Spaceborne bistatic synthetic aperture radar attitude guidance method and system

Abstract

The application provides a spaceborne bistatic synthetic aperture radar attitude guiding method and system, and the method comprises the following steps: S1, calculating a main star attitude angle according to a main star full zero Doppler guiding law; S2, calculating wave foot information of an earth surface according to the main star attitude angle and synthetic aperture radar characteristic parameters; S3, calculating a secondary star pitch axis and roll axis guiding angle by using a first coincidence condition based on the wave foot information and a secondary star orbit; and S4, calculating a secondary star yaw axis guiding angle by using a second coincidence condition based on the wave foot information and the secondary star orbit.

Description

Technical Field

[0001] This invention relates to the field of satellite observation technology, and more specifically, to a spaceborne bistatic synthetic aperture radar attitude guidance method and system. Background Technology

[0002] Spaceborne bistatic synthetic aperture radar has significant application value and strategic importance. Compared with traditional monostatic synthetic aperture radar, spaceborne bistatic synthetic aperture radar can not only meet diverse application needs, but also has many advantages, such as the ability to detect target areas from multiple angles, enhance anti-jamming capabilities, provide flexible system configuration, and have stronger stealth capabilities.

[0003] In bistatic synthetic aperture radar (SAR) systems, the transmitting and receiving platforms are located on separate platforms, and this separate operation presents new technical challenges. Traditional methods involve each platform performing its own attitude guidance, resulting in energy dispersion, increased difficulty in ground signal processing, and challenges to engineering implementation. Therefore, how to utilize the orbital separation characteristics of satellites in a bistatic SAR system and the radar parameters to design a satellite attitude guidance method that ensures the wave feet of the two radars coincide on the Earth's surface, thereby achieving high-quality radar imaging, has become an urgent problem to be solved.

[0004] Patent document CN105539884A (application number: CN201610081901.7) discloses a guidance law design method for satellites when performing yaw attitude control. The goal of this method is to ensure that the satellite target attitude change is within the controllable range of the reaction wheel. It is not applicable to synthetic aperture radar satellite attitude guidance missions.

[0005] Patent document CN104390649A (application number: CN201410431322.1) discloses a satellite attitude guidance method and device in a sea surface solar flare observation mode. This method can periodically sample and upload planned attitude data to the satellite according to observation requirements, and reconstruct the attitude data on the satellite, saving upload resources and on-board computing resources. This method focuses on satellite attitude representation and does not provide specific methods for satellite attitude guidance.

[0006] Patent document CN103674033A (application number: CN201310687784.5) discloses a satellite attitude guidance method for spaceborne synthetic aperture radar (SAR). This method first establishes a two-body model and a SAR model, and then calculates the normal direction of the zero Doppler plane where the satellite is located based on the two-body model and the SAR model, thereby calculating the satellite's yaw guidance angle. This method only considers single-satellite SAR attitude guidance and does not completely eliminate the Doppler center frequency, thus limiting its application scenarios.

[0007] Patent document CN104375511A (application number: CN201410588779.3) discloses a yaw guidance method for geosynchronous orbit SAR satellites. This method selects a suitable strategy for attitude beam coordination guidance based on the system's antenna scanning capability, attitude control capability, and attitude sensor installation layout. However, this guidance method focuses on satisfying single-satellite attitude guidance constraints and cannot be extended to attitude guidance missions of spaceborne bistatic synthetic aperture radar satellites.

[0008] Patent document CN104730506A (application number: CN201510098357.2) discloses a zero-Doppler attitude guidance method for synthetic aperture radar (SAR) satellites. This guidance method uses instantaneous spacecraft orbital information to eliminate the coupling of the Doppler center frequency with the Earth's rotation and the satellite's orbital ellipse in the range and azimuth directions, achieving a Doppler center frequency close to zero Hertz for the SAR beam center echo. However, this method focuses on the complete elimination of the Doppler center frequency of a single satellite, and the algorithm cannot be extended to attitude guidance missions of spaceborne bistatic SAR.

