On-orbit singularity analysis method for single-frame control moment gyroscope cluster

Abstract

The embodiment of the present disclosure relates to a single-frame control moment gyro group on-orbit singularity analysis method. The method comprises the following steps: obtaining an angular momentum vector of an SGCMG based on each frame angle of the SGCMG, and establishing an angular momentum output equation of the SGCMG group; obtaining an output torque equation of the SGCMG group and a torque matrix of the SGCMG group according to the angular momentum output equation; obtaining a singular standard of the SGCMG by using the torque matrix and a singular frame angle of a singular state, and judging whether the frame angle is a singular point; if the frame angle is the singular point, judging the type and the singular direction of the singular point; drawing according to the singular direction and the angular momentum output equation; constructing a PID control model of the SGCMG, and simulating and analyzing a singular influence mechanism of the SGCMG group. According to the embodiment of the present disclosure, the visual drawing and the mechanism simulation running can directly show the singular point space distribution and changes of the SGCMG after the abnormality, and the spacecraft attitude maneuvering performance changes of the SGCMG group under external force interference can be accurately obtained.

Description

Technical Field

[0001] This disclosure relates to the field of control moment gyroscope technology, and more particularly to an on-orbit singularity analysis method for a single-frame control moment gyroscope group. Background Technology

[0002] With increasing demands for space exploration, space missions requiring spacecraft to perform are becoming increasingly complex. These missions typically require spacecraft to possess large-angle attitude maneuvering capabilities, thus placing higher demands on attitude actuators. Single-Gimbal Control Moment Gyroscopes (SGCMGs), with their superior torque amplification capabilities, are frequently used as actuators for large-angle attitude maneuvers in spacecraft. They achieve attitude control through angular momentum exchange, offering advantages such as high control precision, large torque output, and long service life. Generally, three or more SGCMGs are used in a specific configuration to achieve torque output along the roll axis, pitch axis, and yaw axis. The pentagonal pyramid configuration is widely used due to its technological maturity.

[0003] However, as an inherent problem of SGCMG, singularities directly affect the torque output of the actuator. The lack of singularity visualization and comprehensive analysis of the impact domain when SGCMG fails in orbit has a certain impact on the timely analysis and handling of anomalies in the attitude control system of in-orbit spacecraft and its mission use.

[0004] Therefore, it is necessary to improve one or more of the problems existing in the above-mentioned related technical solutions.

[0005] It should be noted that this section is intended to provide background or context for the technical solutions of this disclosure as set forth in the claims. The description herein does not constitute an admission that it is prior art simply because it is included in this section. Summary of the Invention

[0006] The purpose of this disclosure is to provide an on-orbit singularity analysis method for a single-frame control moment gyroscope group, thereby overcoming, at least to some extent, one or more problems caused by the limitations and defects of related technologies.

[0007] According to an embodiment of this disclosure, an on-orbit singularity analysis method for a single-frame control moment gyroscope group is provided, the method comprising:

[0008] The angular momentum vector of the SGCMG is obtained based on each frame angle of the SGCMG, and the angular momentum output equation of the SGCMG group is established based on the angular momentum vector of the SGCMG; wherein, the SGCMG group includes several SGCMGs.

[0009] The output torque equation and torque matrix of the SGCMG group are obtained based on the angular momentum output equation of the SGCMG group.

[0010] Using the moment matrix of the SGCMG group and the singular frame angles of the singular states, the singularity criterion of the SGCMG is obtained, and it is determined whether each frame angle of the SGCMG is a singular point based on the singularity criterion of the SGCMG.

[0011] If the frame angle of the SGCMG is the singular point, then the type of the singular point is determined according to the singular point judgment matrix, and the singular direction of the singular point is obtained.

[0012] Visualization is performed based on the singular direction and the angular momentum output equation of the SGCMG group;

[0013] A PID control model for the SGCMG is constructed, and the singular influence mechanism of the SGCMG group is simulated and analyzed by combining the output torque equation of the SGCMG group.

[0014] In one embodiment of this disclosure, the angular momentum output equation of the SGCMG group is:

[0015] (1)

[0016] In the formula, h is the angular momentum vector and ,in, h i For the first i The angular momentum of an SGCMG n Let A be the total number of SGCMGs; let B be the first installation matrix of the SGCMG group; and let B be the second installation matrix of the SGCMG group. The frame angle is a sine diagonal matrix. The frame angle cosine diagonal matrix; This is the frame angle vector.

