CN112666892A - Shafting measuring device based on mixed four-point method and shafting profile online reconstruction method - Google Patents

Shafting measuring device based on mixed four-point method and shafting profile online reconstruction method Download PDF

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CN112666892A
CN112666892A CN202011538645.2A CN202011538645A CN112666892A CN 112666892 A CN112666892 A CN 112666892A CN 202011538645 A CN202011538645 A CN 202011538645A CN 112666892 A CN112666892 A CN 112666892A
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displacement sensor
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安冬
张倩
王赛男
邵萌
须颖
杨奕潇
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Shenyang Jianzhu University
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Abstract

本发明公开的基于混合四点法的轴系测量装置及轴系廓形在线重构方法，通过定向等距的移动固定架，对待测轴的轴向多个截面进行改进的四点误差分离技术，提取每个被测截面中心的相对变化量，然后根据各个截面的中心位置拟合空间中线，并以萃取的空间中线作为基准轴线将各个截面分离的圆度误差组合到轴线上，进而实现回转轴系的轮廓重构，再进一步的进行形状误差评定。本发明能实现在测量过程中准确分离回转轴系的径向误差运动和倾斜误差运动，从而确保重构轴系廓形的精确度。

Figure 202011538645

The shafting measuring device based on the hybrid four-point method and the on-line reconstruction method of the shafting profile disclosed by the invention are a four-point error separation technology that improves the axial multiple sections of the shaft to be measured by orienting the equidistant moving fixed frame. , extract the relative variation of the center of each measured section, then fit the space center line according to the center position of each section, and use the extracted space center line as the reference axis to combine the roundness errors of each section separation on the axis, and then realize the return The contour reconstruction of the rotating shaft system is carried out, and then the shape error evaluation is further carried out. The invention can realize the accurate separation of the radial error motion and the inclination error motion of the rotary shafting during the measurement process, thereby ensuring the accuracy of the reconstructed shafting profile.

Figure 202011538645

Description

Shafting measuring device based on mixed four-point method and shafting profile online reconstruction method
Technical Field
The invention belongs to the technical field of precision shafting test technology and online measurement equipment detection, and particularly relates to a shafting measuring device based on a mixed four-point method and an online reconstruction method of shafting profile.
Background
The precise rotating shaft system is used as the core of a high-precision machine tool, the running state of the precise rotating shaft system directly influences the machining precision, the precise rotating shaft system is firstly studied to clearly know and describe the space motion state of the precise rotating shaft system, for the machine tool, the deviation of a machine tool rotating shaft can generate important influence on the machining precision, and the precise rotating shaft system can directly determine factors such as the shape, the roughness and the like of a machined part. Therefore, the rotation precision has direct influence on the operation of the precision rotating shaft system, and the accurate measurement and the accurate extraction of the rotation error of the precision rotating shaft system are very important research means. In a plurality of form and position error detection projects specified by the national standard, the detection of the cylindricity error of a cylindrical workpiece is a very important component, at present, instruments mainly used for measuring the cylindricity in China are a roundness measuring instrument, a three-coordinate measuring machine and the like, and the precision measuring instruments are expensive in cost, have high requirements on measuring environments and are difficult to meet the requirements of on-line measurement.
In order to realize the precise measurement of the shape error of the precise rotating shaft system, the precision of the measuring instrument is hardly improved, so that a plurality of scholars at home and abroad develop a new method for researching a measuring method with higher precision. The cylindrical part can be regarded as an overlap of countless cross-section circles along the axial direction, the online reconstruction of the cylinder can be realized by a cross-section method theoretically, and the error shape of the cylinder is combined by using a limited number of roundness errors on the cross section [ Lishengyi, Daisei and other works. The basic idea of Error Separation Techniques (ESTs) as a basic method for precision measurement is as follows: the characteristic that the contour surface of a measured workpiece is uniquely determined is utilized, a plurality of measuring heads are simultaneously applied to measure the surface of the measured part, a measured signal simultaneously contains the shape error of the measured part and the motion error of a measuring mechanism, the shape error of the measured part and the motion error of the measuring mechanism are separated by adopting a certain mathematical algorithm, and then the shape error of the measured workpiece and the motion error of the measuring mechanism are obtained, so that the precision measurement of the shape error of the measured part [ Bufu Hua, Li Shu, Zulon, etc.. the design of a cylindricity high-precision measuring system [ J ]. Loyang institute of technology (Nature science edition), 2002 (1): 46-48]. The traditional three-point method is also the key for measuring the shape error of a shafting part because the inherent first-order harmonic suppression problem [ Lishenyi, Daisei and the like, precision and ultra-precision machining in-place detection and error separation technology [ M ]. national defense science and technology university press, 2007] exists, the first-order harmonic component of the shafting gyration error and the shape error is difficult to accurately separate, the evaluation of the section roundness error cannot be influenced, the bending of the central axis of a cylinder is hidden, the difficulty is caused to the online reconstruction of the cylinder, and the evaluation of the cylindricity is also influenced.
Disclosure of Invention
In order to solve the technical problems, the invention provides a shafting measuring device based on a mixed four-point method and an online reconstruction method of shafting profile, which can realize the separation and removal of guide rail errors, shafting motion errors and inclination errors introduced by a measuring system from measured signals, thereby obtaining accurate signals to be measured and ensuring the accuracy of a reconstructed cylinder.
The invention relates to a shafting measuring device based on a mixed four-point method, which comprises: the device comprises a motor, a guide rail mechanism, a sliding frame, a fixed frame, a first displacement sensor, a second displacement sensor, a third displacement sensor and a fourth displacement sensor; the guide rail mechanism comprises two parallel guide rails, the sliding frame is arranged on the two guide rails in a spanning mode and can move along the guide rails, the fixed frame is arranged at the top of the sliding frame, and the fixed frame is provided with a circular arc-shaped through hole; the output shaft of the motor is connected with one end of the shaft to be measured, and the other end of the shaft to be measured penetrates through the through hole; the first displacement sensor, the second displacement sensor and the third displacement sensor are arranged on the fixed frame and positioned on a first plane, and the first plane is vertical to the axis of the shaft to be measured; the probes of the first displacement sensor, the second displacement sensor and the third displacement sensor are positioned in the through hole and positioned on the same concentric circumference of the shaft to be measured, the fourth displacement sensor is arranged on the fixed frame, and the probe of the fourth displacement sensor and the probe of the first displacement sensor are positioned on a straight line parallel to the axis of the shaft to be measured.
