An Adaptive Relative Navigation Method for Satellite Formations Based on Variable Universe Fuzzy Logic

Abstract

The present invention discloses a satellite formation adaptive relative navigation method based on variable universe fuzzy logic. The method includes the following steps: establishing a state model of the satellite formation relative navigation system by using the inter-satellite linear Clohessy-Wilshire relative dynamics; establishing an inter-satellite non-linear measurement model of the system through the geometric relationship between the inter-satellite relative position and the measurement; establishing an uncertain system noise and measurement noise interference model in the satellite formation relative navigation system; after setting the corresponding parameters of the filter and the variable universe adaptive fuzzy logic system, estimating the three-axis relative motion state by using the variable universe fuzzy logic adaptive EKF to complete the satellite formation adaptive relative navigation. Without significantly increasing the computational amount, the present invention makes up for the disadvantages of the existing satellite formation adaptive relative navigation methods, such as the decrease in relative navigation accuracy and the slow convergence speed under uncertain noise interference.

Description

Technical Field

[0001] The present invention belongs to the technical field of relative navigation in space, and particularly relates to a satellite formation adaptive relative navigation method based on variable universe fuzzy logic. Background Art

[0002] A high-precision and high-reliability satellite formation relative navigation method is a basic prerequisite for a satellite formation to perform various operations such as formation configuration maintenance and reconstruction, and to ensure the smooth implementation of complex space missions, which has received unprecedented attention from major space powers.

[0003] Relative navigation generally refers to the process of determining the relative position and velocity between satellite formations: first, using the inter-satellite dynamics model for prediction; then, using the relative navigation sensors between satellites to obtain measurement information such as relative distance and line of sight; finally, updating the relative state in the filtering algorithm. However, traditional navigation filtering algorithms need to assume in advance that the system and measurement noise covariance matrices can be accurately known. Otherwise, inaccurate noise covariance matrices often lead to a significant decline in filtering performance, or even cause filtering divergence. In practical applications, the operating space environment of satellite formations and the interference received by relative navigation sensors are complex and variable, and it is almost impossible to obtain an accurate noise covariance matrix. Therefore, how to achieve reliable, stable and high-precision satellite formation adaptive relative navigation under unknown or time-varying interference of the noise covariance matrix is an urgent issue to be studied.

[0004] At present, the adaptive filtering methods in satellite formation relative navigation can be mainly divided into four categories: correlation function method, maximum likelihood estimation method, Bayesian method, and covariance matching method. The literature [ZANNI L, LE BOUDEC J Y, CHERKAOUI R, et al. A Prediction-Error Covariance Estimator for Adaptive Kalman Filtering in Step-Varying Processes: Application to Power-System State Estimation[J]. IEEE Transactions on Control Systems Technology, 2017, 25(5): 1683-1697.] considers the autocorrelation function of the filter output or innovation by analyzing the historical data of multiple past measurement residuals, but it can only be applied to time-invariant linear Kalman filters, with poor real-time performance and a narrow application range. The literature [HU G, GAO B, ZHONG Y, et al. Unscented kalman filter with process noise covariance estimation for vehicular ins / gps integration system[J]. Information Fusion, 2020, 64: 194-204.] transforms the adaptive filtering problem into the problem of differentiating the covariance matrix of measurement residuals with respect to unknown noise parameters under the maximum likelihood criterion, but it requires a large data window, with a slow filtering convergence speed and is prone to divergence. The literature [Yulong Huang, Yonggang Zhang, Zhemin Wu, et al. A Novel Adaptive Kalman Filter With Inaccurate Process and Measurement Noise Covariance Matrices[J]. IEEE Transactions on Automatic Control, 2018, 63(2): 594-601.] estimates the unknown noise covariance matrix while approximately solving the state to be estimated locally by choosing a suitable prior distribution for the noise covariance matrix, but this type of method requires too many assumptions, has certain limitations, and a large amount of computation.The last covariance matching method is widely used in adaptive filters. Its basic idea is how to adaptively adjust the noise covariance matrix according to the matching degree between the theoretical innovation and the actual innovation. Compared with the first three types of methods, it has low computational complexity and good adaptability. However, the key lies in how to design the adaptive adjustment method of the noise covariance matrix, which is also the current research focus.

