A decoupling method for dynamic effects of movable flexible components

Abstract

The present invention provides a decoupling method for the dynamic effects of movable flexible components, which can reduce the difficulty of dynamic modeling of control-oriented systems and effectively support control system design and simulation. The method of the present invention is a decoupling analysis method for the dynamic effects of large movable flexible components, which solves the problem of dynamic modeling of large flexible spacecraft for control. Combined with the urgent engineering demand of space relative navigation in my country's aerospace engineering, a decoupling analysis method for the dynamic effects of large movable flexible components is provided. This method can effectively reduce the workload of dynamic modeling of spacecraft under any configuration of flexible components, and support the dynamic modeling of variable-configuration spacecraft systems for control system design.

Description

Technical Field

[0001] The invention relates to the technical field of complex spacecraft dynamics, and in particular to a method for decoupling dynamic effects of a movable flexible component. Background Art

[0002] In recent years, with the rapid development of space technology, the structural scale of some spacecraft has become larger and larger, and such large-scale spacecraft have begun to face complex tasks such as on-orbit assembly, construction, and configuration change. Such large-scale spacecraft will inevitably have large flexible components, and such flexible components are required to be drivable and have multi-degree-of-freedom movable configuration change capabilities, such as large flexible solar sails and large antennas. The large-scale development of spacecraft is one of the important directions for the development of future spacecraft. Large movable flexible components, low system structural stiffness, and high flexibility are typical characteristics of future large-scale spacecraft.

[0003] Large movable flexible components on spacecraft have a huge impact on the overall dynamics and control of the system. How to avoid or reduce the vibration of large flexible components is an important part of ensuring the stable attitude control of the system. However, large flexible components have low structural fundamental frequency, high structural flexibility, and are also drivable components. This greatly increases the difficulty of control-oriented system dynamics modeling. It is necessary to study how to develop control-oriented dynamics modeling methods that not only reflect the dynamic characteristics of large flexible components, but also meet the control system's overall variable configuration simulation requirements for spacecraft. At the same time, it is also necessary to ensure that the system dynamics model is small in scale and can achieve stable convergence of simulation calculations. Therefore, the dynamics and control of spacecraft containing large flexible components still require many key technologies to be tackled.

[0004] Large flexible components will significantly change the system configuration during the driving process, and the dynamic characteristics of the system will also change significantly during the system configuration change process. Therefore, the system dynamics model should also reflect the changes in the dynamic characteristics of the configuration change process to support the control system design and simulation. During the activity of flexible components, the system dynamics model will change with the driving degrees of freedom and angle. The dynamic influence of flexible components is closely coupled with the driving degrees of freedom and angle, which makes the dynamic modeling of variable configuration spacecraft for control purposes extremely complex. Summary of the invention

[0005] In view of this, the present invention provides a decoupling method for the dynamic influence of a movable flexible component, which can reduce the difficulty of control-oriented system dynamics modeling and effectively support control system design and simulation.

[0006] The present invention provides a method for decoupling the dynamic influence of a movable flexible component, comprising the following steps:

[0007] Step 1, establishing a rigid-flexible coupling dynamic equation of a flexible component with dual driving degrees of freedom;

[0008] Step 2: propose the rigid-flexible coupling influence of components on the spacecraft and establish the flexible coupling coefficient formula of the dual-axis drive freedom component;

[0009] Step 3, based on the flexible coupling coefficient formula, define a flexible simulation coefficient expression, and establish a transformation relationship between the flexible coupling coefficient and the flexible simulation coefficient;

[0010] Step 4, expand the established flexible simulation coefficient transformation relationship, perform variable separation, and establish a flexible coupling coefficient expression consisting of a dual-axis driving angle and a constant simulation coefficient term without an angle;

[0011] Step 5, process the coordinate system according to the description of the driving angle of the flexible component to further obtain the expression form relative to the spacecraft control coordinate system, and realize the decoupling analysis of the dynamic influence of the movable flexible component.