[0009] In view of the above shortcomings, this invention proposes a spaceborne bistatic synthetic aperture radar attitude guidance method, which takes into account the complete elimination of the Doppler center frequency of the primary satellite while ensuring that the payload fields of view of the primary and secondary satellites overlap on the Earth's surface, thereby maximizing the echo energy acquired by the receiving device and improving the radar imaging quality. Summary of the Invention

[0010] To address the shortcomings of existing technologies, the purpose of this invention is to provide a spaceborne bistatic synthetic aperture radar attitude guidance method and system.

[0011] According to one aspect of the present invention, a spaceborne bistatic synthetic aperture radar attitude guidance method includes:

[0012] Step S1: Calculate the attitude angle of the primary star according to the primary star's all-zero Doppler guidance law; Step S2: Calculate the wave foot information of the Earth's surface based on the main star's attitude angle and synthetic aperture radar characteristic parameters; Step S3: Based on the wave foot information and the secondary star orbit, calculate the pitch and roll axis guidance angles of the secondary star using the first coincidence condition; Step S4: Based on the wave foot information and the secondary star orbit, calculate the secondary star yaw axis guidance angle using the second coincidence condition.

[0013] Preferably, in step S2, the synthetic aperture radar characteristic parameters include a first characteristic vector and a second characteristic vector, wherein the first characteristic vector is the center beam vector, the second characteristic vector is the far-end incident vector, and the angle between the incident vector and the Z-axis of the satellite system is defined as... .

[0014] Preferably, in step 2, calculating the wave foot information of the Earth's surface includes: Sub-step S2.1: Calculate the first eigenvector of the synthetic aperture radar of the primary satellite. Second eigenvector Representation under this system:

[0015] Sub-step S2.2: Calculate the first eigenvector based on the coordinate transformation. Second eigenvector Representation in Earth-fixed coordinate system:

[0016] Among them, matrix This represents the rotation matrix from the inertial frame to the Earth-fixed frame, obtained through the Earth orientation parameters; This represents the rotation matrix from the primary star's orbital frame to the inertial frame, which can be determined based on the primary star's position. The rotation matrix from the primary star's body system to its orbital system is obtained by converting the yaw and pitch angles determined in step S1. Sub-step S2.3: Calculate the intersection points of the first and second eigenvectors with the Earth:

[0017] in, Indicates the position of the primary star in the Earth-fixed system, parameter and Calculated using the following formula:

[0018]

[0019]

[0020]

[0021] In the formula, and These represent the Earth's equatorial radius and polar radius, respectively. These represent the coordinates of the primary star in the Earth-solid system. This represents the coordinates of the first eigenvector in the Earth-fixed coordinate system. This represents the coordinates of the second eigenvector in the Earth-fixed coordinate system. The specific calculation formula is shown in sub-step S2.2.

[0022] Preferably, in step S3, the first coincidence condition is that the center beam vectors of the synthetic aperture radar of the primary and secondary stars coincide with the intersection point of the Earth.

[0023] Preferably, in step S3, the pitch and roll angles of the auxiliary satellite are calculated using the first coincidence condition. The specific calculation steps are as follows:

[0024] in, , , These are the vectors between the first eigenvector of the auxiliary star to the primary star and Earth, respectively, and can be determined by the following formula:

[0025] in, This indicates the position of the secondary star in the inertial coordinate system. P0 is represented in the auxiliary star orbit system, where P0 is the intersection of the first eigenvector and Earth; The rotation matrix of the auxiliary star from the orbital frame to the inertial frame can be calculated from the position of the satellite in the inertial coordinate system, and ||·|| represents the norm of the vector.

[0026] Preferably, the second coincidence condition is that the intersection of the far-end incident vector of the auxiliary star and the Earth lies on the line connecting the intersections of the first and second characteristic vectors of the primary star and the Earth.

