Variable damping semi-active vibration suppression method for flexible mechanical arm in aerospace space

Abstract

The invention discloses an aerospace space flexible mechanical arm variable damping semi-active vibration suppression method which comprises the steps that S1, a semi-active structure is designed, and a semi-active vibration suppression system is obtained; s2, theoretical analysis is conducted on the designed semi-active structure, and the force transmission relation and the kinematics relation between the magneto-rheological damper and the flexible mechanical arm system are deduced; s3, designing and solving a mechanical model of the magnetorheological damper; s4, dynamic modeling is conducted on the space flexible mechanical arm system through a finite element method; and S5, giving a specific mechanism of the method after numerical simulation and experiments of various working conditions, and obtaining an optimal vibration suppression effect. According to the method, the adaptability is higher, and the method is particularly suitable for vibration suppression of complex flexible structures such as space flexible mechanical arms.

Description

Technical Field

[0001] The present invention relates to the technical field of space manipulators, and in particular to a variable damping semi-active vibration suppression method for a flexible aerospace manipulator arm. Background Art

[0002] Space manipulators are core equipment for space station construction, maintenance, and deep space exploration missions, and are widely used for complex operations such as spacecraft docking and cargo handling. Unlike rigid manipulators, space manipulators are typically designed as flexible systems with low damping and stiffness. This flexible structure not only offers advantages such as low energy consumption, lightweight design, and reduced contact impact, but also exhibits good adaptability in dynamic environments.

[0003] However, the low damping characteristics of flexible systems make them prone to elastic vibrations. Due to the long decay time of these vibrations, these vibrations can significantly affect the positioning accuracy of the manipulator's end effector. This accuracy issue can cause delays in space missions, or even render some missions impossible to complete. Therefore, effectively suppressing elastic vibrations and improving the positioning accuracy of space manipulators has become a crucial step in ensuring their stable operation.

[0004] At present, there are relatively few studies on the vibration suppression technology of flexible arms, and the technical development in related fields is still insufficient. The existing relevant patents and literature are as follows: (1) Invention patent CN201710021067.7 provides an active vibration controller for a flexible manipulator, including a sensing piezoelectric sheet, a driving piezoelectric sheet, a data acquisition board, a driving power supply, a driving power supply system and a host computer. This invention solves the problems of the traditional passive or active control method of the flexible manipulator by positioning and installing the sensing piezoelectric sheet and the driving piezoelectric sheet on the flexible manipulator. It has the advantages of simple structure and high piezoelectric efficiency. According to the different detection data of the sensing piezoelectric sheet, the position of the vibrating driving piezoelectric sheet is different, and the vibration suppression effect is better. (2) Invention patent CN202110336162.2 provides a parametric resonance vibration reduction method for a flexible manipulator based on modal interaction. This method extracts the parameter excitation that can cause the resonance of the flexible manipulator, discretizes the flexible manipulator, and establishes a nonlinear dynamic model. Based on the modal interaction principle, the control equation of the parametric resonance vibration absorber is constructed, the steady-state response is analyzed, and the approximate analytical solution is solved. Finally, by analyzing the influence of the vibration absorber control coefficient, the optimal control coefficient is selected to suppress the parametric resonance of the flexible manipulator. (3) Invention patent CN201911358470.4 provides a vibration reduction method for a single-link flexible manipulator. This method is based on the Euler-Bernoulli beam theory, establishes the bending vibration equation of the flexible manipulator in a generalized coordinate system, and establishes a dynamic model through finite segment discretization. By analyzing the model, the natural vibration frequency is obtained, and the current signal time is designed according to the different motion stages of the manipulator to realize vibration reduction trajectory planning. Combined with the command shaping and trajectory generation feedforward method, simple operation and obvious vibration reduction effect are achieved, which has good industrial application value. However, the above methods are all insufficient when applied to the field of vibration suppression of space flexible manipulators. Method (1) belongs to the active vibration control method. The system requires corresponding sensors and actuators. In the field of vibration suppression of space manipulators, there are problems with the reliability of the method. At the same time, the weight of the space manipulator itself and its load are huge, and the energy consumed by the actuator is much greater than that of the industrial manipulator. Method (2) uses the vibration absorption principle and has an excellent suppression effect on a single vibration frequency. However, the working conditions of the space manipulator are complex and often involve multi-frequency vibrations, so the effect of the vibration absorption method is greatly reduced. Method (3) starts from the aspect of trajectory planning and weakens the generation of vibration from the source. It can be used together with the method of the present invention. However, relying solely on the trajectory planning method cannot effectively suppress the elastic vibration generated by unknown external disturbances.

