CN113009819B - Force control-based elliptical vibration cutting machining method - Google Patents

Force control-based elliptical vibration cutting machining method Download PDF

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CN113009819B
CN113009819B CN202110182746.9A CN202110182746A CN113009819B CN 113009819 B CN113009819 B CN 113009819B CN 202110182746 A CN202110182746 A CN 202110182746A CN 113009819 B CN113009819 B CN 113009819B
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elliptical vibration
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CN113009819A (en
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张臣
霍建强
石晗
于福航
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Nanjing University of Aeronautics and Astronautics
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Nanjing University of Aeronautics and Astronautics
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    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
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Abstract

The invention discloses a force control-based elliptical vibration cutting method, which takes a force control strategy as a core and combines a data synchronous acquisition technology to realize elliptical vibration self-adaptive tracking cutting. The force control strategy is a double closed-loop control strategy based on an outer loop impedance controller and an inner loop position controller, the outer loop impedance controller gives out ideal output displacement of the tool nose of the elliptical vibration cutting device according to a force signal and a displacement signal, and the inner loop position controller carries out multi-point sampling in each elliptical vibration period to compensate the output displacement error of the tool nose. The invention can judge the cutting position according to the cutting force information, realize the tracking cutting processing of the surface profile of the workpiece, simultaneously compensate the processing error caused by clamping or tool setting, and improve the processing precision.

Description

Force control-based elliptical vibration cutting machining method
Technical Field
The invention relates to an elliptical vibration cutting machining method, in particular to an elliptical vibration cutting machining method based on force control.
Background
The elliptical vibration cutting processing technology has attracted extensive attention and research as a processing technology with excellent performance for processing difficult-to-process materials and surface microtexture. The elliptical vibration cutting is a motion of adding an elliptical track in a cutting motion, in an elliptical cutting period, a part of time is cut and processed by a cutter, and the other part of the cutter is separated from cutting chips, so that the effects of reducing friction force and cutting heat can be achieved, meanwhile, the cutter abrasion is delayed, the service life of the cutter is prolonged, and the processing quality is also improved. In order to effectively improve the machining performance of the elliptical vibration cutting device, it is important to develop a control system suitable for the device. The elliptical vibration cutting control system is divided into an open-loop control system and a closed-loop control system, the open-loop control system is high in working frequency and quick in response, and a feedback loop based on sensor detection does not exist, so that the motion precision is not very high. The elliptical vibration cutting device under closed-loop control has accurate motion and good comprehensive performance, but a high-precision sensor needs to be installed.
The existing elliptical vibration control method is a position control method, utilizes position information of a tool nose of an elliptical vibration cutting device acquired by a capacitive displacement sensor and combines different control algorithms to realize closed-loop control. The research of three-dimensional elliptical vibration cutting motion control aims at designing a hardware platform of a three-dimensional elliptical vibration cutting control system aiming at a self-developed three-dimensional elliptical vibration cutting device, establishing a control system model, and selecting an optimal PID control algorithm from a plurality of control methods to carry out cutting processing experiments. The study of the three-dimensional elliptical vibration auxiliary cutting device and the control researches a fuzzy self-adaptive sliding mode control method, a dynamic model of a three-dimensional elliptical vibration cutting system is established based on a wiener algorithm and an improved memetic algorithm, a sliding mode function and a sliding mode control law are designed in a targeted manner, a three-dimensional elliptical vibration auxiliary cutting processing experiment platform is established, and the performances of the device and the control system are verified from three aspects of cutting traces, surface roughness and cutter abrasion of a workpiece. The design of the RBF self-adaptive sliding mode controller of the three-dimensional elliptical vibration cutting system is combined with the advantages of the two methods of the radial basis function neural network self-adaptive control and the sliding mode control, the radial basis function neural network self-adaptive sliding mode controller for the three-dimensional elliptical vibration auxiliary cutting system is provided, and simulation results show that the controller can tend to be stable in a short time and has strong robustness.
Disclosure of Invention
The purpose of the invention is as follows: the invention aims to provide the force control-based elliptical vibration cutting machining method which can realize tracking cutting machining of the surface profile of a workpiece, compensate machining errors caused by clamping or tool setting and improve machining precision.