[0005] Existing literature has designed many adaptive adjustment methods for the noise covariance matrix based on various methods such as optimization methods, neural networks, and fuzzy logic systems under the basic idea of covariance matching. Among them, by using the fuzzy logic system, an adaptive law can be directly constructed based on the human intuition of understanding the EKF principle, which can very simply and efficiently express expert experience without introducing various assumptions, and has good robustness and adaptability. In the literature [Al-sudany H N, Lantos B. Comparison of Adaptive Fuzzy EKF and Adaptive Fuzzy UKF for State Estimation of UAVs Using Sensor Fusion[J]. Periodica Polytechnica Electrical Engineering and Computer Science, 2022, 66(3): 215-266.], the fuzzy logic system is introduced into EKF and UKF respectively, and a multiplicative factor form is used to adjust the noise covariance matrix parameters. Finally, it is tested in actual navigation data, and better relative navigation filtering results are obtained compared with the original method. Considering the limitations of the multiplicative factor adjustment, the literature [Fraser C T, Ulrich S. Adaptive extended Kalman filtering strategies for spacecraft formation relative navigation[J]. Acta Astronautica, 2021, 178: 700-721.] adopts a multiplicative factor adjustment form, and realizes the adaption of noise parameters by designing multiple fuzzy logic systems to adjust the noise covariance matrix, further reducing the computational complexity. However, the fuzzy logic systems adopted in the above works are all fixed-universe fuzzy logic systems, and the selection of parameters such as the system universe range, membership function, and defuzzification method depends very much on the designer's own experience. Summary of the Invention

[0006] Aiming at the deficiencies of the existing technology, the purpose of the present invention is to provide a satellite formation adaptive relative navigation method based on variable universe fuzzy logic, so as to solve the problem that the relative navigation accuracy is significantly reduced or even diverges under the uncertain interference of system and measurement noise in the existing methods.

[0007] To achieve the above object, the technical solution adopted by the present invention is as follows:

[0008] A satellite formation adaptive relative navigation method based on variable universe fuzzy logic of the present invention includes the following steps:

[0009] Step 1: Use the Clohessy-Wilshire relative motion equation linearized between satellites to establish a system state model for the recursion of relative states.

[0010] Step 2: Establish a measurement model for the satellite formation adaptive relative navigation system, and substitute the relative states between satellites into the measurement model to obtain the relative distance and line-of-sight angle measurements between satellites.

[0011] Step 3: Establish an uncertain system noise and measurement noise interference model in the system.

[0012] Step 4: Set the relevant parameters of the filter and the relevant parameters of the variable universe adaptive fuzzy logic system.

[0013] Step 5: Solve the three-dimensional position and velocity state information of the slave satellite relative to the master satellite through the variable universe fuzzy logic adaptive EKF algorithm to complete the satellite formation adaptive relative navigation.

[0014] Furthermore, in Step 1, the relative motion state of the satellite formation evolves according to the following state model of the satellite formation adaptive relative navigation system:

[0015] ;

[0016] Wherein, is the system state variable, which is the three-axis components of the relative position and velocity of the slave satellite relative to the master satellite defined in the master satellite orbit coordinate system; is the orbital angular velocity of the master satellite; is the sampling time interval; is the system noise, expressed as the unmodeled perturbation interference.

[0017] Furthermore, in Step 2, establish the following measurement model for the satellite formation adaptive relative navigation system to obtain the relative distance and line-of-sight angle measurements between satellites:

[0018] ;

[0019] Wherein, the relative distance between satellites; is the pitch angle between satellites; is the azimuth angle between satellites; is the measurement noise.

[0020] Furthermore, in step three, the established uncertain system noise and measurement noise interference models are as follows:

[0021] ;

[0022] ;

[0023] wherein, is the probability; and are the actual system noise covariance matrix and the actual measurement noise covariance matrix respectively, and satisfy:

[0024] ;

[0025] wherein, is the th filtering time; is the total filtering time; is the nominal system noise covariance matrix; is the nominal measurement noise covariance matrix.