The invention discloses an online reconstruction method of shafting profile based on a mixed four-point method, which adopts a shafting measuring device based on the mixed four-point method as claimed in claim 1 to collect data, and comprises the following steps:
step 1: dividing the shaft to be measured into L measuring sections with the distance of d along the axial direction, moving the sliding frame to make the first plane align with the L sections respectively, and when the moving frame is positioned on the corresponding measuring section, rotating the shaft to be measured for a circle and acquiring the distance from the probe head to the shaft surface as measuring data through the first displacement sensor, the second displacement sensor, the third displacement sensor and the fourth displacement sensor:
step 2: performing discrete Fourier transform on the measurement data acquired in the step 1, and acquiring an s-order harmonic component of a roundness profile of a measurement section aligned with the first plane by a section three-point roundness error separation method, wherein s is 0,2,3, …, N-2, and N represents the total measurement point number of each sensor; obtaining a first-order harmonic component of a radial error separation result by an axial two-point method;
and step 3: calculating 1 order harmonic component of roundness profile of L measurement sections on the measured shaft:
and 4, step 4: and fitting a space central line according to 1-order harmonic components of the roundness profiles of the L positioning measurement sections, combining the profile shapes of all the sections to the space central line, and reconstructing the cylindrical profile of the shaft to be measured.
The shafting measuring device based on the mixed four-point method and the shafting profile online reconstruction method have the following beneficial effects:
1. the invention provides a mixed four-point method-based online reconstruction method of shafting profile, which is matched with a shafting measuring device to carry out data measurement: specifically, 4 displacement sensors are adopted to measure two axial sections and carry out error separation, and the relative change quantity of the centers of two section circles is extracted, and because the relative change is the inherent characteristic between the two sections and cannot be changed along with the rotation of a shaft system, the centers of all the section circles of a measured body are fitted, and the spatial central line of a reconstruction cylinder can be obtained.
2. The invention considers that the radial rotation error has no repeatability under the condition that the shafting rotates every week, and the three-point roundness error separation method and the axial two-point method are combined, so that the section roundness error, the radial error and the inclination error caused by the shafting rotation motion and the guide rail error motion in the error separation process can be accurately separated, which is one of innovations of a mixed four-point method different from the existing error separation method.
3. In the invention, because the error motion caused by shafting rotation and the error motion of the guide rail can be accurately separated in the error separation process, the guide rail in the measurement system has no overhigh precision requirement, and the online measurement cost of the shafting is greatly reduced, which is the application value of the invention.
4. The measuring method of the invention combines the error shapes of the cylinders by adopting the roundness error on each section and taking the fitted space central line as a reference. Therefore, the evaluation of the error of the cylindrical shape can be carried out more comprehensively, comprising the following steps: errors such as roundness, cylindricity and bus straightness, and the like, so the method has wider application.
In summary, the roundness error of each section is obtained by a section three-point roundness error separation method, the relative variation of the centers of two section circles is extracted by an axial two-point method, and the centers of the section circles are fitted to be used as the reference axis of a reconstruction cylinder, so that the problem that the variation of the centers of the section circles cannot be extracted due to first-order harmonic suppression is solved, the roundness profiles of the sections are combined by using the fitted central line as the reference axis, innovation from evaluation of single section roundness error variation to evaluation of whole cylinder profile error variation is realized, and a more comprehensive shape error evaluation method is provided.
Drawings
FIG. 1 is a schematic diagram of a shafting measuring device based on a hybrid four-point method according to an embodiment of the present invention;
FIG. 2 is a top view of a shafting measuring device based on a hybrid four-point method according to an embodiment of the present invention;
FIG. 3 is a flow chart of the online reconstruction method of the shafting profile based on the mixed four-point method;
FIG. 4 is a schematic diagram of the measurement of the shafting measuring device based on the mixed four-point method;
fig. 5 is a schematic diagram of the influence of the guide rail error motion and the shafting error motion on the measurement process.
Detailed Description
The invention relates to a shafting measuring device based on a mixed four-point method, which comprises: the device comprises a motor, a guide rail mechanism, a sliding frame 9, a fixed frame 8, a first displacement sensor 1, a second displacement sensor 2, a third displacement sensor 3 and a fourth displacement sensor 4. The guide rail mechanism comprises two parallel guide rails 10, the sliding frame 9 is arranged on the two guide rails 10 in a crossing mode and can move along the guide rails 10, the fixed frame 8 is arranged at the top of the sliding frame 9, and the fixed frame 8 is provided with a circular arc-shaped through hole. The output shaft of the motor is connected with one end of the shaft to be measured 5, and the other end of the shaft to be measured 5 is arranged in the through hole in a penetrating mode. The first displacement sensor 1, the second displacement sensor 2 and the third displacement sensor 3 are arranged on the fixed frame 8 and are positioned on a first plane, and the first plane is vertical to the axis of the shaft to be measured; the probes of the first displacement sensor 1, the second displacement sensor 2 and the third displacement sensor 3 are positioned in the through hole and positioned on the same concentric circumference of the shaft to be measured, the fourth displacement sensor 4 is arranged on the fixed frame 8, and the probe of the fourth displacement sensor 4 and the probe of the first displacement sensor 1 are positioned on a straight line parallel to the axis of the shaft to be measured. The guide rail 10 is provided with scale marks. The fixed frame 8 is provided with 4 sensor positioning holes, and the first displacement sensor 1, the second displacement sensor 2, the third displacement sensor 3 and the fourth displacement sensor 4 are arranged in the corresponding sensor positioning holes and fixed through stop screws. The 4 sensors were located within 25 μm of the measured shaft surface. The included angle between the first displacement sensor 1 and the second displacement sensor 2 is 60 degrees, and the included angle between the first displacement sensor 1 and the third displacement sensor 3 is 120 degrees.
As shown in fig. 1 and 2, in the present embodiment, the rotating shaft system of the electric spindle 7 is taken as an example, and the rotating shaft system of the electric spindle is complicated in structure, so that the rotating shaft cannot be directly measured. In this embodiment, the connecting rod 6 is respectively connected to the electric spindle 7 and the shaft to be measured 5, and the roundness error of the rotating shaft of the electric spindle is detected by performing online reconstruction on the shaft to be measured 5.