[0012] Wherein, in the step 1, the rigid-flexible coupling dynamic equation of the flexible component with dual driving degrees of freedom is as follows:

[0013]

[0014]

[0015]

[0016]

[0017] Wherein, equation (1) is the translational motion equation of the system center of mass, equation (2) is the rotational motion equation of the system around the center of mass, (3) is the control equation of the flexible attachment, and (4) is the vibration equation of the flexible attachment; equations (1) and (2) are based on the attitude control coordinate system of the assembly; i is the number of the flexible attachment carried by the system, and equation (3) is based on the local coordinate system of the flexible attachment; n is the total number of flexible attachments contained in the assembly corresponding to the working condition;

[0018] ω s is the angular velocity array of the assembly, measured in b frame, ∈3×1;

[0019] is the antisymmetric matrix of the angular velocity array, ∈3×3;

[0020] M is the total mass matrix of the assembly, ∈3×3;

[0021] I s is the inertia matrix of the assembly relative to the system center of mass, measured in the b system, ∈3×3;

[0022] P s is the external force matrix acting on the assembly, measured in b system, ∈3×1;

[0023] T s is the external moment matrix acting on the assembly, measured in b system, ∈3×1;

[0024] T ai is the control torque array acting on the nth flexible attachment, measured in the local coordinate system of the flexible attachment, ∈3×1;

[0025] I ai is the inertia matrix of the i-th flexible attachment relative to its respective connection point, measured in the local coordinate system of the flexible attachment, ∈3×3;

[0026] ω ai is the rotational angular velocity array of the i-th flexible attachment, measured in the local coordinate system of the flexible attachment, ∈3×1;

[0027] Ω ai is the diagonal matrix of modal frequencies of the i-th flexible attachment, ∈m×m;

[0028] η ai is the modal coordinate matrix of the i-th flexible attachment, ∈m×1;

[0029] ζ ai is the modal damping coefficient of the i-th flexible attachment;

[0030] F tci is the flexible coupling coefficient matrix of the i-th flexible attachment vibration to the assembly translation, ∈3×m;

[0031] F sci is the flexible coupling coefficient matrix of the i-th flexible attachment vibration to the assembly rotation, ∈3×m;

[0032] F si is the flexible coupling coefficient matrix of the vibration of the i-th flexible attachment to its own rotation, ∈3×m;

[0033] R sai is the rigid coupling coefficient matrix between the rotation of the i-th flexible attachment and the rotation of the assembly, ∈3×3;

[0034] m is the modal order.

[0035] Among them, in the step 2, the rigid-flexible coupling influence term of the component on the spacecraft is proposed, and the flexible coupling coefficient formula of the dual-axis drive freedom component is established. The specific steps are as follows:

[0036] Step 2.1, define the system control coordinate system fixed to the spacecraft, recorded as the {0} system; establish the accessory installation coordinate system fixed to the spacecraft, recorded as the {1} system; establish the accessory body coordinate system fixed to the first-level driving degree of freedom accessory, recorded as the {2} system; establish the accessory body coordinate system fixed to the second-level driving degree of freedom accessory, recorded as the {3} system, and the two levels of driving degrees of freedom revolve around the x-axis of the {2} system and the y-axis of the {3} system respectively;

[0037] Step 2.2, according to the definition of the coordinate system, the rotation from the {1} system to the {3} system is a two-degree-of-freedom rotation around x first and then around y. Let β be the rotation angle of the flexible attachment from the {2} system to the {3} system around y. The rotation matrix around y of the {3} system back to the {2} system is: α is the rotation angle around x from the {1} system to the {2} system. The {2} system rotates back to the {1} system with the rotation attitude matrix around x. The finite element system to the {3} system is a constant matrix {1} system to system control system is a constant matrix The posture matrix relationship is:

[0038] For the rigid-flexible coupling dynamic equation of the system, the rigid-flexible coupling influence of its components on the spacecraft is F tci 、F sci ; According to the structural finite element method, the expression of the attachment flexibility coupling coefficient is obtained as follows:

[0039]

[0040] According to the above attitude matrix, the attachment flexibility coupling coefficient is expanded as follows:

[0041]

[0042] in D i is the rigid body modal matrix of the flexible attachment structure; m i is the mass matrix of the flexible attachment structure; Φ i It is the vibration mode array of the flexible accessory structure.

[0043] Among them, in the step 3, based on the flexible coupling coefficient formula, the flexible simulation coefficient expression is defined, and the transformation relationship between the flexible coupling coefficient and the flexible simulation coefficient is established. The specific steps are as follows:

[0044] Step 3.1, define the accessory simulation coefficients Ft and Fa as follows:

[0045]

[0046] Among them, m represents the flexible mode order, and the simulation coefficient is expanded into component form as follows:

[0047]

[0048] Step 3.2, for the expression of the accessory flexible coupling coefficient, expand it according to the definition of the simulation coefficient and obtain the following expression:

[0049]

[0050]

[0051]

[0052] Among them, R3 and Ftci have the same form; is a constant, is a variable, is a variable; Described as a constant in the control system, In the T-frame fixed connection system {2}, it is described as a constant and the above equations are substituted into the solution of the flexibility simulation coefficient.

