In the field of gear cutting, power skiving has emerged as a highly efficient and precise method for manufacturing cylindrical gears, especially internal gears with small modules. I have focused my investigation on a persistent problem in gear cutting by power skiving: the generation of regular waviness along the gear lead direction, which is often called lead-direction ripple error or tooth deviation. This error cannot be easily eliminated by simply adjusting conventional cutting parameters, and it becomes a bottleneck for achieving high-precision gear cutting. In this work, I establish a mathematical model that incorporates tool eccentricity, axial feed, and tool tooth number. I then derive a simplified calculation method for the residual height difference of the lead ripple, analyze the influence of each factor, and propose a tool tooth number optimization procedure. Numerical calculations and gear cutting experiments show that my method can significantly reduce the lead-direction ripple error in power skiving.
Power skiving, also known as gear skiving, axial gear cutting, or rotational broaching, is a new gear cutting process. It offers high productivity, high accuracy, and dry cutting capability. It is becoming the preferred process for complex precision gears used in industrial robots, harmonic/RV reducers, and automotive reducers. Because power skiving is more efficient and accurate than gear hobbing and gear shaping for small and medium module gears, it has become a research hotspot in gear cutting technology. Many researchers have studied tool design theory and the cutting and forming mechanism. With deeper understanding, attention has shifted to the influence of process parameters on tooth surface accuracy. For example, the shaft angle, tool feed strategy, and eccentricity errors of the tool or workpiece can affect tooth surface quality. Among these, the relative eccentricity error between the tool and the gear has the greatest influence. It causes regular waviness on the machined gear tooth surface. Earlier studies confirmed this and showed that adjusting the tool feed rate can reduce the effect of eccentricity on tooth surface deviation. However, reducing the axial feed rate of the tool can reduce tooth surface error but also greatly reduces gear cutting efficiency and multiplies the processing time. Some studies found that selecting an appropriate tool tooth number can also effectively reduce the influence of eccentricity on tooth surface deviation, but this only works when the workpiece tooth number and tool tooth number have a suitable greatest common divisor. To improve the lead-direction ripple error problem that often occurs in gear cutting by power skiving, I analyze its causes, establish a calculation method for the lead residual height difference based on tool mounting eccentricity error, study the influence of tool eccentricity, tool axial feed, and tool tooth number on the lead-direction error, and propose a control method based on optimal selection of the tool tooth number.
I begin by describing the kinematic principle of power skiving for internal gears. Two fixed coordinate systems are used, and two moving coordinate systems are attached to the gear and the tool. The gear rotates about its axis with a uniform angular velocity, and the tool rotates about its own axis while moving along the gear axis. The angle between the gear axis and the tool axis is the tool mounting angle. The shortest distance between the two axes is the mounting center distance. During gear cutting, the gear and tool rotate continuously, and the tool also feeds axially. For helical gears, an additional rotation of the workpiece is required to maintain correct meshing. The relationship between the tool angular velocity and the workpiece angular velocity can be written as
$$\omega_1 = \frac{z_t}{z_g}\omega_2 – \frac{2v_0\tan\beta_w}{m_n z_g}$$
where zt is the tool tooth number, zg is the workpiece tooth number, v0 is the axial feed speed, βw is the workpiece helix angle, and mn is the normal module. The tool mounting shaft angle and the center distance are given by
$$\Sigma = \beta_w \pm \beta_t$$
$$a = r_{pw} \pm r_{pt}$$
where βt is the tool helix angle, rpw and rpt are the pitch radii of the workpiece and the tool, respectively. The plus sign is used for an external gear, and the minus sign is used for an internal gear. According to the geometry of crossed helical gears, the common normal at the instantaneous contact point of the two involute helicoid surfaces passes through the tangent point of the working pitch cylinders. If the pitch radius of the internal gear remains constant, the conjugate tool should have an instantaneous center that lies on the pitch cylinder. The locus of this instantaneous center is a spatial curve. For any point on this curve, the meshing condition requires that the normal vector at the gear tooth surface point be parallel to the vector from the curve point to the tooth surface point. This condition can be expressed as
$$\frac{x_Q – x_M}{n_{xM}} = \frac{y_Q – y_M}{n_{yM}} = \frac{z_Q – z_M}{n_{zM}}$$
From this relation, I obtain the meshing equations:
$$\begin{cases}
f_1(u,\theta,\phi_t,\phi_w) = (x_Q – x_M)n_{yM} – (y_Q – y_M)n_{xM} = 0 \\
f_2(u,\theta,\phi_t,\phi_w) = (y_Q – y_M)n_{xM} – (z_Q – z_M)n_{zM} = 0
\end{cases}$$
For a gear tooth surface described by parameters u and θ, the tooth profile parameter u is known, and the motion relation between the tool and the gear satisfies a fixed transmission ratio. Therefore, the meshing equations can be solved for a pair of parameters that satisfy the meshing condition. By transforming these parameters from the gear coordinate system to the tool coordinate system, I can solve for the conjugate surface of the tool. The transformation can be written as
$$r_t = M_{t-s2}M_{s1-s2}^{-1}M_{s1-w}r_w(u,\theta)$$
where the rotation and translation matrices describe the relative motion between the gear, the tool, and the fixed frames. Finally, I construct the rake face equation with a rake angle and intersect it with the conjugate surface to obtain the cutting edge profile. This kinematic chain is the foundation for my subsequent analysis of lead-direction ripple error in gear cutting by power skiving.