[0027] Preferably, in step S4, the auxiliary star yaw axis guidance angle is calculated using the second coincidence condition, and the specific steps include: Step S4.1: Calculation Unit vector of the connecting line :

[0028] in, This indicates that P2 is represented in the secondary star orbit system. It is the intersection of the second eigenvector and the Earth; Step S4.2: Vector In the auxiliary star system The projection of the plane onto the X-axis has a component of 0. Based on this condition, we obtain the equation regarding the yaw angle:

[0029] in, Representation matrix The element in the i-th row and j-th column, The rotation matrix from the primary system to the orbital system of the auxiliary star; Step S4.3: The yaw angle from step S4.2... The equation simplifies to the following form:

[0030] And calculate the yaw angle of the auxiliary star:

[0031] in:

[0032] In the formula, a, b, and c are the corresponding coefficients of the simplified equation for the yaw angle.

[0033] According to another aspect of the present invention, a spaceborne bistatic synthetic aperture radar attitude guidance system includes: Module M1: Calculate the attitude angle of the primary star based on the primary star's all-zero Doppler guidance law; Module M2: Calculates wave foot information on the Earth's surface based on the host star's attitude angle and synthetic aperture radar characteristic parameters; Module M3: Based on wave foot information and the secondary star orbit, calculate the pitch and roll axis guidance angles of the secondary star using the first coincidence condition; Module M4: Based on the wave foot information of the auxiliary star orbit, the yaw axis guidance angle of the auxiliary star is calculated using the second coincidence condition.

[0034] Preferably, in module M2, the synthetic aperture radar characteristic parameters include a first characteristic vector and a second characteristic vector, wherein the first characteristic vector is the center beam vector, and the second characteristic vector is the far-end incident vector, and the angle between the incident vector and the Z-axis of the satellite system is defined as... .

[0035] Preferably, in module M2, calculating the wave foot information of the Earth's surface includes: Submodule M2.1: Calculates the first eigenvector of the synthetic aperture radar of the primary satellite. Second eigenvector Representation under this system:

[0036] Submodule M2.2: Calculates the first eigenvector based on coordinate transformation. Second eigenvector Representation in Earth-fixed coordinate system:

[0037] Among them, matrix This represents the rotation matrix from the inertial frame to the Earth-fixed frame, obtained through the Earth orientation parameters; This represents the rotation matrix from the primary star's orbital frame to the inertial frame, which can be determined based on the primary star's position. The rotation matrix from the primary star's body system to its orbital system is obtained by converting the yaw and pitch angles determined by module M1. Submodule M2.3: Calculates the intersection of the first and second eigenvectors with the Earth.

[0038] in, Indicates the position of the primary star in the Earth-fixed system, parameter and Calculated using the following formula:

[0039]

[0040]

[0041]

[0042] In the formula, and These represent the Earth's equatorial radius and polar radius, respectively. These represent the coordinates of the primary star in the Earth-solid system. This represents the coordinates of the first eigenvector in the Earth-fixed coordinate system. This represents the coordinates of the second eigenvector in the Earth-fixed coordinate system.

[0043] In module M3, the first overlap condition is that the center beam vectors of the synthetic aperture radars of the primary and secondary stars coincide with the intersection point of the Earth. In module M3, the pitch and roll angles of the auxiliary satellite are calculated using the first coincidence condition. The specific calculation steps are as follows:

[0044] in, , , These are the vectors between the first eigenvector of the auxiliary star to the primary star and Earth, respectively, and can be determined by the following formula:

[0045] in, This indicates the position of the secondary star in the inertial coordinate system. P0 is represented in the auxiliary star orbit system, where P0 is the intersection of the first eigenvector and Earth; The rotation matrix of the auxiliary star from the orbital frame to the inertial frame can be calculated from the position of the satellite in the inertial coordinate system, and ||·|| represents the norm of the vector. The second coincidence condition is that the intersection of the far-end incident vector of the auxiliary star and the Earth lies on the line connecting the intersections of the first and second characteristic vectors of the primary star and the Earth. In module M4, the auxiliary star yaw axis guidance angle is calculated using the second coincidence condition. The specific steps include: Module M4.1: Calculation Unit vector of the connecting line :