[0053] Among them, in step 4, the established flexible simulation coefficient transformation relationship is expanded to separate variables, and a flexible coupling coefficient expression consisting of a dual-axis driving angle and a constant simulation coefficient term without an angle is established. The specific steps are as follows:

[0054] Step 4.1, expand the translational flexibility coupling coefficient Ftci and separate the variables as follows:

[0055]

[0056]

[0057] Pick:

[0058]

[0059]

[0060] in

[0061] Step 4.2, expand the rotational flexibility coupling coefficient Fsci = R1 + R2 + R3 and separate the variables, and expand the first term R1 as follows:

[0062]

[0063]

[0064] Defining rp Ι =[0 -rp3 rp2], rp ΙΙ=[rp3 0 -rp1], rp ΙΙΙ =[-rp2 rp1 0], then:

[0065]

[0066] The third term R3 is expanded as follows:

[0067]

[0068]

[0069] The second term R2 is expanded as follows:

[0070]

[0071]

[0072]

[0073] After simplification, we get the simplified form of Fsci:

[0074]

[0075] Where: Bs = [B1 B2 B3 B4 B5 B6 B7 B8 B9].

[0076] Wherein, in step 5, the established flexible coupling coefficient expression is processed according to the flexible component driving angle description coordinate system to further obtain the expression form relative to the spacecraft control coordinate system. The specific steps are as follows:

[0077] Step 5.1, set The first-stage drive angle and the second-stage drive angle of the accessory described by the control system are set to be integer multiples of 90° from the spacecraft {1} system to {0}. The posture matrix composed of 1, 0, and -1 elements is obtained

[0078]

[0079]

[0080]

[0081]

[0082] From this we get

[0083]

[0084] in,

[0085]

[0086] Step 5.2, rearrange the driving angle terms from the first-order terms to the second-order terms of trigonometric functions. Elements of the form:

[0087]

[0088] That is, the 6th element is moved forward to the 4th position, correspondingly:

[0089] becomes:

[0090]

[0091] becomes:

[0092]

[0093] The expression of the coupling coefficient after sorting is as follows:

[0094]

[0095]

[0096] In the formula, and is the constant matrix of the dual-degree-of-freedom flexible component that is independent of the driving angle, It is a variable matrix related to the dual-degree-of-freedom driving angle of the flexible component, reflecting the dynamic coupling effect of the flexible component on the spacecraft at different driving angles, and realizing decoupling analysis.

[0097] Beneficial effects:

[0098] 1. The method of the present invention is a decoupling analysis method for the dynamic effects of large movable flexible components, which solves the problem of dynamic modeling of large flexible spacecraft for control. Combined with the urgent engineering demand of space relative navigation in my country's aerospace engineering, a decoupling analysis method for the dynamic effects of large movable flexible components is given. This method can effectively reduce the workload of spacecraft dynamic modeling under any configuration of flexible components, and support the dynamic modeling of variable configuration spacecraft systems for control system design.

[0099] 2. The present invention obtains the simulation coefficient of the flexible component without angle by decoupling the dynamic coupling effect from the driving angle, and further can obtain the dynamic coupling effect of the flexible component on the system at any angle during the driving process. The present invention establishes a rigid-flexible coupling coefficient model of a large flexible component that can be driven with two degrees of freedom, and proposes a dynamic influence analysis method for decoupling the component from the driving angle. Specifically, a rigid-flexible coupling dynamic model of a spacecraft with a flexible attachment is established, and an expression for the flexible coupling coefficient between the two-degree-of-freedom flexible attachment and the spacecraft is further given. The expression reflects the transformation relationship from the finite element coordinate system of the flexible component structure to the spacecraft control coordinate system.