The technical scheme is as follows: the invention provides an elliptical vibration cutting machining method based on force control, which comprises the following steps:
(1) inputting the actual cutting force detected by a piezoelectric dynamic force sensor and the actual output position of the tool nose of the elliptical vibration cutting device detected by a capacitive displacement sensor into an environmental parameter estimator, and estimating the current environmental stiffness coefficient and the undeformed environmental position by the environmental parameter estimator;
(2) according to the two environment parameters obtained by estimation, obtaining an ideal output position of the tool nose of the elliptical vibration cutting device by utilizing an expected position generation model;
(3) inputting the difference value between the actual cutting force and the expected cutting force and the ideal output position obtained in the step (2) into an improved impedance model, calculating the position deviation needing to be compensated according to the improved impedance model, and interfering the ideal output position by the position deviation so as to form a new expected output position which is input into the inner ring position controller currently;
(4) the difference value of the new expected output position and the actual output position is input into an inner ring position controller for position closed loop control, and the inner ring position controller outputs a sinusoidal voltage value to drive the elliptical vibration cutting device to carry out vibration cutting.
Further, the environment parameter estimator adopts a forgetting factor recursive least square method, and the formula is as follows:
Figure BDA0002940586450000021
Figure BDA0002940586450000022
γ(k)＝1/[1+xT(k)P(k-1)x(k)]
wherein k represents the current time, k-1 represents the last time,
Figure BDA0002940586450000023
keis the environmental stiffness coefficient, xeIs an undeformed environmental position;
Figure BDA0002940586450000024
x is the actual output position of the tip of the elliptical vibration cutting device detected by the capacitive displacement sensor, faThe cutting force in the cutting depth direction at the cutter point of the elliptical vibration cutting device is detected by a piezoelectric dynamic force sensor,
Figure BDA0002940586450000025
P11、P12、P21、P22are respectively covarianceThe value of each entry of the matrix P, λ is the forgetting factor,
at the beginning of the algorithm
Figure BDA0002940586450000026
The initial value of P is obtained by the first two data points, which is solved directly by the following equation
Figure BDA0002940586450000027
Figure BDA0002940586450000028
Wherein the content of the first and second substances,
Figure BDA0002940586450000031
further, the expected position generation model is derived through a dynamic impedance model and an environment model, wherein the dynamic impedance model and the environment model are respectively adopted
Figure BDA0002940586450000032
And fa＝ke(xa-xe)，
Wherein m, b and k respectively represent a target inertia coefficient, a target damping coefficient and a target stiffness coefficient in the impedance model.
Figure BDA0002940586450000033
Respectively showing the expected position, speed and acceleration of the knife tip of the elliptical vibration cutting device which are to be input into the inner ring position controller,
Figure BDA0002940586450000034
the ideal output position, ideal speed and ideal acceleration of the cutting edge of the elliptical vibration cutting device are respectively shown, and e is the difference between the expected cutting force and the actual cutting force, namely, e ═ fi-fa，fiThe expected cutting force in the cutting depth direction at the cutter point of the elliptical vibration cutting device is obtained by
Figure BDA0002940586450000035
And fa＝ke(xa-xe) A differential equation for the force deviation is obtained:
Figure BDA0002940586450000036
wherein the content of the first and second substances,
Figure BDA0002940586450000037
and
Figure BDA0002940586450000038
the first derivative and the second derivative of the expected cutting force in the cutting depth direction at the tool tip of the elliptical vibration cutting device.