The fixed frame 8 can move along the guide rail 10 in the Z direction under the support of the sliding frame 9. The electric spindle 7 is connected with the shaft to be measured 5 through the connecting rod 6 and drives the shaft to be measured 5 to rotate. The guide rail 10 is axially parallel to the shaft 5 to be measured. When the measurement is started, the fixed frame 8 moves to the axial end of the connecting rod 6 along the guide rail 10Z under the support of the sliding frame 9, so that the fixed frame 8 is located at a first position, and the sliding frame 9 moves a distance d each time along the Z direction. In the specific implementation, the online reconstruction method of the shafting profile based on the mixed four-point method obtains the roundness error of each section through a section three-point roundness error separation method, extracts the relative variation of the centers of two section circles through an axial two-point method, and fits the centers of the section circles to be used as the reference axis of a reconstruction cylinder, so that the problem that the variation of the centers of the section circles cannot be extracted due to first-order harmonic suppression is solved, the roundness error of each section is combined by using the fitted center line as the reference axis, the innovation from the roundness variation of a single section to the variation of the profile of the whole cylinder is realized, and a more comprehensive shape error evaluation method is provided.
As shown in fig. 3, a flow chart of an online reconstruction method of a shafting profile based on a mixed four-point method is shown, and the method adopts the shafting measurement device based on the mixed four-point method to acquire data, and specifically includes:
step 1: dividing a measured shaft into L measuring sections with the distance of d along the axial direction, moving the sliding frame to enable the first plane to be aligned with the L sections respectively, enabling the shaft to be measured to rotate for a circle when the moving frame is positioned on the corresponding section, and acquiring the distance between a probe and the shaft surface as measuring data through the first displacement sensor, the second displacement sensor, the third displacement sensor and the fourth displacement sensor, wherein the step 1 comprises the following steps:
step 1.1, arranging a first displacement sensor, a second displacement sensor and a third displacement sensor on a fixed frame according to a three-point roundness error separation method, enabling the three sensors to be positioned on a first plane, enabling the fourth displacement sensor to be positioned on a second plane, enabling the first plane and the second plane to be parallel and enabling the distance between the first plane and the second plane to be d,
step 1.2, defining the length direction of the guide rail as the Z-axis direction, the vertical direction as the Y-axis direction, the direction vertical to the Y-axis in the end face of the shaft to be measured as the X-axis direction, and transferring the first displacementThe angle between the sensor and the X axis is 0, and the included angle between the second displacement sensor and the first displacement sensor is beta1The included angle between the third displacement sensor and the first displacement sensor is beta2As shown in fig. 4;
step 1.3, dividing a measured shaft into L measuring sections with the distance d along the axial direction; defining the mapping sequence P-1, 2, …, L; the position measurement refers to a measurement cross section which is aligned with the first plane after the fixing frame moves a distance d along the Z-axis direction from the end part of the shaft to be measured;
step 1.4, initializing P to 1;
step 1.5, moving the fixing frame along the Z-axis direction to enable a first plane of the fixing frame to be aligned with a P-th measuring section of the shaft to be measured;
step 1.6, driving the shaft to be measured to rotate for a circle by a motor, and acquiring measurement data of the shaft to be measured rotating for a circle on the Pth measurement section through a first displacement sensor, a second displacement sensor and a third displacement sensor; the fourth displacement sensor collects the measurement data of one rotation on the P +1 th measurement section to complete the measurement of the P-th positioning;
and step 1.7, repeating the step 1.5 and the step 1.6, and measuring the P +1 positioning until the L-th positioning data measurement is finished, so as to obtain the L positioning measurement data.
Step 2: performing discrete Fourier transform on the measurement data acquired in the step 1, and acquiring an s-order harmonic component of a roundness profile of a measurement section aligned with the first plane by a section three-point roundness error separation method, wherein s is 0,2,3, …, N-2, and N represents the total measurement point number of each sensor; obtaining a first-order harmonic component of a radial error separation result by an axial two-point method, wherein the step 2 comprises the following steps:
step 2.1, the measurement data collected by the first displacement sensor, the second displacement sensor and the third displacement sensor during the P-th position measurement is mi(lPK), wherein i ═ 1,2,3, k ═ 0,1, …, N-1; n represents the total number of points measured for each sensor, and the angular interval between samples is 2 pi/N, lPThe position is expressed as the axial positioning of the P-th positioning position of the axis to be measured; k represents the serial number of the measuring point of each sensor;
step 2.2, extracting by using a three-point roundness error separation methodTaking the s-harmonic component R (l) of the roundness profile on the P-th measured section of the axis to be measuredPS), s ═ 0,2,3, N-2 denotes the order of the harmonic component;
the measurement data acquired by the first displacement sensor, the second displacement sensor and the third displacement sensor are weighted and combined by using a three-point roundness error separation method, and the calculation is as follows:
Figure BDA0002854298230000071
due to the included angle beta between the first displacement sensor 1 and the X-axis1Since 0 is obtained, the weighting coefficients output by the first displacement sensor 1, the second displacement sensor 2, and the third displacement sensor 3 are: a is1＝1,a2＝-sinβ2/sin(β2-β1)，a3＝sinβ1/sin(β2-β1) Thus, the radial motion error of the shafting in the weighted signal and the linear motion error of the guide rail are removed, and the separation of the roundness error is realized;
when the harmonic order s is not equal to 1, N-1, all s satisfy the condition that G (l, s) is not equal to 0, the roundness error of the measured section can be obtained by performing inverse discrete Fourier transform on the harmonic component R (l, s) of the roundness of the measured section, and the time delay phase shift property of the Fourier transform is applied, so that the method can be obtained:
Figure BDA0002854298230000081
wherein the transfer function is:
Figure BDA0002854298230000082
in the formula, s (s ═ 0, 1.., N-1) is the order of harmonic waves, and G (l, s) is the transfer function of a three-point roundness error separation method; r (l, s) ═ DFT [ R (l, k) ] represents the s harmonic component of the measured section roundness. Substituting the weighting function coefficients into the error separation transfer function equation (3), which is in discrete form:
Figure BDA0002854298230000083
by selecting the appropriate p1,p2So that p is1,p2When the greatest common factor with N is 1, the transfer function g(s) ≠ 0(s ═ 0,2, …, N-2) except for s ═ 1 and N-1, and the harmonic component R (l, k) ═ IFFT [ R (l, s) of the measured cross-sectional roundness profile can be obtained]。
It has been proved that, when the harmonic order s is 1 and s is N-1, the transfer function G(s) is 0 and G (s-1) is 0, the three-point error separation technique cannot effectively obtain the harmonic component R (l, s) of the roundness profile of the measured section, and thus cannot accurately obtain the roundness error R (l, k) of the measured section, which makes it difficult to separate the axle roundness error and the gyration error, so-called "first harmonic suppression". Therefore, the three-point roundness error separation method has an inherent first-order harmonic suppression problem, and has no influence on the error separation process, but has a very important influence on the evaluation of the bending change of the central axis of the cylinder, and causes difficulty on the online reconstruction of the cylinder. The first harmonic component of the roundness profile of the measured section is the eccentric error of the least square circle center of the measured section, so the eccentric coordinate of the measured section is calculated, and the eccentric vector of the roundness profile of the measured section, namely the first harmonic component, is further solved.