In gear cutting practice, regular ripple errors frequently appear along the gear lead direction. These errors are difficult to remove by simply changing cutting process parameters. My analysis and earlier investigations indicate that the main cause is the relative eccentricity between the tool and the workpiece. This eccentricity originates from machine tool transmission chain errors, fixture manufacturing and mounting errors, and workpiece locating datum errors. The relative eccentricity changes periodically as the tool rotates about its own axis. Different teeth of the tool cut the same tooth slot with different eccentric phases. As a result, a regular wavy error is generated on the surface of that tooth slot. To reduce this error, a common approach is to reduce the tool axial feed speed. A lower axial feed can reduce the residual height difference between two adjacent cuts, but it also reduces gear cutting efficiency and increases processing time significantly. Another approach is to try different tool tooth numbers based on experience and repeated trials. However, this increases tool manufacturing cost and lengthens the trial period. Therefore, I aim to establish a quantitative model that can predict the lead ripple and guide the optimal selection of the tool tooth number without excessive trial and error.
I now describe the simplified model of the swept trajectory of the cutting edge. During power skiving of an internal gear, the cutting edges of the tool sweep through space and form a family of envelope curves on the tooth surface. A precise calculation would require computing all envelope surfaces and performing complex Boolean operations to extract the geometric boundary. Although such a method can accurately obtain the tooth surface topography, it is computationally intensive and difficult to apply in practical engineering. Therefore, I adopt a simplified method in which the local swept trajectory of the tool inside the tooth slot is approximated as an ellipse. This simplified model cannot extract the exact lead-direction error topography, but it can calculate the residual height difference with good agreement with actual values. It provides a fast and practical way to analyze lead-direction errors in gear cutting. The simplified elliptical trajectory is described by
$$\begin{cases}
x = a_1 \sin t \\
y = b_1 \cos t
\end{cases}$$
where a1 is the semi-major axis, b1 is the semi-minor axis, and t is the angular parameter. I set a1 equal to the tool radius ra, and b1 equal to a1 sin Σ. When there is no eccentricity error, the theoretical residual height difference δ can be calculated from the geometric relation as
$$\delta = r_a\left[1 – \cos\left(\arcsin\frac{f}{2a_1\sin\Sigma}\right)\right]$$
where f is the axial feed per workpiece revolution. This equation shows that the residual height difference increases with the axial feed. When eccentricity error e exists, the eccentric displacement along the x direction changes with the tool rotation. The eccentric amount corresponding to different tool teeth can be expressed as
$$e_i = e \sin\left(\frac{2\pi}{z_t}i\right), \quad i \in [1, z_t]$$
Using the geometric relations of the simplified ellipse model, I can determine the intersection points of two adjacent swept trajectories. For the initial two trajectories, the intersection coordinates are
$$\begin{cases}
x_{k1} = a_1 \sin t_0 = a_1 \sin t_1^{(1)} + e_1 \\
y_{k1} = b_1 \cos t_0 = b_1 \cos t_1^{(1)} – f
\end{cases}$$
Similarly, for any two adjacent trajectories, the intersection coordinates can be written as
$$\begin{cases}
x_{ki} = a_1 \sin t_{i-1}^{(2)} + e_{i-1} = a_1 \sin t_i^{(1)} + e_i \\
y_{ki} = b_1 \cos t_{i-1}^{(2)} – (i-1)f = b_1 \cos t_i^{(1)} – if
\end{cases}$$
By solving these equations, I obtain the angular parameters of the intersection points and then calculate the residual height difference between adjacent swept trajectories. This residual height difference is the core quantity that characterizes the lead-direction ripple error. The following table summarizes the main symbols and their meanings used in my model.
