CN113326627B - Harmonic drive hysteresis stiffness modeling method based on genetic characteristics - Google Patents

Harmonic drive hysteresis stiffness modeling method based on genetic characteristics Download PDF

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CN113326627B
CN113326627B CN202110645176.2A CN202110645176A CN113326627B CN 113326627 B CN113326627 B CN 113326627B CN 202110645176 A CN202110645176 A CN 202110645176A CN 113326627 B CN113326627 B CN 113326627B
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陈满意
宋港
邱临风
张�杰
杨燃
张瀚
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Institute Of Advanced Research Wuhan University Of Technology Shangyu District Shaoxing City
Zhejiang Laifual Harmonic Drive Co ltd
Wuhan University of Technology WUT
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Zhejiang Laifual Harmonic Drive Co ltd
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Abstract

The invention relates to a harmonic drive hysteresis stiffness modeling method based on genetic characteristics, which comprises the steps of determining a stiffness function according to a hysteresis stiffness relation between torque and torsion angle of harmonic drive; establishing a friction function according to the genetic characteristic of the harmonic gear transmission, and determining the friction function as the integral of the torsion angle and the time under the genetic effect; correcting discontinuity of the friction function to obtain a friction correction function; and determining a harmonic drive hysteresis rigidity model according to the rigidity function and the friction correction function. According to the invention, by establishing a more accurate harmonic drive dynamic model, namely a harmonic drive hysteresis rigidity model, the dynamic analysis precision and the transmission performance of harmonic gear drive are improved, and the dynamic control precision of the harmonic reducer is improved.

Description

Harmonic drive hysteresis stiffness modeling method based on genetic characteristics
Technical Field
The application relates to the technical field of harmonic gear transmission, in particular to a harmonic transmission hysteresis stiffness modeling method based on genetic characteristics.
Background
The harmonic reducer has a hysteresis phenomenon in the gear transmission process, generally, the hysteresis phenomenon means that the state of the system is not only related to the input of the current system, but also has different results due to different paths of the past input process, namely, the state of the system depends on one property of the historical state of the system.
As shown in fig. 1, when the torque T transmitted by the harmonic reducer increases, the torsion angle θ increases, and the harmonic gear drive stiffness also increases; when the transmission torque is reduced to 0, the torsion angle does not return to the original point, and with the forward and reverse loading and unloading of the load at the output end, a hysteresis stiffness curve generates a loop, which means that the same output torque can correspond to different torsion angles, wherein delta theta is called hysteresis loss, and the harmonic transmission stiffness has certain hysteresis characteristics. The hysteresis rigidity phenomenon of harmonic transmission can cause the reduction of the transmission precision of a system, cause the change of the rotating speed of an output shaft of a harmonic reducer and directly influence the energy conversion of the harmonic transmission.
At present, when dynamically modeling harmonic drive, stiffness and friction phenomena are generally considered, coefficients of the harmonic drive are defined as constants or piecewise constants, and parameter identification is carried out through experiments. Since the harmonic drive is a flexible drive, the motion state of the system is related to the previous state, the stiffness and damping of the system are nonlinear dynamic functions with respect to time, and the simplification can cause the dynamic accuracy of the drive system to be reduced, so that the dynamic accuracy of the drive system controlled by the existing harmonic drive dynamic model is not high, and the inventor considers that the existing harmonic drive dynamic model needs further improvement.
Disclosure of Invention
In order to overcome the problems in the prior art, the harmonic drive hysteresis stiffness modeling method based on the genetic characteristics is provided, and by establishing a more accurate harmonic drive dynamic model, namely a harmonic drive hysteresis stiffness model, the dynamic analysis precision and the transmission performance of harmonic gear drive are improved, and the dynamic control precision of a harmonic speed reducer is improved.