[0017] In one embodiment of this disclosure, the formula for the frame angle sine diagonal matrix is:

[0018] (2)

[0019] The formula for the cosine diagonal matrix of the frame angle is:

[0020] (3)

[0021] In the formula, the frame angle vector , For the frame angle, n For the total number of SGCMG, (·) T This is the transpose of the matrix.

[0022] In one embodiment of this disclosure, the output torque equation of the SGCMG group is:

[0023] (4)

[0024] In the formula, Let be the moment matrix of the SGCMG group, and ; for The derivative with respect to time, i.e. ; Let H be the derivative of H with respect to time.

[0025] In one embodiment of this disclosure, the formula for the singularity criterion of the SGCMG is:

[0026] (5)

[0027] In the formula, u is the unit direction vector of the control torque. It is a singular moment matrix. For the first i The singular moment matrix of each SGCMG Singular frame angles of singular states.

[0028] In one embodiment of this disclosure, the singularity judgment matrix is:

[0029] (6)

[0030] In the formula, N is the null space matrix, and P is the projection matrix.

[0031] In one embodiment of this disclosure, the formula for the projection matrix is:

[0032] (7)

[0033] In the formula, For the first i Angular momentum of an SGCMG in a singular state; u s It is a singular direction; It is a diagonal matrix.

[0034] In one embodiment of this disclosure, the PID control model of the SGCMG includes:

[0035] Controller module, actuator module, and attitude dynamics module;

[0036] The controller module generates control commands based on the spacecraft's attitude deviation and sends the control commands to the actuator module;

[0037] The actuator module receives the control command and calculates the control command torque. Based on the control command torque, it calculates the angular velocity of the SGCMG low-speed frame using the manipulation rate to generate the actual control torque.

[0038] The attitude dynamics module establishes an attitude dynamics model of a rigid spacecraft in the principal axis coordinate system of inertia and outputs the spacecraft's attitude.

[0039] In one embodiment of this disclosure, the mathematical model of the controller module is as follows:

[0040] (8)

[0041] In the formula, T cx For the control torque of the spacecraft's rolling shaft, T cy For the spacecraft pitch axis control torque, T cz For the yaw axis control torque of the spacecraft; Kp x This is the proportional coefficient for the spacecraft's rolling axis. Kp y This is the pitch axis scaling factor for the spacecraft. Kp z This is the yaw axis scaling factor for the spacecraft; Ki x The integral coefficient of the spacecraft's rolling axis. Ki y The integral coefficient of the spacecraft's pitch axis. Ki z The integral coefficient of the spacecraft's yaw axis; Kd x For the differential coefficient of the spacecraft's rolling axis, Kd y For the differential coefficients of the spacecraft's pitch axis, Kd z The differential coefficient of the spacecraft's yaw axis; Im x For the spacecraft's rolling axis inertia, Im y For the pitch axis inertia of the spacecraft, Im z This refers to the yaw axis inertia of the spacecraft. This refers to the angular deviation of the spacecraft's rolling axis. This refers to the angular deviation of the spacecraft's pitch axis. This refers to the angular deviation of the spacecraft's yaw axis. The rate of change of the spacecraft's rolling axis angle. The rate of change of the spacecraft's pitch axis angle. The rate of change of the yaw axis angle of the spacecraft; Let be the derivative of the angular deviation of the spacecraft's rolling axis with respect to time. Let be the derivative of the angle deviation of the spacecraft's pitch axis with respect to time. This is the derivative of the yaw angle deviation of the spacecraft with respect to time. The orbital angular velocity, h x Let ω be the angular momentum along the spacecraft's rolling axis. h y Let angular momentum be along the spacecraft's pitch axis. h z Let ω be the angular momentum along the spacecraft's yaw axis, with subscript ω. x The spacecraft's rolling axis in the principal axis coordinate system of inertia, subscript y The spacecraft pitch axis in the principal axis coordinate system of spacecraft inertia, with subscript... z This is the spacecraft yaw axis in the spacecraft's principal axis coordinate system of inertia.

[0042] In one embodiment of this disclosure, the mathematical model of the actuator module is as follows:

[0043] (9)

[0044] In the formula, A Adj This is a vector adjustment matrix; The first adjustment coefficient, This is the second adjustment coefficient; I3 is the singularity metric; I3 is the 3×3 identity matrix.

