# Modification and Optimization of Cycloidal Gear Tooth Profile Based on Machining Error Compensation
## 4 Modifying the Parameter Optimization  
### Optimization Model  
(1) Machining error compensation model  
The machining error is analyzed to obtain the coordinate equation for the tooth profile of the cycloidal gear after compensating for the machining error, as shown in Formula (4). The coordinate equation of the tooth profile of the cycloidal gear after the combined modification of equidistant and radial movement modification can be obtained as follows [35]:  
\[ \begin{split} x_{3}&=\left[{r_{p}}^{\prime}-{r_{ rp}}^{\prime}S^{-0.5}\right]\cos(1-i_{H})\varphi-\frac{\sigma}{{r_{p}}^{\prime}} \left[{r_{p}}^{\prime}-z_{p}{r_{p}}^{\prime}S^{-0.5}\right]\cos i_{H}\varphi \\ y_{3}&=\left[{r_{p}}^{\prime}-{r_{rp}}^{\prime}S^{- 0.5}\right]\sin(1-i_{H})\varphi+\frac{\sigma}{{r_{p}}^{\prime}}\left[{r_{p}}^ {\prime}-z_{p}{r_{rp}}^{\prime}S^{-0.5}\right]\sin i_{H}\varphi\end{split}  \tag{18}\]  
where \({r_{p}}^{\prime}=r_{p}+\Delta r_{p}\), \({r_{rp}}^{\prime}=r_{rp}+\Delta r_{rp}\). The curve modified by equidistant and radial movement modification cannot completely coincide with the compensated tooth profile, and there is a specific deviation. When  
Figure 10: Simulation results of tooth profile variation: (**a**) statistics of tooth profile variation; (**b**) normal fitting of tooth profile variation. \(\varphi\) turns \(2\pi\), a complete cycloidal gear tooth can be formed. For example, the interval \([0,2\pi]\) is divided into \(m-1\) equal parts. After error compensation, the tooth profile coordinates \((x_{2i},y_{2i})(i=1,2,\cdots m)\) of the cycloidal gear can be obtained according to Formula (4). At the same time, the tooth profile coordinates \((x_{3i},y_{3i})(i=1,2,\cdots m)\) of the cycloidal gear after the combined modification of equidistant and radial movement can be obtained from (18). There is a specific deviation between the coordinates of the two tooth profiles, in which its degree can be measured by the average value of the deviation of m points, as shown in Formula (19). The smaller the deviation degree, the higher the fitting degree. \[D\big{(}\Delta r_{p},\Delta r_{rp}\big{)}=\frac{1}{m}\sum_{i=1}^{m}\sqrt{\big{(} x_{3i}-x_{2i}\big{)}^{2}-\big{(}y_{3i}-y_{2i}\big{)}^{2}} \tag{19}\]  
(2) Carrying model  
After modifying the cycloidal gear by the combined modification of equidistant and radial movement, the multi-teeth meshing condition no longer exists without considering the elastic deformation of the cycloidal pin wheel. However, when a certain pair of cycloidal gear teeth come into contact, there are different degrees of initial normal backlash between other cycloidal gear teeth. As shown in Formula (20) [36], the initial normal backlash in the normal direction of the i-th pair of cycloidal gears and the pin teeth caused by equidistant modification and radial movement modification is:  
\[\Delta(\varphi)_{i}=\Delta r_{rp}\Big{(}1-\sin\varphi_{i}S^{-1/2}\Big{)}- \Delta r_{p}\Big{(}1-K_{1}\cos\varphi_{i}-\Big{(}1-K_{1}{}^{2}\Big{)}^{1/2} \sin\varphi_{i}\Big{)}S^{-1/2} \tag{20}\]  
The loaded tooth surface will undergo different degrees of elastic deformation during transmission. In the RV transmission, since the pin teeth are half buried in the pin holes of the pin gear housing, there is no bending deformation, and elastic deformation mainly considers contact deformation. Therefore, the total deformation comprises the contact deformation between the cycloidal gear teeth and the pin teeth, and between the pin teeth and the pin gear housing.