A harmonic drive hysteresis stiffness modeling method based on genetic characteristics comprises the following steps:
determining a stiffness function according to a hysteresis stiffness relationship between a torque and a torsion angle of the harmonic drive;
establishing a friction function according to the genetic characteristics of harmonic gear transmission, and determining the friction function as the integral of the torsion angle and time under the genetic effect;
correcting discontinuity of the friction function to obtain a friction correction function;
and determining a harmonic drive hysteresis rigidity model according to the rigidity function and the friction correction function.
Optionally, a friction function is established according to the genetic characteristic of the harmonic gear transmission, and the friction function is determined as an integral of a torsion angle and time under the genetic effect, specifically as follows:
Figure BDA0003108965550000021
wherein
Figure BDA0003108965550000022
Is a genetic factor, u represents the duration of the genetic action; t represents the current time; θ(s) represents the torsion angle of the state at the s-th time from the 0-th time point at which the system starts to operate to the present time point t.
Alternatively, the genetic element
Figure BDA0003108965550000023
And the torsion angle θ(s) satisfy the following relationship:
Figure BDA0003108965550000024
θ(s)＝Csin(ws)；
the friction function obtained by the formula conversion is as follows:
Figure BDA0003108965550000025
optionally, the discontinuity of the friction function is corrected to obtain a friction correction function, where the friction correction function is as follows:
Figure BDA0003108965550000031
optionally, the stiffness function is determined as f (θ), where f (θ) is an nth order polynomial function, and typically, N is 3 or 5.
Optionally, the stiffness function f (θ) is defined as a fifth-order polynomial function, and a harmonic drive hysteresis stiffness model is determined according to the stiffness function and the friction correction function, specifically as follows:
Figure BDA0003108965550000032
wherein a, b, c represent the coefficients of a fifth order polynomial function.
Optionally, the method further includes:
the method comprises the steps of testing forward loading and reverse loading of a harmonic reducer to obtain experimental data of torque and a torsion angle;
and fitting the experimental data by adopting a least square method, and identifying the parameters of the harmonic drive hysteresis stiffness model.
Optionally, the test of forward loading and backward loading is performed on the harmonic reducer to obtain experimental data of torque and torsion angle, and the process is as follows:
the input end of the harmonic gear is fixed, the load end carries out forward loading to rated torque, the load end carries out gradual unloading, then the load end carries out backward loading to the rated torque and carries out gradual unloading, the change of the torque in the loading/unloading process is recorded through a torque sensor, the change condition of a corresponding torsion angle is obtained through an angle sensor, and experimental data of the torque and the torsion angle are uploaded to an upper computer.
Optionally, fitting the experimental data by using a least square method specifically includes:
using least squares and based on discrete experimental data points (theta)i,
Figure BDA0003108965550000041
) Establishing a minimum torque error objective function:
Figure BDA0003108965550000042
wherein, thetaiTwist angle representing the ith experimental data point;
Figure BDA0003108965550000043
torque experimental value, T (θ), representing the ith experimental data pointi) The theoretical value of the torque corresponding to the ith torsion angle is shown; epsilon (theta)i) An error value between the experimental value of torque and the theoretical value of torque is indicated.
Optionally, the identifying of the parameters of the harmonic drive hysteresis stiffness model specifically includes:
definition recognition function J sigmai(ε(θi))2J is a function of the parameters a, B, c, a, B to be fitted, J (a, B, c, a, B) Σi(ε(θi))2Wherein a, B and c represent coefficients of a stiffness function f (theta), A represents the memory intensity of the genetic factor, and B represents the speed of memory decay;
solving min sigmai(ε(θi))2) And determining the corresponding obtained fitting parameters a, B, c, A and B as the parameters of the harmonic drive hysteresis stiffness model.
The beneficial technical effects that this application includes are as follows:
the harmonic drive hysteresis stiffness model consists of a stiffness function and a friction function, and the friction function is defined as a function related to a torsion angle at the past moment according to the genetic characteristic of harmonic gear drive, so that the dynamic stiffness characteristic of the harmonic gear drive is better described; the discontinuity problem of the friction function is corrected, so that the deviation between the established rigidity hysteresis model curve and the theoretical rigidity hysteresis model curve is not too large, the hysteresis rigidity error is compensated, the finally obtained harmonic drive hysteresis rigidity model is more accurate, the dynamic analysis precision and the transmission performance of harmonic drive are improved, and an effective mathematical model basis is provided for the control of the harmonic reducer.