[0026] Furthermore, in step four, the output dynamic adjustment scaling factors and of the variable universe adaptive fuzzy logic system are both 1, , and the input dynamic adjustment factors are as follows:

[0027] ;

[0028] wherein, ; ; The universe of discourse is divided into five levels, and the five levels are negative large (NB), negative medium (NM), zero (ZE), positive medium (PM), and positive large (PB); The negative large (NB) and positive large (PB) are set as S-type membership functions, and the remaining levels adopt Gaussian-type membership functions; The Mamdani inference method is selected as the fuzzy inference mechanism, the centroid method is selected as the defuzzification method, and according to the basic idea of the covariance matching method that the theoretical innovation and the actual innovation need to match, the following IF-THEN form of fuzzy inference rules are designed :

[0029] .

[0030] Furthermore, in step five, the adjustment coefficients and Online adjustment of the estimated measurement noise covariance matrix and the estimated prediction error covariance matrix :

[0031] ;

[0032] ;

[0033] wherein, and are proportionality coefficients and satisfy and , ensuring and are positive definite; finally, the innovation and the Kalman gain are recalculated to achieve relative state update and state estimation covariance matrix update, and the relative position and velocity of the satellite formation are calculated from the estimated relative state, completing the satellite formation adaptive relative navigation:

[0034] ;

[0035] ;

[0036] ;

[0037] ;

[0038] wherein, is the Jacobian matrix of the measurement model , is the measurement residual.

[0039] Compared with the prior art, the beneficial effects of the present invention:

[0040] The present invention introduces a variable universe adaptive fuzzy logic system and constructs a new satellite formation adaptive relative navigation method based on the EKF. This method calculates the matching degree and divergence obtained from the theoretical innovation and the actual innovation in the filtering process, and then designs multiple single-input single-output variable universe adaptive fuzzy logic systems according to the basic idea of covariance matching for online adjustment and estimation of the prediction error covariance matrix and the measurement noise covariance matrix, so as to eliminate the gap between the theoretical innovation and the actual innovation to improve the filtering performance of the EKF and achieve satellite formation adaptive relative navigation. Compared with the existing adaptive relative navigation methods, without significantly increasing the computational complexity, the relative navigation accuracy and the adaptability of the system are effectively improved under the interference of uncertain system noise and measurement noise, and it can meet the requirements of satellite formation tasks. The overall idea of the invention is novel, with strong innovation, and has high engineering application prospects for future satellite formation tasks. Description of the Drawings

[0041] Figure 1 is the flowchart of the steps for the satellite formation adaptive relative navigation of the present invention;

[0042] Figure 2 is the schematic diagram of the simulation scenario for the satellite formation adaptive relative navigation of the present invention;

[0043] Figure 3 is the schematic diagram of the measurement model for the satellite formation adaptive relative navigation of the present invention;

[0044] Figure 4 is the schematic diagram of the principle of the variable universe adaptive fuzzy logic system of the present invention;

[0045] Figure 5 is the curve graph of the input-output membership function of the present invention;

[0046] Figure 6 is the three-axis relative position error graph of the satellite formation adaptive relative navigation of the present invention;

[0047] Figure 7 is the three-axis relative velocity error graph of the satellite formation adaptive relative navigation of the present invention;

[0048] Figure 8 is the three-axis position root mean square error graph of the satellite formation adaptive relative navigation of the present invention;

[0049] Figure 9 is the three-axis velocity root mean square error graph of the satellite formation adaptive relative navigation of the present invention. Detailed Embodiment

[0050] For the convenience of those skilled in the art to understand, the present invention will be further described in detail below in conjunction with specific embodiments and the drawings. The content mentioned in the embodiments does not limit the present invention.

[0051] Embodiment 1:

[0052] A satellite formation adaptive relative navigation method based on variable universe fuzzy logic of the present invention, the process of which can refer to Figure 1 , and specifically includes the following steps:

[0053] Step 1: As shown in Figure 2 , define the primary satellite orbit coordinate system LVLH, which is the reference coordinate system for satellite formation relative navigation. According to the Clohessy-Wilshire motion equation linearized between satellites, construct the discretized system state model as follows:

[0054] (1);

[0055] Among them, is a system state variable, which is the three-axis components of the relative position and velocity of the slave satellite relative to the master satellite defined in the master satellite orbit coordinate system LVLH; is the orbital angular velocity of the master satellite; is the sampling time interval; is the system noise, which can be expressed as the unmodeled perturbation interference; represents the filtering moment.

[0056] Step 2: As Figure 3 shown, is the inter-satellite relative distance; is the pitch angle between satellites; is the azimuth angle between satellites. According to Figure 3 the relative position between the master satellite and the slave satellite and the measurement

[0057] (2);

[0058] Among them, is the measurement noise.

