Influences of the workpiece pose errors on gear skiving accuracy

This paper investigates the influence of the workpiece pose errors on gear skiving accuracy. First, based on the theory of homogeneous coordinate transformation, the mapping relationship between the tooth flank of the workpiece and the skiving tool is established according to the principle of gear skiving. Second, the workpiece pose errors are defined, and the relationship between each workpiece pose error and cutting depth is analyzed. According to the kinematic chain of the machine tool, a method for measuring the workpiece pose errors based on the actual inverse kinematic model is proposed. Third, the tooth deviations caused by individual workpiece pose errors and the coupling workpiece pose errors are studied. Then, the sensitive errors are obtained by numerical simulation. Finally, the numerical analysis is verified by gear skiving experiments, and the proposed method can be used to identify the possible workpiece pose errors. Furthermore, the particle swarm optimization algorithm is introduced to guide the workpiece setting pose errors in the gear skiving process.


Introduction
Gear skiving is a modern and productive method for machining internal and external cylindrical gears. In recent years, gear skiving has attracted the attention of researchers and manufacturers. It has better efficiency and accuracy than the traditional gear hobbing and gear shaping of small module gears and can machine the internal gear without a relief groove. Moreover, gear skiving technology aims to be used in the gear finishing process to reduce the gear distortion errors caused by heat treatment. Therefore, the potential advantages of gear skiving drive some well-known machine tool manufacturers to produce high-end gear skiving machines [1][2][3][4].
Many scholars have made progress on the design and manufacture technologies of the skiving tool and cutting mechanism. A variety of new high-performance skiving tools have been developed, such as super-cylindrical skiving tool [5][6][7], assembled skiving tool [8][9][10], and super-skiving tool designed by MHI [1]. Chen et al. [11] researched the basic principle of the skiving technology and proposed a design method of error-free spur skiving tool for internal helical gear. Then, Guo et al. [12,13] established the mathematical models for designing the spur and helical skiving tool and researched the cutting mechanism of gear skiving to optimize the tool design. In addition, the skiving strategies have also been optimized to improve the skiving tool life, machining accuracy, and efficiency. Marx and Vogel [14] proposed a semicompleting skiving machining strategy. Kreschel and Kobialka [15] proposed a multiple-feed skiving technology. Guo et al. [16] combined the characteristics of skiving strategies in [14,15] and proposed a multiple-step feed machining technology to effectively reduce the cutting force and improve the tool life and the cutting performance. Uriu et al. [17] analyzed the relationship between the shaft angle and the cutting speed to determine the tool setting parameters, which can effectively improve the accuracy of the tooth profile of the tool.
Although skiving technologies have developed rapidly in recent years, the research on the influence of geometric errors on the accuracy of gear skiving is sparse. Most of the existing researches focused on tool eccentricity error. Guo et al. [18] analyzed the influences of tool setting on gear skiving accuracy on profile accuracy, the spiral deviation, and tooth surface quality. Ren et al. [19] studied the influence of different cutter eccentricities on tooth surface roughness under different feed rates by numerical analysis. Yang et al. [20] established the tooth flank equation with errors and the mapping relationship between the geometric pose errors and profile normal errors. The sensitivity of profile normal errors to pose errors is analyzed. However, there is no detailed study on the influence of workpiece pose errors caused by setting, cutting force, and heat, on the gear skiving accuracy. In general, gear skiving is usually used for machining gears with a small module (less than 5 mm). The smaller the gear module, the more significant the influence of the workpiece pose errors on the tooth deviations. Therefore, it is necessary to study the relationship between workpiece pose errors and gear skiving accuracy.
In this paper, the influences of the workpiece pose errors on gear skiving accuracy are studied, including profile deviations, helix deviations, and pitch deviations. Firstly, the mapping relationship between the tooth flank of the workpiece and the skiving tool is established. Secondly, the pose errors of the workpiece are identified, and the regulations of the cutting depth under different pose errors in the gear skiving process are analyzed. Then, a six-step method for measuring the workpiece pose errors based on actual inverse kinematics is proposed. Thirdly, the influences of individual workpiece pose error and the coupling workpiece pose errors on skiving accuracy are studied. Finally, a series of experiments are performed to verify the numerical analysis, and the particle swarm optimization (PSO) algorithm is used to predict the maximum workpiece setting pose errors in the gear skiving process.

