# Simulation and Compensation of Axial Geometric Errors for Cycloidal Gears Based on Form Grinding

Authors: Chuang Jiang; Huiliang Wang; Tianhao Han; Xing Liu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

To increase quality, reduce cycloidal gear noise, and avoid unnecessary vibration and shock, a compensation of axial geometric errors method is proposed based on the cycloidal gear form grinding. In the process of machining cycloidal gears, the relative position relationship between the grinding wheel and workpiece is affected by geometric errors of the motion axes, which has serious effects on the surface accuracy of the cycloidal gears. Combined with cycloidal gear form grinding kinematic principles, a geometric error model for each axis of a four-axis computer numerical control form grinding machine is established. By changing the compensation value of the geometrical errors on six degrees of freedom, the error of the cycloid gear tooth surface machined is obtained. Based on a sensitivity analysis of geometrical errors of each axis, the corrections are determined through an optimization process that targets the minimization of the tooth flank errors. The geometric errors of each axis of the cycloid gear grinding machine are compensated, and then, the cycloid gears produced by the machine are processed. Through the processing experiment, the error data of the actual processing before and after the compensation are compared, which indicates that the machining accuracy of the cycloid gear grinding machine is obviously improved. It has an important guiding significance in improving the precision and performance of large CNC form gear grinding machines.


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## Body

## 1. Introduction

Cycloidal gears are used in rotating vector reducers, instruments and apparatus, planetary gearboxes, and other mechanical transmission devices. A rotary vector reducer is usually referred to as an RV reducer. Compared with mechanical transmissions such as friction, hydraulic pressure, and belts, cycloidal gear transmission efficiency is higher and has a wide range. At the same time, the life of the cycloidal gears is longer, its transmission ratio is stable, and its work reliability is also higher. Therefore, cycloidal gears have necessary and optional applications in various machinery manufacturing industries such as RV reducer, planetary gearbox, and construction machinery. As an important part widely used in mechanical transmission, the quality and accuracy of cycloidal gears during design and processing will directly affect the quality and reliability of mechanical products. To meet the demand for these kinds of gears, the simulation and compensation of axial geometric errors for cycloidal gear grinding are applied to the form grinding machine to increase its load capacity and reduce the transmission error of the RV reducer device. Therefore, how to improve the machining accuracy and optimize the design of cycloidal gears has always been a research hot spot. The problem has been paid close attention to by the business community and academia of various countries [1, 2].In recent years, with the rising labor costs and the continuous adjustment of the industrial structure, the development trend of the manufacturing industry has become an intelligent manufacturing technology with robots as the core, which has a very important role in promoting the upgrading and transformation of the industry. The RV reducer is a very important mechanical device widely used in the robot industry. At each arm joint of the robot, the main shaft and the arm are connected, which can reduce the rotation speed of the arm and increase the torque. During the process, the precision of the cycloidal gears is closely related to the precision and quality of the RV reducer. One of the most important parts of the RV reducer is the cycloid gear. As an important core component in the RV reducer, the cycloid gears determine the rotation accuracy of the whole reducer, so the machining accuracy of the cycloid gears is also the same as that of the robot important indicators for design and research. In this case, it is very important to manufacture and process the cycloid gear grinding machine based on the forming method. It is of great significance to improve the accuracy of the cycloidal gears [3–5]. Therefore, the research on the precision of the cycloid gear grinding machine is also the focus.Schultschik [6] puts forward the concept of machine tool space error for the first time, using the closed-loop constraints of the machine tool's mechanical structure. Denavit et al. [7] proposed the ideal dynamic model of multiaxis machine tools based on a standard homogeneous transformation matrix. Paul [8] incorporated robot mathematics, programming, and control into the basic manipulator’s idea of the reference frame and modified the model. Hocken [9] proposed the static positioning error modeling of the measuring machine by the matrix transformation method and measured the three-dimensional measuring machine using the three-dimensional measuring technology. Since the 1980s, scholars have begun to study the machining error caused by the spatial error of the machine tool and established a spatial error model including the manufacturing geometric error [10]. Kim [11] established an error compensation model for a three-axis machine tool, through which the axis motion deviation can be predicted within a certain accuracy range. Soons et al. [12] introduced the model of the rotation axis in the rigid body kinematics model and conducted a comprehensive error model study of multi-axis machine tools including translation and rotation axes. Rahman et al. [13] used the homogeneous coordinate transformation matrix to study the quasi-static error of the machine tool, including the thermal error and geometric error of the rotary axis and the translation axis. Flourssen et al. [14] proposed a new scheme of the three-dimensional hemisphere to identify the error of the rotary axis in a five-axis machine tool by using three-dimensional length measurements. Abbaszadeh et al. [15] classified the geometric errors of multiple motion axes in the machine tool into static errors and motion errors. Tsutsumi et al. [16,17] conducted a mathematical analysis of the influence of installation deviation on the parameter identification of the simplified model of the rotating shaft and simulated the simplified model identification algorithm, which proved the reliability of the simplified method. Zargarbashihe and Mayer [18] analyzed the static and kinematic geometric errors of the first rotary axis of a five-axis machine tool, and mainly studied the five motion errors of the first rotary axis. Lei et al. [19] analyzed the motion error of five-axis machine tools, designed a measurement scheme that is not disturbed by the linear axis error model, and modeled the backlash error and servo mismatch. Sharif and Ibaraki et al. [20] studied the detection scheme of completing all measurements in one installation without considering the geometric error of the translation axis, reducing the uncertain identification results caused by the need for multiple installations and multiple separate modeling. Ibaraki et al. [21] used the probe to measure the deviation between the actual motion position and the theoretical position of the rotary axis of the machine tool during the machining process through the spatial position relationship among the static errors of each axis of the five-axis machine tool. Lee et al. [22] carried out the quintic polynomial modeling of the rotating shaft by limiting some polynomial coefficients, which simplified the identification difficulty and improved the accuracy of error estimation. Tsutsumi et al. [23] proposed two error identification schemes of different coordinate systems for double turntable machine tools, respectively, using a Cartesian coordinate system and cylindrical coordinate system to conduct experiments on the static error of double turntables. Mchichi and Mayer [24] analyzed the inherent errors of the machine tool and identified and analyzed the errors through standard ball arrays and special probes. Fan et al. [25] focused on the unified kinematic modeling method of three-axis machine tools, modularized each kinematic pair model, and studied the commonalities in modeling of three-axis machine tools with four different structural forms. The concept of exponential product and spiral in robotics is used to model the kinematics of the 5-axis machine tool. It is pointed out that the method does not need to establish a fixed coordinate system for each motion pair [26]. An innovative geometric error compensation method for machining with non-rotary cutters is proposed, in which the cutter rotation angle is considered for both modeling and compensation of geometric errors, and the instantaneous ideal contact point is calculated based on the worm grinding of face gears generation process [27–29]. The insufficient machining accuracy of cycloid gear will lead to the increase in transmission error of gear pair and the stress concentration in meshing, which will lead to bending fatigue and contact fatigue, and other problems. These problems will make gear wear and the machine operating results of deviation, unable to achieve the desired effect, and even cause safety accidents in serious cases [30, 31].This work aims to establish a model of axial geometric errors for cycloidal gear grinding machines based on form grinding. To improve the machining accuracy of machine tools to a large extent, the geometric error compensation technology of machine tools is indispensable, and how to accurately establish the motion error model of machine tools is one of the key issues especially in the cycloidal gear grinding machine. After establishing the numerical control model of the geometric errors of the cycloidal gear grinding machine, the theoretical tooth surface of the cycloidal gears can be compared with the error tooth surface through the actual tooth surface point coordinates. Then, the actual geometric errors of the machine tool are measured by the error measuring tool, and the error is compensated. Finally, experimental results will show that this method has application value in improving the accuracy of machining the cycloidal gears.The outline of the remainder of this study is as follows. In Section2, a topological structure analysis of machine tool multibody system based on the gear form grinding is developed. Geometric error modeling of cycloidal gear grinding machine is the main subject of Section 3, Section 4, and Section 5. The cycloidal gears processed and numerical examples are illustrated in Section 6. Finally, some conclusions are drawn in Section 7.


