior estimates of each mode, thereby obtaining the overall estimate of the system. This soft fusion retains the advantages of multi-model parallelism: it takes into account multiple hypotheses when there is uncertainty, and it favors a more suitable single mode when the posterior mode probability is high.

[0189] The weighted average of the intra-mode covariance reflects the internal uncertainty of each filtering mode, while the inter-mode root mean square error reflects the inconsistency between estimates from different modes. Combining these two results in the fused posterior covariance of the integrated navigation model, thus providing a complete quantification of the overall estimation uncertainty. This is crucial for consistency testing and subsequent filter decisions.

[0190] A robust adaptive fusion filtering astronomical attitude determination system is provided for executing a robust adaptive fusion filtering astronomical attitude determination method as described in any of the above, comprising an integrated navigation model construction module, a robust adaptive parallel sub-filter module, a mode probability update module, and a fusion output module.

[0191] The integrated navigation model construction module is used to construct an integrated navigation model that includes the inertial navigation error state equation of the starlight and inertial integrated navigation system described by Lie group and the observation equation of the starlight and inertial integrated navigation system described by Lie group.

[0192] The robust adaptive parallel sub-filter module is used to calculate the observation residuals and observation residual covariance of the corresponding mode; based on the observation residuals and observation residual covariance of the corresponding mode, robust mode and adaptive mode parallel sub-filters are performed, and the posterior state and covariance of the two modes are updated.

[0193] The mode probability update module is used to update the mode probability based on the updated posterior state and covariance under the two modes;

[0194] The fusion output module is used to fuse the updated posterior states and covariances under the two modes to obtain the fusion output of the parallel sub-filter. The fusion output of the parallel sub-filter is used as the final estimate of the starlight and inertial integrated navigation system based on the Lie group description, and the output includes astronomical attitude determination results including azimuth, pitch and roll angles.

[0195] This invention establishes a geometrically consistent starlight and inertial navigation system model based on Lie group description. Addressing the issue of inconsistent variance estimation in traditional EKF systems in nonlinear systems, this method reconstructs the starlight and inertial navigation model based on Lie group and Lie algebra theory, establishing the inertial navigation error state equation. Based on the gyroscope drift correction method, and using the difference between the INS and CNS output attitude angles as the observable, the system observation equation is established. By maintaining the orthogonality constraint of the rotation matrix through Lie group exponential mapping, consistency between state propagation and error update is achieved. This solves the long-term error accumulation problem caused by EKF linearization at the modeling level, fundamentally ensuring the geometric consistency of state estimation on the rotating manifold, overcoming the theoretical deficiencies of traditional EKF, effectively solving the problem of inconsistent state variance estimation for unobservable errors, and significantly improving the system's stability and accuracy.

[0196] Furthermore, to address the issues of non-Gaussian noise and observation outliers in complex dynamic environments, this paper integrates the Sage-Husa adaptive strategy with a robust filtering framework, introducing an adaptive forgetting factor and a robust smoothing factor to construct a robust adaptive fusion filtering algorithm based on Lie group description. The algorithm first mixes the mode estimates based on the mode transition probability and the posterior of the previous time step to obtain the initial mixture value for each mode. Then, it uses the Sage-Husa adaptive strategy in parallel to recursively estimate the observation noise covariance online, simultaneously calculating the normalized residuals and constructing channel weights using the Huber function to correct the observation covariance. Under each mode, it calculates the gain based on the corrected covariance and updates the state and covariance. Finally, it updates the mode posterior probability using the observation likelihood of each mode. Finally, it weights and fuses the posterior probabilities of each mode according to the posterior probability to obtain the final estimate. This algorithm can adjust the noise covariance in real time in dynamically changing environments, and has strong anti-interference and adaptive capabilities. It effectively overcomes the dependence of traditional Kalman filters on noise statistical assumptions, and extends the classical optimal estimation theory to more challenging non-Gaussian and time-varying noise environments. It effectively solves the problems of complex noise characteristics and measurement field values ​​in complex dynamic environments, and significantly improves the robustness and accuracy of starlight and inertial navigation in complex dynamic environments, thereby achieving high-precision astronomical attitude determination.