[0005] To this end, the present invention uses magnetorheological dampers (MRDs) as controllable damping elements to semi-actively suppress vibrations in a flexible space manipulator. Similar literature to the underlying theory of this invention includes: Patent CN201310244097.6, which relates to a flexible manipulator vibration reduction device and method based on controllable stiffness and damping, comprising a flexible manipulator, a magnetorheological damping device, and a linear guide assembly. The magnetorheological damping device consists of a magnetic circuit core, an excitation coil, a coil baffle, a magnetorheological elastomer, and a liquid damper. Its stiffness and damping can be controlled by adjusting the excitation current to meet internal resonance requirements and dissipate vibration energy. However, this invention primarily addresses the vibration suppression of a cantilever beam under a linear guide and does not consider the actual operating conditions of a flexible aerospace manipulator operating through joint rotation. Therefore, the vibration suppression conditions studied differ from the actual operating conditions of a space manipulator. Summary of the Invention

[0006] In view of the shortcomings of the prior art, the purpose of the present invention is to provide a variable damping semi-active vibration suppression method for aerospace flexible manipulators, which is used to solve some difficulties in suppressing elastic vibration of aerospace flexible manipulators at the current stage. For example: (1) The reliability of the vibration reduction method, that is, the working environment of the space flexible manipulator is harsh, such as extreme temperature changes, solar radiation pressure, electromagnetic interference, etc. These nonlinear disturbances have different performances, and the reliability of traditional vibration suppression methods faces huge challenges; (2) Excessive energy consumption, that is, the energy of the space station is limited, and the space manipulator uses traditional methods, such as active control, which consumes a lot of energy; (3) High cost and complexity, that is, the traditional vibration suppression system requires additional sensors, actuators and computing resources, which greatly increases the hardware cost, complexity and maintenance difficulty of the system; (4) The diversity of working conditions, that is, the space manipulator will face working conditions such as high speed and light load or low speed and heavy load, and the vibration frequency range during operation is relatively wide, which will lead to some passive control methods, such as vibration absorption technology, having insufficient vibration reduction effects. In order to achieve the above-mentioned purpose and other advantages of the present invention, a variable damping semi-active vibration suppression method for aerospace flexible manipulators is provided, comprising:

[0007] S1. Design a semi-active structure to obtain a semi-active vibration suppression system;

[0008] S2. Conduct theoretical analysis on the designed semi-active structure and derive the force transmission relationship and kinematic relationship between the magnetorheological damper and the flexible robotic arm system;

[0009] S3. Design and solve the mechanical model of magnetorheological damper;

[0010] S4. Dynamic modeling of the spatial flexible manipulator system using the finite element method;

[0011] S5. After numerical simulation and experiments of various working conditions, the specific mechanism of the method is given to obtain the best vibration suppression effect.

[0012] Preferably, the semi-active structure includes a rigid joint, a flexible arm fixedly connected to the rigid joint, a load fixedly connected to the flexible arm, a connector connected to the flexible arm, and an MR damper movably connected to the connector.

[0013] Preferably, the derivation of the force transmission relationship and kinematic relationship between the magnetorheological damper and the flexible robotic arm system in step S2 specifically includes:

[0014] The damper piston displacement s is derived p The lateral displacement u of the flexible arm at the hinge position D relationship;

[0015] Derived piston speed and lateral deformation velocity relationship;

[0016] Derivation of the damping force F p and the damping force F D force transmission relationship.

[0017] Preferably, the mechanical model in step S3 specifically includes:

[0018] Design a nonlinear dual viscosity model;

[0019] By outputting the damping force F p About piston speed The MRD mechanical model was fitted based on the experimental test curve.

[0020] Preferably, the step S4 specifically includes the following steps:

[0021] S41. Select the Euler-Bernoulli beam as the physical model of the flexible manipulator.

[0022] S42. Calculate the kinetic energy and potential energy of the system, including the kinetic energy of the rigid joint, the kinetic energy of the flexible arm, the kinetic energy of the end load, and the elastic potential energy of the arm;

[0023] S43. Use the finite element method to discretize the flexible arm model, divide the arm into N units, with a total of N+1 nodes, and each node has two degrees of freedom, namely lateral deformation u and rotation;

[0024] S44. Derive the kinetic equation of the i-th unit;

[0025] S45. Derive the complete system dynamics equation.

