## 5 Results and Discussions
### Effect of Manufacturing Errors  
According to Section 2, the MEs considered in this study include the total profile deviation of the cycloid gear, and the total profile deviation, pitch deviation, and center circle diameter deviation of the pin. To study their effects on the transmission accuracy, a specific case is applied for the proposed calculation method. The MEs in this case are \(F_{\alpha}^{\rm c}=\pm 3\) um, \(F_{\rm p}=15\) um, \(\delta_{\rm RP}=4\) um, and the upper and lower deviations of \(F_{\alpha}^{\rm P}\) are zero and \(-2\) um, respectively. The pitch deviation is usually considered to be the long-term components of the MEs [38], and the linear function [11] and the sine function [38; 39; 40] are generally used to simulate the cumulative pitch deviation (\(F_{\rm p}\)) in published works. However, these two functions ignore the effect of the randomness of the pitch deviation on the contact state of the gear teeth. Thus, a sine function and a random variable that obeys the normal distribution are superimposed to simulate the cumulative pitch deviation in this study. The expectation and the standard deviation of the normal distribution are \(0\) um and \(2\) um, respectively. The cumulative pitch deviation of the pins used in this study is shown in Figure 9. Additionally, it is assumed that the profile deviation of each discrete element on the tooth surface is randomly distributed in its tolerance zone. To ensure the lubrication requirements and the compensation for MEs, the compound modification of the negative isometric and negative offset is adopted based on the results described in Section 5.1, and the PM parameters of the cycloid gear are \(\Delta R_{\rm rp}=-0.004\) mm and \(\Delta R_{\rm p}=-0.017\) mm. Figure 10 shows the calculation results of the TE and backlash considering different types of MEs. The figure shows that the pitch deviation (\(F_{\rm p}\)) not only changes the curve waveforms of the TE and backlash, but also the main source of their peak-to-peak value. The profile deviation (\(F_{\alpha}\)) produces slight fluctuations in the curves of the TE and backlash  
Figure 9: Cumulative pitch deviation. compared with the results without MEs and the results considering the pin center circle diameter deviation (\(\delta_{\text{RP}}\)). This is mainly because the contact element on each slice has changed under the influence of the profile deviation, which leads to changes in the contact stiffness and the tooth deflection under a load. The fluctuations of the TE and backlash also cause the PPTE and PPB to increase, but the increase is very small compared to the results considering the pitch deviation. In addition, the profile deviation and the pin center circle diameter deviation mainly affect the MTE and MB. According to the above analysis, the MTE and MB are mainly related to the initial gap between the tooth surfaces. For the profile deviation, the initial gap mainly depends on the upper deviation of \(F_{\alpha}\). Since the upper deviations of \(F_{\alpha}^{\text{c}}\) and \(F_{\alpha}^{\text{P}}\) have a similar effect on the initial gap, only the influence of the upper deviation of \(F_{\alpha}^{\text{c}}\) on the TE and backlash is considered in this study. In the above case, the upper and lower deviations of \(F_{\alpha}^{\text{c}}\) are 3 \(\upmu\)m and \(-\)3 \(\upmu\)m, respectively. To quantitatively study the influence of \(F_{\alpha}\) on the MTE and MB, the above-mentioned \(F_{\alpha}^{\text{c}}\) are moved equivalently to obtained different upper deviations, and the TE and the backlash for different \(F_{\alpha}^{\text{c}}\) and \(\delta_{\text{RP}}\) are calculated (see Figure 11). According to Figure 11, the absolute values of the MTE and MB decrease with the increase in the upper deviation of \(F_{\alpha}^{\text{c}}\), and the values increase with the increase in \(\delta_{\text{RP}}\).