A flexible robot arm main resonance control method based on nonlinear feedback and time delay feedback

Abstract

The application discloses a flexible mechanical arm main resonance control method based on nonlinear feedback and time delay feedback, and mainly comprises the following steps: discretizing the transverse deformation of the flexible mechanical arm by using the assumed mode method to obtain the vibration equation of the flexible mechanical arm; introducing a linear feedback control term, a nonlinear feedback control term and a time delay feedback control term to constitute a control signal model of a vibration absorber; constructing the control signal model of the vibration absorber; establishing a flexible mechanical arm system vibration control model of the vibration absorber by using Kane equation; applying a multi-scale method to solve the approximate analytical solution of the system vibration equation to obtain a single mode solution and a double mode solution of the system steady-state response; and finally, under the condition that the flexible mechanical arm system vibration control model of the vibration absorber coexists with the internal resonance condition and the main resonance condition, the main resonance of the flexible mechanical arm is controlled, and the mode amplitude of the first order mode of the flexible mechanical arm in the main resonance is well inhibited.

Description

TECHNICAL FIELD

[0001] The application provides a flexible manipulator main resonance control method based on nonlinear feedback and time delay feedback, and the main resonance control method based on nonlinear feedback and time delay feedback is proposed for the nonlinear dynamic characteristic problem of 1:1 main resonance of the flexible manipulator based on the saturation vibration reduction mechanism of internal resonance. BACKGROUND

[0002] With the advancement of industrialization, the flexible manipulator is gradually developing in the direction of light weight and heavy load, but the flexible characteristics of the flexible manipulator are more and more obvious, and the negative response caused by the flexibility cannot be ignored. The main resonance phenomenon caused by the periodic external excitation frequency or the periodic variable topology frequency of the flexible manipulator being equal to or close to the natural frequency of a certain order mode of the flexible manipulator is the most serious. In the main resonance state, the flexible manipulator will produce considerable vibration in a very short time, and once the amplitude exceeds the limit that can be borne by the structure, the rapid deterioration of the mechanism operation performance and the failure of the structure will be caused, so it is necessary to study the main resonance control of the flexible manipulator.

[0003] The saturation phenomenon is a special phenomenon in a nonlinear vibration system, and the excess energy is transmitted to the absorber through 1:2 internal resonance. When saturation occurs, with the increase of the amplitude of the external excitation, the amplitude of the controlled mode will not continue to increase with the increase of the amplitude of the external excitation, and the excess energy will be transmitted to the absorber. On the basis of the saturation principle, the application innovatively proposes a flexible manipulator main resonance control method with nonlinear feedback and time delay feedback, the energy transmission capacity of the flexible manipulator and the main resonance absorber is effectively improved by constructing controllable nonlinear coupling terms, and on this basis, a controllable time delay feedback term is further introduced, so that the time for the flexible manipulator to enter the saturation state is greatly shortened, the saturation amplitude is further reduced, the vibration reduction frequency band is widened, and the main resonance of the flexible manipulator is more effectively suppressed. SUMMARY

[0004] The application solves the technical problems in the prior art, and provides a flexible manipulator main resonance control method based on nonlinear feedback and time delay feedback, which is based on the internal resonance principle, the nonlinear saturation control principle and the time delay feedback control principle, and can effectively suppress the main resonance of the flexible manipulator.

[0005] The technical scheme for solving the technical problems of the application is as follows: a flexible manipulator main resonance control method based on nonlinear feedback and time delay feedback, and the method comprises the following specific steps:

[0006] Step one: make assumptions for the flexible manipulator and each joint of the system: the flexible manipulator is uniform and isotropic; all joints are rigid; the structure of the flexible manipulator conforms to the characteristics of the Euler-Bernoulli beam; and the longitudinal deformation of the flexible manipulator is not considered, and the deformation only occurs in the transverse direction.

[0007] Step two: discretize the transverse bending deformation of the flexible manipulator by using the assumed mode method:

[0008]

[0009] In formula (1), w(x, t) is the transverse deformation of the flexible manipulator, is the i-th order mode function, q i (t) represents the modal coordinates.

[0010] Step three: construct the absorber control signal. Introduce a position feedback coefficient to adjust the output stiffness of the motor, introduce a velocity feedback coefficient to adjust the damping of the absorber, and form a linear control term; introduce a nonlinear feedback control term to nonlinearly couple the response signals of the flexible manipulator and the absorber; introduce a time delay feedback control term to feedback the response signal of the absorber after a certain time delay; and finally, the linear feedback control term, the nonlinear feedback control term, and the time delay feedback control term constitute the control signal model of the absorber.

