## 3 Calculation Principle of Transmission Error and Backlash  
As shown in Figure 4, the TE and backlash of the cycloid-pin pair are calculated with an iterative process. Firstly, the error tooth surface model proposed in Section 2 is generated based on the input gear data, PMs, MEs, and AEs. Figure 5a shows the tooth contact model of the cycloid gear and pin. \(X_{\rm f}O_{\rm f}Y_{\rm f}\) is the fixed coordinate system established at the center of the pin center circle (\(O_{\rm f}\)), and the center of the cycloid gear is defined as \(O_{\rm c}\). The eccentric distance between \(O_{\rm f}\) and \(O_{\rm c}\) is \(e\). It is well known that a cycloid gear not only rotates counterclockwise with the input crankshaft around \(O_{\rm f}\), but also rotates clockwise around \(O_{\rm c}\) to drive the output mechanism to rotate. The rotation angles of the cycloid gear around \(O_{\rm f}\) and \(O_{\rm c}\) are defined as \(\theta_{\rm f}\) and \(\theta_{\rm c}\), respectively, and the rotation step sizes of the cycloid gear around \(O_{\rm f}\) and \(O_{\rm c}\) are defined as \(\Delta\theta_{\rm f}\) and \(\Delta\theta_{\rm c}\), respectively. At the first position of the cycloid gear, \(\theta_{\rm f}\) and \(\theta_{\rm c}\) are initialized to zero. To calculate the TE and backlash corresponding to each input angle of the gear pair, the cycloid gear is first rotated \(\theta_{\rm f}\) (\(\theta_{\rm f}=\theta_{\rm f}+\Delta\theta_{\rm f}\)) counterclockwise around \(O_{\rm f}\) and \(\theta_{\rm c}\) (\(\theta_{\rm c}=\theta_{\rm f}/I\)) clockwise around \(O_{\rm c}\) with coordinated transformation, when the cycloid gear and the pins are in the theoretical meshing position. \(I\) is the transmission ratio of the cycloid-pin gear pair. The coordinate transformation matrices of the cycloid gear rotating around \(O_{\rm f}\) and \(O_{\rm c}\) (\(\mathbf{M}_{\rm of}\) and \(\mathbf{M}_{\rm oc}\)) are given as follows:  
\[\mathbf{M}_{\rm of}=\left[ \begin{array}{cccc}1&0&0&e\cos\theta_{\rm f}\\ 0&1&0&e\sin\theta_{\rm f}\\ 0&0&1&0\\ 0&0&0&1\end{array} \right], \tag{14}\]  
\[\mathbf{M}_{\rm oc}=\left[ \begin{array}{cccc}\cos\theta_{\rm c}&-\sin\theta_{ \rm c}&0&e\cos\theta_{\rm f}-e\cos(\theta_{\rm f}+\theta_{\rm c})\\ \sin\theta_{\rm c}&\cos\theta_{\rm c}&0&e\sin\theta_{\rm f}-e\sin(\theta_{\rm f}+ \theta_{\rm c})\\ 0&0&1&0\\ 0&0&0&1\end{array} \right]. \tag{15}\]
As shown in Figure 5a, when the cycloid gear is in the theoretical meshing position, there is a different initial clearance (\(\cdots\), \(c_{t-2}\), \(c_{t-1}\), \(c_{t}\), \(c_{t+1}\), \(c_{t+2}\), \(\cdots\)) between the cycloid gear and each pin due to the influences of the PMs, MEs, and AEs. In published works, the slicing method has been widely used to calculate the mesh stiffness of a helical gear pair [32; 33; 34; 35; 36]. According to the error tooth surface model proposed in Section 2, the cycloid gear and pin can also be regarded as composed of \(n_{1c}\) and \(n_{1p}\) slices, respectively, and each slice is composed of \(n_{2c}\) and \(n_{2p}\) elements (see Figure 2a). Thus, as shown in Figure 5b, the cycloid gear and pin can also be regarded as composed of \(n_{1c}\) and \(n_{1p}\) slices, respectively. Figure 4: Flow chart of the calculation process of TE and backlash. Figure 5: Tooth contact analysis: (**a**) tooth contact model of cycloid gear and pin, (**b**) initial position of each slice along tooth width, and (**c**) balance position of each slice along tooth width. the clearance \(c_{t}\) is the minimum clearance in all slices (\(\cdots\), \(S_{\text{p}}^{i-1}\), \(S_{\text{p}}^{i}\), \(S_{\text{p}}^{i+1}\), \(\cdots\), and, \(\cdots\), \(S_{\text{c}}^{i-1}\), \(S_{\text{c}}^{i}\), \(S_{\text{c}}^{i+1}\), \(\cdots\)) along the tooth width direction. Since each group of slices (\(S_{\text{p}}^{i}\) and \(S_{\text{c}}^{i}\)) is also composed of multiple discrete elements, the clearance (\(c_{\text{p}}^{i}+c_{\text{c}}^{i}\)) between each group of slices is the minimum clearance of all elements on the two slices.