Figure BDA0002940586450000039
Performing laplace transform on the above equation to obtain a first derivative and a second derivative of a difference e between the expected cutting force and the actual cutting force, respectively:
Figure BDA00029405864500000310
wherein, E(s), Fi(s)、Xi(s) are e and f, respectivelyi、xiThe Ralsberg transform of (1). The force tracking cutting error at steady state is:
Figure BDA00029405864500000311
if the force tracking deviation e is to be eliminatedssThe molecule of the above formula should be 0,
Figure BDA00029405864500000312
after laplace inverse transformation is carried out on the above formula, a relational expression between the ideal output position of the knife edge of the elliptical vibration cutting device and the expected cutting force is obtained, namely, an expected position generation model is as follows:
Figure BDA00029405864500000313
further, the improved impedance model is as follows:
Figure BDA0002940586450000041
in the formula, kpIs a proportionality coefficient, kiFor the integral coefficient, the force tracking cutting error at steady state is:
Figure BDA0002940586450000042
comparison
Figure BDA0002940586450000043
And
Figure BDA0002940586450000044
the improved impedance model has smaller force tracking error than the original impedance model and can reach a steady state more quickly,
Figure BDA0002940586450000045
the formula is a continuous equation of an improved impedance model, the discretization of the impedance model is needed to be written into a real-time controller in practical application, an incremental method is adopted for the discretization of the model, and the time taken for the outer-ring impedance controller to circulate once is assumed to be T
xa(t)＝xa(k)
xi(t)＝xi(k)
e(t)＝e(k)
Figure BDA0002940586450000046
Figure BDA0002940586450000047
Figure BDA0002940586450000048
Substituting the above 6 formulas
Figure BDA0002940586450000049
Equation, the following equation can be obtained:
Figure BDA00029405864500000410
by subtracting the two formulas, a discretization model of the expected position of the cutting edge of the elliptical vibration cutting device, which is to be input into the inner ring position controller, can be obtained:
Figure BDA0002940586450000051
furthermore, the inner ring position controller adopts fuzzy PID control, the controller inputs an expected position which is to be input into the inner ring position controller and an actual output position detected by the capacitance type displacement sensor, and the proportional coefficient, the integral coefficient and the differential coefficient in the PID algorithm are adjusted in real time according to the current position deviation and the difference value of the current position deviation and the position deviation at the last moment, so that the defect that the traditional PID cannot optimize parameters in real time is overcome.
Further, the method uses a device consisting of an outer loop impedance controller and an inner loop position controller, wherein the outer loop impedance controller comprises three parts: an environmental parameter estimator, a desired location generation model, an improved impedance model.
Further, the inner loop position controller employs a fuzzy PID algorithm.
The method adopts four parallel threads to realize the force control-based elliptical vibration cutting strategy, and the specific mode is as follows:
the four threads are an environment parameter estimation thread, an expected position generation thread, an impedance control thread and a position control thread respectively. The system distributes processing time among each thread according to set cycle time, the piezoelectric dynamic force sensor samples a plurality of points in a single vibration period, but only one point is input into a force control strategy after processing, so the cycle time of the first three threads is determined according to the vibration period when the elliptical vibration cutting device works, the position control thread performs multiple sampling and closed-loop adjustment in the single vibration period, and the cycle time interval is smaller than that of the other three threads. In addition, the cycle of data acquisition should be delayed for a certain time before the cycle begins, and the cycle is restarted at the peak point.
Different variables need to be shared between threads. The variables needing to be shared between the environment parameter estimation thread and the expected position generation thread are an environment rigidity coefficient and an undeformed environment position, the variables needing to be shared between the expected position generation thread and the impedance control thread are ideal output positions of the knifepoint of the elliptical vibration cutting device, the variables needing to be shared between the impedance control thread and the position control thread are expected positions of the knifepoint of the elliptical vibration cutting device, which are input into an inner ring position controller, and the variables needing to be shared among the threads further comprise an actual output position of the knifepoint of the elliptical vibration cutting device detected by a capacitive displacement sensor and a cutting force in the cutting depth direction of the knifepoint of the elliptical vibration cutting device detected by a piezoelectric dynamic force sensor.
The method can identify the surface condition of the workpiece and the clamping error, and has strong self-adaptive processing capability. The method can judge the environmental condition according to the real-time cutting force information to obtain the position of the cutter actually cutting into the workpiece, and can realize the purposes of tracking cutting processing and compensating environmental errors by combining the advantages of position control. The implementation of the method of the invention requires selecting a real-time controller with parallel advantages to execute the program, and a real-time operating system running on the real-time controller can ensure high certainty of the control algorithm.
Has the advantages that: the invention can judge the cutting position according to the cutting force information, can realize real-time tracking cutting processing and has high processing precision.
Drawings
FIG. 1 is a schematic diagram of a force control based elliptical vibration machining strategy.
Detailed Description
The elliptical vibration cutting system based on force control in the embodiment comprises an outer ring impedance controller and an inner ring position controller, wherein the outer ring impedance controller generates cutting position compensation quantity according to cutting force deviation and inputs the cutting position compensation quantity into the inner ring position controller, so that closed-loop adjustment and control of the cutter point position of the elliptical vibration cutting device are realized. Wherein, outer loop impedance controller contains three parts: an environment parameter estimator, a desired position generation model, an improved impedance model, and an inner loop position controller using a fuzzy PID algorithm (fuzzy PID controller), as shown in fig. 1.