## 2. Topological Structure Analysis of Machine Tool Multi-Body System

The main research is based on the cycloidal gear form grinding machine. This method of forming a tooth surface can ensure that when grinding two or three cycloidal gears used in the same RV reducer, the parameters such as the tooth shape of the processed cycloidal gears are the same. Compared with the method of grinding gears by the generating grinding method, the form gear grinding method has greater advantages in terms of grinding accuracy, grinding efficiency, cost, and machine tool structure. The correction grinding wheel of the cycloidal gear form grinding machine uses a diamond wheel so that the cycloidal gear within 500 mm can be ground. Gear form grinding is arranged based on the universal CNC gear form grinding machine, which has four digital servo closed-loop controlled axes: three rectilinear motions (axisX, Y, and Z) and one rotational motion (axis C) defined in Figure 1. SP1 and SP2 are spindles of the grinding wheel and diamond wheel, respectively.Figure 1
Schematic diagram of the structure and motion axis of the cycloidal gear grinding machine.Based on the 3D model of the cycloidal gear grinding machine, the kinematic analysis of the gear grinding system of the research object is carried out according to the theory of multibody system, and the relations between the adjacent low-sequence bodies of the moving parts are determined. The topology diagram of the multibody system and the coordinate system of the machine tool are shown in Figure2. The machine tool bed is taken as the matrix B1. The gear grinding system can be divided into two motion chains, for example, the workpiece motion chain extending from the machine tool bed to the worktable and the tool motion chain extending from the fuselage to the spindle, wherein the workpiece motion chain contains the X-axis of the moving body, B3 as C-axis of the moving body, and B4 as the moving parts of the workpiece. On the other hand, the tool motion chain includes B5 as the Z-axis of the moving body and B6 as the moving body of the spindle for the cutting tool.Figure 2
Topology of the multibody system of the machine tool.In the multibody system of gear grinding, it is also necessary to establish a fixed coordinate system for each moving body. The fixed coordinate system is based on the fixed body of the Earth asB0. The machine tool body coordinate system is taken as the reference coordinate system, and the subcoordinate system of each part is established for other moving bodies according to their own serial numbers in the topological structure. Through the multibody system theory, the kinematic equations of each axis of the machine tool can be established by using the relations among the coordinate systems.


## 3. Geometric Error Modeling of Cycloidal Gear Grinding Machine

In the process of cycloidal gear grinding, the machining accuracy of cycloidal gears will be directly affected by the motion axis of the machine tool. According to the above analysis of cycloidal wheel gear grinding machine in a total of three axes, respectively, there are two straight axisX and Z and a rotation axis C of the workpiece, so the geometric errors of machine tools include the level of each axis straightness error, vertical straightness error, linear positioning error, deflection angle error, back dip angle error, and the cross error. Then, there is the vertical error between the X-axis and the Z-axis in the Y direction and the vertical error between the X-axis and the Z-axis. Therefore, in the cycloidal gear grinding machine, there are three axes in the six degrees of freedom of 18 geometric errors and three-axis perpendicularity errors, a total of 21 geometric errors.


### 3.1. Establishing Coordinate System

The space position relation between the coordinate systemScxc,yc,zc of the C-axis and the workpiece coordinate system Sgxg,yg,zg is determined by the transformation matrix MgC. Since the workpiece is fixed on the C-axis turntable and rotates around its axial direction, the coordinate system of the workpiece moving body and the C-axis moving body can be regarded as the whole one body. There is no relative movement between the two bodies, so the position and displacement transformation matrix between the two can be obtained by the unit matrix:(1)MgC=1000010000100001.When all the moving bodies are in theoretical motion, there is no motion error between them. Based on the movement of theX-axis, the rotation angle of the C-axis along the Z direction of the gear form grinding machine is expressed by γ. The theoretical coordinate transformation of rotating the moving body C-axis to the moving body X-axis is determined by the transformation matrix MXC, so it can be expressed as(2)MXC=cosγ−sinγ00sinγcosγ0000100001.From above, we found that the second coordinate transformation matrixMCX from the X-axis to the C-axis is obtained and is numerically represented as MCX=MXC−1. The theoretical coordinate transformation from machine tool lathe bed coordinate system Sfxf,yf,zf to X-axis coordinate system Sxxx,yx,zx is determined by the transformation matrix MXF. In the ideal state, the X-axis of the moving body moves in the X direction along the machine tool guide rail relative to the fuselage coordinate system. When the moving distance is x, the homogeneous coordinate transformation between the X-axis coordinate system and the fuselage coordinate system MFX can be expressed as(3)MFX=100x010000100001.Similarly, the homogeneous coordinate transformation matrixMXF from the fuselage to the X-axis coordinate system is obtained and is numerically represented as MXF=MFX−1. The theoretical coordinate transformation from Z-axis coordinate system to the fuselage coordinate system Sf is determined by the transformation matrix MFZ. In the theoretical state, the Z-axis of the moving body moves along the Z direction of the machine tool guide rail relative to the fuselage coordinate system. When the moving distance is z, the homogeneous coordinate transformation between the Z-axis coordinate system and the fuselage coordinate system is expressed as a matrix MFZ. That is, it can be expressed as(4)MFZ=10000100001z0001.The theoretical coordinate transformation from the grinding wheel coordinate systemSwxw,yw,zw to the Z-axis coordinate system is determined by the transformation matrix MZW. Since the grinding wheel is fixedly connected to the skateboard of the Z-axis, it can be regarded as the Z-axis coordinate system of the moving body that coincides with the coordinate system of the grinding wheel of the moving body. There is no relative motion between the two. Therefore, MZW is also the identity matrix:(5)MZW=1000010000100001.So far, the theoretical homogeneous coordinate system transformation matrix between each coordinate system of cycloid gear grinding machine gear grinding system has been obtained, and the second coordinate transformation matrix between the coordinate system of moving body grinding wheel and the coordinate system of moving body workpiece in the ideal state can be obtained by multiplying them asMgW:(6)MgW=MgC⋅MCX⋅MXF⋅MFZ⋅MZW.


