CN111015737A - Vibration reduction method of single-connecting-rod flexible mechanical arm - Google Patents

Vibration reduction method of single-connecting-rod flexible mechanical arm Download PDF

Info

Publication number
CN111015737A
CN111015737A CN201911358470.4A CN201911358470A CN111015737A CN 111015737 A CN111015737 A CN 111015737A CN 201911358470 A CN201911358470 A CN 201911358470A CN 111015737 A CN111015737 A CN 111015737A
Authority
CN
China
Prior art keywords
mechanical arm
vibration
section
acceleration
flexible mechanical
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Pending
Application number
CN201911358470.4A
Other languages
Chinese (zh)
Inventor
马鹏宇
李永超
任豪
汪学斌
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Changan University
Original Assignee
Changan University
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Changan University filed Critical Changan University
Priority to CN201911358470.4A priority Critical patent/CN111015737A/en
Publication of CN111015737A publication Critical patent/CN111015737A/en
Pending legal-status Critical Current

Links

Images

Classifications

    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J19/00—Accessories fitted to manipulators, e.g. for monitoring, for viewing; Safety devices combined with or specially adapted for use in connection with manipulators
    • B25J19/0091—Shock absorbers
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1628—Programme controls characterised by the control loop
    • B25J9/163—Programme controls characterised by the control loop learning, adaptive, model based, rule based expert control
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1656—Programme controls characterised by programming, planning systems for manipulators
    • B25J9/1664—Programme controls characterised by programming, planning systems for manipulators characterised by motion, path, trajectory planning

Abstract

The invention discloses a vibration reduction method of a single-connecting-rod flexible mechanical arm. And under a generalized coordinate system, establishing a bending vibration equation based on the Euler-Bernoulli beam theory. And carrying out parameter discretization through finite segmentation to establish a flexible mechanical arm dynamic model. And analyzing the dynamic model to obtain characteristic parameters such as the natural vibration frequency of the mechanical arm. The mechanical arm movement is set to be three-stage movement of acceleration, uniform speed and deceleration, current signal time of different movement stages of the control motor is designed according to the natural frequency of the mechanical arm, movement track planning and movement parameter configuration are carried out on the flexible mechanical arm, and vibration reduction of transverse movement is achieved. On the basis of analyzing the command shaping vibration suppression method, the invention combines the track generation feedforward method, has simple and quick command operation, does not need to change the structure of the mechanical arm, has obvious vibration reduction effect and has better application value in the aspect of vibration reduction of the rotary motion of the industrial flexible mechanical arm.