The elliptical vibration cutting machining method based on force control comprises the following steps:
(1) inputting the actual cutting force detected by a piezoelectric dynamic force sensor and the actual output position of the tool nose of the elliptical vibration cutting device detected by a capacitive displacement sensor into an environmental parameter estimator, and estimating the current environmental stiffness coefficient and the undeformed environmental position by the environmental parameter estimator;
(2) according to the two environment parameters obtained by estimation, obtaining an ideal output position of the tool nose of the elliptical vibration cutting device by utilizing an expected position generation model;
(3) inputting the difference value between the actual cutting force and the expected cutting force and the ideal output position obtained in the step (2) into an improved impedance model, calculating the position deviation needing to be compensated according to the improved impedance model, and interfering the ideal output position by the position deviation so as to form a new expected output position which is input into the inner ring position controller currently;
(4) the difference value of the new expected output position and the actual output position is input into an inner ring position controller for position closed loop control, and the inner ring position controller outputs a sinusoidal voltage value to drive the elliptical vibration cutting device to carry out vibration cutting.
The model and the specific method adopted by the four parts are as follows:
1.1 Environment parameter estimator
The environmental parameter estimator of the force control machining strategy adopts a forgetting factor recursion least square method, environmental parameters can change slightly in the elliptical vibration cutting process, the forgetting factor is reasonably set to reduce the information quantity of old data, the effectiveness of new data is improved, and the environmental parameters are identified in real time. The environment parameter estimator in the method is as follows:
Figure BDA0002940586450000071
wherein k represents the current time, k-1 represents the last time,
Figure BDA0002940586450000072
keis the environmental stiffness coefficient, xeIs an undeformed environmental position;
Figure BDA0002940586450000073
x is the actual output position of the tip of the elliptical vibration cutting device detected by the capacitive displacement sensor, faThe cutting force in the cutting depth direction at the cutter point of the elliptical vibration cutting device is detected by a piezoelectric dynamic force sensor,
Figure BDA0002940586450000074
P11、P12、P21、P22the values of each item of the covariance matrix P are respectively, and λ is a forgetting factor.
At the beginning of the algorithm
Figure BDA0002940586450000075
The initial value of P is obtained by the first two data points, which is solved directly by the following equation
Figure BDA0002940586450000076
Wherein the content of the first and second substances,
Figure BDA0002940586450000077
1.2 expected position Generation model
The expected position generating model of the force control machining strategy is obtained by derivation of a dynamic impedance model and an environment model, wherein the dynamic impedance model and the environment model respectively adopt formulas (3) and (4)
Figure BDA0002940586450000078
fa＝ke(xa-xe) (4)
Wherein m, b and k respectively represent a target inertia coefficient, a target damping coefficient and a target stiffness coefficient in the impedance model.
Figure BDA0002940586450000079
Respectively showing the expected position, speed and acceleration of the knife tip of the elliptical vibration cutting device which are to be input into the inner ring position controller,
Figure BDA00029405864500000710
the ideal output position, ideal speed and ideal acceleration of the cutting edge of the elliptical vibration cutting device are respectively shown, and e is the difference between the expected cutting force and the actual cutting force, namely, e ═ fi-fa，fiThe expected cutting force in the cutting depth direction at the cutter point of the elliptical vibration cutting device. Obtaining a differential equation about the force deviation through (3) and (4):
Figure BDA0002940586450000081
wherein the content of the first and second substances,
Figure BDA0002940586450000082
and
Figure BDA0002940586450000083
the first derivative and the second derivative of the expected cutting force in the cutting depth direction at the tool tip of the elliptical vibration cutting device. Laplace transform is carried out on the formula to obtain
Figure BDA0002940586450000084
Wherein, E(s), Fi(s)、Xi(s) are e and f, respectivelyi、xiThe Ralsberg transform of (1). The force tracking cutting error at steady state is:
Figure BDA0002940586450000085
if the force tracking deviation e is to be eliminatedssThe molecule of formula (7) should be 0,
Figure BDA0002940586450000086
after Laplace inverse transformation is carried out on the above formula, a relational expression between the ideal output position of the knife edge of the elliptical vibration cutting device and the expected cutting force is obtained, namely the expected position generation model is
Figure BDA0002940586450000087
The model can obtain an ideal output position corresponding to a time-varying expected cutting force, timely adjusts the output of the elliptical vibration cutting device, and has higher applicability.