### 3.2. Homogeneous Coordinate Transformation with Geometric Errors

The coordinate transformation fromX-axis coordinate system Sx to C-axis coordinate system Sc with geometric error is expressed as MCXe. In the actual machining process, the actual coordinate transformation from C-axis to X-axis coordinate system is equivalent to the superposition of an error matrix of the C-axis relative to the X-axis coordinate system. On the homogeneous coordinate transformation matrix from the coordinate system Sc to the coordinate system Sx under the state of X-axis movement RXC,(7)MXCe=MXC×RXC.RXC is the error matrix generated when the moving body moves from the C-axis to the X-axis coordinate system, including the angular displacement error around the Z-axis εzc, straightness error along X-axis, Y-axis, and Z-axis (δxc, δyc, and δzc), and rotation angle error along Y-axis and X-axis (εyc and εxc). So we get the transformation matrix as follows:(8)RXC=1−εzcεycδxcεzc1−εxcδyc−εycεxc1δzc0001.The actual coordinate transformation fromC-axis to X-axis is obtained by inserting the error matrix into the formula:(9)MXCe=MXC×RXC,=cosγ−sinγ00sinγcosγ0000100001×1−εzcεycδxcεzc1−εxcδyc−εycεxc1δzc0001,=cosγ−εzcsinγ−εzccosγ−sinγεyccosγ+εxcsinγδxccosγ−εycsinγsinγ+εzccosγ−εzcsinγ+cosγεycsinγ−εxccosγδxcsinγ+εyccosγ−εycεxc1δzc0001.Similarly, coordinate transformation fromX-axis coordinate system Sx to C-axis coordinate system Sc with geometric error is described as MCXe, which is the inverse matrix MXCe, so it can be numerically represented as follows:(10)MCXe=MXCe−1.The actual coordinate system transformation from fuselage coordinate systemSf to X-axis coordinate system Sx with geometric error is expressed as MXFe. In the actual processing machine operation, RFX is expressed in terms of the geometric error term 3×3 order rotation matrix, so(11)RFX=1−εzxεyxεzx1−εxx−εyx−εxx1.PFX in terms of the geometric error is described by the term 3×1 order displacement vector, so(12)PFX=x+δx⋅sx.Since theX-axis of the moving body moves x distance along the X-axis direction of its coordinate system, then the above vector is expressed as x=x,0,0T. The motion error of the X-axis of the moving body is described in terms of δx=δxx,δyx,δzxT in the X, Y, and Z directions of the machine tool coordinate system, respectively. The perpendicularity error of the X-axis of a moving body is expressed as sx=0,syx,szxT in the X, Y, and Z directions.In the actual state, the coordinate transformation of the geometric error term coordinate systemSx to the fuselage coordinate system Sf can be represented as(13)MFXe=RFXPFX01×311×1.The ideal coordinate transformation including the geometric error termX-axis to the fuselage coordinate system Sf can be obtained by substituting the matrix MFXe. This includes the linear displacement error δxx along the X-axis, the straightness error δyx and δzx, the yaw angle error εyx and εzx in the X-Y plane and the X-Z plane, and the rotation error εxx on the X-axis, so(14)MFXe=1−εzxεyxx+δxxεzx1−εxxδyx−εyxεxx1δzx0001.Similarly, coordinate transformation from the machine tool coordinate system to theX-axis coordinate system with geometric error is expressed as MXFe, that is to say, it is the inverse of MFXe. Therefore, it can be represented by an inverse matrix as follows:(15)MXFe=MFXe−1.The coordinate transformation from the coordinate systemSz to the fuselage coordinate system Sf with geometric error is expressed as MFZe. Similarly, when the moving body moves a distance z along the Z direction with the grinding wheel, the homogeneous coordinate transformation matrix from the Z-axis coordinate system to the machine tool coordinate system can be obtained. It includes the linear displacement error δzz along with the z-axis term, straightness error δyz and δzx in the Y-Z plane and X-Z plane, yaw angle error εyz and εxz in the Y-Z plane and X-Z plane, and Z-axis transverse rotation error εzz and can be expressed as(16)MFZe=1−εzzεyzδxzεzz1−εxzδyz−εyzεxz1z+δzz0001.Since there is no relative movement among the workpiece, the turntable, the grinding wheel, and theZ-axis, it can be regarded as the workpiece coordinate system Sg coinciding with the C-axis coordinate system Sc, the grinding wheel coordinate system Sw, and the coordinate system Sz. Therefore, there is no relative position error and rotation angle error between matrix MgCe and MZWe. Then, it can also be viewed as the identity matrix, which is(17)MgCe=MZWe=1000010000100001.So far, the coordinate transformation matrix from the moving body grinding wheel coordinate systemSw containing geometric errors to the moving body workpiece coordinate system Sg can be obtained by multiplying them MgWe, so it is represented as(18)MgWe=MgFe⋅MFWe=MgCe⋅MCXe⋅MXFe⋅MFZe⋅MZWe.


### 3.3. Geometric Error Model of Gear Grinding Machine

The coordinate transformation matrix from the grinding wheel coordinate system to the workpiece coordinate systemSg, which contains the geometric error, is expressed as MgWe. It can also be viewed as the homogeneous coordinate transformation matrix between these two coordinates in the ideal state MgW. An actual matrix containing its rotation angle error and relative position error is superimposed on the grinding wheel coordinate system Sw to the workpiece coordinate system Sg, which is expressed as EgW. So the geometric error can be obtained by the equation as follows:(19)MgWe=EgW⋅MgW.As mentioned above, when small quantities of second order and above are ignored, the hypothesis principle of minimum error can be used as the basis andEgW is represented as(20)EgW=1−ηzηypxηz1−ηxpy−ηyηx1pz0001,where px, py, and pz are errors of relative displacement form the grinding wheel coordinate system Sw to the workpiece coordinate system Sg along the X, Y, and Z directions, respectively; ηx, ηy, and ηz are errors of rotation angle between the grinding wheel coordinate system Sw and the workpiece coordinate system Sg along the X direction, Y direction, and Z direction, respectively.Taking into account the difficulty of calculating errors of relative position and rotation angle, we can use MATLAB software to calculate each matrix between the grinding wheel coordinate system and the workpiece coordinate system inX, Y, and Z directions. The gear form grinding machines involve two processes, namely, gear grinding and wheel dressing, so they must be equipped with four axes to satisfy the demands of these two processes, respectively. So the geometric error model of the cycloid CNC gear grinding machine can be obtained as follows:(21)δx=δxx+δzx+δxc⋅cosγ−δyx+δyz+δyc⋅sinγ+x⋅εzx+εzz⋅sinγ−z⋅εyz⋅cosγ+εxz⋅sinγ,δy=δxx+δxz+δxc⋅sinγ+δyx+δyz+δyc⋅cosγ+x⋅εzx+εzz⋅cosγ+z⋅εxy+εxz⋅cosγ−εyz⋅sinγ,δz=δzx+δzz−x⋅−εyx−εyz,εx=−εyx+εyc+εyz⋅sinγ+εxx+εxz+εxc⋅cosγ,εy=εyx+εyz+εyc⋅cosγ+εxx+εxz+εxc⋅sinγ,εz=εzx+εzz+εzc.


