CN108620955B - Measurement and Identification Method of Rotary Axis Error of Machine Tool Based on Monocular Vision - Google Patents

Abstract

The invention relates to a monocular vision-based method for measuring and identifying errors of machine tool rotary axes, belonging to the field of machine tool geometric error measurement, and relates to a method for measuring and identifying geometric errors of machine tool rotary axes using monocular vision. This method uses a monocular camera to collect positioning target images with coded marking points during the intermittent and continuous two rotations of the rotary shaft, and reconstructs the coordinates of the coded marking points in the machine tool coordinate system after image processing for intermittent rotation. Using the corresponding relationship between the theoretical point and the actual point coordinates in the error matrix model, the independent position error of the machine tool is solved, and then the comprehensive error of the machine tool is obtained through image processing obtained by continuous rotation shooting, and finally the independent position error of the machine tool is eliminated in the comprehensive error of the machine tool Get the machine tool dependent position error. This method effectively solves the problems of cumbersome measurement process and complex identification model, and uses monocular vision to realize non-contact measurement, with low equipment cost and high equipment robustness.

Description

Measurement and Identification Method of Rotary Axis Error of Machine Tool Based on Monocular Vision

technical field

The invention belongs to the field of geometric error measurement of machine tools, and relates to a method for measuring and identifying the geometric error of a rotary axis of a machine tool by using monocular vision.

Background technique

With the rapid development of aviation, aerospace, nuclear power, metallurgy and other high-tech fields and the rapid improvement of processing levels, the demand for high-end CNC machine tools and equipment has increased rapidly, and higher requirements have been placed on the processing accuracy of CNC machine tools. Five-axis linkage CNC machining is the main processing method for continuous, smooth, and complex surfaces. Compared with traditional three-axis CNC machining, the machining accuracy, quality and efficiency have been greatly improved, and the rotary axis is an important part of five-axis CNC machine tools. Its geometric error forms a complex spatial error through multi-axis coupling during machining, which seriously affects the machining accuracy of the machine tool. Therefore, it is of practical significance to study the measurement and identification of the rotary axis error in the five-axis linkage CNC machine tool. At present, the main machine tool error detection methods include: standard part measurement, laser interferometer, ballbar, plane grating, R-test, etc. Among them, the visual measurement method has the advantages of low cost, simple structure, and high measurement accuracy, which can realize non-destructive detection of machine tool errors. Contact measurement, therefore, it is particularly important to invent a rotary axis error detection and identification technology that is easy to operate, low in measurement cost, and high in measurement accuracy.

The patent No. CN 105371793 A invented by Zhou Xiangdong, Tang Xiaoqi and others from Huazhong University of Science and Technology is "a method for measuring the geometric error of the rotation axis of a five-axis machine tool by one-time clamping". Based on the spatial error model of the mechanism, it can be calculated by one-time clamping measurement. Twelve geometric error parameters of the two rotation axes of the five-axis CNC machine tool, but this method is a contact measurement method, and requires multiple probes to touch different measurement surfaces of the cube reference specimen one by one, and the operation process is relatively cumbersome. The patent No. CN 102744648 A invented by Lin Jing and Wang Xiufeng of Xi'an Ruite Rapid Manufacturing Engineering Research Co., Ltd. is "A Method for Measuring and Separating Errors of Rotary Tables of CNC Machine Tools". The metal ring in the center, combined with the error decoupling method, can complete the non-contact measurement of the geometric error of the machine tool. This method is simple in structure and easy to operate, but it is more suitable for the error detection of the linear axis, and the cost is high, and it is not suitable for the rotary axis. Error measurement and identification.

Contents of the invention

The technical problem to be solved by the present invention is to overcome the problems of the prior art and to invent a method for measuring and identifying the error of the rotary axis of the machine tool based on monocular vision. This method uses a monocular vision system combined with a positioning target to detect the error of the rotary axis of a five-axis CNC machine tool, and realizes the separation of multiple errors in a single measurement of the rotary axis. First, install a positioning target containing coded marking points on the machine tool workbench and index the rotary axis, collect the images of the marked points during the intermittent rotation of the machine tool every certain index value, and use the camera imaging model to reconstruct the coded marking point machine tool Coordinates in the coordinate system, bring the coordinates of the measurement points into the error matrix model to solve the independent position error of the machine tool; then set the machine tool to rotate continuously for one circle, control the camera to collect the image of the marker point at the same division value, similarly, the machine tool can be obtained Comprehensive errors, elimination of independent position errors of machine tools is machine-dependent position errors. This method uses coded marking points combined with monocular vision to realize the detection and identification of CNC machine tool rotary axis errors, which effectively solves the problems of cumbersome measurement process and complex identification models. The identification model is simple and non-contact measurement, easy to operate, low equipment cost and high robustness.