1.3 improved impedance model
The improved impedance model of the force control processing strategy is characterized in that a PI compensation link is added in the traditional impedance model, the response and the interference of an outer loop can be quickly corrected, the force tracking steady-state error is further reduced, and the improved impedance model is as follows:
Figure BDA0002940586450000088
in the formula, kpIs a proportionality coefficient, kiFor the integral coefficient, the force tracking cutting error at steady state is:
Figure BDA0002940586450000089
comparing (7) and (11), the improved impedance model has a smaller force tracking error than the original impedance model and can reach a steady state faster.
(10) The formula is a continuous equation of an improved impedance model, the discretization of the impedance model is needed to be written into a real-time controller in practical application, an incremental method is adopted for the discretization of the model, and the time taken for the outer-ring impedance controller to circulate once is assumed to be T
Figure BDA0002940586450000091
Substituting the 6 expressions in (12) into the expression (10) can obtain the following expression:
Figure BDA0002940586450000092
and (3) obtaining a discretization model of the expected position of the tool nose of the elliptical vibration cutting device, which is input into the inner ring position controller, by differentiating the two equations in (13):
Figure BDA0002940586450000093
1.4 inner ring position control
The inner ring position controller of the force control processing strategy adopts fuzzy PID control, the input of the controller is 1.3, the expected position to be input into the inner ring position controller and the actual output position detected by a capacitance displacement sensor are obtained, and the proportional coefficient, the integral coefficient and the differential coefficient in a PID algorithm are adjusted in real time according to the current position deviation and the difference value of the current position deviation and the position deviation at the last moment, so that the defect that the traditional PID can not optimize parameters in real time is overcome.

Claims (4)

1. A force control-based elliptical vibration cutting machining method is characterized in that: the method comprises the following steps:
(1) inputting the actual cutting force detected by a piezoelectric dynamic force sensor and the actual output position of the tool nose of the elliptical vibration cutting device detected by a capacitive displacement sensor into an environmental parameter estimator, and estimating the current environmental stiffness coefficient and the undeformed environmental position by the environmental parameter estimator;
(2) according to the two environment parameters obtained by estimation, obtaining an ideal output position of the tool nose of the elliptical vibration cutting device by utilizing an expected position generation model;
(3) inputting the difference value between the actual cutting force and the expected cutting force and the ideal output position obtained in the step (2) into an improved impedance model, calculating the position deviation needing to be compensated according to the improved impedance model, and interfering the ideal output position by the position deviation so as to form a new expected output position which is input into the inner ring position controller currently;
(4) the difference value of the new expected output position and the actual output position is input into an inner ring position controller for position closed-loop control, and the inner ring position controller outputs a sine voltage value to drive an elliptical vibration cutting device to carry out vibration cutting;
the environment parameter estimator adopts a forgetting factor recursive least square method, and the formula is as follows:
Figure RE-FDA0003411609060000011
Figure RE-FDA0003411609060000012
γ(k)＝1/[1+xT(k)P(k-1)x(k)]
wherein k represents the current time, k-1 represents the last time,
Figure RE-FDA0003411609060000013
keis the environmental stiffness coefficient, xeIs an undeformed environmental position;
Figure RE-FDA0003411609060000014
x is the actual output position of the tip of the elliptical vibration cutting device detected by the capacitive displacement sensor, faThe cutting force in the cutting depth direction at the cutter point of the elliptical vibration cutting device is detected by a piezoelectric dynamic force sensor,
Figure RE-FDA0003411609060000015
P11、P12、P21、P22respectively, the value of each item of the covariance matrix P, lambda is a forgetting factor,
at the beginning of the algorithm
Figure RE-FDA0003411609060000016
The initial value of P is obtained by the first two data points, which is solved directly by the following equation
Figure RE-FDA0003411609060000017
Figure RE-FDA0003411609060000018
Wherein the content of the first and second substances,
Figure RE-FDA0003411609060000021
