Combined navigation method combining starlight refraction positioning with inertial navigation in near space

Abstract

The invention provides an integrated navigation method combining starlight refraction positioning with inertial navigation in a near space, and belongs to the technical field of starlight refraction navigation positioning. The method comprises the following steps: firstly, determining a state model of starlight refraction positioning in a near space by taking an error equation of an inertial navigation system of a near space aircraft as a state equation; designing a near space starlight refraction positioning algorithm, identifying a refraction star by using a star sensor on the aircraft, acquiring a refraction angle as a measurement, and establishing a measurement model of starlight refraction positioning in a near space; and obtaining the optimal estimation value of the system state variable by using an extended Kalman filter (EKF) method. The method has the advantages of being suitable for near space, high in reliability, high in autonomy and the like, and can be finally used for pure astronomical positioning and astronomical / inertial integrated navigation.

Description

Technical Field

[0001] This invention belongs to the field of starlight refraction combined navigation and positioning technology, and particularly relates to a combined navigation method that combines starlight refraction positioning with inertial navigation in near space. Background Technology

[0002] Near space, also known as near-space, transverse region, or suborbital region, lies between aviation and spaceflight. Its range extends from approximately 20 km to 100 km above the Earth's surface, roughly encompassing the upper stratosphere (20 km-55 km), mesosphere (55 km-85 km), and lower thermosphere (85 km-100 km). Essentially, it corresponds to the "intermediate atmosphere" region in space science, situated between the highest altitude of current aircraft and the lowest orbital altitude of current spacecraft. Near-space vehicles refer to aircraft capable of residing in and performing missions within near space.

[0003] The star-light refraction indirect horizon-sensitive autonomous astronomical navigation method, developed in the 1980s, is a low-cost navigation and positioning scheme. It only requires observing refracted starlight using a star sensor and combining this with an atmospheric refraction model to achieve high-precision navigation and positioning. Furthermore, the measurement accuracy of current star sensors is much higher than that of Earth sensors, resulting in a significant improvement in accuracy compared to direct horizon-sensitive navigation methods. Therefore, the star-light refraction indirect horizon-sensitive navigation method is simple in structure, low in cost, and can achieve high navigation accuracy, making it a promising astronomical navigation method.

[0004] Currently, the navigation technologies used in near-space vehicles mainly include satellite navigation, inertial navigation, and celestial navigation.

[0005] The high efficiency of satellite navigation systems is widely recognized. However, when applied to near-space vehicles, the high temperatures and pressures generated on the vehicle's surface during high-speed atmospheric flight cause ionization of surrounding gas molecules, severely interfering with satellite signals and resulting in significant signal degradation. A 2007 report from the European Space Agency and European countries' Atmospheric Reentry Experiment (AREA) project showed that satellite signals can experience severe attenuation, even complete interruption, during high-speed flight. Therefore, if near-space vehicles heavily rely on satellite navigation systems, it is difficult to achieve stable and reliable autonomous navigation, and their accuracy and reliability cannot be guaranteed.

[0006] Inertial navigation technology offers a high degree of autonomy and is used in various near-space vehicles. The US X-43A experimental aircraft incorporated inertial sensor components, while Germany's SHEFEX-2 experimental aircraft employed the iIMU-FCAI-MDS inertial navigation system. However, Chinese researchers believe that near-space vehicles, due to their high speeds, will experience significantly higher centripetal and Coriolis accelerations than conventional aircraft, exacerbating the error divergence of inertial navigation systems. Therefore, when applied to near-space vehicles, relying solely on inertial navigation systems will introduce substantial errors, necessitating external correction of the inertial navigation measurement results using other navigation methods.

[0007] Celestial navigation is immune to electromagnetic interference and has strong anti-jamming capabilities, overcoming the fatal weakness of satellite navigation in today's harsh electronic warfare environment. NASA recognized early on the importance of hypersonic vehicles utilizing celestial navigation technology for autonomous navigation, emphasizing its strategic role within the national-level positioning, navigation, and timing (PNT) system. However, due to the relatively low frequency of observations and navigation calculations in celestial navigation systems, a more continuous navigation system is needed to integrate with it to meet the requirements of near-space vehicles.