[0045] In one embodiment of this disclosure, the mathematical model of the attitude dynamics module is as follows:

[0046] (10)

[0047] In the formula, For disturbance torque; The angular acceleration of the spacecraft's rolling axis. Let be the angular acceleration of the spacecraft's pitch axis. This is the angular acceleration of the spacecraft's yaw axis.

[0048] The technical solutions provided by the embodiments of this disclosure may include the following beneficial effects:

[0049] In the embodiments of this disclosure, the above-described single-frame control moment gyroscope group on-orbit singularity analysis method can, on the one hand, visually display the spatial distribution and changes of singular points after some SGCMG anomalies through visualization drawing, and also roughly analyze the changes in its attitude maneuvering performance based on the angular momentum output; on the other hand, through mechanism simulation operation, the changes in the spacecraft's attitude maneuvering performance under external force interference can be accurately obtained. Attached Figure Description

[0050] The accompanying drawings, which are incorporated in and form part of this specification, illustrate embodiments consistent with this disclosure and, together with the description, serve to explain the principles of this disclosure. It is obvious that the drawings described below are merely some embodiments of this disclosure, and those skilled in the art can obtain other drawings based on these drawings without any inventive effort.

[0051] Picture 1 This diagram illustrates the steps of the on-orbit singularity analysis method for a single-frame control moment gyroscope group in an exemplary embodiment of this disclosure.

[0052] Picture 2 A diagram illustrating the configuration of the pentagonal pyramid SGCMG group in an exemplary embodiment of this disclosure is shown. Detailed Implementation

[0053] Exemplary embodiments will now be described more fully with reference to the accompanying drawings. However, these exemplary embodiments can be implemented in many forms and should not be construed as limited to the examples set forth herein; rather, they are provided so that this disclosure will be more comprehensive and complete, and will fully convey the concept of the exemplary embodiments to those skilled in the art. The described features, structures, or characteristics may be combined in any suitable manner in one or more embodiments.

[0054] Furthermore, the accompanying drawings are merely illustrative diagrams of embodiments of this disclosure and are not necessarily drawn to scale. The same reference numerals in the drawings denote the same or similar parts, and therefore repeated descriptions of them will be omitted. Some block diagrams shown in the drawings are functional entities and do not necessarily correspond to physically or logically independent entities.

[0055] This example implementation provides an on-orbit singularity analysis method for a single-frame control moment gyroscope group. (Reference) Picture 1 As shown, the on-orbit singularity analysis method for a single-frame control moment gyroscope group may include steps S101 to S106.

[0056] Step S101: Obtain the angular momentum vector of the SGCMG based on each frame angle of the SGCMG, and establish the angular momentum output equation of the SGCMG group based on the angular momentum vector of the SGCMG; wherein, the SGCMG group includes several SGCMGs.

[0057] Step S102: Obtain the output torque equation of the SGCMG group and the torque matrix of the SGCMG group based on the angular momentum output equation of the SGCMG group;

[0058] Step S103: Using the moment matrix of the SGCMG group and the singular frame angles of the singular states, obtain the singularity criterion of the SGCMG, and determine whether each frame angle of the SGCMG is a singular point based on the singularity criterion of the SGCMG.

[0059] Step S104: If the frame angle of the SGCMG is the singular point, then determine the type of the singular point according to the singular point judgment matrix, and obtain the singular direction of the singular point;

[0060] Step S105: Visualize the output equation of the angular momentum of the SGCMG group based on the singular direction;

[0061] Step S106: Construct a PID control model for the SGCMG group, and perform a singular influence mechanism simulation analysis on the SGCMG group by combining the output torque equation of the SGCMG group.

[0062] Using the above-mentioned on-orbit singularity analysis method for single-frame control moment gyroscope groups, on the one hand, through visualization drawing, not only can the spatial distribution and changes of singular points after some SGCMG anomalies be displayed intuitively, but also the changes in attitude maneuvering performance can be roughly analyzed based on the angular momentum output; on the other hand, through mechanism simulation operation, the changes in the spacecraft's attitude maneuvering performance under external force interference can be accurately obtained.

[0063] Below, we will refer to Picture 1 to Picture 2 The steps of the on-orbit singularity analysis method for the single-frame control moment gyroscope group described in this example embodiment will be explained in more detail.