[0197] Spacecraft simulation experiments were conducted in MATLAB, with both process noise and observation noise set to white noise. The standard deviation of process noise was set to 10″, and the standard deviation of observation noise to 100″, linearly increasing to 200″ over the entire simulation time T=500s. This progressively increasing noise model simulated the changes in observation noise of the star sensor in a complex measurement environment. The initial standard deviation of attitude estimation deviation was set to 10″. Simultaneously, measurement field values ​​with an amplitude 15 times the original noise intensity were added at simulation times t=150s, 300s, and 450s to simulate abnormal situations encountered by the star sensor.

[0198] The schematic diagram of the actual attitude angles of the simulated spacecraft during motion is shown below. Figure 2a As shown in the diagram, the simulated spacecraft motion star sensor measurement values ​​are illustrated below. Figure 2b As shown.

[0199] After generating the simulated data, the observed data were processed using the traditional EKF method, the robust KF method, and the proposed method, respectively. The actual values, observed values, traditional EKF estimates, robust KF estimates, and proposed method estimates of the simulated attitude angles were plotted, as shown in the figure below. Figures 3 to 5 As shown.

[0200] Depend on Figures 3 to 5 It can be seen that after processing with three filters, the smoothness of the attitude angle observation values ​​is significantly improved, the error with the actual values ​​is significantly reduced, the three filtered estimated value curves are closer to the actual value curves, and the robustness and adaptability of this method are more significant when facing abnormal noise and measurement field interference.

[0201] To more intuitively evaluate the estimation accuracy of the traditional EKF method, the robust KF method, and our proposed method, we plotted the estimation errors of the three attitude angles after applying the three filtering methods, as shown in the following figure. Figure 6 As shown. From Figure 6 It can be seen that the estimation errors of the measurement outliers at t=150s, 300s, and 450s by the robust KF method and this method are significantly smaller than those by the traditional EKF method. Furthermore, the adaptability of this method to changing noise and its robustness in dealing with outliers are better than those of the robust KF method throughout the simulation time.

[0202] The RMS error accuracy evaluation of the three methods during the simulation time is shown in Table 1:

[0203] Table 1

[0204]

[0205] In summary, the robust adaptive fusion filtering astronomical attitude determination method and system provided by this invention has strong adaptability, robustness and accuracy when dealing with complex noise and measurement field values, and can achieve high-precision astronomical attitude determination of starlight and inertial navigation in complex dynamic environments.

[0206] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A robust adaptive fusion filtering astronomical attitude determination method, characterized in that: Includes the following steps: S1: Construct a combined navigation model that includes the state equation of the inertial navigation error of the starlight and inertial navigation system based on the description of Lie groups and the observation equation of the starlight and inertial navigation system based on the description of Lie groups; S2: Based on the pre-set mode transition probability and initial mode probability of the integrated navigation model, calculate the initial mixed state value and initial mixed covariance of the robust mode and adaptive mode. Then, make predictions based on the initial mixed state value and initial mixed covariance of the robust mode and adaptive mode to obtain the observation residual and observation residual covariance of the corresponding mode. S3: Based on the observation residuals and observation residual covariance of the corresponding modes, the adaptive mode combines the Sage-Husa adaptive strategy, and the robust mode combines the Huber robust estimation strategy to perform parallel sub-filtering of the two modes. When the adaptive mode sub-filters, the Kalman gain is calculated based on the measurement noise covariance matrix estimated at the previous time step, and the posterior state and covariance of the adaptive mode are updated. When the robust mode sub-filters, the normalized residuals of each observation channel are calculated, the weights of each observation channel are calculated according to the Huber cost function and a weight matrix is ​​constructed, the equivalent weighted observation residual covariance is calculated using the weight matrix, and then the robust Kalman gain is calculated to update the posterior state and covariance of the robust mode. S4: Update the mode probabilities based on the updated posterior states and covariances under the two modes; S5: Based on the updated mode probabilities, the updated posterior states and covariances of the two modes are fused to obtain the fused output of the parallel sub-filter. The fused output of the parallel sub-filter is used as the final estimate of the starlight and inertial integrated navigation system based on the Lie group description. The output includes astronomical attitude determination results including azimuth, pitch and roll angles.

2. The robust adaptive fusion filtering astronomical attitude determination method according to claim 1, characterized in that: In step S1, the state equation for the inertial navigation error of the starlight and inertial integrated navigation system described by the Lie group is Equation (1): （1）； in: The derivative of the inertial navigation error state vector. Represents the system state transition matrix. This represents the inertial navigation error state vector. Represents the noise transition matrix. This represents the process noise vector.

