Flexible mechanical arm main resonance control method based on nonlinear feedback and time delay feedback

Abstract

The invention discloses a flexible mechanical arm main resonance control method based on nonlinear feedback and time delay feedback, and the method mainly comprises the steps: carrying out the discretization of the transverse deformation of a flexible mechanical arm through employing a hypothesis modal method, and obtaining a vibration equation of the flexible mechanical arm; a linear feedback control item, a nonlinear feedback control item and a time-delay feedback control item are introduced to form a control signal model of the vibration absorber. Constructing a vibration absorber control signal model; a Kane equation is used for establishing a flexible mechanical arm system vibration control model including the vibration absorber; solving an approximate analytic solution of the system vibration equation by applying a multi-scale method to obtain a single-mode solution and a dual-mode solution of the system steady-state response; and finally, under the coexistence state of the internal resonance condition and the main resonance condition of the flexible mechanical arm system in which the vibration absorber is included, the main resonance of the flexible mechanical arm is controlled, and the modal amplitude of the first-order modal of the main resonance of the flexible mechanical arm is well inhibited.

Description

Technical Field

[0001] The present invention proposes a main resonance control method of a flexible robotic arm based on nonlinear feedback and time-delay feedback. Based on the saturation vibration reduction mechanism of internal resonance, a main resonance control method of a flexible robotic arm based on nonlinear feedback and time-delay feedback is proposed to effectively suppress the main resonance of the flexible robotic arm. Background Art

[0002] With the advancement of industrialization, flexible manipulators are gradually developing in the direction of light weight and heavy load, but the flexibility of flexible manipulators is becoming more and more obvious, and the negative response brought by flexibility cannot be ignored. The main resonance phenomenon caused when the periodic external excitation frequency of the flexible manipulator or its periodic variable topological frequency is equal to or close to the natural frequency of a certain modal order of the flexible manipulator itself is the most serious. In the main resonance state, the flexible manipulator will produce considerable vibration in a very short time. Once the amplitude exceeds the limit that the structure can withstand, it will lead to a rapid deterioration of the mechanism's operating performance and structural failure. Therefore, it is very necessary to study the main resonance control of the flexible manipulator.

[0003] Saturation is a unique phenomenon in nonlinear vibration systems, which transfers excess energy to the vibration absorber through a 1:2 internal resonance. When saturation occurs, as the external excitation amplitude increases, the amplitude of the controlled mode will not continue to increase with the increase in the external excitation amplitude, and the excess energy will be transferred to the vibration absorber. Based on the saturation principle, the present invention innovatively proposes a main resonance control method for a flexible manipulator with nonlinear feedback and time-delay feedback, and effectively improves the energy transfer capacity between the flexible manipulator and the main resonance vibration absorber by constructing a controllable nonlinear coupling term; on this basis, a controllable time-delay feedback term is further introduced, thereby greatly shortening the time for the flexible manipulator to enter the saturation state, further reducing the saturation amplitude, widening the vibration reduction frequency band, and more effectively suppressing the main resonance of the flexible manipulator. Summary of the invention

[0004] The technical problem to be solved by the present invention is to propose a main resonance control method of a flexible robotic arm based on nonlinear feedback and time-delay feedback in view of the shortcomings of the prior art. The method is based on the internal resonance principle, nonlinear saturation control principle and time-delay feedback control principle, and can effectively suppress the main resonance of the flexible robotic arm.

[0005] The technical solution of the present invention to solve the technical problem is: a flexible manipulator main resonance control method based on nonlinear feedback and time-delay feedback, the specific steps of the method are as follows:

[0006] Step 1: Make assumptions about the flexible robot arm and each joint of the system: the flexible robot arm is uniform and the material is isotropic; all joints are rigid; the structure of the flexible robot arm conforms to the characteristics of the Euler-Bernoulli beam; the longitudinal deformation of the flexible robot arm is not considered, and deformation only occurs in the lateral direction.