[0064] In one embodiment, the angular momentum output equation of the SGCMG group is:

[0065] (1)

[0066] In the formula, h is the angular momentum vector and ,in, h i For the first i The angular momentum of an SGCMG Let A be the total number of SGCMGs; let B be the first installation matrix of the SGCMG group; and let B be the second installation matrix of the SGCMG group. The frame angle is a sine diagonal matrix. The frame angle cosine diagonal matrix; This is the frame angle vector.

[0067] Specifically, let the SGCMG group be composed of... n Composed of identical SGCMGs, the first i The angular momentum of each SGCMG is hi Establish angular momentum column vectors The total angular momentum output of the SGCMG group can be expressed as:

[0068] (1)

[0069] In the above formula, h is the angular momentum vector and ,in, h i For the first i The angular momentum of an SGCMG n Let A be the total number of SGCMGs; let B be the first installation matrix of the SGCMG group; and let B be the second installation matrix of the SGCMG group. The frame angle is a sine diagonal matrix. The frame angle cosine diagonal matrix; This is the frame angle vector.

[0070] In one embodiment, the formula for the sine diagonal matrix of the frame angle is:

[0071] (2)

[0072] The formula for the cosine diagonal matrix of the frame angle is:

[0073] (3)

[0074] In the formula, the frame angle vector , For the frame angle, n For the total number of SGCMG, (·) T This is the transpose of the matrix.

[0075] Specifically, assuming the pentagonal pyramidal SGCMG group consists of 6 SGCMGs (a~f), such as Picture 2 As shown, five SGCMG frames (a~e) are axially symmetrically located on the side of the pentagonal pyramid, forming an angle with the yaw axis. The base of the pyramid is a regular pentagon, and the f-th SGCMG frame axis is perpendicular to the roll-pitch plane. The angular momentum of a single SGCMG is h = 25 Nms, and the maximum angular velocity of the low-speed frame is... =20° / s, roll and pitch maneuver angles are both 45°, SGCMG group installation matrices A and B are respectively:

[0076]

[0077] In one embodiment, the output torque equation of the SGCMG group is:

[0078] (4)

[0079] In the formula, Let be the moment matrix of the SGCMG group, and ; for The derivative with respect to time, i.e. ; Let H be the derivative of H with respect to time.

[0080] Specifically, the rotational speed of each gyroscope frame in the SGCMG group The resulting combined gyroscopic torque T c It can be represented as:

[0081] (4)

[0082] in, The moment matrix of the SGCMG group and the system frame angle vector Related; The rotational speed vector of each SGCMG frame in the SGCMG group is The derivative with respect to time, i.e. ,akin, H is the derivative of H with respect to time in formula (1).

[0083] In one embodiment, the formula for the singularity criterion of the SGCMG is:

[0084] (5)

[0085] In the formula, u is the unit direction vector of the control torque. It is a singular moment matrix. For the first i The singular moment matrix of each SGCMG Singular frame angles of singular states.

[0086] Specifically, from a mathematical perspective, the frame angle combination in the singular state is: The corresponding moment matrix If the columns represent the directions of each SGCMG, then the following equation holds:

[0087] (5)

[0088] In the above formula, u is the unit direction vector of the given control torque. In numerical calculation, u is initialized as the unit outward normal vector of the sphere. With formula (4) The calculation method is the same, only the angle is different. Different. If formula (5) is satisfied, it indicates that the system is in a singularity, and the current control torque direction is recorded as the singular direction u. s .

[0089] In one embodiment, the singularity determination matrix is:

[0090] (6)

[0091] In the formula, N is the null space matrix, and P is the projection matrix.

[0092] The formula for the projection matrix is:

[0093] (7)

[0094] In the formula, For the first i Angular momentum of an SGCMG in a singular state; u s It is a singular direction; It is a diagonal matrix.

[0095] Specifically, when Q is a positive definite matrix and P is a positive definite matrix, all elements in the SGCMG group are in the singular direction u s The projection on reaches its maximum value, becoming a saturated singularity (i.e., an external singularity), and the singularity is distributed on the momentum volume envelope; when Q is a positive definite matrix and P is an indefinite matrix, all elements in the SGCMG group are in the singular direction u. s The projections on all reach extreme values ​​(maximum or minimum), which are manifest singularities, and these manifest singularities are distributed inside the momentum volume; when Q is an indeterminate matrix, the frame angles of the SGCMG group are in the combination of singular frame angles. Within the domain, it can be reconstructed as a latent singularity, and the latent singularities are distributed inside the momentum volume. See Table 1 for details.