Principle of gear skiving
During the gear skiving processing, the tool and workpiece are placed so that their rotating axes form a crossed-axes angle. They rotate synchronously to generate a sliding motion at the contact point, thereby performing cutting [1].

Motion model of gear skiving
The kinematical principle of gear skiving is shown in Fig. 1. There is a shaft angle Σ between the tool axis and the workpiece axis. The tool and workpiece rotate around their axes with angular velocities ω 1 and ω 2 , respectively. At the same time, the workpiece has an axial feed f along the workpiece axis. Therefore, the angular velocity ω 2 can be obtained as follows: Fig. 2 The definition of the coordinate system where the z t and z g are the tooth number of the tool and workpiece, respectively, and p is the helix parameter which can be expressed as where p z is the helix lead on the reference cylinder of the workpiece, p z = d g tan g .

Coordinate system definition
is the fixed coordinate system of the tool that rotates around the Z t -axis.
is the fixed coordinate system of the workpiece that rotates around the Z g -axis.
are rigidly attached to the tool and workpiece, respectively. The initial positions of S 1 and S 2 have coincided with S t and S g , respectively. S g′ (O g′ -X g′ Y g′ Z g′ ) is the coordinate system with pose errors attached to the workpiece.
According to the homogeneous coordinate transformation, the transformation matrices from S t to S g can be obtained: where M g2 is the transfer matrix from S 2 to S g , M 21 is the transfer matrix from S 1 to S 2 , and M 1t is the transfer matrix from S t to S 1 . The details of the transfer matrix are shown as follows: where 1 and 2 are the rotation angles of the tool and workpiece around the Z t -axis and Z g -axis, respectively; 1 = 1 t ; 2 = 2 t ; and t is the cutting time. The shaft angle Σ =|β 1 ± β 2 |, h = ft is the width of the workpiece, and a is the center distance. Let

Identification and analysis of workpiece pose errors
The workpiece pose errors are generated during the workpiece setting process, including 3 position deviation errors and 3 angle deviation errors, as shown in Table 1. According to the research by Ding et al. [21], the Motion-first model and Error-first model can be used to analyze the geometric error of the rotary axis. The Error-first model is preferred for the modeling and definition of geometric errors of the rotary axis. Therefore, the Error-first model will be used to define and identify the workpiece pose errors in the gear skiving system.

Identification of workpiece pose errors
As shown in Fig. 3, the coordinate system of the workpiece without pose errors S g is the theoretical coordinate system, and the coordinate system of the workpiece with pose error S g′ is the actual coordinate system. It is assumed that all the pose errors are small, and each error is relatively   Table 1 shows the definition of workpiece setting pose errors. For simplicity, abbreviate cosine and sine as "C" and "S," so the transfer matrix from S g′ to S g can be expressed as Assuming the angle deviation errors xy , xz , zy are small, cos( ) ≈ 1 and sin( ) ≈ , where = xy , xz , zy . By ignoring high-order infinitesimals, the transfer matrix of Eq. (4) can be represented by It can be known that the position deviation error gz and the angle deviation error xy only affect the initial position of the workpiece because the direction of the Z-axis keeps unchanged during the rotation of the workpiece [21]. Therefore, the main workpiece pose errors that affect the skiving accuracy are gx , gy , xy , xz , and Eq. (4) can be represented by