[0007] Step 2: Use the assumed modal method to discretize the lateral bending deformation of the flexible manipulator:

[0008]

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

[0010] Step 3: Construct the vibration absorber control signal. Introduce the position feedback coefficient to adjust the output stiffness of the motor, introduce the speed feedback coefficient to adjust the damping of the vibration absorber, and form a linear control term; introduce a nonlinear feedback control term to nonlinearly couple the flexible manipulator response signal with the vibration absorber response signal; introduce a time-delay feedback control term to feedback the vibration absorber response signal after a certain time lag; 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 vibration absorber.

[0011] Step 4: According to the Kane equation, establish the dynamic model after the introduction of the vibration absorber; take the first-order modal response of the flexible manipulator and absorber rocker response is the generalized coordinate; based on the internal resonance principle, by adjusting the position feedback gain coefficient k of the vibration absorber motor p , so that the natural frequency of the vibration absorber and the first-order modal natural frequency of the flexible robotic arm have a commonality relationship of 1:2, forming an internal resonance with the flexible robotic arm.

[0012] Step 5: Introduce weak harmonic excitation into the flexible robotic arm system, use the multi-scale method to obtain the approximate analytical solution of the dynamic equation, and obtain the single-mode solution and dual-mode solution of the system modal response.

[0013] Step 6: Based on the steady-state solution obtained in step 5, analyze the internal resonance principle of the flexible robotic arm system containing a vibration absorber without external excitation.

[0014] Step 7: Perform stability analysis on the steady-state solution obtained by the multi-scale method in step 5 to obtain the control parameter range that makes the system stable.

[0015] Step 8: Based on the parameter range for stabilizing the system obtained in step 7, design a vibration absorber for controlling the main resonance of the flexible robotic arm according to actual working conditions and requirements.

[0016] Preferably, among the modes of bending deformation of the flexible manipulator, the first mode has a larger vibration energy in the lateral vibration, so only the first mode (n=1) that 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 formed by a linear control term, a nonlinear control term and a time-delay control term:

[0019]

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

[0021] In the vibration control of the system, when the vibration energy dissipates, the vibration absorber returns to a stable position and can be used as the initial zero position. Therefore, the target angular displacement in equation (3) is and target angular velocity are all zero. Formula (3) becomes:

[0022]

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

[0024]

[0025] In the formula,

[0026] Preferably, a weak external excitation is applied to the end of the flexible robotic arm. The dynamic equations of the system after external excitation are shown in Equations (7) and (8). The modal solutions of the steady-state response of the flexible manipulator are obtained by the multi-scale method, as shown in Equations (9) to (13).

[0027]

[0028]

[0029] In the formula,

[0030] Single mode solution:

[0031]

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

[0033] Bimodal solution:

[0034]

[0035] Preferably, the modal amplitude a of the nonlinear solution of the flexible manipulator is 1N It has nothing to do with the external excitation amplitude, but only with the internal and external resonance tuning coefficients, the vibration absorber control signal and the structural parameters of the system.

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

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

[0038] (1) Based on the principle of internal resonance saturation control, this invention is different from most traditional passive control methods and active control methods. It innovatively proposes a new type of flexible manipulator main resonance vibration reduction technology, introduces nonlinear feedback control and time-delay feedback control into the vibration absorber at the same time, and constructs a vibration absorber with both nonlinear terms and time-delay control terms. It accelerates the time for the flexible manipulator to enter the saturation state and further reduces the amplitude of the flexible manipulator after entering the saturation state.

