# Modification and Optimization of Cycloidal Gear Tooth Profile Based on Machining Error Compensation
## 2 Theoretical Analysis of Machining Error
### Machining Error Compensation  
The cycloidal gear mathematical model can facilitate the quantitative analysis of the cycloidal gear. The standard cycloidal gear profile equation is given as follows [13]:  
\[ \begin{array}{l}x=\left(r_{p}-r_{rp}S^{-0.5}\right)\cos(1-i_{H})\varphi- \left(a-K_{1}r_{rp}S^{-0.5}\right)\cos i_{H}\varphi\\ y=\left(r_{p}-r_{rp}S^{-0.5}\right)\sin(1-i_{H})\varphi+\left(a-K_{1}r_{rp}S^{- 0.5}\right)\sin i_{H}\varphi\end{array}  \tag{1}\]  
where \(r_{p}\) is the radius of the center circle of the pin tooth (mm), \(r_{rp}\) is the radius of the pin tooth (mm), \(i_{H}\) is the relative transmission ratio of the cycloidal gear and the pin wheel, \(\varphi\) is the rotation angle of the rotating arm \(\overline{O_{p}O_{c}}\) relative to the center of a specific pin tooth (rad) (as shown in Figure 4), \(a\) is the eccentric distance of the center of the cycloidal gear relative to the center of the pin tooth center circle (mm), \(K_{1}\) is the curate ratio (the ratio of short epicycloid amplitude to epicycloid amplitude and \(K_{1}=r_{p}^{\prime}/r_{p}=az_{p}/r_{p}\)), and \(S=1+K_{1}^{2}-2K_{1}\,cos\varphi\). Figure 4: Schematic diagram of a standard cycloidal tooth profile. Figure 3: Schematic diagram of the generation method. Figure 5 shows the tooth profile variation caused by the grinding wheel radius error during the cycloidal gear machining process. Since, in principle, the change of the grinding wheel grinding radius error is similar to the change of the pin tooth radius, the pin tooth radius in the \(x\) and \(y\) directions is differentiated to obtain the coefficients \(k_{1x}\) and \(k_{1y}\) of the variation of the cycloidal gear profile in the \(x\) and \(y\) directions caused by the grinding wheel grinding radius error, as shown in Formula (2). Similarly, the error items, such as the distance error from the center of the grinding wheel to the center of the spindle center, the eccentricity error of the machine tool spindle, the transmission ratio error of the machine tool, and the eccentricity error of the installation of the cycloidal gear blank, are differentiated to obtain the variation coefficients of each error item on the \(x\) and \(y\) axes, as shown in Table 1.  
\[ \begin{array}{l}k_{1x}=\frac{\partial x}{\partial r_{rp}}=\{K_{1}\cos(i_{H }\varphi)-\cos[(1-i_{H})\varphi]\}S^{-0.5}\\ k_{1y}=\frac{\partial y}{\partial r_{rp}}=\{-K_{1}\sin(i_{H}\varphi)-\sin[(1-i_ {H})\varphi]\}S^{-0.5}\end{array}  \tag{2}\]  
where \(\alpha=(1-i_{H})\varphi\), \(\beta=i_{H}\varphi\), \(T_{i}=K_{i}-\cos\varphi\), and \(\lambda=r_{rp}/r_{p}\). Since these errors are small, their superposition can be linearly approximated. The comprehensive variation of the cycloidal tooth profile is equal to the sum of each variation as shown in Figure 6; that is:  
\[ \begin{array}{l}g_{x}(\varphi,\xi)=\Delta x=k_{1x}\xi_{1}+k_{2x}\xi_{2}+k_{3 x}\xi_{3}+k_{4x}\xi_{4}+k_{5x}\xi_{5}\\ g_{y}(\varphi,\xi)=\Delta y=k_{1y}\xi_{1}+k_{2y}\xi_{2}+k_{3y}\xi_{3}+k_{4y}\xi_ {4}+k_{5y}\xi_{5}\end{array}  \tag{3}\]