[0100] 3. The flexible coupling coefficient formula for decoupling the flexible attachment and the two-degree-of-freedom driving angle proposed in the present invention can obtain the flexible coupling coefficient of the flexible component and the spacecraft according to different driving angles, thereby realizing the decoupling of the flexible coupling effect of the component and the driving angle, and can be used for system dynamics modeling of large movable flexible components under any configuration. BRIEF DESCRIPTION OF THE DRAWINGS

[0101] Figure 1 It is a schematic diagram of the coordinate system definition of the present invention. DETAILED DESCRIPTION

[0102] The present invention is described in detail below with reference to the accompanying drawings and embodiments.

[0103] The basic idea of ​​the decoupling analysis method of the dynamic influence of a large movable flexible component of the present invention is to establish a rigid-flexible coupling coefficient model of a large flexible component that can be driven with two degrees of freedom, and propose a dynamic influence analysis method of decoupling the component from the driving angle, which includes the following steps:

[0104] Step 1, establish a rigid-flexible coupling dynamics model of the system (rigid-flexible coupling dynamics equation of a flexible component with dual driving degrees of freedom);

[0105] Step 2: propose the rigid-flexible coupling influence of components on the spacecraft and establish the flexible coupling coefficient formula of the dual-axis drive freedom component;

[0106] Step 3, based on the flexible coupling coefficient formula, define a flexible simulation coefficient expression, and establish a transformation relationship between the flexible coupling coefficient and the flexible simulation coefficient;

[0107] Step 4, expand the established flexible simulation coefficient transformation relationship, perform variable separation, and establish a flexible coupling coefficient expression consisting of a dual-axis driving angle and a constant simulation coefficient term without an angle;

[0108] Step 5, process the coordinate system according to the description of the driving angle of the flexible component to further obtain the expression form relative to the spacecraft control coordinate system, and realize the decoupling analysis of the dynamic influence of the movable flexible component.

[0109] Specifically, in step 1, the rigid-flexible coupling dynamic equation of the flexible component with dual driving degrees of freedom established by the present invention is as follows:

[0110]

[0111]

[0112]

[0113]

[0114] Wherein, equation (1) is the system center of mass translation equation, equation (2) is the system rotation equation around the center of mass, (3) is the flexible attachment control equation, and (4) is the flexible attachment vibration equation. Equations (1) and (2) are based on the attitude control coordinate system of the assembly. i is the number of the flexible attachment of the system, and (3) is based on the local coordinate system of the flexible attachment. n is the total number of flexible attachments contained in the assembly corresponding to the working condition.

[0115] ω s is the angular velocity array of the assembly, measured in b frame, ∈3×1;

[0116] is the antisymmetric matrix of the angular velocity array, ∈3×3;

[0117] M is the total mass matrix of the assembly, ∈3×3;

[0118] I s is the inertia matrix of the assembly relative to the system center of mass, measured in the b system, ∈3×3;

[0119] P s is the external force matrix acting on the assembly, measured in b system, ∈3×1;

[0120] T s is the external moment matrix acting on the assembly, measured in b system, ∈3×1;

[0121] T ai is the control torque array acting on the nth flexible attachment, measured in the local coordinate system of the flexible attachment, ∈3×1;

[0122] I ai is the inertia matrix of the i-th flexible attachment relative to its respective connection point, measured in the local coordinate system of the flexible attachment, ∈3×3;

[0123] ω ai is the rotational angular velocity array of the i-th flexible attachment, measured in the local coordinate system of the flexible attachment, ∈3×1;

[0124] Ω aiis the diagonal matrix of modal frequencies of the i-th flexible attachment, ∈m×m;

[0125] η ai is the modal coordinate matrix of the i-th flexible attachment, ∈m×1;

[0126] ζ ai is the modal damping coefficient of the i-th flexible attachment, which is generally taken as 0.005;

[0127] F tci is the flexible coupling coefficient matrix of the i-th flexible attachment vibration to the assembly translation, ∈3×m;

[0128] F sci is the flexible coupling coefficient matrix of the i-th flexible attachment vibration to the assembly rotation, ∈3×m;

[0129] F si is the flexible coupling coefficient matrix of the vibration of the i-th flexible attachment to its own rotation, ∈3×m;

[0130] R sai is the rigid coupling coefficient matrix between the rotation of the i-th flexible attachment and the rotation of the assembly, ∈3×3;

[0131] m is the modal order.