[0008] Therefore, a single navigation mode is insufficient for real-time and precise navigation, failing to meet the needs of near-space vehicles. This has led to the development of combined navigation methods. Combining individual navigation methods in pairs can leverage their strengths and compensate for their weaknesses, improving navigation performance. The X-51A, developed by the United States in 2013, employs a combined navigation technology of inertial and satellite navigation; the SHEFEX-2 experimental spacecraft launched by Germany in 2012 uses a combined navigation method of inertial, satellite, and celestial navigation. However, as mentioned above, satellite navigation is susceptible to interference, and the celestial navigation component used on the SHEFEX-2 did not employ a high-precision, high-cost solution. Its celestial navigation system primarily operates within an altitude range above near-space, and is only used for attitude correction, failing to fully utilize its capabilities in near-space.

[0009] For carriers located in near space, when using star sensors to observe stars refracted through the atmosphere, the starlight reaching the carrier is starlight that has entered the atmosphere but has not exited it. This renders the traditional starlight refraction model inapplicable, bringing new challenges to the application of the indirect sensitive horizon positioning method based on starlight refraction within the atmosphere. Summary of the Invention

[0010] To address the aforementioned technical problems, this invention proposes a combined navigation method that integrates starlight refraction positioning with inertial navigation in near space. Based on a novel measurement model, it is applicable to the navigation and positioning of near space vehicles.

[0011] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0012] A combined navigation method integrating starlight refraction positioning and inertial navigation in near space includes the following steps:

[0013] Step S110: Using the error equation of the inertial navigation system of the near-space vehicle as the state equation, the state model of the navigation model is obtained.

[0014] Step S120: Calculate the refraction angle based on the star map obtained by the star sensor to obtain the measurement model of the starlight refraction model in the atmosphere.

[0015] Step S130: Linearize and discretize the model from steps S110 and S120;

[0016] In step S140, the measurement model is processed using the extended Kalman filter method. The processed measurement model is then combined with the state equation from step S110. Different combinations of navigation, including pure inertial navigation, astronomical attitude determination, starlight refraction positioning, and altimeter assistance, are used to obtain the aircraft's speed, position, and attitude information.

[0017] A computing device includes: at least one processor and a memory storing program instructions; when the program instructions are read and executed by the processor, the computing device performs a combined navigation method of starlight refraction positioning combined with inertial navigation in near space.

[0018] A readable storage medium storing program instructions, which, when read and executed by a computing device, cause the computing device to perform a combined navigation method of starlight refraction positioning combined with inertial navigation in near space.

[0019] A computer program product includes a computer program that, when executed by a processor, implements a combined navigation method that combines starlight refraction positioning with inertial navigation in near space.

[0020] The beneficial effects of this invention are as follows:

[0021] (1) A near-space starlight refraction positioning algorithm was designed, and a new measurement model for the starlight refraction navigation and positioning system in the atmosphere was established.

[0022] (2) The method proposed in this invention can provide new application scenarios and more accurate estimation results compared with existing integrated navigation methods, which makes the method able to provide high-precision navigation and positioning for near-space aircraft.

[0023] (3) Through simulation experiments on the navigation model proposed in this invention, it is found that the method proposed in this invention can achieve high-precision positioning under different conditions, and the error is controlled within the design range, showing good robustness. Attached Figure Description

[0024] Figure 1 This is a schematic diagram illustrating the principle of starlight refraction when the carrier is inside the atmosphere.

[0025] Figure 2 This is a flowchart of the combined navigation method of starlight refraction positioning and inertial navigation in near space according to the present invention;

[0026] Figure 3 The image shows the fitted surface obtained using bivariate fitting.

[0027] Figure 4(a) shows the fitted surface of the position error in the X-axis direction, and Figure 4(b) shows the fitted surface of the position error in the Y-axis direction. Detailed Implementation

[0028] Exemplary embodiments of the present disclosure will now be described in more detail with reference to the accompanying drawings. While exemplary embodiments of the present disclosure are shown in the drawings, it should be understood that the present disclosure may be implemented in various forms and should not be limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the disclosure to those skilled in the art.

[0029] Figure 2 This is a flowchart of the combined navigation method of starlight refraction positioning and inertial navigation in near space according to the present invention. Figure 2 As shown, the method includes:

[0030] Step S110: Using the error equation of the inertial navigation system of the near-space vehicle as the state equation, the state model of the navigation model is obtained.

[0031] Step S120: Calculate the refraction angle based on the star map obtained by the star sensor to obtain the measurement model of the starlight refraction model in the atmosphere.