3. The robust adaptive fusion filtering astronomical attitude determination method according to claim 1, characterized in that: In step S1, the observation equation of the starlight and inertial combined navigation system described by the Lie group is Equation (2): （2）； in: Represents the observation vector. Represents the observation matrix. This represents the attitude error angle transformation matrix. This indicates observation noise.

4. The robust adaptive fusion filtering astronomical attitude determination method according to claim 1, characterized in that: The method for calculating the mixed initial value and mixed initial covariance of the robust mode and adaptive mode in step S2 is as follows: S211: Calculation mode according to formula (3) From pattern Mixed weights: （3）； in: Representation pattern From pattern Mixed weights, Indicates from pattern Currently in mode The transition probability, express The filter is in mode at any given time. The posterior pattern probability, This represents all possible source modes of the filter from the previous time step. Indicates from pattern Currently in mode The transition probability, express The filter is in mode at any given time. posterior pattern probability; S212: Pattern-based From pattern The mixed weights are constructed according to equation (4). exist Initial state mixture at time: (4); in: Representation pattern exist The mixed initial values ​​of the state at time 10:

00. Representation pattern exist The posterior state estimate vector at time 1; S213: Pattern-based exist The initial mixed state value at time t is calculated according to equation (5). exist Mixed initial covariance at time: (5); in: Representation pattern exist The mixed initial covariance at time 1, Representation pattern exist The posterior covariance at time 1, This indicates the matrix transpose.

5. A robust adaptive fusion filtering astronomical attitude determination method according to claim 4, characterized in that: In step S2, prediction is performed based on the mixed initial values ​​and mixed initial covariance of the states under both robust and adaptive modes. The method for obtaining the observation residuals and observation residual covariance of the corresponding modes is as follows: S214: Change the mode exist The initial mixed state value and initial mixed covariance at time step are advanced by one sampling time, and the pattern is obtained according to equation (6). exist Prior state vector and prior covariance at each time step: (6); in: Representation pattern exist Prior state vector at time step This indicates that a starlight and inertial navigation system based on the Lie group description is in... The state transition matrix at time t, Representation pattern exist Prior covariance at time, Representation pattern exist The system process noise covariance matrix at time t; S215: Based on current observation vectors and patterns exist Given the prior state vector and prior covariance at time step, calculate the observation residuals and observation residual covariance according to equation (7): (7); in: express Time Mode The observation residuals express The observation vector at time t, express The observation matrix at time, express Time Mode The observed residual covariance express The transpose of the observation matrix at time t. express Time Mode The observation noise covariance matrix.

6. The robust adaptive fusion filtering astronomical attitude determination method according to claim 5, characterized in that: The method for adaptive mode sub-filtering and updating the posterior state and covariance in step S3 is as follows: S311: The adaptive Kalman gain is calculated based on the estimated value of the observation noise covariance matrix according to equation (8): （8）； in: Indicates in Adaptive Kalman gain at time step Indicates adaptive mode in Prior covariance at time, Indicates adaptive mode in The estimated value of the observation noise covariance matrix at time t; S312: Mode based on adaptive Kalman gain according to equation (9) exist Posterior state and covariance update at time step: （9）； in: Indicates adaptive mode in The posterior state vector at time t. Indicates adaptive mode in Prior state vector at time step express Observation residuals in time-adaptive mode Indicates adaptive mode in The posterior covariance at time 1, Represents the identity matrix; S313: Introduce an adaptive forgetting factor and update it according to Equation (10) using an exponentially weighted average. Estimates of the observation noise covariance matrix at time step: (10); where: Indicates adaptive mode in The estimated value of the observation noise covariance matrix at time t. Represents the adaptive forgetting factor. express The transpose matrix of the observation residuals in the time-adaptive mode.