The technical solution adopted in the present invention is a method for measuring and identifying errors of machine tool rotary axes based on monocular vision, which is characterized in that the method uses a monocular camera to collect information with The positioning target image of the coded mark point, after image processing of intermittent rotation, reconstructs the coordinates of the coded mark point in the machine tool coordinate system, and uses the correspondence between the theoretical point and the actual point coordinates in the error matrix model to solve the independent position error of the machine tool, Then, the comprehensive error of the machine tool can be obtained through image processing obtained by continuous rotation shooting, and finally the independent position error of the machine tool is eliminated in the comprehensive error of the machine tool to obtain the dependent position error of the machine tool; the specific steps of the method are as follows:

The first step is to install the experimental device

The experimental device consists of a monocular camera 1, a positioning target 2, a machine tool table 3, a system mounting frame 4, and a fixture 5. The monocular camera 1 is installed on the system mounting frame 4 vertically to the machine tool table 3. Clamp it on the machine tool workbench 3 to prepare for calibrating the monocular camera; Fixture 5 is installed on the machine tool workbench 3 to facilitate the replacement and clamping of the positioning target 2 and the checkerboard calibration plate, and reset the C-axis of the machine tool’s rotary axis to be tested to zero ;

The positioning target 2 is a photolithographic glass plate, and the upper surface is photolithographically engraved with the first, second, third and fourth circular coding points 69, 71, 73, 75 distributed in a diamond shape, and the central coding point 119 located in the center of the image , and establish the machine tool coordinate system with the center code point 119 as the origin;

The second step, camera calibration

The present invention calibrates the internal parameters and distortion parameters of the monocular camera based on the Zhang’s calibration method combined with a high-precision checkerboard calibration board, and selects a point coordinate in space as (X _p , Y _p , Z _p ), which projects the point coordinates on the image plane is (x, y), considering the distortion phenomenon of the actual imaging comprehensively, the expression of the nonlinear perspective projection model of the camera is as follows:

Among them, f is the focal length of the monocular camera, α _x ＝f/dx and α _y ＝f/dy are defined as the normalized focal lengths on the x and y axes respectively, and (x ₀ , y ₀ ) is the origin of the pixel coordinate system Coordinates, M ₀ composed of four parameters α _x , α _y , u ₀ , v ₀ is the camera internal parameter matrix, R is a 3×3 unit rotation orthogonal matrix, t is a translation vector, and the M ₁ formed is the camera External parameter matrix; that is, the relationship between the machine tool coordinate system and the pixel coordinate system is determined by the internal parameters α _x , α _y , u ₀ , v ₀ and external parameters R, t; the point (x, y) is the ideal position of the projected point, ( x′, y′) are the actual coordinates considering the distortion, δ _x and δ _y are the nonlinear distortion values of the horizontal and vertical axes, r is the distance between the projected point and the origin in the pixel coordinate system, k ₁ , k ₂ and k ₃ are the first-order, second-order and third-order radial distortion coefficients respectively, p ₁ and p ₂ are the first-order and second-order centrifugal distortion coefficients; the internal parameters α _x , α _y , u ₀ , v can be obtained by calibrating the monocular camera ₀ and distortion parameters k ₁ , k ₂ , k ₃ , p ₁ , p ₂ , and then determine the camera imaging model; after the monocular camera 1 is calibrated, replace the checkerboard calibration plate with a positioning target 2, and fix it on the machine tool table 3 up;