[0032] Step S130: Linearize and discretize the model from steps S110 and S120;

[0033] In step S140, the measurement model is processed using the extended Kalman filter method. The processed measurement model is then combined with the state equation from step S110. By fusing different combined navigation methods, including pure inertial navigation, astronomical attitude determination, starlight refraction positioning, and altimeter assistance, the velocity, position, and attitude information of the aircraft are obtained.

[0034] Furthermore, in step S110, the error equation of the inertial navigation system is given by the state equation, where the state variables are...

[0035] ,

[0036] The state model can be represented as:

[0037] (1)

[0038] in, Represents the platform misalignment angles in three directions. Represents the velocity error in three directions. Represents the errors in longitude, latitude, and altitude. This represents the constant drift of the gyroscope in three directions. This represents the constant bias of the accelerometer in three directions. and This represents the output of the gyroscope and accelerometer in the body coordinate system. and Represents the angular velocity of the geographic coordinate system and the Earth coordinate system relative to the inertial coordinate system. The transformation matrices representing the ontological coordinate system and the geographic coordinate system, M1 and M2 are respectively:

[0039] (2)

[0040] (3)

[0041] in, and These represent the principal radii of curvature of the meridian and the lateral meridian, respectively. It is the longitude of the aircraft. It refers to the altitude of the aircraft. It is the altitude of the aircraft relative to the ground surface. and These represent the velocity components of the aircraft in the east and north directions, respectively.

[0042] Furthermore, in step S120, the relationship between the refraction angle of the star captured by the star sensor of the spacecraft flying within the atmosphere and various parameters is as follows:

[0043] (4)

[0044] (5)

[0045] (6)

[0046] (7)

[0047] (8)

[0048] (9)

[0049] The parameters in the formula are as follows Figure 1 As shown, Figure 1 In the figure, A represents the altitude of the refracted light rays observed from the aircraft above the Earth's surface (i.e., apparent altitude). The foot of the perpendicular from the spacecraft to the apparent position of the star is given by G, where G is the lowest point on the starlight path, O is the Earth's center, and S is the spacecraft's position. The Gladstone-Dale constant in the Gladstone-Dale law, which describes the relationship between refractive index and atmospheric density, is taken as 2.25 × 10⁻⁶ in the simulation experiment. -7 ; The atmospheric density at the location of the aircraft; For height The atmospheric density at that location can generally be measured. and The specific value; H is the average radius of the Earth, which is taken as 6371.0088 km in the simulation experiment; H is the height of the density scale. The altitude of the aircraft is 25km in the simulation experiment; The refraction height is the distance at which light rays are closest to Earth. The apparent height is the height of the refracted light rays observed from the ground surface when viewed from the carrier. For refractive index, Atmospheric density, For the angle of refraction, Angular distance of starlight; The vector pointing from the Earth's center to the spacecraft; It is a star vector.

[0050] Solving the above formulas simultaneously, we can obtain the solution. This leads to another expression for the angle of refraction R:

[0051] (10)

[0052] The numerical calculation results are subjected to bivariate polynomial fitting to obtain R0, and The approximate relationship is:

[0053] (11)

[0054] The sum of squared errors for evaluating the fitting results is 2.2548 × 10⁻⁶. 3 The coefficient of determination is 0.9997, and the root mean square error is 1.2049. The fitted surface is as follows: Figure 3 As shown.

[0055] Before studying the pure astronomical positioning model, a preliminary analysis is performed. Ignoring altimeter errors, for a star sensor installed along the X-axis, solving the simultaneous equations yields the range of change in the refraction angle of a certain star when the spacecraft moves 114m to 342m along the X-axis. The range of the refraction angle is 1-3 arcseconds; the range of the refraction angle when moving 28m-83m along the Y direction is... The range is 1 to 3 arcseconds. For a star sensor installed along the Y-axis, solving the simultaneous equations yields the range of change in the refraction angle of a certain star when the spacecraft moves 184m to 552m along the X-axis. The range of the refraction angle is 1-3 arcseconds; the range of the refraction angle when moving 37m-110m along the Y direction is... The values ​​range from 1 to 3 arcseconds. It can be concluded that if the star sensor is installed in the same direction as the spacecraft's motion, the position error is relatively insensitive. If the star sensor is installed perpendicular to the spacecraft's motion, the position error is more sensitive to the refraction angle of the star captured by the star sensor. The fitted images are shown in Figure 4(a) and Figure 4(b).