[0096] Table 1. Exotic Properties of SGCMG

[0097]

[0098] The above formulas are used to calculate the output angular momentum and all singularities of the SGCMG group. Then, a scatter plot is plotted using the `scatterplot3d` function in R to visualize the three-dimensional spatial distribution of angular momentum and singularities. This allows for analysis of the changes in angular momentum and singularity distribution after the SGCMG anomaly, and a rough calculation of the changes in control performance based on angular momentum changes. If the first... i For each SGCMG anomaly, simply change its corresponding output angular momentum. h i Simply set it to zero; the entire calculation method remains unchanged.

[0099] In one embodiment, the SGCMG's PID (Proportion Integral Derivative) control model includes: a controller module, an actuator module, and an attitude dynamics module. The controller module generates control commands based on the spacecraft's attitude deviation and sends these commands to the actuator module. The actuator module receives the control commands, calculates the control command torque, and uses the control command torque to calculate the SGCMG's low-speed frame angular velocity using the manipulation rate, generating the actual control torque. The attitude dynamics module establishes a rigid body spacecraft attitude dynamics model in the principal axis coordinate system of inertia and outputs the spacecraft's attitude.

[0100] Specifically, a proportional-integral-derivative (PID) control model for the SGCMG system was established, and the impact of anomalies in some SGCMG groups on the spacecraft's attitude control was accurately simulated and analyzed.

[0101] The control model consists of three modules: controller, actuator, and attitude dynamics. A PID control model for a satellite was constructed using MATLAB / SIMULINK; this model does not consider actuators other than the SGCMG.

[0102] Controller module. It generates control commands based on attitude deviations and transmits them to the actuators (SGCMG system). The mathematical model is as follows:

[0103] (8)

[0104] in, T cx For the control torque of the spacecraft's rolling shaft, T cy For the spacecraft pitch axis control torque, T cz For the yaw axis control torque of the spacecraft; Kp x This is the proportional coefficient for the spacecraft's rolling axis. Kp y This is the pitch axis scaling factor for the spacecraft. Kp z This is the yaw axis scaling factor for the spacecraft; Ki x The integral coefficient of the spacecraft's rolling axis. Ki y The integral coefficient of the spacecraft's pitch axis. Ki z The integral coefficient of the spacecraft's yaw axis; Kd x For the differential coefficient of the spacecraft's rolling axis, Kdy For the differential coefficients of the spacecraft's pitch axis, Kd z The differential coefficient of the spacecraft's yaw axis; Im x For the spacecraft's rolling axis inertia, Im y For the pitch axis inertia of the spacecraft, Im z This refers to the yaw axis inertia of the spacecraft. This refers to the angular deviation of the spacecraft's rolling axis. This refers to the angular deviation of the spacecraft's pitch axis. This refers to the angular deviation of the spacecraft's yaw axis. The rate of change of the spacecraft's rolling axis angle. The rate of change of the spacecraft's pitch axis angle. The rate of change of the yaw axis angle of the spacecraft; Let be the derivative of the angular deviation of the spacecraft's rolling axis with respect to time. Let be the derivative of the angle deviation of the spacecraft's pitch axis with respect to time. This is the derivative of the yaw angle deviation of the spacecraft with respect to time. The orbital angular velocity, , , r The orbital radius of the spacecraft (in km); h x Let ω be the angular momentum along the spacecraft's rolling axis. h y Let angular momentum be along the spacecraft's pitch axis. h z Let ω be the angular momentum along the spacecraft's yaw axis, with subscript ω. x The spacecraft's rolling axis in the principal axis coordinate system of inertia, subscript y The spacecraft pitch axis in the principal axis coordinate system of spacecraft inertia, with subscript... z Let be the spacecraft's yaw axis in the principal axis coordinate system of inertia. Its value is the product of the moment of inertia and the angular velocity along the corresponding axis, i.e. , , ; Rolling shaft angular velocity Pitch axis angular velocity yaw axis angular velocity They are respectively , , .