Regulations between cutting depth and workpiece pose errors
According to the principle of gear skiving, skiving can be regarded as the relative movement of the tool and the workpiece. The cutting edge of the tool can be regarded as a curve on the rake face. The tool continuously moves to envelop the tooth flank. Therefore, the target tooth flank is an involute spiral surface, which can be expressed as where r b is the base circle radius of the workpiece, u is the involute parameter, and is the rotation angle of the (4) transverse tooth profile of the target tooth flank around the Z g -axis.
The coordinates of the point on the cutting edge of the tool are considered a function of the independent variable u , so the expression of the point on the cutting edge in S t can be obtained as follows: When the workpiece pose errors are zero, combining Eqs. When the workpiece pose errors are not zero, substituting Eq. (6) into Eq. (9), the swept target tooth flank in S g ′ , can be expressed as: where = gx , gy and = xz , zy .
Therefore, the trajectory of the skiving process can be obtained. The cutting trajectory of gear skiving in a single tooth flank is shown in Fig. 4(a). It can be observed that the actual cutting trajectory deviates from the theoretical cutting trajectory due to the workpiece pose errors. The actual flank becomes twisted and reduces the skiving accuracy, as shown in Fig. 4(b). Furthermore, the cutting depth variation Δd, the difference between the actual cutting depth and the theoretical one, can be expressed as: Using Eq. (11), when all the position deviation errors are set as 0.05 mm and all the angle deviation errors are set as 0.05°, the regulations of the cutting depth Δd change with the workpiece width h and the tooth number of workpiece z g is studied.
As shown in Fig. 5, the influence of the position deviation errors on the cutting depth is a near-fixed value when z g = 1.
Significantly, when it is affected by the position deviation error in the x direction, the error value is almost completely mapped on the cutting depth. However, the angle deviation errors present a slope of tan in affecting the cutting depth, which increases the cutting depth with the increase of the workpiece width. The regulation of Δd changing with the workpiece width under the coupling pose errors is shown in Fig. 6. Due to the influence of the four pose errors, the cutting depth increases with the increase of the workpiece width. The change of Δd under the coupling errors is approximately equal to the linear superposition of the change of each pose error in Fig. 5. In addition, it can be seen from Figs. 5 and 6 that the curve is not a smooth line but a wavy line. Our analysis is that because the gear skiving is a continuous chip-breaking cutting, the gear flank will have tiny ripples. Moreover, the results of the numerical simulation are visible to match the gear skiving principle well.
To study the regulations of the cutting depth Δd change with the tooth number of the workpiece z g , the position of the workpiece width h = 15 mm is selected in this paper. As shown in Fig. 7(a), the influences of the position and angle deviation errors on the cutting depth change periodically with the change of tooth number in a cosine or sine curve. In Fig. 7(b), the angle deviation errors have a much smaller impact on Δd than the position deviation