The third step, mark point recognition

Index the rotary axis, collect the image of the mark point during the intermittent rotation of the machine tool every certain index value, and use the camera imaging model to reconstruct the coordinates of the code mark point in the machine tool coordinate system; the present invention uses the circular code mark point Locate the target 2 to represent the movement position of the machine tool. The center of the ring code is a black circle mark point, and the outer ring is a concentric ring area, which is used to represent the identity information of the ring code, which is called a code belt; the ring area is divided according to the angle. There are 10 segments, each segment corresponds to a binary bit; the background color is white and black, and the corresponding binary codes are "0" and "1"; starting from the center of the marked point, scan the coding belt in a certain direction until the black code belt is scanned It is recorded as 1, and the white code band is marked as 0; if the code band is not scanned, start scanning again from the center; after scanning for one week, the code value sequence of the entire code point is read out, forming a binary sequence, and finally converted into Decimal integers, so as to obtain the code value of the code point; after decoding, according to the code value of different code mark points, and then use the internal and external parameters of the calibrated camera imaging model, the three-dimensional coordinates of each corresponding mark point can be reconstructed, and through the coordinate system benchmark Conversion, and finally the three-dimensional coordinates of the coding mark points in the machine tool coordinate system can be obtained as (X _w , Y _w , Z _w );

The fourth step, rotary axis error measurement and identification

The rotary axis error of CNC machine tools mainly has two kinds of errors, which are machine tool independent position error (PIGE) and machine tool dependent position error (PDGE). The former is mainly the assembly error of the machine tool and has nothing to do with the machine tool command position, so it is a constant, and the latter is mainly The geometric error caused by the manufacturing defects of the parts and components of the CNC machine tool is related to the command position of the machine tool, so it is a function of the variable and the rotation angle. The combination of the two can express the comprehensive error of the machine tool rotary axis as:

e＝e _PIGE +e _PDGE (θ) (2)

First, identify the independent position error of the rotary axis of the machine tool. When the machine tool is controlled to move intermittently to a certain position, the comprehensive error term of the rotary axis only contains e _PIGE , and there are 4 connection errors, including 2 linear position errors and 2 corner error, then its error matrix can be expressed as

Among them, ε _xc and ε _yc are the linear position errors between the actual center and the ideal center of the C-axis in the X and Y directions; δ _xc and δ _yc are the rotation angle errors between the actual axis and the ideal axis of the C-axis around the X-axis and the Y-axis. Since the error matrix T contains only 4 unknown quantities, that is, through the graduation of the rotary axis, at the 4 angles of 0°, 90°, 180° and 270° of the machine tool coordinate system, the encoding mark is collected according to the monocular camera The image of the point, using the coordinates in the pixel coordinate system and the camera imaging model, respectively reconstructs the coordinates of the coding mark point in the machine tool coordinate system as The machine tool independent position error of the rotary axis of the CNC machine tool can be obtained by using the equation obtained from formula (4). Set the theoretical point coordinates of the four angles in the machine tool coordinate system as The solution formula is:

Then identify the position error of the rotary axis of the machine tool, and control the machine tool to run at a certain speed. At this time, the comprehensive error term of the rotary axis is e _PIGE + e _PDGE (θ), and its error matrix can be expressed as:

Among them, Δε _xc and Δε _yc are the linear position errors between the actual center and the ideal center of the C-axis in the X and Y directions; Δδ _xc and Δδ _yc are the rotation angle errors between the actual axis of the C-axis and the ideal axis around the X-axis and the Y-axis. The four errors are the machine tool-dependent position errors of the rotary axis of the machine tool; the machine-tool rotary axis machine-dependent position errors have been identified under static conditions before, that is, ε _xc , ε _yc , δ _xc , and δ _yc are known quantities, then T It still contains 4 unknown quantities, and also triggers the camera to shoot at the 4 angles of 0°, 90°, 180° and 270° in the machine tool coordinate system and reconstructs the coordinates of the coded marker points respectively, which are recorded as The solution formula is:

That is, the comprehensive error of the rotary axis of the CNC machine tool is obtained under quasi-static work, and the known machine tool-dependent position error is subtracted, which is the machine tool-dependent position error of the rotary axis.

The advantage of the present invention is that the detection and identification of the rotary axis error of the CNC machine tool is realized by using the coding mark points combined with monocular vision, which effectively solves the problems of cumbersome measurement process and complex identification model, and realizes non-contact measurement by monocular vision, which has a simple structure , easy to operate, low equipment cost and high equipment robustness. The identification model of this method is simple and non-contact measurement. The detection and identification of the rotary axis error of the CNC machine tool is realized by using the coded marking points combined with the monocular vision. The equipment cost is low and the robustness is high.