[0105] Actuator module. Based on the control command torque, the angular velocity of the SGCMG low-speed frame is calculated using the control law. This generates actual control torque, specifically in the form of:

[0106] (9)

[0107] Where C is the moment matrix of the SGCMG group, i.e., in equation (4) (The same applies below); A Adj This is a vector adjustment matrix; , This is the adjustment coefficient; I3 is the singularity metric; I3 is the 3×3 identity matrix.

[0108] Attitude Dynamics Module. Establishes an attitude dynamics model of the rigid body spacecraft in the principal axis coordinate system, and outputs the spacecraft's attitude. The mathematical model is as follows:

[0109] (10)

[0110] in, To mitigate the disturbance torque, a torque on the order of 10⁻⁴ Nm is typically selected. , , Angular accelerations of the roll axis, pitch axis, and yaw axis.

[0111] By using visualization technology to analyze the singularity distribution, angular momentum output performance, and singularity avoidance of the SGCMG swarm, and by using PID control system simulation to analyze the impact of the SGCMG swarm output on the attitude maneuver performance of the satellite's roll axis, pitch axis, and yaw axis after some anomalies, we can provide a theoretical basis and technical support for the on-orbit application of spacecraft using the SGCMG swarm.

[0112] Using the above-mentioned on-orbit singularity analysis method of single-frame control moment gyroscope group, we first use visualization technology to analyze the distribution of singular points and changes in angular momentum output performance of the SGCMG group from a data perspective. Then, through PID control system simulation, we analyze the impact of the SGCMG group output on the attitude maneuver performance of the spacecraft's roll axis, pitch axis, and yaw axis after some anomalies from a mechanistic perspective.

[0113] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature. In the description of embodiments of this disclosure, "a plurality of" means two or more, unless otherwise explicitly specified.

[0114] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of this disclosure. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. In addition, those skilled in the art can combine and integrate the different embodiments or examples described in this specification.

[0115] Other embodiments of this disclosure will readily occur to those skilled in the art upon consideration of the specification and practice of the invention disclosed herein. This application is intended to cover any variations, uses, or adaptations of this disclosure that follow the general principles of this disclosure and include common knowledge or customary techniques in the art not disclosed herein. The specification and examples are to be considered exemplary only, and the true scope and spirit of this disclosure are indicated by the appended claims.

Claims

1. A method for on-orbit singularity analysis of a single-frame control moment gyroscope group, characterized in that, The method includes: The angular momentum vector of the SGCMG is obtained based on each frame angle of the SGCMG, and the angular momentum output equation of the SGCMG group is established based on the angular momentum vector of the SGCMG; wherein, the SGCMG group includes several SGCMGs. The output torque equation and torque matrix of the SGCMG group are obtained based on the angular momentum output equation of the SGCMG group. Using the moment matrix of the SGCMG group and the singular frame angles of the singular states, the singularity criterion of the SGCMG is obtained, and it is determined whether each frame angle of the SGCMG is a singular point based on the singularity criterion of the SGCMG. If the frame angle of the SGCMG is the singular point, then the type of the singular point is determined according to the singular point judgment matrix, and the singular direction of the singular point is obtained. Visualization is performed based on the singular direction and the angular momentum output equation of the SGCMG group; A PID control model for the SGCMG is constructed, and the singular influence mechanism of the SGCMG group is simulated and analyzed in conjunction with the output torque equation of the SGCMG group. The PID control model of the SGCMG includes: Controller module, actuator module, and attitude dynamics module; The controller module generates control commands based on the spacecraft's attitude deviation and sends the control commands to the actuator module; The actuator module receives the control command and calculates the control command torque. Based on the control command torque, it calculates the angular velocity of the SGCMG low-speed frame using the manipulation rate to generate the actual control torque. The attitude dynamics module establishes an attitude dynamics model of a rigid spacecraft in the principal axis coordinate system of inertia and outputs the attitude of the spacecraft. The mathematical model of the controller module is as follows: （8） In the formula, T cx For the control torque of the spacecraft's rolling shaft, T cy For the spacecraft pitch axis control torque, T cz For the yaw axis control torque of the spacecraft; Kp x This is the proportional coefficient for the spacecraft's rolling axis. Kp y This is the pitch axis scaling factor for the spacecraft. Kp z This is the yaw axis scaling factor for the spacecraft; Ki x The integral coefficient of the spacecraft's rolling axis. Ki y The integral coefficient of the spacecraft's pitch axis. Ki z The integral coefficient of the spacecraft's yaw axis; Kd x For the differential coefficient of the spacecraft's rolling axis, Kd y For the differential coefficients of the spacecraft's pitch axis, Kd z The differential coefficient of the spacecraft's yaw axis; Im x For the spacecraft's rolling axis inertia, Im y For the pitch axis inertia of the spacecraft, Im z This refers to the yaw axis inertia of the spacecraft. This refers to the angular deviation of the spacecraft's rolling axis. This refers to the angular deviation of the spacecraft's pitch axis. This refers to the angular deviation of the spacecraft's yaw axis. The rate of change of the spacecraft's rolling axis angle. The rate of change of the yaw axis angle of the spacecraft; Let be the derivative of the angular deviation of the spacecraft's rolling axis with respect to time. Let be the derivative of the angle deviation of the spacecraft's pitch axis with respect to time. This is the derivative of the yaw angle deviation of the spacecraft with respect to time. The orbital angular velocity, h x Let ω be the angular momentum along the spacecraft's rolling axis. h y Let angular momentum be along the spacecraft's pitch axis. h z Let ω be the angular momentum along the spacecraft's yaw axis, with subscript ω. x The spacecraft's rolling axis in the principal axis coordinate system of inertia, subscript y The spacecraft pitch axis in the principal axis coordinate system of spacecraft inertia, with subscript... z This is the spacecraft yaw axis in the spacecraft's principal axis coordinate system of inertia.