Step 2.3, performing discrete Fourier transform on data collected by the first displacement sensor and the fourth displacement sensor in the P-th positioning measurement data, and acquiring a first-order harmonic component of a radial error separation result by an axial two-point method, specifically:
and (3) rotating the shaft to be measured for one circle at the P-th position measurement, and respectively acquiring data of the measured section P and the P +1 for one circle by using the first displacement sensor and the fourth displacement sensor:
Figure BDA0002854298230000091
Figure BDA0002854298230000092
in the formula: r (l)p,k)，r(lp+1K) k is 0,2,3 …, and N-2 are respectively roundness profiles of the P th and P +1 th measured positions of the axis to be measured; epsilonx(lpK) is the run-out of the first displacement sensor at the P-th position measured by the X-direction error motion without tilt error at the shaft end; d × β (l)pK) is the runout of the X-direction error motion caused by the inclination error at the shaft end of the first displacement sensor on the P +1 th positioning position;
Figure BDA0002854298230000093
respectively causing the X-direction deviation of the fixed frame on the P-th and P + 1-th position measuring positions of the shaft to be measured in the linear motion process of the sliding frame; m is1(lp,k)，m4(lpAnd k) respectively represents the measured data of one rotation of the shaft to be measured, which are collected by the first displacement sensor and the fourth displacement sensor at the P-th positioning position. Fig. 5 is a schematic diagram showing the influence of the guide rail error motion and the shafting error motion on the measurement process.
It should be explained here that, firstly, the first displacement sensor 1 is repeatedly used, and is respectively applied to the three-point roundness error separation method and the axial two-point method of the measured section, so that the number of sensors is reduced, and the uncertain influence caused by a large number of sensors in the error separation process is reduced; secondly, the first displacement sensor 1 is located at the shaft end where the tilt error is zero, see fig. 2.
And (3) performing discrete Fourier transform on two sides of the formula (5.1) and the formula (5.2) to obtain a first-order harmonic component of a section radial error separation result:
M1(lp,1)＝R(lp,1)+Ex(lp,1) (6.1)
M4(lp,1)＝R(lp+1,1)+Ex(lp,1)+d×Φ(1) (6.2)
in the formula, R (l)p,1)＝DFT[r(lp,1)]A 1 st harmonic component of the profile of the measured cross section for the P-th position finding; ex(lp,1)＝DFT[εx(lp,1)]1 order harmonic component of radial motion error of the measured P-th position measuring section; dxΦ (1) ═ DFT [ d × β (l)p,1)]Is the 1 st harmonic component of X-direction motion error caused by shafting inclination error.
The displacement signal of the fixing frame 8 in the X-axis direction is a constant in a period caused by the P-th and P + 1-th position measurement of the shaft 5 to be measured in the linear motion process of the guide rail 10, and the first harmonic component of the constant is zero after discrete Fourier transform, namely the constant is zero
Figure BDA0002854298230000101
Therefore, the linear motion error of the guide rail 10 is removed in the equations (6.1) and (6.2).
And step 3: calculating 1 order harmonic component of roundness profiles of L measuring sections on the measured shaft, wherein the step 3 comprises the following steps:
step 3.1, initializing P to 1;
step 3.2, the fourier series form of the roundness profile of the measurement section of the shaft to be measured at the P-th position can be written as formula (7):
Figure BDA0002854298230000102
in the formula, r0(α) is the dc harmonic component of the roundness profile of the measured section, which is also the radius of the profile of the measured section; r is1＝a1cosα+b1sin alpha is the first harmonic component R (l) of the measured cross-sectional profileP,1)，(a1,b1) The eccentric coordinates are expressed as the center of a circle of the measured section profile and the center of a least square circle;
Figure BDA0002854298230000103
the harmonic component of the roundness profile of the measured section is the roundness profile error of each section;
order to
Figure BDA0002854298230000104
Representing the known harmonic components, equation (7) can be written as:
r(α)＝r1(α)+r2(α) (8)
step 3.3, the measurement data of the first displacement sensor and the second displacement sensor are as follows:
m1(l,k)＝r(l,k)+εx(l,k) (9.1)
m2(l,k)＝r(l,k+p1)+εx(l,k)×cosβ1+εy(l,k)×sinβ1 (9.2)
the discrete form is as follows:
Figure BDA0002854298230000105
Figure BDA0002854298230000106
wherein k is 0,1, and N-1 represents the measurement point of each sensor, and N represents the total measurement point; alpha 2 pi/N is a sampling interval and is the number of spacing points between the first displacement sensor and the X axis, namely p1α is the angle between the first displacement sensor and the X-axis; p is a radical of2＝β2/α(p2N-1) is the number of spaced points between the second displacement sensor and the X-axis;
step 3.4, transforming according to the formulas (10.1) and (10.2), and knowing that:
a1 cos(kα)+b1 sin(kα)+εx(k)＝m1(k)-r2(k)＝h1 (11.1)
Figure BDA0002854298230000111