Description

Vibration reduction method of single-connecting-rod flexible mechanical arm
Technical Field
The invention belongs to the technical field of mechanical vibration and vibration control of flexible mechanical arms, and particularly relates to a vibration reduction method of a single-connecting-rod flexible mechanical arm.
Background
With the development of the robot technology, the wide development of the mechanical arm meets the requirements of industrial production quality and efficiency, and simultaneously shows the development trend of high speed and high precision. Compared with a heavy mechanical arm with a large volume, the light and flexible mechanical arm has more excellent characteristics and performance, and the flexibility and the working space expansibility in the movement process are increased. Meanwhile, because the damping of the mechanical arm is small, the elastic deformation of the component exists in the working process, the vibration can last for a long time, the normal work is influenced, and the tail end of the mechanical arm can be accurately positioned difficultly. Therefore, there is a need to analyze and research a vibration control method of a flexible manipulator of a robot to ensure real-time, efficient and precise control of the motion of the flexible manipulator.
Current vibration control methods have made significant progress in the general vibration of flexible robotic arms that are weak in flexibility and can be linearized. However, in many practical industrial systems, the mathematical model of the controlled object or process is difficult to determine, and even if the mathematical model is obtained under a certain condition, the dynamic parameters thereof are continuously changed based on the structure of the model after the operating conditions and some factors are changed.
Disclosure of Invention
The invention aims to provide a vibration reduction method of a single-connecting-rod flexible mechanical arm, which aims to solve the problems.
In order to achieve the purpose, the invention adopts the following technical scheme:
a vibration reduction method of a single-connecting-rod flexible mechanical arm comprises the following steps:
step 1: under a generalized coordinate system, a bending vibration equation is established based on an Euler-Bernoulli beam theory, parameter discretization is carried out through finite segmentation, and a flexible mechanical arm dynamic model is established;
step 2: analyzing the dynamic model to obtain characteristic parameters such as the natural vibration frequency of the mechanical arm and the like, thereby obtaining a transfer function of the mechanical arm;
and step 3: setting the motion of the mechanical arm into three-stage motion of acceleration, uniform speed and deceleration, designing current signal time for controlling different motion stages of a motor according to the natural frequency of the mechanical arm, and planning a motion track and formulating motion parameters of the flexible mechanical arm;
and 4, step 4: and verifying the stability of the vibration control of the mechanical arm.
Further, step 1 specifically includes:
for variable cross-section, the cross-sectional area A (x) and the bending stiffness EI (x) are functions related to an axial coordinate x, based on the Euler-Bernoulli beam theory, simplifying a balance equation, neglecting high-order micro-scale, establishing a bending vibration equation,
Figure BDA0002336575840000021
in the formula: rho-is the density of the flexible mechanical arm;
y (x, t) -is the transverse vibration displacement of the mechanical arm;
the cross-sectional beam area a (x) and the moment of inertia of the cross-section to the neutral axis i (x) are both continuously variable in the axial direction of the beam with respect to x, and are expressed as follows:
Figure BDA0002336575840000022
wherein: a. the0＝b0h0，
Figure BDA0002336575840000023
m represents different section variation types of the beam;
in the formula: l-is the total length of the variable cross-section beam;
c0-the gradient coefficient of the width or height of the beam cross section along the axis;
b0,h0-width and height of the cross section of the beam at x-0, respectively;
A0,I0-the cross-sectional area of the beam at x-0 and the moment of inertia about the neutral axis, respectively;
based on the finite element method, the non-uniform variable-section beam is cut and divided into a plurality of beam sections which are continuously connected, when the divided small sections are enough, the length of each section of beam is very small, and the small sections of beams are regarded as uniform beams with equal sections;
let the length of the i-th (i ═ 1,2,3 … n) beam be liThe product of the elastic modulus and the section moment of inertia is (EI)iLinear density of (rho A)iThe flexural stiffness and linear density of the micro-segment beam are expressed as follows:
Figure BDA0002336575840000024
at this time, the equation of the ith section of free vibration on the beam is as follows:
Figure BDA0002336575840000031
according to the form of the solution of the high-order differential equation, the solution of the equation is:
yi(x,t)＝Yi(x)Qi(t)
wherein: qi(t) ═ sin (ω t + Φ), ω is the circular frequency of the beam lateral vibration,
Figure BDA0002336575840000036
is determined by initial conditions; y isi(x) Is the mode function of the i-th section of the beam; can be set as follows:
Yi(x)＝Aisinψi+Bicosψi+Cisinhψi+Dicoshψi
wherein: psii＝μi(x-xi-1)，xi-1≤x≤xi，i＝1,2,3…n，x0＝0，Ai、Bi、Ci、DiIs the coefficient to be determined; in addition, the method can be used for producing a composite material
Figure BDA0002336575840000032
In the formula: omega-variable cross-section beam transverse vibration circle frequency;
in the same way, the mode function of the i +1 th section on the beam is set as follows:
Yi+1(x)＝Ai+1sinψi+1+Bi+1cosψi+1+Ci+1sinhψi+1+Di+1coshψi+1
the orthogonality condition of the main vibration mode of the transverse vibration of the flexible mechanical arm is determined by vibration mechanics
Figure BDA0002336575840000033
Multiplying equal sign of formula by Yj(x) And is integrated to obtain
Figure BDA0002336575840000034
Obtaining a vibration differential equation of the flexible mechanical arm under the action of the rotation torque according to the formula, considering the structural damping of the system, and assuming that the modal damping ratio of the system is zetaiAnd written in input-output form
Figure BDA0002336575840000035
Further, step 2 specifically includes:
considering the moment acting on the flexible mechanical arm as a concentrated moment, the moment m (x, t) can be written as:
m(x,t)＝M(t)δ(x-ξ)
in the formula, delta (·) is a dirac function, and ξ is the distance from the stress point to the motor axis point O;
the center of mass of the arm is at a distance ξ from the axis
Figure BDA0002336575840000041
In the formula, r is the radius of the shaft sleeve; l is the total length of the variable cross-section beam; b0, b1The fixed end x of the beam is equal to 0 and the cantilever end x is equal to the height of the cross section at the position L; b2Is the length of the mechanical arm clamp;
assuming a modal damping ratio of the system as ζiThen the above equation becomes the following form:
Figure BDA0002336575840000042
the integral to the right of the above equation is reduced in its entirety as follows:
Figure RE-GDA0002383822230000043
wherein: b isi＝Yi′(x2)-Yi′(x1)。
Thus, the equation is reduced to:
Figure BDA0002336575840000044
selecting a generalized modal coordinate function as a state vector of a state space expression, namely:
Figure BDA0002336575840000045
taking the terminal vibration displacement y of the flexible mechanical arm as the output response of the system, and converting the above formula into a form of a space state expression to obtain:
Figure BDA0002336575840000046
wherein:
Figure BDA0002336575840000047
the matrix I is an identity matrix;
Figure BDA0002336575840000051
is a diagonal matrix;
R＝diag[ζ1ω1,ζ2ω2,…ζnωn]is a diagonal matrix;
B＝[0,Bi]T；
C＝diag[Y1(L),Y2(L),…,Yn(L),0]
Figure BDA0002336575840000052
obtaining a state space expression of the vibration of the flexible mechanical arm, and solving a transfer function of the flexible mechanical arm system according to the state space expression:
G(s)＝C(sI-A)-1B
the vibration transfer function of the flexible mechanical arm can be obtained by the formula.
Further, step 3 specifically includes:
at the time when t1 is 0s, an acceleration signal, i.e. with amplitude a, is applied to the controlled object1When the pulse signal is generated, the system generates oscillation response of corresponding amplitude; after a vibration of a system inherent period, an opposite acceleration signal with amplitude A is applied to the controlled object at T2 ═ T2Generating an oscillation response in response to the amplitude; thus, after the combined action superposition of the impulse response is completed, the oscillation response of the system is enabled to be t2After the moment, the two phases are mutually offset; if an acceleration signal is applied to the controlled object, the amplitude is A1Of the pulse signal of (a), through n system natural periods of vibration, i.e. t2＝nTPeriod of vibrationThen applying a reverse acceleration signal to the controlled object, i.e. with amplitude A2The residual vibration will also be eliminated;
selecting and using a three-section type motion track, dividing the motion track of the mechanical arm into 3 stages of motion states, namely acceleration, constant speed, deceleration and stop, and assuming that the time of the corresponding acceleration section is T1, the end time of the constant speed section is T2 and the end time of the deceleration section is T3; setting the acceleration a1 and the acceleration a2 of the deceleration segment and the highest speed v of the system operation to make the acceleration period T1 and the deceleration period T3-T2 equal to the natural vibration period of the system, namely T1＝v/a1＝T， T3-T2＝v/a2T; taking the acceleration a of the acceleration section1And acceleration a of the deceleration section2The consistency is kept between the first and the second,the system input acceleration and velocity signals are described by the following equations:
acceleration:
Figure BDA0002336575840000053
speed:
Figure BDA0002336575840000061
determining the running time of each motion stage of the 3-segment motion trail by adjusting the motion acceleration of the mechanical arm and the maximum speed which can be reached by the system;
and calculating and adjusting to enable the movement time of the acceleration stage and the movement time of the deceleration stage to be equal to the natural vibration period of a controlled object of the system, so that the vibration of the system is eliminated when the acceleration stage is finished, and the vibration of the system is also eliminated when the deceleration stage is finished.
Further, in step 3, the natural frequency of the flexible mechanical arm is measured through modeling simulation or experiments, the current signal time for controlling different motion stages of the motor is designed according to the natural period of the mechanical arm, and the acceleration and deceleration time is integral multiple of the natural period.
Compared with the prior art, the invention has the following technical effects:
the invention provides a vibration reduction method of a single-link flexible mechanical arm, which is used for establishing a bending vibration equation based on an Euler-Bernoulli beam theory under a generalized coordinate system. And carrying out parameter discretization through finite segmentation, and establishing a flexible mechanical arm dynamic model by using a Lagrange equation. And analyzing the dynamic model to obtain characteristic parameters such as the natural vibration frequency of the mechanical arm. The motion of the mechanical arm is set to be three-stage motion with acceleration, uniform speed and deceleration, current signal time of the control motor in different motion stages is designed according to the natural frequency of the mechanical arm, motion trail planning and motion parameter configuration are carried out on the flexible mechanical arm, and accurate and stable control over the flexible mechanical arm is achieved.
Through establishing and analyzing a flexible mechanical arm dynamic model, the control motor is controlled in different motion stages according to the natural frequency of the mechanical armDesigning current signal time, configuring the motion track of the flexible mechanical arm into a 3-section motion state, configuring motion acceleration and speed values to enable the motion time of an acceleration and deceleration section to be equal to integral multiple of the inherent period of the mechanical arm, and tAcceleration/deceleration＝n·TControlled object(n is a positive integer greater than zero), the residual vibration of the system can be greatly reduced, and the stable control of the residual vibration of the flexible mechanical arm system can be realized. Acceleration and deceleration time tAcceleration/deceleration＝TControlled objectAnd the restraining quantity of the residual vibration of the system is more than 85%, so that the control system is more practical and accurate.
Through modeling, simulation and experimental research, the reasonability and correctness of the flexible mechanical arm motion parameter configuration vibration attenuation method are proved, and the method is simple and easy to implement compared with the existing vibration attenuation method and has practical value in the field of engineering application.
Drawings
FIG. 1: flexible mechanical arm rotation motion deformation schematic diagram
FIG. 2: schematic diagram of variable cross-section flexible mechanical arm
FIG. 3: flexible mechanical arm finite segment segmentation schematic diagram
FIG. 4: mechanical arm example model and size
FIG. 5: flexible mechanical arm vibration mode
FIG. 6: acceleration signal of system
FIG. 7: system speed signal
FIG. 8: simulink simulation program diagram
FIG. 9: flexible mechanical arm residual vibration schematic diagram
FIG. 10: residual vibration diagram of the applied vibration damping method.
Detailed Description
The invention is further described below with reference to the accompanying drawings:
referring to fig. 1 to 10, the present invention provides a vibration damping method for a single link flexible manipulator, which establishes a bending vibration equation based on the Euler-Bernoulli beam theory in a generalized coordinate system. And carrying out parameter discretization through finite segmentation, and establishing a flexible mechanical arm dynamic model by utilizing a Lagrange's equation. And analyzing the dynamic model to obtain characteristic parameters such as the natural vibration frequency of the mechanical arm. The motion of the mechanical arm is set to be three-stage motion with acceleration, uniform speed and deceleration, current signal time of different motion stages of the control motor is designed according to the natural frequency of the mechanical arm, motion trail planning and motion parameter configuration are carried out on the flexible mechanical arm, and accurate and stable control over the flexible mechanical arm is achieved. The method comprises the following steps:
the method comprises the following steps: and (3) modeling the dynamics of the flexible mechanical arm.
For variable cross-section, the cross-sectional area A (x) and the bending stiffness EI (x) are functions of an axial coordinate x, based on Euler-Bernoulli beam theory, simplifying a balance equation, neglecting high-order micro-scale, establishing a bending vibration equation,
Figure BDA0002336575840000081
in the formula: rho-is the density of the flexible mechanical arm;
y (x, t) -is the transverse vibration displacement of the mechanical arm;
referring to fig. 1 and 2, the cross-sectional area a (x) of the variable cross-section beam and the moment of inertia of the cross-section to the neutral axis i (x) are each a function that varies continuously with respect to x in the axial direction of the beam, as expressed below:
Figure BDA0002336575840000082
wherein: a. the0＝b0h0，
Figure BDA0002336575840000083
m represents different section variation types of the beam, 1 or 2 can be taken, if m is 1, the width of the cross section of the beam is a fixed value, and the height is linearly changed along the axis; if m 2, both the width and the height of the beam cross-section vary linearly along the axis.
In the formula: l-is the total length of the variable cross-section beam;
c0-the gradient coefficient of the width or height of the beam cross section along the axis;
b0,h0-width and height of the cross section of the beam at x-0, respectively;
A0,I0the cross-sectional area of the beam at x-0 and the moment of inertia about the neutral axis, respectively
Referring to fig. 3, based on the finite element method, the non-uniform variable cross-section beam is cut and divided into a plurality of beam sections which are connected in series, when the divided small sections are enough, the length of each section of beam is very small, and then the small sections of beams can be regarded as uniform constant cross-section beams.