[0132] In step 2, the rigid-flexible coupling influence term of the component on the spacecraft is proposed, and the flexible coupling coefficient formula of the dual-axis drive freedom component is established. The specific steps are as follows:

[0133] Step 2.1, define the system control coordinate system fixed to the spacecraft, recorded as {0} system; establish the accessory installation coordinate system fixed to the spacecraft, recorded as {1} system; establish the accessory body coordinate system fixed to the first-level driving degree of freedom accessory, recorded as {2} system; establish the accessory body coordinate system fixed to the second-level driving degree of freedom accessory, recorded as {3} system. The coordinate system definitions are as follows: Figure 1 As shown, the two levels of driving freedom are respectively around the x-axis of the {2} system and the y-axis of the {3} system.

[0134] Step 2.2, according to the definition of the coordinate system, the rotation from the {1} system to the {3} system is a two-degree-of-freedom rotation around x first and then around y. Let β be the rotation angle of the flexible attachment from the {2} system to the {3} system around y, so the rotation matrix around y when the {3} system is transformed back to the {2} system is: α is the rotation angle around x from the {1} system to the {2} system, then the {2} system is transformed back to the {1} system with the rotation attitude matrix around x The finite element system to the {3} system is a constant matrix {1} system to system control system is a constant matrix The posture matrix relationship is:

[0135] For the rigid-flexible coupling dynamic equation of the system, the rigid-flexible coupling influence of its components on the spacecraft is F tci 、F sci .

[0136] According to the structural finite element method, the expression of the attachment flexibility coupling coefficient can be obtained as follows:

[0137]

[0138] Further according to the above attitude matrix, the attachment flexibility coupling coefficient is expanded as follows:

[0139]

[0140] in:

[0141]

[0142]

[0143]

[0144] D i Rigid body modal matrix of flexible attachment structure; m i is the mass matrix of the flexible attachment structure; Φ i It is the vibration mode array of the flexible accessory structure.

[0145] In step 3, based on the flexible coupling coefficient formula, the flexible simulation coefficient expression is defined to establish a transformation relationship between the flexible coupling coefficient and the flexible simulation coefficient. The specific steps are as follows:

[0146] Step 3.1, define the accessory simulation coefficients Ft and Fa as follows:

[0147]

[0148] Among them, m represents the flexible mode order, and the simulation coefficient is expanded into component form as follows:

[0149]

[0150] Step 3.2, for the expression of the accessory flexible coupling coefficient, expand it according to the definition of the simulation coefficient and obtain the following expression:

[0151]

[0152]

[0153]

[0154] Among them, R3 and Ftci have the same form.

[0155] and, is a constant, is a variable, is a variable; Described as a constant in the control system, In the T-frame fixed connection system {2}, it is described as a constant and the above equations are substituted into the solution of the flexibility simulation coefficient.

[0156] In step 4, the established flexible simulation coefficient transformation relationship is expanded to separate variables, and a flexible coupling coefficient expression consisting of a dual-axis driving angle and a constant simulation coefficient term without an angle is established. The specific steps are as follows:

[0157] Step 4.1, expand the translational flexibility coupling coefficient Ftci and separate the variables as follows:

[0158]

[0159]

[0160] Pick:

[0161]

[0162] in

[0163] Step 4.2, expand the rotational flexibility coupling coefficient Fsci = R1 + R2 + R3 and separate the variables, and expand the first term R1 as follows:

[0164]

[0165]

[0166] Defining rp Ι =[0 -rp3 rp2], rp ΙΙ =[rp3 0 -rp1], rp ΙΙΙ =[-rp2 rp1 0], then:

[0167]

[0168] The third term R3 is expanded as follows:

[0169]

[0170]

[0171] The second term R2 is expanded as follows:

[0172]

[0173]

[0174]

[0175] To simplify further, let

[0176]

[0177]

[0178]

[0179]

[0180]

[0181]

[0182]

[0183]

[0184] Then we finally get the simplified form of Fsci:

[0185]

[0186] Where: Bs = [B1 B2 B3 B4 B5 B6 B7 B8 B9].

[0187] In step 5, the established flexible coupling coefficient expression is processed according to the flexible component drive angle description coordinate system to further obtain the expression form relative to the spacecraft control coordinate system. The specific steps are as follows:

[0188] Step 5.1, the formula of the flexible coupling coefficient has been obtained as follows:

[0189] Ftai=At 3×8 ·η 8×1

[0190]

[0191] However, α and β are both angles described in the initial time {1} system. If the accessory drive angle input must describe the rotation angle in the spacecraft control system, the transformation matrix needs to be considered The sign of η changes, so it is necessary to further change the flexible coupling coefficient obtained in step (4) to achieve the description coordinate system of the transformation angle.