in the formula, m1(k),m2(k) The output signals from which the 1 st harmonic component and the N-1 st harmonic component are removed for the first and second displacement sensors are known discrete sequences; r is2(k),r2(k+p1) Removing 1 st order harmonic component and N-1 st order harmonic component for first and second displacement sensorsA roundness error sequence measuring three-point roundness error separation; the right side of the equation is a known quantity, and the right side of the equation (11.1) is h1Equation (11.2) is given as h on the right2；
The shafting radial rotation error is expressed by epsilon (k), and the component epsilon of the shafting radial rotation error on X, Y axesx(k),εy(k)]Can be expressed as:
εx(k)＝ε(k)×cos(kα) (12.1)
εy(k)＝ε(k)×sin(kα) (12.2)
the coupling type (11.1), (11.2), (12.1) and (12.2) can eliminate the component of the shafting radial rotation error in the formulas (12.1) and (12.2) on the XY axis, and can obtain:
h2(k)cos(kα)-h1sin(kα)＝b1cos(2kα)-a1sin(2kα) (13)
step 3.5, obtaining the eccentric coordinates (a) of the circle center and the least square circle center of the measured section profile1,b1) Multiplying the two sides of the equation (13) by trigonometric functions sin (2k alpha) and cos (2k alpha) respectively, and summing in a measurement period, according to the orthogonality of the trigonometric functions, obtaining:
Figure BDA0002854298230000112
Figure BDA0002854298230000113
step 3.6, obtaining a first-order harmonic component R (l) of the roundness profile of the measured cross section at the P-th position measuring position on the shaft 5 to be measured according to the following formulaP,1)：
R(lP,1)＝r1(α)＝a1cosα+b1cosα (15)
Step 3.7, moving the sliding frame 9 to enable the fixing frame 8 to be positioned at the P +1 th position of the shaft 5 to be measured, and repeatedly executing the step 3.2 to obtain the first-order harmonic component R (l) of the roundness profile of the section to be measured at the P +1 th position on the shaft to be measuredP+1,1)；
Step 3.8, obtaining a first-order harmonic component of the cylinder inclination error at the P-th position-measuring right section on the axis to be measured according to the following formula:
d×Φ(1)＝M4(lP,1)-M1(lP,1)-R(lP+1,1)+R(lP,1) (16)
step 3.9, when the measured body position 5 is positioned at the P +1 th measuring position, obtaining the first harmonic component R (l) of the roundness profile of the measured section at the P +2 th measuring position according to the formula (17)P+2,1)：
R(lP+2,1)＝M4(lP+1,1)-M1(lP+1,1)-d×Φ(1)+R(lP+1,1) (17)
Step 3.10, based on the N-1 th harmonic component R (l)P+1N-1) is the first harmonic component R (l)P+1And 1) obtaining the N-1 order harmonic component of the roundness profile of the measured section of the shaft 5 to be measured at the P-th position by using the complex conjugate;
and 3.11, assigning the P +1 to the P, and repeatedly executing the step 3.9 until the first-order harmonic component and the (N-1) th-order harmonic component of the roundness profile of the L measured sections are obtained.
It should be noted here that the first harmonic component of the roundness profile of the measured cross section can be iteratively calculated from the P +2 th positioning by calculating the first harmonic component of the roundness profile of the measured cross section at the P th positioning and the P +1 th positioning, and then the first harmonic component of the roundness profile of the measured cross section can be measured by the equation (17).
And 4, step 4: fitting a space central line according to 1-order harmonic components of the L measured section roundness profiles, combining the shapes of the section profiles onto the space central line, and reconstructing a cylindrical profile, wherein the step 4 comprises the following steps of:
step 4.1, the s-order harmonic component R (l) is subjected to the equation (18)PS) (s ═ 0,1,2, …, N-1) by inverse discrete fourier transform to obtain the profile shape r (l) of the P-th measured cross section of the axis to be measuredP,k)：
r(lP,k)＝IDFT[R(lP,s)]，k＝0,1,2,…,N-1 (18)
Step 4.2, after obtaining each harmonic component of the L positioning measurement cross section shapes, repeating the step 4.1 to obtain L positioning cross section profile shapes;
step 4.3, center vector 2R (l) of each section profileP1)/N, P is 1,2, …, L is fitted to a spatial central line as a reference axis of L measured cross sections, and the profile of the shaft 5 to be measured is reconstructed, thereby realizing the evaluation of the shape error of the shaft 5 to be measured.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the scope of the present invention, which is defined by the appended claims.

Claims (9)