[0056] Neglecting altimeter error, what is the change in the refraction angle of a certain star in the X-axis direction sensor when the latitude changes by 0.1 degrees? The change in the refraction angle of a star in the Y-axis direction sensor is approximately 1.3615 arcseconds. Approximately 8.9570 arcseconds. The change in the refraction angle of a star in the X-axis direction sensor when the longitude changes by 0.1 degrees. The change in the refraction angle of a certain star in the Y-axis direction sensor is approximately 15.8119 arcseconds. It is approximately 0.3721 arcseconds.

[0057] From equation (7), if we take Represented by the coordinates (x, y, z) of the aircraft in the inertial coordinate system, and... Use the spacecraft coordinates (x, y, z) and the star vector of the star. Then, equation (7) can be further expressed as:

[0058] (12)

[0059] By capturing celestial objects in the X and Y directions at a given location and utilizing the refraction angles of the two stars, the spacecraft's coordinates in an inertial coordinate system can be calculated, thus achieving purely astronomical positioning. Due to the spacecraft's flight altitude... Since the distance is fixed at 25km, the following equation can be derived:

[0060] (13)

[0061] in, It is the angle of starlight refraction measured when the spacecraft observed the first star. It is the angle of starlight refraction measured by the spacecraft when observing the second star. It is the star vector of the first star. It is the star vector of the second star.

[0062] If spherical coordinates are used for calculation, Let R be the denoted R, then the system of equations (13) can be expressed as:

[0063] (14)

[0064] in, It is the azimuth angle, which is the angle rotated counterclockwise from the positive x-axis in the xOy plane. It is the polar angle, that is, the angle measured downwards from the positive z-axis. This is achieved by solving for the polar angle in the spherical coordinate system. and The values ​​can be converted into the aircraft's position coordinates in the inertial coordinate system, and these old and new coordinates can be used for subsequent error calculations.

[0065] The measurement model in integrated navigation is described below: Using methods such as centroid extraction and star map recognition, the pixel coordinates of non-refracting and refracting stars in the star map identified by the star sensor are obtained. Based on these pixel coordinates, the unit starlight vector of the refracting star in the star sensor coordinate system can be expressed, thus yielding the expression for the refraction angle—a crucial component of the measurement model. Based on the coordinates of the non-refracting stars in both the star sensor and inertial coordinate systems, the transformation matrix from the star sensor coordinate system to the inertial coordinate system can be calculated. According to the calculation By combining the basic navigation star catalog with the simulated star map, a non-refracted star map can be obtained. By comparing the captured star map from the star sensor at a specific moment with the simulated star map, refracted stars in the captured star map can be identified, and the pixel coordinates of the refracted stars can be obtained. , where n represents the total number of refracted stars in the field of view of the star sensor.

[0066] Based on the pixel coordinates of the refracted star, the unit starlight vector of the refracted star in the star sensor coordinate system is: The unit starlight vector of the non-refracting star in the star sensor coordinate system of the simulated star map is: We can obtain:

[0067] (15)

[0068] in, fov is the field of view of the star sensor. and It is the number of pixels of the star sensor CCD.

[0069] The starlight refraction angle R can be expressed as:

[0070] (16)

[0071] In this application, the measuring angle is defined as:

[0072] (17)

[0073] When near-space vehicles employ an inertial / astronomical combined navigation method, only attitude information is available as an observation. The star sensor output represents the vehicle's attitude, and the observation in the measurement equations is the converted digital platform misalignment angle. The measurement equation is:

[0074] (18)

[0075] in, ; It is a 3×3 identity matrix. It is the state vector, that is, the system measurement information vector at time t. , It is the noise measured by the star sensor.

[0076] Furthermore, in step S130, the Extended Kalman Filter (EKF) is a commonly used filtering method for linearizing nonlinear systems. To estimate state variables using EKF filtering, the state equations and measurement models must be linearized and discretized to obtain the first-order linearized matrix of the state equations. and the first-order linearized matrix of the measurement model .