[0046] in, This indicates that P2 is represented in the secondary star orbit system. It is the intersection of the second eigenvector and the Earth; Module M4.2: Vectors In the auxiliary star system The projection of the plane onto the X-axis has a component of 0. Based on this condition, we obtain the equation regarding the yaw angle:

[0047] in, Representation matrix The element in the i-th row and j-th column, The rotation matrix from the primary system to the orbital system of the auxiliary star; Module M4.3: This module will update the yaw angle information in Module M4.2. The equation simplifies to the following form:

[0048] And calculate the yaw angle of the auxiliary star:

[0049] in:

[0050] In the formula, a, b, and c are the corresponding coefficients of the simplified equation for the yaw angle.

[0051] Compared with the prior art, the present invention has the following beneficial effects: 1. The attitude guidance law for spaceborne bistatic synthetic aperture radar proposed in this invention can achieve the overlap of the payload fields of view of the primary and secondary satellites on the Earth's surface, thereby enabling the receiving device to obtain the maximum echo energy and improve the radar imaging quality.

[0052] 2. The attitude guidance results of the spaceborne bistatic synthetic aperture radar proposed in this invention are all given explicitly, which has the characteristics of simple form and small amount of calculation, and is easy to implement on the satellite and operate autonomously.

[0053] 3. The attitude guidance law for spaceborne bistatic synthetic aperture radar proposed in this invention only requires the position and velocity of the primary satellite and the position and velocity of the secondary satellite at the calculation time, and can adapt to any formation configuration, thus having higher practicality. Attached Figure Description

[0054] Other features, objects, and advantages of the present invention will become more apparent from the following detailed description of non-limiting embodiments with reference to the accompanying drawings: Figure 1 A schematic diagram of the field of view of a synthetic aperture radar is shown. Figure 2 A spatial schematic diagram of a bistatic synthetic aperture radar system is shown. Figure 3 A schematic diagram of the attitude guidance calculation steps is shown. Detailed Implementation

[0055] The present invention will now be described in detail with reference to specific embodiments. These embodiments will help those skilled in the art to further understand the present invention, but do not limit the invention in any way. It should be noted that those skilled in the art can make several changes and improvements without departing from the concept of the present invention. These all fall within the protection scope of the present invention.

[0056] A spaceborne bistatic synthetic aperture radar attitude guidance method, such as Figure 3 As shown, it includes: Step S1: Calculate the primary star's attitude angles according to the primary star's zero-Doppler guidance law. For ease of calculation, the Euler angles from the satellite's own frame to the inertial frame are used as the satellite's attitude, and zero-Doppler guidance is achieved through two-dimensional attitude guidance. Specific steps include: Step S1.1: Calculate the yaw angle of the primary star:

[0057] Among them, subscript The parameters representing the primary star, These represent the orbital inclination, perigee argument, and true perigee angle of the primary star, respectively. Indicates the revisit cycle.

[0058] Step S1.2: Calculate the elevation angle of the primary star:

[0059] Step S2: Calculate the wave foot information on the Earth's surface based on the primary satellite's attitude angle and the synthetic aperture radar (SAR) characteristic parameters. The SAR characteristic parameters include a first characteristic vector and a second characteristic vector. The first characteristic vector is the center beam vector, and the second characteristic vector is the far-end incident vector. The angle between this vector and the satellite's Z-axis is defined as... The wave foot information on the Earth's surface is calculated based on the position and attitude of the primary star. The specific steps include: Step S2.1: Calculate the first eigenvector of the synthetic aperture radar of the primary satellite. Second eigenvector Representation under this system:

[0060] Step S2.2: Calculate the first eigenvector based on the coordinate transformation. Second eigenvector Representation in Earth-fixed coordinate system:

[0061] Among them, matrix The rotation matrix from the inertial frame to the Earth-fixed frame can be obtained from the Earth orientation parameters; The rotation matrix from the orbital frame to the inertial frame can be determined based on the position of the primary star; The rotation matrix representing the system to the orbital system can be obtained by converting the yaw and pitch angles determined in step S1.