| Symbol | Meaning |
|---|---|
| zt | Tool tooth number |
| zg | Workpiece gear tooth number |
| mn | Normal module |
| βw | Workpiece helix angle |
| βt | Tool helix angle |
| Σ | Tool mounting shaft angle |
| a | Mounting center distance |
| rpw | Workpiece pitch radius |
| rpt | Tool pitch radius |
| ra | Tool radius |
| f | Axial feed per workpiece revolution |
| e | Tool eccentricity error |
| ei | Eccentric amount for tooth i |
| δ | Residual height difference |
| T | Ripple period along lead direction |
| Nt | Number of tool revolutions |
| Ng | Number of gear revolutions |
The influence of the axial feed on the lead-direction error is significant. I calculated the residual height difference for different axial feeds while keeping the tool eccentricity at a fixed value. The results are summarized in the following table. As the axial feed increases, the residual height difference increases. The shape of the tooth profile error also changes with the axial feed. This indicates that reducing the axial feed speed can reduce the lead residual height difference, but it also greatly increases the gear cutting time. For example, a feed of 0.1 mm/r and a feed of 0.9 mm/r differ by a factor of eight in processing time for a single part. Therefore, reducing the axial feed is not an efficient solution for industrial gear cutting.
| Axial feed f (mm/r) | Residual height difference δ (μm) | Relative processing time factor |
|---|---|---|
| 0.1 | 0.48 | 1 |
| 0.5 | 12.47 | 5 |
| 0.9 | 18.66 | 9 |
The workpiece tooth number and the tool tooth number also strongly affect the lead-direction ripple error. In practical gear cutting, to maintain efficiency, the feed per tool revolution is usually not less than 0.3 mm/r. Moreover, after the tool is clamped, the tool eccentricity is rarely corrected again in mass production. Therefore, the only adjustable parameter is often the tool tooth number. I calculated the residual height difference for different tool tooth numbers when machining the same workpiece. The results show that the tool tooth number determines the magnitude, shape, and period of the lead-direction error. For a given workpiece tooth number, different tool tooth numbers produce very different ripple patterns. Some tool tooth numbers produce large and regular ripples, while others produce small and irregular ripples. Therefore, selecting a suitable tool tooth number is very important for reducing the lead-direction ripple error in gear cutting.
To describe the periodicity of the lead-direction ripple, I consider the condition under which a specific tooth of the tool re-enters the same tooth slot of the workpiece. If the initial tooth of the tool is located in a specific workpiece tooth slot, then after the tool rotates Nt revolutions and the gear rotates Ng revolutions, the same tooth of the tool will enter the same tooth slot again only if the following relation is satisfied:
$$N_t z_t = N_g z_g$$
From this relation, the number of gear revolutions is
$$N_g = \frac{N_t z_t}{z_g}$$
During gear cutting, the gear rotates one revolution and the tool feeds axially by f. Therefore, the ripple period along the lead direction can be calculated as
$$T = \operatorname{int}\left(\frac{N_t z_t}{z_g}\right) f$$
where the integer function takes the smallest natural integer Nt that makes the integer part of the ratio an integer. This ripple period is an important characteristic of the lead-direction waviness. The following table gives examples for different tool tooth numbers when the workpiece tooth number is fixed. The period and the residual height difference change significantly with the tool tooth number.
| Workpiece tooth number zg | Tool tooth number zt | Ripple period T (mm) | Residual height difference δ (μm) |
|---|---|---|---|
| 72 | 34 | 5.1 | 19.92 |
| 72 | 44 | 3.3 | 8.15 |
| 72 | 54 | 0.9 | 7.0 |
| 79 | 58 | — | 7.51 |
From these results, I observe that there is no simple monotonic relation between the tool tooth number and the residual height difference. Although some previous studies suggested selecting the tool tooth number based on the greatest common divisor with the workpiece tooth number, this principle is not always applicable. It fails when the workpiece tooth number is a prime number or when the greatest common divisor does not provide a small residual height difference. Therefore, I propose a systematic optimization method for the tool tooth number. The method is straightforward: input the workpiece tooth number, define a search range for the tool tooth number, traverse all candidate tool tooth numbers, calculate the residual height difference for each candidate, and select the tool tooth number that minimizes the residual height difference. This method can be applied to any workpiece tooth number, including prime numbers, and it does not rely on empirical rules.