[0011] Step four: establish the dynamic model after introducing the absorber according to the Kane equation; take the first-order modal response of the flexible manipulator and the response of the absorber pendulum as the generalized coordinates; based on the internal resonance principle, adjust the position feedback gain coefficient k p of the absorber motor, so that the natural frequency of the absorber and the first-order modal natural frequency of the flexible manipulator have a 1:2 common relationship, and form internal resonance with the flexible manipulator.

[0012] Step five: introduce a weak harmonic excitation into the flexible manipulator system, and use the multiple scale method to obtain the approximate analytical solution of the dynamic equation, and obtain the single-mode solution and the double-mode solution of the modal response of the system.

[0013] Step six: analyze the internal resonance principle of the flexible manipulator system with the absorber under no external excitation according to the steady-state solution obtained in step five.

[0014] Step seven: perform stability analysis on the steady-state solution obtained by the multiple scale method in step five, and obtain the control parameter range that makes the system stable.

[0015] Step eight: according to the parameter range that makes the system stable obtained in step seven, design the absorber for controlling the main resonance of the flexible manipulator according to the actual working conditions and requirements.

[0016] Preferably, in each order mode of the bending deformation of the flexible manipulator, the first order mode has greater vibration energy in the lateral vibration, so that only the first order mode (n=1) which plays a major role in the deformation is considered, and the deflection of the flexible manipulator is:

[0017]

[0018] Preferably, the control model of the vibration absorber is composed of a linear control term, a nonlinear control term and a time delay control term:

[0019]

[0020] In formula (3), is the initial position of the pendulum rod, is the initial angular velocity, k p is the position feedback coefficient of the vibration absorber, k d is the velocity feedback coefficient, e1 represents the nonlinear feedback gain coefficient, represents the second derivative of the flexible manipulator mode function, g1 represents the time delay feedback gain coefficient, and λ represents the time delay amount, and

[0021] In the vibration control of the system, the vibration absorber returns to the stable position after the vibration energy is dissipated, which can be used as the initial zero position, so that the target angular displacement and the target angular velocity in formula (3) are both zero. Formula (3) becomes:

[0022]

[0023] Preferably, the vibration equation of the system is established by the Kane equation The final simplified result of the equation is shown in formula (5) and formula (6).

[0024]

[0025] In the formula,

[0026] Preferably, a weak external excitation The dynamic equation of the system after the external excitation is shown in formula (7) and formula (8). The modal solution of the steady-state response of the flexible manipulator is obtained by the multi-scale method, and is shown in formula (9) to formula (13).

[0027]

[0028]

[0029] In the formula,

[0030] Single mode solution:

[0031]

[0032] a2=a 2L = 0 (10)

[0033] Double mode solution:

[0034]

[0035] Preferably, the modal amplitude a of the nonlinear solution of the flexible manipulator 1N and the external excitation amplitude, only related to the internal-external resonance tuning coefficient, the absorber control signal and the structural parameters of the system.

[0036] Preferably, by analyzing the stability of the steady-state solution of the system in step seven, according to the actual working conditions and needs, the appropriate absorber control parameters are selected.

[0037] The advantages of the present application compared with the prior art are:

[0038] (1) The present application is based on the internal resonance saturation control principle, which is different from most traditional passive control methods and active control methods, and innovatively proposes a new type of flexible manipulator main resonance vibration reduction technology, which introduces nonlinear feedback control and time delay feedback control into the absorber at the same time, and constructs an absorber with nonlinear term and time delay control term. The time for the flexible manipulator to enter the saturation state is accelerated, and the amplitude of the flexible manipulator after entering the saturation state is further reduced.