7. A robust adaptive fusion filtering astronomical attitude determination method according to claim 5, characterized in that: The method for robust mode sub-filtering and updating the posterior state and covariance in step S3 is as follows: S321: Implement robust mode in The first moment The residual components of each observation channel are normalized according to their prediction variance using equation (11) to obtain the robust model in Normalized residuals at time: （11）； in: Indicates robust mode in The first moment Normalized residuals of each observation channel, Indicates robust mode in Normalized residuals at time, express The observation residuals of the time-robust model at the 1st moment The components of each observation channel, express The diagonal component of the covariance of the observation residuals of the time-robust model. This represents diagonal matrix operations. Indicates the number of observation channels; S322: Calculate the weight of each channel based on the Huber cost function according to equation (12): (12); in: Indicates robust mode in The first moment The weights of each observation channel, The threshold representing the Huber cost function; S323: Introduce a robust smoothing factor. According to formula (13), the weights of each channel are smoothed by a small exponential smoothing to obtain the smoothing weights of each observation channel in the robust mode. （13）； in: Indicates robust mode in The first moment Smoothing weights for each observation channel, Indicates robust mode in The first moment The weights of each observation channel, Indicates the robust smoothing factor; S324: Assemble the smoothing weights of all robust mode observation channels into a weight diagonal matrix according to equation (14): （14）； in: Represents the robust mode weight diagonal matrix; S325: Based on the robust mode weight diagonal matrix, calculate the equivalent weighted observation residual covariance according to equation (15), and based on the equivalent weighted observation residual covariance, calculate the robust Kalman gain according to equation (16): （15）； （16）； in: Indicates robust mode in Equivalent weighted observation residual covariance at time step express Observation residual covariance of time-robust mode Indicates robust Kalman gain. Indicates robust mode in Prior covariance at time, Indicates robust mode in The equivalent weighted observation residual covariance inverse matrix at time points; S326: Based on robust Kalman gain, the robust mode posterior state and covariance are updated according to equation (17): （17）； in: Indicates robust mode in The posterior state vector at time t. Indicates robust mode in Prior state vector at time step Indicates robust mode in The posterior covariance at time 1, express The observation noise covariance matrix of the time-robust model. This represents the robust Kalman gain transpose matrix.

8. The robust adaptive fusion filtering astronomical attitude determination method according to claim 7, characterized in that: The method for updating the pattern probability in step S4 is as follows: S411: According to equation (18), map the posterior mode probability of the previous time step to the prior mode probability of the current time step using the preset transition probability: （18）； in: express The filter is in mode at any given time. The prior pattern probability, express The filter is in mode at any given time. posterior pattern probability; S412: Calculate the pattern according to equation (19) Likelihood of the next observation vector: （19）； in: express Always in mode The likelihood of the observed vector. Represents pi (π). This represents the operation of the exponential function. Indicates the dimension of the observation vector; S413: According to Bayes' theorem, based on equation (20), the current pattern is... The likelihood of the observed vector is multiplied by the prior mode probability and normalized to obtain the current filter mode. The posterior pattern probability is used to update the pattern probability: （20）； in: express The filter is in mode at any given time. The posterior pattern probability.

9. A robust adaptive fusion filtering astronomical attitude determination method according to claim 8, characterized in that: In step S5, the updated posterior states and covariances under the two modes are fused according to equation (21) to obtain the fused output of the integrated navigation model: （21）； in: This represents the fused posterior state vector output of the integrated navigation model. Representation pattern exist The posterior state vector at time t. This represents the fusion posterior covariance output of the integrated navigation model. Representation pattern exist The posterior covariance at time.

10. A robust adaptive fusion filtering astronomical attitude determination system, used to execute a robust adaptive fusion filtering astronomical attitude determination method as described in any one of claims 1 to 9, characterized in that, It includes a combined navigation model construction module, a robust adaptive parallel sub-filter module, a mode probability update module, and a fusion output module; The integrated navigation model construction module is used to construct an integrated navigation model that includes the inertial navigation error state equation of the starlight and inertial integrated navigation system described by Lie group and the observation equation of the starlight and inertial integrated navigation system described by Lie group. The robust adaptive parallel sub-filter module is used to calculate the observation residuals and observation residual covariance of the corresponding mode; based on the observation residuals and observation residual covariance of the corresponding mode, robust mode and adaptive mode parallel sub-filters are performed, and the posterior state and covariance of the two modes are updated. The mode probability update module is used to update the mode probability based on the updated posterior state and covariance under the two modes; The fusion output module is used to fuse the updated posterior states and covariances under the two modes to obtain the fusion output of the parallel sub-filter. The fusion output of the parallel sub-filter is used as the final estimate of the starlight and inertial integrated navigation system based on the Lie group description, and the output includes astronomical attitude determination results including azimuth, pitch and roll angles.

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