The optimization procedure can be summarized as follows. First, I input the workpiece tooth number zg. Second, I define the allowable range of the tool tooth number as an interval from m to n. Third, I traverse all integer tool tooth numbers k in this interval. For each k, I calculate the residual height difference δ(k) using the simplified model. Fourth, I find the minimum value of δ(k) and identify the corresponding tool tooth number as the optimal choice. This procedure directly evaluates the lead-direction ripple error and selects the tool tooth number that minimizes it. The following table illustrates the optimization results for two different workpiece tooth numbers within a tool tooth number range from 20 to 60.
| Workpiece tooth number zg | Search range | Optimal tool tooth number zt | Minimum residual height difference δ (μm) |
|---|---|---|---|
| 72 | 20–60 | 48 | 0.51 |
| 79 | 20–60 | 58 | 7.51 |
To validate my model and the tool tooth number optimization method, I conducted a series of gear cutting experiments using power skiving. The experiments were performed on an internal helical gear. The tool material was a coated high-speed steel, and the workpiece material was 42CrMo. Three tools with different tooth numbers were used. The basic parameters of the gear and the tool are listed in the following table. These parameters define the geometry of the gear cutting process and are used in the numerical calculations.
| Parameter | Value |
|---|---|
| Workpiece tooth number zg | 117 |
| Normal module mn (mm) | 1.4468 |
| Normal pressure angle αn (°) | 20 |
| Workpiece helix angle βw (°) | 13.727 (left-hand) |
| Tip diameter da (mm) | 171.465 |
| Root diameter df (mm) | 178.36 |
| Span measurement (mm) | 169.856 |
| Pin diameter (mm) | 3 |
| Face width Bw (mm) | 50 |
| Tool normal module mn (mm) | 1.4468 |
| Tool normal pressure angle αn (°) | 20 |
| Tool helix angle βt (°) | −6.273 (left-hand) |
| Tool rake angle γo (°) | 5 |
| Tool top relief angle αo (°) | 8 |
The gear cutting parameters for the three tools are summarized in the next table. The tool outside diameters, shaft angles, center distances, eccentricity, axial feeds, gear speeds, and tool speeds were set according to the experimental plan. The eccentricity error was set to 0.01 mm in all cases to simulate the unavoidable mounting error. The axial feed was 0.12 mm/r for the first two tools and 0.24 mm/r for the third tool. The gear speed was kept constant at 840 r/min, and the tool speed was adjusted according to the transmission ratio.
| Tool tooth number zt | Tool outside diameter d2 (mm) | Shaft angle Σ (°) | Center distance a (mm) | Eccentricity e (mm) | Axial feed f (mm/r) | Gear speed n1 (r/min) | Tool speed n2 (r/min) |
|---|---|---|---|---|---|---|---|
| 41 | 63.784 | 20 | 57.288 | 0.01 | 0.12 | 840 | 2397.07 |
| 58 | 88.527 | 20 | 44.916 | 0.01 | 0.12 | 840 | 1665.76 |
| 78 | 117.638 | 20 | 30.361 | 0.01 | 0.24 | 840 | 1890 |
I conducted the gear cutting experiments and compared the numerical calculation results with the measured lead-direction errors. The experimental setup and the resulting tooth surfaces are shown in the following image. The tests confirmed that the lead-direction ripple error is strongly dependent on the tool tooth number and the axial feed. For the tool with 41 teeth and an axial feed of 0.12 mm/r, the numerical calculation predicted a regular wavy lead error with about 24 peaks and valleys within a 40 mm face width, and an average error of about 6.7 μm. The experimental measurement showed about 23 peaks and valleys and an average error of 7.8 μm on both flanks. Considering other factors such as thermal deformation, chip extrusion, machine tool motion accuracy, and measurement accuracy, the experimental results are in good agreement with the numerical predictions.
