## 3 Calculation Principle of Transmission Error and Backlash
The contact width and depth are given as follows:  
\[B_{n}^{i}=\frac{B_{\text{p}}}{n_{\text{1p}}}, \tag{18}\]  
\[\delta_{n}^{i}=\rho_{\text{c}}^{n_{\text{c}}i_{\text{c}}j_{\text{c}}}+\rho_{ \text{p}}^{n_{\text{p}}i_{\text{p}}j_{\text{p}}}-d_{\text{cp}}, \tag{19}\]  
where \(d_{\text{cp}}\) is the distance between the centers of curvature corresponding to the two control points, which can be obtained based on the coordinates, radius of curvature, and normal vector at the element control point, as well as the axis direction vector of the approximate cylinder described in Section 2. When the contact depth \(\delta_{n}^{i}\) is greater than zero, the two elements are in contact. Otherwise, they are not involved in contact. As shown in Figure 6c,d, when the misalignment deviations (\(\varphi\) and \(\gamma\)) are considered, the axis of cycloid gear is not parallel to the axis of the pin. To calculate the contact depth in these two cases, the two cylindrical surfaces are discretized into \(n_{\text{e}}\) arcs. It is assumed that there are \(n_{\text{ec}}\) pairs of arcs that meet the above contact condition, and the mean of their contact depth is approximated to the contact depth of the two discrete elements. The contact width and depth are given as follows:  
\[B_{n}^{i}=\frac{B_{\text{p}}n_{\text{ec}}}{n_{\text{1p}}n_{\text{e}}}, \tag{20}\]  
\[\delta_{n}^{i}=\sum_{j=1}^{n_{\text{ec}}}\delta_{\text{cp}}^{l}/n_{\text{ec}}\,, \tag{21}\]  
where \(\delta_{\text{cp}}^{l}\) is the contact depth of the arcs \(\Gamma_{\text{p}}^{l}\) and \(\Gamma_{\text{c}}^{l}\). Figure 6: Four contact status for any two elements on each group of slices: (**a**) elements separation without \(\varphi\) and \(\gamma\), (**b**) elements contact without \(\varphi\) and \(\gamma\), (**c**) elements separation considering \(\varphi\) and \(\gamma\), (**d**) elements contact considering \(\varphi\) and \(\gamma\). The Hertzian contact stiffness of any two elements is given as follows [13]:  
\[K_{n}^{i}=\frac{F_{n}^{i}}{\delta_{n}^{i}}=\frac{\pi B_{n}^{i}}{2\left[\frac{1-v_ {\mathrm{c}}^{2}}{E_{\mathrm{c}}}\left(\ln\frac{4\rho_{\mathrm{c}}^{\mathrm{c} _{\mathrm{c}_{\mathrm{c}_{\mathrm{c}_{\mathrm{c}_{\mathrm{c}_{\mathrm{c}_{ \mathrm{c}_{\mathrm{c}_{\mathrm{c}_{\mathrm{c}_{\mathrm{c}}}}}}}}}}}-\frac{1}{2}} }{\frac{1-v_{\mathrm{c}}^{2}}{B_{n}^{i}}-\frac{1}{2}\right)+\frac{1-v_{ \mathrm{c}}^{2}}{E_{\mathrm{p}}}\left(\ln\frac{4\rho_{\mathrm{p}}^{\mathrm{p} \mathrm{p}\mathrm{p}}}{B_{n}^{i}}-\frac{1}{2}\right)\right], \tag{22}\]  
where \(E_{\mathrm{c}}\) and \(E_{\mathrm{p}}\) are the modulus of elasticity, and \(v_{\mathrm{c}}\) and \(v_{\mathrm{p}}\) are the Poisson's ratios, where the subscripts c and p represent cycloid gear and pin, respectively. \(B_{n}^{i}\) is the contact width and \(b_{n}^{i}\) is the semi-Hertzian contact width, which can be calculated with:  
Figure 5:0  
Figure 5:1  
Figure 5:2  
where \(E^{*}\) and \(\rho^{*}\) are the equivalent elasticity modulus and the equivalent radius of curvature, respectively. The total output torque generated by the cycloid-pin gear pair is given as follows:  
Figure 5:3  
where \(z_{\mathrm{pc}}\) is the number of contacted pins and \(L_{n}^{i}\) is the arm of force of the slice \(S_{\mathrm{c}}^{i}\) (see Figure 5a), which is given as follows:  
Figure 5:4  
where in the vector \(\mathbf{PQ}\), \(\mathrm{P}\) is the control point of the contact element \(\Sigma_{\mathrm{c}}^{n_{\mathrm{c}}\mathrm{c}_{\mathrm{c}}\mathrm{c}_{\mathrm{ c}}}\) on the slice \(S_{\mathrm{c}}^{i}\). The coordinates of \(\mathrm{P}\) can be obtained using Equation (6), and \(\mathrm{Q}\) is the corresponding point of \(\mathrm{P}\) on the axis of the cycloid gear. The coordinates of \(\mathrm{Q}\) are given as follows:  
\[\left[ \begin{array}{cc}X_{\mathrm{Q}}^{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}}}&1\end{array}