[0192] set up The first-stage drive angle and the second-stage drive angle of the attachment described by the control system are set to be integer multiples of 90° from the spacecraft {1} system to {0}. The attitude matrix consists of 1, 0, -1 as elements. Then:

[0193]

[0194]

[0195]

[0196]

[0197] therefore:

[0198]

[0199]

[0200] in,

[0201]

[0202] Step 5.2, rearrange the driving angle terms from the first-order terms to the second-order terms of trigonometric functions. Elements of the form:

[0203]

[0204] That is, the 6th element is moved forward to the 4th position, correspondingly:

[0205] becomes:

[0206]

[0207] becomes:

[0208]

[0209] Therefore, the expression of the coupling coefficient after sorting is as follows:

[0210]

[0211]

[0212] In the formula, and is the constant matrix of the dual-freedom flexible component that is independent of the driving angle, It is a variable matrix related to the dual-degree-of-freedom driving angle of the flexible component, which can reflect the dynamic coupling effect of the flexible component on the spacecraft at different driving angles, and realizes a decoupling analysis method for the dynamic effect of large movable flexible components.

[0213] In summary, the above are only preferred embodiments of the present invention and are not intended to limit the protection scope of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for decoupling the dynamic influence of a movable flexible component, characterized in that: The steps include: Step 1, establishing a rigid-flexible coupling dynamic equation of a flexible component with dual driving degrees of freedom; Step 2: propose the rigid-flexible coupling influence of components on the spacecraft and establish the flexible coupling coefficient formula of the dual-axis drive freedom component; Step 3, based on the flexible coupling coefficient formula, define a flexible simulation coefficient expression, and establish a transformation relationship between the flexible coupling coefficient and the flexible simulation coefficient; Step 4, expand the established flexible simulation coefficient transformation relationship, perform variable separation, and establish a flexible coupling coefficient expression consisting of a dual-axis driving angle and a constant simulation coefficient term without an angle; Step 5, processing is performed according to the description coordinate system of the flexible component driving angle, and the expression form relative to the spacecraft control coordinate system is further obtained to realize the decoupling analysis of the dynamic influence of the movable flexible component; In step 1, the rigid-flexible coupling dynamic equation of the flexible component with dual driving degrees of freedom is as follows: Wherein, equation (1) is the translational motion equation of the system center of mass, equation (2) is the rotational motion equation of the system around the center of mass, (3) is the control equation of the flexible attachment, and (4) is the vibration equation of the flexible attachment; equations (1) and (2) are based on the attitude control coordinate system of the assembly; i is the number of the flexible attachment carried by the system, and equation (3) is based on the local coordinate system of the flexible attachment; n is the total number of flexible attachments contained in the assembly corresponding to the working condition; ω s is the angular velocity array of the assembly, measured in b frame, ∈3×1; is the antisymmetric matrix of the angular velocity array, ∈3×3; M is the total mass matrix of the assembly, ∈3×3; I s is the inertia matrix of the assembly relative to the system center of mass, measured in the b system, ∈3×3; P s is the external force array acting on the assembly, measured in b system, ∈3×1; T s is the external moment matrix acting on the assembly, measured in b system, ∈3×1; T ai is the control torque array acting on the nth flexible attachment, measured in the local coordinate system of the flexible attachment, ∈3×1; I ai is the inertia matrix of the i-th flexible attachment relative to its respective connection point, measured in the local coordinate system of the flexible attachment, ∈3×3; ω ai is the rotational angular velocity array of the i-th flexible attachment, measured in the local coordinate system of the flexible attachment, ∈3×1; Ω ai is the diagonal matrix of modal frequencies of the i-th flexible attachment, ∈m×m; η ai is the modal coordinate matrix of the i-th flexible attachment, ∈m×1; ζ ai is the modal damping coefficient of the i-th flexible attachment; F tci is the flexible coupling coefficient matrix of the i-th flexible attachment vibration to the assembly translation, ∈3×m; F sci is the flexible coupling coefficient matrix of the i-th flexible attachment vibration to the assembly rotation, ∈3×m; F si is the flexible coupling coefficient matrix of the vibration of the i-th flexible attachment to its own rotation, ∈3×m; R sai is the rigid coupling coefficient matrix between the rotation of the i-th flexible attachment and the rotation of the assembly, ∈3×3; m is the modal order; In step 2, the rigid-flexible coupling influence term of the component on the spacecraft