[0059] Step 3: According to the established satellite formation adaptive relative navigation system state model and measurement model, add the corresponding uncertain noise interference, and model the system noise and the measurement noise as "Gaussian noise" with a mean of zero, an unknown and time-varying covariance matrix, and and are uncorrelated, in the form of:

[0060] (3);

[0061] (4);

[0062] Among them, is the probability; and are the actual system noise covariance matrix and the actual measurement noise covariance matrix respectively, and satisfy:

[0063] (5);

[0064] Among them, is the th filtering moment; is the total filtering time; is the nominal system noise covariance matrix; is the nominal measurement noise covariance matrix.

[0065] Step 4: Set the initial parameters of the filter, including the initial filtering state and the initial state error covariance matrix , and the two satisfy the following relationship:

[0066] (6);

[0067] wherein, is the initial state vector; denotes taking the expectation. In this embodiment: the initial state ; the initial state error covariance matrix ;

[0068] The system noise parameter and the measurement noise parameter respectively include: setting the nominal system noise covariance matrix and the nominal noise covariance matrix . Besides being used for the filter, both are also used in Equation (5) in Step 3 to simulate the interference of the uncertain system noise and the measurement noise. In this embodiment: the nominal system noise covariance matrix; the nominal measurement noise covariance matrix ; wherein, diag(·) represents forming a diagonal matrix from a vector.

[0069] As Figure 4 shown is the basic principle block diagram of the single-input single-output variable universe adaptive fuzzy logic system for adjusting the prediction error covariance matrix and the measurement noise covariance matrix . The input of the system needs to go through three main steps, namely fuzzification, fuzzy inference, and defuzzification, to obtain the corresponding output.

[0070] In this embodiment, the basic input universe and the output universe of all single-input single-output variable universe adaptive fuzzy logic systems can both be set as: . To ensure that each input can be within its basic input universe, the input quantization factor of the variable universe adaptive fuzzy logic system for adjusting the prediction error covariance matrix is set as , while the input quantization factor of the variable universe adaptive fuzzy logic system for adjusting the noise covariance matrix , and are respectively set as , , . Meanwhile, the corresponding proportionality coefficients are respectively set as , , and 。

[0071] As Figure 5 shown, during the fuzzification process, the universe of discourse is divided into five levels: Negative Big (NB), Negative Medium (NM), Zero (ZE), Positive Medium (PM), and Positive Big (PB). Specifically, in the input universe of discourse, it is evenly divided. In order to ensure that the output has a sufficiently high adjustment accuracy around zero, the distribution around the zero point of the output universe of discourse is denser, that is, it shows a non-uniform division method. At the same time, the corresponding membership functions are set: NB and PB are S-shaped membership functions, and the remaining levels use Gaussian membership functions.

[0072] The variable universe strategy of the variable universe adaptive fuzzy logic system refers to dynamically adjusting the fuzzy universe by introducing a dynamic adjustment scaling factor under the condition that the fuzzy division of the basic universe of the system remains unchanged. For the basic universes of input and output, they have the following forms respectively:

[0073] (7);

[0074] Among them, and are the to-be input and output of the variable universe adaptive fuzzy logic system respectively. In this embodiment, the output dynamic adjustment scaling factors and are both set to 1. Considering that the system needs to perform stretching and transformation on all the peak points of the membership functions corresponding to the input variables or the universe of discourse of fuzzy singleton elements during the calculation process, which will consume a large amount of computing power of the on-board computer, and The input dynamic adjustment scaling factors , will adopt a simple proportional scaling factor:

[0075] (8);

[0076] Among them, ; is a very small positive number; in this embodiment, , 。

[0077] According to the basic idea of the covariance matching method that the theoretical innovation and the actual innovation need to match, the following 5 IF-THEN form fuzzy inference rules are designed :

[0078] (9);

[0079] Among them, is the system input, the corresponding output.

[0080] The Mamdani inference method, which is a commonly used one, is selected for the fuzzy inference mechanism. Combining the above five fuzzy rules, the inference calculation from input to output of the variable universe adaptive fuzzy logic system is completed.