Measurement of workpiece pose errors
To measure the workpiece pose errors quickly, a method for measuring the defined workpiece pose errors gx , gy , zy , xz based on the actual inverse kinematic model (AIKM) is proposed in this paper. The schematic of the gear skiving machine tool used in this paper is shown in Fig. 8(a). To make the measurement more convenient, the workpiece origin coordinate O g is set as 0 0 0 1 T , which is also the reference origin of the machine tool. The proposed method uses the tool as the measurement reference in this paper. It is noted that the proposed method employs an on-machine measurement system, which is different from the online measurement system. Therefore, according to Figs. 2 and 8, the transformation matrices from S t to S g can be expressed as According to the kinematic chain of the machine tool shown in Fig. 8(b), substituting Eq. (6) into Eq. (12), the transformation matrices from S t to S g can be obtained as follows: where T represents the translation matrix, R represents the rotation matrix. x, y, and z represent the position coordinate value in the X-axis, Y-axis, and Z-axis, respectively. h is the workpiece width, and a, b, and c represent the angle coordinate value around the A-axis, B-axis, and C-axis, respectively; L is the swing length from the O t to the origin of the A-axis coordinate system. Due to an individual angle deviation error, at least two position information is required to be measured. The proposed measurement exploits the serial orthogonal configuration of the machine tool and involves the six measurement steps shown in Fig. 9(a)-(f). For the first measurement process, the machine tool is positioned at a = b = c = 0 • , and x = 0 as the initial position, as shown in Fig. 9(a). In this case, the position of the tool plane is t = 0 0 0 1 T , which is defined with respect to the tool . Then, the tool plane is in contact with the workpiece plane through moving the Y-axis and Z-axis, and the position of the tool plane is g = P gx P gy 0 1 T , which is defined with respect to the workpiece coordinate system (S g (O g -X g Y g Z g )). It is noted that the coordinate shown in the numerical control (NC) system at this time is x 1 y 1 z 1 . Using Eq. (13), the equation g = gt t can be derived as where the symbol "*" indicates that the item is a parameter that does not need to be considered. P gx and P gy are the position of the tool in the X and Y directions of the workpiece coordinate system (S g (O g -X g Y g Z g )), respectively, which do not affect the calculation result. Therefore, in order to avoid too many symbols in this paper, all the steps use P gx and P gy to represent the position information of the tool in S g that does not affect the calculation results. And the P gz in the subsequent steps is used to represent the position of the tool in the Z direction.
According to Eq. (14), the following equation is obtained: For the second measurement process, the initial position is the same as the first process. However, the tool plane t = 0 0 0 1 T is in contact with the workpiece plane g = P gx −P gy 0 1 T , as shown in Fig. 9(b). In this case, the coordinate shown in the NC system is x 2 y 2 z 2 . Using Eq. (13), the equation g = gt t can be derived as According to Eq. (16), the following equation is obtained: For the third measurement process, the initial position is the same as the first process. However, the tool plane t = 0 −r t 0 1 T touches the side of the workpiece g = P gx r g P gz 1 T , as shown in Fig. 9(c), where r t and r g are the radius of the tool and workpiece, respectively. In this case, the coordinate shown in the NC system is x 3 y 3 z 3 . Using Eq. (13), the equation g = gt t can be derived as According to Eq. (18), the following equation is obtained: For the fourth measurement process, the initial position is set as a = b = 0 • , c = 90 • , and x = 0 . Furthermore, the tool plane t = 0 0 0 1 T is in contact with the workpiece plane g = P gx P gy 0 1 T , as shown in Fig. 9(d). In this case, the coordinate shown in the NC system is x 4 y 4 z 4 . Using Eq. (13), the equation g = gt t can be derived as According to Eq. (20), the following equation is obtained: For the fifth measurement process, the initial position is the same as the fourth process. However, the tool plane t = 0 0 0 1 T is in contact with the workpiece plane g = −P gx P gy 0 1 T , as shown in Fig. 9(e). In this case, the coordinate shown in the NC system is x 5 y 5 z 5 . Using Eq. (13), the equation g = gt t can be derived as According to Eq. (22), the following equation is obtained: Fig. 9 The measuring method in six steps: (a) the first measurement process, (b) the second measurement process, (c) the third measurement process, (d) the fourth measurement process, (e) the fifth measurement process, and (f) the sixth measurement process For the sixth measurement process, the initial position is the same as the fourth process. However, the tool plane t = 0 −r t 0 1 T touches the side of the workpiece g = r g P gy P gz 1 T , as shown in Fig. 9(f). In this case, the coordinate shown in the NC system is x 6 y 6 z 6 . Using Eq. (13), the equation g = gt t can be derived as According to Eq. (24), the following equation is obtained: Solving Eqs. (15), (17), (19), (21), (23), and (25), the main workpiece pose errors can be obtained as follows: Since xz and zy are small, xz zy ≈ 0 . Equation (29) can be represented as: According to Eqs. (26), (27), (28), and (30), and the measuring position where x = 0, it can be found that the angle deviation errors can be considered constants, and the position deviation errors are linearly related to the angle deviation errors and the Z-axis coordinate. In this (29) gx = y 6 − r t − r g − (z 6 − h) zy + (x 6 − gy ) xz zy (30) gx = y 6 − r t − r g − (z 6 − h) zy paper, measurement tests are performed on the gear skiving machine tool at several positions. Table 2

Relationship between workpiece pose errors and gear skiving accuracy
To further study the relationship between workpiece pose errors and gear skiving accuracy, the position deviation error along the X g -axis is set as gx = 0.05 mm , respectively, and the position deviation error along the Y g -axis is set as gy = 0.05 mm . And the angle deviation error around the X g -axis is set as zy = 0.05 • , and the angle deviation error around the Y g -axis is set as xz = 0.05 • , respectively.