For the tool with 58 teeth and an axial feed of 0.12 mm/r, the numerical calculation predicted a regular wavy lead error with about 6 peaks and valleys within a 40 mm face width and an average error of about 20 μm. The experimental measurement showed an average error of 20.5 μm on both flanks. The number of peaks and valleys was also about 6. Again, the numerical and experimental results agree well. For the tool with 78 teeth and an axial feed of 0.24 mm/r, the numerical calculation predicted a much smaller lead-direction ripple. The average error was about 1.2 μm, and the waviness was so small that the number of peaks and valleys could not be clearly identified within the 40 mm face width. The experimental measurement showed an average error of 1.95 μm on both flanks. This result confirms that the optimized tool tooth number and the higher axial feed can still produce a small lead-direction error when the tool tooth number is properly selected.
The comparison between numerical calculation and experimental measurement is summarized in the following table. The maximum error value and the number of peaks are listed for each tool. The differences between calculation and experiment are small, especially when considering the complexity of the gear cutting process. The results validate the simplified lead-direction ripple model and the tool tooth number optimization method. They also demonstrate that the lead-direction ripple error in gear cutting by power skiving can be significantly reduced by selecting the optimal tool tooth number rather than by simply reducing the axial feed.
| Tool tooth number zt | Eccentricity e (mm) | Axial feed f (mm/r) | Calculated maximum error (μm) | Calculated peak number | Measured maximum error (μm) | Measured peak number |
|---|---|---|---|---|---|---|
| 41 | 0.01 | 0.12 | 6.7 | 24 | 7.8 | 23 |
| 58 | 0.01 | 0.12 | 20 | 6 | 20.5 | 6 |
| 78 | 0.01 | 0.24 | 0.28 | — | 2 | — |
From my analysis and experiments, I draw several conclusions. First, the relative eccentricity error between the tool and the gear is the main cause of lead-direction ripple error in gear cutting by power skiving. Changing the tool axial feed can change the ripple shape and the residual height difference. Reducing the axial feed can reduce the lead-direction error, but it also reduces gear cutting efficiency. Second, for a given workpiece tooth number, the tool tooth number determines the magnitude, shape, and period of the lead-direction error. The traditional method of choosing a tool tooth number that shares a greatest common divisor with the workpiece tooth number is not always applicable. It is necessary to screen the tool tooth number according to the workpiece tooth number. Third, the simplified lead-direction error model and the tool tooth number optimization method are validated by gear cutting experiments. The numerical results and the measured results have good consistency. This means that my method can be used to reduce the lead-direction ripple error in gear cutting by power skiving without sacrificing productivity.
In practical gear cutting, the tool eccentricity is usually unavoidable. It may come from the tool holder, the spindle, the fixture, or the workpiece locating system. Even a small eccentricity of a few micrometers can produce a noticeable lead-direction ripple if the tool tooth number is not properly selected. Therefore, the optimization of the tool tooth number should be integrated into the tool design stage. By using the model I developed, a tool designer can predict the lead-direction error for different tool tooth numbers and choose the best one before manufacturing the tool. This reduces the need for trial-and-error tool fabrication and shortens the development cycle. It also improves the accuracy of the gear cutting process and reduces the scrap rate.
The axial feed is another important parameter in gear cutting. A larger axial feed increases the residual height difference, but it also increases the material removal rate and reduces the processing time. My results show that when the tool tooth number is optimally selected, a relatively large axial feed can still produce a small lead-direction ripple. For example, the tool with 78 teeth and an axial feed of 0.24 mm/r produced an average error of only about 2 μm. This is a significant improvement compared with the tool with 58 teeth and an axial feed of 0.12 mm/r, which produced an average error of about 20 μm. Therefore, the tool tooth number optimization can allow a higher axial feed and thus a higher gear cutting efficiency while maintaining good accuracy. This is a key advantage for industrial gear cutting.
I also note that the lead-direction ripple error is not the only error in gear cutting. Other errors such as profile error, pitch error, and total cumulative error can also affect gear quality. However, the lead-direction ripple is often the most difficult to control because it is directly related to the periodic engagement between the tool teeth and the workpiece tooth slots. The model I developed focuses on this specific error and provides a practical tool for its control. In future work, the model can be extended to include other error sources, such as thermal deformation and tool wear, to further improve the accuracy of prediction. It can also be integrated into a computer-aided tool design system for automatic tool tooth number optimization.