## 3.1. Establishing Coordinate System

The space position relation between the coordinate systemScxc,yc,zc of the C-axis and the workpiece coordinate system Sgxg,yg,zg is determined by the transformation matrix MgC. Since the workpiece is fixed on the C-axis turntable and rotates around its axial direction, the coordinate system of the workpiece moving body and the C-axis moving body can be regarded as the whole one body. There is no relative movement between the two bodies, so the position and displacement transformation matrix between the two can be obtained by the unit matrix:(1)MgC=1000010000100001.When all the moving bodies are in theoretical motion, there is no motion error between them. Based on the movement of theX-axis, the rotation angle of the C-axis along the Z direction of the gear form grinding machine is expressed by γ. The theoretical coordinate transformation of rotating the moving body C-axis to the moving body X-axis is determined by the transformation matrix MXC, so it can be expressed as(2)MXC=cosγ−sinγ00sinγcosγ0000100001.From above, we found that the second coordinate transformation matrixMCX from the X-axis to the C-axis is obtained and is numerically represented as MCX=MXC−1. The theoretical coordinate transformation from machine tool lathe bed coordinate system Sfxf,yf,zf to X-axis coordinate system Sxxx,yx,zx is determined by the transformation matrix MXF. In the ideal state, the X-axis of the moving body moves in the X direction along the machine tool guide rail relative to the fuselage coordinate system. When the moving distance is x, the homogeneous coordinate transformation between the X-axis coordinate system and the fuselage coordinate system MFX can be expressed as(3)MFX=100x010000100001.Similarly, the homogeneous coordinate transformation matrixMXF from the fuselage to the X-axis coordinate system is obtained and is numerically represented as MXF=MFX−1. The theoretical coordinate transformation from Z-axis coordinate system to the fuselage coordinate system Sf is determined by the transformation matrix MFZ. In the theoretical state, the Z-axis of the moving body moves along the Z direction of the machine tool guide rail relative to the fuselage coordinate system. When the moving distance is z, the homogeneous coordinate transformation between the Z-axis coordinate system and the fuselage coordinate system is expressed as a matrix MFZ. That is, it can be expressed as(4)MFZ=10000100001z0001.The theoretical coordinate transformation from the grinding wheel coordinate systemSwxw,yw,zw to the Z-axis coordinate system is determined by the transformation matrix MZW. Since the grinding wheel is fixedly connected to the skateboard of the Z-axis, it can be regarded as the Z-axis coordinate system of the moving body that coincides with the coordinate system of the grinding wheel of the moving body. There is no relative motion between the two. Therefore, MZW is also the identity matrix:(5)MZW=1000010000100001.So far, the theoretical homogeneous coordinate system transformation matrix between each coordinate system of cycloid gear grinding machine gear grinding system has been obtained, and the second coordinate transformation matrix between the coordinate system of moving body grinding wheel and the coordinate system of moving body workpiece in the ideal state can be obtained by multiplying them asMgW:(6)MgW=MgC⋅MCX⋅MXF⋅MFZ⋅MZW.


## 3.2. Homogeneous Coordinate Transformation with Geometric Errors

The coordinate transformation fromX-axis coordinate system Sx to C-axis coordinate system Sc with geometric error is expressed as MCXe. In the actual machining process, the actual coordinate transformation from C-axis to X-axis coordinate system is equivalent to the superposition of an error matrix of the C-axis relative to the X-axis coordinate system. On the homogeneous coordinate transformation matrix from the coordinate system Sc to the coordinate system Sx under the state of X-axis movement RXC,(7)MXCe=MXC×RXC.RXC is the error matrix generated when the moving body moves from the C-axis to the X-axis coordinate system, including the angular displacement error around the Z-axis εzc, straightness error along X-axis, Y-axis, and Z-axis (δxc, δyc, and δzc), and rotation angle error along Y-axis and X-axis (εyc and εxc). So we get the transformation matrix as follows:(8)RXC=1−εzcεycδxcεzc1−εxcδyc−εycεxc1δzc0001.The actual coordinate transformation fromC-axis to X-axis is obtained by inserting the error matrix into the formula:(9)MXCe=MXC×RXC,=cosγ−sinγ00sinγcosγ0000100001×1−εzcεycδxcεzc1−εxcδyc−εycεxc1δzc0001,=cosγ−εzcsinγ−εzccosγ−sinγεyccosγ+εxcsinγδxccosγ−εycsinγsinγ+εzccosγ−εzcsinγ+cosγεycsinγ−εxccosγδxcsinγ+εyccosγ−εycεxc1δzc0001.Similarly, coordinate transformation fromX-axis coordinate system Sx to C-axis coordinate system Sc with geometric error is described as MCXe, which is the inverse matrix MXCe, so it can be numerically represented as follows:(10)MCXe=MXCe−1.The actual coordinate system transformation from fuselage coordinate systemSf to X-axis coordinate system Sx with geometric error is expressed as MXFe. In the actual processing machine operation, RFX is expressed in terms of the geometric error term 3×3 order rotation matrix, so(11)RFX=1−εzxεyxεzx1−εxx−εyx−εxx1.PFX in terms of the geometric error is described by the term 3×1 order displacement vector, so(12)PFX=x+δx⋅sx.Since theX-axis of the moving body moves x distance along the X-axis direction of its coordinate system, then the above vector is expressed as x=x,0,0T. The motion error of the X-axis of the moving body is described in terms of δx=δxx,δyx,δzxT in the X, Y, and Z directions of the machine tool coordinate system, respectively. The perpendicularity error of the X-axis of a moving body is expressed as sx=0,syx,szxT in the X, Y, and Z directions.In the actual state, the coordinate transformation of the geometric error term coordinate systemSx to the fuselage coordinate system Sf can be represented as(13)MFXe=RFXPFX01×311×1.The ideal coordinate transformation including the geometric error termX-axis to the fuselage coordinate system Sf can be obtained by substituting the matrix MFXe. This includes the linear displacement error δxx along the X-axis, the straightness error δyx and δzx, the yaw angle error εyx and εzx in the X-Y plane and the X-Z plane, and the rotation error εxx on the X-axis, so(14)MFXe=1−εzxεyxx+δxxεzx1−εxxδyx−εyxεxx1δzx0001.Similarly, coordinate transformation from the machine tool coordinate system to theX-axis coordinate system with geometric error is expressed as MXFe, that is to say, it is the inverse of MFXe. Therefore, it can be represented by an inverse matrix as follows:(15)MXFe=MFXe−1.The coordinate transformation from the coordinate systemSz to the fuselage coordinate system Sf with geometric error is expressed as MFZe. Similarly, when the moving body moves a distance z along the Z direction with the grinding wheel, the homogeneous coordinate transformation matrix from the Z-axis coordinate system to the machine tool coordinate system can be obtained. It includes the linear displacement error δzz along with the z-axis term, straightness error δyz and δzx in the Y-Z plane and X-Z plane, yaw angle error εyz and εxz in the Y-Z plane and X-Z plane, and Z-axis transverse rotation error εzz and can be expressed as(16)MFZe=1−εzzεyzδxzεzz1−εxzδyz−εyzεxz1z+δzz0001.Since there is no relative movement among the workpiece, the turntable, the grinding wheel, and theZ-axis, it can be regarded as the workpiece coordinate system Sg coinciding with the C-axis coordinate system Sc, the grinding wheel coordinate system Sw, and the coordinate system Sz. Therefore, there is no relative position error and rotation angle error between matrix MgCe and MZWe. Then, it can also be viewed as the identity matrix, which is(17)MgCe=MZWe=1000010000100001.So far, the coordinate transformation matrix from the moving body grinding wheel coordinate systemSw containing geometric errors to the moving body workpiece coordinate system Sg can be obtained by multiplying them MgWe, so it is represented as(18)MgWe=MgFe⋅MFWe=MgCe⋅MCXe⋅MXFe⋅MFZe⋅MZWe.