Drawings
Fig. 1 is a hysteresis stiffness curve of a harmonic reducer provided in the background art.
FIG. 2 is a flowchart of a method of an embodiment of a harmonic drive hysteresis stiffness modeling method based on genetic characteristics provided by the present invention;
FIG. 3 is a graphical illustration of stiffness hysteresis curves provided by the present invention when the friction function is not modified;
FIG. 4 is a schematic block diagram of one embodiment of a test platform provided by the present invention;
FIG. 5 is a graphical representation of experimental data points provided by the present invention;
fig. 6 is a schematic diagram for plotting the stiffness hysteresis curve provided by the present invention.
Detailed Description
The present application is described in further detail below with reference to figures 2-6.
The embodiment of the application discloses a harmonic drive hysteresis stiffness modeling method based on genetic characteristics, and with reference to fig. 2, the method comprises the following steps:
and S1, determining a rigidity function according to the hysteresis rigidity relation between the torque and the torsion angle of the harmonic drive.
In this embodiment, the relationship between the torque T and the corresponding twist angle θ of the harmonic reducer drive is generally defined by the nonlinear equation T ═ f (θ), where f (θ) is a base stiffness function determined from the harmonic gear characteristics.
In the present embodiment, the stiffness function f (θ) is defined as a fifth-order polynomial function passing through the origin, and in other embodiments, the stiffness function f (θ) can be defined as a third-order polynomial function or the like; specifically, the stiffness function f (θ) is expressed as follows:
f(θ)＝aθ5+bθ3+cθ
wherein a, b and c represent fifth-order polynomial function coefficients.
And S2, establishing a friction function according to the genetic characteristics of the harmonic gear transmission, and determining the friction function as the integral of the torsion angle and the time under the genetic effect.
It should be noted that the genetic characteristic means that the system output depends not only on the actual state of the system, but also on all the states that the system passes through. The harmonic gear drive torsion angle depends not only on the magnitude of the transmitted torque but also on the direction of change of the transmitted torque. In the present embodiment, the friction function is defined as the value of the torsion angle θ before time t, which is determined by the genetic characteristics of the harmonic gear drive
Figure BDA0003108965550000061
In the formula
Figure BDA0003108965550000062
Represents a genetic element; θ(s) represents a torsion angle of the harmonic reducer at the s-th time from 0 time when the system starts to operate to the current time t; set at t0The genetic effect before an instant < t is ignored, the origin of time is set to t0When 0, the friction function is determined as follows:
Figure BDA0003108965550000063
it should be noted that, the genetic effect of the harmonic gear transmission system is set to disappear gradually after a period of time, the genetic factor should meet the condition of continuous reduction,
Figure BDA0003108965550000064
expressed as an exponential function:
Figure BDA0003108965550000065
wherein A, B represents the function parameter of genetic factor, A represents memory intensity, B represents the speed of memory decay, u is tableThe genetic factor function may reflect the dynamic stiffness characteristics of a harmonic gear drive, indicating the duration of genetic action, i.e., the length of time a condition has elapsed from occurrence to the current time. s represents the time when a state occurs between 0 and the current time t at which the system starts operating.
Based on the genetic factors, the genetic role of the state at time s in the process can be characterized specifically as:
Figure BDA0003108965550000066
to take into account the state of the entire system before time t, assuming that the drive torque is continuous and periodic, the hysteresis stiffness curve converges to a centrosymmetric hysteresis loop, and further, the torsion angle should satisfy the relationship: θ (t) ═ csin (wt), where C denotes the magnitude of the torsion angle and w denotes the angular velocity of the harmonic gear.