[0039] (2) Compared with fully active and passive vibration absorbers, the semi-active vibration absorber designed in the present invention has a simple structure and good system stability, and can achieve wide-band vibration absorption by adjusting control parameters. In the constructed vibration absorber control signal, the natural frequency and damping of the vibration absorber are adjusted by linear feedback items. The coupling relationship between the vibration absorber and the flexible robotic arm is enhanced by nonlinear feedback control items, allowing the energy of the flexible robotic arm to flow into the vibration absorber faster. The time-delay feedback control item is used to further shorten the time for the flexible robotic arm to enter the saturation state, thereby reducing the saturation amplitude of the flexible robotic arm. The control parameters can be flexibly adjusted according to changes in working conditions, with a wider vibration reduction frequency band and good stability. BRIEF DESCRIPTION OF THE DRAWINGS

[0040] Figure 1 It is a schematic diagram of the overall structure of the flexible mechanical arm system incorporating the vibration absorber in the present invention;

[0041] Figure 2 It is a feedback control principle diagram of the vibration absorber motor in the present invention;

[0042] Figure 3The diagram is a schematic diagram of the internal resonance principle of the energy exchange between the flexible mechanical arm and the vibration absorber in the present invention;

[0043] Figure 4 is the modal amplitude of the flexible manipulator according to the present invention and the external excitation amplitude f0 * The change curve of

[0044] Figure 5 is the saturation amplitude of the flexible manipulator according to the present invention and the speed feedback gain coefficient k d The change curve of

[0045] Figure 6 is a curve showing the variation of the saturation amplitude of the flexible mechanical arm with the nonlinear feedback gain coefficient e1 in the present invention;

[0046] Figure 7 is the modal response curve of the flexible manipulator under different nonlinear feedback gain coefficients e1 in the present invention;

[0047] Figure 8 The curve of the change of the saturation amplitude of the flexible manipulator with time lag λ under different time lag feedback gain coefficients g1 in the present invention;

[0048] Fig. 9 It is a comparison diagram of the end response of the main resonance of the flexible mechanical arm without the vibration absorber and in the optimal state of the vibration absorber in the present invention.

[0049] Figure 1 The symbols are explained as follows:

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

[0051] Figure 2 The symbols are explained as follows:

[0052] w D represents the end response of the flexible robotic arm, represents the second derivative of the vibration mode function of the flexible manipulator, e1 represents the coupling feedback gain coefficient of the vibration absorber motor, k p Represents the vibration absorber motor position feedback gain coefficient, k d Indicates the vibration absorber motor speed feedback gain coefficient, represents the desired angular displacement of the rigid strut, represents the angular displacement of the actual absorber rocker, τ3 represents the absorber motor control torque, g1 represents the time-delay feedback gain coefficient, λ represents the time-delay amount, and DETAILED DESCRIPTION

[0053] The present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments, but the implementation of the present invention is not limited thereto.

[0054] The present invention proposes a main resonance control method for a flexible robotic arm based on nonlinear feedback and time-delay feedback. A nonlinear dynamic model of the flexible robotic arm taking into account a vibration absorber is established through the Kane equation. The main resonance control of the flexible robotic arm is achieved by combining the internal resonance principle, nonlinear saturation control principle and time-delay feedback control principle.

[0055] The flexible manipulator system with vibration absorber is subjected to dynamic modeling, internal resonance principle analysis and feedback control principle analysis. In order to more clearly illustrate the main resonance control method proposed in the present invention, a simplified model is used as follows: Figure 1 As shown, the flexible robotic arm system model is replaced by the actual one that takes into account the vibration absorber.

[0056] The specific steps of this method are as follows:

[0057] Step 1: Make assumptions about the flexible robot arm and each joint of the system: the robot arm is uniform and the material is isotropic; all joints are rigid; the structure of the flexible robot arm conforms to the characteristics of the Euler-Bernoulli beam; the longitudinal deformation of the flexible robot arm is not considered, and deformation only occurs in the lateral direction.

[0058] Step 2: Use the assumed modal method to discretize the lateral bending deformation of the flexible manipulator:

[0059]

[0060] In formula (1), w(x, t) is the lateral deformation of the flexible manipulator, represents the i-th order vibration mode function, q i (t) represents the modal coordinate.