Influence of individual pose error
In the process of gear skiving, the workpiece pose errors will cause the actual tooth flank to deviate from the theoretical tooth flank, affecting the accuracy of the skived gear. To analyze the coordinate points on any tooth flank of the skived gear, it is necessary to rotate the ith enveloped tooth flank to the initial tooth flank (i = 1) in the workpiece coordinate system to evaluate the tooth deviation. The tooth deviation evaluation equation of any tooth flank can be obtained as follows: where Then, the tooth deviations including profile form deviation f f , profile slope deviation f H , helix form deviation f f , helix slope deviation f H , single flank deviation f pt , and cumulative pitch deviation F p can be derived by where r ′ g is the actual tooth flank and r g is the theoretical tooth flank.  Fig. 10. The normal distance |PB| from any point P on the actual tooth flank to the theoretical tooth flank represents the tooth deviation. Therefore, the normal evaluation equation of any point on the actual tooth flank can be obtained  Table 3. The positive deviation is taken as a positive value and the negative deviation as a negative value.
As shown in Figs. 11 and 12, the influences of individual workpiece position deviation error on the tooth deviation are analyzed. In the gear accuracy evaluation, the maximum error on the right and left flanks are selected as the evaluation value of tooth deviation. If the absolute value of the maximum tooth deviation is less than 5 μm, it will be considered an insensitive error with respect to pose error. Otherwise, it is considered a sensitive error. It can be found that the tooth deviation evaluated from the maximum error of each tooth flank presents regular cosine or sine curves. With the change of the phase angle of the position deviation error,    Fig. 12 Influence of individual workpiece position deviation error ( gy = 0.05 mm ) on tooth deviation the number of teeth corresponding to the maximum tooth deviation is also changing. In Figs. 11 and 12, the maximum tooth deviation of f f , f f , f H , and f pt is less than 5 μm, so it can be considered that these tooth deviations are insensitive to workpiece position deviation errors. However, the maximum tooth deviation of f H and F p is more than 5 μm, so they are sensitive to workpiece position deviation errors.
As shown in Figs. 13 and 14, the influences of individual workpiece angle deviation error on the tooth deviation are analyzed. The tooth deviation curves also show regular cosine or sine curves, and the curves also change with the phase angle of angle deviation error. It can be found that f H , f f , f f , and f pt are insensitive to workpiece angle deviation errors. However, f H and F p are sensitive to workpiece angle deviation errors. Table 4 shows the maximum tooth deviation on the right flank. In summary, the profile slope deviation f H and cumulative pitch deviation F p are sensitive to workpiece position deviation errors. The helix slope deviation f H and cumulative pitch deviation F p are sensitive to workpiece angle deviation errors. The diversity of the tooth deviation on both flanks is determined by the phase angle of workpiece pose errors. The conclusions can be used to determine the existence of pose errors and the phase angle of the pose errors.

Relationship between individual pose error and coupling pose errors
In this section, the pose errors are applied together to study the coupling effect. In Fig. 15, the influence of the coupling pose errors on tooth deviation is analyzed. The tooth deviation curves show regular cosine or sine curves, and the analyzed results show that f H , f H , f pt , and F p are sensitive to coupling pose errors. The maximum deviation values are 22 μm, 36 μm, 9 μm, and 204 μm, respectively. From the above analysis, it can be concluded that the workpiece pose errors mainly affect the profile slope deviation f H , helix slope deviation f H , and cumulative pitch deviation F p . Therefore, this paper mainly focuses on studying the relationship between individual pose error and coupling pose errors on these three deviations.
For simplicity, f C (f H , f H , F p ) is used to represent the influence of the coupling pose errors on tooth deviation, f I (f H , f H , F p ) is used to represent the influence of individual pose error on tooth deviation, and ∑ f I (f H , f H , F p ) is used to represent the linear superposition of influence of individual error on the deviation. According to [24,25], the influences of the coupling errors on deviation are regarded as the linear superposition of influence of individual error on deviation under different errors when machining the bevel gear and grinding gear. Therefore, assuming that f C (f H , f H , F p ) is also a linear superposition of each f I (f H , f H , F p ) in the gear skiving system, and then calculating the difference between f C (f H , f H , F p ) and f I (f H , f H , F p ) , the following equations can be obtained: is very small; and the coefficient k i (i = 1, 2, 3) is basically equal to 1. Therefore, the influences of the coupling errors on deviation can be approximately regarded as the linear superposition of the influences of the individual error on deviation under different errors in the gear skiving process. This rule is also consistent with the regulations of the cutting depth Δd in "Regulations between cutting depth and workpiece pose errors". Moreover, this method can also determine whether position errors or angle deviation errors and whether each pose error has the same phase. It is beneficial to trace and compensate for the pose error of the gear skiving process.

