## 3 Calculation Principle of Transmission Error and Backlash
Then the cycloid gear is rotated counterclockwise by \(\Delta\theta_{\text{c}}\) around \(O_{\text{c}}\) to approach the pin. As the cycloid gear continues to rotate around \(O_{\text{c}}\), the clearances \(c_{\text{p}}^{i}+c_{\text{c}}^{i}\) are continuously reduced and contact deflection even occurs (see Figure 5c). In the initial stage of contact between the cycloid gear and pin, the total output torque generated by the gear pair is not enough to balance the external load, and the iterative process is continued until the difference between the total output torque and the external load converge within 1%. To improve the calculation efficiency, the rotation step size (\(\Delta\theta_{\text{c}}\)) is adjusted according to the bisection method. When the total output torque of the gear pair is balanced with the external load, the rotation angle of the cycloid gear around \(O_{\text{c}}\) is defined as \(\theta_{\text{c}3}\). The TE is given as follows:  
\[TE=\theta_{\text{f}}/I-\theta_{\text{c}3}. \tag{16}\]  
In contrast to the TE, the calculation of the backlash does not consider the influence of the tooth deflection under a load. When the cycloid gear is rotated to the theoretical meshing position, the cycloid gear is rotated clockwise and counterclockwise around \(O_{\text{c}}\), and the rotation angles (\(\theta_{\text{c}1}\) and \(\theta_{\text{c}2}\)) of the cycloid gear around \(O_{\text{c}}\) at the initial contact in the two directions are obtained. The backlash is given as follows:  
\[Backlash=\theta_{\text{c}2}-\theta_{\text{c}1}. \tag{17}\]  
According to the above detailed description of the calculation process of the TE and backlash, the determination of the contact between the cycloid gear and pin and the calculation of the output torque generated by the cycloid-pin gear pair in the iteration process are the keys to calculating the TE and backlash. According to the principle of the slicing method, it is assumed that each slice along the tooth width is independent and that the load on one slice cannot convey deflections to the neighboring slices [37]. Additionally, due to the geometry of a cycloid-pin gear pair, the contact deflection is much larger than the tooth bending and shear deflections, and it is regarded as local deflection [13]. Thus, only the contact deflection between the tooth surfaces is considered in this study, and it is feasible to ignore the influence of the load deflection between the adjacent slices along the tooth width. As mentioned above, each slice in Figure 5b is composed of multiple discrete elements, and each discrete element is approximately represented as a cylindrical surface. The position and the size of these cylindrical surfaces have been determined according to the description in Section 2. Thus, the contact of each group of slices can be converted into the contact of the corresponding equivalent cylindrical surfaces. There are four possible contact status for any two elements on each group of slices. As shown in Figure 6a,b, when the misalignment deviations (\(\varphi\) and \(\gamma\)) are not considered, the axes of the two elements (\(\Sigma_{\text{c}}^{n_{\text{c}}i_{\text{c}}i_{\text{c}}}\) and \(\Sigma_{\text{p}}^{n_{\text{p}}i_{\text{p}}j_{\text{p}}}\)) are parallel to each other. The contact width is equal to the element width, and the contact depth is equal to the difference between the sum of the radii of curvature and the distance between the centers of curvature corresponding to the two control points.