[0061] Among the various modes of deformation of the flexible manipulator, the first-order mode (n=1) has a larger energy. Considering only the first-order mode that plays a major role in the deformation, the deflection of the flexible manipulator is:

[0062] w(x,t)=φ1(x)q1(t) (2)

[0063] Step 3: Construct the vibration absorber control signal. The control principle diagram of the vibration absorber motor is as follows: Figure 2As shown. The position feedback coefficient is introduced to adjust the output stiffness of the vibration absorber motor, and the speed feedback coefficient is introduced to adjust the damping of the vibration absorber to form a linear feedback control item; the nonlinear feedback control item is introduced to nonlinearly couple the response signal of the flexible manipulator with the response signal of the vibration absorber; the time-delay feedback control item is introduced to lag the response signal of the vibration absorber before feedback; finally, the linear control item, the nonlinear control item and the time-delay control item constitute the control signal model of the vibration absorber;

[0064]

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

[0066] In the vibration control of the system, when the vibration energy dissipates, the vibration absorber returns to a stable position and can be used as the initial zero position. Therefore, the target angular displacement in equation (3) is and target angular velocity are all zero. Formula (3) becomes:

[0067]

[0068] Step 4: Based on the Kane equation, according to the model assumptions of step 1, such as Figure 1 As shown in Figure 1, the dynamic model of the system after the introduction of the vibration absorber and the coordinate system of each component are established. Assuming that the vibration absorber is a concentrated mass system, the mass of the vibration absorber pendulum is equivalent to the mass of the end ball.

[0069] Since the flexible manipulator is light in weight, the rotation angle change caused by its lateral deformation is ignored in the dynamic modeling process. Figure 1 The system shown in the figure is subjected to velocity analysis, and generalized velocity is introduced. Figure 1 The deflection speed and angular speed of each component of the system are shown in Table 1.

[0070] Table 1

[0071]

[0072] The first-order modal response of the flexible manipulator and absorber rocker response is the generalized coordinate, the first-order modal damping of the flexible manipulator is f1, according to the Kane equation Substituting the generalized inertial force, generalized active force and the control signal of the vibration absorber, the dynamic equation of the system is obtained:

[0073]

[0074] In the formula,

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

[0076]

[0077] Based on the principle of internal resonance, by adjusting the position feedback gain coefficient k of the vibration absorber motor p , making the natural frequency of the vibration absorber and the first-order modal natural frequency of the flexible manipulator have a commonality relationship of 1:2.

[0078] Step 5: Apply weak external excitation to the end of the flexible robotic arm According to equations (5) and (6) in step 4, the dynamic equation of the system after applying external excitation is:

[0079]

[0080]

[0081] In the formula,

[0082] In order to facilitate the subsequent solution of equations (8) and (9), the parameters are dimensionless. 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 multi-scale method is used to solve the approximate analytical solutions of equations (10) and (11), and a small parameter 0<ε<<1 is introduced:

[0085]

[0086] When the flexible manipulator has the main resonance, the external excitation is considered to be a weak forced excitation, that is, Omit ε 2 and higher order terms and extract ε 0 , ε 1 The coefficient expression of can be obtained as follows.

[0087] ε 0 The equation of order:

[0088]

[0089] ε 1 The equation of order:

[0090]

[0091] Assume that the general solution of equation (13) and equation (14) is:

[0092]

[0093] remember Substituting equations (17) to (19) into equation (16), we can obtain:

[0094]

[0095] In the formula, represents the correction coefficient of X2 after the nonlinear feedback gain coefficient term is taken into account, 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] Tuning parameters σ1 and σ2 are introduced. σ1 represents the deviation of the internal resonance frequency, which is related to the structural design and control strategy of the vibration absorber. σ2 represents the deviation of the main resonance frequency, which is related to the vibration reduction frequency band of the vibration absorber. σ1 represents the modal frequency ω of the vibration absorber. f2 Deviation from the first-order frequency of the flexible manipulator modal ω f1 The degree of external excitation frequency ω is represented by σ2. f3 Deviation from the first-order frequency of the flexible manipulator modal ω f1 degree.

