## 3 Error Sensitivity Analysis of the TE Caused by Components of the Cycloidal Gear Transmission
### Principle of Error Sensitivity Analysis Based on Tayler Expansion  
The TE of the cycloidal gear transmission can be represented as  
\[\Delta\theta=\Delta\theta_{j}(\delta_{1},\delta_{2},\cdots,\delta_{j}), \tag{18}\]  
where \(\delta_{1},\delta_{2},\cdots,\delta_{j}\) are errors, clearances, and modifications in Equations (5)-(13). The corresponding relationship between \(\theta_{j}\) and \(\delta_{j}\) is shown in Table 1.  
When a main factor \(\delta_{j}\) has a small deviation \(\Delta\delta_{j}\), Equation (18) can be expanded based on Tayler expansion and its high-order items can be omitted [28]. Equation (18) can be written as  
\[ \begin{array}{ll}\Delta\theta&=\Delta\theta_{j}(\delta_{1}+\Delta\delta_{1 },\delta_{2}+\Delta\delta_{2},\cdots,\delta_{j}+\Delta\delta_{j})\\ &=\Delta\theta_{j}(\delta_{1},\delta_{2},\cdots,\delta_{j})+(\frac{\partial \Delta\theta_{1}}{\partial\theta_{1}}\Delta\delta_{1}+\frac{\partial\Delta \theta_{2}}{\partial\theta_{2}}\Delta\delta_{2}+\cdots+\frac{\partial\Delta \theta_{j}}{\partial x_{j}}\Delta\delta_{j}).\end{array}  \tag{19}\]  
The deviation of the TE \(\Delta\theta=\Delta\theta_{j}(\delta_{1},\delta_{2},\cdots,\delta_{j})\) can be represented as  
\[\delta_{\Delta\theta}=\frac{\partial\Delta\theta_{1}}{\partial\delta_{1}} \Delta\delta_{1}+\frac{\partial\Delta\theta_{2}}{\partial\delta_{2}}\Delta \delta_{2}+\cdots+\frac{\partial\Delta\theta_{j}}{\partial\delta_{j}}\Delta \delta_{j}. \tag{20}\]  
A sensitivity index is defined as  
\[S_{j}=\frac{\partial\Delta\theta_{j}/\partial\delta_{j}}{\partial\Delta\theta _{0}/\partial\delta_{0}}, \tag{21}\]  
where \(\partial\Delta\theta_{0}/\partial\delta_{0}\) is a reference input error parameter. Influences of TEs due to different error factors \(\Delta\delta_{j}\) on the TE \(\Delta\theta\) are analyzed by using a sensitivity index matrix, and the error factor that causes the maximum TE can be determined [29]. When the value of \(\delta_{j}\) is determined, the partial derivative \(\partial\Delta\theta_{j}/\partial\delta_{j}\) is a constant. The sensitivity index can be written as  
\[S_{j}=\frac{S_{j}}{g_{0}}, \tag{22}\] \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Geometric TE** \(\theta_{j}\) & \(\theta_{a}\) & \(\theta_{b}\) & \(\theta_{c}\) & \(\theta_{d}\) & \(\theta_{e}\) & \(\theta_{f}\) & \(\theta_{g}\) & \(\theta_{h}\) & \(\theta_{i}\) \\ \hline
**Main factors** \(\delta_{j}\) & \(\Delta r_{rp}\) & \(\Delta r_{p}\) & \(\delta_{r_{p}}\) & \(\delta_{r_{p}}\) & \(\delta_{j}\) & \(F_{r_{1}}\) & \(\delta_{l_{\Sigma}}\) & \(\Delta F_{p}\) & \(\delta\Delta r_{rp}\) and \(\delta\Delta r_{p}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main factors of the geometric transmission error (TE). where \(g_{j}=\partial\Delta\theta_{j}/\partial\delta_{j}\) and \(g_{0}=\partial\Delta\theta_{0}/\partial\delta_{0}\). By substituting Equations (21) and (22) into Equation (20), Equation (20) can be written as  
\[\delta_{\Delta\theta}=g_{1}\Delta\delta_{1}+g_{2}\Delta\delta_{2}+\cdots+g_{ j}\Delta\delta_{j}. \tag{23}\]  
Error coefficients of error factors are obtained by substituting Equations (5)-(13) into Equations (18)-(22), as shown in Table 2. By substituting the parameters of the initial RV speed reducer prototype into error coefficients, numerical solutions of sensitivity indices can be obtained, as shown in Table 3. The initial RV speed reducer prototype contains a tooth profile modification due to the lubrication requirement. Assume the numerical solution of the pinwheel center circle radius error \(\delta_{a}\) as the basic number that is equal to 1; other items are changed to a multiple of \(\delta_{a}\). Hence, the sensitivity index of the TE caused by components of the cycloidal gear transmission can be deduced, as shown in Table 2.  
The sensitivity index of the eccentricity error is the smallest one that is only 0.024% of the sensitivity index of the reference value, as shown in Table 2. The influence on the TE caused by the eccentricity error is neglected in this study.