is proposed, and the flexible coupling coefficient formula of the dual-axis drive freedom component is established. The specific steps are as follows: Step 2.1, define the system control coordinate system of the fixed spacecraft, recorded as {0} system; establish the accessory installation coordinate system fixed to the spacecraft, recorded as {1} system; Establish an accessory body coordinate system fixedly connected to the first-level driving degree of freedom accessory, denoted as {2} system; Establish an accessory body coordinate system fixedly connected to the second-level driving degree of freedom accessory, denoted as the {3} system, and the two levels of driving degrees of freedom revolve around the x-axis of the {2} system and the y-axis of the {3} system respectively; Step 2.2, according to the definition of the coordinate system, the rotation from the {1} system to the {3} system is a two-degree-of-freedom rotation around x first and then around y. Let β be the rotation angle of the flexible attachment from the {2} system to the {3} system around y. The rotation matrix around y of the {3} system back to the {2} system is: α is the rotation angle around x from the {1} system to the {2} system. The {2} system rotates back to the {1} system with the rotation attitude matrix around x. The finite element system to the {3} system is a constant matrix {1} system to system control system is a constant matrix The posture matrix relationship is: For the rigid-flexible coupling dynamic equation of the system, the rigid-flexible coupling effect of its components on the spacecraft is F tci 、F sci ; According to the structural finite element method, the expression of the attachment flexibility coupling coefficient is obtained as follows: According to the above attitude matrix, the attachment flexibility coupling coefficient is expanded as follows: in D i is the rigid body modal matrix of the flexible attachment structure; m i is the mass matrix of the flexible attachment structure; Φ i is the vibration mode array of the flexible attachment structure; In step 3, based on the flexible coupling coefficient formula, the flexible simulation coefficient expression is defined to establish a transformation relationship between the flexible coupling coefficient and the flexible simulation coefficient. The specific steps are as follows: Step 3.1, define the accessory simulation coefficients Ft and Fa as follows: Among them, m represents the flexible mode order, and the simulation coefficient is expanded into component form as follows: Step 3.2, for the expression of the accessory flexible coupling coefficient, expand it according to the definition of the simulation coefficient and obtain the following expression: Among them, R3 and Ftci have the same form; is a constant, is a variable, is a variable; Described as a constant in the control system, In the T-frame fixed connection system {2}, it is described as a constant, and the above equations are substituted to solve the flexibility simulation coefficient; In step 4, the established flexible simulation coefficient transformation relationship is expanded to separate variables, and a flexible coupling coefficient expression consisting of a dual-axis driving angle and a constant simulation coefficient term without an angle is established. The specific steps are as follows: Step 4.1, expand the translational flexibility coupling coefficient Ftci and separate the variables as follows: Pick: Where At=[A1 A2 A3 A4 A5 A6 A7 A8], Step 4.2, expand the rotational flexibility coupling coefficient Fsci = R1 + R2 + R3 and separate the variables, and expand the first term R1 as follows: Defining rp Ι =[0-rp3 rp2], rp ΙΙ =[rp30-rp1], rp ΙIΙ =[-rp2 rp1 0], then: The third term R3 is expanded as follows: The second term R2 is expanded as follows: After simplification, we get the simplified form of Fsci: Where: Bs = [B1 B2 B3 B4 B5 B6 B7 B8 B9]; In step 5, the established flexible coupling coefficient expression is processed according to the flexible component drive angle description coordinate system to further obtain the expression form relative to the spacecraft control coordinate system. The specific steps are as follows: Step 5.1, set The first-stage drive angle and the second-stage drive angle of the accessory described by the control system are set to be integer multiples of 90° from the spacecraft {1} system to {0}. The posture matrix composed of 1, 0, and -1 elements is obtained From this we get in, Step 5.2, rearrange the driving angle terms from the first-order terms to the second-order terms of trigonometric functions. Elements of the form: That is, the 6th element is moved forward to the 4th position, correspondingly: becomes: becomes: The expression of the coupling coefficient after sorting is as follows: In the formula, and is the constant matrix of the dual-freedom flexible component that is independent of the driving angle, It is a variable matrix related to the dual-degree-of-freedom driving angle of the flexible component, reflecting the dynamic coupling effect of the flexible component on the spacecraft at different driving angles, and realizing decoupling analysis.

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