[0081] After the above two processes of fuzzification and fuzzy inference, the fuzzy expression form of the output has been obtained. However, as Figure 4 shown, this fuzzy quantity still needs to go through the defuzzification process to obtain its precise output expression. In this embodiment, the centroid method will be used for defuzzification, that is, the centroid of the area enclosed by the membership function and the abscissa is taken as the output value, and its characteristic is that it can consider all the information of the fuzzy quantity. For the discrete domain case with output quantization levels, the specific calculation method is as follows:

[0082] (10);

[0083] Among them, represents the determined value of the output; represents the membership function; represents the element of the fuzzy set.

[0084] Step Five: Use the variable universe fuzzy logic adaptive EKF to calculate the three-dimensional position and velocity state information of the slave satellite relative to the master satellite, and complete the adaptive relative navigation of the satellite formation. The specific process is as follows:

[0085] Utilize the filtering state and the state error covariance matrix obtained at the previous moment, and substitute them into the state model of the satellite formation adaptive relative navigation system (Equation (1)) for one-step recursion to obtain the one-step predicted state and the predicted error state covariance matrix : Among them, is the Jacobian matrix of the system model :

[0086] (11);

[0087] Substitute the one-step predicted state into the measurement model of the satellite formation adaptive relative navigation system (Equation (2)) to obtain the theoretical measurement value , from which the measurement residual and the theoretical innovation can be calculated. Among them, is the Jacobian matrix of the measurement model :

[0088] (12);

[0089] Obtained through The measurement residuals from times of observations are used to calculate the actual innovation :

[0090] (13);

[0091] Wherein, is the filtering time; , is the number of the latest measurement residuals; According to actual test experience, in this embodiment is taken as .

[0092] Based on the theoretical innovation and the actual innovation calculate the covariance matching degree and the divergence . In the covariance matching method, the matching degree and the divergence can measure the filtering performance of the current EKF at the filtering time , and are defined as follows:

[0093] (14);

[0094] (15);

[0095] Wherein, represents taking the trace of the matrix.

[0096] The diagonal elements of the matching degree and the divergence are respectively input into the single-input single-output variable universe adaptive fuzzy logic system for adjusting the diagonal elements of the measurement noise covariance matrix and for adjusting the prediction error covariance matrix . Corresponding to and in Equation (8), the corresponding adjustment coefficients and are obtained.

[0097] Using the obtained adjustment coefficients and to online adjust the estimated measurement noise covariance matrix and the estimated prediction error covariance matrix , there are the following adjustment forms:

[0098] (16);

[0099] (17);

[0100] Among them, and are proportionality coefficients and satisfy and , ensuring and are positive definite.

[0101] Then, the innovation and the Kalman gain are recalculated to achieve relative state update and state estimation covariance matrix update. The specific calculation process is as follows:

[0102] (18);

[0103] (19);

[0104] (20);

[0105] (21).

[0106] Step Five The above process needs to be continuously repeated until the relative navigation time of the entire satellite formation ends. From the relative state estimated at each filtering moment , the relative position and velocity of the satellite formation can be calculated, and the adaptive relative navigation of the satellite formation is completed.

[0107] In the specific embodiment of the present invention, the relevant calculation conditions and technical parameters are as follows:

[0108] 1) Orbit parameters of the main satellite: semi-major axis 7087297.6770 m, eccentricity 0.0015, orbital inclination 98.1847°, right ascension of the ascending node 189.8909°, argument of perigee 0.0000°, true anomaly 0.0000°;

[0109] 2) Orbit parameters of the slave satellite: semi-major axis 7087297.5560 m, eccentricity 0.0015, orbital inclination 98.1853°, right ascension of the ascending node 189.8914°, argument of perigee 1.0975°, true anomaly 358.9035°;

[0110] 3) Monte Carlo simulation is performed 100 times, and the simulation time is 500 s.

[0111] Based on the satellite formation adaptive relative navigation method of the present invention and the above-set calculation conditions and parameters, a numerical simulation experiment is carried out. As Figure 6 and Figure 7 are the three-axis relative position error and relative velocity error of the satellite formation adaptive relative navigation under uncertain noise interference in a certain simulation experiment, Figure 8 andFigure 9 It is the root mean square error of the three-axis relative position and velocity of satellite formation adaptive relative navigation in 100 Monte Carlo simulation experiments under uncertain noise interference. It can be seen from this that although affected by uncertain noise interference, the relative navigation accuracy and convergence speed of the present invention can also reach a relatively high level, with good stability and high reliability, meeting the task requirements of satellite formation relative navigation.