Let the length of the i-th (i ═ 1,2,3 … n) beam be liThe product of the elastic modulus and the moment of inertia of the cross section (bending stiffness) is (EI)iLinear density of (rho A)iThe bending stiffness and linear density of the micro-segment beam can thus be expressed as follows:
Figure BDA0002336575840000084
at this time, the equation of the ith section of free vibration on the beam is as follows:
Figure BDA0002336575840000085
according to the form of the solution of the higher mathematic higher-order differential equation, the solution of the equation is:
yi(x,t)＝Yi(x)Qi(t)
wherein: qi(t) ═ sin (ω t + Φ), ω is the circular frequency of the beam lateral vibration,
Figure BDA0002336575840000091
is determined by the initial conditions. Y isi(x) Is the modal function of the i-th section of the beam. Can be set as follows:
Yi(x)＝Aisinψi+Bicosψi+Cisinhψi+Dicoshψi
wherein: psii＝μi(x-xi-1)，xi-1≤x≤xi，i＝1,2,3…n，x0＝0，Ai、Bi、Ci、DiIs the undetermined coefficient. In addition, the method can be used for producing a composite material
Figure BDA0002336575840000092
In the formula: omega-variable cross section beam transverse vibration circle frequency.
In the same way, the mode function of the i +1 th section on the beam is set as follows:
Yi+1(x)＝Ai+1sinψi+1+Bi+1cosψi+1+Ci+1sinhψi+1+Di+1coshψi+1
by vibration mechanics[6]The orthogonality condition of the main vibration mode of the transverse vibration of the flexible mechanical arm is
Figure BDA0002336575840000093
Multiplying equal sign of formula by Yj(x) And is integrated to obtain
Figure BDA0002336575840000094
According to the formula, a vibration differential equation of the flexible mechanical arm under the action of the rotation torque can be obtained, and because the damping force exists in the system in the practical application process of the flexible mechanical arm system, the structural damping of the system is considered, and the modal damping ratio of the system is assumed to be zetaiAnd written in input-output form
Figure BDA0002336575840000095
Step two: and analyzing the dynamic model to obtain a transfer function.
Now considering the moment acting on the flexible robot arm as the collective moment, the moment m (x, t) can be written as:
m(x,t)＝M(t)δ(x-ξ)
in the formula, δ (·) is a dirac function, and ξ is a distance from the force bearing point to the motor axis O point.
The moment of the mechanical arm is generated by the inertia force, the equivalent action point of the inertia force is the mass center of the mechanical arm, and the distance ξ between the mass center of the mechanical arm and the axis of the mechanical arm shown in figure 2 is known from mathematical knowledge
Figure BDA0002336575840000101
In the formula, r is the radius of the shaft sleeve; l is the total length of the variable cross-section beam; b0, b1The fixed end x of the beam is equal to 0 and the cantilever end x is equal to the height of the cross section at the position L; b2Is the robot arm gripper length.
Since the flexible mechanical arm system has damping force in the system in the practical application process, the structural damping of the system is also considered, and the modal damping ratio of the system is assumed to be zetaiThen the above equation becomes the following form:
Figure BDA0002336575840000102
the integral to the right of the above equation is reduced in its entirety as follows:
Figure RE-GDA0002383822230000103
wherein: b isi＝Yi′(x2)-Yi′(x1)。
Therefore, equation (3.20) can be simplified to:
Figure BDA0002336575840000104
selecting a generalized modal coordinate function as a state vector of a state space expression, namely:
Figure BDA0002336575840000105
in order to research the vibration phenomenon of the flexible mechanical arm, the terminal vibration displacement y of the flexible mechanical arm is used as the output response of the system, and the above formula is converted into a form of a spatial state expression to obtain:
Figure BDA0002336575840000106
wherein:
Figure BDA0002336575840000107
the matrix I is an identity matrix;
Figure BDA0002336575840000111
is a diagonal matrix;
R＝diag[ζ1ω1,ζ2ω2,…ζnωn]is a diagonal matrix;
B＝[0,Bi]T；
C＝diag[Y1(L),Y2(L),…,Yn(L),0]
Figure BDA0002336575840000112
through the above series of calculation derivation, a state space expression of the flexible mechanical arm vibration is obtained, and therefore, a transfer function of the flexible mechanical arm system can be solved:
G(s)＝C(sI-A)-1B
the vibration transfer function of the flexible mechanical arm can be obtained by the formula.
Step three: and (4) preparing a motion track and motion parameters of the flexible mechanical arm.
At t1An acceleration signal with amplitude A is applied to the controlled object at the time of 0s1When the pulse signal is generated, the system generates oscillation response of corresponding amplitude; after a vibration of a system's natural period, then at t2Applying an opposite acceleration signal, i.e. with amplitude A, to the controlled object at time T2Pulse signal ofAn oscillatory response responsive to amplitude is also produced. After the superposition of the combined action of the impulse response is finished, the oscillation response of the system can be enabled to be t2The time and the later cancel each other. If an acceleration signal is applied to the controlled object, the amplitude is A1Of the pulse signal, through n system natural periods of oscillation, i.e. t2＝nTPeriod of vibrationThen applying a reverse acceleration signal to the controlled object, i.e. with amplitude A2Will also be eliminated