[0097]

[0098] When internal resonance is formed, ω f1 ≈2ω f2 When , in formula (20) and formula (21), we extract Coefficient. According to the condition of eliminating long-term terms, the solvability condition can be obtained:

[0099]

[0100] A1 and A2 in equations (23) and (24) are expressed in polar coordinates as follows:

[0101]

[0102] Where a1, a2, α1, and α2 are all real functions of the slow-varying time T1 and can be determined by equations (23) and (24).

[0103] Convert equation (23) and equation (24) into an autonomous system and introduce:

[0104] γ1=2σ1T1+2α2-α1 (26)

[0105]

[0106] Substituting equation (25) into equation (23) and equation (24), separating the real part and the imaginary part, and combining equation (26) and equation (27), we can obtain:

[0107]

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

[0109]

[0110]

[0111] In the formula,

[0112]

[0113] In the steady-state response of the main resonance, because the motion of the system has reached a stable state, the steady-state parameter a i , γ i are all constants, i.e., a i ′=0,γ i ′=0. Two cases are discussed below.

[0114] The first is the single-mode solution a1≠0, a2=0, which is the linear solution of the main resonance response of the flexible manipulator:

[0115]

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

[0117] The second is the dual-mode solution a1≠0, a2≠0, which is the nonlinear solution of the main resonance response of the flexible manipulator:

[0118]

[0119] From the above analysis, it can be seen that under the single-mode solution, the modal amplitude of the flexible manipulator is linearly related to the external excitation amplitude; under the dual-mode solution, the modal amplitude of the flexible manipulator under external excitation is linearly related to the modal amplitude of the external excitation. It has nothing to do with the internal and external resonance tuning coefficients, the vibration absorber control signal and the structural parameters of the system.

[0120] Step 6: Based on the steady-state solution obtained in step 5, the internal resonance principle of the flexible manipulator system with a vibration absorber is analyzed without external excitation. According to equations (28) and (29) in step 5, when the external excitation F = 0 and the time delay λ = 0, only the condition of the existence of internal resonance is considered, that is, in the case of no damping (ζ1 = ζ2 = 0), it can be obtained:

[0121]

[0122] In the formula E is the integral constant, which is proportional to the initial vibration energy of the system. From formula (38), we can see that when C>0, the absolute values ​​of a1 and a2 have a maximum value. At the same time, in order to ensure that the sum on the right side of the equal sign is a constant, the increase and decrease of the two values ​​are exactly opposite. This phenomenon can be understood as when C>0, the vibration absorber mode Flexible robot arm mode An internal resonance is established between the two modes, and the vibration energy in the flexible manipulator can flow between the two modes. The energy conversion relationship is as follows: Figure 3 shown.

[0123] Step 7: Perform stability analysis on the steady-state solution obtained by the multi-scale method in step 5. Convert equations (28) to (31) into the rectangular coordinate system:

[0124]

[0125] In the formula, p n and q n is a real function of T1, and

[0126]

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

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

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

[0130]

[0131] After stability analysis of the Jacobin matrix, the velocity feedback gain coefficient k of the internal resonance absorber can be obtained: d , the parameter range of the nonlinear coupling coefficient e1, the time-delay feedback gain coefficient g1 and the time-delay amount λ, and then the vibration absorber controller for controlling the main resonance of the flexible manipulator is designed according to this stable range. Selecting the appropriate parameter range according to the flexible manipulator system actually controlled can eliminate the unstable area of ​​system vibration or expand the range of the stable area.

[0132] Figure 4 These are the variation curves of the modal amplitude of the flexible manipulator and the modal amplitude of the vibration absorber with the external excitation amplitude under the feedback control without time lag and the feedback control with time lag based on the introduction of linear feedback and nonlinear feedback control. Figure 4 In the figure, the red line indicates the modal response amplitude of the flexible manipulator, and the blue line indicates the modal response amplitude of the vibration absorber; the solid line indicates that the system response of this section is stable, and the dotted line indicates that the system response of this section is unstable. Without introducing time-delay feedback control, based on the principle of internal resonance and nonlinear saturation control, by adjusting appropriate parameters, under the action of the vibration absorber, as the external excitation amplitude increases, the modal amplitude of the flexible manipulator will enter the saturation state after a period of time, and the excess energy will enter the vibration absorber; on this basis, after introducing time-delay feedback control, selecting appropriate time-delay feedback control parameters, the saturation amplitude of the flexible manipulator will be further reduced, and the external excitation amplitude that causes the flexible manipulator to enter the saturation amplitude will also be further reduced, widening the vibration reduction frequency band.