[0039] (2) Compared with the full active and passive absorber, the semi-active absorber designed in the present application has simple structure and good system stability, and can realize wide frequency absorption by adjusting the control parameters; in the constructed absorber control signal, the natural frequency and damping of the absorber are adjusted by the linear feedback term; through the nonlinear feedback control term, the coupling relationship between the absorber and the flexible manipulator is enhanced, so that the energy of the flexible manipulator can flow into the absorber faster; through the time delay feedback control term, the time for the flexible manipulator to enter the saturation state is further shortened, and the saturation amplitude of the flexible manipulator is reduced; the control parameters can be flexibly adjusted according to the working conditions, and have wider vibration reduction frequency band and good stability. BRIEF DESCRIPTION OF DRAWINGS

[0040] Figure 1 The whole structure of the flexible manipulator system in the present application is shown in the figure;

[0041] Figure 2 The feedback control principle diagram of the absorber motor in the present application is shown in the figure;

[0042] Figure 3The internal resonance principle diagram of the energy exchange between the flexible manipulator and the vibration absorber in the application;

[0043] Figure 4 The modal amplitude curve of the flexible manipulator in the application with the change of the external excitation amplitude f0 * ;

[0044] Figure 5 The saturation amplitude curve of the flexible manipulator in the application with the change of the velocity feedback gain coefficient k d ;

[0045] Figure 6 The saturation amplitude curve of the flexible manipulator in the application with the change of the nonlinear feedback gain coefficient e1

[0046] Figure 7 The modal response curve of the flexible manipulator in the application under different nonlinear feedback gain coefficients e1

[0047] Figure 8 The saturation amplitude curve of the flexible manipulator in the application with the change of the time delay amount λ under different time delay feedback gain coefficients g1

[0048] Figure 9 The end response comparison diagram of the main resonance of the flexible manipulator without the action of the vibration absorber and under the optimal state of the vibration absorber in the application.

[0049] Figure 1 The symbols in the application are as follows:

[0050] 1 represents a rigid arm motor, 2 represents a rigid manipulator, 3 represents a flexible manipulator motor, 4 represents a flexible manipulator, 6 represents a vibration absorber system, 6 represents a vibration absorber motor, 7 represents an equivalent vibration absorber end mass ball, 8 represents a vibration absorber motor swing rod,

[0051] Figure 2 The symbols in the application are as follows:

[0052] w D represents the end response of the flexible manipulator, represents the second differential of the flexible manipulator mode function, e1 represents the coupling feedback gain coefficient of the vibration absorber motor, k p represents the position feedback gain coefficient of the vibration absorber motor, k d represents the velocity feedback gain coefficient of the vibration absorber motor, represents the expected rigid strut angular displacement, represents the actual angular displacement of the vibration absorber swing rod, τ3 represents the control torque of the vibration absorber motor, g1 represents the time delay feedback gain coefficient, λ represents the time delay amount, and DETAILED DESCRIPTION:

[0053] The application will be described in further detail below with reference to the drawings and specific embodiments, but the application is not limited to the embodiments.

[0054] The application provides a flexible robot arm main resonance control method based on nonlinear feedback and time delay feedback.

[0055] The flexible robot arm system with the absorber is modeled, the internal resonance principle is analyzed, and the feedback control principle is analyzed. Figure 1 In order to more clearly describe the main resonance control method provided by the application, a simplified model as shown in

[0056] The method comprises the following steps:

[0057] Step one: assumptions are made for the flexible robot arm and each joint of the system: the robot arm is uniform and isotropic in material; all joints are rigid; the structure of the flexible robot arm meets the characteristics of the Euler-Bernoulli beam; the longitudinal deformation of the flexible robot arm is not considered, and the deformation only occurs in the transverse direction.

[0058] Step two: the transverse bending deformation of the flexible robot arm is obtained by using the assumed mode method:

[0059]

[0060] In formula (1), w(x, t) is the transverse deformation of the flexible robot arm, φn(x) represents the nth mode function, and q i (t) represents the modal coordinate.

[0061] Among the modal deformations of the flexible robot arm, the first modal deformation (n = 1) has a large energy, and only the first modal deformation which plays a main role in the deformation is considered, so the deflection of the flexible robot arm is:

[0062] w(x, t) = φ1(x)q1(t) (2)

[0063] Step three: the control signal of the absorber is constructed. The control principle diagram of the absorber motor is as shown in Figure 2The position feedback coefficient is introduced to adjust the output stiffness of the vibration absorber motor, the velocity feedback coefficient is introduced to adjust the damping of the vibration absorber, a linear feedback control term is formed; a nonlinear feedback control term is introduced to nonlinearly couple the flexible manipulator response signal and the vibration absorber response signal; a time delay feedback control term is introduced to feedback the response signal of the vibration absorber after a certain time delay; finally, the linear control term, the nonlinear control term and the time delay control term form the control signal model of the vibration absorber;

