CN116242300A - Method for identifying installation errors of rotary parts based on elliptical error model - Google Patents
Method for identifying installation errors of rotary parts based on elliptical error model Download PDFInfo
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- G01—MEASURING; TESTING
- G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
- G01B21/00—Measuring arrangements or details thereof, where the measuring technique is not covered by the other groups of this subclass, unspecified or not relevant
- G01B21/22—Measuring arrangements or details thereof, where the measuring technique is not covered by the other groups of this subclass, unspecified or not relevant for measuring angles or tapers; for testing the alignment of axes
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Abstract
The invention provides a method for identifying the installation error of a rotary part based on an elliptical error model, and the analysis result can judge the geometric eccentricity, the phase position, the deflection angle of the end face and the direction of the rotary part at the same time, thereby providing conditions for the accurate detection or the processing of the next step. The detection principle is that a displacement sensor (such as a contact scanning probe, a non-contact optical probe and the like) is utilized to measure a circle of complete contour line on the radial direction of a revolving body part to obtain an original relative data set, the contour deformation data set is obtained through wavelet decomposition, and the relative data set for optimization is obtained through further processing. Due to the influence of deflection of the rotary body part, the measurement section which is theoretically circular is changed into an ellipse, and the short axis length b of the ellipse section is easy to know to be consistent with the actual radius of the measured rotary body part. And optimizing the objective function by utilizing an improved Gaussian-Newton method in combination with the data set for optimization, and adjusting the threshold value and the increment coefficient of each parameter of the increment matrix according to different iteration convergence speeds of the optimization parameters.
Description
Technical Field
The invention belongs to the technical field of installation and positioning of rotary workpieces, and particularly relates to a rotary part installation error identification method based on an elliptical error model.
Background
Common rotation parts in engineering mainly comprise gears, spline shafts, mounting centers and the like, and are used for transmission, positioning and clamping. When the rotary part is mounted on the turntable, geometric eccentric errors are generated due to gaps between the axis of the rotary part and the axis of the turntable, and parallelism errors, namely deflection errors, are generated due to inclination angles between the end face and the mounting face of the rotary part. The geometric eccentricity error and yaw error are collectively referred to as mounting errors. In high-precision machining and detection, extremely high requirements are placed on accurate installation and positioning of a workpiece, for example, high-precision detection of gears is generally carried out on a measuring instrument in a constant temperature chamber and is clamped by using a center, and currently, the installation error of the gears is generally reduced by high precision of center manufacturing, but the installation error is gradually increased along with the increase of clamping times; more importantly, gears are required to be installed on different machine tools when undergoing different procedures in the manufacturing process, and the installation errors caused by the installation errors directly affect the tooth surface machining precision. Particularly, in the case of a large-sized rotary part, even a small installation error may increase an end machining or detection error due to an excessively large radius. The installation errors of the revolving body parts are caused by a plurality of reasons, and the reasons mainly come from the reference misalignment during mechanical hoisting and the inaccuracy of a detection instrument during manual verification. In the actual production process at present, the standard sphere is mostly used for identifying the center parameter of the revolving body part by adopting a small circular arc method or the gauge block is used for identifying the standard gauge block method, and the methods can only identify one kind of geometric errors or deflection errors and have lower precision. In the actual production process, the current more method is to use absolute measurement data to carry out ellipse fitting or parameter optimization, for example, some scholars propose to scan the profile of the measured rotator part by using a high-precision motion axis of a measuring instrument to obtain the absolute distance from a point on the profile to a rotation center so as to detect the installation error, but the accurate acquisition of the measurement data of the method is provided that the center axis of the clamped measured rotator part has no error, the application scene has limitation, and the acquired data has difference from the theoretical requirement. Compared with the absolute measurement data, the profile relative change data in the process of rotating the rotary part for one circle is easy to obtain, and the profile relative change data can be divided into radial profile relative change data of the rotary part and relative distance change data between the radial profile and the sensor, and can be obtained by measuring the contact type sensor and the non-contact type sensor respectively. How to use the relative measurement data to identify the installation error is a worth solving problem.