Further, the friction function is converted as follows:
Figure BDA0003108965550000071
wherein
Figure BDA0003108965550000072
Represents the derivative of the twist angle θ (t); it should be noted that, in the following description,
Figure BDA0003108965550000073
to test the torque forward unloading/reverse loading,
Figure BDA0003108965550000074
the torque is unloaded in reverse/loaded in forward direction for the experiment.
And S3, correcting the discontinuity of the friction function to obtain a friction correction function.
It should be noted that z (t) defined in step S2 is a piecewise function, and is discontinuous at the critical point of the piecewise function
Figure BDA0003108965550000075
When, when
Figure BDA0003108965550000076
In that
Figure BDA0003108965550000077
And
Figure BDA0003108965550000078
the formula transformation of the friction function z (T) in the process of transformation between the two can lead to T (theta, T) being at
Figure BDA0003108965550000079
Discontinuity, as shown in fig. 3, may cause misalignment between points a and B, and misalignment between points C and D, and finally may cause the variation trend of the hysteresis stiffness curve of the model to not conform to the variation trend of the hysteresis stiffness curve of the harmonic gear transmission, and the hysteresis stiffness error is too large. Due to the genetic characteristic of the friction function, the difference between two end points is unequal, and if the problem of unequal difference is solved by adopting an addition and subtraction method, the deviation between the finally established stiffness hysteresis model curve and the theoretical stiffness hysteresis curve is overlarge. For the problem of the friction function, further, the friction function needs to be corrected, and the obtained friction correction function is specifically as follows:
Figure BDA00031089655500000710
to be noted, when
Figure BDA00031089655500000711
When cos (wt) ═ 0, point a and point B coincide with each other, and point C and point D coincide with each other; while at the twist angle theta equal to 0,
Figure BDA00031089655500000712
the change trend of the rigidity hysteresis curve is met, and the requirement of T (theta, T) continuity can be met.
And S4, determining a harmonic drive hysteresis rigidity model according to the rigidity function and the friction correction function.
It should be noted that, because of the stiffness hysteresis phenomenon in harmonic gear transmission, the hysteresis stiffness relationship between torque and torsion angle is expressed as being composed of two parts, namely an elastic term and a friction term, where the elastic term is a stiffness function f (θ) and the friction term is z. According to the established rigidity function and the friction correction function, the harmonic drive hysteresis rigidity model is obtained as follows:
Figure BDA0003108965550000081
the harmonic drive hysteresis stiffness model of the embodiment is composed of a stiffness function and a friction function, and the friction function is defined as a function related to a torsion angle at the past moment according to the genetic characteristic of harmonic gear drive, so that the dynamic stiffness characteristic of the harmonic gear drive is better described; the discontinuity problem of the friction function is corrected, so that the deviation between the established rigidity hysteresis model curve and the theoretical rigidity hysteresis model curve is not too large, the hysteresis rigidity error is compensated, the finally obtained harmonic drive hysteresis rigidity model is more accurate, the dynamic analysis precision and the transmission performance of harmonic drive are improved, and an effective mathematical model basis is provided for the control of the harmonic reducer.
In this embodiment, the harmonic drive hysteresis stiffness modeling method based on genetic characteristics further includes the following steps:
and S5, acquiring experimental data of the torque and the torsion angle by testing the forward loading and the reverse loading of the harmonic reducer.
In this embodiment, the test platform is of a modular structure and can be used for testing the performance of speed reducers of different models, referring to fig. 4, the test platform comprises a driving motor 1, an input torque measurement system 2, an input rotation angle measurement system 3, a harmonic speed reducer 4, an output rotation angle measurement system 5 and an output torque/load control system 6, in this embodiment, a high-precision double-range torque sensor is adopted in the input torque measurement system 2, and the torque sensor is used for accurately measuring the torque of each working condition; the input torque measuring system 2 and the input rotation angle measuring system 3 adopt grating angle sensors which measure input and output torsion angles.