Referring to fig. 6 and 7, the three-stage motion trajectory is selected and used, the motion trajectory of the mechanical arm is divided into 3 stages of motion states, which are acceleration, uniform speed, deceleration and stop, respectively, and it is assumed that the corresponding acceleration stage time is T1The ending time of the uniform velocity section is T2The end time of the deceleration section is T3. Acceleration a of acceleration section is reasonably set1And acceleration a of the deceleration section2And the maximum speed v at which the system operates, such that the acceleration period T1And a deceleration period time T3-T2Equal to the natural vibration period of the system, i.e. T1＝v/a1＝T，T3-T2＝v/a2T. Acceleration a of acceleration section in the embodiment of the invention1And acceleration a of the deceleration section2In accordance, the system input acceleration and velocity signals may be described by the following equations:
acceleration:
Figure BDA0002336575840000113
speed:
Figure BDA0002336575840000121
when the method for suppressing vibration is performed by using the speed configuration method of the 3-segment motion trajectory, under the motion condition that the use requirement can be realized by the operation of the mechanical arm, obviously, as long as the acceleration of the acceleration and the deceleration of the operation of the mechanical arm and the speed and the motion displacement of the uniform speed stage when the speed reaches the maximum are determined, the operation time of each motion stage of the 3-segment motion trajectory can be determined accordingly.
The movement time of the acceleration stage and the movement time of the deceleration stage are equal to the natural vibration period of the controlled object of the system through calculation and adjustment, so that the vibration of the system can be eliminated when the acceleration stage is finished, and the vibration of the system can also be eliminated when the deceleration stage is finished, so that the residual vibration of the system can be inhibited after the whole movement of the system is finished. Step four: and verifying the stability of the vibration control of the mechanical arm.
Referring to fig. 4, an example arm model was calculated for a material density of ρ 7850kg/m3The elastic modulus is E ═ 2.07X 1011N/m2The diameter d of the mechanical arm fixing device is 30 mm. In the paper, only the first-order vibration mode of the single-link flexible mechanical arm is researched, and the natural circular frequency of the object, which is obtained by calculation according to the dynamic modeling of the step three and the mechanical arm example, is omega-24.798 rad/s
The cross-sectional area a (l) and the moment of inertia i (l) of the end of the arm are:
Figure BDA0002336575840000122
Figure BDA0002336575840000123
regularizing the main vibration mode according to the knowledge in vibration mechanics, namely:
Figure BDA0002336575840000124
wherein y (x) is ═ C [ (cos γ x-cosh γ x) + α (sin γ x-sinh γ x) ];
Figure BDA0002336575840000125
Figure BDA0002336575840000126
Figure BDA0002336575840000131
the coefficient C to be determined in the mode shape function y (x) can be solved from the above formulas to-4.0383
And (3) bringing the parameter C into the mode shape function Y (x), solving the mode shape function value of the tail end of the flexible mechanical arm and the derivative value of the mode shape function value, wherein x is L, and the following steps are included:
Y(L)＝1.1038
and simultaneously, the parameters of a matrix B in the space state equation can be obtained:
B＝Y′(x2)-Y′(x1)＝Y′(d)-Y′(0)＝0.4052
the respective coefficient matrices of the state space equation can thus be obtained:
Figure BDA0002336575840000132
Figure BDA0002336575840000133
finally, the transfer function g(s) of the flexible arm system is obtained. Namely, the input is a mechanical arm angular acceleration signal, and the output is a transfer function of the vibration displacement of the tail end of the mechanical arm
Figure BDA0002336575840000134
Referring to fig. 8, in the three-stage trajectory velocity configuration vibration study Simulink simulation model, a velocity input signal of the system is constructed by summing two Ramp modules and two Saturation modules, the input signal is applied to the system model obtained by solving, a vibration output signal of the flexible manipulator is obtained by running simulation, and a displacement signal and an acceleration signal corresponding to the velocity of the system are also comprehensively observed. ,
referring to FIG. 9, acceleration is accomplished at the robot arm,After three-stage motion of uniform speed and deceleration, 4.25x10 still exists in the system-3m, the vibration of the system is not eliminated after the movement is finished and is adjusted for 3s, which indicates that the speed setting parameters cannot realize the suppression of the vibration; on the other hand, the maximum vibration displacement during the movement for the case of the acceleration time of 0.35s was 5.21 × 10-3m。
Referring to fig. 10, when the vibration reduction method is applied and the parameter setting is performed, it can be visually seen that the residual vibration of the system after the system movement is finished is suppressed, and the residual vibration is 0.13 × 10-3m, is reduced by 93.48% with respect to the amount of vibration during movement, so that the residual vibration of the system can be said to be substantially completely suppressed, and the maximum vibration displacement during movement is also reduced by 19.68% compared to the previous case.
Meanwhile, only one vibration peak is observed in the acceleration section and the deceleration section, which can well explain that the three-section type motion track speed configuration vibration suppression method provided by the paper is very effective, the method is simple and easy to implement, and the system vibration can be well eliminated at the moment when the motion of the mechanical arm is finished, so that the working efficiency of the mechanical arm is improved, and the mechanical arm can work more accurately and stably.