Description of drawings

Figure 1 is a model diagram of the error measurement and identification of the rotary axis of the machine tool based on monocular vision. Among them, 1-monocular camera, 2-positioning target, 3-machine tool table, 4-system mounting frame, 5-fixture.

Figure 2 is an image of the coded markers on the positioning target. Among them, 69, 71, 73, and 75 are the first, second, third, and fourth circular coding points respectively, 119 is the central coding point of the image center, and X and Y are the coordinate axes of the machine tool coordinate system.

Figure 3 is a schematic diagram of the error measurement and identification of the rotary axis of the machine tool. Among them, ε _xc , ε _yc are the linear position errors between the actual center and the ideal center of the C-axis in the X direction and the Y direction, respectively, and δ _xc , δ _yc are the rotation angle errors of the actual axis and the ideal axis of the C-axis around the X-axis and the Y-axis respectively , C axis is the theoretical rotary axis of the machine tool, and C' axis is the actual rotary axis of the machine tool.

Figure 4 is a flow chart of the error measurement and identification of the rotary axis of the machine tool.

Detailed ways

The specific embodiments of the present invention will be described in detail below in conjunction with the technical solutions and accompanying drawings.

Figure 1 is a model diagram of the error measurement and identification of the rotary axis of the machine tool based on monocular vision. In the present invention, the positioning target 2 is a photolithographic glass plate with a thickness of 3mm, and the upper surface is photoengraved with four circular code points distributed in a diamond shape and one code point located in the center, which is clamped on the machine tool workbench 3 during measurement. , the monocular camera 1 is used to collect the image of the positioning target 2 with coded marking points in the process of intermittent and continuous two rotations of the rotary shaft, collect the image of the positioning target with coded marking points, and after the image processing of the intermittent rotation, Reconstruct the coordinates of the coding mark points in the machine tool coordinate system, use the correspondence between the theoretical point and the actual point coordinates in the error matrix model to solve the independent position error of the machine tool, and then obtain the comprehensive error of the machine tool through image processing of continuous rotation acquisition, and finally The machine tool dependent position error can be obtained by eliminating the machine tool independent position error in the machine tool comprehensive error. The flow of the method is shown in Figure 4, and the specific steps are as follows:

The first step, the installation of the experimental device

The experimental device platform is composed of a monocular camera 1, a characteristic target 2, a machine tool workbench 3, a system mounting frame 4 and a fixture 5. The checkerboard calibration plate is installed on the machine tool workbench 3 through the fixture 5, and then the monocular camera 1 The vertical machine tool table 3 is installed on the system mounting frame 4, and the C-axis of the rotary axis of the machine tool to be tested is reset to zero.

The second step, camera calibration

This method calibrates the internal parameters and distortion parameters of the monocular camera based on Zhang’s calibration method combined with a high-precision checkerboard calibration board. After calibration, the internal parameters α _x , α _y , u ₀ , v ₀ , external parameters R, t, and distortion parameters k ₁ , k ₂ , k ₃ , p ₁ , p ₂ , and then determine the camera imaging model;

The third step, mark point feature recognition

In order to accurately characterize the movement position of the machine tool, a positioning target with a ring-coded marking point is used, as shown in Figure 2. Divide the area of the circular coding belt into 10 segments on average according to the angle, and each segment corresponds to a binary bit. Scan the coding belt in a certain direction during image processing, and record the black code belt as 1 and the white code belt as 0. After scanning for one week , to obtain the binary sequence of the entire coding point, which is converted into a decimal integer to be the code value. Using the code values of different coding points and the camera imaging model, the three-dimensional coordinates of the coding point in the machine tool coordinate system can be reconstructed as (X _w , Y _w , Z _w );

The fourth step, rotary axis error measurement and identification

Figure 3 is a schematic diagram of the error measurement and identification of the rotary axis of the machine tool. In this method, according to the number of unknown quantities in the error matrix model, the indexing value of the C-axis of the machine tool is set to 90°, that is, during the intermittent rotation of the machine tool, Collect an image at four angles of 0°, 90°, 180° and 270°, and reconstruct the 3D coordinates of the coded points containing PIGE At the same time, in the machine tool coordinate system established with the center point of the first image as the origin, the three-dimensional coordinates of the theoretical point are obtained according to the positional relationship of the coded points on the high-precision positioning target Using formulas (3) and (4), according to the corresponding relationship between the theoretical point and the actual point coordinates in the error matrix model, the machine tool independent position error (PIGE) is solved;

After returning the C-axis of the machine tool to zero, control the machine tool to rotate continuously for one round, and also trigger the camera to collect images every 90° of the index value, and restore the three-dimensional coordinates of the coding point after subsequent image processing Through the formulas (5) and (6), the comprehensive error of the machine tool including PIGE and PDGE can be solved. At this time, PIGE is the known quantity solved in the previous step. Finally, the independent position error (PIGE) of the machine tool can be obtained by eliminating the comprehensive error of the machine tool (PIGE). Dependent Position Error (PDGE).