[0064]

[0065] In formula (3), is the initial position of the pendulum rod, is the initial angular velocity, k p is the position feedback coefficient of the vibration absorber, k d is the velocity feedback coefficient, e1 represents the nonlinear feedback gain coefficient, represents the second derivative of the flexible manipulator mode function, g1 represents the time delay feedback gain coefficient, and λ represents the time delay amount, and

[0066] In the vibration control of the system, when the vibration energy is dissipated, the vibration absorber returns to the stable position as the initial zero position, so the target angular displacement and the target angular velocity in formula (3) are zero. Formula (3) becomes:

[0067]

[0068] Step four: based on the Kane equation, according to the model assumption of step one, as shown in Figure 1 , the dynamic model of the system after introducing the vibration absorber and the coordinate system of each component are established. It is assumed that the vibration absorber is a concentrated mass system, and the mass of the pendulum rod of the vibration absorber is equivalent to the mass of the small ball at the end.

[0069] Because the flexible manipulator is light in mass, the change of the rotation angle caused by the lateral deformation is ignored in the dynamic modeling process. The velocity analysis is performed on the system as shown in Figure 1 Figure 1 The partial velocity and the partial angular velocity of each component of the system as shown in table 1.

[0070] Table 1

[0071]

[0072] Taking the first order mode response of the flexible manipulator and the pendulum rod response of the vibration absorber as the generalized coordinates, the first order mode damping of the flexible manipulator is f1, and the Kane equation is ​Substitute the generalized inertial force, the generalized active force and the control signal of the absorber into the equation, and the dynamic equation of the system is obtained:

[0073]

[0074] where,

[0075] The vibration frequencies of the flexible manipulator and the absorber can be obtained by formula (5) and formula (6):

[0076]

[0077] Based on the principle of internal resonance, by adjusting the position feedback gain coefficient k p of the absorber motor, the natural frequency of the absorber and the first-order modal natural frequency of the flexible manipulator are in the ratio of 1:2.

[0078] Step five: apply a weak external excitation at the end of the flexible manipulator According to formula (5) and formula (6) in step four, the dynamic equation of the system after applying the external excitation is:

[0079]

[0080]

[0081] where,

[0082] For the convenience of solving formula (8) and formula (9) later, the parameters in them are dimensionless. τ = ω R t, λ = ω R λ, X i = Y i (i = 1, 3), X2 = Y2l R , the dimensionless expression can be obtained as follows:

[0083]

[0084] The approximate analytical solution of formula (10) and formula (11) is solved by using the multi-scale method, and a small parameter 0 < ε << 1 is introduced:

[0085]

[0086] When the flexible manipulator occurs principal resonance, the external excitation is considered to be a weak forced excitation, that is, set ε 2 and higher order terms are ignored, and the expression of the coefficient of ε 0 , ε 1 is extracted. The following formula can be obtained.

[0087] ε 0 Equation of order:

[0088]

[0089] ε 1 Equation of order:

[0090]

[0091] Let the general solution of equation (13) and equation (14) be:

[0092]

[0093] Let Substitute equation (17) to equation (19) into equation (16), we can get:

[0094]

[0095] In the formula, X2 represents the correction coefficient of X2 after taking into account the nonlinear feedback gain coefficient term, G1 = g1 · K1 represents the correction value of the dimensionless time delay feedback gain coefficient g1, cc represents the conjugate of the remaining terms in this formula, and NST represents the term unrelated to the long-term term.

[0096] Introduce tuning parameters σ1 and σ2. σ1 represents the deviation value of the internal resonance frequency, which is related to the structural design and control strategy of the absorber, and σ2 represents the deviation value of the main resonance frequency, which is related to the damping frequency band of the absorber. Use σ1 to represent the degree to which the modal frequency ω f2 of the flexible manipulator deviates from the first order frequency ω f1 of the flexible manipulator mode. Use σ2 to represent the degree to which the external excitation frequency ω f3 deviates from the first order frequency ω f1 of the flexible manipulator mode.

[0097]

[0098] When the internal resonance is formed, that is, ω f1 ≈2ω f2 In equation (20) and equation (21), respectively, extract the coefficients. According to the condition of eliminating the long-term term, we can get the solvability condition:

[0099]

[0100] Express A1 and A2 of equation (23) and equation (24) in the form of polar coordinates as follows:

[0101]

[0102] where a1, a2, a1, a2 are real functions of the slow time T1, which can be determined by equations (23) and (24).