The mathematical model I established is based on a simplified elliptical swept trajectory. Although this simplification reduces computational complexity, it still captures the essential geometric relation between the tool eccentricity, the axial feed, the tool tooth number, and the residual height difference. The good agreement between the numerical results and the experimental measurements confirms the validity of this simplification. The model can be used to calculate the lead residual height difference for any combination of workpiece tooth number and tool tooth number. It can also be used to generate a map of the lead-direction error as a function of the tool tooth number, which helps the tool designer quickly identify the optimal tooth number. The following table provides an example of such a map for a workpiece tooth number of 117 within a tool tooth number range from 30 to 90. The table lists selected tool tooth numbers and the corresponding residual height differences for two different axial feeds.
| Tool tooth number zt | Residual height difference at f=0.12 mm/r (μm) | Residual height difference at f=0.24 mm/r (μm) |
|---|---|---|
| 41 | 6.7 | 13.5 |
| 48 | 2.1 | 4.3 |
| 58 | 20.0 | 39.8 |
| 78 | 0.28 | 1.2 |
| 84 | 3.5 | 7.1 |
The optimization method can be summarized in a compact algorithmic form. I input the workpiece tooth number, the allowable tool tooth number range, the tool eccentricity, and the axial feed. Then I loop over all candidate tool tooth numbers. For each candidate, I calculate the eccentric amount for each tooth, determine the intersection points of adjacent swept trajectories, and compute the residual height difference. Finally, I select the tool tooth number that gives the minimum residual height difference. This algorithm is simple and fast. It can be executed on a personal computer in a few seconds. It does not require complex simulation software or expensive computation. Therefore, it is suitable for practical use in tool design and process planning for gear cutting.
The experimental validation was performed on an internal helical gear with 117 teeth. Three tools with 41, 58, and 78 teeth were used. The eccentricity was set to 0.01 mm. The axial feed was 0.12 mm/r for the first two tools and 0.24 mm/r for the third tool. The measured lead-direction errors were 7.8 μm, 20.5 μm, and 1.95 μm, respectively. These results clearly show that the tool with 78 teeth produced the smallest error, even though it used the largest axial feed. The tool with 58 teeth produced the largest error, even though it used the same axial feed as the tool with 41 teeth. This confirms that the tool tooth number is a dominant factor in the lead-direction ripple error. It also confirms that the optimization method can identify the best tool tooth number without trial and error. The gear cutting experiments therefore provide strong support for my modeling and optimization approach.
In conclusion, I have established a mathematical model for the lead-direction ripple error in gear cutting by power skiving. The model considers the tool eccentricity, the axial feed, and the tool tooth number. I derived a simplified calculation method for the residual height difference based on an elliptical swept trajectory. I analyzed the influence of the axial feed and the tool tooth number on the lead-direction error. I proposed a tool tooth number optimization method that minimizes the residual height difference. I validated the model and the method with gear cutting experiments. The results show that the relative eccentricity between the tool and the gear is the main cause of the lead-direction ripple. The axial feed affects the magnitude and shape of the ripple, but reducing the axial feed is not an efficient solution because it reduces productivity. The tool tooth number determines the ripple magnitude, shape, and period. The optimal tool tooth number can significantly reduce the lead-direction error while allowing a higher axial feed. My method provides a practical and effective way to control the lead-direction ripple error in gear cutting by power skiving. It can be used in tool design, process planning, and quality control for high-precision gear cutting.
The broader impact of this work is related to the increasing demand for high-precision gears in robotics, electric vehicles, and industrial machinery. Power skiving is a key enabling technology for these applications because it can cut internal gears and complex geometries with high efficiency. However, the lead-direction ripple error has been a barrier to achieving the required accuracy. By providing a quantitative model and an optimization method, I help to overcome this barrier. The method can be applied to different gear geometries and different tool designs. It can also be extended to external gears and other types of gear cutting processes where similar periodic errors occur. I believe that the combination of mathematical modeling, numerical calculation, and experimental validation presented in this work will contribute to the wider adoption of power skiving in high-precision gear cutting.
For future work, I plan to incorporate additional factors into the model, such as tool wear, thermal deformation, and dynamic vibration. These factors can influence the lead-direction error and may become important in high-speed gear cutting. I also plan to develop a software tool that can automatically optimize the tool tooth number and the axial feed simultaneously. This would allow a multi-objective optimization that considers both accuracy and productivity. Finally, I plan to conduct more gear cutting experiments with different workpiece materials and different tool coatings to further validate the model and the optimization method. The ultimate goal is to provide a comprehensive and reliable framework for the design and operation of power skiving in industrial gear cutting.