2. The on-orbit singularity analysis method for a single-frame control moment gyroscope group according to claim 1, characterized in that, The angular momentum output equation of the SGCMG group is: （1） In the formula, h is the angular momentum vector and , [·] T Let be the transpose of the matrix, where h i For the first i The angular momentum of an SGCMG n Let A be the total number of SGCMGs; let B be the first installation matrix of the SGCMG group; and let B be the second installation matrix of the SGCMG group. The frame angle is a sine diagonal matrix. The frame angle cosine diagonal matrix; This is the frame angle vector.

3. The on-orbit singularity analysis method for a single-frame control moment gyroscope group according to claim 2, characterized in that, The formula for the sine diagonal matrix of the frame angle is: （2） The formula for the cosine diagonal matrix of the frame angle is: （3） In the formula, the frame angle vector , For the frame angle, n For the total number of SGCMG, (·) T This is the transpose of the matrix.

4. The on-orbit singularity analysis method for a single-frame control moment gyroscope group according to claim 3, characterized in that, The output torque equation of the SGCMG group is: （4） In the formula, Let be the moment matrix of the SGCMG group, and ; for The derivative with respect to time, i.e. ; Let H be the derivative of H with respect to time.

5. The on-orbit singularity analysis method for a single-frame control moment gyroscope group according to claim 4, characterized in that, The formula for the singularity criterion of SGCMG is: （5） In the formula, u is the unit direction vector of the control torque. It is a singular moment matrix. For the first i The singular moment matrix of each SGCMG The singular frame angle is a singular state.

6. The on-orbit singularity analysis method for a single-frame control moment gyroscope group according to claim 5, characterized in that, The judgment matrix for the singular point is: （6） In the formula, N is the null space matrix, and P is the projection matrix, N T It is the transpose of N.

7. The on-orbit singularity analysis method for a single-frame control moment gyroscope group according to claim 6, characterized in that, The formula for the projection matrix is: （7） In the formula, For the first i Angular momentum of an SGCMG in a singular state; u s It is a singular direction; It is a diagonal matrix.

8. The on-orbit singularity analysis method for a single-frame control moment gyroscope group according to claim 1, characterized in that, The mathematical model of the actuator module is as follows: （9） In the formula, A Adj This is a vector adjustment matrix; The first adjustment coefficient, This is the second adjustment coefficient; I3 is the singularity metric; I3 is the 3×3 identity matrix.

9. The on-orbit singularity analysis method for a single-frame control moment gyroscope group according to claim 1, characterized in that, The mathematical model of the attitude dynamics module is as follows: （10） In the formula, For disturbance torque; The angular acceleration of the spacecraft's rolling axis. Let be the angular acceleration of the spacecraft's pitch axis. This is the angular acceleration of the spacecraft's yaw axis.

Images