[0133] Step 8: Based on the parameter range for stabilizing the system obtained in step 7, design a vibration absorber for controlling the main resonance of the flexible robotic arm according to actual working conditions and requirements.

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

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

[0136] When the speed feedback gain coefficient k is given d After the nonlinear coupling coefficient e1 is added, the amplitude of the main resonance mode of the flexible manipulator changes with the hysteresis λ under different time-delay feedback gain coefficients g1 as shown in the figure below: Figure 8 As shown in the figure. After the time-delay parameter is introduced, the modal amplitude of the main resonance of the flexible manipulator starts from the modal amplitude value without adding time-delay, and gradually decreases to a minimum amplitude point as the time-delay amount λ increases; when the time-delay amount λ continues to increase, the modal amplitude of the main resonance of the flexible manipulator increases again. When the time-delay amount λ is located on the left side of the horizontal coordinate of the minimum amplitude point, the modal amplitude of the main resonance of the flexible manipulator is negatively correlated with the time-delay amount λ; when the time-delay amount λ is located on the right side of the horizontal coordinate of the minimum amplitude point, the modal amplitude of the main resonance of the flexible manipulator is positively correlated with the time-delay amount λ. Therefore, the optimal value of the time-delay amount λ is at the minimum vibration reduction point. At the same time, as the time-delay feedback gain coefficient g1 increases, the minimum amplitude point of the main resonance modal 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 modal amplitude of the main resonance of the flexible manipulator, but at the same time, the value of the time-delay amount λ also needs to be near the minimum amplitude point to play the role of the main resonance vibration absorber.

[0137] Fig. 9 This is a comparison chart of the main resonance modal responses of the flexible robotic arm with and without a vibration absorber after the vibration absorber control parameters are selected according to actual working conditions and needs.

Claims

1. A method for controlling the main resonance of a flexible manipulator based on nonlinear feedback and time-delay feedback, the specific steps of the method are as follows: Step 1: Make assumptions about the flexible robot arm and each joint of the system: the robot arm is uniform and the material is isotropic; all joints are rigid; the structure of the flexible robot arm conforms to the characteristics of the Euler-Bernoulli beam; the longitudinal deformation of the flexible robot arm is not considered, and deformation only occurs in the lateral direction. Step 2: Use the assumed modal method to discretize the lateral bending deformation of the flexible manipulator: In formula (1), w(x, t) is the lateral deformation of the flexible manipulator, represents the i-th order vibration mode function, q i (t) represents the modal coordinate. Step 3: Construct the vibration absorber control signal. Introduce the position feedback coefficient to adjust the output stiffness of the motor, introduce the speed feedback coefficient to adjust the damping of the vibration absorber, and form a linear control term; introduce a nonlinear feedback control term to nonlinearly couple the flexible manipulator response signal with the vibration absorber response signal; introduce a time-delay feedback control term to feedback the vibration absorber response signal after a certain time lag; 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 vibration absorber. Step 4: According to the Kane equation, establish the dynamic model after the introduction of the vibration absorber; take the first-order modal response of the flexible manipulator and absorber rocker response is the generalized coordinate; based on the internal resonance principle, by adjusting the position feedback gain coefficient k of the vibration absorber motor p , so that the natural frequency of the vibration absorber and the first-order modal natural frequency of the flexible robotic arm have a commonality relationship of 1:2, and the flexible robotic arm forms an internal resonance. Step 5: Introduce weak harmonic excitation into the flexible robotic arm system: Use the multi-scale method to obtain the approximate analytical solution of the system dynamics equation, and obtain the single-mode solution and dual-mode solution of the system modal response. Step 6: Based on the approximate analytical solution obtained in step 5, analyze the internal resonance principle of the flexible robotic arm system containing a vibration absorber without external excitation. Step 7: Perform stability analysis on the steady-state solution obtained by the multi-scale method in step 5 to obtain the control parameter range that makes the system stable. Step 8: Based on the parameter range for stabilizing the system obtained in step 7, design a vibration absorber for controlling the main resonance of the flexible robotic arm according to actual working conditions and requirements.