[0077] (19)

[0078] (20)

[0079] in, It is a 6×6 identity matrix. yes The state transition matrix is ​​a matrix that defines the coupling relationships between various state variables at any given time. It is the (k-1)th time. It is the sampling period. It is the gravitational constant. y and z represent the positions of the spacecraft in the x, y, and z directions, respectively. It is the distance from the spacecraft to the origin of the coordinate system, i.e. , It is the state vector at time k. It describes how the observations depend on the state vector. With time A function relating to the relationship. It is a state vector The predicted value (prior estimate), that is, obtained from the state equation... Time prediction The state at any given moment;

[0080] ,

[0081] in, It is the unit vector indicating the direction of the navigation starlight. It is the observation matrix.

[0082] Step S140 may include: The filtering process of the extended Kalman filter method includes:

[0083] Updated in time:

[0084] The state-one-step prediction equation for the extended Kalman filter method is:

[0085] (twenty one)

[0086] in, It is the posterior estimate of the state at time k-1.

[0087] One-step prediction mean square error:

[0088] (twenty two)

[0089] in, It is the prediction error covariance at the current time (i.e., the k-th time). It is the covariance of the estimation error at the previous time step (i.e., the (k-1)th time step). It is the process noise covariance.

[0090] Measurement Update:

[0091] The filter gain equation for the extended Kalman filter method is:

[0092] (twenty three)

[0093] in, It is a measurement Jacobian matrix. It is the measurement noise covariance. It is the Kalman gain.

[0094] The state estimation equation for the extended Kalman filter method is:

[0095] (twenty four)

[0096] in, It is the posterior estimate of the state at time k. It is the actual observation value at the k-th time.

[0097] Estimate mean square error:

[0098] (25)

[0099] The spacecraft's position and velocity are obtained by applying an extended Kalman filter to the combined inertial and starlight refraction navigation system.

[0100] The present invention also provides a computing device, comprising: at least one processor and a memory storing program instructions; when the program instructions are read and executed by the processor, the computing device causes the computing device to perform a combined navigation method of starlight refraction positioning combined with inertial navigation in near space.

[0101] The present invention also provides a readable storage medium storing program instructions, which, when read and executed by a computing device, cause the computing device to perform a combined navigation method of starlight refraction positioning and inertial navigation in near space.

[0102] The present invention also provides a computer program product, including a computer program that, when executed by a processor, implements a combined navigation method that combines starlight refraction positioning with inertial navigation in near space.

[0103] The contents not described in detail in this specification are prior art known to those skilled in the art. Those skilled in the art will readily understand that the above descriptions are merely specific embodiments of the present invention and are not intended to limit the invention. Any modifications, equivalent substitutions, and improvements 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 combined navigation method using starlight refraction positioning and inertial navigation in near space, characterized in that, Includes the following steps: Step S110: Using the error equation of the inertial navigation system of the near-space vehicle as the state equation, the state model of the navigation model is obtained. Step S120: Calculate the refraction angle based on the star map obtained by the star sensor to obtain the measurement model of the starlight refraction model in the atmosphere. Step S130: Linearize and discretize the model from steps S110 and S120; In step S140, the measurement model is processed using the extended Kalman filter method. The processed measurement model is then combined with the state equation from step S110. Different combinations of navigation, including pure inertial navigation, astronomical attitude determination, starlight refraction positioning, and altimeter assistance, are used to obtain the aircraft's speed, position, and attitude information.

2. The combined navigation method of starlight refraction positioning and inertial navigation in near space according to claim 1, characterized in that, Step S110 includes: the error equation of the inertial navigation system is a state equation, in which the state variables are: ， The state model is represented as: (1) in, Represents the platform misalignment angles in three directions. Represents the velocity error in three directions. Represents the errors in longitude, latitude, and altitude. This represents the constant drift of the gyroscope in three directions. This represents the constant bias of the accelerometer in three directions. and This represents the output of the gyroscope and accelerometer in the body coordinate system. and Represents the angular velocity of the geographic coordinate system and the Earth coordinate system relative to the inertial coordinate system. The transformation matrices representing the ontological coordinate system and the geographic coordinate system, M1 and M2 are respectively: (2) (3) in, and These represent the principal radii of curvature of the meridian and the zonal circle, respectively. Represents the longitude of the aircraft. It refers to the altitude of the aircraft. It refers to the local altitude of an aircraft relative to the Earth's surface. These are the velocity components of the aircraft in the east and north directions, respectively.

3. The combined navigation method of starlight refraction positioning and inertial navigation in near space according to claim 1, characterized in that, Step S120 includes: using a method of centroid extraction and star map recognition, the non-refracting and refracting stars in the star map identified by the star sensor are obtained, and their star point pixel coordinates are obtained; based on the star point pixel coordinates, the unit starlight vector of the refracting star in the star sensor coordinate system is expressed, and then the expression of the refraction angle, a key part of the measurement model, is obtained.