[0103] Substituting equations (23) and (24) into the autonomous system, we have

[0104] γ1= 2σ1T1+ 2α2- a1 (26)

[0105]

[0106] Substituting equation (25) into equations (23) and (24), and separating the real and imaginary parts, and combining equations (26) and (27), we have

[0107]

[0108] a2' = Z4a1a2sinγ1- Z5a2sin(ω f2 λ)- Z6a2 (29)

[0109]

[0110]

[0111] where

[0112]

[0113] In the steady-state response of the primary resonance, because the motion of the system has reached a steady state, the steady-state parameters a i , γ i are constants, i.e., a i ' = 0, γ i ' = 0. We discuss two cases.

[0114] The first case is the single-mode solution a1≠ 0, a2= 0, i.e., the linear solution of the primary resonance response of the flexible manipulator:

[0115]

[0116] a2= a 2L = 0 (34)

[0117] The second case is the double-mode solution a1≠ 0, a2≠ 0, i.e., the nonlinear solution of the primary resonance response of the flexible manipulator:

[0118]

[0119] From the above analysis, it can be seen that the modal amplitude of the flexible manipulator is linearly related to the modal amplitude of the external excitation in the single-mode solution; in the double-mode solution, the modal amplitude of the flexible manipulator under the action of the external excitation is related to the modal amplitude of the external excitation and is irrelevant to the external excitation, but is related to the internal and external resonance tuning coefficients, the absorber control signal and the structural parameters of the system.

[0120] Step six: According to the steady-state solution obtained in step five, the internal resonance principle of the flexible manipulator system containing the absorber without external excitation is analyzed. According to formula (28) and formula (29) in step five, when the external excitation F = 0 and the time lag λ = 0, only the condition for the existence of internal resonance is considered, that is, in the case of no damping (ζ1 = ζ2 = 0), the following can be obtained:

[0121]

[0122] In the formula E is an integral constant, which is proportional to the initial vibration energy of the system. From formula (38), it can be seen that when C > 0, the absolute values of a1 and a2 have a maximum value, and at the same time, in order to ensure that the sum on the right side of the equation is a constant, the increasing and decreasing states of the two values are exactly opposite. This phenomenon can be understood as follows: when C > 0, internal resonance is established between the absorber modal and the flexible manipulator modal , and the vibration energy in the flexible manipulator can flow between the two modes, and the energy conversion relationship is as shown in Figure 3 .

[0123] Step seven: Stability analysis is performed on the steady-state solution obtained by the multi-scale method in step five. Convert formula (28) to formula (31) to the rectangular coordinate system:

[0124]

[0125] In the formula, p n and q n are real functions of T1, and

[0126]

[0127] According to the above equation, the stability of the equilibrium solution can be analyzed. By perturbation analysis of formula (39) to formula (42), the perturbation equation can be obtained as follows:

[0128] {△p1′,△q1′,△p2′,△q2′} T =[J]{△p1,△q1,△p2,△q2} T (44)

[0129] where T represents the transpose matrix, and [J] is the Jacobian matrix. The Jacobian matrix corresponding to equations (39) to (42) is:

[0130]

[0131] After the stability analysis of the Jacobin matrix, the parameter range of the velocity feedback gain coefficient k d , the nonlinear coupling coefficient e1, the time delay feedback gain coefficient g1, and the time delay amount λ of the inner resonance absorber can be obtained. Then, the absorber controller for controlling the main resonance of the flexible manipulator is designed according to the stability range. According to the actual control of the flexible manipulator system, the appropriate parameter range can eliminate the unstable region of the system vibration or expand the range of the stable region.

[0132] Figure 4 The curves of the mode amplitude of the flexible manipulator and the mode amplitude of the absorber with the amplitude of the external excitation under the conditions of no time delay feedback control and time delay feedback control based on the introduction of linear feedback and nonlinear feedback control. Figure 4 In the figure, the red line represents the mode response amplitude of the flexible manipulator, and the blue line represents the mode response amplitude of the absorber; the solid line indicates that the system response in this section is stable, and the dashed line indicates that the system response in this section is unstable. Without introducing time delay feedback control, based on the principle of inner resonance and nonlinear saturation control, by adjusting appropriate parameters, under the action of the absorber, as the amplitude of the external excitation increases, the mode amplitude of the flexible manipulator will enter a saturated state after a period of time, and the excess energy will enter the absorber. On this basis, after introducing time delay feedback control, by selecting appropriate time delay feedback control parameters, the saturated amplitude of the flexible manipulator will be further reduced, and the amplitude of the external excitation at which the flexible manipulator enters the saturated amplitude will also be further reduced, thereby widening the vibration reduction frequency band.