2. The main resonance control method of a flexible manipulator based on the combination of nonlinear feedback and time-delay feedback as claimed in claim 1 is characterized in that In step 2, among the various modes of bending deformation of the flexible manipulator, the first-order mode has a larger vibration energy in the lateral vibration, so only the first-order mode (n=1) that plays a major role in the deformation is considered, and the deflection of the flexible manipulator is: w(x,t)=φ1(x)q1(t) (2).

3. The main resonance control method of a flexible manipulator based on nonlinear feedback and time-delay feedback as claimed in claim 1 is characterized in that In step 3, the control signal model of the vibration absorber is constructed by linear feedback control term, nonlinear feedback control term and time-delay feedback control term: In formula (3), is the starting position of the pendulum, is the initial angular velocity, k p is the vibration absorber position feedback coefficient, k d is the speed feedback coefficient, e1 represents the nonlinear feedback gain coefficient, represents the second derivative of the vibration function of the flexible manipulator, g1 represents the time-delay feedback gain coefficient, λ represents the time-delay amount, and In the vibration control of the system, when the vibration energy dissipates, the vibration absorber returns to a stable position and can be used as the initial zero position. Therefore, the target angular displacement in equation (3) is and target angular velocity are all zero. Formula (3) becomes: 。 4. The main resonance control method of a flexible manipulator based on nonlinear feedback and time-delay feedback as claimed in claim 1 is characterized in that In step 4, by adjusting the position feedback gain coefficient k in the vibration absorber motor control signal p , so that the natural frequency of the flexible manipulator and the vibration frequency of the vibration absorber satisfy the commonality relationship of 2:1, and the vibration frequency is shown in formula (7); the vibration control equation of the system is established using the Kane equation, and the final simplified results are shown in formulas (5) and (6). In the formula, .

5. The main resonance control method of a flexible manipulator based on nonlinear feedback and time-delay feedback as claimed in claim 1 is characterized in that In step 5, a weak external excitation is applied to the end of the flexible robotic arm. The system vibration equations are obtained as shown in Equations (8) and (9), and the single-mode solution and dual-mode solution of the steady-state response are obtained by the multi-scale method, as shown in Equations (10) to (13). In the formula, Single mode solution: <h2 style=";text-align:left;direction:ltr">a2=a<h2 style=";text-align:left;direction:ltr"> 2L <h2 style=";text-align:left;direction:ltr"> =0 (11) Bimodal solution: 。 6. The method for controlling the main resonance of a flexible mechanical arm based on nonlinear feedback and time-delay feedback as claimed in claim 1 is characterized in that In step six, based on the steady-state solution obtained in step five, the internal resonance principle of the flexible manipulator system containing a vibration absorber without external excitation is analyzed.

7. The method for controlling the main resonance of a flexible mechanical arm based on nonlinear feedback and time-delay feedback as claimed in claim 1 is characterized in that In step seven, a stability analysis is performed on the steady-state solution obtained by the multi-scale method in step five to obtain the control parameter range that makes the system stable.

8. The method for controlling the main resonance of a flexible mechanical arm based on nonlinear feedback and time-delay feedback as claimed in claim 1 is characterized in that In step eight, based on the parameter range for stabilizing the system obtained in step seven, a vibration absorber for controlling the main resonance of the flexible manipulator is designed according to actual working conditions and requirements.

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