Claims (5)

1. A vibration reduction method of a single-connecting-rod flexible mechanical arm is characterized by comprising the following steps:
step 1: under a generalized coordinate system, a bending vibration equation is established based on an Euler-Bernoulli beam theory, parameter discretization is carried out through finite segmentation, and a flexible mechanical arm dynamic model is established;
step 2: analyzing the dynamic model to obtain characteristic parameters such as the natural vibration frequency of the mechanical arm and the like, thereby obtaining a transfer function of the mechanical arm;
and step 3: setting the motion of the mechanical arm into three-stage motion of acceleration, uniform speed and deceleration, designing current signal time for controlling different motion stages of a motor according to the natural frequency of the mechanical arm, and planning a motion track and formulating motion parameters of the flexible mechanical arm;
and 4, step 4: and verifying the stability of the vibration control of the mechanical arm.
2. The vibration reduction method of the single-link flexible mechanical arm according to claim 1, wherein the step 1 specifically comprises the following steps:
for variable cross-section, the cross-sectional area A (x) and the bending stiffness EI (x) are functions related to an axial coordinate x, based on the Euler-Bernoulli beam theory, simplifying a balance equation, neglecting high-order micro-scale, establishing a bending vibration equation,
Figure FDA0002336575830000011
in the formula: rho-is the density of the flexible mechanical arm;
y (x, t) -is the transverse vibration displacement of the mechanical arm;
the cross-sectional beam area a (x) and the moment of inertia of the cross-section to the neutral axis i (x) are both continuously variable in the axial direction of the beam with respect to x, and are expressed as follows:
Figure FDA0002336575830000012
wherein: a. the0＝b0h0，
Figure FDA0002336575830000013
m represents different section variation types of the beam;
in the formula: l-is the total length of the variable cross-section beam;
c0-the gradient coefficient of the width or height of the beam cross section along the axis;
b0,h0-width and height of the cross section of the beam at x-0, respectively;
A0,I0-the cross-sectional area of the beam at x-0 and the moment of inertia about the neutral axis, respectively;
based on the finite element method, the non-uniform variable-section beam is cut and divided into a plurality of beam sections which are continuously connected, when the divided small sections are enough, the length of each section of beam is very small, and the small sections of beams are regarded as uniform beams with equal sections;
let the length of the i-th (i ═ 1,2,3 … n) beam be liThe product of the elastic modulus and the section moment of inertia is (EI)iLinear density of (rho A)iThe flexural stiffness and linear density of the micro-segment beam are expressed as follows:
Figure FDA0002336575830000021
at this time, the equation of the ith section of free vibration on the beam is as follows:
Figure FDA0002336575830000022
according to the form of the solution of the high-order differential equation, the solution of the equation is:
yi(x,t)＝Yi(x)Qi(t)
wherein: qi(t) ═ sin (ω t + Φ), ω is the circular frequency of the beam lateral vibration,
Figure FDA0002336575830000023
is determined by initial conditions; y isi(x) Is the mode function of the i-th section of the beam; can be set as follows:
Yi(x)＝Aisinψi+Bicosψi+Cisinhψi+Dicoshψi
wherein: psii＝μi(x-xi-1)，xi-1≤x≤xi，i＝1,2,3…n，x0＝0，Ai、Bi、Ci、DiIs the undetermined coefficient; in addition, the method can be used for producing a composite material
Figure FDA0002336575830000024
In the formula: omega-variable cross-section beam transverse vibration circle frequency;
in the same way, the mode function of the i +1 th section on the beam is set as follows:
Yi+1(x)＝Ai+1sinψi+1+Bi+1cosψi+1+Ci+1sinhψi+1+Di+1coshψi+1
the orthogonality condition of the main vibration mode of the transverse vibration of the flexible mechanical arm is determined by vibration mechanics
Figure FDA0002336575830000025
Multiplying equal sign of formula by Yj(x) And is integrated to obtain
Figure FDA0002336575830000026
Figure FDA0002336575830000031
Obtaining a vibration differential equation of the flexible mechanical arm under the action of the rotation torque according to the formula, considering the structural damping of the system, and assuming that the modal damping ratio of the system is zetaiAnd written in input-output form
Figure FDA0002336575830000032
3. The vibration reduction method of the single-link flexible mechanical arm according to claim 1, wherein the step 2 specifically comprises the following steps:
considering the moment acting on the flexible mechanical arm as a concentrated moment, the moment m (x, t) can be written as:
m(x,t)＝M(t)δ(x-ξ)
in the formula, delta (·) is a dirac function, and ξ is the distance from the stress point to the motor axis point O;
the center of mass of the arm is at a distance ξ from the axis
Figure RE-FDA0002383822220000033
In the formula, r is the radius of the shaft sleeve; l is the total length of the variable cross-section beam; b0,b1The fixed end x of the beam is equal to 0 and the cantilever end x is equal to the height of the cross section at the position L; b2Is the length of the mechanical arm clamp;
assuming a modal damping ratio of the system as ζiThen the above equation becomes the following form:
Figure RE-FDA0002383822220000034
the integral to the right of the above equation is reduced in its entirety as follows:
Figure RE-FDA0002383822220000035
wherein: b isi＝Yi′(x2)-Yi′(x1)；
Thus, the equation is reduced to:
Figure RE-FDA0002383822220000036
selecting a generalized modal coordinate function as a state vector of a state space expression, namely:
Figure RE-FDA0002383822220000041
taking the terminal vibration displacement y of the flexible mechanical arm as the output response of the system, and converting the above formula into a form of a space state expression to obtain:
Figure RE-FDA0002383822220000042
wherein:
Figure RE-FDA0002383822220000043
the matrix I is an identity matrix;
Figure RE-FDA0002383822220000044
is a diagonal matrix;
R＝diag[ζ1ω1,ζ2ω2,…ζnωn]is a diagonal matrix; b ═ 0, Bi]T；
C＝diag[Y1(L),Y2(L),…,Yn(L),0]
Figure RE-FDA0002383822220000045
Obtaining a state space expression of the vibration of the flexible mechanical arm, and solving a transfer function of the flexible mechanical arm system according to the state space expression:
G(s)＝C(sI-A)-1B
the vibration transfer function of the flexible mechanical arm can be obtained by the formula.
4. The vibration reduction method of the single-link flexible mechanical arm according to claim 1, wherein the step 3 specifically comprises the following steps:
at the time when t1 is 0s, an acceleration signal, i.e. with amplitude a, is applied to the controlled object1When the pulse signal is generated, the system generates oscillation response of corresponding amplitude; after a system natural period of vibration, an opposite acceleration signal with amplitude A is applied to the controlled object at T2 ═ T2Generating an oscillation response in response to the amplitude; thus, after the combined action superposition of the impulse response is completed, the oscillation response of the system is enabled to be t2After the moment, the two phases are mutually offset; if an acceleration signal is applied to the controlled object, the amplitude is A1Of the pulse signal of (a), through n system natural periods of vibration, i.e. t2＝nTPeriod of vibrationThen applying a reverse acceleration signal to the controlled object, i.e. with amplitude A2The residual vibration will also be eliminated;
selecting and using a three-section type motion track, dividing the motion track of the mechanical arm into 3 stages of motion states, namely acceleration, constant speed, deceleration and stop, and assuming that the time of the corresponding acceleration section is T1, the end time of the constant speed section is T2 and the end time of the deceleration section is T3; setting the acceleration a1 and the acceleration a2 of the deceleration segment and the highest speed v of the system operation to make the acceleration period T1 and the deceleration period T3-T2 equal to the natural vibration period of the system, namely T1＝v/a1＝T，T3-T2＝v/a2T; taking the acceleration a of the acceleration section1And acceleration a of the deceleration section2In agreement, the system input acceleration and velocity signals are described by the following equations:
acceleration:
Figure FDA0002336575830000051
speed:
Figure FDA0002336575830000052
determining the running time of each motion stage of the 3-segment motion trail by adjusting the motion acceleration of the mechanical arm and the maximum speed which can be reached by the system;
the movement time of the acceleration stage and the movement time of the deceleration stage are equal to the natural vibration period of the controlled object of the system through calculation and adjustment, so that the vibration of the system is eliminated when the acceleration stage is completed, and the vibration of the system is also eliminated when the deceleration stage is completed.
5. The vibration damping method for the single-link flexible mechanical arm according to claim 1, wherein in step 3, the natural frequency of the flexible mechanical arm is measured through modeling simulation or experiment, and the time of the current signal for controlling different motion phases of the motor is designed according to the natural period of the mechanical arm, and the acceleration and deceleration time is an integral multiple of the natural period.
CN201911358470.4A 2019-12-25 2019-12-25 Vibration reduction method of single-connecting-rod flexible mechanical arm Pending CN111015737A (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN201911358470.4A CN111015737A (en) 2019-12-25 2019-12-25 Vibration reduction method of single-connecting-rod flexible mechanical arm