[0062] Step S2.3: Calculate the intersection points of the first and second eigenvectors with the Earth:

[0063] Among them, parameters and Calculated using the following formula:

[0064]

[0065]

[0066]

[0067] In the formula, and These represent the Earth's equatorial radius and polar radius, respectively, taken as 6378.137 km and 6356.752 km; These represent the coordinates of the primary star in the Earth-solid system, which can be obtained through coordinate transformation.

[0068] Step S3: Based on the secondary star's orbit, calculate the elevation and roll guidance angles of the secondary star using the first coincidence condition. The first coincidence condition is that the intersection point of the synthetic aperture radar center beam vectors of the primary and secondary stars with the Earth's orbit is coincident. Using the first coincidence condition, calculate the elevation and roll angles of the secondary star:

[0069] in, , , These are the vectors between the first eigenvector of the auxiliary star to the primary star and Earth, respectively, and can be determined by the following formula:

[0070] Step S4: Based on the secondary star's orbit, calculate the secondary star's yaw axis guidance angle using the second coincidence condition. The second coincidence condition is that the intersection of the secondary star's far-end incident vector and Earth lies on the line connecting the intersections of the primary star's first and second eigenvectors and Earth. The specific steps for calculating the secondary star's yaw axis guidance angle using the second coincidence condition include: Step S4.1: Calculation Unit vector of the connecting line :

[0071] Step S4.2: Vector In the auxiliary star system The projection of the plane onto the X-axis has a component of 0. Based on this condition, we obtain the equation regarding the yaw angle:

[0072] in, Representation matrix The i Line number j The elements of the column. The rotation matrix contains the yaw angle to be calculated.

[0073] Step S4.3: The yaw angle from step S4.2... The equation simplifies to the following form

[0074] And calculate the yaw angle of the auxiliary star:

[0075] The specific calculation methods for L1 and L2 are shown in the formulas below:

[0076] In the formula, a, b, and c are the coefficients of the simplified equation in step S4.3.

[0077] The present invention also provides a spaceborne bistatic synthetic aperture radar attitude guidance system. The spaceborne bistatic synthetic aperture radar attitude guidance system can be implemented by executing the process steps of the spaceborne bistatic synthetic aperture radar attitude guidance method. That is, those skilled in the art can understand the spaceborne bistatic synthetic aperture radar attitude guidance method as a preferred embodiment of the spaceborne bistatic synthetic aperture radar attitude guidance system.

[0078] Those skilled in the art will understand that, besides implementing the system and its various devices, modules, and units provided by this invention in the form of purely computer-readable program code, the same functions can be achieved entirely through logical programming of the method steps, making the system and its various devices, modules, and units of this invention function in the form of logic gates, switches, application-specific integrated circuits, programmable logic controllers, and embedded microcontrollers. Therefore, the system and its various devices, modules, and units provided by this invention can be considered as a hardware component, and the devices, modules, and units included therein for implementing various functions can also be considered as structures within the hardware component; alternatively, the devices, modules, and units for implementing various functions can be considered as both software modules implementing the method and structures within the hardware component.

[0079] Specific embodiments of the present invention have been described above. It should be understood that the present invention is not limited to the specific embodiments described above, and those skilled in the art can make various changes or modifications within the scope of the claims, which do not affect the essence of the present invention. Unless otherwise specified, the embodiments and features described in this application can be arbitrarily combined with each other.