## 3.3. Geometric Error Model of Gear Grinding Machine

The coordinate transformation matrix from the grinding wheel coordinate system to the workpiece coordinate systemSg, which contains the geometric error, is expressed as MgWe. It can also be viewed as the homogeneous coordinate transformation matrix between these two coordinates in the ideal state MgW. An actual matrix containing its rotation angle error and relative position error is superimposed on the grinding wheel coordinate system Sw to the workpiece coordinate system Sg, which is expressed as EgW. So the geometric error can be obtained by the equation as follows:(19)MgWe=EgW⋅MgW.As mentioned above, when small quantities of second order and above are ignored, the hypothesis principle of minimum error can be used as the basis andEgW is represented as(20)EgW=1−ηzηypxηz1−ηxpy−ηyηx1pz0001,where px, py, and pz are errors of relative displacement form the grinding wheel coordinate system Sw to the workpiece coordinate system Sg along the X, Y, and Z directions, respectively; ηx, ηy, and ηz are errors of rotation angle between the grinding wheel coordinate system Sw and the workpiece coordinate system Sg along the X direction, Y direction, and Z direction, respectively.Taking into account the difficulty of calculating errors of relative position and rotation angle, we can use MATLAB software to calculate each matrix between the grinding wheel coordinate system and the workpiece coordinate system inX, Y, and Z directions. The gear form grinding machines involve two processes, namely, gear grinding and wheel dressing, so they must be equipped with four axes to satisfy the demands of these two processes, respectively. So the geometric error model of the cycloid CNC gear grinding machine can be obtained as follows:(21)δx=δxx+δzx+δxc⋅cosγ−δyx+δyz+δyc⋅sinγ+x⋅εzx+εzz⋅sinγ−z⋅εyz⋅cosγ+εxz⋅sinγ,δy=δxx+δxz+δxc⋅sinγ+δyx+δyz+δyc⋅cosγ+x⋅εzx+εzz⋅cosγ+z⋅εxy+εxz⋅cosγ−εyz⋅sinγ,δz=δzx+δzz−x⋅−εyx−εyz,εx=−εyx+εyc+εyz⋅sinγ+εxx+εxz+εxc⋅cosγ,εy=εyx+εyz+εyc⋅cosγ+εxx+εxz+εxc⋅sinγ,εz=εzx+εzz+εzc.


## 4. Simulation of Geometric Error

### 4.1. Analysis of the Influence of Geometric Error

To reflect the influence of the geometric errors of each axis of the cycloid gear grinding machine, it is also necessary to find the relationship between the geometric errors of the machine tool and the tooth surface errors after the completion of the geometric error model. First, according to the parameters of a cycloidal gear in Table1, the grid spatial coordinate data of cycloidal gear tooth surface are calculated, and the calculated coordinate data are imported into MATLAB for tooth surface simulation.Table 1
Cycloidal gear parameters.
ParameterSymbolNumericalNumber of teethZa39Pin gear teethZb40EccentricityA1.5 mmNeedle wheel radiusRz82 mmShort of coefficientK10.731707Radius of needle gear sleeverz3.5 mmTooth widthB13 mmUsing the control variable method under the condition that the geometric errors of each axis of the machine tool enhance one other and cancel the other ones, it is assumed that only the displacement error of each axis is 0.005 mm or the rotation error 0.01°. Finally, the errors of other directions are zero, and the tooth surface of the error simulation is shown in Figure3.Figure 3
Six degrees of freedom error tooth surface point simulation. (a) Simulation of the tooth surface with displacement error in theX direction δx. (b) Simulation of the tooth surface with X direction rotation error εx. (c) Simulation of the tooth surface with displacement error in Y direction δy. (d) Simulation of the tooth surface with Y direction rotation error εy. (e) Simulation of the tooth surface with displacement error in Z direction δz. (f) Simulation of the tooth surface with Z direction rotation error εz.
(a)(b)(c)(d)(e)(f)Above all, we can intuitively see the influence of the error components of the geometric errors of the machine tool on the tooth surface in the six degrees of freedom between the grinding wheel and the workpiece coordinate system. Through the simulation of the tooth surface, we can compensate for the geometric errors of each axis of the machine tool by measuring the parameters of the processed cycloidal gear.


### 4.2. Relationship between Geometric Errors and Tooth Surface Errors

According to the YK7350 B cycloidal gear grinding machine discussed in this study, the coordinate analysis of the spatial position relationship between the grinding wheel and workpiece is shown in Figure4.Figure 4
Spatial position relationship between grinding wheel and workpiece.Coordinate systemsSFxF,yF,zF and Sgxg,yg,zg, whose relative positions are described by the machine tool coordinate systems, are rigidly connected to the grinding wheel and work gear, respectively. When the coordinate system of the two moving bodies is without error, the actual processed cycloidal gear coincides with the theoretical cycloidal gear at the same time. Therefore, the gear cross-section zg0 is coincident with ZFi. The actual tooth surface equation is the profile equation of the grinding wheel in the coordinate system of the machine tool, which can be expressed as(22)xg0=RzsinθZa−K1ZbsinZbθZa+rzK1sinZbθ/Za−sinθ/Za1+K12−2K1cosθ,yg0=RzcosθZa−K1ZbcosZbθZa−rz−K1cosZbθ/Za+cosθ/Za1+K12−2K1cosθ,zg0=zFi,where Rz and rz are the radius of the needle wheel and the needle gear sleeve, respectively； Za and Zb are the number of work gear teeth and needle wheel teeth, respectively; K1 is the short amplitude coefficient; and θ is the rotation angle of the workpiece.By rotating the coordinate of the cycloid equation of the first tooth surface, the cycloid equation of the numberi tooth surface can be determined by(23)xgi=xg0cos2πi/Za−yg0sin2πi/Za,ygi=xg0sin2πi/Za−yg0cos2πi/Za,zgi=zg0.In the processing of cycloidal gears form grinding machine, the movement of each axis determines the trajectory of the grinding wheel. If the dressing error of the grinding wheel is also ignored, then the coordinate vector of the dressed axial wheel profile can be expressed as(24)rqη=xqη,yqη,0,1T,where η is the axial parameter of the grinding wheel surface.The rotary surface of the grinding wheel is the track surface swept by the axial profile of the grinding wheel when it rotates around its axis. In the coordinate system of grinding wheel, the rotation parameter of grinding wheel is expressed byφ, and then, the coordinate vector of grinding wheel can be expressed as(25)rwη,φ=Mwqφrqη.Among them,(26)Mwqφ=cosφsinφ0001000−sinφcosφ00001.Therefore, the unit normal vector of the wheel surface can be expressed as(27)nwη,φ=∂rwη,φ/∂η×∂rwη,φ/∂φ∂rwη,φ/∂η×∂rwη,φ/∂φ.In the ideal case, the equation of grinding wheel surface is jointly expressed by the unit normal vectornwη,φ and coordinate vector rwη,φ. According to the homogeneous coordinate transformation matrix between the cycloidal gear coordinate system and the grinding wheel coordinate system in the ideal state, the grinding wheel surface in the workpiece coordinate system can be expressed as(28)rgη,φ,γ,x=MgW⋅rwη,φ,ngη,φ,γ,x=MgW⋅nwη,φ.In the actual machining process, the grinding wheel determines the position relationship between its coordinate systemSw and the workpiece coordinate system Sg. The geometric error of each axis of the machine tool also determines the position of the grinding wheel, and all the influence of the geometric error of each axis of the machine tool is used G=x1,x2,…,xmT; then, the actual grinding wheel surface equation can be expressed as(29)regη,φ,γ,x,G=MgWe⋅rwη,φ,negη,φ,γ,x,G=MgWe⋅nwη,φ.When the parameters of the grinding wheel are known, the grinding contact point is defined as a radial vectorrg from the origin of the workpiece coordinate system to the surface of the grinding wheel. Therefore, the contact conditions in the ideal state and the actual condition are, respectively, as follows:(30)fη,φ,γ,x=kg×rg+kg⋅ng,feη,φ,γ,x=kg×rg+kg⋅neg.Here, the machining parameterγ is represented by constant γk, and contact conditions between the grinding wheel and workpiece will lead to f=0. While these conditions are true, there remains the only parameter concerning the grinding wheel rotation profile φ and grinding wheel axial profile parameters η in the equation. Here, the range of η is known, that is, let η=ηj=1,2,3,…,n. According to the contact conditions between the grinding wheel and the workpiece, we can figure out the corresponding angle φj. So the coordinate vector of the contact point is obtained, and the unit normal vector is calculated out, too. Because it is through the profile parameters and contact conditions f=0 to find out φ, φ can be represented as η. Therefore, the contact line between the grinding wheel and the workpiece at the end of the k can be expressed as:(31)rkgη=rkjgη,φj,nkgη=nkjgη,φj.In the process of gear processing, the contact line between the grinding wheel and the workpiece together is the tooth surface processed by the machine tool, and the theoretical tooth surface and actual tooth surface composed of discrete contact lines can be, respectively, expressed as(32)rgη=rkgη,ngη=nkgη,regη=rekgη,negη=nekgη.In summary, the relationship between geometric errors of each axis and tooth surface errors of the CNC gear grinder can be established, and its model can be expressed as follows:(33)δFg=δgx,δgy,δgz,0T=rekjg−rkjg,εFg=εgx,εgy,εgz,0T=nekjg−nkjg,where k=1,2,…,λ and j=1,2,…,n.