[0026] Compared with the prior art, the present invention has the following beneficial effects: the present invention addresses the elastic vibration problem of aerospace space manipulators in complex working environments and proposes an effective vibration suppression method, which aims to effectively suppress the elastic vibration of the space manipulator while ensuring the reliability of the system, saving space energy, and reducing maintenance costs. The present invention adopts variable damping technology to effectively suppress the elastic vibration of the flexible manipulator system. By establishing a finite element dynamic model of the flexible manipulator system, a reasonable damper mechanical model is proposed to describe the dissipation process of vibration energy in the system. At the same time, the vibration suppression phenomenon is analyzed through numerical simulation, and the vibration suppression mechanism of the semi-active control system is deeply studied. Ultimately, the effectiveness of the present invention is verified, and a significant improvement in vibration suppression efficiency is achieved. By accurately simulating and analyzing the vibration characteristics of the flexible manipulator system, the method of the present invention can provide an effective vibration suppression theoretical basis for the system regardless of the working conditions of the space manipulator. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] Figure 1 A schematic diagram of a physical model of a flexible manipulator according to the variable damping semi-active vibration suppression method of an aerospace flexible manipulator of the present invention;

[0028] Figure 2 Schematic diagram of the i-th unit in the xOy coordinate system according to the first embodiment of the variable damping semi-active vibration suppression method of the aerospace flexible manipulator of the present invention;

[0029] Figure 3 Schematic diagram of the process of the variable damping semi-active vibration suppression method of the aerospace flexible manipulator according to the present invention;

[0030] Figure 4 Schematic diagram of the overall structure of a space manipulator containing an MR damper according to the variable damping semi-active vibration suppression method of an aerospace flexible manipulator of the present invention;

[0031] Figure 5 A distribution diagram of geometric parameters of a semi-active structure in an embodiment of the variable damping semi-active vibration suppression method for an aerospace flexible manipulator according to the present invention;

[0032] Figure 6 A comparison diagram of various parts before and after vibration in an embodiment of the variable damping semi-active vibration suppression method for aerospace flexible manipulator according to the present invention;

[0033] Figure 7 A force transmission relationship diagram of a semi-active structure in an embodiment of the variable damping semi-active vibration suppression method for an aerospace flexible manipulator according to the present invention;

[0034] Figure 8Schematic diagram of a damper force-velocity curve under experimental test in an embodiment of the variable damping semi-active vibration suppression method for aerospace flexible manipulator according to the present invention;

[0035] Figure 9 A comparison diagram of a curve after mechanical model fitting and an actual mechanical curve of the variable damping semi-active vibration suppression method for aerospace flexible manipulators according to the present invention;

[0036] Figure 10 One of the two typical working conditions of the space flexible manipulator in the embodiment of the variable damping semi-active vibration suppression method of the aerospace space flexible manipulator according to the present invention: a trapezoidal velocity curve diagram;

[0037] Figure 11 One of the two typical working conditions of the space flexible manipulator in the embodiment of the variable damping semi-active vibration suppression method of the aerospace space flexible manipulator according to the present invention: a fifth-order polynomial trajectory diagram;

[0038] Figure 12 4. This is a diagram showing the vibration suppression results of a space manipulator when the working condition is a trapezoidal velocity curve in an embodiment of the variable damping semi-active vibration suppression method for a space flexible manipulator according to the present invention;

[0039] Figure 13 4. This is a diagram showing the vibration suppression results of a space manipulator when the working condition is a fifth-order polynomial trajectory in an embodiment of the variable damping semi-active vibration suppression method for an aerospace flexible manipulator according to the present invention;

[0040] Figure 14 Graph showing the time-domain vibration response of the system when the current is 0.0A, 0.1A, and 0.2A in an embodiment of the variable damping semi-active vibration suppression method for aerospace flexible manipulator according to the present invention;

[0041] Figure 15 Graphs showing the time-domain vibration responses of the system when the currents are 0.3A, 0.4A, and 0.5A in an embodiment of the variable damping semi-active vibration suppression method for aerospace flexible manipulators according to the present invention;

[0042] Figure 16 Graph showing the time-domain vibration responses of the system when the currents are 0.6A, 0.7A, and 0.9A in an embodiment of the variable damping semi-active vibration suppression method for aerospace flexible manipulators according to the present invention. DETAILED DESCRIPTION

[0043] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0044] Reference Figure 1 A variable damping semi-active vibration suppression method for a flexible aerospace manipulator comprises:

[0045] S1. Design a semi-active structure to obtain a semi-active vibration suppression system; Figure 4 As shown, the semi-active structure includes a rigid joint, a flexible arm, an end load, an MR damper and an adapted connection.

