# Modification and Optimization of Cycloidal Gear Tooth Profile Based on Machining Error Compensation
## 2 Theoretical Analysis of Machining Error
### Sensitivity Analysis  
The sensitivity analysis aims to analyze the influence of the change rate of various machining errors on the variation of the cycloidal gear profile. It expresses the different degrees of influence and the changing rules of each machining error on the variation of the cycloidal gear profile. Accordingly, the most sensitive error item can be obtained, and the numerical method can perform a sensitivity analysis on each error term. Assuming the machining errors \(e=[e_{1},e_{2},e_{3},e_{4},e_{5}]\) that affect the cycloidal gear profile as the design variables, the function \(f(e)=f(e_{1},e_{2},e_{3},e_{4},e_{5})\) is the normal variation of the cycloidal gear profile, and the sensitivity of the design variables corresponding to this function is:  
\[\
abla f=\left[\frac{\partial f}{\partial e_{1}}\frac{\partial f}{\partial e _{2}},\frac{\partial f}{\partial e_{3}},\frac{\partial f}{\partial e_{4}}, \frac{\partial f}{\partial e_{5}}\right] \tag{5}\]  
As shown in Figure 7, the variation of the cycloidal tooth profile caused by the machining error in the \(x\) and \(y\) directions is converted into the variation in the normal direction of the cycloid through the geometric relationship [29], which can facilitate the comparison of sensitivities, namely:  
\[\Delta N=\Delta X\cos\theta+\Delta Y\sin\theta \tag{6}\]  
where \(\cos\theta=-[K_{1}\text{cos}i_{H}\varphi-\cos(1-i_{H})\varphi]S^{-1/2}\), \(\sin\theta=[K_{1}\text{cos}i_{H}\varphi+\cos(1-i_{H})\varphi]S^{-1/2}\).  
Similarly, the variation coefficients in the \(x\) and \(y\) directions are converted into those in the normal direction; that is, the sensitivity coefficients of each processing error item, as shown in the following formula:  
\[k_{N}=k_{x}\cos\theta+k_{y}\sin\theta \tag{7}\]  
The design parameters of the RV reducer cycloidal gear used in this paper are shown in Table 2. These parameters are the parameters of the Nabtesco model RV-40E reducer.  
Table 3 shows the variation range of each machining error item. The data in Table 3 comes from the work of references [27; 30], and fine-tuning is carried out on this basis; the information is not from the experimental data.  
Figure 6: Schematic diagram of the tooth profile variation.  
In summary, the sensitivity of the cycloidal gear profile to each machining error item can be obtained, as shown in Figure 8. Figure 8 shows that the transmission ratio error of the machine tool has the maximum influence on the variation in the cycloidal gear profile.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Pin tooth center circle radius \(r_{p}\) (mm) & 64 \\ Pin tooth radius \(r_{rp}\) (mm) & 3 \\ Number of pin teeth \(z_{p}\) & 40 \\ Eccentricity \(a\) (mm) & 1.3 \\ Cycloidal gear width \(b\) (mm) & 8.9 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Cycloidal gear design parameters.  
Figure 8: Sensitivity of cycloidal gear profile to various machining error items: (**a**) the transmission ratio error of the machine tool; (**b**) other error sources.  
Figure 7: Schematic diagram of the conversion of the normal variation of the cycloidal gear profile.