[0112] The above-disclosed is only the preferred embodiment of the present invention. It should be pointed out that for those of ordinary skill in the art, without departing from the principle of the present invention, several improvements and modifications can be made, and these improvements and modifications should also be regarded as the protection scope of the present invention.

Claims

1. A satellite formation adaptive relative navigation method based on variable universe fuzzy logic, characterized in that It includes the following steps: Step 1: Establish a system state model using the inter-satellite linearized Clohessy-Wilshire relative motion equation for the recursion of relative states; Step 2: Establish a measurement model for the satellite formation adaptive relative navigation system, and substitute the relative states between satellites into the measurement model to obtain the inter-satellite relative distance and line-of-sight angle measurements; Step 3: Establish an uncertain system noise and measurement noise interference model in the system; Step 4: Set the relevant parameters of the filter and the relevant parameters of the variable universe adaptive fuzzy logic system; Calculate the matching degree and divergence obtained from the theoretical innovation and actual innovation in the filtering process, and then design multiple single-input single-output variable universe adaptive fuzzy logic systems according to the basic idea of covariance matching for online adjustment and estimation of the prediction error covariance matrix and the measurement noise covariance matrix; Step 5: Solve the three-dimensional position and velocity state information of the slave satellite relative to the master satellite through the variable universe fuzzy logic adaptive EKF algorithm to complete the satellite formation adaptive relative navigation.

2. The satellite formation adaptive relative navigation method based on variable universe fuzzy logic according to claim 1, characterized in that In Step 1, the relative motion state of the satellite formation evolves according to the following state model of the satellite formation adaptive relative navigation system: ； Among them, is a system state variable, which is the three-axis components of the relative position and velocity of the slave star relative to the master star defined in the master star orbit coordinate system; is the orbital angular velocity of the master star; is the sampling time interval; is the system noise, which is expressed as an unmodeled perturbation disturbance.

3. The satellite formation adaptive relative navigation method based on variable universe fuzzy logic according to claim 1, characterized in that In Step 2, establish the following measurement model for the satellite formation adaptive relative navigation system to obtain the inter-satellite relative distance and line-of-sight angle measurements: ； Among them, Inter-satellite relative distance; is the elevation angle between satellites; is the azimuth angle between satellites; is the measurement noise.

4. A satellite formation adaptive relative navigation method based on variable universe fuzzy logic according to claim 1, characterized in that In Step 3, the established uncertain system noise and measurement noise interference model is as follows: ； ； wherein, is the probability; and are the actual system noise covariance matrix and the actual measurement noise covariance matrix respectively, and satisfy: ； Among them, is the th filtering moment; is the total filtering time; is the nominal system noise covariance matrix; is the nominal measurement noise covariance matrix.

5. A satellite formation adaptive relative navigation method based on variable universe fuzzy logic according to claim 1, characterized in that In step 4, the output of the variable universe adaptive fuzzy logic system dynamically adjusts the scaling factor and both are 1, , and the input dynamic adjustment factor is as follows: ； Among them, ; ; The universe of discourse is divided into five levels, namely negative large (NB), negative medium (NM), zero (ZE), positive medium (PM), and positive large (PB); The negative large (NB) and positive large (PB) are set as S-type membership functions, and the remaining levels use Gaussian membership functions; The Mamdani inference method is selected as the fuzzy inference mechanism, and the centroid method is selected as the defuzzification method. According to the basic idea of the covariance matching method that the theoretical innovation and the actual innovation need to match, the following IF-THEN form of fuzzy inference rules are designed : 。 6. The satellite formation adaptive relative navigation method based on variable universe fuzzy logic according to claim 1, characterized in that In step five, in the variable universe fuzzy logic adaptive EKF algorithm, the adjustment coefficients and are used to online adjust the estimated measurement noise covariance matrix and the estimated prediction error covariance matrix : ； ； Among them, and are proportionality coefficients and satisfy and , ensuring and are positive definite; finally, the innovation and the Kalman gain are recalculated to achieve relative state update and state estimation covariance matrix update, and the relative position and velocity of the satellite formation are calculated from the estimated relative state, completing the adaptive relative navigation of the satellite formation: ； ； ； ； Among them, is the Jacobian matrix of the measurement model , and is the measurement residual.

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