[0046] S2. Conduct theoretical analysis on the designed semi-active structure and derive the force transmission relationship and kinematic relationship between the magnetorheological damper and the flexible manipulator system; e.g. Figure 5 As shown in Figure 2, the planar geometric parameters of the semi-active structure include: the length of the rigid link L1, the axis distance between the MR damper and the arm L2, and the axis distance between the hinge 2 and the arm L3. When the flexible arm vibrates, a lateral displacement u will occur at the hinge 2. D , the vibration energy is transmitted through the rigid connecting rod, the damper starts to work, and its piston displacement is s p , this situation is like Figure 6-7 shown.

[0047] Furthermore, the damper piston displacement s is derived p The lateral displacement u of the flexible arm at the hinge position D The relationship is as follows:

[0048]

[0049] Derived piston speed and lateral deformation velocity The relationship is as follows:

[0050]

[0051] Derivation of the damping force F p and the damping force F D The force transmission relationship, such as Figure 7 shown.

[0052]

[0053] k1(u D )、k2(u D ) are all related to the lateral displacement u D The associated proportionality coefficient represents the scaling effect of the semi-active structure on the damping force.

[0054] S3. Design and solve the mechanical model of the magnetorheological damper; the mechanical model can accurately describe the mechanical properties of the damping fluid before and after yielding. Figure 8-9 , specifically including the following steps:

[0055] According to its constitutive model, the designed nonlinear dual viscosity model is as follows:

[0056]

[0057] Where Cτ(I C ) can be written as τ1(I C ), yield coefficient V2=τ1(I C ) / γ p , γ p is the shear rate, and it is only necessary to obtain the flow coefficient V1 (I C ) and yield coefficient V2(I C ), the mechanical expression of the MR damper can be obtained.

[0058] By outputting the damping force F p About piston speed The MRD mechanical model is fitted with the experimental test curve, and the specific expressions of its coefficients are as follows:

[0059] V2(I C )=-261.5I C 5 +877.5I C 4 -1259.4I C 3 +798.7I C 2 -41.9I C +14.5

[0060] V1(I C )=12.4I C 5 -13.9I C 4 -4.0I C 3 +6.55I C 2 +0.937I C +0.385.

[0061] S4. Use the finite element method to perform dynamic modeling of the space flexible manipulator system; use the finite element method to describe the elastic displacement function and construct the dynamic model of the space flexible manipulator. The specific process is as described above. This modeling is different from the traditional space flexible manipulator modeling and can more accurately describe the vibration of the flexible manipulator system under semi-active control. The role of the MR damper is added to the dynamic model of the flexible manipulator to obtain the rigid-flexible coupling dynamic model of the semi-active control system. Because the diameter of the hinge is much smaller than the length of the arm, the input damping force Considered as a concentrated load, the added force vector is as follows:

[0062]

[0063] Where x0 represents the point of action, ranging from [0,L e ], can be taken as 0, is the Dirac delta function, F i δ is the force on the i-th element, with the value of i being determined based on the hinge position. It is important to note that while the damper is added as an external force, substituting the damper's mechanical model into the equation actually affects the parameters of the damping matrix in the finite element equations and should not be treated as an external force for numerical solutions.

[0064] Further, such as Figure 1-2 , step S4 specifically includes the following steps:

[0065] Step S4-1, select the Euler-Bernoulli beam as the physical model of the flexible manipulator: In the figure, the XOY coordinate system is a fixed coordinate system, and the xOy coordinate system is a follower coordinate system that is always tangent to the root axis of the flexible manipulator. The angle between the XOY coordinate system and the xOy coordinate system represents the rotation angle θ(t) of the rigid joint, τ(t) is the torque on the rigid joint, and J h is the moment of inertia of the rigid joint, R is the radius of the rigid joint, L is the length from point O of the rigid joint to the end of the undeformed flexible arm, m is the mass of the end load, ρ is the material density of the flexible arm, E is the Young's modulus of the flexible arm's material, I is the moment of inertia of the area, and A is the cross-sectional area of the arm. When the rigid joint rotates, the flexible arm rotates with it. The lateral deformation at x at time t is written as u(x, t). This deformation is relative in the xOy coordinate system. Considering this along with the arc length generated by the rotation angle θ(t), the absolute displacement at x at time t can be calculated.