Claims

1. A spaceborne bistatic synthetic aperture radar attitude guidance method, characterized in that, include: Step S1: Calculate the attitude angle of the primary star according to the primary star's all-zero Doppler guidance law; Step S2: Calculate the wave foot information of the Earth's surface based on the main star's attitude angle and synthetic aperture radar characteristic parameters; Step S3: Based on the wave foot information and the secondary star orbit, calculate the pitch and roll axis guidance angles of the secondary star using the first coincidence condition; Step S4: Based on the wave foot information and the secondary star orbit, calculate the secondary star yaw axis guidance angle using the second coincidence condition.

2. The method according to claim 1, characterized in that, In step S2, the synthetic aperture radar characteristic parameters include a first characteristic vector and a second characteristic vector. The first characteristic vector is the center beam vector, and the second characteristic vector is the far-end incident vector. The angle between the incident vector and the Z-axis of the satellite's own system is defined as... .

3. The method according to claim 2, characterized in that, Step 2, calculating the wave foot information of the Earth's surface, includes: Sub-step S2.1: Calculate the first eigenvector of the synthetic aperture radar of the primary satellite. Second eigenvector Representation under this system: Sub-step S2.2: Calculate the first eigenvector based on the coordinate transformation. Second eigenvector Representation in Earth-fixed coordinate system: Among them, matrix This represents the rotation matrix from the inertial frame to the Earth-fixed frame, obtained through the Earth orientation parameters; This represents the rotation matrix from the primary star's orbital frame to the inertial frame, which can be determined based on the primary star's position. The rotation matrix from the primary star's body system to its orbital system is obtained by converting the yaw and pitch angles determined in step S1. Sub-step S2.3: Calculate the intersection points of the first and second eigenvectors with the Earth: in, Indicates the position of the primary star in the Earth-fixed system, parameter and Calculated using the following formula: In the formula, and These represent the Earth's equatorial radius and polar radius, respectively. These represent the coordinates of the primary star in the Earth-solid system. This represents the coordinates of the first eigenvector in the Earth-fixed coordinate system. This represents the coordinates of the second eigenvector in the Earth-fixed coordinate system.

4. The method according to claim 1, characterized in that, In step S3, the first coincidence condition is that the center beam vectors of the synthetic aperture radar of the primary and secondary stars coincide with the intersection point of the Earth.

5. The method according to claim 1, characterized in that, In step S3, the pitch and roll angles of the auxiliary satellite are calculated using the first coincidence condition. The specific calculation steps are as follows: in, , , These are the vectors between the first eigenvector of the auxiliary star to the primary star and Earth, respectively, and can be determined by the following formula: in, This indicates the position of the secondary star in the inertial coordinate system. P0 is represented in the auxiliary star orbit system, where P0 is the intersection of the first eigenvector and Earth; The rotation matrix of the auxiliary star from the orbital frame to the inertial frame can be calculated from the position of the satellite in the inertial coordinate system, and ||·|| represents the norm of the vector.

6. The method according to claim 1, characterized in that, The second coincidence condition is that the intersection of the far-end incident vector of the auxiliary star and Earth lies on the line connecting the intersections of the first and second characteristic vectors of the primary star and Earth.

7. The method according to claim 5, characterized in that, In step S4, the auxiliary star yaw axis guidance angle is calculated using the second coincidence condition. The specific steps include: Step S4.1: Calculation Unit vector of the connecting line : in, This indicates that P2 is represented in the secondary star orbit system. It is the intersection of the second eigenvector and the Earth; Step S4.2: Vector In the auxiliary satellite system The projection of the plane onto the X-axis has a component of 0. Based on this condition, we obtain the equation regarding the yaw angle: in, Representation matrix The element in the i-th row and j-th column, The rotation matrix from the primary system to the orbital system of the auxiliary star; Step S4.3: The yaw angle from step S4.2... The equation simplifies to the following form: And calculate the yaw angle of the auxiliary star: in: In the formula, a, b, and c are the corresponding coefficients of the simplified equation for the yaw angle.