[0133] Step eight: According to the parameter range obtained in step seven that makes the system stable, design the absorber for controlling the main resonance of the flexible manipulator according to the actual working conditions and requirements.

[0134] Figure 5 The velocity feedback gain coefficient k d has an influence on the mode amplitude of the main resonance of the flexible manipulator. From Figure 5 it can be seen that when the damping of the main resonance absorber increases, the saturated amplitude a1 increases, and the vibration suppression effect of the saturation control weakens; when the damping is small, such as when the velocity feedback gain coefficient k d ∈[0, 0.04], the change of the damping has little effect on the change of the saturated amplitude. Therefore, it can be obtained that by changing the value of the velocity feedback gain k d , the vibration energy can be dissipated by adjusting the damping of the absorber, but excessive damping of the absorber will hinder the transfer of vibration energy between the flexible manipulator and the absorber.

[0135] Figure 6 The influence curve of nonlinear feedback gain coefficient e1 on the saturation mode amplitude of the main resonance of the flexible manipulator. From Figure 6 It can be seen that, given the velocity feedback gain k d , as e1 increases, the saturation amplitude a1 decreases, and the saturation amplitude suppression effect of the main resonance control is enhanced, but the degree of effect enhancement gradually weakens as e1 increases, and the best effect is concentrated in the [0, 0.02] section. When e1 > 0.02, the saturation amplitude suppression effect of the main resonance control has no obvious effect. As shown in Figure 7 , under the condition that the external excitation amplitude is constant, when the flexible manipulator reaches the saturation state of the main resonance, the saturation amplitude of the steady-state response also gradually decreases as the nonlinear feedback gain coefficient increases, and the time for the main resonance to enter saturation is also gradually shortened.

[0136] When the given velocity feedback gain coefficient k d and the nonlinear coupling coefficient e1 are given, the image of the main resonance mode amplitude of the flexible manipulator varying with the time delay λ under different time delay feedback gain coefficients g1 is shown in Figure 8 . When the time delay parameter is introduced, the main resonance mode amplitude of the flexible manipulator gradually decreases from the mode amplitude value without time delay to a minimum amplitude point as the time delay λ increases; when the time delay λ continues to increase, the main resonance mode amplitude of the flexible manipulator also increases. When the time delay λ is located to the left of the minimum amplitude point horizontal coordinate, the main resonance mode amplitude of the flexible manipulator is negatively correlated with the time delay λ; when the time delay λ is located to the right of the minimum amplitude point horizontal coordinate, the main resonance mode amplitude of the flexible manipulator is positively correlated with the time delay λ. Therefore, the optimal time delay λ value is at the minimum damping point. At the same time, as the time delay feedback gain coefficient g1 increases, the minimum amplitude point of the main resonance mode amplitude of the flexible manipulator moves to the lower left of the image, indicating that the increase of the time delay feedback gain coefficient g1 will reduce the main resonance mode amplitude of the flexible manipulator, but at the same time, the time delay λ value also needs to be near the minimum amplitude point to play the effect of the main resonance absorber.

[0137] Figure 9 The comparison chart of the main resonance mode response of the flexible manipulator with and without the absorber after selecting the absorber control parameters according to the actual working conditions and needs.