Disclosure of Invention
The invention aims to provide a method for identifying the installation error of a rotary part based on an elliptical error model, and the analysis result can be used for judging the geometric eccentricity, the phase of the rotary part, the deflection angle and the direction of the end face at the same time and guiding the correction of the installation error, so that conditions are provided for the accurate detection or the processing of the next step. The detection principle is that a displacement sensor (such as a contact scanning probe, a non-contact optical probe and the like) is utilized to measure a circle of complete contour line on the radial direction of a revolving body part to obtain an original relative data set, the contour deformation data set is obtained through wavelet decomposition, and the relative data set for optimization is obtained through further processing. Due to the influence of deflection of the rotary body part, the measurement section which is theoretically circular is changed into an ellipse, and the short axis length b of the ellipse section is easy to know to be consistent with the actual radius of the measured rotary body part. An elliptical installation error model is established through an elliptical standard equation and coordinate system transformation, an objective function with installation error parameters is deduced according to the model, the objective function is optimized by utilizing an improved Gaussian-Newton method in combination with a data set for optimization, and thresholds and increment coefficients of all parameters of an increment matrix are adjusted according to different iteration convergence speeds of optimization parameters. The final installation error parameters obtained by iterative optimization are respectively the ellipse circle center parameters Q (x e ,y e ) The long axis direction parameter θ, the long axis length parameter a, refer to the circle radius R. The circle center parameter of the ellipse is used for representing the magnitude and the phase of the geometric installation error; the minor axis of the ellipse is the deflection axis, and the deflection angle can be calculated according to the ratio of the minor axis to the major axis
Is of a size of (a) and (b).
In order to achieve the above purpose, the present invention provides the following technical solutions: a method for identifying installation errors of rotary parts based on an elliptical error model comprises the following steps:
the first step: the method comprises the steps of installing a rotary part on a workbench, checking the clamping condition of a clamp of the rotary workbench and the rotary part to be detected, and firstly ensuring the coaxiality between the central axis of the rotary part to be detected and the central axis of the rotary workbench and the parallelism between the end face of the rotary part to be detected and the end face of the rotary workbench as much as possible by a manual calibration mode; after the inspection is finished, installing measuring equipment, placing the measuring equipment at any position on a profile to be scanned, and scanning the complete contour of the rotary part by combining the rotary interpolation motion of the rotary shaft to obtain an original relative data set X= { X 1 ,x 2 ,x 3 ...x n Wherein n is the number of elements of the original relative dataset and x j Is the original relative data, wherein j=1 to n;
and a second step of: in order to accurately obtain the contour shape information reflecting the scanned contour of the revolving body part, the noise, roughness and waviness medium-high frequency components in the original relative data set need to be removed, and the low frequency components reflecting the contour are left; therefore, the original relative data set is subjected to frequency component separation by utilizing a wavelet decomposition method, and the low-frequency component is reconstructed to obtain the profile-changing data set X c ＝{x c1 ,x c2 ,x c3 ...x cm -wherein m is the number of elements of the profile-change dataset and x ci Is profile change data, wherein i=1 to m;
and a third step of: deforming a profile into a dataset X c Determining the included angle between the connecting line of each measuring point and the origin and the X axis in a 360-degree equal dividing mode according to time sequence
The calculation formula is as follows:
processing the profile-change dataset: determining each measuring point x by the relative value of the measured data ci Corresponding length variation x ci-L The calculation formula is as follows:
in formula (2), min (·) is a minimum function, x cmin Is X c Minimum in the dataset; the relative data set that is ultimately applied to the installation error parameter optimization is
Fourth step: points (x) on a standard elliptic equation in a rectangular coordinate system z ,y z ) Obtaining points (x, y) on an elliptical detection model with installation errors through coordinate transformation, wherein the formula is as follows:
if the measured data is the relative distance variation between the profile of the rotary part and the axis of the rotary table, and the radial profile is relative variation data, part of the parameters in the formula (3) are further written as:
if the measured data is the relative distance variation between the profile of the rotary part and the sensor, and the relative distance variation between the radial profile and the sensor, the partial parameters in the formula (3) are further written as follows:
in the formulas (3), (3-1) and (3-2), x e ,y e For the circle center parameter of the ellipse to be optimized, theta is the included angle between the major axis of the ellipse to be optimized and the x axis of the rectangular coordinate system, R is the radius of the reference circle to be optimized, a is the major axis parameter of the ellipse to be optimized, b is the radius value of the measured rotator part, delta is the standard ellipse angle parameter, delta epsilon [0 ],2π]；
fifth step: the deformation finishing of the step (3) is carried out to obtain
Eliminating the parameter delta and establishing a least square objective function f target 2 Obtaining:
the optimization target is
Sixth step: setting theta, x e ,y e A, R is reasonably initial, and the formula (6) is iteratively optimized by adopting a modified Gaussian-Newton method, wherein the specific process is that
1) Define variables A, B, d_A, d_B as
2)f target For the parameters theta, x respectively e ,y e Derivation of a, R
3) Building jacobian gradient matrix as
T is matrix transposition; wherein the method comprises the steps of
The delta matrix can be expressed as
In the formula (8), f= [ f 1 ,f 2 ,...,f m ] T Wherein f i To be used in
Substitution into an objective function f target The obtained function value;
4) Updating optimized parameter values
In the formula (9), p is the iteration number; g i (i=1 to 5) is an increment coefficient, and is required to be according to an increment matrix parameter [ delta theta ]; Δx e ；Δy e ；Δa；Δb]Is reasonably selected;
5) Iteration end condition: judging whether the variables in the increment matrix are not greater than the set threshold value respectively or not, wherein
If yes, the optimization solution is finished; otherwise, continuing to execute from the process 3);
seventh step: by using the optimized parameters theta, x obtained by iteration e ,y e A, R calculates the installation error as
Geometric eccentricity:
phase angle beta=tan -1 (y e /x e ) The method comprises the steps of carrying out a first treatment on the surface of the Face deflection angle->
Yaw axis phase angle α= (θ±pi/2);
eighth step: and adjusting the installation position of the workpiece according to the calculated deflection angle and the geometric error, and eliminating the installation error.
Compared with the prior art, the invention has the following beneficial effects:
the invention adopts relative measurement data aiming at the installation error identification of the rotary part, obtains an original relative data set by measuring any complete contour shape on the rotary part, obtains a contour deformation data set by utilizing a wavelet decomposition method, and obtains a relative data set for optimization by comparison and difference processing. The absolute distance information of the profile and the axis of the rotary part is not needed, the data acquisition is easier, the detection mode is simple, and complex path planning is not needed.
The invention is different from the traditional fitting mode, and the installation deflection angle and the geometric eccentricity error of the revolving body part can be simultaneously analyzed and calculated through one-time data processing. Fitting errors can be avoided to the greatest extent by using the improved target optimization method, and accurate actual installation errors are obtained.
According to the method, different thresholds are set for the increment matrix of the tested swivel body part installation error optimization parameter to serve as iteration identification stopping conditions, and increment coefficients are introduced, so that the solving efficiency can be improved.
The radius value of the measured rotator part is used as the short axial length of the error ellipse, so that the short axial length does not participate in optimization iteration, and the singular of the jacobian matrix is effectively avoided; and setting an intermediate parameter reference circle radius R, and converting the relative measurement data into absolute measurement data containing an unknown radius R to optimize.
The installation error of the revolving body part is directly obtained by solving parameters such as the circle center, the long axis phase angle and the like of the fitting ellipse, and the rounding error of the traditional fitting mode is effectively reduced.
The installation error detection instrument can adopt all sensors capable of measuring relative displacement information, and the installation error detection method can be suitable for all rotary workpieces with complete contour lines and any type of relative measurement data.
The invention is mainly aimed at detecting the installation error under the relative measurement data, but can be used for detecting the installation error under the absolute measurement data, and the invention only needs to replace R with the absolute measurement value, thereby having wide universality.