According to the characteristics of non-contact measurement of monocular vision, the present invention uses a simple error identification model to solve the problems of cumbersome measurement process and complex identification model, and has the advantages of simple structure and convenient operation. The detection and identification of machine tool rotary axis errors has the characteristics of good equipment economy and high measurement stability.

Claims (1)

1. a kind of measurement of machine tool rotary axis error and discrimination method based on monocular vision, it is characterized in that, this method uses monocular Camera rotating shaft interval, twice in succession turn round during, acquisition have coded markings point positioning target image, by pair The image procossing of intermittent rotary rebuilds coordinate of the coded markings point under lathe coordinate system, utilizes mathematical point and actual point coordinate Corresponding relationship in error matrix model solves lathe independent position error, using what is shot to continuous rotation Image procossing can acquire lathe composition error, and lathe independent position error is finally eliminated in lathe composition error can be obtained lathe Rely on location error；Specific step is as follows for method:

The first step installs experimental provision

Experimental provision is by monocular camera (1), positioning target (2), platen (3), system mounting rack (4) and fixture (5) group At；The vertical platen (3) of monocular camera (1) is mounted on system mounting rack (4), then gridiron pattern scaling board is clamped in Platen is prepared on (3) for camera calibration；Fixture (5) is mounted on platen (3), positioning target is facilitated (2) with the replacement clamping of gridiron pattern scaling board；

Positioning target (2) is photoetching glass plate, and upper surface photoetching has the first circle codification point (69) of the distribution that assumes diamond in shape, the second circle Shape encoded point (71), third circle codification point (73), the 4th circle codification point (75) and centrally located centre code point It (119), is and with centre code point (119) that origin establishes lathe coordinate system；Positioning target (2) is photoetching glass plate, upper surface Photoetching has the first circle codification point (69), the second circle codification point (71), third circle codification point (73), for the distribution that assumes diamond in shape 4 circle codification points (75) and centre code point (119) positioned at picture centre, and be that origin is established with centre code point (119) Lathe coordinate system, four additional circle codification point are used for the motion information of the accurate transmission lathe；Gridiron pattern is demarcated after calibration is good Plate is replaced with positioning target (2), and fixed on platen (3), and lathe rotating shaft C axis to be measured is zeroed；

Second step, camera calibration

The present invention demarcates the intrinsic parameter and distortion ginseng of monocular camera according to Zhang Shi standardization combined high precision gridiron pattern scaling board Number, selection some coordinates in space are (X_p, Y_p, Z_p), subpoint coordinate is (x, y) in picture plane, comprehensively considers actual imaging Distortion phenomenon uses the non-linear perspective projection model expression of video camera are as follows:

Wherein, f is the focal length of monocular camera, α_x=f/dx and α_y=f/dy is respectively defined as the normalization focal length on two axis of x, y, (x₀, y₀) it is pixel coordinate system origin, by α_x、α_y、u₀、v₀The M that four parameters are constituted₀For camera Intrinsic Matrix, R 3 × 3 unit rotating orthogonal battle array, t are translation vector, the M of composition₁For Camera extrinsic matrix number；I.e. by intrinsic parameter α_x、α_y、u₀、v₀ And outer parameter R, t determines the relationship of lathe coordinate system and pixel coordinate system；Point (x, y) is the ideal position of subpoint, (x ', y ') For the actual coordinate for considering distortion, δ_x、δ_yFor the nonlinear distortion variate of horizontal axis and the longitudinal axis, r is subpoint and original under pixel coordinate system The distance of point, k₁、k₂With k₃Respectively single order, second order and three rank coefficient of radial distortion, p₁、p₂It is single order, second order centrifugal distortion system Number；Intrinsic parameter α can be obtained by demarcating monocular camera_x、α_y、u₀、v₀And distortion parameter k₁、k₂、k₃、p₁、p₂, and then determine camera at As model；After monocular camera (1) calibration is good, gridiron pattern scaling board is replaced with positioning target (2), and in platen (3) Upper fixation；