Claims

1. A flexible robot arm main resonance control method based on nonlinear feedback and time delay feedback, the method comprising the following steps: Step one: make assumptions about the flexible robot arm and each joint of the system: the robot arm is uniform and isotropic in material; all joints are rigid; the structure of the flexible robot arm meets the characteristics of the Euler-Bernoulli beam; and the longitudinal deformation of the flexible robot arm is not considered, and the deformation only occurs in the transverse direction; Step two: use the assumed mode method to discretize the transverse bending deformation of the flexible robot arm: In formula (1), w(x, t) is the lateral deformation of the flexible manipulator, φ i (x) represents the i-th order mode function, q i (t) represents the modal coordinate; Step three: construct the absorber control signal, introduce a position feedback coefficient to adjust the output stiffness of the motor, introduce a velocity feedback coefficient to adjust the damping of the absorber, form a linear control term; introduce a nonlinear feedback control term to nonlinearly couple the response signals of the flexible robot arm and the absorber; introduce a time delay feedback control term to feedback the response signal of the absorber after a certain time delay; and finally form an absorber control signal model composed of the linear feedback control term, the nonlinear feedback control term, and the time delay feedback control term; Step four: According to Kane equation, the dynamic model after introducing the vibration absorber is established; the first order modal response of the flexible manipulator and the response of the swing rod of the vibration absorber is the generalized coordinate; based on the principle of internal resonance, by adjusting the position feedback gain coefficient k p of the motor of the vibration absorber, the natural frequency of the vibration absorber and the first order modal natural frequency of the flexible manipulator are in the public relation of 1:2, and the flexible manipulator forms internal resonance; Step five: introduce a weak harmonic excitation into the flexible robot arm system: use the multiple scale method to obtain an approximate analytical solution of the system dynamics equation, and obtain a single-mode solution and a double-mode solution of the system modal response; Step six: analyze the internal resonance principle of the flexible robot arm system with the absorber under no external excitation according to the approximate analytical solution obtained in step five; Step seven: perform stability analysis on the steady-state solution obtained by the multiple scale method in step five to obtain the control parameter range that makes the system stable; Step eight: design the absorber for controlling the main resonance of the flexible robot arm according to the parameter range that makes the system stable obtained in step seven, and according to the actual working conditions and requirements.

2. The method of claim 1, wherein the method is characterized by In step two, among the various modes of bending deformation of the flexible robot arm, the first mode has relatively large vibration energy in transverse vibration, so only the first mode (n = 1) that plays a major role in deformation is considered, and the deflection of the flexible robot arm is: w(x, t) = φ1(x)q1(t) (2). 3.The flexible manipulator primary resonance control method based on nonlinear feedback and time-delay feedback according to claim 1, characterized in that In step three, the control signal model of the absorber is composed of the linear feedback control term, the nonlinear feedback control term, and the time delay feedback control term: in formula (3), is the initial position of the pendulum, is the initial angular velocity, k p is the absorber position feedback coefficient, k d is the velocity feedback coefficient, e1 represents a nonlinear feedback gain coefficient, represents the second differential of the flexible robot arm mode function, g1 represents a time delay feedback gain coefficient, λ represents a time delay amount, and In the vibration control of the system, the absorber returns to the stable position after the vibration energy is dissipated, which can be used as the initial zero position, so the target angular displacement and the target angular velocity in equation (3) are both zero; Equation (3) becomes:

4. The method of claim 1, wherein the method is characterized by In step four, the position feedback gain coefficient k in the absorber motor control signal is adjusted p The natural frequency of the flexible manipulator and the vibration frequency of the absorber satisfy the commensurate relationship of 2:1, and the vibration frequency is shown in formula (7); the vibration control equation of the system is established by using Kane equation, and the final simplified result is shown in formula (5) and formula (6); In the formulae, 5. The method of claim 1, wherein the method is characterized by In step five, a weak external excitation is applied at the end of the flexible manipulator The system vibration equation is obtained as shown in equation (8) and equation (9); In the formulae, When the flexible arm is in the primary resonance, the external excitation is considered as a weak forced excitation. The approximate analytical solutions of Eqs. (8) and (9) are obtained by using the method of multiple scales, and a small parameter 0 < ε << 1 is introduced, i.e. set The single-mode and double-mode solutions of the steady-state response are obtained by using the method of multiple scales, as shown in Eqs. (10)-(14). The single-mode solution is as follows: a2 = a 2L = 0 (11) The double-mode solution is as follows:

6. The method of claim 1, wherein the method is characterized by In step six, the internal resonance principle of the flexible robot arm system with the absorber under no external excitation is analyzed according to the steady-state solution obtained in step five.

7. The method of claim 1, wherein the method is characterized by In step seven, the stability of the steady-state solution obtained by the multiple scale method in step five is analyzed to obtain the control parameter range that makes the system stable.

8. The method of claim 1, wherein the method is characterized by In step eight, the absorber for controlling the main resonance of the flexible robot arm is designed according to the parameter range that makes the system stable obtained in step seven, and according to the actual working conditions and requirements.

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