1.基于混合四点法的轴系测量装置，其特征在于，包括：电机、导轨机构、滑动架、固定架、第一位移传感器、第二位移传感器、第三位移传感器和第四位移传感器；所述导轨机构包括两根平行的导轨，所述滑动架跨设在两根导轨上可沿导轨移动，所述固定架安装在滑动架顶部，固定架上设有圆弧形的通孔；所述电机的输出轴与待测轴的一端连接，待测轴的另一端穿设在通孔中；所述第一位移传感器、第二位移传感器和第三位移传感器安装在固定架上且位于第一平面上，所述第一平面与待测轴的轴线垂直；第一位移传感器、第二位移传感器和第三位移传感器的探针位于通孔内且位于与待测轴同心的同一圆周上，所述第四位移传感器安装在固定架上，第四位移传感器的探针和第一位移传感器的探针位于与待测轴的轴线平行的直线上。1. A shafting measurement device based on a hybrid four-point method, characterized in that it comprises: a motor, a guide rail mechanism, a sliding frame, a fixed frame, a first displacement sensor, a second displacement sensor, a third displacement sensor and a fourth displacement sensor; The guide rail mechanism includes two parallel guide rails, the sliding frame is mounted on the two guide rails and can move along the guide rails, the fixed frame is installed on the top of the sliding frame, and the fixed frame is provided with an arc-shaped through hole; The output shaft of the motor is connected with one end of the shaft to be measured, and the other end of the shaft to be measured is penetrated in the through hole; the first displacement sensor, the second displacement sensor and the third displacement sensor are installed on the fixing frame and are located in the first displacement sensor. On a plane, the first plane is perpendicular to the axis of the axis to be measured; the probes of the first displacement sensor, the second displacement sensor and the third displacement sensor are located in the through hole and on the same circumference concentric with the axis to be measured, The fourth displacement sensor is mounted on the fixing frame, and the probe of the fourth displacement sensor and the probe of the first displacement sensor are located on a straight line parallel to the axis of the shaft to be measured. 2.如权利要求1所述的基于混合四点法的轴系测量装置，其特征在于，所述固定架上设有4个传感器定位孔，第一位移传感器、第二位移传感器、第三位移传感器和第四位移传感器设置在相应的传感器定位孔内，并通过止动螺丝固定。2. The shafting measuring device based on the hybrid four-point method according to claim 1, wherein the fixing frame is provided with 4 sensor positioning holes, a first displacement sensor, a second displacement sensor, a third displacement sensor The sensor and the fourth displacement sensor are arranged in the corresponding sensor positioning holes, and are fixed by a set screw. 3.如权利要求1所述的基于混合四点法的轴系测量装置，其特征在于，所述导轨上设有刻度线。3 . The shafting measurement device based on the hybrid four-point method according to claim 1 , wherein a scale line is provided on the guide rail. 4 . 4.基于混合四点法的轴系廓形在线重构方法，其特征在于，采用权利要求1所述的基于混合四点法的轴系测量装置进行数据采集，包括：4. the shafting profile online reconstruction method based on the hybrid four-point method, is characterized in that, adopts the shafting measuring device based on the hybrid four-point method according to claim 1 to carry out data acquisition, comprising: 步骤1：将待测轴沿轴向划分为L个间距为d的测量截面，移动滑动架使第一平面分别与L个截面对齐，移动架定位在相应的测量截面时待测轴旋转一周并通过第一位移传感器、第二位移传感器、第三位移传感器和第四位移传感器采集探针距轴面的距离作为测量数据：Step 1: Divide the axis to be measured into L measurement sections with a distance of d in the axial direction, move the sliding frame so that the first plane is aligned with the L sections respectively, and when the moving frame is positioned at the corresponding measurement section, the axis to be measured rotates once and The distance between the probe and the axial plane is collected by the first displacement sensor, the second displacement sensor, the third displacement sensor and the fourth displacement sensor as the measurement data: 步骤2：对步骤1采集的测量数据进行离散傅里叶变换，通过截面三点圆度误差分离法获取与第一平面对齐的测量截面的圆度轮廓的s阶谐波分量，s＝0,2,3,…,N-2，N表示每个传感器的测量总点数；通过轴向两点法获取径向误差分离结果的一阶谐波分量；Step 2: Discrete Fourier transform is performed on the measurement data collected in step 1, and the s-order harmonic component of the roundness profile of the measurement section aligned with the first plane is obtained by the section three-point roundness error separation method, s=0, 2,3,…,N-2, N represents the total number of measurement points of each sensor; the first-order harmonic component of the radial error separation result is obtained by the axial two-point method; 步骤3：计算被测轴上L个测量截面的圆度轮廓的1阶谐波分量：Step 3: Calculate the 1st-order harmonic components of the roundness profile of the L measurement sections on the measured axis: 步骤4：根据L个测位测量截面的圆度轮廓的1阶谐波分量拟合空间中线，将各截面轮廓形状组合到空间中线上，重构待测轴的圆柱轮廓。Step 4: Fit the space center line according to the first-order harmonic components of the roundness contours of the L positional measurement sections, combine the contour shapes of each section to the space center line, and reconstruct the cylindrical contour of the axis to be measured. 5.如权利要求4所述的基于混合四点法的轴系廓形在线重构方法，其特征在于，步骤1包括：5. the shafting profile online reconstruction method based on the hybrid four-point method as claimed in claim 4, is characterized in that, step 1 comprises: 步骤1.1、在固定架上根据三点圆度误差分离法布置第一位移传感器、第二位移传感器和第三位移传感器，使三个传感器位于第一平面上，第四位移传感器位于第二平面上，第一平面和第二平面相平行且间距为d，Step 1.1. Arrange the first displacement sensor, the second displacement sensor and the third displacement sensor on the fixed frame according to the three-point circularity error separation method, so that the three sensors are located on the first plane, and the fourth displacement sensor is located on the second plane. , the first plane and the second plane are parallel and the distance is d, 步骤1.2、定义导轨的长度方向为Z轴方向，竖直方向为Y轴方向，待测轴的端面中与Y轴方向垂直的方向为X轴方向，第一位移传感器与X轴角度为0，第二位移传感器与第一位移传感器的夹角为β1，第三位移传感器与第一位移传感器的夹角为β2 Step 1.2. Define the length direction of the guide rail as the Z-axis direction, the vertical direction as the Y-axis direction, the direction perpendicular to the Y-axis direction in the end face of the axis to be measured, and the X-axis direction. The first displacement sensor and the X-axis angle are 0, The angle between the second displacement sensor and the first displacement sensor is β 1 , and the angle between the third displacement sensor and the first displacement sensor is β 2 步骤1.3、将被测轴沿轴向划分为L个间距为d的测量截面；定义测位序列P＝1,2,…,L；测位指的是固定架从待测轴端部沿Z轴方向移动距离d后与第一平面对齐的测量截面；Step 1.3. Divide the measured shaft into L measurement sections with a distance of d along the axial direction; define the positioning sequence P=1,2,...,L; the positioning refers to the fixed frame from the end of the shaft to be measured along Z The measurement section aligned with