4. The combined navigation method of starlight refraction positioning and inertial navigation in near space according to claim 3, characterized in that, Let the unit starlight vector of the i-th refracting star in the star sensor coordinate system be . The unit starlight vector of the non-refracting star in the star sensor coordinate system of the simulated star map is: The angle of refraction is expressed as: (16) The measured angle is defined as: (17) In the formula, The refraction height is the distance at which light rays are closest to Earth. The Gladstone-Dale constant in the Gladstone-Dale law, which describes the relationship between refractive index and atmospheric density; The atmospheric density at the location of the aircraft; The average radius of the Earth; The vector pointing from the Earth's center to the spacecraft; , Star vector; The measured values ​​of the measurement equation are obtained by converting the digital platform misalignment angle. The measurement equation is: (18) in, ; It is a 3×3 identity matrix. It is the state vector, that is, the system measurement information vector at time t; , It is the noise measured by the star sensor.

5. The combined navigation method of starlight refraction positioning and inertial navigation in near space according to claim 4, characterized in that, The relationship between the refraction angle of the star captured by the star sensor and various parameters is as follows: (4) (5) (6) (7) (8) (9) in, The Gladstone-Dale constant in the Gladstone-Dale law, which describes the relationship between refractive index and atmospheric density; The atmospheric density at the location of the aircraft; For height Atmospheric density at that location; H is the average radius of the Earth; H is the height of the density scale. The altitude of the aircraft; The refraction height is the distance at which light rays are closest to Earth. The apparent height is the height of the refracted light rays observed from the ground surface when viewed from the carrier. For refractive index, Atmospheric density, For the angle of refraction, Angular distance of starlight; The vector pointing from the Earth's center to the spacecraft; It is a star vector.

6. The combined navigation method of starlight refraction positioning and inertial navigation in near space according to claim 5, characterized in that, Step S130 includes: The first-order linearized matrix of the state equation is obtained. and the first-order linearized matrix of the measurement model : (19) (20) in, It is a 6×6 identity matrix. yes The state transition matrix is ​​a matrix that defines the coupling relationships between various state variables at any given time. It is the (k-1)th time. It is the sampling period. It is the gravitational constant. y and z are the positions of the spacecraft in the x, y, and z directions, respectively. It is the distance from the spacecraft to the origin of the coordinate system, i.e. , It is the state vector at time k. It describes how the observations depend on the state vector. With time A function relating to the relationship. It is a state vector The predicted value, that is, from the state equation... Time prediction The state at any given moment; The observation matrix is ​​as follows: ， in, It is the unit vector indicating the direction of the navigation starlight. It is the observation matrix.

7. A combined navigation method for starlight refraction positioning and inertial navigation in near space according to claim 6, characterized in that, Step S140 includes: The filtering process of the extended Kalman filter method includes: Updated in time: The state-one-step prediction equation for the extended Kalman filter method is: (21) in, It is the posterior estimate of the state at time k-1; One-step prediction mean square error: (22) in, It is the prediction error covariance at the current time, i.e., the k-th time. It is the estimation error covariance of the previous time step, i.e., the (k-1)th time step. It is the process noise covariance; Measurement Update: The filter gain equation for the extended Kalman filter method is: (23) in, It is a measurement Jacobian matrix. It is the measurement noise covariance. It is the Kalman gain; The state estimation equation for the extended Kalman filter method is: (24) in, It is the posterior estimate of the state at time k. It is the actual observation value at time k; Estimate mean square error: (25)。 8. A computing device, characterized in that, include: At least one processor and a memory storing program instructions; When the program instructions are read and executed by the processor, the computing device performs the combined navigation method of starlight refraction positioning combined with inertial navigation in near space as described in any one of claims 1-7.

9. A readable storage medium storing program instructions, characterized in that, When the program instructions are read and executed by the computing device, the computing device performs the combined navigation method of starlight refraction positioning combined with inertial navigation in near space as described in any one of claims 1-7.

10. A computer program product, comprising a computer program, characterized in that, When the computer program is executed by the processor, it implements the combined navigation method of starlight refraction positioning combined with inertial navigation in near space as described in any one of claims 1-7.

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