It should be noted that the testing principle of the testing platform is as follows: the upper computer drives the motor to run at a certain speed by controlling the frequency converter, a torque sensor in the input torque measuring system can measure input torque, an angle encoder in the input corner measuring system can record the input angle of the harmonic reducer, after power passes through the harmonic reducer, the output corner measuring system can record the output angle, and the output torque/load control system can measure output torque or can provide load through the magnetic powder brake; after experimental data such as the torsion angle, the torque and the like are collected, the experimental data are transmitted to an upper computer for analysis and processing.
The specific test method comprises the following steps: after gear backlash is eliminated in the gear steering direction, the input end of the harmonic gear is fixed, forward loading is carried out on the load end to rated torque, unloading is carried out step by step, then reverse loading is carried out to rated torque and unloading is carried out step by step, the change of the torque in the loading/unloading process is recorded through a torque sensor, and the change condition of a corresponding torsion angle is obtained through an angle sensor.
It should be noted that, before the experiment, it should be checked whether the input end of the harmonic reducer is completely fixed, and the gap in the transmission system is eliminated, and the initial torsion angle is returned to zero in the upper computer system. Loading at a load end through a magnetic powder brake, slowly loading to rated torque from 0, then slowly unloading, and finally carrying out reverse loading/unloading; the torque and the torsion angle are recorded in the upper computer through corresponding sensors.
And S6, fitting the experimental data by adopting a least square method, and identifying the parameters of the harmonic drive hysteresis stiffness model.
In the present embodiment, setting
Figure BDA0003108965550000091
For the experimental torque value, T (θ) is the theoretical torque value, and ∈ (θ) is the torque error, then:
Figure BDA0003108965550000092
further, discrete experimental data points (θ) obtained from the testi,
Figure BDA0003108965550000093
) And establishing a minimum torque error objective function by adopting a least square method:
Figure BDA0003108965550000101
wherein, thetaiTwist angle representing the ith experimental data point;
Figure BDA0003108965550000102
torque experimental value, T (θ), representing the ith experimental data pointi) The theoretical value of the torque corresponding to the ith torsion angle is shown; epsilon (theta)i) An error value between the experimental value of torque and the theoretical value of torque is indicated.
Further, an identification function J sigma is definedi(ε(θi))2J is a function of the parameters a, B, c, a, B to be fitted, J (a, B, c, a, B) Σi(ε(θi))2(ii) a Solving min sigmai(ε(θi))2) I.e. the minimum value of J (a, B, c, a, B), the corresponding obtained a, B, c, a, B fitting parameters are determined as the parameters of the harmonic drive hysteresis stiffness model. In this embodiment, the measured experimental data points are imported into the MATLAB, and fitting is performed on the MATLAB by using a least square method, and an appropriate parameter is selected as an initial iteration point to fit an unknown parameter in the harmonic drive hysteresis stiffness model of this embodiment, so as to obtain an optimal solution.
In the embodiment, the performance of the harmonic drive hysteresis stiffness model of the embodiment is further analyzed, referring to fig. 5, an experimental data point is drawn by taking the torque T as an abscissa and the rotation angle θ as an ordinate; after parameter identification, 5 undetermined parameters required by the rigidity model are obtained, and the corrected rigidity hysteresis curve of the speed reducer refers to fig. 6.
It should be understood that, the sequence numbers of the steps in the foregoing embodiments do not imply an execution sequence, and the execution sequence of each process should be determined by its function and inherent logic, and should not constitute any limitation to the implementation process of the embodiments of the present invention.

