## 3 Calculation Principle of Transmission Error and Backlash
\right]^{\mathrm{T}}=\mathbf{M}_{\mathrm{ ae}}\left[ \begin{array}{cc}e\cos\theta_{\mathrm{f}}&e\sin\theta_{\mathrm{f}}&B_{ \mathrm{c}}-B_{\mathrm{c}}(i_{\mathrm{c}}-1/2)\big{/}n_{\mathrm{1c}}&1\end{array}  \right]^{\mathrm{T}}. \tag{28}\]  
Thus, the vector \(\mathbf{PQ}\) is given as follows:  
\[\mathbf{PQ}=\left[\begin{array}{cc}X_{\mathrm{Q}}^{n_{\mathrm{c}}\mathrm{c }_{\mathrm{c}}\mathrm{c}_{\mathrm{c}}}-X_{\mathrm{c}}^{n_{\mathrm{c}}\mathrm{c }_{\mathrm{c}}\mathrm{c}_{\mathrm{c}}}&Y_{\mathrm{Q}}^{n_{\mathrm{c}}\mathrm{c }_{\mathrm{c}}\mathrm{c}_{\mathrm{c}}}&Z_{\mathrm{Q}}^{n_{\mathrm{c}}\mathrm{c }_{\mathrm{c}}\mathrm{c}_{\mathrm{c}}}-Z_{\mathrm{cbacklash, respectively, and the term \"MCS\" refers to the maximum contact stress on the tooth surface. As shown in Figure 7a, the backlash is always equal to zero during the entire meshing cycle, which is because the influence of the PMs, MEs, AEs, and the tooth deflection under a load are not considered when calculating the backlash. Additionally, according to Figure 7a,b, it is not difficult to find that both the TE and the number of contact pins change periodically with \(\theta_{\text{f}}\), and the maximum number and the minimum number of contact pins are 20 and 19, respectively. Theoretically, when the tooth surfaces are perfect, half of the pins are always in contact with the cycloid gear during the entire meshing cycle. However, due to the influence of the tooth deflection under a load, the tooth pair in the critical contact state loses contact. The MTE and PPTE are \(-4.5206\) arcsec and \(0.0531\) arcsec, respectively. According to Equation (16), due to the tooth deflection under a load, the actual rotation angle of the cycloid gear around \(O_{\text{c}}\) is greater than its theoretical value, resulting in a negative mean value. Additionally, both the contact stiffness and the load shared on each pin are different at the different positions of \(\theta_{\text{f}}\), which leads to the different tooth deflection under a load. This is the reason why the TE has small fluctuations between its peaks and troughs. During the transmission of the cycloid-pin gear pair, the position of the contact point on the cycloid tooth surface changes with the rotation of the cycloid gear. However, according to Equations (22) and (23), the contact stiffness has a nonlinear relationship with the radius of curvature of the cycloid gear and the contact force. As shown in Figure 2a, the tooth surface of the cycloid gear is discrete with the same arc length along the tooth profile, and Figure 7c shows the relationship between the contact stiffness and the contact force at different arc lengths of the single cycloid tooth profile. This shows that the contact stiffness is different at the same arc length position under different loads. This is because the semi-Hertzian contact width increases with the increase in the load, which increases the contact stiffness. Additionally, the contact stiffness is different at different arc length positions under the same load, especially in the transition region of the concave and convex tooth profiles. This is because the radius of curvature of the cycloid gear varies greatly in the transition region, because the radius of curvature at the transition point is theoretically infinite. Figure 7d shows the load distribution on the pins when \(\theta_{\text{f}}=60^{\circ}\), \(120^{\circ}\), and \(180^{\circ}\). The load distribution on the pins shows a trend of first increasing and then decreasing. In addition, there is a significant decrease in the load on one of the pins near the maximum load region, such as the 25 th pin, when \(\theta_{\text{f}}=180^{\circ}\). This is because the corresponding contact point of this pin is located near the transition region between the convex and concave tooth surfaces of cycloid gear, and the radius of curvature at the contact point is too large, resulting in a significant reduction in the contact stiffness.