8. A spaceborne bistatic synthetic aperture radar attitude guidance system, characterized in that, include: Module M1: Calculate the attitude angle of the primary star based on the primary star's all-zero Doppler guidance law; Module M2: Calculates wave foot information on the Earth's surface based on the host star's attitude angle and synthetic aperture radar characteristic parameters; Module M3: Based on wave foot information and the secondary star orbit, calculate the pitch and roll axis guidance angles of the secondary star using the first coincidence condition; Module M4: Based on wave foot information and the secondary star's orbit, calculates the secondary star's yaw axis guidance angle using the second coincidence condition.

9. The system according to claim 8, characterized in that, In module M2, the synthetic aperture radar characteristic parameters include a first characteristic vector and a second characteristic vector. The first characteristic vector is the center beam vector, and the second characteristic vector is the far-end incident vector. The angle between the incident vector and the satellite's Z-axis is defined as... .

10. The system according to claim 8, characterized in that, In module M2, wave foot information of the Earth's surface is calculated, including: Submodule M2.1: Calculates the first eigenvector of the synthetic aperture radar of the primary satellite. Second eigenvector Representation under this system: Submodule M2.2: Calculates the first eigenvector based on coordinate transformation. Second eigenvector Representation in Earth-fixed coordinate system: Among them, matrix This represents the rotation matrix from the inertial frame to the Earth-fixed frame, obtained through the Earth orientation parameters; This represents the rotation matrix from the primary star's orbital frame to the inertial frame, which can be determined based on the primary star's position. The rotation matrix from the primary star's body system to its orbital system is obtained by converting the yaw and pitch angles determined by module M1. Submodule M2.3: Calculates the intersection of the first and second eigenvectors with the Earth. in, Indicates the position of the primary star in the Earth-fixed system, parameter and Calculated using the following formula: In the formula, and These represent the Earth's equatorial radius and polar radius, respectively. These represent the coordinates of the primary star in the Earth-solid system. This represents the coordinates of the first eigenvector in the Earth-fixed coordinate system. This represents the coordinates of the second eigenvector in the Earth-fixed coordinate system; In module M3, the first overlap condition is that the center beam vectors of the synthetic aperture radars of the primary and secondary stars coincide with the intersection point of the Earth. In module M3, the pitch and roll angles of the auxiliary satellite are calculated using the first coincidence condition. The specific calculation steps are as follows: in, , , These are the vectors between the first eigenvector of the auxiliary star to the primary star and Earth, respectively, and can be determined by the following formula: in, This indicates the position of the secondary star in the inertial coordinate system. P0 is represented in the auxiliary star orbit system, where P0 is the intersection of the first eigenvector and Earth; The rotation matrix of the auxiliary star from the orbital frame to the inertial frame can be calculated from the position of the satellite in the inertial coordinate system, and ||·|| represents the norm of the vector. The second coincidence condition is that the intersection of the far-end incident vector of the auxiliary star and the Earth lies on the line connecting the intersections of the first and second characteristic vectors of the primary star and the Earth. In module M4, the auxiliary star yaw axis guidance angle is calculated using the second coincidence condition. The specific steps include: Module M4.1: Calculation Unit vector of the connecting line : in, This indicates that P2 is represented in the secondary star orbit system. It is the intersection of the second eigenvector and the Earth; Module M4.2: Vectors In the auxiliary satellite system The projection of the plane onto the X-axis has a component of 0. Based on this condition, we obtain the equation regarding the yaw angle: in, Representation matrix The element in the i-th row and j-th column, The rotation matrix from the primary system to the orbital system of the auxiliary star; Module M4.3: This module will update the yaw angle information in Module M4.

2. The equation simplifies to the following form: And calculate the yaw angle of the auxiliary star: in: In the formula, a, b, and c are the corresponding coefficients of the simplified equation for the yaw angle.

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