### 4.3. Simulation Analysis of Error Sensitivity

Morris method is used to calculate the mean of the basic effects for each input parameterμi and the standard deviation σi as a sensitivity index.(34)μi=1SN∑j=1SNEEi,σi=1SN∑j=1SNEEij−μi2.The mean value and standard deviation of the basic effect of each error term of geometric errors can be obtained, and the obtained mean value and standard deviation are normalized as(35)Mμi=1m∑i=1mμi,Mσi=1m∑i=1mσi.Finding the normalized meanMμi and the standard deviation Mσi can be used as a sensitivity index to identify the sensitivity of geometric errors. Here, the normalized mean value Mμi the larger the error, the greater the influence of the error on the tooth surface error. And the standard deviation Mσi the larger the error, the greater the coupling effect between the error and other error terms. The sensitivity index of machine tool geometric error on tooth surface error is shown in Figure 5.Figure 5
Graph of sensitivity index of tooth surface error component. (a) Sensitivity index diagram of tooth surface error componentδgx. (b) Sensitivity index diagram of tooth surface error component εgx. (c) Sensitivity index diagram of tooth surface error component δgy. (d) Sensitivity index diagram of tooth surface error component εgy. (e)Sensitivity index diagram of tooth surface error component δgz. (f)Sensitivity index diagram of tooth surface error component εgz.
(a)(b)(c)(d)(e)(f)


## 4.1. Analysis of the Influence of Geometric Error

To reflect the influence of the geometric errors of each axis of the cycloid gear grinding machine, it is also necessary to find the relationship between the geometric errors of the machine tool and the tooth surface errors after the completion of the geometric error model. First, according to the parameters of a cycloidal gear in Table1, the grid spatial coordinate data of cycloidal gear tooth surface are calculated, and the calculated coordinate data are imported into MATLAB for tooth surface simulation.Table 1
Cycloidal gear parameters.
ParameterSymbolNumericalNumber of teethZa39Pin gear teethZb40EccentricityA1.5 mmNeedle wheel radiusRz82 mmShort of coefficientK10.731707Radius of needle gear sleeverz3.5 mmTooth widthB13 mmUsing the control variable method under the condition that the geometric errors of each axis of the machine tool enhance one other and cancel the other ones, it is assumed that only the displacement error of each axis is 0.005 mm or the rotation error 0.01°. Finally, the errors of other directions are zero, and the tooth surface of the error simulation is shown in Figure3.Figure 3
Six degrees of freedom error tooth surface point simulation. (a) Simulation of the tooth surface with displacement error in theX direction δx. (b) Simulation of the tooth surface with X direction rotation error εx. (c) Simulation of the tooth surface with displacement error in Y direction δy. (d) Simulation of the tooth surface with Y direction rotation error εy. (e) Simulation of the tooth surface with displacement error in Z direction δz. (f) Simulation of the tooth surface with Z direction rotation error εz.
(a)(b)(c)(d)(e)(f)Above all, we can intuitively see the influence of the error components of the geometric errors of the machine tool on the tooth surface in the six degrees of freedom between the grinding wheel and the workpiece coordinate system. Through the simulation of the tooth surface, we can compensate for the geometric errors of each axis of the machine tool by measuring the parameters of the processed cycloidal gear.