Claims (5)

1. A harmonic drive hysteresis stiffness modeling method based on genetic characteristics is characterized by comprising the following steps: the method comprises the following steps:
determining a stiffness function according to a hysteresis stiffness relationship between a torque and a torsion angle of the harmonic drive;
establishing a friction function according to the genetic characteristics of harmonic gear transmission, and determining the friction function as the integral of the torsion angle and time under the genetic effect;
correcting discontinuity of the friction function to obtain a friction correction function;
determining a harmonic drive hysteresis stiffness model according to the stiffness function and the friction correction function;
establishing a friction function according to the genetic characteristics of harmonic gear transmission, and determining the friction function as the integral of the torsion angle and time under the genetic effect, wherein the method specifically comprises the following steps:
Figure FDA0003532264240000011
wherein
Figure FDA0003532264240000012
Is a genetic factor, u represents the duration of the genetic action; t represents the current time; θ(s) represents a torsion angle of a state at the s-th time from the 0-th time when the system starts to operate to the current time t;
the genetic factor
Figure FDA0003532264240000013
And the torsion angle θ(s) satisfy the following relationship:
Figure FDA0003532264240000014
θ(s)＝Csin(ws)；
the friction function obtained by the formula conversion is as follows:
Figure FDA0003532264240000015
wherein A, B represents the function parameter of the genetic factor, A represents the memory intensity, B represents the speed of memory decay; c represents the magnitude of the torsion angle; w represents the angular velocity of the harmonic gear;
Figure FDA0003532264240000016
represents the derivative of θ (t);
correcting discontinuity of the friction function to obtain a friction correction function, wherein the friction correction function is as follows:
Figure FDA0003532264240000021
the stiffness function f (theta) is defined as a fifth-order polynomial function, and a harmonic drive hysteresis stiffness model is determined according to the stiffness function and the friction correction function, and the method specifically comprises the following steps:
Figure FDA0003532264240000022
wherein a, b, c represent the coefficients of a fifth order polynomial function.
2. The method of modeling harmonic drive hysteresis stiffness based on genetic characteristics of claim 1, wherein: the method further comprises the following steps:
the method comprises the steps of testing forward loading and reverse loading of a harmonic reducer to obtain experimental data of torque and a torsion angle;
and fitting the experimental data by adopting a least square method, and identifying the parameters of the harmonic drive hysteresis stiffness model.
3. The method of modeling harmonic drive hysteresis stiffness based on genetic characteristics of claim 2, wherein: the method comprises the following steps of testing forward loading and backward loading of the harmonic reducer to obtain experimental data of torque and a torsion angle, wherein the process comprises the following steps:
the input end of the harmonic gear is fixed, the load end carries out forward loading to rated torque, the load end carries out gradual unloading, then the load end carries out backward loading to the rated torque and carries out gradual unloading, the change of the torque in the loading/unloading process is recorded through a torque sensor, the change condition of a corresponding torsion angle is obtained through an angle sensor, and experimental data of the torque and the torsion angle are uploaded to an upper computer.
4. The method of modeling harmonic drive hysteresis stiffness based on genetic characteristics of claim 2, wherein: fitting the experimental data by using a least square method, which comprises the following steps:
using least squares and based on discrete experimental data points
Figure FDA0003532264240000031
Establishing a minimum torque error objective function:
Figure FDA0003532264240000032
wherein, thetaiTwist angle representing the ith experimental data point;
Figure FDA0003532264240000033
torque experimental value, T (θ), representing the ith experimental data pointi) The theoretical value of the torque corresponding to the ith torsion angle is shown; epsilon (theta)i) An error value between the experimental value of torque and the theoretical value of torque is indicated.
5. The method of modeling harmonic drive hysteresis stiffness based on genetic characteristics of claim 4, wherein: the parameters of the harmonic drive hysteresis stiffness model are identified as follows:
definition recognition function J ═ Σi(ε(θi))2J is a function of the parameters a, B, c, a, B to be fitted, J (a, B, c, a, B) Σi(ε(θi))2Wherein a, B and c represent coefficients of a stiffness function f (theta), A represents the memory intensity of the genetic factor, and B represents the speed of memory decay;
solving for min (Sigma)i(ε(θi))2) And determining the corresponding obtained fitting parameters a, B, c, A and B as the parameters of the harmonic drive hysteresis stiffness model.
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《谐波齿轮传动非线性动力学建模及仿真研究》;罗阳等;《机械传动》;20210430;第58-63页 *

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