Drawings
FIG. 1 is a schematic diagram of a physical error model of the present invention;
FIG. 2 is a schematic diagram of parameters of an elliptical error model according to the present invention;
FIG. 3 is a schematic diagram of a measurement point conversion when the objective function of the present invention is established, wherein (a) in FIG. 3 is a schematic diagram of a measurement point before conversion, and (b) in FIG. 3 is a schematic diagram of a measurement point after conversion;
FIG. 4 is a schematic diagram of a relative data set for optimization obtained after actual measurement data processing according to the present invention;
fig. 5 is a flow chart of the execution of the method of the present invention.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, not all embodiments of the present invention. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
Referring to fig. 1-5, the present invention provides a technical solution: a method for identifying installation errors of rotary parts based on an elliptical error model comprises the following steps:
the first step: the method comprises the steps of installing a rotary part on a workbench, checking the clamping condition of a clamp of the rotary workbench and the rotary part to be tested, and firstlyThe coaxiality of the central axis of the to-be-detected rotary part and the central axis of the rotary table and the parallelism of the end face of the to-be-detected rotary part and the end face of the rotary table are ensured as much as possible by a manual calibration mode, and measuring equipment is installed around the rotary part after the inspection is finished. FIG. 1 is a schematic diagram of a solid of revolution component after generating geometric eccentricity and yaw error, wherein L 2 For the axis of the rotary table, L 1 For the axis of the rotary body part to be measured, the contour actually measured by the measuring equipment is an actually measured ellipse in the figure, and the actually measured data is an original relative data set X= { X 1 ,x 2 ,x 3 ...x n Wherein n is the number of elements of the original relative dataset and x j Is the original relative data, where j=1 to n. In fig. 1, OQ is the geometric eccentricity of the rotary part at the measured position, b is the measured minor axis radius of the ellipse, a is the measured major axis radius of the ellipse,
the yaw angle is the yaw angle, and the short axis is the yaw axis.
And a second step of: in order to accurately acquire contour shape information reflecting the scanned contour of the rotary part, it is necessary to remove middle and high frequency components such as noise, roughness, waviness and the like in the original relative data set, and leave low frequency components reflecting the contour. Therefore, the original relative data set is subjected to frequency component separation by utilizing a wavelet decomposition method, and the low-frequency component is reconstructed to obtain the profile-changing data set X c ＝{x c1 ,x c2 ,x c3 ...x cm Wherein m is the number of elements of the profile-change dataset, x ci Is profile change data, wherein i=1 to m;
and a third step of: as shown in FIG. 2, the profile is transformed into a dataset X c Determining the included angle between the connecting line of each measuring point and the origin and the X axis in a 360-degree equal dividing mode according to time sequence
The calculation formula is as follows:
processing the profile-change dataset: determining each measuring point x by measuring data relative value ci Corresponding length variation x ci-L (x shown in FIG. 2) ci-L The calculation formula is as follows for the length variation of the radial profile of the revolving body part under the relative variation data:
in formula (2), min (·) is a minimum function, x cmin Is X c Minimum in the dataset. The relative data set that is ultimately applied to the installation error parameter optimization is
Fourth step: as shown in fig. 3, a point (x z ,y z ) Obtaining points (x, y) on an elliptical detection model with installation errors through coordinate system conversion, wherein the formula is as follows:
if the measured data is the relative distance variation between the profile of the rotary part and the axis of the rotary table (radial profile relative variation data, such as using a contact scanning probe), part of the parameters in equation (3) may be further written as
If the measured data is the relative distance variation between the profile of the solid of revolution and the sensor (the relative distance variation data between the radial profile and the sensor, such as by using a non-contact optical probe), part of the parameters in equation (3) can be further written as
In the formula (3), x e ,y e For the circle center parameter of the ellipse to be optimized, θ is the included angle between the major axis of the ellipse to be optimized and the x axis of the rectangular coordinate system, R is the radius of the reference circle to be optimized, a is the major axis parameter of the ellipse to be optimized, b is the radius value of the measured rotator part, and the parameter is schematically shown in fig. 2. Delta is a standard ellipse angle parameter, delta is 0,2 pi]As shown in fig. 3 (a).
Fifth step: the deformation finishing of the step (3) is carried out to obtain
Eliminating the parameter delta and establishing a least square objective function f target 2 Obtaining:
the optimization target is
Sixth step: setting theta, x e ,y e A, R is reasonably initial, and the formula (6) is iteratively optimized by adopting a modified Gaussian-Newton method, wherein the specific process is that
1) Define variables A, B, d_A, d_B as
2)f target For the parameters theta, x respectively e ,y e Derivation of a, R
3) Building jacobian gradient matrix as
T is matrix transposition; wherein the method comprises the steps of
The delta matrix can be expressed as +.>
In the formula (8), f= [ f 1 ,f 2 ,...,f m ] T Wherein f i To be used in
Substitution into an objective function f target The function value obtained.