[0066] Step S4-2, calculate the kinetic energy and potential energy of the system:

[0067] The expression of the kinetic energy T1 of the rigid joint is:

[0068]

[0069] The expression of the kinetic energy T2 of the flexible arm is:

[0070]

[0071] The expression of the kinetic energy T3 of the end load is:

[0072]

[0073] The expression of the elastic potential energy U of the arm is:

[0074]

[0075] Step S4-3, the finite element method is used to discretize the flexible arm model, and the arm is divided into N units, with a total of N+1 nodes. Each node has two degrees of freedom, namely the lateral deformation u and the angle μ. Each unit has four degrees of freedom, and the unit length L e =L / N, Figure 2 This is a schematic diagram of the system's i-th unit in the xOy coordinate system. The details are as follows:

[0076] Model the i-th unit and define the unit displacement array:

[0077] δ e =[u1,μ1,u2,μ2] T

[0078] Assume that the lateral deformation of any point in the i-th element is It can be expressed as follows using shape functions and element displacement arrays:

[0079]

[0080] The specific expression of the shape function is as follows:

[0081]

[0082] Step S4-4, deriving the dynamic equation of the i-th unit.

[0083] u(x,t) in the xOy coordinate system and the lateral deformation of any point in the i-th unit The relationship is as follows:

[0084]

[0085] Kinetic energy T2 of the i-th unit (i) The expression is as follows:

[0086]

[0087] The expression of the kinetic energy T3 of the end load is as follows:

[0088]

[0089] The potential energy U of the i-th unit (i) The expression is as follows:

[0090]

[0091] The Lagrangian function L of the i-th unit (i) =T2 (i) -U(i) Substituting into the Lagrange equation:

[0092]

[0093] By calculating and arranging the formulas, the dynamic model of the i-th unit is obtained as follows:

[0094]

[0095] Where,

[0096]

[0097] In step S4-5, the complete system dynamics equation is derived as follows:

[0098]

[0099] Where,

[0100] set up but

[0101] set up but set up

[0102] but

[0103]

[0104] in, is a 1×(2N+2) row matrix, is a (2N+2)×(2N+2) matrix, δ(t), is a column vector of (2N+2)×1, and at the same time, is the zero vector.

[0105] S5. After numerical simulation and experiments of various working conditions, the specific mechanism of the method is given to obtain the best vibration suppression effect. The working conditions include the path planning of the rotating joint. There are generally two methods: trapezoidal speed curve, see Figure 10 and polynomial function traces, see Figure 11 .

[0106] The specific mechanism of the semi-active vibration suppression method is: as the current increases, the system's vibration suppression performance goes from good to bad; and there is always an optimal control current range that makes the semi-active system's vibration suppression performance reach the optimal level.

[0107] The simulation results of the semi-active control system of the present invention ( Figure 14 、 Figure 15 and Figure 16 ) is discussed in detail, where the target angle θ of the trapezoidal velocity curve is 120°, the angular velocity ω is 3° / s, and the angular acceleration α is 15° / s 2 , the end load of the system is 1000kg.

[0108] Figure 14 The time-domain vibration responses of the lateral displacement u(L, t) of the space manipulator's end are shown for currents of 0.0A, 0.1A, and 0.2A, respectively. The system's vibration response without semi-active control is also shown for comparison. It can be seen that the system's vibration response decreases significantly under semi-active control, and the end's lateral displacement u(L, t) decreases as the current increases. These results indicate that within the current range of 0.0A to 0.2A, the higher the current, the better the system's vibration suppression performance.

[0109] Figure 15 The time-domain vibration response of the lateral displacement u(L,t) of the end of the spatial manipulator is shown for currents of 0.3A, 0.4A, and 0.5A, respectively. As the current increases, the maximum absolute value of the lateral displacement u(L,t) decreases, and the vibration decay time increases. The maximum values of |u(L,t)| are 54.298mm, 48.783mm, and 44.818mm for currents of 0.3A, 0.4A, and 0.5A, respectively. These results indicate that within the current range of 0.3A to 0.5A, whether the system's vibration suppression effectiveness improves with increasing current depends on the priority of the maximum displacement value and the vibration decay time in the vibration suppression requirements. If reducing the maximum displacement value is more pressing, increasing the current will result in better vibration suppression.