the first plane after moving the distance d in the axial direction; 步骤1.4、初始化P＝1；Step 1.4, initialize P=1; 步骤1.5、沿Z轴方向移动固定架，使固定架的第一平面与待测轴的第P个测量截面对齐；Step 1.5. Move the fixing frame along the Z-axis direction to align the first plane of the fixing frame with the P-th measurement section of the axis to be measured; 步骤1.6、由电机驱动待测轴旋转一周，通过第一位移传感器、第二位移传感器和第三位移传感器采集待测轴在第P测量截面上旋转一周的测量数据；第四位移传感器采集在第P+1测量截面上旋转一周的测量数据，完成第P测位的测量；Step 1.6, the shaft to be measured is driven by the motor to rotate once, and the first displacement sensor, the second displacement sensor and the third displacement sensor are used to collect the measurement data of the shaft to be measured on the P-th measurement section for one rotation; The measurement data of one rotation on the P+1 measurement section is completed, and the measurement of the P-th position is completed; 步骤1.7、重复步骤1.5和步骤1.6，进行P+1测位的测量，直到完成第L测位的数据测量，获取L个测位的测量数据。Step 1.7: Repeat step 1.5 and step 1.6 to measure the P+1 position until the data measurement of the Lth position is completed, and obtain the measurement data of the L position. 6.如权利要求4所述的基于混合四点法的轴系廓形在线重构方法，其特征在于，步骤2包括：6. The shafting profile online reconstruction method based on the hybrid four-point method as claimed in claim 4, wherein step 2 comprises: 步骤2.1、第P测位时第一位移传感器、第二位移传感器和第三位移传感器采集的测量数据为mi(lP,k)，其中i＝1,2,3，k＝0,1,…,N-1；N表示每个传感器的测量总点数，采样的角间隔为α＝2π/N，lP＝P×d表示为待测轴第P测位的轴向定位；k表示每个传感器的测量点序号；Step 2.1. The measurement data collected by the first displacement sensor, the second displacement sensor and the third displacement sensor during the P-th position measurement is m i (l P ,k), where i=1,2,3, k=0,1 ,...,N-1; N represents the total number of measurement points of each sensor, the sampling angular interval is α=2π/N, and l P =P×d represents the axial positioning of the P-th position of the axis to be measured; k represents The measurement point serial number of each sensor; 步骤2.2、利用三点圆度误差分离方法提取待测轴的第P测位的测量截面上的圆度轮廓的s阶谐波分量R(lP,s),s＝0,2,3...,N-2表示谐波分量的阶数；Step 2.2, using the three-point roundness error separation method to extract the s-order harmonic component R(l P ,s) of the roundness profile on the measurement section of the P-th position of the axis to be measured, s=0,2,3. ..,N-2 represents the order of harmonic components; 步骤2.3、对第P测位的测量数据中第一位移传感器和第四位移传感器所采集的数据进行离散傅里叶变换，通过轴向两点法获取径向误差分离结果的一阶谐波分量M1(lp,1)和M4(lP,1)。Step 2.3, perform discrete Fourier transform on the data collected by the first displacement sensor and the fourth displacement sensor in the measurement data of the P-th position, and obtain the first-order harmonic component of the radial error separation result by the axial two-point method M 1 (I p ,1) and M 4 (I P ,1). 7.如权利要求6所述的基于混合四点法的轴系廓形在线重构方法，其特征在于，所述步骤2.3具体为：7. The shafting profile on-line reconstruction method based on hybrid four-point method as claimed in claim 6, is characterized in that, described step 2.3 is specifically: 在第P测位将待测轴旋转一周，第一位移传感器和第四位移传感器分别采集测量截面P和P+1旋转一周的数据：In the P-th position measurement, the axis to be measured is rotated once, and the first displacement sensor and the fourth displacement sensor respectively collect the data of one rotation of the measurement section P and P+1:
Figure RE-FDA0002949712630000031
Figure RE-FDA0002949712630000031
Figure RE-FDA0002949712630000032
Figure RE-FDA0002949712630000032
 对式(5.1)和式(5.2)两边进行离散傅里叶变换，可获得截面径向误差分离结果的一阶谐波分量：Discrete Fourier transform is performed on both sides of Equation (5.1) and Equation (5.2) to obtain the first-order harmonic components of the separation result of the cross-section radial error: M1(lp,1)＝R(lp,1)+Εx(lp,1) (6.1)M 1 (lp ,1) = R( lp ,1)+E x ( lp ,1) (6.1) M4(lp,1)＝R(lp+1,1)+Εx(lp,1)+d×Φ(1) (6.2)M 4 (l p ,1)=R(l p+1 ,1)+Ε x (l p ,1)+d×Φ(1) (6.2) 式中，R(lp,1)＝DFT[r(lp,1)]为第P测位的测量截面的轮廓的1阶谐波分量；Εx(lp,1)＝DFT[εx(lp,1)]为第P测位被测截面径向运动误差的1阶谐波分量；d×Φ(1)＝DFT[d×β(lp,1)]为轴系倾斜误差引起的X向运动误差的1阶谐波分量。In the formula, R(l p ,1)=DFT[r(l p ,1)] is the first-order harmonic component of the profile of the measurement section of the P-th location; E x (l p ,1)=DFT[ε x (l p ,1)] is the first-order harmonic component of the radial motion error of the measured section at the P-th position; d×Φ(1)=DFT[d×β(l p ,1)] is the shafting inclination The 1st harmonic component of the error-induced X-direction motion error. 8.如权利要求4所述的基于混合四点法的轴系廓形在线重构方法，其特征在于，步骤3包括：8. the shafting profile online reconstruction method based on the hybrid four-point method as claimed in claim 4, is characterized in that, step 3 comprises: 步骤3.1、初始化P＝1；Step 3.1, initialize P=1; 步骤3.2、所述待测轴第P测位的测量截面的圆度轮廓的傅里叶级数形式可写为式(7)：Step 3.2, the Fourier series form of the circularity profile of the measurement section of the P-th position of the axis to be measured can be written as formula (7):
Figure RE-FDA0002949712630000041
Figure RE-FDA0002949712630000041
 式中，r0(α)为被测截面圆度轮廓的直流谐波分量，也是被测截面轮廓的半径；r1＝a1cosα+b1sinα为被测截面轮廓的一阶谐波分量R(lP,1)，(a1,b1)表示为被测截面轮廓的圆心与最小二乘圆心的偏心坐标；
Figure RE-FDA0002949712630000042
为被测截面圆度轮廓的谐波分量，即各截面的圆度轮廓误差；In the formula, r 0 (α) is the DC harmonic component of the roundness profile of the measured section, and is also the radius of the measured section profile; r 1 =a 1 cosα+b 1 sinα is the first-order harmonic component of the measured section profile R(l P ,1), (a 1 ,b 1 ) are expressed as the eccentric coordinates of the center of the measured section profile and the center of the least squares circle;
Figure RE-FDA0002949712630000042
is the harmonic component of the roundness profile of the measured section, that is, the roundness profile error of each section; 令