4) Updating optimized parameter values
In the formula (9), p is the iteration number; g i (i=1 to 5) is an increment coefficient, and is required to be according to an increment matrix parameter [ delta theta ]; Δx e ；Δy e ；Δa；Δb]Different change speeds are reasonably selected.
5) Iteration end condition: and judging whether all variables in the increment matrix are not larger than a set threshold value.
If yes, the optimization solution is finished; otherwise, execution continues from process 3).
Seventh step: according to the graphs (1) and (2), the optimization parameters theta, x obtained by iteration are utilized e ,y e A, R calculates the installation error as
Geometric eccentricity:
phase angle: beta=tan -1 (y e /x e )
End face deflection angle:
yaw axis phase angle: alpha= (θ±pi/2)
Eighth step: adjusting the installation position of the rotary table according to the determined deflection angle and geometric error, and eliminating the installation error; during actual measurement, the sensor measuring head, the measured piece and the measuring equipment are free from interference in the measuring process, and the sensor measuring data are effective.
Example verification is as follows:
the measuring sensor adopts a scanning Ma Bosi G25 measuring head with the radius of 3mm, and the measured data are the relative change data of the radial profile of the measured rotator part. The radius of the measured rotator part is 200mm, so that the radius b=200 mm of the elliptical short axis in the installation error model. To compare the optimization effect, the installation error preset values are shown in table 1.
TABLE 1 preset value of measured swivel body part mounting error
The original relative data set is processed to obtain the relative data set for optimization
As shown in fig. 4. Setting increment coefficients to g respectively 1 ＝1000,g 2 ＝g 3 ＝g 4 ＝g 5 =1; the increment matrix threshold values are epsilon respectively 1 ＝10^-8,ε 2 ＝ε 3 ＝ε 4 ＝ε 5 ＝10^-3。
The iteration preliminary set values are shown in Table 2, i.e
The setting of the initial value is arbitrary within a reasonable range.
Table 2 initial value of optimization iteration
| Geometric eccentricity X in X direction e | Y-direction geometric eccentricity Y e | Long axis phase angle theta | Radius of reference circle R | Major axis radius a | |
| | 0mm | 0° | 250mm | 240mm |
And finally solving the experiment to obtain each optimized parameter, and the iteration times and time are shown in table 3.
TABLE 3 solving results
The installation error parameters were calculated from table 3 as:
geometric eccentricity:
phase angle: beta=tan -1 (y e /x e )＝tan -1 (0.05/0.009)＝79.796°
End face deflection angle:
yaw axis phase angle: α= (θ±pi/2) = (25±90) °
As shown in Table 3 and the calculation results, the geometrical eccentricity and the end face deflection angle of the invention are basically consistent with the preset values, the time consumption is 0.42s, and the invention has extremely high precision and efficiency.
The detection and calculation method can realize high-precision calculation of all parameters of the installation error of the revolving body part. The detection method uses the relative measurement data, iterates by using a target fitting method, and effectively solves the problem of installation error detection when the current absolute measurement data is difficult to acquire and the relative measurement data is easy to acquire. The method has simple detection action, the available sensor is any type of displacement sensor such as a non-contact sensor or a contact scanning measuring head, the solving efficiency is high, and the overall working efficiency is not reduced.
The measurement evaluation of the invention can be used as a preparation step before the detection or the processing of the revolving body parts to be programmed and written into a software system, thereby realizing the automatic and rapid installation error detection and laying an important 'high-precision' foundation for the subsequent operation.
Although embodiments of the present invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made therein without departing from the principles and spirit of the invention, the scope of which is defined in the appended claims and their equivalents.