## 4.2. Relationship between Geometric Errors and Tooth Surface Errors

According to the YK7350 B cycloidal gear grinding machine discussed in this study, the coordinate analysis of the spatial position relationship between the grinding wheel and workpiece is shown in Figure4.Figure 4
Spatial position relationship between grinding wheel and workpiece.Coordinate systemsSFxF,yF,zF and Sgxg,yg,zg, whose relative positions are described by the machine tool coordinate systems, are rigidly connected to the grinding wheel and work gear, respectively. When the coordinate system of the two moving bodies is without error, the actual processed cycloidal gear coincides with the theoretical cycloidal gear at the same time. Therefore, the gear cross-section zg0 is coincident with ZFi. The actual tooth surface equation is the profile equation of the grinding wheel in the coordinate system of the machine tool, which can be expressed as(22)xg0=RzsinθZa−K1ZbsinZbθZa+rzK1sinZbθ/Za−sinθ/Za1+K12−2K1cosθ,yg0=RzcosθZa−K1ZbcosZbθZa−rz−K1cosZbθ/Za+cosθ/Za1+K12−2K1cosθ,zg0=zFi,where Rz and rz are the radius of the needle wheel and the needle gear sleeve, respectively； Za and Zb are the number of work gear teeth and needle wheel teeth, respectively; K1 is the short amplitude coefficient; and θ is the rotation angle of the workpiece.By rotating the coordinate of the cycloid equation of the first tooth surface, the cycloid equation of the numberi tooth surface can be determined by(23)xgi=xg0cos2πi/Za−yg0sin2πi/Za,ygi=xg0sin2πi/Za−yg0cos2πi/Za,zgi=zg0.In the processing of cycloidal gears form grinding machine, the movement of each axis determines the trajectory of the grinding wheel. If the dressing error of the grinding wheel is also ignored, then the coordinate vector of the dressed axial wheel profile can be expressed as(24)rqη=xqη,yqη,0,1T,where η is the axial parameter of the grinding wheel surface.The rotary surface of the grinding wheel is the track surface swept by the axial profile of the grinding wheel when it rotates around its axis. In the coordinate system of grinding wheel, the rotation parameter of grinding wheel is expressed byφ, and then, the coordinate vector of grinding wheel can be expressed as(25)rwη,φ=Mwqφrqη.Among them,(26)Mwqφ=cosφsinφ0001000−sinφcosφ00001.Therefore, the unit normal vector of the wheel surface can be expressed as(27)nwη,φ=∂rwη,φ/∂η×∂rwη,φ/∂φ∂rwη,φ/∂η×∂rwη,φ/∂φ.In the ideal case, the equation of grinding wheel surface is jointly expressed by the unit normal vectornwη,φ and coordinate vector rwη,φ. According to the homogeneous coordinate transformation matrix between the cycloidal gear coordinate system and the grinding wheel coordinate system in the ideal state, the grinding wheel surface in the workpiece coordinate system can be expressed as(28)rgη,φ,γ,x=MgW⋅rwη,φ,ngη,φ,γ,x=MgW⋅nwη,φ.In the actual machining process, the grinding wheel determines the position relationship between its coordinate systemSw and the workpiece coordinate system Sg. The geometric error of each axis of the machine tool also determines the position of the grinding wheel, and all the influence of the geometric error of each axis of the machine tool is used G=x1,x2,…,xmT; then, the actual grinding wheel surface equation can be expressed as(29)regη,φ,γ,x,G=MgWe⋅rwη,φ,negη,φ,γ,x,G=MgWe⋅nwη,φ.When the parameters of the grinding wheel are known, the grinding contact point is defined as a radial vectorrg from the origin of the workpiece coordinate system to the surface of the grinding wheel. Therefore, the contact conditions in the ideal state and the actual condition are, respectively, as follows:(30)fη,φ,γ,x=kg×rg+kg⋅ng,feη,φ,γ,x=kg×rg+kg⋅neg.Here, the machining parameterγ is represented by constant γk, and contact conditions between the grinding wheel and workpiece will lead to f=0. While these conditions are true, there remains the only parameter concerning the grinding wheel rotation profile φ and grinding wheel axial profile parameters η in the equation. Here, the range of η is known, that is, let η=ηj=1,2,3,…,n. According to the contact conditions between the grinding wheel and the workpiece, we can figure out the corresponding angle φj. So the coordinate vector of the contact point is obtained, and the unit normal vector is calculated out, too. Because it is through the profile parameters and contact conditions f=0 to find out φ, φ can be represented as η. Therefore, the contact line between the grinding wheel and the workpiece at the end of the k can be expressed as:(31)rkgη=rkjgη,φj,nkgη=nkjgη,φj.In the process of gear processing, the contact line between the grinding wheel and the workpiece together is the tooth surface processed by the machine tool, and the theoretical tooth surface and actual tooth surface composed of discrete contact lines can be, respectively, expressed as(32)rgη=rkgη,ngη=nkgη,regη=rekgη,negη=nekgη.In summary, the relationship between geometric errors of each axis and tooth surface errors of the CNC gear grinder can be established, and its model can be expressed as follows:(33)δFg=δgx,δgy,δgz,0T=rekjg−rkjg,εFg=εgx,εgy,εgz,0T=nekjg−nkjg,where k=1,2,…,λ and j=1,2,…,n.


## 4.3. Simulation Analysis of Error Sensitivity

Morris method is used to calculate the mean of the basic effects for each input parameterμi and the standard deviation σi as a sensitivity index.(34)μi=1SN∑j=1SNEEi,σi=1SN∑j=1SNEEij−μi2.The mean value and standard deviation of the basic effect of each error term of geometric errors can be obtained, and the obtained mean value and standard deviation are normalized as(35)Mμi=1m∑i=1mμi,Mσi=1m∑i=1mσi.Finding the normalized meanMμi and the standard deviation Mσi can be used as a sensitivity index to identify the sensitivity of geometric errors. Here, the normalized mean value Mμi the larger the error, the greater the influence of the error on the tooth surface error. And the standard deviation Mσi the larger the error, the greater the coupling effect between the error and other error terms. The sensitivity index of machine tool geometric error on tooth surface error is shown in Figure 5.Figure 5
Graph of sensitivity index of tooth surface error component. (a) Sensitivity index diagram of tooth surface error componentδgx. (b) Sensitivity index diagram of tooth surface error component εgx. (c) Sensitivity index diagram of tooth surface error component δgy. (d) Sensitivity index diagram of tooth surface error component εgy. (e)Sensitivity index diagram of tooth surface error component δgz. (f)Sensitivity index diagram of tooth surface error component εgz.
(a)(b)(c)(d)(e)(f)


## 5. Geometric Error Measurement and Compensation

A laser interferometer is a length and angle measurement instrument. Using the laser wavelength as the standard, two beams of coherent light in the process of transmission will form interference and get the light and dark stripes on the laser. The wavelength of the laser is relatively stable, and the distance between each fringe is one wavelength of the laser. Through these stripes, the laser head can obtain the moving distance of the measured object.According to the formula,(36)N=∫0tΔfdt.Due to theΔf=2v/cf and the v=dl/dt, f=c/c, so(37)N=∫0tΔfdt=2λ∫0tdl=2lλ.The measured distancel is obtained and numerically represented as(38)l=λ2N,where N is the number of the cumulative pulse; λ is the laser wavelength； c is the speed of light； and v is the moving speed of the moving mirror.In actual machine tool error measurement work, a Renishaw dual-frequency laser interferometer is used. In addition to the principle of light interference, it also uses the frequency change generated by the Doppler effect to detect the position of each axis of the machine tool. After the laser head is electrified, a beam of light is divided into two beams of coherent light through the spectroscope. The polarization plane is perpendicular to the incident plane. A beam of light is reflected through the spectroscope and radiated to the fixed mirror to form F1, which is the reference light. Another beam of light in the incident plane is transmitted through the spectroscope to the moving mirror to form F2, which is the measurement light. The actual measurement is shown in Figure6.Figure 6
Actual measurement of straight axis geometric error.


### 5.1. Measuring Principle and Measurement of Rotating Axis

The principle of measuring the error of the rotation axis of the CNC machine tool is the same as that of measuring the error of the linear axis. It only needs to change that the linear length change of the moving mirror is changed to the angle change of the laser angle mirror by using the trigonometric function. The Renishaw XR20-W nonlinear rotary shaft calibration device is used in the actual measurement. The high-precision angle mirror on the device also has a high-precision small motor. The XR20-W is mounted at the center of the machine tool rotary shaft turntable, and the angle mirror small turntable can be precisely rotated through the internal high-precision grating system and drive.Similarly, using the frequency difference generated by light interference and the Doppler effect, the displacement differenceΔL of the mirror can be obtained. According to the optical path principle of measurement, the relative optical path change of measuring arm 2 and measuring arm 1 is ΔL=s×sinθ. If the nominal spacing s between two mirrors in an angular mirror is yes, then the measuring system software can calculate the angle error by arcsine as follows:(39)θ=arcsinΔLs,where s is the parameter of the nominal spacing between two mirrors.