the expected position generation model is obtained by derivation through a dynamic impedance model and an environment model, wherein the dynamic impedance model and the environment model are respectively adopted
Figure RE-FDA0003411609060000022
And fa＝ke(xa-xe)，
Wherein m, b and k respectively represent a target inertia coefficient, a target damping coefficient and a target rigidity coefficient in the impedance model, and xa、
Figure RE-FDA0003411609060000023
Respectively representing the expected position, speed, acceleration, x of the cutting edge of the elliptical vibration cutting device to be input into the inner ring position controlleri、
Figure RE-FDA0003411609060000024
The ideal output position, ideal speed and ideal acceleration of the cutting edge of the elliptical vibration cutting device are respectively shown, and e is the difference between the expected cutting force and the actual cutting force, namely, e ═ fi-fa，fiThe expected cutting force in the cutting depth direction at the cutter point of the elliptical vibration cutting device is obtained by
Figure RE-FDA0003411609060000025
And fa＝ke(xa-xe) A differential equation for the force deviation is obtained:
Figure RE-FDA0003411609060000026
wherein the content of the first and second substances,
Figure RE-FDA0003411609060000027
and
Figure RE-FDA0003411609060000028
the first derivative and the second derivative of the expected cutting force in the cutting depth direction at the cutter point of the elliptical vibration cutting device,
Figure RE-FDA0003411609060000029
first and second derivatives of the difference e between the desired cutting force and the actual cutting force,
laplace transform of the above equation yields:
Figure RE-FDA00034116090600000210
wherein, E(s), Fi(s)、Xi(s) are e and f, respectivelyi、xiThe Ralsberg transform of (1); the force tracking cutting error at steady state is:
Figure RE-FDA00034116090600000211
if the force tracking deviation e is to be eliminatedssThe molecule of the above formula should be 0,
Figure RE-FDA00034116090600000212
after laplace inverse transformation is carried out on the above formula, a relational expression between the ideal output position of the knife edge of the elliptical vibration cutting device and the expected cutting force is obtained, namely, an expected position generation model is as follows:
Figure RE-FDA00034116090600000213
the improved impedance model is as follows:
Figure RE-FDA0003411609060000031
in the formula, kpIs a proportionality coefficient, kiFor the integral coefficient, the force tracking cutting error at steady state is:
Figure RE-FDA0003411609060000032
comparison
Figure RE-FDA0003411609060000033
And
Figure RE-FDA0003411609060000034
the improved impedance model has smaller force tracking error than the original impedance model and can reach a steady state more quickly,
Figure RE-FDA0003411609060000035
the formula is a continuous equation of an improved impedance model, the discretization of the impedance model is needed to be written into a real-time controller in practical application, an incremental method is adopted for the discretization of the model, and the time taken for the outer-ring impedance controller to circulate once is assumed to be T
xa(t)＝xa(k)
xi(t)＝xi(k)
e(t)＝e(k)
Figure RE-FDA0003411609060000036
Figure RE-FDA0003411609060000037
Figure RE-FDA0003411609060000038
Substituting the 6 formulas into
Figure RE-FDA0003411609060000039
Equation, the following equation can be obtained:
Figure RE-FDA00034116090600000310
Figure RE-FDA00034116090600000311
by subtracting the above two equations, a discretized model of the desired position of the cutting edge of the elliptical vibration cutting apparatus to be input to the inner ring position controller can be obtained:
Figure RE-FDA0003411609060000041
2. the force control-based elliptical vibration cutting machining method according to claim 1, characterized in that: the inner ring position controller adopts fuzzy PID control, the controller inputs an expected position which should be input into the inner ring position controller and an actual output position detected by the capacitance displacement sensor, and the proportional coefficient, the integral coefficient and the differential coefficient in the PID algorithm are adjusted in real time according to the current position deviation and the difference value of the current position deviation and the position deviation at the last moment, so that the defect that the traditional PID can not optimize parameters in real time is overcome.
3. The force control-based elliptical vibration cutting machining method according to claim 1, characterized in that: the device used by the method consists of an outer ring impedance controller and an inner ring position controller, wherein the outer ring impedance controller comprises three parts: an environmental parameter estimator, a desired location generation model, an improved impedance model.
4. The force control-based elliptical vibration cutting machining method according to claim 3, characterized in that: the inner ring position controller employs a fuzzy PID algorithm.
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