[0110] Figure 16 The time-domain vibration responses of the lateral displacement u(L,t) of the end of the spatial manipulator are shown for currents of 0.6A, 0.7A, and 0.9A, respectively. The figure also shows the system vibration response when the damper is treated as an ideal rigid support for comparison. In this case, as the current increases, the maximum value of |u(L,t)| does not decrease significantly, but the vibration decay time increases, and the vibration response trend approaches that of an ideal rigid support. These results indicate that within the current range of 0.6A to 0.9A, the higher the current, the worse the system's vibration suppression performance.

[0111] Combined with the above analysis results, it can be clearly seen that the variable damping semi-active vibration suppression method for aerospace flexible manipulators proposed in the present invention has demonstrated significant technical advantages in practical applications. First, the semi-active vibration suppression method proposed in the present invention can always find a reasonable control current range under different working conditions, so that the semi-active structure can effectively suppress the elastic vibration of the space flexible manipulator. Secondly, the vibration suppression method of the present invention is applicable to a variety of vibration working conditions, including wide-band, multi-band and resonant frequency, and has good vibration suppression effects. In addition, while ensuring a good vibration suppression effect, the method of the present invention also has good system stability and excellent energy-saving performance. In summary, the present invention provides an effective method for suppressing elastic vibrations of aerospace flexible manipulators, which makes up for the shortcomings of existing methods and provides technical guarantees for high-precision, high-reliability and low-energy vibration control of space flexible mechanisms.

[0112] The number of devices and processing scales described herein are intended to simplify the description of the present invention, and the application, modification, and variation of the present invention will be apparent to those skilled in the art. Although the embodiments of the present invention have been disclosed above, they are not limited to the applications listed in the specification and embodiment. They can be applied to various fields suitable for the present invention. For those skilled in the art, additional modifications can be easily implemented. Therefore, the present invention is not limited to the specific details and illustrations shown and described herein without departing from the general concept defined by the claims and their equivalents.

Claims

1. A variable damping semi-active vibration suppression method for a flexible aerospace manipulator, characterized in that: The following steps are involved: S1. Design a semi-active structure to obtain a semi-active vibration suppression system; S2. Conduct theoretical analysis on the designed semi-active structure and derive the force transmission relationship and kinematic relationship between the magnetorheological damper and the flexible robotic arm system; S3. Design and solve the mechanical model of magnetorheological damper; S4. Dynamic modeling of the flexible robotic arm system in space using the finite element method; S5. After numerical simulation and experiments of various working conditions, the specific mechanism of the method is given to obtain the best vibration suppression effect.

2. The variable damping semi-active vibration suppression method for aerospace flexible manipulator according to claim 1, characterized in that: The semi-active structure includes a rigid joint, a flexible arm fixedly connected to the rigid joint, a load fixedly connected to the flexible arm, a connecting piece connected to the flexible arm, and an MR damper movably connected to the connecting piece.

3. The variable damping semi-active vibration suppression method for aerospace flexible manipulator according to claim 2, characterized in that: The derivation of the force transmission relationship and kinematic relationship between the magnetorheological damper and the flexible robotic arm system in step S2 specifically includes: The damper piston displacement s is derived p The lateral displacement u of the flexible arm at the hinge position D relationship; Derived piston speed and lateral deformation velocity relationship; Derivation of the damping force F p and the damping force F D force transmission relationship.

4. The variable damping semi-active vibration suppression method for aerospace flexible manipulator according to claim 1, characterized in that: The mechanical model in step S3 specifically includes: Design a nonlinear dual viscosity model; By outputting the damping force F p About piston speed The MRD mechanical model was fitted based on the experimental test curve.

5. The variable damping semi-active vibration suppression method for aerospace flexible manipulator according to claim 4, characterized in that: The step S4 specifically includes the following steps: S41. Select the Euler-Bernoulli beam as the physical model of the flexible manipulator. S42. Calculate the kinetic energy and potential energy of the system, including the kinetic energy of the rigid joint, the kinetic energy of the flexible arm, the kinetic energy of the end load, and the elastic potential energy of the arm; S43. Use the finite element method to discretize the flexible arm model, divide the arm into N units, with a total of N+1 nodes, and each node has two degrees of freedom, namely lateral deformation u and rotation; S44. Derive the kinetic equation of the i-th unit; S45. Derive the complete system dynamics equation.

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