### 5.2. Measurement Geometric Error of Each Axis

To investigate the response of each axis to the geometric error that is independently changed and the corresponding geometric error that is obtained, various error data of the machine tool are obtained through the actual measurement, which provide a data basis for the subsequent geometric error compensation of the CNC machine tool. Figure7 shows the geometric error of actual measurement by a laser interferometer.Figure 7
Positioning errors of each axis by the laser interferometer. (a) The position error of theX-axis. (b) The position error of the Y-axis. (c) The position error of the Z-axis. (d) The position error of the C-axis.
(a)(b)(c)(d)


### 5.3. Compensation Result of Geometric Error

From the measurement analysis curve of theC-axis after compensation, we can see that the angle error and repeated angle error of the machine tool are significantly reduced (as shown in Figure 8), and the performance of the machine tool is significantly improved. By measuring the geometric error data of each axis of the gear grinding machine and analyzing the compensation values of the error compensation points of each motion axis, a machine tool error compensation table is established. Using software compensation, the error compensation value of each compensation point is written into the program through Siemens 840D system to compensate for the error of each axis of the machine tool. After the error compensation, the machine tool is run again and the error value of each motion axis of the machine tool is measured again. Geometric errors are significantly reduced.Figure 8
Positioning errors after each axis of the machine tool compensated. (a) Positioning error is compensated byX-axis. (b) Positioning error is compensated by Y-axis. (c) Positioning error is compensated by Z-axis. (d) Positioning error is compensated by C-axis.
(a)(b)(c)(d)


## 5.1. Measuring Principle and Measurement of Rotating Axis

The principle of measuring the error of the rotation axis of the CNC machine tool is the same as that of measuring the error of the linear axis. It only needs to change that the linear length change of the moving mirror is changed to the angle change of the laser angle mirror by using the trigonometric function. The Renishaw XR20-W nonlinear rotary shaft calibration device is used in the actual measurement. The high-precision angle mirror on the device also has a high-precision small motor. The XR20-W is mounted at the center of the machine tool rotary shaft turntable, and the angle mirror small turntable can be precisely rotated through the internal high-precision grating system and drive.Similarly, using the frequency difference generated by light interference and the Doppler effect, the displacement differenceΔL of the mirror can be obtained. According to the optical path principle of measurement, the relative optical path change of measuring arm 2 and measuring arm 1 is ΔL=s×sinθ. If the nominal spacing s between two mirrors in an angular mirror is yes, then the measuring system software can calculate the angle error by arcsine as follows:(39)θ=arcsinΔLs,where s is the parameter of the nominal spacing between two mirrors.


## 5.2. Measurement Geometric Error of Each Axis

To investigate the response of each axis to the geometric error that is independently changed and the corresponding geometric error that is obtained, various error data of the machine tool are obtained through the actual measurement, which provide a data basis for the subsequent geometric error compensation of the CNC machine tool. Figure7 shows the geometric error of actual measurement by a laser interferometer.Figure 7
Positioning errors of each axis by the laser interferometer. (a) The position error of theX-axis. (b) The position error of the Y-axis. (c) The position error of the Z-axis. (d) The position error of the C-axis.
(a)(b)(c)(d)


## 5.3. Compensation Result of Geometric Error

From the measurement analysis curve of theC-axis after compensation, we can see that the angle error and repeated angle error of the machine tool are significantly reduced (as shown in Figure 8), and the performance of the machine tool is significantly improved. By measuring the geometric error data of each axis of the gear grinding machine and analyzing the compensation values of the error compensation points of each motion axis, a machine tool error compensation table is established. Using software compensation, the error compensation value of each compensation point is written into the program through Siemens 840D system to compensate for the error of each axis of the machine tool. After the error compensation, the machine tool is run again and the error value of each motion axis of the machine tool is measured again. Geometric errors are significantly reduced.Figure 8
Positioning errors after each axis of the machine tool compensated. (a) Positioning error is compensated byX-axis. (b) Positioning error is compensated by Y-axis. (c) Positioning error is compensated by Z-axis. (d) Positioning error is compensated by C-axis.
(a)(b)(c)(d)


## 6. Experimental

### 6.1. Cycloidal Gear Grinding

The cycloidal gears processed by the gear grinder are experimentally ground, and the cycloidal gear processing by the CNC gear grinder before and after the geometric error compensation is carried out. Grinding is performed on the gear form grinding according to the parameters of the cycloidal gear (in Table2). The actual grinding process is shown in Figure 9.Table 2
Error comparison before and after compensation.
ItemsError before compensation (μm)Error after compensation (μm)Accuracy of ascensionLeft tooth surfaceFP55.718.18–5Right tooth surfaceFP44.1μm23.2μm7–5Left tooth surfacefP13.44.86–4Right tooth surfacefP17.48.27–5Tooth profile deviationFa566388–5Figure 9
Actual processing of cycloidal gear.


### 6.2. Detection and Measurement

The installation and measurement processes of each part in the actual measurement of cycloidal gear are shown in Figure10.Figure 10
Actual measurement of cycloidal gear.The measuring equipment uses the Gleason 650GMS gear detection analyzer, which can measure gears up to 650 mm in diameter, and the measuring speed is increased by 25%. The gear detection analyzer is also equipped with a Renishaw SP80H 3D scanning probe, which can carry out combined measurement of measuring needles of various sizes and extended lengths, with high speed and high-precision measurement capability. The measurement results are shown in Figures11 and 12.Figure 11
Machine tool compensates cycloidal gear deviation before machining.Figure 12
Machined cycloidal gear deviation after machine tool compensation.The test results are shown in Table2, indicating that the machining accuracy of the cycloid gear grinding machine is significantly improved.


## 6.1. Cycloidal Gear Grinding

The cycloidal gears processed by the gear grinder are experimentally ground, and the cycloidal gear processing by the CNC gear grinder before and after the geometric error compensation is carried out. Grinding is performed on the gear form grinding according to the parameters of the cycloidal gear (in Table2). The actual grinding process is shown in Figure 9.Table 2
Error comparison before and after compensation.
ItemsError before compensation (μm)Error after compensation (μm)Accuracy of ascensionLeft tooth surfaceFP55.718.18–5Right tooth surfaceFP44.1μm23.2μm7–5Left tooth surfacefP13.44.86–4Right tooth surfacefP17.48.27–5Tooth profile deviationFa566388–5Figure 9
Actual processing of cycloidal gear.


## 6.2. Detection and Measurement

The installation and measurement processes of each part in the actual measurement of cycloidal gear are shown in Figure10.Figure 10
Actual measurement of cycloidal gear.The measuring equipment uses the Gleason 650GMS gear detection analyzer, which can measure gears up to 650 mm in diameter, and the measuring speed is increased by 25%. The gear detection analyzer is also equipped with a Renishaw SP80H 3D scanning probe, which can carry out combined measurement of measuring needles of various sizes and extended lengths, with high speed and high-precision measurement capability. The measurement results are shown in Figures11 and 12.Figure 11
Machine tool compensates cycloidal gear deviation before machining.Figure 12
Machined cycloidal gear deviation after machine tool compensation.The test results are shown in Table2, indicating that the machining accuracy of the cycloid gear grinding machine is significantly improved.


## 7. Conclusions

In this study, the geometric error model of the CNC gear grinding machine is established by a multibody motion system. The theoretical tooth surface and error tooth surface are simulated, and the compensation of the axial geometric error method is used to research the sensitivity of the influence of geometric error of each axis on the tooth surface. Then, the geometric error of each axis of the machine tool is measured and compensated. The accuracy of the machine tool is verified through the gear processing experiment, which provided a reliable guarantee for the manufacture of high-precision cycloid gear. As the numerical examples show, compensation of the axial geometric error method can improve tooth quality. An additional advantage of the approach is to provide the user with valuable information about the grinding machine and machining capacity.



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*Source: 4804498-2022-04-21.xml*