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN201911358470.4A CN111015737A (en) 2019-12-25 2019-12-25 Vibration reduction method of single-connecting-rod flexible mechanical arm

Publications (1)

Publication Number Publication Date
CN111015737A true CN111015737A (en) 2020-04-17

Family

ID=70213285

Family Applications (1)

Application Number Title Priority Date Filing Date
CN201911358470.4A Pending CN111015737A (en) 2019-12-25 2019-12-25 Vibration reduction method of single-connecting-rod flexible mechanical arm

Country Status (1)

Country Link
CN (1) CN111015737A (en)

Cited By (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112356020A (en) * 2020-09-21 2021-02-12 中国人民解放军军事科学院国防科技创新研究院 Variable cross-section flexible arm feedback control line length positioning method and device and medium
CN113246175A (en) * 2021-05-07 2021-08-13 配天机器人技术有限公司 Robot end vibration classification method, robot and storage medium
CN114001193A (en) * 2021-09-18 2022-02-01 上海华兴数字科技有限公司 Start-stop vibration reduction control method and device for operation machinery and electronic equipment
CN116021555A (en) * 2023-03-29 2023-04-28 之江实验室 Vibration absorption control method and device for flexible joint mechanical arm
CN113246175B (en) * 2021-05-07 2024-05-03 配天机器人技术有限公司 Robot terminal vibration classification method, robot and storage medium

Citations (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN102825613A (en) * 2012-09-17 2012-12-19 北京航空航天大学 Actively vibration damping method and device based on controllable local degree of freedom

Patent Citations (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN102825613A (en) * 2012-09-17 2012-12-19 北京航空航天大学 Actively vibration damping method and device based on controllable local degree of freedom

Non-Patent Citations (1)

* Cited by examiner, † Cited by third party
Title
梁开旭: "机器人单连杆柔性机械臂减振方法研究", 《中国优秀硕士学位论文全文数据库 信息科技辑》 *

Cited By (7)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112356020A (en) * 2020-09-21 2021-02-12 中国人民解放军军事科学院国防科技创新研究院 Variable cross-section flexible arm feedback control line length positioning method and device and medium
CN112356020B (en) * 2020-09-21 2021-06-04 中国人民解放军军事科学院国防科技创新研究院 Variable cross-section flexible arm feedback control line length positioning method and device and medium
CN113246175A (en) * 2021-05-07 2021-08-13 配天机器人技术有限公司 Robot end vibration classification method, robot and storage medium
CN113246175B (en) * 2021-05-07 2024-05-03 配天机器人技术有限公司 Robot terminal vibration classification method, robot and storage medium
CN114001193A (en) * 2021-09-18 2022-02-01 上海华兴数字科技有限公司 Start-stop vibration reduction control method and device for operation machinery and electronic equipment
CN114001193B (en) * 2021-09-18 2024-03-15 上海华兴数字科技有限公司 Start-stop vibration reduction control method and device for working machine and electronic equipment
CN116021555A (en) * 2023-03-29 2023-04-28 之江实验室 Vibration absorption control method and device for flexible joint mechanical arm

Similar Documents

Publication Publication Date Title
CN111015737A (en) Vibration reduction method of single-connecting-rod flexible mechanical arm
Mohamed et al. Command shaping techniques for vibration control of a flexible robot manipulator
Shan et al. Modified input shaping for a rotating single-link flexible manipulator
Park et al. Design of learning input shaping technique for residual vibration suppression in an industrial robot
Moallem et al. An integral manifold approach for tip-position tracking of flexible multi-link manipulators
Eppinger et al. Introduction to dynamic models for robot force control
CN109927032A (en) A kind of mechanical arm Trajectory Tracking Control method based on High-Order Sliding Mode observer
Zhu et al. Simulation studies of tip tracking control of a single-link flexible robot based on a lumped model
Ata et al. Dynamic analysis of a two-link flexible manipulator subject to different sets of conditions
Sariyildiz et al. Robust position control of a novel series elastic actuator via disturbance observer
Zimmert et al. 2-DOF control of a fire-rescue turntable ladder
CN109815637A (en) A kind of emulation mode calculating complete soft Manipulator Dynamics response
Rostami Kandroodi et al. Control of flexible joint manipulator via reduced rule-based fuzzy control with experimental validation
Alandoli et al. A review study on flexible link manipulators
El-Badawy et al. Nonlinear modeling and control of flexible-link manipulators subjected to parametric excitation
Fenili The rigid-flexible robotic manipulator: Nonlinear control and state estimation considering a different mathematical model for estimation
CN107511830A (en) A kind of series parallel robot in five degrees of freedom controller parameter adaptively adjusts implementation method
Sayahkarajy et al. Mixed sensitivity H2/H∞ control of a flexible-link robotic arm
Choi et al. Unconstrained and constrained motion control of a planar two‐link structurally flexible robotic manipulator
Qiang et al. Residual vibration suppression using off-line learning input shaping method for a flexible joint robot
Duarte et al. Modeling and dual loop sliding mode control of a two flexible-link robot to reduce the transient response
Duarte et al. Modeling and centralized sliding mode control of a two-flexible-link robot
Mejerbi et al. Influence of the motion profile on the performance of a flexible arm
Duarte et al. Modeling and sliding mode control of a single-link flexible robot to reduce the transient response
Berninger et al. The influence of structural dynamics on cascaded joint position control of a flexible beam with a compliant gear

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination
RJ01 Rejection of invention patent application after publication
RJ01 Rejection of invention patent application after publication

Application publication date: 20200417