Figure RE-FDA0002949712630000043
表示已知的谐波分量，式(7)可写为：make
Figure RE-FDA0002949712630000043
Representing the known harmonic components, equation (7) can be written as: r(α)＝r1(α)+r2(α) (8)r(α)=r 1 (α)+r 2 (α) (8) 步骤3.3、第一位移传感器和第二位移传感器的测量数据如下：Step 3.3, the measurement data of the first displacement sensor and the second displacement sensor are as follows: m1(l,k)＝r(l,k)+εx(l,k) (9.1)m 1 (l,k)=r(l,k)+ε x (l,k) (9.1) m2(l,k)＝r(l,k+p1)+εx(l,k)×cosβ1+εy(l,k)×sinβ1 (9.2)m 2 (l,k)=r(l,k+p 1 )+ε x (l,k)×cosβ 1 +ε y (l,k)×sinβ 1 (9.2) 其离散形式如下：Its discrete form is as follows:
Figure RE-FDA0002949712630000044
Figure RE-FDA0002949712630000044
Figure RE-FDA0002949712630000045
Figure RE-FDA0002949712630000045
 式中，k＝0,1,...,N-1表示每个传感器的测样点，N表示测量总点数；α＝2π/N为采样间隔，p1＝β1/α(p1＝0,1,...,N-1)为第一位移传感器与X轴之间的间隔点数，即p1α为第一位移传感器与X轴之间的角度；p2＝β2/α(p2＝0,1,...,N-1)为第二位移传感器与X轴之间的间隔点数；In the formula, k=0,1,...,N-1 represents the measurement points of each sensor, N represents the total number of measurement points; α=2π/N is the sampling interval, p 1 =β 1 /α(p 1 =0,1,...,N-1) is the number of interval points between the first displacement sensor and the X axis, that is, p 1 α is the angle between the first displacement sensor and the X axis; p 2 =β 2 / α(p 2 =0,1,...,N-1) is the number of interval points between the second displacement sensor and the X-axis; 步骤3.4、根据式(10.1)和(10.2)进行变换，并将式(11.1)和(11.2)右边命为h1,h2，可知：Step 3.4. Transform according to formulas (10.1) and (10.2), and name the right sides of formulas (11.1) and (11.2) as h 1 , h 2 , it can be known that: a1cos(kα)+b1sin(kα)+εx(k)＝m1(k)-r2(k)＝h1 (11.1)a 1 cos(kα)+b 1 sin(kα)+ε x (k)=m 1 (k)-r 2 (k)=h 1 (11.1)
Figure RE-FDA0002949712630000051
Figure RE-FDA0002949712630000051
 步骤3.5、通过计算可获得被测截面轮廓的圆心与最小二乘圆心的偏心坐标(a1,b1)：Step 3.5. The eccentric coordinates (a 1 , b 1 ) of the center of the measured section profile and the center of the least squares circle can be obtained by calculation:
Figure RE-FDA0002949712630000052
Figure RE-FDA0002949712630000052
Figure RE-FDA0002949712630000053
Figure RE-FDA0002949712630000053
 步骤3.6、根据下式可获得待测轴上第P测位的测量截面的圆度轮廓的1阶谐波分量R(lP,1)：Step 3.6, according to the following formula, the first-order harmonic component R(l P ,1) of the roundness profile of the measurement section at the P-th position on the axis to be measured can be obtained: R(lP,1)＝r1(α)＝a1cosα+b1cosα (15)R(l P ,1)=r 1 (α)=a 1 cosα+b 1 cosα (15) 步骤3.7、移动滑动架使第一平面位于待测轴的第P+1测位，重复执行步骤3.2，获得待测轴上第P+1测位的测量截面的圆度轮廓的1阶谐波分量R(lP+1,1)；Step 3.7, move the carriage so that the first plane is located at the P+1th position of the axis to be measured, and repeat step 3.2 to obtain the first-order harmonic of the roundness profile of the measurement section at the P+1th position on the axis to be measured component R(l P+1 ,1); 步骤3.8、联立公式6.1和6.2获得待测轴上第P测位的第二平面处圆柱倾斜误差的一阶谐波分量：Step 3.8. Combine formulas 6.1 and 6.2 to obtain the first-order harmonic component of the cylinder tilt error at the second plane of the P-th position on the axis to be measured: d×Φ(1)＝M4(lP,1)-M1(lP,1)-R(lP+1,1)+R(lP,1) (16)d×Φ(1)=M 4 (l P ,1)-M 1 (l P ,1)-R(l P+1 ,1)+R(l P ,1) (16) 步骤3.9、待测轴于第P+1测位时，根据式(17)获得第P+2测位被测截面圆度轮廓的一阶谐波分量R(lP+2,1)：Step 3.9. When the axis to be measured is positioned at the P+1th position, obtain the first-order harmonic component R(l P+2 ,1) of the roundness profile of the measured section at the P+2th position according to formula (17): R(lP+2,1)＝M4(lP+1,1)-M1(lP+1,1)-d×Φ(1)+R(lP+1,1) (17)R(l P+2 ,1)=M 4 (l P+1 ,1)-M 1 (l P+1 ,1)-d×Φ(1)+R(l P+1 ,1) (17 ) 步骤3.10、根据第N-1阶谐波分量R(lP+1,N-1)是一阶谐波分量R(lP+1,1)的共轭复数，可获得待测轴的第P测位的测量截面的圆度轮廓的N-1阶谐波分量；Step 3.10. According to the N-1 order harmonic component R(l P+1 ,N-1) is the complex conjugate of the first order harmonic component R(l P+1 ,1), the first order of the axis to be measured can be obtained. The N-1 order harmonic component of the roundness profile of the measuring section of the P-positioning; 步骤3.11、将P+1赋值给P，重复执行步骤3.9，直至获得L个测位的测量截面的圆度轮廓的1阶谐波分量和第N-1阶谐波分量。Step 3.11: Assign P+1 to P, and repeat step 3.9 until the 1st-order harmonic components and the N-1th-order harmonic components of the roundness profile of the L measured cross-sections are obtained. 9.如权利要求4所述的基于混合四点法的轴系廓形在线重构方法，其特征在于，步骤4包括：9. The method for on-line reconstruction of the shafting profile based on the hybrid four-point method as claimed in claim 4, wherein step 4 comprises: 步骤4.1、利用式(18)对s阶谐波分量R(lP,s)(s＝0,1,2,…,N-1)进行离散傅里叶逆变换，得到待测轴的第P测位的测量截面的轮廓形状r(lP,k)：Step 4.1. Use formula (18) to perform inverse discrete Fourier transform on the s-order harmonic component R(l P ,s) (s=0,1,2,...,N-1) to obtain the first The profile shape r(l P ,k) of the measuring section of the P-positioning: r(lP,k)＝IDFT[R(lP,s)]，k＝0,1,2,…,N-1 (18)r(l P ,k)=IDFT[R(l P ,s)], k=0,1,2,...,N-1 (18) 步骤4.2、获得L个测位的测量截面形状的各谐波分量后，重复执行步骤4.1，以获取L个测位横截面轮廓形状；Step 4.2: After obtaining the harmonic components of the L measured cross-sectional shapes, repeat step 4.1 to obtain the L measured cross-sectional profiles; 步骤4.3、将各个截面轮廓的中心向量2R(lP,1)/N,P＝1,2,…,L拟合为一条空间中线，作为L个测量截面的基准轴线，重构待测轴的轮廓，从而实现对待测轴的形状误差评定。Step 4.3. Fit the center vector 2R(l P ,1)/N, P=1,2,...,L of each section profile as a space center line, which is used as the reference axis of the L measurement sections to reconstruct the axis to be measured , so as to realize the shape error evaluation of the axis to be measured.
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