Third step, reference point identifying

Rotating shaft is indexed, acquisition label point image during lathe was every certain scale division value intermittent rotary one week, The coordinate under coded markings point lathe coordinate system is rebuild using camera imaging model；The present invention is using annulus coded markings point Positioning target characterizes machine tool motion position, and annulus encoding centre is dark circles mark point, outer ring is concentric annular regions, is used for Characterize the identity information of annulus coding, referred to as coding-belt；The circle ring area is equally divided into 10 sections, every section correspondence one according to angle Binary digit；Background colour is white and black, and corresponding binary coding is " 0 ", " 1 "；From the mark point center of circle, according to Certain orientation scanning encoding band, scanning are denoted as 1 to black code band, and white code band is denoted as 0；If not scanning coding-belt, It is rescaned since center；After run-down, the code value sequence of entire encoded point is all read, and forms a binary system Sequence is eventually converted into decimal integer, to obtain the code value of encoded point；After decoding, according to the code of different coding mark point Value recycles the camera imaging model inside and outside parameter demarcated, can rebuild the three-dimensional coordinate of each correspondence markings point, and pass through Coordinate system Reference Transforming is crossed, the three-dimensional coordinate that can finally obtain the coded markings point under lathe coordinate system is (X_w, Y_w, Z_w)；

4th step, error of rotary axle measurement and identification

Rotating shaft of numerical control machine error is that lathe independent position error and lathe rely on position and miss respectively there are mainly two types of error Difference, the former is mainly the rigging error of lathe and unrelated with lathe command position, so being constant, the latter is mainly derived from numerical control Geometric error caused by the manufacturing defect of lathe parts and components, it is related to lathe command position, so being variable and angle of revolution One function, the two combine and can indicate the composition error of machine tool rotary axis are as follows: e=e_PIGE+e_PDGE(θ) (2)

Lathe rotating shaft lathe independent position error is recognized first, when controlling lathe intermittent movement to a certain position, is returned at this time The composition error item of shaft contains only e_PIGE, and connect error and share 4, including 2 linear position errors, 2 angular errors, Then its error matrix is represented by

Wherein, ε_xc、ε_ycFor C axis practical center and the linear position error of desired center in the x direction and the y direction；δ_xc、δ_ycFor C axis The angular errors of practical axis and ideal axis around X-axis and Y-axis；Due to containing only 4 unknown quantitys in error matrix T, by right The indexing of rotating shaft is acquired according to monocular camera and is encoded under 0 ° of lathe coordinate system, 90 °, 180 ° and 270 ° of 4 angles The image of mark point is rebuild coded markings point lathe respectively and is sat using the coordinate and camera imaging model under pixel coordinate system Mark system under coordinate beIt can be acquired using the equation that formula (4) obtains The lathe independent position error of rotating shaft of numerical control machine；Set the mathematical point coordinates of lower 4 angles of lathe coordinate system asSolution formula are as follows:

Lathe rotating shaft lathe is recognized again and relies on location error, and control lathe is run with certain speed, at this time the synthesis of rotating shaft Error term is e_PIGE+e_PDGE(θ), then its error matrix indicates are as follows:

Wherein, Δ ε_xc、Δε_ycFor C axis practical center and the linear position error of desired center in the x direction and the y direction；Δδ_xc、 Δδ_ycIt is the practical axis of C axis and ideal axis around the angular errors of X-axis and Y-axis, above-mentioned 4 errors are that required lathe relies on position Set error；Lathe rotating shaft lathe independent position error, i.e. ε are recognized in a static condition before_xc、ε_yc、δ_xc、δ_ycFor The amount of knowing then still contains 4 unknown quantitys, touches under 0 °, 90 °, 180 ° and 270 ° of 4 angles equally in lathe coordinate system in T It sends out camera to shoot and rebuild coded markings point coordinate respectively, be denoted asIt solves Formula are as follows:

The composition error that rotating shaft of numerical control machine is acquired under quasi-static work subtracts known lathe independent position error i.e. Location error is relied on for the lathe of rotating shaft.