Experiment and analysis
As shown in Fig. 18, three typical machining workpiece pose errors are selected to process on the gear skiving machine tool to verify the proposed numerical analysis method. In order to reduce the influence of other factors, it is necessary to control the process of the experiment. The geometric accuracy of the gear skiving machine tool is checked and compensated for to reduce the influence of irrelevant errors. The machine tool is idling for 2 h before gear skiving, reducing the influence of thermal error. Moreover, the machining parameters used in each group of experiments are set to be the same so that the main factor that affects the gear skiving accuracy in the machining process is the workpiece pose errors.
According to the setting value in Table 5, first of all, the workpiece pose errors can be adjusted manually by a dial gauge. Then, the proposed measurement method in "Measurement of workpiece pose errors" is used to measure the workpiece pose errors to further verify whether the error value in the experiments meets the requirements. The basic parameters of the gear skiving used in the experiment are consistent with Table 3. After skiving, the teeth of the workpiece are sorted sequentially; numbers 1, 23, and 47 are selected to measure in the gear measuring center, as shown in Fig. 18. According to the analysis result in "Identification and analysis of workpiece pose errors", the profile slope deviation f H , helix slope deviation f H , and cumulative pitch deviation F p are sensitive to the workpiece pose errors. Therefore, these deviations are analyzed and measured. Table 6 shows the comparison between numerical results and experiment results on the selected right tooth flanks. Since the on-site processing environment may affect the gear skiving, the actual measurement results are slightly larger than the numerical analysis results. However, the overall trend of the measurement accuracy is consistent with the numerical results. According to Table 6, in actual skiving, f H is sensitive to position deviation errors, f H is sensitive to angle deviation errors, and F p is sensitive to position deviation errors and angle deviation errors. The experimental results verify the numerical simulation results. Therefore, this method can be used to trace the position and angle deviation errors of the workpiece and guide the setting requirements of the workpiece in gear skiving.
Particle swarm optimization (PSO) is an optimization algorithm for simulating the foraging behavior of birds, which is widely used in engineering problems to find the optimal solution [26]. In this study, the PSO algorithm is used to further study the relationship between the workpiece pose errors and the gear skiving accuracy to assist the proposed method to find the maximum allowable workpiece pose errors in gear skiving under different conditions.
In the actual gear skiving process, most of the processing gear module affected by the existing technology is small, and the maximum diameter of the workpiece that can be skived is also limited. Therefore, in two cases, the optimal global prediction of the workpiece pose errors is based on the PSO algorithm: one of the modules is 2, the tooth number of the workpiece ranges is 60-120, and the other of the tooth number is 70, and the module range is 0.5-3.
After previous analysis, it shows that f H , f H , and F p are more sensitive to the workpiece pose errors. Therefore, we consider these three deviations as the target. To reduce the calculation amount, the position deviation errors and the angle deviation errors in one direction are considered, respectively, which is also more in line with the actual machining situation. Then, the individual position deviation error and angle deviation error are randomly set as the initial value of the PSO algorithm. Through the evaluation of calculation accuracy, the required workpiece pose errors are finally obtained. The detailed process of the prediction is shown in Fig. 19.
In this paper, the machining accuracy in ISO 5 [23] is taken as the optimization target, and the prediction results  Fig. 17 The corresponding coefficient between coupling errors and linear superposition of individual error  Fig. 20. In Fig. 20(a), if the number of the module does not change, the position deviation error increases with the increase of the tooth number. In contrast, the angle deviation error remains unchanged. If the number of gears does not change, the position deviation error increases step by step with the increase of module, while the angle deviation error remains unchanged; the detail is described in Fig. 20(b). Therefore, it can be known that the control of angle deviation error is relatively strict in the skiving process, which is conducive to ensuring that the helix slope deviation f H is within the allowable range. In contrast, the acceptable range of the position deviation error can be appropriately increased with an increase in the number of gears or modules. This is because as the number of gears or modules increases, the diameter of the gear also increases so that the position deviation error will have a relatively small impact on the accuracy of the gear skiving. The actual processing has verified this problem to obtain the same results. Therefore, the method mentioned in this paper can be used to guide and quickly trace the source of the workpiece pose errors in the actual gear skiving process.  Data availability The authors confirm that the data supporting the findings of this study are available within the article.

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Competing interests
The authors declare no competing interests.