Claims (1)
1. The method for identifying the installation errors of the rotary parts based on the elliptical error model is characterized by comprising the following steps of:
the first step: the method comprises the steps of installing a rotary part on a workbench, checking the clamping condition of a clamp of the rotary workbench and the rotary part to be detected, and firstly ensuring the coaxiality between the central axis of the rotary part to be detected and the central axis of the rotary workbench and the parallelism between the end face of the rotary part to be detected and the end face of the rotary workbench as much as possible by a manual calibration mode; after the inspection is finished, installing measuring equipment, placing the measuring equipment at any position on a profile to be scanned, and scanning the complete contour of the rotary part by combining the rotary interpolation motion of the rotary shaft to obtain an original relative data set X= { X 1 ,x 2 ,x 3 ...x n Wherein n is the number of elements of the original relative dataset and x j Is the original relative data, wherein j=1 to n;
and a second step of: in order to accurately obtain the contour shape information reflecting the scanned contour of the revolving body part, the noise, roughness and waviness medium-high frequency components in the original relative data set need to be removed, and the low frequency components reflecting the contour are left; therefore, the original relative data set is subjected to frequency component separation by utilizing a wavelet decomposition method, and the low-frequency component is reconstructed to obtain the profile-changing data set X c ＝{x c1 ,x c2 ,x c3 ...x cm -wherein m is the number of elements of the profile-change dataset and x ci Is profile change data, wherein i=1 to m;
and a third step of: deforming a profile into a dataset X c Determining the included angle between the connecting line of each measuring point and the origin and the X axis in a 360-degree equal dividing mode according to time sequence
The calculation formula is as follows:
processing the profile-change dataset: determining each measuring point x by the relative value of the measured data ci Corresponding length variation x ci-L The calculation formula is as follows:
in formula (2), min (·) is a minimum function, x cmin Is X c Minimum in the dataset; the relative data set that is ultimately applied to the installation error parameter optimization is
Fourth step: points (x) on a standard elliptic equation in a rectangular coordinate system z ,y z ) Obtaining points (x, y) on an elliptical detection model with installation errors through coordinate transformation, wherein the formula is as follows:
if the measured data is the relative distance variation between the profile of the rotary part and the axis of the rotary table, and the radial profile is relative variation data, part of the parameters in the formula (3) are further written as:
if the measured data is the relative distance variation between the profile of the rotary part and the sensor, and the relative distance variation between the radial profile and the sensor, the partial parameters in the formula (3) are further written as follows:
in the formulas (3), (3-1) and (3-2), x e ,y e For the circle center parameter of the ellipse to be optimized, θ is the included angle between the major axis of the ellipse to be optimized and the x axis of the rectangular coordinate system, R is the radius of the reference circle to be optimized, a is the major axis parameter of the ellipse to be optimized, b is the radius value of the measured rotator part, δ is the standard ellipse angle parameter, δ is [0, 2pi ]]；
Fifth step: the deformation finishing of the step (3) is carried out to obtain
Eliminating the parameter delta and establishing a least square objective function f target 2 Obtaining:
the optimization target is
Sixth step: setting theta, x e ,y e A, R is reasonably initial, and the formula (6) is iteratively optimized by adopting a modified Gaussian-Newton method, wherein the specific process is that
1) Define variables A, B, d_A, d_B as
2)f target For the parameters theta, x respectively e ,y e Derivation of a, R
3) Building jacobian gradient matrix as
T is matrix transposition; wherein the method comprises the steps of
i=1 to m; the delta matrix can be expressed as
In the formula (8), f= [ f 1 ,f 2 ,...,f m ] T Wherein f i To be used in
Substitution into an objective function f target The obtained function value;
4) Updating optimized parameter values
In the formula (9), p is the iteration number; g i (i=1 to 5) is an increment coefficient, and is required to be according to an increment matrix parameter [ delta theta ]; Δx e ；Δy e ；Δa；Δb]Variation of (2)The speed is reasonably selected;
5) Iteration end condition: judging whether the variables in the increment matrix are not greater than the set threshold value respectively or not, wherein
If yes, the optimization solution is finished; otherwise, continuing to execute from the process 3);
seventh step: by using the optimized parameters theta, x obtained by iteration e ,y e A, R calculates the installation error as
Geometric eccentricity:
phase angle beta=tan -1 (y e /x e ) The method comprises the steps of carrying out a first treatment on the surface of the Face deflection angle->
Yaw axis phase angle α= (θ±pi/2);
eighth step: and adjusting the installation position of the workpiece according to the calculated deflection angle and the geometric error, and eliminating the installation error.
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